STAT
05tcentennial
VECTOR ANALYSIS
'Bicentennial publications
With the approval of the President and Fellows
of Yale University, a series of volumes has been
prepared by a number of the Professors and In
structors, to be issued in connection with the
Bicentennial Anniversary, as a partial indica
tion of the character of the studies in which the
University teachers are engaged.
This series of volumes is respectfully dedicated to
«raDuate$ of tfc
'
/
VECTOR ANALYSIS^
A TEXT-BOOK FOR THE USE OF STUDENTS
OF MATHEMATICS AND PHYSICS
FOUNDED UPON THE LECTURES OF
J. WILLARD GIBBS, PH.D., LL.D.
Formerly Professor of Mathematical Physics in Yale University
BY
EDWIN BIDWELL WILSON, PH.D.
Professor of Vital Statistics in
Harvard School of Public Health
NEW HAVEN
YALE UNIVERSITY PRESS
Copyright, 1901 and 1929
BY YALE UNIVERSITY
Published, December, 1901
Second Printing, January ', 19/3
Third Printing, July, 1916
fourth Printing^ April, 1922
Fifth Printing, October, 1925
Sixth Printing, April, 1020
Seventh Printing, October, 1951
Eighth Printing, April, 1943
Ninth Printing, April, 1947
All rights reserved. This book may not be re
produced, in whole or in part, in any form, ex
cept by written permission from the publishers.
PRINTED IN THE UNITED STATES OF AMERICA
PKEFACB BY PROFESSOR GIBBS
SINCE the printing of a short pamphlet on the Elements of
Vector Analysis in the years 1881-84, — never published, but
somewhat widely circulated among those who were known to
be interested in the subject, — the desire has been expressed
in more than one quarter, that the substance of that trea
tise, perhaps in fuller form, should be made accessible to
the public.
As, however, the years passed without my finding the
leisure to meet this want, which seemed a real one, I was
very glad to have one of the hearers of my course on Vector
Analysis in the year 1899-1900 undertake the preparation of
a text-book on the subject.
I have not desired that Dr. Wilson should aim simply
at the reproduction of my lectures, but rather that he should
use his own judgment in all respects for the production of a
text-book in which the subject should be so illustrated by an
adequate number of examples as to meet the wants of stu
dents of geometry and physics.
J. WILLARD GIBBS.
YALE UNIVERSITY, September, 1901.
GENERAL PREFACE
WHEN I undertook to adapt the lectures of Professor Gibbs
on VECTOR ANALYSIS for publication in the Yale Bicenten
nial Series, Professor Gibbs himself was already so fully
engaged upon his work to appear in the same series, Elementary
Principles in Statistical Mechanics, that it was understood no
material assistance in the composition of this book could be
expected from him. For this reason he wished me to feel
entirely free to use my own discretion alike in the selection
of the topics to be treated and in the mode of treatment.
It has been my endeavor to use the freedom thus granted
only in so far as was necessary for presenting his method in
text-book form.
By far the greater part of the material used in the follow
ing pages has been taken from the course of lectures on
Vector Analysis delivered annually at the University by
Professor Gibbs. Some use, however, has been made of the
chapters on Vector Analysis in Mr. Oliver Heaviside's Elec
tromagnetic Theory (Electrician Series, 1893) and in Professor
Foppl's lectures on Die Maxwell' sche Theorie der Electricitdt
(Teubner, 1894). My previous study of Quaternions has
also been of great assistance.
The material thus obtained has been arranged in the way
which seems best suited to easy mastery of the subject.
Those Arts, which it seemed best to incorporate in the
text but which for various reasons may well be omitted at
the first reading have been marked with an asterisk (*). Nu
merous illustrative examples have been drawn from geometry,
mechanics, and physics. Indeed, a large part of the text has
to do with applications of the method. These applications
have not been set apart in chapters by themselves, but have
x GENERAL PREFACE
been distributed throughout the body of the book as fast as
the analysis has been developed sufficiently for their adequate
treatment. It is hoped that by this means the reader may be
better enabled to make practical use of the book. Great care
has been taken in avoiding the introduction of unnecessary
ideas, and in so illustrating each idea that is introduced as
to make its necessity evident and its meaning easy to grasp.
Thus the book is not intended as a complete exposition of
the theory of Vector Analysis, but as a text-book from which
so much of the subject as may be required for practical appli
cations may be learned. Hence a summary, including a list
of the more important formulae, and a number of exercises,
have been placed at the end of each chapter, and many less
essential points in the text have been indicated rather than
fully worked out, in the hope that the reader will supply the
details. The summary may be found useful in reviews and
for reference.
The subject of Vector Analysis naturally divides itself into
three distinct parts. First, that which concerns addition and
the scalar and vector products of vectors. Second, that which
concerns the differential and integral calculus in its relations
to scalar and vector functions. Third, that which contains
the theory of the linear vector function. The first part is
a necessary introduction to both other parts. The second
and third are mutually independent. Either may be taken
up first. For practical purposes in mathematical physics the
second must be regarded as more elementary than the third.
But a student not primarily interested in physics would nat
urally pass from the first part to the third, which he would
probably find more attractive and easy than the second.
Following this division of the subject, the main body of
the book is divided into six chapters of which two deal with
each of the three parts in the order named. Chapters I. and
II. treat of addition, subtraction, scalar multiplication, and
the scalar and vector products of vectors. The exposition
has been made quite elementary. It can readily be under
stood by and is especially suited for such readers as have a
knowledge of only the elements of Trigonometry and Ana-
GENERAL PREFACE xi
lytic Geometry. Those who are well versed in Quaternions
or allied subjects may perhaps need to read only the sum
maries. Chapters III. and IV. contain the treatment of
those topics in Vector Analysis which, though of less value
to the students of pure mathematics, are of the utmost impor
tance to students of physics. Chapters V. and VI. deal with
the linear vector function. To students of physics the linear
vector function is of particular importance in the mathemati
cal treatment of phenomena connected with non-isotropic
media ; and to the student of pure mathematics this part of
the book will probably be the most interesting of all, owing
to the fact that it leads to Multiple Algebra or the Theory
of Matrices. A concluding chapter, VII., which contains the
development of certain higher parts of the theory, a number
of applications, and a short sketch of imaginary or complex
vectors, has been added.
In the treatment of the integral calculus, Chapter IV.,
questions of mathematical rigor arise. Although modern
theorists are devoting much time and thought to rigor, and
although they will doubtless criticise this portion of the book
adversely, it has been deemed best to give but little attention
to the discussion of this subject. And the more so for the
reason that whatever system of notation be employed ques
tions of rigor are indissolubly associated with the calculus
and occasion no new difficulty to the student of Vector
Analysis, who must first learn what the facts are and may
postpone until later the detailed consideration of the restric
tions that are put upon those facts.
Notwithstanding the efforts which have been made during
more than half a century to introduce Quaternions into
physics the fact remains that they have not found wide favor.
On the other hand there has been a growing tendency espe
cially in the last decade toward the adoption of some form of
Vector Analysis. The works of Heaviside and Foppl re
ferred to before may be cited in evidence. As yet however
no system of Vector Analysis which makes any claim to
completeness has been published. In fact Heaviside says :
"I am in hopes that the chapter which I now finish may
xii GENERAL PREFACE
serve as a stopgap till regular vectorial treatises come to be
written suitable for physicists, based upon the vectorial treat
ment of vectors" (Electromagnetic Theory, Vol. I., p. 305).
Elsewhere in the same chapter Heaviside has set forth the
claims of vector analysis as against Quaternions, and others
have expressed similar views.
The keynote, then, to any system of vector analysis must
be its practical utility. This, I feel confident, was Professor
Gibbs's point of view in building up his system. He uses it
entirely in his courses on Electricity and Magnetism and on
Electromagnetic Theory of Light. In writing this book I
have tried to present the subject from this practical stand
point, and keep clearly before the reader's mind the ques
tions: What combinations or functions of vectors occur in
physics and geometry ? And how may these be represented
symbolically in the way best suited to facile analytic manip
ulation ? The treatment of these questions in modern books
on physics has been too much confined to the addition and
subtraction of vectors. This is scarcely enough. It has
been the aim here to give also an exposition of scalar and
vector products, of the operator y, of divergence and curl
which have gained such universal recognition since the ap
pearance of Maxwell's Treatise on Electricity and Magnetism,
of slope, potential, linear vector function, etc., such as shall
be adequate for the needs of students of physics at the
present day and adapted to them.
It has been asserted by some that Quaternions, Vector
Analysis, and all such algebras are of little value for investi
gating questions in mathematical physics. Whether this
assertion shall prove true or not, one may still maintain that
vectors are to mathematical physics what invariants are to
geometry. As every geometer must be thoroughly conver
sant with the ideas of invariants, so every student of physics
should be able to think in terms of vectors. And there is
no way in which he, especially at the beginning of his sci
entific studies, can come to so true an appreciation of the
importance of vectors and of the ideas connected with them
as by working in Vector Analysis and dealing directly with
GENERAL PREFACE xiii
the vectors themselves. To those that hold these views the
success of Professor Foppl's Vorlesungen uber Technische
Mechanik (four volumes, Teubner, 1897-1900, already in a
second edition), in which the theory of mechanics is devel
oped by means of a vector analysis, can be but an encour
aging sign.
I take pleasure in thanking my colleagues, Dr. M. B. Porter
and Prof. H. A. Bumstead, for assisting me with the manu
script. The good services of the latter have been particularly
valuable in arranging Chapters III. and IV* in their present
form and in suggesting many of the illustrations used in the
work. I am also under obligations to my father, Mr. Edwin
H. Wilson, for help in connection both with the proofs and
the manuscript. Finally, I wish to express my deep indebt
edness to Professor Gibbs. For although he has been so
preoccupied as to be unable to read either manuscript or
proof, he has always been ready to talk matters over with
me, and it is he who has furnished me with inspiration suf
ficient to carry through the work.
EDWIN BIDWELL WILSON.
YALE UNIVERSITY, October, 1901.
PREFACE TO THE SECOND EDITION
THE only changes which have been made in this edition are
a few corrections which my readers have been kind enough to
point out to me.
E. B. W.
TABLE OF CONTENTS
PAGE
PREFACE BY PROFESSOR GIBBS • vii
GENERAL PREFACE ix
CHAPTER I
ADDITION AND SCALAR MULTIPLICATION
ARTS.
1-3 SCALARS AND VECTORS 1
4 EQUAL AND NULL VECTORS 4
5 THE POINT OF VIEW OF THIS CHAPTER 6
6-7 SCALAR MULTIPLICATION. THE NEGATIVE SIGN .... 7
8-10 ADDITION. THE PARALLELOGRAM LAW 8
11 SUBTRACTION 11
12 LAWS GOVERNING THE FOREGOING OPERATIONS .... 12
13-16 COMPONENTS OF VECTORS. VECTOR EQUATIONS .... 14
17 THE THREE UNIT VECTORS 1, j, k 18
18-19 APPLICATIONS TO SUNDRY PROBLEMS IN GEOMETRY. . . 21
20-22 VECTOR RELATIONS INDEPENDENT OF THE ORIGIN ... 27
23-24 CENTERS OF GRAVITY. BARYCENTRIC COORDINATES . . 39
25 THE USE OF VECTORS TO DENOTE AREAS 46
SUMMARY OF CHAPTER i 51
EXERCISES ON CHAPTER i . . 52
CHAPTER II
DIRECT AND SKEW PRODUCTS OF VECTORS
27-28 THE DIRECT, SCALAR, OR DOT PRODUCT OF TWO VECTORS 55
29-30 THE DISTRIBUTIVE LAW AND APPLICATIONS 58
31-33 THE SKEW, VECTOR, OR CROSS PRODUCT OF TWO VECTORS 60
34-35 THE DISTRIBUTIVE LAW AND APPLICATIONS 63
36 THE TRIPLE PRODUCT A* B C 67
XVI
CONTENTS
ARTS. PAGE
37-38 THE SCALAR TRIPLE PRODUCT A* B X C OR [ABC] . . 68
39-40 THE VECTOR TRIPLE PRODUCT A X (B X C) 71
41-42 PRODUCTS OF MORE THAN THREE VECTORS WITH APPLI
CATIONS TO TRIGONOMETRY 75
43-45 RECIPROCAL SYSTEMS OF THREE VECTORS 81
46-47 SOLUTION OF SCALAR AND VECTOR EQUATIONS LINEAR IN
AN UNKNOWN VECTOR 87
48-50 SYSTEMS OF FORCES ACTING ON A RIGID BODY .... 92
51 KINEMATICS OF A RIGID BODY 97
52 CONDITIONS FOR EQUILIBRIUM OF A RIGID BODY ... 101
53 RELATIONS BETWEEN TWO RIGHT-HANDED SYSTEMS OF
THREE PERPENDICULAR UNIT VECTORS 104
54 PROBLEMS IN GEOMETRY. PLANAR COORDINATES . . . 106
SUMMARY OF CHAPTER n 109
EXERCISES ON CHAPTER n 113
CHAPTER III
THE DIFFERENTIAL CALCULUS OF VECTORS
55-56 DERIVATIVES AND DIFFERENTIALS OF VECTOR FUNCTIONS
WITH RESPECT TO A SCALAR VARIABLE 115
57 CURVATURE AND TORSION OF GAUCHE CURVES .... 120
58-59 KINEMATICS OF A PARTICLE. THE HODOGRAPH . . . 125
60 THE INSTANTANEOUS AXIS OF ROTATION 131
61 INTEGFATION WITH APPLICATIONS TO KINEMATICS . . . 133
62 SCALAR FUNCTIONS OF POSITION IN SPACE 136
63-67 THE VECTOR DIFFERENTIATING OPERATOR V • • • • 138
68 THE SCALAR OPERATOR A • V 147
69 VECTOR FUNCTIONS OF POSITION IN SPACE 149
70 THE DIVERGENCE V* AND THE CURL VX 150
71 INTERPRETATION OF THE DIVERGENCE V 152
72 INTERPRETATION OF THE CURL V X 155
73 LAWS OF OPERATION OF V> V * > V X 157
74-76 THE PARTIAL APPLICATION OF V- EXPANSION OF A VEC
TOR FUNCTION ANALOGOUS TO TAYLOR'S THEOREM.
APPLICATION TO HYDROMECHANICS 159
77 THE DIFFERENTIATING OPERATORS OF THE SECOND ORDER 166
78 GEOMETRIC INTERPRETATION OF LAPLACE'S OPERATOR
V* V AS THE DISPERSION 170
SUMMARY OF CHAPTER in 172
EXERCISES ON CHAPTER in 177
CONTENTS
xvii
CHAPTER IV
THE INTEGRAL CALCULUS OF VECTORS
ARTS. PAGE
79-80 LINE INTEGRALS OF VECTOR FUNCTIONS WITH APPLICA
TIONS 179
81 GAUSS'S THEOREM 184
82 STOKES'S THEOREM 187
83 CONVERSE OF STOKES'S THEOREM WITH APPLICATIONS . 193
84 TRANSFORMATIONS OF LINE, SURFACE, AND VOLUME IN
TEGRALS. GREEN'S THEOREM 197
85 REMARKS ON MULTIPLE-VALUED FUNCTIONS 200
86-87 POTENTIAL. THE INTEGRATING OPERATOR " POT " . . 205
88 COMMUTATIVE PROPERTY OF POT AND V 211
89 REMARKS UPON THE FOREGOING 215
90 THE INTEGRATING OPERATORS "NEW," "LAP," " MAX " 222
91 RELATIONS BETWEEN THE INTEGRATING AND DIFFER
ENTIATING OPERATORS 228
92 THE POTENTIAL " POT " is A SOLUTION OF POISSON'S
EQUATION 230
93-94 SOLENOIDAL AND IRROTATIONAL PARTS OF A VECTOR
FUNCTION. CERTAIN OPERATORS AND THEIR INVERSE . 234
95 MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, AND
MAXWELLIANS 240
96 CERTAIN BOUNDARY VALUE THEOREMS 243
SUMMARY OF CHAPTER iv 249
EXERCISES ON CHAPTER iv 255
CHAFIER V
LINEAR VECTOR FUNCTIONS
97-98 LINEAR VECTOR FUNCTIONS DEFINED 260
99 DYADICS DEFINED 264
100 ANY LINEAR VECTOR FUNCTION MAY BE REPRESENTED
BY A DYADIC. PROPERTIES OF DYADICS .... 266
101 THE NONION FORM OF A DYADIC 269
102 THE DYAD OR INDETERMINATE PRODUCT OF TWO VEC
TORS IS THE MOST GENERAL. FUNCTIONAL PROPERTY
OF THE SCALAR AND VECTOR PRODUCTS 271
108-104 PRODUCTS OF DYADICS 276
105-107 DEGREES OF NULLITY OF DYADICS 282
108 THE IDEMFACTOR 288
XV111
CONTENTS
ARTS. PAGE
109-110 RECIPROCAL DYADICS. POWERS AND ROOTS OF DYADICS 290
111 CONJUGATE DYADICS. SELF-CONJUGATE AND ANTI-
SELF-CONJUGATE PARTS OF A DYADIC 294
112-114 ANTI-SELF-CONJUGATE DYADICS. THE VECTOR PROD
UCT. QUADRANTAL VER8ORS 297
115-116 REDUCTION OF DYADICS TO NORMAL FORM .... 302
117 DOUBLE MULTIPLICATION OF DYADICS 306
118-119 THE SECOND AND THIRD OF A DYADIC . . ... 310
120 CONDITIONS FOR DIFFERENT DEGREES OF NULLITY . 313
121 NONION FORM. DETERMINANTS 315
122 INVARIANTS OF A DYADIC. THE HAMILTON-CAYLEY
EQUATION .319
SUMMARY OF CHAPTER v 321
EXERCISES ON CHAPTER v 329
CHAPTER VI
ROTATIONS AND STRAINS
123-124 HOMOGENEOUS STRAIN REPRESENTED BY A DYADIC . 332
125-126 ROTATIONS ABOUT A FIXED POINT. VERSORS . . . 334
127 THE VECTOR SEMI-TANGENT OF VERSION 339
128 BlQUADRANTAL VERSORS AND THEIR PRODUCTS . . . 343
129 CYCLIC DYADICS 347
130 RIGHT TENSORS 351
131 TONICS AND CYCLOTONICS 353
132 REDUCTION OF DYADICS TO CANONICAL FORMS, TONICS,
CYCLOTONICS, SIMPLE AND COMPLEX SHEARERS . . 356
SUMMARY OF CHAPTER vi 368
CHAPTER VII
MISCELLANEOUS APPLICATIONS
136-142 QUADRIC SURFACES 372
143-146 THE PROPAGATION OF LIGHT IN CRYSTALS .... 392
147-148 VARIABLE DYADICS 403
149-157 CURVATURE OF SURFACES 411
158-162 HARMONIC VIBRATIONS AND BIVECTORS .... 426
VECTOR ANALYSIS
VECTOR ANALYSIS
CHAPTER I
ADDITION AND SCALAR MULTIPLICATION
1.] IN mathematics and especially in physics two very
different kinds of quantity present themselves. Consider, for
example, mass, time, density, temperature, force, displacement
of a point, velocity, and acceleration. Of these quantities
some can be represented adequately by a single number —
temperature, by degrees on a thermometric scale ; time, by
years, days, or seconds ; mass and density, by numerical val- .
ues which are wholly determined when the unit of the scale
is fixed. On the other hand the remaining quantities are not
capable of such representation. Force to be sure is said to be
of so many pounds or grams weight; velocity, of so many
feet or centimeters per second. But in addition to this each
of them must be considered as having direction as well as
magnitude. A force points North, South, East, West, up,
down, or in some intermediate direction. The same is true
of displacement, velocity, and acceleration. No scale of num
bers can represent them adequately. It can represent only
their magnitude, not their direction.
2.] Definition : A vector is a quantity which is considered
as possessing direction as well as magnitude.
Definition : A scalar is a quantity which is considered as pos
sessing magnitude but no direction.
2 VECTOR ANALYSIS
The positive and negative numbers of ordinary algebra are the
typical scalars. For this reason the ordinary algebra is called
scalar algebra when necessary to distinguish it from the vector
algebra or analysis which is the subject of this book.
The typical vector is the displacement of translation in space.
Consider first a point P (Fig. 1). Let P be displaced in a
straight line and take a new position Pf.
This change of position is represented by the
line PP. The magnitude of the displace
ment is the length of PP1; the direction of
it is the direction of the line PP1 from P to
P1. Next consider a displacement not of one,
but of all the points in space. Let all the
points move in straight lines in the same direction and for the
same distance D. This is equivalent to shifting space as a
rigid body in that direction through the distance D without
rotation. Such a displacement is called a translation. It
possesses direction and magnitude. When space undergoes
a translation T, each point of space undergoes a displacement
equal to T in magnitude and direction; and conversely if
the displacement PP' which any one particular point P suf
fers in the translation T is known, then that of any other
point Q is also known : for Q Q' must be equal and parallel
to PP.
The translation T is represented geometrically or graphically
by an arrow T (Fig. 1) of which the magnitude and direction
are equal to those of the translation. The absolute position
of this arrow in space is entirely immaterial. Technically the
arrow is called a stroke. Its tail or initial point is its origin;
and its head or final point, its terminus. In the figure the
origin is designated by 0 and the terminus by T. This geo
metric quantity, a stroke, is used as the mathematical symbol
for all vectors, just as the ordinary positive and negative num
bers are used as the symbols for all scalars.
ADDITION AND SCALAR MULTIPLICATION 3
* 3.] As examples of scalar quantities mass, time, den
sity, and temperature have been mentioned. Others are dis
tance, volume, moment of inertia, work, etc. Magnitude,
however, is by no means the sole property of these quantities.
Each implies something besides magnitude. Each has its
own distinguishing characteristics, as an example of which
its dimensions in the sense well known to physicists may
be cited. A distance 3, a time 3, a work 3, etc., are very
different. The magnitude 3 is, however, a property common
to them all — perhaps the only one. Of all scalar quanti-
tities pure number is the simplest. It implies nothing but
magnitude. It is the scalar par excellence and consequently
it is used as the mathematical symbol for all scalars.
As examples of vector quantities force, displacement, velo
city, and acceleration have been given. Each of these has
other characteristics than those which belong to a vector pure
and simple. The concept of vector involves two ideas and
two alone — magnitude of the vector and direction of the
vector. But force is more complicated. When it is applied
to a rigid body the line in which it acts must be taken into
consideration; magnitude and direction alone do not suf
fice. And in case it is applied to a non-rigid body the point
of application of the force is as important as the magnitude or
direction. Such is frequently true for vector quantities other
than force. Moreover the question of dimensions is present
as in the case of scalar quantities. The mathematical vector,
the stroke, which is the primary object of consideration in
this book, abstracts from all directed quantities their magni
tude and direction and nothing but these ; just as the mathe
matical scalar, pure number, abstracts the magnitude and
that alone. Hence one must be on his guard lest from
analogy he attribute some properties to the mathematical
vector which do not belong to it ; and he must be even more
careful lest he obtain erroneous results by considering the
4 VECTOR ANALYSIS
vector quantities of physics as possessing no properties other
than those of the mathematical vector. For example it would
never do to consider force and its effects as unaltered by
shifting it parallel to itself. This warning may not be
necessary, yet it may possibly save some confusion.
4.] Inasmuch as, taken in its entirety, a vector or stroke
is but a single concept, it may appropriately be designated by
one letter. Owing however to the fundamental difference
between scalars and vectors, it is necessary to distinguish
carefully the one from the other. Sometimes, as in mathe
matical physics, the distinction is furnished by the physical
interpretation. Thus if n be the index of refraction it
must be scalar ; m, the mass, and £, the time, are also
scalars ; but /, the force, and a, the acceleration, are
vectors. When, however, the letters are regarded merely
as symbols with no particular physical significance some
typographical difference must be relied upon to distinguish
vectors from scalars. Hence in this book Clarendon type is
used for setting up vectors and ordinary type for scalars.
This permits the use of the same letter differently printed
to represent the vector and its scalar magnitude.1 Thus if
C be the electric current in magnitude and direction, C may
be used to represent the magnitude of that current ; if g be
the vector acceleration due to gravity, g may be the scalar
value of that acceleration ; if v be the velocity of a moving
mass, v may be the magnitude of that velocity. The use of
Clarendons to denote vectors makes it possible to pass from
directed quantities to their scalar magnitudes by a mere
change in the appearance of a letter without any confusing
change in the letter itself.
Definition : Two vectors are said to be equal when they have
the same magnitude and the same direction.
1 This convention, however, is by no means invariably followed. In some
instances it would prove just as undesirable as it is convenient in others. It is
chiefly valuable in the application of vectors to physics.
ADDITION AND SCALAR MULTIPLICATION 5
The equality of two vectors A and B is denoted by the
usual sign =. Thus A = B
Evidently a vector or stroke is not altered by shifting it
about parallel to itself in space. Hence any vector A = PPr
(Fig. 1) may be drawn from any assigned point 0 as origin ;
for the segment PPf may be moved parallel to itself until
the point P falls upon the point 0 and P' upon some point T.
In this way all vectors in space may be replaced by directed
segments radiating from one fixed point 0. Equal vectors
in space will of course coincide, when placed with their ter
mini at the same point 0. Thus (Fig. 1) A = PP\ and B = Q~Qf,
both fall upon T = ~OT.
For the numerical determination of a vector three scalars
are necessary. These may be chosen in a variety of ways.
If r, </>, 0 be polar coordinates in space any vector r drawn
with its origin at the origin of coordinates may be represented
by the three scalars r, </>, 6 which determine the terminus of
the vector. r~(r,*,0).
Or if #, y9 z be Cartesian coordinates in space a vector r may
be considered as given by the differences of the coordinates a/,
y'i zf of its terminus and those #, y, z of its origin.
r~ (xr — x,yr — y,zr — z).
If in particular the origin of the vector coincide with the
origin of coordinates, the vector will be represented by the
three coordinates of its terminus
r -(*',*,', *')•
When two vectors are equal the three scalars which repre
sent them must be equal respectively each to each. Hence
one vector equality implies three scalar equalities.
6 VECTOR ANALYSIS
Definition : A vector A is said to be equal to zero when its
magnitude A is zero.
Such a vector A is called a null or zero vector and is written
equal to naught in the usual manner. Thus
A = 0 if A = 0.
All null vectors are regarded as equal to each other without
any considerations of direction.
In fact a null vector from a geometrical standpoint would
be represented by a linear segment of length zero — that is to
say, by a point. It consequently would have a wholly inde
terminate direction or, what amounts to the same thing, none at
all. If, however, it be regarded as the limit approached by a
vector of finite length, it might be considered to have that
direction which is the limit approached by the direction of the
finite vector, when the length decreases indefinitely and ap
proaches zero as a limit. The justification for disregarding
this direction and looking upon all null vectors as equal is
that when they are added (Art. 8) to other vectors no change
occurs and when multiplied (Arts. 27, 31) by other vectors
the product is zero.
5.] In extending to vectors the fundamental operations
of algebra and arithmetic, namely, addition, subtraction, and
multiplication, care must be exercised riot only to avoid self-
contradictory definitions but also to lay down useful ones.
Both these ends may be accomplished most naturally and
easily by looking to physics (for in that science vectors con
tinually present themselves) and by observing how such
quantities are treated there. If then A be a given displace
ment, force, or velocity, what is two, three, or in general x
times A? What, the negative of A? And if B be another,
what is the sum of A and B ? That is to say, what is the
equivalent of A and B taken together ? The obvious answers
to these questions suggest immediately the desired definitions.
ADDITION AND SCALAR MULTIPLICATION 1
Scalar Multiplication
6.] Definition: A vector is said to be multiplied by a
positive scalar when its magnitude is multiplied by that scalar
and its direction is left unaltered
Thus if v be a velocity of nine knots East by North, 2 £ times
v is a velocity of twenty-one knots with the direction still
East by North. Or if f be the force exerted upon the scale-
pan by a gram weight, 1000 times f is the force exerted by a
kilogram. The direction in both cases is vertically down
ward.
If A be the vector and x the scalar the product of x and A is
denoted as usual by
x A or A x.
It is, however, more customary to place the scalar multiplier
before the multiplicand A. This multiplication by a scalar
is called scalar multiplication, and it follows the associative law
x (y A) = (x y) A = y (x A)
as in ordinary algebra and arithmetic. This statement is im
mediately obvious when the fact is taken into consideration
that scalar multiplication does not alter direction but merely
multiplies the length.
Definition : A unit vector is one whose magnitude is unity.
Any vector A may be looked upon as the product of a unit
vector a in its direction by the positive scalar A, its magni
tude.
A = A a = a A.
The unit vector a may similarly be written as the product of
A by I/A or as the quotient of A and A.
1 A
a = ^ A = -I'
A A
8 VECTOR ANALYSIS
7.] Definition : The negative sign, — prefixed to a vector
reverses its direction but leaves its magnitude unchanged.
For example if A be a displacement for two feet to the right,
— A is a displacement for two feet to the left. Again if the
stroke A~B be A, the stroke B A, which is of the same length
as A £ but which is in the direction from B to A instead of
from A to 5, will be — A. Another illustration of the use
of the negative sign may be taken from Newton's third law
of motion. If A denote an "action," — A will denote the
" reaction." The positive sign, + , may be prefixed to a vec
tor to call particular attention to the fact that the direction
has not been reversed. The two signs + and — when used
in connection with scalar multiplication of vectors follow the
same laws of operation as in ordinary algebra. These are
symbolically
+ + = + ; +- = -; - + = -; = +;
— (ra A) = m (— A).
The interpretation is obvious.
Addition and Subtraction
8.] The addition of two vectors or strokes may be treated
most simply by regarding them as defining translations in
space (Art. 2), Let S be one vector and T the other. Let P
be a point of space (Fig. 2). The trans
lation S carries P into P1 such that the
line PP1 is equal to S in magnitude and
direction. The transformation T will then
carry P1 into P11 — the line P' P" being
parallel to T and equal to it in magnitude.
FIG. 2. Consequently the result of S followed by
T is to carry the point P into the point
P". If now Q be any other point in space, S will carry Q
into Q' such that Q~Qr = S and T will then carry Qf into Q"
ADDITION AND SCALAR MULTIPLICATION
such that Q'Q" = T. Thus S followed by T carries Q into Q".
Moreover, the triangle Q Qf Q" is equal to PP'P". For
the two sides Q Qf and Q' Q", being equal and parallel to S
and T respectively, must be likewise parallel to P P1 and
P'P" respectively which are also parallel to S and T. Hence
the third sides of the triangles must be equal and parallel
That is
Q Q" is equal and parallel to PP".
As Q is any point in space this is equivalent to saying that
by means of S followed by T all points of space are displaced
the same amount and in the same direction. This displace
ment is therefore a translation. Consequently the two
translations S and T are equivalent to a single translation R.
Moreover
if S = PP' and T = P' P", then R = PP".
The stroke R is called the resultant or sum of the two
strokes S and T to which it is equivalent. This sum is de
noted in the usual manner by
R = S + T.
From analogy with the sum or resultant of two translations
the following definition for the addition of any two vectors is
laid down.
Definition : The sum or resultant of two vectors is found
by placing the origin of the second upon the terminus of the
first and drawing the vector from the origin of the first to the
terminus of the second.
9.] Theorem. The order in which two vectors S and T are
added does not affect the sum.
S followed by T gives precisely the same result as T followed
by S. For let S carry P into P' (Fig. 3) ; and T, P' into P".
S + T then carries P into P". Suppose now that T carries P
into P'". The line PP'" is equal and parallel to PP". Con-
10 VECTOR ANALYSIS
sequently the points P, P'9 Pff, and Pm lie at the vertices of
a parallelogram. Hence
pm pn js equal an(J par-
allel to PP. Hence S
carries P"f into P". T fol
lowed by S therefore car
ries P into P" through P\
whereas S followed by T
carries P into P" through
Pm. The final result is in
either case the same. This may be designated symbolically
by writing
R = S + T = T + S.
It is to be noticed that S = PP1 and T = PPm are the two sides
of the parallelogram pprp»p"' which1 have the point P as
common origin ; and that JL=PP" is the diagonal drawn
through P. This leads to another very common way of
stating the definition of the sum of two vectors.
If two vectors be drawn from the same origin and a parallelo
gram be constructed upon them as sides, their sum will be that
diagonal which passes through their common origin.
This is the well-known " parallelogram law " according to
which the physical vector quantities force, acceleration, veloc
ity, and angular velocity are compounded. It is important to
note that in case the vectors lie along the same line vector
addition becomes equivalent to algebraic scalar addition. The
lengths of the two vectors to be added are added if the vectors
have the same direction ; but subtracted if they have oppo
site directions. In either case the sum has the same direction
as that of the greater vector.
10.] After the definition of the sum of two vectors has
been laid down, the sum of several may be found by adding
together the first two, to this sum the third, to this the fourth,
and so on until all the vectors have been combined into a sin-
ADDITION AND SCALAR MULTIPLICATION 11
gle one. The final result is the same as that obtained by placing
the origin of each succeeding vector upon the terminus of the
preceding one and then drawing at once the vector from
the origin of the first to the terminus of the last. In case
these two points coincide the vectors form a closed polygon
and their sum is zero. Interpreted geometrically this states
that if a number of displacements R, S, T • • • are such that the
strokes R, S, T • • • form the sides of a closed polygon taken in
order, then the effect of carrying out the displacements is nil.
Each point of space is brought back to its starting point. In
terpreted in mechanics it states that if any number of forces
act at a point and if they form the sides of a closed polygon
taken in order, then the resultant force is zero and the point
is in equilibrium under the action of the forces.
The order of sequence of the vectors in a sum is of no con
sequence. This may be shown by proving that any two adja
cent vectors may be interchanged without affecting the result.
To show
Let A = 0 A, B = A B, C = B C, D = 0 D, E = D E.
Then
_
Let now B C1 = D. Then C! B C D is a parallelogram and
consequently Cf D = C. Hence
OJ£ = A + B + D + C + E,
which proves the statement. Since any two adjacent vectors
may be interchanged, and since the sum may be arranged in
any order by successive interchanges of adjacent vectors, the
order in which the vectors occur in the sum is immaterial.
11.] Definition : A vector is said to be subtracted when it
is added after reversal of direction. Symbolically,
A - B = A + (- B).
By this means subtraction is reduced to addition and needs
12
VECTOR ANALYSIS
no special consideration. There is however an interesting and
important way of representing the difference of two vectors
geometrically. Let A = 'OA, B = 0IT(Fig. 4). Complete
the parallelogram of which A and B
are the sides. Then the diagonal
~OG = C is the sum A + B of the
two vectors. Next complete the
parallelogram of which A and — B
= OB' are the sides. Then the di
agonal 02) = !) will be the sum of
A and the negative of B. But the
segment OD is parallel and equal
to BA. Hence BA may be taken as the difference to the two
vectors A and B. This leads to the following rule : The differ
ence of two vectors which are drawn from the same origin is
the vector drawn from the terminus of the vector to be sub
tracted to the terminus of the vector from which it is sub
tracted. Thus the two diagonals of the parallelogram, which
is constructed upon A and B as sides, give the sum and dif
ference of A and B.
12.] In the foregoing paragraphs addition, subtraction, and
scalar multiplication of vectors have been defined and inter
preted. To make the development of vector algebra mathe
matically exact and systematic it would now become necessary
to demonstrate that these three fundamental operations follow
the same formal laws as in the ordinary scalar algebra, al
though from the standpoint of the physical and geometrical
interpretation _of vectors this may seem superfluous. These
laws are
m (n A) = n (m A) = (m n} A,
(A + B) + C = A+ (B + C),
II A + B r, B + A,
IIIa (m + n) A = m A + n A,
m (A + B) = m A + m B,
III, - (A + B) = - A - B.
ADDITION AND SCALAR MULTIPLICATION 13
10 is the so-called law of association and commutation of
the scalar factors in scalar multiplication.
I6 is the law of association for vectors in vector addition. It
states that in adding vectors parentheses may be inserted at
any points without altering the result.
11 is the commutative law of vector addition.
IIIa is the distributive law for scalars in scalar multipli
cation.
III6 is the distributive law for vectors in scalar multipli
cation.
Ill, is the distributive law for the negative sign.
The proofs of these laws of operation depend upon those
propositions in elementary geometry which have to deal with
the first properties of the parallelogram and similar triangles.
They will not be given here; but it is suggested that the
reader work them out for the sake of fixing the fundamental
ideas of addition, subtraction, and scalar multiplication more
clearly in mind. The result of the laws may be summed up
in the statement :
The laws which govern addition, subtraction, and scalar
multiplication of vectors are identical with those governing these
operations in ordinary scalar algebra.
It is precisely this identity of formal laws which justifies
the extension of the use of the familiar signs =, +, and —
of arithmetic to the algebra of vectors and it is also this
which ensures the correctness of results obtained by operat
ing with those signs in the usual manner. One caution only
need be mentioned. Scalars and vectors are entirely different
sorts of quantity. For this reason they can never be equated
to each other — except perhaps in the trivial case where each is
zero. For the same reason they are not to be added together.
So long as this is borne in mind no difficulty need be antici
pated from dealing with vectors much as if they were scalars.
Thus from equations in which the vectors enter linearly with
14 VECTOR ANALYSIS
scalar coefficients unknown vectors may be eliminated or
found by solution in the same way and with the same limita
tions as in ordinary algebra; for the eliminations and solu
tions depend solely on the scalar coefficients of the equations
and not at all on what the variables represent. If for
instance
aA + &B + cC + dD = 0,
then A, B, C, or D may be expressed in terms of the other
three
as D = --:OA + &B + cC).
a
And two vector equations such as
3 A+ 4B=E
and 2 A + 3 B = F
yield by the usual processes the solutions
A=3E-4F
and B = 3 F - 2 E.
Components of Vectors
13.] Definition : Vectors are said to be collinear when
they are parallel to the same line; coplanar, when parallel
to the same plane. Two or more vectors to which no line
can be drawn parallel are said to be non-collinear. Three or
more vectors to which no plane can be drawn parallel are
said to be non-coplanar. Obviously any two vectors are
coplanar.
Any vector b collinear with a may be expressed as the
product of a and a positive or negative scalar which is the
ratio of the magnitude of b to that of a. The sign is positive
when b and a have the same direction ; negative, when they
have opposite directions. If then OA = a, the vector r drawn
ADDITION AND SCALAR MULTIPLICATION 15
from the origin 0 to any point of the line 0 A produced in
either direction is
r = x a. (1)
If x be a variable scalar parameter this equation may there
fore be regarded as the (vector) equation of all points in the
line OA. Let now B be any point not
upon the line OA or that line produced •
in either direction (Fig. 5).
Let OB = b. The vector b is surely
not of the form x a. Draw through B Flo 5 "
a line parallel to OA and Let R be any
point upon it. The vector BE is collinear with a and is
consequently expressible as #a. Hence the vector drawn
from 0 to R is
0~E=0~B + ITR
or r = b + #a. (2)
This equation may be regarded as the (vector) equation of
all the points in the line which is parallel to a and of which
B is one point.
14.] Any vector r coplanar with two non-collinear vectors
a and b may be resolved into two components parallel to a
and b respectively. This resolution may
be accomplished by constructing the par
allelogram (Fig. 6) of which the sides are
parallel to a and b and of which the di
agonal is r. Of these components one is
x a ; the other, y b. x and y are respec
tively the scalar ratios (taken with the
proper sign) of the lengths of these components to the lengths
of a and b, Hence
r = x a + y b (2)'
is a typical form for any vector coplanar with a and b. If
several vectors rx, r2, r3 • • • may be expressed in this form as
16
VECTOR ANALYSIS
their sum r is then
rl = xl a + yl b,
r2 = #2 a + 2/2 b,
r3 = xz a + 2/3 b.
+ (ft + ft + ft + •••) »•
This is the well-known theorem that the components of a
sum of vectors are the sums of the components of those
vectors. If the vector r is zero each of its components must
be zero. Consequently the one vector equation r = 0 is
equivalent to the two scalar equations
y\ + ft + ft + • • • =
(3)
15.] Any vector r in space may be resolved into three
components parallel to any three given non-coplanar vectors.
Let the vectors be a, b,
and c. The resolution
may then be accom
plished by constructing
the parallelepiped (Fig.
7) of which the edges
are parallel to a, b, and
c and of which the di
agonal is r. This par-
allelopiped may be
drawn easily by passing
three planes parallel re
spectively to a and b, b and c, c and a through the origin 0
of the vector r ; and a similar set of three planes through its
terminus It. These six planes will then be parallel in pairs
FIG. 7.
ADDITION AND SCALAR MULTIPLICATION 17
and hence form a parallelepiped. That the intersections of
the planes are lines which are parallel to a, or b, or c is
obvious. The three components of r are x a, y b, and zc;
where x, y, and z are respectively the scalar ratios (taken with
the proper sign) of the lengths of these components to the
length of a, b, and c. Hence
r = # a + 7/b + zc (4)
is a typical form for any vector whatsoever in space. Several
vectors rlf r2, r3 . . . may be expressed in this form as
rx = xl a + yl b + zl c,
r2 = #2 a + y2 b + *2 c»
1*3 = XZ a + 2/3 b
Their sum r is then
1 =rl + r2 + F3 + * ' ' = 0*1 + *2 + XZ + ' • 0 a
+ (2/i + 2/2+ 3/3 + •••)!>
+ Ol +^2 + ^3+ "O0-
If the vector r is zero each of its three components is zero.
Consequently the one vector equation r = 0 is equivalent to
the three scalar equations
xl + #2 + #3 + • - • = 0 v
2/i + 2/2 + 2/3 + • • • = ° y r = 0. (5)
*i + *2 + % + • • • = 0 /
Should the vectors all be coplanar with a and b, all the com
ponents parallel to c vanish. In this case therefore the above
equations reduce to those given before.
16.] If two equal vectors are expressed in terms of the
same three non-coplanar vectors, the corresponding scalar co
efficients are equal.
18 VECTOR ANALYSIS
Let r = r',
r' = x9 a + y1 b + z' c,
'
Then x = x', y = y
For r - r' = 0 = (x - xf) a + (y - y') b + (* - z1) c.
Hence x - *' = 0, y - y' = 0, z - *' = 0.
But this would not be true if a, b, and c were coplanar. In
that case one of the three vectors could be expressed in terms
of the other two as
c = m a + n b.
Then r = #a + y b + s c = (a + m z) a + (y + TI z) b,
r' = x! a + y1 b + z;c = (x1 + m z') a + (y' + n z') b,
r — r' = [(x + m z ) — (x' + m z')] a,
Hence the individual components of r — r' in the directions
a and b (supposed different) are zero.
Hence x + mz = xr + mzr
y -f n z = yf + n z1.
But this by no means necessitates x, y, z to be equal respec
tively to x\ y\ z1. In a similar manner if a and b were col-
linear it is impossible to infer that their coefficients vanish
individually. The theorem may perhaps be stated as follows :
. In case two equal vectors are expressed in terms of one vector,
or two non-collinear vectors, or three non-coplanar vectors, the
corresponding scalar coefficients are equal. But this is not ne
cessarily true if the two vectors be collinear ; or the three vectors,
coplanar. This principle will be used in the applications
(Arts. 18 et seq.).
The Three Unit Vectors i, j, k.
17.] In the foregoing paragraphs the method of express
ing vectors in terms of three given non-coplanar ones has been
explained. The simplest set of three such vectors is the rect-
ADDITION AND SCALAR MULTIPLICATION 19
angular system familiar in Solid Cartesian Geometry. This
rectangular system may however be either of two very distinct
types. In one case (Fig. 8, first part) the Z-axis l lies upon
that side of the X Y- plane on which rotation through a right
angle from the X-axis to the F-axis appears counterclockwise
or positive according to the convention adopted in Trigonome
try. This relation may be stated in another form. If the X
axis be directed to the right and the F-axis vertically, the
^-axis will be directed toward the observer. Or if the X-
axis point toward the observer and the F-axis to the right,
the ^-axis will point upward. Still another method of state-
Z
,,k
Right-handed
FIG. 8.
Left-handed
ment is common in mathematical physics and engineering. If
a right-handed screw be turned from the Xaxis to the F-
axis it will advance along the (positive) Z-axis. Such a sys
tem of axes is called right-handed, positive, or counterclock
wise.2 It is easy to see that the F-axis lies upon that side of
the ^X-plane on which rotation from the ^-axis to the X-
axis is counterclockwise ; and the X-axis, upon that side of
1 By the X-, Y-, or Z-axis the positive half of that axis is meant. The X Y-
plane means the plane which contains the X- and Y-axis, i. e., the plane z = 0.
2 A convenient right-handed system and one which is always available consists
of the thumb, first finger, and second finger of the right hand. If the thumb and
first finger be stretched out from the palm perpendicular to each other, and if the
second finger be bent over toward the palm at right angles to first finger, a right-
handed system is formed by the fingers taken in the order thumb, first finger,
second finger.
20 VECTOR ANALYSTS
the F^-plane on which rotation from the F-axis to the Z-
axis is counterclockwise. Thus it appears that the relation
between the three axes is perfectly symmetrical so long as the
same cyclic order XYZXY is observed. If a right-handed
screw is turned from one axis toward the next it advances
along the third.
In the other case (Fig. 8, second part) the ^-axis lies upon
that side of the X F-plane on which rotation through a right
angle from the JT-axis to the F-axis appears clockwise or neg
ative. The F-axis then lies upon that side of the ^X-plane
on which rotation from the ^-axis to the X-axis appears
clockwise and a similar statement may be made concerning
the X-axis in its relation to the F^-plane. In this case, too,
the relation between the three axes is S3rmmetrical so long
as the same cyclic order X YZX Y is preserved but it is just
the opposite of that in the former case. If a fe/Mianded screw
is turned from one axis toward the next it advances along
the third. Hence this system is called left-handed, negative,
or clockwise.1
The two systems are not superposable. They are sym
metric. One is the image of the other as seen in a
mirror. If the JT- and F-axes of the two different systems be
superimposed, the ^-axes will point in opposite directions.
Thus one system may be obtained from the other by reversing
the direction of one of the axes. A little thought will show
that if two of the axes be reversed in direction the system will
not be altered, but if all three be so reversed it will be.
Which of the two systems be used, matters little. But in
asmuch as the formulae of geometry and mechanics differ
slightly in the matter of sign, it is advisable to settle once for
all which shall be adopted. In this book the right-handed or
counterclockwise system will be invariably employed.
1 A left-handed system may be formed by the left hand just as a right-handed
one was formed by the right.
ADDITION AND SCALAR MULTIPLICATION 21
Definition : The three letters i, j, k will be reserved to de
note three vectors of unit length drawn respectively in the
directions of the JT-, T-, and Z- axes of a right-handed rectan
gular system.
In terms of these vectors, any vector may be expressed as
r = xi + y] + zk. (6)
The coefficients xy y, z are the ordinary Cartesian coordinates
of the terminus of r if its origin be situated at the origin of
coordinates. The components of r parallel to the X-, F-, and
^f-axes are respectively
x i, y j, z k.
The rotations about i from j to k, about j from k to i, and
about k from i to j are all positive.
By means of these vectors i, j, k such a correspondence is
established between vector analysis and the analysis in Car
tesian coordinates that it becomes possible to pass at will
from either one to the other. There is nothing contradic
tory between them. On the contrary it is often desirable
or even necessary to translate the formulae obtained by
vector methods into Cartesian coordinates for the sake of
comparing them with results already known and it is
still more frequently convenient to pass from Cartesian
analysis to vectors both on account of the brevity thereby
obtained and because the vector expressions show forth the
intrinsic meaning of the formulae.
Applications
*18.J Problems in plane geometry may frequently be solved
easily by vector methods. Any two non-collinear vectors in
the plane may be taken as the fundamental ones in terms of
which all others in that plane may be expressed. The origin
may also be selected at pleasure. Often it is possible to
22 VECTOR ANALYSIS
make such an advantageous choice of the origin and funda
mental vectors that the analytic work of solution is materially
simplified. The adaptability of the vector method is about
the same as that of oblique Cartesian coordinates with differ
ent scales upon the two axes.
Example 1 : The line which joins one vertex of a parallelo
gram to the middle point of an opposite side trisects the diag
onal (Fig. 9).
Let A BCD be the parallelogram, BE the line joining the
vertex B to the middle point E of the side
AD, R the point in which this line cuts the
diagonal A C. To show A R is one third of
FlG 9 AC. Choose A as origin, A B and AD as the
two fundamental vectors S and T. Then
A C is the sum of S and T. Let AR = R. To show
R = 1 (S + T).
-
where x is the ratio of ER to EB — an unknown scalar.
And R = y (S + T),
where y is the scalar ratio of A R to A C to be shown equal
to.
Hence \ T + x (S -i T) = y (S + T)
or * S + 1 (1 - X) T = y S + y T.
Hence, equating corresponding coefficients (Art. 16),
2 (1 - x) = y.
ADDITION AND SCALAR MULTIPLICATION 23
From which y = « .
Inasmuch as x is also «- the line j&2? must be trisected as
o
well as the diagonal A C.
Example 2 : If through any point within a triangle lines
be drawn parallel to the sides the sum of the ratios of these
lines to their corresponding sides is 2.
Let ABC be the triangle, R the point within it. Choose
A as origin, A B and A C as the two fundamental vectors S
and T. Let
AR = R = w S + 7i T. (a)
m S is the fraction of A B which is cut off by the line through
R parallel to A C. The remainder of A B must be the frac
tion (1 — m) S. Consequently by similar triangles the ratio of
the line parallel to A C to the line A C itself is (1 — ra).
Similarly the ratio of the line parallel to A B to the line A B
itself is (1 — n ). Next express R in terms of S and T — S the
third side of the triangle. Evidently from (a)
R = (m + ri) S + n (T - S).
Hence (m + ri) S is the fraction of A B which is cut off by the
line through R parallel to B C. Consequently by similar tri
angles the ratio of this line to BC itself is (m + n). Adding
the three ratios
(1 - m) + (1 - n) + (m + ri) = 2,
and the theorem is proved.
Example 3 : If from any point within a parallelogram lines
be drawn parallel to the sides, the diagonals of the parallelo
grams thus formed intersect upon the diagonal of the given
parallelogram.
Let A B CD be a parallelogram, R a point within it, KM
and LN two lines through R parallel respectively to AB and
24 VECTOR ANALYSIS
AD, the points K, Z, M, N lying upon the sides DA, AS,
B C, CD respectively. To show that the diagonals KN and
LM of the two parallelograms KRND and LBME meet
on A C. Choose A as origin, A B and A D as the two funda
mental vectors S and T. Let
R = AB = m S 4- ft T,
and let P be the point of intersection of KN with LM.
Then KN=KR + BN = m S + (1 - rc) T,
=(1 -m) S + 7i T,
Hence P = n T + x [m S + (1 — n) T],
and P = m S + y [(1 - m) S + n T].
Equating coefficients,
x m = m + y (1 — m)
By solution, ;
m + n — 1
m
~ m + n — 1
Substituting either of these solutions in the expression for P,
the result is
P^-^-^S + T),
which shows that P is collinear with A C.
* 19.] Problems in three dimensional geometry may be
solved in essentially the same manner as those in two dimen
sions. In this case there are three fundamental vectors in
terms of which all others can be expressed. The method of
solution is analogous to that in the simpler case. Two
ADDITION AND SCALAR MULTIPLICATION 25
expressions for the same vector are usually found. The co
efficients of the corresponding terms are equated. In this way
the equations between three unknown scalars are obtained
from which those scalars may be determined by solution and
then substituted in either of the expressions for the required
vector. The vector method has the same degree of adapta
bility as the Cartesian method in which oblique axes with
different scales are employed. The following examples like
those in the foregoing section are worked out not so much for
their intrinsic value as for gaining a familiarity with vectors.
Example 1 : Let A B CD be a tetrahedron and P any
point within it. Join the vertices to P and produce the lines
until they intersect the opposite faces in A\ B', C1, Df. To
show
PA' PB' PC1 PD'
A~Af TTB' ~C~Of
' "
Choose A as origin, and the edges A J?, A C, AD as the
three fundamental vectors B, C, D. Let the vector A P be
P = A P=IE + raC + 7i D,
Also A' = A A' = A B + BA'.
The vector BA1 is coplanar with WC = C — B and BD
D — B. Hence it may be expressed in terms of them.
A' = B + ^1(C-B)+y1(D~B).
Equating coefficients Jcl m = xv
Hence &., =
PA' _ V
ZZ7 ~~&
1 I + m + n
PA* JL-1
and — "• ^7 '
26 VECTOR ANALYSIS
In like manner A B' = #2 C + y2 D
and A B' = ^t £ + B B' = B + &2 (P - B).
Hence o;2C + y2D = B + A:2(ZB + mC + ^D-B
and 0 = 1 + *, (J - 1),
Hence
2 -i __
' -
and
In the same way it may be shown that
PC' .PL1
CC* 3D'
Adding the four ratios the result is
•i d JL vn -4- <w ^ _L 7 J_ w -I- 77 1
Example % : To find a line which passes through a given
point and cuts two given lines in space.
Let the two lines be fixed respectively by two points A
and B, C and D on each. Let 0 be the given point. Choose
it as origin and let
C = ~OC, D=d~D.
Any point P of A B may be expressed as
P= OP= 0~A + xA£ = A + x (B- A).
Any point Q of CD may likewise be written
If the points P and Q lie in the same line through 0, P and
are collinear That is
ADDITION AND SCALAR MULTIPLICATION
27
Before it is possible to equate coefficients one of the four
vectors must be expressed in terms of the other three.
Then P = A + x (B - A)
— & Tf _1_ ( 1 A _J_ m Tl _1_ >w P I^^T
Hence 1 — x = z y /,
x = zy m,
0 = z [1 + y (n - 1)J.
Hence
m
x =
y =
2 =
i
i- »
_________ •
I + m
Substituting in P and ft
I A+ m B
+ m
ft =
Either of these may be taken as defining a line drawn from 0
and cutting A B and CD.
Vector Relations independent of the Origin
20.] Example 1 : To divide a line A B in a given ratio
m : n (Fig. 10).
Choose any arbitrary point 0 as
origin. Let OA = A and OB = B.
To find the vector P = ~OP of which
the terminus P divides AB in the
ratio m : n.
m
B
FIG. 10.
That is,
P =
B = A
-f- 7i
n A + m B
n
(B - A).
(7)
28 VECTOR ANALYSIS
The components of P parallel to A and B are in inverse ratio
to the segments A P and PB into which the line A B is
divided by the point P. If it should so happen that P divided
the line AB externally, the ratio A P / PE would be nega
tive, and the signs of m and n would be opposite, but the
formula would hold without change if this difference of sign
in m and n be taken into account.
Example 2 : To find the point of intersection of the medians
of a triangle.
Choose the origin 0 at random. Let A BC be the given
triangle. Let 0~A = A, ()B = B, and "00 = C. Let Af,£',C'
be respectively the middle points of the sides opposite the
vertices A, B, (7. Let M be the point of intersection of the
medians and M = 0 M the vector drawn to it. Then
and
~< = B
Assuming that 0 has been chosen outside of the plane of the
triangle so that A, B, C are non-coplanar, corresponding coeffi
cients may be equated.
Hence x = y — - •
9 3
Hence M =4 (A + B + C).
ADDITION AND SCALAR MULTIPLICATION 29
The vector drawn to the median point of a triangle is equal
to one third of the sum of the vectors drawn to the vertices.
In the problems of which the solution has just been given
the origin could be chosen arbitrarily and the result is in
dependent of that choice. Hence it is even possible to disre
gard the origin entirely and replace the vectors A, B, C, etc.,
by their termini A, B, C, etc. Thus the points themselves
become the subjects of analysis and the formulae read
n A + m B
m + n
and M=~(A + B + C).
This is typical of a whole class of problems soluble by vector
methods. In fact any purely geometric relation between the
different parts of a figure must necessarily be independent
of the origin assumed for the analytic demonstration. In
some cases, such as those in Arts. 18, 19, the position of the
origin may be specialized with regard to some crucial point
of the figure so as to facilitate the computation ; but in many
other cases the generality obtained by leaving the origin un-
specialized and undetermined leads to a symmetry which
renders the results just as easy to compute and more easy
to remember.
Theorem : The necessary and sufficient condition that a
vector equation represent a relation independent of the origin
is that the sum of the scalar coefficients of the vectors on
one side of the sign of equality is equal to the sum of the
coefficients of the vectors upon the other side. Or if all the
terms of a vector equation be transposed to one side leaving
zero on the other, the sum of the scalar coefficients must
be zero.
Let the equation written in the latter form be
30 VECTOR ANALYSIS
Change the origin from 0 to 0' by adding a constant vector
B = OO1 to each of the vectors A, B, C, D ---- The equation
then becomes
a (A 4- B) + 6 (B + B) + c (C + B) + d (D + R) + • - • = 0
If this is to be independent of the origin the coefficient of B
must vanish. Hence
That this condition is fulfilled in the two examples cited
is obvious.
if
m + n
If M = \ (A f B + C),
m + n m + n
l
3
* 21.] The necessary and sufficient condition that two
vectors satisfy an equation, in which the sum of the scalar
coefficients is zero, is that the vectors be equal in magnitude
and in direction.
First let a A + 6 B = 0
and a + 6 = 0.
It is of course assumed that not both the coefficients a and b
vanish. If they did the equation would mean nothing. Sub
stitute the value of a obtained from the second equation into
the first.
-&A + 6B = 0.
Hence A = B.
ADDITION AND SCALAR MULTIPLICATION 31
Secondly if A and B are equal in magnitude and direction
the equation
A-B = 0
subsists between them. The sum of the coefficients is zero.
The necessaiy and sufficient condition that three vectors
satisfy an equation, in which the sum of the scalar coefficients
is zero, is that when drawn from a common origin they termi
nate in the same straight line.1
First let aA + 6B + cC = 0
and a + b + c = 0.
Not all the coefficients a, J, c, vanish or the equations
would be meaningless. Let c be a non-vanishing coefficient.
Substitute the value of a obtained from the second equation
into the first.
or
Hence the vector which joins the extremities of C and A is
collinear with that which joins the extremities of A and B.
Hence those three points -4, -B, C lie on a line. Secondly
suppose three vectors A= OA, B = OB,G= 00 drawn from
the same origin 0 terminate in a straight line. Then the
vectors
AB = B - A and A~C = C - A
are collinear. Hence the equation
subsists. The sum of the coefficients on the two sides is
the same.
The necessary and sufficient condition that an equation,
in which the sum of the scalar coefficients is zero, subsist
1 Vectors which have a common origin and terminate in one line are called by
Hamilton " termino-collinear:'
82 VECTOR ANALYSIS
between four vectors, is that if drawn from a common origin
they terminate in one plane.1
First let a A + 6B + cC + dV = 0
and a + b + c + d = Q.
Let d be a non-vanishing coefficient. Substitute the value
of a obtained from the last equation into the first.
or d (D - A) = 6 (A - B) + c (A - C).
The line A D is coplanar with A B and A C. Hence all four
termini A, B, (7, D of A, B, C, D lie in one plane. Secondly
suppose that the termini of A, B, C, D do lie in one plane.
Then AZ) = D - A, ~AC = C - A, and ~AB = B - A are co
planar vectors. One of them may be expressed in terms of
the other two. This leads to the equation
/ (B - A) + m (C - A) + n (D - A) = 0,
where /, m, and n are certain scalars. The sum of the coeffi
cients in this equation is zero.
Between any five vectors there exists one equation the sum
of whose coefficients is zero.
Let A, B, C,D,E be the five given vectors. Form the
differences
E-A, E--B, E-C, E-D.
One of these may be expressed in terms of the other three
- or what amounts to the same thing there must exist an
equation between them.
ft (E - A) + / (E - B) + m (E - C) + n (E - D) = 0.
The sum of the coefficients of this equation is zero.
1 Vectors which have a common origin and terminate in one plane are called
by Hamilton " termino-complanar."
ADDITION AND SCALAR MULTIPLICATION 33
*22.] The results of the foregoing section afford simple
solutions of many problems connected solely with the geo
metric properties of figures. Special theorems, the vector
equations of lines and planes, and geometric nets in two and
three dimensions are taken up in order.
Example 1: If a line be drawn parallel to the base of a
triangle, the line which joins the opposite vertex to the inter
section of the diagonals of the
trapezoid thus formed bisects the
base (Fig. 11).
Let ABC be the triangle, ED
the line parallel to the base CB,
G the point of intersection of the
diagonals EB and DC of the tra
pezoid CBDE, and Fthe intersec
tion of A G with CB. To show FI(J n
that F bisects CB. Choose the
origin at random. Let the vectors drawn from it to the
various points of the figure be denoted by the corresponding
Clarendons as usual. Then since ED is by hypothesis paral
lel to CB, the equation
E - D = n (C - B)
holds true. The sum of the coefficients is evidently zero as
it should be. Rearrange the terms so that the equation
takes on the form
E — nC = "D — 7i B.
The vector E — n C is coplanar with E and C. It must cut
the line EC. The equal vector D — 7&B is coplanar with D
and B. It must cut the line DB. Consequently the vector
represented by either side of this equation must pass through
the point A. Hence
E — 7iC = D — ?iB = #A.
34 VECTOR ANALYSIS
However the points E, 0, and A lie upon the same straight
line. Hence the equation which connects the vectors E,C,
and A must be such that the sum of its coefficients is zero.
This determines x as 1 — n.
Hence B - »C = D - »B = (1 - w) A.
By another rearrangement and similar reasoning
E + 7i B =D + 7iC= (1 + n)Qt.
Subtract the first equation from the second :
n (B + C) = (1 + n) G - (1 - n) A.
This vector cuts EC and AQ. It must therefore be a
multiple of F and such a multiple that the sum of the coeffi
cients of the equations which connect B, C, and F or 0, A,
and F shall be zero.
Hence n (B + C) = (1 + »)G - (1 - ») A = 2 nf.
„•
Hence F =
and the theorem has been proved. The proof has covered
considerable space because each detail of the reasoning has
been given. In reality, however, the actual analysis has con
sisted of just four equations obtained simply from the first.
Example % : To determine the equations of the line and
plane.
Let the line be fixed by two points A and B upon it. Let
P be any point of the line. Choose an arbitrary origin.
The vectors A, B, and P terminate in the same line. Hence
aA + 6B
and a + I + p = 0.
™ ,
Therefore P =
a + b
ADDITION AND SCALAR MULTIPLICATION 35
For different points P the scalars a and b have different
values. They may be replaced by x and y, which are used
more generally to represent variables. Then
x + y
Let a plane be determined by three points -4, B, and C.
Let P be any point of the plane. Choose an arbitrary origin.
The vectors A, B, C, and P terminate in one plane. Hence
6B + cC
and a + b + c+p = Q.
aA + 6B + cC
Therefore P =
-f c
As a, 6, c, vary for different points of the plane, it is more
customary to write in their stead x, yt z.
+ y + z
Example 3 ; The line which joins one vertex of a com
plete quadrilateral to the intersection of two diagonals
divides the opposite sides har
monically (Fig. 12).
Let A, B, C, D be four vertices
of a quadrilateral. Let A B meet
CD in a fifth vertex E, and AD
meet BC in the sixth vertex F.
Let the two diagonals AC and p 12
BD intersect in G. To show
that FG intersects A B in a point i" and CD in a point E1'
such that the lines AB and (7I> are divided internally at
E1 and 2?" in the same ratio as they are divided externally
by E. That is to show that the cross ratios
86 VECTOR ANALYSIS
Choose the origin at random. The four vectors A, B, C, D
drawn from it to the points A, B, C, D terminate in one
plane. Hence
and a + b + e + d = 0.
Separate the equations by transposing two terms :
Divide : 0 =
a + c = — (b + d).
• cC 6B +
a + c b + d
aA + d'D __ 6B + cC
a + d b + c
(a + C)G — (a + d)F cC — di
In like manner F =
Form:
(a + c) - (a + d) " (a + c) — (a + d)
(a + c)Q — (a + rf)F cC —
or
c — a c — a
Separate the equations again and divide :
aA + EB _ cC +
a -f b c + d
(6)
Hence 2? divides A B in the ratio a : b and CD in the ratio
c / d. But equation (a) shows that JEff divides C D in the
ratio — c:d. Hence E and E" divide CD internally and
externally in the same ratio. Which of the two divisions is
internal and which external depends upon the relative signs
of c and d. If they have the same sign the internal point
of division is E; if opposite signs, it is E1'. In a similar way
E1 and E may be shown to divide A B harmonically.
Example 4 - To discuss geometric nets.
By a geometric net in a plane is meant a figure composed
of points and straight lines obtained in the following manner.
Start with a certain number of points all of which lie in one
ADDITION AND SCALAR MULTIPLICATION 37
plane. Draw all the lines joining these points in pairs.
These lines will intersect each other in a number of points.
Next draw all the lines which connect these points in pairs.
This second set of lines will determine a still greater number
of points which may in turn be joined in pairs and so on.
The construction may be kept up indefinitely. At each step
the number of points and lines in the figure increases.
Probably the most interesting case of a plane geometric net is
that in which four points are given to commence with.
Joining these there are six lines which intersect in three
points different from the given four. Three new lines may
now be drawn in the figure. These cut out six new points.
From these more lines may be obtained and so on.
To treat this net analytically write down the equations
= 0 (c)
and a + b + c + d = Q
which subsist between the four vectors drawn from an unde
termined origin to the four given points. From these it is
possible to obtain
a A + 6B cC + dD
Tjl
a + b c + d
A + cC Z>B + dD
a + c b + d
A + dJ) &B + cC
a + d b + c
by splitting the equations into two parts and dividing. Next
four vectors such as A, D, E, F may be chosen and the equa
tion the sum of whose coefficients is zero may be determined.
This would be
— aA + dV + (a + b) E-f (a + c) P = 0.
By treating this equation as (c) was treated new points may
be obtained*
38 VECTOR ANALYSIS
— a A + dD (a + 6)E + (a + c)F
H =
1 =
— a + d 2a + b + c
aA + (a + ft)E __ <?D+ (a + c)F
a 4- c + d
(a + 6) E
c a + b + d
Equations between other sets of four vectors selected from
A, B, C, D, E, F, 0 may be found ; and from these more points
obtained. The process of finding more points goes forward
indefinitely. A fuller account of geometric nets may be
found in Hamilton's " Elements of Quaternions," Book I.
As regards geometric nets in space just a word may be
said. Five points are given. From these new points may be
obtained by finding the intersections of planes passed through
sets of three of the given points with lines connecting the
remaining pairs. The construction may then be carried for
ward with the points thus obtained. The analytic treatment
is similar to that in the case of plane nets. There are
five vectors drawn from an undetermined origin to the given
five points. Between these vectors there exists an equation
the sum of whose coefficients is zero. This equation may be
separated into parts as before and the new points may thus
be obtained.
+ 6B cC + dD +
then F =
a + b c + d + e
A + cC 6B + dV + e
a + b b + d + c
are two of the points and others may be found in the same
way. Nets in space are also discussed by Hamilton, loc. cit.
ADDITION AND SCALAR MULTIPLICATION 39
Centers of Gravity
* 23.] The center of gravity of a system of particles may
be found very easily by vector methods. The two laws of
physics which will be assumed are the following:
1°. The center of gravity of two masses (considered as
situated at points) lies on the line connecting the two masses
and divides it into two segments which are inversely pro
portional to the masses at the extremities.
2°. In finding the center of gravity of two systems of
masses each system may be replaced by a single mass equal
in magnitude to the sum of the masses in the system and
situated at the center of gravity of the system.
Given two masses a and b situated at two points A and B.
Their center of gravity G is given by
where the vectors are referred to any origin whatsoever.
This follows immediately from law 1 and the formula (7)
for division of a line in a given ratio.
The center of gravity of three masses a, J, c situated at the
three points -4, B, C may be found by means of law 2. The
masses a and b may be considered as equivalent to a single
mass a + b situated at the point
a A + &B
a + b
Then G = (a + 6) "A + 6B + c C
a -f- b
TT aA-h&B-f-cC
Hence G = •
a -f b + c
40 VECTOR ANALYSIS
Evidently the center of gravity of any number of masses
a, &, c, d, ... situated at the points A, B, C, D, ... may
be found in a similar manner. The result is
aA + ftB + cO + rfD + ... ^
a + b + c + d + ...
Theorem 1 : The lines which join the center of gravity of a
triangle to the vertices divide it into three triangles which
are proportional to the masses at the op
posite vertices (Fig. 13). Let A, B, C
be the vertices of a triangle weighted
with masses a, &, c. Let G be the cen
ter of gravity. Join A, B, C to G and
produce the lines until they intersect
the opposite sides in Af, B\ C1 respectively. To show that
the areas
G B C : G C A : G A B : A B C = a : b : c : a + b + c .
