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PROPERTY OF
1
if
* 8 i 7
»*iib
ARTtS 5i C I !■ N T I A VERITAS
VECTOR ANALYSIS
The author will be grateful to any reader
who will call his attention to an error in this
book, or who will suggest new problems,
changes in the text, or additions thereto.
Careful consideration will be given to such
suggestions and criticisms, as it is the author's
desire to make his book as nearly perfect as
possible. Communications of this nature
should be addressed to Dr. J. G. Coffin, 199
Elizabeth Ave., Hempstead, N. Y.
VECTOK ANALYSIS
AN INTRODUCTION
TO
VECTOR-METHODS
a
AND THEIR VARIOUS APPLICATIONS
TO
PHYSICS AND MATHEMATICS
BY
JOSEPH GEORGE COFFIN, B.S., Ph.D.
(MASS. INST. TECH. '98 AND CLARK UNIVERSITY *03)
BX-ASSOCIATE PROFESSOR OP PHYSICS AT THE COLLEGE OP THU
CITY OF NEW YORK
CONSULTING AND DEVELOPMENT ENGINEER WITH THE
GENERAL BAKING COMPANY
SECOND EDITION
NEW YORK
JOHN WILEY & SONS, Inc.
London : CHAPMAN 6 HALL, Limited
Copyright, 1909, 1911,
BT
J. G. COFFIN
Printed in U. 8. A.
Stmbopc Pitt*
P.H. OILSON COHFABT
•OS TQM. U.f.A-
3-30
U^vr :c "<. i- .. ^ : ' /ii '' %, J
/
I . -■
»
. - /V. */V
PREFACE
Ever since the development of Quaternion analysis, by
Sir William Rowan Hamilton, and of the " Ausdehnungs-
lehre," by Grassmann, there has been a growing feeling that
the older and more common processes of analysis were in
some way artificial and complex.
This fact exists, for it is such, because these newer
methods and ideas apply more naturally, more simply and
more directly to many of the conceptions of geometry,
mechanics and mathematical physics, than those long
accepted.
Why then have these admitted advantages not led to a more
universal adoption of these methods ? The answer seems to be
that the required change of ideas, of manner of thought and
of notation, was too radical. It is well known that changes
evolve slowly, and although to many, evolution is far too slow
a process, the only way to proceed is to aid to the best of
one's ability in bringing about the desired result.
One who has studied and labored over the applications
of mathematical analysis to physical and geometrical prob-
lems, naturally has reluctance to discard the old familiar
looking formulae and start anew in an unknown and radically
different language.
However great the skill and ingenuity shown by the
pioneer in solving problems by Quaternions, there was
always left the thought to the unbiased student that a lack
of parallelism existed between the old and the new methods
of treatment. Such a lack undoubtedly does exist, but it is
only during the last few years that a method has been
VI PREFACE.
evolved which avoids this fatal defect. It is chiefly through
the labors of Gibbs and Heaviside that an analysis has
been perfected which not only does away with the unnec-
essary complexity and artificiality of other analyses but
offers a strictly natural and therefore as direct and simple a
substitute as possible, and, at the same time in no wise is at
variance, but runs parallel to them.
This new, yet old method is Vector Analysis; it com-
bines within itself most of the advantages of both Quater-
nions and of Cartesian Analysis.
The adoption of Vector Analysis is urged on the grounds
of naturalness, simplicity and directness; with it the true
meaning of processes and results is brought out as clearly
as possible, and desirable abbreviation is obtained.
It is admitted, that to a straight and clear thinker, almost
any notation or mathematical method suffices, and to such
a one, changes in notation or method may appear hardly
worth while. He has already attained one of the results
which, perforce, follow the intelligent assimilation of a
vector method of thinking. To him there is left but the
attainment of a simple notation which is the logical accom-
paniment of clear thought. A few examples of vector con-
centration are to be found in the exercises of the last chapter
of this book. But the sole use of vector notation, without
the insight and clear conceptions which should obtain at the
same time, is without any value whatsoever, vitiates the
vector point of view, and is contrary to the spirit of it.
It is almost unnecessary to state that the mind of the
physicist ought to be of the visual type so well exemplified in
the mind of Faraday. He should see the lines of force
emerging from the magnet; see that they are continuous
within the metal; follow them, in his mind's eye, as
they are displaced by various causes; he should have some
sort of a visual conception of the manner in which the
electro-magnetic waves are traveling through the ether
PREFACE. vu
around him; to him the divergence and the divergence
theorem should have a simple meaning.
To a mind other than this, the study of mathematical
physics must be merely a series of analytical transforma-
tions without the vitality of their visual significance. To
purely analytical minds, as distinct from the visual or intu-
itive type, the methods of Vector Analysis reduce to little
more than an analytical shorthand. To the intuitive mind,
however, they are illuminative and simplifying, allowing
the mind to grasp and the hand to write the essential facts
and transformations, unembarrassed with the generally unde-
sirable complexity of Cartesian symbolism. It is impossible
to study and to apply Vector Analysis to problems and not
to have one's ideas and thought made clearer and better by
the labor involved.
There are very good reasons for all these advantages. In
Nature we are confronted with quantities called scalars
which have size or magnitude only, and also with other
quantities called vectors which have direction as well as
magnitude. In order to manipulate vector quantities by
the older methods, they were decomposed into three com-
ponents along three arbitrary axes and the operations made
upon these components. Is it not evident that the bringing
in of three arbitrary axes is an artificial process, and that
the decomposition of the vector into components along these
axes is also artificial, unnatural even? Why not go directly
to the vector itself and manipulate it without axes and
without components? To do this, is possible, and in the
following pages an attempt is made to show how it may be
done.
There is still another ground for urging more extended
study of Vector Analysis than now obtains. So many physi-
* cists of renown have been converted to its methods and use
i that to ignore their leadership is an impossibility. When
such men as Lorentz, Foppl, Heaviside, Bucherer, Gibbs,
viii PRKFACK.
Abraham, Bjerknes, Sommerfeld, Cohn and many others are
converted to its use, it is high time that the student famil-
iarize himself at least with vector notation, even if not to
become an expert in its use.
No one can deny the vast improvement that has taken
place, in recent years, in our conceptions of physical pro-
cesses; and few will deny that a large part of this improve-
ment has been due to the ideas introduced with the advent
of vector methods of thought.
That Lagrange reduced all of mechanics to a purely ana-
lytical basis without, as he boasts, necessitating diagrams,
is certainly a wonderful accomplishment. Yet how much
clearer and more elegant if the equations become alive with
meaning, if to the algebraic transformations a mental pic-
ture of what is taking place is obtained!
Maxwell gave a splendid reference in favor of the new
methods when he said, in speaking about the motion of
the top, " Poinsot has brought the subject under the power
of a more searching analysis than that of the calculus, in
which ideas take the place of symbols and intelligible propo-
sitions supersede equations."
Vector Analysis has the advantages of Lagrange's ana-
lytical method as well as those of the idealogical method
of Poinsot.
The writer does not, in any way, urge the rejection of
anything of value in any method whatsoever. It is not
well nor is it intended that the methods of Vector Analysis
should be essentially different from those to which the
student is supposed to be accustomed. In fact, it has been
the aim throughout this book to evolve an analysis to which
all the knowledge of the reader can be immediately applied
and to so expound this analysis, that Cartesian equations may
be immediately written in vector notation and conversely.
There is still another important advantage, which shoulc
not be overlooked, that is, vector notation just as vecto
i
PREFACE. ix
thought, is entirely independent of any choice of axes, or
planes of reference, and yet the transformation of the vector
equations into other systems, requiring these axes or planes
is always extremely easy. To prove that a natural invari-
ant is invariant to a change of axes, has always appeared
to the writer an extremely foolish operation and a waste
of time. This is not saying, that in a mathematical theory
of invariants such a property of an algebraic expression is
not instructive or interesting. But to say, for example, that
the properties of the lines of force which cut a set of equipo-
tential surfaces at right angles, (i.e., the lines F =-VF)
may be dependent upon the particular set of axes used to
investigate them, is a waste of time to say the least. How
can a truth vary with the language used in expressing it?
No attempt at mathematical rigor is made. Such refine-
ments serve only to conceal the simplicity of fact, which it is
the aim of these pages to elucidate. The appearance of
extended proofs, the writer considers to be entirely out of
place in a book of this kind. On the other hand, no one is
more in favor of mathematical rigor than he; the point is
simply to eliminate discussions whose presence would lead
the attention astray from the main ideas of the argument.
In any case, whenever a demonstration does not satisfy the
fastidious, the results may be found more rigorously, if not
more clearly, established in works devoted to mathemat-
ically rigorous demonstrations.
The student will find with a little study that he may easily
take down lectures, given in Cartesian notation, directly
into vector notation. Serious trial will convince him that time
is gained and what is still more important, that equations
will be, must be, understood if this is done. It is by pre-
cisely such a process that the writer familiarized himself
with the subject.
The notation adopted is that of Prof. Willard Gibbs, one
of the too few great American physicists and mathema-
X PREFACE.
ticians. The reasons leading to this choice are fully set
forth in the Appendix.
The first part of the book is devoted to a concise treat-
ment of the fundamental principles of the subject, the
remaining chapters, to the application of the analysis to the
beginnings of mathematical physics, including geometry,
mechanics, magnetism, electricity, heat and hydrodynamics.
It was found necessary to omit many beautiful applications
in elasticity, electron theory and other parts of physics in
order to keep the size of the volume within bounds.
The student who takes up the later chapters, is supposed to
be familiar, to a certain extent, with the subjects therein
contained, and these chapters are intended to show the
beginner how to translate and demonstrate the theorems
into the new calculus. The writer therefore makes this his
apology for a certain necessary lack of logical sequence in the
treatment of. the various subjects.
The treatment of alternating currents and allied subjects
has been omitted, because in practically every modern book-
on the subject the notation of the special vector method
employed, is fully explained in some part of it.
It is hoped that but few errors still remain in the text.
The author, alone, corrected the proof, but numerous
equations and special difficulties met with in printing in
a new notation, rendered the corrections very difficult and
laborious.
The copy has been read by Prof. Saurel, professor of
mathematics in the College of the City of New York, and
the author wishes to acknowledge here his indebtedness for
the kindness as well as for many valuable suggestions.
A detailed list of works on Quaternions is rendered unnec-
essary by Professor Macfarlane's " Bibliography " published
by the " Association for the Promotion of the Study of Qua-
ternions and Allied Mathematics," Dublin, 1904, but a list of
works which have been especially consulted is appended to
PREFACE. jd
the preface, and the writer here acknowledges his obligations
to all of them. If this book succeeds in making plain the
author's particular point of view; in simplifying ideas, or in
causing simple ideas to seem clearer than before, he will feel
amply repaid for any pains taken in producing what was to
him a labor of love.
J. G. Coffin.
New York, April 9, 1909.
PREFACE TO SECOND EDITION
In this new edition a number of small errors which are
peculiarly difficult of discovery in a work involving so
many different kinds of type have been corrected. The
sincere thanks of the writer are due to the large number of
correspondents who have greatly helped him in this revision.
The author is glad to be able to state that to his knowledge
but one theoretical error has been discovered up to the present
time.
Certain portions have been rewritten and fourteen pages
of notes have been added to the appendix.
In particular a short digression on different varieties of
vectors; certain additional definitions of differential geometry
with reference to curves in space which seemed interesting
and useful; the demonstration of Frenet's valuable formulae
for space curves; an interesting example of vector reasoning
as applied to the solution of the differential equation of
motion of an electron in a magnetic field; two new proofs
of Stokes' Theorem not found as far as we know in any
treatise of vector analysis; an additional proof of Gauss's
Theorem; and proofs of two theorems in integration analo-
gous to the Divergence Theorem.
xii PREFACE.
Both the publisher and the writer are delighted with the
reception accorded this little book in this country and
abroad.
The writer is of the opinion that a great many results of
mathematical physics are elementary and easily understood
by the student if explained in the right way, and the student
thereby finds himself in a position to go right ahead in the
more difficult extensions, when he comes to them. This
book was written with that end in view. It is practically
an elementary course in mathematical physics.
He also hopes that not only will this volume help the
student to an acquisition of the fundamentals of Vector
Analysis, but that also, and not least, it will awaken in him a
desire for further study in that most beautiful and extensive
of all branches of study, — Mathematical Physics.
He believes that in this country there is a wealth of mate-
rial for the making of brilliant investigators in this line, if
they are encouraged to approach the higher branches with-
out the fear that it is beyond their capabilities.
He therefore makes a plea for the encouragement of stu-
dents having ability in this direction so that soon it can no
longer be said that we are not up to the standard of the in-
vestigators of the Old World. True, they had a long start
and we have been handicapped, but we hope in the course
of a few years to be abreast of them.
J. G. Coffin.
New York, June, 1911.
CONTENTS.
CHAPTER I.
Elementary Operations of Vector Analysis.
or. PAOB
1. Definitions — Vector — Scalar 1
2. Graphical Representation of a Vector 1
3. Equality of Vectors — Negative Vector — Unit Vector —
Reciprocal Vector 2
4. Composition of Vectors — Addition and Subtraction — Vector
Sum as an Integration 4
5. Scalar and Vector Fields — Point-Function — Definition of
Lame* — Continuity of Scalar and Vector Functions 6
6. Decomposition of Vectors 8
7. The Unit Vectors 1 J k 9
8. Vector Equations — Equations of Straight Line and Plane. . 11
9. Condition that Three Vectors Terminate in a Straight Line —
Examples 13
10. Equation of a Plane 16
11. Plane Passing through Ends of Three Given Vectors ■».... 16
12. Condition that Four Vectors Terminate in a Plane 18
13. To Divide a Line in a Given Ratio — Centroid 18
14. Relations Independent of the Origin — General Condition. . . 21
Exercises and Problems 22
CHAPTER II.
Scalar and Vector Products op Two Vectors.
15. Scalar or Dot Product — Laws of the Scalar Product 28
16. Line-Integral of a Vector 31
17. Surface-Integral of a Vector 32
18. Vector or Cross Product — Definition 34
19 Distributive Law of Vector Products — Physical Proof 35
.90. Cartesian Expansion of the Vector Product 38
xiii
av CONTENTS.
AST. FAGS
21. Applications to Mechanics — Moment 39
22. Motion of a Rigid Body 41
23. Composition of Angular Velocities 41
and Problems 43
CHAPTER in.
Vector and Scalar Products of Three Vectors.
24. Possible Combinations of Three Vectors 48
26. Triple Scalar Product V - a-(b*c) 48
26. Condition that Three Vectors lie in a Plane — Manipula-
tion of Scalar Magnitudes of Vectors 50
27. Triple Vector Product q — a*(b*c) — Expansion and Proof. 51
28. Demonstration by Cartesian Expansion 53
29. Third Proof 54
30. Products of More than Three Vectors 55
31. Reciprocal System of Vectors 57
32. Plane Normal to a and Passing through End of b — Plane
through Ends of Three Given Vectors — Vector Perpen-
dicular from Origin to a Plane 58
33. Line through End of b Parallel to a 60
34. Circle and Sphere 61
34a. Resolution of System of Forces Acting on a Rigid Body —
Central Axis — Minimum Couple 63
Exercises and Problems .66
CHAPTER IV.
Differentiation of Vectors.
35. Two Ways in which a Vector may Vary — Differentiation with
Respect to Scalar Variables 70
36. Differentiation of Scalar and Vector Products 72
37. Applications to Geometry — Tangent and Normal 73
38. Curvature — Osculating Plane — Tortuosity — Geodetic Lines
on a Surface 76
CONTENTS,
ABT. PACB
39. Equations of Surfaces — Curvilinear Codrdinates — Ortho-
gonal System 79
40. Applications to Kinematics of a Particle — Holographs —
Equations of Hodographs 80
41. Integration with Respect to a Scalar Variable — Orbit of a
Planet — Harmonic Motion — Ellipse 83
42. Hodograph and Orbit under Newtonian Forces 87
43. Partial Differentiation — Origin of the Operator V 90
Exercises and Problems 91
CHAPTER V.
The Differential Operators.
d d d
dx dy dz
44. Scalar and Vector Fields 94
45. Scalar and Vector Functions of Position — Mathematical
and Physical Discontinuities 95
46. Potential — Level or Equipotential Surfaces — Relation
between Force and Potential 98
47. V applied to a Scalar Function — Gradient — Independence
of Axes — Fourier's Law 102
48. V applied to Scalar Functions — Effect of V on Scalar
Product 104
49. The Operator 8 ^ V, or Directional Derivative — Total Deriva-
tive 106
50. Directional Derivative of a Vector — V applied to a Vector
Point-Function 107
51. Divergence — The Operator V- 109
52. The Divergence Theorem — Examples — Equation of Flow
of Heat 112
53. Equation of Continuity — Solenoidal Distribution of a Vector 116
54. Curl — The Operator V* — Example of Curl 117
55. Motion of Rotation without Curl — Irrotational Motion 119
56. V, V% V* applied to Various Functions — Proofs of Formulae 120
57. Expansion Analogous to Taylor's Theorem 124
58. Stokes' Theorem 124
59. Condition for Vanishing of the Curl — Conservative System
of Forces Vtt
XVi CONTENTS.
ART. PAGB
60. Condition for a Perfect Differential 129
61. Expression for Taylor's Theorem — The Operator e*' v ( ). 131
62. Euler's Theorem on Homogeneous Functions 131
63. Operators Involving V Twice — Possible Combinations —
The Operator V 2 -V "V 133
64. Differentiation of r m by V* 135
Exercises and Problems 136
CHAPTER VI.
Applications to Electrical Theory.
65. Gauss's Theorem — Solid Angle — Gauss's Theorem for the
Plane — Second Proof 138
66. The Potential Function — Poisson's and Laplace's Equations
— Harmonic Function 143
67. Green's Theorems 148
68. Green's Formulae — Green's Function 148
69. Solution of Poisson's Equation — The Integrating Operator
—///.4* ■-
70. Vector-Potential 153
71. Separation of a Vector-Function into Solenoidal and Lamellar
Components — Other Systems of Units 154
72. Energy in Terms of Potential 156
73. Energy in Terms of Field Intensity 157
74. Surface and Volume Density in Terms of Polarization 159
75. Electro-Magnetic Field — Maxwell's Equations 160
76. Equation of Propagation of Electro-Magnetic Waves 163
77. Poynting's Theorem — Radiant Vector 164
78. Magnetic Field due to a Current 165
79. Mechanical Force on an Element of Current 167
80. Theorem on Line Integral of the Normal Component of a
Vector Function 168
81. Electric Field at any Point due to a Current 170
82. Mutual Energy of Circuits — Inductance — Neumann's
Integral 171
83. Vector-Potential of a Current — Mutual Energy of Systems
of Conductors — Integration Theorem 173
84. Mutual and Self-Energies of Two Circuits 175
Exercises and Problems 176
CONTENTS. xvu
CHAPTER VII.
Applications to Dynamics, Mechanics and Hydrodynamics.
ABT. PAOB
85. Equations of Motion of a Rigid Body — D'Alembert's Equa-
tion — Equations of Translation — Motion of Center of
Mass 178
86. Equations of Rotation — Kinetic Energy of Rotation —
Moment of Inertia 180
87. Linear Vector-Function — Instantaneous Axis 182
88. Motion of Rotation under No Forces — Poinsot Ellipsoid —
Moments and Products of Inertia — Coordinates of a
Linear Vector-Function — Principal Moments of Inertia —
Principal Axes 184
89. Geometrical Representation of the Motion — Invariable
Plane — Invariable Axis 191
90. Polhode and Herpolhode Curves — Permanent Axes —
Equations of Polhode and Herpolhode 192
91. Moving Axes and Relative Motion — Theorem of Coriolis. 194
92. Transformation of Equations of Motion — Centrifugal Couple
— Gyroscope 198
93.. Eider's Equations of Motion 199
94. Analytical Solution of Euler's Equations under No Impressed
Forces 200
95. Hamilton's Principle — Lagrangian Function 202
96. Extension of Vector to More than Three Dimensions —
Definitions 204
97. Lagrange's Generalized Equations of Motion — The Oper-
ator VL =» Contains the Whole of Mechanics 205
98. Hydrodynamics — Fundamental Equations — Equation of
Continuity — Euler's Equations of Motion of a Fluid . . . 207
99. Transformations of the Equations of Motion 211
100. Steady Motion — Practical Application 212
101. Vortex Motion — Non-creatable in a Frictionless System —
Helmholtz's Equations 212
102. Circulation — Definition 214
103. Velocity-Potential — Circulation Invariable in a Friction-
less Fluid 216
Exercises and Problems 217
CONTENTS.
APPENDIX.
Notation and Formula.
Various Notations in Use 221
Hamilton 22 1
Heaviside 221
Grassmann 221
Gibbs 222
Comparison of Formulae in Different Notations. 222
Notation of this Book 224
Formula.
Resume" of the Principal Formulae of Vector Analysis 229
Vectors 229
Vector and Scalar Products — Products of Two Vectors 230
Products of Three Vectors 231
Differentiation of Vectors 233
The Operator V, del 233
Linear Vector Function 237
Note on Different Varieties of Vectors 240
Definitions of the Normal, Normal Plane, Principal Normal, Bi-
nomial and Rectifying Plane for a Space Curve , . 242
Frenet's Formulas for a Space Curve 244
Motion of an Electron in a Uniform Magnetic Field 245
Two Proofs of Stokes' Theorem 249
Proof of Gauss's Theorem 251
Other Integration Theorems 252
Index 255
BIBLIOGRAPHY
Works Specially Consulted in the Preparation of this Book,
Appell. Traits de Mecanique
Rationelle.
Bjerknes. Vorleeungen iiber Hy-
drodynamischen Fernkrafte.
Bucherer. Elemente der Vek-
tor-Analysis.
Burnsidb and Panton. Theory
of Equations.
Clifford. Elements of Dynamic.
Drude. Theory of Optics.
Emtage. Introduction to the
Mathematical Theory of Elec-
tricity and Magnetism.
Fehr. Mlthode Vectorielle de
Grassmann.
Fischer. Vektordifferentiation
und Vektorintegration.
Fo'ppl. MaxwelTsche Theorie
der Elektricitat.
. Vorleeungen uber Tech-
nische Mechanik.
4Gans. Einfuhrung in die Vek-
toranalysis.
Gibbs. Collected Papers.
Heaviside. Electrical Papers.
' . Electro-magnetic Theory.
Henrici and Turner. Vectors
and Rotors with Applications.
Ibbbtson. Mathematical Theory
of Elasticity.
Jaumank. Bewegungslehre.
Jolt. Manual of Quaternions.
Kelland and Tait. Introduc-
tion to Quaternions.
Kirchhoff. Vorlesungen iiber
Mathematische Physik.
Lagrange. Mecanique Analy-
tique.
Love. Theory of Elasticity.
Maxwell. Electricity and Mag-
netism.
McAulay. Utility of Quaternions
in Physics.
. Octonibns.
Minchin. Treatise on Statics.
Pierce, B. O. Elements of the
Theory of the Newtonian Po-
tential Function.
Poinsot. Theorie Nouvelle de la
Rotation des Corps.
Routh. Rigid Dynamics.
Steinmetz. Alternating Current
Phenomena.
Tait. Dynamics.
. Quaternions.
Walton. Problems in Mechan-
ics.
Webster. The Dynamics of a
Particle and of Rigid, Elastic
and Fluid Bodies.
The Theory of Electricity
and Magnetism.
Williamson and Tarleton. Ele-
mentary Treatise on Dynamics.
Wilson, Gibbs-. Vector An-
alysis.
xix
SUGGESTIONS FOR WRITING VECTOR ANALYSIS
ON THE BOARD
A number of inquiries have come in asking how to write
vectors on the blackboard. It seems that the bold-faced
type or Clarendon is perfectly satisfactory as far as print is
concerned, but it is impracticable to produce such a differ-
ence in chalk-written symbols. To a great extent these same
troubles also occur in manuscript.
There are several methods of differentiating vectors from
purely scalar symbols which have proved satisfactory.
The notation given in the text is entirely practicable and
definite. That is, if a, b or r denote vectors in any discussion,
let a , bo or r denote their magnitudes and a h b\ or n denote
their directions or unit vectors along a, 6 or r respectively.
Thus a = aoai.
There is here a slight chance of ambiguity in the equation
a = a\i + Oij + ajc
where the i-component of a might be confounded with the
unit vector along a. The writer does not consider this a
serious objection. Like Tait we say that anybody finding
difficulty with this small matter has begun the study of
vectors too soon!
Another method is to place a line or dash over the vector-
symbol. So that if a denotes a vector, then a is its magnitude,
Oi is its direction and ai is its t-component.
Still another method to which the writer is very partial,
having been brought up on Hamilton's notation, is to reserve
the Greek alphabet for vectors.
So that if a is any vector, a or a is its magnitude, and a>
is its direction, while ai 9 a% } a 8 are its i-, j-, fc-components.
xxi
xxu
SUGGESTIONS
After reading the book notices and reviews we are still of
the same opinion as to the essential superiority of Gibb's
notation over others, notwithstanding the criticisms of it
which we expected. We have never claimed that a*b was a
symmetrical function, but we do claim that thenotation of both
a*b and a*b is symmetrical. The minus sign in ct*b = — b*a,
does not make the notation unsymmetrical.
Almost simultaneously with this text was issued a vector
analysis by the Italian mathematicians Burali-Forti and
Marcolongo.
These gentlemen have invented still another notation
which is similar to ours but which employs the X (large cross)
for a scalar product and an inverted V (a) for a vector
product. With the symmetry of their notation we are in
favor, but why introduce any more notations when there
are already so many to pick from?
This question of notation, which has nothing to do with
the spirit of the method, is for each individual to solve for
himself. We have employed what we believe to be the sim-
plest and best and we have presented at length our argu-
ments in favor of it.
J. G. COFFIN, 1911
VECTOR ANALYSIS
CHAPTER I.
ELEMENTARY OPERATIONS OF VECTOR ANALYSIS.
Definitions.
1. A Vector is a directed segment of a straight line on which
are distinguished an initial and a terminal point. A vector
thus has a magnitude and a direction. Any quantity which
can be represented by such a segment may be called a vector
quantity. The importance of this generalized conception is
easily understood when it is considered that motion or dis-
placement, velocity, acceleration, force, electric current,
magnetic flux, lines of force, stresses and strains due to any
cause, flow of heat and of fluids, all involve two parts, i.e.,
magnitude and direction. All such quantities are vector
quantities.
A Scalar on the other hand is any quantity which although
having magnitude does not involve direction. For example,
mass, density, temperature, energy, quantity of heat, electric
charge, potential, ocean depths, rainfall, numerical statistics
such as birth rates, mortality or population, are all scalar
quantities.
A scalar, then, reduced to its simplest terms is merely a
number and as such obeys all the laws of ordinary alge-
braic analysis. A vector, however, involving direction in
addition to its numerical magnitude has an analysis pecul-
iar to itself, the laws of which are to be derived.
2. Graphical Representation of a Vector. Any vector
quantity may be represented graphically by an arrow.
1
VECTOR ANALYSIS.
The tail of the arrow, 0, is called the origin; the head, A, 1
is called the end or terminus.
Symbolically a vector may be denoted by two letters,
the first one indicating the origin, the second one the
terminus.
A small arrow is often placed over these letters to indi-
cate more exactly that the quantity considered is a vector.
Thus, OA denotes the vector beginning at 0, ending at A,
and pointing in the direction from to A. This notation
while useful is at times cumbersome. Hence more usually
a vector will be denoted by a single letter, which involving
more than a mere scalar is printed differently to distinguish
it from purely scalar quantities, i.e., in Bold-faced Type.
Thus the vector a* means the going of the distance OA
in the direction to A from any point in space <w origin.
Fig. 1.
3. Equality of Vectors. All lines having the same length
or magnitude and the same sense are equal vectors whatever
their origin may be. Thus in Fig. 1, OA and 0' A' are equal
vectors.
Negative Vector. The vector 0" A" having the same
length and direction as a but the opposite sense, is defined
* The terms Step, Stroke, or Directed Magnitude are sometimes used
^a synonyms of Vector.
)
VECTOR ANALYSIS. 8
as the negative of a and is written — a. Evidently also OA
is the negative of 0"A".
Unit Vector. The directional part of any vector a may
be concisely represented by a vector having the same sense
and direction as a but of unit length. Such a vector is
called a unit vector and will be denoted by adding the
suffix 1 to the symbol representing the vector. Thus & x is
a vector having the same direction as a, but of unit
length.
The length of a vector is termed its magnitude, size, or
its absolute value. Sometimes, also, the term tensor is
used. The magnitude of a vector a will be written a,
using the same letter as that which denotes the vector but
printed in italic type. It will be sometimes convenient also
to denote the magnitude of a by adding the subscript to a
thus:
a = a.
The vector a then may be considered as one, a times as
long as a t and hence we may write:
a = as. x or = a a t . (1)
Any vector then may be represented by the product of its
unit vector into its magnitude as in (1).
The expression m a denotes a vector m times as long as a,
having the same direction but m times its magnitude. The
multiplier — 1 from what has been said about negative
vectors, reverses a vector.
Parallel vectors whatever their magnitude are said to be
coUinear.
Reciprocal Vector. The vector parallel to a but whose
length is the reciprocal of the length of a is said to be the
reciprocal of a.
So that if a = aa t
I = a-* = 5i- (2)
a a
VECTOR ANALYSIS.
Composition of Vectors.*
4. Addition and Subtraction. To obtain graphically the
sum of the two vectors a and b, draw b starting from the
end of a; the line joining the
origin of a with the end of b
is the sum in question. In other
words, it is the diagonal of the
parallelogram of which the two
vectors a and b are the sides.
Evidently the sum (a + b) is
the same as (b + a). If there
are more than two vectors to be
added, the sum of the first two
may be taken and the third added
to it as above, then to the resultant add the next one and
so on. A moment's consideration of Fig. 3 will show that
Fia. 2.
if we draw the vectors one after the other in a chain, each
new one from the end of the last one drawn, the line joining
See Appendix, p. 240, Note on Different Varieties of Vectors.
VECTOR ANALYSIS. 6
the origin of the first one to the terminus of the last one is
the vector sum of them all.
A consideration of Fig. 3 will also show that the order in
which they are taken is immaterial. The same construc-
tion then is used to find the sum of any number of vectors
as is used in finding the resultant of the forces which would
be represented by these vectors. Hence the importance of
vectors in mechanics.
To subtract two vectors, add to the first the second one
reversed. The extension of these rules to both positive
and negative is obvious.
\
t
Fiq. 4.
Vector Sam as an Integration. Any curve may be consid-
ered to be built up of an infinite number of infinitely short
vectors, their directions being at every point along the
tangent to the curve.
The sum of such a series of vectors differs in no way
from the sum of a finite number of finite vectors. If da
represents any one of these small vectors, then by adding
them all the resultant AB is obtained. The operation of
6 VECTOR ANALYSIS.
adding this infinite number of infinitesimal vectors may be
represented by an integration sign thus:
AB
= J^da. (3)
If the curve is a closed one, whether a plane curve or not,
then A and B coincide and AB = or /da around a closed
path is zero.
Scalar and Vector Fields and their Addition.
5. Point-Function. Definition of Lame'. If for every
position of a point in a region of space a quantity has one
or more definite values assigned to it, it is said to be a func-
tion of the point, or more concisely, a point-function. We
may have both scalar and vector point-functions.
As an Example of a Scalar Point-Function* consider the
potential at any point due to any distribution of matter
M x and let its value be V v Now consider the potential at
the same point P due to any other distribution of matter
M 2 and let its value be V 2 . Then the potential at P due
to both masses together is simply V x + V 2 . This value is
found by adding together the two scalars V x and V 2 .*
* Perhaps the following example of scalar field will be clearer to
some minds. Consider a point P and let it be illuminated by a source
of light M ,. Evidently every point in the vicinity of the source is illu-
minated to a greater or lesser extent according to its distance from the
source. The illumination or intensity of light at all points of the space
considered may be represented by a scalar point-function. Let now
another source of light M t be brought into the space under considera-
tion. This source produces a certain intensity of illumination at
every point of the space, of course including the point P. The total
amount of illumination now received at the point P is the scalar sum
of the amounts it receives from each individual source. This is true
of every other point in the field. So that in general in order to find the
illumination at any point due to separate sources, one simply adds the
VECTOR ANALYSIS. 7
Practical Definition of Continuity of a Scalar Point func-
tion. If, as we go from any point in space to any near
adjacent point, the magnitude of the scalar point-function
Pio. 5.
suffers no abrupt change, the function is said to be con*
tinuous.
Aa an Example of a Vector Point-Function consider the
force of attraction at any point P due to the attraction of
the mass M,. This force is evidently a vector quantity, as
it has a definite magnitude and a definite direction, so that
its representation requires the use of a vector at P; let F,
be this vector. Similarly let F, be the vector representing
the force at P due to the matter M ,. Then the force at P
due to the combined action of M, and M , is the vector sum
of F, and F, and must be obtained by the laws of vector addi-
tion; t.e. p the parallelogram law. If we go from the point
P to another point Q in space the magnitudes and direc-
valuea of the separate intensities at the point due to the separate
sources respectively. This constitutes an addition of scalar fields.
ITie field* are here tcalar fields because we are considering only the
ammmt* of the illumination received at any point.
8 VECTOR ANALYSIS.
tions of these forces F, and F, at Q, and hence, in general,
their sum, F, + F„ at Q undergo changes.
Fro. 6.
Practical Definition of the Continuity of a Vector Point
Function. If, as we go from any point in space to any near
adjacent point, the direction as well as the magnitude of
the vector point-function suffers no abrupt change, the
function is said to be continuous.
6. Decomposition of Vectors Into Components. From £ 4
it is evident that any vector q may be considered as the
sum of any number of compo-
nent vectors, which when joined
end to end, as in vector addi-
tion, the first one begins at the
origin of q, and the last one ends
at the terminus of q. Thus: ,
q = a + b + c + d + e.
These vectors need not lie in
one plane. Vectors all of which
lie in or parallel to the same plane are said to be coplanar.
In particular it is often convenient to decompose a vector
Fro. 7.
VECTOR ANALYSIS.
9
into two or three components at right angles to each other;
two in case all the vectors under consideration are coplanar;
three, when they are not coplanar.
7. The Three Unit Vectors i ] k. Consider the right-handed
Cartesian system of axes. The three unit vectors along the
xyz axes are called i j k respectively. It is evident that
any vector r is equivalent to a certain vector OA along OX,
plus a vector AB along OZ, plus a vector BC along OY.
Fia. 8.
In other words, if x y z denote the magnitudes of these
vectors respectively, we may write for any vector r whose
components are x, y, z }
r = xi + y\ +zk. <£\
10
VECTOR ANALYSIS.
xi, y j, and zk are the three projections of r along the three
axes respectively. If a, /?, y be the direction angles of any
vector parallel to OC, then evidently
x — r cos a,
y = r cos /?,
z = r cos j\
(5)
This decomposition of a vector into two or three rectan-
gular components is of the utmost importance and is the
connecting link between the two or three dimensional Car-
tesian and Vector Analyses, respectively.
If two vectors are given,
a = a t i + a 2 j + a,k f
5 = ^1+6^+ bjt,
their sum is evidently
(a + b) = (a, + 6,) i + (a 2 + 6 2 ) j +(a 8 + & 8 ) k. (6)
This may be extended to any number of vectors and
shows that the components of the sum are equal to the
sums of the components, so that
ga = igo, + j£a,+ kga,
(7)
This theorem is of use in the composition of forces. It is
possible to resolve any vector r into three components par-
allel to any three non-coplanar vectors; and such a resolu-
tion is easily seen to be unique. Practically, in order to
find the rectangular components of a vector, equations (5)
are employed, so that
r = r (i cos a + j cos /? + k cos y). (8)
If we divide through by the magnitude of r there remains
H-r.
i cos a + j cos (i + k cos p,
(9)
VECTOR ANALYSIS.
11
so that the rectangular components of a unit vector are
always its direction cosines.
By inspection of Fig. 8 it is evident that
r* = x 2 + y 1 + #.
Fiq. 9.
Fig. 10.
Vector Equations.
8. Equations of the Straight Line and Plane. Let r be a
variable vector, with origin at 0, and s a variable scalar;
it is then evident on inspection (Fig # 9) that
r = sa (10)
is the equation of a straight line passing through the origin
and parallel to a. It is also easily seen (Fig. 10) that
r = b + «a (11)
12
VECTOR ANALYSIS.
is the equation of the straight line through the terminus of
b and parallel to a. By means of equation (11) the equa-
tion of a line passing through the ends of any two given
vectors a and b may easily be derived.
S
Fig. 11.
The vector AB is (b — a), hence by equation (11) the
line through the terminus of a parallel to (b — a) is
r = a + t (b - a),
where t is a scalar variable. These equations may be put
into the easily remembered forms
r = tb + (1 -<)a,
and by analogy
r = sa + (1 -s)b. (12)
It is evident that if the directions of the coordinate axes
be taken along a and b, then the magnitudes of a and b
VECTOR ANALYSIS. 18
respectively are the intercepts the line makes with these
axes, the corresponding Cartesian equation being
£ + {-!.
a b
All problems in line geometry are now readily solvable.
If all the lines of the problem lie in one plane, two, and only
two, arbitrary non-parallel vectors are chosen and all others
expressed in terms of them. For a problem in three dimen-
sions all the lines are expressed in terms of three, and only
three, arbitrary non-coplanar vectors.
9. Condition that Three Vectors should Terminate In the
Same Straight Line. Putting equation (12) in the form
«a+(l-s)b-r =
it is seen that in the linear relation connecting three vectors
which end in the spme straight line the sum of the coeffi-
cients is equal to zero. Or in other words, if
xa + yb + zc => (13)
and x + y + z = 0,
the three vectors a, b, and c necessarily end in the same
straight line, and are said to be termino-collinear.
Example. As a simple example of the general method of
procedure, let us prove that the diagonals of a parallelo-
gram meet in a point which bisects them both. Take the
origin at the corner 0, and write down the equation of the
diagonals OC and AB in terms of a and b, the vectors OA
and AB. Notice that the origin may be chosen arbitrarily
and hence may be taken so as to simplify the equations.
Very often, however, it is better not to place the origin
at any special or definite point, so that more symmetry is
produced in the resulting equations.
14
VECTOR ANALYSIS.
The equation of OC is
r = 8 (a + b), by equation (10)
and that of AB is
r = t a + (1 - t) b, by equation (12)
where s and t are variable scalars. For intersection, both
equations must be satisfied by the same value of r; hence,
equating,
s (a + b) = la + (1 -Ob. (14)
Fig. 12
This vector equation is actually equivalent to two scalar
equations and suffices to determine s and t, for the vector r
to the point of intersection is uniquely determined in terms
of the vectors a and b, so that the scalar coefficients of these
vectors on both sides of equation (14) must be respectively
equal. The coefficients of a give
a = t
and of b 8 = (1 — t).
This makes 8 = t = £, and the vector to the point of inter-
section is then, by substituting this value for 8 in r= 8 (a -f b),
6b - i(a + b) - I OC.
This principle is applicable to any kind of line problem
in two or three dimensions. The method of equating the
VECTOR ANALYSIS.
15
coefficients of the same vector on both sides of an equation
is analogous to the conditions for equality of two complex
imaginary expressions; that is, if
s + it = s' + it',
then 8 = s' and t = t'.
Example. As an example of the symmetrical method to
prove that the medians of a triangle meet in a single point
which trisects each of them. Choose any point not in the
plane of the triangle for origin, and define the triangle by
the three vectors a, b, and c from the origin to its vertices
A, B, and C We choose the origin out of the plane of
the triangles so that we may use three independent vectors
instead of but two, as would be necessary if the origin were
taken in the same plane.
Then OA' = J (b + c),
OB' = i (c + a),
OC = J (a + b),
so that the equation of
AA' is r = xa + (1 - z) £ (b + c), (a)
BB'ia r = i/b+(l - y )j(c + a) f (6)
CC is r = «c + (1 - z) i (a + b). (c)
Equate the coefficients in (a) and (b) for intersection,
of a, x - i (1 - y),
of b f 4 (1 - x) - y,
of c f I (1 - x) = i (1 - y),
so that x = y = J and the vector to their point of intersec-
tion is
3
This is evidently the point of intersection of the third line
with either of the first two, by symmetry. It is also the
16
VECTOR ANALYSIS.
v c
Fiq. 13.
mean point of A, B f and C, as explained below. It is the
point of trisection, because adding to a, $ of A A' we obtain
the same result, thus:
OA' - i (b + c) f
and OD - a + if- a + ft (b + c)] - a + b + c .
Fig. 14.
By choosing the origin at one of the vertices the sym-
metry is lost but a gain in directness and shortness is made.
In problems involving algebraic coefficients instead of nu-
merical ones the symmetrical method is generally preferable.
VECTOR ANALYSIS.
