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Welcome back to recitation.

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In this video, I'd like us to
work on the following problem.

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Which is we begin with a
vector field, capital F,

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that is z*x*i plus
z*y*j plus x*k.

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And we're going to
look at the curve

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C that is a helix, that we can
describe by the parameter t.

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So we'll describe it as
cosine t comma sine t comma t.

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And we're interested
in the portion

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of the helix that goes from
(1, 0, 0) to (minus 1, 0, pi).

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And I'd like you to do two
things with this problem.

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The first thing
I'd like you to do

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is I'd like you to sketch
the curve that is carved out

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when you follow the t
values that will start you

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at (1, 0, 0) and will finish
you up at (minus 1, 0, pi).

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The second thing I
would like you to do

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is I would like you to compute
the line integral F dot dr

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over that portion of the helix.

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So there are two
parts to this problem.

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Why don't you pause the video,
work on these two parts,

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and then when you're
feeling comfortable

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seeing the solution,
bring the video back up

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and I'll show you how I did it.

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OK, welcome back.

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So again what we're interested
in doing in this problem is

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first, understanding
what the curve looks

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like that we want to take
this line integral over.

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And then actually computing
the line integral.

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So we have this
C that is a helix

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and it's described by
cosine t, sine t, t.

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And in fact-- well, I won't
say any more about this helix.

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But it actually
should remind you

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when you see the picture of some
portion of what DNA looks like.

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It's going to spiral
around, just the way

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some little side of-- if you
take DNA, how it spirals up,

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it's going to be the
boundary of some of that.

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So we'll see that momentarily.

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And if you notice, the
first thing that's helpful

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if you want to
sketch the curve, is

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that t-- I know immediately
what the parameters are in t.

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t is ranging from 0 to pi.

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So I know already exactly
what I want to do.

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And in order to draw this
curve, what I'm going to do

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is I'm going to give myself
a frame of reference.

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Because otherwise
it's going to be

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really hard to draw this curve.

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And the frame of reference
is the following.

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All of these points, all
of the points cosine t,

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sine t, t, lie-- in the x-y
distance from the origin,

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they lie at a radius 1.

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So in terms of x
and y, they're all

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going to sit on the boundary
of a cylinder of radius 1.

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So let me draw
what I mean by that

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and then we'll see
where the points are.

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So let me actually come
write right over here.

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So my first part
sketching the curve,

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the first thing I'm going to
do is give myself a cylinder.

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Which I'll show you is
of radius 1 momentarily.

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And I'm going to actually say
this is the z-axis, coming

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straight through the middle.

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And then the y-axis is going
to come out the side, as usual.

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And the x-axis is going to
come down in this direction.

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OK, so this cylinder I'm
thinking of, it has radius 1.

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So at any given z
value, any fixed z

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value that I intersect
with the cylinder gives

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a circle of radius 1.

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Now if you notice
again, what I was

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trying to explain
from over here,

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is that if I don't look at
the z component, obviously

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the cosine t, sine
t is something

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that's on the unit circle
if I ignore the z component.

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And so that's how I know that
these x and y values here

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are just going to
lie on the cylinder,

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because they're always
distance 1 from the z-axis,

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at any given height.

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So that's the first
thing I observe.

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The second thing I observe
is that I mentioned

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t goes from 0 to pi, and that's
exactly the z values also.

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So the z values are
going from 0 to pi,

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so I know my first value is
going to be on the xy-plane.

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And my last value is going to
be on the z equals pi plane.

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And then the last
thing to observe

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is that what is being carved
out in the x and y components?

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Well, it's really
exactly what you

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would do if you were trying
to parameterize a circle.

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But you're
parameterizing a circle

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and instead of just
drawing it always

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on the same z
plane-- so xy-plane,

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z equals a constant-- you're
parameterizing that circle

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and you're also moving up.

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And so the first value (1, 0,
0) is happening somewhere here.

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And the last value is
happening at negative 1, 0, pi.

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And so I have to go sort of
in the backwards direction.

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So here's my x equals
negative 1, 0, pi.

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And so I'm kind of behind.

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You should think
of this as being--

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let me see if I can draw
this-- this is behind the pi--

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maybe I should draw
it a little further.

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If this is the pi height, it
was more like right there.

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It's behind the z-axis here.

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It's on the other side.