The last proportion between ABC and a + b + c comes
from compounding the first three. It is, however, useful in
the demonstration.
ABC AA' A G . GA' b + c
+ 1.
Hence
GBC~ GA! ' CTA' G~Af
ABC a + b + c
In a similar manner
and
GBC a
BCA a + I + c
GCA~ b
CAB _ a + b + c
GAB ~ ~~c '
Hence the proportion is proved.
Theorem 2 : The lines which join the center of gravity of
a tetrahedron to the vertices divide the tetrahedron into four
ADDITION AND SCALAR MULTIPLICATION 41
tetrahedra which are proportional to the masses at the oppo
site vertices.
Let -4, B, C, D be the vertices of the tetrahedron weighted
respectively with weights a, &, c, d. Let 0 be the center of
gravity. Join A, B, C, D to G and produce the lines until
they meet the opposite faces in A', B\ G\ D'. To show that
the volumes
BCDG:CDAG:DABG:ABCG:ABCD
BCDA
BCDG
In like manner
and
and
A A'
A
G
. j_
G
A' b + c + d ,
1
GA'
G
a
A>
+ b +
GA' a
c + d
CDA
G
a
a + 6 + c + d
CD
DA
AB
BG
a -\
-b
b
+ c + d
DA
AB
B
C
C
G
«4
-b
c
+ c + d
ABCD d
which proves the proportion.
* 24.] By a suitable choice of the three masses, a, J, c lo
cated at the vertices A, B, (7, the center of gravity G may
be made to coincide with any given point P of the triangle.
If this be not obvious from physical considerations it cer
tainly becomes so in the light of the foregoing theorems.
For in order that the center of gravity fall at P, it is only
necessary to choose the masses a, 6, c proportional to the
areas of the triangles PEG, PCA^ and PAB respectively.
Thus not merely one set of masses a, &, c may be found, but
an infinite number of sets which differ from each other only
by a common factor of proportionality. These quantities
42 VECTOR ANALYSIS
a, 6, c may therefore be looked upon as coordinates of the
points P inside of the triangle ABC. To each set there
corresponds a definite point P, and to each point P there
corresponds an infinite number of sets of quantities, which
however do not differ from one another except for a factor
of proportionality.
To obtain the points P of the plane ABC which lie outside
of the triangle ABC one may resort to the conception of
negative weights or masses. The center of gravity of the
masses 2 and — 1 situated at the points A and B respectively
would be a point G dividing the line A B externally in the
ratio 1 : 2. That is
Any point of the line A B produced may be represented by
a suitable set of masses a, b which differ in sign. Similarly
any point P of the plane ABC may be represented by a
suitable set of masses a, 6, c of which one will differ in sign
from the other two if the point P lies outside of the triangle
ABC. Inasmuch as only the ratios of a, 6, and c are im
portant two of the quantities may always be taken positive.
The idea of employing the masses situated at the vertices
as coordinates of the center of gravity is due to Mobius and
was published by him in his book entitled " Der barycentrische
Calcul" in 1827. This may be fairly regarded as the starting
point of modern analytic geometry.
The conception of negative masses which have no existence
in nature may be avoided by replacing the masses at the
vertices by the areas of the triangles GBC, GO A, and
GAB to which they are proportional. The coordinates of
a point P would then be three numbers proportional to the
areas of the three triangles of which P is the common vertex ;
and the sides of a given triangle ABC, the bases. The sign
of these areas is determined by the following definition.
ADDITION AND SCALAR MULTIPLICATION 43
Definition: The area ABC of a triangle is said to be
positive when the vertices A, B, C follow each other in the
positive or counterclockwise direction upon the circle de
scribed through them. The area is said to be negative when
the points follow in the negative or clockwise direction.
Cyclic permutation of the letters therefore does not alter
the sign of the area.
Interchange of two letters which amounts to a reversal of
the cyclic order changes the sign.
A CB = BA 0 = CBA = -A B C.
If P be any point within the triangle the equation
PAB+PBC+PCA=ABC
must hold. The same will also hold if P be outside of the
triangle provided the signs of the areas be taken into con
sideration. The areas or three quantities proportional to
them may be regarded as coordinates of the point P.
The extension of the idea of " barycentric " coordinates to
space is immediate. The four points A, B, C, D situated at
the vertices of a tetrahedron are weighted with mass a, J, c, d
respectively. The center of gravity G is represented by
these quantities or four others proportional to them. To
obtain points outside of the tetrahedron negative masses
may be employed. Or in the light of theorem 2, page 40,
the masses may be replaced by the four tetrahedra which
are proportional to them. Then the idea of negative vol
umes takes the place of that of negative weights. As this
idea is of considerable importance later, a brief treatment of
it here may not be out of place.
Definition : The volume A B CD of a tetrahedron is said
to be positive when the triangle ABC appears positive to
44 VECTOR ANALYSIS
the eye situated at the point D. The volume is negative
if the area of the triangle appear negative.
To make the discussion of the signs of the various
tetrahedra perfectly clear it is almost necessary to have a
solid modeL A plane drawing is scarcely sufficient. It is
difficult to see from it which triangles appear positive and
which negative. The following relations will be seen to
hold if a model be examined.
The interchange of two letters in the tetrahedron A BCD
changes the sign.
ACBD = CBAD=BACD=DBCA
The sign of the tetrahedron for any given one of the pos
sible twenty-four arrangements of the letters may be obtained
by reducing that arrangement to the order A B C D by
means of a number of successive interchanges of two letters.
If the number of interchanges is even the sign is the same
as that of A B CD ; if odd, opposite. Thus
If P is any point inside of the tetrahedron A B CD the
equation
ABCP-BCDP+ CDAP-DABP=ABCD
holds good. It still is true if P be without the tetrahedron
provided the signs of the volumes be taken into considera
tion. The equation may be put into a form more symmetri
cal and more easily remembered by transposing all the terms
to one number. Then
The proportion in theorem 2, page 40, does not hold true
if the signs of the tetrahedra be regarded. It should read
BCDG:CDGA:DGAB:GABC:ABCD
ADDITION AND SCALAR MULTIPLICATION 45
If the point G- lies inside the tetrahedron a, J, c, d repre
sent quantities proportional to the masses which must be
located at the vertices A,B,C,D respectively if G is to be the
center of gravity. If G lies outside of the tetrahedron they may
still be regarded as masses some of which are negative — or
perhaps better merely as four numbers whose ratios determine
the position of the point Gr. In this manner a set of "bary-
centric " coordinates is established for space.
The vector P drawn from an indeterminate origin to any
point of the plane A B C is (page 35)
aA + yB + zC
x + y + z
Comparing this with the expression
aA + &B + cC
a + b + c
it will be seen that the quantities x, y, z are in reality nothing
more nor less than the barycentric coordinates of the point P
with respect to the triangle ABO. In like manner from
equation
__#A + yB + 2C + wD
x + y + z + w
which expresses any vector P drawn from an indeterminate
origin in terms of four given vectors A, B, C, D drawn from
the same origin, it may be seen by comparison with
+ &B + c C + rfD
0 =
a + b + c + d
that the four quantities x, y, 2, w are precisely the bary
centric coordinates of P, the terminus of P, with respect to
the tetrahedron A B CD. Thus the vector methods in which
the origin is undetermined and the methods of the " Bary
centric Calculus " are practically co-extensive.
It was mentioned before and it may be well to repeat here
46
VECTOR ANALYSIS
that the origin may be left wholly out of consideration and
the vectors replaced by their termini. The vector equations
then become point equations
x A + y B 4- z 0
and
x + y + z
xA + yB + zC + wD
• w.
At 0
x + y + z
This step brings in the points themselves as the objects of
analysis and leads still nearer to the " Barycentrische Calcul "
of Mobius and the "Ausdehnungslehre " of Grassmann.
The Use of Vectors to denote Areas
25.] Definition: An area lying in one plane MN and
bounded by a continuous curve PQR which nowhere cuts
itself is said to appear positive from the point 0 when the
letters PQR follow each
other in the counterclockwise
or positive order; negative,
when they follow in the
negative or clockwise order
(Fig. 14).
It is evident that an area
can have no determined sign
per se, but only in reference
to that direction in which its
boundary is supposed to be traced and to some point 0 out
side of its plane. For the area P R Q is negative relative to
PQR; and an area viewed from 0 is negative relative to the
same area viewed from a point Of upon the side of the plane
opposite to 0. A circle lying in the X F-plane and described
in the positive trigonometric order appears positive from every
point on that side of the plane on which the positive £axis
lies, but negative from all points on the side upon which
ADDITION AND SCALAR MULTIPLICATION 47
the negative ^-axis lies. For this reason the point of view
and the direction of description of the boundary must be kept
clearly in mind.
Another method of stating the definition is as follows : If
a person walking upon a plane traces out a closed curve, the
area enclosed is said to be positive if it lies upon his left-
hand side, negative if upon his right. It is clear that if two
persons be considered to trace out together the same curve by
walking upon opposite sides of the plane the area enclosed
will lie upon the right hand of one and the left hand of the
other. To one it will consequently appear positive ; to the
other, negative. That side of the plane upon which the area
seems positive is called the positive side ; the side upon
which it appears negative, the negative side. This idea is
familiar to students of electricity and magnetism. If an
electric current flow around a closed plane curve the lines of
magnetic force through the circuit pass from the negative to
the positive side of the plane. A positive magnetic pole
placed upon the positive side of the plane will be repelled by
the circuit.
A plane area may be looked upon as possessing more than
positive or negative magnitude. It may be considered to
possess direction, namely, the direction of the normal to the
positive side of the plane in which it lies. Hence a plane
area is a vector quantity. The following theorems concerning
areas when looked upon as vectors are important.
Theorem 1 : If a plane area be denoted by a vector whose
magnitude is the numerical value of that area and whose
direction is the normal upon the positive side of the plane,
then the orthogonal projection of that area upon a plane
will be represented by the component of that vector in the
direction normal to the plane of projection (Fig. 15).
Let the area A lie in the plane MN. Let it be projected
orthogonally upon the plane M ' N'. Let M N&nd M* Nr inter-
48
VECTOR ANALYSIS
sect in the line I and let the diedral angle between these
two planes be x. Consider first a rectangle PQJRS in MN
whose sides, PQ, RS and QR, SP are respectively parallel
and perpendicular to the line /. This will project into a
rectangle P'Q'R'S1 in M'N'. The sides P'Qf and JR'S'
will be equal to PQ and US; but the sides Q1 R' and S'P'
will be equal to QR and SP multiplied by the cosine of #,
the angle between the planes. Consequently the rectangle
At'
FIG. 15.
Hence rectangles, of which the sides are respectively
parallel and perpendicular to I, the line of intersection of the
two planes, project into rectangles whose sides are likewise
respectively parallel and perpendicular to I and whose area is
equal to the area of the original rectangles multiplied by the
cosine of the angle between the planes.
From this it follows that any area A is projected into an
area which is equal to the given area multiplied by the cosine
of the angle between the planes. For any area A may be di
vided up into a large number of small rectangles by drawing a
series of lines in MN parallel and perpendicular to the line I.
ADDITION AND SCALAR MULTIPLICATION 49
Each of these rectangles when projected is multiplied by the
cosine of the angle between the planes and hence the total
area is also multiplied by the cosine of that angle. On the
other hand the component A' of the vector A, which repre
sents the given area, in the direction normal to the plane
MfNf of projection is equal to the total vector A multiplied
by the cosine of the angle between its direction which is
the normal to the plane M ^and the normal to M'Nr. This
angle is x ; for the angle between the normals to two planes
is the same as the angle between the planes. The relation
between the magnitudes of A and A' is therefore
A1 = A cos x,
which proves the theorem.
26.] Definition : Two plane areas regarded as vectors are
said to be added when the vectors which represent them are
added.
A vector area is consequently the sum of its three com
ponents obtainable by orthogonal projection upon three
mutually perpendicular planes. Moreover in adding two
areas each may be resolved into its three components, the
corresponding components added as scalar quantities, and
these sums compounded as vectors into the resultant area.
A generalization of this statement to the case where the three
planes are not mutually orthogonal and where the projection
is oblique exists.
A surface made up of several plane areas may be repre
sented by the vector which is the sum of all the vectors
representing those areas. In case the surface be looked upon
as forming the boundary or a portion of the boundary of a
solid, those sides of the bounding planes which lie outside of
the body are conventionally taken to be positive. The vec
tors which represent the faces of solids are always directed
out from the solid, not into it
4
50 VECTOR ANALYSIS
Theorem 2 : The vector which represents a closed polyhedral
surface is zero.
This may be proved by means of certain considerations of
hydrostatics. Suppose the polyhedron drawn in a body of
fluid assumed to be free from all external forces, gravity in
cluded.1 The fluid is in equilibrium under its own internal
pressures. The portion of the fluid bounded by the closed
surface moves neither one way nor the other. Upon each face
of the surface the fluid exerts a definite force proportional
to the area of the face and normal to it. The resultant of all
these forces must be zero, as the fluid is in equilibrium. Hence
the sum of all the vector areas in the closed surface is zero.
The proof may be given in a purely geometric manner.
Consider the orthogonal projection of the closed surface upon
any plane. This consists of a double area. The part of the
surface farthest from the plane projects into positive area ;
the part nearest the plane, into negative area. Thus the
surface projects into a certain portion of the plane which is
covered twice, once with positive area and once with negative.
These cancel each other. Hence the total projection of a
closed surface upon a plane (if taken with regard to sign) is
zero. But by theorem 1 the projection of an area upon a
plane is equal to the component of the vector representing
that area in the direction perpendicular to that plane. Hence
the vector which represents a closed surf ace has no component
along the line perpendicular to the plane of projection. This,
however, was any plane whatsoever. Hence the vector is
zero.
The theorem has been proved for the case in which the
closed surface consists of planes. In case that surface be
1 Such a state of affairs is realized to all practical purposes in the case of a
polyhedron suspended in the atmosphere and consequently subjected to atmos
pheric pressure. The force of gravity acts but is counterbalanced by the tension
in the suspending string.
ADDITION AND SCALAR MULTIPLICATION 51
curved it may be regarded as the limit of a polyhedral surface
whose number of faces increases without limit. Hence the
vector which represents any closed surface polyhedral or
curved is zero. If the surface be not closed but be curved it
may be represented by a vector just as if it were polyhedral.
That vector is the limit l approached by the vector which
represents that polyhedral surface of which the curved surface
is the limit when the number of faces becomes indefinitely
great.
SUMMARY OF CHAPTER I
A vector is a quantity considered as possessing magnitude
and direction. Equal vectors possess the same magnitude
and the same direction. A vector is not altered by shifting it
parallel to itself. A null or zero vector is one whose mag
nitude is zero. To multiply a vector by a positive scalar
multiply its length by that scalar and leave its direction
unchanged. To multiply a vector by a negative scalar mul
tiply its length by that scalar and reverse its direction.
Vectors add according to the parallelogram law. To subtract
a vector reverse its direction and add. Addition, subtrac
tion, and multiplication of vectors by a scalar follow the same
laws as addition, subtraction, and multiplication in ordinary
algebra. A vector may be resolved into three components
parallel to any three non-coplanar vectors. This resolution
can be accomplished in only one way.
r = x* + yb + zc. (4)
The components of equal vectors, parallel to three given
non-coplanar vectors, are equal, and conversely if the com
ponents are equal the vectors are equal. The three unit
vectors i, j, k form a right-handed rectangular system. In
1 This limit exists and is unique. It is independent of the method in which
the polyhedral surface approaches the curved surface.
52 VECTOR ANALYSIS
terms of them any vector may be expressed by means of the
Cartesian coordinates #, y, z.
r = xi + yj+zk. (6)
Applications. The point which divides a line in a given
ratio m : n is given by the formula
(7)
m + n
The necessary and sufficient condition that a vector equation
represent a relation independent of the origin is that the sum
of the scalar coefficients in the equation be zero. Between
any four vectors there exists an equation with scalar coeffi
cients. If the sum of the coefficients is zero the vectors are
termino-coplanar. If an equation the sum of whose scalar
coefficients is zero exists between three vectors they are
termino-collinear. The center of gravity of a number of
masses a, &, c • • • situated at the termini of the vectors
A, B, C • • • supposed to be drawn from a common origin is
given by the formula
A vector may be used to denote an area. If the area is
plane the magnitude of the vector is equal to the magnitude
of the area, and the direction of the vector is the direction of
the normal upon the positive side of the plane. The vector
representing a closed surface is zero.
EXERCISES ON CHAPTER I
1. Demonstrate the laws stated in Art. 12.
2. A triangle may be constructed whose sides are parallel
and equal to the medians of any given triangle.
ADDITION AND SCALAR MULTIPLICATION 53
3. The six points in which the three diagonals of a com*
plete quadrangle l meet the pairs of opposite sides lie three
by three upon four straight lines.
4. If two triangles are so situated in space that the three
points of intersection of corresponding sides lie on a line, then
the lines joining the corresponding vertices pass through a
common point and conversely.
5. Given a quadrilateral in space. Find the middle point
of the line which joins the middle points of the diagonals.
Find the middle point of the line which joins the middle
points of two opposite sides. Show that these two points are
the same and coincide with the center of gravity of a system
of equal masses placed at the vertices of the quadrilateral.
6. If two opposite sides of a quadrilateral in space be
divided proportionally and if two quadrilaterals be formed by
joining the two points of division, then the centers of gravity
of these two quadrilaterals lie on a line with the center of
gravity of the original quadrilateral. By the center of gravity
is meant the center of gravity of four equal masses placed at
the vertices. Can this theorem be generalized to the case
where the masses are not equal ?
7. The bisectors of the angles of a triangle meet in a
point.
8. If the edges of a hexahedron meet four by four in three
points, the four diagonals of the hexahedron meet in a point.
In the special case in which the hexahedron is a parallelepiped
the three points are at an infinite distance
9. Prove that the three straight lines through the middle
points of the sides of any face of a tetrahedron, each parallel
to the straight line connecting a fixed point P with the mid
dle point of the opposite edge of the tetrahedron, meet in a
1 A complete quadrangle consists of the six straight lines which may he passed
through four points no three of which are collinear. The diagonals are the lines
which join the points of intersection of pairs of sides
54 VECTOR ANALYSIS
point E and that this point is such that PE passes through
and is bisected by the center of gravity of the tetrahedron.
10. Show that without exception there exists one vector
equation with scalar coefficients between any four given
vectors A, B, C, D.
11. Discuss the conditions imposed upon three, four, or
five vectors if they satisfy two equations the sum of the co
efficients in each of which is zero.
CHAPTER II
DIRECT AND SKEW PRODUCTS OF VECTORS
Products of Two Vectors
27.] THE operations of addition, subtraction, and scalar
multiplication have been defined for vectors in the way
suggested by physics and have been employed in a few
applications. It now becomes necessary to introduce two
new combinations of vectors. These will be called products
because they obey the fundamental law of products ; i. e., the
distributive law which states that the product of A into the
sum of B and C is equal to the sum of the products of A into
B and A into C.
Definition : The direct product of two vectors A and B is
the scalar quantity obtained by multiplying the product of
the magnitudes of the vectors by the cosine of the angle be
tween them.
The direct product is denoted by writing the two vectors
with a dot between them as
A-B.
This is read A dot B and therefore may often be called the
dot product instead of the direct product. It is also called
the scalar product owing to the fact that its value is sca
lar. If A be the magnitude of A and B that of B, then by
definition
A-B = ^£cos (A,B). (1)
Obviously the direct product follows the commutative law
A-B = B'A. (2)
56 VECTOR ANALYSIS
If either vector be multiplied by a scalar the product is
multiplied by that scalar. That is
(x A) • B = A • (x B) = x (A • B).
In case the two vectors A and B are collinear the angle be
tween them becomes zero or one hundred and eighty degrees
and its cosine is therefore equal to unity with the positive or
negative sign. Hence the scalar product of two parallel
vectors is numerically equal to the product of their lengths.
The sign of the product is positive when the directions of the
vectors are the same, negative when they are opposite. The
product of a vector by itself is therefore equal to the square
of its length
A.A=^42. (3)
Consequently if the product of a vector by itself vanish the
vector is a null vector.
In case the two vectors A and B are perpendicular the
angle between them becomes plus or minus ninety degrees
and the cosine vanishes. Hence the product A • B vanishes.
Conversely if the scalar product A • B vanishes, then
A B cos (A, B) = 0.
Hence either A or B or cos (A, B) is zero, and either the
vectors are perpendicular or one of them is null. Thus the
condition for the perpendicularity of two vectors, neither of
which vanishes, is A • B = 0.
28.] The scalar products of the three fundamental unit
vectors i, j, k are evidently
i»i = j»j = k«k = l, (4)
i . j = j . k = k . i = 0.
If more generally a and b are any two unit vectors the
product
a • b = cos (a, b).
DIRECT AND SKEW PRODUCTS OF VECTORS 57
Thus the scalar product determines the cosine of the angle
between two vectors and is in a certain sense equivalent to
it. For this reason it might be better to give a purely
geometric definition of the product rather than one which
depends upon trigonometry. This is easily accomplished as
follows : If a and b are two unit vectors, a • b is the length
of the projection of either upon the other. If more generally
A and B are any two vectors A • B is the product of the length
of either by the length of projection of the other upon it.
From these definitions the facts that the product of a vector
by itself is the square of its length and the product of two
perpendicular vectors is zero follow immediately. The trigo
nometric definition can also readily be deduced.
The scalar product of two vectors will appear whenever the
cosine of the included angle is of importance. The following
examples may be cited. The projection of a vector B upon a
vector A is
A«B AB
A = — — A a cos (A, B) = B cos (A, B) a, (5)
A • A
where a is a unit vector in the direction of A. If A is itself a
unit vector the formula reduces to
(A-B) A = £cos (A,B) A.
If A be a constant force and B a displacement the work done
by the force A during the displacement is A • B. If A repre
sent a plane area (Art. 25), and if B be a
vector inclined to that plane, the scalar prod
uct A • B will be the volume of the cylinder
of which the area A is the base and of
which B is the directed slant height. For
the volume (Fig. 16) is equal to the base FlG
A multiplied by the altitude h. This is
the projection of B upon A or B cos (A, B). Hence
v = A h = A B cos (A, B) = A • B.
58 VECTOR ANALYSIS
29.] The scalar or direct product follows the distributive
law of multiplication. That is
(A + B) .C = A-C + B.C. (6)
This may be proved by means of projections. Let C be equal
to its magnitude C multiplied by a unit vector c in its direc
tion. To show
(A + B) • (<7c) = A • (0o) + B. (0o)
or (A + B) • c = A • c + B • c.
A • c is the projection of A upon c ; B • c, that of B upon c ;
(A + B) • c, that of A + B upon c. But the projection of the
sum A + B is equal to the sum of the projections. Hence
the relation (6) is proved. By an immediate generalization
(A + B + ...)'(P + a+-") = A-P + A.Q+...
+ B.P + B.Q + ... <ey
The scalar product may be used just as the product in ordi
nary algebra. It has no peculiar difficulties.
If two vectors A and B are expressed in terms of the
three unit vectors i, j, k as
A = ^[1i + ^2j + ^8k,
and B = ^ i + JB2 j + £B k,
then A- B = (Al i + Az j + AB k) . (^i + £2 j + ^k)
= Al Bl i . i + Al £2 i • j + Al B% i • k
By means of (4) this reduces to
A-* = A1£l + A2E, + ABJB,. (7)
If in particular A and B are unit vectors, their components
Al,A^,A3 and B19S29SB are the direction cosines of the
lines A and B referred to X, Y, Z.
DIRECT AND SKEW PRODUCTS OF VECTORS 59
Al = cos (A, JT), A<i = cos (A, F), Az = cos (A, ^f),
^! = cos (B, JT), .#2 = cos (B, T), £3 = cos (B, £).
Moreover A • B is the cosine of the included angle. Hence
the equation becomes
cos (A, B) = cos (A, X) cos (B, X) + cos (A, T) cos (B, T)
+ cos (A,£) cos (B,Z).
In case A and B are perpendicular this reduces to the well-
known relation
0 = cos (A, JT) cos (B, X) + cos (A, Y) cos (B, F)
+ cos (A,^) cos (B,£)
between the direction cosines of the
line A and the line B.
30.] If A and B are two sides 0 A
and OB of a triangle OAB, the third o
side A£isG = -B-JL (Fig. 17). PlG' 17*
C*C = (B-A). (B-A)=B-B
or (7 2 = A2 + J52-2 A^cos(AB).
That is, the square of one side of a triangle is equal to the
sum of the squares of the other two sides diminished by twice
their product times the cosine of the angle between them.
Or, the square of one side of a triangle is equal to the sum of
the squares of the other two sides diminished by twice the
product of either of those sides by the projection of the other
upon it — the generalized Pythagorean theorem.
If A and B are two sides of a parallelogram, C = A + B
and D = A — B are the diagonals. Then
C.C = (A + B).(A + B)=A.A + 2A.B + B.B,
D.D=(A-B).(A-B)=A-A-2A.B + B.B,
C-C + D.D = 2(A-A + B«B),
or a2 + 7)2 = 2 (A*
60 VECTOR ANALYSIS
That is, the sum of the squares of the diagonals of a parallelo
gram is equal to twice the sum of the squares of two sides.
In like manner also
or C*-D2 = 4A£ cos (A, B).
That is, the difference of the squares of the diagonals of a
parallelogram is equal to four times the product of one of the
sides by the projection of the other upon it.
If A is any vector expressed in terms of i, j, k as
A = Al i + A2 j + AB k,
then A • A = A* = A* + A* + A*. (8)
But if A be expressed in terms of any three non-coplanar unit
vectors a, b, c as
+ 2 J c b«c + 2 c a e «a
A2 = a? + 62 + c2 + 2 a b cos (a, b) + 2 b c cos (b, c)
+ 2 ca cos (c, a).
This formula is analogous to the one in Cartesian geometry
which gives the distance between two points referred to
oblique axes. If the points be xv yv zv and #2, yv z% the
distance squared is
D2 = (*2 - x^ + (ya - yi)2 + (z2 - zj*
+ 2 (a, - xj (ya - 2/0 cos (X, Y)
+ 2 (yt- ft) («,-*!> cos (F.S)
+ 2 (z2 -24) (x2 - xj cos (^,-T).
31.] Definition: The skew product of the vector A into
the vector B is the vector quantity C whose direction is the
normal upon that side of the plane of A and B on which
DIRECT AND SKEW PRODUCTS OF VECTORS 61
rotation from A to B through an angle of less than one
hundred and eighty degrees appears positive or counter
clockwise ; and whose magnitude is obtained by multiplying
the product of the magnitudes of A and B by the sine of the
angle from A to B.
The direction of A x B may also be defined as that in
which an ordinary right-handed
screw advances as it turns so as c= AXB
to carry A toward B (Fig. 18).
The skew product is denoted by
a cross as the direct product was ^
by a dot. It is written FIG. 18.
C = A x B
and read A cross B. For this reason it is often called the cross
product. More frequently, however, it is called the vector prod
uct, owing to the fact that it is a vector quantity and in con
trast with the direct or scalar product whose value is scalar.
The vector product is by definition
C = A x B = ^J5sin (A,B)c, (9)
when A and B are the magnitudes of A and B respectively and
where c is a unit vector in the direction of C. In case A and
B are unit vectors the skew product A X B reduces to the
unit vector c multiplied by the sine of the angle from A to B.
Obviously also if either vector A or B is multiplied by a scalar
x their product is multiplied by that scalar.
0 A) X B = A X (zB) = xC.
If A and B are parallel the angle between them is either zero
or one hundred and eighty degrees. In either case the sine
vanishes and consequently the vector product A X B is a null
vector. And conyersely if A X B is zero
A B sin (A, B) = 0.
62 VECTOR ANALYSIS
Hence A or B or sin (A, B) is zero. Thus the condition for
parallelism of two vectors neither of which vanishes is A X B
= 0. As a corollary the vector product of any vector into
itself vanishes.
32.] The vector product of two vectors will appear wher
ever the sine of the included angle is of importance, just as
the scalar product did in the case of the cosine. The two prod
ucts are in a certain sense complementary. They have been
denoted by the two common signs of multiplication, the dot
and the cross. In vector analysis they occupy the place held
by the trigonometric functions of scalar analysis. They are
at the same time amenable to algebraic treatment, as will be
seen later. At present a few uses of the vector product may
be cited.
If A and B (Fig. 18) are the two adjacent sides of a parallel
ogram the vector product
C = A x B = A B sin (A, B) c
represents the area of that parallelogram in magnitude and
direction (Art. 25). This geometric representation of A X B
is of such common occurrence and importance that it might
well be taken as the definition of the product. From it the
trigonometric definition follows at once. The vector product
appears in mechanics in connection with couples. If A and
— A are two forces forming a couple, the moment of the
couple is A X B provided only that B is a vector drawn from
any point of A to any point of — A. The product makes its
appearance again in considering the velocities of the individ
ual particles of a body which is rotating with an angular ve
locity given in magnitude and direction by A. If R be the
radius vector drawn from any point of the axis of rotation A
the product A X & will give the velocity of the extremity of
B (Art. 51). This velocity is perpendicular alike to the axis
of rotation and to the radius vector B.
DIRECT AND SKEW PRODUCTS OF VECTORS 63
33.] The vector products A X B and B x A are not the
same. They are in fact the negatives of each other. For if
rotation from A to B appear positive on one side of the plane
of A and B, rotation from B to A will appear positive on the
other. Hence A X B is the normal to the plane of A and B
upon that side opposite to the one upon which B x A is the
normal. The magnitudes of A X B and B X A are the same.
Hence
AxB = -BxA. (10)
The factors in a vector product can be interchanged if and only
if the sign of the product be reversed.
This is the first instance in which the laws of operation in
vector analysis differ essentially from those of scalar analy
sis. It may be that at first this change of sign which must
accompany the interchange of factors in a vector product will
give rise to some difficulty and confusion. Changes similar to
this are, however, very familiar. No one would think of inter
changing the order of x and y in the expression sin (x — y)
without prefixing the negative sign to the result. Thus
sin (y — x) = — sin (x — y),
although the sign is not required for the case of the cosine,
cos (y — x) = cos ( x — y).
Again if the cyclic order of the letters ABC in the area of a
triangle be changed, the area will be changed in sign (Art.
25).
AB C = -ACB.
In the same manner this reversal of sign, which occurs
when the order of the factors in a vector product is reversed,
will appear after a little practice and acquaintance just as
natural and convenient as it is necessary.
34.] The distributive law of multiplication holds in the
case of vector products just as in ordinary algebra — except
64
VECTOR ANALYSTS
that the order of the factors must be carefully maintained
when expanding.
A very simple proof may be given by making use of the ideas
developed in Art. 26. Suppose that C
is not coplanar with A and B. Let A
and B be two sides of a triangle taken
in order. Then — (A + B) will be the
third side (Fig. 19). Form the prism
of which this triangle is the base and
of which C is the slant height or edge.
The areas of the lateral faces of this
prism are
A x C, B x C, — (A -f B) x C.
The areas of the bases are
5 (A x B) and - - (A x B).
But the sum of all the faces of the prism is zero; for the
prism is a closed surface. Hence
4
FIG. 19.
AxC + BxC-(A + B)xC = 0,
or A X C + B X C = (A + B) X C. (11)
The relation is therefore proved in case C is non-coplanar
with A and B. Should C be coplanar with A and B, choose D,
any vector out of that plane. Then C + D also will lie out of
that plane. Hence by (11)
A X (C + D) + B X (C + D) = (A + B) x (C + D).
Since the three vectors in each set A, C, D, and B, C, D, and
A + B, C, D will be non-coplanar if D is properly chosen, the
products may be expanded.
DIRECT AND SKEW PRODUCTS OF VECTORS 65
AxC + AxD-fBxC + BxD
= (A + B) x C + (A + B) x D.
But by (11) AxD + BxD = (A + B)xD.
Hence AxC + BxC = (A + B)xC.
This completes the demonstration. The distributive law holds
for a vector product. The generalization is immediate.
(A + B+---)x(P + a + ---) = AxP + Axa + --- (11)'
+ B x P + B x <J + •••
35.] The vector products of the three unit vectors i, j, k are
easily seen by means of Art. 17 to be
ixi = jxj = kxk = 0,
ixj=-j xi = k, (12)
jxk = — k x j = i,
kxi = — ixk=j.
The skew product of two equal l vectors of the system i, j, k
is zero. The product of two unequal vectors is the third taken
with the positive sign if the vectors follow in the cyclic order
i j k but with the negative sign if they do not.
If two vectors A and B are expressed in terms of i, j, k,
their vector product may be found by expanding according
to the distributive law and substituting.
A x B ' = (Al i + -42 j + ^3k) x (^i + £2j + £3*)
= Al£lixi + AlB2ixj + AlBzixTt
+ A2£l j x i + AI £2 j x j + AZ BB j x k,
+ Az S1 k x i + AB BZ k x j + Az Bz k x k.
Hence A x B = (A^B^ - AB£2) i + (AZB1 -A,BB)j
4- (A, £z - A2 BJ k.
1 This follows also from the fact that the sign is changed when the order of
factors is reversed. Hence j X j = — j Xj=0.
5
66 VECTOR ANALYSIS
This may be written in the form of a determinant as
Ax B =
The formulae for the sine and cosine of the sum or dif
ference of two angles follow immediately from the dot and
cross products. Let a and b be two unit vectors lying in the
i j-plane. If x be the angle that a makes with i, and y the
angle b makes with i, then
a =
Hence
If
Hence
Hence
Hence
a • b :
a • b :
cos (y — x) :
V
a.V:
cos (y + x) :
a x b :
a x b =
sin (y — x) -
axb' =
ax b' =
sin (y + x) -
cos x i + sin x j,
cos y i -f sin y j,
cos (a, b) = cos (y — x),
cos x cos y + sin x sin y.
cos y cos x + sin y sin x.
cos y i — sin y j,
cos (a, b') = cos (y + x).
cos y cos x — sin y sin x.
k sin (a, b) = k sin (y — x),
k (sin y cos x — sin x cos y).
sin y cos x — sin x cos y.
k sin (a, b') = k sin (y + x)9
k (sin y cos x + sin x cos y).
sin y cos x + sin # cos y.
If /, m, 7i and Z', w', TI' are the direction cosines of two
unit vectors a and a' referred to JT, F, £, then
a = li + m j + 7i k,
m'j
a • a' = cos (a, a') = IV + m mr + n nf,
as has already been shown in Art. 29. The familiar formula
for the square of the sine of the angle between a and a' may
be found.
DIRECT AND SKEW PRODUCTS OF VECTORS 67
a x a' = sin (a, a') e = (mnf — m' ri) i + (n V — n? I) j
+ (Jm'-f'm) k,
where e is a unit vector perpendicular to a and a'.
(a x a') • (a x a') = sin2 (a, af ) e • e = sin2 (a, a').
sin2 (a, a') = (mn'—mrn)*+ (nl'—nf /)2 +(lm' — I'm)*.
This leads to an easy way of establishing the useful identity
= (72 + w2 + 7i2) (V* + m'2 + n'*) — (ll' + mm' + n n')2.
Products of More than Two Vectors
36.] Up to this point nothing has been said concerning
products in which the number of vectors is greater than
two. If three vectors are combined into a product the result
is called a triple product. Next to the simple products
A-B and AxB the triple products are the most important.
All higher products may be reduced to them.
The simplest triple product is formed by multiplying the
scalar product of two vectors A and B into a third C as
(A-B) C.
This in reality does not differ essentially from scalar multi
plication (Art. 6). The scalar in this case merely happens to
be the scalar product of the two vectors A and B. Moreover
inasmuch as two vectors cannot stand side by side in the
form of a product as BC without either a dot or a cross to
unite them, the parenthesis in (A»B) C is superfluous. The
expression « ^ n
cannot be interpreted in any other way * than as the product
of the vector C by the scalar A«B.
i Later (Chap. V.) the product BC, where no sign either dot or cross occurs,
will be defined. But it will be seen there that (A.B) C and A-(B C) are identical
and consequently no ambiguity can arise from the omission of the parenthesis.
68 VECTOR ANALYSIS
37.] The second triple product is the scalar product of
two vectors, of which one is itself a vector product, as
A-(BxC) or (AxB>C.
This sort of product has a scalar value and consequently is
often called the scalar triple prod
uct. Its properties are perhaps most
easily deduced from its commonest
geometrical interpretation. Let A, B,
and C be any three vectors drawn
from the same origin (Fig. 20).
Then BxC is the area of the par
allelogram of which B and C are two adjacent sides. The
scalar . * (14)
will therefore be the volume of the parallelepiped of which
BxC is the base and A the slant height or edge. See Art. 28.
This volume v is positive if A and BxC lie upon the same
side of the B C-plane ; but negative if they lie on opposite
sides. In other words if A, B, C form a right-handed or
positive system of three vectors the scalar A* (BxC) is posi
tive; but if they form a left-handed or negative system, it
is negative.
In case A, B, and C are coplanar this volume will be
neither positive nor negative but zero. And conversely if
the volume is zero^the three edges A, B, C of the parallelo-
piped must lie in one plane. Hence the necessary and suffi
cient condition for the coplanarity of three vectors A, B, C none
of which vanishes is A-(BxC) = 0. As a corollary the scalar
triple product of three vectors of which two are equal or
collinear must vanish ; for any two vectors are coplanar.
The two products A«(BxC) and (AxB)-C are equal to the
same volume v of the parallelepiped whose concurrent edges
are A, B, C. The sign of the volume is the same in both
cases. Hence (AxB).c = A.(BxC) = ,. (14)
DIRECT AND SKEW PRODUCTS OF VECTORS 69
This equality may be stated as a rule of operation. The dot
and the cross in a scalar triple product may be interchanged
without altering the value of the product.
It may also be seen that the vectors A, B, C may be per
muted cyclicly without altering the product
A-(BxC) = B-(CxA) = C-(AxB). (15)
For each of the expressions gives the volume of the same
parallelepiped and that volume will have in each case the
same sign, because if A is upon the positive side of the B C-
plane, B will be on the positive side of the C A-plane and C
upon the positive side of the A B-plane. The triple product
may therefore have any one of six equivalent forms
A<BxC) = B-(CxA) =: C.(AxB) (35)'
= (AxB)-C = (BxC)-A = (CxA)-B
If however the cyclic order of the letters is changed the
product will change sign.
A-(BxC) = - A<CxBV (16)
This may be seen from the figure or from the fact that
BxC = — CxB.
Hence : A scalar triple product is not altered by interchanging
the dot or the cross or by permuting cyclicly the order of the
vectors, but it is reversed in sign if the cyclic order be changed.
38.] A word is necessary upon the subject of parentheses
in this triple product. Can they be omitted without am
biguity ? They can. The expression
A-BxC
can have only the one interpretation
A<BxC).
For the expression (A-B)xC is meaningless. It is impos
sible to form the skew product of a scalar A«B and a vector
70 VECTOR ANALYSIS
C. Hence as there is only one way in which A»BxC may
be interpreted, no confusion can arise from omitting the
parentheses. Furthermore owing to the fact that there are
six scalar triple products of A, B, and C which have the same
value and are consequently generally not worth distinguish
ing the one from another, it is often convenient to use the
symbol
[ABC]
to denote any one of the six equal products.
[A B C] = A.BxC = B*CxA = C'AxB
= AxB.C = BxC-A = CxA-B
then [A B C] = - [A 0 B]. (16)'
The scalar triple products of the three unit vectors i, j, k
all vanish except the two which contain the three different
vectors.
[ijk] = _[ikj] = l. (17)
Hence if three vectors A, B, C be expressed in terms of i, j, k
as
B = ^ i + A, j + £8 k,
C = C'1i+C'2j + C'3k,
then [ABC] =A1£ZC3 + £, C2 A.+
This may be obtained by actually performing the multiplica
tions which are indicated in the triple product. The result
may be written in the form of a determinant.1
[A B C] =
-4i
\
B
1 This is the formula given in solid analytic geometry for the volume of a
tetrahedron one of whose vertices is at the origin. For a more general formula
see exercises.
DIRECT AND SKEW PRODUCTS OF VECTORS 71
If more generally A, B, C are expressed in terms of any three
non-coplanar vectors a, b, c which are not necessarily unit
vectors,
A = ax a + a2 b + «8 c
B = &! a + &2 b + J8 c
C = cl a
c2 b
where a^ #2, #3," ftp &2,
stants, then
[A B 0] = (al &2 CB +
are certain con
[a b c].
or
[A B C] =
[a be]
'
(19)'
39.] The third type of triple product is the vector product
of two vectors of which one is itself a vector product. Such
are
Ax(BxC) and (AxB)xC.
The vector Ax(BxC) is perpendicular to A and to (BxC).
But (BxC) is perpendicular to the plane of B and C. Hence
Ax (BxC), being perpendicular to (BxC) must lie in the
plane of B and C and thus take the form
Ax(BxC) = x B + y C,
where x and y are two scalars. In like manner also the
vector (AxB)xC, being perpendicular to (AxB) must lie
in the plane of A and B. Hence it will be of the form
(AxB)xC = ra.A + n B
where m and n are two scalars. From this it is evident that
in general
(AxB)xC is not equal to Ax(BxC).
The parentheses therefore cannot be removed or inter
changed. It is essential to know which cross product is
72 VECTOR ANALYSIS
formed first and which second. This product is termed the
vector triple product in contrast to the scalar triple product.
The vector triple product may be used to express that com
ponent of a vector B which is perpendicular to a given vector
A. This geometric use of the product is valuable not only in
itself but for the light it sheds
AXB
AXB*
B
upon the properties of the product.
Let A (Fig. 21) be a given vector
and B another vector whose com
ponents parallel and perpendicular
to A are to be found. Let the
components of B parallel and per-
A X (AXB) pendicular to A be B'and B" re-
2i spectively. Draw A and B from a
common origin. The product AxB
is perpendicular to the plane of A and B. The product
Ax (AxB) lies in the plane of A and B. It is furthermore
perpendicular to A. Hence it is collinear with B". An
examination of the figure will show that the direction of
Ax (AxB) is opposite to that of B". Hence
Ax(AxB) = — cB",
where c is some scalar constant.
Now Ax (AxB) = - A* B sin (A, B) V
but - c B"-^= - c B sin (A, B) b",
if b" be a unit vector in the direction of B".
Hence c — A2 — A* A.
Hence B" = - Ax(AxB) . (20)
The component of B perpendicular to A has been expressed
in terms of the vector triple product of A, A, and B. The
component B' parallel to A was found in Art 28 to be
DIRECT AND SKEW PRODUCTS OF VECTORS 73
B'=£?A (21)
B = B'+B» = £?A-±^>. (22)
A«A A«A
40.] The vector triple product Ax (BxC) may be expressed
as the sum of two terms as
Ax(BxC)=A-C B-A-B C
In the first place consider the product when two of the
vectors are the same. By equation (22)
A-A B = A-B A - Ax(AxB) (22)
or Ax(AxB) = A*B A - A- A B (23)
This proves the formula in case two vectors are the same.
To prove it in general express A in terms of the three
non-coplanar vectors B, C, and BxC.
A = bE + cC + a (BxC), (I)
where #, &, c are scalar constants. Then
Ax(BxC) = SBx(BxC) + cCx(BxC) (II)
+ a (BxC)x(BxC).
The vector product of any vector by itself is zero. Hence
(BxC)x(BxC) = 0
Ax(BxC) = 6Bx(BxC) + c Cx(BxC). (II)'
By (23) Bx(BxC) = B-C B - B-B C
Cx(BxC) = - Cx(CxB) = - C-B C + C-C B.
Hence Ax(BxC) = [(&B-C + cC-C)B- (6B-B + cC«B)C]. (II)"
But from (I) A-B = JB«B + cC-B + a (BxC>B
and A-C = b B-C + c C*C + a (BxC)-O.
By Art. 37 (BxC)-B = 0 and (BxC)-C = 0.
Hence A-B = JB-B + cC-B,
A-C = 5B-C + cC-C.
74 VECTOR ANALYSIS
Substituting these values in (II)",
Ax(BxC) = A.C B - A.B C. (24)
The relation is therefore proved for any three vectors A, B, C.
Another method of giving the demonstration is as follows.
It was shown that the vector triple product Ax(BxC) was
of the form
Ax(BxC) = #B + yC.
Since Ax(AxC) is perpendicular to A, the direct product of
it by A is zero. Hence
A-[Ax(BxC)] = a; A*B + yA«C = 0
and x : y = A*C : — A»B.
Hence Ax(BxC) = n (A-0 B - A-B C),
where n is a scalar constant. It remains to show n = 1.
Multiply by B.
Ax(BxC>B = n (A-C B.B-A-B C-B).
The scalar triple product allows an interchange of dot and
cross. Hence
Ax(BxC>B = A<BxC)xB = - A-[Bx(BxC)],
if the order of the factors (BxC) and B be inverted.
-A-[Bx(BxC)] = -A-[B.C B-B.BC]
= — B-C A«B + B-B A«C.
Hence n = 1 and Ax(BxC) = A.C B — A-B C. (24)
From the three letters A, B, C by different arrangements,
four allied products in each of which B and C are included in
parentheses may be formed. These are
Ax(BxC), Ax(CxB), (CxB)xA, (BxC)xA.
As a vector product changes its sign whenever the order of
two factors is interchanged, the above products evidently
satisfy the equations
Ax (BxC) = - Ax(CxB) = (CxB)xA = - (BxC)xA.
DIRECT AND SKEW PRODUCTS OF VECTORS 75
The expansion for a vector triple product in which the
parenthesis comes first may therefore be obtained directly
from that already found when the parenthesis comes last.
(AxB)xC = - Cx(AxB) = - C-B A + C«A B.
The formulae then become
Ax(BxC) = A-C B - A.B C (24)
and (AxB)xC = A*C B - C-B A. ' (24)'
These reduction formulae are of such constant occurrence and
great importance that they should be committed, to memory.
Their content may be stated in the following rule. To expand
a vector triple product first multiply the exterior factor into the
remoter term in the parenthesis to form a scalar coefficient for
the nearer one, then multiply the exterior factor into the nearer
term in the parenthesis to form a scalar coefficient for the
remoter one, and subtract this result from the first.
41.] As far as the practical applications of vector analysis
are concerned, one can generally get along without any
formulae more complicated than that for the vector triple
product. But it is frequently more convenient to have at
hand other reduction formulae of which all may be derived
simply by making use of the expansion for the triple product
Ax(BxC) and of the rules of operation with the triple pro
duct A«BxC.
To reduce a scalar product of two vectors each of which
is itself a vector product of two vectors, as
(AxB>(CxD).
Let this be regarded as a scalar triple product of the three
vectors A, B, and CxD — thus
AxB-(CxD).
Interchange the dot and the cross.
76 VECTOR ANALYSIS
AxB.(CxD) = A-Bx(CxD)
Bx(CxD) = B-D C - B-C D.
Hence (AxB>(CxD) = A-C B-D - A-D B.C. (25)
This may be written in determinantal form.
(25)'
If A and D be called the extremes ; B and C the means ; A
and C the antecedents: B and D the consequents in this
product according to the familiar usage in proportions, then
the expansion may be stated in words. The scalar product
of two vector products is equal to the (scalar) product of the
antecedents times the (scalar) product of the consequents
diminished by the (scalar) product of the means times the
(scalar) product of the extremes.
To reduce a vector product of two vectors each of which
is itself a vector product of two vectors, as
(AxB)x(CxD).
Let CxD = E. The product becomes
(AxB)xE = A-E B - B-E A.
Substituting the value of E back into the equation :
(AxB)x(CxD) = (A-CxD)B - (B-CxD) A. (26)
Let F = AxB. The product then becomes
Fx(CxD) = F«D C — F-C D
(AxB)x(CxD) = (AxB-D)C - (AxB-C) D. (26)'
By equating these two equivalent results and transposing
all the terms to one side of the equation,
[B C D] A - [C D A] B + [D A B] C - [A B C] D - 0. (27)
This is an equation with scalar coefficients between the four
vectors A, B, C, D. There is in general only one such equa-
DIRECT AND SKEW PRODUCTS OF VECTORS 11
tion, because any one of the vectors can be expressed in only
one way in terms of the other three : thus the scalar coeffi
cients of that equation which exists between four vectors are
found to be nothing but the four scalar triple products of
those vectors taken three at a time. The equation may also
be written in the form
[A B C] D = [B C D] A + [C A D] B + [A B D] C. (27)'
More examples of reduction formulae, of which some are
important, are given among the exercises at the end of the
chapter. In view of these it becomes fairly obvious that
the combination of any number of vectors connected in
any legitimate way by dots and crosses or the product of any
number of such combinations can be ultimately reduced to
a sum of terms each of which contains only one cross at most.
The proof of this theorem depends solely upon analyzing the
possible combinations of vectors and showing that they all
fall under the reduction formulae in such a way that the
crosses may be removed two at a time until not more than
one remains.
* 42.] The formulae developed in the foregoing article have
interesting geometric interpretations. They also afford a
simple means of deducing the formulae of Spherical Trigo
nometry. These do not occur in the vector analysis proper.
Their place is taken by the two quadruple products,
(AxB>(CxD) = A-C B-D - B-C A-D (25)
and (AxB)x(CxD) = [ACD] B - [BCD] A
= [ABD] C - [ABC] D, (26)
which are now to be interpreted.
Let a unit sphere (Fig. 22) be given. Let the vectors
A, B, C, D be unit vectors drawn from a common origin, the
centre of the sphere, and terminating in the surface of the
sphere at the points A,B, (7, D. The great circular arcs
78
VECTOR ANALYSIS
FIG. 22.
AB, A C) etc., give the angles between the vectors A and B,
A and C, etc. The points A, B, C, D determine a quadrilateral
upon the sphere. A C and BD are one
pair of opposite sides ; A D and B C> the
other. A B and CD are the diagonals.
(AxB).(CxD) = A-C B-D - A-D B-C
AxB = sin (A, B), CxD = sin (C, D).
The angle between AxB and CxD is the
angle between the normals to the AB-
and CD-planes. This is the same as
the angle between the planes themselves. Let it be denoted
by x. Then
(AxB). (CxD) = sin (A,B) sin (C,D) cos a:.
The angles (A, B), (C, D) may be replaced by the great
circular arcs AB, CD which measure them. Then
(AxB).(CxD) = sin A B sin CD cos#,
A-C B-D- A.D B*C = cos AC cosBD - cos AD cos BC.
Hence
sin A B sin CD cos x = cos A C cos B D — cos AD cos B C.
In words : The product of the cosines of two opposite sides
of a spherical quadrilateral less the product of the cosines of
the other two opposite sides is equal to the product of the
sines of the diagonals multiplied by the
cosine of the angle between them. This
theorem is credited to Gauss.
Let A, B, C (Fig. 23) be a spherical tri
angle, the sides of which are arcs of great
circles. Let the sides be denoted by a, 6, c
respectively. Let A, B, C be the unit vectors
drawn from the center of the sphere to the points -A, B, C.
Furthermore let pa, pb, pe be the great circular arcs dropped
FIG. 23.
DIRECT AND SKEW PRODUCTS OF VECTORS 79
perpendicularly from the vertices -4, J9, C to the sides a, 6, £.
Interpret the formula
(AxB)-(CxA) = A-C B-A - B.C A-A.
(AxB) = sin (A, B) = sin c, (CxA) = sin (C, A) = sin 6.
Then (AxB) • (CxA) = sin c sin b cos #,
where x is the angle between AxB and CxA. This
angle is equal to the angle between the plane of A, B and the
plane of C, A. It is, however, not the interior angle A which
is one of the angles of the triangle : but it is the exterior
angle 180° — A, as an examination of the figure will show.
Hence
(AxB). (CxA) = sin c sin b cos (180° — A)
= — sin c sin 6 cos A
A»C B«A — B«C A- A = cos & cos c — cos a 1.
By equating the results and transposing,
cos a = cos 6 cos c — sin 6 sin c cos A
cos 6 = cos c cos a — sin c sin a cos B
cos c = cos a cos 6 — sin a sin 6 cos C.
The last two may be obtained by cyclic permutation of the
letters or from the identities
(BxC).(AxB) = B-A C'B - C-A,
(CxAHBxC) = C-B A.C - B-C.
Next interpret the identity (AxB)x(CxD) in the special
cases in which one of the vectors is repeated.
(AxB)x(AxC) = [A B C] A.
Let the three vectors a, b, c be unit vectors in the direction of
BxC, CxA, AxB respectively. Then
AxB = c sin c, AxC = — b sin 6
(AxB)x(AxC) = — cxb sin c sin & = A sin c sin 6 sin A
[A B C] = (AxB)-C = c«C sin c = cos (90° —pc) sin c
[ABC] A = sin c sin pc A.
80 VECTOR ANALYSIS
By equating the results and cancelling the common factor,
sin^c = sin b sin A
sin^?a = sin c sin B
sin pb = sin a sin C.
The last two may be obtained by cyclic permutation of the
letters. The formulae give the sines of the altitudes of the
triangle in terms of the sines of the angle and sides. Again
write
(AxB)x(AxC) = [ABC]A
(BxC)x(BxA) = [BCA]B
(CxA)x(CxB) = [CAB]C.
Hence sin c sin b sin A = [A B C]
sin a sin c sin B = [B C A]
sin b sin a sin C = [C A B].
The expressions [ABC], [BCA], [CAB] are equal. Equate
the results in pairs and the formulae
sin b sin A = sin a sin B
sin c sin B = sin b sin C
sin a sin C = sin c sin A
are obtained. These may be written in a single line.
sin A sin B sin C
sin a sin b sin c
The formulae of Plane Trigonometry are even more easy to
obtain. If A B C be a triangle, the sum of the sides taken
as vectors is zero — for the triangle is a closed polygon.
From this equation
a + b-f c = 0
almost all the elementary formulae follow immediately. It
is to be noticed that the angles from a to b, from b to c, from
DIRECT AND SKEW PRODUCTS OF VECTORS 81
o to a are not the interior angles A, B, (7, but the exterior
angles 180° -A, 180° - B, 180° - C.
— a = b + c
a»a = (b + c)*(b + c) = b-b + c-c + 2 b«c.
If a, J, c be the length of the sides a, b, c, this becomes
c2 = a2 + 62 - 2 a 6 cos C.
The last two are obtained in a manner similar to the first
one or by cyclic permutation of the letters.
The area of the triangle is
^axb = ^bxc = 2cxa =
2 a b sin C = % b c sin A = ^ c a sin B.
If each of the last three equalities be divided by the product
£ a b c, the fundamental relation
sin A sin B sin 0
is obtained. Another formula for the area may be found from
the product
(bxc)«(bxc) = (cxa)-(axb)
2 Area (6 c sin A} = (c a sin B) (a b sin (7)
a2 sin -Z?sin C
2 Area =
sin A
Reciprocal Systems of Three Vectors. Solution of Equations
43.] The problem of expressing any vector r in terms of
three non-coplanar vectors a, b, c may be solved as follows.
Let
82 VECTOR ANALYSIS
where a, J, c are three scalar constants to be determined
Multiply by • b x c.
r.bxc = a a»bxc + 6 b«bxc + cc-bxo
or [rbc] = a [a be].
In like manner by multiplying the equation by • c x a and
. a X b the coefficients b and c may be found.
[r c a] = I [b c a]
[r a b] = c [c a b]
Hence r = a+b+, (28)
[be a] [c a b]
The denominators are all equal. Hence this gives the
equation
[a b c] r — [b c r] a + [c r a] b - [r a b] c = 0
which must exist between the four vectors r, a, b, c.
The equation may also be written
r»b x c r»o x a, r«a x b
r = -r . - a + r v .. b + e
[abe] [abc] [a be]
bxc cxa, axb
or r = r» r _ a + r • r b + r • o.
[abc] [abc] [abc]
The three vectors which appear here multiplied by !•, namely
bxc cxa axb
_ * _ - > _
[a be] [a b c] [a b c]
are very important. They are perpendicular respectively to
the planes of b and c, c and a, a and b. They occur over and
over again in a large number of important relations. For
this reason they merit a distinctive name and notation.
Definition : The system of three vectors
b x c ^ cxa axb
[abc]' [abc] ' [abc]
DIRECT AND SKEW PRODUCTS OF VECTORS 83
which are found by dividing the three vector products bxc,
c x a, a x b of three non-coplanar vectors a, b, c by the scalar
product [abc] is called the reciprocal system to a, b, c.
The word non-coplanar is important. If a, b, c were co
planar the scalar triple product [a b c] would vanish and
consequently the fractions
bxc cxa axb
j j ________
[a be] [a b c] [a b c]
would all become meaningless. Three coplanar vectors have
no reciprocal system. This must be carefully remembered.
Hereafter when the term reciprocal system is used, it will be
understood that the three vectors a, b, c are not coplanar.
The system of three vectors reciprocal to system a, b, c
will be denoted by primes as a', b', c',
,_bxc? h, _ c x a , , __ a x b (29)
""[abc]' [abc]' ~[a¥c]'
The expression for r reduces then to the very simple form
r = r-a' a + r-b'b + r.c' c. (30)
The vector r may be expressed in terms of the reciprocal
system a', b', c' instead of in terms of a, b, c. In the first
place it is necessary to note that if a, b, c are non-coplanar,
a', b', c' which are the normals to the planes of b and c,
c and a, a and b must also be non-coplanar. Hence r may
be expressed in terms of them by means of proper scalar
coefficients #, y, z.
r — x a' + ?/b' + z c'
Multiply successively by -a, -b, -c. This gives
[a b c] r-a = x [b c a], x = r-a
[abc]r-b = y [cab], y = r-b
[a b c] r-c = z [a b c], z = r-c
Hence r = r-a a ' + r-b b ' + r-c c '. (31)
84 VECTOR ANALYSIS
44.] If a', V, c' be the system reciprocal to a, b, c the
scalar product of any vector of the reciprocal system into the
corresponding vector of the given system is unity ; but
the product of two non-corresponding vectors is zero. That is
a'.a = bM>=:c'.c = l (32)
a'.b = a'.c = b'-a = b'*c = c'-a = c'-b = 0.
This may be seen most easily by expressing a', V, c' in
terms of themselves according to the formula (31)
r = r«aa' + r*bb' + r»cc'.
Hence a' = a'-aa' + a'«b V + a'«cc'
b' = b'.aa' + b'-bb' + bW
c' = c'«aa' + c'-bb' + c'.cc'.
Since a', b', c' are non-coplanar the corresponding coeffi
cients on the two sides of each of these three equations must
be equal. Hence from the first
1 = a '*a 0 = a'-b 0 = a' c.
From the second 0 = b'«a l=b'«b 0 = b'«e.
From the third 0 = c'«a 0 = c'«b l = c'«o.
This proves the relations. They may also be proved
directly from the definitions of a', b', c'.
bxc bxc«a [be a]
a'» a = • a = = = 1
[abc] [abc] [abc]
bxc bxc-b 0
a' . b = • b = = =0
[abc] [abc] [abc]
and so forth.
Conversely if two sets of three vectors each, say A, B, C,
and a, b, c, satisfy the relations
A«a = B«b = C«c = 1
A-b = A«c = B«a = B-c = C«a = C«b = 0
DIRECT AND SKEW PRODUCTS OF VECTORS 85
then the set A, B, C is the system reciprocal to a, b, c.
By reasoning similar to that before
A = A-a a' + A«b b' + A-c c'
B = B-a a' + B-b b' + B-c c'
C = C«aa' + C-bb' + C-c c'.
Substituting in these equations the given relations the re
sult is
A = a', B = b', C = c'.
Hence
Theorem : The necessary and sufficient conditions that the
set of vectors a', b', c' be the reciprocals of a, b, c is that
they satisfy the equations
a'.a = b'.b = c'.c = l (32)
a'-b = a'-c = b'«a = b'.c = c'-a = c'.b = 0.
As these equations are perfectly symmetrical with respect
to a', b', c' and a, b, c it is evident that the system a, b, c may
be looked upon as the reciprocal of the system a', b', c' just
as the system a', b', c' may be regarded as the reciprocal of
a, b, c. That is to say,
Theorem: If a', b', c' be the reciprocal system of a, b, c,
then a, b, c will be the reciprocal system of a', V, c'.
V x c' c' x a' a' x b' (29V
- — • b=- - - . v /
[a'b'e'] [a'b'c'] [a'b'c']
These relations may be demonstrated directly from the
definitions of a', b', c'. The demonstration is straightfor
ward, but rather long and tedious as it depends on compli
cated reduction formulae. The proof given above is as short
as could be desired. The relations between a', b',c'and
a, b, c are symmetrical and hence if a', b', c' is the reciprocal
system of a, b, c, then a, b, c must be the reciprocal system of
86
VECTOR ANALYSIS
45.] Theorem : If a', V, c' and a, b, c be reciprocal systems
the scalar triple products [a'b'c'] and [a b c] are numerical
reciprocals. That is
t.'b'.']=[i
[a'bV] [abc]=l
xc cxa axb"|
[a "be] [abc] [abc] J
[bxc cxa axb].
(33)
But
Hence
Hence
~[abc]3
[bxc cxa axb] = (bxc)x(cxa>(axb).
(bxc) x (cxa) = [abc]c.
[bxc cxa axb] = [abc] c-axb = [abc]2.
1 1
[a'bV] =
[abc]:
[abc]2 =
[abc]
(33)'
By means of this relation between [a' b' c'] and [a b c] it
is possible to prove an important reduction formula,
(P.axE)(A«BxC) =
P-A P.B p.c
Q.A Q.B a-c
B*A £*B £*C
(34)
which replaces the two scalar triple products by a sum of
nine terms each of which is the product of three direct pro
ducts. Thus the two crosses which occur in the two scalar
products are removed. To give the proof let P, ft, B be
expressed as
P = P-A A' + P.B B' + P.C C'
Then
But
B = B-A A' + B.B B' + B«C C'.
P.A P.B P.C
[POB] =
a-A Q.B a-c
R-A R.B R.C
1
[ABC]'
[A'B'C'].
[A'B'C'J =
DIRECT AND SKEW PRODUCTS OF VECTORS 87
Hence [PQE] [ABC] =
P.A P-B P-C
Q.A a-B Q.C
R.A R.B B*C
The system of three unit vectors i, j, k is its own reciprocal
system.
jxki kxi ,, i x j
J'k==k- (35)
For this reason the primes i', j', k' are not needed to denote
a system of vectors reciprocal to i, j, k. The primes will
therefore be used in the future to denote another set of rect
angular axes i, j, k , just as X* , F', Z* are used to denote a
set of axes different from X, F, Z.
The only systems of three vectors which are their own reciprocals
are the right-handed and left-handed systems of three unit
vectors. That is the system i, j, k and the system i, j, — k.
Let A, B, C be a set of vectors which is its own reciprocal.
Then by (32)
A«A = B-B = CNC = 1.
Hence the vectors are all unit vectors.
A-B = A-C = 0.
Hence A is perpendicular to B and C.
B-A = B-C = 0.
Hence B is perpendicular to A and C.
C-A =C.B = O.
Hence C is perpendicular to A and B.
Hence A, B, C must be a system like i, j, k or like i, j, — k.
* 46.] A scalar equation of the first degree in a vector r is
an equation in each term of which r occurs not more than
once. The value of each term must be scalar. As an exam
ple of such an equation the following may be given.
a a-bxr + 6(oxd)»(exr) + c f«r + d = 0,
88 VECTOR ANALYSIS
where a, b, c, d, e, f are known vectors ; and a, &, c, d, known
scalars. Obviously any scalar equation of the first degree in
an unknown vector r may be reduced to the form
r-A = a
where A is a known vector ; and a, a known scalar. To ac
complish this result in the case of the given equation proceed
as follows.
a axbor + "b (cxd)xe-r + c f»r + d = 0
{a axb + b (cxd)xe + c f}«r = — d.
In more complicated forms it may be necessary to make use
of various reduction formulae before the equation can be made
to take the desired form,
]>A = a.
As a vector has three degrees of freedom it is clear that one
scalar equation is insufficient to determine a vector. Three
scalar equations are necessary.
The geometric interpretation of the equa
tion
r.A => a (36)
is interesting. Let r be a variable vector
(Fig. 24) drawn from a fixed origin. Let
A be a fixed vector drawn from the same
origin. The equation then becomes
r A cos (r,A) = a,
a
or T cos (r,A) =— ,
if r be the magnitude of r ; and A that of A. The expression
r cos (r, A)
is the projection of r upon A. The equation therefore states
that the projection of r upon a certain fixed vector A must
DIRECT AND SKEW PRODUCTS OF VECTORS 89
always be constant and equal to a/ A. Consequently the ter
minus of r must trace out a plane perpendicular to the vector
A at a distance equal to a/ A from the origin. The projec
tion upon A of any radius vector drawn from the origin to a
point of this plane is constant and equal to a/ A. This gives
the following theorem.
Theorem : A scalar equation in an unknown vector may be
regarded as the equation of a plane, which is the locus of the
terminus of the unknown vector if its origin be fixed.
It is easy to see why three scalar equations in an unknown
vector determine the vector completely. Each equation de
termines a plane in which the terminus of r must lie. The
three planes intersect in one common point. Hence one vec
tor r is determined. The analytic solution of three scalar
equations is extremely easy. If the equations are
r«A = a
r-B = b (37)
r-C = c9
it is only necessary to call to mind the formula
r = r.A A' + r-BB' + r-C C'.
Hence r = a A' + 6 B' + c C'. (38)
The solution is therefore accomplished. It is expressed in
terms A', B', C' which is the reciprocal system to A, B, C. One
caution must however be observed. The vectors A, B, C will
have no reciprocal system if they are coplanar. Hence the
solution will fail. In this case, however, the three planes de
termined by the three equations will be parallel to a line.
They will therefore either not intersect (as in the case of the
lateral faces of a triangular prism) or they will intersect in a
common line. Hence there will be either no solution for r or
there will be an infinite number.
90 VECTOR ANALYSIS
From four scalar equations
r-A = a
r.B = 6 (39)
r«C = c
r«D =d
the vector r may be entirely eliminated. To accomplish this
solve three of the equations and substitute the value in the
fourth.
r = aA' + 6B' + cC'
a A'«D + &B'.D + cC'-D = d
or a [BCD] + b [CAD] + c [ABD] = d [ABC]. (40)
* 47.] A vector equation of the first degree in an unknown
vector is an equation each term of which is a vector quantity
containing the unknown vector not more than once. Such
an equation is
(AxB)x(Cxr) + D ET + n r + F =0,
where A, B, C, D, E, F are known vectors, n a known scalar,
and r the unknown vector. One such equation may in gen
eral be solved for r. That is to say, one vector equation is in
general sufficient to determine the unknown vector which is
contained in it to the first degree.
The method of solving a vector equation is to multiply it
with a dot successively by three arbitrary known non-coplanar
vectors. Thus three scalar equations are obtained. These
may be solved by the methods of the foregoing article. In the
first place let the equation be
A a«r + B b»r + C c-r = D,
where A, B, C, D, a, b, c are known vectors. No scalar coeffi
cients are written in the terms, for they may be incorporated in
the vectors. Multiply the equation successively by A', B', C'.
It is understood of course that A, B, C are non-coplanar.
DIRECT AND SKEW PRODUCTS OF VECTORS 91
a-r = D-Ar
b-r = D-B'
c-r = D-C'.
But r = a' a-r + b' b-r + c' c-r.
Hence r = D-A' a' + D-B' b' + D-C' c'.
The solution is therefore accomplished in case A, B, C are non-
coplanar and a, b, c also non-coplanar. The special cases in
which either of these sets of three vectors is coplanar will not
be discussed here.
The most general vector equation of the first degree in an
unknown vector r contains terms of the types
A a-r, n r, Exr, D.
That is it will contain terms which consist of a known
vector multiplied by the scalar product of another known vec
tor and the unknown vector ; terms which are scalar multi
ples of the unknown vector; terms which are the vector
product of a known and the unknown vector ; and constant
terms. The terms of the type A a-r may always be reduced
to three in number. For the vectors a, b, c, • • • which are
multiplied into r may all be expressed in terms of three non-
coplanar vectors. Hence all the products a-r, b-r, or, • • •
may be expressed in terms of three. The sum of all terms of
the type A a-r therefore reduces to an expression of three
terms, as
A a-r + B b-r + C c-r.
The terms of the types n r and Exr may also be expressed
in this form.
n r = 7i a' a-r + n b'b-r + n c'c-r
Exr = Exa' a-r + Exb' b-r+Exc' c-r.
Adding all these terms together the whole equation reduces
to the form
L a-r + M b-r + N c-r = K.
92 VECTOR ANALYSIS
This has already been solved as
r = K.L' a' + K-M' b' + XJT c'.
The solution is in terms of three non-coplanar vectors a', V, cf.
These form the system reciprocal to a, b, c in terms of which
the products containing the unknown vector r were expressed.