17
the
iin
i
r
i
i
10. Equation of a Plane. The vector to any point in the
plane determined by the vectors a and b and passing through
the origin is evidently
r = a a + t b, (14)
where * and t are two independent scalar variables. If the
origin be removed to the origin of a vector c, through the
terminus of which the plane parallel to a and b passes,
then the vector to any point P in the plane is now given by
r = c + «a + <b. (15)
11. To find the equation of a plane passing through the ends
of the three non-coplanar vectors a, b,and c, notice that the vec-
tors (a — c) and (b — c) evidently lie in the plane. By employ-
ing the previous equation (15), the equation may be written
r = c + s (a — c) + t (b — c),
# Fig. 15.
which may be put into easily remembered form, analogous
to equation (12)
r = sa + tb + (1 - s - c. (16)
It is evident that if the directions of the coordinate axes
be taken along a, b, and c, then the intercepts made by the
VECTOR ANALYSIS.
plane with these axes are the lengths of a, b, and c respec-
tively, the corresponding Cartesian equation being
•- + * + -
. 1.
12. Condition that Four Vectors Terminate In the Same
Plane. Rearranging equation (16),
s a + t b + (1 — a - c — r => 0,
it is seen that whenever there is a linear relation between
any four vectors they terminate in one and the same plane
if the Bum of the coefficients is zero. Or in other words, if
x* + yb + zc + wd =
and x + y + z + w = 0, (17)
a, b, c, and d terminate in the same plane and are said to be
termino-coplanar.
13. To Divide a line In a Given Ratio. Centrold. To find
the value of a vector which divides the distance between
two points A and £ in a given ratio, m to n say, it is simply
VECTOR ANALYSIS.
19
necessary to express the vector r in the form, evident on
inspection,
r = a +
m
m + n
(b - a) =
na -f mb
m + n
(18)
Fig. 17.
It is a well-known result in mechanics that the center of
gravity of two masses m t and m 2 divides the line joining
them inversely as these masses, so that by (18)
r e m t a, + m&i
mrfiiij,
m,
is the vector to their center of mass or their centroid. If
now there is a third point a 3 with mass m 3 added to the
20
VECTOR ANALYSIS.
system, the new centroid will be that of the two masses r 2
with mass (m^ + m 3 ) and a, with mass m,, or again by (18),
r f s K-fm a )r a -fm,a 8 == m 1 a t +m a a a +wA = Sma (19)
The generalization is immediate. If M = 2 m denotes the
total mass of the system of particles and r the vector to
their center of mass,
M r « 2 m a. (20)
If the masses form a continuous body, the formula
becomes
SU pdv
(21)
where p is the density and dv is the element of volume.
The integrations are taken throughout the volume.
If f = xi + y\ + zk
and • an = x n i -f t/nj + 3*k,
formula (20) breaks up into the three well-known ones for the
three coordinates of the center of mass,
MX = XTnXrti
i
n
My ^^myn,
(22)
i
n
M*=£
WWn.
Similarly (21) gives three of the form
Jll xpdxdydz
* " err ,etc-
llj pdxdydz
(23)
VECTOR ANALYSIS.
21
14. Relations Independent of the Origin. That the center
of gravity and therefore all the formulae just derived are
independent of the origin may be shown by the following
reasoning.
Taking the origin at 0, the vector to R, the center of
gravity of the two masses m and n is, by (18),
ma + nb
m + n
(24)
Now change the origin to 0', the new vectors to the
masses being a' and b' and the vector to the first origin
from the new one being c.
Fig. 19.
The vector to the center of gravity from 0' is now given by
ma' + wb'
r* -
m -f n
But since a' = a + c and b' = b + c, this equation may be
written
i> _ *» (c + a) + n (c + b) ^ ma + nb
m + n m -\- n
22 VECTOR ANALYSIS.
which says that the new center of gravity is the same as
before, as r 7 = r + c.
It will be noticed on writing (24) in the form
(m + n)r — ma — n b =
that the algebraic sum of the scalar coefficients is zero.
This leads to the
General Condition for a Relation Independent of the
Origin. The necessary and sufficient condition that a linear
vector equation represent a relation independent of the
origin is that the sum of the scalar coefficients of the equa-
tion be equal to zero. Let the equation be
m^ + mjBL 2 -f • • • = 0. (25)
Change the origin from to O f by adding a constant
vector 1, the distance from to 0' , to each of the vectors,
*i> a 2> *•> e ^ c »J the equation then becomes
™>i (*! + 1) + m 2 (a a +. 1) + • - • -
or m^ + m s a s + • • • + 1 (m l + m 2 + • • • ) = 0.
If this is to be independent of the origin, i.e., the same as
(25), the coefficient of 1 must vanish, or
Wi 4- ^2 -f- • • • = 0.
EXERCISES AND PROBLEMS.
1. Prove that the vectors
± a ± b ± c
when drawn from a common origin terminate at the vertices of a
parallelopiped.
2. A person traveling eastward at a rate of 3 miles an hour finds
that the wind seems to blow directly from the north; on doubling
his speed it appears to come from the northeast. Find the vector
wind velocity.
3. A ship whose head is pointing due south is steaming across a
current running due west ; at the end of two hours it is found that
the ship h&a gone 36 miles in the direction 15° west of south. Find
the velocities of the ship and current, graphically and analytically.
VECTOR ANALYSIS. 23
4. A weight W hangs by a string and is pushed aside by a hori-
zontal force until the string makes an angle of 45° with the vertical.
Find the horizontal force and the tension of the string.
5. A vector r is the resultant of two vectors a and b which make
angles of 30° and 45° with it on opposite sides. How large are the
latter vectors?
6. A car is running at 14 miles an hour and a man jumps from
it with a velocity of 8 feet per second in a direction making an angle
of 30° with the direction of the car's motion. What is his velocity
relative to the ground ?
7. Verify, by drawing, the truth of the laws of association and
commutation, taking a number of vectors, a, b, c, d, etc., to scale,
and show that the resultant is independent of the order of addition
or subtraction.
8. Given the vector
r =- a,l + aj
derive the vector of same length perpendicular to it through the
origin.
Derive the vector perpendicular to the one you find. Compare
with the original one.
9. Find the relative motion of two particles moving with the
same speed v, one of which describes a circle of radius a while the
other moves along a diameter.
10. Two particles move with speeds v and 2 v respectively in
opposite directions, in the circumference of a circle. In what
positions is their relative velocity greatest and least, and what
values has it at those positions?
11. Draw the vectors
a = 61- 4] + 10k
b 61+ 4J-10k
c = 41- 6]-10k
d - 10 J + 4 k
Find their sum graphically and analytically.
12. The equation
(r - a)o = (r - b)
represents the plane bisecting at right angles the line AS.
24
VECTOR ANALYSIS.
13. Find the equation of the locus of a point equidistant from
two fixed planes.
14. The line which joins one vertex of a parallelogram to the
middle point of an opposite side trisects the diagonal.
16. To find a line which passes through a given point and cuts
two given lines in space.
16. If
ia+yb-0
and x + y -
show that a and b are equal in magnitude and direction. Or what
is the same thing, that measured from the same origin, a and b end
at the same point.
17. If
xa + t/b + ze -
and x + y + z =-
show that a, b, and c terminate in the same straight line ; they are
then said to be te rmi no-coll i near.
18. If
xa-ft/b + sc-fwd—
and x + y + z + w~*Q
show that a, b, c, and d terminate in the same plane ; they are then
said to be termino-coplanar.
19. A triangle may be constructed whose sides are equal and
parallel to the medians of any given triangle.
20. 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 joining 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.
21. Discuss the conditions imposed upon three, four, or five
vectors if they satisfy two equations, the sum of the coefficients in
each of which is zero.
22. Take a number of points at random on a sheet of paper,
assigning arbitrary masses to them. Verify by drawing that their
center of mass is independent of the origin chosen in finding it.
VECTOR ANALYSIS. 25
23. If a system of masses, each mass concentrated at a point, be
divided into a number of partial systems, and each of these be
replaced by its resultant mass, then the new system has the same
center of mass as the original one.
24. A cardboard square is bent along a diagonal until the two
parts are at right angles. Find the position of the center of gravity.
26. Forces acting at a point are represented by OA , OB, OC,
. . . , ON. Show that if they are in equilibrium is the centroid
of the points A, B,C, . . . , N.
26. The middle points of the lines which join the points of
bisection of the opposite sides of a quadrilateral coincide whether
the four sides be in the same plane or not.
27. The bisectors of the angles of a triangle meet in a point which
trisects each of them.
Employ unit vectors along two of the sides as independent
vectors. The bisectors are then a, -f b„ etc.
28. If two forces acting at a point are represented by the
vectors n a and b their resultant is represented in magnitude and
direction by the vector (n + 1) OG, the point G being taken on AB
so that BG - nAG.
This allows the resultant of two forces to be drawn knowing one
and part of another.
29. If two forces are equal to n.OA and m.OB, the resultant
BO n
passes through the point G determined so that — — — and is
AG m
equal to (m + n) OG in magnitude.
30. Forces P„ F„ . . . , F n , acting in a plane at are in equi-
librium. Any transversal cuts their lines of action in points L,, L„
. . . , L„ ; and a length OLi is positive when in the same direction as
— >
0F it Prove that
££ - 0.
*40L
31. Show that the resultant of any number of concurrent forces,
F l9 Pi, F», . . . may be found thus: measure off any lengths /„ /,,
l t , . . . from the point of meeting along them respectively; place at
F F
the ends of these lines particles of masses proportional to — * , —
26 VECTOR ANALYSIS.
F
--*■ , . . . ; let be the center of gravity of these particles; then 00
is the line of action of the resultant of the given forces and its mag-
nitude is
OG
*sf
32. A particle placed at is acted upon by forces represented in
magnitudes and directions by the lines 0A lt 0A t , . . . 0A n , which
join to any fixed points A lf A t , . . . A n ) where must be placed
so that the magnitude of the resultant force may be constant ?
Am. If r represent the magnitude of the resultant, may be
•»
placed anywhere on a sphere of radius - described around the cen-
n
troid of the fixed points as center.
88. ABCD is a quadrilateral of which A and C are opposite
vertices. Two forces acting at A are represented by the sides AB
and AD; two at C by CB and CD. Prove that the resultant is
represented in magnitude and direction by four times the line
joining the middle points of the diagonals of the quadrilateral.
34. Show that the resultant of the three vector diagonals of a
parallelopiped meeting at a point is represented by twice the
diagonal of the parallelopiped drawn from the same point.
35. If through any point within a parallelogram, parallels be
drawn to the sides, the corresponding diagonals of the two new
parallelograms thus formed and of the original one meet in a point.
36. The middle points P, Q, R of the diagonals of any complete
quadrilateral ABCDEF are colli near.
37. Any point is joined to the vertices of a parallelogram;
show that the sum of the vectors to the vertices is four times the
vector to the intersection of the diagonals.
What conclusion do you derive from this fact ?
38. ABCDEF A is a regular hexagon. Show that the resultant
of the forces represented by A B, 2 AC, 3 AD, 4 A E t 5 A F is repre-
sented by a vector of magnitude n/351 AB, and find its direction.
39. ABCDEF A is a regular hexagon. Find the resultant of the
forces represented by the lines AB, AC, AD, AE, AF.
40. is any point in the plane of a triangle ABC, and D, E, F
are the middle points of the sides. Show that the system of forces
OA, OB, OC is equivalent to the system 0D t OE, OF.
VECTOR ANALYSIS. 27
41. ABC is a triangle with a right angle at A ; AD is the per-
pendicular on BC. Prove that the resultant of forces — acting
AB
along AB and — - acting along AC is — - acting along AD.
AC AD
42. P„ P„ . . . P n are points which divide the circumference of
a circle into n equal parts. If a particle G lying on the circum-
ference be acted upon by forces represented by GP lt GP t , . . . GP n ,
show that the magnitude of the resultant is constant wherever G is
taken on that circumference.
It is n X OG, being the center of the circle.
48. If be the center of the circumscribed circle of a triangle
ABC, and L the intersection of the perpendiculars from the vertices
on the sides, prove that the resultant of forces represented by LA,
LB, I/J will be represented in magnitude and direction by 2 LO.
44. D is & point in the plane of the triangle ABC, and / is the
center of its inscribed circle. Show that the resultant of the vectors
aAD, bBD, cCD is (a + b + c) ID, where a, b, c are the lengths of
the sides of the triangle.
46. The chords APB and CPD of a circle intersect at right
angles. Show that the resultant of PA, PB, TC, and PD is repre-
sented by twice the vector PO, where is the center of the circle.
46. Prove that the mean center of a tetrahedron is (a) the inter-
section of bisectors of opposite edges; (b) the intersection of lines
joining the vertices to the mean points of the opposite faces. Show
that the former lines bisect one another, and that the latter quad-
risect one another.
47. A, B, and C being three given points in a plane show that
any point in this plane can be made their centroid by giving suit-
able weights to these points.
48. Show that the medians of a triangle intersect in a point
which is the mean center of the vertices A, B, C with weights
1, 1, 1; that the altitudes intersect in a point which is the cen-
troid of the vertices with weights, tan A, tan B, tan C, respectively;
that the bisectors intersect in a point which is the centroid of the
vertices with weights equal to the lengths of the opposite sides.
CHAPTER n.
SCALAR AND VECTOR PRODUCTS OF TWO VECTORS.
The Scalar or Dot Product.
15. The Scalar Product of two vectors a and b, denoted
by a-b, Sab, ab or (ab) by various writers, is a scalar de-
fined by the equation
a-b = a b cos (ab) = b-a.
(26)
B C
Fig. 20.
This equation shows that the scalar product may be looked
upon as the product of the length of one of the two veo- /
tors multiplied by the projection of the other upon it, or
OAxOD = OBX OC.
Evidently, if the two vectors a and b are perpendicular
to each other cos (ab) = and their scalar product is zero.
The condition, then, of perpendicularity of two finite vectors
is that their scalar product be zero.
Or, if a-b = 0, then a -L b. (27)
28
VECTOR ANALYSIS. 29
If a and b are parallel vectors, cos (ab) = 1 and
a-b = a b,
and in particular if b = a,
a*a = a 2 .
The scalar product of a vector into itself is often written
as the square of the vector, thus,
a*a = a 2 .
In general, to obtain the magnitude of a vectorial expression
it is only necessary to square it, and the result is the square
of its absolute value or magnitude.
The Scalar Product Obeys the Ordinary Laws of Multipli-
cation. Consider the two vectors c and d as well as their
sum (c + d). Consider also their projections upon any
other vector b.
Fig. 21.
The projection of c on b is 0E y the projection of d on
b is EF, the projection of (c + d) on b is OF; hence
c-b + d-b = (c + d)-b = b-(c + d). (28)
This result is easily extended to the scalar product of the
sums of any number of vectors.
The application of these results to the unit vectors i, j ,
and k is of great importance, giving immediately
M = j.j = kk = i 2 - j s = k 2 = 1,
i.j = j.i - j.R = kj = k-i = ik - 0- <?ft\
80
VECTOR ANALYSIS.
If the two vectors a and b be given in terms of their
coordinates,
a = a t i + a 2 j + a 8 k
and
b = 6J + 6J + & 8 k,
then, by (28) and (29),
ab = (o t i + a 2 l + a 3 k)-(b t i + 6 2 j + 6 8 k)
= a x b x + a 2 b 2 + ajb 3 . (30)
If a x and b t are unit vectors, their projections on the three
axes are equal to their direction cosines; and since in this
case a^fy = cos (a^), then, by (30),
a^bx = cos (a^j) = cos (a t i) cos (b t i) + cos (aj) cos (bj)
+ cos (a t k) cos (b t k),
the familiar formula of Cartesian geometry for the angle
between two lines in terms of their direction cosines.
Fig. 22.
The well-known and useful formula giving directly the
magnitude of the resultant of any two vectors in terms of
their magnitudes and the angle between them, may be
derived in the following manner. In the triangle ABC
c = a + b.
VECTOR ANALYSIS.
81
Squaring to find its magnitude,
c«c = c 2 » c 2 - (a + b)-(a + b) = a»a + 2a-b + b-b
or c 2 — a 3 + 2 a 6 cos (ab or ) + b 2
and c 2 -» a 2 — 2 a 6 cos (<f>) + ft 2 ,
where <£ is the supplement to the angle between a and b.
16. line-Integral of a Vector. The scalar product plays
a very important r61e in mechanics and physics. For ex-
ample, the work done by a force F in the* displacement dt
is by definition
Fdr cos (Fdr) = F«dr.
If the force is known in direction and magnitude for every
point of its path, the work
** done in overcoming the forces
from A to B may be found by
evaluating the integral
W
-s:
F-dr.
(31)
This is called the line-integral
of the vector F along the curve
AB. The term "line-integral
of a vector along a curve" thus '
denotes the integral of the
tangential component along it,
unless expressly stated other-
Fig. 23. wise -
If q denotes the vector
velocity at any point of a fluid, the integral
C =
lq»dr
82 VECTOR ANALYSIS.
over any path in the fluid is called the circulation along
that path. If e denote the electric force at any point in
space, the integral
-/-
taken along any path gives the electro-motive force along
that path. This kind of an integral is thus of great impor-
tance in all branches of physics,
17. Surface-Integral of a Vector. As another example,
imagine a surface S drawn in any vector field; for example,
in a moving fluid. Let q be the vector velocity, determinate
at every point in the region considered. The lines of flow of
the fluid are therefore known and may be drawn. The
■■mount of liquid which passes outward through the ele-
VECTOR ANALYSIS. 88
ment dS in unit time at any point on the surface is the
outward normal component of q multiplied by the area dS,
or
q cos (nq) dS = q-n dS
where n is the unit outward drawn normal to dS. The
total outward flux through the surface is, then, the surface
integral.
Total Flux
= C Tq-n dS (32)
taken over the surface in question. It may easily be seen
that in this example the vector q may be any physical
vector such as electric force, magnetic force, gravitational
force, or flux of heat, and others.
The term surface-integral of a vector over any surface
will in the following denote the integral of the outward
normal component over the surface, unless otherwise ex-
pressly stated in the context.
The surface integral (32) expresses a very simple fact.
If, for instance, we know the motion of every part of a
fluid, it should be possible, at least theoretically, to find
out how much of the fluid leaves or enters a given region
by considering how much passes through every part of the
bounding surface of the region and adding the results
together. To find the amount passing through any ele-
ment of the surface we must evidently consider only the
normal component of the current of fluid. The tangential
component of the current does not pass through the surface.
The integral is the mathematical expression of this concep-
tion and represents the total outward flux through the
surface S. Of course if the flow is inwards the result will
be negative, and if as much flows outwards through one
portion of the surface as there flows inwards elsewhere the
result will be zero.
84
VECTOR ANALYSIS.
Hie Vector or Cross Product.
18. The vector product of two vectors a and b is a vector^
written a*b (in distinction from a*b, the dot product), also
Vab or [ab] by different authors, and is defined by the
equation
a*b -» € a b sin (ab) = — b*a, (33)
where € is a vector, normal to the plane of a and b and so
directed that as you turn the first named vector a into the
second one b, € points in the direction that a right-handed
screw (cork-screw) would progress if turned in this same
i *a6 siiifab)=axb
Fig. 25.
manner. In other words, a*b is a vector perpendicular to
both a and b and whose magnitude may be represented by
the area of the parallelogram of which a and b are the adja-
cent sides. The sense of this vector is purely conventional
but is taken to conform with the more usual system of axes,
i.e., the right-handed one.
According to this convention if the factor b came first
instead of a, in the product, the only difference would be
in the reversal of the sense of €, so that
a*b = — b*a»
VECTOR ANALYSIS. 35
It is in this change of sign, when the order of the factors is
changed, that the vector product differs from the product
of ordinary algebraic or scalar quantities. It is therefore
necessary when manipulating vector products to preserve
the order of the factors unchanged or, at every change of
order, to introduce a minus sign as a factor.
In particular if a and b be finite vectors and
a*b = 0, then a || b
as the sine of their included angle must be zero. This, then,
is the condition for parallelism of the two vectors a and b.
Since any vector is parallel to itself,
a*a = 0. (34)
Remembering that the unit vectors i, j, and k are mutually
perpendicular, it follows immediately from the definition
that
j*k = i = — k«j f
kxi - j = - ixfc, (35)
|xj = k - - j*i.
Notice the cyclical order of the factors in the above equa-
tions.
We have also, by (34),
i»i = jxj = k*k = 0.
19. Distributive Law for Vector Products. It is obvious
from the definition of a*b that
t a*b = a'*b, (36)
where a' is the component of a J_ to b. Because in Fig. (26),
as a' and b are in the same plane as a and b, € is the same as
before, and ao'=ao sin </>. We may also say that the vector
product of b with the component of a parallel to b is zero.
So that in any vector product we may, if we wish, replace
one of the vectors by its normal component to the other, and
vice versa, without changing the value of the product.
86
VECTOR ANALYSIS.
Keeping this in mind, we may prove that the distributive
law holds for vector products, or, in symbols, that
(a + b)*c = a*c + b*c. (37)
where a and b are any two vectors.
Let c be drawn (Fig. 27) -L to
the plane of the paper at and
towards the reader. Let a' and
b' be the components of a and
b -L to c and hence lying in the
plane of the paper. The vectors ^
Fia. 26.
Fig. 27.
a'*c and b'*c will also lie in the plane of the paper perpen-
dicular to a' and b' respectively.
Since
6/ • 7T
c sin —
A'B* _ (b'»c) fl » = 2 = V_ =
OA'
(a'xc).
n a
ac sin
the triangles OAB and OA'B' are similar, hence
OB
OB'
m
and OB' is -L to OB. Consequently
OB' = (a' + b^c = OA' + A'B' = a'*c + b'«c.
We may now replace a' and b' by a and b according to (36)
above, so that
(a 4- b)*c = a*c 4- b*c.
* See equation (1) for notation.
VECTOR ANALYSIS.
87
If c itself be considered to be made up of two vectors
e and f, then by the same reasoning
a*(e + f ) = a*e + a*f
and b*(e + f)= b*e + b*f,
so that
(a + b)*(e+ f)= axe + a*f + b*e + b*f (38)
and so on for any number of vectors.
Physical Proof of the Distributive Law. — It is interest-
ing to prove the distributive law for vector products by
means of the following hydrostatic theorem. It is well
Fia. 28.
known that any closed polyhedral surface immersed in a
fluid is in equilibrium under the normal hydrostatic pres-
sures exerted upon its faces by the liquid. These pressures
produce forces normal to the faces of the polyhedron which
are proportional to their areas and may therefore be repre-
sented by vectors perpendicular to them, the length of
each one being proportional to the total pressure on the
face to which it is perpendicular. The condition for equi-
librium is then that the sum of these vectors be zero, i.e.,
that they have no resultant. This result is seen to be also
88 VECTOR ANALYSIS.
true for any curved surface by considering it as the limiting
case of a polyhedron with an infinite number of infinitely
small, plane facets. Let a, b and — (a + b) be the three
sides of a triangle, taken in order. Form the prism of
which this triangle and any third vector c is the slant height
or edge. The areas of the lateral faces of this prism are
respectively, viewing them from the outside,
a«c b*c and — (a -f b)*c;
the areas of the end faces are similarly
\ a*b and — \ a*b.
Now by the preceding hydrostatic theorem the vector sum
of the faces of any closed surface is zero, hence
a*c + b»c — (a + b)*c + i a»b — i a*b = 0,
giving again (a + b)*c = a*c + b*c.
This proof, which is given purely for its physical interest,
amounts to saying that the vector area of any closed surface
is equal to zero. The relation holds, however nearly par-
allel to the plane of a and b, c may be. It may also be
shown to hold when c lies in the plane of a and b. Con-
versely, assuming that the distributive law holds, the
hydrostatic theorem employed in the above proof follows
immediately.*
20. Cartesian Expansion for the Vector Product. It is
often convenient to express a vector product in terms of the
components of its vectors.
Let a = aj -h a 2 j + 03k,
b = b t i + 6 2 j + 6 3 k;
then a*b = (a,i + aj + djkM&J + 6 2 j + bjk)
which by the extension of (38) becomes
a*b - (a 2 6 f - ajb 2 ) I + (a^ - a t b M ) j + (a t 6 2 - a 2 b t ) k. (39)
* Still another proof of the distributive law may bo found in Foppl :
Einfuhrung in die MaxwelTsche Theorie der Elektricitat, pp. 16 and 17.
VECTOR ANALYSIS.
39
This expression may be conveniently condensed into the
determinant
i j k
axb = ai a% a% (40)
h\ bt bt
This is a useful mnemonic form for the vector product. As
previously stated, if the vector product is zero the vectors,
if finite, are parallel. This condition in terms of their
projections on the axes is given by noticing that in (39) the
three coefficients of i v j, and k must separately vanish, or
again from the determinant form by noticing that two
rows must be proportional, or that
fli = a* _ a*
&i b% bt
a well known result.
If ^ and b x are unit vectors «i*bi is the sine of their
included angle 0; the quantities a if a*, a*, and 6i, 6*, bt,
being then their direction cosines respectively. Squaring
formula (39) there results
sin* = (ajh — ajb*)* + (ajbx — aj>t) 2 + (aJh — a&i) 1 .
If we express the distributive law in the determinant
form we obtain the following addition theorem in determi-
nants of the third order.
i j k
di a* at
(bi + ft) (b, + c t ) (h + c)
1 j k
1 j k
=
a\ a* a%
+
ai a* at
b\ 62 bt
C\ C* Ct
21. Application to Mechanic*. Moment. The moment of
a force F about a point is defined as the product of the
force into its perpendicular distance from the point 0, or in
symbols, by
FXUA =FXrsin0. (41)
* Conversely, assuming the addition theorem tot detaxrs&b&sta^ >&&
distributive law of vector products follows imnxedvateVj .
40
VECTOR ANALYSIS.
This moment, in the figure, is right-handed about a vector
perpendicular to the paper and pointing directly up, so
that a vector of magnitude F X UA in this direction would
represent the moment of F about in a very convenient
manner. According to this convention the vector
M = r*F (42)
represents in magnitude and direction the moment of F
about 0, where r is the vector to the point of application
of the force. If the force F is the resultant of a number
of forces F lf F 2 . . . acting at the same point of application,
then by (38)
rxF = r^F^ F 3 + . . .) = r-Fj-h r*F 2 + . . .
or, the moment of the resultant of any number of forces
about a point is equal to the sum
of the separate moments. This
theorem also shows that moments
obey the parallelogram law. Con-
versely, assuming the truth of this
theorem of moments, the distribu-
tive law for vector products is a
necessary consequence.
If F have components X Y Z,
r components x y z, and M com-
ponents M x My M ty the moment
of F about the origin may be
immediately written down by (40)
Faintfxr
=rsintfxF
M = r*F =
So that
1
x
X
J
y
Y
k
z
Z
= i(yZ -zY) + i(zX -xZ) +k(xY-yX)
M x
My
M,
(yz
(zX
(xY
zY),
xZ),
(43)
VECTOR ANALYSIS.
41
22. Motion of a Rigid Body. Consider the motion of
rotation of a rigid body about an axis, with a constant
angular velocity o>. A velocity of rotation being of neces-
sity about some axis, it is convenient to represent this kind
of motion by a vector whose magnitude is proportional to
the angular velocity and whose direction coincides with the
axis pf rotation. Its direction and that of the correspond-
ing rotation may be simply represented
by the symbol in Fig. 30.
Notice that this is also the relation
existing between direction of current
and corresponding magnetic field.
Choose an origin on the axis of
rotation Fig. 31 and consider a point
P anywhere in the body, to find the
velocity of the point P. Let P be
determined by the radius vector r drawn to it from the
origin. The velocity q of P is at right angles to o> and
to r, its magnitude being given by the expression
q = to X r sin 0,
as is easily seen in the figure. In other words q is. repre-
sented not only in magnitude but in direction as well, by
q = «*r. (44)
Fig. 30.
23. Composition of Angular Velocities. Since angular
velocities may be represented by vectors let us see whether
they compound according to the parallelogram law. To
prove this definitely, let the body have several angular
velocities o>,, o> 2 , co, . . . about axes passing through the
origin. Then the linear velocities of P separately due to
these are ^
q 2 = «vr*
42
VECTOR ANALYSIS.
and hence the velocity of P due to them all acting simulta-
neously is
<I = <Ii + <h + <b + • • • ■» <»i*r + a> 2 *r + oyr + • • •
= (a>! + a> 2 + a, + • • • )*r,
or the resultant velocity q of P is the same as if the body
rotated with an angular velocity a about an axis through
given in magnitude and direction by
o) = a>j + a> 2 + a, +
This proves the above statement.
2*.
Fig. 31.
If the body have in addition to its angular velocity a
velocity of translation q t the resultant velocity q of the
point P is simply
q = q< + <»*. (45)
VECTOR ANALYSIS 48
In the case that q* is J- to o», there must be a line of points
which are instantaneously at rest. This line is determined
by the condition q = or
r*<» = q t , (46)
which is a straight line parallel to o>. Change the origin to a
point 0' on this line, the expression for the velocity reduces to
the form q = ^
where r 7 is the vector from 0' to any point in the body. If q<
is not -L to o), decompose it into two components qf and q?
such that q< = q«' + qt".
Let q«' be || to a, and qf -L to a, we may then proceed as
before with q<". It is thus seen that the most general motion
possible of a rigid body is that of a rotation about a certain
axis and a velocity of translation along it; in other words, a
screw motion.
If o) and qt are variable this holds true at any instant,
although the direction and pitch of the screw motion may be
rapidly and of course continuously changing. The axis of
rotation about which a rigid body is rotating at any given
instant is called the Instantaneous axis of rotation. If the
body has one point fixed the velocity qt is zero and the instan-
taneous axis of rotation always passes through this fixed
point. The equation of the instantaneous axis is then
given by the condition that
n
r*c& = qr.
EXERCISES AND PROBLEMS.
1. Show that the two vectors
a = 9 i + j - 6 k
and b = 4i-6j+5k
are at right angles to each other.
2. The coordinates of two points are (3, 1, 2) and (2, — 2, 4);
find the cosine of the angle between the vectors joining these points
to the origin.
44 VECTOR ANALYSIS.
3. Write out in the form
a = a,i + aj + ajk
several pairs of mutually perpendicular vectors.
4. Write out in the form
a — Oji + ad + a,k
the expressions for several unit vectors.
6. Find a vector in the ij-plane which has the same length as the
vector
a«4i-2j+3k.
Find a vector in the jk-plane having the same length, and the same
j-projection as a.
6. Let a and b be two unit vectors lying in the lj -plane. Let a
be the angle that a makes with 1, and /? the angle b makes with 1;
then
a — 1 cos a + j sin a,
b = 1 cos p + j sin p.
Form the dot and cross products and show that the addition
theorems for the cosine and sine follow from their interpretation.
7. Let
a = a x \ + (hi + ajk
and b - 6J + 6 2 j + 6 3 k
be the unit vectors to the points A and B. Find the distance between
A and B and its^ direction cosines in terms of a u a 2 , a, and b l9 b a , b % .
8. Three vectors of lengths a, 2 a, 3 a meet in a point and are
directed along the diagonals of the three faces of a cube, meeting at
the point. Determine the magnitude of their resultant. Find
the resultant in the form
r = xl + y i + zk
and from this calculate its magnitude.
9. The sum of the squares of the diagonals of a parallelogram is
equal to the sum of the squares of the sides.
10. Parallelograms upon the same base and between same
parallels are equal in area.
11. The squares of the sides of any quadrilateral exceed the
squares of the diagonals by four times the square of the line which
joins the middle points of the diagonals.
VECTOR ANALYSIS. 45
12. Under what conditions will the resultant of a system of
vectors of magnitudes 7, 24, and 25 be equal to zero ?
13. Three vectors of lengths a, a, and a V2 meet in a point and
are mutually at right angles. Determine the magnitude of the
resultant and the angles between its direction and that of each
component.
14. ABC is a triangle, and P any point in BC. If PQ represent
the resultant of the forces represented by AP, PB, BC, show that
the locus of Q is a straight line parallel to BC.
16. The angle in a semicircle is a right angle.
Take equation of circle
r» - 2 a-r = 0,
factor with r and interpret.
16. If two circles intersect, the line joining the centers is per-
pendicular to the line joining the points of intersection.
17. O is a fixed point, AB a given straight line. A point Q ifl
taken in the line OP drawn to a point P in AB, such that
OP • OQ - fc 2 (a const.).
To find the locus of Q.
Application to problems in Inversion.
18. If any line pass through the centroid of a number of points,
the sum of the perpendiculars on this line from the different points,
measured in the same direction, is zero.
Application to method of Least Squares.
19. Write out the vector product of the two vectors
a= 61 + 0.3J- 5k
and b - 0.1 i - 4.2 j + 2.5k
and show by calculation that the resulting vector is perpendicular
to each of the constituent vectors of the product.
80. Find the area of the triangle determined by the two vectors
a = 31 + 4J
and b 5 1 + 7 J.
I
46 VECTOR ANALYSIS.
21. Find the area of the parallelogram determined by the vectors
a - i + 2J +3k
and b 3i-2J+k.
22. Express the relations between the sides and opposite angles
of a triangle.
In any triangle of vector sides a, b, c,
a - b - c,
take the vector product of a with this and interpret.
28. By means of the equation of § 20 find the sine of the angle
between the two vectors
a -3i + J +2k
and b-2i-2J+4k.
24. Show that the equation of a line perpendicular to the two
vectors b and c is
r - a + x b*c.
26. Find the perpendicular from the origin on the line
a*(r - b) - 0.
26* Derive an expression for the area of a square of which
r - a x i + aj
is the semi-diagonal.
27. If the middle point of one of the non-parallel sides of a
trapezoid be joined to the extremities of the opposite side, a triangle
is obtained whose area is one-half of that of the trapezoid.
28. Find the relations between two right-handed systems of
three mutually perpendicular unit vectors. See Gibbs-Wilson,
p. 104.
29. Given c = a -f b.
Expand the right-hand side of each of the equations
c»c = (a + b)»c,
c-c -= (a + b)»(a +b),
und give the geometric interpretation oi \tofc t«s*&\u
VECTOR ANALYSIS. 47
80. Given r = xa + yb + zc
where a b c are three non-coplanar vectors. Expand the right-
hand side of the equation
r-r - (xa +yb + 3c)»Cca + yb + sc)
and give the geometric interpretation of the result.
31. Show that the work done by a force during a displacement
is equal to the sum of the quantities of work performed by its
components during the displacement.
82. A fluid is flowing across a plane surface with a uniform
velocity which is represented in magnitude and direction by the
vector q. If n is the unit normal to the plane, show that the
volume of the fluid that passes through the unit area of the plane
in unit time isq«n.
83. Show that a system of forces represented in magnitude,
direction, and position by the successive sides of a plane polygon
is equivalent to a couple whose moment is equal to twice the
area of the polygon.
84. If be any point whatever, either in the plane of the tri-
angle ABC or out of that plane, the squares of the sides of the tri-
angle fall short of three times the squares of the distances of the
angular points from 0, by the square of three times the distance
of the mean point from 0.
35. The sum of the squares of the distances of any point from
the angular points of the triangle exceeds the sum of the squares
of its distances from the middle points of the sides by the sum of
the squares of half the sides.
36. Show that
(a-b)*(a + b) = 2a*b.
and give its geometric interpretation,
37. Show that
(a - b).(a + b) = a 2 - b 2
and interpret.
CHAPTER III.
VECTOR AND SCALAR PRODUCTS INVOLVING
THREE VECTORS.
24. From the three vectors a, b, and c the following com-
binations may be derived:
1. a (Jfcfi) (a vector) 4. a (b?£) (not defined)
2. a»(b*c) (a scalar) 5. &*(brc) (absurd) (47)
3. a*(b*c) (a vector) 6. a*(tvp) (absurd).
Of these six expressions, 5 and 6 are meaningless and
absurd, because they are the scalar product and vector
product, respectively, of a vector (a) and a scalar (b»c), and
such products require a vector on each side, of the dot or cross.
As to 4, since no definition of the product of two vectors with-
out a dot or a cross has been made, it is as yet meaningless.
In this book we shall not consider such products. We
shall consider in detail the three remaining triple products.
The first one of these, a (b»c), is simply the vector a multiplied
by the scalar quantity (b-c) and is a vector in the same direc-
tion as a, but be cos (be) times longer. This triple product,
then, offers no new difficulties, and means
a (b«c) = a X be cos (be).
25. The Triple Product V = a»(b*c) is a scalar and rep-
resents the volume of a parallelopiped of which the three
conterminous edges are a, b, and c. This is easily seen to
be the case, as b*c is the area of the base represented by
a vector US -L to this base; the scalar product of a and the
vector US will be this area multiplied by the projection of
the slant height a along it, or, in other words, the volume.
As evidently this volume, V, may be obtained by forming the
48
VECTOR ANALYSIS.
49
vector products of any two of the three vectors a, b, and c
(thus giving the area of one of the faces) and forming the
scalar product of this vector-area with the remaining third
vector, it follows that
V = b-(c*a)= c-(a*b)= a»(b*c).
If the vectors (ca), (a*b), and (b*c) are taken so that they
form an acute angle with b, c, and a, respectively, then the
volume is to be considered positive, the cosine term in the
x
V
Fig. 32.
scalar product being positive. Otherwise the volume is to be
considered negative. Of course the inversion of the factors
in the vector products should change the sign, by (33), so
that we have
7 =
b»(c*a)
c-(a*b)
a-(b*c)
(c*a)»b
(a*b)-c
(b*c)«a
b-(a*c) =
c«(b*a) =
a»(c*b) =
— (a*c)*b
— (b*a)-c
— (c*b)«a.
(48)
By a consideration of these equalities the following laws
may be seen to hold:
1. The sign of the scalar triple product is unchanged as
long as the cyclical order of the factors is unchanged.
50
VECTOR ANALYSIS.
2. For every change of cyclical order a minus sign is
introduced.
3. The dot and the cross may be interchanged ad libitum.
The equalities (48) are called by Heavlside the Parallelo-
piped Law.
The product a»(b*c) may be written in terms of the com-
ponents of its vectors along any three rectangular Cartesian
axesas M-i - >>■ ■■■■'■ ' * 11 ' , ■i'^'i < M»'*>- ^- *i
a-(b*c) = a^lbfy - 6 3 c 2 ) + a 2 (b& - bfyi+a^lb^ -6,^)
= o t
b,
b,
+ o,
fc,
ft,
+ <h
*>.
6,
c,
c,
c»
Cl
Cl
c»
6.
6, *>,
Ci
c 2
c 8
(49)
This is the familiar determinant expression for the volume
of a parallelopiped with one corner at the origin.
The parallelopiped principle, then, expresses the fact that as
long as the cyclical order of the rows is unchanged the deter-
minant is also unchanged, but that every interchange of
cyclical order introduces a minus sign as a factor. To the
student familiar with determinants this is a well known
property. Conversely, assuming this property of a deter-
minant as proven, the equations (48) immediately follow.
The twelve expressions (48) are often written in one, as
[abc], a special symbol of abbreviation taken from Grass-
mann.
26. Condition that Three Vectors Lie in One Plane.
Should the three finite non-parallel vectors a, b, and c lie
in one plane the volume of the parallelopiped they deter-
mine is zero. Hence the condition that the three non-
parallel vectors should lie m a plane is that
[abc] = 0. (50)
VECTOR ANALYSIS. 51
In the expression [abc], if any two of the vectors are parallel
the volume of the parallelopiped is again evidently zero.
Hence, in general, [aab] — 0. (51)
To look at it in another way, we may put, by (48)
a»(a*b) = (a*a)»b,
and as a*a = 0, then a-(a*b) = 0, so that in a triple scalar
product if any two of the vectors are parallel their triple
scalar product is zero.
In the determinant (49) above, this corresponds to having
any two rows proportional to each other, the result being,
as is well known, identically zero.
The parenthesis in an expression such as a*(b*c) is in
reality unnecessary, as its only other interpretation (a«b)*c
is without meaning, being the vector product of a scalar
(a-b) and a vector c. The parentheses are introduced,
however, when by so doing the interpretation is made easier.
Scalar magnitudes of the vectors* it is important to
remember, which occur in any kind of scalar or vector
products may be placed in any part of the expression as
factors.
For example,
a«(b*c) = aa^(b*c)
= abc a^ibfcj (52)
= b&i»(cb l *oc l ) 9 etc., etc.
27. The Triple Product q = a*(b*c) is a vector. In this
expression the parenthesis, or some separating symbol, is
necessary, as a*(b*c) ^ (a*b)*c. The sign of this product
changes every time the order of the factors a and (b*c) is
changed in a*(b*c), or whenever the order of the factors b
and c is changed in (b*c). The vector product being always
perpendicular to both of its components, q is perpendicular
to a as well as to b*c, hence
q-a = and q»(b*c)= 0. (531
52 VECTOR ANALYSIS.
Equation (53) shows that q lies in the same plane as b and
c, either by (50) or by seeing that it is perpendicular to a line
which is itself perpendicular to b and c. It is important that
this result be clearly visualized. The habit of visualization
should be cultivated, as it is of great importance to the student
whatever kind of analysis he be using, but particularly so in
this. To a purely analytical mind vector analysis offers but
few advantages.