* SUNDRY APPLICATIONS OF PKODUCTS
Applications to Mechanics
48.] In the mechanics of a rigid body a force is not a
vector in the sense understood in this book. See Art. 3.
A force has magnitude and direction ; but it has also a line
of application. Two forces which are alike in magnitude
and direction, but which lie upon different lines in the body
do not produce the same effect. Nevertheless vectors are
sufficiently like forces to be useful in treating them.
If a number of forces fx, f2, f3, ---act on a body at the
same point 0, the sum of the forces added as vectors is called
the resultant R.
E = f1 + f2 + f8 + ...
In the same way if f x, f 2, f 8 • • • do not act at the same point
the term resultant is still applied to the sum of these forces
added just as if they were vectors.
B = f1 + fa + f8 + ... (41)
The idea of the resultant therefore does not introduce the
line of action of a force. As far as the resultant is concerned
a force does not differ from a vector.
Definition: The moment of a force f about the point 0 is
equal to the product of the force by the perpendicular dis
tance from 0 to the line of action of the force. The moment
however is best looked upon as a vector quantity. Its mag
nitude is as defined above. Its direction is usually taken to
DIRECT AND SKEW PRODUCTS OF VECTORS 93
be the normal on that side of the plane passed through the
point 0 and the line f upon which the force appears to pro
duce a tendency to rotation about the point 0 in the positive
trigonometric direction. Another method of defining the
moment of a force t = PQ about the point 0 is as follows :
The moment of the force f = PQ about the point 0 is equal
to twice the area of the triangle 0 PQ. This includes at once
both the magnitude and direction of the moment (Art. 25).
The point P is supposed to be the origin ; and the point Q,
the terminus of the arrow which represents the force f. The
letter M will be used to denote the moment. A subscript will
be attached to designate the point about which the moment is
taken.
The moment of a number of forces f x, f2, • • • is the (vector)
sum of the moments of the individual forces.
If
This is known as the total or resultant moment of the forces
* v *&••••
49.] If f be a force acting on a body and if d be the vector
drawn from the point 0 to any point in the line of action of
the force, the moment of the force about the point 0 is the
vector product of d into f .
Mo W = dxf (42)
For dxf = d f sin (d, f) e,
if e be a unit vector in the direction of dxf.
•
dxf = dsm (d, f)/e.
Now d sin (d, f) is the perpendicular distance from 0 to f.
The magnitude of dxf is accordingly equal to this perpen
dicular distance multiplied by/, the magnitude of the force.
94 VECTOR ANALYSIS
This is the magnitude of the moment MO {f} . The direction
of dxf is the same as the direction of the moment. Hence
the relation is proved.
Mo {f} = dxf.
The sum of the moments about 0 of a number of forces
f p f 2, • • • acting at the same point P is equal to the moment
of the resultant B of the forces acting at that point. For let
d be the vector from 0 to P. Then
Mo {f x> = dxf l
Mo {fa| = dxfa
+ ... (43)
= dx(£1 + fa + -..)=dxB
The total moment about 0 f of any number of forces f x, f 2, • • •
acting on a rigid body is equal to the total moment of those
forces about 0 increased by the moment about 0' of the
resultant BO considered as acting at 0.
M<x {f i> f 2 >•••} = Mo {f r f 2, • • •} + Mo< {Bo \. (44)
Let dj, d2, • • • be vectors drawn from 0 to any point in
f r f2, • • • respectively. Let d/, d2', • • • be the vectors drawn
from Of to the same points in f x, f 2, • • • respectively. Let o
be the vector from 0 to_0'. Then
d^d/H-c, d2 = d2' + c, •••
Mo {f i, f2, • • •} = dxxf j + d2xf2 + • • •
Mo'{f!,f2, •••J=d1'xf1 + da'xfa + ...
^ = (dx - c)xf ! + (d2 - c)xf a + • - .
= dxxf x+d2xf 2 + ---- cx(f j + f a + . . .)
But — c is the vector drawn from 0' to 0. Hence — c x f,
is the moment about 0' of a force equal in magnitude and
parallel in direction to f 1 but situated at 0. Hence
DIRECT AND SKEW PRODUCTS OF VECTORS 95
fa + ...) = - cxBo = Mo' {Bo}.
Hence MO/ {f x, f2, • • •} = M0 {fr fa ;,•••} + MCX {Bo |. (44)
The theorem is therefore proved.
The resultant £ is of course the same at all points. The
subscript 0 is attached merely to show at what point it is
supposed to act when the moment about Of is taken. For
the point of application of E affects the value of that moment.
The scalar product of the total moment and the resultant
is the same no matter about what point the moment be taken.
In other words the product of the total moment, the result
ant, and the cosine of the angle between them is invariant
for all points of space.
E • MO' {f i, f 2 »•••}= B • MO {f ! , f 2 9 • • •}
where Of and 0 are any two points in space. This -important
relation follows immediately from the equation
Mo' {*i, fa, • • •} = Mo {fj , f2, • • •} + Mo' {Eo}.
For E.Mo'if!,f2, •••}=*• M0 {f^, • • •} + E- M0' {B0}.
But the moment of E is perpendicular to E no matter what
the point 0 of application be. Hence
E-MO* IEO} = o
and the relation is proved. The variation in the total
moment due to a variation of the point about which the
moment is taken is always perpendicular to the resultant.
50.] A point Or may be found such that the total moment
about it is parallel to the resultant. The condition for
parallelism is
{fx, fa, •-•}=<)
=0
96 VECTOR ANALYSIS
where 0 is any point chosen at random. Replace Mo'{Eo}
by its value and for brevity omit to write the f v f 2, • • • in the
braces { }. Then
RxMcy = ExMo - Ex(cxE) = 0.
The problem is to solve this equation for c.
ExMo — E«E c + R.c E = 0.
Now R is a known quantity. Mo is also supposed to be
known. Let c be chosen in the plane through 0 perpen
dicular to E. Then E»c = 0 and the equation reduces to
ExM0 •= E«E c
ExMo
E-E
If c be chosen equal to this vector the total moment about
the point Or, which is at a vector distance from 0 equal to c,
will be parallel to E. Moreover, since the scalar product of
the total moment and the resultant is constant and since the
resultant itself is constant it is clear that in the case where
they are parallel the numerical value of the total moment
will be a minimum.
The total moment is unchanged by displacing the point
about which it is taken in the direction of the resultant.
For Mo' jf !, f2, • • •} = Mo {f ! , f 2, • • •} - cxE.
If c = 0 Of is parallel to E, cxE vanishes and the moment
about Of is equal to that about 0. Hence it is possible to
find not merely one point Or about which the total moment
is parallel to the resultant ; but the total moment about any
point in the line drawn through 0' parallel to E is parallel
to E. Furthermore the solution found in equation for c is
the only one which exists in the plane perpendicular to E —
unless the resultant E vanishes. The results that have been
obtained may be summed up as follows :
DIRECT AND SKEW PRODUCTS OF VECTORS 97
If any system of forces f 19 f 2, ••• whose resultant is not
zero act upon a rigid body, then there exists in space one
and only one line such that the total moment about any
point of it is parallel to the resultant. This line is itself
parallel to the resultant. The total moment about all points
of it is the same and is numerically less than that about any
other point in space.
This theorem is equivalent to the one which states that
any system of forces acting upon a rigid body is equivalent
to a single force (the resultant) acting in a definite line and
a couple of which the plane is perpendicular to the resultant
and of which the moment is a minimum. A system of forces
may be reduced to a single force (the resultant) acting at any
desired point 0 of space and a couple the moment of which
(regarded as a vector quantity) is equal to the total moment
about 0 of the forces acting on the body. But in general the
plane of this couple will not be perpendicular to the result
ant, nor will its moment be a minimum.
Those who would pursue the study of systems of forces
acting on a rigid body further and more thoroughly may
consult the Traite de Mecanique Rationnelle l by P. APPELL.
The first chapter of the first volume is entirely devoted to
the discussion of systems of forces. Appell defines a vector
as a quantity possessing magnitude, direction, and point of
application. His vectors are consequently not the same as
those used in this book. The treatment of his vectors is
carried through in the Cartesian coordinates. Each step
however may be easily converted into the notation of vector
analysis. A number of exercises is given at the close of
the chapter.
51.] Suppose a body be rotating about an axis with a con
stant angular velocity a. The points in the body describe
circles concentric with the axis in planes perpendicular to
1 Paris, Gauthier-Villars et Fils, 1893.
7
98
VECTOR ANALYSIS
FIG. 25.
the axis. The velocity of any point in its circle is equal
to the product of the angular velocity and the radius of the
circle. It is therefore equal to the product of the angular
velocity and the perpendicular dis
tance from the point to the axis.
The direction of the velocity is
perpendicular to the axis and to
the radius of the circle described
by the point.
Let a (Fig. 25) be a vector drawn
along the axis of rotation in that
direction in which a right-handed
screw would advance if turned in
the direction in which the body is
rotating. Let the magnitude of a
be a, the angular velocity. The vector a may be taken to
represent the rotation of the body. Let r be a radius vector
drawn from any point of the axis of rotation to a point in the
body. The vector product
axr = a rsin(a,r)
is equal in magnitude and direction to the velocity v of the
terminus of r. For its direction is perpendicular to a and r
and its magnitude is the product of a and the perpendicular
distance r sin (a, r) from the point to the line a. That is
v = axr. (45)
If the body be rotating simultaneously about several axes
ai* a2> aa •' ' which pass through the same point as in the
case of the gyroscope, the velocities due to the various
rotations are
vi -=a1xr1
v8 = a8xr8
DIRECT AND SKEW PRODUCTS OF VECTORS 99
where rx, r2, r3, • • • are the radii vec tores drawn from points
on the axis a19 a2, a3, • • • to the same point of the body. Let
the vectors rx, r2, r8, • • • be drawn from the common point of
intersection of the axes. Then
TJ = ra = r8 = • • • = r
and
v = vt + v2 + v3 + • • • == axxr + a2xr + a8xr + • • •
This shows that the body moves as if rotating with the
angular velocity which is the vector sum of the angular
velocities a19 a2, a8, • • • This theorem is sometimes known
as the parallelogram law of angular velocities.
It will be shown later (Art.) 60 that the motion of any
rigid body one point of which is fixed is at each instant of
time a rotation about some axis drawn through that point.
This axis is called the instantaneous axis of rotation. The
axis is not the same for all time, but constantly changes its
position. The motion of a rigid body one point of which is
fixed is therefore represented by
v = axr (45)
where a is the instantaneous angular velocity; and r, the
radius vector drawn from the fixed point to any point of the
body.
The most general motion of a rigid body no point of which
is fixed may be treated as follows. Choose an arbitrary
point 0. At any instant this point will have a velocity v0.
Relative to the point 0 the body will have a motion of rotation
about some axis drawn through 0. Hence the velocity v of
any point of the body may be represented by the sum of
V0 the velocity of 0 and axr the velocity of that point
relative to 0.
v = v0 + axr. (46)
100 VECTOR ANALYSIS
In case v0 is parallel to a, the body moves around a and
along a simultaneously. This is precisely the motion of a
screw advancing along a. In case v0 is perpendicular to a, it
is possible to find a point, given by the vector r, such that
its velocity is zero. That is
This may be done as follows. Multiply by xa.
(axr)xa = — v0xa
or a«a r — a-r a = — v0xa.
Let r be chosen perpendicular to a. Then a»r is zero and
a»a r = — v0 x a
f = - v0 x a
a«a
The point r, thus determined, has the property that its veloc
ity is zero. If a line be drawn through this point parallel to
a, the motion of the body is one of instantaneous rotation
about this new axis.
In case v0 is neither parallel nor perpendicular to a it may
be resolved into two components
v — v ' 4- v "
•n — vn r »n
which are respectively parallel and perpendicular to a.
v = v0' + v0" + axr
A point may now be found such that
v0" = — axr.
Let the different points of the body referred to this point be
denoted by r'. Then the equation becomes
v = v0' + axr'. (46)'
The motion here expressed consists of rotation about an axis
a and translation along that axis. It is therefore seen that
the most general motion of a rigid body is at any instant
DIRECT AND SKEW PRODUCTS OF VECTORS 101
the motion of a screw advancing at a certain rate along a
definite axis a in space. The axis of the screw and its rate
of advancing per unit of rotation (i. e. its pitch) change from
instant to instant.
52.] The conditions for equilibrium as obtained by the
principle of virtual velocities may be treated by vector
methods. Suppose any system of forces f x, f 2, • • • act on a
rigid body. If the body be displaced through a vector dis
tance D whether this distance be finite or infinitesimal the
work done by the forces is
The total work done is therefore
W^^il + D.f2 + ...
If the body be in equilibrium under the action of the forces
the work done must be zero.
W= D-fj + D-f2 + • • • = D-Cfj + f2 + • • 0 = D.E = 0.
The work done by the forces is equal to the work done by
their resultant. This must be zero for every displacement
D. The equation
D-E = 0
holds for all vectors D. Hence
E = 0.
The total resultant must be zero if the body be in equilibrium.
The work done by a force f when the rigid body is dis
placed by a rotation of angular velocity a for an infinitesimal
time t is approximately
a-dxf t,
where d is a vector drawn from any point of the axis of rota
tion a to any point of f. To prove this break up f into two
components f ', f " parallel and perpendicular respectively to a.
a-dxf = a-dxf ' + a-dxf ".
102 VECTOR ANALYSIS
As f ' is parallel to a the scalar product [a d f '] vanishes.
a-dxf = a-dxf ".
On the other hand the work done by t" is equal to the work
done by f during the displacement. For f ' being parallel to
a is perpendicular to its line of action. If h be the common
vector perpendicular from the line a to the force f ", the work
done by f " during a rotation of angular velocity a for time
t is approximately
The vector d drawn from any point of a to any point of f may
be broken up into three components of which one is h, another
is parallel to a, and the third is parallel to f ". In the scalar
triple product [adf] only that component of d which is
perpendicular alike to a and f " has any effect. Hence
W= a-hxf " t = a-dxf ' tf = a-dxf t.
If a rigid body upon which the forces f v f 2, • • • act be dis
placed by an angular velocity a for an infinitesimal time t
and if dx, d2, • • • be the vectors drawn from any point 0 of
a to any points of f v f 2, • - • respectively, then the work done
by the forces f v f2, - • • will be approximately
W= (a-djXfj + a«d2xf2 + • • •) t
= a.(d1xf1 + d2xf 2 + .-.)*
= a.M0{f1,f2,...} t.
If the body be in equilibrium this work must be zero.
Hence a*M0 \tl9 f2, • • •} t = 0.
The scalar product of the angular velocity a and the total
moment of the forces tv f2, • • • about any point 0 must be
zero. As a may be any vector whatsoever the moment itself
must vanish.
Mo {fr f r - • •} = 0.
DIRECT AND SKEW PRODUCTS OF VECTORS 103
The necessary conditions that a rigid body be in equilib
rium under the action of a system of forces is that the result
ant of those forces and the total moment about any point in
space shall vanish.
Conversely if the resultant of a system of forces and the
moment of those forces about any one particular point in space
vanish simultaneously, the body will be in equilibrium.
If E = 0, then for any displacement of translation D
D«E = o.
JF=D-f1 + D.f2 + ... = 0
and the total work done is zero, when the body suffers any
displacement of translation.
Let Mo {fp f2> • • •} be zero for a given point 0. Then for
any other point O1
Mo< {fx, f2, • • -\ = Mo 1 flf f2, - • •} + M0' {Bo}-
But by hypothesis E is also zero. Hence
Hence
where a is any vector whatsoever. But this expression is
equal to the work done by the forces when the body is rotated
for a time t with an angular velocity a about the line a
passing through the point O1. This work is zero.
Any displacement of a rigid body may be regarded as a
translation through a distance D combined with a rotation
for a time t with angular velocity a about a suitable line a in
space. It has been proved that the total work done by the
forces during this displacement is zero. Hence the forces
must be in equilibrium. The theorem is proved.
104 VECTOR ANALYSIS
Applications to Geometry
53.] Relations between two right-handed systems of three
mutually perpendicular unit vectors. — Let i, j, k and i', j', k'
be two such systems. They form their own reciprocal systems.
Hence
r = ™/I+r*^,tr^*v, (47)
and r = r«i' i' + r«j' j' + r«k' k'.
From this
/ i' = i'-i i + i'.j j + i'-k k = a1 i + a2 j + a3 k
I k' = k'-i i + k'-j j + k'-k k = cl i + c2 j + c3 k.
The scalarsflj, a2, a3; blt Z>2, b 3; cv 02, c3 are respectively the
direction cosines of i'; j'; k' with respect to i, j, k.
That is
<&]_ — cos (i', i) a2 = cos (i', j) a3 = cos (i', k)
0j "==- COS (j ) l) Ot^'==- COS (j i j) t>3 —- COS (J , Kj ( 4o)
cx = cos (k', i) c2 = cos (k', j) c3 = cos (k', k).
In the same manner
^ i = i-i'i' + i-j'j7 + i-k' k'= ^ i' + \ y + GI k'
j j - j-i'i' + j-j'j' + j-k'k' = aa i' + 62 j' + c2 k' (47)"
( k = k.i'i' + k.j'j' + k-k'k' = aa if + J8 J' + C3 k/
»!/!_/ "1 9i 9i 9
t fcf •!? — I — — /t ^ I n & j /• 4
and ) j.j = 1 = «22 + J22 + c22 (49)'
( k-k = 1 = as2 + 632 + c32
and ] j'.k' = 0 = \ cj + &2 c2 + bB cs (50)
I i_^ •! f\
\ K •! = U = Cj ttj -f- C<£ dy -f- C3 a%
DIRECT AND SKEW PRODUCTS OF VECTORS 105
and \ j-k = 0 = a* a9 + K I* + c0 c, (50)'
j.k = 0 = #2 a3 + 62 63 +
and
But
Hence
k' = i'xj'= (a2 53 - a3 62) i
yl "2
= (a2 &3 - a3 62),
(51)
(52)
Or
Co =
and similar relations may be found for the other six quantities
av a2, a3 ; bv &2, &3. All these scalar relations between the
coefficients of a transformation which expresses one set of
orthogonal axes X1, F', Z* in terms of another set JT, F, Z are
important and well known to students of Cartesian methods.
The ease with which they are obtained here may be note
worthy.
A number of vector relations, which are perhaps not so well
known, but nevertheless important, may be found by multi
plying the equations
i' = al i + a2 j + a3 k
in vector multiplication.
&! k' — Cj j ' = a3 j — a2 k.
(53)
The quantity on either side of this equality is a vector. From
its form upon the right it is seen to possess no component in
106 VECTOR ANALYSIS
the i direction but to lie wholly in the jk-plane ; and from
its form upon the left it is seen to lie in the j'k '-plane.
Hence it must be the line of intersection of those two planes.
Its magnitude is V af + a£ or V b^ + c^. This gives the
scalar relations
af + a* = V + *!2 = 1 - a*.
The magnitude 1 — a^ is the square*of the sine of the angle
between the vectors i and i'. Hence the vector
^k'-cj'^sj-aak (53)
is the line of intersection of the j'k'- and jk-planes, and
its magnitude is the sine of the angle between the planes.
Eight other similar vectors may be found, each of which gives
one of the nine lines of intersection of the two sets of mu
tually orthogonal planes. The magnitude of the vector is in
each case the sine of the angle between the planes.
54.] Various examples in Plane and Solid Geometry may
be solved by means of products.
Example 1 : The perpendiculars from the vertices of a trian
gle to the opposite sides meet in a point. Let A B 0 be the
triangle. Let the perpendiculars from A to BC and from B
to CA meet in the point 0. To show 0 0 is perpendicular
to A B. Choose 0 as origin and let OA = A, OB = B, and
=C. Then
= C-B,
By hypothesis
A.(C - B) = 0
and B<A - C) = 0.
Add; C<B - A) = 0,
which proves the theorem.
Example 2 : To find the vector equation of a line drawn
through the point B parallel to a given vector A.
DIRECT AND SKEW PRODUCTS OF VECTORS 107
Let 0 be the origin and B the vector OS. Let £ be the ra
dius vector from 0 to any point of the required line. Then
E — B is parallel to A. Hence the vector product vanishes.
Ax(B-B) = 0.
This is the desired equation. It is a vector equation in the
unknown vector B. The equation of a plane was seen (page
88) to be a scalar equation such as
B«C = c
in the unknown vector B.
The point of intersection of a line and a plane may be
found at once. The equations are
( Ax(B - B) = 0
i B-C = e
AxB = AxB
A-C B - C-B A = (AxB)xC
A-C B - c A = (AxB)xC
Hence (AxB)xC + c A .
A-C
The solution evidently fails when A»C = 0. In this case how
ever the line is parallel to the plane and there is no solution ;
or, if it lies in the plane, there are an infinite number of solu
tions.
Example 3: The introduction of vectors to represent planes.
Heretofore vectors have been used to denote plane areas of
definite extent. The direction of the vector was normal to
the plane and the magnitude was equal to the area to be re
presented. But it is possible to use vectors to denote not a
plane area but the entire plane itself, just as a vector represents
a point. The result is analogous to the plane coordinates of
analytic geometry. Let 0 be an assumed origin. Let M N be
a plane in space. The plane MN is to be denoted b^ a vector
108 VECTOR ANALYSIS
whose direction is the direction of the perpendicular dropped
upon the plane from the origin 0 and whose magnitude is the
reciprocal of the length of that perpendicular. Thus the nearer
a plane is to the origin the longer will be the vector which
represents it.
If r be any radius vector drawn from the origin to a point
in the plane and if p be the vector which denotes the plane,
then
r-p = 1
is the equation of the plane. For
r»p = r cos (r, p) p.
Now p, the length of p is the reciprocal of the perpendicular
distance from 0 to the plane. On the other hand r cos (r, p)
is that perpendicular distance. Hence r«p must be unity.
If r and p be expressed in terms of i, j, k
r = #i + yj + zk
p = ui + vj + wit
Hence r»p = xu + yv + zw = 'L.
The quantities u, v, w are the reciprocals of the intercepts of
the plane p upon the axes.
The relation between r and p is symmetrical. It is a rela
tion of duality. If in the equation
r-p = 1
r be regarded as variable, the equation represents a plane p
which is the locus of all points given by r. If however p be
regarded as variable and r as constant, the equation repre
sents a point r through which all the planes p pass. The
development of the idea of duality will not be carried out.
It is familiar to all students of geometry. The use of vec
tors to denote planes will scarcely be alluded to again until
Chapter VII.
DIRECT AND SKEW PRODUCTS OF VECTORS 109
SUMMARY OF CHAPTER II
The scalar product of two vectors is equal to the product
of their lengths multiplied by the cosine of the angle between
them.
A-B = A B cos (A, B) (1)
A-B = B.A (2)
A.A = ^«. (3)
The necessary and sufficient condition for the perpendicularity
of two vectors neither of which vanishes is that their scalar
product vanishes. The scalar products of the vectors i, j, k
are
^=J!Uk!=o (4)
A.B = A1B1 + A,, B2 + AS Bz (7)
H = A* = A* + A* + A*. (8)
If the projection of a vector B upon a vector A is Bf,
-R' — A'B A (*\
XA ' '
The vector product of two vectors is equal in magnitude to
the product of their lengths multiplied by the sine of the an
gle between them. The direction of the vector product is the
normal to the plane of the two vectors on that side on which
a rotation of less than 180° from the first vector to the second
appears positive.
AxB = A B sin (A, B) c. (9)
The vector product is equal in magnitude and direction to the
vector which represents the parallelogram of which A and B
are the two adjafcent sides. The necessary and sufficient con
dition for the parallelism of two vectors neither of which
110
VECTOR ANALYSIS
vanishes is that their vector product vanishes. The com
mutative laws do not hold.
AxB =
AxB = -BxA
ixi =jxj = kxk =
ixj =— jxi = k
jxk = — kxj =i
kxi = — ixk = j
£2) i + (Aa Bl - A1
(10)
(12)
AxB =
Bn Bo
(13)
(13)'
The scalar triple product of three vectors [A B C] is equal
to the volume of the parallelepiped of which A, B, C are three
edges which meet in a point.
[AB C] = A-BxC = B.CxA = C-AxB
= AxB-C = BxC«A = Cx A-B
[ABC] =- [A OB].
(15)'
(16)'
The dot and the cross in a scalar triple product may be inter
changed and the order of the letters may be permuted cyclicly
without altering the value of the product ; but a change of
cyclic order changes the sign.
[ABC] =
(18)'
[ABC] =
[a be]
(19)'
DIRECT AND SKEW PRODUCTS OF VECTORS 111
If the component of B perpendicular to A be B",
B,, = _AX(AXB)
A*A
Ax(BxC) = A-C B - A-B C (24)
(AxB)xC = A-C B - C-B A (24)'
(AxB>(CxD) = A.C B-D - A-D B-C (25)
(AxB)x(CxD) = [A CD] B- [BCD] A
= [ABD] C-[ABC] D. (26)
The equation which subsists between four vectors A, B, C, D
is
[BCD] A-[CDA]B + [DAB] C- [ABC] D = 0. (27)
Application of formulae of vector analysis to obtain the for
mulae of Plane and Spherical Trigonometry.
The system of vectors a', V, c' is said to be reciprocal to the
system of three non-coplanar vectors a, b, c
bxc cxa axb
when a'= _ . ., b' = =— -=> *' = ——• (29)
[a be] [abc] [abc]
A vector r may be expressed in terms of a set of vectors and
its reciprocal in two similar ways
r = r.a' a + r.V b + r-c' c (30)
r = r-aa' + r.bb' + r.cc'. (31)
The necessary and sufficient conditions that the two systems of
non-coplanar vectors a, b, c and a', b', c' be reciprocals is that
a'.a = V»b = c'«c = 1
a'.b = a'-c = b'.c = b'.a = c'-a = e'-b = 0.
If a', b', c' form a system reciprocal to a, b, c ; then a, b, c will
form a system reciprocal to a', b', c'.
112 VECTOR ANALYSIS
P.A P.B P.C
a-A ft-B a-c
R.A B-B R-C
[PaK][ABC] =
(34)
The system i, j, k is its own reciprocal and if conversely a
system be its own reciprocal it must be a right or left handed
system of three mutually perpendicular unit vectors. Appli
cation of the theory of reciprocal systems to the solution of
scalar and vector equations of the first degree in an unknown
vector. The vector equation of a plane is
r-A = a. (36)
Applications of the methods developed in Chapter II., to the
treatment of a system of forces acting on a rigid body and in
particular to the reduction of any system of forces to a single
force and a couple of which the plane is perpendicular to that
force. Application of the methods to the treatment of
instantaneous motion of a rigid body obtaining
v = v0 + a x r (46)
where v is the velocity of any point, v0 a translational veloc
ity in the direction a, and a the vector angular velocity of ro
tation. Further application of the methods to obtain the
conditions for equilibrium by making use of the principle of
virtual velocities. Applications of the method to obtain
the relations which exist between the nine direction cosines
of the angles between two systems of mutually orthogonal
axes. Application to special problems in geometry including
the form under which plane coordinates make their appear
ance in vector analysis and the method by which planes (as
distinguished from finite plane areas) may be represented
by vectors.
DIRECT AND SKEW PRODUCTS OF VECTORS 113
EXERCISES ON CHAPTER II
Prove the following reduction formulae
1. Ax{Bx(CxD)} = [ACD]B-A-BCxD
= B«D AxC — B-C AxD.
2. [AxB CxD ExF] = [ABD] [CEF]- [ABC] [DBF]
= [ABE] [FCD] - [ABF] [BCD]
= [CD A] [BEF] - [CDB] [AEF].
3. [AxB BxC CxA] = [ABC]2.
P.A P.B P
4 [PQE] (AxB) =
Q.B Q
R»A R«B R
5. Ax(BxC) + Bx(CxA) + Cx(AxB) = 0.
6. [AxP Bxtt CxR] + [Axtt BxR CxP]
+ [AxR BxP Cxtt] = 0.
7. Obtain formula (34) in the text by expanding
[(AxB)xP].[Cx(ttxR)]
in two different ways and equating the results.
8. Demonstrate directly by the above formulae that if
a', V, c' form a reciprocal system to a, b, c; then a, b, c form
a system reciprocal to a', b', c'.
9. Show the connection between reciprocal systems of vec
tors and polar triangles upon a sphere* Obtain some of the
geometrical formulae connected with polar triangles by inter
preting vector formulae such as (3) in the above list.
10. The perpendicular bisectors of the sides of a triangle
meet in a point.
11. Find an expression for the common perpendicular to
two lines not lying in the same plane.
114
VECTOR ANALYSIS
12. Show by vector methods that the formulae for the vol
ume of a tetrahedron whose four vertices are
IS
13. Making use of formula (34) of the text show that
[abo] = a be
N
1
n
m
n
1
I
m
I
1
where a, &, c are the lengths of a, b, c respectively and where
I = cos (b, c), m = cos (c, a), n = cos (a, b).
14. Determine the perpendicular (as a vector quantity)
which is dropped from the origin upon a plane determined by
the termini of the vectors a, b, c. Use the method of solution
given in Art. 46.
15. Show that the volume of a tetrahedron is equal to one
sixth of the product of two opposite edges by the perpendicu
lar distance between them and the sine of the included angle.
16. If a line is drawn in each face plane of any triedral angle
through the vertex and perpendicular to the third edge, the
three lines thus obtained lie in a plane.
CHAPTER III
THE DIFFERENTIAL CALCULUS OF VECTORS
Differentiation of Functions of One Scalar Variable
55.] IF a vector varies and changes from r to r' the incre
ment of r will be the 'difference between r' and r and will be
denoted as usual by A r.
Ar = r'-r, (1)
where A r must be a vector quantity. If the variable r be
unrestricted the increment A r is of course also unrestricted :
it may have any magnitude and any direction. If, however,
the vector r be regarded as a function (a vector function) of
a single scalar variable t the value of A r will be completely
determined when the two values t and tf of £, which give the
two values r and r', are known.
To obtain a clearer conception of the quantities involved
it will be advantageous to think of the vector r as drawn
from a fixed origin 0 (Fig. 26). When
the independent variable t changes its
value the vector r will change, and as t
possesses one degree of freedom r will
vary in such a way that its terminus
describes a curve in space, r will be
the radius vector of one point P of
the curve ; r', of a neighboring point Pf. A r will be the
chord PP1 of the curve. The ratio
Ar
A*
FIG. 26.
116 VECTOR ANALYSIS
will be a vector collinear with the chord P Pf but magnified
in the ratio 1 : A t. When A t approaches zero Pf will ap
proach P, the chord PP1 will approach the tangent at P, and
the vector
Ar ... rfr
— — will approach —
i\ t (t t
which is a vector tangent to the curve at P directed in that
sense in which the variable t increases along the curve.
If r be expressed in terms of i, j, k as
r = rx i + r2 j + rz k
the components rv r2, r3 will be functions of the scalar t.
r' = (r1 + Arj)i+ (^2 + Ar2)j + (r3 + Ar3)k
Ar = r' — r = Arx i + Ar2j + Ar3k
A r _ A ?*! . A r2 . A r8
^1 " J+ k
and
Hence the components of the first derivative of r with re
spect to t are the first derivatives with respect to t of the
components of r. The same is true for the second and higher
derivatives.
. . ~
i j __ _ f • , _ 3
*
_ __ _ _
dt*~ dt* dt* dt*
(2)'
dn r dn r, . dnrfl dnr*
- — - l i j __ ? i _i __ ? v
dtn dt» dtn J dt*
In a similar manner if r be expressed in terms of any three
non-coplanar vectors a, b, c as
r = aa + &b + cc
dnr dna dnb dnc
THE DIFFERENTIAL CALCULUS OF VECTORS 117
Example : Let r = a cos t + b sin t.
The vector r will then describe an ellipse of which a and b
are two conjugate diameters. This may be seen by assum
ing a set of oblique Cartesian axes X, Y coincident with a
and b. Then
X = a cos t, Y = 6 sin t,
which is the equation of an ellipse referred to a pair of con
jugate diameters of lengths a and b respectively.
dr
-3— = — a sin t + b cos t.
a t
Hence — = a cos (t + 90°) + b sin (t + 90°).
The tangent to the curve is parallel to the radius vector
for 0 + 90°). 2r
= — (a cos t + b sin t).
The second derivative is the negative of r. Hence
is evidently a differential equation satisfied by the ellipse.
Example : Let r = a cosh t + b sinh t.
The vector r will then describe an hyperbola of which a and
b are two conjugate diameters.
dr
— = a sinh t + b cosh t,
dt
and -— - = a cosh t + b sinh t.
Hence — — = r
d t*
is a differential equation satisfied by the hyperbola.
118 VECTOR ANALYSIS
56.] A combination of vectors all of which depend on the
same scalar variable t may be differentiated very much as in
ordinary calculus.
d
For
(a + Aa) . (b
A(a-b) = (a + Aa) • (b + Ab) - a-b
Ab Aa Aa-Ab
= a • — • H -- • b + - 1 - -
-
A* A* A*
Hence in the limit when A t = 0,
d_
dt
_(a.bxc) = a-b
dt v \d t
X [b X
The last three of these formulae may be demonstrated exactly
as the first was.
The formal process of differentiation in vector analysis
differs in no way from that in scalar analysis except in this
one point in which vector analysis always differs from scalar
analysis, namely : The order of the factors in a vector product
THE DIFFERENTIAL CALCULUS OF VECTORS 119
cannot be changed without changing the sign of the product.
Hence of the two formulae
d
and
the first is evidently incorrect, but the second correct. In
other words, scalar differentiation must take place without
altering the order of the factors of a vector product. The
factors must be differentiated in situ. This of course was to
be expected.
In case the vectors depend upon more than one variable
the results are practically the same. In place of total deriva
tives with respect to the scalar variables, partial derivatives
occur. Suppose a and b are two vectors which depend on
three scalar variables #, y, z. The scalar product a»b will
depend upon these three variables, and it will have three
partial derivatives of the first order.
The second partial derivatives are formed in the same way.
52
-
9y \3x5y
120 VECTOR ANALYSTS
Often it is more convenient to use not the derivatives but
the differentials. This is particularly true when dealing with
first differentials. The formulas (3), (4) become
d (a • b) = da, • b + a • db, (3)'
d (a X b) = ds, x b + a x db, (4)'
and so forth. As an illustration consider the following
example. If r be a unit vector
r«r = 1.
The locus of the terminus of r is a spherical surface of unit
radius described about the origin, r depends upon two vari
ables. Differentiate the equation.
Hence r • d r = 0.
Hence the increment di of a unit vector is perpendicular to
the vector. This can be seen geometrically. If r traces a
sphere the variation d r must be at each point in the tangent
plane and hence perpendicular to r.
*57.j Vector methods may be employed advantageously
in the discussion of curvature and torsion of curves. Let r
denote the radius vector of a curve
where f is some vector function of the scalar t. In most appli
cations in physics and mechanics t represents the time. Let
s be the length of arc measured from some definite point of
the curve as origin. The increment A r is the chord of the
curve. Hence A r / A s is approximately equal in magnitude
to unity and approaches unity as its limit when A s becomes
infinitesimal. Hence d r / d s will be a unit vector tangent to
the curve and will be directed toward that portion of the
THE DIFFERENTIAL CALCULUS OF VECTORS 121
curve along which s is increasing (Fig. 27). Let t be the
unit tangent UAt
The curvature of the curve is the
limit of the ratio of the angle through
which the tangent turns to the length
of the arc. The tangent changes by the increment At. As t
is of unit length, the length of A t is approximately the angle
through which the tangent has turned measured in circular
measure. Hence the directed curvature C is
LIM = t =
As=0 As ds ds*
The vector C is collinear with A t and hence perpendicular to
t; for inasmuch as t is a unit vector At is perpendicular
to t.
The tortuosity of a curve is the limit of the ratio of the
angle through which the osculating plane turns to the length
of the arc. The osculating plane is the plane of the tangent
vector t and the curvature vector C. The normal to this
planei8 N = txC.
If c be a unit vector collinear with C
n = t x c
will be a unit normal (Fig. 28) to the osculating plane and
the three vectors t, c, n form an i, j, k system,
that is, a right-handed rectangular system.
Then the angle through which the osculating
plane turns will be given approximately by
A n and hence the tortuosity is by definition _
d n / d s.
From the fact that t, c, n form an i, j, k system of unit
vectors
122 VECTOR ANALYSIS
t» t = c • c = n«n = 1
and t«c = c»n = n»t = 0.
Differentiating the first set
t-dt = c»dc = n»dn = 0,
and the second
t* do + rft« c =c»dn + dc«n = n«dt + dn«t==0.
But d t is parallel to c and consequently perpendicular to n.
n- dt = 0.
Hence d n • t = 0.
The increment of n is perpendicular to t. But the increment
of n is also perpendicular to n. It is therefore parallel to c.
As the tortuosity is T = dn/ds, it is parallel to dn and hence
to c.
The tortuosity T is
~ds^
d*r d*r
T — v
* j O •» «*
d s*
i \
VC»C/
The first term of this expression vanishes. T moreover has
been seen to be parallel to C = d2r/ds2. Consequently the
magnitude of T is the scalar product of T by the unit vec
tor c in the direction of C. It is desirable however to have
the tortuosity positive when the normal n appears to turn in
the positive or counterclockwise direction if viewed from
that side of the n c-plane upon which t or the positive part
of the curve lies. With this convention d n appears to move
in the direction — c when the tortuosity is positive, that is, n
turns away from c. The scalar value of the tortuosity will
therefore be given by — c • T.
THE DIFFERENTIAL CALCULUS OF VECTORS 123
1 dr d2r d 1
— c • T = — c • — x —
But c is parallel to the vector d2 i/d s2. Hence
dr
ds ds2~~
And c is a unit vector in the direction C. Hence
C
~
Hence T. -c-T = - . x
(12)
Or r= . (13)
The tortuosity may be obtained by another method which
is somewhat shorter if not quite so straightforward.
t»c = c»n = n»t = 0.
Hence dt»c = — dc»t
dc • n = — dn • c
dn*t = — dt»n.
Now d t is parallel to c ; hence perpendicular to n. Hence
d t • n = 0. Hence dn»t = 0. Butdnis perpendicular to n.
Hence d n must be parallel to c. The tortuosity is the mag
nitude of dn/ds taken however with the negative sign
because d n appears clockwise from the positive direction of
the curve. Hence the scalar tortuosity T may be given by
dn dc
r=- — .c = n.— , (14)
ds ds
r = txc-^-C, (14)'
as
124
VECTOR ANALYSIS
C
c =
dc
ds'
V c«c
dC , d /
!•; C ---VC.C
ds ds
C-C
But
t x c • C = 0.
t x c • — A/C • C
C-C
m
1 =
_ dC
T~
ITTc '
(13)
ds*
In Cartesian coordinates this becomes
T=
d x
dy
dz
ds
ds
ds
d*x
d*y
d*z
~<n*
d^*
ds*
a x
ds y
d* z
d^
d^
dT*
(13)'
Those who would pursue the study of twisted curves and
surfaces in space further from the standpoint of vectoi-s will
find the book " Application de la Methode Vectorielle de Grass-
maun d la Greometrie Infinitesimale"1 by FEHB extremely
1 Paris, Carre et Naud, 1899.
THE DIFFERENTIAL CALCULUS OF VECTORS 125
helpful. He works with vectors constantly. The treatment
is elegant. The notation used is however slightly different
from that used by the present writer. The fundamental
points of difference are exhibited in this table
HI X a2 ~ Oi «2]
ax • a2 x a3 = [ax a2 a3] ~ [ax a2 aj.
One used to either method need have no difficulty with the
other. All the important elementary properties of curves
and surfaces are there treated. They will not be taken
up here.
* Kinematics
58.] Let r be a radius vector drawn from a fixed origin to
a moving point or particle. Let t be the time. The equation
of the path is then
The velocity of the particle is its rate of change of position.
This is the limit of the increment A r to the increment A t.
LIM f A r"| d r
V = A * .
This velocity is a vector quantity. Its direction is the
direction of the tangent of the curve described by the par
ticle. The term speed is used frequently to denote merely
the scalar value of the velocity. This convention will be
followed here. Then
.-£, (16)
if s be the length of the arc measured from some fixed point
of the curve. It is found convenient in mechanics to denote
differentiations with respect to the time by dots placed over
the quantity differentiated. This is the oldfliixional notation
126 VECTOR ANALYSIS
introduced by Newton. It will also be convenient to denote
the unit tangent to the curve by t. The equations become
-•-T. <">
v = v t. (17)
The acceleration is the rate of change of velocity. It
is a vector quantity. Let it be denoted by A. Then by
definition
LIM A v d v
_ _
-At=OA7-rf7 =
dv d
and
Differentiate the expression v = v t.
dv d(vt) dv dt
A - — __ — - v ' . — _ * I nj __ . -
dt ~ dt dt dt
dv dzs~
dt dt d s
_ — _ _ _ = C t?
d t ds d t
where C is the (vector) curvature of the curve and v is the
speed in the curve. Substituting these values in the equation
the result is
A = s t + v* C.
The acceleration of a particle moving in a curve has there
fore been broken up into two components of which one is paral
lei to the tangent t and of which the other is parallel to the
curvature C, that is, perpendicular to the tangent. That this
resolution has been accomplished would be unimportant were
THE DIFFERENTIAL CALCULUS OF VECTORS 127
it not for the remarkable fact which it brings to light. The
component of the acceleration parallel to the tangent is equal
in magnitude to the rate of change of speed. It is entirely
independent of what sort of curve the particle is describing.
It would be the same if the particle described a right line
with the same speed as it describes the curve. On the other
hand the component of the acceleration normal to the tangent
is equal in magnitude to the product of the square of the
speed of the particle and the curvature of the curve. The
sharper the curve, the greater this component. The greater
the speed of the particle, the greater the component. But the
rate of change of speed in path has no effect at all on this
normal component of the acceleration.
If r be expressed in terms of i, j, k as
r = # i + y} '+ z k,
v = V ** + y* + *2, (16)'
A = v = r = ii + yj + * k, (18)'
x x + i/ i/ + z %
A = v=s = y *
V x2 + y* + z2
From these formulae the difference between st the rate of
change of speed, and A = r, the rate of change of velocity,
is apparent. Just when this difference first became clearly
recognized would be hard to say. But certain it is that
Newton must have had it in mind when he stated his second
law of motion. The rate of change of velocity is proportional
to the impressed force ; but rate of change of speed is not.
59.] The hodograph was introduced by Hamilton as an
aid to the study of the curvilinear motion of a particle.
With any assumed origin the vector velocity r is laid off.
The locus of its terminus is the hodograph. In other words,
the radius vector in the hodograph gives the velocity of the
128 VECTOR ANALYSIS
particle in magnitude and direction at any instant. It is
possible to proceed one step further and construct the hodo
graph of the hodograph. This is done by laying off the
vector acceleration A = r from an assumed origin. The
radius vector in the hodograph of the hodograph therefore
gives the acceleration at each instant.
Example 1 : Let a particle revolve in a circle (Fig. 29)
of radius r with a uniform
fV * ^-r^ angular velocity a. The
speed of the particle will then
be equal to
v = a r.
Let r be the radius vector
drawn to the particle. The
velocity v is perpendicular to r and to a. It is
f = v = a x r.
The vector v is always perpendicular and of constant magni
tude. The hodograph is therefore a circle of radius v = a r.
The radius vector r in this circle is just ninety degrees in
advance of the radius vector r in its circle, and it conse
quently describes the circle with the same angular velocity
a. The acceleration A which is the rate of change of y is
always perpendicular to v and equal in magnitude to
A = a v = a2 r.
The acceleration A may be given by the formula
r = A = axv = ax(axr) = a«r a — a-a r.
But as a is perpendicular to the plane in which r lies, a • r = 0.
Hence •• 9
r = A = — a»a r = — a2 r.
The acceleration due to the uniform motion of a particle in
a circle is directed toward the centre and is equal in magni
tude to the square of the angular velocity multiplied by the
radius of the circle.
THE DIFFERENTIAL CALCULUS OF VECTORS 129
Example 2: Consider the motion of a projectile. The
acceleration in this case is the acceleration g due to gravity.
r = A = g.
The hodograph of the hodograph reduces to a constant
vector. The curve is merely a point. It is easy to find
the hodograph. Let v0 be the velocity of the projectile
in path at any given instant. At a later instant the velocity
will be
v = v0 + t g.
Thus the hodograph is a straight line parallel to g and pass
ing through the extremity of v0. The hodograph of a
particle moving under the influence of gravity is hence a
straight line. The path is well known to be a parabola.
Example 3 : In case a particle move under any central
acceleration
r = A = f(r).
The tangents to the hodograph of r are the accelerations r!
But these tangents are approximately collinear with the
chords between two successive values r and f0 of the radius
vector in the hodograph. That is approximately
A*
Multiply by rx. r x r = r x .
Since r and r are parallel
r x (r - r0) = 0.
Hence r x r = r x r0 .
But J r x f is the rate of description of area. Hence the
equation states that when a particle moves under an ac
celeration directed towards the centre, equal areas are swept
over in equal times by the radius vector.
9
130 VECTOR ANALYSIS
Perhaps it would be well to go a little more carefully into
this question. If r be the radius vector of the particle in
its path at one instant, the radius vector at the next instant
is r + A r. The area of the vector of which r and r + A r are
the bounding radii is approximately equal to the area of the
triangle enclosed by r, r + A r, and the chord A r. This
area is
The rate of description of area by the radius vector is
consequently
LIM irx(r+ Ar) Lm 1 AT 1
A* = 02 A* ~A*-=02 A*~2r
Let r and r0 be two values of the velocity at two points
P and P0 which are near together. The acceleration r0 at P0
is the limit of
r — r0 _ A r
A* " A * *
A * * "
Break up the vector — ^- = ?^IlI? into two components one
A t A t
parallel and the other perpendicular to the acceleration r0.
Ar.
if n be a normal to the vector if0. The quantity x ap
proaches unity when A t approaches zero. The quantity y
approaches zero when A t approaches zero.
Ar = r-r0 = #A*r0 + yA*n.
Hence r x (r - r0) = x A* r x r0 + y A* r x n.
r x (f - r0) = r x r - (r0 + ^ A M x f0.
THE DIFFERENTIAL CALCULUS OF VECTORS 131
Hence
Ar
rxr-r0xf0=— xr0A* + zA*rxr0 + yA* rxn.
/A 6
But each of the three terms upon the right-hand side is an
infinitesimal of the second order. Hence the rates of descrip
tion of area at P and Pd differ by an infinitesimal of the
second order with respect to the time. This is true for any
point of the curve. Hence the rates must be exactly equal
at all points. This proves the theorem.
60.] The motion of a rigid body one point of which is
fixed is at any instant a rotation about an instantaneous axis
passing through the fixed point.
Let i, j, k be three axes fixed in the body but moving in
space. Let the radius vector r be drawn from the fixed point
to any point of the body. Then
But d r = (d r • i) i + (d r • j) j 4- (d r • k) k.
Substituting the values of d r • i, d r • j, d r • k obtained from
the second equation
dr = (xi • di+ yi • d j + 2 i • d k) i
j • di + yj *dj + zj •
But i • j =j • k = k • i = 0.
Hence i • dj +j • di = Q or j-c?i = — i • dj
j.dk + k.dj = 0 or k.dj = — j-dk
k.di + i.rfk = 0 or i«dk = — k • di.
Moreover i.i=j .j=k«k = l.
Hence i • d i = j • d j = k • d k = 0.
132 VECTOR ANALYSIS
Substituting these values in the expression for d r.
dr = (zi'dk — yj«di)i+(«j«di — s k
+ (y k . d j - x i • d k) k.
This is a vector product.
dr = (Wj i + i«dkj + j«dik)x(>i + yj + 2; k).
Let d j . d k . d i
— k-r«l+|-i«J+J'iik-
Then . d r
r = ;n=axr-
This shows that the instantaneous motion of the body is one
of rotation with the angular velocity a about the line a.
This angular velocity changes from instant to instant. The
proof of this theorem fills the lacuna in the work in Art. 51.
Two infinitesimal rotations may be added like vectors.
Let ax and a2 be two angular velocities. The displacements
due to them are
dl r = ax x r d t,
d2 r = a2 x r d t.
If r be displaced by a, it becomes
T + d1T = T + a,1xrdt.
If it then be displaced by a2, it becomes
r 4- d r = r + dl r + % x [r + (ax x r) d t] d t.
Hence d r = aj x r d t + a2 x r d t + a2 x (ax x r) (d £)2.
If the infinitesimals (d t)2 of order higher than the first be
neglected,
d r = ax x r d t + a2 x r d t = (ax + a2) x r d t,
which proves the theorem. If both sides be divided by d t
. dr
r = — = (a1 + a2) x r.
THE DIFFERENTIAL CALCULUS OF VECTORS 133
This is the parallelogram law for angular velocities. It
was obtained before (Art. 51) in a different way.
In case the direction of a, the instantaneous axis, is con
stant, the motion reduces to one of steady rotation about a.
r = a x r.
The acceleration r = axr + axr = axr + ax (axr).
As a does not change its direction a must be collinear with
a and hence a x r is parallel to a x r. That is, it is perpen
dicular to r. On the other hand ax (a x r) is parallel to r.
Inasmuch as all points of the rotating body move in con
centric circles about a in planes perpendicular to a, it is
unnecessary to consider more than one such plane.
The part of the acceleration of a particle toward the centre
of the circle in which it moves is
a x (a x r).
This is equal in magnitude to the square of the angular
velocity multiplied by the radius of the circle. It does not
depend upon the angular acceleration a at all. It corresponds
to what is known as centrifugal force. On the other hand
the acceleration normal to the radius of the circle is
axr.
This is equal in magnitude to the rate of change of angular
velocity multiplied by the radius of the circle. It does not
depend in any way upon the angular velocity itself but only
upon its rate of change.
61.] The subject of integration of vector equations in which
the differentials depend upon scalar variables needs but a
word. It is precisely like integration in ordinary calculus.
If then d r = d s,
r = s + C,
134 VECTOR ANALYSIS
where C is some constant vector. To accomplish the integra
tion in any particular case may be a matter of some difficulty
just as it is in the case of ordinary integration of scalars.
Example 1 : Integrate the equation of motion of a
projectile.
The equation of motion is simply
which expresses the fact that the acceleration is always ver
tically downward and due to gravity.
r = g t + b,
where b is a constant of integration. It is evidently the
velocity at the time t = 0.
r = ig*2 + b* + c.
c is another constant of integration. It is the position vector
of the point at time t = 0. The path which is given by this
last equation is a parabola. That this is so may be seen by
expressing it in terms of x and y and eliminating t.
Example % : The rate of description of areas when a par
ticle moves under a central acceleration is constant.
r = f(r).
Since the acceleration is parallel to the radius,
r x r = 0.
But r x r = — (r x r).
a L>
For — (r x f ) = r x f + r x r.
u/ t
Hence — (r x r) = 0
CL t
and r x f = C,
which proves the statement.
THE DIFFERENTIAL CALCULUS OF VECTORS 135
Example 3 : Integrate the equation of motion for a particle
moving with an acceleration toward the centre and equal to
a constant multiple of the inverse square of the distance
from the centre.
^2
Given
Then r x r = 0.
Hence r x r = C.
Multiply the equations together with x.
r xC -1 -1 (
— ^- = — rx (rxr)= -jjj- {r.r r - r-r r}.
r • r = r2.
Differentiate. Then r • r = r r.
Hence *2L = £ _ L r
o o *••
Each side of this equality is a perfect differential.
Integrate. Then r x C = £ + e I,
c* r
where e I is the vector constant of integration, e is its magni
tude and I a unit vector in its direction. Multiply the equa
tion by r • .
r • r x C r • r
+ e r • I.
But
c* r *
r • r x C r x r • C C • C
136 VECTOR ANALYSIS
T f C • C
p = — s" and cos u = cos (r, I).
c
Then p = r + e r cos u.
Or p
r =
1 + e cos w
This is the equation of the ellipse of which e is the eccentri
city. The vector I is drawn in the direction of the major
axis. The length of this axis is
It is possible to cany the integration further and obtain
the time. So far merely the path has been found.
Scalar Functions of Position in Space. The Operator V
62. ] A function V (x, y, z) which takes on a definite scalar
value for each set of coordinates #, y, z in space is called a
scalar function of position in space. Such a function, for ex
ample, is
V O, y, z) == x2 + y* + z2 = r\
This function gives the square of the distance of the point
(x, y, z) from the origin. The function V will be supposed to
be in general continuous and single-valued. In physics scalar
functions of position are of constant occurrence. In the
theory of heat the temperature T at any point of a body is a
scalar function of the position of that point. In mechanics
and theories of attraction the potential is the all-important
function. This, too, is a scalar function of position.
If a scalar function V be set equal to a constant, the equa
tion
V(x,y,z)=c. (20)
defines a surface in space such that at every point of it the
function V has the same value c. In case V be the tempera-
THE DIFFERENTIAL CALCULUS OF VECTORS 137
ture, this is a surface of constant temperature. It is called an
isothermal surface. In case V be the potential, this surface of
constant potential is known as an equipotential surface. As
the potential is a typical scalar function of position in space,
and as it is perhaps the most important of all such functions
owing to its manifold applications, the surface
V O, y,z)=c
obtained by setting V equal to a constant is frequently spoken
of as an equipotential surface even in the case where V has
no connection with the potential, but is any scalar function
of positions in space.
The rate at which the function V increases in the X direc
tion — that is, when x changes to x + A x and y and z remain
constant — is
LIM [" F" (a? + A a, y, g) - T (x, y, z) "1
Aa = 0 L A x J*
This is the partial derivative of Fwith respect to x. Hence
the rates at which V increases in the directions of the three
axes X, Y) Z are respectively
3V 3V 3V
~Wx' Ty* Tz
Inasmuch as these are rates in a certain direction, they may
be written appropriately as vectors. Let i, j, k be a system
of unit vectors coincident with the rectangular system of
axes X, Y) Z. The rates of increase of V are
3V 3V 3V
1JZ* J5? ' ~3~z'
The sum of these three vectors would therefore appear to be
a vector which represents both in magnitude and direction
the resultant or most rapid rate of increase of V. That this
is actually the case will be shown later (Art. 64).
138 VECTOR ANALYSIS
63.] The vector sum which is the resultant rate of increase
of Fis denoted by VF
V V represents a directed rate of change of V — a directed
or vector derivative of F^ so to speak. For this reason VF
will be called the derivative of V; and F, the primitive of
VF. The terms gradient and slope of F are also used for
V F. It is customary to regard V as an operator which obtains
a vector V F from a scalar function F of position in space.
This symbolic operator V was introduced by Sir W. R.
Hamilton and is now in universal employment. There
seems, however, to be no universally recognized name l for it,
although owing to the frequent occurrence of the symbol
some name is a practical necessity. It has been found by
experience that the monosyllable del is so short and easy to
pronounce that even in complicated formulae in which V occurs
a number of times no inconvenience to the speaker or hearer
arises from the repetition. V F is read simply as " del F."
Although this operator V has been defined as
v=i*+ji-+k*
dx dy 9z
1 Some use the term Nabla owing to its fancied resemblance to an Assyrian
harp. Others have noted its likeness to an inverted A and have consequently
coined the none too euphonious name Ailed by inverting the order of the letters in
the word Delta. Foppl in his Einfuhrung in die Maxwell1 sche Theorie der Elec-
tricitdt avoids any special designation and refers to the symbol as "die Operation
V.v How this is to be read is not divulged. Indeed, for printing no particular
name is necessary, but for lecturing and purposes of instruction something is re
quired—something too that does not confuse the speaker or hearer even when
often repeated.
THE DIFFERENTIAL CALCULUS OF VECTORS 139
so that it appears to depend upon the choice of the axes, it
is in reality independent of them. This would be surmised
from the interpretation of V as the magnitude and direction
of the most rapid increase of V. To demonstrate the inde
pendence take another set of axes, i', j', k' and a new set of
variables #', y'^ zf referred to them. Then V referred to this
system is
v/ = i'?T7 + J' o^7 + k'^T7 (22)'
a x a y d z
By making use of the formulae (47)' and (47)", Art. 53, page
104, for transformation of axes from i, j, k to i', j', k' and by
actually carrying out the differentiations and finally by
taking into account the identities (49) and (50), V may
actually be transformed into V.
The details of the proof are omitted here, because another
shorter method of demonstration is to be given.
64] Consider two surfaces (Fig. 30)
and
y,z)=c
V (x, y, z) = c + d c,
upon which V is constant and which are moreover infinitely
near together. Let #, y, z be a given point upon the surface
V=c. Let r denote the ra
dius vector drawn to this
point from any fixed origin.
Then any point near by in
the neighboring surface V
— c + d c may be represented
by the radius vector r + d r.
The actual increase of Ffrom
the first surface to the second
is a fixed quantity dc. The rate of increase is a variable
FIG. 30.
140 VECTOR ANALYSIS
quantity and depends upon the direction dr which is fol
lowed when passing from one surface to the other. The rate
of increase will be the quotient of the actual increase d c and
the distance V d r • d r between the surfaces at the point
x, y, z in the direction d r. Let n be a unit normal to the
surfaces and d n the segment of that normal intercepted
between the surfaces, n d n will then be the least value for
d r. The quotient .
\/d r • d r
will therefore be a maximum when d r is parallel to n and
equal in magnitude of d n. The expression
is therefore a vector of which the direction is the direction of
most rapid increase of Fand of which the magnitude is the
rate of that increase. This vector is entirely independent of
the axes JT, Y, Z. Let d c be replaced by its equal d V which
is the increment of F^in passing from the first surface to the
second. Then let V V be defined again as
Vr=4^n. (24)
d n
From this definition, V V is certainly the vector which
gives the direction of most rapid increase of V and the rate
in that direction. Moreover VFis independent of the axes.
It remains to show that this definition is equivalent to the one
first given. To do this multiply by • d r.
dV
VF.dr = — -n.dr. (25)
d n
n is a unit normal. Hence n • d r is the projection of d r on
n and must be equal to the perpendicular distance d n between
the surfaces.
THE DIFFERENTIAL CALCULUS OF VECTORS 141
dV
dT = —-dn = dV (25)'
dn
5V 5V 5V
But =7r- -=— -z—
dx dy 5z
where (d x? + (d y)2 + (d *)2 = d r • d r.
If dr takes on successively the values i dx, j dy, kdz the
equation (25)' takes on the values
5V
ids= ~—dx
d x
sv
dy = — dy (26)
9V
If the factors rf a;, rf y, rf 0 be cancelled these equations state
that the components VF • i, VF* j, VF- k of VF in the
i, j, k directions respectively are equal to
3V '5V 5_V
5x 5y* 5z'
VF=(VF. i)i + (Vr-j)j + (VT. k)k.
Henceby(26) VF= i |^ + j |T+ k |£
The second definition (24) has been reduced to the first
and consequently is equivalent to it.
*65.] The equation (25)' found above is often taken as a
definition of V V. According to ordinary calculus the deriv
ed y
ative — - satisfies the equation
d x
, dy
dx — = dy»
dx
142 VECTOR ANALYSIS
Moreover this equation defines dy / dx. In a similar manner
it is possible to lay down the following definition.
Definition: The derivative V^ of a scalar function of
position in space shall satisfy the equation
for all values of d r.
This definition is certainly the most natural and important
from theoretical considerations. But for practical purposes
either of the definitions before given seems to be better.
They are more tangible. The real significance of this last
definition cannot be appreciated until the subject of linear
vector functions has been treated. See Chapter VII.
The computation of the derivative V of a function is most
frequently carried on by means of the ordinary partial
differentiation.
Example 1 :
Let
V(x,y,z)
= r =
V** + yz+z*.
v
.9r
. 9r
Sr
V
d x
. _j
J%+
9z'
Va*
+ y* + z*
+ k-
z
'+ y't + z*
(ix + jy + kz)
The derivative of r is a unit vector in the direction of r.
This is evidently the direction of most rapid increase of r
and the rate of that increase.
THE DIFFERENTIAL CALCULUS OF VECTORS 143
Example % : Let
T V1F
1 X
Hence
T (xz + yz + «2)* (xz + yz + 22)*
-k 2 + g2)t'
_1 1
1 -r -r
and V~==7 - ^i = ~T'
r (r • r)§ r3
The derivative of 1/r is a vector whose direction is that
of — r, and whose magnitude is equal to the reciprocal of the
square of the length r.
Example 3: V rn — n rn~2 r = n r* — •
i>r
The proof is left to the reader.
Example 4 •' Let F(#, y, z) = log y#2 + y*.
V log V^Tp = i TT— 5 + j 2f 2 + ° k
22 22
If r denote the vector drawn from the origin to the point
, y, z) of space, the function V may be written as
2/1 *) = log Vr.r-(k.r)2
and ix + )y = T — k k«r.
Hence V log V^ + y ""
r • r —
T - k k«r
(r-kk.r).(r-kk.r)
144 VECTOR ANALYSIS
There is another method of computing V which is based
upon the identity
Example 1 : Let V = Vr«r = r.
d V = =^ =
Hence
v r »r
-
V i>r r
Example 2 : Let V = r • a, where a is a constant vector.
d F=dr.a = dr'Vr.
Hence V V = a.
Example 3: Let F= (rxa) • (rxb), where a and b are
constant vectors.
V = r»r a»b — r-a r«b.
dV = 2cZr*r a-b — dr-a r-b — dr»b r-a = di • V Fl
Hence V F == 2 r a-b — a r.b — b r^a
Vr= (ra-b-ar-b) + (ra-b -br.a)
= bx(rxa) -fax (rxb).
Which of these two methods for computing V shall be
applied in a particular case depends entirely upon their
relative ease of execution in that case. The latter method is
independent of the coordinate axes and may therefore be
preferred. It is also shorter in case the function Fcan be
expressed easily in terms of r. But when V cannot be so
expressed the former method has to be resorted to.
*66.] The great importance of the operator V in mathe
matical physics may be seen from a few illustrations. Sup
pose T (#, y, z) be the temperature at the point #, y, z of a
THE DIFFERENTIAL CALCULUS OF VECTORS 145
heated body. That direction in which the temperature de
creases most rapidly gives the direction of the flow of heat.
V T, as has been seen, gives the direction of most rapid
increase of temperature. Hence the flow of heat f is
f = _& vr,
where k is a constant depending upon the material of the
body. Suppose again that V be the gravitational potential
due to a fixed body. The force acting upon a unit mass at
the point (#, y, z) is in the direction of most rapid increase of
potential and is in magnitude equal to the rate of increase
per unit length in that direction. Let F be the force per unit
mass. Then
F = VF.
As different writers use different conventions as regards the
sign of the gravitational potential, it might be well to state
that the potential Preferred to here has the opposite sign to
the potential energy. If W denoted the potential energy of
a mass m situated at #, y, z, the force acting upon that mass
would be
F = - VfF.
In case V represent the electric or magnetic potential due
to a definite electric charge or to a definite magnetic pole re
spectively the force F acting upon a unit charge or unit pole
as the case might be is
F = - VF.
The force is in the direction of most rapid decrease of
potential. In dealing with electricity and magnetism poten
tial and potential energy have the same sign ; whereas in
attraction problems they are generally considered to have
opposite signs. The direction of the force in either case is in
the direction of most rapid decrease of potential energy. The
difference between potential and potential energy is this.
10
146 VECTOR ANALYSIS
Potential in electricity or magnetism is the potential energy
per unit charge or pole ; and potential in attraction problems
is potential energy per unit mass taken, however, with the
negative sign.
*67.] It is often convenient to treat an operator as a
quantity provided it obeys the same formal laws as that
quantity. Consider for example the partial differentiators
!_ A !..
9x' 3y' 3z
As far as combinations of these are concerned, the formal laws
are precisely what they would be if instead of differentiators
three true scalars
a, 6, c
were given. For instance
the commutative law
99 d 9
— — = —-— *, a 0 = 0 a,
Sx3y 3ySx
the associative law
5 9 3\ 3 3 3
and the distributive law
3 f 3 3\ 33 33
— ( -—+_-)=_--_ + ._ -- a(b + c) = a
3x\3y 3zJ 3x3y dxdz
hold for the differentiators just as for scalars. Of course such
formulae as
3 3
where u is a function of x cannot hold on account of the
properties of differentiators. A scalar function u cannot be
placed under the influence of the sign of differentiators.
Such a patent error may be avoided by remembering that an
operand must be understood upon which 3/3 # is to operate.
THE DIFFERENTIAL CALCULUS OF VECTORS 147
In the same way a great advantage may be obtained by
looking upon
V-if +jf + kf
3x dy dz
as a vector. It is not a true vector, for the coefficients
.£., JL, A
P# dy dz
are not true scalars. It is a vector differentiator and of
course an operand is always implied with it. As far as formal
operations are concerned it behaves like a vector. For
instance
V (u + v) = V u + V v,
V(ttfl) = (Vtt) v + ^(Vtf),
c V u = V (c u),
if w and v are any two scalar functions of the scalar variables
#, y, 2 and if c be a scalar independent of the variables with
regard to which the differentiations are performed.
68.] If A represent any vector the formal combination
A. Vis
A.V = Al/-x+A2/- + Asj-, (27)
provided A = Al i + A^ j + A% k.
This operator A • V is a scalar differentiator. When applied
to a scalar function V (x, y, z) it gives a scalar.
<^r-A+^+^- (28)
Suppose for convenience that A is a unit vector a.
(a.V)F=a1I+a2r+a8r (29)
148 VECTOR ANALYSIS
where av a^aB are the direction cosines of the line a referred
to the axes Jf, F, Z. Consequently (a • V) V appears as the
well-known directional derivative of V in the direction a.
This is often written
3V 3V , 3F, 3V
T^^+^-^sT- (29)
It expresses the magnitude of the rate of increase of V in
the direction a. In the particular case where this direction is
the normal n to a surface of constant value of F, this relation
becomes the normal derivative.
if n x, n 2, n 3 be the direction cosines of the normal.
The operator a • V applied to a scalar function of position
V yields the same result as the direct product of a and the
vector V V.
(a.V)F=a.(VF). (30)
For this reason either operation may be denoted simply by
a- VF
without parentheses and no ambiguity can result from the
omission. The two different forms (a • V) Fand a- (V F)
may however be interpreted in an important theorem.
(a • V) F is the directional derivative of F in the direction
a. On the other hand a • ( V V) is the component of V F in
the direction a. Hence : The directional derivative of F in
any direction is equal to the component of the derivative
VFin that direction. If Fdenote gravitational potential the
theorem becomes : The directional derivative of the potential
in any direction gives the component of the force per unit
mass in that direction. In case Fbe electric or magnetic
potential a difference of sign must be observed.
THE DIFFERENTIAL CALCULUS OF VECTORS 149
Vector Functions of Position in Space
69.] A vector function of position in space is a function
V (x, y, z)
which associates with each point x, y, z in space a definite
vector. The function may be broken up into its three com
ponents
V (x, y, z) = F! (x, y,z)i+ F2 (x, y, z) j + F3 (a?, y, z) k.
Examples of vector functions are very numerous in physics.
Already the function VF has occurred. At each point of
space V F has in general a definite vector value. In mechan
ics of rigid bodies the velocity of each point of the body is a
vector function of the position of the point. Fluxes of heat,
electricity, magnetic force, fluids, etc., are all vector functions
of position in space.
The scalar operator a • V may be applied to a vector func
tion V to yield another vector function.
Let V = Fi (x, y,z) i + F2 (as, y, z) j + F3 (x, y, z) k
and a = a1 i + a2 j + a3 k.
Then a-V = «i^ + a3^+-af3
(a.V)V = (a.V) F! i + (a.V) F2j + (a.V)F, k
9V 3V 9V\
-
(
a
9V, 9V2 3V2
150 VECTOR ANALYSIS
This may be written in the form
Hence (a • V) V is the directional derivative of the vector
function V in the direction a. It is possible to write
(a • V) V = a - V V
without parentheses. For the meaning of the vector symbol
V when applied to a vector function V has not yet been
defined. Hence from the present standpoint the expression
a • V V can have but the one interpretation given to it by
(a • V) V.
70.] Although the operation V V has not been defined and
cannot be at present,1 two formal combinations of the vector
operator V and a vector function V may be treated. These
are the (formal) scalar product and the (formal) vector prod
uct of V into V. They are
T <82>
and VxV = i+]-+kxV. (88)
V • V is read del dot V; and V x V, del cross V.
0 0 0
The differentiators — , — , — , being scalar operators, pass
by the dot and the cross. That is
«
(32)'
Qy 3z
(88)'
These may be expressed in terms of the components F", PI. V,
ofV.
i A definition of V V will be given in Chapter VII. ,
THE DIFFERENTIAL CALCULUS OF VECTORS 151
Now
Then
dx 9x
Sx
3y dy 9y'~5y
8V_9V} 5V, 3F
3* 3* h3« Jl ~3«
i.fl-
3V
(34)
Hence V • V =
Moreover i >
•rf + ^7 + 3T- (32)"
This may be written in the form of a determinant
VxV=
i j k
333
(33)"
152 VECTOR ANALYSIS
It is to be understood that the operators^are to be applied to
the functions Vv F"2, F3 when expanding the determinant.
From some standpoints objections may be brought forward
against treating V as a symbolic vector and introducing V • V
and V x V respectively as the symbolic scalar and vector
products of V into V. These objections may be avoided by
simply laying down the definition that the symbols V • and
V x, which may be looked upon as entirely new operators
quite distinct from V, shall be
and VxV = ix+jx-4-kx-. (33)'
But for practical purposes and for remembering formulae it
seems by all means advisable to regard
3 5 3
as a symbolic vector differentiator. This symbol obeys the
same laws as a vector just in so far as the differentiators
333
^ — ' T~~' T~ °^)ey ^e same *aws ^ or(^nary scalar quantities.
71.] That the two functions V • V and V x V have very
important physical meanings in connection with the vector
function V may be easily recognized. By the straight
forward proof indicated in Art. 63 it was seen that the
operator V is independent of the choice of axes. From this
fact the inference is immediate that V • V and V x V represent
intrinsic properties of V invariant of choice of axes. In order
to perceive these properties it is convenient to attribute to the
function V some definite physical meaning such as flux or
flow of a fluid substance. Let therefore the vector V denote
THE DIFFERENTIAL CALCULUS OF VECTORS 153
at each point of space the direction and the magnitude of the
flow of some fluid. This may be a material fluid as water
or gas, or a fictitious one as heat or electricity. To obtain as
great clearness as possible let the fluid be material but not
necessarily restricted to incompressibility like water.
Then
= i~+j. — + k • *I
dx 3y dz
is called the divergence of V and is often written
V« V= div V.
The reason for this term is that V«V gives at each point the
rate per unit volume per unit time at which fluid is leaving
that point — the rate of diminution of density. To prove
this consider a small cube of matter (Fig. 31). Let the edges
of the cube be dx, dy, and dz respectively. Let
V (x, y, z) = Vl (x9 y,z)i+ V^ (xy y, z) j + F3 (x, y, z) k.
Consider the amount of fluid which passes through those
faces of the cube which are parallel to the F^-plane, i. e.
perpendicular to the X
axis. The normal to the
face whose x coordinate is
the lesser, that is, the nor
mal to the left-hand face
of the cube is — i. The flux
of substance through this
face is
xy2
FIG. 31.
-i.V (x,y,z) dy dz.
The normal to the oppo- z
site face, the face whose
x coordinate is greater by the amount dx, is + i and the flux
through it is therefore
164 VECTOR ANALYSIS
r 3v i
i • V (x + dx, y, z) dy dz = i • V(#, y, z) + — dx dy dz
3V
= i • V (xy y, z) dy dz + i • - dx dy dz.
c) x
The total flux outward from the cube through these two
faces is therefore the algebraic sum of these quantities. This
is simply
3V , , . 3^
i • -=— dx dy dz = -^ — - dx dy dz.
9 x 9 x
In like manner the fluxes through the other pairs of faces of
the cube are
3V,,, j,,c)V
i • ^ — dx dy dz and k • - — dx dy dz.
9 y 9 z
The total flux out from the cube is therefore
/. 3V t 3V t , 3V\
( i • — + j • — + k . — ) dx dy dz.
\ 9x dy 9zJ
This is the net quantity of fluid which leaves the cube per
unit time. The quotient of this by the volume dx dy dz of
the cube gives the rate of diminution of density. This is
V.T.I. + , .
9x dy 9z 9x dy 9z
Because V • V thus represents the diminution of density
or the rate at which matter is leaving a point per unit volume
per unit time, it is called the divergence. Maxwell employed
the term convergence to denote the rate at which fluid ap
proaches a point per unit volume per unit time. This is the
negative of the divergence. In case the fluid is incompressible,
as much matter must leave the cube as enters it. The total
change of contents must therefore be zero. For this reason
the characteristic differential equation which any incompres
sible fluid must satisfy is
THE DIFFERENTIAL CALCULUS OF VECTORS 155
where V is the flux of the fluid. This equation is often
known as the hydrodynamic equation. It is satisfied by any
flow of water, since water is practically incompressible. The
great importance of the equation for work in electricity is due
to the fact that according to Maxwell's hypothesis electric dis
placement obeys the same laws as an incompressible fluid. If
then D be the electric displacement,
div D = V • D = 0.
72.] To the operator V X Maxwell gave the name curl.
This nomenclature has become widely accepted.
V x V = curl V.
The curl of a vector function V is itself a vector function
of position in space. As the name indicates, it is closely
connected with the angular velocity or spin of the flux at
each point. But the interpretation of the curl is neither so
easily obtained nor so simple as that of the divergence.
Consider as before that V represents the flux of a fluid.
Take at a definite instant an infinitesimal sphere about any
point (#, y, z). At the next instant what has become of the
sphere ? In the first place it may have moved off as a whole
in a certain direction by an amount d r. In other words it
may have a translational velocity of dr/dt. In addition to
this it may have undergone such a deformation that it is no
longer a sphere. It may have been subjected to a strain by
virtue of which it becomes slightly ellipsoidal in shape.
Finally it may have been rotated as a whole about some
axis through an angle dw. That is to say, it may have an
angular velocity the magnitude of which is dw/dt. An
infinitesimal sphere therefore may have any one of three
distinct types of motion or all of them combined. First, a
translation with definite velocity. Second, a strain with three
definite rates of elongation along the axes of an ellipsoid.
156 VECTOR ANALYSIS
Third, an angular velocity about a definite axis. It is this
third type of motion which is given by the curl. In fact,
the curl of the flux V is a vector which has at each point of
space the direction of the instantaneous axis of rotation at
that point and a magnitude equal to twice the instantaneous
angular velocity about that axis.
The analytic discussion of the motion of a fluid presents
more difficulties than it is necessary to introduce in treating
the curl. The motion of a rigid body is sufficiently complex
to give an adequate idea of the operation. It was seen (Art.
51) that the velocity of the particles of a rigid body at any
instant is given by the formula
v = v0 + a x r.
curl v = Vxv = Vxv0 + Vx(axr).
Let a = al i + a% j + a3 k
r = r1i + r2j + r3k=:;ri + 2/j-fzk
expand V X (a x r) formally as if it were the vector triple
product of V, a, and r. Then
V x v = V x v0 + (V - r) a - (V • a) r.
v0 is a constant vector. Hence the term V x v0 vanishes.
V . r = — + ^ + — = 3.
3 x 3 y 3 z
As a is a constant vector it may be placed upon the other side
of the differential operator, V • a = a • V.
/ 3 3 3\
a - Vr=( ai^ + a2£j-+ as^ Jr = a1i
Hence Vxv = 3a — a = 2a.
Therefore in the case of the motion of a rigid body the curl
of the linear velocity at any point is equal to twice the
angular velocity in magnitude and in direction.
THE DIFFERENTIAL CALCULUS OF VECTORS 157
V x v = curl v = 2 a,
a = ^Vxv=| curl v.
v = v0 + ^ (V x v) x r = v0 + \ (curl v) x r. (34)
The expansion of V x (a x r) formally may be avoided by
multiplying a x r out and then applying the operator V X to
the result.
73.] It frequently happens, as in the case of the applica
tion just cited, that the operators V>V% V X, have to be
applied to combinations of scalar functions, vector functions,
or both. The following rules of operation will be found
useful. Let u, v be scalar functions and u, v vector func
tions of position in space. Then
V(t6 + t?) = Vw + Vfl (35)
V.(u + v) = V.u + V-v (36)
Vx(u + v) = Vxu + Vxv (37)
V (u v) = v V u + u V v (38)
V • (w v) = V M • v + M V • v (39)
v (40)
(41)
+ v x (V x u) + u x (V x v)1
V.(uxv)=v.Vxu — u-Vxv (42)
Vx (uxv) = v.Vu~vV-u-u.Vv + uV.v.1 (43)
A word is necessary upon the matter of the interpretation
of such expressions as
V u v, V u • v, V u x v.
The rule followed in this book is that the operator V applies
to the nearest term only. That is,
1 By Art. 69 the expressions v • V n and n • V v me to be interpreted as
(V • V)uand (u* V)v-
158 VECTOR ANALYSIS
V uv = (V u) v
V u • v = ( V u) • v
V u x v = ( V u) x v.
If V is to be applied to more than the one term which follows
it, the terms to which it is applied are enclosed in a paren
thesis as upon the left-hand side of the above equations.
The proofs of the formulae may be given most naturally
by expanding the expressions in terms of three assumed unit
vectors i, j, k. The sign 2 of summation will be found con
venient. By means of it the operators V> V*» A x take the
form
The summation extends over #, y, z.
To demonstrate Vx (wv)
^ 9 x
Hence Vx (wv) = Vwxv + ^Vxv.
To demonstrate
V (u • v) = v • V u + u • V v + v x ( V x u) + n x ( V x v).
THE DIFFERENTIAL CALCULUS OF VECTORS 159
^ . 3 u ^ . 3 v
V(u.v) = 2^-v+^^.-
Now
,3u
^ 3 u . ^ . 9 u
2v._1 = vx(Vxu) + 2v.i-
or IE v • — i = v x ( V x u) + v • V u.
** 9 x
3 v
In like manner T u • ;r- * = u x ( V x v) + u • V v.
d x
Hence V(u»v)=vVu +
+ v x (V x n) + u x (V x v).
The other formulae are demonstrated in a similar manner.
71] The notation l
V(u-v)u (44)
will be used to denote that in applying the operator V to the
product (u • v), the quantity u is to be regarded as constant.
That is, the operation V is carried out only partially upon
the product (u • v). In general if V is to be carried out
partially upon any number of functions which occur after
it in a parenthesis, those functions which are constant for the
differentiations are written after the parenthesis as subscripts.
Let M = U i
1 This idea and notation of a partial V so to speak may be avoided by means
of the formula 41. But a certain amount of compactness and simplicity is
lost thereby. The idea of V (u • v)u is surely no more complicated than u • V v or
v X (V X n).
160 VECTOR ANALYSIS
then n-v = M101 + u^v2 + uz
and V (u'V)
3^0
But
and V(u.v)T =
Hence V(u-v) rrr^j Vwx +
But V(u-v)n = w1Vi?1 + ^aVi?a + w8V«?8 (44)'
and V(u«v)v = ^j V^x + i;2 V^2 + ^3 V^3.
Hence V (u- v) = V (u- v)u + V (u- v)v. (45)
This formula corresponds to the following one in the nota
tion of differentials
d (u • v) = d (u • v)u + d (u • v)T
or d (u • v) = u • d v + d n • v.
The formulae (35)-(43) given above (Art. 73) may be
written in the following manner, as is obvious from analogy
with the corresponding formulae in differentials :
V (u + v*) = V (u + v\ + V (u + v)9 (35)'
V. (u + v) = V- (u + v)u + V- (u + v)v (36)'
V x (u + v) = V x (u + v)u + V x (u + v)y (37)'
THE DIFFERENTIAL CALCULUS OF VECTORS 161
V (u v) = V (u v\ + V (u v\ (38)'
V- (u v) = V- (u v). + V- O v)r (39)'
V x (u v) = V x (i* v)a + V x (u v)v (40)'
V (u- v) = V (u. v)u + V (u. v)v (41)'
V • (u x v) = V • (u x v)a + V • (u x v)y (42)'
V x (u x v) = V x (u x v)a + V x (u x v)v. (43)'
This notation is particularly useful in the case of the
scalar product u^v and for this reason it was introduced.
In almost all other cases it can be done away without loss of
simplicity. Take for instance (43)'. Expand V x (u x v)u
formally.
V x (u x v)u = (V • v) u — (V • u) v,
where it must be understood that u is constant for the differ
entiations which occur in V. Then in the last term the
factor u may be placed before the sign V. Hence
V X (uX v)u = u V * v — u- Vv.
In like manner V x (u x v)v = v • V u — v V • u.
Hence Vx(uxv)=vVu — v V • u — u • V v + u V • v.
75.] There are a number of important relations in which
the partial operation V (u • v)u figures.
u x (V x v) = V (u • v)a - u • V v, (46)
or V(u-v)u = u. Vv + u x (V x v), (46)'
or u • V v = V (u • v)u + (V x v) x u. (46)"
The proof of this relation may be given by expanding in
terms of i, j, k. A method of remembering the result easily
is as follows. Expand the product
u x (V x v)
ll
162 VECTOR ANALYSIS
formally as if V, u, v were all real vectors. Then
ux(Vxv)=u.vV — u • V v.
The second term is capable of interpretation as it stands.
The first term, however, is not. The operator V has nothing
upon which to operate. It therefore must be transposed so
that it shall have u • v as an operand. But u being outside
of the parenthesis in u x (V x v) is constant for the differen
tiations. Hence
u • v V = V (u • v)u
and u x (V x v) = V (u • v)u — u • V v. (46)
If u be a unit vector, say a, the formula
a-Vv = V(a»v)a + (V x v) x a (47)
expresses the fact that the directional derivative a» V v of a
vector function v in the direction a is equal to the derivative
of the projection of the vector v in that direction plus the
vector product of the curl of v into the direction a.
Consider the values of v at two neighboring points.
v (x, y, z)
and v (x + dx, y + dy> z + dz)
d v = v (x + dx, y + dy, z + dz) — v (#, y, z).
Let v = v{i + v2 j + v8k
dv = dvli + dv%j + dv3k.
But by (25)' dv1 = dr*
d v% = dr»
dv% = dr
Hence d v s= d r • (V vl i + V v2 j + V vz k).
Hence d v = d r • V v
By (46)" d v = V (rfr • v)dr + (V x v) x dr. (48)
THE DIFFERENTIAL CALCULUS OF VECTORS 163
Or if V0 denote the value of v at the point (#, y, z) and v the
value at a neighboring point
v = v0 + V (d r • v)dr + (V x v) x dr. (49)
This expression of v in terms of its value v0 at a given point,
the dels, and the displacement d r is analogous to the expan
sion of a scalar functor of one variable by Taylor's theorem,
/<*>=/(*•>+ .TOO **
The derivative of (r • v) when v is constant is equal to v.
That is V (r • v)v = v.
For V (r • v)v = v • V r - (V x r> x v,
9
v« Vr = vli + v%j + 08k = v,
V x r = 0.
Hence V (r • v)v = v.
In like manner if instead of the finite vector r, an infinitesimal
vector d r be substituted, the result still is
V (d r • v)v = v.
V/fllO*^
By (49) v = V0 + V (d r • v)dr + (V x v) x d r
V (d r • v) = V (d r • v)d; + V (d r • vV
Hence V (d r • v)dr = V (d r • v) — v.
Substituting :
v = ^vo + ^V(dr.v) + ^(Vxv)xdr. (50)
This gives another form of (49) which is sometimes more
convenient It is also slightly more symmetrical.
164 VECTOR ANALYSIS
* 76.] Consider a moving fluid. Let v (#, T/, 3, t) be the
velocity of the fluid at the point (#, y, z) at the time t. Sur
round a point (a;0, y0, z0) with a small sphere.
dr •dr = c2.
At each point of this sphere the velocity is
v = v0 + d r • V v.
In the increment of time B t the points of this sphere will have
moved the distance
The point at the center will have moved the distance
The distance between the center and the points that were
upon the sphere of radius d r at the commencement of the
interval $ t has become at the end of that interval S t
To find the locus of the extremity of drr it is necessary to
eliminate d r from the equations
c2 = d r • d r.
The first equation may be solved for d r by the method of
Art. 47, page 90, and the solution substituted into the second.
The result will show that the infinitesimal sphere
has been transformed into an ellipsoid by the motion of the
fluid during the time 8 1.
A more definite account of the change that has taken place
may be obtained by making use of equation (50)
THE DIFFERENTIAL CALCULUS OF VECTORS 165
v = iv0 + |v(rfr.v) + 2-(Vx v) xdr,
v = v0+i[V(dr.v)-v0] + |(Vxv)xdr;
S
or of the equation (49)
v = v0+ V(dr-v)dr+(Vx v)xrfr,
v = v0 + [V (dr • v)dr + I (V x v) x d r]+ ^ (V x v) x d r.
The first term v0 in these equations expresses the fact that
the infinitesimal sphere is moving as a whole with an instan
taneous velocity equal to V0. This is the translational element
of the motion. The last term
^(Vxv)xdr = £ curl v x d r
shows that the sphere is undergoing a rotation about an
instantaneous axis in the direction of curl v and with an angu
lar velocity equal in magnitude to one half the magnitude of
curl v. The middle term
or v(dr.v)dr-£ (Vx v) x dr
expresses the fact that the sphere is undergoing a defor
mation known as homogeneous strain by virtue of which it
becomes ellipsoidal. For this term is equal to
dx V^j + dy V#2 + dz
if Vj, v2, vs be respectively the components of v in the direc
tions i, j, k. It is fairly obvious that at any given point
(#o> 2A» zo) ^ set of three mutually perpendicular axes i, j, k
may be chosen such that at that point V^, V#2, V#3 are re-
166 VECTOR ANALYSIS
spectively parallel to them. Then the expression above
becomes simply
dx*»ii+dy^i + dzs^.
dx y 9y 9^
The point whose coordinates referred to the center of the
infinitesimal sphere are
dx, dy, dz
is therefore endowed with this velocity. In the time S t it
will have moved to a new position
The totality of the points upon the sphere
goes over into the totality of points upon the ellipsoid of
which the equation is
dx2 dy2 dz*
y
The statements made before (Art. 72) concerning the three
types of motion which an infinitesimal sphere of fluid may
possess have therefore now been demonstrated.
77.] The symbolic operator V may be applied several times
in succession. This will correspond in a general way to
forming derivatives of an order higher than the first. The
expressions found by thus repeating V will all be independ
ent of the axes because V itself is. There are six of these
dels of the second order.
Let V (#, y, z) be a scalar function of position in space.
The derivative VF is a vector function and hence has a curl
and a divergence. Therefore
V-VF, VxVF
THE DIFFERENTIAL CALCULUS OF VECTORS 167
are the two derivatives of the second order which may be
obtained from V.
V-VF=div VF (51)
V x VF=curl VF. (52)
The second expression V x V V vanishes identically. That is,
the derivative of any scalar function V possesses no curl. This
may be seen by expanding V x V V in terms of i, j, k. All
the terms cancel out. Later (Art. 83) it will be shown con
versely that if a vector function W possesses no curl, i. e. if
V x W = curl W = 0, then W = VF,
W is the derivative of some scalar function F.
The first expression V • V F when expanded in terms of
i, j, k becomes
Symbolically, V • V = — ^ + —-5 + -r— „ •
0 'y»2 O <i/2 O /v 2
The operator V • V is therefore the well-known operator of
Laplace. Laplace's Equation
becomes in the notation here employed
V-VF=0. (53)'
When applied to a scalar function F the operator V • V yields
a scalar function which is, moreover, the divergence of the
derivative.
Let T be the temperature in a body. Let c be the con
ductivity, p the density, and k the specific heat. The
flow f is
168 VECTOR ANALYSIS
The rate at which heat is leaving a point per unit volume per
unit time is V • f. The increment of temperature is
rfr=-^-V.f dt.
p K
d-£ = -^.VT.
at p K
This is Fourier's equation for the rate of change of tempera
ture.
Let V be a vector function, and Vv Vv Vz its three com
ponents. The operator V • V of Laplace may be applied to V.
v.vv = v-vr1i + v-vr2j + v.vr3k (54)
If a vector function V satisfies Laplace's Equation, each of
its three scalar components does. Other dels of the second
order may be obtained by considering the divergence and curl
of V. The divergence V • V has a derivative
VV-V = VdivV. (55)
The curl V X V has in turn a divergence and a curl,
and V • V x V, VxVxV.
V - V x V = div curl V (56)
and V x V x V = curl curl V. (57)
Of these expressions V • V x V vanishes identically. That is,
the divergence of the curl of any vector is zero. This may be
seen by expanding V • V x V in terms of i, j, k. Later (Art.
83) it will be shown conversely that if the divergence of a
vector function W vanishes identically, i. e. if
V • W = div W = 0, then W = V x V = curl V,
W is the curl of some vector function V.
THE DIFFERENTIAL CALCULUS OF VECTORS 169
If the expression V x (V x V) were expanded formally
according to the law of the triple vector product,
Vx(VxV) = V-VV-V.VV.
The term V ' V V is meaningless until V be transposed to
the beginning so that it operates upon V.
VxVxV = VV.V-V-VV, (58)
or curl curl V = V div V - V • VV. (58)'
This formula is very important. It expresses the curl of the
curl of a vector in terms of the derivative of the divergence
and the operator of Laplace. Should the vector function V
satisfy Laplace's Equation,
V • VV = 0 and
curl curl V = V div V.
Should the divergence of V be zero,
curl curl V = — V • VV.
Should the curl of the curl of V vanish,
V div V = V • VV.
To sum up. There are six of the dels of the second order.
V-VT, VxVF,
V-VV, V V • V, V • V x V, V x V x V.
Of these, two vanish identically.
VxVr=0, V-VxV = 0.
A third may be expressed in terms of two others.
VxVxV = VV.V-V.VV. (58)
The operator V • V is equivalent to the operator of Laplace.
170 VECTOR ANALYSIS
* 78.] The geometric interpretation of V« Vw is interesting.
It depends upon a geometric interpretation of the second
derivative of a scalar function u of the one scalar variable x.
Let ui be the value of u at the point xt. Let it be required
to find the second derivative of u with respect to x at the
point x0. Let xl and x2 be two points equidistant from #0.
That is, let
Xn *"•" XQ — — XQ ~~~ «*/ — &t
•*• a ^^ ni
o °
Then - ^
is the ratio of the difference between the average of u at the
points xl and #2 and the value of u at x0 to the square of the
distance of the points xv #3 from x0. That
d*u
. LIM.
is easily proved by Taylor's theorem.
Let u be a scalar function of position in space. Choose
three mutually orthogonal lines i, j, k and evaluate the
expressions
Let o?2 and a?! be two points on the line i at a distance a from
x0 ; #4 and #3, two points on j at the same distance a from
•s0 > #e and #6, two points on k at the same distance a from x0.
— = — u~
?^_. LIM ._2 :
THE DIFFERENTIAL CALCULUS OF VECTORS 171
Add:
-LIM r
~a=OL
As V and V- are independent of the particular axes chosen,
this expression may be evaluated for a different set of axes,
then for still a different one, etc. By adding together all
these results
u\ + u% + * • • 6 n terms
-
a== a
Let n become infinite and at the same time let the different
sets of axes point in every direction issuing from #0. The
fraction
u\ + U2 + ' * ' ^ n terms
6 n
then approaches the average value of u upon the surface of a
sphere of radius a surrounding the point x0. Denote this
by ua.
= a
V • V u is equal to six times the limit approached by the ratio
of the excess of u on the surface of a sphere above the value
at the center to the square of the radius of the sphere. The
same reasoning holds in case u is a vector function.
If u be the temperature of a body V-V u (except for a
constant factor which depends upon the material of the
172 VECTOR ANALYSIS
body) is equal to the rate of increase of temperature (Art.
77). If V»V^is positive the average temperature upon a
small sphere is greater than the temperature at the center.
The center of the sphere is growing warmer. In the case
of a steady flow the temperature at the center must remain
constant. Evidently therefore the condition for a steady
flow is
V • V u = 0.
That is, the temperature is a solution of Laplace's Equation.
Maxwell gave the name concentration to — V • V u whether
u be a scalar or vector function. Consequently V • V u may
be called the dispersion of the function u whether it be scalar
or vector. The dispersion is proportional to the excess of
the average value of the function on an infinitesimal surface
above the value at the center. In case u is a vector function
the average is a vector average. The additions in it are
vector additions.
SUMMARY OF CHAPTER III
If a vector r is a function of a scalar t the derivative of
r with respect to t is a vector quantity whose direction is
that of the tangent to the curve described by the terminus
of r and whose magnitude is equal to the rate of advance of
that terminus along the curve per unit change of t. The
derivatives of the components of a vector are the components
of the derivatives.
dnr dnr,. d» r» . d* r~
= i H j H k ( 2V
dt* dtn dt* J dt*
A combination of vectors or of vectors and scalars may be
differentiated just as in ordinary scalar analysis except that
the differentiations must be performed in situ.
d
d~t(
},»
d* ^
a • b) = -— • b + t
d t
Xb)=^Xb +
db
*5T
db
a x 511
THE DIFFERENTIAL CALCULUS OF VECTORS 173
(3)
(4)
or d (a • b) = d a • b + a • d b, (3)'
d(axb) = daxb + axdb, (4)'
and so forth. The differential of a unit vector is perpendicu
lar to that vector.
The derivative of a vector r with respect to the arc s of
the curve which the terminus of the vector describes is
the unit tangent to the curves directed toward that part of the
curve along which $ is supposed to increase.
r."- <"
The derivative of t with respect to the arc * is a vector whose
direction is normal to the curve on the concave side and
whose magnitude is equal to the curvature of the curve.
The tortuosity of a curve in space is the derivative of the
unit normal n to the osculating plane with respect to the
arc s.
^n_^_/rfr <?2r _1 _ \
~ ds~ ds\dsX ds*' VCTC/
The magnitude of the tortuosity is
r=
rdr d*r cZ3r"|
L^s ^T2 rf^J
174 VECTOR ANALYSIS
If r denote the position of a moving particle, t the time,
v the velocity, A the acceleration,
*-*---*
The acceleration may be broken up into two components of
which one is parallel to the tangent and depends upon the
rate of change of the scalar velocity v of the particle in its
path, and of which the other is perpendicular to the tangent
and depends upon the velocity of the particle and the curva
ture of the path.
A = s t + 02 C. (19)
Applications to the hodograph, in particular motion in a
circle, parabola, or under a central acceleration. Application
to the proof of the theorem that the motion of a rigid body
one point of which is fixed is an instantaneous rotation about
an axis through the fixed point.
Integration with respect to a scalar is merely the inverse
of differentiation. Application to finding the paths due to
given accelerations.
The operator V applied to a scalar function of position in
space gives a vector whose direction is that of most rapid
increase of that function and whose magnitude is equal to
the rate of that increase per unit change of position in that
direction
THE DIFFERENTIAL CALCULUS OF VECTORS 175
The operator V is invariant of the axes i, j, k. It may be
denned by the equation
n, (24)
or W-dT = dV. (25)'
Computation of the derivative V V by two methods depend
ing upon equations (21) and (25)'. Illustration of the oc
currence of V in mathematical physics.
V may be looked upon as a fictitious vector, a vector
differentiator. It obeys the formal laws of vectors just in
so far as the scalar differentiators of 51 5 x> "9 / d y, 9 1 3 z obey
the formal laws of scalar quantities
A-VF=^+^i^'l7 <28>
If a be a unit vector a • V V is the directional derivative of V
in the direction a.
a.VF = (a-V) F=a«(VF). (30)
If V is a vector function a • VV is the directional derivative
of that vector function in the direction a.
- _ ,-J~ + k.^-, (32)'
3 x 9 y 3 z
VxV=ix|^ + jx — + kx— , (33)'
3x 3y 3 z
V.V= ^— J + ^ + ^— 8, (32)''
3 x 3 y 3 z
176 VECTOR ANALYSIS
Proof that V • V is the divergence of V and V x V, the curl
of V.
V • V = div V,
V X V = curl V.
V O + t;) = V u + Vtf, (35)
V • (u + v) = V • u + V • v, (36)
Vx(u + v)=Vxn + Vxv, (37)
V (u v) = v V u + u V v, (38)
V • (u v) = V u • v + u V • v, (39)
V x (u v) = V u x v + u V x v, (40)
V(n»v)=vVu + U'Vv + vx ( V x n)
+ n x (V x v), (41)
V • (n x v) = v Vxu-u-Vxv, (42)
Vx (u x v) =v .Vu — v V» u — u • Vv + uV* v. (43)
Introduction of the partial del, V (u • v)u, in which the dif
ferentiations are performed upon the hypothesis that u is
constant.
u x (V x v) = V (u • v)u — n • V v. (46)
If a be a unit vector the directional derivative
a • V v = V (a • v)a + (V x v) x a. (47)
The expansion of any vector function v in the neighborhood
of a point (x# y# z0) at which it takes on the value of v0 is
v = v0 + V (d r • v)dr + (V x v) x dr, (49)
or v = \ v0 + V (d r . v) + \ (V x v) x d r. (50)
Application to hydrodynamics.
The dels of the second order are six in number.
THE DIFFERENTIAL CALCULUS OF VECTORS 177
V x VF= curl VF= 0, (52)
x)2F" wv &v
V-VJ^vV^f^+^ + Vp (51)
d x2 d y2 9 z2
V» V is Laplace's operator. If V»VF=0, V satisfies La
place's Equation. The operator may be applied to a vector.
VV. V = VdivV, (55)
V • V x V = div curl V = 0, (56)
Vx VxV=curlcurlV = VV.V- V. VV. (58)
The geometric interpretation of V • V as giving the disper
sion of a function.
EXERCISES ON CHAPTER III
1. Given a particle moving in a plane curve. Let the
plane be the ij-plane. Obtain the formulae for the compo
nents of the velocity parallel and perpendicular to the radius
vector r. These are
rp £kxr,
where 0 is the angle the radius vector r makes with i, and k
is the normal to the plane.
2. Obtain the accelerations of the particle parallel and
perpendicular to the radius vector. These are
Express these formulae in the usual manner in terms of x
and y.
12
178 VECTOR ANALYSIS
3. Obtain the accelerations of a moving particle parallel
and perpendicular to the tangent to the path and reduce the
results to the usual form.
4. If r, </>, 0 be a system of polar coordinates in space,
where r is the distance of a point from the origin, </> the
meridianal angle, and 6 the polar angle ; obtain the expressions
for the components of the velocity and acceleration along the
radius vector, a meridian, and a parallel of latitude. Reduce
these expressions to the ordinary form in terms of #, y, z.
5. Show by the direct method suggested in Art. 63 that
the operator V is independent of the axes.
6. By the second method given for computing V find
the derivative V of a triple product [a be] each term of which
is a function of #, y, z in case
a = (r • r) r, b = (r • a) e, c = r x t,
where d, e, f are constant vectors.
7. Compute V • V F when Fis r2, r, -, or -r •
, r r*
8. Compute V • V V, VV • V, and V x V x V when V is
equal to r and when V is equal to -j> and show that in these
cases the formula (58) holds.
9. Expand V x V V and V • V x V in terms of i, j, k and
show that they vanish (Art. 77).
10. Show by expanding in terms of i, j, k that
Vx VxV=VV. V-V« VV.
11. Prove A.V(7-W) = VA.VW+ WA- VV,
and
(VxV) x W=Vx (Vx
CHAPTER IV
THE INTEGRAL CALCULUS OF VECTORS
79.] Let W (#, y, z) be a vector function of position in
space. Let C be any curve in space, and r the radius vector
drawn from some fixed origin to the points of the curve.
Divide the curve into infinitesimal elements dr. From the
sum of the scalar product of these elements d r and the value
of the function W at some point of the element —
thus 2 W • d r.
The limit of this sum when the elements dr become infinite
in number, each approaching zero, is called the line integral of
W along the curve C and is written
'.dr.
and dT = i dx + j dy + k dz,
r r
I W • dr = i [W+dx -\-W*dy +W%dz\. (1)
t/ (7 t/ C7
The definition of the line integral therefore coincides with
the definition usually given. It is however necessary to
specify in which direction the radius vector r is supposed to
describe the curve during the integration. For the elements
d r have opposite signs when the curve is described in oppo-
180 VECTOR ANALYSIS
site directions. If one method of description be denoted by
C and the other by — (7,
/W • d r = -- I W • d r.
-G J c
In case the curve C is a closed curve bounding a portion of
surface the curve will always be regarded as described in
such a direction that the enclosed area appears positive
(Art. 25).
If f denote the force which may be supposed to vary from
point to point along the curve (7, the work done by the force
when its point of application is moved from the initial point
r0 of the curve C to its final point r is the line integral
ff . dr= f'f • dr.
J c J r0
Theorem : The line integral of the derivative V F' of a
scalar function V(x,y, z) along any curve from the point
r0 to the point r is equal to the difference between the values
of the function F '(#, y, z) at the point r and at the point r0.
That is,
Vr.dr = F(r) - F(r0) = V(x,y,z) - V(x«y»*d.
o
By definition d r • V F" = d V
fdV= F(r) - F(r0) = Ffey,^) - V(x»y«z.). (2)
Theorem : The line integral of the derivative V F" of a
single valued scalar function of position V taken around a
closed curve vanishes.
The fact that the integral is taken around a closed curve
is denoted by writing a circle at the foot of the integral sign.
To show
(3)
THE INTEGRAL CALCULUJS OF VECTORS 181
The initial point r0 and the final point r coincide. Hence
Hence by (2) fvF.dr = 0.
Jo
Theorem : Conversely if the line integral of W about every
closed curve vanishes, W is the derivative of some scalar
function V (x, y, z) of position in space.
Given
J o
To show W = V V.
Let r0 be any fixed point in space and r a variable point.
The line integral
J
di
is independent of the path of integration C. For let any two
paths C and Cf be drawn between r0 and r. The curve which
consists of the path C from r0 to r and the path — Cf from r
to r0 is a closed curve. Hence by hypothesis
/W*cZr+ fw.dr = 0,
j «/ —c'
/W«dr = — / W*dr.
-c' J c'
Hence / W • d r = / W • dr.
J c J c'
Hence the value of the integral is independent of the path
of integration and depends only upon the final point r.
182 VECTOR ANALYSIS
The value of the integral is therefore a scalar function of
the position of the point r whose coordinates are x, y, z.
£
Let the integral be taken between two points infinitely near
together.
y,z).
But by definition V V • d r = d V.
Hence W
The theorem is therefore demonstrated.
80.] Let f be the force which acts upon a unit mass near
the surface of the earth under the influence of gravity. Let
a system of axes i, j, k be chosen so that k is vertical. Then
The work done by the force when its point of application
moves from the position r0 to the position r is
w
= I f*dT = I — # k • d r = — I gdz.
J r J r J r
Hence w = — g (z — z0) = g (z0 — z).
The force f is said to be derivable from a force-function V
when there exists a scalar function of position V such that
the force is equal at each point of the derivative VF.
Evidently if V is one force-function, another may be obtained
by adding to V any arbitrary constant. In the above ex
ample the force-function is
V=w = g(zQ-z).
Or more simply V = — g z.
The force is f = VF=-0k.
THE INTEGRAL CALCULUS OF VECTORS 183
The necessary and sufficient condition that a force-function
V (z, y, z) exist, is that the work done by the force when its
point of application moves around a closed circuit be zero.
The work done by the force is
w = I f • d r .
If this integral vanishes when taken around every closed
contour
And conversely if f = V V
the integral vanishes. The force-function and the work done
differ only by a constant.
V = w + const
In case there is friction no force-function can exist. For the
work done by friction when a particle is moved around in a
closed circuit is never zero.
The force of attraction exerted by a fixed mass M upon
a unit mass is directed toward the fixed mass and is propor
tional to the inverse square of the distance between the
masses.
M
f = -c-r.
r6
This is the law of universal gravitation as stated by Newton.
It is easy to see that this force is derivable from a force-
function V. Choose the origin of coordinates at the center
of the attracting mass M. Then the work done is
M
? — r • d r.
But r • d r = r d r,
r dr
rr dr M 1)
= -c$r I — =-cM j --- }•
J r r2 I r r0 3
184 VECTOR ANALYSIS
By a proper choice of units the constant c may be made
equal to unity. The force-function V may therefore be
chosen as
If there had been several attracting bodies
the force-function would have been
M <
where rr r2, r8, • • • are the distances of the attracted unit
mass from the attracting masses Mv M%y MB • • •
The law of the conservation of mechanical energy requires
that the work done by the forces when a point is moved
around a closed curve shall be zero. This is on the assump
tion that none of the mechanical energy has been converted
into other forms of energy during the motion. The law of
conservation of energy therefore requires the forces to be
derivable from a force-function. Conversely if a force-
function exists the work done by the forces when a point is
carried around a closed curve is zero and consequently there
is no loss of energy. A mechanical system for which a force-
function exists is called a conservative system. From the
example just cited above it is clear that bodies moving under
the law of universal gravitation form a conservative system —
at least so long as they do not collide.
81.] Let W (x, y, z) be any vector function of position in
space. Let S be any surface. Divide this surface into in
finitesimal elements. These elements may be regarded as
plane and may be represented by infinitesimal vectors of
which the direction is at each point the direction of the
normal to the surface at that point and of which the magni
tude is equal to the magnitude of the area of the infinitesimal
THE INTEGRAL CALCULUS OF VECTORS 185
element. Let this infinitesimal vector which represents the
element of surface in magnitude and direction be denoted by
d a. Form the sum
which is the sum of the scalar products of the value of W
at each element of surface and the (vector) element of
surface. The limit of this sum when the elements of sur
face approach zero is called the surface integral of W over
the surface $, and is written
(4)
The value of the integral is scalar. If W and da be ex
pressed in terms of their three components parallel to i, j, k
or d a = dy dz i -f dz dx j + dx dy k,
(5)
The surface integral therefore has been defined as is cus
tomary in ordinary analysis. It is however necessary to
determine with the greatest care which normal to the surface
d a is. That is, which side of the surface (so to speak) the
integral is taken over. For the normals upon the two sides
are the negatives of each other. Hence the surface integrals
taken over the two sides will differ in sign. In case the
surface be looked upon as bounding a portion of space d a
is always considered to be the exterior normal.
If f denote the flux of any substance the surface integral
f.rfa
s
186 VECTOR ANALYSIS
gives the amount of that substance which is passing through
the surface per unit time. It was seen before (Art. 71) that
the rate at which matter was leaving a point per unit
volume per unit time was V • f . The total amount of mat
ter which leaves a closed space bounded by a surface S per
unit time is the ordinary triple integral
(6)
Hence the very important relation connecting a surface in
tegral of a flux taken over a closed surface and the volume
integral of the divergence of the flux taken over the space
enclosed by the surface —
/»/»/»
CO
Written out in the notation of the ordinary calculus this
becomes
I I \Xdy dz + Ydzdx + Zdxdy~\
3Y,
where X, F, Z are the three components of the flux f . The
theorem is perhaps still more familiar when each of the three
components is treated separately.
(8)'
This is known as Gauss's Theorem. It states that the surface
integral (taken over a closed surface) of the product of a
function X and the cosine of the angle which the exterior
normal to that surface makes with the X-axis is equal to
the volume integral of the partial derivative of that function
THE INTEGRAL CALCULUS OF VECTORS 187
with respect to x taken throughout the volume enclosed by
that surface.
If the surface S be the surface bounding an infinitesimal
sphere or cube
ff f-da = V-f dv
where d v is the volume of that sphere or cube. Hence
V.f = ^ fff-da. (9)
dv J J a
This equation may be taken as a definition of the divergence
V • f . The divergence of a vector function f is equal to the
limit approached by the surface integral of f taken over a sur
face bounding an infinitesimal body divided by that volume
when the volume approaches zero as its limit. That is
V.f= ,A -- f-da. (10)
dv±Q dvJJs
From this definition which is evidently independent of the
axes all the properties of the divergence may be deduced. In
order to make use of this definition it is necessary to develop
at least the elements of the integral calculus of vectors before
the differentiating operators can be treated. This definition
of V • f consequently is interesting more from a theoretical
than from a practical standpoint.
82.] Theorem : The surface integral of the curl of a vector
function is equal to the line integral of that vector function
taken around the closed curve bounding that surface.
f f
J J
V x W-da= w-dr. (11)
8 J O
This is the celebrated theorem of Stokes. On account of its
great importance in all branches of mathematical physics a
number of different proofs will be given.
188 VECTOR ANALYSIS
First Proof : Consider a small triangle 1 23 upon the surface
S (Fig. 32). Let the value of W at the vertex 1 be W0.
Then by (50), Chap. III., the value at any neighboring point is
W = ~{ W0 + V (W* 8 r) + (V x W) x 8 r j ,
where the symbol 8 r has been introduced for the sake of dis
tinguishing it from d r which is to be used as the element of
integration. The integral of W taken around the triangle
FIG. 32.
Cw-dr=l fwo-dr + g fv(W-Sr).<Zr
+ 5 f (V x W) x Sr-dr.
«/ A
The first term I fwo.dr = iwo. Cdr
2 JA JA
vanishes because the integral of d r around a closed figure, in
this case a small triangle, is zero. The second term
g fv(W-Sr).dr
J A
vanishes by virtue of (3) page 180. Hence
THE INTEGRAL CALCULUS OF VECTORS 189
Cw*di = l fvxWxSr-dr.
JA J A
Interchange the dot and the cross in this triple product.
V xW-Sr x dr.
=| J
When dr is equal to the side 12 of the triangle, Sr is also
equal to this side. Hence the product
Sr x di
vanishes because 8 r and d r are collinear. In like manner
when dr is the side 31, 8r is the same side 13, but taken
in the opposite direction. Hence the vector product vanishes.
When dr is the side #5, Sr is a line drawn from the vertex
1 at which W= W0 to this side S3. Hence the product 8 r x d r
is twice the area of the triangle. This area, moreover, is the
positive area 1 % 3. Hence
|£r x dr =
where d a denotes the positive area of the triangular element
of surface. For the infinitesimal triangle therefore the
relation
= V x W
holds.
Let the surface 8 be divided into elementary triangles.
For convenience let the curve which bounds the surface
be made up of the sides of these triangles. Perform the
integration
fw-dr
J A
around each of these triangles and add the results together.
2/1
a JA
190
VECTOR ANALYSIS
The second member ]£ V x W • d a
3
is the surface integral of the curl of W.
2 V x W-rfa=JJv x W
In adding together the line integrals which occur in the first
member it is necessary to notice that all the sides of the ele
mentary triangles except those which lie along the bounding
curve of the surface are traced twice in opposite directions.
Hence all the terms in the sum
which arise from those sides of the triangles lying within the
surface S cancel out, leaving in the sum only the terms
which arise from those sides which make up the bounding
curve of the surface. Hence the sum reduces to the line in
tegral of W along the curve which bounds the surface S.
= fw
Jo
Hence
V x W • d a = W • d r.
= f
Jo
FIG. 33.
Second Proof : Let C be any closed
contour drawn upon the surface S
(Fig. 33). It will be assumed that C
is continuous and does not cut itself.
Let Cr be another such contour near
to C. Consider the variation S which
takes place in the line integral of W
in passing from the contour C to the
contour C".
THE INTEGRAL CALCULUS OF VECTORS 191
£ /V.dr = f
t/ t/ 0
S fw»dr = f
But d(W-
and
Hence J*W '&dT= Cw*dST=f d(W*Sr) -- CdW *ST.
The expression d (W • 8 r) is by its form a perfect differential.
The value of the integral of that expression will therefore be
the difference between the values of W • d r at the end and at
the beginning of the path of integration. In this case the
integral is taken around the closed contour C. Hence
/^
Jc
Hence
and S fw-rfr= fsw.dr- f
9W J J
But d W = -K— d a? + -7T— d y +
PW 3W 3W
or d W = -^ — i • d! r + ^ — j • d r + -^— k • d r,
& x d y d z
and
v x & y
192 VECTOR ANALYSIS
Substituting these values
dT i.Br-~ -8r i-dr
x ox
+ similar terms in y and z. [
But by (25) page 111
Hence sfw»dr=/ j i x ^ — «Srxdr
+ similar terms in y and z | .
or 8 f W • d r = f V x W • 8 r x d r.
In Fig. 33 it will be seen that d r is the element of arc
along the curve C and 8 r is the distance from the curve C to
the curve Cr. Hence 8 r X d r is equal to the area of an ele
mentary parallelogram included between C and Cf upon the
surface S. That is
S fw-dr= fv x W'da.
Let the curve C starting at a point 0 in £ expand until it
coincides with the contour bounding S. The line integral
will vary from the value 0 at the point 0 to the value
/
t/O
THE INTEGRAL CALCULUS OF VECTORS 193
taken around the contour which bounds the surface S. This
total variation of the integral will be equal to the sum of the
variations 8
Or f W«dr= ff Vx W-da. (11)
83.] Stokes's theorem that the surface integral of the curl
of a vector function is equal to the line integral of the func
tion taken along the closed curve which bounds the surface
has been proved. The converse is also true. If the surface
integral of a vector function U is equal to the line integral of the
function W taken around the curve bounding the surface and if
this relation holds for all surfaces in space, then TT is the curl of
W. That is
if f fll. da = f W»dr, thenU=Vx W. (12)
Form the surface integral df the difference between IT and
V x W.
// (tf~Vx W)*da=f W*dr - f W-dr = 0,
or f f (TI- V x W)-da = 0.
Let the surface S over which the integration is performed be
infinitesimal. The integral reduces to merely a single term
(U_V x
As this equation holds for any element of surface d a, the
first factor vanishes. Hence
IT- V x W = 0.
Hence IT = V x W.
The converse is therefore demonstrated.
13
194 VECTOR ANALYSIS
A definition of V x W which is independent of the axes
i, j, k may be obtained by applying Stokes's theorem to an in
finitesimal plane area. Consider a point P. Pass a plane
through P and draw in it, concentric with P, a small circle of
area d a.
Vx W.da=f W*dT. (13)
When d a has the same direction as V X W the value of the
line integral will be a maximum, for the cosine of the angle
between V x W and d a will be equal to unity. For this
value of da,
=rfaIM0 F/V f W-rfrl (13)'
rfa=:0 Lda.dajo J
Hence the curl V x W of a vector function W has at each
point of space the direction of the normal to that plane in
which the line integral of W taken about a small circle con
centric with the point in question is a maximum. The mag
nitude of the curl at the point is equal to the magnitude of
that line integral of maximum value divided by the area of
the circle about which it is taken. This definition like the
one given in Art. 81 for the divergence is interesting more
from theoretical than from practical considerations.
Stokes's theorem or rather its converse may be used to de
duce Maxwell's equations of the electro-magnetic field in a
simple manner. Let E be the electric force, B the magnetic
induction, H the magnetic force, and C the flux of electricity
per unit area per unit time (i. e. the current density).
It is a fact learned from experiment that the total electro
motive force around a closed circuit is equal to the negative
of the rate of change of total magnetic induction through
the circuit. The total electromotive force is the line integral
of the electric force taken around the circuit. That is
E«dr.
THE INTEGRAL CALCULUS OF VECTORS 195
The total magnetic induction through the circuit is the sur
face integral of the magnetic induction B taken over a surface
bounded by the circuit. That is
B • d*.
i
Experiment therefore shows that
or /E-dr=/l— B • d a.
J o J J a
Hence by the converse of Stokes's theorem
V x E = - B, curl E = - B.
It is also a fact of experiment that the work done in carry
ing a unit positive magnetic pole around a closed circuit is
equal to 4?r times the total electric flux through the circuit.
The work done in carrying a unit pole around a circuit is
the line integral of H around the circuit. That is
The total flux of electricity through the circuit is the
surface integral of C taken over a surface bounded by the
circuit. That is „ „
///•*••
Experiment therefore teaches that
= 47r C f
J J s
196 VECTOR ANALYSIS
By the converse of Stokes's theorem
V x H = 4 TT C.
With a proper interpretation of the current C, as the dis
placement current in addition to the conduction current,
an interpretation depending upon one of Maxwell's primary
hypotheses, this relation and the preceding one are the funda
mental equations of Maxwell's theory, in the form used by
Heaviside and Hertz.
The theorems of Stokes and Gauss may be used to demon
strate the identities.
V • V x W = 0, div curl W = 0.
Vx VF=0, curl VF=0.
According to Gauss's theorem
'VX Wdv=
According to Stokes's theorem
f fvxW-da = CW'dr.
Hence
fffv-VxWdtf= Cw*dr.
Apply this to an infinitesimal sphere. The surface bounding
the sphere is closed. Hence its bounding curve reduces to a
point ; and the integral around it, to zero.
V-VxWdv = fw-dr = 0,
J o
V • V x W = 0.
THE INTEGRAL CALCULUS OF VECTORS 197
Again according to Stokes's theorem
ffvxvr.<2a = fvr-dr.
Apply this to any infinitesimal portion of surface. The curve
bounding this surface is closed. Hence the line integral of
the derivative VF" vanishes.
V x
As this equation holds for any d a, it follows that
Vx VF=0.
In a similar manner the converse theorems may be
demonstrated. If the divergence V • TT of a vector function
TJ is everywhere zero, then TT is the curl of some vector
function W.
• TJ = V x W*
If the curl V x II of a vector function TT is everywhere zero,
then U is the derivative of some scalar function F",
84.] By making use of the three fundamental relations
between the line, surface, and volume integrals, and the
dels / viz. :
, (2)
JYv x W-rfa= f W.rfr, (11)
(7)
it is possible to obtain a large number of formulae for the
transformation of integrals. These formulae correspond to
198 VECTOR ANALYSIS
those connected with u integration by parts " in ordinary
calculus. They are obtained by integrating both sides of the
formulae, page 161, for differentiating.
First V (u v) = u V v + v V u.
C C C
Jc ~JcV ( T J GV
Hence I % V v • di = [uv] — \ vV u* dx. (14)
r
The expression [u v]
represents the difference between the value of (u v) at r, the
end of the path, and the value at r0, the beginning of the path.
If the path be closed
f^Vvdr = - C 0 V u*dr. (14)'
Jo Jo
Second V x (u v) = u V x v + V u x v.
f* f* f* (* (* f*
I I Vx(wv)*^a=/ / ^Vxv»rfa+/ I Vwxv-da.
J J S J J S J J 8
Hence
f* f* f* f* f*
I I V^xvda=l uv • dr— I I wVxvda, (15)
J J a Jo J J a
&Vxv«da= / uv • dr — I I V?txv«rfa, (15)'
Jo J J a
Third Vx (wV/y)^^VxV^ + V^xV'2;«
But V x V v = 0
Hence V x (u V v) = V u x V v,
or
THE INTEGRAL CALCULUS OF VECTORS 199
f* f* f* f*
J J S J J 8
Hence
f* f* f* ,-y , P ^-J 7 S-4 />V
IIVO'AVt/ It I 1 \ /
J J s Jo Jo
Fourth V • (u v) = u V • v + V w • v.
/»/»/» r r r C C C *
JJJV.(«T)^=JJJ«VT^+JJJ V«-vd
Hence
^ v • a —
or
C C C ^7u»v dv= I I Mvda— rrr^V'V^i;, (17);
Fifth v(V^xv) = VXV^*v — v^-VXv.
V (V M x v) = — V ^ •' V x v,
Hence
rfv^xvrfa = — fffv^«vxvdi;. (18)
In all these formulae which contain a triple integral the
surface $ is the closed surface bounding the body throughout
which the integration is performed.
Examples of integration by parts like those above can be
multiplied almost without limit. Only one more will be
given here. It is known as Greens Theorem and is perhaps
the most important of all. If u and v are any two scalar
functions of position,
200 VECTOR ANALYSIS
V • (^ V fl) = V ^ • V tf + 24 V« V ^
V O V u) = V u • V v + v V • V u>
J J J ^u^vclv==J J J V- (uvv)dv— C f Cu^
Hence
/ / /V^-Vtfdfl=/ /^VvcU— / r/^V-Vvdi?,
= / / ^ V ^ • d a — f j I v^*V udv. (19)
By subtracting these equalities the formula (20)
/ / / (^ V • V ^ — v V • V w) ^ ^ = / / (^ V t> — v V ^) • ^ a.
is obtained. By expanding the expression in terms of i, j, k
the ordinary form of Green's theorem may be obtained. A
further generalization due to Thomson (Lord Kelvin) is the
following :
/ / lw^/u*Vvdv=l I uwVv*d&— I I I u\
= / I vwVU"d* — I I I v\? • [w^ u^ dv, (21)
where w is a third scalar function of position.
The element of volume dv has nothing to do with the scalar
function v in these equations or in those that go before. The
use of v in these two different senses can hardly give rise to
any misunderstanding.
* 85.] In the preceding articles the scalar and vector func
tions which have been subject to treatment have been sup-
THE INTEGRAL CALCULUS OF VECTORS 201
posed to be continuous, single-valued, possessing derivatives
of the first two orders at every point of space under consider
ation. When the functions are discontinuous or multiple-
valued, or fail to possess derivatives of the first two orders
in certain regions of space, some caution must be exercised in
applying the results obtained.
Suppose for instance
VF-
The line integral
— y dx
Introducing polar coordinates
x = r cos 6,
y = r sin 0,
7 V • d r = I d 0.
Form the line integral from the point ( + 1,0) to the point
(—1, 0) along two different paths. Let one path be a semi
circle lying above the JT-axis ; and the other, a semicircle
lying below that axis. The value of the integral along the
first path is
/»-*•
along the second path, I d 6 — — TT.
From this it appears that the integral does not depend merely
upon the limits of integration, but upon the path chosen,
202 VECTOR ANALYSIS
the value along one path being the negative of the value
along the other. The integral around the circle which is a
closed curve does not vanish, but is equal to ± 2 TT.
It might seem therefore the results of Art. 79 were false
and that consequently the entire bottom of the work which
follows fell out. This however is not so. The difficulty is
that the function
—1 V
F=tan ^-
x
is not single-valued. At the point (1,1), for instance, the
function V takes on not only the value
-i TT
F= tan l = -r>
4
but a whole series of values
7T
-+&7T,
where k is any positive or negative integer. Furthermore at
the origin, which was included between the two semicircular
paths of integration, the function V becomes wholly inde
terminate and fails to possess a derivative. It will be seen
therefore that the origin is a peculiar or singular point of the
function V. If the two paths of integration from (+ 1, 0) to
(—1,0) had not included the origin the values of the integral
would not have differed. In other words the value of the
integral around a closed curve which does not include the
origin vanishes as it should.
Inasmuch as the origin appears to be the point which
vitiates the results obtained, let it be considered as marked
by an impassable barrier. Any closed curve 0 which does
not contain the origin may be shrunk up or expanded at will ;
but a closed curve 0 which surrounds the origin cannot be
so distorted as no longer to enclose that point without break
ing its continuity. The curve C not surrounding the origin
THE INTEGRAL CALCULUS OF VECTORS 203
may shrink up to nothing without a break in its continuity ;
but C can only shrink down and fit closer and closer about
the origin. It cannot be shrunk down to nothing. It must
always remain encircling the origin. The curve C is said to
be reducible ; (7, irreducible. In case of the function F, then,
it is true that the integral taken around any reducible circuit
C vanishes; but the integral around any irreducible circuit C
does not vanish.
Suppose next that V is any function whatsoever. Let all
the points at which V fails to be continuous or to have con
tinuous first partial derivatives be marked as impassable
barriers. Then any circuit 0 which contains within it no
such point may be shrunk up to nothing and is said to be
reducible; but a circuit which contains one or more such
points cannot be so shrunk up without breaking its continuity
and it is said to be irreducible. The theorem may then be
stated: The line integral of the derivative VF" of any function
V vanishes around any reducible circuit C. It may or may not
vanish around an irreducible circuit In case one irreducible
circuit C may be distorted so as to coincide with another
irreducible circuit C without passing through any of the
singular points of V and without breaking its continuity,
the two circuits are said to be reconcilable and the values of
the line integral of V F about them are the same.
A region such that any closed curve C within it may be
shrunk up to nothing without passing through any singular
point of V and without breaking its continuity, that is, a
region every closed curve in which is reducible*, is said to be
acyclic. All other regions are cyclic.
By means of a simple device any cyclic region may be ren
dered acyclic. Consider, for instance, the region (Fig. 34) en
closed between the surface of a cylinder and the surface of a
cube which contains the cylinder and whose bases coincide
with those of the cylinder. Such a region is realized in a room
204 VECTOR ANALYSIS
in which a column reaches from the floor to the ceiling. It
is evident that this region is cyclic. A circuit which passes
around the column is irreducible. It cannot be contracted to
nothing without breaking its continuity. If
£~^x / now a diaphragm be inserted reaching from
the surface of the cylinder or column to the
surface of the cube the region thus formed
bounded by the surface of the cylinder, the
surface of the cube, and the two sides of the
diaphragm is acyclic. Owing to the inser
tion of the diaphragm it is no longer possible
to draw a circuit which shall pass completely around the cyl
inder — the diaphragm prevents it. Hence every closed cir
cuit which may be drawn in the region is reducible and the
region is acyclic.
In like manner any region may be rendered acyclic by
inserting a sufficient number of diaphragms. The bounding
surfaces of the new region consist of the bounding surfaces of
the given cyclic region and the two faces of each diaphragm.
In acyclic regions or regions rendered acyclic by the fore
going device all the results contained in Arts. 79 et seq.
hold true. For cyclic regions they may or may not hold
true. To enter further into these questions at this point is
unnecessary. Indeed, even as much discussion as has been
given them already may be superfluous. For they are ques
tions which do not concern vector methods any more than the
corresponding Cartesian ones. They belong properly to the
subject of integration itself, rather than to the particular
notation which may be employed in connection with it and
which is the primary object of exposition here. In this
respect these questions are similar to questions of rigor.
THE INTEGRAL CALCULUS OF VECTORS 205
The Integrating Operators. The Potential
86.] Hitherto there have been considered line, surface,
and volume integrals of functions both scalar and vector.
There exist, however, certain special volume integrals which,
owing to their intimate connection with the differentiating
operators V, V», Vx, and owing to their especially frequent
occurrence and great importance in physics, merit especial
consideration. Suppose that
^0** Vv *a)
is a scalar function of the position in space of the point
For the sake of definiteness V may be regarded as the
density of matter at the point (#2, yv 22). In a homogeneous
body V is constant. In those portions of space in which no
matter exists V is identically zero. In non-homogeneous dis
tributions of matter V varies from point to point; but at
each point it has a definite value.
The vector
r2 = z2i + y2j + *2k,
drawn from any assumed origin, may be used to designate
the point (#2, y2, z2). Let
On yi. *i)
be any other fixed point of space, represented by the vector
drawn from the same origin. Then
r2 - rx = O2 - »!>! + (y2 - yi) j + (z2 - *j) k
is the vector drawn from the point (xv yv Zj) to the point
(#2> IJy 2 2)- AS ^s vec^or occurs a large number of times
in the sections immediately following, it will be denoted by
ri2 = r2 ~~ ri-
206 VECTOR ANALYSIS
The length of r12 is then r12 and will be assumed to be
positive.
-i2 = V r12 • r12 = V (*2 - x^ + (y2 - ^)2 + 02 - ^)2.
Consider the triple integral
The integration is performed with respect to the variables
^2> ^2> ^ 2 — that is, with respect to the body of which V
represents the density (Fig. 35). During
the integration the point (xv yv z^ re
mains fixed. The integral / has a definite
value at each definite point (xv yv zj.
It is a function of that point. The in-
FIG. 3o. .
terpretation of this integral / is easy, if
the function V be regarded as the density of matter in space.
The element of mass dm at (#2, y2, z2) is
dm — V (#2, y2, 2!2) dx^ dyz dz% = Vdv.
The integral / is therefore the sum of the elements of mass
in a body, each divided by its distance from a fixed point
r
J
dm
This is what is termed the potential at the point (xv yv
due to the body whose density is
The limits of integration in the integral / may be looked at
in either of two ways. In the first place they may be
regarded as coincident with the limits of the body of which
V is the density. This indeed might seem the most natural
set of limits. On the other hand the integral / may be
THE INTEGRAL CALCULUS OF VECTORS 207
regarded as taken over all space. The value of the integral
is the same in both cases. For when the limits are infinite
the function V vanishes identically at every point (#2, y2, 22)
situated outside of the body and hence does not augment
the value of the integral at all. It is found most convenient
to consider the limits as infinite and the integral as extended
over all space. This saves the trouble of writing in special
limits for each particular case. The function Vot itself then
practically determines the limits owing to its vanishing iden
tically at all points unoccupied by matter.
87.] The operation of finding the potential is of such
frequent occurrence that a special symbol, Pot, is used for it.
Pot r=fff V^ y" *2> rf*2 dy^ dzy (22)
The symbol is read "the potential of V." The potential,
Pot V, is a function not of the variables #.2, yv z2 with
regard to which the integration is performed but of the point
(xv y^ Zj) which is fixed during the integration. These
variables enter in the expression for r12. The function V
and Pot V therefore have different sets of variables.
It may be necessary to note that although V has hitherto
been regarded as the density of matter in space, such an
interpretation for V is entirely too restricted for convenience.
Whenever it becomes necessary to form the integral
i" <22>'
of any scalar function V, no matter what V represents, that
integral is called the potential of V. The reason for calling
such an integral the potential even in cases in which it has
np connection with physical potential is that it is formed
according to the same formal law as the true potential and
208 VECTOR ANALYSIS
by virtue of that formation has certain simple rules of opera
tion which other types of integrals do not possess.
Pursuant to this idea the potential of a vector function
W O2, y2, z2)
may be written down.
Pot W =W (*2 y* *2) dx, rfy. rf,r (23)
In this case the integral is the sum of vector quantities
and is consequently itself a vector. Thus the potential of a
vector function W is a vector function, just as the potential
of a scalar function V was seen to be a scalar function of posi
tion in space. If W be resolved into its three components
W O2, 2/2, z2) = i X O2, yv z2) + j T <>2, yv z2)
+ kZ <>2, yv z2)
Pot W = i Pot X + j Pot Y+ k Pot Z. (24)
The potential of a vector function W is equal to the vector
sum of the potentials of its three components X, Y, Z.
The potential of a scalar function V exists at a point
(xv yv zp) when and only when the integral
taken over all space converges to a definite value. If,
for instance, V were everywhere constant in space the in
tegral would become greater and greater without limit as
the limits of integration were extended farther and farther
out into space. Evidently therefore if Jhe potential is to exist
F must approach zero as its limit as the point (#2, yv 32)
recedes indefinitely. A few important sufficient conditions
for the convergence of the potential may be obtained by
transforming to polar coordinates. Let
THE INTEGRAL CALCULUS OF VECTORS 209
x = r sin 6 cos fa
y—r sin 6 sin fa
z = r cos 0,
dv = r2sm0 dr dO d<f>.
Let the point (xv yv ^) which is fixed for the integration
be chosen at the origin. Then
ri2 = r
and the integral becomes
or simply PotF= CCCVrsmff dr d0 dfa
If the function V decrease so rapidly that the product
Vr*
remains finite as r increases indefinitely, then the integral con
verges as far as the distant regions of space are concerned.
For let
r = 00
dr d0d<f>
r = 00
dr d0 d<f>
= QO
Hence the triple integral taken over all space outside of a
sphere of radius R (where R is supposed to be a large quan
tity) is less than %TT* K jR, and consequently converges as far
as regions distant from the origin are concerned.
14
210 VECTOR ANALYSIS
If the function V remain finite or if it become infinite so
weakly that the product
Vr
remains finite when r approaches zero, then the integral converges
as far as regions near to the origin are concerned. For let
Vr<K
f CCrrsmddr d0 d<f> < C C fadr d0 d<f>.
r =
C C C
dO d<t> =
Hence the triple integral taken over all space inside a sphere
of radius R (where R is now supposed to be a small quantity)
is less than 2 Tr2 K R and consequently converges as far as
regions near to the origin which is the point (xv yv Zj) are
concerned.
If at any point (x2, y2, z2) not coincident with the origin,
i. e. the point (xx, yv zx), the function V becomes infinite so
weakly that the product of the value 0/V at a point near to
(X2> J2> Z2) ty the square of the distance of that point from
(x2, y2, z2) remains finite as that distance approaches zero, then
the integral converges as far as regions near to the point (x2, y2, z2)
are concerned. The proof of this statement is like those given
before. These three conditions for the convergence of the
integral Pot V are sufficient. They are by no means neces
sary. The integral may converge when they do not hold.
It is however indispensable to know whether or not an integral
under discussion converges. Unless the tests given above
show the convergence, more stringent ones must be resorted
to. Such, however, will not be discussed here. They belong
to the theory of integration in general rather than to the
THE INTEGRAL CALCULUS OF VECTORS
211
theory of the integrating operator Pot. The discussion of
the convergence of the potential of a vector function W re
duces at once to that of its three components which are scalar
functions and may be treated as above.
88.] The potential is a function of the variables xv yv zl
which are constant with respect to the integration. Let the
value of the potential at the point (xv yv z^ be denoted by
The first partial derivative of the potential with respect to xl
is therefore
LIM ^[
The value of this limit may be determined by a simple
device (Fig. 36). Consider
the potential at the point
due to a certain body T. This
is the same as the potential at
the point
FlG- 36-
due to the same body T displaced in the negative direction by
the amount A xr For in finding the potential at a point P
due to a body T the absolute positions in space of the body
T and the point P are immaterial. It is only their positions
relative to each other which determines the value of the poten
tial. If both body and point be translated by the same
amount in the same direction the value of the potential is un
changed. But now if T be displaced in the negative direction
by the amount A»#, the value of Fat each point of space is
changed from
v C*2> y* **) to v 0*2
where A#2 = A xr
212 VECTOR ANALYSIS
Hence
[Pot V(xvyvz^ + AX,, yt, *, = [Pot F<>2 + A a;2,y2,
Hence LlM j [Pot HX. + A »..,,, .t- [Pot
A #! = 0 /
It will be found convenient to introduce the limits of
integration. Let the portion of space originally filled by the
body T be denoted by M ; and let the portion filled by the
body after its translation in the negative direction through
the distance A xl be denoted by M' . The regions M and M1
overlap. Let the region common to both be M ; and let the
remainder of M be m; the remainder of M1 , m1 . Then
Pot V (a, + A *„ y r *2) f ^
d rrr
J J J m'
^
Pot
/»/»/» "\F ( W (II * *\ /*/*/* 1^ f V tl 9 ^
I I I r I O/ft« t/n« ^O/, ill *V«*/O1 V91 " <) J 1
= / / / I *y rft>,+ / / / ^_Mrfr2.
J J J M ^j2 J J Jm T YL
Hence (25) becomes, when A ^j is replaced *by its equal Aic2,
t As all the following potentials are for the point arlf yi, «i the bracket and
indices have been dropped.
THE INTEGRAL CALCULUS OF VECTORS 213
+
Or,
my(*
C f r
J J J
'
A £2
,
"
A a;2==0
LIM
^,
r12
v yy g2
= rrr
jJJ j
LIM (
r12
^
9 X
when A ^! approaches zero as its limit the regions mand m;,
which are at no point thicker than A #, approach zero ; M '
and Jf both approach -Jf as a limit.
t There are cases in which this reversal of the order in which the two limits
are taken gives incorrect results. This is a question of double limits and leads to
the mazes of modern mathematical rigor.
J If the derivative of Fis to exist at the surface bounding T the values of the
function V must diminish continuously to zero upon the surface. If Fchanged
suddenly from a finite value within the surface to a zero value outside the de
rivative QVlS^i would not exist and the triple integral would be meaningless.
For the same reason V is supposed to be finite and continuous at every point
within the region T.
214 VECTOR ANALYSIS
Then if it be assumed that the region T is finite and that V
vanishes upon the surface bounding T
T/nvr rrr V(Y <>/ z \
i <t\ jvi I I . I V \ ^o* £fo» ^9y 7 /\
A^ojjjOT riaAa;2 J«i =
Consequently the expression for the derivative of the poten
tial reduces to merely
3 Pot F r r r i 3 F 3 F
— = 1/1 — dv* = Pot •
d xl J J J M r12 3#2 3^2
^%^ partial derivative of the potential of a scalar function V
is equal to the potential of the partial derivative of V.
The derivative V of the potential ofVis equal to the potential
of the derivative V V.
VPotF=PotVF (27)
This statement follows immediately from the former. As
the V upon the left-hand side applies to the set of vari
ables xv y^ Zj, it may be written Vr In like manner the
V upon the right-hand side may be written V2 to call atten
tion to the fact that it applies to the variables #2, y2, z2 of F.
Then Va Pot F= Pot V2 F (27)'
To demonstrate this identity V may be expanded in terms of
.3 PotF .3 PotF . 3 PotF
I i I lr
* ^ J * T- *• ^
SV
+jPot -l-
3
THE INTEGRAL CALCULUS OF VECTORS 215
As i, j, k are constant vectors they may be placed under
the sign of integration and the terms may be collected. Then
by means of (26)
The curl V X and divergence V • of the potential of a vector
function W are equal respectively to the potential of the curl and
divergence of that function.
V, x Pot W = Pot V2 x W,
(28)
or curl Pot W = Pot curl W
and Vj • Pot W = Pot V2 • W,
or div Pot W = Pot div W.