As q lies in the plane of b and c it is possible to express q in
the form
q=* sb- yc,
where z and y are scalar multipliers. Let us try to determine
the quantities x and y. Since q is perpendicular to a,
a»q = xa-b — i/a»c=0
and, therefore,
x : y = a-c : a»b or x = n a*c,
y = na*b f
where n is a scalar factor of proportionality. So that
q = a*(b*c) = n [b(a«c) - c(a-b)]. (54)
We shall now prove that n is independent of the magni-
tudes and inclinations of the vectors a, b, and c. It is inde-
pendent of their magnitudes because they may be taken out
by (52) as scalar coefficients and eliminated from the equa-
tion. Since we are dealing with the mutual relations between
any three vectors, we may choose one of them arbitrarily.
Let that one be a. Let us now replace one of the remaining
vectors, c, for instance, by the sum of two other vectors d
and e. Then
a*(b*(d + e)) = n [b a-(d + e)- (d + e) a-b],
or
a*(b*d) + a*(b*e) = n [b a-(d + e)— (d-f e) a-b],
or finally,
n'[b a*d- d a»b]+n''[b a-e-e a«b>=n\b a^d-V^-^-Ve^a-bl
/
VECTOR ANALYSIS. 53
If d and e have been chosen so that b, d, and e are not co-
planar, then we may equate coefficients of the vectors b, d, and
e on both sides. Thus
n'a-d + n"a«e = na»(d + e).
n'a-b = na»b,
n" a»b = n a*b,
which necessitates that
n' = n" = n.
The coefficient n in equation (54) is thus independent of
a, b, and c, and is, therefore, a numerical constant. To find
its value we are now at liberty to consider a special case. Let
a, b, c be unit vectors. Let a = c and let b be perpendicu-
lar to c. This is the equivalent to writing a = k. b = j,
c = k, We then have for equation (54), 3- u
* kx(j*k) = n [j(k-k)- k(k-j)].
but j*k = i and k*i = j,
k-k = 1 and k»j = 0:
therefore the equation reduces to
j — n(l)l hence n = 1.
We have thus proved the very important relation
a*(b*c) = b(a»c) — c(a»b), (55)
which should be memorized.
28. Demonstration by Cartesian Expansion. A demon-
stration of this equation may also be obtained by expanding
in terms of the Cartesian components of the vectors.
This method is a very useful one when no other demonstra-
tion readily offers itself, but generally (not in this case)
has the disadvantage of being long and cumbersome. No
better examples of the concentration of the vector notation
may be found than by carrying through a wwctferet *& s>nh&>l
transformations. On account of the impoYWiofc ol \>afc ^ojaa*-
54
VECTOR ANALYSIS.
tion (55), and also to give an example of the expansion
method in general, its demonstration by this method will be
carried out.
As the components of (b*c) are
we may write, by (40),
&A
W
and
b t b 2
c x c 2
, see (49)
a*(b*c) =
i
J
k
°i °i <h
bj>*
bj>x
6,6,
Cfr
<Vi
C|C,
= i [a a (6 l c a -6,c 1 )-a,(6^ l -6 1 c,)]
+j [thtPfy-bfJ-^bfr-bft)]
+k[a 1 (6 s c 1 -6 1 c B )-a 2 (6 2 c B -6 1 c,)].
The terms may now be rearranged into
a*(b*c) = lb t (o^ + a 2 c 2 + a s c s )
4- j& 2 (°i c i + ^ + a s c s)
+ k 6 S (a^ + a 2 c 2 + a^)
— i c x (fljby + a 2 6 2 + Oj&j)
— j c 2 (a l 6 l + aA| + «A)
— kc, (a^ + a 2 b 2 + Ogb s ).
The new underlined terms have been added and subtracted.
The first three lines are
(I&i + j& 2 + kfc s ) (a-c)= b(a-c),
the last three are
-(ici + jc 2 + kc,) (a-b)-- c(a-b),
hence
a*(b*c) = b(a»c) — c(a«b).
28. Third Proof. That n = 1 in (54) may also be proved
as follows: Consider first the triple vector product in which
two of the vectors are the same,
b* (b*c) = n (b b*c — c b-b) .
VECTOR ANALYSIS. 65
Taking the scalar product of this and c, or, in shorter language,
applying c dot (o) to it, we obtain
c.b*(b*c)= n[(b-c) 2 - Vc 2 ].
But by an interchange of the dot and the cross and one
change of cyclical order the left-hand side becomes
(c*b).(b*c) - - (b*c).(bxc) - - (frc) 2 . (66)
We know, however, that
(b.c) 2 + (b*c)*= bV, (56a)
as by definition
6V cos 2 (be) + fcV sin 2 (be) = 6V
is equivalent to it; hence the right hand of the first equation
is nothing more than— n (b*c) 2 , and comparing with (56) we
see that n must be unity. The theorem is thus true, when
two of the vectors are the same. Consider now the general
case,
a*(b*c) = n (b a«c — c a-b). (57)
Apply b dot to it, obtaining
b*a*(b*e) «= n (b«b a»c — b-c a«b)
= na«(cb»b- bb«c),
which as we have just proved may be written
= na{- b«(b*c)].
But on the left-hand side we have
b«a*(b*c) = — a«[b*(b*c)],
by an interchange of dot and cross and one of cyclical order.
Comparing the last two equations we see that n — 1 in
general.
The parenthesis in a*(b*c) is necessary, for (a*b)xc is quite
different from the first expression, as one may readily see by
expanding the two, or by the reasoning of § 27.
90. Products of More than Three Vectors. In practical
applications to physics more complicated products than those
of three vectors seldom arise. Whenever they do,\fcfc^ \as2"j
56
VECTOR ANALYSIS.
be reduced by successive applications of the preceding prin-
ciples. In any case they represent extremely complicated
Cartesian expansions. As an example of such a reduction,
consider the scalar expression containing four vectors,
(a*bHc*d).
Interchange the first cross and dot and expand the vector
triple product, which will give,
(a*b)»(c*d)= a-b«(c*d)
= a»(c b«d — d b»c)
= a»c b«d — a«d b»c
a«c b«c
a-d b-d
(58)
This formula will be used in the deduction of Stokes' Theo-
rem in § (58).
Again consider the quadruple vector product (a*b)*(c*d),
which may be expanded by (55),
(a*b)*(c*d) = c (a*b)«d — d (a«b)»c
orinto = b (c*d)«a — a (od)-b, (59)
taking in the first case (a*b), c, and d as the three vectors of
a triple product, and in the second case a, b, and (c*d). By
subtracting these two equal expressions from each other we
have
a b-(od)- b a*(c*d) + c d-(a*b)- d c*(a*b)=0, (60)
an important relation holding between any four vectors.
Putting d = r, this equation may be written
r [abc] = [rbc]a + [rca]b + [rab]c,
so that
[abc] [abc] [abc]
c,
(61)
or
r-
b»c
[abc]
a+ r«T=^b -t- r«
\abc\
\aY»c\
c,
(62)
VECTOR ANALYSIS. 57
an important and useful formula which gives the coefficients
necessary to express r in terms of any three arbitrary vectors
not lying in the same plane. This expansion is under these
conditions always possible as explained in § (7).
31. Reciprocal System of Vectors. The three vectors
** *• and -3^-, (63)
[abc] [abc] [abc]
perpendicular respectively to the planes of b and c, c and a,
and a and b, occur frequently in important relations and are
said to be the system reciprocal to a, b, and c. They have
peculiar and interesting properties which the student will
find fully demonstrated in another work.*
It will be noticed that only two kinds of products of vectors
have been defined, i.e., the scalar product and the vector
product. One should carefully remember as well that the
scalar product and the vector product have been defined in
terms of two simple vectors, but that instead of simple vec-
tors any expression which is itself a vector may be used in
place of the simple vectors to form these products. If this is
carefully kept in mind it will make clear that in vector analy-
sis certain combinations of symbols are meaningless.
For example, a*b being a vector, it may be used in con-
junction with another simple vector e, or another vector
product c*d, to form new scalar products and vector prod-
ucts, such as
(a*b)-e and (a*b)-(c*d),
(a*b)*e and (a*b)*(c*d),
or even [(a*b)*(c*d)M(e*b)*(g*h)], etc.,
which are all legitimate expressions. But, on the other
hand, neither a-b, nor (a*b)»(c*d), nor (a*b)»e may be
used again to form either scalar or vector products because
they are merely scalars.
* Gibbe- Wilson, Vector Analyst, pp. Sfc-Sfc.
58
VECTOR ANALYSIS.
Equations of Plane, Line, and Sphere.
32. The Plane Perpendicular to a and Passing through the
Terminus of b. Let r be the radius vector to any point in it.
The projection of r upon a is evidently constant and equal
to the projection of b upon a as long as the terminus of r is
in the plane; this condition is expressed by the equation
a»r = constant = a«b, (64)
or a*(r — b)= 0,
which is, therefore, the equation of the plane. It also states
that (r — b) is perpendicular to a and hence parallel to the
plane which is an evident truth and could be used to derive
the equation of the plane. If the origin is in the plane, b =
and a-r = (65)
is the equation, which is otherwise evident, as r is then always
perpendicular to a. If the equation of a plane is desired, the
plane being parallel to two given vectors c and d and passing
through the terminus of b, simply remember that c*d is a
vector perpendicular to the plane, and putting c*d in place
of a above, its equation is
(odHr-b^-Q. (06)
VECTOR ANALYSIS.
59
If the equation of a plane passing through the ends of three
given vectors a, b, and c Is desired, remember that the vec-
tors (r — a), (a — b), and (b — c) lie in the same plane and
express this fact by (50), giving
(r-a).(a-b).(b-c)=0,
or expanding
r-(a-b -f b-c +- c-a) = a-(a-b + b-c + c-a) 1
or (67)
4>-(r — a) «=0, where + = (a-b + b-c + c>a).J
Fio. 34.
Comparing this last equation with (64), we see that + is a
vector perpendicular to the plane.
To and the vector-perpendicular p from the origin to the
plane. Referring to the plane in Fig. 33, the equation o(
which is
a-(r-b)=0,
60
VECTOR ANALYSIS.
let.r become perpendicular to the plane and hence some
multiple of a, say x a, then
a«(ia- b) = 0,
a«b
a*
and p = a;a = a^ = -a-b = a" 1 a-b. (68a)
or
xa? = a-b;
xa =
then x
?^ = Ia.b = a- 1 a.b.
This result is also evident on inspection. For a-b is the
projection of b on a multiplied by the magnitude of a; hence,
to obtain the value of the projection we must divide by the
magnitude of a, so that directly
a-b
p = ^^ — a" 1 a-b.
a
In equation (67) the perpendicular p is then
p = fy-i ^.a.
(686)
33. The equation of a straight line through the end of b
parallel to a is, since a and (r — b) are always parallel,
a*(r-b)=0. (69)
This is compatible with the
equation derived previously,
r = b + xa,
for applying a* to it, we obtain
69 in the form
a*r = a*b.
Again, the equation of a line
perpendicular to c and d and
passing through the end of b
is, because c*d is parallel to a
in the above equation,
(odHr-b^-Q. (70)
Fig. 35.
VECTOR ANALYSIS
61
34. Equations of the Circle and the Sphere. In a circle or
sphere, with origin at the center, the length of the vector to
the surface from the origin is constant and equal to the
radius. Hence
r — a, (not r=a)
or i* - a 2 , (71)
is the equation of the circle or of the sphere according as
two or three dimensional space is considered.
Fia. 36.
If the origin is removed to at a distance — c from 0, then
(71) becomes
(r - c) 2 - a',
or r 3 — 2 r-c = a 2 — c 2 = const. (72)
If c = a, that is, if the origin is on the circumference of the
circle, the equation reduces to
r 3 - 2 r-a = 0. (73)
This, in polar codrdinates, is nothing mote \tawi
r = 2 a cos 0,
J52
VECTOR ANALYSIS.
where is the angle between r and any predetermined radius,
from which 6 is measured.
In rectangular coordinates (73) considered as the equation
of a sphere is written immediately
x 3 + t/ 2 + # = 2 (xa t + ya 2 + zaj,
Fig. 37.
where a lf a 2 , a 3 are the projections of any chosen radius along
the three axes. If the plane of yz be taken perpendicular to
this radius, a 2 and a 3 are zero, so that
£* 4- y 2 4- & = 2 a x x.
If we drop the 2-coordinate the equation reduces to that of a
circle tangent at the origin to the y axis and with its center
on the x axis.
The equation with origin at center may be put in the form
or
r 3 - a 2 = 0,
(r-a)-(r + a)=0,
which says, see Fig. 38, that the two lines AD and DB are
always at right angles, a familiar result. Such illustrations
*nay be multiplied indefinitely and sWn \X» s»» "«Vta ^VasSx
VECTOR ANALYSIS. 63
equations may be written down to fit almost any condi-
tions. When translated into their Cartesian equivalents
they give familiar forms.
Such books as Tait, "Quaternions," Kelland and Tait,
"Introduction to Quaternions/' may be consulted with ad-
vantage at this point by the student. In these books the
whole treatment of the line, plane, circle, sphere, and conic
sections, with few exceptions, is one of vector analysis pure and
Fig. 38.
simple. The occurrence of a quaternion is a very rare event.
The only difference to be noted particularly in reading these
works is that the scalar product has the opposite sign to ours,
and that
a«b is written S ab
and a*b is written V ab.
34a. Resolution of a System of Forces Acting on a Rigid
Body. Consider any point as origin. This point may be
anywhere, even outside of the body. The system of forces
64
VECTOR ANALYSIS.
Pn ^2> Fa> etc «> acting on the body is equivalent to a single
force R at 0, where
R=2 F
and a couple whose strength is
C = £a*F,
where a^ a 2 . . . are the vectors to the forces F u F, . . .
from 0.
To prove this, consider one of the forces F acting at P; we
may introduce the zero system + F — F at without alter-
Pia. 39.
ing in any way the effect of F on the body. By combining
— F with the F at P we get a couple of strength a*F,
leaving a force F acting at 0. We may do the same for each
of the forces F u F 2 , etc., so that finally we have
(a) all the forces F lf . . . F 2 acting at
and (b) an equal number oi coupVea a v *F v , a^F ^ * % %
*
VECTOR ANALYSIS. 66
Combine all the forces into a resultant force, R =2)F
and all the couples in a resultant couple C = ]£}a*F, which
proves the theorem. Such a resolution may be made for any
point whatever.
Central Axis. Couple a Minimum for Points on this Axis.
In general the axis of the couple C derived above is not
parallel to the resultant force R.
Let us inquire if there are any points for which an analo-
gous resolution will give a couple whose axis is parallel to R.
Notice that whatever point 0' is chosen R is the same vector.
Let 0' be such a point and let 00' = r, and for 0' the couple
would be, since O'P = a — r,
Xa - r)«F.
Then the condition that this couple be parallel to R becomes
£(a - r)*F - x R = JJa-F - r*£F,
so that
xR = C- r*R.
To find x, multiply by -R~ l , because R~ l is parallel to R,
and the triple scalar product term vanishes.
x - CR- 1 .
Hence
r*R = C-R(C.R- 1 )
= R*(C*R- 1 ) = (R- l *C)*R,
a linear equation for r in terms of R and C (given vectors by
last theorem). There is then a line of points for which this
kind of resolution is possible. This line is called the Central
Axis of the System, which is then reduced to a force R and a
couple about R of a certain magnitude = xR = R C-R" 1
= Ri(C-Rx). (a)
Considering the equation of the central axis
r*R = (R l *C)*R,
66
VECTOR ANALYSIS.
it is seen that one of the values of r is (R" f «C), and since
this last vector is evidently perpendicular to R, it must be the
line ON; so that the equation may also be written
r = R- l *C + t/R = ON + yR,
where R~ l *C is the normal vector from to the central
axis.
Fio. 39a.
By (a) we see that the couple about the central axis is the
component of the couple at any other point along R, and
hence is always less than for any other point; so that it is a
minimum and equal to
C«Rj along R.
It is, however, the same, i.e., constant for ott points on the
central axis.
EXERCISES AND PROBLEMS.
1. Prove the following formulae :
a*jb*(c*d)j - [acd] b - a-b c*d
= b«d a*c — b«c a*d
[a*b c«d e*f] - [abd] [cef] - [abc] [def]
- [abe] [fed] - [abf] [ecd]
- tcda\\bet\ - \pttt\fe4Q,
VECTOR ANALYSIS.
67
2. Prove
[a*b b*c c*a] — [abc] 1
and interpret the result by determinants.
3. Show that
p-a p-b p
[pqr] (a*b) - q-a q*b q
r-a r*b r
4. Show that
a*(b*c) + b*(c*a) + c*(a*b) - 0.
5. Show that
[a*p b*q c*r] + [a*q b*r c*p] + [a*r b*p c*q] - 0.
6. Prove that
(a*b)-(c*d) - (a*c)(b-d) - (a*d)(b*c)
and that
(a*b)*(c*d) - b [acd] - a [bed]
- c [abd] - d [abc].
7. Deduce the fundamental formulae of spherical trigonometry
from the equations
(a*b)-(c*d) — a»c b-d - a*d b-c,
(a*b)*(c*d) = [acd] b - [bed] a
- [abd] c - [abc] d.
Make the vectors unit vectors and take an origin at center of sphere
of unit radius, thus making all the vectors terminate upon the sur-
face of the sphere.
8. Show that the components of b parallel and perpendicular to
a are respectively
b = a— = aiai-b
and
b" - -
a*a
a»(a«b)
a*a
- a 1 x(a l «b).
9. The second vector a may be omitted from «*^ -V \$y ^Kaq
it be omitted in ar l *(a -f b) or in a*(a -V \jT>— Yc t
68 VECTOR ANALYSIS.
10. The perpendiculars from the vertices of a triangle to the
sides opposite meet in a point.
11. Find the point of intersection of a line and a plane and dis-
cuss the result.
12. The perpendicular bisectors of the sides of a triangle meet
in a point.
13. Find an expression for the common perpendicular to two
lines not lying in the same plane.
14. Determine the vector-perpendicular drawn from the origin
to the plane determined by the three points a, b, c.
15. Find the equation of a plane passing through a given point c
and parallel to each of two given straight lines b' and b".
Ans. (r - c)-b'*b" - 0.
16. Find the length of the common perpendicular to each of two
given straight lines parallel to b t and b, and passing through ^ and
a 2 respectively, and show that it is
d - y (b^),
where y- (a i~ a » ),( W .
Uvb 2 ) 2
17. Find the equation of the line of intersection of two planes.
18. Deduce the Cartesian equation for the volume of the tetra-
hedron whose vertices are
a, b, c, d,
where a — a x \ -f a 2 J + 03k, etc.
19. Deduce the Cartesian equation for the area of the triangle
whose vertices are
a - a x \ + oj + 03k,
b - 6J + 6 2 J + 6 3 k,
c — c x \ + c 2 j + c 3 k.
20. Find by translating into Cartesian notation that the volume
of a pyramid, of which the vertex is a given point (xyz) and the
base a triangle formed by joining three given points aoo, obo, ooe
in the rectangular coordinate axes, is
V-iabc(- + *■ + *• -lV
\a b c J
VECTOR ANALYSIS. 69
21. Show how to determine the directions of two vectors of given
magnitude so that their resultant shall be of given magnitude and
direction. When is this impossible ?
22. The moment of the force AB about the line CD is six times
the volume of the tetrahedron ABCD divided by the number of
units of length in CD.
Find vector moment at C, and then the component of this along
CD.
23. The laws of refraction of light from a medium of index fi into
one of index pf are comprised in the relation
where n, a, and a' are unit vectors along the normal, the incident
and refracted rays respectively.
24. Write out the equations of problems 1 to 8 inclusive in
Cartesian notation.
25. If r = a + b x is the equation of a straight line, b being a
unit vector, prove that the line through the origin perpendicular
to it is
r = y (a — a*b b)
and that its length is
Va 2 - (a-b)*.
26. The equation of the plane through the origin perpendicular
to the vector a may be written in either of the forms
a«r = a~ l -r = (r + a) = (r— a) .
27. Let r = b + x d be the equation of a straight line. For
what values of x does it meet the sphere r 2 = c 2 ? By the theorem
on the coefficients of an equation as related to the roots, derive the
theorems for the product and sum of the intercepts respectively.
28. Derive the equation of the sphere (or circle) from the equa-
tion
(r — c) = a.
This equation states that the end of r must remain at a constant
distance a from the end of c, hence, etc. . . .
29. Prove that if the sum and difference of two forces are at
right angles the two forces are equal.
80. Prove that if the lengths of the s\im and &Stet«&R& rea^Rfc-
tivelyof two farces are equal, the two forces ate a\. t\&& «&.&*&«
CHAPTER IV.
DIFFERENTIATION OF VECTORS.
35. Two Ways in which a Vector may Vary. . If a small
vector d a be added to the vector a, the result in general will
be a new vector differing by a small amount from a not only
in length but also in direction.
If the small vector which is added be perpendicular to a,
then the length of a will remain unchanged.
4
4
t
4
If the small vector which is added be parallel to a, then its
direction will remain unchanged.
These three cases are shown in Fig. 40.
Differentiation with Respect to Scalar Variables. Let the
vector a (I) be & function oi a aca\ai \unaic\fe t. 1^ other
70
VECTOR ANALYSIS.
71
words, let its length and direction be known and determinate
as soon as a value of t is given.
Let OA x be the value of a (t) when t = t x , and let OA 2 be
the new value of a (t) when / = t 2 . Then the change in a(t)
due to the change in t of t 2 — t x is
A x A 2 = a(t 2 )-a(t x ).
Dividing this equation by t 2 — t x in
order to find the rate of change and
making t 2 — t x infinitely small, we
define
*L = Urn a ('»>-» ft ) .
dt (,-drO t 2 — t x
If < 2 — t x = A, where A is some small
scalar, this may be written in the more
familiar form,
d* = Jim ag + »-a(Q #
eft A* o h
Evidently the rate change of the
vector a with respect to t is made up
vectorially of the three rates of change
of its components
(74)
(75)
da x
da,
dO;
along i, ^ along j, and =» along k,
Fig. 41.
or
da _ dc^t ,
(ft eft
da
<ft J + (ft K *
(76)
Precisely the same reasoning holds for the rate of change
da
with respect to t of the vector representing i~,
written
or
cPa
d_(da\
dt \dt ) " dP
Similarly for the higher derivatives we may then write
dt» dl» dt n 3 dt*
<3^
72 VECTOR ANALYSIS.
The vectors i j k are to be considered constant in length
(being unit vectors) and in direction (being along fixed axes)
in all of these differentiations.
If -— be denoted by p, then
dt J r
^ = pa = p (a t i + o 2 j + a,k)= pa x \ + pa J + pa,k
and (78)
--5 =pn a = pn( ai j + a j +Ojk) = p n a 1 i + p^ J + p^k.
It will be noticed that the operator p = — acts like a
CM
scalar multiplier,
36. Differentiation of Scalar and Vector Products. The
differential of a«b, for instance, is defined just as in scalar
calculus as
(a + da)*(b + db)- ab = d (a-b).
Expanding and neglecting small quantities of the second
order there remains
d (a-b) = a-b + da-b + a*db — a«b
= da-b + a-db;
hence, dividing through by dt,
d t K x da K , db
— (a-D) = — — D + a«— — »
dt dt dt
or p (a*b) = p a»b 4- a-p b = b-p a + p b*a. (79)
In a similar manner
d / \ da «_ . db
_ (a . b) = _, b+a ,_,
or p (a*b) = p a*b 4- a*p b. (80)
The differentiations then take place very much as they do
in scalar calculus, but with this \mpottax& &&feTOu&,ttva& the
VECTOR ANALYSIS. 73
order of the factors must remain unchanged in all expressions
where a change in order of the vectors, as previously ex-
plained, would not be allowable. For example, in (79) the
order is immaterial, while in (80) it is essential.
The formula
|[..(b*)]-f.(b«) + ..(f<) + ..(b.f)
or p[a*(b*c)] = pa-(b*c) + a*(pb*c) + a-(b*pc) (81)
and |[a« (be)] = f .<b-c> + a* (f -c)+ » (b-f )
or p[a*(b*c)] = pa*(b*c) + a*(p b*c) + a*(b*pc)
are in the same way easily seen to be true results.
It is instructive to notice the manner in which the opera-
tor p operates in turn on each one of the factors.
If a vector is to remain constant in length, then
a = const.
or a«a = const.;
hence a*da = 0,
or da is perpendicular to a, as is geometrically evident by
considering
a*a = a 2 = const.
as the equation of a sphere or circle.
37. Applications to Geometry. We shall obtain some
interesting and useful results, as well as a clearer insight into
the calculus of vectors, by the following applications to
geometry.
Let a variable vector r be drawn from a fixed origin 0. We
shall assume that the terminus of r can be located as soon as
a value of t, an independent scalar variable, is given. By a
slight extension of mathematical nomenclature and symbols
we shall express this result by writing
r=f(t), W
74
VECTOR ANALYSIS.
reading it as: r equals a vector function of t. To indicate the
vector character of the function, f is printed in bold-faced
type.
As t varies continuously, the terminus of r describes some
curve or curves in space, depending upon whether ris a single-
valued or multi-valued function of t.
Fig. 42.
We assume in the following that the function f is a con-
tinuous and single-valued function of the independent scalar
variable t.
Let t = $ be the distance along the curve
r = f «
from any point P on the curve. The increment dr is evi-
dv
dently a vector along th£ curve, and of length ds, hence —
ds
is a unit vector tangent to the curve at the point under con-
sideration, M, when M' has approached indefinitely near to
dv
M. For convenience we shall write — =t, where t is a unit
ds " — •
vector al ong the tangent to the curve, or as we call it the unit
tangent,
■»■«"
and
dr
da
VECTOR ANALYSIS.
ds ds as
= M + t 2 \ + tjk.
75
V
i)
So that
. __ dx a _ dy . __dz
**'&' h ~d^' h ~dl
are the direction cosines of the tangent.
Tangent and Normal. The equation of the tangent line
at r is then the equation of a line through the terminus of r
dv
and parallel to t= — , and by (69) is written
as
(r-|)»£-0, ■> (83)
where § is the variable vector to this line from o. .
Expanding this vector product by means of (40) we
obtain the familiar Cartesian equations
1 J k
<*-*,) (</-*,) (*-<•)
dx dy dz
ds
ds ds
50
or making the three components along i, j, and k equal to
zero.
dx dy dz
ds ds ds
(84)
The plane normal to the curve at r is, by (64),
<r-{)-£ = 0,
or expanding in its Cartesian form,
(86)
fr-'^+o-wfc + fr-wV*- **
76
VECTOR ANALYSIS.
38. Curvature.* Consider three adjacent points on the
curve, M l9 M 2 , and Af 8 ; the unit tangents through M x and Af,
and through M 2 and M s differ only in direction, hence the
vector added to the first one to obtain the second one is
at right angles to both and therefore measures the angle
Fiq. 43.
through which it is turned in going from M x to Af r By
definition the curvature is defined to be the magnitude of — .
as
It is convenient to call the vector — = — the vector-
as as*
curvature, c, as it has the same magnitude as — and, being
as
normal to the tangent, points towards the center of curva-
ture. The vector-curvature, being perpendicular to two
consecutive tangents, lies in their plane. The radius of
curvature p has a length inversely proportional to the mag-
nitude of the curvature, but points in the same direction,
and hence may be written
' - c_l - (sT • (87)
* See Appendix, p. 242, tot otto fafanftoom.
(o'J-
VECTOR ANALYSIS.
77
Osculating Plane. The plane containing two consecu-
tive tangents is called the osculating plane. If | be the
vector to any point in this plane from 0, since the three
vectors t, — , and (r — |) lie in it, its equation may be
as
written down at once by (50) as
(r -{,.(«.£)-<>.
<'-*>-(5-S)-°-
(88)
Fig. 44.
This by (49) may be written in the familiar form of a
determinant
ds 2
m dx
* J ds ds>
0.
(89)
/
t,-'
dz
d&
78 VECTOR ANALYSIS.
Tortuosity. A twisted curve in space twists in two dis-
tinct ways. Any small portion of the curve lies in its
osculating plane at that point, and this small portion of
the curve has a curvature as described above. As we go
along the curve, however, the osculating plane turns through
a certain angle; the limit of the ratio of the angle turned
through by the osculating plane to the arc traversed to pro-
duce that change is called the tortuosity.
Hence if n be a unit normal to the osculating plane,
ds
where ds is the magnitude of the arc.
Geodetic Lines on a Surface. The differential equation
to a geodetic line on a surface may be obtained in the fol-
lowing simple manner from the definition:
A geodetic line is a curve on a surface, the osculating
plane of the curve being everywhere normal to the surface.
It is the curve a stretched string would lie along if the
surface were a perfectly smooth one, the reaction of the sur-
face to the pressure of the string being everywhere along the
normal to the surface where it is in contact with the string.
Let t be the unit tangent to the geodetic, let n be the
unit normal to the curve, and let m = n*t be a unit vector
lying therefore in the surface normal to n and to t.
The osculating plane is determined by t and dt which lie
in it by definition. If the curve is a geodetic, the normal
to this plane t*dt lies in the surface, and is hence perpen-
dicular to n.
Expressing this fact,
n .(t*dt)=0.
Since t lies along dv (§ 37) and dt lies along <Pr (87), this
equation becomes
n-(dr*cPr) = 0, (89a)
I which is the differential equation, to ttte £pata&&.
VECTOR ANALYSIS.
If the surface is of the form
V (r) — const.,
1
79
N
■• K
then VV is a vector normal to it. See (106). But VV has direction
cosines proportional to
dV dV
dV
dx dy dz
and
n is along I ^ + J ^ ,+ k ^
dx
dz
Hence (89a) becomes, by (49),
dV
dV
dV
dx
*
dz
dx
dy
dz
<Px
<Py
d*z
30. Equations of Surfaces. Curvilinear Coordinates. The
equation
r - f (u, v), (90)
where u and v are two independent scalar variables, repre-
sents a surface.
If a particular value u x be given to u while v is unre-
stricted,
r = f (u lt v)
being of the form (82), is some curve lying wholly on the
surface. If a particular value v x be given to v while u is
unrestricted,
r - f (u, v,)
is some other curve lying wholly on the surface. These
two curves intersect at the point or points r determined by
the equation
80 VECTOR ANALYSIS.
We may then determine a point on the surface by giving
particular values u, and ti„ say, to u and v. This point will
be found at the intersection of the two curves
r = f (u u v) and r = f (u, «,).
Curvilinear coordinates is the name given to these vari-
ables u and v, such a series of curves divides the surface
up into a network of curvilinear quadrilaterals, the angles
of which may have any value. In the particular case that
these curves cut each other always at right angles they
Fra. 45.
are said to form an Orthogonal System of curves. When
the two systems of curves divide the surface up into
infinitesimal square elements they are said to form an
Isothermal System. Such systems are of the greatest impor-
tance in mathematical physics. The student should consult
on this subject an excellent book by Fehr, " Applications
de la Methodo Vectorielle a la Geom£trie Infinitesimals "
(Carre' et Naud, Paris, 1899). His notation is different
from ours, and is fully explained in his introduction, but
his methods are quite similar.
40. Applications to the Kinematics of a Particle. Let
the independent variable ( now denote the time; then— — v
ib the vector velocity along the curvet — t(ft. ^otinhMmk
VECTOR ANALYSIS. 81
dt
— is no longer a unit vector, as here dt 7* da, the element
dt
of arc, but the direction is still along the tangent. If t is
the unit tangent to the curve, then
v = vt,
where v, the magnitude of v, is called the " speed."
The acceleration
-j- is the increase of speed along the curve. Speed is here
dt
used to denote the velocity irrespective of its direction.
And by (87)
dt SSB dt L d8 s=cv
dt** ds dt '
... a _*!t + *:-*t + £, (92)
dt dt P
or the acceleration of a particle on a curve may be resolved
into two components at right angles to each other, one
fit)
— increasing the linear speed along the curve, the other
dt
v 2
one i£c, or — , where p is the vector radius of curvature and
P r
is directed towards the center of curvature, merely changing
the direction of the motion.
Hodograph. The hodograph is a curve obtained in any
given case of motion of a particle, by laying off from an
arbitrary origin vectors equal to the velocities of the par-
ticle for all points of the path. The locus of the extremities
of these vectors is the hodograph.
When a particle describes a curve, therc Sa Wu&cl *. ^rivefc.
related to it simultaneously describing the \vodo^«^- T \N£a»
82
VECTOR ANALYSIS.
conception was introduced by Hamilton, and is an efficient
aid to the study of curvilinear motion.
Evidently the hodograph itself may have a hodograph,
and this perhaps another, depending upon the complexity
of the motion, and so on.
The hodograph of a particle at rest is a point at the arbi-
trary origin.
The hodograph of uniform motion in a straight line is a
point at the end of a vector of length equal to the velocity.
Fio. 46.
The hodograph of uniformly accelerated motion in a
straight line is another straight line parallel to the first,
described with uniform speed by the hodograph variable.
The hodograph of uniform motion In a circle is another
circle, since the speed is constant, of radius equal to the
speed. The vector velocity in the circle is always perpen-
dicular to the radius vector in the original path. Evidently
the points P and P' move around their origins with the
same angular velocities. The velocity of P' in general is
the rate of change of v, and hence is the acceleration
of P.
VECTOR ANALYSIS. 88
Since the two circles must be described in the same
times,
2tvt __ 2m .
v a
hence
a = —
r
a familiar result.
Equation of the Hodograph. If
r - f (0
be the equation of the path described by a particle, con-
taining not merely the form of the path but the law of its
description as well, then
£-r«)
is the equation of the hodograph and the law of its descrip-
tion. Again
is the hodograph of the hodograph, and so on.
41. Integration with Respect to Scalar Variables. (Recon-
sult paragraphs 4 and 16.)
The inverse of differentiation offers merely the difficul-
ties of scalar integration. The constants of integration,
however, are constant vectors. As a simple example con-
sider the motion of a particle under constant acceleration,
under gravity for instance. The differential equation of the
motion may be written
<Pr
where a is a constant vector. Integrating once,
84
VECTOR ANALYSIS.
where v is a constant vector, as it is a vector equation and
dt
v is determined by the value of — when t = 0. Integrat-
ed
ing again we obtain
r = iat 2 + v t + s ,
(93)
where again s is a second constant vector and whose value
is that of r when t = 0. Equation (93) gives the value of
r at any time L The equation says that starting from the
point s f (t.e. at S) f r, the vector to any point of the path, may
Fig. 46a.
be found by adding to s the vector sum of the two motions
v t and £ a t 2 . The terminus of r evidently describes a
parabola passing through s , because the coordinates of
any point on the curve referred to the oblique axes parallel
to v and a are proportional to the first power and the
second power of the same quantity t, respectively.
Orbit of a Planet. Central Acceleration. As another
example, consider the motion of a particle under a central
acceleration; that is, one always directed towards or away
from a fixed point, the exact law of the force of attraction
as a function of the distance being \eit \iifc&AYmvh»to. Tha
VECTOR ANALYSIS.
85
planetary motions are of this description. In this case the
differential equation of the motion is
dt
i= r i/( r >-
(94)
As the acceleration is always along and therefore parallel
to the radius vector, the product
— *r = 0.
dP
This may be written — {— *r J = 0,
Fiq. 47.
for, carrying out the differentiations, we obtain from this
last
<Pr m , dv dv n
dt 2 dt dt
of which the second term on the left is zero, because any
vector product containing parallel vectors is zero. Hence,
integrating,
dt
— *r = const, vector = c, (95)
dt
where c is a vector perpendicular to the ptaoe oi x mA'
to
ax
86 VIXTOR ANALYSIS.
But — is the vector velocity along the tangent and r is
at
the radius vector, so that by § 18 the above equation is
twice the area swept out by the radius vector in unit time.
We obtain the result, then, that under any central accelera-
tion the rate of description of areas is a constant, and the
orbit lies in a plane perpendicular to a constant vector c.
Harmonic Motion. Equation of Ellipse. As another ex-
ample to integrate
fr + »* - 0, (96)
which is the equation of a central acceleration proportional
to the distance from the center. Such motions take place
wherever Hooke's Law is followed.
We know that the two solutions of the scalar equation
are r = a cos mt and r = 6 sin mt,
and that the complete solution is the sum of these two. If
we replace the arbitrary constants a and 6 by the arbitrary
constant vectors a and b, obtaining
r = a cos mt + b sin mt, (97)
it is easily seen, by differentiation, that this equation is the
complete solution of the vector differential equation
— + m 2 r = 0.
By an extension of this process, which is easily seen to hold,
we may then state the rule for the solution of linear differ-
ential equations to any order with constant coefficients: Find
the solutions, assuming the vector variable to be a scalar variable,
multiply these each by an arbitrary vector and add. The resuli
unH ' &e (he complete vector solution.
VECTOR ANALYSIS.
87
These arbitrary vectors are to be determined from the
initial or final conditions of the problem exactly in the same
manner as we do with scalar equations. Equation (97) is
the composition of two simple harmonic motions along direc-
tions determined by a and b, and is easily seen to represent in
general an ellipse inscribable in the parallelogram whose
sides are 2 a and 2 b respectively.
r = a cos mt -f b sin mt
is therefore one form of the equation of an ellipse, if m is real.
42. Hodograph and Orbit under Newtonian Forces. As
another example in vector differentiation and integration
consider the case of motion under a force directed along the
radius vector and inversely proportional to the square of
the distance in magnitude. This is the ordinary planetary
motion. We are to solve the differential equation
dPr = mr t
de r 2
Multiplying by r*, we have at once
cPr
(a)
and hence
de
<*»\
* This equation states that the acceleration Vb daswtoA waXwwd^
le. the forces are repulsive. For attractive tarcsa dtaawgs ^ ^ "' m -
88
VECTOR ANALYSIS.
where c is a constant vector. This last equation states that
the rate of description of areas is constant, as above.
Lemma in Differentiation. Consider now the identity
r = rr lt
>V # nw^'*^
then dv — rdr x + r x dr,
» «■*.
hence multiplying by ry,
rydr = r rydr\,
an t.h«.t. r.x/JlT. = — — .
a result we might have written down immediately by similar
triangles.
Multiply this last result by ry, obtaining from the left-
hand side of equation
TfirfdTj - T x {T^dT x ) - dT x (T X .T X ) = - dt v
since r x and dv x are perpendicular and iyr\ = 1, so that
dr t
= _ ry (r»dr)
r 2
and
dt
'±M
Now the parenthesis on the right is a constant vector for
central forces by (6), so that, by (a),
j
(Pr „ rye dr.
d* 2 r 2 d*
and integrating,
dr
dt
<c = d — m r„
(c)
wAe/e rfwa constant vector petpeivd\c\3\ac to c, %a\& vmchV^
multiplying by c«
VECTOR ANALYSIS
89
Multiplying equation (c) by - * we obtain
1 Idr \ dr 1 dt 1 1 A 1
c Ictt / d/ c dt c c c
This becomes, since c-- = 1, and because by (6) -j- is
normal to c ( or to -J ,
•I = c-^d - mc-i* ri = ^ x (d - mrO , (<©
the equation to the hodograph.
Since c and d are constant vectors, c~ ! *d is a constant
vector normal to c, and hence lies in the plane of the orbit;
m c -1 *!", is a vector constant in length, so that the extremity
of -7- lies in a circle drawn around the point c~*xd as center
dt y
and in the plane of the orbit. This length, since c and i\ are
perpendicular, is 2 .
c
To obtain the equation to the path multiply the equation
of the hodograph by r*,
r* -j- = c = r*(c~ l *d) — mT*(c~ l *r x ).
Expanding the two triple vector products and remembering
(6),
c = c^r-d — wrc 1 .
Multiplying by c»,
So that
r =
c*— r«d — mr.
<?
m
d cos a — m ,
d
— COfctt
m
90 VECTOR ANALYSIS.
the polar equation of a conic; the angle a being measured
from the line d.
Comparing with
I
r =
1 — e cos a
the general polar equation of a conic where the focus is the
origin, where I is the semi latus rectum, and where e is the
eccentricity, we find that the path of the orbit is a conic of
eccentricity — ; that d is along the major axis and that the
tn
magnitude of the major axis is *
m 2 - <P
43. Partial Differentiation. When any vector is a func-
tion of more than one scalar variable it can be differentiated
partially with respect .to each one, the remaining variables
being considered constant during the differentiations. Such
partial differential coefficients are written just as in ordinary
scalar calculus as — , -^ , etc., where x, z, . . . are the inde-
ox dz
pendent scalar variables. The total change in a due to simul-
taneous changes in the variables dx, dy, dz, . . . is written
da = ^dx+ P^dy+ ^ dz. (98)
ox dy dz
Symbolically we may write this as
da = (j^dx+ £-dy+ £ dz\a, (99)
where the expression in parentheses is to be considered as
a differential operator to be applied to a, as in (78).