These relations may be proved in a manner analogous to the
above. It is even possible to go further and form the dels
of higher order
v • v Pot r= Pot v • VF; (30)
Uf>iac<3*
V- V Pot W = Pot V • V W, (31)
V V • Pot W = Pot VV • W, (32)
V x V x Pot W= Pot V x V x W. (33)
The dels upon the left might have a subscript 1 attached to
show that the differentiations are performed with respect to
the variables xv yv zv and for a similar reason the dels upon
the right might have been written with a subscript 2. The
results of this article may be summed up as follows:
Theorem: The differentiating operator V and the integrating
operator Pot are commutative.
*89.] In the foregoing work it has been assumed that the
region T was finite and that the function Fwas everywhere
finite and continuous inside of the region T and moreover
decreased so as to approach zero continuously at the surface
bounding that region. These restrictions are inconvenient
216 VECTOR ANALYSIS
and may be removed by making use of a surface integral.
The derivative of the potential was obtained (page 213) in
essentially the form
otr_ r r r I SV
xl J J J ,f r122x2
-LJ-LJjl J. § § § * \«^o i ••• *V2'^2'**'2^ 7
a V2
12
LIM 1 rrr Tr(ga>yffga)fgr
ri2
Let d a be a directed element of the surface $ bounding the
region J!f. The element of volume dvz in the region mr is
therefore equal to
dt?2=A#2 i • da.
Hence - I I I ,^2^-^2^2112)
L f f f
«2J J Jm-
= T f V
J J
r!2
The element of volume d v2 ^n *^e regi°n m ^s equal to
di,2 = -Aa;2i.da.
Hence * /Tf
J\X^J J Jm
«•>
Consequently
— i-da. (34)
^ r!2
THE INTEGRAL CALCULUS OF VECTORS 217
The volume integral is taken throughout the region M with
the understanding that the value of the derivative of V at
the surface S shall be equal to the limit of the value of that
derivative when the surface is approached from the interior
of M. This convention avoids the difficulty that arises in
connection with the existence of the derivative at the surface
S where V becomes discontinuous. The surface integral is
taken over the surface S which bounds the region.
Suppose that the region M becomes infinite. By virtue of
the conditions imposed upon V to insure the convergence of
the potential
Vr* < K.
Let the bounding surface S be a sphere of radius £, a quan
tity which is large.
i • d a < R2 d 6 d<f>.
<//§»*-«•»§•.
s
The surface integral becomes smaller and smaller and ap
proaches zero as its limit when the region M becomes infinite.
Moreover the volume integral
JLJT^
remains finite as M becomes infinite. Consequently provided
V is such a function that Pot V exists as far as the infinite
regions of space are concerned, then the equation
=
holds as far as those regions of space are concerned.
Suppose that V ceases to be continuous or becomes infinite
at a single point (x^ yv z^) within the region T. Surround
218 VECTOR ANALYSIS
this point with a small sphere of radius R. Let S denote the
surface of this sphere and M all the region T not included
within the sphere. Then
r r r i 9V r r v
=JJj«^^dv* + JJ*-^1'
By the conditions imposed upon V
Vr<K
V .
//>" <//.*
d6 d^
Consequently when the sphere of radius R becomes smaller
and smaller the surface integral may or may not become zero.
Moreover the volume integral
1 3V .
may or may not approach a limit when E becomes smaller
and smaller. Hence the equation
SPotF SV
has not always a definite meaning at a point of the region
T at which V becomes infinite in such a manner that the
product Vr remains finite.
If, however, V remains finite at the point in question so
that the product Vr approaches zero, the constant K is zero
and the surface integral becomes smaller and smaller as R
approaches zero. Moreover the volume integral
THE INTEGRAL CALCULUS OF VECTORS 219
approaches a definite limit as R becomes infinitesimal. Con
sequently the equation
5 Pot V _ p dV
— 7\ — A Ob
holds in the neighborhood of all isolated pointe at which V
remains finite even though it be discontinuous.
Suppose that V becomes infinite at some single point
(iC2, y2, 22) not coincident with (x^ yv z^). According to the
conditions laid upon V
VI* < K,
where I is the distance of the point (z2, y2, z2) from a point
near to it. Then the surface integral
V .
» r!2
need not become zero and consequently the equation
5PotF SV
= Pot TT
need not hold for any point (a?r yv z^) of the region. But
if V becomes infinite at #2, y2, z2 in such a manner that
VI <K,
then the surface integral will approach zero as its limit and
the equation will hold.
Finally suppose the function V remains finite upon the
surface S bounding the region jT, but does not vanish there.
In this case there exists a surface of discontinuities of V.
Within this surface V is finite ; without, it is zero. The
surface integral
F.
220 VECTOR ANALYSTS
does not vanish in general. Hence the equation
SPotF 9V
—= --- = Pot -^r—
dX1 v%i
cannot hold.
Similar reasoning may be applied to each of the three
partial derivatives with respect to xv yv zr By combining
the results it is seen that in general
Vj PotF= Pot V2F+ f f Z da. (35)
Let F be any function in space, and let it be granted that
Pot F exists. Surround each point of space at which V
ceases to be finite by a small sphere. Let the surface of the
sphere be denoted by S. Draw in space all those surfaces
which are surfaces of discontinuity of V. Let these sur
faces also be denoted by S. Then the formula (35) holds
where the surface integral is taken over all the surfaces
which have been designated by S. If the integral taken
over all these surfaces vanishes when the radii of the spheres
above mentioned become infinitesimal, then
^ (27)'
This formula
V1PotF=PotV2F.
will surely hold at a point (xx, yv Zj) if V remains always
finite or becomes infinite at a point (x2, y2, z2) so that the
product V 1 remains finite, and if V possesses no surfaces of
discontinuity, and if furthermore the product V r3 remains finite
as r becomes infinite.1 In other cases special tests must be
applied to ascertain whether the formula (27) ' can be used
or the more complicated one (35) must be resorted to.
1 For extensions and modifications of this theorem, see exercises.
THE INTEGRAL CALCULUS OF VECTORS 221
The relation (27) is so simple and so amenable to trans
formation that V will in general be assumed to be such a
function that (27) holds. In cases in which V possesses a
surface S of discontinuity it is frequently found convenient
to consider V as replaced by another function V which has
in general the same values as Fbut which instead of possess
ing a discontinuity at S merely changes very rapidly from
one value to another as the point (#2, y2, 22) passes from one
side of S to the other. Such a device renders the potential
of V simpler to treat analytically and probably conforms to
actual physical states more closely than the more exact
conception of a surface of discontinuity. This device prac
tically amounts to including the surface integral in the
symbol Pot VF:
In fact from the standpoint of pure mathematics it is
better to state that where there exist surfaces at which the
function V becomes discontinuous, the full value of Pot V V
should always be understood as including the surface integral
//.£
in addition to the volume integral
>VF
SSSr-
U *J *J '10
2'
12
In like manner Pot V • W, Pot V X W, New V • W and other
similar expressions to be met in the future must be regarded
as consisting not only of a volume integral but of a surface
integral in addition, whenever the vector function W possesses
a surface of discontinuities.
It is precisely this convention in the interpretation of
formulae which permits such simple formulae as (27) to hold
in general, and which gives to the treatment of the integrat
ing operators an elegance of treatment otherwise unobtainable.
222 VECTOR ANALYSIS
The irregularities which may arise are thrown into the inter
pretation, not into the analytic appearance of the formulse.
This is the essence of Professor Gibbs's method of treatment.
90.] The first partial derivatives of the potential may also
be obtained by differentiating under the sign of integration.1
Q2> 3/21*2) _ , , ,
CCC
=Jj J «*>-
^ rrr (*,-*!> r^y,,*,)
i "^^^ V[(*a-*1)H(y2^1)2+(v-^)T 8 (37)
In like manner for a vector function W
S Pot W /* /* ' /* «*, «~ .. «~ ~
p ^ ""I I / . /r/- „ \9 i /.. .. \2 i /^. ~ \2^ia 2 «y !
Or
and ^!W= / / / *"•-,-'" " d,r (38)'
12
But 2 -! 2 -x 2 - ! = 12.
1 If an attempt were made to obtain the second partial derivatives in the same
manner, it would be seen that the volume integrals no longer converged.
THE INTEGRAL CALCULUS OF VECTORS 223
Hence V Pot F = /// ^f- d vr (39)
In like manner
^,, (40)
and V • Pot W = ^ *
= fff
These three integrals obtained from the potential by the
differentiating operators are of great importance in mathe
matical physics. Each has its own interpretation. Conse
quently although obtained so simply from the potential each
is given a separate name. Moreover inasmuch as these
integrals may exist even when the potential is divergent,
they must be considered independent of it. They are to
be looked upon as three new integrating operators defined
each upon its own merits as the potential was defined.
Let, therefore,
(42)
12
12
.3
r 12
= Max W. (44)
If the potential exists, then
V Pot F= New F
VxPotW = LapW (45)
V-PotW = MaxW.
The first is written New V and read « The Newtonian of V!9
224 VECTOR ANALYSIS
The reason for calling this integral the Newtonian is that if
V represent the density of a body the integral gives the force,
of attraction at the point (x^ yv Zj) due to the body. This
will be proved later. The second is written Lap W and
read "the Laplacian of W." This integral was used to a
considerable extent by Laplace. It is of frequent occurrence
in electricity and magnetism. If W represent the current
C in space the Laplacian of C gives the magnetic force at the
point (xv yv zj due to the current. The third is written
Max W and read " the Maxwellian of W." This integral was
used by Maxwell. It, too, occurs frequently in electricity
and magnetism. For instance if W represent the intensity
of magnetization I, the Maxwellian of I gives the magnetic
potential at the point (x^ yv z^) due to the magnetization.
To show that the Newtonian gives the force of attraction
according to the law of the inverse square of the distance.
Let dm<i be any element of mass situated at the point
f°rce at (xv Vv zi) due to dm is equal to
in magnitude and has the direction of the vector r12 from the
point (xv yv zj to the point (#2, y2, z2). Hence the force is
Integrating over the entire body, or over all space according
to the convention here adopted, the total force is
where V denotes the density of matter.
THE INTEGRAL CALCULUS OF VECTORS 225
The integral may be expanded in terms of i, j, k,
12
The three components may be expressed in terms of the po
tential (if it exists) as
12
(42)'
It is in this form that the Newtonian is generally found in
books.
To show that the Laplacian gives the magnetic force per
unit positive pole at the point (xv yv z^) due to a distribution
W (#2, y<p z2) °f electric flux. The magnetic force at (xv yv «x)
due to an element of current d C2 is equal in magnitude to
the magnitude d C% of that element of current divided by the
square of the distance r12 ; that is
dC*
«*2
T 12
The direction of the force is perpendicular both to the vector
element of current dC2 and to the line r12 joining the points.
The direction of the force is therefore the direction of the
vector product of r12 and dC2. The force is therefore
3
12
r
T
15
226 VECTOR ANALYSIS
Integrating over all space, the total magnetic force acting at
the point (x^ yv z^) upon a unit positive pole is
c r rri2 x d C2 r r r*™ x w 7
/// JVJ -J J J ^ ".-
This integral may be expanded in terms of i, j, k. Let
W (xv yv *2) = i X(xv yv z^ + j Y (x^ yv z^)
4- k^O2, yv z%).
ri2=(^2-^i) i+(y«-yi)J+ («a-*i)k-
The i, j, k components of Lap W are respectively
C.-^^
(43)'
In terms of the potential (if one exists) this may be written
3 Pot Z S Pot Y
i • Lap W = — g g
r=lP^_a|2t^ (43),,
To show that if I be the intensity of magnetization at the
point (xv %>*2)> that is, if I be a vector whose magnitude is
equal to the magnetic moment per unit volume and whose
THE INTEGRAL CALCULUS OF VECTORS 227
direction is the direction of magnetization of the element d v%
from south pole to north pole, then the Maxwellian of I is the
magnetic potential due to the distribution of magnetization.
The magnetic moment of the element of volume d t>2 is I d v%.
The potential at (xvyv 24) due to this element is equal to its
magnetic moment divided by the square of the distance r12
and multiplied by the cosine of the angle between the direc
tion of magnetization I and the vector r12. The potential is
therefore
r12»I dv%
Integrating, the total magnetic potential is seen to be
12
This integral may also be written out in terms of x, y, z.
Let-
*ia • I = Oa - xi) A + (y a ~ Vi) B + (*2 - *i) &
If instead of xv yv zl the variables x^ y, z; and instead of
xv y& zz ^e variables %, ?;, f be used1 the expression takes
oq the form given by Maxwell.
According to the notation employed for the Laplacian
Max w -fff (*.-»i
(44)'
1 Maxwell : Electricity and Magnetism, Vol. II. p. 9.
228 VECTOR ANALYSIS
The Maxwellian of a vector function is a scalar quantity.
It may be written in terms of the potential (if it exists) as
SPotF
Max W = -= -- + — = - + — = -- (44)"
dxl 3yl 9zl
This form of expression is much used in ordinary treatises
upon mathematical physics.
The Newtonian, Laplacian, and Maxwellian, however, should
not be associated indissolubly with the particular physical
interpretations given to them above. They should be looked
upon as integrating operators which may be applied, as the
potential is, to any functions of position in space. The New
tonian is applied to a scalar function and yields a vector
function. The Laplacian is applied to a vector function
and yields a function of the same sort. The Maxwellian
is applied to a vector function and yields a scalar function.
Moreover, these integrals should not be looked upon as the
derivatives of the potential. If the potential exists they
are its derivatives. But they frequently exist when the
potential fails to converge.
91.] Let V and W be such functions that their potentials
exist and have in general definite values. Then by (27) and
(29)
V • V PotF= V • Pot VF = Pot V • VF.
But by (45) V Pot V = New F,
and V.Pot VF=Max VF.
Hence V. V PotF= V. NewF= Max VF
= PotV.VF (46)
By (27) and (29) V V • Pot W = V Pot V. W= Pot V V • W.
But by (45) V • Pot W = Max W,
and by (45) V Pot V« W = New V. W.
THE INTEGRAL CALCULUS OF VECTORS 229
Hence V V • Pot W = V Max W = New V • W
= Pot VV.W (47)
By (28) V x V x Pot W = V x Pot V x W
= Pot V x V x W.
But by (45) V x Pot W = Lap W,
and V x Pot V x W = Lap V x W.
Hence V x V x Pot W = V x Lap W = Lap V x W
= Pot V x V x W. (48)
By (56), Chap. III. V - V x Pot W = 0,
or V • Pot V x W = 0.
Hence V • Lap W = Max V x W = 0. (49)
And by (52), Chap. III. V x V PotF= 0,
or VxPotVF=0.
Hence V x New V = Lap V V = 0. (50)
And by (58), Chap. III. V x V x W = VV • W - V • V W,
V.VW = VV-W — VxVxW.
Hence V • V Pot W — New V • W — Lap V X W, (51)
or V • V Pot W = V Max W — V X Lap W.
These formulae may be written out in terms of curl and
div if desired. Thus
div New V = Max V F, (46)'
V Max W = New div W (47)'
curl Lap W = Lap curl W (48)'
div Lap W = Max curl W = 0 (49)'
curl New V = Lap V F = 0 (50)'
V • V Pot W = New div W — Lap curl W. (51)'
230 VECTOR ANALYSIS
Poisson's Equation
92.] Let V "be any function in space such that the potential
PotF
has in general a definite value. Then
V • V PotF= - 4 TrF, (52)
c>2PotF 32PotF 32PotF
This equation is known as Poisson's Equation.
The integral which has been defined as the potential is a
solution of Poisson's Equation. The proof is as follows.
Vx . Vx Pot r= Vj . New F= Max V2 r= T f C *™'^*V dvv
The subscripts 1 and ^ have been attached to designate
clearly what are variables with respect to which the differen
tiations are performed.
V1.V1PotF=V1.NewF=ff TVJ--. V2Fdva.
But Vj — = - V2 —
rvt ru
and V2 • (v V2 — ^ = V2 — • V2F+ V Va . Va — •
THE INTEGRAL CALCULUS OF VECTORS 231
Hence - V2 — - V2 V = V V2 • V2 — - V2. ( V V2 — \
r!2 r!2 \ r!2/
m v,— .v.r=rv. . v2 — + v..(V v — V
y 13 r!2 \ W
Integrate :
But V2 • V2 — = 0.
That is to say — satisfies Laplace's Equation. And by (8)
Hence Vj • Vx Pot V = f f f Vx - - • V2 Vd v2 (53)
=rr ^ vt— .rfa.
J J s 7*12
The surface integral is taken over the surface which bounds
the region of integration of the volume integral. This is
taken " over all space." Hence the surface integral must be
taken over a sphere of radius R, a large quantity, and R must
be allowed to increase without limit. At the point (xr y^z^)^
however, the integrand of the surface integral becomes in
finite owing to the presence of the term
232 VECTOR ANALYSIS
Hence the surface S must include not only the surface of the
sphere of radius J2, but also the surface of a sphere of radius
R', a small quantity, surrounding the point (x^y^z^) and Bf
must be allowed to approach zero as its limit.
As it has been assumed that the potential of V exists, it is
assumed that the conditions given (Art. 87) for the existence
of the potential hold. That is
< -fiT, when r is large
Vr < K, when r is small.
Introduce polar coordinates with the origin at the point
(#i> #i» «i)« Then r12 becomes simply r
and Vx — = - Va — = -^ *
l'ii '12 **
Then for the large sphere of radius R
1 r
V, — . da = — r r2sm0 d0 dd>.
*3
!2
4*3
Hence the surface integral over that sphere approaches zero
as its limit. For
Hence when -R becomes infinite the surface integral over the
large sphere approaches zero as its limit.
For the small sphere
1 r
Vt — • d a = -- 5 r2 sin 0 d 0 d 6.
<r 7*v
r!2
Hence the integral over that sphere becomes
THE INTEGRAL CALCULUS OF VECTORS 233
Let V be supposed to be finite and continuous at the point
(xvl/vzi) which has been selected as origin. Then for the
surface integral V is practically constant and equal to its
value
V (*ii Vv *i)
at the point in question.
sintf d
f fs
- f /Vsi
Hence - sintf d8 d< = -
when the radius Rf of the sphere of integration approaches
zero as its limit. Hence
v> • v°
- ff, rv- •<• =-4 *F <68)1
and V- VPotF=-47rF. (52)
In like manner if W is a vector function which has in
general a definite potential, then that potential satisfies Pois-
son's Equation.
V • V Pot W = - 4 TT W. (52)'
The proof of this consists in resolving W into its three com
ponents. For each component the equation holds. Let
v-
V. VPotF=-47r F,
V • V Pot Z = — 4 TT Z.
Consequently
V • V Pot (JTi + Fj + Zk) = - 4 TT (JTi + Fj +
234 VECTOR ANALYSIS
Theorem : If V and W are such functions of position in space
that their potentials exist in general, then for all points at which
V and W are finite and continuous those potentials satisfy
Poisson's Equation,
V- VPot r=-4irF; (52)
V • V Pot W = - 4 TT W. (52)'
The modifications in this theorem which are to be made at
points at which V and W become discontinuous will not be
taken up here.
93.] It was seen (46) Art. 91 that
V • VPotF = V- NewT=Max VF1
Hence V • New V = - 4 TT V (53)
or Max VF=-47rF.
In a similar manner it was seen (51) Art. 91 that
V • V Pot W = V Max W — V x Lap W
= New V • W — Lap V x W.
Hence V Max W - V x Lap W = - 4 TT W, (54)
or New V . W - Lap V x W = - 4-rr W. (54)'
By virtue of this equality W is divided into two parts.
W = -7— Lap V x W — 7— New V- W. (55)
4-7T 4?T
Let W = W! + W2,
where W t = -r— Lap V x W = - — Lap curl W (56)
4-rr 4?r
-: — NewV- W = — 7
4-7T 4-7T
and W« = — -: — NewV- W = — 7— Newdiv W. (57)
- -
THE INTEGRAL CALCULUS OF VECTORS 235
Equation (55) states that any vector function W multiplied
by 4 TT is equal to the difference of the Laplacian of its curl
and the Newtonian of its divergence. Furthermore
V • W, = — V • Lap V x W = -7— V- V x Lap Wr
4?r 4-7T
But the divergence of the curl of a vector function is zero.
Hence V.W1 = divW1 = 0 (58)
V x W2 = - j— Vx New V • W2 = - — VxV Max W2.
But the curl of the derivative of a scalar function is zero.
Hence V x W2 = curl W2 = 0. (59)
Consequently any vector function W which has a potential
may be divided into two parts of which one has no divergence
and of which the other has no curl. This division of W into
two such parts is unique.
In case a vector function has no potential but both its curl
and divergence possess potentials, the vector function may be
divided into three parts of which the first has no divergence ;
the second, no curl; the third, neither divergence nor curl.
Let W = — Lap V x W - — New V • W + W«. (55)'
4 7T 4 7T
As before
— V • Lap V x W = T— V • V x Pot V x W = 0
4?r 4-7T
— 1 —1
and -— V x New V • W — - — VxV Pot V • W = 0.
4-7T 4 7T
The divergence of the first part and the curl of the second
part of W are therefore zero.
236 VECTOR ANALYSIS
— V x Lap VxW = — VxVxPotVxW
4?r 4-7T
= — VV • Pot V x W - ~ V • V Pot V X W.
4 7T 4?T
-- VV • P ot V x W = -- V P o t V • V x W = 0 ,
4?r
for V • V x W = 0.
Hence ^— V • VPot V x W = V x W.
4-7T
Hence — V x Lap VxW = VxW = VxWl.
4?r
The curl of W is equal to the curl of the first part
-r— Lap V x W
47T
into which W is divided. Hence as the second part has no
curl, the third part can have none. Moreover
-— V • New V • W = V • W — V • W2.
T: 7T
Thus the divergence of W is equal to the divergence of
the second part
- — New V • W.
4?r
into which W is divided. Hence as the first part has no
divergence the third can have none. Consequently the third
part W3 has neither curl nor divergence. This proves the
statement.
By means of Art. 96 it may be seen that any function W3
which possesses neither curl nor divergence, must either
THE INTEGRAL CALCULUS OF VECTORS 237
vanish throughout all space or must not become zero at
infinity. In physics functions generally vanish at infinity.
Hence functions which represent actual phenomena may be
divided into two parts, of which one has no divergence and
the other no curl.
94.] Definition : A vector function the divergence of which
vanishes at every point of space is said to be solenoidal. A
vector function the curl of which vanishes at every point of
space is said to be irrotational.
In general a vector function is neither solenoidal nor irrota
tional. But it has been shown that any vector function which
possesses a potential may be divided in one and only one
way into two parts Wv W2 of which one is solenoidal and
the other irrotational. The following theorems may be stated.
They have all been proved in the foregoing sections.
With respect to a solenoidal function Wv the operators
— Lap and V X or curl
4?r
are inverse operators. That is
— Lap V x Wi = V x -j— Lap Wi = Wr (60)
4?r 4-rr
Applied to an irrotational function W2 either of these opera
tors gives zero. That is
— Lap W2 = 0 , V x W2 = 0. (61)
With respect to an irrotational function W2, the operators
- — New and — v • or — div
4?r
are inverse operators. That is
_ _L New V • W2 = - V • -i- New W2 = W2. (62)
4 7T 4 7T
238 VECTOR ANALYSIS
With respect to a scalar function V the operators
— V • or — div and - — New,
4-7T
and also — -= — Max and V
4?r
are inverse operators. That is
-V.-i NewF= V (63)
4 7T
and ~--Max VF= V.
4?r
TFttA respect to a solenoidal function Wx the operators
- — Pot and V x V x or curl curl
47T
are inverse operators. That is
— Pot V x V x Wx = V x V x — Pot Wi = Wr (64)
4?r 4?r
With respect to an irrotational function W2 the operators
— Pot and — VV •
4?r
are inverse operators. That is
_ _L Pot VV . W2 = - VV . -L Pot W2 = W2. (65)
With respect to any scalar or vector function V, W the
operators
— Pot and — V • V
4-7T
are inverse operator*. That is
THE INTEGRAL CALCULUS OF VECTORS 239
_ JL Pot v • v v= - v • v -i- Pot F= v
4?r 4?r
and - ,— Pot V • V W = - V • V -^- Pot W = W. (66)
4?r 4?r
With respect to a solenoidal function Wx the differentiating
operators of the second order
— V • V and V X V x
are equivalent
- V • V Wx = V x V x Wr (67)
With respect to an irrotational function W2 the differentiat
ing operators of the second order
V • V and V V •
are equivalent That is
V- VW2 = V V-W2. (68)
By integrating the equations
4^^=- V.NewF
and 4 TT W = V x Lap W - V Max W
by means of the potential integral Pot
4<7rPotF=:-Pot V • New F= - Max New F (69)
4 TT Pot W = Pot V x Lap W - Pot V Max W
4 TT Pot W = Lap Lap W - New Max W. (70)
Hence for scalar functions and irrotational vector functions
— - — New Max
4«7T
is an operator which is equivalent to Pot. For solenoidal vector
functions the operator ^
— - — Lap Lap
240 VECTOR ANALYSIS
gives the potential. For any vector function the first operator
gives the potential of the irrotational part; the second^ the
potential of the solenoidal part.
*95.] There are a number of double volume integrals which
are of such frequent occurrence in mathematical physics as
to merit a passing mention, although the theory of them will
not be developed to any considerable extent. These double
integrals are all scalar quantities. They are not scalar func
tions of position in space. They have but a single value.
The integrations in the expressions may be considered for
convenience as extended over all space. The functions by
vanishing identically outside of certain finite limits deter
mine for all practical purposes the limits of integration in
case they are finite.
Given two scalar functions Z7, V of position in space.
The mutual potential or potential product, as it may be called,
of the two functions is the sextuple integral
Pot
(71)
One of the integrations may be performed
,yi,^) PotVdv,
(*2' y* *2> Pot Ud **• (T2)
In a similar manner the mutual potential or potential product
of two vector functions W, W" is
(71)'
This is also a scalar quantity. One integration may be car
ried out
THE INTEGRAL CALCULUS OF VECTORS 241
Pot (W, W") =w'(xv yv «,) . Pot W" dvt
The mutual Laplacian or Laplacian product of two
vector functions W ', W" of position in space is the sextuple
integral
Lap(W',W")
=ffffffw' (*!' yi' *i) • ;nr x w" (** y* *«) <*»i <*"2-
(73)
One integration may be performed.
Lap (W, W") = f ( f W" (^2, ya, *2) • Lap W rf va
(T4)
v y i» *i) • LaP w" d »r
The Newtonian product of a scalar function F, and a vector
function W of position in space is the sextuple integral
rf*2.
(75)
By performing one integration
New ( F, W) =///W (*2, y2, *2) • New Frf t,a. (76)
In like manner the Maxwellian product of a vector function
W and a scalar function F of position in space is the
integral
Max (W,F) =/////JV(*i^*i) J- ' W0r2,2/2,*2)rf W
(77)
16
242 VECTOR ANALYSIS
One integration yields
Max (W, F) =fff V(xv yv zj Max W d v 1 = - New ( F, W).
(78)
By (53) Art. 93.
4?r UPotr = - (V • New CO PotF.
V • [New U Pot F] = (V • New V) Pot F + (New IT) • V Pot F.
-(V.New£OPotF=-V.[NewPTotF]+NewtT.NewF
Integrate :
47r f f |VpotFdi>=- f f fv. [Ne
+ C f CtfewU- NewFdv.
4-Tr Pot IT, F)= f New£T. NewFdv
, F)= f f f
- T f
Pot F New Z7« rf a. (79)
The surface integral is to be taken over the entire surface S
bounding the region of integration of the volume integral.
As this region of integration is " all space," the surface S may
be looked upon as the surface of a large sphere of radius R.
If the functions U and F vanish identically for all points out
side of certain finite limits, the surface integral must vanish.
Hence
4 TT Pot ( U, F) = f f fNew U • New Vd v. (79)'
By (54) Art. 93,
47rW". PotW' = V x Lap W" • Pot W
- V Max W" • Pot W'.
THE INTEGRAL CALCULUS OF VECTORS 243
But V . [Lap W" x Pot W'] = Pot W' • V x Lap W"
- Lap W" • V x Pot W',
and V • [Max W" Pot W] = Pot W • V Max W"
+ Max W" V • Pot W.
Hence V x Lap W" • Pot W = V • [Lap W" x Pot W]
+ Lap W" • Lap W',
and V Max W" • Pot W = V - [Max W" Pot W]
- Max W" Max W'.
Hence substituting:
4 ?r W" • Pot W = Lap W' • Lap W' + Max W' Max W"
+ V • [Lap W" X Pot W']
-V • [Max W" Pot W'].
Integrating .-
4 TT Pot (W, W") = ff f Lap W'« Lap W" dv
r c r
+ / / I Max W' Max W d v (80)
J J J
— I I PotW' x Lap W" da— / / Max W"PotWWa.
If now W' and W" exist only in finite space these surface
integrals taken over a large sphere of radius B must vanish
and then
4 TT Pot (W, W") = f f fLap W' • Lap W" d v
+ 11 fMax W' Max W" d v. (80)'
J J J
* 96.] There are a number of useful theorems of a function-
theoretic nature which may perhaps be mentioned here owing
244 VECTOR ANALYSIS
to their intimate connection with the integral calculus of
vectors. The proofs of them will in some instances be given
and in some not. The theorems are often useful in practical
applications of vector analysis to physics as well as in purely
mathematical work.
Theorem : If V (#, y, z) be a scalar function of position
in space which possesses in general a definite derivative V V
and if in any portion of space, finite or infinite but necessarily
continuous, that derivative vanishes, then the function V is
constant throughout that portion of space.
Given VF=0.
To show F= const.
Choose a fixed point (#15 yv zj in the region. By (2) page
180
y> * V F. d r = V(x, y,z)-V (xv yv zj.
u ft* *i
But fvr.dr =f<) . dr = 0.
Hence F(#, y, z) = V (xv yv zj = const.
Theorem : If F" (#, y, 2;) be a scalar function of position
in space which possesses in general a definite derivative V V ;
if the divergence of that derivative exists and is zero through
out any region of space,1 finite or infinite but necessarily
continuous ; and if furthermore the derivative V V vanishes
at every point of any finite volume or of any finite portion of
surface in that region or bounding it, then the derivative
vanishes throughout all that region and the function V re
duces to a constant by the preceding theorem.
1 The term throughout any region of space must be regarded as including the
boundaries of the region as well as the region itself.
THE INTEGRAL CALCULUS OF VECTORS 245
Given V • V V= 0 for a region T,
and V F= 0 for a finite portion of surface S.
To show J^= const.
Since V Evanishes for the portion of surface S, Vis certainly
constant in S. Suppose that, upon one side of S and in the
region T, V were not constant. The derivative V V upon
this side of S has in the main the direction of the normal to
the surface S. Consider a sphere which lies for the most
part upon the outer side of S but which projects a little
through the surface S. The surface integral of VF over
the small portion of the sphere which projects through the
surface S cannot be zero. For, as V V is in the main normal
to S9 it must be nearly parallel to the normal to the portion
of spherical surface under consideration. Hence the terms
VT- da,
in the surface integral all have the same sign and cannot
cancel each other out. The surface integral of V V over
that portion of S which is intercepted by the spherical sur
face vanishes because V V is zero. Consequently the surface
integral of V V taken over the entire surface of the spherical
segment which projects through S is not zero.
But f r vr-da= f r fv. vrd*=o.
Hence f /Vr« da = 0.
It therefore appears that the supposition that V is not
constant upon one side of S leads to results which contradict
the given relation V • V V — 0. The supposition must there
fore have been incorrect and V must be constant not only in
S but in all portions of space near to $ in the region T. By
246 VECTOR ANALYSIS
an extension of the reasoning V is seen to be constant
throughout the entire region T.
Theorem : If V (x, y, z) be a scalar function of position in
space possessing in general a derivative V V and if through
out a certain region 1 T of space, finite or infinite, continuous
or discontinuous, the divergence V • V V of that derivative
exists and is zero, and if furthermore the function V possesses
a constant value c in all the surfaces bounding the region
and V (x, y, z) approaches c as a limit when the point (x, y, z)
recedes to infinity, then throughout the entire region T the
function V has the same constant value c and the derivative
W vanishes.
The proof does not differ essentially from the one given
in the case of the last theorem. The theorem may be gen
eralized as follows :
Theorem: If V(x,y, z) be any scalar function of position
in space possessing in general a derivative W; if U (x, y, z)
be any other scalar function of position which is either posi
tive or negative throughout and upon the boundaries of a
region T, finite or infinite, continuous or discontinuous; if
the divergence V • [ U V V~\ of the product of U and V V
exists and is zero throughout and upon the boundaries of T
and at infinity ; and if furthermore V be constant and equal
to c upon all the boundaries of T and at infinity ; then the
function V is constant throughout the entire region T and
is equal to c.
Theorem : If V (#, y, z) be any scalar function of position
in space possessing in general a derivative V V ; if through
out any region T of space, finite or infinite, continuous or
discontinuous, the divergence V • V V of this derivative exists
and is zero ; and if in all the bounding surfaces of the region
T the normal component of the derivative VF" vanishes and
at infinite distances in T (if such there be) the product
1 The region includes its boundaries.
THE INTEGRAL CALCULUS OF VECTORS 247
r2 9 Vj 3 r vanishes, where r denotes the distance measured
from any fixed origin ; then throughout the entire region T
the derivative V Evanishes and in each continuous portion
of T V is constant, although for different continuous portions
this constant may not be the same.
This theorem may be generalized as the preceding one
was by the substitution of the relation V • ( U V F) = 0 for
V-VF=Oand Ur*3V/3r = 0 for r^SV/Sr = 0.
As corollaries of the foregoing theorems the following
statements may be made. The language is not so precise
as in the theorems themselves, but will perhaps be under
stood when they are borne in mind.
If V U = V V, then U and V differ at most by a
constant.
If V-V£7=V.VF and if VZ7 = VF in any finite
portion of surface S, then V U = V V at all points and V
differs from V only by a constant at most.
If V.VJ7= V- VF and if V= V in all the bounding
surfaces of the region and at infinity (if the region extend
thereto), then at all points £7 and Fare equal.
If V • V £7 = V • V F and if in all the bounding surfaces
of the region the normal components of VZ7 and VFare
equal and if at infinite distances r2(3 U/Sr — 9 F/5r) is
zero, then V ?7and V Fare equal at all points of the region
and U differs from F only by a constant.
Theorem : If W' and W" are two vector functions of position
in space which in general possess curls and divergences ; if
for any region I7, finite or infinite but necessarily continuous,
the curl of W' is equal to the curl of W" and the divergence
of W' is equal to the divergence of W"; and if moreover
the two functions W' and W" are equal to each other at
every point of any finite volume in T or of any finite surface
in Tor bounding it; then W' is equal to W" at every point
of the region T.
248 VECTOR ANALYSIS
Since V x W' = V x W", V x (W' - W") = 0. A vec
tor function whose curl vanishes is equal to the derivative *
of a scalar function V (page 197). Let VF=W' — W".
Then V • V V= 0 owing to the equality of the divergences.
The theorem therefore becomes a corollary of a preceding one.
Theorem : If W' and W" are two vector functions of posi
tion which in general possess definite curls and divergences ;
if throughout any aperiphractic* region T, finite but not
necessarily continuous, the curl of W' is equal to the curl of
W" and the divergence of W' is equal to the divergence of
W"; and if furthermore in all the bounding surfaces of the
region T the tangential components W7 and W" are equal;
then W; is equal to W" throughout the aperiphractic region T.
Theorem: If W' and W" are two vector functions of posi
tion in space which in general possess definite curls and
divergences ; if throughout any acyclic region T, finite but not
necessarily continuous, the curl of W' is equal to the curl
W" and the divergence of W' is equal to the divergence of
W"; and if in all the bounding surfaces of the region T the
normal components of W' and W" are equal ; then the func
tions W' and W" are equal throughout the region acyclic T.
The proofs of these two theorems are carried out by means
of the device suggested before.
Theorem: If W' and W" are two vector functions such
that V • V W' and V • V W" have in general definite values
in a certain region T, finite or infinite, continuous or discon
tinuous ; and if in all the bounding surfaces of the region
and at infinity the functions W' and W" are equal ; then W'
is equal to W" throughout the entire region T.
The proof is given by treating separately the three com
ponents of W'and W".
1 The region T may have to be made acyclic by the insertion of diaphragms.
2 A region which encloses within itself another region is said to be periphrac-
tic. If it encloses no region it is aperiphractic.
THE INTEGRAL CALCULUS OF VECTORS 249
SUMMARY OF CHAPTER IV
The line integral of a vector function W along a curve C is
defined as
f Wdr=f [Widx + W^dy + W.dz]. (1)
J c J c
The line integral of the derivative V V of a scalar function
V along a curve C from r0 to r is equal to the difference
between the values of V at the points r and r0 and hence the
line integral taken around a closed curve is zero ; and con
versely if the line integral of a vector function W taken
around any closed curve vanishes, then W is the derivative
V V of some scalar function V.
fri
«/ TO
f
J
(2)
(3)
and if C W • dr = 0, then W = VF.
Jo
Illustration of the theorem by application to mechanics.
The surface integral of a vector function W over a surface
S is defined as
= ff
Gauss's Theorem| : The surface integral of a vector func-
tiorTtaken over a closed surface is equal to the volume
integral of the divergence of that function taken throughout
the volume enclosed by that surface
250 VECTOR ANALYSIS
= f f {Xdydz+ Ydz dx + Zdxdy], (8)
if X, I7, Z be the three components of the vector function W.
Stokes's Theorem: The surface integral of the curl of a
vector function taken over any surface is equal to the line
integral of the function taken around the line bounding the
surface. And conversely if the surface integral of a vector
function TJ taken over any surface is equal to the line integral
of a function W taken around the boundary, then U is the
curl of W.
//,VxW.*.=/oW.*r, (11)
and if ffjj • da =f W • rfr, then TI = V x W. (12)
Application of the theorem of Stokes to deducing the
equations of the electro-magnetic field from two experimental
facts due to Faraday. Application of the theorems of Stokes
and Gauss to the proof that the divergence of the curl of
a vector function is zero and the curl of the derivative of
a scalar function is zero.
Formulae analogous to integration by parts
I w V v • di = \u v~]T — / v V u • rf r, (14)
f f r r r
J J 8 t/O J J S
cc ^ c r
I t vwXv/y«aa=ioi6v'y«ar = — I v V u • a r, (16)
*/ */ 8 */° t/O
THE INTEGRAL CALCULUS OF VECTORS 251
I C CuV *vdv = I I uv • d&- ii f V u*vdv, (17)
/Y vu x v . da=- r r A/!* . v x * dv. (i8>
Green's Theorerii:
/ / I V u • V v dv = / f ttV-y -da — f T TwV -V v dv
= if v V ^ • d a — I I I vV *V udv, (19)
.Vi;-i;V.Vw)rfi;= f C (uVv — t?Vw).rfa. (20)
__ — -
Kelvin's generalization:
i I Tw^7u^vdv= I I w^Vv-rfa— // / ^
= / / i? w V i^ • da — T T TV V • [w V w] rf v. (21)
The integrating operator known as the potential is defined
by the equation
Pot r= V(xyvZ^ dxt dy2 dzy (22)
Pot w =*?yy* ^^2 ^y2 ^^- (23)
VPot T=PotVF; (27)
V x Pot W = Pot V x W, (28)
V • Pot W = Pot V • W, (29)
V • V Pot F= Pot V • VF, (30)
252 VECTOR ANALYSIS
V • V Pot W = Pot V • V W, (31)
VV • Pot W = Pot VV • W, (32)
V x V x Pot W = Pot V x V x W. (33)
The integrating operator Pot and the differentiating operator
V are commutative.
The three additional integrating operators known as the
Newtonian, the Laplacian, and the Maxwellian.
__ __ f f *19 ' \€Ay*» fO? ^9 / •• •» ^ /• i **•*.
New T= / / / -^ — ^- ^-^- rf^2 dy2 dz2. (42)
f* f* f* Y ^ \XT ^7* ?y 2 ^
Lap W = I I / — o 2' 2' 2 ^^2 ^2/2 dzv (43)
j J J r°12
Max W= I I I — q 2> 2* 2 rf^2 rfy« d^2.
If the potential exists these integrals are related to it as fol
lows:
V Pot F= New V,
V x Pot W = Lap W, (45)
The interpretation of the physical meaning of the Newtonian
on the assumption that V is the density of an attracting
body, of the Laplacian on the assumption that W is electric
flux, of the Maxwellian on the assumption that W is the
intensity of magnetization. The expression of these integrals
or their components in terms of a?, y, % ; formulae (42)', (43)',
(44)' and (42)", (43)", (44)".
V • New F= Max V F, (46)
V Max W = New V • W, (47)
V x Lap W = Lap V x W, (48)
THE INTEGRAL CALCULUS OF VECTORS 253
V • Lap W = Max V x W = 0, (49)
V x New V = Lap V V= 0, (50)
V • V Pot W = New V ' W — Lap V X W
= V Max W - V x Lap W. (51)
The potential is a solution of Poisson's Equation. That is,
V. VPotF = -47rF; (52)
and V. VPotF=-47rW. (52)'
F= V.NewF, (53)
W = -A — Lap V x W — - — New V • W. (55)
4-7T 4 7T
Hence W is divided into two parts of which one is
solenoidal and the other irrotational, provided the potential
exists. In case the potential does not exist a third term W3
must be added of which both the divergence and the curl
vanish. A list of theorems which follow immediately from
equations (52), (52)', (53), (55) and which state that certain
integrating operators are inverse to certain differentiating
operators. Let V be a scalar function, Wx a solenoidal vector
function, and W2 an irrotational vector function. Then
— Lap V x Wj = V x — Lap Wl = Wr (60)
47T 4 7T
— Lap W2 = 0, V x W2 = 0 (61)
4 7T
--— New V. W, = - V •— New W2 = W2. (62)
47T 4?T
254 VECTOR ANALYSIS
f_V- -A -NewF = V
l (63)
- — Max VF= V.
4 7T
— - Pot V x V x Wt = V x V x Pot Wx = W, (64)
47T
--1- Pot VV • W2 = - V • V - - Pot W2 = W2. (65)
4 7T 4 7T
(66)
[--- Pot v. vr=- v-v -
4?r 4?r
1 !
L- -r - Pot V . V W = - V • V — Pot W = W.
4?r 4?r
-V-VWj^VxVxWj (67)
V • V W2 = VV . W2 (68)
4 TT Pot V = - Max New V (69)
4 TT Pot W = Lap Lap W - New Max W. (70)
Mutual potentials Newtonians, Laplacians, and Maxwellians
may be formed. They are sextuple integrals. The integra
tions cannot all be performed immediately ; but the first three
may be. Formulae (71) to (80) inclusive deal with these inte
grals. The chapter closes with the enunciation of a number
of theorems of a function-theoretic nature. By means of
these theorems certain facts concerning functions may be
inferred from the conditions that they satisfy Laplace's equa
tion and have certain boundary conditions.
Among the exercises number 6 is worthy of especial atten
tion. The work done in the text has for the most part assumed
that the potential exists. But many of the formulce connecting
Newtonians, Laplacians, and Maxwellians hold when the poten
tial does not exist. These are taken up in Exercise 6 referred to.
THE INTEGRAL CALCULUS OF VECTORS 255
EXERCISES ON CHAPTER IV
I.1 If V is a scalar function of position in space the line
integral
is a vector quantity. Show that
That is ; the line integral of a scalar function around a
closed curve is equal to the skew surface integral of the deriv
ative of the function taken over any surface spanned into
the contour of the curve. Show further that if V is constant
the integral around any closed curve is zero and conversely
if the integral around any closed curve is zero the function V
is constant.
Hint : Instead of treating the integral as it stands multiply
it (with a dot) by an arbitrary constant unit vector and thus
reduce it to the line integral of a vector function.
2. If W is a vector function the line integral
=/w
J c
x dr
is a vector quantity. It may be called the skew line integral
of the function W. If c is any constant vector, show that if
the integral be taken around a closed curve
H • c = / / (cV«W — c«VW)'da = c«/ Wxdr,
1 The first four exercises are taken from Foppl's Einfiihrung in die Max-
well'sche Theorie der Electricitat where they are worked out.
256 VECTOR ANALYSIS
and H-c = c. ]JJ8 V • Wda- J Js V (W • d a) j
In case the integral is taken over a plane curve and the
surface S is the portion of plane included by the curve
Show that the integral taken over a plane curve vanishes
when W is constant and conversely if the integral over any
plane curve vanishes W must be constant.
3. The surface integral of a scalar function V is
This is a vector quantity. Show that the surface integral
of V taken over any closed surface is equal to the volume
integral of W taken throughout the volume bounded by
that surface. That is
Hence conclude that the surface integral over a closed sur
face vanishes if V be constant and conversely if the surface
integral over any closed surface vanishes the function V must
be constant.
4. If W be a vector function, the surface integral
T= f C d&x W
may be called the skew surface integral. It is a vector
quantity. Show that the skew surface integral of a vector
THE INTEGRAL CALCULUS OF VECTORS 257
function taken over a closed surface is equal to the volume
integral of the vector function taken throughout the volume
bounded by the surface. That is
Hence conclude that the skew surface integral taken over
any surface in space vanishes when and only when W is an
irrotational function. That is, when and only when the line
integral of W for every closed circuit vanishes.
5. Obtain some formulae for these integrals which are
analogous to integrating by parts.
6. The work in the text assumes for the most part that the
potentials of Fand W exist. Many of the relations, however,
may be demonstrated without that assumption. Assume that
the Newtonian, the Laplacian, the Maxwellian exist. For
simplicity in writing let
Then New V = Vt Pn V(xv yy *2) d t>2, (81)
Lap W =i^12 X W (xv yv «2) dvv (82)
Max W =fffvipu «W (z2, yv z^dvv (83)
(84)
c c r
-JJJr^rdvr
17
258 VECTOR ANALYSIS
By exercise (3)/// V2 (Pit v) dv*
It can be shown that if V is such a function that New V
exists, then this surface integral taken over a large sphere of
radius R and a small sphere of radius R* approaches zero
when R becomes indefinitely great; and Rf, indefinitely
small. Hence
or NewF=PotVF. (85)
Prove in a similar manner that
Lap W = Pot V x W, (86)
Max W = Pot V • W. (87)
By means of (85), (86), (87) it is possible to prove that
V x Lap W = Lap V x W,
V-New F=Max VF,
V Max W = New V • W.
Then prove
/*/*/* f* f* f*
VxLapW=i / I^12VV-W di?2 — I l f^12V-VWdi
%} <J *J *J J *J
and V Max W = f/JJPii V V • W d vv
Hence V x Lap W - V Max W = -ffffv V ' V W d v
Hence V x Lap W - V Max W = 4 TT W. (88)
7. An integral used by Helmholtz is
THE INTEGRAL CALCULUS OF VECTORS 259
or if W be a vector function
H (W) =/// »« W d "2" <9°)
Show that the integral converges if V diminishes so rapidly
that
K
when r becomes indefinitely great.
Vtf(F) = #(VF) = New(r2 F), (91)
V • # (W) = # (V • W) = Max (r2 W), (92)
V x H (W) = 5" (V x W) = Lap (r2 W), (93)
=Jff (V. VP) = Max(r2VF) = 2 Pot F (94)
. (95)
H ( F) = - -L Pot Pot PI (96)
J 7T
^ (W) = - -?- Pot Pot W. (97)
2 7T
~2W = VxVx^T(W) + VV.^r (W). (98)
8. Give a proof of Gauss's Theorem which does not depend
upon the physical interpretation of a function as the flux of a
fluid. The reasoning is similar to that employed in Art. 51
and in the first proof of Stokes's Theorem.
9. Show that the division of W into two parts, page 235,
is unique.
10. Treat, in a manner analogous to that upon page 220,
the case in which V has curves of discontinuities.
CHAPTER V
LINEAR VECTOR FUNCTIONS
97.] AFTER the definitions of products had been laid down
and applied, two paths of advance were open. One was
differential and integral calculus ; the other, higher algebra
in the sense of the theory of linear homogeneous substitutions.
The treatment of the first of these topics led to new ideas
and new symbols — to the derivative, divergence, curl, scalar
and vector potential, that is, to V, V», Vx, and Pot with the
auxiliaries, the Newtonian, the Laplacian, and the Maxwellian.
The treatment of the second topic will likewise introduce
novelty both in concept and in notation — the linear vector
function, the dyad, and the dyadic with their appropriate
symbolization.
The simplest example of a linear vector function is the
product of a scalar constant and a vector. The vector r'
T' = CT (1)
is a linear function of r. A more general linear function
may be obtained by considering the components of r individ
ually. Let i, j, k be a system of axes. The components of
r are
i • r, j • r, k • r.
Let each of these be multiplied by a scalar constant which
may be different for the different components.
cl i • r, c2 j • r, c3 k • r.
LINEAR VECTOR FUNCTIONS 261
Take these as the components of a new vector r'
r' = i (Cji-^ + j (caj-r) + k (c8k-r). (2)
The vector r' is then a linear function of r. Its components
are always equal to the corresponding components of r each
multiplied by a definite scalar constant.
Such a linear function has numerous applications in geom
etry and physics. If, for instance, i, j, k be the axes of a
homogeneous strain and cv c2, c3, the elongations along these
axes, a point
r = ix + j y + bz
becomes r' = i cl x -f j c2 y + k c3 z,
or r ' = i cl i • r + j <?2 j • r + k c3 k • r.
This sort of linear function occurs in the theory of elasticity
and in hydrodynamics. In the theory of electricity and
magnetism, the electric force E is a linear function of the
electric displacement D in a dielectric. For isotropic bodies
the function becomes merely a constant
But in case the body be non-isotropic, the components of the
force along the different axes will be multiplied by different
constants kv &2, &3. Thus
E = i%1i*D + j£2j .D + k&gk-D.
The linear vector function is indispensable in dealing with
the phenomena of electricity, magnetism, and optics in non-
isotropic bodies.
98.] It is possible to define a linear vector function, as has
been done above, by means of the components of a vector.
The most general definition would be
262 VECTOR ANALYSIS
Definition : A vector r' is said to be a linear vector func
tion of another vector r when the components of r' along
three non-coplanar vectors are expressible linearly with scalar
coefficients in terms of the components of r along those same
vectors.
If r = XB, + yb + zc, where [abc] ^ 0,
and r' = #'a + y'b + z'c,
and if xf = alx + bly + clzf
yr = a^x + 62y + c2z, (3)
zf = azx + lzy + c3z,
then r' is a linear function of r. (The constants a^ lv cv
etc., have no connection with the components of a, b, c par
allel to i, j, k.) Another definition however is found to be
more convenient and from it the foregoing may be deduced.
Definition : A continuous vector function of a vector is
said to be a linear vector function when the function of the
sum of any two vectors is the sum of the functions of those
vectors. That is, the function /is linear if
/(r1 + r2)=/(r1)+/(ra). (4)
Theorem : If a be any positive or negative scalar and if /
be a linear function, then the function of a times r is a times
the function of r.
/0»r) = a/(r), (5)
And hence
/(a1r1 + a2r2 + a3r3 + .-•)
= «i f<Ji) + <**f (ra)+ «8/(*8) + ' ' ' (5)
The proof of this theorem which appears more or less
obvious is a trifle long. It depends upon making repeated
use of relation (4).
LINEAR VECTOR FUNCTIONS 263
Hence /(2r) = 2/(r).
In like manner / (n r) = nf (r)
where n is any positive integer.
Let m be any other positive integer. Then by the relation
just obtained
Hence / (.i) =/( i r )=-?./ (,).
\ w / \ m / m
That is, equation (5) has been proved in case the constant a
is a rational positive number.
To show the relation for negative numbers note that
/(0)=/(0 + 0) = 2/(0).
Hence /(0) = 0.
But /(O) =/(r-r) =/(r+(-r)) =/(r)
Hence r=
To prove (5) for incommensurable values of the constant
a, it becomes necessary to make use of the continuity of the
function /. That is
Let x approach the incommensurable number a by passing
through a suite of commensurable values. Then
Hence *****. + ( xi} = a
x = a J v ' ~
264 VECTOR ANALYSIS
LlM (ar)=ar.
# = a v
Hence /(") = «/ 00
which proves the theorem.
Theorem: A linear vector f unction /(r) is entirely deter
mined when its values for three non-coplanar vectors a, b, c are
known.
Let l=/(a),
m=/(b),
n=/(c).
Since r is any vector whatsoever, it may be expressed as
r = #a + yb + 3C.
Hence / (r) = x 1 + y m + z n.
99.] In Art. 97 a particular case of a linear function was
expressed as
r' = i cl i • r + j c2 j • r + k c3 k • r.
For the sake of brevity and to save repeating the vector r
which occurs in each of these terms in the same way this
may be written in the symbolic form
In like manner if ap a2, a8 • • • be any given vectors, and bp b2,
b3, • • • another set equal in number, the expression
r' = a! bx • r + a2 b2 • r + a3b3 • r + - • • (6)
is a linear vector function of r ; for owing to the distributive
character of the scalar product this function of r satisfies
relation (4). For the sake of brevity r' may be written sym
bolically in the form
r' = (ai bx + aa b2 + a3 b3 + ••.)• r. (6)'
LINEAR VECTOR FUNCTIONS 265
No particular physical or geometrical significance is to be
attributed at present to the expression
(a^ + a^ + agbg + .••)• (7)
It should be regarded as an operator or symbol which -con
verts the vector r into the vector r' and which merely
affords a convenient and quick way of writing the relation
(6).
Definition : An expression a b formed by the juxtaposition
of two vectors without the intervention of a dot or a cross is
called a dyad. The symbolic sum of two dyads is called a
dyadic binomial ; of three, a dyadic trinomial ; of any num
ber, a dyadic polynomial. For the sake of brevity dyadic
binomials, trinomials, and polynomials will be called simply
dyadics. The first vector in a dyad is called the antecedent ;
and the second vector, the consequent. The antecedents of a
dyadic are the vectors which are the antecedents of the
individual dyads of which the dyadic is composed. In like
manner the consequents of a dyadic are the consequents of
the individual dyads. Thus in the dyadic (7) ap a2, a3 • • • are
the antecedents and br b2, b3 • • - the consequents.
Dyadics will be represented symbolically by the capital
Greek letters. When only one dyadic is present the letter
0 will generally be used. In case several are under consid
eration other Greek capitals will be employed also. With
this notation (7) becomes
and (6)' may now be written briefly in the form
r' = d> • r. (8)
By definition 0 • r = aj bx • r + a2 b2 • r + a3 b3 • r +
The symbol <P-r is read 0 dot r. It is called the direct
product of 0 into r because the consequents bj, b2, b3 • - • are
266 VECTOR ANALYSIS
multiplied into r by direct or scalar multiplication. The
order of the factors 0 and r is important. The direct
product of r into 0 is
r • <P = r . (aa ^ + a2 b2 + a3 b 3 + • . • )
= r . ax bj + r • a2 b2 + r - a3 b3 + . . . (9)
Evidently the vectors 0 • r and r • 0 are in general different.
Definition : When the dyadic 0 is multiplied into r as 0 • r,
0 is said to be a pref actor to r. When r is multiplied in 0 as
r • <#, 0 is said to be a post/actor to r.
A dyadic 0 used either as a pref actor or as a postf actor to a
vector r determines a linear vector function of r. The two linear
vector functions thus obtained are in general different from
one another. They are called conjugate linear vector func
tions. The two dyadics
^ajbj + ajbg + agbg + ...
and ¥ = bx ax + b2 a2 + b3 a3 + • • • ,
each of which may be obtained from the other by inter
changing the antecedents and consequents, are called conjitr
gate dyadics. The fact that one dyadic is the conjugate of
another is denoted by affixing a subscript C to either.
Thus ¥ = 0C 0 = ¥c.
Theorem: A dyadic used as a postf actor gives the same
result as its conjugate used as a prefactor. That is
r • 0 = 0C • r. (9)
100.] Definition : Any two dyadics 0 and W are said to
be equal
when 0 • r = W • r for all values of r,
or when r • 0 = r • W for all values of r, (10)
or when B • 0 • r = B • W • r for all values of s and r.
LINEAR VECTOR FUNCTIONS 267
The third relation is equivalent to the first. For, if the
vectors 0 • r and W • r are equal, the scalar products of any
vector s into them must be equal. And conversely if the
scalar product of any and every vector s into the vectors 0 • r
and ¥*T are equal, then those vectors must be equal. In
like manner it may be shown that the third relation is equiva
lent to the second. Hence all three are equivalent.
Theorem : A dyadic 0 is completely determined when the
values 0.a, 0.b, 0.c,
where a, b, c are any three non-coplanar vectors, are known.
This follows immediately from the fact that a dyadic defines
a linear vector function. If
0 . r = 0.(#a + 2/b + zc)==#'a + ?/*b-M»c,
Consequently two dyadics 0 and W are equal provided equa
tions (10) hold for three non-coplanar vectors r and three
non-coplanar vectors s.
Theorem : Any linear vector function / may be represented
by a dyadic 0 to be used as a prefactor and by a dyadic ¥,
which is the conjugate of 0, to be used as a postfactor.
The linear vector function is completely determined when
its values for three non-coplanar vectors (say i, j, k) are
known (page 264). Let
/
Then the linear function / is equivalent to the dyadic
to be used as a postfactor; and to the dyadic ¥
¥= <P(7 = ia + jb + kc,
to be used as a prefactor.
268 VECTOR ANALYSIS
The study of linear vector functions therefore is identical
with the study of dyadics.
Definition : A dyad a b is said to be multiplied by a scalar
a when the antecedent or the consequent is multiplied by
that scalar, or when a is distributed in any manner between
the antecedent and the consequent. If a = a' a11
a (ab) = (a a) b = a (a b) = (a' a) (a" b).
A dyadic 0 is said to be multiplied by the scalar a when
each of its dyads is multiplied by that scalar. The product
is written
a 0 or <Pa.
The dyadic a $ applied to a vector r either as a prefactor or
as a postfactor yields a vector equal to a times the vector
obtained by applying 0 to r — that is
(a 0) • r = a (0 • r).
Theorem : The combination of vectors in a dyad is distrib
utive. That is
(a + b) c = a c + b c ...
and a (b + c) = ab + ac.
This follows immediately from the definition of equality of
dyadics (10). For
[(a + b) c] • r = (a + b) c • r = a c • r + b c • r = (a c + b c) • r
and
[a(b + c)] «r = a (b + c) «r = ab -r + ac- r = (ab + ac) • r.
Hence it follows that a dyad which consists of two factors,
each of which is the sum of a number of vectors, may be
multiplied out according to the law of ordinary algebra
— except that the order of the factors in the dyads must be
maintained.
LINEAR VECTOR FUNCTIONS 269
bn+ ... (11)'
+ cl-f cm-f cn+ • • •
The dyad therefore appears as a product of the two vectors of
which it is composed, inasmuch as it obeys the characteris
tic law of products — the distributive law. This is a justifi
cation for writing a dyad with the antecedent and conse
quent in juxtaposition as is customary in the case of products
in ordinary algebra.
The N onion Form of a Dyadic
10L] From the three unit vectors i, j, k nine dyads may
be obtained by combining two at a time. These are
ii, ij, ik,
ji, jj, jk, (12)
ki, kj, kk.
If all the antecedents and consequents in a dyadic 0 be ex
pressed in terms of i, j, k, and if the resulting expression be
simplified by performing the multiplications according to the
distributive law (11)' and if the terms be collected, the dyadic
0 may be reduced to the sum of nine dyads each of which is
a scalar multiple of one of the nine fundamental dyads given
above.
0 = an ii + a12ij + a13ik
+ «2iJi +a22 jj + a23jk (13)
+ a31ki + a32kj + a33kk.
This is called the nonion form of 0.
Theorem : The necessary and sufficient condition that two
dyadics 4> and W be equal is that, when expressed in nonion
270 VECTOR ANALYSIS
form, the scalar coefficients of the corresponding dyads be
equal.
If the coefficients be equal, then obviously
<P. r= W . r
for any value of r and the dyadics by (10) must be equal.
Conversely, if the dyadics 0 and W are equal, then by (10)
s • 0 • r = s • W • r
for all values of s and r. Let s and r each take on the values
i,j,k. Then (14)
i . d> . i = i - ¥ • i, i . 0 . j = i . W • j, i • 0 • k = i • iT • k
j . 0.i=j. W.i, j. </».j =j. ¥. j, j. <P.k = j. ?T.k
k. <P-i = k. ?F.i, k- 0- j = k- ?F.j, k • 0- k = k • r.k.
But these quantities are precisely the nine coefficients in the
expansion of the dyadics 0 and W. Hence the corresponding
coefficients are equal and the theorem is proved.1 This
analytic statement of the equality of two dyadics can some
times be used to greater advantage than the more fundamental
definition (10) based upon the conception of the dyadic as
defining a linear vector function.
Theorem : A dyadic 0 may be expressed as the sum of nine
dyads of which the antecedents are any three given non-
coplanar vectors, a, b, c and the consequents any three given
non-coplanar vectors 1, m, n.
Every antecedent may be expressed in terms of a, b, c ;
and every consequent, in terms of 1, m, n. The dyadic may
then be reduced to the form
0 = an al + &12 am + a13 an
+ a21 bl + «22 bin + a23 bn (15)
-f- fflai c 1 + a32 c m + ^33 c n.
1 As a corollary of the theorem it is evident that the nine dyads (12) are in
dependent. None of them may be expressed linearly in terms of the others.
LINEAR VECTOR FUNCTIONS 271
This expression of <P is more general than that given in
(13). It reduces to that expression when each set of vectors
a, b, c and 1, m, n coincides with i, j, k.
Theorem : Any dyadic <# may be reduced to the sum of
three dyads of which either the antecedents or the consequents,
but not both, may be arbitrarily chosen provided they be non-
coplanar.
Let it be required to express 4> as the sum of three dyads
of which a, b, c are the antecedents. Let 1, m, n be any other
three non-coplanar vectors. 0 may then be expressed as in
(15). Hence
0 = a (an 1 + «12 m + a13 n) + b (a21 1 + »22 m + «23 n)
+ c Osi 1 + «32 m + a32 n),
or <P = aA + bB + cC. (16)
In like manner if it be required to express $ as the sum of
three dyads of which the three non-coplanar vectors 1, m, n are
the consequents
0 = Ll + Mm + Nn, (16)'
where L = an a + a2l b + a31 c,
M = a12a + «22b + a32 c>
N = alB a + a23 b + aZB c.
The expressions (15), (16), (16)' for 0 are unique. Two equal
dyadics which have the same three non-coplanar ante
cedents, a, b, c, have the same consequents A, B, C - - these
however need not be non-coplanar. And two equal dyadics
which have the same three non-coplanar consequents 1, m, n,
have the same three antecedents.
102. ] Definition: The symbolic product formed by the juxta
position of two vectors a, b without the intervention of a dot
or a cross is called the indeterminate product of the two vectors
a and b.
272 VECTOR ANALYSIS
The reason for the term indeterminate is this. The two
products a • b and a x b have definite meanings. One is a
certain scalar, the other a certain vector. On the other hand
the product ab is neither vector nor scalar — it is purely
symbolic and acquires a determinate physical meaning only
when used as an operator. The product a b does not obey
the commutative law. It does however obey the distributive
law (11) and the associative law as far as scalar multiplication
is concerned (Art 100).
TJieorem : The indeterminate product a b of two vectors is
the most general product in which scalar multiplication is
associative.
The most general product conceivable ought to have the
property that when the product is known the two factors are
also known. Certainly no product could be more general.
Inasmuch as scalar multiplication is to be associative, that is
a (ab) = (a a) b = a (a b) = (a* a) (a"b),
it will be impossible to completely determine the vectors a
and b when their product a b is given. Any scalar factor
may be transferred from one vector to the other. Apart from
this possible transference of a scalar factor, the vectors com
posing the product are known when the product is known. In
other words —
Theorem : If the two indeterminate products a b and a' b'
are equal, the vectors a and a', b and b' must be collinear and
the product of the lengths of a and b (taking into account the
positive or negative sign according as a and b have respec
tively equal or opposite directions to a' and b') is equal to the
product of the lengths of a' and b'.
Let a = al i + &2 j + a3 k,
b = ll i + &2 j + 63 k,
LINEAR VECTOR FUNCTIONS 273
a' = a1' i + a2' j + a3' k,
v = Vi + yj + &,'*•
Then &b = a1b1 ii + a^^ ij + a1b3 ik
a2&3 jj + a263 jk
a,&, kj + a363 kk.
and a'V = «»,'&/ ii + a/V ij + aj'63' ik
+ <V ji + «,'&,' jj + a263 jk
+ o8'6j' ki + o,' 6t' kj + a3'&3' kk.
Since ab = a'b' corresponding coefficients are equal. Hence
a1:a2:as = a1':a2':a3',
which shows that the vectors a and a' are collinear.
And &1:»,:68 = V-»«/- V.
which shows that the vectors b and V are collinear.
But al bl = a/ &/.
This shows that the product of the lengths (including sign)
are equal and the theorem is proved.
The proof may be carried out geometrically as follows.
Since ab is equal to a'V
ab • r = a'b' • r
for all values of r. Let r be perpendicular to b. Then b • r
vanishes and consequently V«r also vanishes. This is true
for any vector r in the plane perpendicular to b. Hence b and
b' are perpendicular to the same plane and are collinear. In
like manner by using a b as a postf actor a and a' are seen
to be parallel. Also
ab-b = a'b'-b,
which shows that the products of the lengths are the same.
18
274 VECTOR ANALYSIS
The indeterminate product ab imposes Jive conditions upon
the vectors a and b. The directions of a and b are fixed and
likewise the product of their lengths. The scalar product
a • b, being a scalar quantity, imposes only one condition upon
a and b. The vector product a x b, being a vector quantity,
imposes three conditions. The normal to the plane of a and
b is fixed and also th'e area of the parallelogram of which they
are the side. The nine indeterminate products (12) of i,j, k
into themselves are independent. The nine scalar products
are not independent. Only two of them are different.
and i.j=j.i=j.k = k»j=k«i = i«k = 0.
The nine vector products are mot independent either; for
ixi = jxj = kxk = 0,
and ixj = — jxi, jxk= — kxj, kxi — — ixk.
The two products a • b and a x b obtained respectively from
the indeterminate product by inserting a dot and a cross be
tween the factors are functions of the indeterminate product.
That is to say, when ab is given, a • b and a x b are determined.
For these products depend solely upon the directions of a and b
and upon the product of the length of a and b, all of which
are known when ab is known. That is
if ab = a'b', a • b = a' • b' and a x b = a' x b'. (17)
It does not hold conversely that if a • b and a x b are known
a b is fixed ; for taken together a • b and a X b impose upon the
vectors only four conditions, whereas a b imposes five. Hence
a b appears not only as the most general product but as the
most fundamental product. The others are merely functions
of it. Their functional nature is brought out clearly by the
notation of the dot and the cross.
LINEAR VECTOR FUNCTIONS 275
Definition: A scalar known as the scalar of 0 may be ob
tained by inserting a dot between the antecedent and conse
quent of each dyad in a dyadic. This scalar will be denoted
by a subscript S attached to 0. l
If <P = a1b1 + a2b2 + a3b3 + ...
08 = &1 • bx + a2 • b2 + a3 . b3 + . . • (18)
In like manner a vector known as the vector of 0 may be
obtained by inserting a cross between the antecedent and con
sequent of each dyad in 0. This vector will be denoted by
attaching a subscript cross to 0.
0X = aj x bx + a2 x b2 + a3 x b3 + . - • (19)
If 0 be expanded in nonion form in terms of i, j, k,
0s = an + a^ + aBZ, (20)
#x = (« 28 - a32) * + 0*31 - a!3) J + (^12 - a2l) k- (21)
Or 0S = i- 0-i + j- <P«j + k. (?-k, (20)'
<?x =(j . (P-k-k* ^.j) i+ (k- (P-i-i. (P.k) j
+ (i- 0-j-j.0.i)k. (21)'
In equations (20) and (21) the scalar and vector of 0 are
expressed in terms of the coefficients of 0 when expanded
in the nonion form. Hence if 0 and W are two equal
dyadics, the scalar of 0 is equal to the scalar of ¥ and the
vector of 0 is equal to the vector of ¥.
If 0 = W, 0S = ¥s and 0X = yx. (22)
From this it appears that 0S and 0X are functions of 0
uniquely determined when 0 is given. They may sometimes
be obtained more conveniently from (20) and (21) than from
(18) and (19), and sometimes not.
1 A subscript dot might be used for the scalar of * if it were sufficiently distinct
and free from liability to misinterpretation.