Origin of the Operator Del (V). The operator (99) has
the form of a scalar product, the two constituent vectors
of which are
( ox dy dz)
* See Appendix, p. 245, for Path Described V^ ioi¥b^TO&V&%Tta&P
yrm Magnetic Field
VECTOR ANALYSIS.
91
If the single symbol V (read del) be used to denote the
first expression
'=(*^+J#: +k£V (100)
v=
Bx dy
the equation may be written
da = (V-dr) a. (101)
We are thus led naturally to the consideration of the
properties of this symbolic vector V.
PROBLEMS AND EXERCISES.
1. If r«dr — 0, show that r — const.
If r*dr - 0, show that r,« const.
If r»dr*<Pr — 0, show that r*dr has a fixed direction and that r
is always parallel to a fixed plane.
2. Show that
and that
dr
dr
T
r,«dr — — »dr.
•dr.
3. Given a particle moving in a plane curve, in the plane of 1J,
obtain the components of — along and perpendicular to the
dt
radius vector.
They are
— r. and r — JL p..
dt l dt l
A unit line -L to r t is k*r t , where k is normal to the plane.
4. Obtain similarly by differentiation of
dt dt dt
the accelerations along r t and perpendicular to r t .
They are
92 VECTOR ANALYSIS.
5. If r, <f> t be a system of polar coordinates in space, where r is
the distance of a point from the origin, </> the meridional angle, and
6 the polar angle, obtain the expressions for the components of the
velocity along the radius vector, the meridian, and a parallel of
latitude.
6. Find the accelerations along the same directions in the problem
above.
Express them in Cartesian form.
7. The curve
p = a cos / + b sin t
represents an ellipse of which a and b are conjugate radii. The
vector
r — — = — a sin t + b cos t
dt
a cos
(H +bsin (H
is the radius conjugate to r, and parallel to the tangent at r.
8. The parallelogram determined by the conjugate radii of an
ellipse is constant in area.
[r*r / = const.].
9. An elliptical helix is represented by
p = a cos / + b sin t + ct.
10. Show that the tangent line and the osculating plane of any
curve p — f (a) may be respectively written in the forms,
p - r + xr* f
p = r + xt* + yr",
where r'= — and r" = — _ , and
ds ds 2
x and y are variable scalars.
11. Find the tortuosity, T, of any curve where T is defined as the
rate change of the normal n to the osculating plane with respect to
the arc ds.
/rfP x ^rcPr\
= /dn\ _ \ds*ds 2 'ds*) ^
ds 2 ds*
Express this in Cartesian notation.
12. Find the curvature of a circular YifcVix. ¥\xA VJfta tox\,>H£&3
fa circular helix.
VECTOR ANALYSIS.
93
13. The equation
p = x«K0 + a,
where a is a constant vector, represents a cone standing on the
curve p -= +(/) with its vertex at the extremity of a.
14. The equation
p - 4>(J) + xa,
where a is a constant vector, represents a cylinder standing on the
curve r — +(/) and having its generators parallel to a.
15. Prove that the acceleration of a particle moving in a circle
with uniform speed is given by
dP r*
16. Write out the equations of the hodograph for uniformly
accelerated motion in a straight line.
17. Find the hodograph to the motion
¥= ±mt >
(a)
where the acceleration varies as the distance from the origin.
The solution of (a) being that of ( — ± m 2 j p — 0, that is
p — A cos ml + B sin mt,
and p - Ae™* + Ber™*.
Interpret these equations and those of the resulting hodographs.
18. Show that if the hodograph be a circle, and the acceleration
be directed to a fixed point, the orbit must be a conic section, which
is limited to being a circle if the acceleration follow any other law
than the inverse square.
19. In the hodograph corresponding to acceleration /(p) directed
towards a fixed center, the curvature is inversely as ^/(p).
20. Show directly without analysis that
p r,
and hence that
dp,
dl
CHAPTER V.
THE DIFFERENTIAL OPERATORS.
*
The Vector Operator V (read del). This sign is some-
times called " nabla " (Heaviside) and also " atled," which
is "delta" (a) reversed. The term "del" is, however,
well worthy of adoption, as it is short, easy to pronounce
and conflicts with no other terminology. As V is the most
important differential operator in mathematical physics its
properties will be studied in detail.
Definition. V is defined by the equation
We have already come across the scalar differential oper-
ator p on page 72. The paragraphs concerning p should
be consulted at this point. As by its definition V is made
up of three symbolic components along the three axes
i j k, the symbolic magnitudes of them being — , — , and —
ox oy bz
respectively, it may be looked upon as a symbolic vector
itself. This view of V as a vector, is important and of great
help in the comprehension of what follows. The employment
of V in the treatment of the physical properties of space is
of the most frequent occurrence. It is, therefore, extremely
desirable to have a geometric or visual representation of
such physical properties in space, or fields, as they are
called, and of the effect of operators upon them.
\,
44. Scalar and Vector Fields. Reconsult § 5 at this point.
Definition. If to every point in a region, finite or not,
there corresponds a definite value of some physical property,
the region so defined is called a jicld. SfacflaXdi \Jaia ^ro^erty
94
VECTOR ANALYSIS. 96
be a scalar one the field is called a scalar field. As examples
of such may be mentioned the temperature at any given
instant, at all points of a body; or the density at all points;
or the potential at all points, due to electrical, magnetic, or
gravitational matter respectively.
On the other hand, if the property is a vector one it is
said to be a vector field. As examples of these are the
velocity at all* points of a fluid; the electrical, magnetic, or
gravitational intensity (of force) at all points of a region due
to electrical, magnetic, or gravitational matter, respectively.
45. Scalar and Vector Functions of Position. Assume
any arbitrary origin and from it draw a variable radius
vector r. This vector r may extend to and determine any
point in space. By the term "value of r" is meant the
" position of the terminus of r." Now, if to every value of
r there corresponds a definite scalar quantity V, V is said to
be a scalar point-function of r and is written
V=/(r). (103)
If to every value of r there corresponds a definite vector
quantity F, F is said to be a vector point-function of r and
is written
F = f(r). (104)
[y=/(r) and F _^f^r))are thus the functional representa-
tions of scalar and vector fields respectively.
Mathematical and Physical Discontinuities. The functions
met with in physics are almost always continuous and
single-valued except perhaps at isolated points, lines or sur-
faces finite in number. If not, they can be made so by
various devices, such as by inserting diaphragms to prevent
passing into a region by two or more different paths, etc.
The functions dealt with, in what follows, are supposed to
be of this description.
The most common kinds of discontinulWsfc \Xv*X> c&^rot y&l
mathematics are those in which, eittveT tYvfc ns\wr> \\s^ A
96
VECTOR ANALYSIS.
a function suffers an abrupt change, or where the rate
of change of the function abruptly takes on a new value
as the independent variable is continuously increased or
diminished.
Graphically these mean a break in the curve, or a sudden
change in the direction of the curve representing the
function, as in Fig. 48 (a), at P and at Q. In nature such
discontinuities do not take place. For example, the tem-
(a)
Fig. 48.
perature cannot have one value on one side of a surface and
another value on the other side, where the two sides of the
surface are infinitely near to each other. In reality there
is a continuous but very rapid change in the temperature
from its value on one side to its value on the other as we
pass through the surface. Besides, infinitely thin surfaces
do not exist except in our imagination.
If the temperature gradient in a body has one definite
value and seems to change abruptly to another value quite
different from the first, we know that in reality there is a
very rapid but finite rate of change of the gradient at the
place in question. This absence of discontinuity in any
natural function is indicated in Fig. 48 (6), which shows the
continuous function, which to a\\ mtente %xA v\xrp$&&
VECTOR ANALYSIS.
w
replaces the discontinuous function (a). It is for this
reason that the usual attention will not be paid to the con-
sideration of discontinuities in what follows. All natural
functions being in reality continuous, such consideration
is physically superfluous. We may state the same idea
explained above, by saying that on sufficient magnification
of finite amount, all curves representing natural phenomena
will be found to be continuous.
(6)
Fig. 48.
We do not, however, wish to convey the idea that the
study of discontinuities is unimportant, as on the contrary
the mathematical results derived from their study are of
the greatest importance, and teach us how to attack prob-
lems involving sudden natural changes, or as we might call
them, " apparent discontinuities " in physical functions.
In fact the methods generally employed are to assume
them to be actual mathematical discontinuities and treat
them as such. The point we wish to make is that in the
general analytical expression of natural phenomena it is
unnecessary to complicate the formulae by the separate
consideration of discontinuities, but to let the student
treat them by the recognized mathematical n\fctA\&d& ^V&s&»
ever it is convenient or necessary to do ao.
98 VECTOR ANALYSIS.
46. The Potential. For the sake of definiteness we shall
consider the potential due to electrical matter. The whole
argument applies almost identically to magnetic or gravita-
tional potential; to the distribution of temperature in a body,
or to the velocity-potential in moving fluids, etc., etc.
Definition. The potential at any point in space due to a
distribution of electrical matter may be defined as the work
done on a unit positive quantity of electricity as it is brought
by any path from infinity to that point. As like charges re-
pel each other, it will require positive work to be done on the
unit charge to bring it in the neighborhood of any positive
distribution of electricity, and hence the potential around
such a distribution will be positive, increasing as the points are
taken nearer and nearer to it. It is evident, also, that the
forces acting on the unit charge are repelling forces and that
they act in the direction opposite to the increase of potential.
For instance in the electric field due to the charge + q on
a small sphere, the unit positive charge at P is repelled by
a force acting radially outward, of amount calculated by
Coulomb's Law
r 3
where r is the distance from P to the center of the sphere.
The force F evidently becomes greater as P approaches the
sphere, and work has to be done upon the unit charge in order
to make it do so.
Level or Equipotentlal Surfaces. Let all the points having
the same potential be found, or, in other words, find all those
points which require the same amount of work to be ex-
pended upon the unit positive charge to bring it from infinity
up to them. If C be this amount of work and V be the
potential function, then the equation to the locus defined by
these points will be
V(r) = C,
where r is measured from some arbitrary on©&.
VECTOR ANALYSIS.
99
Find similarly all points which require a small amount
more of work C + dC; the equation to this locus will be
V (r) - C + dC.
In the special case of the sphere, Fig. 48A, these points will
lie on spheres concentric with the charged sphere.
Fig. 48A.
These equations define surfaces which are called Level or
Equipotential Surfaces of the function V(r). Let many such
surfaces be constructed and let the quarvtvivea qI ^<3tV otv-
ployed in reaching the successive ones &\iter \yj eqvxo\, ot\wx&»*
100
VECTOR ANALYSIS.
It requires no work to earry the unit charge from one point
to another having the same potential, for by definition it
requires the same amount of work to bring the unit charge
from infinity to either of these points by any path, and we
may choose the path leading to the second point to pass
Fig. 49. Showing Lines of Force and Equi potential Surfaces Around
a Charged Conductor.
through the first point. Hence the work done in going from
the first point to the second point must be zero.
Relation, between Force and Potential. Consider in par-
tico)ar two adjacent level surfaces, the difference in poten-
tial between them being dV. Ttoa m«»n& ftssfc '■& wopC-osa
VECTOR ANALYSIS.
101
dV units of work to carry the unit charge from one of the
surfaces to the other in any manner.
Since the amount of work is constant in going from one
level surface to the next one, the greatest forces will be
encountered in going by the shortest path from one to the
other, so that at any given point P t the maximum force is
along the common perpendicular to the two surfaces. Com-
paring the forces at different points P x or P 2 these maximum
forces F t or F 2 will be found greater the nearer the surfaces
are together. Hence the forces in the field are normal to the
level surfaces and are inversely proportional to their distance
apart. But V increases most rapidly along the normal to a
level surface and its rate of increase is greatest where the sur-
faces lie closest together. We are therefore led to expect a
relation between the force at a point and Wvfc x^Xfe o\ Ywst^aafc
of V at the same point. All this is concAS^ TC\>reefc\&fc& > « , S
\
102 VECTOR ANALYSIS.
writing for the work done in going from one surface to the
next,
dV - F-dn,
where dn is the normal distance between the two surfaces, so
that
F = 5^(~n). (105)
an
where n is the unit normal pointing in the direction of increas-
ing potential. This important equation states that the force
at any point is normal to the level surface passing through that
point; opposite to the direction of fastest increase in_V, and
equal in magnitude to this fastest rate of increase.
Thus a knowledge of the potential everywhere gives a
knowledge of the forces everywhere not only in magnitude but
in direction as well.
A scalar point-function, as it does not involve direction, is
clearly simpler of representation on a diagram than a vector
one. The potential function is very useful for this reason, as
a complete knowledge of its value everywhere immediately
gives us a complete knowledge of the forces everywhere.
Thus the comparatively simple scalar function intrinsically
contains all that we wish to know about the comparatively
more complicated vector function. This property alone is
sufficient to justify its invention and use.
47. V Applied to a Scalar Point-Function. Gradient or
Slope of a Scalar Point-Function.
Definition. The vector f perpendicular to the level surface
\ at any point, equal in magnitude to the fastest rate of increase
of V, and pointing in the direction of this fastest increase,
is called the gradient or the slope of V at that point and
is written
grad V or ^ope V,
preferably the first.
VECTOR ANALYSIS.
108
Grad V Independent of Choice of Axes. The force F acting
on the unit charge is, by the above definition in connection
with § 46, evidently equal to — grad V, but as F is entirely
independent of any choice of axes, so is — grad V independ-
ent of them.
It remains to show that the operator V applied to V gives
+dr
the grad as defined above. The work done on a unit charge
>
as it is carried from M to M f is, by § 16, where MM' = dr,
— F-dr = grad V-dr = dV.
Considering now V as a function of x y z,
dV = — dx + - - dy + -!- dz,
ox ay dz
SO that
grad 7-dr = ft |X + j |Z + k^(ldx + j dy + kdz)
= VV-dr.
104 VECTOR ANALYSIS.
As this equation is true whatever path dv is taken between
the two surfaces,
grad V = VV. (106)
Thus the application of the operator V to a scalar point
function is a vector which gives its rate of most rapid
increase in magnitude and direction. The significance and
importance of this operator is now easily understood.
The vector vF is often called after Lame* the first differ-
ential parameter of V.
Fourier's Law. If instead of potential we consider tem-
perature, the level surfaces are then isothermal surfaces,
and the V of the temperature function gives the rate of
the most rapid increase of the temperature in magnitude
and direction. As the flow of heat takes place in the direc-
tion of most rapid decrease, q, the intensity of flow, is given
b y q = - kvO,
where k is a characteristic of the medium at the point in
question and called its conductivity, and is the tempera-
ture at any point. This is called Fourier's Law for the
flow of heat.
48. Illustrations of the Application of V to Scalar Func-
\ "\ tions of Position. By means of ordinary partial differentia-
tion on the functions
r = (x 3 + f + s 2 )*
and r»= (x 2 + f + Z 2 )*
the following important results are obtained:
= xi + yj + zk = r = r
(z* + if + **)* r
* G. Lam6. Lecons sur lea coordoiuieea curvilignes et leurs di verses
applications. Paris, 1859.
VECTOR ANALYSIS.
105
and
Vr n =
-('^ + '^ + k i)" ,+ » ,+ ^'
(x* + y* + *)».
nr n " ly 7r = nr""" 1 r t = nf 1 " 3 r.
(108)
So that V differentiates, the function r n similarly to the
scalar differentiation of u n by — >
—zL- = nu n * — •
dx dx
A shorter method for obtaining this formula will be given
in §49.
The particular cases
Vr = r. and V - = — £
r r 3
(109)
are important results.
^va Application of V to a Scalar Product. It is frequently
necessary to apply V to a scalar product such as a»b.
Remembering the laws for the ordinary differentiation of
scalar products and the definition of V it is easily shown
that
V (a-b) = V (a-b) + V 6 (a.b),
where the subscript affixed to the V shows which vector is
considered variable, i.e., the one it acts upon.
The notation
V a (a-b) = V(a-b) 6 (110)
is also sometimes used, where the subscript in the second term
shows which vector is considered constant &\iYvxv%\X\fc &&**-
entiation. V a ( ), then, from this point ol nww Sd&rsw&b*
106
VECTOR ANALYSIS.
tiates partially, the vector a alone being considered variable
in the parenthesis. In particular, if the vector c&, say, of
the scalar product »»r. is a constant vector, we obtain the
important result
V r (wr) = *, (111)
for
V (c*r) = Vr(<»-r) = V r (^x + w,y + io^s)
= i u) x + j a) 2 + k a) a = c&.
k
49. The Scalar Operator s t *v or Directional Derivative.
Since VV is a vector, its scalar product with any other vector
s t may be taken, and by definition this would give the com-
ponent of the magnitude of VV in the direction of that vec-
tor. This is ordinarily written
dV BV , dV , dV
d8 ax by dz
where «„ « 2 , *, are the components of the unit vector s u
or what is the same thing, the direction cosines of the direc-
tion Sj.
This expression may be looked upon as the operation of
~ d t d t d d
>l dx
(112)
upon V, which is the familiar directional derivative of V in
the direction s t or, otherwise,
<W)F-VVV (-£*)
(113)
so that the directional derivative in any direction is the com-
ponent of the magnitude of the gradient in that direction.
The directional derivative then applied to the potential func-
tion gives the component of the force in the direction in which
it is taken.
VECTOR ANALYSIS.
107
Total Derivative of a Function. If s t be replaced by the
small vector dr, the operator dr-V applied to a scalar point
function gives the total derivative of the function because
dr-W = (dr-V) v = (Jz dx + J- d y+'& dz ) V = dV. (114)
This is the «ame thing as the directional derivative along
(dr), multiplied by dr, or
dtS7V = dr ((dr) r V7) . (115)
We may now obtain equation (108) for the differentiation
of r n by means of the identity (114)
dr-V( )=d r ( ),
as djr n = nr"~ l dr = nr* -1 r,«dr,
so that the factor of -dr, nf"' 1 r, is the Vr n
and Vr n = nr"' 1 r, = nr B ~*r
as before.
f'JJM. The Scalar Operator 8,-V Applied to a Vector. The
operator s,*V may be also applied to a vector point-function
F, giving as a result a new vector function, thus:
(Sj.V) F = w (IF, + jF, + kF s )
= i WF, + j s,-vF, + k 8,-vF,
+ j(
dF,
s *ir +Sj
dF.^ dF.
dF
dF„
dx " "* By ' "' dz
+ a,
+ *S
')
)
(116)
This is the directional derivative of the vector function F in
the direction s v It is also the vector whose ro\K$Qt&R&& *sfe
the directional derivatives of the component <&^«
108
VECTOR ANALYSIS.
The parentheses may be omitted in s^vCF) and the ex-
pression written simply s^vF, but it does not mean that
(s,-v)F = s^vF), because vF has no significance in this
analysis. s^vF may then be interpreted as nothing else
but (vV) F.
We may prove that the component along any constant
vector a of the directional derivative along b, of a vector
function F, is the directional derivative along b of the
component of F along a; that is, ^
ai-(bi-VjpF) = bi-V/p(arF).
And even without restriction to unit vectors that
a.(b.VjpF) = b.Vjp(a.F). (117)
This follows directly because V differentiating F alone,
a- may be placed after the V. Also because (116)
a-(b-VF) = a-i b-V/<\ + aj b-VF 2 + a-k b-VF t
= ajb-VFj + a 2 b-VF 2 4- aJb^F s
= b-V F ( ai F t + a 2 F 2 + a,F t )
= b«V^(a-F).
Applying «-V to r, the radius vector, gives
2 d
dx . . d
dz
. OX , • OW , , OZ
dx dy oz
= i w l + j w 2 + k Q) s = tt.
This expression should not be confounded with n*T7t, where
the r is the magnitude of r. Since Vr = r t the value of this
last expression would be
co.vr = c&»r x .
Combining «»Vr = » with equation (111), we see that
fjlt£
VECTOR ANALYSIS. 109
^^ The Operation V on a Vector Point-Function. Any vector
point-function F may be resolved into three components
along i j and k so that
F - i F x + j F 2 + k F r
F v F 2t and F s are scalar functions of x y z, or of r. Con-
sidering V as a vector, the product vF can have no meaning
unless the definition of the product of two vectors a and b be
extended so that the product ab (without dot or cross) shall
have a meaning.* But the scalar product of V and a vector ' ^
F and the vector product of V and a vector F may be found
by rules already given. The two expressions V-F and V*F
are of such importance that special names have been given to
them.
•^51. Divergence. The operator V*( ) or div ( ) [read
del dot ( ) or divergence of ( )] when applied to the
function F gives a scalar which in Cartesian notation is
In order to obtain a physical interpretation of this quantity
consider any vector field, the field of force due to an electri-
cal distribution for instance. The convention usually adopted
is that from every unit positive charge there originate 4 n
lines of force and into every negative unit charge there end
4 n lines of force. The exact number of lines of force that
issue from unit charge which convention has adopted, is of
absolutely no consequence in this argument, and the new
system which assumes the unit charge to give rise to but one
* This has not been done in this book, although Professor Gibbs has
achieved beautiful results in his researches using this extended defini-
tion. The product ab he calls a dyad, fee OfoWWtaafe* NwXwv
AnaJyab, Chapter V.
110
VECTOR ANALYSIS.
line may be adopted if desirable. If an element of volume
be considered, for example a small parallelopiped with its
sides dx dy dz parallel to the axes ofxy and z respectively,
the amount of electrical matter within it may evidently be
measured by finding the excess of the lines which come out
of it, over those which go into it. For every unit of positive
Fig. 52.
electricity there would emanate 4 n lines outward, and for
every negative unit 4 n lines would enter into it. Consider-
ing these lines to cancel each other when going in opposite
directions it is easily seen that the algebraic sum of the
charges within the box may be found in amount and sign
by an examination of the lines which leave or enter the box.
Hence the lines which diverge itom \>&fe &«ibk&Xi ^wKlVs**
VECTOR ANALYSIS.
Ill
measure of the positive charge within it. If the charge is
negative, lines will end inside of the box, and therefore will
converge into it.
To obtain an analytical expression for this quantity resolve ,
F, or the flux of force, as it is called, into its three components
parallel to i j and k. The flux into the face parallel to the
2/z-plane nearest the origin is F x dy dz, the flux out of the
opposite face is
BFj
dx
(
F x + ^^ dx J dy dz,
so that the amount which comes out in excess of that which
goes in, as far as the x-component of F is concerned, is
(an \ «a rp
F x + -T- 1 dx\dy dz — F x dy dz = -r- 1 dxdydz.
Similarly, for the other two components, which are obtained
in the same manner,
-7-* dz dy dz,
-r- 1 dx dy dz,
dz * '
so that the total amount of the flux F which diverges from
the box dx dy dz is
f -
(
dx dy
W* +
dF,
dz
J dxdydz.
Dividing by dx dy dz, the element of volume, to obtain the
amount of flux which would come out of a unit volume under
the same conditions, there remains precisely
L divF = vF = ^»+^ + ^»
dx
*»
dz
(120)
Strictly then the term divergence means the \ttimta& <& Xxbm^
which diverge per unit volume.
112 VECTOR ANALYSIS.
If the operator V* be applied to the vector function repre*
senting the flux of heat or the velocity of a fluid, it will give
by exactly the same reasoning the rate at which heat is issu-
ing from a point per unit volume or the rate at which the
fluid is originating at a point per unit volume. In the case of
heat, if the divergence exists and is positive, there must be at
the point in question a source of heat, heat actually created,
or else at the point where the heat is leaving the temperature
must be diminishing.
In the case of fluids, if the divergence exists and is positive,
there must be either a source of fluid, fluid actually created,
or else the density of the body at the point must be diminish-
ing. If the divergence is negative, the opposite conditions
hold in both the above examples. For instance, if the diver-
gence of heat is negative, or in other words, if it converges,
there must be a sink of heat, heat actually destroyed, anni-
hilated, or else the temperature at the place must be rising,
etc. In the case of electricity, the existence of a positive
divergence proves the existence of positi ve electrical matter
at the point . The negative of divergence is sometimes called
convergence. It is better, however, to retain but one of the
terms and use the negative sign to indicate convergence.
52. The Divergence Theorem.* This important theorem
has a significance almost axiomatic when considered in the
light of the foregoing. Consider any closed surface S, Fig. 24,
lying in a vector field q, the velocity, say, in a moving incom-
pressible fluid. It is evident that the excess of fluid which
comes out, over that which goes in, may be measured in two
distinct ways: first, by finding the total outward normal flux
over the surface, or second, by going throughout the interior
and taking the algebraic sum of the sources and sinks or diver-
* For a rigid mathematical proof see A. G. Webster, Electricity
and Magnetism, pp. 60-62; Dynamics, pp. 340-342; also R. Cans,
Bin f ah rung in die Vektoranalysis, pp. 2&-S&. §*& k^auSY^ \>. ^SZ^
for other theorems analogous to the DWct%«w» TV««t«nv. feia^,
opell, Trait6 do Mgcaniquo rt-UouneWe, tome\lVx\K V
VECTOR ANALYSIS.
113
gences for every infinitesimal volume element contained within
the surface. In symbols this is most conveniently expressed
as
/
U-i
S?~
C f n*q dS= J J fv-q dv,
utwarcTdTB
(121)
where n is the ouMttra-drawn unit normal, dS the element of
surface, dv the element of volume.
In words this reads: In a vector field the surface integral
of the outward flux {i.e., normal component of flux, see § 17)
oyer any closed surface S is equal to the volume integral of
the divergence taken throughout the volume enclosed by S.
This is the divergence theorem.
From a mathematical point of view this demonstration
may not be considered rigorous, but the ideas that this inter-
pretation gives should be clearly understood by every student.
In Cartesian dress this theorem becomes
/ / [ft cos (nx) + £2 cos ( n u) + ft cos (nz)]dS
-sm
<?2i + *& + ?&
dx By dz
\dzdydz.
(122)
The idea of divergence is evidently independent of any
choice of axes since none are required for its conception.
Considered as the result of operating by V« it is invariant to
change of axes because V has been shown to be invariant.
Its invariant character may also be directly proved, as usual,
by a transformation of axes, but this is a long and unnecessary
process.
Examples. In particular
dx By , dz
dz dy
dz
V&&\
114 VECTOR ANALYSIS.
Let us apply this result, using r for q in the divergence
theorem. We obtain immediately
f f n-r dS = 3 f / Cdv = 3 X vol.
Or in other words, .th jge times the volume included bv anv
closed surfacej sjiht.ftip^ hv multiplying every element of
surface by the perpendicular from the origin to its plane and
lidding the results
nXspliere, for instance, taking the origin at the center, r is
perpendicular to every element of surface and is of constant
length, therefore n-r = r,
3 X vol sphere = r / f dS = 4 ttt 8 ,
a true result.
To obtain Vi\. By (123) and J28),
V-r = 3 = V.(rr,) = rV-r, + r^Vr. = rV-i^ + ^ (by (112))
• • dr
= rV.r t + 1.
Hence v^r,= -.
r
Equation of the Flow of Heat. As another example of the
use of the Divergence Theorem (121) consider the general
laws of thermal flow. Consider a volume of matter through
which heat is flowing, and consider a surface S drawn any-
where in this space. Let q be the flux of heat, or in other
words, the amount of heat which crosses unit area drawn
normally to the lines of flow per unit time; q is also called
the heat current-density.
The amount of heat which escapes through the surface
in any time is furnished at the expense of the material
inside that surface which must then be cooling off at a
certain rate.
By Fourier's Law, § 47, the heat flows in the direction of
greatest decrease in temperature, 8, atvCi ^f\\Xv fewvctasuafc?
VECTOR ANALYSIS.
115
proportional to a property of the material through which it
is flowing, called its heat or thermal conductivity k. So that,
q = - &V0 .
The coefficient k may vary from point to point of the
medium, and may be also a function of the temperature.
In most practical applications it is assumed to be constant.
If there are no sources nor sinks of heat within the sur-
face any elementary volume dv is cooling at some rate
dO
— — . The amount of heat which leaves this elementary
ot
volume in unit time must then be, if p be its density and e
its specific heat,
et r
For the whole volume, S, the heat lost, which must be equal
to that passing through the surface, is
By the Divergence Theorem the surface integral is
f I I V-qdv,
J J JyoI
and since q = — kVO,
So that
V-q =— V«fcV0.
f ff -*L cp dv= f ff -V-kVOdv.
J J JtoI °t J J Jjol
Since this equation holds whatever surface be considered,
the integrands are equal everywhere and
116 +* VECTOR ANALYSIS.
If A: be assumed constant this becomes
" *™
ot cp
= a*7*0, where a a = - • (107)
cp
This is the general differential equation for the flow of
heat in a body.
If the steady state is reached, that is, if the temperatures
are everywhere constant (this does not mean the same
everywhere), it becomes
V*0 = 0,
independently of the values of k, p, and c; that is, the
distribution of temperature follows the same law as the
distribution of potential according to Laplace's Equation
(157). So that, what is true about the potential, is under
analogous conditions true of temperature, and the two
subjects, temperature distribution and potential, become
identical in mathematical treatment.
53. Equation of Continuity. Considering again a moving
liquid, if there are no sources nor sinks of the fluid in the
region considered, then the equation
V-q = div q = (124)
expresses the condition that the fluid does not concentrate
towards nor expand from any point, as this is the only
remaining way by which more liquid can leave any small
closed surface than can enter it, or conversely. In other
words, it means incompressibility. This equation is called
the equation of continuity. It is of great importance in
electricity, as according to the theory of Maxwell the electric
displacement behaves like an incompressible fluid. If the
divergence does exist it means Wv&\> aX» Vtafe ^o\w\, ^mvtered
VECTOR ANALYSIS.
U7
there must be a source of lines of force or what is the same
thing, electricity.
Solenoldal Distribution of a Vector. Should the divergence
of a vector function be zero everywhere, then always as much
vector flux enters any volume element as leaves it, or in
other words, the lines of vector flux cannot end nor begin in
free space. They must then form closed curves or end at
infinity. Such a vector distribution is called solenoidal. For
example, the motion of any incompressible fluid such as
water, gives a velocity distribution which is solenoidal.
54. Curl. The Operator V* applied to F or curl F (read
del cross F or curl of F), also sometimes written in German
books, rot F (read rotation of F), is a new vector derived from
F. Like V it is invariant to choice of axes and has an im-
portant significance in physics. It may be defined by the
equation
i i k
V*F = curl F =
dx
F>
J
d_
dz
^3
\ dy dz J \dz dx )
+"(£
%)■ «»
The new vector, curl F, has components
($-¥)■ (
M
dz
Mm.
dx
Mf
MA
along the three axes. When applied to a vector function
V* gives a result independent of the axes because V itself is
independent of them. We may say in general that all com-
binations of V with vector or scalar point-functions give
results independent of any choice of axes. By a direct
transformation from one set of axes to auot\v«t ^^ tnnj \st«w
the invariant property of these operator axA Vtao& ^foskaa^
118 VECTOR ANALYSIS.
any lingering doubt in the mind of the skeptic. To a physi-
cist, however, to say that the operations of V upon any func-
tions are dependent upon a choice of axes, is like saying that
the physical properties of any medium depend upon the lan-
guage in which you express them. For instance, we have
shown (§ 47) that V F, where V is the potential say, gives rise
to a vector showing the direction in which V changes most
rapidly and its magnitude.
What have axes to do with such a result? It is true
whatever kind of coordinates are used, however placed, or
even if none are used at all.
We are here dealing with the properties themselves, and
not with any particular method of representing them. It is
in this respect that the analysis of vectors is extremely useful,
as by its intelligent study clear conceptions must necessarily
be obtained.
Example of Curl. In order to give an idea of the meaning
of the V* or curl of a vector function, consider the general
motion of a rigid body. We have seen (§ 22) that the motion
may be resolved into a velocity of translation q of the origin
chosen arbitrarily and an angular velocity of rotation eo
about a line passing through this origin. The velocity q of
any point r is then given by
q = q + **r,
where q and eo are the same for all points in the body at any
given instant. Taking the curl of this equation, dr, in other
words, applying or operating with V*, we obtain
V*q = Vxq + V*(c**r).
Since q is a constant throughout the body V*q = and
Vxq = V*(c»*r).
In this product V differentiates r alone because a is a con-
stant throughout the body at any Wms. litottaxtataA^ut
VECTOR ANALYSIS. 119
value of this expression expand the triple vector product by
(55), considering V as an ordinary vector,
Vx(w*r)= *(Vr-r)— r(Vr-w).
As V cannot act upon », we interpret the last term as(w-V r ;r,
but V-r = 3 by (123) and (<o-V) r = <o by (118),
V*q = 3a>— a> = 2a>,
and <o = \ Vxq = \ curl q. (126)
See also equation 131.
We see then that when a rigid system is in motion the
operator V* applied to its velocity-function gives twice its
angular velocity in magnitude and direction. We may
then write q = q + i curl q*r. -
Consider now a very small portion of a fluid such that the
portion may be considered to move as a rigid body for the
instant; it is fairly evident that the curl of the velocity there
would give similarly twice its angular velocity of rotation.
The curl or V* is an operator such that when applied to any
velocity-function it gives twice the angular velocity of rota-
tion at any point in direction and magnitude.
55. Motion of Rotation which has No Curl. Irrotational
Motion. A clearer idea of curl may perhaps be given by a
consideration of the following two possible motions of a fluid
about an axis. Considering Fig. 53, if the infinitesimal
portions of the fluid, indicated by short straight lines, move
from position 1 to position 2, as indicated, then evidently
every elementary portion of the fluid has rotated by the
same amount and the operator Vx would give this rotation
multiplied by 2. On the contrary, if the infinitesimal ele-
ments in moving about the axis O do not rotate Vsvafc tcto&&.
facing one way as in B, then the cur\ oi swta. a. mtfC\OTLHro&&
120
VECTOR ANALYSIS.
be zero.* Superficially, however, to the eye the two motions
here described would look the same. If we assume that the
molecules of iron are free to rotate we may realize these two
motions. If a piece of iron were rotated in a strong magnetic
field, the molecules constantly pointing in the fixed direction
of magnetic induction, we should obtain a motion such as B,
B
Fig. 53.
while if rotated in a non-magnetic field the motion would be
similar to A.
Any motion which has a curl is said to be rotational or wr-
tical; if it has no curl it is called irrotational or non-vortical.
Any motion represented by a function whose curl is zero is
one in which the infinitesimal elements do not rotate, and
conversely.
56. V, V* and V* Applied to Various Functions. It is fre-
quently necessary in many cases to apply the operators
formed with V to combinations of scalar and vector func-
tions. The following rules will be found useful for reference
* A rigid body whose elementary parts move with it as in A , Fig. 53,
might be called atomically-rigid; a rigid body whose elementary parts
move as in B would then be noiv-atomically-rigid. A calculation was
made to see whether the difference in the moment of inertia of these
two kinds of motion could be observed in the case of iron. But mole-
culea are bo small that the calculated difference in moment of inertia
could not be observed by the moat reratiiweXstaoxatorg \o£&qaA&.
VECTOR ANALYSIS.
121
(127)
Let u and v be scalar point-functions; u and v vector point-
functions.
Then
V (u + v) = Vu + Vv,
v-(u + v)= y-u + v-v,
V (uv) = vVw + wW
V'(ttV)* 7tt«v + i* V-V — *\
V*(arV) = vu«v + u v*v. — \
w N 4
V(u-v)= u-vv + u*(v*v) + v-vu + v*(V*u)
V v (u-v)+ V tt (u-v).
(128)
(129)
V(u*v) = Y257*u — u-V*v. (130)
V x (u*v) = u (V„»«v) — v (V w *u)
= uV,«v + v-V tt u — vV tt -u — u*V,v. (131)
The convention here used, is that the operator V applies to
the nearest term when there are no parentheses or else the
variables on which it operates are indicated by subscripts to
it. So that, for example,
Vuu means (Vu) v and not V(uv),
and Vtt-v means (Vu)-v. In this last case it could mean
nothing else, as in V(u-v), u*\ being a scalar product of a
scalar and a vector can have no meaning.
In the above fundamental formulae subscripts have been
used to.render ambiguity impossible.
Methods of Proof of the Formulae. These formulae may all
be verified by expanding the quantities in terms of their com-
ponents along i j k, differentiating, and rearranging.
If we remember that the symbolic components of V,
3 d d
— > — t — obey with the components of any other vector
ox ay oz
or of any other V all the laws of common algebra, we should
expect V to obey the same laws as any other vector va. ^arc&r
binatiofl with vectors or other V'a. ^NVfti \*toa& vcl xxbbA *^ca
122 VECTOR ANALYSIS.
majority of the formulae on page 121 can be written down at
once without relying upon the demonstration outlined above.
It is evident at once from the definition of V that
V(u + v) = Vu + W.
Take the formula for instance
V'(uv) = Vu-v + W*v.
The V is supposed to operate upon both u and v. All the
possible combinations of V and a dot with u and v are formed
which can have a meaning, letting V act once upon each
variable. In the above example, as u is a scalar, V can act
on it only as Vu. The dot, which is as yet unemployed, is
used in forming the scalar product of Vw, a vector, with v,
another vector. As yet V has not differentiated v. In the
second term the only way it can act on v is by forming a
scalar product, giving V- v, which multiplied by u is ttV # v,
the correct result.
As another example, consider the expansion for V*(u*v).
We expand this triple vector product as usual, considering V
to be an ordinary vector,
V*(u*v) = u (V w .v)— v (V^u),
where be it remembered that V is to differentiate both u and v
in each of the terms. This is here specifically indicated by
the use of subscripts. From u (V w -v) can be formed u(V,-v)
and (v-Vu)u only; so that
u(7 w 'V) = u (V„iv) + (v.V u ) u
similarly from v (V^u), v (V„«u) and (u«V„) v can be
derived, so that
v (V tt0 .u) = v (V u -u) + (u.V„) v.
The single subscripts as here used are not necessary accord-
ing to the convention explained above. We then have
V*(U*V) = UV-V -V V'VU — Y^u — u^tn, (132)
VECTOR ANALYSIS. 123
where the V on the left is to operate on both u and v, while
on the right it operates only on the vector following it. This
kind of notation is exactly similar to
dx
(uv)= {h u ) v + u {i; v )
du , dv
dx dx
Consider the expression v*(V u *u) in which V is to act upon u
alone. Expanding,
Vx(V u xu) = V u (u*v) — u (V tt -v)
= V u (u.v)-(v.V u )u.
Similarly
u*(V^v) = V„ (u«v)— (u«V„) v.
Adding the two equations, we combine V u (u-v) -f V„ (u»v)
into V(u-v) by definition, hence equation (129).
The notation
V(u*v) = V u (u-v) + V v (u*v)
is strictly analogous to
d (u*v) — d u (u-v) + d v (u«v),
which corresponds to partial differentiation, and is true for
the same reasons.
We may write also
V*(t* v) = Vu*(u v) + V„-(u v),
or V*(u*v) = V u *(u*v) + V^(u*v), etc.
The process outlined above will always lead to correct
results. It is something more than a help to the memory.
A general rigid mathematical proof of its validity has been
given.*
* 8ee to this effect Joly. Manual oi QpatoimoTBiV.'Vb.
124
VECTOR ANALYSIS.
57. Expansion Analogous to Taylor's Theorem. Expand-
ing as a triple vector product and assuming that V jjg te on v
alone, we have
Ux(v*v) = V„ (u-v) — v (V„.u)
and = V, (u.«v)— u«vv,
or u-vv = V„ (u*v) — u*(V*v). (133)
If u = r l9 a unit vector, and v = q, then (133) becomes
«VVq = V ff (r^q) — r^ curl q, (134)
and states that the directional derivative of a vector func-
tion q in the direction r t is equal to the derivative of the pro-
jection of q in that direction plus the vector product of the
curl of q into that direction.
Multiplying the directional derivative by dr , we obtain the
difference in q due to a displacement dr in the direction r t ;
this gives then, if q r is the value of q at the end of r and q r+dr
is the value of q at the end of r + dr,
dq = dr-vq = q r+dr - q r = V fl (dr-q)- dr-(y^q).
So that
or
flr+dr = <I r + V « (*""<l) + (V^)^r, (135)
q (r + dr) = q (r) 4- Vjdr-q (r)]+[v*q (r)>dr.
This equation is analogous to the expansion of a function
by means of Taylor's theorem.
58. Theorem Due to Stokes. The line-integral of a vector
function F around any closed contour is equal to the surface
integral of the curl of that function over any surface of which
the contour is a bounding ecjge.
In symbols this is
/
F-dr -
= f f n-curl F dS.
J J Cap
(136)
VECTOR ANALYSIS.
125
This important theorem may be demonstrated in a number
of different ways. The following is a demonstration given
by Helmholtz depending upon the variation of a line-integral.
The principle of commutativity of d with d and f is all
that is needed to assume here. Consider the line-integral
J of the vector point-function F (r) along the path ACB.
J - f A F-dr (path ACB).