276 VECTOR ANALYSIS
Products of Dyddics
103.] In giving the definitions and proving the theorems
concerning products of dyadics, the dyad is made the under
lying principle. What is true for the dyad is true for the
dyadic in general owing to the fact that dyads and dyadics
obey the distributive law of multiplication.
Definition: The direct product of the dyad ab into the
dyad c d is written , ,x , ,.
(ab)« (cd)
and is by definition equal to the dyad (b • c) a d,
(ab)-(cd) = a(b.c)d = b-c ad.1 (23)
That is, the antecedent of the first and the consequent of the
second dyad are taken for the antecedent and consequent
respectively of the product and the whole is multiplied by
the scalar product of the consequent of the first and the
antecedent of the second.
Thus the two vectors which stand together in the product
(ab). (cd)
are multiplied as they stand. The other two are left to form
a new dyad. The direct product of two dyadics may be
defined as the formal expansion (according to the distributive
law) of the product into a sum of products of dyads. Thus
*=(a1b1 + a2b2 + a3b3 + ...)
and r^CCjdj + c2d2 + c3d3 + -..)
d>. ?T=(a1b1 +a2b2 + a3b3 + •••)•
(c^j + c2d2 + C3d3 +•••)
= a1b1«c1d1 + a1b1*e2da + axbx • C3d3 + • • •
+ a2b2-c1d1 +a2b2.c2d2 + a2b2 • C3d3 + • • • (23)
+ agbg-c^ + a3b3-c2d2 + a3b3« c3d3 H
+ •
1 The parentheses may be omitted in each of these three expressions.
LINEAR VECTOR FUNCTIONS 277
x ajdj + bj • c2 ax d2 -f bx • c3 ax d3 + • ••
l a2dx + b2 -c2 a2d2 + b2 • c3 a^j d3 -f- • ••
b3.c2 a3d2 + b3-c3 a3d3
(23)"
The product of two dyadics 0 and W is a dyadic 0 • W.
Theorem : The product 0 • W of two dyadics (P and W when
regarded as an operator to be used as a prefactor is equiva
lent to the operator W followed by the operator 0.
Let £=&.¥.
To show Q • r = d> • ( W • r),
or ((?• W)*T = 0- (^'0- (24)
Let ab be any dyad of <? and c d any dyad of W.
(ab • cd) • r = b • c (ad • r) = (b • c) (d • r) a,
ab • (c d • r) = a b • c (d • r) = (b • c) (d • r) a,
Hence (a b • c d) • r = a b • (c d • r).
The theorem is true for dyads. Consequently by virtue of
the distributive law it holds true for dyadics in general.
If r denote the position vector drawn from an assumed origin
to a point P in space, r' = W • r will be the position vector of
another point P', and r" = (^•(3r«r) will be the position
vector of a third point Pn. That is to say, W defines a trans
formation of space such that the points P go over into the
points Pf. 0 defines a transformation of space such that the
points Pf go over into the points P". Hence W followed by
0 carries P into Pff. The single operation 0 • W also carries
PintoP".
Theorem: Direct multiplication of dyadics obeys the dis*
tributive law. That is
278 VECTOR ANALYSIS
0 • ( ¥ + ¥f) = 0 . W + 0 . ¥'
and (0f + 0) • W = 0f . ¥ + 0 . W. (25)
Hence in general the product
(4>+ 4>' + 4>" + ...).( W+ ¥'+ ?F"+...)
may be expanded formally according to the distributive law.
Theorem : The product of three dyadics <P, W, Q is associa-
tive. Thatis (t.r).o= t.(¥ma> (26)
and consequently either product may be written without
parentheses, as 0 . V . Q, (26)'
The proof consists in the demonstration of the theorem for
three dyads ab, cd, ef taken respectively from the three
dyadics 4>, ¥, Q.
(ab»cd) «ef = (b«c) ad« ef = (b«c) (d- e) af,
ab» (cd«ef) = (d^e) ab -cf = (d« e) (b «c) af.
The proof may also be given by considering 0, W, and Q
as operators
Let
Let
Again {^.(f. J»)} . r = *. [(f. J2).r].
Hence {(* • F) • Q\ -r = {(?• (V • £T)\ «r
for all values of r. Consequently
LINEAR VECTOR FUNCTIONS 279
The theorem may be extended by mathematical induction
to the case of any number of dyadics. The direct product
of any number of dyadics is associative. Parentheses may
be inserted or omitted at pleasure without altering the result.
It was shown above (24) that
(<P • T) • r = 0 . ( ¥ - r) = <P • V • r. (24)'
Hence the product of two dyadics and a vector is associative.
The theorem is true in case the vector precedes the dyadics
and also when the number of dyadics is greater than two.
But the theorem is untrue when the vector occurs between
the dyadics. The product of a dyadic, a vector, and another
dyadic is not associative.
(#.r). V ± 0.(r- ¥). (27)
Let ab be a dyad of $, and c d a dyad of ¥.
(a b • r) • c d = b • r (a • c d) = (b • r) (a • c) d,
ab • (r • c d) = ab • d (r • c) = b • d (r • c) a
Hence (ab • r) • c d •£ ab • (r • cd).
The results of this article may be summed up as follows :
Theorem: The direct product of any number of dyadics
or of any number of dyadics with a vector factor at either
end or at both ends obeys the distributive and associative
laws of multiplication — parentheses may be inserted or
omitted at pleasure. But the direct product of any number
of dyadics with a vector factor at some other position than at
either end is not associative — parentheses are necessary to
give the expression a definite meaning.
Later it will be seen that by making use of the conjugate
dyadics a vector factor which occurs between other dyadics
may be placed at the end and hence the product may be
made to assume a form in which it is associative.
280 VECTOR ANALYSIS
104.] Definition: The skew products of a dyad ab into
a vector r and of a vector r into a dyad ab are defined
respectively by the equations
(ab) x r = a(b x r),
rx(ab) = (r x a)b.
The skew product of a dyad and a vector at either end is a
dyad. The obvious extension to dyadics is
rrrajbj x r + a2 b2 xr + a3b3 x r + ...
r x 0 = r x (aa ba + a2 b2 + a3 b3 + . . .) (28)'
= r x ajbj + r x a2b2 + r x a3b3 + ...
Theorem: The direct product of any number of dyadics
multiplied at either end or at both ends by a vector whether
the multiplication be performed with a cross or a dot is
associative. But in case the vector occurs at any other
position than the end the product is not associative. That is,
(rx <P) • Sr = rx(0.y)=rx <P • ¥,
(<P • ?F) xr=(P.(?P'xr) = <P.?rxr,
(r x #) • s = r x ( 0 • s) = r x <P • s, (29)
r . (0 x s) = (r • </>) x s = r • <P x s,
rx($xs) = (rx $)xs = rx $xs,
but !P •(rX^)^(Sjr-r) X*.
Furthermore the expressions
s • r x <P and <P x r • s
can have no other meaning than
s • r x <P = s • (r x <P),
LINEAR VECTOR FUNCTIONS 281
since the product of a dyadic 0 with a cross into a scalar s • r
is meaningless. Moreover since the dot and the cross may
be interchanged in the scalar triple product of three vectors
it appears that
s • r x ^ = (s x r) • 0,
<p x r • s = 4> • (r x s), (31)
and 0-(r x 5F) = (</> x r) • V.
The parentheses in the following expressions cannot be
omitted without incurring ambiguity.
<p.(r x s) ± (0-r) x s,
(sx r). 0*sx(r-0), (31)'
(0-r) x ¥ * 0 x(r. £>).
The formal skew product of two dyads a b and c d would be
(ab) x (cd) = a(b x c)d.
In this expression three vectors a, b x c, d are placed side
by side with no sign of multiplication uniting them. Such
an expression
rst (32)
is called a triad ; and a sum of such expressions, a triadic.
The theory of triadics is intimately connected with the theory
of linear dyadic functions of a vector, just as the theory of
dyadics is connected with the theory of linear vector functions
of a vector. In a similar manner by going a step higher
tetrads and tetradics may be formed, and finally polyads and
polyadics. But the theory of these higher combinations of
vectors will not be taken up in this book. The dyadic
furnishes about as great a generality as is ever called for in
practical applications of vector methods.
282 VECTOR ANALYSIS
Degrees of Nullity of Dyadics
105.] It was shown (Art. 101) that a dyadic could always
be reduced to a sum of three terms at most, and this reduction
can be accomplished in only one way when the antecedents
or the consequents are specified. In particular cases it may
be possible to reduce the dyadic further to a sum of two
terms or to a single term or to zero. Thus let
<P = al + bm + cn.
If 1, m, n are coplanar one of the three may be expressed
in terms of the other two as
1 = x m + y n.
Then $ = a#m + ayn + bm + cn,
0 = (a# + b)m + (ay + c)n.
The dyadic has been reduced to two terms. If 1, m, n were
all collinear the dyadic would reduce to a single term and if
they all vanished the dyadic would vanish.
Theorem : If a dyadic 0 be expressed as the sum of three
terms
<p = al + bm + en
of which the antecedents a, b, c are known to be non-coplanar,
then the dyadic may be reduced to the sum of two dyads
when and only when the consequents are coplanar.
The proof of the first part of the theorem has just been
given. To prove the second part suppose that the dyadic
could be reduced to a sum of two terms
$ = dp + eq
and that the consequents 1, m, n of 0 were non-coplanar.
This supposition leads to a contradiction. For let 1', m', n'
be the system reciprocal to 1, m, n. That is,
mx n n x 1 1 x m
_
=
[Tmn]'
LINEAR VECTOR FUNCTIONS 283
The vectors 1', mf, n' exist and are non-coplanar because
1, m, n have been assumed to be non-coplanar. Any vector r
may be expressed in terms of them as
r = xlf + ym' + zn/
<p.r = (al + bm + en) • (xl1 + ym' + zn').
But 1 • 1' = m • m' = n • n' = 1,
and 1 • m' = 1' • m = m • n' = m' • n = n • 1' = n' • 1 = 0.
Hence $ • r = x a + y b + z,e.
By giving to r a suitable value the vector d> • r may be made
equal to any vector in space.
But 0 • r = (dp + e q) - r = d (p • r) + e (q • r).
This shows that 0 • r must be coplanar with d and e. Hence
0 • r can take on only those vector values which lie in the
plane of d and e. Thus the assumption that 1, m, n are non-
coplanar leads to a contradiction. Hence 1, m, n must be
coplanar and the theorem is proved.
Theorem : If a dyadic 0 be expressed as the sum of three
terms
of which the antecedents a, b, c are known to be non-coplanar,
the dyadic 0 can be reduced to a single dyad when and only
when the consequents 1, m, n are collinear.
The proof of the first part was given above. To prove
the second part suppose <P could be expressed as
Let
284 VECTOR ANALYSIS
From the second equation it is evident that W used as a
postfactor for any vector
r = x a' + y b' + zc',
where a', V, c' is the reciprocal system to a, b, c gives
From the first expression
r • 0 = 0.
Hence #lxp+ymxp + znxp
must be zero for every value of r, that is, for every value of x,
y, z. Hence
1 x p = 0, mxp = 0, nxp = 0.
Hence 1, m, and n are all parallel to p and the theorem has
been demonstrated.
If the three consequents 1, m, n had been known to be non-
coplanar instead of the three antecedents, the statement of
the theorems would have to be altered by interchanging the
words antecedent and consequent throughout. There is a fur
ther theorem dealing with the case in which both antecedents
and consequents of 0 are coplanar. Then 0 is reducible to
the sum of two dyads.
106.] Definition: A dyadic which cannot be reduced to
the sum of fewer than three dyads is said to be complete. A
dyadic which may be reduced to the sum of two dyads, but
cannot be reduced to a single dyad is said to be planar. In
case the plane of the antecedents and the plane of the con
sequents coincide when the dyadic is expressed as the sum of
two dyads, the dyadic is said to be uniplanar. A dyadic
which may be reduced to a single dyad is said to be linear.
In case the antecedent and consequent of that dyad are col-
LINEAR VECTOR FUNCTIONS 285
linear, the dyadic is said to be unilinear. If a dyadic may be
so expressed that all of its terms vanish the dyadic is said to
be zero. In this case the nine coefficients of the dyadic as
expressed in nonion form must vanish.
The properties of complete, planar, uniplanar, linear, and
unilinear dyadics when regarded as operators are as follows.
Let
s = 0 • r and t = r • <P.
If 0 is complete s and t may be made to take on any desired
value by giving r a suitable value.
As 0 is complete 1, m, n are non-coplanar and hence have a
reciprocal system lf, m', n'.
s = 0 . (xlf + ymf + zn') =#a + yb + zc.
In like manner a, b, c possess a system of reciprocals a', V, c'.
yb' + zc') • 0 = xl + ym + zn.
A complete dyadic 0 applied to a vector r cannot give zero
unless the vector r itself is zero.
If 0 is planar the vector s may take on any value in the plane
of the antecedents and t any value in the plane of the consequents
of 0 ; but no values out of those planes. The dyadic 0 when
used as a prefactor reduces every vector r in space to a vector
in the plane of the antecedents. In particular any vector r
perpendicular to the plane of the consequents of 0 is reduced
to zero. The dyadic 0 used as a postfactor reduces every
vector r in space to a vector in the plane of the consequents
of <P. In particular a vector perpendicular to the plane of
the antecedents of 0 is reduced to zero. In case the dyadic
is uniplanar the same statements hold.
If 0 is linear the vector s may take on any value collinear
with the antecedent of 0 and t any value collinear with the con-
286 VECTOR ANALYSIS
sequent of 0 ; but no other values. The dyadic 0 used as a
prefactor reduces any vector r to the line of the antecedent
of 0. In particular any vectors perpendicular to the con
sequent of 0 are reduced to zero. The dyadic 0 used as a
postfactor reduces any vector r to the line of the consequent
of 0. In particular any vectors perpendicular to the ante
cedent of 0 are thus reduced to zero.
If 0 is a zero dyadic the vectors s and t are loth zero no
matter what the value of r may be.
Definition : A planar dyadic is said to possess one degree of
nullity. A linear dyadic is said to possess two degrees of
nullity. A zero dyadic is said to possess three degrees of nul
lity or complete nullity.
107.] Theorem : The direct product of two complete dyadics
is complete; of a complete dyadic and a planar dyadic,
planar ; of a complete dyadic and a linear dyadic, linear.
Theorem: The product of two planar dyadics is planar
except when the plane of the consequent of the first dyadic
in the product is perpendicular to the plane of the antece
dent of the second dyadic. In this case the product reduces
to a linear dyadic — and only in this case.
Let 0 — B,I\)I + a2b2,
Q = 0 . W.
The vector s = W • r takes on all values in the plane of Cj
and c2
The vector sf = 0 • s takes on the values
g' = 0 . s = x (bx • cx) ax + y (bx • c2) ax
+ x (b2 ' Cl) a2 + y 0>2 • C) E2>
s' = \x (bj • cx) + y (bj • c2)} ax + {x (b2 • cx) + y (b2 • c2)} a2.
LINEAR VECTOR FUNCTIONS 287
Let s' = x'&i + y'a2,
where x1 = x (bx • Cj) + y (bl • c2),
and y1 — x (b2 • Cj) 4- y (b2 • c2).
These equations may always be solved for x and y when
any desired values x1 and y' are given — that is, when s' has
any desired value in the plane of EJ and a2 — unless the
determinant
V^ bj-c,
b2 • cx b2 • c2
But by (25), Chap. II., this is merely the product
0>i x t>2) • (ci x C2) = °-
The vector \ x b2 is perpendicular to the plane of the con
sequents of <P; and cl x c2, to the plane of the antecedents of
¥. Their scalar product vanishes when and only when the
vectors are perpendicular — that is, when the planes are per
pendicular. Consequently s' may take on any value in the
plane of ax and a2 and 0 • ¥ is therefore a planar dyadic
unless the planes of bx and b2, cx and c2 are perpendicular.
If however bx and b2, Cj and c2 are perpendicular sf can take
on only values in a certain line of the plane of ax and a^ and
hence <P • W is linear. The theorem is therefore proved.
Theorem : The product of two linear dyadics is linear
except when the consequent of the first factor is perpen
dicular to the antecedent of the second. In this case the
product is zero — and only in this case.
Theorem : The product of a planar dyadic into a linear is
linear except when the plane of the consequents of the
planar dyadic is perpendicular to the antecedent of the linear
dyadic. In this case the product is zero — and only in this
case.
Theorem: The product of a linear dyadic into a planar
dyadic is linear except when the consequent of the linear
288 VECTOR ANALYSIS
dyadic is perpendicular to the plane of the antecedents of
the planar dyadic. In this case the product is zero — and
only in this case.
It is immediately evident that in the cases mentioned the
products do reduce to zero. It is not quite so apparent that
they can reduce to zero in only those cases. The proofs are
similar to the one given above in the case of two planar
dyadics. They are left to the reader. The proof of the
first theorem stated, page 286, is also left to the reader.
The Idemfactor;1 Reciprocals and Conjugates of Dyadics
108.] Definition : If a dyadic applied as a pre factor or as
a postf actor to any vector always yields that vector the
dyadic is said to be an idemf actor. That is
if 0 • r = r for all values of r,
or if r • 0 = r for all values of r,
then 0 is an idemfactor. The capital I is used as the sym
bol for an idemfactor. The idemfactor is a complete dyadic.
For there can be no direction in which I • r vanishes.
Theorem : When expressed in nonion form the idemfactor is
I = ii + jj + kk. (33)
Hence all idemfactors are equal.
To prove that the idemfactor takes the form (33) it is
merely necessary to apply the idemfactor I to the vectors
i, j, k respectively. Let
1 = an ii+ «12 ij + a13ik
ki + a32 kj + a33 kk.
1 In the theory of dyadics the idemfactor I plays a role analogous to unity in
ordinary algebra. The notation is intended to suggest this analogy.
LINEAR VECTOR FUNCTIONS 289
I . i = an i + a21 j 4- a31 k.
If I-i = i,
an = 1 and a21 = a31 = 0.
In like manner it may be shown that all the coefficients
vanish except an, a22, a33 all of which are unity. Hence
I = ii + jj + kk. (33)
Theorem : The direct product of any dyadic and the idem-
factor is that dyadic. That is,
0 • I = 0 and 1-0 = 0.
For (0 • I) • r = 0 • (I • r) = 0 • r,
no matter what the value of r may be. Hence, page 266,
In like manner it may be shown that I • 0 = 0.
Theorem: If a', V, c' and a, b, c be two reciprocal systems
of vectors the expressions
I = aa' + bb' + cc', (34)
I = a'a + b'b + c'c
are idemfactors.
For by (30) and (31) Chap. II.,
r = r»aa' + r*bb' + r «cc',
and r = r»a'a + r'b'b + r.c'c.
Hence the expressions must be idemfactors by definition.
Theorem : Conversely if the expression
0 = al + bm + en
is an idemfactor 1, m, n must be the reciprocal system of
a, b, c.
19
290 VECTOR ANALYSIS
In the first place since (P is the idemfactor, it is a complete
dyadic. Hence the antecedents a, b, c are non-coplanar and
possess a set of reciprocals a', b', c'. Let
r = #a' + y V + 20'.
By hypothesis r • $ = r.
Then r • <P = xl + ym-\-zn = xsi' + y'b' + zc'
for all values of r, that is, for all values of x, y, z. Hence the
corresponding coefficients must be equal. That is,
Theorem : If (Pand ¥ be any two dyadics, and if the product
<? • W is equal to the idemf actor ; l then the product W • 0,
when the factors are taken in the reversed order, is also
equal to the idemfactor.
Let 0 • V = L
To show W • 0 = L
r . (0 . W) - 0 = (r • 0) • ( W • (?) = r • 0.
This relation holds for all values of r. As 0 is complete r • 0
must take on all desired values. Hence by definition
W • 0 = I.
If the product of two dyadics is an idemfactor, that product
may be taken in either order.
109.] Definition: When two dyadics are so related that
their product is equal to the idemfactor, they are said to be
1 This necessitates both the dyadics * and V to be complete. For the product
of two incomplete dyadics is incomplete and hence could not be equal to the
idemfactor.
LINEAR VECTOR FUNCTIONS 291
reciprocals.1 The notation used for reciprocals in ordinary
algebra is employed to denote reciprocal dyadics. That is,
if 0.y=I, 0 = ?F-i = 1 and 5T= 0-i=L (35)
W 0
Theorem: Reciprocals of the same or equal dyadics are
equal.
Let 0 and W be two given equal dyadics, <J>~1 and JT"1
their reciprocals as defined above. By hypothesis
0= W,
and W. ¥~i = l.
To show 0-1 = ¥~i.
0. 0-1 = 1= ¥• ¥-1.
As 0=¥, 0.0~i=0.¥-\
0-1.0 = I,
I.0-i = 0-i = I. W~i = ¥-1.
Hence 0-i = ¥~\
The reciprocal of 0 is the dyadic whose antecedents are the
reciprocal system to the consequents of 0 and whose conse
quents are the reciprocal system to the antecedents of 0.
If a complete dyadic 0 be written in the form
0 = al + bm + en,
its reciprocal is 0"1 = 1' a' + m' V + n' c'. (36)
For (al + bm + cn) • (1'a' + n'V + n'c') =aa' + bV + ce'.
Theorem : If the direct products of a complete dyadic 0
into two dyadics W and Q are equal as dyadics then W and Q
1 An incomplete dyadic has no (finite) reciprocal.
292 VECTOR ANALYSIS
are equal. If the product of a dyadic 0 into two vectors
r and s (whether the multiplication be performed with a dot
or a cross) are equal, then the vectors r and s are equal.
That is,
if 0 - ¥ = d> • J2, then ¥ = Q,
and if 0 • r = 0 • s, then r = s, (37)
and if 0 x r = 0 x s, then r = s.
This may be seen by multiplying each of the equations
through by the reciprocal of 0,
0-1.0. W = ¥ = 0-i • 0 • Q = £,
0-i . 0 . r = r = 0~! • 0 • s = s,
0-1. 0xr = IXr=0"1- 0X8 = 1X8.
To reduce the last equation proceed as follows. Let t be
any vector,
t»Ixr = t«Ixs,
t • I = t.
Hence t x r = t x s.
As t is any vector, r is equal to s.
Equations (37) give what is equivalent to the law of can-
celation for complete dyadics. Complete dyadics may be
canceled from either end of an expression just as if they
were scalar quantities. The cancelation of an incomplete
dyadic is not admissible. It corresponds to the cancelation
of a zero factor in ordinary algebra.
110.] Theorem: The reciprocal of the product of any
number of dyadics is equal to the product of the reciprocals
taken in the opposite order.
It will be sufficient to give the proof for the case in which
the product consists of two dyadics. To show
LINEAR VECTOR FUNCTIONS 293
0 . V • 5F-1 • 0~l = 0 • ( ?F • y-1} . 0~l= 0. 0-l = I.
Hence (0 • ?F) • ( JF-1 • 0-1) = I.
Hence 0 • ?T and W~l • (P"1 must be reciprocals. That is,
The proof for any number of dyadics may be given in the
same manner or obtained by mathematical induction.
Definition : The products of a dyadic <P, taken any number
of times, by itself are called powers of 0 and are denoted in
the customary manner.
0 . 0 = 0*,
0 . 0 . 0 = 0 . 02 — 0^
and so forth.
Theorem : The reciprocal of a power of <P is the power of
the reciprocal of <P.
(0»)-i = (0-1)" = 0-» (37)
The proof follows immediately as a corollary of the preced
ing theorem. The symbol <P"n may be interpreted as the
nth power of the reciprocal of 0 or as the reciprocal of
the nth power of 0.
If 0 be interpreted as an operator determining a trans
formation of space, the positive powers of 0 correspond to
repetitions of the transformation. The negative powers of 0
correspond to the inverse transformations. The idemfactor
corresponds to the identical transformation — that is, no trans
formation at all. The fractional and irrational powers of CP
will not be defined. They are seldom used and are not
single-valued. For instance the idemfactor I has the two
square roots ±1. But in addition to these it has a doubly
infinite system of square roots of the form
<P = -ii + jj + kk.
294 VECTOR ANALYSIS
Geometrically the transformation
is a reflection of space in the j k-plane. This transformation
replaces each figure by a symmetrical figure, symmetrically
situated upon the opposite side of the j k-plane. The trans
formation is sometimes called perversion. The idemfactor
has also a doubly infinite system of square roots of the form
Geometrically the transformation
r'= V.T
is a reflection in the i-axis. This transformation replaces each
figure by its equal rotated about the i-axis through an angle
of 180°. The idemfactor thus possesses not only two square
roots ; but in addition two doubly infinite systems of square
roots ; and. it will be seen (Art. 129) that these are by no
means all.
111.] The conjugate of a dyadic has been defined (Art. 99)
as the dyadic obtained by interchanging the antecedents and
consequents of a given dyadic and the notation of a subscript
C has been employed. The equation
r . 0 = 4>0 • r (9)
has been demonstrated. The following theorems concerning
conjugates are useful.
Theorem : The conjugate of the sum or difference of two
dyadics is equal to the sum or difference of the conjugates,
(d>± T)0=0C± Wc.
Theorem : The conjugate of a product of dyadics is equal
to the product of the conjugates taken in the opposite order.
LINEAR VECTOR FUNCTIONS 295
It will be sufficient to demonstrate the theorem in case
the product contains two factors. To show
(d>.T)c=Wc.0Ct (40)
(0 . W)c. r = r • (0 • W) = (r • 4>) • 5F,
r . 0 = <PC . r,
(r . <P) . W = Vc. (r • <P) = ^ • <^. r.
Hence (4> • V)c=¥c.4>c.
Theorem : The conjugate of the power of a dyadic is the
power of the conjugate of the dyadic.
This is a corollary of the foregoing theorem. The expression
0nc may be interpreted in either of two equal ways.
Theorem : The conjugate of the reciprocal of a dyadic is
equal to the reciprocal of the conjugate of the dyadic.
= ^ (42)
For (@~l)c • ®c = (& • 0~1)c = Ic = I-
The idemfactor is its own conjugate as may be seen from
the nonion form.
I = ii + j j + kk
Hence (^c)"1 *®c
Hence C^)'1 = (*"%
The expression ^c-1 may therefore be interpreted in either
of two equivalent ways — as the reciprocal of the conjugate
or as the conjugate of the reciprocal.
Definition: If a dyadic is equal to its conjugate, it is said
to be self -conjugate. If it is equal to the negative of its con-
296 VECTOR ANALYSIS
jugate, it is said to be anti-self -conjugate. For se//-conjugate
dyadics.
r • 0 = 0 • r, 0 = 0C.
For anti-self-conjugate dyadics
r • 0 = — 0 • r, 0 = — 00.
Theorem : Any dyadic may be divided in one and only one
way into two parts of which one is self-conjugate and the
other anti-self-conjugate.
For 0=5(0+0,) + 2(0-0c). (43)
But (& + 0c)c = 0c+<pcc=: 4>c + d>,
and (0 - 0<,)c = $€ - <PCC = 0<, _ 0.
Hence the part |(0 + &c) is self-conjugate; and the part
|(^~ 4>c), anti-self-conjugate. Thus the division has been
accomplished in one way. Let
and
Suppose it were possible to decompose 0 in another way
into a self-conjugate' and an anti-self-conjugate part. Let
then
0 = (0' + J2) + (0"-£).
Where (0' + 0) = (0' + £), = 0'0 + £c = 0f + ^
Hence if (0; -f J2) is self-conjugate, fi is self-conjugate.
Hence if (0" — J2) is anti-self-conjugate £ is anti-self-
conjugate.
LINEAR VECTOR FUNCTIONS 297
Any dyadic which is both self-conjugate and anti-self-conju
gate is equal to its negative and consequently vanishes.
Hence Q is zero and the division of 0 into two parts is
unique.
Anti-self-conjugate Dyadics. The Vector Product
112.] In case 0 is any dyadic the expression
gives the anti-self-conjugate part of 0. If 0 should be en
tirely antinself-con jugate 0 is equal to 0". Let therefore 0n
be any anti-self-conjugate dyadic,
Suppose <P = al-hbm-fcn,
0 — $c = al — la + bm — mb + cn — nc,
20" • r = a 1 • r — la»r-hbm»r — mb»r + cn»r — n c • r.
But a 1 • r — 1 a • r = — (a x 1) x r,
bm • r — mb • r = — (b x m) x r,
c n • r — n c • r = — (cxn)xr.
Hence 0" • r = — ~ (a x 1 + b x m 4- c x n) x r.
But by definition <Px = axl-fbxm4-cxn.
Hence 0" • r = - ~ 0X x r,
r • 0" = 0"c . r = - 0" • r = I0X x r = - \ r x 0X.
The results may be stated in a theorem as follows.
Theorem : The direct product of any anti-self-conjugate
dyadic and the vector r is equal to the vector product of
minus one half the vector of that dyadic and the vector r.
VECTOR ANALYSIS
Theorem : Any anti-self-conjugate dyadic <P!f possesses one
degree of nullity. It is a uniplanar dyadic the plane of
whose consequents and antecedents is perpendicular to <PX",
the vector of <P.
This theorem follows as a corollary from equations (44).
Theorem : Any dyadic 0 may be broken up into two parts
of which one is self-conjugate and the other equivalent to
minus one half the vector of 0 used in cross multiplication.
<p . r = <Pr • r — ^ <PX x r,
or symbolically $ • = 0' • — \ #x X. (45)
113.] Any vector c used in vector multiplication defines a
linear vector function. For
cx(r + s)=cxr + cxs.
Hence it must be possible to represent the operator c x as a
dyadic. This dyadic will be uniplanar with plane of its
antecedents and consequents perpendicular to c, so that it
will reduce all vectors parallel to c to zero. The dyadic may
be found as follows
By (31) I- (c x !) = (! x c) - 1,
(I x c) • r = \ (I x c) • I } • r = {I - (c x I)} • r
= I • (c x I) • r = (c x I) • r.
Hence c x r = (I x c) • r = (e x I) • r,
and r x c = r • (I x c) = r • (c x I). (46)
This may be stated in words.
LINEAR VECTOR FUNCTIONS 299
Theorem : The vector c used in vector multiplication with
a vector r is equal to the dyadic I x c or c X I used in direct
multiplication with r. If c precedes r the dyadics are to be
used as prefactors ; if c follows r, as postfactors. The dyadics
I X c and c X I are anti-self-conjugate.
In case the vector c is a unit vector the application of the
operator c X to any vector r in a plane perpendicular to c is
equivalent to turning r through a positive right angle about
the axis c. The dyadic c X I or I x c where c is a unit vector
therefore turns any vector r perpendicular to c through a
right angle about the line c as an axis. If r were a vector
lying out of a plane perpendicular to c the effect of the dyadic
I X c or c x I would be to annihilate that component of r which
is parallel to c and turn that component of r which is perpen
dicular to c through a right angle about c as axis.
If the dyadic be applied twice the vectors perpendicular to
r are rotated through two right angles. They are reversed in
direction. If it be applied three times they are turned through
three right angles. Applying the operator I x c or c X I four
times brings a vector perpendicular to c back to its original
position. The powers of the dyadic are therefore
(I x c)2 = (c x I)2 = - (I - cc),
(I x c)3 = (c x I)3 = - I x c = - c x I,
(47)
(I x c)4 = (c x I)4 = I - c c,
(I x c)5 = (c x I)6 = I x c = c x I.
It thus appears that the dyadic I x c or c x I obeys the same
law as far as its powers are concerned as the scalar imaginary
V — 1 in algebra.
The dyadic Ixc orcxlisa quadrantal versor only for
vectors perpendicular to c. For vectors parallel to c it acts
as an annihilator. To avoid this effect and obtain a true
300 VECTOR ANALYSIS
quadrantal versor for all vectors r in space it is merely neces
sary to add the dyad c c to the dyadic I X c or c X I.
If X = Ixc + cc = cxI + cc,
X2 = — I + 2cc,
X3 = — iXc + cc, (48)
The dyadic X therefore appears as a fourth root of the
idemfactor. The quadrantal versor X is analogous to the
imaginary V — 1 of a scalar algebra. The dyadic X is com
plete and consists of two parts of which I x c is anti-self-
conjugate ; and c c, self-conjugate.
114.] If i, j, k are three perpendicular unit vectors
Ixi = ixl = kj — jk,
I xj-j x I = ik-ki, (49)
Ixk=k x I=ji — ij,
as may be seen by multiplying the idemfactor
into i, j, and k successively. These expressions represent
quadrantal versors about the axis i,j, k respectively combined
with annihilators along those axes. They are equivalent,
when used in direct multiplication, to i x, jx, k X respectively,
jj,
The expression (I x k)4 is an idemfactor for the plane of i and
j, but an annihilator for the direction k. In a similar man
ner the dyad k k is an idemfactor for the direction k, but an
LINEAR VECTOR FUNCTION'S 301
annihilate! for the plane perpendicular to k. These partial
idemfactors are frequently useful.
If a, b, c are any three vectors and a', V, c' the reciprocal
system,
aa' + bb'
used as a prefactor is an idemfactor for all vectors in the
plane of a and b, but an annihilator for vectors in the direc
tion c. Used as a postfactor it is an idemfactor for all vectors,
in the plane of a' and V, but an annihilator for vectors in the
direction c'. In like manner the expression
cc'
used as a prefactor is an idemfactor for vectors in the direction
c, but for vectors in the plane of a and b it is an annihilator.
Used as a postfactor it is an idemfactor for vectors in the
direction c', but an annihilator for vectors in the plane of a'
and V, that is, for vectors perpendicular of c.
If a and b are any two vectors
(a x b) x I = I x (a x b) = ba - ab. (50)
For
{(a x b) x I}«r = (a x b) x r = ba«r — ab T = (ba — ab>r.
The vector a x b in cross multiplication is therefore equal to
the dyadic (b a — a b) in direct multiplication. If the vector
is used as a prefactor the dyadic must be so used.
(a xb) x r = (b a — a b) • r,
r x (a x b) = r - (ba - ab). (51)
This is a symmetrical and easy form in which to remember
the formula for expanding a triple vector product.
302 VECTOR ANALYSIS
Reduction of Dyadics to Normal Form
115.] Let 0 be any complete dyadic and let r be a unit
vector. Then the vector r'
is a linear function of r. When r takes on all values consis
tent with its being a unit vector — that is, when the terminus
of r describes the surface of a unit sphere, — the vector r'
varies continuously and its terminus describes a surface. This
surface is closed. It is in fact an ellipsoid.1
Theorem : It is always possible to reduce a complete dyadic
to a sum of three terms of which the antecedents among
themselves and the consequents among themselves are mutu
ally perpendicular. This is called the normal form of 0.
<P = ai'i + bj'j + ck'k.
To demonstrate the theorem consider the surface described
by
r' = 0-r.
As this is a closed surface there must be some direction of r
which makes r' a maximum or at any rate gives r' as great
a value as it is possible for r' to take on. Let this direction
of r be called i, and let the corresponding direction of r' —
the direction in which r' takes on a value at least as great as
any — be called a. Consider next all the values of r which
lie in a plane perpendicular to i. The corresponding values
of r' lie in a plane owing to a fact that (P • r is a linear vector
1 This may be proved as follows :
r' = * • r, r^*-1 - r/ = rl.*c -1.
Hence r.r=l=: r'. (*e-i.* -1)- r' = r' • V • r'.
By expressing ¥ in nonion form, the equation r' • ¥ • r' = 1 is seen to be of the second
degree. Hence r' describes a quadric surface. The only closed quadric surface
is the ellipsoid.
LINEAR VECTOR FUNCTIONS 303
function. Of these values of r' one must be at least as great
as any other. Call this b and let the corresponding direction
of r be called j. Finally choose k perpendicular to i and j
upon the positive side of plane of i and j. Let c be the
value of r' which corresponds to r = k. Since the dyadic 0
changes i, j, k into a, b, c it may be expressed in the form
<P = ai + bj + ck.
It remains to show that the vectors a, b, c as determined
above are mutually perpendicular.
r' = (ai + bj + ck)-r,
drf = (ai -f- bj + ck) -dr,
r'« drr = r'« ai • di + r' • bj • di + r'- ck- dr.
When r is parallel to i, r' is a maximum and hence must be
perpendicular to dir. Since r is a unit vector di is always
perpendicular to r. Hence when r is parallel to i
r'«b j-dr + r'«c k«dr = 0.
If further dr is perpendicular to j, r'»c vanishes, and if
dr is perpendicular to k, r'«b vanishes. Hence when r is
parallel to i, r' is perpendicular to both b and c. But when
r is parallel to i, r' is parallel to a. Hence a is perpendicular
to b and c. Consider next the plane of j and k and the
plane of b and c. Let r be any vector in the plane of j and k.
r' = (bj + ck)«r,
drr = (bj + ck) • dr,
r'-dr' = r'-b j • dr -f r'«c k-dr.
When r takes the value j, r' is a maximum in this plane and
hence is perpendicular to drf. Since r is a unit vector it is
304 VECTOR ANALYSIS
perpendicular to dr. Hence when r is parallel to j, dr
is perpendicular to j, and
Hence r' • c is zero. But when r is parallel to j, r' takes the
value b. Consequently b is perpendicular to c.
It has therefore been shown that a is perpendicular to b and
c, and that b is perpendicular to c. Consequently the three
antecedents of 0 are mutually perpendicular. They may be
denoted by i', j ', k'. Then the dyadic $ takes the form
4> = ai'i +bj'j +ck'k, (52)
where a, J, c are scalar constants positive or negative.
116.] Theorem: The complete dyadic <? may always be
reduced to a sum of three dyads whose antecedents and
whose consequents form a right-handed rectangular system
of unit vectors and whose scalar coefficients are either all
positive or all negative.
& = ± (ai'i + fcj'j + ck'k). (53)
The proof of the theorem depends upon the statements
made on page 20 that if one or three vectors of a right-handed
system be reversed the resulting system is left-handed, but
if two be reversed the system remains right-handed. If then
one of the coefficients in (52) is negative, the directions of the
other two axes may be reversed. Then all the coefficients
are negative. If two of the coefficients in (52) are negative,
the directions of the two vectors to which they belong may
be reversed and then the coefficients in 0 are all positive.
Hence in any case the reduction to the form in which all
the coefficients are positive or all are negative has been
performed.
As a limiting case between that in which the coefficients
are all positive and that in which they are all negative comes
LINEAR VECTOR FUNCTIONS 305
the case in which one of them is zero. The dyadic then
takes the form
<P = ai'i +&j'j (54)
and is planar. The coefficients a and b may always be taken
positive. By a proof similar to the one given above it is
possible to show that any planar dyadic may be reduced to
this form. The vectors i'andj'are perpendicular, and the
vectors i and j are likewise perpendicular.
It might be added that in case the three coefficients a, &, c
in the reduction (53) are all different the reduction can be
performed in only one way. If two of the coefficients (say
a and 6) are equal the reduction may be accomplished in an
infinite number of ways in which the third vector k' is always
the same, but the two vectors i', j' to which the equal coeffi
cients belong may be any two vectors in the plane per
pendicular to k. In all these reductions the three scalar
coefficients will have the same values as in any one of them.
If the three coefficients a, 6, c are all equal when $ is reduced
to the normal form (53), the reduction may be accomplished
in a doubly infinite number of ways. The three vectors
i', j', k' may be any right-handed rectangular system in
space. In all of these reductions the three scalar coefficients
are the same as in any one of them. These statements will
not be proved. They correspond to the fact that the ellipsoid
which is the locus of the terminus of r' may have three
different principal axes or it maybe an ellipsoid of revolution,
or finally a sphere.
Theorem : Any self -con jugate dyadic may be expressed in
the form 0 = aii + &jj + ckk (55)
where a, &, and c are scalars, positive or negative.
Let <P = ai'i -f Jj'j +ck'k, (52)
+ 6jj' + ckk',
20
306 VECTOR ANALYSIS
0.0c=a*i'if + &2j'j' +cak'k
jj + c2kk.
Since 0 = 0^
0* 0C= 0C. 0= 0*.
I — * * + j j + k k — n' + j' j '
If i and i' were not parallel (&2 — a2!) would annihilate
two vectors i and i' and hence every vector in their plane.
(0* — a2 I) would therefore possess two degrees of nullity
and be linear. But it is apparent that if a, 6, c are different
this dyadic is not linear. It is planar. Hence i and i' must
be parallel. In like manner- it may be shown that j and j ',
k and k' are parallel. The dyadic 0 therefore takes the form
0= aii + bjj + ckk
where a, J, c are positive or negative scalar constants.
Double Multiplication *
117.] Definition : The double dot product of two dyads is
the scalar quantity obtained by multiplying the scalar product
of the antecedents by the scalar product of the consequents.
The product is denoted by inserting two dots between the
ab:cd = a-c b«d. (56)
This product evidently obeys the commutative law
ab:cd = cd:ab,
1 The researches of Professor Gibbs upon Double Multiplication are here
printed for the first time.
LINEAR VECTOR FUNCTIONS 307
and the distributive law both with regard to the dyads and
with regard to the vectors in the dyads. The double dot
product of two dyadics is obtained by multiplying the prod
uct out formally according to the distributive law into the
sum of a number of double dot products of dyads.
If <p = *l}>l + a2b2 + a3b3+ ...
and W = G! dx + c2 d2 + c8 d3 + • • •
= a1b1:o1d1 4- a1b1:c2d2 + a1b1:c3d3 + • • •
+ aab^Cjdj + a2b2:c2d2 + a2b2:c3d3 + ••• (56)'
+ a3b3:c1d1 + a3b3:c2d2 + a3b3 :c3ds + . . •
+ aa»c1 bg-d! -f d2«c2 b2«d2 + a2-c3 b2»d3 + •••
+ a3-c1 bg.djH-a3.C2 b3-d2 + a3.c3 b3.d3 + ---
+ ............... (66)"
Definition: The double cross product of two dyads is the
dyad of which the antecedent is the vector product of the
antecedents of the two dyads and of which the consequent is
the vector product of the consequent of the two dyads. The
product is denoted by inserting two crosses between the
dyads
ab£cd = axc b x d. (57)
This product also evidently obeys the commutative law
ab £ cd = cd * ab,
308 VECTOR ANALYSIS
and the distributive law both with regard to the dyads and
with regard to the vectors of which the dyads are composed.
The double cross product of two dyadics is therefore defined
as the formal expansion of the product according to the
distributive law into a sum of double cross products of
dyads.
If <P = a1b1 + a2b2 + a3b3 + ...
and *F = cl&1 + c2d2 + C3d3 + •••
0 * y = (a^ + a2b2 + a3b3 + • • •) x (c^ + C2d2
+ c3a3 + ...)
= a1b1 * Ojdj + axbx * c2 d2 + ajbj * c3d3 + • ••
+ a2b2 * cjdj + a2b2 £ c2d2 + a2b2 * c3d3 + ... (57)'
+ aab3 x M! + a3b3 * c2d2 + a3b3 ^ c3d3 + •••
c! b2xd1-fa2xc2 b2xd2 + a2xc3 ba
+ a3xc3 b3xdx +a3xc2 bgXdj +a3xc3 b3
+ ............ ... (57)"
Theorem : The double dot and double cross products of
two dyadics obey the commutative and distributive laws of
multiplication. But the double products of more than two
dyadics (whenever they have any meaning) do not obey the
associative law.
d> : ¥— W :0
$>*¥=¥*$ (58)
(<P * T) I Q * 0 I (^x£)-
The theorem is sufficiently evident without demonstration.
LINEAR VECTOR FUNCTIONS 309
Theorem : The double dot product of two fundamental
dyads is equal to unity or to zero according as the two
dyads are equal or different.
ij:ki = i-k j «i = 0.
Theorem: The double cross product of two fundamental
dyads (12) is equal to zero if either the antecedents or the
consequents are equal. But if neither antecedents nor con
sequents are equal the product is equal to one of the funda
mental dyads taken with a positive or a negative sign.
That is
ij *ik =ix i j x k = 0
ij *ki =i x k j x i = +jk.
There exists a scalar triple product of three dyads in
which the multiplications are double. Let <P, 5T, Q be any
three dyadics. The expression
0 * WiQ
is a scalar quantity. The multiplication with the double
cross must be performed first. This product is entirely in
dependent of the order in which the factors are arranged or
the position of the dot and crosses. Let ab, cd, and ef be
three dyads,
ab*cd:ef=[ace] [bdf]. (59)
That is, the product of three dyads united by a double cross
and a double dot is equal to the product of the scalar triple
product of the three antecedents by the scalar triple product
of the three consequents. From this the statement made
above follows. For if the dots and crosses be interchanged
or if the order of the factors be permuted cyclicly the two
scalar triple products are not altered. If the cyclic order of
310 VECTOR ANALYSIS
the factors is reversed each scalar triple product changes
sign. Their product therefore is not altered.
118.] A dyadic 0 may be multiplied by itself with double
cross. Let
<P = al + bm + en
0 * 0 = (al + bm + en) * (al-f bm + en)
ss= a x a 1 x 1 + a x b Ixm + axc Ixn
i
+bxa mxl+bxb m x m + b x c mxn
+ cxa nxl + cxb n x m + c x c n x n.
The products in the main diagonal vanish. The others are
equal in pairs. Hence
0£<P = 2(bxc mxn+cxa nxl-faxb Ixm). (60)
If a, b, c and 1, m, n are non-coplanar this may be written
+ b/m' + c'n'> (60)'
The product fl> £ $ is a species of power of 0. It may be re
garded as a square of 0 • The notation $2 will be employed
to represent this product after the scalar factor 2 has been
stricken out.
0*0
02=— ^— = (bxc mxn + cxa nxl + axb Ixm) (61).
J
The triple product of a dyadic 0 expressed as the sum of
three dyads with itself twice repeated is
</>*$: 0 = 2 $2: <P
<P2:0=(bxc mxn-fexa nxl + axb Ixm)
: (al + bm + en).
In expanding this product every term in which a letter is
repeated vanishes. For a scalar triple product of three vec-
LINEAR VECTOR FUNCTIONS 311
tors two of which are equal is zero. Hence the product
reduces to three terms only
02:0=[bca] [mnl] + [cab] [nlm] + [abc] [linn]
or 02 : 0 = 3 [a b c] [Imn]
0*0:0 = 6 [abc] pmn].
The triple product of a dyadic by itself twice repeated is
equal to six times the scalar triple product of its antecedents
multiplied by the scalar triple product of its consequents.
The product is a species of cube. It will be denoted by 08
after the scalar factor 6 has been stricken out.
0*0:0
(62)
119.] If 02 be called the second of 0 ; and 08, the third of
0, the following theorems may be stated concerning the
seconds and thirds of conjugates, reciprocals, and products.
Theorem : The second of the conjugate of a dyadic is equal
to the conjugate of the second of that dyadic. The third of
the conjugate is equal to the third of the dyadic.
<*,).= <».),
Theorem: The second and third of the reciprocal of a
dyadic are equal respectively to the reciprocals of the second
and third.
<*-'), = (*,)-!=*,•*
(f1). = (*,)->-*.-»
Let 0 = al + bm + cn
<p-1 = l'a' + m'b' + n'c' (36)
a'l' + b'm' + c'n'
n']
812 VECTOR ANALYSTS
(*«) -1 — [a' b' c'] [!' m' n'] (1 a + m b + n c)
[a b c] [1 m n] '
But [a'b'c'] [a be] = 1 and [1'm'n'] [Imn] = 1.
Hence (0,)-* = (0-1)., = 0,-*.
08=[abc] [Imn],
[abc] [Imn]'
C^-Oa = IX W] [1'm'n'].
Hence (0,)-1 = (*-»), = 0,-*.
Theorem: The second and third of a product are equal
respectively to the product of the seconds and the product of
the thirds.
(f.f), = *,.*,
(0. *•),= *, ^3-
Choose any three non-coplanar vectors 1, m, n as consequents
of 0 and let 1', m', n' be the antecedents of W.
<P = al + b m + en,
?T = l'd + m'e + n'f,
0 • r = ad + be + cf,
( 0 . W\ =bxc exf + cxa fxd + axb dxe,
<P2 = bxc mxn-fcxa nxl + axb Ixm,
?T2 = m' x n' e x f + n' x 1' f x d + 1' x m' dxe.
Hence 02«5P*2 = bxc exf + cxa fxd + axb dxe.
Hence (#• 5T)2 = <?2 • ?F2 .
(^. JT)8= [abc] [def]
LINEAR VECTOR FUNCTIONS 313
08 = [abc] [Imn],
rg = P'm'n'] [defj.
Hence 08 ¥z= [a be] [def].
Hence (0.F),= 08y,.
Theorem : The second and third of a power of a dyadic are
equal respectively to the powers of the second and third of
the dyadic.
(*"), = W=0,"
(0')8 = W = 03B'
Theorem : The second of the idemfactor is the idemfactor.
The third of the idemfactor is unity.
I« = I
1=1 (6T)
lg — 1.
Theorem: The product of the second and conjugate of
a dyadic is equal to the product of the third and the
idemfactor.
0a. 0,= 08I, (68)
<P2 = b x c mxn + cxa nxl + axb Ixm,
0C;= la + mb + nc,
<P2 • $<, = [1 m n] (b x c a + c x a b + a x b c).
The antecedents a, b, c of the dyadic 0 may be assumed to
be non-coplanar. Then
(b x c a + c x a b + a x b c) = [ab c] (a' a + V b + c' c)
= [abc] I.
Hence 02 • <&c = ^>3 1 .
120.] Let a dyadic 0 be given. Let it be reduced to the
sum of three dyads of which the three antecedents are
non-coplanar.
314 VECTOR ANALYSIS
0 = al + b m + cn,
02 = b x c mxn + cxa nxl + axb 1 x m,
[Imn].
Theorem: The necessary and sufficient condition that a
dyadic 0 be complete is that the third of 0 be different from
zero.
For it was shown (Art. 106) that both the antecedents and
the consequents of a complete dyadic are non-coplanar.
Hence the two scalar triple products which occur in 08
cannot vanish.
Theorem: The necessary and sufficient condition that a
dyadic $ be planar is that the third of 0 shall vanish but the
second of <P shall not vanish.
It was shown (Art. 106) that if a dyadic 0 be planar its con
sequents 1, m, n must be planar and conversely if the conse
quents be coplanar the dyadic is planar. Hence for a planar
dyadic <P8 must vanish. But $2 cannot vanish. Since a,
b, c have been assumed non-coplanar, the vectors b x c, c x a,
a x b are non-coplanar. Hence if 02 vanishes each of the
vectors mxn, nxl, Ixm vanishes — that is, 1, m, n are col-
linear. But this is impossible since the dyadic 0 is planar
and not linear.
Theorem: The necessary and sufficient condition that a
non-vanishing dyadic be linear is that the second of 0, and
consequently the third of 0, vanishes.
For if 0 be linear the consequents 1, m, n, are collinear.
Hence their vector products vanish and the consequents of
<P2 vanish. If conversely <P2 vanishes, each of its consequents
must be zero and hence these consequents of 0 are collinear.
The vanishing of the third, unaccompanied by the vanish
ing of the second of a dyadic, implies one degree of nullity.
The vanishing of the second implies two degrees of nullity.
LINEAR VECTOR FUNCTIONS 315
The vanishing of the dyadic itself is complete nullity. The
results may be put in tabular form.
08^0, 0 is complete.
<P3 = 0, #2 * 0, 0 is planar. (69)
08 = 0, </>2 = 0, <P * 0, 0 is linear.
It follows immediately that the third of any anti-self-conjugate
dyadic vanishes; but the second does not. For any such
dyadic is planar but cannot be linear.
Nonion Form. Determinants.1 Invariants of a Dyadic
121.] If 0 be expressed in nonion form
0 = au ii + a12 i j + a18 ik (13)
+ a81ki + a32kj + a33kk.
The conjugate of <P has the same scalar coefficients as 0, but
they are arranged symmetrically with respect to the main
diagonal. Thus
(70)
The second of $ may be computed. Take, for instance, one
term. Let it be required to find the coefficient of ij in C?2.
What terms in 0 can yield a double cross product equal to
ij? The vector product of the antecedents must be i and
the vector product of the consequents must be j. Hence the
antecedents must be j and k ; and the consequents, k and i.
These terms are
021J1 x«33kk = -a2la33iJ
a31 k i J a23 j k = a 31 a23 i j.
1 The results hold only for determinants of the third order. The extension to
determinants of higher orders is through Multiple Algebra.
316 VECTOR ANALYSIS
Hence the term in i j in $2 is
This is the first minor of a19 in the determinant
at
*12
ai
a*
This minor is taken with the negative sign. That is, the
coefficient of i j in 02 is what is termed the cof actor of the
coefficient of i j in the determinant. The cofactor is merely
the first minor taken with the positive or negative sign
according as the sum of the subscripts of the term whose
first minor is under consideration is even or odd. The co
efficient of any dyad in 02 is easily seen to be the cofactor of
the corresponding term in $. The cofactors are denoted
generally by large letters.
is the cofactor of a*
33
n
is the cofactor of a12.
is the cofactor of a32.
With this notation the second of 0 becomes
ik
kk
(71)
The value of the third of <P may be obtained by writing
as the sum of three dyads
0 = (an i + a21 j + asl k) i + (a12 i + a22 j + a32 k) j
+ (a13i + a23j + a33k)k
LINEAR VECTOR FUNCTIONS 317
^3 = [Oil i + «21 J + "31 k) («21 * + «22 J + a33k)
This is easily seen to be equal to the determinant
a1
«2
aQ
ai
az
a*
(72)
For this reason 03 is frequently called the determinant of 0
and is written
<P3 = I 0 I (72)'
The idea of the determinant is very natural when 0 is
regarded as expressed in nonion form. On the other hand
unless 0 be expressed in that form the conception of $3,
the third of $, is more natural.
The reciprocal of a dyadic in nonion form may be found
most easily by making use of the identity
02.</>c=03I (68)
or
or
Hence 0"1 =
an ai2 ais
a21 «22 a23
#01 #00 &OQ
Ol oa oo
(73)
318 VECTOR ANALYSIS
If the determinant be denoted by D
(73),
If ¥ is a second dyadic given in nonion form as
+ 631ki + &32kj + &33kk,
the product 0 • W of the two dyadics may readily be found
by actually performing the multiplication
0 . ¥ = On 6U + a12 621 + a18 681) ii + (au 612 + ala622
+ « 6 i J + a 6 + a 6 + « ik
632) k J + (a31 612 + a32 623 + a33 &33> k k'
: W = an in + a12 612 + o18 J18
621
31
Since the third or determinant of a product is equal to the
product of the determinants, the law of multiplication of
determinants follows from (65) and (74).
LINEAR VECTOR FUNCTIONS
319
"11
"21
a.
«i
a22 a23
Ojrtn dinn
an 619 + «io &,
"12
22
a32 622
"11
"31
23
&12 »13
&22 623
&32 633
!3
"11 "13
*21 &13
K31 613
"21'
^31^
a
*32 (
4- a
33
12
22
32
!3
23
(76)
The rule may be stated in words. To multiply two deter
minants form the determinant of which the element in the
mth row and nth column is the sum of the products of the
elements in the rath row of the first determinant and nth
column of the second.
If 0 = al
<?2 = bxc mxn + cxa nxl + axb Ixm.
Then
I 02I =(^2)3= [bxc cxa axb] [mxn nxl Ixm]
Hence I <P 2 I = (<P2)3 = [a b c]2 [1 m n]2 = <P32.
Hence
n
22
33
*ia
"22
a2
a«
(77)
The determinant of the cofactors of a given determinant of
the third order is equal to the square of the given determinant.
122.] A dyadic 0 has three scalar invariants — that is
three scalar quantities which are independent of the form in
which ^ is expressed. These are
the scalar of <P, the scalar of the second of <P, and the third
or determinant of 0. If 0 be expressed in nonion form these
quantities are
320
VECTOR ANALYSIS
(78)
*11
hi
32
*18
33
No matter in terms of what right-handed rectangular system
of these unit vectors 0 may be expressed these quantities are
the same. The scalar of 0 is the sum of the three coefficients
in the main diagonal. The scalar of the second of 0 is the
sum of the first minors or cofactors of the terms in the
main diagonal. The third of 0 is the determinant of the
coefficients. These three invariants are by far the most
important that a dyadic 0 possesses.
Theorem : Any dyadic satisfies a cubic equation of which
the three invariants 0S, 0%& (P3 are the coefficients.
By (68) (0-xI\*(0-xY)c = (0-xl\
#n — x a12 a13
21 #22 X #23
#31 #32 #33 X
Hence (# — x I)3 = 0Z — x 02S + x2 0S — x*
as may be seen by actually performing the expansion.
(<p __ x i)2 . f<p _ x !)<; = 0Z — x 0^ + x2 0S — x*.
This equation is an identity holding for all values of the
scalar x. It therefore holds, if in place of the scalar x, the
dyadic 0 which depends upon nine scalars be substituted.
That is
But the terms upon the left are identically zero. Hence
LINEAR VECTOR FUNCTIONS 321
This equation may be called the Hamilton-Cayley equation.
Hamilton showed that a quaternion satisfied an equation
analogous to this one and Cayley gave the generalization to
matrices. A matrix of the Tith order satisfies an algebraic
equation of the nth degree. The analogy between the theory
of dyadics and the theory of matrices is very close. In fact,
a dyadic may be regarded as a matrix of the third order and
conversely a matrix of the third order may be looked upon as
a dyadic. The addition and multiplication of matrices and
dyadics are then performed according to the same laws. A
generalization of the idea of a dyadic to spaces of higher
dimensions than the third leads to Multiple Algebra and the
theory of matrices of orders higher than the third.
SUMMARY OF CHAPTER V
A vector r' is said to be a linear function of a vector r
when the components of r' are linear homogeneous functions
of the components of r. Or a function of r is said to be a
linear vector function of r when the function of the sum of
two vectors is the sum of the functions of those vectors.
»(ri + ra)=f(r1) + f(ra). (4)
These two ideas of a linear vector function are equivalent.
A sum of a number of symbolic products of two vectors,
which are obtained by placing the vectors in juxtaposition
without intervention of a dot or cross and which are called
dyads, is called a dyadic and is represented by a Greek
capital. A dyadic determines a linear vector function of
a vector by direct multiplication with that vector
0 = &1 bx + a2 b2 + a3 b3 + • • - (7)
0 - r = ax bj • r + a2 b2 • r + a3 b3 • r H (8)
21
322 VECTOR ANALYSIS
Two dyadics are equal when they are equal as operators
upon all vectors or upon three non-coplanar vectors. That
is, when
<P • i = W • r for all values or for three non-
coplanar values of r, (10)
or r • 0 = r • ¥ for all values or for three non-
coplanar values of r,
or s • 0 • r = s • W • r for all values or for three non-
coplanar values of r and s.
Any linear vector function may be represented by a dyadic.
Dyads obey the distributive law of multiplication with
regard to the two vectors composing the dyad
(a + b + c+ •••) (1 + m + n + ...) = al + am + an+ •
+ bl + bm + bn +
+ cl + cm + en +
(11)'
Multiplication by a scalar is associative. In virtue of these
two laws a dyadic may be expanded into a sum of nine terms
by means of the fundamental dyads,
ii, ij, ik,
ji, Jj, Jk, (12)
ki, kj, kk,
as 0 = anii + a12 i j + a18ik,
= «ai J * + <*22 J J + <*23 J k» (13)
= a31 k i + a82 k j + a33 k k.
If two dyadics are equal the corresponding coefficients in
their expansions into nonion form are equal and conversely
LINEAR VECTOR FUNCTIONS 323
Any dyadic may be expressed as the sura of three dyads of
which the antecedents or the consequents are any three
given non-coplanar vectors. This expression of the dyadic is
unique.
The symbolic product ab known as a dyad is the most
general product of two vectors in which multiplication by a
scalar is associative. It is called the indeterminate product.
The product imposes five conditions upon the vectors a and
b. Their directions and the product of their lengths are
determined by the product. The scalar and vector products
are functions of the indeterminate product. A scalar and
a vector may be obtained from any dyadic by inserting a dot
and a cross between the vectors in each dyad. This scalar
and vector are functions of the dyadic.
0* = »i • *! + aa • ba + a8 • b8 + • • • (18)
0X = &1 x bj + a2 x b2 + a3 x b3 + • • • (19)
0, = i-0.i + j*0-j + k'0*k (20)
= an -f a22 + #339
0X = (j . 0 . k - k • 0 • j) i + (k • 0 • i - i • 0 • k) j
+ (i- 0-j -j • 0«i) k (21)
The direct product of two dyads is the dyad whose ante
cedent and consequent are respectively the antecedent of the
first dyad and the consequent of the second multiplied by
the scalar product of the consequent of the first dyad and
the antecedent of the second.
JL
(ab) • (c d) = (b . c) a/ *T (23)
The direct product of two dyadics is the formal expansion,
according to the distributive law, of the product into the
324 VECTOR ANALYSIS
sum of products of dyads. Direct multiplication of dyadics
or of dyadics and a vector at either end or at both ends obeys
the distributive and associative laws of multiplication. Con
sequently such expressions as
Q.W.T, s.0-?7*, s.^.^.r, $>.W.Q (24)-(26)
may be written without parentheses; for parentheses may
be inserted at pleasure without altering the value of the
product. In case the vector occurs at other positions than
at the end the product is no longer associative.
The skew product of a dyad and a vector may be defined
by the equation
(ab) x r = a b x r,
r x (ab) = r x a b. (28)
The skew product of a dyadic and a vector is equal to the
formal expansion of that product into a sum of products of
dyads and that vector. The statement made concerning the
associative law for direct products holds when the vector is
connected with the dyadics in skew multiplication. The
expressions
r x 0 • ?F, 0 • ^ x r, r x $ • s, r • $ x s, r x <P x s (29)
may be written without parentheses and parentheses may be
inserted at pleasure without altering the value of the product.
Moreover
s • (r x <P) = (s x r) - <P, (<P x r) • s = 0 • (r x s),
<p.(rx ?P) = (0 x r) • W. (31)'
But the parentheses cannot be omitted.
The necessary and sufficient condition that a dyadic may
be reduced to the sum of two dyads or to a single dyad or
to zero is that, when expressed as the sum of three
dyads of which the antecedents (or consequents) are known
LINEAR VECTOR FUNCTIONS 325
to be non-coplanar, the consequents (or antecedents) shall
be respectively coplanar or collinear or zero. A complete
dyadic is one which cannot be reduced to a sum of fewer
than three dyads. A planar dyadic is one which can be
reduced to a sum of just two dyads. A linear dyadic is one
which can be reduced to a single dyad.
A complete dyadic possesses no degree of nullity. There
is no direction in space - for which it is an annihilator. A
planar dyadic possesses one degree of nullity. There is one
direction in space for which it is an annihilator when used as
a prefactor and one when used as a postfactor. A linear
dyadic possesses two degrees of nuljity. There are two
independent directions in space for which it is an annihilator
when used as a prefactor and two directions when used as a
postfactor. A zero dyadic possesses three degrees of nullity
or complete nullity. It annihilates every vector in space.
The products of a complete dyadic and a complete, planar,
or linear dyadic are respectively complete, planar, or linear.
The products of a planar dyadic with a planar or linear dyadic
are respectively planar or linear, except in certain cases where
relations of perpendicularity between the consequents of the
first dyadic and the antecedents of the second introduce one
more degree of nullity into the product. The product of a
linear dyadic by a linear dyadic is in general linear ; but in
case the consequent of the first is perpendicular to the ante
cedent of the second the product vanishes. The product of
any dyadic by a zero dyadic is zero.
A dyadic which when applied to any vector in space re
produces that vector is called an idemfactor. All idemfactors
are equal and reducible to the form
I = ii + jj + kk. (33)
Or I = aa' + bb' + cc'. (34)
The product of any dyadic and an idemfactor is that dyadic.
326 VECTOR ANALYSIS
If the product of two complete dyadics is equal to the idem-
factor the dyadics are commutative and either is called
the reciprocal of the other. A complete dyadic may be
canceled from either end of a product of dyadics and vectors
as in ordinary algebra ; for the cancelation is equivalent to
multiplication by the reciprocal of that dyadic. Incomplete
dyadics possess no reciprocals. They correspond to zero in
ordinary algebra. The reciprocal of a product is equal to the
product of the reciprocals taken in inverse order.
(0. 5F)-1 = 5F-1 • 0-i. (38)
The conjugate of a dyadic is the dyadic obtained by inter
changing the order of the antecedents and consequents. The
conjugate of a product is equal to the product of the con
jugates taken in the opposite order.
(0. 9%= Wc. 0C. (40)
The conjugate of the reciprocal is equal to the reciprocal of
the conjugate. A dyadic may be divided in one and only
one way into the sum of two parts of which one is self-
conjugate and the other anti-self-conjugate.
Any anti-self-conjugate dyadic or the anti-self-conjugate
part of any dyadic, used in direct multiplication, is equivalent
to minus one-half the vector of that dyadic used in skew
multiplication.
T=-j0xxr,
(44)
A dyadic of the form c X I or I x c is anti-self-conjugate and
used in direct multiplication is equivalent to the vector o
used in skew multiplication.
LINEAR VECTOR FUNCTIONS 327
Also c x r = (I x c) • r = (c x I) • r, (46)
c x <P = (I x c) • 0 = (c x I) • 0.
The dyadic c X I or I x c, where c is a unit vector is a quad-
ran tal versor for vectors perpendicular to c and an annihilator
for vectors parallel to c. The dyadic Ixc + ccisa true
quadrantal versor for all vectors. The powers of these dyadics
behave like the powers of the imaginary unit V^l, as may
be seen from the geometric interpretation. Applied to the
unit vectors i, j, k
I x i = i x I = kj - j k, etc. (49)
The vector a x b in skew multiplication is equivalent to
(a x b) X I in direct multiplication.
(ax b) x 1 = 1 x (ax b)=ba-ab (50)
(a x b) x r = (b a — a b) • r
r x (a x b) = r • (b a - ab). (51)
A complete dyadic may be reduced to a sum of three
dyads of which the antecedents among themselves and the
consequents among themselves each form a right-handed
rectangular system of three unit vectors and of which the
scalar coefficients are all positive or all negative.
0=± (ai'i + ftj'j + ck'k). (53)
This is called the normal form of the dyadic. An incom
plete dyadic may be reduced to this form but one or more of
the coefficients are zero. The reduction is unique in case
the constants a, 6, c are different. In case they are not
different the reduction may be accomplished in more than
one way. Any self-conjugate dyadic may be reduced to
the normal form
4> = aii + 6jj + ckk, (55)
in which the constants a, S, c are not necessarily positive.
328 VECTOR ANALYSIS
The double dot and double cross multiplication of dyads
is defined by the equations
ab:cd = a«c b.d, (56)
ab£cd = axc bxd. (57)
The double dot and double cross multiplication of dyadics
is obtained by expanding the product formally, according to
the distributive law, into a sum of products of dyads. The
double dot and double cross multiplication of dyadics is com
mutative but not associative.
One-half the double cross product of a dyadic 0 by itself
is called the second of 0. If
<P2=i <Px <P = b xc mxn + cxa nxl+axb Ixm. (61)
One-third of the double dot product of the second of 0 and 0
is called the third of 0 and is equal to the product of the
scalar triple product of the antecedents of 0 and the scalar
triple product of the consequent of 0.
0a = \0$ 0: <P=[abc] [Imn]. (62)
The second of the conjugate is the conjugate of the second.
The third of the conjugate is equal to the third of the
original dyadic. The second and third of the reciprocal are
the reciprocals of the second and third of the second and
third of a dyadic. The second and third of a product are the
products of the seconds and thirds.
(*c\ = (*.)*
(65)
LINEAR VECTOR FUNCTIONS 329
The product of the second and conjugate of a dyadic is equal
to the product of the third and the idemfactor.
^^c=^1 (68)
The conditions for the various degrees of nullity may be
expressed in terms of the second and third of 0.
4>3 * 0, 0 is complete
08 = 0, <P2 * 0, 0 is planar (69)
<P3 = 0, $2 = 0, 0 * 0, 0 is linear.
The closing sections of the chapter contain the expressions
(70)-(78) of a number of the results in nonion form and the
deduction therefrom of a number of theorems concerning
determinants. They also contain the cubic equation which is
satisfied by a dyadic 4>.
03 _ Qa 02 + 0^ 03 + ^ [ _ 0 (79)
This is called the Hamilton-Cayley equation. The coeffi
cients 0S, <P<iS, and 03 are the three fundamental scalar in
variants of <P.
EXERCISES ON CHAPTER V
1. Show that the two definitions given in Art. 98 for
a linear vector function are equivalent
2. Show that the reduction of a dyadic as in (15) can be
accomplished in only one way if a, b, c, 1, m, n, are given.
3. Show (<P x a)c= - a x (1>C.
4. Show that if <Pxr= ¥ XT for any value of r different
from zero, then 0 must equal ?P— unless both 0 and ¥ are
linear and the line of their consequents is parallel to r.