The possibility of computing this integral in general
depends entirely on the path ACB, and with this under-
Fig. 54.
standing it is perfectly definite. It is now required to find
the variation in this integral when the path ACB is varied
into an adjacent one AEB infinitesimally close to ACB
but differing from it in an arbitrary mawiuet . Thfc\?«<2>^^<»
however, are to begin at A and end a\» B , V«o fexsA v^vefcA
AwhUM
126
VECTOR ANALYSIS.
on the contour. Taking the variation of the integral we
obtain
dJ = d r B F.dr= f B d(F.dr)= ( B dF-dr+ f* F-ddr.
J A J A J A J A
This becomes, by an interchange of d and d in the last
integral and with an integration by parts,
dJ - F.<Jr / B + f B (8 F.rfr - dF-dr).
The integrated term is zero, for since the limits A and B are
fixed, there can be no variation d r at these points. Remem-
bering also (§ 50) that
«JF = <Ji--VfF and dF = dr.V F F,
d J - f B dr-(dr-VF ) F - dr*{dr -Vf) F
= r B 3r-V/p(rfr.F)-rfr.VF(^r.F), using (117)
where V acts on F alone.
By (58) this may be written as
dJ = rVrxdr).(V*F).
Or more directly by (129),
dr-VF = VF(*r.F)- <Jr*(V><F),
hence
rfr.(5r.VF)= dr-VF(^r.F)- dr.<Jr*(VxF),
But by (117) the left side is dr^7 F (rfr-F).
Hence
dtS7 F {dt-F)- rfr.VF(^r-F)= £r*dr.V*F.
Referring to the figure \t is sieetv that $r*dr is the vector
area of the infinitesimal surface tonneA \*j fc* «cA to ^ **
VECTOR ANALYSIS. 127
that calling n the unit normal to the elementary area dS, we
may write <Jr*dr = n dS, so that
gj = f B n.(v*F)dS.
= f B n .(v*F)
Another infinitesimal transformation is now made to a new
curve AGB having the same fixed ends A and B and so on,
until the movable path has swept over the surface included
between the limiting curves / and //. Let now the sum of
all the variations in J be added together. The result will be
equal to the difference between J x and J 2) the values of J for
the extreme paths; heiuce
J 2 - J x - lim J <JJ= f Tn.(VxF) dS.
But — J x is the line-integral from A to £ along BCA, and
therefore J 2 — J x is the value of the line-integral around the
contour AD BCA. So that
/F-dr = f f (n-curl F) dS. (136)
«/ «/cap
This is Stokes' Theorem.*
59. Condition for the Vanishing of the Curl of a Vector-
Function. In the above demonstration, if the value of the
integral J is the same whatever path is taken between A and
B, and if this is true wherever the points A and B are taken,,
it follows that J x always equals J 2 , so that for any surface S
u
n»curl F dS = and hence V*F = curl F = (137)
5
must be true everywhere. In this case the value of / depends
only upon the position of the ends of the path and in no wise
* See also for other demonstrations of Stokes' Theorem,
Bucherer, Elemente der Vektor-Analysis, pp. 42-44.
Gibbe-Wilson, Vector Analysis, pp. 188-190.
Gmdb: Einfuhrung in die Vektorana\yB\&, pp. ^&-3&„
See Appendix, p. 249, for two other proofe.
- -^ J -
128 VECTOR ANALYSIS.
upon the shape of it. Conversely, if the curl F — in a
region, then the line-integral of F between any two points
A and B in the region is independent of the path chosen
between them. In this case if one end it of a curve is fixed,
the value of the integral / F-dr is simply a function of its
upper limit B whatever the path from A to B may be. Let
<f> denote this scalar function. The integrand F»dr must
then be a perfect differential and hence of the form d<f> f which
by (114) is the same as
dcf> = d r*v<£,
so that for all values of dr,
dr-v<£ = dr-F
and hence F = v<£. (138)
Or in other words, if F has no curl, it is the rate of fastest
increase V<£ or grad $ of a scalar function
4>s-f * ptdr + 4>A>
where <j> A is a constant.
The scalar function <f> thus determined is called the poten-
tial of F. As we have seen before, a scalar function divides
space up into shells or laminae by means of its level surfaces.
The vector-function F = v<£ derived from such a function
is for this reason said to be a lamellar vector (Maxwell). The
curl of a lamellar vector is then always zero,
or curl (V<£) = 0. (139)
Conservative System of Forces. By Stokes' Theorem (136)
we see that if the line-integral of F-dr, i.e., the work around
any closed path, always vanishes, then the forces have no
curl; and also that in this case \tafc iotofca ycl \3&a fold axe
VECTOR ANALYSIS. 129
derivable from a potential function <£. Such a system of forces
is a Conservative System, and we may define such a system:
When the forces acting on a system of bodies are of such a
nature that the algebraic total of the work done in performing
any series of displacements which bring the system back to
its original configuration is nil, the system of forces is said to
be conservative. The condition, then, for a system of forces
F to be conservative is
curl F = v*F = 0.
60. Condition for a Perfect Differential. If the total
differential dcf> of a scalar point-function <f>(r) = C be taken,
we obtain, by (f'49% the equation
tfy = dr*v<£ - 0.
This is of the form rfr-f (r) — 0,
which in general is not a perfect differential. In Cartesian
notation this equation becomes, if f l9 f 2 , / 8 are the compo-
nents of f along the three axes,
f x dx + f 2 dy + fjdz = 0,
and our problem is to find the condition for integrability of it.
Assume that, by multiplying this differential equation by
some scalar factor //, it may be made a perfect differential, or
that for every value of dr
fi(dr4) — dr*/i f = dr-v<£.
/i is called an integrating factor. In this case, then,
fi f = V<£.
In order to eliminate <f> take the curl of this equation, because
we know that the curl of a lamellar vector (139) is zero,
rn ar~"« — -•-■ —
130 VECTOR ANALYSIS.
which may be expanded as
V*(/* f) = /*V*f + (V//)*f = 0.
Applying f • to eliminate the second term, because a*a«b =
in general, there remains
f.(^Vxf)= //(f.Vxf)=0.
So that finally we find that the condition that the equation
defined by f-rfr= should be integrable is that f and its curl,
V«f , shall be at right angles or that the curl vanish. To put
this discussion in a more familiar form, we have proved that
the condition of integrability of
F-rfr = Xdx + Ydy + Zdz
is that F-V*F = (140)
or
*@-S)* '(£-©+«£-$-*
a well-known result.
If the
curl F = 0,
this equation is evidently satisfied.
If the curl F is not zero, the equation says that it must be
everywhere perpendicular to F.
For example,
yz dx -f zx dy -f xy dz
has no curl and is therefore derivable from a single primitive,
i.e.,
xyz = const.
On the other hand,
axf&dx + b£x*dy + cx*]pdz
has a curl, but this curl,
2&(cy - bz) i + 2 \p(az - cx'i \ -V 7, #<J>x - oa^Vw,
and the vector F,
VECTOR ANALYSIS.
ai?s?\ + b#&\ + cx*y>k }
131
are at right angles, and this equation is also derivable from a
single primitive, i.e.,
- + - + - = const.
x y z
111. Taylor's Theorem. The Operator «••*( ). Taylor's
Theorem is often written concisely as
f(*+h,y+k,M+l)-f(xyz) + ±^
+
• •
If the components of r be x, y, z, and those of c be h, k, I, it
may still further be condensed into
/(r +€) = /(r) + cV/(r) + 1 (cV)V(r) + 1 (cV)V(r) + . . .
Remembering the expansion for e*,
1! 2! 3!
we may write the last equation symbolically in the still
shorter form,
/(r + c) - 6«'V(r), (141)
so that the symbolic differential operator e*' v acting on any
function /(r) gives its value when r becomes r -f €.
62. Euler's Theorem on Homogeneous Functions. We
may employ this equation to demonstrate * AafcfcVo^ ^^rtcsksl
due to Elder and known by his nams. k ta&s&srcL ^ A
132
VECTOR ANALYSIS.
degree n in a variable r is said to be homogeneous when r
occurs the same number of times in every term of it. It is
one such that
^(or)= a n <f>(r), (142)
where a is any constant. Apply Taylor's Theorem to the
homogeneous function <£(r) and let c = g r, where g is a small
scalar multiplier. Then
<? r ' v <t>(r) - *(r + g r) - «r(l + g)] - (1 + * )«#<r),
and hence
0(r) + ^ r-V^(r) + £- (r.V)^(r) + . . .
- [l + ng + n( *~ 1 V + ' ' •]*«.
Subtracting <£(r) from both sides and dividing through by
g there remains
r-V^(r) + JL (r.V)^(r) + . . . = In + ^=^ g+- . ."L(r).
This equation being true for an infinite number of values of
g, we may equate the coefficients of the same powers of g on
both sides of the equation, giving
n^(r)= r-V0(r),
n(n- lW(r) = (r.V)^(r),
n(n - l)(n - 2)<£(r) - (r.V)ty(r).
(143)
The first of equations (143) is known as Euler's Theorem
on homogeneous functions, which in terms of x y z becomes
nd>(x yz)=x-J-+y-3- + z-3->
^ v * ' dx By dz
(144)
The remaining equations are extensions of the theorem,
involving derivatives of higher oidera.
VECTOR ANALYSIS. 188
Operators Involving V Twice.
63. Possible Expressions Containing V Twice. Given a
scalar point-function V and a vector function F, the follow-
ing six combinations, involving V twice, are possible ones:
1° V- W S V 2 V (div grad V, a acaUr)
2° V*W (curigradKEO)
3° jy*V)F (V*F. a veotor).
4° V(V»F) (grad div P, a vector).
5° V*(V*F) (div curl P = 0).
6° V«(VxF) (curl curl P = curl* P, a veotor.)
Two of these expressions vanish identically,
V* Y V = 0, curl (grad V) =0
because any ve ctor product containing two like vecto rs is
zero, and
V» VxF = 0, div (curl P) =0
beca use any tr iple scalar-product with two l ike vecto rs is also
zero. These two results may be proved, if not sufficiently
evident, by expanding according to the ordinary rules.
The 6th may be expanded into
V-(V*F) = V(V-F)- F(V-V) (145)
so that the 6 th - 4 th - 3 d -
Equation (145) is sometimes written as
curl (curl F) = curl 3 F = grad div F — V*F. (146)
This last important equation would be, as it is written,
rather difficult to remember, but the advantage of retaining
the notation in dels is made evident by the previous equation
which may always be written out according to the rules for
the expansion of a triple vector-product. As wuo>\5wsc ^xaxs*-
pie of the advantage of the symbolic notaWon \\. \s> ^fcsfea&3
IT '
184
VECTOR ANALYSIS.
easy to remember whether it is curl grad^ V or grad curl V,
which is identically zero. For when they are written in
terms of V, i.e., V*VF and V(V*V), one is evidently zero and
the other can have no meaning at all. In other words, the
del-notation when Interpreted according to the ordinary rules
of vector products, leads us to correct results Independently
of any physical or other considerations.
The Operator V 2 or V«V. (Read del square of . . .)
Operating on
dx By 02
with V-, or in other words taking the div of the grad of V, we
obtain,
V .v7 a w s ?r + w +? r a ^ + * + *>, (147)
dx 2 dif bz* \b& dy*
the well known Laplacian operator, which when equated to
zero is satisfied by the potential function in free space.
It is evident on inspection that
V.(VF) = (V-V) F= V 8 V.
Since the curl of grad V, or in symbols V*VV, is zero iden-
tically there can be no ambiguity whatever when V is twice
applied to a scalar-function. When V is applied to a vector
function, it means that it is applied to the three scalar-func-
tion components of the vector function, and hence offers no
new difficulty.
If VF _
then
V»F t -0
VF, = 0.
that is the three components oi ^F ^aXM^li^a^^wjc^tvQiu
VECTOR ANALYSIS.
185
Since
V*VV = = curl grad V,
it follows that the vector W is a lamellar vector, by (§ 59).
Since V-V*F = = div. curl F,
it follows that the curl of any vector is a solenoidal vector,
by (§ 53).
64. Differentiation of the Scalar Function f 1 by V*. We
proved equation (108), that
Vr m = mr" 1 " 1 r t = mr™^* r.
Take the divergence (V-) of this vector.
V*Vr m = W"-* m fr" 1 " 2 V*r + r*Vr m " f |
- m [3 r^ 1 - 3 * r(m - 2)r m " f {,
because by (112),
so that
and finally
V*r = 3 and r*V = r iyV = r li-
ar
W" = m(m + l)?^" 1 .
(148)
The two values of m which will satisfy the differential equa-
tion
W»_ 0,
are easily seen to be m = and m = — 1, so that the scalar
function - satisfies Laplace's equation, or
^-=0.
r
This may also be shown, of course, by direct differentiation
of the function - •
r
186
VECTOR ANALYSIS.
EXERCISES AND PROBLEMS.
1. Prove that V«F is an operator independent of choice of axes,
by actually carrying out the transformation to a new set of axes.
If the coordinates of the new set be denoted by primes, it should
be found that
BX } BY | BZ ^ BX' j BY' } BZ'
Bx By Bz Bx 9 B%f Bd
where X,Y,Z, are the components of a vector function F, and where
X'jY'jZ' are its components referred to the new axes.
2. Prove directly by a change of axes that
V*F or CurlF
is invariant to that change.
3. Prove that
Va-r- a,
Vr « r lf
Vr 2 - Vr-r- 2r,
where a is a constant vector. These follow from the relation
d ( ) - dr.V ( ).
4. Prove that
Vt- 3,
V-r,--,
r
V(a*r) - 0.
*7* 1
6. Verify that
V-VF
-(*.+ * +
\Bx* By*
B*)*'
where
F
-ix + jr + kz.
6. Prove that
(a-V)V = a-(W)
and state the resulting theorem. Apply this to a simple problem
in potential.
VECTOR ANALYSIS.
18T
7. Show that
„/l\ a*r a*ri
b-V ( a-V
K)-
3a*rb*r , a*b
7 h "T f
r 5 r 8
where a and b are constant vectors.
8. Show by direct expansion that
V*W =
and V*V*V= 0.
9. find the resultant attraction at the origin of the masses 12,
16, and 20 units respectively concentrated at the ends of the vectors
a - 3 i + 4 J, b - - 5 I + 12 J, and c - 8 1 - 6 J.
10. From the expression for the attraction at a point P due to a
mass M, its density p at any point being a point-function of r,
- fff PJ r
vol
deduce the ordinary Cartesian expressions for the component
attractions along the axes.
XL What theorems do equations (111), (117), (118) express?
Write them out in Cartesian notation.
IS, Explain paragraphs 46 and 47, considering instead of
potential
(a) Temperature distribution in a body.
(6) Velocity distribution in a fluid.
-w#-
CHAPTER VI.
APPLICATIONS TO ELECTRICAL THEORY.
65. Gauss's Theorem. Solid Angle. Consider a point 0,
and any small area in space dS. Join every point in the
boundary of the area dS to 0, thus forming a small cone. We
define as the solid angle subtended by the area dS at 0, the
value of — , where dS is the area cut out, by the cone, on
r^
any sphere of radius r, described about as center. This is
numerically equal to the area cUo, cut out by the same oone on
Fig. 55.
the sphere of unit radius, described about 0. The dimen-
sions of solid angle are evidently zero. Since the total area
of a unit sphere is equal to 4 n, this is also the solid angle sub-
tended by the whole of space or by any surface which com-
pletely surrounds the point 0. dS evidently subtends the
same solid angle at O as dS. Calling n the unit normal to dS,
its sense being taken in any conventional manner previously
agreed upon (outward from the surface in the following),
dS =idScoa^n^,
VECTOR ANALYSIS. 189
according as it makes an acute or an obtuse angle with r,
respectively.
Now because dw and dS are parallel sections of the same
cone we mav write
dS = r>d<o,
so that the solid angle
d<o=m = ± dScos ( rn) . (149)
r 2 r 2
If a point is chosen inside any closed surface, any small
cone with vertex at will cut out through the surface
always once more than it cuts into it, but if the point be out-
side the surface it will cut in as many times as it cuts out. If
n be chosen positive when drawn outwards, then the angle
between n and r will be acute wherever the cone cuts out,
obtuse wherever it cuts in. Thus an elementary cone when
its vertex is inside a closed surface contributes an element of
solid angle, +<ho. As for example, in the figure the solid
angles + dw and — dw, due to 1 and 2, annul each other, leav-
ing -I- dw due to 3. When the vertex is outside, the resulting
solid angle is zero; as for example, the solid angles —dw and
4- dw, due to 4 and 5, completely annul each other. So also
do the elements at 6 and 7 and 8 and 9 in pairs. If we inte-
grate or sum up all the solid angles due to all the elementary
areas dS over the whole surface S, we shall obtain 4 it for the
sum, if the point is inside and zero if the point is outside.
Expressing these results in symbols we have
inside of 8 (150)
outside of S.
These results, which are purely mathematical, vtfe Vs&nrtcl
Gauss's Theorem.
140
VECTOR ANALYSIS.
Gauss's Theorem for the Plane. In a plane we may obtain
an analogous theorem, i.e., the plane angle subtended by a
closed contour in a plane at a point is 2 n or according as
the point is inside or outside of the closed contour. In the
figure (56) consider a point A connected to by a radius
vector which starts from B and moves once around the con-
tour until it reaches B again. Evidently whatever the shape
Fig. 56.
of the contour may be, the radius vector r has made but one
revolution about and therefore covered an arc equal to 2 n
on the unit circle about 0. In the case that the contour is
completely outside of the same reasoning shows that the
radius vector when it reaches B has not rotated around at all.
In a plane the angle d<f> subtended by an arc dv at a point
is, using a notation similar to (159),
, , . dr cos (rn) , n*dr
VECTOR ANALYSIS.
141
and the integrals are
or =
according as the point is inside or outside of the contour.
We may apply analogous reasoning to Gauss's Theorem
in space. Think of the solid angle subtended by an exten-
sible sheet which gradually is extended completely in any
manner around any point 0, or which is made to form a
closed surface of any shape completely outside of 0, and the
two results of equation (150) will become visually self-
evident.
The importance of this theorem in physics is the fact
that the surface-integral of the normal-component of the
very important vector-function varying inversely as the
square of the distance r from a point is in symbols
and Gauss's Theorem reduces these integrals.
Second Proof of Gauss's Theorem. We may obtain another
proof of Gauss's Theorem from the following physical con-
siderations. The field of force around a point at which is
concentrated a unit of matter, electrical say, is by Coulomb's
Law
F oc ?J. (151)
r
This means that the vector F is directed radially outwards,
but that its magnitude varies inversely as r 2 . Consider two
closed surfaces S t and S 2 , the first surrounding the point
and the other lying completely outside of 0. About draw
two spheres, one of which is completely outside of both of the
two surfaces S x and S 2 , and the other of a stcv^YY feTt&\u^\%&K&.
so that it does not touch either S v ox S v Svksfc >tafe ^^
142
VECTOR ANALYSIS.
tude of F falls off exactly as fast as the areas of the spheres
increase with increasing radius, the surface integral of this
vector over each of the spheres is the same. Or, in other
\
■•. \
Fig. 57.
words, the flux that gets through one reaches the other.
Evidently the same amount of flux must have passed through
the surface S i} so that we may write for the flux
C Cr^Lds^A. C f r . r ds^— f fds
J J 8 X T 2 RxJ t/sphere R\J J
of rad. R x
Rl
,4 TlR? = \lt.
VECTOR ANALYSIS.
143
As for the surface S 2 , since no flux is gained or lost in going
from sphere R x to sphere R 2 , whatever flux went into S 2 must
also have come out of it again, and therefore
Hence the theorem.*
J Js~? X
dS = 0.
66. The Potential. Poisson's and Laplace's Equations.
From the definition of the potential in § 46 as the work done
on a unit positive charge in bringing it from infinity up to
the point at which the potential is desired, we may show that
the scalar function — is the potential function correspond-
T
ing to the inverse square, or Coulomb's (in gravitation, New-
ton's) law of force Foe ^.
Consider a quantity of positive matter, m, at 0.
The potential at any point P, or in other words the work
done on a unit positive charge in bringing it from infinity to
—io~+
Fig. 58.
P, is equal to the line-integral of the force function F from P
to oo along the path PQ traversed by the unit charge.
V P = fVdr = - f^ iydr, see {16
1 r
which may be written, because V - = — -J by (109),
r "r
* 8ee Appendix, p. 251, for another piool ol Owrf%t^«w^
144
VECTOR ANALYSIS.
This last expression is the total derivative of — , so that the
integral is independent of the path and depends only upon
the limits, that is, upon the starting point and ending point,
giving
V p -2/ r '.2L.
Should there be other masses present, the potential function
Fig. 59.
due to them all is the sum of the separate functions due to
each, or
y..«L + 2h + . .._£«., (163)
where r Pa is the distance from the point at which the potential
is to be found to the mass m«. If the masses instead of being
at discrete points form a continuous distribution, the sum-
mation becomes a volume integraV, dm, \ta& &«(£&&X» <&\&a&&, k
VECTOR ANALYSIS. 145
becomes pdv, where p is the volume density of the matter
under consideration. We have then,
F- fff *5- fff^. (164)
The integral thus defined which is to be taken over the
volume occupied by the masses may be shown to be finite,
continuous, uniform, as well as its first derivatives. It
vanishes itself at infinity to the first order, and its first
derivatives to the second order.
A system of forces for which the line-integral between any
two points is independent of the path is called a Conserva-
tive System.
If we multiply Gauss's Integrals by m, a mass concentrated
at the point 0, we shall obtain
ffsf
'-Yn-^dS^Aitm. (155)
This integral states that the outward normal component of
flux of force (according to the inverse square or Newtonian
law) through a closed surface surrounding is 4 it times the
amount of matter within; any matter lying outside of the
surface contributing nothing. As every element of mass m
contributes 4 itm we have the proposition that the outward
flux of force through any closed surface due to any distribu-
tion is 4tt times the total amount of matter within the sur-
face. It is in this form that Gauss's Integral is usually
given, but evidently, from what precedes, it is a geometrical
theorem rather than an electrical or gravitational one.
The force at any point due to any distribution of matter
is — grad V or — VV, by § 47, where
dv
J J J*> r
is the potential function due to the distribution Tfcfc ^sg^
oo denotes that the integral is to V» tebYen on*k M5a»n^aR^»
Iff.
146 VECTOR ANALYSIS.
of space. This is equivalent to integrating over the matter
alone, as wherever there is no matter p = and the integral
contributes nothing. So that we may write (155) as
- f f n-W dS = 4n f f f pdv,
pdv being the total quantity of matter within S.
By means of the divergence theorem (121) the surface inte-
gral above may be transformed into a volume integral taken
throughout the volume enclosed by S.
- r fn.wds =- r rfv.vydv=4^ r r r^dv.
As this equality holds whatever surface S is taken, it follows
that the integrands are everywhere equal and
V-VF = TPV = - 4 up. (156)
This is Poisson's Equation.
In free space where o = this becomes
V 3 !^ 0, (157)
which is Laplace's Equation.
We may interpret these equations as follows: Every
quantity of matter emits lines of force, 4 n lines per unit quan-
tity. This numeric 4 n is purely conventional and appears
because the intensity at unit distance from unit charge is
defined as unity ; and since unit intensity corresponds to
one line per unit area, there must be 4 n lines emitted in order
to have one for each of the 4 n units of area in the surface of
the unit sphere. So then, if a surface of volume dv be drawn
around a point where the density is p, the lines passing
through the surface are equal to 4 n times the quantity of
matter p dv within it, or
div F (due to volume dv) = ±np dv,
and per unit volume
divF = 4*p.
VECTOR ANALYSIS.
147
Now if F have a potential, that is, if F can be represented
as the grad or V of some scalar function W, then putting
F = VJT,
V. VW = VW - 4 tcp.
This equation is true for the potential W due to attracting
matter.
In the case of repelling forces, since the force is opposite
to the direction of increase in the scalar function, we may
place W = — V, and we may write as before
^7=- 4itp.
In free space where there is no density p the equation becomes
div F - 0,
which says that the lines of force are solenoidally distributed,
that is, the flux takes place in unbroken continuous paths,
and hence cannot end nor begin at any point of space devoid
of matter.
The reason for the term (div) divergence is evident from
the foregoing.
Harmonic Function. A function which in a region is
single-valued, continuous, and satisfies Laplace's equation is
said to be harmonic in that region.
A Spherical Harmonic of degree n is any homogeneous
(143) harmonic point-function of space. That is, if V sat-
isfies the equations
V'F-O
and
r.VV =nV,
it is a spherical harmonic of degree n. The study and use of
such functions is of great importance in all branches of math-
ematical physics.
148 VECTOR ANALYSIS.
67. Green's Theorems. Two theorems due to Green of
very important application in theoretical physics follow
immediately by an application of the divergence theorem,
J Js J J JtoI
to the function W = UVV, where U and V are two scalar
point-functions which with their derivatives are uniform and
continuous in the space considered. Applying V* to W,
V-W = V.([/VF) = UV*V + VC/«W.
Substituting in the equation above
f fn.lF7VdS= C f fuv % Vdv + f f CvU-Wdv.
(158)
Similarly, by symmetry, putting for W = W U, we have
Cfn-WUdS = f f fw*Udv+ f f CvU-Wdv.
Subtracting these two equations there remains
f fn.(UW- Wt/)AS = f f f (!F7*V -W*U) dv.
(159)
The surface integrals are to be taken over the surfaces bound-
ing the region under consideration, and the volume integrals
throughout the volumes enclosed by these surfaces. Equa-
tions (158) and (159) are called Green's Theorem in its first
and second forms respectively.
68. Green's Formulae. Apply Green's Theorem in its
second form to two functions U and V. Let U be the func-
tion U = — , and let V be the potential due to any distribu-
r
tion of matter. The region to be considered is the space
lying between the infinite sphere, S*, w*>j «rcA*.<s» & ^\as^
VECTOR ANALYSIS.
149
surround the distribution, and the infinitesimal sphere of
radius t, surrounding 0, the point from which r is measured.
The equation
f fn-(UW - W U) dS - f f f(W*V- TV U) dv
i, since V*t/ ■= V i - 0, by (148)
/H-"-'»H/J? w ) t
The surface integral is to be taken over the bounding sur-
faces S to the region. The infinite sphere contributes nothing,
as at infinity -W and W- become zero to the third
r r
order, both containing r* in the denommaVrar . ^«t ^flofe wo^
150 VECTOR ANALYSIS.
sphere about the first part of the integral may be trans-
formed,
f fn.IvVdSf- rri(n.V7)e 2 do> = £ f fn-Wdu),
where dto is the solid angle subtended by an element of the
small sphere at 0, and, where £, its small radius, is constant
during integration. As e becomes smaller and smaller and
because n-VV, the normal force on the surface, is finite, the
integral vanishes in the limit.
Considering the second part of the surface integral over
the small sphere, we may write
- f 77 n-W±)dS =- V f(du) = - V X 4*,
1 n i*
because + n-V -dS = -V dS = solid angle due to dS. As
r r 3
the radius of the small sphere diminishes V approaches V©,
its value at 0.
So that finally,
Region Surfaces (161)
the surface integral being taken over the bounding surfaces
S and the volume integral over the region bounded by them,
shaded in the figure.
If V 2 V is equal to zero in the region considered, the poten-
tial function at any point is
^-nfSriv™-™;)"' (162)
which shows that it is completely determined everywhere if
the values of the potential V and of its normal derivative n«W
are known over the bounding suxta£fc& S>» \^ *&«> \sv»X\Kt
VECTOR ANALYSIS.
151
producing this potential and distributed in any manner
within S be taken out and replaced by a surface density
of matter a on S of amount
4 n \ r }
(163)
the potential at will be exactly the same as before, because
by substituting this value for a in
we obtain equation (162). We shall call this distribution
an Equivalent Layer. But in general this will not neces-
sarily make the surface an equipotential surface.
If the point is inside the surface S, a similar deduction
gives the formula
(164)
where n is to be drawn as the external normal to the region
in which lies. With this convention the two formulae due to
Green (161) and (164) are identical in form.
Green's Function. Adding together Green's equation (160),
which may be written
= f C f(UV*V- W*U) dv- f f n.([/W- WU)dS,
and (161), which hold under the same conditions, we obtain
4 " r ' -fff{° - §w-fffw
~JS[[" ~ r) W " ^ l" " l ^' &- <5SB *
■iaHu
152 VECTOR ANALYSIS.
This equation is of especial importance in the theories
of light and electricity. The quantity (u ) which ap-
pears in the integral is sometimes known as Green's Function.
09. Solution of Poisson's Equation. Equation (161) states
that if the quantity V'V is known throughout a region
bounded by any surface S, and if the quantities V and W
are known at all points of the surface, then V is completely
determined within the surface. Allow the surface S to
recede to infinity so that we are now considering the poten-
tial in the whole of space. Then in the equation the surface
integral contributes nothing, as all the quantities multiplied
by dS approach zero to a sufficiently high order. There
remains, then, in the limit only
v -—hSfSJr*- <166)
Now, by Poisson's Equation, V satisfies the relation
So that, by (166), if the value of p, the density, be given at
every point in space, V is determined by the integral
SSSjf-
which is therefore a solution of Poisson's Equation.
The Integrating Operator Pot. The operation of finding
the potential due to a distribution whose density is defined
everywhere by the scalar function p(t) plays such an impor-
tant r61e in mathematical physics that Prof. Gibbs has given
to this operation a special name and defines
(read potential of p).
The sign oo indicates t\iatl\ve\\m\\,«»m^ be taken over the
whole of space, as wherever lYiera Sa xvo TO&Xtat ^tafe \x&k©&.
VECTOR ANALYSIS.
153
contributes nothing. We shall call the operation indicated
by the above equation " the potential of p" even when p
does not represent a volume density.
Considering again equation (166)
and using the notation (168), we see that we may write
V ^--i-potV 2 ^.
4 TZ
(169)
So that the application of
4tt
to a function nullifies the effect of V 2 on that same function,
or, in other words,
pot ( ) is the inverse operator to V 2 ( ).
4;r
70. Vector-Potential. In the same way that the potential
due to the scalar function p is formed, that is,
y-v*,-SSSs*-
we may define the potential of the vector function
where p lf p 2 and p if are given scalar functions of r, as
V = potp
J J Joo r
dv
-1 f f f&dv + i f f fbdv + kf f f&dv.
(read rector-potential of p)
iT«^n
154 VECTOR ANALYSIS.
The vector function V so defined is called the vector'
potential of p. Its three components evidently satisfy the
relations satisfied by the scalar potential, so that we have
V»V - - 4 n p. (170)
In strict analogy with the solution of Poisson's Equation
for a scalar potential we have then for the solution of a vec-
tor-potential
v_-±fff?X dv . (171)
71. Separation of a Vector Point-Function W, which has
a Vector-Potential, into Solenoidal or Rotational and Lamellar
or Irrotational Components. This means that the vector
function W is to be separated into two parts, one of which
has no divergence and the other no curl. We then assume
W - X + Y, (172)
where V-X - and V«Y = 0.
Consider the scalar function <f> and the vector function V,
related to X and Y, respectively, in the following manner:
X - V*V,
Y-- V0.
Then
W= V*V- V0.
If it is possible to determine V and <f> the problem is solved.
To do this take the divergence of W, giving
VAV = - Vty.
So that the solution for <f> is, by (166),
= -2- pot <y.w).
4x
VECTOR ANALYSIS. 155
Similarly, taking the curl of W,
VAV - V*(VxV) = V(V-V)- V'V.
Now since V is as yet undetermined, we may assume that
its divergence is zero or that V-V = 0, hence
V*W = - V 2 V.
So that V*W is 4 it times the vector function of which V is the
vector potential, and by (171)
=» — pot (V*W), which determines V. (174)
47T
Finally, since
W - V*V - V<£,
the decomposition is thus accomplished. This decomposi-
tion is sometimes known as Helmholtz's Theorem.*
Other Systems of Units. The factor 4 k which occurs in
many of these equations is due to the definition of unit quan-
tity of matter. In virtue of this definition it is necessary to
assume that every unit of matter emits 4 n lines of force, so
that, for example, the number of lines cutting through any
closed surface around any amount of matter will be 4 n times
as many as there are units of matter inside. Of late it is the
fashion to eliminate this " eruption of it's" as Heaviside has
it. This may be done in various ways, one of which is to
redefine the unit quantity in such a manner that it emits
but one line of force, in which case the equations
F=»r, divF = 4^ and TPV^-lnp,
IT
become respectively
p^l^fMh^ divF = p and W~- p. (176\
47T r
* Wiaa. Abh. Band 1, p. 101. For a mnemouvc \V no\, wratitast v***^
of Ms theorem, multiply equation (145) by -£- - »ii&\Yk\«ff«fc»** WBtfBB ''
■**■»
156
VECTOR ANALYSIS.
Such a choice of units eliminates 4 n in a number of for-
mulae but introduces it in others. It is nevertheless the most
convenient assumption in the modern theory, where the
energy is located in the space between the acting matter, and
in which action at a distance no longer holds first place. The
potential at a distance r from a mass ra, for example, becomes
1 <m
- — instead of—. The operation of forming the potential,
4tt t r
or pot V, would in this system consist in forming the integral,
** ' -///£•
(177)
and the theorem of Helmholtz would become in this notation
W = - V pot (V-W) + Vxpot (V*W),
(178)
72. Energy of a System in Terms of Potential. Con-
sider two particles of matter acting according to the inverse
m.
*«
m t
Fig. 60a.
square law, trip and mq f respectively, separated by a distance
rpq. In order to bring the mass nip from infinity to its position
an amount of work (§ 66)
tn q
Wpq= r^mp,
PQ
must be expended on the mass m p if the masses repel, or by
trip if they attract. For deftriiteivesa assume the matter to be
repelling.
VECTOR ANALYSIS. 157
The expression above may be written in two ways,
Wpq= V q m p or V p m q
because 7 fl = ^ and V p =^'
r PQ r pq
Similarly if we have any two systems of particles, the
energy obtainable by allowing the two systems to disperse
to an infinite distance apart is
pq— **p+*q
Tpq
where the summation signs extend to every pair of points,
one point from each of the systems. If we consider the
two systems as one, a factor £ must be introduced, as in the
summation every term would appear twice, so that
TT M =is p S,!» (179)
2 ^
represents the mutual potential energy of a single system
of particles. If the system forms a continuous distribution
the summation (179) becomes
where V is the potential function due to the total distribution.
73. Energy of a Distribution in Terms of Field Intensity.
If the distribution consists of a surface and a volume distribu-
tion of surface density a and volume density p, the above
integral takes the form
W = 2 ff VadS + \fff Vpdv * (180)
The integrals are taken over the suYfocfca uA >tarovy^\<s&»
the volumes of the matter under coxva\d^T^\Kau to»^<j&n^
158
VECTOR ANALYSIS.
At a surface distribution there is a discontinuity or change
in the normal component of the force due to the surface dis-
tribution, given by the well-known expression
F n = 4 no = n-V7.
By drawing a surface completely surrounding the surface dis-
tributions we may apply Green's Theorem to the whole of
space outside of these surfaces. Remembering that from the
last equation
a = J- n-V7 and that p = - -^- VV,
47T 47T
the two integrals which are now
TT = -!- f fvn-WdS--±- f f fvVVdv
8 k J Js Sn J J J„
become by Green's Theorem (158)
w -rJIS.^' i ^i-jn. r ' i "- (181)
If the medium is any other than vacuo, the element in the
integral, F 2 dv must be multiplied by a factor e characteristic
of the medium. The energy of the distribution is in this case
w -tni eFidv -
(182)
This may also be written as
w -rJfl- FFd "rJSS , - Fd '- < 183 >
where
F = fr
(184)
is a vector called the Induction.
VECTOR ANALYSIS. 159
74. Expressions for Surface and Volume Densities of a Dis-
tribution in Terms of the Intensity of Polarization. Starting
again (180) with the energy of a surface and volume distribu-
tion of densities a and p, respectively,
w -\ff.™ + iffL r '*-
Let us assume that this energy may be written also as
w —ifff l * l —ifff u * r *-
where I is called the intensity of polarization, H is the field
strength, and V is the potential corresponding to H. Then
8ince v.(yi) = i-vy + yv-i,
the integral may be transformed into
w - Uff*™ dv - Ifff™ *■
Transforming the first integral by the divergence theorem and
comparing this with the expression for the energy in terms of
<r and p,
we see that the polarization I produces a surface density
a - n-I (185)
and a volume density
p=- vl.
Conversely, assuming a distribution to consist of a surface
density a and a volume density p, it is easy to show by rea-
soning backwards that there is a quantity I related to a and p
by the equations
a = n-I and p = — V-I
such that the energy of the distribution may be represented
by the integral throughout the volume,
jf- -| r r (YwdD. via***
160
VECTOR ANALYSIS.
Equations of the Electro-Magnetic Field.
75. Maxwell's Equations. By experiment Faraday showed
that when the magnetic flux through a linear circuit is varied
there is induced in the circuit an electro-motive force. If
the circuit is a closed one this induced electro-motive force
produces a current in it. He also showed that this electro-
motive force is equal to the negative rate of change of the
magnetic flux. The positive direction of rotation in a cir-
cuit is connected with the positive direction of flux through
it, according to the adjoining diagram which symbolizes the
so-called cork-screw rule. If the arrow shows direction of
increase of magnetic flux, the arrow-head in circuit shows
the direction opposite to the induced current. The figure as
drawn shows the direction of the magnetic flux due to the
current in the circuit. By Lenz's law such a magnetic flux
would induce a current opposite to this; hence the negative
sign in equation 187 below. Since elec-
tricity tends to flow from places of high
to places of low potential we may con-
sider this electro-motive force as some-
thing in the nature of an electrostatic
field which is induced in the space by
the varying flux. That is, the electro-
motive force is induced in the space
even when unoccupied by a conductor.
In a conductor this electro-motive force produces a current
and in a non-conductor tends to produce a current. To
obtain the total electro-motive force around any circuit, we
evaluate the line-integral of this electrostatic field F along
that circuit. Then the induced e.m.f. may be written
Fig. 61.
fF-dr = f f (V*F)-ndS,
by Stokes' Theorem. But Yaxad^y s ex^rosascft. ^s^Xfcak
VECTOR ANALYSIS
161
this is equal to the negative of the rate change of the mag-
netic induction 3C through this circuit, so that
f f (VxF).ndS = -^ f fw-ndS= f f~^-ndS.
•/ t/cap Q&J t/cap t/ t/cap dt
As this is true whatever circuit is considered and whatever
cap is taken as long as it is bounded by the circuit we may
write
d3C
V*F --
dt
(187)
By experiment Ampere proved that a current / is equiva-
lent to any magnetic shell of a certain strength which is
bounded by the current. He also showed that a current may
be measured by means of the magnetic field that it produces,
and quantitatively that 4 n times the current in any section
of a conductor is equal to the line integral of the magnetic
force H taken once around any path linked positively with
the conductor. If q be the current density, then, symboli-
cally,
f H.dr=4;r f f q-ndS,
•/ t/ t/cap
where the surface integral is taken over any cap to the sur-
face. Transforming the first integral by Stokes' Theorem,
we have
f H-dr = f f V*H.n dS = 4 tt f f q-n dS,
•J t/ %/cap t/ t/cap
and as this equation is true whatever portion of space is con-
sidered and whatever path is taken around that portion of
space, we may write
V*H = 4 iz q. (188)
In order to explain the effect of an AfceXxcHn^v*^ Vsts*
upon dielectric non-conductors MaxmM assumed X5u&X \fcsfcfc»^
162 VECTOR ANALYSIS.
of a current q there is produced a so-called displacement-
motion or current q' of electricity, which on the release of the
inducing electro-motive force springs back and takes up its
original position. He assumes that this current-displacement
produces the same magnetic effect as would be produced by a
current of density
q'4? < 189 >
4 7T at
where SF = e F
and £ is a constant of the medium called electric inductivity
at every point of the field. We therefore, in considering a
dielectric, introduce this displacement current density instead
of q, giving
VxH - g. (190)
at
This assumption has been completely verified by the experi-
ments of Rowland in America.* If the dielectric is also con-
ducting we retain the term in q, giving
The term
M Bt
q + f f (191)
4 7T at
is called the total current and being equal to a curl is sole-
noidal, and has no divergence, i.e.
-(« + f, S) - °-
This current therefore moves in closed circuits or paths. It
is because of this equation that electricity is said to act like
an incompressible fluid. (See § 53.) According to the Elec-
tron Theory its compressibility is at most one part in a million.
Since repeated also by Crem\e\i axA'fcefc&st \^t»sa^
VECTOR ANALYSIS.
168
The complete system of equations for media at rest holding
in an insulating dielectric are therefore
d&
- V*H,
\
dt
¥
dJC
" dt
- VxF,
in
combination with
(F =
eF
and 3C =
a H.
(192)
t and ft are the electric and magnetic inductivities respec-
tively, and are defined by these equations. They are con-
stants for a given homogeneous isotropic medium, that is,
for non-crystalline media.*
76. Equation of Propagation of Electro-Magnetic Waves.
Let us assume that there are no permanent magnets in the
space considered, or, in other words, that there i& no intrinsic
magnetization, symbolically this is expressed by writing
V-H - div H - 0.