5. Show that if 0 • r = 0 for any three non-coplanar values
of r, then 0 = 0.
330 VECTOR ANALYSIS
6. Prove the statements made in Art. 106 and the con
verse of the statements.
7. Show that if Q is complete and if 0 • Q = W • Q , then
<P and W are equal. Give the proof by means of theory
developed prior to Art. 109.
8. Definition : Two dyadics such that 0 • ?r= ¥• 0 — that
is to say, two dyadics that are commutative — are said to be
homologous. Show that if any number of dyadics are homo
logous to one another, any other dyadics which may be obtained
from them by addition, subtraction, and direct multiplication
are homologous to each other and to the given dyadics. Show
also that the reciprocals of homologous dyadics are homolo
gous. Justify the statement that if 0 • ¥~l or ¥~l • (P,
which are equal, be called the quotient of 0 by ?F, then the
rules governing addition, subtraction, multiplication and
division of homologous dyadics are identical with the rules
governing these operations in ordinary algebra — it being
understood that incomplete dyadics are analogous to zero,
and the idemfactor, to unity. Hence the algebra and higher
analysis of homologous dyadics is practically identical with
that of scalar quantities.
9. Show that (I X c) • 0 — c X $ and (c X I) • & = c X #.
10. Show that whether or not a, b, c be coplanar
abxc+bcxa+caxb = [abc]I
and bxca+cxab+axbc=[abc]L
11. If a, b, c are coplanar use the above relation to prove
the law of sines for the triangle and to obtain the relation
with scalar coefficients which exists between three coplanar
vectors. This may be done by multiplying the equation by a
unit normal to the plane of a, b, and c.
12. What is the-condition which must subsist between the
coefficients in the expansion of a dyadic into nonion form if
LINEAR VECTOR FUNCTIONS 331
the dyadic be self -con jugate ? What, if the dyadic be anti-
self-conjugate ?
13. Prove the statements made in Art. 116 concerning the
number of ways in which a dyadic may be reduced to its
normal form.
14. The necessary and sufficient condition that an anti-
self-conjugate dyadic 0 be zero is that the vector of the
dyadic shall be zero.
15. Show that if 0 be any dyadic the product 0 • <PC is
self-conjugate.
16. Show how to make use of the relation $x = 0 to
demonstrate that the antecedents and consequents of a self
conjugate dyadic are the same (Art. 116).
17. Show that 02 £ <P2 = 2&3 0
and (0 + W\ = </>2 + 4>*V + ¥^
18. Show that if the double dot product 0 : 0 of a dyadic
by itself vanishes, the dyadic vanishes. Hence obtain the
condition for a linear dyadic in the f orin <P2 : 02 = 0.
19. Show that (<P + ef)3 = <P3 + e- 02 • f.
20. Show that (0 + ?T)3 = 08 + <P2 : V + d> : ?F2 + V*
21. Show that the scalar of a product of dyadics is un
changed by cyclic permutation of the dyadics. That is
CHAPTER VI
ROTATIONS AND STRAINS
123.] IN the foregoing chapter the analytical theory of
dyadics has been dealt with and brought to a state of
completeness which is nearly final for practical purposes.
There are, however, a number of new questions which present
themselves and some old questions which present themselves
under a new form when the dyadic is applied to physics
or geometry. Moreover it was for the sake of the applica
tions of dyadics that the theory of them was developed. It is
then the object of the present chapter to supply an extended
application of dyadics to the theory of rotations and strains
and to develop, as far as may appear necessary, the further
analytical theory of dyadics.
That the dyadic $ may be used to deuote a transformation
of space has already been mentioned. A knowledge of the
precise nature of this transformation, however, was not needed
at the time. Consider r as drawn from a fixed origin, and r'
as drawn from the same origin. Let now
r' = 0-r.
This equation therefore may be regarded as defining a trans
formation of the points P of space situated at the terminus of
r into the point P', situated at the terminus of r'. The origin
remains fixed. Points in the finite regions of space remain in
the finite regions of space. Any point upon a line
r = b + x a
becomes a point rf = $ • b + # $ • *•
ROTATIONS AND STRAINS 333
Hence straight lines go over into straight lines and lines
parallel to the same line a go over by the transformation into
lines parallel to the same line 0 • a. In like manner planes
go over into planes and the quality of parallelism is invariant.
Such a transformation is known as a homogeneous strain.
Homogeneous strain is of frequent occurrence in physics. For
instance, the deformation of the infinitesimal sphere in a fluid
(Art. 76) is a homogeneous strain. In geometry the homo
geneous strain is generally known by different names. It is
called an affine collineation with the origin fixed. Or it is
known as a linear homogeneous transformation. The equa
tions of such a transformation are
x1 = a x +
n l2 13
y< =
124.] Theorem : If the dyadic 0 gives the transformation
of the points of space which is due to a homogeneous strain,
02, the second of 0, gives the transformation of plane areas
which is due to that strain and all volumes are magnified by
that strain in the ratio of 03, the third or determinant of 0
to unity.
Let <P = al + bm + cn
r' = <P.r = al-r-f bm »r -f cn«r.
The vectors 1', m', n' are changed by 0 into a, b, c. Hence
the planes determined by m' and n', n' and 1', 1' and m' are
transformed into the planes determined by b and c, c and a,
a and b. The dyadic which accomplishes this result is
$2 =r b x c mxn + cxa nxl + axb Ixm.
Hence if s denote any plane area in space, the transformation
due to 0 replaces s by the area s' such that
334 VECTOR ANALYSIS
It is important to notice that the vector s denoting a plane
area is not transformed into the same vector s' as it would
be if it denoted a line. This is evident from the fact that in
the latter case 0 acts on s whereas in the former case <P2 acts
upon s.
To show that volumes are magnified in the ratio of <PZ to
unity choose any three vectors d, e, f which determine the
volume of a parallelepiped [d e f]. Express 0 with the vec
tors which form the reciprocal system to d, e, f as consequents.
The dyadic <P changes d, e, f into a, b, c (which are different
from the a, b, c above unless d, e, f are equal to 1', m', n').
Hence the volume [d e f ] is changed into the volume [a b c].
08 = [abc][dVf]
[d'e'fr^Cdef].
Hence [a b c] = [d e f] $3.
The ratio of the volume [a b cj to [d e f] is as <P3 is to unity.
But the vectors d, e, f were any three vectors which deter
mine a parallelepiped. Hence all volumes are changed by
the action of 0 in the same ratio and this ratio is as 03 is to 1.
Eotations about a Fixed Point. Versors
125.] Theorem : The necessary and sufficient condition that
a dyadic represent a rotation about some axis is that it be
reducible to the form
0 = i'i+j'j + k'k (1)
where i', j', k' and i, j, k are two right-handed rectangular
systems of unit vectors.
Let r = #i-f-f-3k
ROTATIONS AND STRAINS 335
Hence if C? is reducible to the given form the vectors i, j, k
are changed into the vectors i', j', k' and any vector r is
changed from its position relative to i, j, k into the same posi
tion relative to i',j',k'. Hence by the transformation no
change of shape is effected. The strain reduces to a rotation
which carries i, j, k into i', j', k'. Conversely suppose the
body suffers no change of shape — that is, suppose it subjected
to a rotation. The vectors i, j, k must be carried into another
right-handed rectangular system of unit vectors. Let these
be i', j', k'. The dyadic <P may therefore be reduced to the
form
0 = i'i + j'j+k'k.
Definition : A dyadic which is reducible to the form
i'i + j'j + k'k
and which consequently represents a rotation is called a
versor.
Theorem: The conjugate and reciprocal of a versor are
equal, and conversely if the conjugate and reciprocal of a
dyadic are equal the dyadic reduces to a versor or a versor
multiplied by the negative sign.
Let 0 = i'i+j'j + k'k,
Hence the first part of the theorem is proved. To prove the
second part let
0 = ai + b j + ck,
<pc = i*+j b + kc,
If 4>-i =<PC,
Hence aa4-bb
336 VECTOR ANALYSIS
Hence (Art. 108) the antecedents a, b, c and the consequents
a, b, c must be reciprocal systems. Hence (page 87) they
must be either a right-handed or a left-handed rectangular
system of unit vectors. The left-handed system may be
changed to a right-handed one by prefixing the negative
sign to each vector. Then
#.*rff,tnt«). (iy
The third or determinant of a versor is evidently equal to
unity ; that of the versor with a negative sign, to minus one.
Hence the criterion for a versor may be stated in the form
0 • $„ = I. 3>n = I 0 I = 1 (%\
{/ 3 \ /
Or inasmuch as the determinant of 0 is plus or minus one
if (P* (PC=I, it is only necessary to state that if
C ' 3 ' \. /
$ is a versor.
There are two geometric interpretations of the transforma
tion due to a dyadic 0 such that
0 9 @ __. j 0 = | 0 1 = 1 (3)
0 — (J/ 1 _j_ j' j _j_ k' k) .
The transformation due to 0 is one of rotation combined with
reflection in the origin. The dyadic i'i+j'j + k'k causes a
rotation about a definite axis — it is a versor. The negative
sign then reverses the direction of every vector in space and
replaces each figure by a figure symmetrical to it with respect
to the origin. By reversing the directions of i' and j' the
system i', j', k' still remains right-handed and rectangular,
but the dyadic takes the form
0 = i'i+j'j-k'k,
or <P = (i'i'+j'j'-k'k') .(i'i + j'j + k'k).
ROTATIONS AND STRAINS 337
Hence the transformation due to 0 is a rotation due to
i'i+j'j + k'k followed by a reflection in the plane of i' and
j'. For the dyadic i'i' + j'j' — k'k' causes such a transfor
mation of space that each point goes over into a point sym
metrically situated to it with respect to the plane of i' and j'.
Each figure is therefore replaced by a symmetrical figure.
Definition : A transformation that replaces each figure by
a symmetrical figure is called a perversion and the dyadic
which gives the transformation is called a perversor.
The criterion for a perversor is that the conjugate of a
dyadic shall be equal to its reciprocal and that the determi
nant of the dyadic shall be equal to minus one.
4>.<PC = I, I0I=-1. (3)
Or inasmuch as if 0 • C?c = I, the determinant must be plus
or minus one the criterion may take the form
0 - 0C = I, I 0 I < 0, (3)'
0 is a perversor.
It is evident from geometrical considerations that the prod
uct of two versors is a versor ; of two perversors, a versor ;
but of a versor and a perversor taken in either order, a
perversor.
. 126.] If the axis of rotation be the i-axis and if the angle
of rotation be the angle q measured positive in the positive
trigonometric direction, then by the rotation the vectors
i, j, k are changed into the vectors i',j',k' such that
i' = i
j' = j cos q + k sin y,
k' = — j sin q + k cos q.
The dyadic $ = i'i + j' j + k'k which accomplishes this rota
tion is
338 VECTOR ANALYSIS
0 = ii + cos q (jj + kk) + sin q (k j - jk). (4)
jj +kk = I-ii,
kj-jk = I x i.
Hence 0 = i i + cos q (I — i i) + sin q I x i. ,(5)
If more generally in place of the i-axis any axis denoted
by the unit vector a be taken as the axis of rotation and if as
before the angle of rotation about that axis be denoted by q,
the dyadic 0 which accomplishes the rotation is
0 = a a + cos q (I — a a) + sin q I x a. (6)
To show that this dyadic actually does accomplish the
rotation apply it to a vector r. The dyad a a is an idemfactor
for all vectors parallel to a; but an annihilator for vectors
perpendicular to a. The dyadic I — a a is an idemfactor
for all vectors in the plane perpendicular to a; but an
annihilator for all vectors parallel to a. The dyadic I x a
is a quadrantal versor (Art. 113) for vectors perpendicular
to a; but an annihilator for vectors parallel to a. If then
r be parallel to a
0.r = aa«r = r.
Hence 0 leaves unchanged all vectors (or components of
vectors) which are parallel to a. If r is perpendicular to a
0 . r = cos q r + sin q a x r.
Hence the vector r has been rotated in its plane through the
angle q. If r were any vector in space its component parallel
to a suffers no change ; but its component perpendicular to a
is rotated about a through an angle of q degrees. The whole
vector is therefore rotated about a through that angle.
Let a be given in terms of i, j, k as
alaz ik
ROTATIONS AND STRAINS 339
-r a2ax ji + a22 j j + a2a3 jk
+ azal ki + «8 «2 kj + a32 kk,
I = ii + jj + kk,
I X a = 0ii-a3ij + «2ik,
~a2 ki + ax kj + Okk.
Hence
$ = {&J2 (1 — cos #) + cos #} i i
+ Saia2 (1 — cos 2) ~~ a3 Sin 2} lj
+ {aias (1 — c°s ?) + aa sin ^^ ik
+ {«2a1 (1 — cos^) + a3sin q} ji
+ {«22 C1 - cos 2) + cos q} j j
+ (a2 a3 (1 "" COS 2) — al Sln 2l J *
+ {«3«! (1 — cos ^) — a2 sin q} ki
+ {^3^2 (1 — cos q) 4- ajsin^} kj
+ {«32(1 — cosg) + cosg} kk. (7)
127.] If 0 be written as in equation (4) the vector of <P
and the scalar of 0 may be found.
0X = i x i + cos q (j x j + k x k) + sin q (k x j - j x k)
<PX = — 2 sin q i
<2>s = i - i + cosg (j • j+k -k) +sing (k • j — j -k),
0a = 1 + k cos q.
The axis of rotation i is seen to have the direction of — <PX,
the negative of the vector of 0. This is true in general.
The direction of the axis of rotation of any versor is the
negative of the vector of (P. The proof of this statement
depends on the invariant property of $x. Any versor 0
may be reduced to the form (4) by taking the direction of i
340 VECTOR ANALYSIS
coincident with the direction of the axis of rotation. After
this reduction has been made the direction of the axis is seen
to be the negative of <PX. But <PX is not altered by the
reduction of <P to any particular form — nor is the axis of
rotation altered by such a reduction. Hence the direction of
the axis of rotation is always coincident with — $x, the direc
tion of the negative of the vector of <?.
The tangent of one-half the angle of version q is
sin q * x ,ON
(8)
—
1 + cos q 1 + 4>
s
The tangent of one-half the angle of version is therefore
determined when the values of <#x and <PS are known. The
vector $x and the scalar (Ps, which are invariants of <P, deter
mine completely the versor <?. Let ft be a vector drawn
in the direction of the axis of rotation. Let the magnitude
of ft be equal to the tangent of one-half the angle q of
version.
The vector ft determines the versor <P completely, ft will be
called the vector semi-tangent of version.
By (6) a versor $ was expressed in terms of a unit vector
parallel to the axis of rotation.
<p = a a + cos q (I — a a) + sin q I x a.
Hence if ft be the vector semi-tangent of version
There is a more compact expression for a versor 0 in terms
of the vector semi-tangent of version. Let c be any vector in
space. The version represented by ft carries
c — ft x c into c + ft X c.
ROTATIONS AND STRAINS 341
It will be sufficient to show this in case c is perpendicular to
ft. For if c (or any component of it) were parallel to ft the
result of multiplying by ft x would be zero and the statement
would be that c is carried into c. In the first place the mag
nitudes of the two vectors are equal. For
(c — ft x c) • (c — ft x c) = c • c +ftxc-ftxc — 2c-ftxc
(c + ft x c) • (c + ft x c)-= c«c-hftxc'ftxc + 2c«ftxc
c«c + ftxc«ftxc = c»c + ft«ft c • c — ft • c ft.c.
Since ft and c are by hypothesis perpendicular
c-c + ftxc.ftxc=:c2(l + tan2 \ q).
The term c • ft X c vanishes. Hence the equality. In the
second place the angle between the two vectors is equal to q.
(c — ftxc)»(c + ftxc)_c«c — ft x c • ft x c
c2 (1 + tan2 - q) c* (1 + tan2 i j)
2 2
= cos q
c2 (1 + tan2 i q)
(c — ft x c) x (c + ft x c) _ 2 c x (ft x c)
c2 (1 + tan2 1 2) c2 (1 + tan2 I j)
2 *
2 c2 tan i 2
= sin j.
Hence the cosine and sine of the angle between c — ft X c
and c + ft x c are equal respectively to the cosine and sine of
the angle q : and consequently the angle between the vectors
must equal the angle q. Now
342 VECTOR ANALYSIS
C — ftXC=(I — Ixft)-C
and (c + a x c) = (I + I x ft) • c
(I + I x Q) • (I - I x tt)-1 -(I-Ixft) = I + Ixft.
Multiply by c
(I + I x a) • (I - I x Q)-1 • (c - Q x c) = c + a x c.
Hence the dyadic
0 = (I + I x tt) • (I - I X Q)-1 (10)'
carries the vector c — ft x c into the vector c + ft X c no matter
what the value of c. Hence the dyadic 0 determines the
version due to the vector semi-tangent of version ft.
The dyadic I + 1 x ft carries the vector c — ft x c into
(I + ft.ft)c.
(I + I x ft) • (c — ftxc) = c + ftxc — ftxc — ftx(ftxc)
(I + I X Q) • (c — Q X c) = c + Q ' Q c = (1 + Q ' Q)C.
Hence the dyadic
1 + ft«ft
carries the vector c — ft x c into the vector c, if c be perpen
dicular to ft as has been supposed. Consequently the dyadic
(I + Ixft)2
1 + ft-ft
produces a rotation of all vectors in the plane perpendicular
to ft. If, however, it be applied to a vector x ft parallel to ft
the result is not equal to x ft.
+ IXQ)-(I + IXQ) (I + IXQ) .Q *Q
i + O-O *V-* I + Q.Q v"l + Q-
ROTATIONS AND STRAINS 343
To obviate this difficulty the dyad Q, ft, which is an annihilator
for all vectors perpendicular to ft, may be added to the nu
merator. The versor (P may then be written
ftft+CI + IXft)*
1 + ft-ft
(i + 1 x ft) • (i + 1 x ft) = i + 2 1 x ft + (i x ft) . (i x ft)
(Ixft)-(I xft) = (I xft) x ft = l.ftft-ft.ftl.
Hence substituting :
^(l-ft.ft)I + 2ftft + 2Ixft
1 + ft »ft
This may be expanded in nonion form. Let
(11)
128. ] If a is a unit vector a dyadic of the form
0 = 2aa-I (12)
is a liquadrantal versor. That is, the dyadic 0 turns the
points of space about the axis a through two right angles.
This may be seen by setting q equal to TT in the general
expression for a versor
0 = a a 4- cos q (I — a a) + sin q I x a,
or it may be seen directly from geometrical considerations.
The dyadic <P leaves a vector parallel to a unchanged but re
verses every vector perpendicular to a in direction.
Theorem: The product of two biquadrantal versors is a
versor the axis of which is perpendicular to the axes of the
344 VECTOR ANALYSIS
biquadrantal versors and the angle of which is twice the
angle from the axis of the second to the axis of the first
Let a and b be the axes of two biquadrantal versors. The
product
£=(2bb-I).(2aa-I)
is certainly a versor; for the product of any two versors
is a versor. Consider the common perpendicular to a and b.
The biquadrantal versor 2 a a — I reverses this perpendicular
in direction. (2bb— I) again reverses it in direction and con
sequently brings it back to its original position. Hence the
product Q leaves the common perpendicular to a and b un
changed. Q is therefore a rotation about this line as axis.
The cosine of the angle from a to Q • a is
a • Q • a = 2 b - a b • a - a . a = 2 (b • a)2 - 1 = cos 2 (b, a).
Hence the angle of the versor Q is equal to twice the angle
from a to b.
Theorem : Conversely any given versor may be expressed
as the product of two biquadrantal versors, of which the axes
lie in the plane perpendicular to the axis of the given versor
and include between them an angle equal to one half the
angle of the given versor.
For let Q be the given versor. Let a and b be unit vectors
perpendicular to the axis — J?x of this versor. Furthermore
let the angle from a to b be equal to one half the angle of
this versor. Then by the foregoing theorem
J2=(2bb-I).(2aa-I). (14)
The resolution of versors into the product of two biquad
rantal versors affords an immediate and simple method for
compounding two finite rotations about a fixed point. Let
d> and ¥ be two given versors. Let b be a unit vector per-
ROTATIONS AND STRAINS 345
pendicular to the axes of 0 and W. Let a be a unit vector
perpendicular to the axis of <P and such that the angle from
a to b is equal to one half the angle of 0. Let c be a unit
vector perpendicular to the axis of W and such that the angle
from b to c is equal to one half the angle of ¥. Then
</> = (2bb-I).(2aa-I)
$T=(2ec-I).(2bb-I)
V. 0 = (2 cc - I) • (2 bb - 1)2. (2 aa - I).
But (2 bb — I)2 is equal to the idemfactor, as may be seen from
the fact that it represents a rotation through four right angles
or from the expansion
(2bb-I).(2bb-I) = 4b.b bb-4bb + I = I.
Hence W • <P = (2 c c - I) • (2 a a - I).
The product of W into 0 is a versor the axis of which is
perpendicular to a and c and the angle of which is equal to
one half the angle from a to c.
If 0 and W are two versors of which the vector semi-
tangents of version are respectively QJ and ft^ the vector
semi-tangent of version Q3 of the product ¥ • <P is
q1 + a2+a2xa1
a§~ i-a.-a,
Let 0=(2bb-I) • (2aa-I)
and ¥ = (2 c c - I) . (2 bb - I).
¥. <P = (2cc-I) «(2aa-I).
— 0 iff — (V • <?)„
* ~ yx —
ba -2aa -2b b
x = 4a • b b X a,
346 VECTOR ANALYSIS
05 = 4(a.b)2-l,
?T = 4 c • b cb — 2 b b - 2 c c + 1,
rx = 4 c • b c x b,
?r5 = 4(c.b)2-l
JF • <p = 4 c • a ca — 2 c c — 2 a a + I»
(?F. <p)x = 4 c»a c x a,
(ST. 0)^ = 4 (c-a)2-l.
axb bx c axe
Hence
t = - -, ^ = T — , 3= -
a • b b • c a • c
(bxc) x (axb) [abc] b
J
x Q =
a • b b • c a • b b • c
But [abc] r = bxc a • r + c x a b • r + a x b c • r,
bxc axb axe
b • c a • b a • b b • c
Hence Q2 x Qj = — C^ — ft2 + 8
0 • Q =
a • b b - c
(a x b) • (b x c) _ a • b b • c a • c b • b
2" a-bb-c a«bb«c a*bb*c
Hence — r-^r — = 1 — ft« • Qr
a* b b • c
Q. = . . .
ROTATIONS AND STRAINS 847
This formula gives the composition of two finite rotations.
If the rotations be infinitesimal ftj and Q^ are both infinitesi
mal. Neglecting infinitesimals of the second order the for
mula reduces to
The infinitesimal rotations combine according to the law of
vector addition. This demonstrates the parallelogram law for
angular velocities. The subject was treated from different
standpoints in Arts. 51 and 60.
icSy Right Tensors, Tonics, and Cyclotonics
129.] If the dyadic <P be a versor it may be written in the
form (4)
0 = ii + cos q (jj + kk) + sin q (kj - jk).
The axis of rotation is i and the angle of rotation about that
axis is q. Let ¥ be another versor with the same axis and
an angle of rotation equal to q'.
¥ = ii + cos qf (j j + kk) + sin qr (kj — jk).
Multiplying :
0 . y = ¥ • 0 = i i + cos (g + ?') (j j + k k)
+ Bin(j+9')(kj-jk). (16)
This is the result which was to be expected — the product of
two versors of which the axes are coincident is a versor with
the same axis and with an angle equal to the sum of the
angles of the two given versors.
If a versor be multiplied by itself, geometric and analytic
considerations alike make it evident that
02 = i i + cos 2q (j j + kk) + sin 2 q (k j - j k),
and 4>» = ii + cos nq (j j + kk) + sin nq (kj — j k).
348 VECTOR ANALYSIS
On the other hand let 4>l equal jj + kk; and <P2 equal
kj-jk. Then
<p» = (i i + cos q 0l + sin q $2)n.
The product of ii into either 0l or <P2 is zero and into itself is
ii. Hence
4>n = ii + (cos q d>l + sin q 02)n
0n = ii + cosn q (PS + n cos11^1 q sin # fl^""1 • <P2 + • . •
The dyadic ^ raised to any power reproduces itself. (Pf = <Pr
The dyadic <P2 raised to the second power gives the negative
of <#! ; raised to the third power, the negative of <P2 ; raised
to the fourth power, 0l ; raised to the fifth power, <PZ and so
on (Art. 114). The dyadic 0l multiplied by 02 is equal to
<P2. Hence
<pn = i i + cosn q 0l + n cosn~lq sin q <P%
nfn—l)
V ;»-2
But &n = i i + cos n q 0l + sin n q <Py
Equating coefficients of <Pl and $2 in these two expressions
for 0n
n (n — 1)
cos n q = cosn q — ~~^TI - COS>1 " ? sin2 q + ' ' '
71 (71-1) (71-2)
n"3
sm 7i q = TI cos "^ j sin q -- — : - cosn"# sm^ + • •
o !
Thus the ordinary expansions for cos nq and sin 715 are
obtained in a manner very similar to the manner in which
they are generally obtained.
The expression for a versor may be generalized as follows.
Let a,b, c be any three non-coplanar vectors ; and a', V, c', the
reciprocal system. Consider the dyadic
<p = aa' 4- cos q (bb' + cc') + sin q (cb' — be'). (17)
ROTATIONS AND STRAINS 349
This dyadic leaves vectors parallel to a unchanged. Vectors
in the plane of b and c suffer a change similar to rotation.
Let
r = cos p b + sin p c,
r' = <P • r = cos (p + q) b + sin (p + q) c.
This transformation may be given a definite geometrical
interpretation as follows. The vector r, when p is regarded
as a variable scalar parameter, describes an ellipse of which
b and c are two conjugate semi-diameters (page 117). Let
this ellipse be regarded as the parallel projection of the
unit circle
r = cos p i + sin q j.
That is, the ellipse and the circle are cut from the same
cylinder. The two semi-diameters i and j of the circle pro
ject into the conjugate semi-diameters a and b of the ellipse.
The radius vector r in the ellipse projects into the radius vector
f in the unit circle. The radius vector r' in the ellipse which
is equal to 0 • r, projects into a radius vector r' in the circle
such that
f' = cos (p + q) i + sin (jp + q) j.
Thus the vector r in the ellipse is so changed by the applica
tion of 0 as a prefactor that its projection f in the unit circle
is rotated through an angle q.
This statement may be given a neater form by making use
of the fact that in parallel projection areas are changed in a
definite constant ratio. The vector r in the unit circle may
be regarded as describing a sector of which the area is to the
area of the whole circle as q is to 2 TT. The radius vector f
then describes a sector of the ellipse. The area of this sector
is to the area of the whole ellipse as q is to 2 TT. Hence the
dyadic $ applied as a prefactor to a radius vector r in an ellipse
of which b and c are two conjugate semi-diameters advances
that vector through a sector the area of which is to the area of
350 VECTOR ANALYSIS
the whole ellipse as q is to 2-Tr.1 Such a displacement of the
radius vector r may be called an elliptic rotation through a
sector q from its similarity to an ordinary rotation of which
it is the projection.
Definition : A dyadic 0 of the form
0 = aa' + cos q (bb' + cc') + sin q (c V - be') (17)
is called a cyclic dyadic. The versor is a special case of a
cyclic dyadic.
It is evident from geometric or analytic considerations that
the powers of a cyclic dyadic are formed, as the powers of a
versor were formed, by multiplying the scalar q by the power
to which the dyadic is to be raised.
0n = a a' + cos nq (b b' + c c') + sin nq (c V — b c').
If the scalar q is an integral sub-multiple of 2 TT, that is, if
27T
— = m,
1
it is possible to raise the dyadic 0 to such an integral power,
namely, the power w, that it becomes the idemfactor
0 may then be regarded as the mth root of the idemfactor.
In like manner if q and 2 TT are commensurable it is possible
to raise 0 to such a power that it becomes equal to the idem-
factor and even if q and 2 TT are incommensurable a power of
d> may be found which differs by as little as one pleases from
the idemfactor. Hence any cyclic dyadic may be regarded as
a root of the idemfactor.
1 It is evident that fixing the result of the application of <£ to all radii vectors
in an ellipse practically fixes it for all vectors in the plane of b and c. For any
vector in that plane may be regarded as a scalar multiple of a radius vector of
the ellipse.
ROTATIONS AND STRAINS 351
130.] Definition: The transformation represented by the
<Z> = «ii + &jJ+ckk (18)
where a, 6, c are positive scalars is called a ^rare strain. The
dyadic itself is called a rt^Atf tensor.
A right tensor may be factored into three factors
The order in which these factors occur is immaterial. The
transformation
is such that the i and j components of a vector remain un
altered but the k-component is altered in the ratio of c to 1.
The transformation may therefore be described as a stretch or
elongation along the direction k. If the constant c is greater
than unity the elongation is a true elongation : but if c is less
than unity the elongation is really a compression, for the ratio
of elongation is less than unity. Between these two cases
comes the case in which the constant is unity. The lengths
of the k-components are then not altered.
The transformation due to the dyadic 0 may be regarded
as the successive or simultaneous elongation of the com
ponents of r parallel to i, j, and k respectively in the ratios
a to 1, b to 1, c to 1. If one or more of the constants a, 6, c
is less than unity the elongation in that or those directions
becomes a compression. If one or more of the constants is
unity, components parallel to that direction are not altered.
The directions i, j, k are called the principal axes of the strain.
Their directions are not altered by the strain whereas, if the
constants #, &, c be different, every other direction is altered.
The scalars a, 6, c are known as the principal ratios of
elongation.
In Art. 115 it was seen that any complete dyadic was
reducible to the normal form
352 VECTOR ANALYSIS
where a, J, c are positive constants. This expression may be
factored into the product of two dyadics.
0= ± (ai'i' + ftj'j' + ck'k') • (i'i + j' j + k'k), (19)
or 0= ± (i'i+j'j + k'k) • (aii + 6jj + ckk).
The factor i'i + j'j + k'k
which is the same in either method of factoring is a versor.
It turns the vectors i, j, k into the vectors i', j', k'. The vector
semi-tangent of the versor
ixi'+j xj' + k xk'
i>i + ^ + k'klsi+i.i-+j.j'+k.k"
The other factor
ai'i' + l j'j' + ck'k',
or aii
is a right tensor and represents a pure strain. In the first
case the strain has the lines i', j', k' for principal axes: in
the second, i, j, k. In both cases the ratios of elongation are
the same, — a to 1, b to 1, c to 1. If the negative sign occurs
before the product the version and pure strain must have
associated with them a reversal of directions of all vectors in
space — that is, a perversion. Hence
Theorem: Any dyadic is reducible to the product of a
versor and a right tensor taken in either order and a positive
or negative sign. Hence the most general transformation
representable by a dyadic consists of the product of a rota
tion or version about a definite axis through a definite angle
accompanied by a pure strain either with or without perver
sion. The rotation and strain may be performed in either
order. In the two cases the rotation and the ratios of elonga
tion of the strain are the same ; but the principal axes of the
strain differ according as it is performed before or after the
ROTATIONS AND STRAINS 353
rotation, either system of axes being derivable from the other
by the application of the versor as a prefactor or postfactor
respectively.
If a dyadic 0 be given the product of 0 and its conjugate
is a right tensor the ratios of elongation of which are the
squares of the ratios of elongation of (P and the axes of which
are respectively the antecedents or consequents of 0 accord
ing as 0C follows or precedes 0 in the product.
4>± ± (ai'i + 6 j' j + ck'k),
0C= ±(aii' + 6 jj' + ckk'),
0 . 0C = a» i' i' + 62 j' j' + c2 k' k', (20)
c2kk.
The general problem of finding the principal ratios of elonga
tion, the antecedents, and consequents of a dyadic in its
normal form, therefore reduces to the simpler problem of find
ing the principal ratios of elongation and the principal axes
of a pure strain.
131.] The natural and immediate generalization of the
right tensor
is the dyadic <P = aaa' + &bb' + ccc' (21)
where a, 6, c are positive or negative scalars and where a, b, c
and a', b', c' are two reciprocal systems of vectors. Neces
sarily a, b, c and a', b', c' are each three non-coplanar.
Definition : A dyadic that may be reduced to the form
(21)
is called a tonic.
The effect of a tonic is to leave unchanged three non-
coplanar directions a, b, c in space. If a vector be resolved
into its components parallel to a, b, c respectively these
23
354 VECTOR ANALYSIS
components are stretched in the ratios a to 1, & to 1, c to 1.
If one or more of the constants a, &, c are negative the com
ponents parallel to the corresponding vector a, b, c are re
versed in direction as well as changed in magnitude. The
tonic may be factored into three factors of which each
stretches the components parallel to one of the vectors a, b, c
but leaves unchanged the components parallel to the other
two.
cc') • (aa' + &bb' + ccXa
The value of a tonic 0 is not altered if in place of a, b, c
any three vectors respectively collinear with them be sub
stituted, provided of course that the corresponding changes
which are necessary be made in the reciprocal system a', b', c'.
But with the exception of this change, a dyadic which is
expressible in the form of a tonic is so expressible in only
one way if the constants a, 6, c are different. If two of the
constants say J and c are equal, any two vectors coplanar
with the corresponding vectors b and c may be substituted
in place of b and c. If all the constants are equal the tonic
reduces to a constant multiple of the idemfactor. Any three
non-coplanar vectors may be taken for a, b, c.
The product of two tonics of which the axes a, b, c are the
same is commutative and is a tonic with these axes and
with scalar coefficients equal respectively to the products of
the corresponding coefficients of the two dyadics.
0 = ax a a' + \ b V + ^ c c'
c2cc'
0. y = ¥• <? = a1a2aa' + ^^bV-f c^cc'. (22)
The generalization of the cyclic dyadic
a a' + cos q (b V + c c') + sin q (c b' — b c')
is 0 = a aa' -1- 1 (b V + cc') + c (c V - be'), (23)
ROTATIONS AND STRAINS 355
where a, b, c are three non-coplanar vectors of which ar, V, c'
is the reciprocal system and where the quantities a, 6, c, are
positive or negative scalars. This dyadic may be changed
into a more convenient form by determining the positive
scalar p and the positive or negative scalar q (which may
always be chosen between the limits ± TT) so that
and c=psinq. (24)
That is,
and tan2=. (24y
Then
+ cc') + p sin q (cV — be'). (25)
This may be factored into the product of three dyadics
0= (aaa' + bV + cc') • (a a' + p bV + jpcc')»
{aa' + cos q (b b' 4- o c') + sin q (cV - be')}.
The order of these factors is immaterial. The first is a tonic
which leaves unchanged vectors parallel to b and c but
stretches those parallel to a in the ratio of a to 1. If a is
negative the stretching must be accompanied by reversal
in direction. The second factor is also a tonic. It leaves
unchanged vectors parallel to a but stretches all vectors in
the plane of b and c in the ratio p to 1. The third is a
cyclic factor. Vectors parallel to a remain unchanged ; but
radii vectors in the ellipse of which b and c are conjugate
semi-diameters are rotated through a sector such that the
area of the sector is to the area of the whole ellipse as q to
2 TT. Other vectors in the plane of b and c may be regarded
as scalar multiples of the radii vectors of the ellipse.
356 VECTOR ANALYSIS
Definition : A dyadic which is reducible to the form
<P = a aa' + p cos q (bb' + cc') + p sin q (c V — be'), (25)
owing to the fact that it combines the properties of the
cyclic dyadic and the tonic is called a cyclolonic.
The product of two cyclotonics which have the same three
vectors, a, b, c as antecedents and the reciprocal system
a', b', c' for consequents is a third cyclotonic and is com
mutative.
cc') + pl sinql (cbf — be')
5F = a2aa' +jp2 cos j2 (bbf + cc') + jpa sin q2 (cb' — be')
0. 5P*= W* <? = a1a2 aa' + plp2 cos (ql + ja) (bb' + cc')
+ Pi P* sin (2l + &) (c b' - b c'). (26)
Reduction of Dyadics to Canonical Forms
132.] Theorem : In general any dyadic 0 may be reduced
either to a tonic or to a cyclotonic. The dyadics for which
the reduction is impossible may be regarded as limiting cases
which may be represented to any desired degree of approxi
mation by tonics or cyclotonics.
From this theorem the importance of the tonic and cyclo
tonic which have been treated as natural generalizations of
the right tensor and the cyclic dyadic may be seen. The
proof of the theorem, including a discussion of all the
special cases that may arise, is long and somewhat tedious.
The method of proving the theorem in general however is
patent. If three directions a, b, c may be found which are
left unchanged by the application of $ then <P must be a
tonic. If only one such direction can be found, there exists
a plane in which the vectors suffer a change such as that due
to the cyclotonic and the dyadic indeed proves to be such.
ROTATIONS AND STRAINS 357
The question is to find the directions which are unchanged
by the application of the dyadic 0.
If the direction a is unchanged, then
0 • a = a a (27)
or (0 — al).a = 0.
The dyadic 0 — a I is therefore planar since it reduces vectors
in the direction a to zero. In special cases, which are set
aside for the present, the dyadic may be linear or zero. In
any case if the dyadic
<P-aI
reduces vectors collinear with a to zero it possesses at least
one degree of nullity and the third or determinant of <P
vanishes.
(0-aI)8 = 0. (28)
Now (page 331) (0 + W)z = <PB + <P2 : W + 0 : W^ + ¥z.
Hence (4> - a I)8 = <PZ - a <Z>2 : 1 + a2 0 : ^ - a3 18
I2 = I and I3 = 1.
But 0 : 1 = 0
Hence the equation becomes
a3 - a2 0a + a 0^ -03 = 0. (29)
The value of a which satisfies the condition that
is a solution of a cubic equation. Let x replace a. The
cubic equation becomes
x* - x* d>3 + x 02S - 08 = 0. (29)
\
358 VECTOR ANALYSIS
Any value of x which satisfies this equation will be such
that
(*-aI), = 0. (28)'
That is to say, the dyadic 0 — x I is planar. A vector per
pendicular to its consequents is reduced to zero. Hence 0
leaves such a direction unchanged. The further discussion
of the reduction of a dyadic to the form of a tonic or a cyclo-
tonic depends merely upon whether the cubic equation in x
has one or three real roots.
133.] Theorem : If the cubic equation
x* - x* 4>s + x 02* - 08 = 0 (29)'
has three real roots the dyadic <P may in general be reduced
to a tonic.
For let x = a, x = &, x = c
be the three roots of the equation. The dyadics
<P — a I, 0 — 61, <P — cl
are in general planar. Let a, b, c be respectively three
vectors drawn perpendicular to the planes of the consequents
of these dyadics.
b = 0, (30)
(0-cI).c = 0.
Then <P • a = a a,
</>-b = &b, (30)'
<p . c = cc.
If the roots a, &, c are distinct the vectors a, b, c are non-
coplanar. For suppose
c = ma + ?ib
ROTATIONS AND STRAINS 359
m $ • a — raca-ffl>0«b — n c b = 0.
But 0 • a = a a, 0 • b = 6 b.
Hence m (a — c) a + n (b — c) b = 0,
and m(a — c) = 0, n(b — c) = 0.
Hence m = 0 or a = c, TI = 0 or b = c.
Consequently if the vectors a, b, c are coplanar, the roots are
not distinct; and therefore if the roots are distinct, the
vectors a, b, c are necessarily non-coplanar. In case the roots
are not distinct it is still always possible to choose three
non-coplanar vectors a, b, c in such a manner that the equa
tions (30) hold. This being so, there exists a system a', b', c'
reciprocal to a, b, c and the dyadic which carries a, b, c into
a a, b b, c c is the tonic
Theorem : If the cubic equation
x* - x* 4>a + x d>2S - 03 = 0 (29)'
has one real root the dyadic 0 may in general be reduced to
a cyclotonic.
The cubic equation has one real root. This must be posi
tive or negative according as <PB is positive or negative. Let
the root be a. Determine a perpendicular to the plane of
the consequents of 4> — a I.
(<P-aI) .a = 0.
Determine a' also so that
a'. (0- a I) = 0
and let the lengths of a and a' be so adjusted that a' • a = l.
This cannot be accomplished in the special case in which a
360 VECTOR ANALYSIS
and a; are mutually perpendicular. Let b be any vector in
the plane perpendicular to a'.
a' • (0 - a I) - b = 0.
Hence (<P — al)«b is perpendicular to a'. Hence <P»b is
perpendicular to a'. In a similar manner <P2» b, $3»b, and
<P~l • b, 0~2«b, etc., will all be perpendicular to a' and lie in
one plane. The vectors <P • b and b cannot be parallel or 0
would have the direction b as well as a unchanged and
thus the cubic would have more than one real root.
The dyadic 0 changes a, 0 • b, b into 0 • a, </>2 • b, <P • b re
spectively. The volume of the parallelepiped
[<p.a </>2«b </>.b] = </>3[a <P-b b]. (31)
But $«a = aa.
Hence a a • (<P2 • b) x (0 • b) = <P3a • (0 • b) x b. (31)'
The vectors <02 • b, $ • b, b all lie in the same plane. Their
vector products are parallel to a' and to each other. Hence
a (02- b) x (</>.b) = 03 ><P«b xb. (31)"
Inasmuch as a and <P3 have the same sign, let
^ = a-i*s. (32)
Let also b1=;r1#-b b2 = /r2 #2 • b? etc. (33)
and b_! — p (&-1 • b b_2 = p2 #~2 • b, etc.
b2 X bx — bx X b,
or (b2 + b) x bx = 0.
The vectors b2 + b and bx are parallel. Let
b2 + b = 27ibr (34)
Then b3 + b1 = 27ib2 b1 + b2 = 2nb3 etc.,
bx -f b_! = 2 n b b_! + b_2 = 2 n b_x etc.
ROTATIONS AND STRAINS 861
Lay off from a common origin the vectors
b, bj, b2, etc., b_ j, b_2, etc.
Since 0 is not a tonic, that is, since there is no direction in
the plane perpendicular to a' which is left unchanged by 0
these vectors bOT pass round and round the origin as m takes
on all positive and negative values. The value of n must
therefore lie between plus one and minus one. Let
n = cos q. (36)
Then b-j + bj = 2 cos q b.
Determine c from the equation
bx = cos q b + sin q c.
Then b_j = cos q b — sin q c.
Let a', bf, c' be the reciprocal system of a, b, c. This is pos
sible since a' was so determined that a' • a = 1 and since
a, b, c are non-coplanar. Let
¥ = cos q (bV + ccO + sin q (c V - be').
Then r«a = 0, ?F.b = b1, ¥.})_l = b.
Hence (a *&' + p ¥) • & = a a = $ • a,
(a aa' + p W) • b = p bx = 0 - b,
(a aa' + p ¥) • b_a =p b = d> . b_r
The dyadic a a a' + p W changes the vectors a, b and b^ into
the vectors 0 - a, 0 • b, and 0 • b_x respectively. Hence
0 = (a aa' + p W) = a aa' + ^ cos j (bb' + cc7)
4- ^? sin q (c V — b o').
The dyadic 0 in case the cubic equation has only one real
root is reducible except in special cases to a cyclotonic.
The theorem that a dyadic in general is reducible to a tonic
or cyclotonic has therefore been demonstrated.
362 VECTOR ANALYSIS
134.] There remain two cases1 in which the reduction
is impossible, as can be seen by looking over the proof. In
the first place if the constant n used in the reduction to cyclo-
tonic form be ± 1 the reduction falls through. In the second
place if the plane of the antecedents of
and the plane of the consequents are perpendicular the
vectors a and a' used in the reduction to cyclotonic form are
perpendicular and it is impossible to determine a' such that
a • a' shall be unity. The reduction falls through.
If n=±l, b_1 + b1 = ±2b.
Let b_1 + b1 = 2b.
Choose c = b1 — b = b — b_r
Consider the dyadic W = a aa' + p (bV + co') 4- p o V
y.a = aa=<P.a,
*P • b — pb + pc — pbi = <P • b,
?p*.o=jt)c=-pb1— JP b = 0 • c.
Hence <P = a aa' + p (b V + cc') + p cbr (37)
The transformation due to this dyadic may be seen best by
factoring it into three factors which are independent of the
order or arrangement
.(aa' + bb' + cc' + cb7).
1 In these cases it will be seen that the cubic equation has three real roots.
In one case two of them are equal and in the other case three of them. Thus
these dyadics may be regarded as limiting cases lying between the cyclotonic in
which two of the roots are imaginary and the tonic in which all the roots are real
and distinct. The limit may be regarded as taking place either by the pure
imaginary part of the two imaginary roots of the cyclotonic becoming zero or by
two of the roots of the tonic approaching each other.
ROTATIONS AND STRAINS 863
The first factor represents an elongation in the direction a in a
ratio a to 1. The plane of b and c is undisturbed. The
second factor represents a stretching of the plane of b and c in
the ratio tp to 1. The last factor takes the form
I + cb'.
(I + oV) • £a = #a,
(I + c V) • x b = x b + x o,
(I + c V) • x c = x c.
A dyadic of the form I + cb' leaves vectors parallel to a and c
unaltered. A vector #b parallel to b is increased by the vec
tor c multiplied by the ratio of the vector # b to b. In other
words the transformation of points in space is such that the
plane of a and c remains fixed point for point but the points
in planes parallel to that plane are shifted in the direction c
by an amount proportional to the distance of the plane in
which they lie from the plane of a and c.
Definition : A dyadic reducible to the form
I + cb'
is called a shearing dyadic or shearer and the geometrical
transformation which it causes is called a shear. The more
general dyadic
<P = a aa' + p (b V + c c;) + o V (37)
will also be called a shearing dyadic or shearer. The trans
formation to which it gives rise is a shear combined with
elongations in the direction of a and is in the plane of b and c.
If n = — 1 instead of n = +1, the result is much the same.
The dyadic then becomes
$ = a aa'-,p (bV + c<0 - c V (37)'
$ = (a aa' + bbr + cc') • {aaf -p (b D' + cc')> • (I + cV).
364 VECTOR ANALYSIS
The factors are the same except the second which now repre
sents a stretching of the plane of b and c combined with a
reversal of all the vectors in that plane. The shearing dyadic
0 then represents an elongation in the direction a, an elonga
tion combined with a reversal of direction in the plane of
b and c, and a shear.
Suppose that the plane of the antecedents and the plane of
the consequents of the dyadic 0—al are perpendicular. Let
these planes be taken respectively as the plane of j and k and
the plane of i and j. The dyadic then takes the form
<p — a I — A j i + B j j + C k i + D k j.
The coefficient B must vanish. For otherwise the dyadic
j — Bk)
is planar and the scalar a + B is a root of the cubic equation.
With this root the reduction to the form of a tonic may be
carried on as before. Nothing new arises. But if B vanishes
a new case occurs. Let
This may be reduced as follows to the form
ab' + bc'
where a • V = a • c' = b • c' = 0 and b • V = 1.
Square W W 2 = A D ki = ac'.
Hence a must be chosen parallel to k ; and c', parallel to i.
The dyadic W may then be transformed into
Then =AD*, V=Ci+Di
A D
b = A j c' = i.
ROTATIONS AND STRAINS 365
With this choice of a, b, V, c' the dyadic ¥ reduces to the
desired form ab'+ be' and hence the dyadic <P is reduced to
0 = al + ab' + bc' (38)
or 0 = aaa' + abb' + ace' + aV + be'.
This may be factored into the product of two dyadics the
order of which is immaterial.
The first factor al represents a stretching of space in all
directions in the ratio a to 1. The second factor
represents what may be called a complex shear. For
r'= IT + ab'«r+ bc'-r= r-t-aV-r + bc'-r.
If r is parallel to a it is left unaltered by the dyadic Q. If
r is parallel to b it is changed by the addition of a term
which is in direction equal to a and in magnitude propor
tional to the magnitude of the vector r. In like manner
if r is parallel to c it is changed by the addition of a term
which in direction is equal to b and which in magnitude is
proportional to the magnitude of the vector r.
£-zb = (I + ab'-f bc').zb= zb + a a
Q *xc = (I + ab' + be') • xc = xc 4- #b.
Definition : A dyadic which may be reduced to the form
<P = aI + ab' + bc' (38)
is called a complex shearer.
The complex shearer as well as the simple shearer men
tioned before are limiting cases of the cyclotonic and tonic
dyadics.
366 VECTOR ANALYSIS
135.] A more systematic treatment of the various kinds
of dyadics which may arise may be given by means of the
Hamilton-Cayley equation
03 _ 0a 02 + 0^ 0 _ 03 i = 0
and the cubic equation in x
x* - 0S x* + <P25 x - 08 = 0. (29)'
If a, &, c are the roots of this cubic the Hamilton-Cayley
equation may be written as
(0 - al) • (<P - JI) • (0 - <?I) = 0. (40)
If, however, the cubic has only one root the Hamilton-Cayley
equation takes the form
(0_al).(02- 2^0082 4> + p*I) = 0. (41)
In general the Hamilton-Cayley equation which is an equa
tion of the third degree in 0 is the equation of lowest degree
which is satisfied by 0. In general therefore one of the above
equations and the corresponding reductions to the tonic or
cyclotonic form hold. In special cases, however, the dyadic
0 may satisfy an equation of lower degree. That equation
of lowest degree which may be satisfied by a dyadic is called
its characteristic equation. The following possibilities occur.
I. (<P - a I) - (0 - b I) • ( 0 - c I) = 0.
II. (0-aI)
III. (
IV. (<P - a !).(</>- 61) = 0.
V. (0-aI)3 = 0.
VI. (<P — al)2 = 0.
VII. (<P-aI) = 0.
ROTATIONS AND STRAINS 367
In the first case the dyadic is a tonic and may be reduced
to the form
6bb' + ccc'.
In the second case the dyadic is a cyclotonic and may be
reduced to the form
d> = a a a' + p cos q (bb' + cc') + p sin q (eb' — be').
In the third case the dyadic is a simple shearer and may be
reduced to the form
</> = aaa' + 6 (bb' + cc') + cb'.
In the fourth case the dyadic is again a tonic. Two of the
ratios of elongation are the same. The following reduction
may be accomplished in an infinite number of ways.
0 = aaa' + b (bb' + cc').
In the fifth case the dyadic is a complex shearer and may be
so expressed that
0= al + ab'-f be'.
In the sixth case the dyadic is again a simple shearer which
may be reduced to the form
4> = al + cb'=a (aa' 4- bb' + cc') + cb'.
In the seventh case the dyadic is again a tonic which may be
reduced in a doubly infinite number of ways to the form
0 = al = a(aa/ + bb' + cc').
These seven are the only essentially different forms which a
dyadic may take. There are then only seven really different
kinds of dyadics — three tonics in which the ratios of elonga
tion are all different, two alike, or all equal, and the cyclo
tonic together with three limiting cases, the two simple and
the one complex shearer.
368 VECTOR ANALYSIS
Summary of Chapter VI
The transformation due to a dyadic is a linear homogeneous
strain. The dyadic itself gives the transformation of the
points in space. The second of the dyadic gives the trans
formation of plane areas. The third of the dyadic gives the
ratio in which volumes are changed.
The necessary and sufficient condition that a dyadic repre
sent a rotation about a definite axis is that it be reducible to
the form
0 = i'i + j'j + k'k (1)
or that 0 • 4>c = I <P3 = + 1 (2)
or that 0 • ®c = I 08 > 0
The necessary and sufficient condition that a dyadic repre
sent a rotation combined with a transformation of reflection
by which each figure is replaced by one symmetrical to it is
that
0 = -(i'i + j'j + k'k) (iy
or that $ • <PC = I* ^3 = — 1
or that 0.00 = 1, 03<0. (3)
A dyadic of the form (1) is called a versor ; one of the form
(1)', a perversor.
If the axis of rotation of a versor be chosen as the i-axis
the versor reduces to
0 = ii + cos q (j j + kk) + sin q (kj - j k) (4)
or 0 = ii + cos q (I — ii) + sin q I x i. (5)
If any unit vector a is directed along the axis of rotation
<p = a a + cos q (I — a a) + sin q 1 x a (6)
The axis of the versor coincides in direction with — 0X.
ROTATIONS AND STRAINS 369
If a vector be drawn along the axis and if the magnitude of
the vector be taken equal to the tangent of one-half the angle
of rotation, the vector determines the rotation completely.
This vector is called the vector semi-tangent of version.
2 (9)
In terms of Q the versor <P may be expressed in a number of
was.
a
dft / a a \
<P = — — + cos q (I - — — ) + sin q I x
a-ft \ a-ay
(10)
or <D = (I + I x ft) • (I - I x Q)-1 (10)'
• J^ + axQ)' (loy,
tf =
1 + Q-ft
If a is a unit vector a dyadic of the form
<P = 2aa-I (11)
is a biquadrantal versor. Any versor may be resolved into
the product of two biquadrantal versors and by means of
such resolutions any two versors may be combined into
another. The law of composition for the vector semi-tangents
of version is
A dyadic reducible to the form
<P = aa' + cos q (bb' + cc') + sin q (cb'-W) (17)
is called a cyclic dyadic. It produces a generalization of
simple rotation — an elliptic rotation, so to speak. The pro-
24
370 VECTOR ANALYSIS
duct of two cyclic dyadics which have the same antecedents
a, b, c and consequents a' b' c' is obtained by adding their
angles q. A cyclic dyadic may be regarded as a root of the
idemfactor. A dyadic reducible to the form
0 = aii +bjj + ckk (18)
where #, &, c are positive scalars is called a right tensor. It
represents a stretching along the principal axis i, j, k in the
ratio a to 1, b to 1, c to 1 which are called the principal ratios
of elongation. This transformation is a pure strain.
Any dyadic may be expressed as the product of a versor,
a right tensor, and a positive or negative sign.
0 = ± (a i'i' + & j' j' + c k' k') (i' i + j' j + k' k)
or <P= ± (i'i + j'j + k'k).(aii + Jjj + ckk). (19)
Consequently any linear homogeneous strain may be regarded
as a combination of a rotation and a pure strain accompanied
or unaccompanied by a perversion.
The immediate generalizations of the right tensor and the
cyclic dyadic is to the tonic
0 = aaa' + &bb' + ccc' (21)
and cyclotonic
cc') + c(cV-bc) (23)
or <P = aaa' + p cos q (bb' + cc')+^sing' (cV — be') (25)
where p = + V 62 + c2 and tan I q = - — -. (24)'
* 2? + 0
Any dyadic in general may be reduced either to the form
(21), and is therefore a tonic, or to the form (25), and is
therefore a cyclotonic. The condition that a dyadic be a
tonic is that the cubic equation
+ 0^ x - <J>3 = 0 (29)'
ROTATIONS AND STRAINS 371
shall have three real roots. Special cases in which the
reduction may be accomplished in more ways than one arise
when the equation has equal roots. The condition that a
dyadic be a cyclotonic is that this cubic equation shall have
only one real root. There occur two limiting cases in which
the dyadic cannot be reduced to cyclotonic form. In these
cases it may be written as
4> =aaa'+jp (bb' + cc') + cb' (37)
and is a simple shearer, or it takes the form
0 = al + ab' + bc' (38)
and is a complex shearer. Dyadics may be classified accord
ing to their characteristic equations
(<P-aI).(0-&I).(<P-cI) =0 tonic
(# — a I) • (<P2 — 2 p cos q 0 + jp2 1) = 0 cyclotonic
(0 — a I) • (# — & I)2 = 0 simple shearer
(0_ <*!)•($ — &I) = 0 special tonic
(0 — a I)8 = 0 complex shearer
(0 — a I)2 = 0 special simple shearer
(0 — a I) = 0 special tonic.
CHAPTER VII
MISCELLANEOUS APPLICATIONS
Quadric Surfaces
136.] If 0 be any constant dyadic the equation
r . 0 . r = const. (1)
is quadratic in r. The constant, in case it be not zero, may
be divided into the dyadic 0 and hence the equation takes
the form
r • 0 • r = 1,
or r • 0 • r = 0. (2)
The dyadic 0 may be assumed to be self -conjugate. For if
W is an anti-self-conjugate dyadic, the product r • W • r is
identically zero for all values of r. The proof of this 'state
ment is left as an exercise. By Art. 116 any self-conjugate
dyadic is reducible to the form
- t Ll ± JJ
- ± ±
If
Hence the equation r • 0 • r = 1
represents a quadric surface real or imaginary.
The different cases which arise are four in number. If the
signs are all positive, the quadric is a real ellipsoid. If one
sign is negative it is an hyperboloid of one sheet; if two are
QUADRIC SURFACES 373
negative, a hyperboloid of two sheets. If the three signs are
all negative the quadric is imaginary. In like manner the
equation
r • 0 • r = 0
is seen to represent a cone which may be either real or
imaginary according as the signs are different or all alike.
Thus the equation
r • 0- • r = const.
represents a central quadric surface. The surface reduces to
a cone in case the constant is zero. Conversely any central
quadric surface may be represented by a suitably chosen self-
conjugate dyadic 0 in the form
r • d> • r = const.
This is evident from the equations of the central quadric
surfaces when reduced to the normal form. They are
#2 7/2 z2
± — ± — ± - = const.
a2 6J c2
The corresponding dyadic <Pis $ = ± — ± — ± — .
a* o* c*
The most general scalar expression which is quadratic in
the vector r and which consequently when set equal to a con
stant represents a quadric surface, contains terms like
r • r, (r • a) (b • r) , r • c, d • e,
where a, b, c, d, e are constant vectors. The first two terms
are of the second order in r ; the third, of the first order ; and
the last, independent of r. Moreover, it is evident that these
four sorts of terms are the only ones which can occur in a
scalar expression which is quadratic in r.
But r • r = r • I • r,
and (r • a) (b • r) = r • a b • r.
374 VECTOR ANALYSIS
Hence the most general quadratic expression may be reduced
to
where 0 is a constant dyadic, A a constant vector, and 0
a constant scalar. The dyadic may be regarded as self-
conjugate if desired.
To be rid of the linear term r • A, make a change of origin
by replacing r by r' — t.
(r'-t). 0- (r'-t) + (r'-t)« A+ C=0
r' . <P * r' — t • $ • r' — r' • <P • t + t • <P • t
+ r'-A-t- A + (7=0.
Since 0 is self-conjugate the second and third terms are
equal. Hence
r' • 0 • r' + 2 r' • (J A - 0 • t) + Cf = 0.
If now 0 is complete the vector t may be chosen so that
IA = 0-t or t = 5 0-1- A.
L L
Hence the quadric is reducible to the central form
r' • 0 • r' = const.
In case 0 is incomplete it is unt'planar or unilinear because
0 is self -con jugate. If A lies in the plane of 0 or in the line
of 0 as the case may be the equation
is soluble for t and the reduction to central form is still pos
sible. But unless A is so situated the reduction is impossible.
The quadric surface is not a central surface.
The discussion and classification of the various non-central
quadrics is an interesting exercise. It will not be taken up
here. The present object is to develop so much of the theory
QUADRIC SURFACES 375
of quadric surfaces as will be useful in applications to mathe
matical physics with especial reference to non-isotropic
media. Hereafter therefore the central quadrics and in par
ticular the ellipsoid will be discussed.
137.] The tangent plane may be found by differentiation.
r • <P • r = 1.
di • 0 • r + r • <t> • di = 0.
Since <P is self-conjugate these two terms are equal and
dr.0-r = 0. (5)
The increment d r is perpendicular to <P • r. Hence 0 • r is
normal to the surface at the extremity of the vector r. Let
this normal be denoted by K and let the unit normal be n.
BT = <P • r (6)
0 • r 0 • r
n =
r) • (0 • r) Vr • #2 • r*
Let p be the vector drawn from the origin perpendicular to
the tangent plane, p is parallel to n. The perpendicular
distance from the origin to the tangent plane is the square
root of p • p. It is also equal to the square root of r • p.
r • p = r cos (r, p) p = p2.
Hence r • p = p • p.
Or Ll! = , . JL = L
p.p p.p
But r»0«r = r«H = l.
Hence inasmuch as p and IT are parallel, they are equal.
0.r = !T=-^-. (T)
p.p
376 VECTOR ANALYSIS
On page 108 it was seen that the vector which has the direc
tion of the normal to a plane and which is in magnitude equal
to the reciprocal of the distance from the origin to the plane
may be taken as the vector coordinate of that plane. Hence
the above equation shows that <P • r is not merely normal to
the tangent plane, but is also the coordinate of the plane.
That is, the length of <P • r is the reciprocal of the distance
from the origin to the plane tangent to the ellipsoid at
the extremity of the vector r.
The equation of the ellipsoid in plane coordinates may be
found by eliminating r from the two equations.
( r • 0 • r = 1,
Hence r • 0 • r = H • 0-1 • 0 • 0-1 • If = JT • 0"1 • H.
Hence the desired equation is
H-0-i-H = l. (8)
*4'+y+"
c2kk.
Let r = #i-t-yj+3k,
and N = ui + v j + wk,
where u, v, w are the reciprocals of the intercepts of the
plane N upon the axes i, j, k. Then the ellipsoid may be
written in either of the two forms familiar in Cartesian
geometry.
or K •0-1.N = a2wa + Z>2 v2 + c2 w2 = 1. (10)
QUADRIC SURFACES 377
138.] The locus of the middle points of a system of
parallel chords in an ellipsoid is a plane. This plane is
called the diametral plane conjugate with the system of
chords. It is parallel to the plane drawn tangent to the
ellipsoid at the extremity of that one of the chords which
passes through the center.
Let r be any radius vector in the ellipsoid. Let n be the
vector drawn to the middle point of a chord parallel to a.
Let r = s + x a.
If r is a radius vector of the ellipsoid
r • 0 • r = (B + x a) • <D • (s + x a) = 1.
Hence s • $ • s + 2 # s • 0 . a + 2? a • # • a = 1.
Inasmuch as the vector s bisects the chord parallel to a the
two solutions of x given by this equation are equal in mag
nitude and opposite in sign. Hence the coefficient of the
linear term x vanishes. ,.
s . 0 . a = 0.
Consequently the vector s is perpendicular to 0 • a. The
locus of the terminus of s is therefore a plane passed through
the center of the ellipsoid, perpendicular to 0 • a, and parallel
to the tangent plane at the extremity of a.
If b is any radius vector in the diametral plane conjugate
with a, _ A
b • 0 • a = 0.
The symmetry of this equation shows that a is a radius
vector in the plane conjugate with b. Let c be a third radius
vector in the ellipsoid and let it be chosen as the line of
intersection of the diametral planes conjugate respectively
with a and b. Then
a . d> . b = 0,
b . 0 • c = 0, (11)
e • 0 • a = 0.
378 VECTOR ANALYSIS
The vectors a, b, c are changed into <P • a, <D • b, <P • c by
the dyadic 0. Let •
a' = 0 • a, V = 0 • b, c' - b • c.
The vectors a', b', c' form the system reciprocal to a, b, c.
For a • a' = a • 0 • a = 1, b • V = b • 0 • b = 1,
c • o' = c • 0 • o = 1,
and a • V = a • 0 • b = 0, b • c' = b • 0 • c = 0,
c • a' = c • 0 • a = 0.
The dyadic 0 may be therefore expressed in the forms
0 = a'a' + b'b' + cV, (12)
and 0"1 = aa + bb + cc.
If for convenience the three directions a, b, c, be called a
system of three conjugate radii vectors, and if in a similar
manner the three tangent planes at their extremities be called
a system of three conjugate tangent planes, a number of
geometric theorems may be obtained from interpreting the
invariants of 0. A system of three conjugate radii vectors
may be obtained in a doubly infinite number of ways.
The volume of a parallelepiped of which three concurrent
edges constitute a system of three conjugate radii vectors is
constant and equal in magnitude to the rectangular parallele
piped constructed upon the three semi-axes of the ellipsoid.
For let a, b, c be any system of three conjugate axes.
0-1 = aa + bb + cc.
The determinant or third of 0"1 is an invariant and inde
pendent of the form in which 0 is expressed.
03-i=[abc]2.
QUADRIC SURFACES 379
But if 0-!:=a2ii + &2jj + c2kk,
Hence [a b c] = a 6 c.
This demonstrates the theorem. In like manner by inter
preting <P3, <Ps~~\ and 0S it is possible to show that:
The sum of the squares of the radii vectors drawn to an
ellipsoid in a system of three conjugate directions is constant
and equal to the sum of the squares of the semi-axes.
The volume of the parallelepiped, whose three concurrent
edges are in the directions of the perpendiculars upon a system
of three conjugate tangent planes and in magnitude equal to
the reciprocals of the distances of those planes from the
center of the ellipsoid, is constant and equal to the reciprocal
of the parallelepiped constructed upon the semi-axes of the
ellipsoid.
The sum of the squares of the reciprocals of the three per
pendiculars dropped from the origin upon a system of three
conjugate tangent planes is constant and equal to the sum of
the squares of the reciprocals^ the semi-axes.
If i, j, k be three mutually perpendicular unit vectors
4>s = i* </> • i + j • <P • j + k • # • k,
tf^-i = i . 0-i . i + j . 0-i . j + k • 0"1 • k.
Let a, b, c be three radii vectors in the ellipsoid drawn
respectively parallel to i, j, k.
a . 0 . a = b
i • 0 • i j
Hence <Pa = - - - + *
a • 0 • a
But the three terms in this expression are the squares of the
reciprocals of the radii vectors drawn respectively in the i, j,
k directions. Hence :
380 VECTOR ANALYSIS
The sum of the squares of the reciprocals of three mutually
perpendicular radii vectors in an ellipsoid is constant. And
in a similar manner: the sum of the squares of the perpen
diculars dropped from the origin upon three mutually perpen
dicular tangent planes is constant.
139.] The equation of the polar plane of the point deter
mined by the vector a is l
s • 0 • a = 1. (13)
For let s be the vector of a point in the polar plane. The
vector of any point upon the line which joins the terminus of
s and the terminus of a is
y s + #a
x + y
If this point lies upon the surface
8*0*8+
x + y x + y
2 x y
If the terminus of s lies in the polar plane of a the two values
of the ratio x:y determined by this equation must be equal
in magnitude and opposite in sign. Hence the term in x y
vanishes.
Hence s • 0 • a = 1
is the desired equation of the polar plane of the terminus
of a.
Let a be replaced by z a. The polar plane becomes
s . (p . z a = 1,
1
or s • (P • a = - •
z
1 It is evidently immaterial whether the central quadric determined by * be
real or imaginary, ellipsoid or hyperboloid.
QUADRIC SURFACES 381
When z increases the polar plane of the terminus of z a
approaches the origin. In the limit when z becomes infinite
the polar plane becomes
s • 0 • a = 0.
Hence the polar plane of the point at infinity in the direction
a is the same as the diametral plane conjugate with a. This
statement is frequently taken as the definition of the diame
tral plane conjugate with a. In case the vector a is a radius
vector of the surface the polar plane becomes identical with
the tangent plane at the terminus of a. The equation
s • <P • a = 1 or s • IT = 1
therefore represents the tangent plane.
The polar plane may be obtained from another standpoint
which is important. If a quadric Q and a plane P are given,
and P = r • c — C = 0,
the equation (r • 0 • r - 1) + k (r • c - C)2 = 0
represents a quadric surface which passes through the curve
of intersection of Q and P and is tangent to Q along that
curve. In like manner if two quadrics Q and Qf are given,
Q = T • 0 • r — 1 = 0
Q' = T* 0'-r-l = 0,
the equation (r • 0 • r — 1) + k (r • #' • r - 1) = 0
represents a quadric surface which passes through the curves
of intersection of Q and Q' and which cuts Q and Qf at no
other points. In case this equation is factorable into two
equations which are linear in r, and which consequently rep
resent two planes, the curves of intersection of Q and Qr
become plane and lie in those two planes.
382 VECTOR ANALYSIS
If A is any point outside of the quadric and if all the tangent
planes which pass through A are drawn, these planes envelop
a cone. This cone touches the quadric along a plane curve —
the plane of the curve being the polar plane of the point A.
For let a be the vector drawn to the point A. The equation
of any tangent plane to the quadric is
s . 0 . r = 1.
If this plane contains A, its equation is satisfied by a. Hence
the conditions which must be satisfied by r if its tangent
plane passes through A are
a • 0 . r = 1,
r • <P • r = 1.
The points r therefore lie in a plane r • (<P • a) = 1 which
on comparison with (13) is seen to be the polar plane of A.
The quadric which passes through the curve of intersection
of this polar plane with the given quadric and which touches
the quadric along that curve is
(r • 0 • r - 1) + k (a • 0 • r - I)2 = 0.
If this passes through the point -4,
(a . 0 . a - 1) + k (a • 0 • a - I)2 = 0.
Hence (r • 0 • r - 1) (a • 0 • a - 1) - (a • 0 • r - I)2 = 0.