Take the curl of the first of these equations (192),
V*— = - Vx(F « V*(V*H) - - V*H + V(V-H).
dt dt
Changing from SF to F and from H to 3C and remembering
thatV-H -0,
« /£ £v*F--v 1 ae.
Differentiating with respect to the time, we obtain
and also
^|1(V*F) = -V»^=V>(V*F)
#
\
(193)
eu Z- 3C = V*3C.
* la order to investigate the form Vhwe «\\»Vwso» V^e.Vs« «*"»»
line media it is necessary to emp\oy the Wtieax ^eXw-to-™*'*^'
164 VECTOR ANALYSIS.
In a similar manner we could show that F, &, H and JC and
their curls also satisfy the equation
& - a'V^. (194)
This is the differential equation for wave motion, one of the
fundamental, partial differential equations of mathematical
physics. It can be shown that the velocity of propagation
is equal to a, and therefore for an electro-magnetic pulse
equal to — = . This turns out to be identical with the
velocity of light in vacuo, as it should be if the ether
is the common medium for the propagation of electrical
waves as well as of light. We now believe in fact, that
light waves and electrical waves are identical.
77. Pointing's Theorem. Radiant Vector. The energy
of the electric field is given by (183) as
w -rM !.'**■
and of the magnetic field, similarly, by
"-Sifff.*"*-
According to Joule the energy due to a current of density q in
the electric field F is
Wj= f f f q-Fdv.
Let us find the variation of the sum of these three with the
time, assuming e and p not to vary; then
£ <F.F - e £-F 2 - 2 £ — -F « 2 F-— = 2 F.(VxH-47rq),
dt dt dt dt *''
-3CH = u-H 2 = 2/< — -H=2H.— = - 2H.(V*F).*
dt ^dt * dt dt v J
* Heaviside introduces a nct\t\o\i& maugcvetic current density q^ in
order to produce a symmetry in tYve eqv\«A!vyaa. T&fc\M& \&x%\&Ntts*
tld then read ^p _ 4 „^.
VECTOR ANALYSIS.
So that the rate change of energy or the activity is
dW
165
dt
~R~ f f f (F#VxH ~~ HVxF ) dr -
By means of (130) and the divergence theorem this may be
written
dW
dt
- T- f f f V.(F*H)di; = J- f Tn.(FxH)d5.
4 7r J J Ju InJJs
Or, in other words, the rate loss of energy per unit volume
may be accounted for, by supposing a flux of energy through
the bounding surface in unit time per unit surface of amount
4;r
(195)
where R, the energy flux, or radiant-vector, is by its form,
a vector product, perpendicular both to F and to H.
78. Magnetic Field Due to a Current. It was proved
experimentally by Ampere that the magnetic scalar potential
O at a point due to a current / whose circuit subtends at the
point a solid angle to is proportional to the product of the cur-
rent and the solid angle. If we so choose our unit of current
that we may write,
we thus define a unit called the electro-magnetic unit of cur-
rent. The magnetic intensity of the field is given, similarly
toF=-V7(106), by
H = - VO = - /Vw.
Its component in any direction h, is by the definition of direc-
tional derivative (§ 49)
H.h 1 = -/h 1 .V(o=-l^ :
I y , J I »lll p
a
^^^P^B
166
VECTOR ANALYSIS.
To find the variation, dco, in the solid angle at a point due
to a small displacement dh of the point, notice that it will be
the same that will take place, supposing the circuit to move
a distance dh in the opposite direction or — dh, the point
remaining fixed. This motion will cause every element dr of
the circuit to describe a small area dr*dh whose component
dr*dh
Fig. 61a.
as a vector along r divided by r 2 is the element of solid angle
d(dco) at the point, due to it. The total change in solid angle
dco, due to the motion of the whole circuit, will be the integral
of this expression around the contour, thus,
d du> = r i ' dr * dh = S h.L « dr = <Jh.dr*V i ,
r 2 r r
whose integral around the circuit, after dividing by dh, is
dh l J t'
VECTOR ANALYSIS. 167
flotha *- H-h, 7^ = /^. Tvlxdr. (196)
dh J r
Since this is true for any direction h u we may write that the
element of magnetic intensity dti, at a point, due to the
element dr of the circuit, is
d\\ = / (viWr= -'-^E . (197)
This expression is determined to any function of r prhs which,
when integrated around the circuit, vanishes. So that to
this extent it is arbitrary. The equation shows that the force
due to an element is perpendicular to the element and to the
radius vector. The radius r being drawn from the point to
the circuit, and the current being positive in the direction of
dr, the order of the factors is taken so as to give the right
direction to H. This is the familiar expression for the mag-
netic intensity at due to dr.
dH = I drain d ± tQ elemeni and ^
70. Mechanical Force on an Element of Circuit. The
magnetic intensity dH, above, is the force with which a unit
positive pole placed at the point would be acted upon in the
field due to the current / in dr. By the principle of equal
action and reaction the element of circuit would be acted
upon by this amount but in the opposite direction. The
force on the element dr due to unit pole at the origin is there-
fore
dF = /dr*^=/<frx4>. (198)
r
So that the force on an element of current is proportional to
the current strength /, to the length of the elemawt dx ^ *xA
to the strength of the field at the e\emfcxv\> V ^afc \a*X«t ^
168
VECTOR ANALYSIS.
proportionality is the sine of the angle between dr and $ and
the force is at right angles to their plane.
Hence the force on the elementary current I'dr* in a field
due to another elementary current /"dr" at whose field at
the element df is by (197) and (198)
r//
dr"*r,
1
where the order of the vector product is reversed to take into
account the change of direction in r, or
cPF= (/WH/"dr"»r.) _ /'/"driftfr"*!-,) (m)
We may resolve this expression immediately into compo-
nents along the radius vector r and along the element dr by
an expansion of the triple vector product, giving
<PF= ££-' drwTr, cos {dr'-dr")-dr" cos (drV)
]
(200)
The X component is
77"
<PF X =
r 2
d^dr" cos (rx) cos (dr*dr")
— cos (dr"x) cos (drV) L etc.
These are well-known results.
80. Theorem on the Line Integral of the Normal Compo-
nent of a Vector Around a Closed Circuit. By means of
Stokes' Theorem the following useful transformation analo-
gous to it may be proved:
fq«dr = f f [(V-q) n - V (q-n)] dS. (201)
O
q is any vector function of r, and n is the unit normal to the
element of surface dS. In this case the line integral of the
normal component of the vector q along a closed circuit is
taken, instead of the more usu«\, \ak^t&s3l <&\s^sc&\Lt«
VECTOR ANALYSIS.
169
The result is a vector one, and it is shown to be expressible
as the surface integral of a certain other vector quantity
related to q, taken over any cap which is bounded by the
Fig. 62.
circuit. Let H be this line integral, and form its scalar
product with the arbitrary constant vector c,
c*H = c- / q*dr — / c-q«<2r = / dr*(c*q).
This last expression being in the form of a tangential line
integral may be transformed by Stokes* Theorem (136) into
c-H = ff [n.V^(cxq)]dS,
t/ t/cap
where the V differentiates q alone. Now
V„*(c*q) = c (V-q)- (c-V) q,
so that
c.H = f f [on (V-q) - n<c^ <&dS>-
--/L V (*H)*
Employing the above theorem (202) and remembering that
r t
170 VFXTOR ANALYSIS.
But (117)
(c-V (r )(q.n)=n.(c.V)q. J
The differentiation refers to q alone, as in the above integral i
the properties of the region are independent of the surface j
considered, and we may substitute (c»V fl )(q*n) for the second •
term in the integral, hence, |
c*H - c- f f [nV,.q - V,(q.n)] dS, j
t/ t/eap .
and finally, since c is an arbitrary vector, j
H= ± fq«dr=± f f [nV-q- v(q-n)]dS, (202) !
J J t/cap I
which sign to take, depends upon the direction of integra-
tion around the contour. This theorem is originally due to
Tait and to McAulay, who gave it in a much more general
form, including Stokes' and other theorems, as special cases.
81. Expression for the Field at any Point in Space Due to
a Current. We may use this theorem to transform the inte-
gral, giving the magnetic force H at any point in space due
to a current of electricity in a closed circuit.
If the magnetic potential is O, / the current in the circuit,
and oj the solid angle subtended by the circuit at the point,
we have, by definition,
H = -VO=- /Vo; =- /V ff ^dS,
t/ t/cap r^
the surface integral being taken over any surface with the
circuit for bounding edge, but not passing through the point.
Now V /Y ^?dS--vrr n-V± dS
t/ t/eap * t/ t/cap ?
VECTOR ANALYSIS.
171
J Jcap \ r) J r
O
- ±/ Tdr*vi, (203)
which result is in agreement with (197). See also (202).
We may thus write
dH = s /dr*V- + * l
r
in Cartesian
dH x = - \dy x (z - z x )- dz x (y - y x )\, etc.
where x lf y lt z Xi are the coordinates of a point in the circuit,
x, y, z those of the point P, so that the small magnetic field
dti due to an element dv is determined to a function 4> prhs
such that when integrated around a closed circuit the result
vanishes. So to a certain extent this resolution of the field
is artificial, and may or may not be the correct one, but in
any case this as well as any other possible resolution will
give the correct value for H above when integrate^ around
a closed circuit ; we have but very scanty knowledge of the
fields due to unclosed circuits.
82. Mutual Energy of Two Circuits. Inductance. Neu-
mann's Integral. Consider two circuits carrying currents /'
and /". The mechanical force on one of them due to the
field of the other is the integral of equation (199) taken once
around each circuit. Since
r r*
we may write
F =
VI" f AWv-xdr")
/'/" f f S V - {dr'-dr") - dt" C it- ^ ^NX.
172
VECTOR ANALYSIS.
Let one circuit be now displaced in any arbitrary manner,
so that any point on it is moved a distance dr'. The work
done in this displacement is
F«<Jr' = /'/" f f S IdV-V ! W.dr") - (dr'-dr") (dr'-V 1\ I .
Fig. 63.
Integrating the second term by parts and remembering
that d dr = d dr,
f(dr'-dr")(dr'.V i)= - tfr'-dr" / - f id ((Jr'-dr"),
the integrated portion vanishes for a closed circuit, hence
F.<Jr' = 77" f n {dt'-dT'')U±\+(?\d(dr'.dr'') I
J, J, r
As the assumed motion of the circuit is arbitrary we may
then find the force in any direction by finding the change in
I
>l» f Cdr'-dt"
(JJ04)
VECTOR ANALYSIS.
173
This integral due to Neumann represents the mutual energy
of the two circuits. When the currents /' and /" are each
unity, the integral gives the mutual-inductance of the cir-
cuits. When taken twice around a single circuit it gives the
self-inductance of the circuit. It is sometimes called the
Electro-Dynamic Potential, as by its variation we obtain
the electro-dynamic forces.
Vector Potential Due to a Current.
83. Mutual Energy of Two Systems of Conductors. If the
magnetic force due to a current be denoted by H, a solenoidal
vector, i.e., V*H = 0, we may write, by means of the theorem
of Helmholtz (175),
«-^-//x^-
But (188)
V*H = 4 n q,
where q is the current density, so that substituting, we obtain
H = V* f f f $dv = V*Q,
where
Qm fffS*.
(205)
Q is called the potential due to the current distribution q, or
the vector-potential belonging to the magnetic force H. The
word potential is used because it is formed in a manner
analogous to the potential due to a scalar distribution of
matter p. Notice also that the force vector H is obtained
from the vector Q in a manner analogous to the way the
force vector F is obtained from the scalar V, where
t/ U *J oo
r
and F = VV.
hence the name vector-potential.
174
VECTOR ANALYSIS.
We may now transform the magnetic energy in terms of
the field H,
into
w -tJSS. Yri —kSSS. n **
w - s//X HV - Q,to -
(206)
Integration Theorem. In general (130) we have, where
H and Q are any two vectors,
V.(H*Q)= Q-VxH - H-VxQ,
the minus sign belonging to the term in which the cyclical
order has been changed. Integrating over all space and
using the divergence theorem (121), S being the bounding
surface,
f f f v,tH "^ dv = f f f (Q' VxH ~ H-V*Q) dv
n.(H*Q)dS.
-si
Substituting (206). in this equation and remembering that
the surface integral vanishes at infinity, because there the
magnetic force vanishes, and also that V*H =* 4 n q there
remains
Now replacing Q by its value, we have finally
w -\fff. SSS.*?*"- <•»>
This sextuple integral covets the ^o\fc <A «$*«*> \mka»*
VECTOR ANALYSIS.
175
Let the only portions of space having any current density
be two closed circuits. The wire forming the circuits may
have a small but finite cross-section.
Place
qdv = I , dr / q'dv' m /"dr", •
where /' and /" are called the currents in the two circuits,
respectively. The integrals then reduce to a double line
integral each integral to be taken once around each of the
circuits, so that
w.-i£ffe?L
(208)
o o
This expression is really identical with Neumann's Integral
(204). The factor £ is due to the fact that by the conven-
tion in (207) the integrals cover each of the circuits twice
and hence would give twice the value of (204).
84. Mutual and Self-Energies of Two Circuits. Each inte-
gral being taken over both circuits, (208) may be broken up
into four parts,
w 11 C C df-dr" .LhC f dr'-dr"
2 J 2 J t r 2 JJ % r
The second and third parts are evidently equal, so that we
may write for their sum
where here, each integral is taken around ita ^<yrra^T^&fc3t»
circuit once.
176
VECTOR ANALYSIS.
If we call the integrals
we may write for the magnetic energy of the field due to both
currents
W m - i LJ> + M l2 IJ 2 + I LJ 2 \
(209)
The integrals L, or L 2 and M l2 are called the self-inductances
and mutual-inductance of the circuits respectively.
EXERCISES AND PROBLEMS.
1. If the line integral of the forces in any field around a closed
contour is zero for any such contour, the forces in the field form a
conservative system.
2. Show that the surface integral of a scalar point-function V
taken over any closed surface is equal to the volume integral of its
grad (VV) taken throughout the volume of that surface ; that is,
J Js J J J vol
Wdv.
8. Show that the line integral of a scalar point-function, V,
around a closed contour is equal to the surface integral of the
vector product of the normal by its gradient taken over any cap
to the contour; that is, prove
fvdr= f f n* W dS.
4. Using the divergence theorem, let q = r lf and prove that the
Dtential of a body may be represented by the surface integral
-us.— -\m
VECTOR ANALYSIS. 177
6. If Pois8on's equation holds,
i.e. 9 V 2 ^ = 4 x P - div P and if V P =
show that the potential of a body in the last example becomes
V = j^ff ri-nVV dS - ^j ' fivn div FdS,
so that if the force at every point of S be known it is possible to
compute the potential.
6. By drawing a small cylindrical box enclosing a portion dS
of a surface charged with a surface density of electricity a, and
making the cylindrical sides everywhere parallel to the lines of
force, show that there is a change in the normal component of the
flux of moment 4 no.
7. The curl of the curl of a solenoidal vector such that the three
functions which give the strengths of its components parallel to
1, j and k satisfy Laplace's Equation, vanishes.
8. If the lines of a vector, F, are all parallel to a plane and the
vector has the same value at all points in any line perpendicular
to the plane, the vector is perpendicular to its curl,
i.e. 9 F- V*F - 0.
9. Compare the results of the last problem with those of § 60.
Can you devise any other functions, the lines of which are every*
where perpendicular to its curl?
10. If the lines of a vector are circles parallel to the ij-plane with
centers on the k axis, and if the intensity of the vector is a function
f (r) of the distance from the k axis, a vector everywhere parallel to
the k axis, of intensity F(r), where f(r) — is a vector poten-
dr
tial-f unction of the original vector. Is the original vector solenoidal ?
CHAPTER VII.
APPLICATIONS TO DYNAMICS, MECHANICS, AND
HYDRODYNAMICS.
Equations of Motion of a Rigid Body.
85. Equations for Translation. D'Alembert's Principle,
upon which Lagrange founded the whole subject of analytical
mechanics, may be written
s(-f- F >*-».
(210)
where dr is any possible arbitrary or virtual displacement
compatible with the constraints imposed upon the system,
and where the 2 sums for all the particles.
In order to deduce the equations of motion of translation
assume the virtual displacement to be the same for all points
of the system, as this is the definition of pure translatory
motion. It then follows, since we may now take the dr from
under 2 sign, that
and since dr is arbitrary, that
(2X1)
which are the ordinary equations:
X("f?-r)-o.
2(-»£- z )-»
See papers by Ziwet and Field in American Mathematical Month-
ly, 1914, pp. 105-113 and by Rees same journal 1923, pp. 290-296 for
interesting vectorial treatments oi YuneniaAica «xAifia\tafeq& tv©d bodies.
YZfc
VECTOR ANALYSIS. 179
Motion of Center of Mass. Let r be the vector to the
center of mass or centroid of the system; then, by the defi-
nition of this point for which (20)
r 2)m - 5)m r,
we have by differentiation
-^Xm-^m-^
bo that finally equation (211) may be written
f?2— 2*. (212)
or, in words, the motion of translation of the centroid of a
system of bodies moves precisely as if all the forces of the
system were applied to the total mass concentrated at that
point. This reduces the problem of the translatory motion
of the system to that of the motion of a single point. An
interesting example of this property is seen in the case of the
motion of a shell which explodes while describing its path
in space. As the resultant of the actions and the reactions
which are produced when the shell explodes is zero, the path
of the center of mass of the fragments is the identical parabola
the center of mass of the shell would have described had
it not exploded. In other words, the path of the center
of mass remains unchanged by the explosion. The center of
mass of a thrown stick describes a smooth parabola, as it
whirls through the air.
The kinetic energy of translation of the body is evidently
given by
r =2-2 m (£H Mi? ' (2i3)
where q is the velocity of the center of mass of the system,
where M = 2 m is its total mass and because all ^QVbta><&^2o&
system have the same velocity.
180
VECTOR ANALYSIS.
86. Equations for Rotation. To deduce the equations of
motion for rotation, let 5co be an elementary rotation, then
dr 8 = 8n*r 8 ,
where dr 8 is an arbitrary possible infinitesimal motion due
to the rotation of any particle m a of the system about some
axis, co. Substituting this in d'Alembert's equation, we
obtain
2) m ^ -3<o<r = V F.&0*r,
and with obvious transformations
2 mdv-W^ = ]£ dco.r*F.
We shall now assume that the particles of the system
rotate about the same axis, so that da shall be the same for
all the particles and may be divided out, and remembering
that (34)
dv dv _ n
dt dt
we obtain ^mv*-^ = T ^mpT *» ^^P
*1 dP dt** dt *4
dP dt
dt
(214)
for the equation of motion of rotation of a system about an
axis. The motion of a rigid system is of course a special
case of this. This equation expands into the familiar Car-
tesian ones,
dt** V dt
(21S)
about the three rectangular axes, by the ordinary rules.
Defining 1W = Y> r*F <gAS\
VECTOR ANALYSIS. 181
as the moment of the applied farces about the axis of rota-
tion, co, and
H = 2 « **¥ < 217 >
^ at
as the moment of momentum about the same axis, the above
equation (214) may be written
^H = M, (218)
at
or even H = M,
or, in words, the rate of increase of angular momentum f
about the axis of rotation of a system is equal to the
moment of the impressed forces about that same axis.
Kinetic Energy of Rotation. Moment of Inertia.* The
kinetic energy of rotation of the system rotating with angular
velocity co is (44)
T = i 2m q 2 = i 2m (<o*r) 2 .
If the system moves as a rigid body all of the co's are the
same, so that
T = i co 2 2m («vr) 2 . (219)
But (to^r) 2 is the perpendicular squared from the point r to
the axis of rotation co (A P in Fig. 31), so that the expression
2m (co^r) 2
means that every elementary mass is to be multiplied by
the square of its distance from the axis of rotation and that
their sum is to be taken. This quantity is called the Mo-
ment of Inertia* of the system about the axis <0i; it evidently
varies with the direction of co. We may then define the
moment of inertia, J, about an axis co, by the equation
/„ = 2m (co^r) 2 = MkJ, (220)
where K, also defined by the above equation, is called the
Radius of Gyration about the axis co. The radius of ^gptfe-
Sion is, therefore, the distance irom ttifc &au& <A to\»&ksbl *fc*
* Or, better, Rotat\ona\Ma»»
f Same as Moment oi Momex&wia-
* > ■»»' =. ' m .-T-'.Si. —
182
VECTOR ANALYSIS.
which, if the total mass M of the system were placed, its
moment of inertia would remain unchanged.
The total kinetic energy of a rigid system moving in any
manner may then be written
r=iJlfq J + i/-« 3 . (221)
M, the mass of the body, is an absolute constant,* but /«., as
stated above, varies with the direction
of the axis about which the system
rotates, and hence the treatment of
rotation is essentially more complicated
than that of pure translation.
We shall treat of the motion of
rotation more in detail, not only for its
intrinsic interest, but also because it
introduces naturally the Linear Vector-
Function and some of its elementary
properties.
87. Linear Vector-Function. Instan-
taneous Axis. Consider a rigid body
of mass M rotating in any manner
about a fixed point. This precludes
any translatory motion of the body
which is now one of pure rotation at
any instant about some axis necessarily
passing through this fixed point. This
Fia. 64.
* In the Electron Theory of Matter the inertia of a particle, at least
in part, is accounted for by the electrical charge which we know the
particle carries. The resistance of an electrical charge to acceleration
is not constant but is a function of the velocity, and theoretically becomes
infinite when its velocity approaches that of light. The apparent inertia
of such a particle is therefore not constant. But for any velocities with
which we are likely to deal in mechanical systems these variations in
inertia are inappreciable. The ordinary equations of mechanics are then
tint approximations only, but for ordinary velocities, up to 10,000 km.
per aec. say, are extremely c\ose to Vtas \.tm\>&.
VECTOR ANALYSIS.
188
axis, which may vary continuously in direction, is called the
Instantaneous Axis of rotation. Let the angular velocity
about this axis at any instant be represented by a vector
of length a = f(t) in the direction of it and in the conven-
tional sense, i.e., that of the motion of progression and direc-
tion of rotation of a corkscrew.
As the velocity of any point r is
dv
4 dt
»*r,
we may write for the moment of momentum H
H = V mr*rr
** dt
==Vmr*(tt"r)=^m(«&r 3 -- r»*r);
(222)
thus H is a vector-function linear in co, $c» say.
This particular function $c» has a number of important
properties which are evident upon inspection. If t and o*
are any two vectors, the following equations hold:
<Kt±<t) = $t±$<t,
(a)
a = const.
<fr*T= (M^T,
(&)
d (♦ t) = 4> dr,
(c)
and
T»<p <r = <r»<p t.
(<*)
(223)
In particular when a linear vector-function has the prop-
erty represented by (223) (d) it is said to be Self-Conjugate.
Form the scalar product of H and a>.
' tt*H = tt4tt = ^tt*^(ttxr))
= 2 m («"!•)• («"!•) = 2m («&*r) a
184 VECTOR ANALYSIS.
Now, since $a is linear in co, the scalar product co«c^co is a
quadratic scalar-function of co, and hence represents when
equated to a constant, a quadric surface.
88. Motion under No Forces. Invariable Plane. Assuming
no applied forces,
5"-°'
so that H = const, vector,
or, in words, under no applied forces the moment of momen-
tum of a rigid system remains constant in magnitude and
direction. It remains perpendicular to the plane, called
Invariable Plane, whose equation is
r-H = const.
Also, since the energy (kinetic) of the system is conserved, it
follows that the moment of inertia /„ about any direction co
is inversely proportional to the square of the radius vector in
the quadric co-c£co = const., because
co*H = co.<t>o> = 2 T = co 2 /*,
so that
T 2 T const. /ooc\
/„= — = — ^- (225)
€0 CO
This equation also says that with a given amount of
energy the body rotates the faster the smaller /«, is; i.e.,
1
(OQC — —•
Polnsot Ellipsoid. Since evidently no finite body has an
infinite or a zero moment of inertia about any axis, the
quadric surface co«c^co = const, must be one the radius vector of
which has a finite minimum and maximum value; that is, it
must be an ellipsoid. This ellipsoid is called the Momenta!
or Poinsot Ellipsoid. Let us consider this surface more in
detail and incidentally diovj \\a e^M^cycLm^^X^'a^fcrai.
VECTOR ANALYSIS.
185
If H lt H 2 , H % are the components of H about the three axes
i, j, and k, then (222)
Hs<|K»=]gm(<»r 3 -r<».r)= H t l + HJ + HJk
= {Hmw^ + if + a?)— Smx^x+ajjiz+avO} i
+ {2ma) 2 (3? + y* + s?) — HmyiajiX+ajiy+tosZ)} j
+ {S?wa> 8 (a? + i/ 3 +2?) — , Lmz(a) l x+(i) 2 y +<*)&)} k
■=» {a^Sm^+s 2 )-- w 2 Jlmxy — cj s 31mxz}i
+ {-w^myx +w 2 l i m(j?+7*)-u) z , Lmyz}\ (226)
+ {—aj^mxz —a) 2 %myz +w 8 Sm(f + t/ , )}k.
Moments and Products of Inertia. Coordinates of a Self-
conjugate Linear Vector-Function. The scalar coefficients
which occur in the above expansion and are reprinted below,
assuming for definiteness A > B > C,
As 2m(i/ J + 2?),
Cs 2m(x 3 + 1/ 2 ),
D = 2m t/2,
B s 2m zx,
F= 31m xy,
(227)
are called the Moments of Inertia about the axes x, t/, and 2,
and Products of Inertia with respect to the planes of yz, zx y
and xy, respectively. The quantities A, B> and C are essen-
tially positive, but Z>, E, and F may be of either sign. They
may be found in any particular case by integration, for
instance, since
dm = pdx dy dz,
and
As f f f(y> + z>) pdxdydz
Body
D = I I I yz pdxdy dz,
Body
(228)
where pis the density at any point oi \tae\iodpj%
186 VECTOR ANALYSIS.
Knowing these six coefficients (A, B, C, D, E, F), the
function ty» is completely determined, and for this reason
they are sometimes called the coordinates of the self-conju-
gate linear vector-function.
Consider now the ellipsoid (224)
«•$« = 2 T = const.
<D-<t>o> = (ofA — to^WiF — w^E
- u^F + <*)* B- uyjj z D (229)
— (ojcdiE — <o 9 ti) 2 D+ <of C,
which may also be written
o).<t>o) = a)? A + oj 2 2 B + (ofC — 2 w^D — 2 w^E— 2w t a) 2 F.
It may be easily seen that in order that T»$<r = <r»^T, it is
necessary that the coefficients D, E, F should occur in pairs as
above. If however they do not occur in pairs, there will
be nine coefficients in (229) all different. In this case the
function is not said to be self-conjugate; but it is still a
linear vector-function. In this last case t.<Jkt is not equal
to <r»<|>T. If we write T.<^<r = c-^'t, <£' is said to be the con-
jugate of <£, and ^ the conjugate of <£'.
Principal Moments of Inertia. Principal Axes. The func-
tion co«<t>co, as is seen by its expansion, is homogeneous in o>,
and the ellipsoid it represents when equated to a positive con-
stant, 2 T, is referred to an origin at its center. We may
now refer the ellipsoid to its three principal axes, its equation
then becoming, as is well known,
Ha* + Ttw* + Uwf = const. (230)
The axes have lengths proportional to > , and — - •
Vj VI v?
The coefficients A, B, and C, in the above equation called
the principal moments of inertia, are the moments of inertia
of the body about these tYtree ptmdpal axes and in general
differ from the values they \i«A\>eXoTfc,\>\&XXtfS3 «^&£h«&;\bl
VECTOR ANALYSIS.
187
the same manner with respect to the new axes. The prod-
ucts of inertia have all vanished. There are thus three
directions in any rigid body for which the products of inertia
when referred to them vanish.
Referred to these axes, since
<».<t> <o = Aa>* 4- Ew* + Vwf,
$ co then becomes ^ co = Za/ji 4- Eoj j 4- Vwjt.
and H - 3^1 + Bi> 2 j 4- (& 3 k. (231)
So that the components of H are
H x = X(o lf
H v - E^
H t = ~C(o v
Looking upon <(> ( ) as an operator, we see from equation
(231 ) that when it is applied to any vector co
cd = w t l 4- to J 4- w s k,
thus $ co = Xcjjl 4- Ew J 4- Tttojt,
it multiplies the components of co by the quantities X, 2?, and
V, respectively. Applying $ again to $(co) we should obtain
from
<►«= Zi^i + Eoj J + Cwjk,
cjx|x0= <|> 2 co = "2 Vl + &*» J + ?V<> (232)
and so on. Defining <t> _1 as that operator which when applied
to ^ annuls its effect, <t> _1 must then evidently divide the com-
ponents of any vector by X, B, and C, respectively, so that
$-*$<o = <o = $- 1 { Xa^i 4- Eu> j + Ua) t k)
so that
= -I^i 4- - En j 4- - £«**
3 B £
= a^i 4- u> 2 \ + a^
^-ia> = ^il 4-^j 4-^k.
I B C
(233)
188
VECTOR ANALYSIS.
Applying $ -1 to this we obtain
£* B 8 C 1
and so on.
Lemma. We shall now show that ty» is perpendicular from
the origin to the tangent plane at co, and that its magnitude
is inversely proportional to the distance from the origin to
this tangent plane, or, in other words, that (<frco)~ l * is the
perpendicular vector from the origin to the tangent plane, at
cd of the quadric
C0.<p<O = 1.
Consider the quadric
cD'<t>co = const.
Differentiate this function, considering co as a variable
Gto«4>co + co-^do) = 2 <to*<^co = 0,
using c and d of equations (223).
\
Fia. 65.
Hence <£co is perpendicular to rfco. But dco is a small vector in
the surface at the extremity of co and therefore lies in the
tangent plane, so that <t>co is perpendicular to this plane. If
o* is the running coordinate of the plane its equation is (64)
cr-c^co= const.,
* Do not confound ♦~ 1 0, ttaa te&vptw&l operator, and [+( )]-*,
the reciprocal of $.
VECTOR ANALYSIS. 18%
and by (68a) the perpendicular vector from the origin to this
plane is
p . s^.«-t (♦.,-..[ (234)
If the constant is unity, p = (^w)" 1 .
Physically this means, referring to (231), that the angular
momentum H = <^a is normal to the tangent plane at co and
inversely proportional to the length of the perpendicular from
the origin on it.
Axes of a Central Quadric. The principal axes of a
central quadric may be defined as those directions for
which the magnitude of the radius vector is a maximum or
a minimum. That is. m2
y CD
is to be a max. or min. subject to the condition that
cD'<(>co = const.
Multiplying the first by an arbitrary multiplier and add-
ing, the condition is obtained by writing the derivative of
cd«(<|><d— >}») = const.
to zero; this is
dco«(<t>co — An) -h co«(c^cto— xdeo) =■ 0.
This becomes, using (223) (c), and (d),
dco-(<t>co — to) =
and since this must be true, independently as to how co
varies, i.e., it is to be a true maximum or minimum, it
follows that ^ = fa
is the condition required.
This is already an interesting result, for it states (234)
that the radius-vector being parallel to e)co, is therefore
for those directions, perpendicular to the surface. We might
have started with this condition as a definition cA ^t«\k^^
190
VECTOR ANALYSIS.
This last equation is by (226) and (227) equivalent to
(A - X)wi - F oh - E coz = 0,
-FtDi + (B-^coi - Da>> = 0, (a)
-Bwi - Do>i + (C- >0w, = 0.
The condition that these equations shall be compatible
for values of io u oh, a)$, other than zero, is that the deter-
minant of the coefficients shall vanish, i.e.,
A — \ —F —E
— F B-\ —D =0. (b)
—E —D C—\
This is a cubic in X, and may be shown to always have
three real roots. Each of these three values for X inserted
into equations (a) will allow for their solution, obtaining
from each of them values for w lf oj 2 and w 9 and hence a
cd = w l i 4- <o 2 j + w 3 k
direction for each X.
There are then always at least three principal axes to a
central quadric surface.
The Principal Axes Intersect Normally. Let X lt X 3 , X 3 be
the roots of the determinantal cubic (b), and tti, <0i, <o 8 the
corresponding axes.
Then ^<o l = ^ 1 <o 1
and <t>co 3 = yJjCOj.
Multiplying the first by ev, and the second by tt t « and
substracting, there results, using (223) (d),
(J,- X 2 ) (<o 1 .co 2 ) = 0,
which means that if X t is not equal to ^, then the two
principal axes <o x and co 3 are perpendicular to each other;
similarly if k* X z and >t 8 ^ X x
we could show that the three principal axes are mutually
perpendicular to each other.
If two roots of the cubic are equal, the position of the
^or/responding axes becomes indeterminate, and it may be
shown that all radii perpen&veuW X.o \tafe &\*fe^\I\^^raL by
VECTOR ANALYSIS. 191
the third root are principal axes of the same length. The
surface is then one of revolution about the determinate
axis. If all three roots are equal, the surface is a sphere,
and any axis is a principal axis. It may also be shown
that the three roots of the cubic are equal to the squares of
the reciprocals of the lengths of the semi-axes, the Cartesian
equation then being
Xi/uj? + ^a; 2 2 + V>3 2 = const.
Comparing with (230) we see that the roots of the deter-
minantal cubic are proportional to the principal moments
of inertia, i = i = As .
1 B C '
89. Geometrical Representation of the Motion. Invariable
Plane. If no impressed forces act upon the rotating body
the equation of motion (218) becomes
the solution of which is
H = 4><o = const, vector, (235)
hence H or <jxo is a vector constant in magnitude and direction
throughout the motion, so that the tangent plane of the
ellipsoid to which it is always perpendicular must remain
fixed in space and is for this reason called the Invariable
Plane. The point where this plane is touched by the ellip-
soid is on the extremity of the instantaneous axis or pole, so
that the ellipsoid is always rolling without sliding on this
plane. In other words, having constructed the ellipsoid of
inertia, and having determined the position of the invariable
tangent plane in space, the motion of the body is the same as
if it were rigidly attached to this ellipsoid which is rolling
without sliding on the invariable plane.
This geometrical condition, in addition to the fact that
the angular velocity of rotation is proportional to the radius
vector to the point of contact of tVve d\VpwA *xA ^sbr^
completely determines the motion.
EjLjE^M^ jdfcyM^-ai ji^a -^*^. J a
192
VECTOR ANALYSIS.
It is easy to see, since the radius vector always passes
through a fixed point, the origin, that it must describe a cone
in space, the vector H being its axis fixed in space. For
this reason, H, is called the Invariable Line.
It also describes a cone in the ellipsoid. The vector H
describes a cone in the ellipsoid because the ellipsoid moves
relatively to it. This description of the motion is due to
Poinsot.
r=0o>
V
v&*
T*5*V
^ct*>
v&a&e
Herpolhode
/ /vvy 1-***" — j
n ~^^/
uone .
Poinsot
\^ y^Polhod*
BUipeoid
^/^Polhode Gone
Fia. 66.
00. Polhode and Herpolhode Curves. If the paths de-
scribed by the point of contact of the ellipsoid and inva-
riable plane be determined on them, for instance by placing
carbon paper between them as they roll on each other, two
curves are obtained: one on the invariable plane, called the
Herpolhode (sinuous path),* and one on the surface of the
ellipsoid, called the Polhode (path of the pole). These curves
are the directing curves of the cones described by the radius
vector co in space and in the ellipsoid, called respectively
Herpolhode Cone and Polhode Cone.
Permanent Axes. It is easy to see that in three cases H and
<o coincide in direction, i.e., when co is perpendicular to the
tangent plane; in this case when both H and co coincide in
* The above name is a misconception, because as a matter of fact the
Herpolhode can be proved to have rtopoVa\»<A *m^\A^*sAW>K*>\&
n ot "sinuous."
VECTOR ANALYSIS.
193
direction along some one of the three principal axes of the
momental ellipsoid these curves reduce to points and the
ellipsoid rotates without rolling, permanently about these
axes. There are then at least three directions at every point
in a body about which if the body be set rotating it will
continue to do so forever. Further consideration shows that
two of these permanent axes are stable and one unstable, this
last being the mean axis. The most stable is the least axis.
Equations of Polhode and Herpolhode Curves. The inter-
section of the cones described in the ellipsoid by the
instantaneous axis with its surface will determine the pol-
hode curves. In the quadric
C0*<|xa = const.
a must always satisfy the condition that the distance, p, to
the tangent plane is constant, or that
&"-
const.,
or (<faO* = -£ *= +«•$» = «o-$(<^«) = co«4> 3 <0 (a quadric),
V
where we define $*( ) = ♦(+( )) etc., see (232),
so that the two equations
<o«<t>co = const. = k
and
(236)
must be simultaneously satisfied. Combining them, we
obtain from
1 k
— <o«<t>o>= -r and
by subtraction
or finally
k&&<o tt«<ba = 0,
p
<3Kl\
194 VECTOR ANALYSIS.
a homogeneous equation of the second degree, and hence a
cone with vertex at the origin. Its equation in Cartesian
coordinates may be immediately written down by (231) and
(232) as
^(r* _*) + ^.(**_!) + m » (? k - |) =0,
or (238)
The intersections of this cone for different values of p with the
ellipsoid tt*$a = k,
give the polhode curves, which are, therefore, twisted curves
of the fourth degree, lying on the momental or Poinsot
ellipsoid.
Since the herpolhode is traced out by the points of contact
of an ellipsoid rotating on its center with an invariable tan-
gent plane, these curves must lie between two concentric
circles on the plane, their centers being at the intersection of
the invariable line H with that plane, and touching them
alternately.
Moving Axes and Relative Motion.
91. Theorem of Coriolis. It is often convenient in dynam-
ics to use axes which themselves move in space and to which
the motions of the body under consideration are referred.
In order to determine at any time the position of the mov-
ing axes, one method is to refer them to axes which remain at
rest throughout the motion. According to this device the
fixed axes are left behind by the moving ones. However, it
is found to be more advantageous to refer the moving axes at
all times to fixed axes instantaneously coinciding with them.
No generality is lost by referring the motion of a body to
moving axes which simply turn about a fixed point in space,
as any motion of translation of the moving axes with refer-
ence to fixed ones may be coiNpe&&%te& tot \*j ^etf*a%\a
VECTOR ANALYSIS.
195
every point of the body considered a motion equal and oppo-
site to that of these moving axes. This condition, then, does
not limit the generality of the choice of moving axes.
Consider any vector OP = r drawn from a fixed origin
and for definiteness let it be the vector to a point P in a
moving body. We shall now consider the motion of the
R
Fig. 67.
point P in two ways. Refer it to two different spaces ini-
tially coincident, one revolving about the axis 01 with an
angular velocity w, the other remaining at rest (or fixed).
Let PR be the motion seen in a time it by an observer
remaining in the fixed space, or, in other words, the absolute
motion in space. Denote this vector PR as dr/„ the sub-
script denoting its reference to fixed space. Consider now
the point P as remaining at rest with reference to the mov-
ing space; it will therefore move relatively to fixed space
with the velocity of the moving space alone and will describe
the path
PQ = (Adt *r = a*r dt,
as exit is the angle described by N P \n \mxx& dlL
196
VECTOR ANALYSIS.
But as the particle P by its own motion actually reaches
the point R, the vector QTt must represent the path of the
particle as seen by an observer moving with the moving space;
in other words, QR = dr^, the subscript denoting its refer-
ence to moving space. From the figure
dr^=dr ffW + <o«rdt.
Dividing through by dt we obtain the very important equa-
tion
\dt) f9 KdtU
(239)
Letting OP—r represent any directed quantity such as
force, velocity, moment of a couple, or angular momentum,
etc., equation (239) shows how to refer them to a moving
space.
The vector r always represents the vector at the beginning
of the motion and referred to either space, as initially they are
both coincident. If r represent a displacement and q the
velocity of the point P,
"HfL
+ »*r.
The acceleration of a body whose motion is known rela-
tively to moving space, and the motion of moving space
known relatively to fixed space, may be obtained by a second
application of this equation.
Replace r by q/ a thus,
•'-(^L + -"-
©L + S* M » + -®L + -*->
+ 2
ma
\dt/m, at
VECTOR ANALYSIS. 197
This equation will readily expand to the familiar ones
referred to Cartesian axes, as ordinarily given, for example:
( "'- , -$i +2 ( < "'f-"-s) + (^-^)-'<"' + ^
+(o l (o i z+a) l w 2 y. (241)
The last three terms may be written out as above by
remembering that
a>x(a>*r) = a(».r) — r «**.
If the point P is attached to the moving space,
(£)__- 0,.»d ^2 ..(f)_-<>,
so that the remaining expressions are the accelerations pro-
duced by the motion of the moving space itself.
Hence the so-called
Acceleration of moving space = -— - x r + a>* (a*r) . (242)
at
If the angular velocity of moving space is constant,
^?=0 and therefore ^?xr = 0.
dt dt
The remaining term »*(<tt*r) is the acceleration produced on
the body by its individual rotation about the axis 01.