By transforming the origin to the point A this is easily seen
to be a cone whose vertex is at that point
140.] Let 0 be any self-conjugate dyadic. It is expres
sible in the form
where A, 5, C are positive or negative scalars. Further-
more let A<B<G
0 - Bl = (<7- B) kk - (5 - A) ii.
QUADRIC SURFACES 383
Let V C — B k = c and V B — A i = a.
Then 0- Bl = cc-aa = \ j(c + a)(c-a)+(c-a)(c+a) J.
Let c + a = p and c — a = q.
Then <P = 51 + £-(pq + qp). (14)
The dyadic $ has been expressed as the sum of a constant
multiple of the idemfactor and one half the sum
pq + qp.
The reduction has assumed tacitly that the constants -4, B, 0
are different from each other and from zero.
This expression for <P is closely related to the circular
sections of the quadric surface
r • 0 • r = 1.
Substituting the value of $, r • 0 • r = 1 becomes
5 r • r + r« p q • r = 1.
Let r • p = n
be any plane perpendicular to p. By substitution
B r.r-ftt q • r — 1 = 0.
This is a sphere because the terms of the second order all
have the same coefficient B. If the equation of this sphere
be subtracted from that of the given quadric, the resulting
equation is that of a quadric which passes through the inter
section of the sphere and the given quadric. The difference
q • r (r • p — n) = 0.
Hence the sphere and the quadric intersect in two plane
curves lying in the planes
q . r = 0 and r • p = n.
384 VECTOR ANALYSIS
Inasmuch as these curves lie upon a sphere they are circles.
Hence planes perpendicular to p cut the quadric in circles.
In like manner it may be shown that planes perpendicular to
q cut the quadric in circles. The proof may be conducted as
follows :
.5 r • r + r • p q • r = 1.
If r is a radius vector in the plane passed through the center
of the quadric perpendicular to p or q, the term r • p q • r van
ishes. Hence the vector r in this plane satisfies the equation
B r-r = l
and is of constant length. The section is therefore a circular
section. The radius of the section is equal in length to the
mean semi-axis of the quadric.
For convenience let the quadric be an ellipsoid. The con
stants A, B, C are then positive. The reciprocal dyadic (P"1
may be reduced in a similar manner.
B
B \B C \A B
Let 1 = 1-* and d = -i.
Then 0'1 - -^ I = f f - dd = \ j (f + d) (f - d)
+ (f-d)(f + d)j
Let £ + d = u and £ — d = v.
Then 0-i = 4 1 + I O v + vu). (15)
£> *
QUADRIC SURFACES 385
The vectors u and v are connected intimately with the cir
cular cylinders which envelop the ellipsoid
r • 0 • r = 1 or N • (t>~1 • N = 1.
For - N • N + N • u v • N = 1.
z>
If now N be perpendicular to u or v the second term, namely,
N • u v • N, vanishes and hence the equation becomes
N - N = B.
That is, the vector N is of constant length. But the equation
is the equation of a cylinder of which the elements and tan
gent planes are parallel to u. If then N • N is constant the
cylinder is a circular cylinder enveloping the ellipsoid. The
radius of the cylinder is equal in length to the mean semi-axis
of the ellipsoid.
There are consequently two planes passing through the
origin and cutting out circles from the ellipsoid. The normals
to these planes are p and q. The circles pass through the
extremities of the mean axis of the ellipsoid. There are also
two circular cylinders enveloping the ellipsoid. The direction
of the axes of these cylinders are n and v. Two elements of
these cylinders pass through the extremities of the mean axis
of the ellipsoid.
These results can be seen geometrically as follows. Pass
a plane through the mean axis and rotate it about that
axis from the major to the minor axis. The section is an
ellipse. One axis of this ellipse is the mean axis of the
ellipsoid. This remains constant during the rotation. The
other axis of the ellipse varies in length from the major to the
minor axis of the ellipsoid and hence at some stage must pass
through a length equal to the mean axis. At this stage of
25
386 VECTOR ANALYSIS
the rotation the section is a circle. In like manner consider
the projection or shadow of the ellipsoid cast upon a plane
parallel to the mean axis by a point at an infinite distance
from that plane and in a direction perpendicular to it. As the
ellipsoid is rotated about its mean axis, from the position in
which the major axis is perpendicular to the plane of projec
tion to the position in which the minor axis is perpendicular
to that plane, the shadow and the projecting cylinder have the
mean axis of the ellipsoid as one axis. The other axis changes
from the minor axis of the ellipsoid to the major and hence at
some stage of the rotation it passes through a value equal to
the mean axis. At this stage the shadow and projecting
cylinder are circular.
The necessary and sufficient condition that r be the major
or minor semi-axis of the section of the ellipsoid r • $ • r = 1
by a plane passing through the center and perpendicular to a
is that a, r, and 0 • r be coplanar.
Let r • <P • r = 1
and r • a = 0.
Differentiate : d r • $ • r = 0,
d r • a = 0.
Furthermore d r • r = 0,
if r is to be a major or minor axis of the section; for r is a
maximum or a mininum and hence is perpendicular to dr.
These three equations show that a, r, and 0 • r are all ortho
gonal to the same vector dr. Hence they are coplanar.
[a r 4> • r] = 0. (16)
Conversely if [a r <P • r] = 0,
dr may be chosen perpendicular to their common plane.
"
QUADRIC SURFACES 387
Hence r is a maximum or a minimim and is one of the prin
cipal semi-axes of the section perpendicular to a.
141.] It is frequently an advantage to write the equation
of an ellipsoid in the form
r • ?T2 • r = 1, (17)
instead of r • <P • r = 1.
'• This may be done ; because if
ii jj kk
* = -2 + ^ + ir>
is a dyadic such that W* is equal to <P. ¥ may be regarded as
a square root of <P and written as $*. But it must be re
membered that there are other square roots of <P — for
example,
and
For this reason it is necessary to bear in mind that the square
root which is meant by <P* is that particular one which has
been denoted by ¥.
The equation of the ellipsoid may be written in the form
or .r. .r = .
Let r' be the radius vector of a unit sphere. The equation of
the sphere is
r' • r' = 1.
388 VECTOR ANALYSIS
If r'= ?T«rit becomes evident that an ellipsoid may be
transformed into a unit sphere by applying the operator ¥
to each radius vector r, and vice versa, the unit sphere may
be transformed into an ellipsoid by applying the inverse oper
ator ¥~l to each radius vector r'. Furthermore if a, b, c are
a system of three conjugate radii vectors in an ellipsoid
a- ?F2 • a = b • §F2 • b = c • ^.0 = 1,
a • ¥2 • b = b • ¥* • c = c • ¥2 • a = 0.
If for the moment a', b', c' denote respectively W • a, W • b,
W • c,
a' • a' = V • V = c' • c' = 1,
a' . V = V • c' = c' • a' = 0.
Hence the three radii vectors a', b', c' of the unit sphere into
which three conjugate radii vectors in the ellipsoid are trans
formed by the operator W ~1 are mutually orthogonal. They
form a right-handed or left-handed system of three mutually
perpendicular unit vectors.
Theorem : Any ellipsoid may be transformed into any other
ellipsoid by means of a homogeneous strain.
Let the equations of the ellipsoids be
r • <P • r = 1,
and r • ¥ • r = 1.
By means of the strain 0* the radii vectors r of the first
ellipsoid are changed into the radii vectors r' of a unit sphere
r'= 01. r, r'.r' = l.
By means of the strain ¥~l the radii vectors r' of this unit
sphere are transformed in like manner into the radii vectors f
of the second ellipsoid. Hence by the product r is changed
into f .
f = r-» . 0» . r. (19)
QUADRIC SURFACES 389
The transformation may be accomplished in more ways
than one. The radii vectors r' of the unit sphere may be
transformed among themselves by means of a rotation with or
without a perversion. Any three mutually orthogonal unit
vectors in the sphere may be changed into any three others.
Hence the semi-axes of the first ellipsoid may be carried over
by a suitable strain into the semi-axes of the second. The
strain is then completely determined and the transformation
can be performed in only one way.
142.] The equation of a family of confocal quadric sur
faces is
— --ir-
a* — n o* — n c* — n
If r • 0 • r = 1 and r • W • r = 1 are two surfaces of the
family,
2 — nl 62 — TII c2— 7&
kk
-
.-.. ^ — Tin C 7l*2
0-1 = (a2- 71^11+ (&2-tti)jj + ((^-w^kk,
y-i = (a2 - 7i2) i i + (&2 - n2) j j + (c2 - n2) k k.
Hence 0-1 - r-1^ (7i2 - 74) (ii + j j + kk)
The necessary and sufficient condition that the two quadrics
r • 0 • r = 1
and r • ¥ • r = 1
be confocal, is that the reciprocals of <P and ¥ differ by a
multiple of the idemfactor
390 VECTOR ANALYSIS
If two confocal quadrics intersect, they do so at right angles.
Let the quadrics be r • <P • r = 1,
and r • ¥ • r = 1.
Let s = <P • r and s' = ¥ • r,
r = 0-1 . s and r = ¥~l • s'.
Then the quadrics may be written in terms of s and s' as
s • 0-1 .s = l,
and s' • W~l . s' = 1,
where by the confocal property,
0-i_ W~* = xl.
If the quadrics intersect at r the condition for perpendicularity
is that the normals d> • r and ¥ • r be perpendicular. That is,
s • s' = 0.
But r = W~l • s' = 4>rl • s = ( r-1 + x I) • s
= ?r-i . s + # s,
x s • s' = s' • W~l • s' - s • r-1 * s'= 1 - s • 5T-1 • s'.
In like manner
• r = 0-1 • • = F-1 • s' = (0-1 - 0 I) • s' = 0-1 • s' - x s'.
X 8 • Sf = S • (P""1 • S' — S • <P~l • S = S • <P~l • S' — 1.
Add: 2 a s • s' = s • (0~l - P"1) • s' = x s • s'.
Hence s • s' = 0,
and the theorem is proved.
If the parameter n be allowed to vary from — oo to + oo the
resulting confocal quadrics will consist of three families of
which one is ellipsoids ; another, hyperboloids of one sheet ;
and the third, hyperboloids of two sheets. By the foregoing
QUADRIC SURFACES 391
theorem each surface of any one family cuts every surface
of the other two orthogonally. The surfaces form a triply
orthogonal system. The lines of intersection of two families
(say the family of one-sheeted and the family of two-sheeted
hyperboloids) cut orthogonally the other family — the family
of ellipsoids. The points in which two ellipsoids are cut by
these lines are called corresponding points upon the two ellip
soids. It may be shown that the ratios of the components of
the radius vector of a point to the axes of the ellipsoid
through that point are the same for any two corresponding
points.
For let any ellipsoid be given by the dyadic
The neighboring ellipsoid in the family is represented by the
dyadic
• • • • 11
" JJ kk
=
a2 — d n b2 — dn c2 — dn
y-\=$-i — ldn.
Inasmuch as 0 and ¥ are homologous (see Ex. 8, p. 330)
dyadics they may be treated as ordinary scalars in algebra.
Therefore if terms of order higher than the first in dn be
omitted, ¥ — 0+&dn.
The two neighboring ellipsoids are then
r • 0 • r = 1,
and r • (# + #2 d n) • f — 1,
By (19) f — (0 + <Z>2 d n)-i • #» • r,
r= I +
f— (I — I<Pdn) T — r — ~
392 VECTOR ANALYSIS
The vectors r and r differ by a multiple of 0 • r which is
perpendicular to the ellipsoid 0. Hence the termini of r and
r are corresponding points, for they lie upon one of the lines
which cut the family of ellipsoids orthogonally. The com
ponents of r and f in the direction i are r • i = x and
dn . dn x
f • i = x = r • i — — i . 0 *r = x — — — -.
2 2 a2
/>» ft fj\
The ratio of these components is - = 1 —
- — — — -
X A a
The axes of the ellipsoids in the direction i are Va2 — d n and
a. Their ratio is
i dn
A/a2 — dn a - i — .. dn x
— — = "l~*'
T ,., V&2 — dn y j V^2 — dn z
In like manner = - and = -.
by c z
Hence the ratios of the components of the vectors r and r
drawn to corresponding points upon two neighboring ellip
soids only differ at most by terms of the second order in d n
from the ratios of the axes of those ellipsoids. It follows
immediately that the ratios of the components of the vectors
drawn to corresponding points upon any two ellipsoids, sepa
rated by a finite variation in the parameter n, only differ at
most by terms of the first order in dn from the ratios of the
axes of the ellipsoids and hence must be identical with them.
This completes the demonstration.
The Propagation of Light in Crystals1
143.] The electromagnetic equations of the ether or of any
infinite isotropic medium which is transparent to electromag
netic waves may be written in the form
1 The following discussion must be regarded as mathematical not physical.
To treat the subject from the standpoint of physics would be out of place here.
THE PROPAGATION OF LIGHT IN CRYSTALS 393
d2V
Pot — + .FD + VF=O, V.D = O (i)
where D is the electric displacement satisfying the hydrody-
namic equation V • D = 0, E a constant of the dielectric meas
ured in electromagnetic units, and V F the electrostatic force
due to the function F. In case the medium is not isotropic the
constant E becomes a linear vector function 0. This function
is self-conjugate as is evident from physical considerations.
For convenience it will be taken as 4 TT <D. The equations
then become
-4-7T0.D + VF=0, V-D=0. (2)
U/ (/
Operate by V x V x.
V x V x Pot — — + 47rVxVx0.D = 0. (3)
CL t
The last term disappears owing to the fact that the curl of
the derivative VF vanishes (page 167). The equation may
also be written as
Pot V x V x -r-y + 47rVxVx<P.D = 0. (3)'
But VxVx=VV.-V.V.
Remembering that V • D and consequently V • — — and
-n _ (t t
V • -V-TT vanish and that Pot V • V is equal to — 4 TT the
a t2
equation reduces at once to
,72 Tk
0.D V V • $ • D, V»D = 0. (4)
dt*
Suppose that the vibration D is harmonic. Let r be the
vector drawn from a fixed origin to any point of space.
394 VECTOR ANALYSIS
Then D = A cos (m • r — n f)
where A and m are constant vectors and n a constant scalar
represents a train of waves. The vibrations take place in
the direction A. That is, the wave is plane polarized. The
wave advances in the direction m. The velocity v of that ad
vance is the quotient of n by m, the magnitude of the vector
m. If this wave is an electromagnetic wave in the medium
considered it must satisfy the two equations of that medium.
Substitute the value of D in those equations.
The value of V • D, V • V $ • D, and VV • 0 • D may be
obtained most easily by assuming the direction i to be coinci
dent with m. m • r then reduces to m i • r which is equal to
m x. The variables y and z no longer occur in D. Hence
D = A cos (m x — n f)
3D
V • D = i • -z— = — i • A m sin (m x — • n f)
d X
V • V d> • D = — m2 0 • A cos (m x — n f)
V V • #• D = — m2i i- 4> • A cos (mx — nf).
Hence V • D = — m • A sin (m • r — nf)
V • V d> • D = — m • m 0 • D
VV* 0.D = -mm. </>.D.
Moreover -j- -^ = — 7ia D.
Hence if the harmonic vibration D is to satisfy the equa
tions (4) of the medium
n2D = m-m <P • D — m m • <P • D (5)
and m • A = 0. (6)
THE PROPAGATION OF LIGHT IN CRYSTALS 395
The latter equation states at once that the vibrations must
be transverse to the direction m of propagation of the waves.
The former equation may be put in the form
D = <P • D - • 4> • D. (5)'
n2 7i2
Introduce s = - •
n
The vector s is in the direction of advance m. The magnitude
of s is the quotient of m by n. This is the reciprocal of the
velocity of the wave. The vector s may therefore be called
the wave-slowness.
D — s - s 0 • D — s s • 0 • D.
This may also be written as
D = — (s x s x 0 • D) = s x (0 • D) x s.
Dividing by the scalar factor cos (m x — n t\
A = sx(0«A)xs = s»s 0 • A — S • • 0 • A. (7)
It is evident that the wave slowness s depends not at all
upon the phase of the vibration but only upon its direction.
The motion of a wave not plane polarized may be discussed by
decomposing the wave into waves which are plane polarized.
144.] Let a be a vector drawn in the direction A of the
displacement and let the magnitude of a be so determined
that a • d> • a = 1. (8)
The equation (7) then becomes reduced to the form
a = sx (<2>*a) Xs = s-s #-a — ss-#-a (9)
a • 0 • a = 1. (8)
These are the equations by which the discussion of the velocity
or rather the slowness of propagation of a wave in different
directions in a non-isotropic medium may be carried on.
a • a = s • s a • 0 • a = s • s. (10)
396 VECTOR ANALYSIS
Hence the wave slowness s due to a displacement in the
direction a is equal in magnitude (but not in direction) to the
radius vector drawn in the ellipsoid a • 0 • a = 1 in that
direction.
axa = 0 = s«s a x <P • a — a x s s • <P . a
0 = s • s(aX # ' a)- # • a — aXs • # • a s • # • a.
But the first term contains <P • a twice and vanishes. Hence
a x s • 0 • a = [a s 0 • a] = 0. (11)
The wave-slowness s therefore lies in a plane with the
direction a of displacement and the normal 0 • a drawn to the
ellipsoid a • <P • a = 1 at the terminus of a. Since s is perpen
dicular to a and equal in magnitude to a it is evidently com
pletely determined except as regards sign when the direction
a is known. Given the direction of displacement the line of
advance of the wave compatible with the displacement is com
pletely determined, the velocity of the advance is likewise
known. The wave however may advance in either direction
along that line. By reference to page 386, equation (11) is seen
to be the condition that a shall be one of the principal axes of
the ellipsoid formed by passing a plane through the ellipsoid
perpendicular to s. Hence for any given direction of advance
there are two possible lines of displacement. These are the
principal axes of the ellipse cut from the ellipsoid a • 0 • a = 1
by a plane passed through the center perpendicular to the
line of advance. To these statements concerning the deter-
minateness of s when a is given and of a when s is given just
such exceptions occur as are obvious geometrically. If a and
0 • a are parallel s may have any direction perpendicular to a.
This happens when a is directed along one of the principal
axes of the ellipsoid. If s is perpendicular to one of the
circular sections of the ellipsoid a may have any direction in the
plane of the section.
THE PROPAGATION OF LIGHT IN CRYSTALS 397
When the direction of displacement is allowed to vary the
slowness s varies. To obtain the locus of the terminus of s, a
must be eliminated from the equation
a = s • s <P • a — SB • 0 • a
or (I - s • s 0 + s s • (P) • a = 0. (12)
The dyadic in the parenthesis is planar because it annihilates
vectors parallel to a. The third or determinant is zero. This
gives immediately
(I — • • • 0 + • • • #) 8 = 0,
or (0-1 - s • s 1 + ss)3 = 0. (13)
This is a scalar equation in the vector s. It is the locus of
the extremity of s when a is given all possible directions. A
number of transformations may be made. By Ex. 19, p. 331,
(<P + ef)8 = <PS + e • <Pa • f = 08 + e • 00-1 • f </>3.
Hence
Dividing out the common factor and remembering that $ is
self-conjugate.
1 + s- (CM-s- si)-1 -8 = 0.
1 +
8-1.8
+ s • - -- -r -8 = 0
S • S I — 8 • 8
8 • S
Hence s • - - -= • s = 0. (14)
1 — 8*8 (P
Let
398 VECTOR ANALYSIS
1 /^_W/_J_\jj + / i uk
i-i.i*-^_-jj [i_gJJ+[ir^J
Let s = xi + yj + zk and s2 = #2 + y* + z2.
Then the equation of the surface in Cartesian coordinates is
20 o
^ j?/2 z*
-72 = 0. (14)'
l-fl i-£_
a2
The equation in Cartesian coordinates may be obtained
rp n f 1 v f rr\ m
directly from
The determinant of this dyadic is
a2 — s2 + x2 x y x z
x y 62 — s2 + y^ y z
x z y z c2 — s2
= 0. (13)'
By means of the relation s2 = x2 + y2 + z2 this assumes the
forms
n n I o i o ~T~ n n ~~" -^
+
"2 2
s2 — c2
or
This equation appears to be of the sixth degree. It is how
ever of only the fourth. The terms of the sixth order cancel
out.
The vector s represents the wave-slowness. Suppose that a
plane wave polarized in the direction a passes the origin at a
THE PROPAGATION OF LIGHT IN CRYSTALS 399
certain instant of time with this slowness. At the end of a
unit of time it will have travelled in the direction s, a distance
equal to the reciprocal of the magnitude of s. The plane will
be in this position represented by the vector s (page 108).
If s = ui + vj + wit
the plane at the expiration of the unit time cuts off intercepts
upon the axes equal to the reciprocals of u, v, w. These
quantities are therefore the plane coordinates of the plane.
They are connected with the coordinates of the points in the
plane by the relation
ux + vy + wz = \.
If different plane waves polarized in all possible different
directions a be supposed to pass through the origin at the
same instant they will envelop a surface at the end of a unit
of time. This surface is known as the wave-surface. The
perpendicular upon a tangent plane of the wave-surface is the
reciprocal of the slowness and gives the velocity with which
the wave travels in that direction. The equation of the wave-
surface in plane coordinates u, v, w is identical with the equa
tion for the locus of the terminus of the slowness vector s.
The equation is
= 0
(15)
where s2 = u2 -f v'2 + w2. This may be written in any of the
forms given previously. The surface is known as FresneVs
Wave-Surface. The equations in vector form are given on
page 397 if the variable vector s be regarded as determining a
plane instead of a point.
145.] In an isotropic medium the direction of a ray of
light is perpendicular to the wave-front. It is the same as
the direction of the wave's advance. The velocity of the ray
400 VECTOR ANALYSIS
is equal to the velocity of the wave. In a non-isotropic
medium this is no longer true. The ray does not travel per
pendicular to the wave-front — that is, in the direction of the
wave's advance. And the velocity with which the ray travels
is greater than the velocity of the wave. In fact, whereas the
wave-front travels off always tangent to the wave-surface, the
ray travels along the radius vector drawn to the point of tan-
gency of the wave-plane. The wave-pknes envelop the
wave-surface; the termini of the rays are situated upon it.
Thus in the wave-surface the radius vector represents in mag
nitude and direction the velocity of a ray and the perpen
dicular upon the tangent plane represents in magnitude and
direction the velocity of the wave. If instead of the wave-
surface the surface which is the locus of the extremity of the
wave slowness be considered it is seen that the radius vector
represents the slowness of the wave; and the perpendicular
upon the tangent plane, the slowness of the ray.
Let v' be the velocity of the ray. Then s • v' = 1 because
the extremity of v' lies in the plane denoted by s. Moreover
the condition that v' be the point of tangency gives d v' per
pendicular to s. In like manner if ar be the slowness of the
ray and v the velocity of the wave, s' • v = 1 and the condition
of tangency gives d s' perpendicular to v. Hence
s • v' = 1 and s' - v = 1, (16)
and s • d v' = 0, v • d s' = 0, v' - d s = 0, s' • d v = 0,
v' may be expressed in terms of a, s, and 0 as follows.
a = s • s <P • a — s s • <P • a,
da = 2s.tfs<P.a — s- ^•arfs + s»s<?-rfa
— sds- & - a. — s s • # • d a.
Multiply by a and take account of the relations a • s = 0 and
a • 4> . d a — 0 and a • a = s • s. Then
THE PROPAGATION OF LIGHT IN CRYSTALS 401
s • d s — a • d s B • 0 • a = 0,
or d s • (s — a s • <P • a) = 0.
But since v' • d s = 0, v' and s — a s • <P • a have the same
direction.
v' = x (s — a s • 0 • a),
s • v' = # (s • s — s • a s • 0 • a) = x s • s.
, s — a s . 0 . a
Hence v' = , (17)
8*8
s . op . a — a • $ • a s • $ • a
v' • <P • a = = 0.
s • s
Hence the ray velocity v' is perpendicular to 0 • a, that is, the
ray velocity lies in the tangent plane to the ellipsoid at the
extremity of the radius vector a drawn in the direction of the
displacement. Equation (17) shows that v' is coplanar with
a and s. The vectors a, s, 0 • a, and v' therefore lie in one
plane. In that plane s is perpendicular to a ; and v;, to 0 • a.
The angle from s to v' is equal to the angle from a to $ • a.
Making use of the relations already found (8) (9) (11)
(16) (17), it is easy to show that the two systems of vectors
a, v', a x v' and 0 • a, s, (<P • a) x s
are reciprocal systems. If 0 • a be replaced by a' the equa
tions take on the symmetrical form
s . a = 0 B • s = a • a a • a' = 1,
vf -af = 0 v'.v'=a'-a' s . v' = 1,
a = s x a' x s a' = v' x a x v' (18)
s = a x v' x a v' = a' x s x a'
a • 0 • a = 1 a' • 0-1 • a' = 1.
Thus a dual relation exists between the direction of displace
ment, the ray-velocity, and the ellipsoid 0 on the one hand ;
26
402 VECTOR ANALYSIS
and the normal to the ellipsoid, the wave-slowness, and the
ellipsoid 0 ~l on the other.
146.] It was seen that if s was normal to one of the cir
cular sections of 0 the displacement a could take place in any
direction in the plane of that section. For all directions in
this plane the wave-slowness had the same direction and the
same magnitude. Hence the wave-surface has a singular
plane perpendicular to s. This plane is tangent to the surface
along a curve instead of at a single point. Hence if a wave
travels in the direction s the ray travels along the elements of
the cone drawn from the center of the wave-surface to this
curve in which the singular plane touches the surface. The
two directions s which are normal to the circular sections of 0
are called the primary optic axes. These are the axes of equal
wave velocities but unequal ray velocities.
In like manner v' being coplanar with a and 0 • a
[4> • a v' a] = [a' v' <P~l • a'] = 0.
The last equation states that if a plane be passed through
the center of the ellipsoid <P~l perpendicular to V, then a'
which is equal to 0 • a will be directed along one of the prin
cipal axes of the section. Hence if a ray is to take a definite
direction a' may have one of two directions. It is more con
venient however to regard v' as a vector determining a plane.
The first equation
[0 . a v' a] = 0
states that a is the radius vector drawn in the ellipsoid 0 to
the point of tangency of one of the principal elements of the
cylinder circumscribed about 0 parallel to v' : if by a principal
element is meant an element passing through the extremities
of the major or minor axes of orthogonal plane sections
of that cylinder. Hence given the direction v' of the ray, the
two possible directions of displacement are those radii vectors
VARIABLE DYADICS 403
of the ellipsoid which lie in the principal planes of the cylin
der circumscribed about the ellipsoid parallel to v'.
If the cylinder is one of the two circular cylinders which
may be circumscribed about 0 the direction of displacement
may be any direction in the plane passed through the center
of the ellipsoid and containing the common curve of tangency
of the cylinder with the ellipsoid. The ray-velocity for all
these directions of displacement has the same direction and
the same magnitude. It is therefore a line drawn to one
of the singular points of the wave-surface. At this singular
point there are an infinite number of tangent planes envelop
ing a cone. The wave-velocity may be equal in magnitude
and direction to the perpendicular drawn from the origin to
any of these planes. The directions of the axes of the two
circular cylinders circumscriptible about the ellipsoid 0 are
the directions of equal ray-velocity but unequal wave-velocity.
They are the radii drawn to the singular points of the wave-
surface and are called the secondary optic axes. If a ray
travels along one of the secondary optic axes the wave planes
travel along the elements of a cone.
Variable Dyadics. The Differential and Integral Calculus
147.] Hitherto the dyadics considered have been constant.
The vectors which entered into their make up and the scalar
coefficients which occurred in the expansion in nonion form
have been constants. For the elements of the theory and for
elementary applications these constant dyadics suffice. The
introduction of variable dyadics, however, leads to a simplifica
tion and unification of the differential and integral calculus of
vectors, and furthermore variable dyadics become a necessity
in the more advanced applications — for instance, in the theory
of the curvature of surfaces and in the dynamics of a rigid
body one point of which is fixed.
404 VECTOR ANALYSIS
Let W be a vector function of position in space. Let r be
the vector drawn from a fixed origin to any point in space.
d r = dx i + dy j + dz k,
3W , 5W 5W
^dx-^ + dy — + dz-Ti— .
$# c?y c/ z
( SW 3W 5W)
Hence d W = d r • H - -- h j -=— + k — - > .
( dx dy dz )
The expression enclosed in the braces is a dyadic. It thus
appears that the differential of W is a linear function of c?r,
the differential change of position. The antecedents are i, j, k,
and the consequents the first partial derivatives of W with re
spect of x, y, z. The expression is found in a manner precisely
analogous to del and will in fact be denoted by V W.
=i- + j + k-. (1)
Then dW = dr.VW. (2)
This equation is like the one for the differential of a scalar
function F.
dV=dr • VF.
It may be regarded as defining VW. If expanded into
nonion form VW becomes
.
VW = 11 —
3x
.5X .9Y 3Z
+ kils-+kjT- + kk-^--,
dz 3 z <y z
if W
VARIABLE DYADICS 405
The operators V • and V x which were applied to a vector
function now become superfluous from a purely analytic
standpoint. For they are nothing more nor less than the
scalar and the vector of the dyadic V W.
div W = V • W = (V W)* (4)
curl W = V x W = (V W)x. (5)
The analytic advantages of the introduction of the variable
dyadic VW are therefore these. In the first place the oper
ator V may be applied to a vector function just as to a scalar
function. In the second place the two operators V • and V x
are reduced to positions as functions of the dyadic V W. On
the other hand from the standpoint of physics nothing is to
be gained and indeed much may be lost if the important in
terpretations of V • W and V x W as the divergence and curl
of W be forgotten and their places taken by the analytic idea
of the scalar and vector of VW.
If the vector function W be the derivative of a scalar
function V^
dW = dVF=e?r« VVF",
where V V F= i i 7—5- + i j = — =- + i k ^ — — ,
<y x <y x c/ y <y x c/ z
Qty 32 y 32 F"
*—*- TT o— T'
dy dx &yz dy d z
+ kj „- + k j -g- + k k
dzdx 9zSy
The result of applying V twice to a scalar function is seen to
be a dyadic. This dyadic is self -conjugate. Its vector V x V V
is zero ; its scalar V • V V is evidently
32F 32V 92V
V-VF= (VVF)*= 0-2 + T-2 + TT
2 * *
406 VECTOR ANALYSIS
If an attempt were made to apply the operator V symboli
cally to a scalar function V three times, the result would be a
sum of twenty-seven terms like
* *
,etc.
r— r, ^r — = — ;r-
c/ x6 v x d y & z
This is a triadic. Three vectors are placed in juxtaposition
without any sign of multiplication. Such expressions will
not be discussed here. In a similar manner if the operator V
be applied twice to a vector function, or once to a dyadic func
tion of position in space, the result will be a triadic and hence
outside the limits set to the discussion here. The operators
V x and V • may however be applied to a dyadic 0 to yield
respectively a dyadic and a vector.
S 0 50 30
V x 0 = i x ^- + j x ^- + k x ^-, (7)
dx Sy 9z
30 30 30
V- 0 = i._ + j. — + k-T-. (8)
3x dy 3z
If 0 = u i + v j + w k,
where u, v, w are vector functions of position in space,
Vx $ = V x u i + Vxvj + Vx w k, (7)f
and V • 0= V • u i + V • v j + V • w k. (8)'
Or if 0 = i u + j v -f k w,
X)v
T— r x* • / w W ^ V \ •I*'** ^ W
Vx 0= i{- ^-) + j^ •=-
*•'-++ <8>"
In a similar manner the scalar operators (a • V) and (V • V)
may be applied to 0. The result is in each case a dyadic,
VARIABLE DYADICS 407
30 30 5d>
(a.V)<P = a.1^ + a2^- + a8^, (9)
32 0 32 $ 32 0
(V . V) <P = ^— f + „ + -™ (10)
o/ z2 c? y* d z2
The operators a • V and V • V as applied to vector func
tions are no longer necessarily to be regarded as single oper
ators. The individual steps may be carried out by means of
the dyadic VW.
(a - V) W = a • (V W) = a • V W,
(V • V) W = V • (V W) = V • V W.
But when applied to a dyadic the operators cannot be inter
preted as made up of two successive steps without making use
of the triadic V 0. The parentheses however may be removed
without danger of confusion just as they were removed in
case of a vector function before the introduction of the dyadic.
Formulae similar to those upon page 176 may be given for
differentiating products in the case that the differentiation
lead to dyadics.
V (u v) = >V u v + u V v,
V(vxw)=Vvxw — Vwxv,
Vx (v x w) = w • V v — V • v w — v • V w + V • w v,
V (v • w) = V v • w + V w • v,
V • (v w) = V • v w + v • V w.
Vx (v w) = V x v w — v x V w,
V . (u #) = V u • <P + u V • 0,
VxVx <P = VV. <P — V • V <P, etc.
The principle in these and all similar cases is that enun
ciated before, namely : The operator V may be treated sym-
408 VECTOR ANALYSIS
bolically as a vector. The differentiations which it implies
must be carried out in turn upon each factor of a product
to which it is applied. Thus
V x (vw) = [V x (v w)]v + [V x (vw)]^
[Vx (vw)]w = V xvw,
[V x (v w)]v = - [v x V w]v = - v x V w.
Hence Vx (v w) = V x v w — v x V w.
Again V (v x w) = [V (v x w)]T + [V (v x w)]^
[V (v x w)]w = V v x w,
[V (v x w)]v = [— V (w x v)]v = — V w x v.
Hence V(vxw) = Vvxw — v w x v.
148.] It was seen (Art. 79) that if C denote an arc of a
curve of which the initial point is r0 and the final point is r
the line integral of the derivative of a scalar function taken
along the curve is equal to the difference between the values
of that function at r and r0.
r* VF= F(r)- F(r0).
In like manner Cd r . V W = W (r) - W (r0),
J c
and Cd r • V W = 0.
Jo
It may be well to note that the integrals
fdr.VW and fvw • dr
are by no means the same thing. VW is a dyadic. The
vector dx cannot be placed arbitrarily upon either side of it.
VARIABLE DYADICS 409
Owing to the fundamental equation (2) the differential di
necessarily precedes V W. The differentials must be written
before the integrands in most cases. For the sake of uni
formity they always will be so placed.
Passing to surface integrals, the following formulae, some
of which have been given before and some of which are new,
may be mentioned.
ff
ax VW= fdr W
ff da. Vx W= fdr*
r/da«Vx0= I dr • <
The line integrals are taken over the complete bounding curve
of the surface over which the surface integrals are taken. In
like manner the following relations exist between volume and
surface integrals.
fff
dv VW=rfa W
///
<*• V x *-
410 VECTOR ANALYSIS
The surface integrals are taken over the complete bounding
surface of the region throughout which the volume integrals
are taken.
Numerous formulae of integration by parts like those upon
page 250 might be added. The reader will rind no difficulty in
obtaining them for himself. The integrating operators may
also be extended to other cases. To the potentials of scalar
and vector functions the potential, Pot </>, of a dyadic may be
added. The Newtonian of a vector function and the Lapla-
cian and Maxwellian of dyadics may be defined.
Pot <? =
New W = // r^^I^> dV
d ,
Max * = •
The analytic theory of these integrals may be developed as
before. The most natural way in which the demonstrations
may be given is by considering the vector function W as the
sum of its components,
W = Xi+ Fj + ^k
and the dyadic 0 as expressed with the constant consequents
i, j, k and variable antecedents u, v, w, or vice versa,
These matters will be left at this point. The object of en
tering upon them at all was to indicate the natural extensions
which occur when variable dyadics are considered. These ex
tensions differ so slightly from the simple cases which have
THE CURVATURE OF SURFACES 411
gone before that it is far better to leave the details to be worked
out or assumed from analogy whenever they may be needed
rather than to attempt to develop them in advance. It is suffi
cient merely to mention what the extensions are and how they
maybe treated.
The Curvature of Surfaces1
149. ] There are two different methods of treating the cur
vature of surfaces. In one the surface is expressed in para-
metic form by three equations
x =/i <X v)» y =/a O> *0» * =/8 <X ")>
or r = f (u, v).
This is analogous to the method followed (Art. 57) in dealing
with curvature and torsion of curves and it is the method
employed by Fehr in the book to which reference was made.
In the second method the surface is expressed by a single
equation connecting the variables x,y,z — thus
, z) = 0.
The latter method of treatments affords a simple application of
the differential calculus of variable dyadics. Moreover, the
dyadics lead naturally to the most important results connected
with the elementary theory of surfaces.
Let r be a radius vector drawn from an arbitrary fixed
origin to a variable point of the surface. The increment d r
lies in the surface or in the tangent plane drawn to the surface
at the terminus of r.
Hence the derivative V^is collinear with the normal to the
surface. Moreover, inasmuch as F and the negative of F when
1 Much of what follows is practically free from the use of dyadics. This is
especially true of the treatment of geodetics, Arts. 155-157.
412 VECTOR ANALYSIS
equated to zero give the same geometric surface, V F may be
considered as the normal upon either side of the surface. In
case the surface belongs to the family defined by
F (#, y, z) = const.
the normal V F lies upon that side upon which the constant
increases. Let V F be represented by N the magnitude of
which may be denoted by N, and let n be a unit normal drawn
in the direction of IT. Then
(1)
If s is the vector drawn to any point in the tangent plane at
the terminus of r, s— r and n are perpendicular. Consequently
the equation of the tangent plane is
(s-r)
and in like manner the equation of the normal line is
(s-r)x VjF=0,
or s = r + & V JP
where k is a variable parameter. These equations may be
translated into Cartesian form and give the familiar results.
150.] The variation dn of the unit normal to a surface
plays an important part in the theory of curvature, dn is
perpendicular to n because n is a unit vector.
THE CURVATURE OF SURFACES 413
-*
N iv2
The dyadic I — nn is an idemf actor for all vectors perpen
dicular to n and an annihilator for vectors parallel to n.
Hence
dn • (I — n n) = d n,
and V^.(I-nn)=0,
N J N N
Hence rf n = — d r • VV .F • (I — nn).
But d r = d r • (I — n n).
Hanco «.-*,. <* ~"> ' V^' P-"). (2)
Let > = (I-..
Then dn = dr« <P. (4)
In the vicinity of any point upon a surface the variation d n of
the unit normal is a linear function of the variation of the
radius vector r.
The dyadic 0 is self -con jugate. For
N4>c= (I - nn), • (VV F)c • (I - nn)^
Evidently (I - n ri)c = (I - n n) and by (6) Art. 147 VV F
is self-conjugate. Hence <PC is equal to 0. When applied to
a vector parallel to n, the dyadic 0 produces zero. It is there
fore planar and in fact uniplanar because self-conjugate. The
antecedents and the consequents lie in the tangent plane to
414 VECTOR ANALYSIS
the surface. It is possible (Art. 116) to reduce 0 to the
form
4> = a i' i' + b j' j' (5)
where i' and j' are two perpendicular unit vectors lying in the
tangent plane and a and b are positive or negative scalars.
dn = dr • (a i'i' + b j'j').
The vectors i', j' and the scalars a, 6 vary from point to point
of the surface. The dyadic C? is variable.
151.] The conic r • 0 • r = 1 is called the indicatrix of the
surface at the point in question. If this conic is an ellipse,
that is, if a and b have the same sign, the surface is convex at
the point ; but if the conic is an hyperbola, that is, if a and b
have opposite signs the surface is concavo-convex. The curve
r . 0 • r = 1 may be regarded as approximately equal to the
intersection of the surface with a plane drawn parallel to the
tangent plane and near to it. If r • 0 • r be set equal to zero
the result is a pair of straight lines. These are the asymp
totes of the conic. If they are real the conic is an hyperbola ;
if imaginary, an ellipse. Two directions on the surface which
are parallel to conjugate diameters of the conic are called con
jugate directions. The directions on the surface which coin
cide with the directions of the principal axes i', j' of the
indicatrix are known as the principal directions. They are a
special case of conjugate directions. The directions upon the
surface which coincide with the directions of the asymptotes
of the indicatrix are known as asymptotic directions. In case
the surface is convex, the indicatrix is an ellipse and the
asymptotic directions are imaginary.
In special cases the dyadic 0 may be such that the coeffi
cients a and b are equal. 0 may then be reduced to the
form
0 = a(i'i' + j'j') (5)'
THE CURVATURE OF SURFACES 415
in an infinite number of ways. The directions i' and j' may be
any two perpendicular directions. The indicatrix becomes a
circle. Any pair of perpendicular diameters of this circle
give principal directions upon the surface. Such a point is
called an umbilic. The surface in the neighborhood of an
umbilic is convex. The asymptotic directions are imaginary.
In another special case the dyadic $ becomes linear and redu
cible to the form <p = a i'i'. (5)"
The indicatrix consists of a pair of parallel lines perpendicular
to i'. Such a point is called a parabolic point of the surface.
The further discussion of these and other special cases will be
omitted.
The quadric surfaces afford examples of the various kinds
of points. The ellipsoid and the hyperboloid of two sheets
are convex. The indicatrix of points upon them is an ellipse.
The hyperboloid of one sheet is concavo-convex. The in
dicatrix of points upon it is an hyperbola. The indicatrix
of any point upon a sphere is a circle. The points are all
umbilies. The indicatrix of any point upon a cone or cylinder
is a pair of parallel lines. The points are parabolic. A sur
face in general may have upon it points of all types — elliptic,
hyperbolic, parabolic, and umbilical.
152.] A line of principal curvature upon a surface is a
curve which has at each point the direction of one of the prin
cipal axes of the indicatrix. The direction of the curve at a
point is always one of the principal directions on the surface at
that point. Through any given point upon a surface two per
pendicular lines of principal curvature pass. Thus the lines
of curvature divide the surface into a system of infinitesi
mal rectangles. An asymptotic line upon a surface is a curve
which has at each point the direction of the asymptotes of the
indicatrix. The direction of the curve at a point is always
one of the asymptotic directions upon the surface. Through
416 VECTOR ANALYSIS
any given point of a surface two asymptotic lines pass. These
lines are imaginary if the surface is convex. Even when real
they do not in general intersect at right angles. The angle
between the two asymptotic lines at any point is bisected by
the lines of curvature which pass through that point.
The necessary and sufficient condition that a curve upon a
surface be a line of principal curvature is that as one advances
along that curve, the increment of d n, the unit normal to the
surface is parallel to the line of advance. For
rfn= 0. dr = (a i'i' + b j'j') • dr
dr — x i' + yj'.
Then evidently d n and d r are parallel when and only when
dr is parallel to i' or j'. The statement is therefore proved.
It is frequently taken as the definition of lines of curvature.
The differential equation of a line of curvature is
dnxdr = 0. (6)
Another method of statement is that the normal to the surface,
the increment d n of the normal, and the element d r of the
surface lie in one plane when and only when the element d r
is an element of a line of principal curvature. The differential
equation then becomes
[n dn rfr] = 0. (7)
The necessary and sufficient condition that a curve upon a
surface be an asymptotic line, is that as one advances along
that curve the increment of the unit normal to the surface is
perpendicular to the line of advance. For
dn = dr • <P
dn • dr = dr • 0 • dr.
If then d n • d r is zero d r • tf> • d r is zero. Hence d r is an
asymptotic direction. The statement is therefore proved. It
THE CURVATURE OF SURFACES 417
is frequently taken as the definition of asymptotic lines. The
differential equation of an asymptotic line is
d n • d r = 0. (8)
153.] Let P be a given point upon a surface and n the
normal to the surface at P. Pass a plane p through n. This
plane p is normal to the surface and cuts out a plane section.
Consider the curvature of this plane section at the point P.
Let n' be normal to the plane section in the plane of the
section, n' coincides with n at the point P. But unless the
plane p cuts the surface everywhere orthogonally, the normal
n' to the plane section and the normal n to the surface will not
coincide, d n and d n' will also be different. The curvature
of the plane section lying in p is (Art. 57).
____
ds d s2
As far as numerical value is concerned the increment of the
unit tangent t and the increment of the unit normal n' are
equal. Moreover, the quotient of d r by d s is a unit vector
in the direction of d n'. Consequently the scalar value of C is
d n1 dr dnf • dr
ds ds ds2
By hypothesis n = n' at P and n«dr = n'-dr = 0,
d (V • d r) = d n' • d r + n' - d2 r = 0.
Hence d n • d r + n • d2 r = d n' • d r + n' • d2 r.
Since n and n' are equal at P,
dn*dr dr • <P • dr dr • <P • dr
Hence C = — — ^— = - j-^ - = — 3 - 3 - (9)
ds2 ds2 dr • dr
27
418 VECTOR ANALYSIS
.
C7 = a - -- — + b
dr • ar ar • ar
Hence tf= a cos2 (i', dr) + & cos2 Q', dr),
or (7= a cos2 (i', rfr) + b sin2 (i', dr). (10)
The interpretation of this formula for the curvature of a
normal section is as follows : When the plane p turns about
the normal to the surface from i' to j', the curvature C of the
plane section varies from the value a when the plane passes
through the principal direction i', to the value b when it
passes through the other principal direction j '. The values
of the curvature have algebraically a maximum and minimum
in the directions of the principal lines of curvature. If a and
b have unlike signs, that is, if the surface is concavo-convex
at Pj there exist two directions for which the curvature of a
normal section vanishes. These are the asymptotic directions.
154.] The sum of the curvatures in two normal sections
at right angles to one another is constant and independent of
the actual position of those sections. For the curvature in
one section is
Cl = a cos2 (i', dr) + b sin2 (i', dr),
and in the section at right angles to this
(72 = a sin2 (i', di) + b cos2 (i', dr).
Hence Ol + C2 = a + b = 4>a (11)
which proves the statement.
It is easy to show that the invariant $%s is equal to the pro
duct of the curvatures a and b of the lines of principal curv
ature.
4>tS = ab
Hence the equation x* — <Pa x + 0^3 = 0 (12)
THE CURVATURE OF SURFACES 419
is the quadratic equation which determines the principal curv
atures a and I at any point of the surface. By means of this
equation the scalar quantities a and b may be found in terms
of F(x, y, z).
. (I-nn)
N
N
(nn • VV.F- nn)^ = (nn • nn • VV^7)^ = (nn
(VV^)« (nn«^
Hence 9. = - -^ —
(nn. VV^)^ = nn: VVJ^=n. VV F • n.
V.V^ VFVF:WF
Hence <^^ = — -- (13)
.
**— T --- ^i -- CIS)'
These expressions may be written out in Cartesian coordinates,
but they are extremely long. The Cartesian expressions for
02/5 are even longer. The vector expression may be obtained
as follows:
(I — nn)2 = nn.
Hence
S-IA\
155.] Given any curve upon a surface. Let t be a unit
tangent to the curve, n a unit normal to the surface and m a
420 VECTOR ANALYSIS
vector defined as n x t. The three vectors n, t, m constitute
an i, j, k system. The vector t is parallel to the element d r.
Hence the condition for a line of curvature becomes
t x d n = 0. (15)
Hence m • d n = 0
d (m • n) = 0 = m • dn + n • d m.
Hence n • d m = 0.
Moreover m • d m = 0.
Hence t x d m = 0, (16)
or dmxdn = Q. (16)'
The increments of m and of n and of r are all parallel in case of
a line of principal curvature.
A geodetic line upon a surface is a curve whose osculating
plane at each point is perpendicular to the surface. That the
geodetic line is the shortest line which can be drawn between
two points upon a surface may be seen from the following
considerations of mechanics. Let the surface be smooth and
let a smooth elastic string which is constrained to lie in the
surface be stretched between any two points of it. The string
acting under its own tensions will take a position of equili
brium along the shortest curve which can be drawn upon the
surface between the two given points. Inasmuch as the
string is at rest upon the surface the normal reactions of the
surface must lie in the osculating plane of the curve. Hence
that plane is normal to the surface at every point of the curve
and the curve itself is a geodetic line.
The vectors t and d t lie in the osculating plane and deter
mine that plane. In case the curve is a geodetic, the normal
to the osculating plane lies in the surface and consequently is
perpendicular to the normal n. Hence
THE CURVATURE OF SURFACES 421
n*tx<2t = 0,
n x t • d i = 0 (17)
or m • d t = 0.
The differential equation of a geodetic line is therefore
[n dr d2r] =0. (18)
Unlike the differential equations of the lines of curvature
and the asymptotic line, this equation is of the second order.
The surface is therefore covered over with a doubly infinite
system of geodetics. Through any two points of the surface
one geodetic may be drawn.
As one advances along any curve upon a surface there is
necessarily some turning up and down, that is, around the
axis m, due to the fact that the surface is curved. There may
or may not be any turning to the right or left. If one advances
along a curve such that there is no turning to the right or
left, but only the unavoidable turning up and down, it is to be
expected that the advance is along the shortest possible route
— that is, along a geodetic. Such is in fact the case. The
total amount of deviation from a straight line is d t. Since n,
t, m form an i, j, k system
I = tt + nn + mm.
Hence dt = tt*dt + nn»dt + mm*dt.
Since t is a unit vector the first term vanishes. The second
term represents the amount of turning up and down; the
third term, the amount to the right or left. Hence m • d t is
the proper measure of this part of the deviation from a
straightest line. In case the curve is a geodetic this term
vanishes as was expected.
156.] A curve or surface may be mapped upon a unit
sphere by the method of parallel normals. A fixed origin is
assumed, from which the unit normal n at the point P of a
422 VECTOR ANALYSIS
given surface is laid off. The terminus Pr of this normal lies
upon the surface of a sphere. If the normals to a surface at all
points P of a curve are thus constructed from the same origin,
the points Pr will trace a curve upon the surface of a unit
sphere. This curve is called the spherical image of the given
curve. In like manner a whole region T of the surface may
be mapped upon a region T1 the sphere. The region T1 upon
the sphere has been called the hodogram of the region T upon
the surface. If d r be an element of arc upon the surface the
corresponding element upon the unit sphere is
dn= 0 • dr.
If da be an element of area upon the surface, the corre
sponding element upon the sphere is d*! where (Art. 124).
d a' = <P2 • d a.
0 = a i'i' + & j'j'
<P2 = a6 i' x j' ifxjf = ab nn.
Hence dd = ab nn • d a. (19)
The ratio of an element of surface at a point P to the area of
its hodogram is equal to the product of the principal radii of
curvature at P or to the reciprocal of the product of the prin
cipal curvatures at P.
It was seen that the measure of turning to the right or left
is m • d t. If then 0 is any curve drawn upon a surface the
total amount of turning in advancing along the curve is the
integral.
r m • dt. (20)
c
For any closed curve this integral may be evaluated in a
manner analogous to that employed (page 190) hi the proof
of Stokes's theorem. Consider two curves C and 0' near
THE CURVATURE OF SURFACES 423
together. The variation which the integral undergoes when
the curve of integration is changed from C to C' is
S f m- dt.
S fm.dt= Cs (m - di)= fSm - dt+ Cm*Sdt
d(m-8t) = dm»St-hm«d 8 1
S Cm* dt= C Sm* dt- C dm* St + C d (m • Si).
The integral of the perfect differential d (m • S t) vanishes
when taken around a closed curve. Hence
S I m«dt= / Sm • dt — I dm • Sk
The idemfactor is I = tt + nn + mm,
8m • (2t = £m • I • e2t = £m • nn • dt,
f or t • c£ t and S m • m vanish. A similar transformation may
be effected upon the term d»m • S t. Then
S / m«dt= /(Sm»n n»^t — rfm»n n-St).
By differentiating the relations m • n = 0 and n • t = 0 it is
seen that
c?m«n=:--m«^n n . dt = — dn • t.
Hence 8 /m«^t=/ (m • Sn t*dn— m • dn t • Sn)
Sim rft=/(mxt-8nx^n)=— I n • Sn x dn.
424 VECTOR ANALYSIS
The differential Sn x dn represents the element of area in
the hodogram upon the unit sphere. The integral
/ n • 8 n x <J n = / n • d a'
represents the total area of the hodogram of the strip of
surface which lies between the curves C and Cf. Let the
curve C start at a point upon the surface and spread out to
any desired size. The total amount of turning which is re
quired in making an infinitesimal circuit about the point is
2 TT. The total variation in the integral is
f 8 fin. rft=f
m.dt-27T.
But if H denote the total area of the hodogram.
Hence / m • d t = 2 TT — -JET,
or iT=27r — fm • rft, (22)
or H+ Cm* dt = 27r.
The area of the hodogram of the region enclosed by any
closed curve plus the total amount of turning along that curve
is equal to 2 TT. If the surface in question is convex the area
upon the sphere will appear positive when the curve upon the
surface is so described that the enclosed area appears positive.
If, however, the surface is concavo-convex the area upon the
sphere will appear negative. This matter of the sign of the
hodogram must be taken into account in the statement made
above.
THE CURVATURE OF SURFACES 425
157.] If the closed curve is a polygon whose sides are
geodetic lines the amount of turning along each side is zero.
The total turning is therefore equal to the sum of the exterior
angles of the polygon. The statement becomes : the sum of
the exterior angles of a geodetic polygon and of the area of
the hodogram of that polygon (taking account of sign) is
equal to 2 TT. Suppose that the polygon reduces to a triangle.
If the surface is convex the area of the hodogram is positive
and the sum of the exterior angles of the triangle is less than
2 TT. The sum of the interior angles is therefore greater than
TT. The sphere or ellipsoid is an example of such a surface.
If the surface is concavo-convex the area of the hodogram is
negative. The sum of the interior angles of a triangle is in
this case less than TT. Such a surface is the hyperboloid of one
sheet or the pseudosphere. There is an intermediate case in
which the hodogram of any geodetic triangle is traced twice in
opposite directions and hence the total area is zero. The sum
of the interior angles of a triangle upon such a surface is equal
to TT. Examples of this surface are afforded by the cylinder,
cone, and plane.
A surface is said to be developed when it is so deformed that
lines upon the surface retain their length. Geodetics remain
geodetics. One surface is said to be developable or applicable
upon another when it can be so deformed as to coincide with
the other without altering the lengths of lines. Geodetics
upon one surface are changed into geodetics upon the other.
The sum of the angles of any geodetic triangle remain un
changed by the process of developing. From this it follows
that the total amount of turning along any curve or the area
of the hodogram of any portion of a surface are also invariant
of the process of developing.
426 VECTOR ANALYSIS
Harmonic Vibrations and Bivectors
158.] The differential equation of rectilinear harmonic
motion is
The integral of this equation may be reduced by a suitable
choice of the constants to the form
x = A sin n t.
This represents a vibration back and forth along the X-axis
about the point x = 0. Let the displacement be denoted by
D in place of x. The equation may be written
D = i A sin n t.
Consider D = i A sin n t cos m x.
This is a displacement not merely near the point x = 0
Or,
but along the entire axis of x. At points x = - , where
in
k is a positive or negative integer, the displacement is at all
times equal to zero. The equation represents a stationary
wave with nodes at these points. At points midway between
these the wave has points of maximum vibration. If the
equation be regarded as in three variables x, y, z it repre
sents a plane wave the plane of which is perpendicular to
the axis of the variable x.
The displacement given by the equation
Dx = i A l cos (m x — n f) (1)
is likewise a plane wave perpendicular to the axis of x but
not stationary. The vibration is harmonic and advances
along the direction i with a velocity equal to the quotient of
HARMONIC VIBRATIONS AND BIVECTORS 427
n by m. If v be the velocity; p the period; and / the wave
length,
n 2?r 2 TT /
v = -, ^p = — , I = — •, v = -. (2)
m n m p
The displacement
D2 = j A 2 cos (m x — nt)
differs from Dj in the particular that the displacement takes
place in the direction j, not in the direction i. The wave as
before proceeds in the direction of x with the same velocity.
This vibration is transverse instead of longitudinal. By a
simple extension it is seen that
D = A cos (m x — n t)
is a displacement in the direction A. The wave advances
along the direction of x. Hence the vibration is oblique to
the wave-front. A still more general form may be obtained
by substituting m • r for m x. Then
D = A cos (m • r — n t). (3)
This is a displacement in the direction A. The maximum
amount of that displacement is the magnitude of A. The
wave advances in the direction m oblique to the displace
ment; the velocity, period, and wave-length are as before.
So much for rectilinear harmonic motion. Elliptic har
monic motion may be defined by the equation (p. 117).
The general integral is obtained as
r = A cos n t + B sin n t.
The discussion of waves may be carried through as pre
viously. The general wave of elliptic harmonic motion
advancing in the direction m is seen to be
428 VECTOR ANALYSIS
D = A cos (m • r — n t) — B sin (m • r — n t). (4)
dV ( }
— = n | A sin (m • r — n t) + B cos (m • r — n t) j (5)
is the velocity of the displaced point at any moment in the
ellipse in which it vibrates. This is of course entirely differ
ent from the velocity of the wave.
An interesting result is obtained by adding up the dis
placement and the velocity multiplied by the imaginary
unit V — 1 and divided by n.
D H -- — = A cos (m • r — n t) — B sin (m • r — n f)
+ V — 1 \ A sin (m • r — n t) + B cos (m • r — n t) }.
The expression here obtained, as far as its form is concerned,
is an imaginary vector. It is the sum of two real vectors of
which one has been multiplied by the imaginary scalar V — 1.
Such a vector is called a bivector or imaginary vector. The
ordinary imaginary scalars may be called biscalars. The use
of bivectors is found very convenient in the discussion of
elliptic harmonic motion. Indeed any undamped elliptic har
monic plane wave may be represented as above by the pro
duct of a bivector and an exponential factor. The real part
of the product gives the displacement of any point and the
pure imaginary part gives the velocity of displacement
divided by n.
159.] The analytic theory of bivectors differs from that of
real vectors very much as the analytic theory of biscalars
differs from that of real scalars. It is unnecessary to have
any distinguishing character for bivectors just as it is need-
HARMONIC VIBRATIONS AND BIVECTORS 429
less to have a distinguishing notation for biscalars. The bi-
vector may be regarded as a natural and inevitable extension
of the real vector. It is the formal sum of two real vectors
of which one has been multiplied by the imaginary unit V — 1-
The usual symbol i will be maintained for V — 1. There is
not much likelihood of confusion with the vector i for the
reason that the two could hardly be used in the same place
and for the further reason that the Italic i and the Clarendon
i differ considerably in appearance. Whenever it becomes
especially convenient to have a separate alphabet for bivec-
tors the small Greek or German letters may be called upon.
A bi vector may be expressed in terms of i, j, k with com
plex coefficients.
If r = TJ + i r2
and ri = xi *
r = # i
or r = #i + yj + z.
Two bivectors are equal when their real and their imaginary
parts are equal. Two bivectors are parallel when one is the
product of the other by a scalar (real or imaginary). If
a bivector is parallel to a real vector it is said to have a real
direction. In other cases it has a complex or imaginary
direction. The value of the sum, difference, direct, skew,
and indeterminate products of two bivectors is obvious with
out special definition. These statements may be put into
analytic form as follows.
Let r = TJ + i r2 and s = s2 + i s2.
Then if r = s, rl = BI and r2 = s2
if r || s r = x s = (xl + i #3) s,
480 VECTOR ANALYSIS
r + s = (r1 + s1) + i(r2 + s2),
r . s = <>! • B! - r2 • s2) + i (rl • s2 + r2 • Sl),
r x s = (rl x BI — r2 x s2) + i (rx x sa + ra x sx)
rs = (rl sl + r2 s2) + i (TI s2 + r2 Sj).
Two bivectors or biscalars are said to be conjugate when
their real parts are equal and their pure imaginary parts
differ only in sign. The conjugate of a real scalar or vector
is equal to the scalar or vector itself. The conjugate of any
sort of product of bivectors and biscalars is equal to the pro
duct of the conjugates taken in the same order. A similar
statement may be made concerning sums and differences.
•
Oi + i r2) • (ri - * r2) = rx • TJ + rt • r2,
Oi + * *2) X (rx - i r2) = 2 i r2 x TV
Ol + * r2> (rl ~ * F2> = (rl rl + F2 r2> + * (r2 Fl - rl r2>-
If the bivector r = TJ 4- i r2 be* multiplied by a root of unity
or cyclic factor as it is frequently called, that is, by an imagi
nary scalar of the form
cos q + i sin q = a + ib, (7)
where a2 + &2 = 1,
the conjugate is multiplied by a — i 6, and hence the four
products
are unaltered by multiplying the bivector r by such a factor.
Thus if
r' = r/ + i r2' = (a + iV) (rx + i r2),
TI • TI + *z • r2' = rx • T! + r2 • r2, etc.
HARMONIC VIBRATIONS AND BIVECTORS 431
160.] A closer examination of the effect of multiplying a
bivector by a cyclic factor yields interesting and important
geometric results. Let
ri' + * ra' = (cos ? + * sin 2) (ri + i ra)- (8)
Then rx' = il cos ^ — r2 sin £,
ra' = r2 cos q + TI sin j.
By reference to Art. 129 it will be seen that the change pro
duced in the real and imaginary vector parts of a bivector by
multiplication with a cyclic factor, is precisely the same as
would be produced upon those vectors by a cyclic dyadic
d> = a a' + cos q (bb' + c c') - sin q (c b' - be')
used as a prefactor. b and c are supposed to be two vectors
collinear respectively with rx and r2. a is any vector not in
their plane. Consider the ellipse of which TJ and r2 are a
pair of conjugate semi-diameters. It then appears that r^
and r2' are also a pair of conjugate semi-diameters of that
ellipse. They are rotated in the ellipse from r2 toward r1$ by
a sector of which the area is to the area of the whole ellipse
as q is to 2 ?r. Such a change of position has been called an
elliptic rotation through the sector q.
The ellipse of which TI and r2 are a pair of conjugate semi-
diameters is called the directional ellipse of the bivector r.
When the bivector has a real direction the directional ellipse
reduces to a right line in that direction. When the bivector
has a complex direction the ellipse is a true ellipse. The
angular direction from the real part TI to the complex part r2
is considered as the positive direction in the directional
ellipse, and must always be known. If the real and imagi
nary parts of a bivector turn in the positive direction in the
ellipse they are said to be advanced ; if in the negative direc
tion they are said to be retarded. Hence multiplication of a
432 VECTOR ANALYSIS
bivector by a cyclic factor retards it in its directional ellipse by
a sector equal to the angle of the cyclic factor.
It is always possible to multiply a bivector by such a cyclic
factor that the real and imaginary parts become coincident
with the axes of the ellipse and are perpendicular.
r = (cos q + i sin q) (a + i b) where a • b = 0.
To accomplish the reduction proceed as follows : Form
r • r = (cos 2 q + i sin 2 q) (a + i b) • (a + i b).
If a • b = 0,
r • r = (cos 2 q + i sin 2 q) (a • a — b • b).
Let r • r = a + i 6,
and tan 2 q = -.
a
With this value of q the axes of the directional ellipse are
given by the equation
a -f i b = (cos q — i sin q) r.
In case the real and imaginary parts a and b of a bivector
are equal in magnitude and perpendicular in direction both a
and b in the expression for r • r vanish. Hence the angle
q is indeterminate. The directional ellipse is a circle. A
bivector whose directional ellipse is a circle is called a circu
lar bivector. The necessary and sufficient condition that a
non-vanishing bivector r be circular is
r • r = 0, r circular.
If r = zi + 2/j + *k,
r . r = x* + y* + z2 = 0.
The condition r • r = 0, which for real vectors implies r = 0,
is not sufficient to ensure the vanishing of a bivector. The
HARMONIC VIBRATIONS AND BIVECTORS 433
bivector is circular, not necessarily zero. The condition that
a bivector vanish is that the direct product of it by its con
jugate vanishes.
Oi + i r2) • (rx - t r2) = rx • rx + r2 • ra = 0,
then F! = r2 = 0 and r = 0.
In case the bivector has a real direction it becomes equal to
its conjugate and their product becomes equal to r • r.
161.] The condition that two bivectors be parallel is that
one is the product of the other by a scalar factor. Any bi-
scalar factor may be expressed as the product of a cyclic
factor and a positive scalar, the modulus of the biscalar. If
two bivectors differ by only a cyclic factor their directional
ellipses are the same. Hence two parallel vectors have their
directional ellipse similar and similarly placed — the ratio of
similitude being the modulus of the biscalar. It is evident
that any two circular bivectors whose planes coincide are
parallel. A circular vector and a non-circular vector cannot
be parallel.
The condition that two bivectors be perpendicular
is r • s = 0,
or rt • «! — r2 • 83 = rx • s2 + r2 • BI = 0.
Consider first the case in which the planes of the bivectors
coincide. Let
r = a (TJ + i r2), s = I (s1 + i g2).
The scalars a and b are biscalars. rx may be chosen perpen
dicular to r2, and sl may be taken in the direction of ra. The
condition r • s = 0 then gives
ra • 82 = 0 and rx • s2 + ra • nl = 0.
28
434 VECTOR ANALYSIS
The first equation shows that r2 and s2 are perpendicular and
hence sl and s2 are perpendicular. Moreover, the second
shows that the angular directions from rx to r2 and from s1 to
s2 are the same, and that the axes of the directional ellipses
of r and s are proportional.
Hence the conditions for perpendicularity of two bivectors
whose planes coincide are that their directional ellipses are
similar, the angular direction in both is the same, and the
major axes of the ellipses are perpendicular.1 If both vectors
have real directions the conditions degenerate into the per
pendicularity of those directions. The conditions therefore
hold for real as well as for imaginary vectors.
Let r and s be two perpendicular bivectors the planes of
which do not coincide. Resolve TI and r2 each into two com
ponents respectively parallel and perpendicular to the plane
of s. The components perpendicular to that plane contribute
nothing to the value of r • s. Hence the components of rx
and r2 parallel to the plane of s form a bivector r' which is
perpendiqular to s. To this bivector and s the conditions
stated above apply. The directional ellipse of the bivector r'
is evidently the projection of the directional ellipse of r upon
the plane of s.
Hence, if two bivectors are perpendicular the directional
ellipse of either bivector and the directional ellipse of the
other projected upon the plane of that one are similar, have
the same angular direction, and have their major axes per
pendicular.
162.] Consider a bivector of the type
where A and m are bivectors and TI is a biscalar. r is the
position vector of a point in space. It is therefore to be con-
1 It should be noted that the condition of perpendicularity of major axes is not
the same as the condition of perpendicularity of real parts and imaginary parts
HARMONIC VIBRATIONS AND BIVECTORS 435
sidered as real, t is the scalar variable time and is also to
be considered as real. Let
A = Aj + i A*p
m = ni + i mj
D =
As has been seen before, the factor (Ax + i Aj) e<(mt*r~nif)
represents a train of plane waves of elliptic harmonic vibra
tions. The vibrations take place in the plane of Aj and A2,
in an ellipse of which Ax and A% are conjugate semi-diam
eters. The displacement of the vibrating point from the
center of the ellipse is given by the real part of the factor.
The velocity of the point after it has been divided by nj
is given by the pure imaginary part. The wave advances
in the direction mr The other factors in the expres
sion are dampers. The factor «•""••* is a damper in the
direction m2. As the wave proceeds in the direction m^ it
dies away. The factor e*** is a damper in time. If na is
negative the wave dies away as time goes on. If n2 is posi
tive the wave increases in energy as time increases. The
presence (for unlimited time) of any such factor in an ex
pression which represents an actual vibration is clearly inad
missible. It contradicts the law of conservation of energy.
In any physical vibration of a conservative system na is ne
cessarily negative or zero.
The general expression (9) therefore represents a train of
plane waves of elliptic harmonic vibrations damped in a
definite direction and in time. Two such waves may be com
pounded by adding the bivectors which represent them. If
the exponent m • r — n t is the same for both the resulting
train of waves advances in the same direction and has the
436 VECTOR ANALYSIS
same period and wave-length as the individual waves. The
vibrations, however, take place in a different ellipse. If the
waves are
the resultant is (A + B) *«««•*—«.
By combining two trains of waves which advance in opposite
directions but which are in other respects equal a system of
stationary waves is obtained.
A e-m*-1V(mi-lr-n0 + A e~m*'r
Ae-^-'e-"" (e<mi*r + $-"i") = 2Acos (n^ • r) e-m*'p e^int
The theory of bivectors and their applications will not be
carried further. The object in entering at all upon this very
short and condensed discussion of bivectors was first to show
the reader how the simple idea of a direction has to give way
to the more complicated but no less useful idea of a directional
ellipse when the generalization from real to imaginary vectors
is made, and second to set forth the manner in which a single
bivector D may be employed to represent a train of plane
waves of elliptic harmonic vibrations. This application of bi
vectors may be used to give the Theory of Light a wonderfully
simple and elegant treatment.1
1 Such use of bivectors is made by Professor Gibbs in his course of lectures on
" The Electromagnetic Theory of Light" delivered biannually at Yale University.
Bivectors were not used in the second part of this chapter, because in the opinion
of the present author they possess no essential advantage over real vectors until
the more advanced parts of the theory, rotation of the plane of polarization by
magnets and crystals, total and metallic reflection, etc., are reached.
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