Since <D«r is perpendicular to o>, a>*(a>*r) is perpendicular both
to a>*r and to co and directed normally towards the axis a.
This is the ordinary Centripetal Acceleration. Because in the
vector product »*r, tTP instead of r may be used (see § 36)
without changing its value, and as co and WP are at right
angles,
a).]?? = o,
this centripetal acceleration may then be written
«*(<D*r) = — «?FTP.
198 VECTOR ANALYSIS.
The term 2<d*[ — ) is called the Compound Centripetal
Kdt/nu
Acceleration of Coriolis.
We may then consider the moving axes to be at rest if to
the actual forces applied to the body fictitious ones bemadded
capable of producing accelerations equal and opposite to the
acceleration of moving space and to the compound centripetal
acceleration. This is the theorem of Coriolis.
92. Transformation of the Equation of Motion. Centri-
fugal Couple. Let us utilize equation (239) to refer the
motion of a rigid body to a space moving with it, or, in other
words, to axes in the body. The equation of motion of a
rigid body about a point referred to fixed space is (218)
dt
If there are no impressed forces, M = and
dt
states that H, the moment of momentum, remains constant
in magnitude and direction in the fixed space, however
peculiar the motion of the body may seem to be.
Employing now the equation of Coriolis, we substitute for
— - ) its equivalent for moving space, obtaining as the equa-
te //a
tion of motion of a rigid body about a fixed point referred to
a space moving with the body
d ^\ + »*H = M. (243)
(
ma
If there are no applied forces, M = 0, and (243) becomes
H*(o is called the Centriiuga\ Couple and, as is evident by
u s form, is perpendicular to bo\\\ » at\AW. "YV^ &*ss* *ngufe
e
VECTOR ANALYSIS. 199
tion then states that the rate of change of the angular momen-
tum H in the body is equal to the centrifugal couple, Hx«. If
the change in H is always normal to itself^ then H is never
increased or decreased in length but only changes in direc-
tion at a rate proportional to H*o>. Thus H describes a cone
in the body, although it remains fixed in fixed space.
If H and <o ever become parallel then H*co vanishes, and in
the body too we have
dt/ms
or, in other words, the body must continue to rotate forever
about the Invariable Line H in fixed space; it is then an
invariable line in the body also. We have seen that there
are at least three such directions, called permanent axes, or
principal axes, for which the above condition is fulfilled. A
symmetrical body supported at its center of mass and rotat-
ing about its axis of symmetry will give this kind of motion.
Gyroscope. The property that a rotating body possesses
of rotating permanently about a principal axis was utilized
by Foucault in the gyroscope. When a symmetrical top is
rapidly spinning in gimbals, it keeps its axis pointing in the
same direction (invariable line) in space, so that if the top is
carried around by the earth's motion, the axis of the top
remaining fixed in fixed space will describe a cone with refer-
ence to the earth (or moving space). By observations on
such an instrument not only can the rotation of the earth
be proved but the latitude of the locality at which the experi-
ment is performed may be determined.
93. Euler's Equations. If co l} (o 2y u) s be the three compo-
nents of <o along the principal axes of a rigid body at a point,
and if U, B, U be the principal moments of inertia about
those same axes, we may write (231)
H = Xaj t l f Eaj 2 i -f (7a>Jc
200
VECTOR ANALYSIS.
There are no products of inertia entering into this equation,
as for the principal axes they vanish (J 89).
Substituting this value for H in (243) there result the
three equations
at
(246)
These are the dynamical equations of Euler for the motion of
a rigid body about a fixed point, referred to axes moving with
the body. Of course
dt
+ »*H = M
is the corresponding vector equation.
94. Analytical Solution of Euler's Equations for Motion
under No Impressed Forces. For convenience we rewrite
the following:
o> = wj + oj 2 ] + <oJk. (a)
H = Xa^i + E(o 2 i + Cwjk = 4><o. (6)
Then
Also
w
<o.
at.
k.
!♦< >-♦!<>•
Euler's equation for this case is
d\\ u d<bo> i
dt to.
(c)
(d)
(•)
(24d)
VECTOR ANALYSIS.
201
Since the right-hand side is perpendicular to <J>«> and to o>,
let us take the scalar products of the equation with <J>« and
with <o respectively. Multiplying with »• we obtain
tt.-2_ = a>«<J>o>*a> = 0.
dt T
which we must integrate.
By differentiating a>«4>a> = const, with respect to the time thus,
dt* dt^ '
which becomes by (e) and (d)
we see that the integral of the equation is
co.<t>co =2T, (247)
where T is an arbitrary constant. Those having read the
preceding pages will recognize in (247) the equation of the
Poinsot Ellipsoid, and in T the kinetic jenergy of the body.
Multiplying the original equation with <J>a>« we h&ve imme-
diately
<!>»•-*— = <J>«>«4>tt*a> = 0,
and by (e) and (d)
i <*«*>)' - 0,
whose integral is
«*>)' - H', (248)
where H 2 is an arbitrary constant. We recognize here the
constancy of H in magnitude only in the body, hence any
change in H must be perpendicular to it. See § 92.
In order to obtain a third integral, multiply (246) by
^- 4 «&, the <|>~ l and the ^ annulling each other.
202
VECTOR ANALYSIS.
From these three equations (247), (248), (249) it is possible to
find co for all time and hence all about the motion. Suppose
it is desired to expand these three integrals into Cartesian
form; we have immediately for (247) and (248)
and
C0.<j>co= 2 T = ~Xw l * + Bo) 2 2 + V(of
(cjxo) 2 - w = n V + 5V + £V-
(250)
(251)
The third is more complicated, but easy,
dm
dt T T
\Z B* V J
I J k
Xio x Eto 3 Cw %
(O l 0) 2 w t
2 B C
(252)
By solving equations (250) and (251) combined with
CO 2 = 0>i 2 + C02 2 + «3 2 ,
for cui, o>2 and co 8 , and substituting in (252), we find
*Tt = V(Xl " w2) (X * " w2) (Xa " ^
where the X's are functions of -A, 25, C, T and i/, an equa-
tion for co in terms of these constants whose general solution
involves elliptic functions.*
05. Hamilton's Principle. Starting again with d'Alem-
bert's equation,
2(™§-f}*-0, (263)
the £r's being any variations consistent with the constraints
imposed upon the system, or, what is the same thing, satisfy
* See article by Professor Greenhill, in fourteenth volume of the
Quarterly Journal, pp. 182 and 2ft5, YSlfc.
VECTOR ANALYSIS.
203
certain equations of condition, we may transform it in the
following manner:
dP di\dt ) dt dt
di\dt ) 2\dt)
Treat each term of (253) in this way; d'Alembert's equation
then becomes
= dT + 2 F.*r,
where T is the kinetic energy due to the velocities of the
masses of the system. As the first term is an exact deriva-
tive, let us integrate with respect to the time from t = ^ to
s -(^)r-X'"( ,r+2F - ,r ) ,fl -
If the positions of the system are given at the times t x and i,,
then the dr's are zero for those times, and the left-hand terms
vanish, leaving
£ 2 (dT + %F.dr\dt-0.
(254)
This equation is true whatever the system of forces is that
acts; if, however, the system is a conservative one, the work
done (by definition) in going from any point to any other
point against these forces is independent of the path chosen,
and is therefore a function solely of the initial and final points
of the path. In this case, then, there must be a scalar point-
function, ( — W say), such that knowing its value everywhere
we can calculate the work done in going from any point to
any other by any path, simply by knowing thfeNfc\N\*&<& — ^ ^
far those points. (Consult § 59.) In ott&x ^OTd&> >&& ^«tf*>
204 VECTOR ANALYSIS.
2?F«(Jr in going from position 1 to position 2 is the differ-
ence in value of the function — W at positions 1 and 2, or
2jF.<Jr = -(W x - W 2 ) = - dW.
In the case of conservative forces, then, 2 Fȣr may be
replaced by — dW and (254) becomes
d f t2 (T-W)dt = 0. (255)
This is Hamilton's Principle.
Lagrangian Function. The function T — W is called the
Lagrangian Function, and is often written L, so that Hamil-
ton's integral becomes
^L dt = 0. (256)
x
96. Extension of the Conception of Vector to More than
Three Dimensions. Certain processes occur in mathematical
physics in which more than three independent variables are
concerned. In such cases as this the vector notation is still
applicable to the manipulation of these quantities. If
q if q 2 , q z . . . be these independent quantities, we conceive of
a vector q,
where l lt i 2 , i 3 . . . are independent unit vectors. By an
extension of the idea of a vector, we are to consider q as a
vector existing in more than three dimensions, as many as
there are q's. An example will show that we are not going
very far beyond the manipulation of ordinary vectors.
Definitions. Consider the generalized vector in n-dimen-
sions,
q= Qih + 92*2 + ' • ' +?nin. (257)
In analogy to V E5 a" i + d"^ + 5~ k
mite ^^l-^...**^ C258)
VECTOR ANALYSIS.
205
If we write q/ for -j** , we may have also
q' - <?/>! + 42% + • • • + 3n'in
and define also V n '
(259)
(260)
Let us apply this notation to the following transformation.
97. Lagrange's General Equations of Motion. By means
of Hamilton's Integral we may deduce Lagrange's Equations.
Let the position r, of any point s of the dynamical (or analo-
gous) system be expressible in terms of the independent
parameters, q l9 q 2 , . . . q n , so that
r «=+«(?i?ift- --?n).
(261)
This means that to every point
q = gA + q 2 \ 2 + • • • + g»i»
in a space of n-dimensions there corresponds a definite value
for all of the parameters of the dynamical system and hence
a definite and determinable configuration of every particle
in the system. If it is possible to find what functions of
the time the q 8 'a are, subject to the dynamical equations
of condition, included in Hamilton's principle, the position
of every particle for all instants will be known. Hence the
simple motion of one point in n-dimensions includes the
problem of the motion of a system of points in three or
less dimensions (or for that matter in more than three).
Differentiating (261),
+■' = t£ = !** *' +#■«»'+•• — *•*■+- < 262 >
at dq t dq 2
Thus every (/>' is expressible as a linear tatt&OTL oil "Oafc ^ ^
1. Ki a*, h
206
VECTOR ANALYSIS.
The kinetic energy function T becomes
Hs ro (£Hs w( *- v ^ (263)
every term of the sum being linear and homogeneous in <f.
The square is a homogeneous quadratic function of the q n s;
often written
T - * Q u q\ 2 + * Q»<? V + • • • + Qtrfi V + • • • (264)
Performing the variation with respect to q' and 9 indicated
by Hamilton's integral (255), and using L = T — W, we have
since
<JL= <?q-V n L + <Jq'.V n 'L,
(<Jq-V n L + <Jq'.V n 'L) <tt = 0.
r
But because
the second term becomes
IM">)^h
which may be integrated by parts, with respect to the time,
into
5q.V n 'L^ 2 -jr 2 dq.|v n 'L,
of which the first term vanishes, as q is fixed at the limits and
hence suffers no variation there, leaving altogether
But as dq is arbitrary, it follows that
at
065)
VECTOR ANALYSIS.
207
This is a vector in n-dimensional space, whose components
must all be zero, for example:
d( d(T-W) \ d(T-W) _ Q
dt\ <*q x ' ) dq t
dt\ dq 2 ' J dq 2
(266)
These are Lagrange's Generalized Equations of Motion.
If then for any system the functions T and W are known,
and it is possible to express them in terms of n-independent
parameters q l9 q 2 , . . . q n , these n-equations (266) make it
possible to determine the values of these parameters for all
time; thus the path of the point
is determined , in n-dimensional space.
Defining the operator T7 ( )>
V< >"(5 v -'- v -) ( '•
V-
all dynamics is included under the Remarkable Formula
L=0. (267)
Hydrodynamics.
08. Fundamental Equations. We shall now derive the
fundamental equations of hydrodynamics for a frictionless
fluid and some of their most important consequences by
means of the previous principles. The directness of attack
and absence of artificiality is especially noticeable in this
application of the vector method.
Let p represent the density of the fluid; we shall assume
that it is a function of the pressure p alone, so that
P-/(P)- (268)
Equation of Continuity. Let q (uv w) tatY&Nt&Nstos <&*S&a
Buid and F(X YZ) be the force per um\» mas» w*aoa%<sa^»
208
VECTOR ANALYSIS.
fluid. Consider a fixed surface S in the fluid. By fixed is
meant that the imagined surface retains its position in space
irrespective of the motion of the fluid itself. The rate of
increase of matter in it is measured by the surface integral
of the flux of the fluid though the surface taken along the
inward drawn normal. No fluid is supposed to be created
nor destroyed inside of the surface. As we use generally the
outward drawn normal n in our formulae, — n will represent
the inward drawn normal, so that we may then write using
the divergence theorem
= ¥tj J Js P V ~J J Jsdt
because m
As this relation holds whatever surface is taken we may
equate the integrands to each other,
dt
+ v.^oq = 0.
(269)
In Cartesian this is
*
This equation is called the equation of continuity. It states
that matter is neither created nor destroyed at any point in
the fluid.
It is convenient here to employ a special notation to be
used when we follow the fluid in its motion as distinct from
considering the fluid as it passes by a fixed region in space.
For example, the rate change of density of a definite portion of
the Quid as it is followed m \t»a mot\ow, symbolized by the
VECTOR ANALYSIS.
209
special notation j~ , is equal to the rate of change of its den-
sity -£ observed as it goes by a fixed point in space plus the
Of
rate of change q- V/o, due to its velocity q, so that
Dt Bt 4 p
(270)
In fact whatever point-function is placed in the parenthesis
— ( )-—() + q-v ( )
This corresponds in the Cartesian notation to
Dt Bt ox By dz
dt dx dt By dt dz dt
We may write the equation of continuity in a slightly differ-
ent form by (128) and using (270),
Bj>
= *£
+ V.(/>q) = ^ + />V-q + q-V^t?
at at
so that
Dj>
Dt
+ pV-q = 0.
(271)
In either form, if the fluid is incompressible, p does not vary
either with time or with position, and hence
V-q = = div q, (272)
which shows that q is then a solenoidal vector and its stream
lines form closed curves or end at infinity, just as a solenoidal
distribution of electrical force acts. In fact the two theories
of Hydrodynamics of incompressible fluids and of Electricity
are identical.
Consider now a small surface always containing the, ocltni
fluid of volume v. This surface may \te &stast\*& *»> ^
moves with the liquid, but it \s sup^ *^ ^° ^ ^^^^
210 VECTOR ANALYSIS.
made up of the same small portions of the fluid with which
it started. For such a surface, evidently the mass is
constant, or
Dt Dt y Dt Dt
Dv Dg
ao that — = - — = V-q by (271). (273)
V p
We may then interpret V*q as the fractional decrease of
density per unit of time, or as the rate of increase of volume
per unit volume, or as the time rate of dilatation, a divergence.
Equation (272) follows also from (273) if p is constant.
Eider's Equations of Motion of a Hold. Consider the
forces acting upon a definite mass of the fluid enclosed in
the surface S.
Let F per unit mass or p F per unit volume be the external
force function, and let p be the pressure function acting
normally over the enclosing surface and along the inwardly
drawn normal. By Newton's law the rate of increase of
momentum (Xpqdv) of the fluid is equal to the applied
forces F acting directly on the mass of the fluid and to the
forces (XpdS) resulting from the pressures acting on the
surrounding surface, or
or fffw ( > * dv > -fff<> F - v *> *■*
D , j x Dq , , D(j>dv)
and the last term vanishes as the mass remains constant
throughout the motion, so that the integral becomes
///©')* ~fff<> F -™*- < 274)
* See note to § 52, p. 252, tot tr&TteloTTna.tkrti o( laat term by diver-
gence theorem.
VECTOR ANALYSIS. 211
As this is true for any volume whatever, the integrands
are equal and we have, using (270),
P S? - '|J + pq ' Vq " P F - V P- (275)
This is Euler's equation of motion which, in connection
with (268), (269) and (270), forms the basis of theoretical
hydrodynamics.
09. Transformation of the Equation of Motion. If we
divide (275) by p and employ the identity (129)
q-Vtf! = V l (q l .q) - q^V^qO,
the subscripts indicating precisely on what the V acts, or
q.Vq ns £ Vq 2 — q*( v *q)>
we may transform this equation into
g - q*(V*q) - F - 3? _ J V q*. (276)
If the externally applied forces have a potential, V, for
instance forces due to gravity or any other conservative
system of forces, then p — __ ^y
If the pressure p at any point depends only upon the
density p, we may define a quantity P such that
p J p
ao that our equation becomes
$9 - q* curl q = -vf V + P+ \ q>) (277)
dt \ 2 /
= VC7,
where U =-(V+P + iq a ).
Referring back to equation (126) where it was shown that
curl q = 2 a>,
where a> is the vorticity or angular velocity of rotation of the
fluid at the point considered, (277) becomes
dt
-2q*» =- vfv -V P •v'^V
212
VECTOR ANALYSIS.
100. Steady Motion. Definition. Steady motion is one in
which F, q, p, and p are independent of the time. If such is
the case and cd = 0, that is, if the motion is non-vortical,
_Vt/ = V(V+P+ *q 2 ) = 0,
or integrating V + P + £ q 2 = const.
If p is constant,
and if there are no applied forces,
F=0
and hence V = const.,
so that
2 + SL — const.
(278?
In other words, where the pressure is great the velocity
must be small, and where the velocity is great the pressure is
small. For example, in a constricted pipe the pressure is
least at the constriction where the velocity of the incom-
pressible fluid necessarily is the greatest. Air pumps and
water meters are constructed on this principle.
101. Vortex Motion. Theorem of Helmholtz. Take the
curl, or apply V* to
g + 2*,— v(v+P + rf).
giving
curl ^ + 2 V* (a*q) = 0, as curl grad s
dt
■a
or - (curl q) + 2 («V«q + q«Va — qV-a — «*Vq) = 0.
dt
.Remembering that
curIq = 2co; that V'<&= V?^A$^ = ^
VECTOR ANALYSIS.
213
as co is a solenoidal vector, i.e., cd = £ curl q, and that (270)
we have
Dm
Dt
dm . „ -_ Dm
dt M Dt
+ »V.q-»»Vq = 0.
This transforms into
£©-?**
(279)
because identically, using (273)
^ Dt
Dm
D /«>
'»(?)
(280)
Dt H Dt p Dt
Hence, also, differentiating (279) again
B> /m\ ID m\ ^,«o D-^
so that if m ever vanishes (279) and (280) likewise vanish,
and similarly all the successive derivatives may be shown to
vanish. Hence if a is ever zero it will always remain zero by
Taylor's theorem, because all of its derivatives vanish at a
certain instant. This theorem due to Helmholtz says that if
no vorticity exists in any incompressible, frictionless fluid at
any time it is impossible to produce any by means of a con-
servative system of forces, and the motion will remain forever
non-vortical.
* If the ether be considered to be a frictionless medium, then a
vortex once set up in it would be indestructible; and conversely, if no
vortices existed, it would be impossible to create any. It is conceivable,
however, that some " Cataclysm " might have rendered the ether tempo-
rarily viscous to some extent. By this we mean that it is conceivable,
for example, that under extraordinary conditions say of temperature the
ether might acquire unusual properties, in which case, if it became
frictionless again after vortical motion had been produced in it while in
this state, such vortical motion would persist totevex. TH»& v^c^aftoss^
is of interest in connection with the vortex-atom ttaoY? o\tga****~
214
VECTOR ANALYSIS.
102. Circulation. The circulation along any path in a
fluid is defined as the line-integral of the velocity along that
path. If <f>AB denote the circulation along the path AB, by
definition
4>ab = J7 q ' dr " (281)
If the path is a closed one, we may express the circulation
around it as a surface integral over any cap bounded by it,
by means of Stokes' Theorem, for
fq*dr = f f n.V*qdS = 2 f f n-adS,
where
2a>=V*q by (126).
Roughly speaking this equation says, if there is a pre-
ponderance of motion of a liquid in one direction or the other
Fig. 68.
around any closed path, that the liquid inside of the closed
path must be rotating.
Consider a tube made up ol \tafc Y\x\fe& oi the vector », and
insider a portion oi it WmfoA Vj Vwo <ra^ ^v *a^ ^>*
VECTOR ANALYSIS. 216
Apply the divergence theorem to this closed surface 8 U S t
and sides. Remembering that <o is solenoidal, we have
Ab the sides contribute nothing to the surface integral
there must be as much flux of ca inward at S, as there is out-
ward at 8 t , or the flux is constant throughout the tube. If
this tube be chosen very small it is called a vortex filament,
and if the section of such a filament be denoted by s the
above result expresses the fact that
s n-to = const., (282)
where it is the normal to the cross-section. This product is
called the strength of the filament. It shows that if at h
finite, or, in other words, if there does exist any vorticity, a
cannot vanish, hence a filament cannot end anywhere in the
fluid. Such filaments must then either form closed curves or
end in the surface of the liquid or at infinity. All vortices,
then, form closed curves in the fluid or else end in the surface.
This also follows from the fact that V-«a = Q,Vo»XS&,*\*>v
adenoidal vector.
216
VECTOR ANALYSIS.
103. Velocity-Potential If cd is zero everywhere, the cir-
culation around any closed curve is zero, hence the circula-
tion from any point A to any other point B is independent
of the path. In this case q*dr is a perfect differential; that
is, it is of the form
dr*q = cUf> = dr* V^,
so that q = V<£. (283)
The velocity q is thus derivable from the function <f> in the
same way (except for sign), that the force is derivable from
the ordinary potential. Accordingly <f> is called the Velocity-
Potential, and is a scalar point-function of the space occupied
by the fluid. All the results of the theory of potential are
therefore directly applicable to the function <j>.
Production of a Vortex Impossible In a Frlcttonless Fluid*
Let us find the time-rate of variation of the circulation
along any path, assuming a velocity-potential to exist. This
path is made up of certain elements of the fluid which are
to be followed in their motion, however distorted the path
may become. Differentiating (281),
D± = D_
Dt DtJ A M Ja Dt Ja Dt
Since the velocity-potential exists,
curl q = 2 a = 0,
and the equation of motion (275) becomes, if W = (
£3 =
Dt
VIP,
V + P)
(284)
so that
also
Dt
q '§i dT= q ' d §? = q " dq = d ^
and hence ^ = f* d ( W A- ^= ^W -V CI* •
Dt
IV
(285)
VECTOR ANALYSIS. 217
If the path is a closed one,
MI-*
as W and *j are scalar point-functions of position and have
identical values at the limits, so that finally
f=0. .286)
Equation (286) then states that the circulation around any
closed curve, formed of a chain of particles of the fluid, cannot
change as these particles are carried about by the liquid. As
we have assumed the circulation to be zero at the beginning
it remains so forever, or, in other words, it is impossible to
create vorticity in frictionless fluid by means of a conserva-
tive system of forces. Also, as it is impossible to conceive
how any system of forces could act on a frictionless fluid
in a non-conservative manner, it follows that it is impos-
sible to create vorticity in any manner in a frictionless
medium.
It was from these peculiar properties of vortices in a
frictionless fluid, discovered by Helmholtz, that Lord Kelvin
was \ed to his Vortex Atom Theory of Matter.
PROBLEMS AND EXERCISES
1. Show that the center of gravity of a system of particles, and
hence of any body, continues to move uniformly in a straight line
when no impressed forces act upon the system.
Find the equation of the path.
2. Show that the total momentum of a system of particles, and
hence of any body, remains constant as lon& && W&ra «xfe to* w^*$ta&.
forces.
218 VECTOR ANALYSIS.
3. Prove that a system of forces acting along and represented
by the sides of a plane polygon taken in order is equivalent to a
couple whose moment is represented by twice the area of the poly-
gon. Extend this to forces acting along a closed plane curve.
4. By means of the theorem (202)
/qxrfr- f f [nV.q - V(q.n)] dS
«/ vcap
o
show that if forces equal in magnitude act everywhere along the
tangents to a plane contour, that the moment of these forces about
any point is measured by twice the area of the contour.
5. If a rigid body has a velocity of translation q< and an
angular velocity of rotation CD, the velocity q at any instant of
a point r in the body may be represented by
q - q< 4- co*r. see § 22.
Show that if q< and CD are constants the path of any point in the
body is a circular helix described with uniform velocity, and find
its equation.
6. Show that two equal rotations in opposite directions about
two parallel axes produce a motion perpendicular to the plane of
the two axes.
7. The motion of a point in a plane being given, refer it to
(a) fixed rectangular vectors in the plane ;
(b) rectangular vectors in the plane, revolving uniformly about a
fixed point.
Translate into Cartesian in both cases.
8. Prove that the central axis of two forces F l and P 3 intersects
the shortest distance between their lines of action and divides it in
the ratio ^ ^ + ^ ^ ^ . ^ ( ^ + ^ ^ ^
6 being the angle between their directions. Also prove that the
moment of the principal couple is
cF y F 2 sin
• •
y/Ff + F 2 2 + 2 F X F 2 cos e
9. Show that
f fn*FdS= f f f VxFdv.
J Js J J Jvol
What conclusion in HydTodynam\ca tofe* ^eaa VV^rasvNsaA^t
VECTOR ANALYSIS.
219
10. Express the following equation in vector notation :
//[{*-*! -«♦
du
dz
Bv
dx
dw
dx
du
cos (nt/)
+ 1-z -r— I cob (nz) \dS
dy
]
An*. J f?7*q).ndS.
11. Express the following equations in vector notation :
du , 0/ „v BU 1 dp
& w * n/ £ \ &U 1 3p
-x- + 2(vf - ui?) - — T^ .
O* 02 p OZ
Let q have components u, v and w, and CD have components
7), f and f.
oq
jlfW.
+ 2«*q - Vt/ Vp.
0» (0
13. Express in vector notation the following equations which
occur in the theory of Elasticity:
&u
a<r
/d 2 i0 d 2 tz; d'uA
u + /t) ^ + "te + V + a?j*
where u, v, w are the components of a vector q, where a is a
scalar variable, and where p, X, and p are constants.
j4na.
^^-U + ^Va + jutV 2 ^
220
VECTOR ANALYSIS.
18* Express the following equations in vector notation:
_vdo__ / afw aft* a^\
3dx "[da* By* Bz* )
Bu . Bu , Bu , Bu
at Bx By dz
p ox
Bv , Bv , Bv , Bv v Bo /B 2 v , B 2 v , B*v
-r- + u- 1- v-z-+ w— — — v(-r-; + -3r: + T^
3* Bx By Bz 3 By \Bx* By* Bz*
)
Bw , Bw , 5m; , Bw v Bo . — ,
Bt Bx By Bz 3 Bz
\Bx* By* Bz*)
p Bz
where u, v, and w are components of the vector q, where X 9 Y,
and Z are components of the vector F, where p and a are scalar
variables, and where v is a scalar constant.
Ana. ^ + q*Vq - £ V<j - vV*q - P - - Vp.
o< 3 f
APPENDIX.
NOTATION AND FORMULA.
The Various Notations.
Whenever new quantities are introduced it is well to have
as simple and as convenient a notation as possible. The
notation devised by the late Professor Willard Gibbs seemed
to us, after much thought on the matter, to be the simplest
and most symmetrical of any of the existing kinds.
Hamilton, the inventor of quaternions, used the letters
S and V for the scalar and vector products respectively of
the vectors that followed them; thus
Sab and Vab
represented respectively the scalar and vector products of
the vectors a and b.
The letter T, standing for tensor, represented the magni-
tude of the vector following it. This notation has many
advantages, but after deliberation it was discarded.
Oliver Heaviside, the English electrician, used a notation
similar to Hamilton's, but rendered it unsymmetrical by
discarding the S for the scalar product while retaining the
V for a vector product. This seemed to us to be a step
backward, although he was followed in its use by Foppl
and Bucherer in Germany and by others.
The disciples of Grassmann, who had devised a notation
of his own, adapted it to the analysis of vectors, and at the
present time the resulting notation has a number of adhe-
rents in Germany and elsewhere. Our main ob^eciva^^Sk
221
222 VECTOR ANALYSIS.
is that it uses different kinds of parentheses to distinguish
the two products, thus preventing the use of these paren-
theses for other purposes. It is also quite cumbersome and
takes much longer to write than any of the other systems,
besides there being a liability of error due to the fact that all
parentheses necessarily look somewhat alike.
Gibbs, on the other hand, puts the distinguishing product
mark between the two vectors instead of in front or around
them.
This is essentially a symmetrical notation, and to our
mind and to many others the best. The two symbols used
to indicate the " variety " of product are the dot (•) and
the cross (*). In order to avoid any confusion with the
ordinary dot and cross used for ordinary products, and a
necessity in any analysis, we have ventured to use a special
dot and a special cross. That is, the dot is above the
writing line and the cross is a small one and when used
is placed in the same position as the dot. Thus
a«b and a*b
are the scalar and vector products of the vectors a and b
respectively.
They are easy to write, easily distinguished and con-
nected with the idea of a product. They do not interfere
with parentheses, neither do they render the use of an ordi-
nary dot (.) or a cross (X) undesirable nor ambiguous in
other parts of the work. They are symmetrically placed.
Comparison of Notations.
A Few Examples of Formulae in the four systems of
notation will render the foregoing clear to the student. We
shall give Hamilton's notation the benefit of our bold-faced
type and avoid the wholesale use of Greek letters which
~*ere employed by turn to xepte^tvX» Ne^tow*
VECTOR ANALYSIS.
ThB formulte are in the order:
1. Gibbs' Notation.
2. Hamilton's Notation.
3. Heaviside's Notation.
4. Gans' Notation. (Grassmannian.)
1. aorao.
2. T&.
2.
3.
4.
a-b = b-a
Sab-Sba
ab - ba
(ab) - (ba)
= ab cos (ab).
Tarbcos(ab).
= ab cos (ab).
= killol cos (ab).
1.
2.
a-b — b-a
Vab - -Vba
— « ah sin (ab).
= t Ta Tb sin (ab).
3. Vab Vba - t ab sin (ab).
4. [at] [ba] - c lalftl sin (ab).
1. a-(b + c) - a-b + a-c. a-b-c - b-o-a.
2. Sa(b + c) - Sab + Sac. SaVbc - SbVca.
3. a(b + c) = ab + ac. aVbc = bVca.
4. («, 6 + e) - <ao) + (ac). (afbc]) - (Hca)).
1. a-(b"C) =ba-c — ca-b.
2. VaVbc - cSab - bSac.
3. VaVbc-bac -cab.
4. [o[bc]] - b(ac) -c(ab).
. „ r-b-c „ , r-c-a . , r-a-b _
1 - r "[ibcl* + iibcT'' + [Scl C -
2. r SrVbc a I SrVca b SrVab
* ' SaVbc SaVbc SaVbc C '
, rVbc rVca b rVab
" aVbc aVbc aVbc '
4 r (»l), , M»l) t Mai.]),.
(a[te]) r (a[taj) T (a(6c])
These examples are sufficient to show the characteristic*
)£ the various notations.
224 VECTOR ANALYSIS.
Notation of this Book.
It seems to be the consensus of opinion that vectors are
best represented by single letters printed in some sort of
bold-faced type. If this is not done, in order to distinguish
a vector from a scalar we are obliged to employ Greek or
other special alphabets and thus deprive ourselves of the
convenience of using that alphabet if desired, and also we
are prevented from using any other letter as a vector.
The Magnitude of a vector is represented by the same
letter as the vector itself but in ordinary or italic type.
A. Unit Vector parallel to any vector is represented by
that vector with the subscript unity. Thus the vector
a
has a magnitude a
and a direction a u
so that we may write a = aa t .
Sometimes the subscript zero to a vector, or particularly
to a vector-expression, will mean that its magnitude alone is
expressed. Thus
(a*b)
denotes the magnitude of a*b.
In order to connect the Analysis of Vectors with Carte-
sian Analysis it is necessary to relate the vector a to its
three components along the three Cartesian axes. We shall
denote these components by adding the subscripts 1, 2, and
3 to the italic letter a.
As mutually perpendicular axes are by far the most im-
portant of any, three unit vectors
i, j, and k
have been universally adopted to represent their directions
in space.
VECTOR ANALYSIS. 225
The vector a then is made up of
a vector along i, of length a v
a vector along j, of length a,,
and a vector along k, of length a^ t
so that
a = aa t = a t i + aj + a s k.
There can be no confusion between the letters a t and a\ 9
for obvious reasons.
A Voluntary Exception to this convention is in the case of
the radius vector r to any point from the origin. Its compo-
nents will be denoted by x, y, and z instead of r v r 2 , and r s in
order to approach more closely to the usual Cartesian char-
acter of the work when translated into that notation. For
this reason we write
r - x i + y j + z k,
and for similar reasons when desirable,
F= Xi +Yj +Zk 9
q = u\ + vj+tok,
a> = £ i + i?j+C k -
The Unit Tangent along and the Unit Normal to a curve
or surface are denoted by the unit vectors t and n. Since
the components of a unit vector are its direction cosines in
t = y + y + tjk
and n = nj + nj + n 8 k,
Jj, $2, and Jj and n lf n?, and r^ are the direction cosines of the
tangent and normal respectively.
The Scalar or Dot Product of two vectors is represented
by placing a dot between them thus:
a-b = b*a = ab cos (ab),
where cos (ab) is the notation used for the cosine of the
angle included by the positive directions oi fc. wAXix
-nfr«W* r " '"•'■■
226 VECTOR ANALYSIS.
The Vector or Cross Product of two vectors is represented
by placing a special cross between them thus:
a*b = — b*a = tab sin (ab).
The unit vector € is a vector perpendicular to the two
vectors in the product and is taken in such a " sense " that
as you turn a into b, c points in the direction a cork-screw
would advance if so rotated.
Then (a^b^ = €
and (a*b) = ab sin (ab).
A Scalar Point-Function is represented by writing the
functional symbol in italic. Thus
V = /(r)
means that V is a scalar point-function of the radius vec-
tor r. That is, for every value of r, V has a determinate
magnitude. This is equivalent to one Cartesian equation.
A Vector Potnt-Functton is represented by writing the
functional symbol in bold-faced type. Thus
F = f (r)
means that F is a vector point-function of the radius vec-
tor r. That is, for every value of r, F has a determinate
magnitude and direction. This is equivalent to three Car-
tesian equations.
A Linear Vector Function, in particular, is represented by
the special symbols
co, + or x-
Velocity and Angular Velocity are represented by the sym-
bols respectively
q and co.
The Scalar Potential Function is represented by the italics
VECTOR ANALYSIS. 227
The Vector Potential is represented by the bold-faced
V.
Electric or Magnetic Intensity or Forces in General are
represented by
F or H.
The Differential Vector Operator V (read del) is equivalent
to
V()a(lf +jA +k A)().
\ dx ay dzf
Besides obeying the laws obeyed by ordinary vectors, it
is a differentiating operator, and the same care should be
taken in its use and interpretation as should be taken with
other differentiating operators.
As Professor Joly* puts it:
" Of course some little care is necessary when V is ex-
pressed in the general form, but it is precisely of the same
kind as the care required to distinguish between
V dx/ \ dx/\ dx) dx\dxj dx\dx J
\dx/ dx \dx/
Del sometimes differentiates partially. Generally a sub-
script attached to it, indicates the variable which it differ*
entiates. Thus tt « k
V a a-D
means that in the scalar product the vector a above is to
be considered variable.
Sometimes the same process may be indicated by writing
as a subscript to the expression the quantity which is to
remain constant during the differentiation. Thus
V(a-b)& is the same as V a (a4>).
* Charles Jasper Joly, Manual of Quaternions, Ma&mttSrack. &. ^a^
1905, Art. 57, p. 75,
3^=
228 VECTOR ANALYSIS.
Which notation is preferable will depend on whether all
but one of the vectors are to be considered variable or
whether all but one are constant, respectively.
Sometimes also the notation
V'(a'-a + a'-b)
is useful, the variables having the same accent as the V,
being alone considered variable during the differentiation.
Gradient (grad), or vector of greatest slope of a scalar
function V, is a vector normal to the level surfaces of the
function V and is given by the operation of V upon the
function thus:
VV = gradient or slope of V = grad V.
The Divergence of a vector function F is given by form-
ing the scalar product of V with the function thus
V-F = divergence of F = div F.
The Curl of a vector function F is obtained by forming
the vector product of V with the function:
V*F = curl F = rotation of F (rot F).
The more important formulae of vector analysis are col-
lected below for reference.
VECTOR ANALYSIS.
229
FORMUUC*
Vectors.
a = oa, = a a l
r = xi +yj +zk.
F= Xi + Y\ +Zk.
= ui +vj + wk.
<o = wj + w 2 j + w s k
= £ i +9 J + C k -
2 a = i 2^ + j 2a 2 + k 2a,.
(1)
(4)
a
- = r t = i cos a + j cos/? + k cos p,
(7)
(2)
(9)
tfhere a, /?, p are the direction angles of the radius vector r.
The equation of a straight line through the terminus of
b and parallel to a is, s being a scalar variable,
r = b + 8 a.
If it passes through the origin,
r = sa.
A line through the ends of a and b
r = sa + (1 —s)b
or r = s b + (1 — s) a.
The condition that three vectors, a, b, and c, should end
in the same straight line is
x a + yb +zc = 0.
x + y +2=0.
* The numbering corresponds V\t\\ \h«A» ol \Jwb \»tS*
(ii)
(10)
(12)
(13)
280
VECTOR ANALYSIS.
The equation of a plane determined by the vectors a
and b, and passing through the terminus of c, is
r = c +sa +tb. (15)
The plane through the origin and parallel to a and b is
r = sa+*b. (14)
The equation of a plane passing through the three points
a, b, and c is
r = sa+tb + (l-s-*)c. (16)
The condition that four vectors, a, b, c, and d, should end
in the same plane is
xa+yb+zc + wd = 0.
x+y+z+w = 0.
(17)
The vector
r =
ma + nb
(18)
m + n
divides the line joining the points a and b in the ratio of
mton.
The vector to the center of gravity of the masses m, at
points a, is
- Ewt a y __ v
r = ^- <«>
If the relation
m 1 a l + m^ + . . . = (25)
is to be independent of the origin chosen, then
Wi + m^ + ...--= 0.
Vector and Scalar Products.
Products of Two Vectors.
a-b = ab cos (ab) = b*a
= a l b i + a 2 b 2 + ajb r
a 2 = a 2 = a, 2 + a 2 3 + V-
a l -b l = cos (,abV
(ax) 2 = a t .a Y - \.
(26)
(30)
VECTOR ANALYSIS.
231
If a.b = 0, then a -Lb. (27)
(a + b)-(c + d) = a-c + a-d + b*c + b*d. (28)
No attention need be paid to the order of the factors.
a»b = € ab sin (ab) = — b*a (33)
- i(<*J>s - <*A) + J (<*A - "A) + k{a t b 2 + ajb t ) (39)
i J k
<*i <h <h ' (40)
6 t b 2 b 9
c is a unit vector, perpendicular to the plane of a and b
and pointing in such a sense that as a is turned towards b
a cork-screw would advance along c.
(a l *b l ) = sin (ab).
a*a = 0. (34)
If a*b=0, then b is parallel to a.
12 = j2 = k 2 _ L j.j = j. k = k>i « o. (29)
M = jxj « k*k - 0. i*j = k, j«k = i, kxi = j. (35)
If a' is the component of a normal to b, then
a*b = a'*b. (36)
(a + b)*(c + d) = a*c + a*d + b*c + b*c. (38)
3reat attention must be paid to the order of the factors.
Products of Three Vectors.
The scalar
«i
a 2 a*
a.(b*c) = (a*b)-c =
6i
b 2 \
Cl
c 2 c s
- [abc] *
(49)
is equal to the volume of the parallelopiped of which a, b,
and c are the three determining edges.
* [abc] represents atf arrangements of the Inptefcc^M^ro^^V^*
having the same cyclical order of factors.
232 VECTOR ANALYSIS.
The vector a*(b*c) = b a-c — c a*b. (55)
Any vector r may be represented in terms of three others
by the formula
r [abc] = [rbc] a + [rca] b + [rab] c. (61)
The plane normal to a and passing through the terminus
of b is
a .(r-b)=0. (64)
The perpendicular from the origin to this plane is
p=a~ l a.b. (68a)
The plane parallel to c and d through the end of b is
(od).(r-b) =0. (70)
The plane through the three points, a, b, and c is
(r - a). (a - b)*(b - c) - f (67)
Dr $-(r — a) =0, where $ = (a*b + b*c + c*a).
The perpendicular from the origin to this plane is
p = $-i 4>. a . (686)
The line through the end of b and parallel to a is
ax(r - b) = 0. (69)
The equation of the sphere (or circle) of radius a with
senter at the origin is
r 2 - a a . (71)
If the origin is at the point c, it becomes
r 2 - 2 r-c = (a 2 - c 2 ) = const. (72)
If the origin lies on the circumference,
r 2 - 2 r-a = 0. (73)
VECTOR ANALYSIS.
233
DIFFERENTIATION OF VECTORS.
dl n dt n dt n dt n
d^.-d^
(77)
d d n
If p = — and p n = — , this may be written
r dt y dt n ' J
p n a = i p n a t + j p n a 2 + k p n a„
p(a-b) = p a»b + a-p b.
(78)
(79)
No attention need be paid to the order of the factors in
a scalar product.
p(a*b) = pa*b + a*p b.
(80)
Great attention must be paid to the order of the factors
in a vector product.
p [a*b*c] = p a*b*c + a-p b*c + a-b*p c,
p [a*(b*c)] = p a*(b*c) + a*(p b*c) + a*(b*p c).
(81)
The Operator V (del).
V = il + jl + k A = de l,
dx dy dz
dr = i dx + j dy + k dz,
dr-V = -^dx + —dy+—dz = d( ),
dx dy dz
VF = i ^ + j ^ + k ^ = grad V (a vector),
dx dy dz
< v ">'-f'
Vr = r„
r r*
Vr" = «r" _1 Vr = nr B-1 r, = nf* - ^,
(102)
(114)
(106)
(109)
b «■■■■
284
VECTOR ANALYSIS.
V(a.b) = V a (a-b) + V*(a-b) = V(a.b) 6 +V(a-b)
(8,-V)y
at
«™-%-Z+*%+*Z
(»i-V) F
(W)F
' V(VF),
^VVXi^+j^+kF,),
a-(b«V/?F) = b-V/?(a»F),
V-F =
dF. , dK , dF
dx
dy
dz
= div F (a scalar).
The Divergence Theorem,
f Ai.qdS = f f f V.q dv,
t/ t/S t/ t/ t/vol.S
(110)
(112)
(113)
(116)
(117)
(119)
(121)
where q is any vector function and n is the externally-
drawn unit normal to dS.
V«F =
-(
i J k
AAA
dx dy dz
Fi F 2 F>
dF 9 dF,
= curl F
dz / *\ dz dx I \dx dy J
dy dz
V*(ca*r) = 2 ca, if o> = const, vector.
(126)
dF,
dy
(126)
General Differentiating Formulae for del.
V (u + v)
V-(u + v)
V*(u + v)
Vu + Vv,
v«u + v-v,
V*U + V*V.
V (ut>) = vVu + wVv,
V-(uv) = Vtt«v + mV«v,
(127)
(128)
VECTOR ANALYSIS.
285
V(u*v) = u-Vv + u*(V*v) + v-Vu + v*(V*u). (129)
V.(u*v)= v«V*u — u*V*v, (130)
V*(ll*V) = u(V w ^v) — v(V w .u)
= uV-v + v-Vu — vV-u — u«Vv. (131)
q(r + dv) - q(r) + V fl (dr-q) + (V*q)xdr. (135)
V(a*r) = a, a = const, vector.
(co.V r )r=<D, (118)
V-r = 3, (123)
V.r,= ^.
r
r r 3
_1 a*r a*r,
r r 8 r 3
K ' r + ^— , a and b const, vectors,
r r 3
V»f (r) = r (r) + ?i^- •
T
Taylor's Theorem.
/(r + €) - e"*/(r). (141)
Stokes' Theorem.
fF-dr - f f n.(V*F)dS. (136)
«/ «/ «scap
O
V«W = V 2 V = div grad V = del square 7 (a scalar)
(147)
b-V
(..vl).
d*v , d'V , a»7
dx 2 dy 2 dz*
> V 2 is called the Laplacian operator.
V*F = V 2 (iF\+ jF 2 +kF,) = iV 2 ^ jV^+kVF.
a 2 F , a 2 F , d 2 F
dx 2
V
236 VECTOR ANALYSIS.
V*W = curl grad 7 = 0,
V-V*F = div curl F = 0,
Vx(VxF) = curl 2 F = V(V-F) - F(V-V) (145)
= grad div F — V'F,
VV m = m(m + l)r*»" a , (148)
V^-O.
r
Gauss' Integral.
f f™dS = 47r or = (150)
according as the origin is taken inside or outside of the
closed surface S.
Laplace's Equation.
V7 = 0. (157)
Poisson's Equation. According to system of units chosen,
it is
V 2 F = -4^ or v 2 7 = -/>. (156)
Its solution is
If V satisfies the equations
V7 =
and r-W = nV,
it is a spherical harmonic of degree n.
Green's Theorems.
f f n-£/WdS= f f f UWdv + f f f (yU-W)dv,
J Ja J J Jfo\ J J Jwoi
(158)
f f n-(UVV-WU)dS= f f f (UW-VV*U)dv.
VECTOR ANALYSIS.
237
Green's Formula.
'--hfff*?*
Region
+ s//"'0- Fv i:h
Surfaces
Pot V =
1
%J •/ «/oo
Vdv
(161)
(168)
— — pot ( ) is the inverse operator to V 2 ( ). (169)
4;r
pot curl ( ) = curl pot ( ), for a vector function.
V pot ( ) = pot V ( ), for a scalar function.
Theorem of Helmholtz.
- — Vpot(V-W) + -i-V*pot(V*W).
4 7r 4 n
(178)
fq*dr - f f [(V.q)n - V(q-n)]dS (201)
O
q = any vector function.
Maxwell's Electro-magnetic Equations for Media at Best.
SF = € F,
^=V*H
dt
dt
- *!? = VxF
3C=^H.
(192)
Linear Vector Function.
If <|> is a linear vector function,
4>(T± <r) = <K±4kt,
4> a t = a 4>t ;
f is said to be the conjugate iwic&QTfc. to ^<
(223)
238
VECTOR ANALYSIS.
If
If
4>' = 4>,
then 4> is said to be self-conjugate*
co = cjj + to j + <oJk.
4>co = Aa^i + £w j + Cwjc, (232)
<j>(<|>co) = <t> 2 co = A^J + B 2 o)J + C"w,k,
(t> n « = A n u) t i + B n wJ + C n a)Jk.
$- l ($a) = co, (233)
Corlolis' Equation. A vector a in fixed space, when
referred to a moving space which has an angular velocity
of rotation co, satisfies the following relation :
and
then
and
If
then
<Stl- (fl + •*
(239)
D'Alembert's Equation.
2(mf-F).ar-0.
(201)
Euler's Equation of Motion of a Rigid Body about a
Fixed Point. j«
=P + »«H - M. (243)
at
Hamilton's Principle.
S f'\T -
W) it - 0.
(256)
Lagrange's Equations of Motion.
at
(266)
or
VL = 0,
(267)
vhere
H^ ; -A
VECTOR ANALYSI&
289
Hydrodynamics.
Equation of Continuity.
d£
dt
+ V.0>q)-O,
or
Dt
+ pV-q = 0.
(269)
(271)
Enter's Equation of Motion of a Fluid.
Circulation along the path AB.
d> = I q*dr.
T AB JA
- Vp. (275)
ADDITIONS TO APPENDIX.
Note to § 4.
^'
P'
Note on Different Varieties of Vectors. — Consider a parti-
cle on the movable platform P. The particle is initially
at A. If the particle remains at rest on the platform while
the platform is displaced uniformly to a new position P' t
the particle will describe the path A A' relatively to the ground.
This motion can be conveniently described by the vector
A A' or more concisely by the single symbol a.
If the platform remains at rest and the particle moves
uniformly to B, the path thus discribed relatively to the
ground is AB } or more shortly b.
Evidently, if the particle moves uniformly from A to B
while the platform moves uniformly from P to P', the
particle will finally end up at B\ and will have described the
path AB uniformly with respect to the ground, and in the
same time. This path is defined to be the sum of the dis-
placements (or vectors) a and b and is written a + b.
(In this case also these displacements may take place
consecutively in either order and the final position of the
particle will be the same.)
Now It Is a fact that forces, velocities and many other
physical quantities obey this same law. Hence they will
obey the consequences of a calculus which follows from the
above, and other definitions cot^XfcT^^m^^^V^RX^.
<2A&
APPENDIX. 241
There is nothing else but convenience which obliges us to
define the sum of two vectors in the above fashion. We
could have defined the sum in other ways, and by other
non-contradictory definitions obtained a consistent analysis.
We choose, of course, the definitions which seem to us most
natural and best suited to our needs.
Free Vectors, Slide or Axial Vectors. — Quite often certain
restrictions must be placed upon our vectors. We may
restrict them to a plane in which they may slide about,
or to remain normal to some plane, or restrict them to
slide back and forth in the lines of which they are segments,
or even attach them to a fixed point allowing no motion
whatsoever.
For example: It is well known that couples may be
represented by vectors perpendicular to their plane and that
the effect of any couple is the same as long as its represen-
tative vector remains parallel to itself, however otherwise
displaced. This is the freest kind of a vector and may be
called a Free Vector.
On the other hand, forces produce the same effect pro-
vided only that they are not displaced out of their line of
action. This kind of a vector with restricted freedom is
called a Slide Vector, or Axial Vector.
Nevertheless, if we disregard for the time the known effect
of displacing a force out of its line of action, i.e., changing its
moment about any given axis, we may consider forces as
free vectors. For instance, in statics, one of the conditions
of equilibrium is that the resultant of all the forces, con-
sidered acting at a common origin, shall vanish. Here we
consider the forces as free vectors for the time being. The
restriction to their line of action is intrinsically contained in
the remaining rule for the vanishing of the resultant moment.
Simultaneous angular velocities compound vectorially,
while finite rotations do not. Thus a rotation of 6 about
an axis followed by a rotation o! 4> &\>ou\» wutfObBt «&&>»»
242 VECTOR ANALYSIS.
not, in general, equal to those rotations taken in reverse
order. And yet a finite rotation has an axis, which is a
direction and an amount or magnitude and is, in that sense, a
vector, but it does not obey the laws of our vector analysis.
However if these rotations were to take place simultane-
ously the resultant rotation would be correctly found by
vector addition as defined above.
It is thus seen that we are endeavoring to deduce a calculus
which coincides as nearly as possible with the fundamental
properties of the majority of quantities to which we apply it.
Having thus constructed a consistent analysis coinciding
as closely as may be with the facts, it can of course be taught
abstractly without reference to them. Later on when phy-
sical quantities are shown to obey the same laws as vectors
have been defined to have, we may employ the results of the
vector calculus to them without further ado.
Personally the writer does not approve of the teaching of
Vector Analysis as an abstract science, nor even as a mathe-
matical subject unless by a teacher who is thoroughly
familiar with the physical results to which it applies and for
which it was designed.
The vector analysis as deduced in this book is that of free
vectors.
Note to § 38.
Normal, Normal Plane, Principal Normal, Binomial,
Rectifying Plane.
Every line perpendicular to the tangent to a curve at the
point of tangency, M } is called a Normal. These normals
lie in a plane called the Normal Plane and from them two are
singled out for special mention: the normal which lies in the
Osculating Plane called the Principal Normal, and the normal
perpendicular to this plane called the Bi normal.
The plane passing through the point of tangency perpen-
dicular to the principal normal \a ^^iX^dL \3cl^ ^^^^jva^Plaxie*
APPENDIX.
243
Thus the tangent, the principal normal and the binormal
form a rectangular system of vectors. Taking the point
of contact as origin, the directions of these vectors may
always be taken so as to form a right-handed system. The
principal normal, however, is always chosen so as to point
towards the center of curvature of the curve.
Let R be the magnitude of the radius of curvature and let
t, v, p be unit vectors along the tangent, principal normal
and binormal respectively. Then
dr .
t = — = r
da
(a)
The vector curvature c ( = p- 1 where p = vector radius of
curvature) is, by definition, the rate of change of the unit
tangent per unit arc or
dr , ,, ,
c = 3- = t' = r" = p- 1 .
ds r
And as its direction is along the principal normal we may
write it, where R is the magnitude of the radius of curvature,
t' « c - ^ = r".
(6)
Hence
R
v - Rt".
As p is a unit vector -L both to t and v, it is
p= T xv = Br'«r". (c)
The direction cosines of these lines are the coefficients of
i, j, k in the following equations.
t = r* = x'i + y'\ + z%
v = ftr" = Rx"\ + Ry"\ + Re"k,
=t«v = fi
= fl
i J k
x y z
~" A. ff ~ n
x y z
y z
y z
\
i + R
i z ,, x ll \ i " VR V^V
— - " i ■--— - T-
244 VECTOR ANALYSIS.
Torsion or Tortuosity is defined on page 78 as
(Here P is the n of page 78.)
The reciprocal of the torsion is a vector called the radius of
torsion S. Hence
T-ST 1 .
Formulae of Frenet for a Space Curve.
In the investigation of the properties of space curves
certain formulae due to Frenet are of fundamental importance.
They enable us to express the derivatives of the unit vectors
along the tangent, principal normal and binormal in terms
of these vectors themselves.
Differentiate (a) giving
t' - r",
which by (6) becomes
<-i-
Differentiate (c) giving
p' =T / *V+TxV' |
which by I becomes
VxV
p'=lJ:+T*v' =TxV / .
tc
From this equation we see that P' is -L to t; and since P is
a unit vector P' is -L to P; it is therefore parallel to v and we
may write
r = J ii
where S is a scalar whose absolute value, as v is a unit vector,
is the radius of torsion, by definition, equation (d).
Again, since v is at right angles to P and t, we may write
APPENDIX,
which equation on differentiating becomes
Using II and I we obtain
VxT BxV
S "*" R
S R
245
m
Equations I, II, III are Frenet's equations; they express
the first derivatives of t, v and P in terms of themselves and
the scalars R and S.
Formula for the Torsion. — From the preceding we can
easily obtain a simple formula for the torsion of a curve.
From II and (c) we have
P' - J - (t«v)' = (flr'xr") /
= fl'r'xr" + «r"*r" + flr'xr'".
The second term is identically zero. By multiplying by
equation (6), v= Rr" we have
P'.v = ^ = i = #r".[#'r'xr" + Rf**"\.
Hence, as the first triple scalar product vanishes
I = T = - #' [r'r"H,
IV
Exercise: Express equations I, II, III, IV in Cartesian
coordinates. Show that for a plane curve the torsion
vanishes, and conversely.
Path Described by an Electron in a Uniform Magnetic Field.
As an interesting application of the vector method con-
sider the motion of an electron in a uniform magnetic field
of strength H. It is well known by the experiments of
^Rowland that a moving electrical charge is equivalent to **
current. Let e be the value of the c\i&x^$, m ^oa Taas* **■
-/—--■ .«..■ - -■ ■I-.V--
246 VECTOR ANALYSIS.
the electron, and v its vector velocity. The force acting on
a linear conductor of length dx is
F = tdrxH.
Let this current i be due to the convection of electricity
carried by the n electrons contained in the element dr, mov-
ing with the common velocity v, then
i dr = ne v.
Hence the force on a single electron is
F = = ev*H.
n
This force produces an acceleration y on the electron, so
that the equation of motion is
my = evxH.
Putting h = — H, the differential equation of the path is
v = v*h. (1)
From the equation we see that the acceleration is normal to
the path, hence the speed is constant. As the acceleration
is also normal to h the velocity component parallel to h is
constant, and hence the whole acceleration is always parallel
to the plane normal to h.
As h is a constant vector, we have from (1)
h.v = h-vxh = = ^(h-v), (2)
so that the angle the path makes with the field is constant.
From (1) again, since v and h and their included angle are
constant, the magnitude of the acceleration is constant.
The radius of curvature of any path is related to the speed
and the normal acceleration by equation (92), page 81,
V s
APPENDIX. 247
Hence
And since the magnitudes of v and v * h are constant so is
the magnitude of p.
por= AriEW (4)
The component of the speed normal to h is
V\ = t>sin(vh).
The radius of curvature of the path in the plane normal to
h is, similarly to (3),
_/ Vi 2 \ _ v* sin 2 (vh) _ v sin (vh) ( .
91 "" V v t K h A " v sin (vh) A sin (v x h) A w
as sin (Vih) = 1, and hence the path is a circle of radius pi.
Thus the motion is completely determined. It is a curve
of constant curvature, described with constant speed whose
projection on a plane normal to the magnetic lines is a circle
of radius p\. The velocity parallel to the field is constant.
It is therefore a right circular helix whose axis is parallel to
the lines of force.
Comparing equations (4) and (5) we see that the radius
of curvature of a curve and that of its projection on a plane
are related by the formula,
Pl =psin 2 (0), (6)
where 6 is the angle between the curve and the normal to
the plane. This holds for every curve, because any small
portion of it may be considered to be a portion of some helix.
This result is due to Euler.
A circular helix is sometimes defined as a curve having (a)
constant curvature and (6) constant torsion. We can also
prove that the above path is a helix by proving (J)). Eopa*-
tion (4) shows that the curvature is constant.
248
VECTOR ANALYSIS.
The torsion T is by IV of Frenet's formulae
T - p* [r' r" r'"] (7)
where the primes denote differentiation with respect to the
arc 8.
Now
and since v is constant
r
T ds v'
and
~ r W ~ »»
V
T'" = 1. .
fl 8
Substituting in (7)
U/ L» *VJ f 2 (v*h)«
By differentiating (1)
v = vxh = (vxh)xh.
Hence
I [Vx(Vxh)H(Vxh)xh]
t* (Vxh) 2 '
which reduced by (58) becomes
v*h
As the two parts of the fraction are separately constant the
torsion is constant.
The result (4) obtained above, otherwise written
7-v = Hp sin (vH),
is of great importance in obtaining a relation between ( — ) and
v for an electron, which in combination with other relations
enables us to determine Wievx septate n^\k»»
APPENDIX. 249
Note to § 58.
Two Proofs of Stokes 9 Theorem.
1° The line integral around the bounding curve, Fig. 54, is
equal to the sum of the line integrals around the elementary
parallelograms into which the surface may be considered
divided. For every side of these parallelograms is integrated
twice and in opposite directions, and the results cancel,
except the sides which coincide with the bounding curve.
Consider an elementary parallelogram ABCD; let AB = dpi
and AD = dp 2 . Let (dF)i be the increment of F along
dpi and (dF) a that along dp 2 . Then
(dF)i = dprVF,
(dF) 2 = dp 2 -VF.
The vector point-function F has at the point A the value F;
at B the value F + (dF)i; at D the value F + (dF) 2 .
The line integral around the parallelogram is then by
definition
F-dpi - [F + (dF) 2 ].dpx - F.dp2 + [F + (dF)!].dp 2
= (dF) r dp 2 - (dF) r df lf
= (dpi-VjOF-dpz- (dprVfOF-dpi,
= (dpi*dp 2 ).(VxF), (by 58),
= (n.VxF)dS.
Because dpi*dp 2 = ndS where n is the unit normal to this
elementary area dS.
The theorem follows immediately.
2° Again by means of the Divergence Theorem which is
the fundamental formula for the transformation of volume
into surface integrals and vice versa, we can deduce Stokes'
Theorem which is the fundamental formula for the trans-
formation of surface into line integrals.
Apply the formula
/TTvf d»= C (W as, v^
250 VECTOR ANALYSIS.
(which is the Divergence Theorem and is to be taken through-
out the volume of, and over the surface of, any closed space)
to an infinitesimal right cylinder, of height h and base whose
area is S. Let ds be an element of the contour of the base,
and Cj a unit vector _L to the base.
Suppose the cylinder so small that F may be considered con-
stant throughout it, so that the two surface integrals over
the two plane ends cancel each other and their sum vanishes.
Replace F in (1) by c^F, then
f f j (CiF) dv = f A 1 * 15 ' 11 *•
By (130) V.(Ci*F) = F-V-q - q.V^F
in which the first term of the right-hand number vanishes,
as Ci is a constant vector; hence
J J V ClVKF(fo= "f /V F - n dS
F.n^ dS* (2)
--//■
But dv = h dS, and dS = h da so that the volume and surface
integrals become surface and line integrals respectively! and
h f f c,.V*F dS = h fp-n*^ ds.
Now suppose the base to be an element of area of a surface
bounded by a closed contour. Then Cj becomes a unit vector
normal to the surface, which we call n, and what was formerly
nxCj in (2) becomes a unit vector in the direction of dr, so
that F.nxd ds = F. dr.
Summing up the elements
%ffnv*F dS = SjF-*.
The summation of the surface integrals means simply that
the integral C C « e JC *
^ J J n.v* F dS
is to be taken over the entire surface.
In summing up the line integrals, the contour of every
elementary area is traced twice, but in opposite directions
except those forming part oi t\ve coxAwm.
* The negative sign indicates re\B&\oi»Aft? <& tori^wwA^
ii tour and the vector n.
APPENDIX.
251
Hence the sum of the line integrals becomes the line inte-
gral around the bounding curve and we have again Stokes'
Theorem.
Note to § 65.
Another Proof of Gauss's Theorem.
By means of the Divergence Theorem an easy proof of
Gauss's theorem may be given.
The problem is to evaluate the integral
over the closed surface S.
This integral may be written
-// n - V (>>
by (109).
There are three cases, according as the origin is without,
within, or on the surface S.
Case I. The origin is without. In this case r can never
be zero, hence because by the Divergence Theorem
ff a »l d8 -fff*;*'
and because
we have
r
by §64,
ss%
*dS = 0.
Case II. The origin is within. Surround the origin by
a small sphere of radius e; then the origin is excluded from
the region bounded by S and S'. Hence the required inte-
gral taken over both surfaces is zero, by Case I, i.e.,
_tt -•-
252
VECTOR ANALYSIS.
But for the sphere, r = «, n«ii = — 1,
ff s 6S - 4rf;
'•/X3*- 4 *-
Hence
/Xt?" 8 - 4 "
Case III. The origin is on the surface. Exclude the
point by an elementary hemisphere.
Then proceeding as in Case II, we find that the required
integral is equal to minus the integral over the hemisphere,
i.e., to 2t.
Note to § 52.
Other Integration Theorems.
We can evaluate two volume integrals in a manner similar
to that given by the Divergence Theorem. First, to find
fff VF *>>
where F is a scalar point-function. Let c be a constant
vector,
fff C ' VF dv = fff Vm ( cF )* m
But by the Divergence Theorem
fffv(cF) dv = /Jn.(cF) dS,
Again
APPENDIX. 263
or Bincc c is perfectly arbitrary,
fffvF*,-JJ«F<m.
=/// v - (F " c >' to -
-ffn-p.c)dS,
-ffc-(n.F)dS.
Hence finally
///v.F*.//n.FdS.
By similar processes, starting with the Divergence Theorem,
which is seen to be a formula of fundamental importance,
many other relations between surface and volume integrals,
and indeed also between line and surface integrals, using
the device employed in the second proof of Stokes' Theorem
above, may be deduced.
*tfj
XH
INDEX.
Accelerated motion* 82.
Acceleration, central, 84.
centripetal, 197.
normal, 81.
of moving space, 197.
radial, 81.
Activity, equation of, 165
Addition of vectors, 4.
Ampere, 161, 165.
Analytic solution of Euler's equa-
tions, 200.
Angle, solid, 138, 166.
Angular velocities, composition of,
41.
Apparent inertia, 182.
Appendix, 221.
Applications to geometry, 73.
to mechanics, 39.
Areal velocity, 86.
Areas, description of, 85.
Axes, moving, 194.
normal to surface for maximum
and minimum, 189.
of a central quadric, 189.
permanent, 193.
principal, 186.
Axis, central, 65.
instantaneous, 43, 188, 194.
of a vector product, 34
Body, rigid, 41, 181.
system of forces on rigid, 63.
Book of Bucherer, 127.
of Fehr, 80.
of Foppl, 38.
of Gans, 112, 127.
of Gibbs- Wilson, 47, 56, 109, 127.
of Joly, 123.
Book of Kelland and Tait, 63.
of Lame, 104.
of Routh, 41, 181.
of Webster, 112.
Calculus of variations, 125.
Cartesian expansion for scalar prod-
uct, 30.
expansion for triple vector prod-
uct, 53.
expansion for vector product, 38.
expansion of divergence, 111.
Center of mass, 19.
of mass, motion of, 179.
Central acceleration, 84.
axis, 65.
quadric, axes of, 189.
Centrifugal couple, 198.
Centripetal acceleration, 197.
Centroid, 18.
Circle, equation of, 61.
Circular motion, 82.
Circulation, 214.
definition, 32.
Collection of formulae, 229.
Collinear vectors, 3.
Combinations of three vectors, 48.
Commutativity of d and J , 125.
Comparison of various notations,
223.
Complex variable, 15.
Components of vector, 8.
Composition of angular velocities, 41.
Compound centripetal acceleration,
198.
Condition for relation to be indepen-
dent of origin, 21.
255
256
INDEX.
Condition of integrability, 130.
of parallelism of vectors, 35
of perpendicularity, 28.
that four vectors terminate in
same plane, 18.
that three vectors end in same
straight line, 13.
that three vectors lie in a plane, 50.
Cone, polhode and herpolhode, 192.
Conic sections, 63.
section orbit of planet, 87.
Conjugate, linear vector-function,
186.
self-, 183.
Conservation of circulation, 216.
Conservation of motion of center of
mass, 179.
of vortex motion, 213.
Conservative system of forces, 129,
145, 203.
Continuity, equation of, 116, 207.
of a scalar point-function, 7.
of a vector point-function, 8.
Convergence, 112.
Coordinates, curvilinear, 79.
isothermal, 80.
of linear vector-function, 185.
orthogonal, 80.
polar, 61.
Coplanar vectors, 8.
Coriolis, 198.
compound acceleration of, 198.
theorem of, 194.
Cork-screw rule, 160, 183.
Coulomb's law, 98, 143.
Couple, minimum, 65.
Cremieu and Pender, 162.
Cross product, 34.
distributive law of, 35.
Curl, example of, 118.
condition of vanishing of, 127.
independent of axes, 117.
of magnetic body, 120.
the operator, 117.
Curvature, 76.
Curve in space, 74.
Curvilinear coordinates, 79,
D'Alembert, 178, 180, 202.
Del, applications of, to scalara, 10C
applied to a vector, 109.
formulas for use of, 121.
rule for use of, 134.
the operator, 94.
Derivative, directional, 106, 107.
total, 107.
Description of areas, 85.
Determinantal cubic, 190.
Determinant form of vector product*
39.
Determinant form of triple scalar
product, 50.
Differential equation of geodetic, 78.
equation of harmonic motion, 86,
operators, 94.
perfect, 129.
Differentiation by V, 121.
by V 1 , 135.
of vectors, 70.
of vector and scalar products, 72.
partial, 90.
with respect to scalar variables,
70.
Directional derivative, 106.
Discontinuities, 95.
Displacement current, 161.
Distributive law for cross products,
35.
law, physical proof of, 37.
Divergence, definition, 109.
Cartesian expansion for, 111.
interpretation of, 146.
physical interpretation of, 109.
theorem, 112.
Division of a line in a given ratio, 18,
Dot or scalar product, 28.
Dyad, reference to, 109.
Dynamical equations of Euler, 200.
Dynamics, 178.
included in a single formula, 207.
Electrical theory* 138.
Electro-magnetic waves, 163.
Electro-motive force definition, 32«
Electro-dynamic potential, 178.
INDEX.
267
Electron theory, 162, 182.
Elementary properties of linear vec-
tor-function, 183.
Ellipse, equation of, 86.
equation relative to focus, 90.
Ellipsoid, moments!, 184.
of Poinsot, 184, 201.
Energy of distribution in terms of
field intensity, 157.
of system in terms of potential,
156.
Equal vectors, 2.
Equation of circle, 61.
of continuity, 208.
of electro-magnetic field, 160.
of hodograph, 83.
of instantaneous axis, 43.
of Lagrange, 205.
of Laplace, 146.
of plane, 17, 58, 59.
of Poisson, 146.
of sphere, 61 .
of straight line, 11, 60.
Equations, dynamical, of Euler, 200.
of hydrodynamics, 207.
of motion, 178.
of polhode and herpolhode curves,
193.
of surfaces, 79.
Equipotential surfaces, 98.
Ether, 213.
Euler 's dynamical equations, 200.
equations of motion of a fluid,
210.
theorem, on homogeneous func-
tions, 131.
Expansion for moment of momen-
tum, 185.
for triple vector product, 53.
Exploding shell, motion of, 179.
Extended vector, definitions, 204.
Extension of vector to n-dimen-
sions, 204.
Faraday, 160.
Fehr, book of, 80.
Field due to a current, 165, 170.
Field, energy in terms of, 157.
intensity, 157.
Fields, addition of, 6.
Filament, vortex, 215.
Fixed point, motion about, 198.
space, 195.
Flow of heat, equation of, 114.
Fluid, Euler 's equations of motion
of a, 210.
Flux of a vector, 33.
of heat, 114.
Poppl, book by, 38.
Force, central, 84.
centrifugal, 197.
for Newtonian law, 143.
Formula for use of V, 121.
principal, of vector analysis, 229.
Foucault gyroscope, 199.
Fourier's law, 104, 114.
Frictionless fluid, 213, 217.
Function, Green's, 148.
Lagrangian, 204.
linear vector, 185.
Fundamental equations of hydro*
dynamics, 207.
Gans, book of, 112, 127.
Gauss's theorem, 138.
theorem for the plane, 140.
theorem, second proof, 141.
General equations of motion of
Lagrange, 205.
Generalized parameters of a system,
205.
Geodetic lines on a surface, 78.
Geometrical representation of mo-
tion of a rigid body, 191.
Geometry, applications to, 73.
Gibbs, Prof. WUlard, 221, 224.
Gibbs- Wilson, book of, 47, 56, 109,
127.
Orad (gradient) of a scalar function,
102.
independent of choice of
103.
Graphical representation of
continuities, 96, 97.
258
INDEX.
Grassmann, 50.
Grassmann 's notation, 221, 223.
Gravity, center of, 20.
Green's formulas, 148.
function, 148.
theorem, 148, 158.
Gyration, radius of, 181.
Gyroscope, 199.
Hamilton's integral, 202, 204, 206.
notation, 63, 221, 223.
principle, 202.
Harmonic function, 147.
motion, differential equation of, 86.
Heaviside, Oliver, 50, 155, 164.
Heaviside's notation, 221, 223.
Helmholtz, 125, 212, 213, 217.
Helmholtz's theorem, 155, 173.
Herpolhode curve, 192.
Hodograph and orbit under New-
tonian forces, 87.
Hodograph, definition, 81.
equation of, 83.
of accelerated motion, 82.
of a particle at rest, 82.
of uniform circular motion, 82.
of uniform motion, 82.
Homogeneous function, 131.
Hydrodynamics, 207.
Incorapresslblllty, condition of, 209.
Independent of the origin, relations
of, 21 .
Inductance, 171.
Induction, vector, 158.
Inertia, apparent, 182.
moment of, 181, 185.
products of, 185.
Instantaneous axis, 43, 183, 194.
Integral due to Neumann, 171, 175.
line of a vector, 31.
of Gauss, 138.
surface, of a vector, 32.
Integrating factor, 129.
operator pot, 152.
Integration, 83.
as a vector sum, 5.
Integration theorem, 174.
Interpretation of products, 57.
Invariable line, 192, 199.
plane, 184, 191.
Inverse square law, 87, 105, 143.
Irrotational motion, 119, 212.
Isothermal surface, 104.
system of curves, 80.
Joly, book of> 123.
Joule, 164.
Kelland and Talt, book of, 63.
Kelvin, Lord, 217.
Kepler, laws of, 86.
Kinematics of a particle, 80.
Kinetic energy of rotation. 181.
of translation, 179.
Lagrange's equations of motion*
205.
Lagrangian function, 204.
Lam6, book of, 104.
definition of, 6.
Lamellar component of a vector
function, 154.
vector, 128, 154.
Laplacian, the, operator, 134.
Laplace's equation, 143.
Law, distributive — for vector prod-
ucts, 35.
of Coulomb, 98.
of Fourier, 104.
of Kepler, 86.
of Lenz, 160.
of Newton, 87, 105, 143.
Laws obeyed by 1 J k, 29, 35.
obeyed by scalar products, 29.
obeyed by triple scalar product,
49.
Layers, equivalent, not equipotential,
151.
Lenz, law of, 160.
Level surfaces, 98.
Linear vector-function, 182, 183,238.
Line integral of a vector, 31.
of normal component, 168.
INDEX.
259
Lines of force, 109.
of vector-function, 32.
Liquid, fundamental equations, 207.
Lord Kelvin, 217.
Magnetic Held due to a current, 165.
field due to element of current,
167.
Magnitude of a vector, definition, 3.
of a vector, 29.
Mathematical and physical discon-
tinuities, 95.
Maxwell, 116, 128.
Maxwell 's equations, 160.
McAulay, theorem of, 170,
Mechanical force on element of cir-
cuit, 167.
Mechanics, 178.
applications to, 39.
Methods of solution of problems, 13.
Minimum couple on central axis, 65.
Moment about an axis, 180.
as vector, 40.
definition, 39.
of inertia, 181, 185.
Momenta! ellipsoid, 184.
Moments of inertia, principal, 186.
Momentum, moment of, 181.
of momentum, 181.
Motion, circular, 82.
harmonic, 86.
irrotational, 119.
of a rigid body, 41.
of center of mass, 179.
Poinsot, 192.
under no forces, 184.
vortex, 212.
Moving axes, 190, 194.
space, 195.
Multiple vector products, 55.
Mutual energy of two circuits, 171,
175.
Mutual inductance, 173, 176.
n-d Intensions, extension of vector
to, 204.
Negative vector, 2.
Neumann's integral, 171, 175.
Newtonian forces, 87.
law of force, 105, 143.
Non-vortical motion, 120, 212.
Normal acceleration, 81.
and tangent, 75.
to tangent plane of quadric, 189.
unit, 78.
Notation of Oans, 223.
of Gibbs, 222, 223.
of Grassmann, 221.
of Hamilton, 221, 223.
of Heaviside, 221, 223.
of this book, 224.
Notations, comparison of, 223.
various, 221.
Operator, integrating pot, 152.
the, curl, 117.
the, V, 94.
the, "p", 72, 73.
involving \7 twice, 133.
( ), 187.
Orbit of a planet, 84.
under Newtonian forces, 87.
Origin of operator \7, 90.
Orthogonal system of curves, 80.
Osculating plane, 77.
Parabola, path of projectile, 179.
Parabolic orbit, 84.
Parallelism, condition of, 35.
condition of, of vectors, 35.
Parallelogram law, 40.
Parallelopiped principle, 50.
Parentheses, 51.
Partial differentiation of vectors,
90.
differentiation with V» 105, 106.
Particle, kinematics of, 80.
Perfect differential, 129.
Permanent axes, 192.
Perpendicular from origin to a plane,
59.
Perpendicularity, condition of, 28.
Physical discontinuities, 95.
proof of distributive law, 37.
260
INDEX.
Plane, equation of, 17, 68, 69.
invariable, 184.
osculating, 77.
passing through end of a vector,
68.
passing through ends of three vec-
tors, 69.
perpendicular to a vector, 68.
through ends of three given vec-
tors, 17.
Planet, orbit of, 84.
Poinsot ellipsoid, 184, 201.
motion, 192.
Point-function, scalar, 6.
vector, 7.
Poisson'8 equation, 143.
equation, solution of, 162.
Polar coordinates, 61 .
Polarization, energy in terms of, 159.
Polhode and herpolhode curves, 192.
Polygon of vectors, 4.
Potential, 6.
definition, 98.
derivatives of, 102, 149.
forces having a, 129, 216.
the, 6, 143.
vector, 153, 173.
velocity, 216.
Poynting's theorem, 165.
Practical application of steady mo-
tion equation, 212.
Principal axes, 186, 189.
axes, at right angles, 190.
moments of inertia, 186.
of D'Alembert, 178, 202.
of Hamilton, 202.
Problems, method of solution, 13.
to Chap. I, 22.
to Chap. II,- 43.
to Chap. Ill, 66.
to Chap. IV, 91.
to Chap. V, 136.
to Chap. VI, 176.
to Chap. VII, 217.
Product, cross, 34.
dot. 28.
scalar, 28.
Product, vector, 34.
Products of inertia, 186.
of two vectors, 28.
of three vectors, 48.
of more than three vectors, 65.
Projectile, path of, 179.
Projections of vector, 29.
Proof of del formulas, 121.
of normality of principal axes, 190.
of expansion of triple vector prod-
uct, 51, 63, 64.
Properties of frictionlees fluid, 213,
217.
Quadric surface, 63, 86, 184.
principal axes of, 189.
tangent plane to, 188.
Quaternions, 63.
Radial acceleration, 81.
Radiant-vector, 166.
Radius of gyration, 181.
Ratio, division of line in given, 18.
Reciprocal operator, 187.
system of vectors, 57.
vector, 3.
Rectangular coordinates, 62.
Relation between any four vectors,
56.
between force and potential, 100.
Relations independent of the origin,
21.
Relative motion, 194.
Remarkable formula, 207.
Remarks on notation, 63,221,223.
Representation of a vector-function,
74.
of vector or cross product, 34.
Resolution of a system of forces, 63.
of a vector-function into solenoidal
and lamellar components, 164.
of velocity, 81.
Resume" of notation of this book,
224.
Rigid body, 41, 181.
body, motion of, 41.
system, 180.
INDEX.
261
Rigidity, various kinds, 120.
Rolling ellipsoid, 191, 193.
Rotation as vector, 41.
equations for, 179.
kinetic energy of, 181.
Routh, book of, 202.
Rowland, 162.
Rule, cork-screw, 160, 183.
Scalar and vector fields, 94.
and vector functions of position,
95.
definition of, 1.
field, 6.
product, 28.
products of three vectors, 48.
products, differentiation of, 72.
variables, integration with respect
to, 83.
Self-conjugate, 183.
inductance, 173, 176.
Sink of heat, 112.
Slope, 102.
Solenoidal component of a vector
function, 154.
distribution, 117.
vector, 117.
Solid angle, 138, 166.
Solution of differential equations,
86.
Source of heat, 112.
Sources and sinks, 114.
Space curve, 74.
Space of TV-dimensions, 204.
Special notation, 204.
Speed, 81.
Sphere, equation of, 61.
Steady motion, 212.
Step, 2.
Stokes' theorem, 124, 160, 161, 168,
214.
Straight line, equation of, 11, 60.
parallel to a given vector, 60.
through end of a vector, 60.
Stroke, 2.
Subtraction of vectors, 4.
Surface, equations of, 79.
Surface integral of a vector, 32.
of revolution, 191.
Surfaces, level or equipotential, 98.
Symmetrical top, 199.
System of forces on a rigid body,
63.
Systems of units, 155.
Talt, 63, 170.
Tangent and normal, 75.
plane to quadric, 188.
unit, 74.
Tangential component in line inte-
gral, 31.
Taylor's theorem, 124, 131, 213.
Tensor of vector, 3.
Termi no-col linear vectors, 13.
Termino-co planar, 18.
Theorem due to Helmholts, 155,
173.
of Coriolis, 194.
of Euler, 131.
of Gauss, 138, 140, 141. ,
of Green, 148.
of McAulay, 170.
of Poynting, 164.
of Stokes, 124, 160, 161, 168, 214.
Third integral of equations of rota-
tion, 201.
Three axes in a rigid body, 193.
Top, 199.
Tortuosity, 78, 92.
Total current, 162.
derivative, 107.
kinetic energy of a system, 182.
Transformation of hydrodynamio
equation, 211.
Translation, equation for, 178.
Triple vector products', 27, 48.
vector product, expansion for, 53.
Tube of vector function, 215.
Uniform motion* 82.
Unit normal, 78.
Units, other systems of, 155.
tangent, 74.
vector, 3.
262
INDEX.
Variable, complex, 15.
Variations, calculus of, 125.
Vector and scalar fields, 94.
Vector, components of, 8.
curvature, 76.
definition of, 1.
equations, 11, 08.
field, 6.
function, representation of, 74.
graphical representation of, 1.
lamellar, 128.
line integral of, 31.
magnitude of, 20.
negative, 2.
perpendicular to a plane, 59.
point-function, 7.
-potential, 153, 173.
product, 34.
products, differentiation of, 72.
products of three vectors, 48.
radiant-, 165.
reciprocal, 3.
sum as an integration, 5.
surface integral of, 32.
-tortuosity, 78.
unit, 3.
velocity, 80.
/ectors, addition and subtrac-
tion, 4.
collinear or parallel, 3.
condition of parallelism of, 35.
Vectors, coplanar, 8.
decomposition of, 8.
differentiation of, 70.
equality of, 2.
products of three, 48.
products of more than three, 55.
reciprocal system of, 57.
termi no-col linear, 13.
termino-coplanar, 18.
the unit, I J k, 9, 29, 85.
Velocity along tangent and normal
of a curve, 81.
angular, 41.
areal, 86.
of electric waves, 164.
Velocity-potential, 216.
Volume of sphere by divergence
theorem, 114.
Vortex-atom theory, 217.
Vortex filament, 215.
motion, 212.
Vorticity, 211.
indestructible, 213.
uncreatable, 216.
Wave equation, 163.
Ways in which a vector may vary
70.
Webster, book of, 112.
Wilson-Gibbs, book of, 109.
Work of a force, 31.
V* '
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