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And so the curve
that's carved out

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is going to look
something like this.

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It's going to come
up through here.

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It's spiraling.

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And then it's
going to go behind,

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on the other side of the
cylinder, and spiral up.

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Now in a perfect world, if
I could draw this actually

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in three dimensions, the way
it's coming up is actually,

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it has to have some
sort of constant rate.

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Because it's always
moving in the z direction

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at a constant rate.

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It's moving in the z
direction linearly.

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Maybe this picture is not
the most perfect picture

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because it looks like it's
going up really fast at the end.

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But it gives us a feel
for how the curve looks.

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If I continued it, it would
come back around to the front

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by the time t went to 2*pi.

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And so this is a spiral.

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It goes around the cylinder,
behind the cylinder.

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And then if I go for
another pi, from pi to 2*pi,

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it's going to go-- continue to
curve around and then come back

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out to the front and be
right above this point.

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So that's the helix.

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That's the shape of the helix.

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So this is an
approximate sketch.

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Good thing I said
sketch the curve.

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So this is a sketch
of the curve.

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And now what we
want to do, again

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as I mentioned, is we want
to compute a line integral,

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we want to compute F
dot dr over this curve.

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So what I'm going
to need is I'm going

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to know that this
is the curve here,

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and I need to understand how to
parameterize F and dr in terms

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of this parameter t.

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So that's what
I'm interested in.

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So let's think about what F
dot dr is in order to do this.

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I think the notation
you've seen from lecture

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is F we usually denote by
capital P, capital Q, capital

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R. And dr we denote
[dx, dy,  dz].

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And so F dot dr, as
we've seen previously,

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is P*dx plus Q*dy plus R*dz.

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So let's see what that is
in the parameters we have.

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So let me first
remind ourselves, x

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in this situation is cosine t.

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y is sine t.

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And z is equal to t.

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Based on how we're
parameterizing the curve.

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And we're interested in the
values of t going from 0 to pi.

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So these are the quantities
that we're going to need.

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Now in order to solve this
problem I need dx, dy, dz.

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I also need P, Q, and R in terms
of these x, y, and z values.

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So let me remind you also--
I'm going to write it over here

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so I don't have to look again--
F is actually z*x comma z*y

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comma x.

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Let me check that.

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Yes, that is what I have.

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So that is my F. So now that
I have it a little closer,

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I'm going to put
it all together.

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And I actually have to move
over to put it together,

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but now the reference is closer.

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So notice first I'm
going to do P*dx.

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So P is z times x.

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And then what is dx?

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Well, dx-- I really need to
figure out what dx/dt is,

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right? dx/dt is negative sine t.

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And so I'm going to replace
the dx by a negative sine t dt.

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I'll be replacing
dy by cosine t dt

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and I'll be replacing dz by dt.

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So I have all the pieces here; I
just have to put them together.

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So let me do that.

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I want to integrate over C
F dot dr. I'm really going

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to be integrating-- let's see.

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Let me come to this side so
I can see everything I need.

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I'm really going
to be integrating--

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I know I'm integrating from
0 to pi in my parameter t.

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My first component of
F I told you was z*x.

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So that's t cosine t, and
then dx is, as I said,

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negative sine t dt.

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So times negative sine t dt.

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That's my first component.

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My next component,
as I said, is Q*dy.

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Q was z*y and so it's t sine t.

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And then dy is cosine t dt.

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Oops, let me put the dt
there, it's a little easier.

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And then I'm going to write
the last component here,

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so we can see it
all in one frame.

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And then the last
thing was R*dz.

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And R is just x.

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So that's cosine t.

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And dz is just dt.

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So there's three components.

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This was the P*dx component,
this is the Q*dy component,

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and this is the R*dz component.

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And so notice, this is great.

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This is why I like this
problem, it's going to be nice.

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Because I've got a t cosine
t times negative sine t,

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and a t sine t times a cosine t.

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And so these two add
up to 0, and so I only

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have to integrate one thing.

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So I only have to integrate
from 0 to pi cosine t dt.

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And so what do I get?

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I get-- this should be sine
t evaluated at 0 and pi.

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And sine of pi I believe is 0.

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And sine of 0 I believe is 0.

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And so I get 0
minus 0, so I get 0.

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So they're actually--
when I compute

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the line integral of F
dot dr over that helix,

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I actually get 0.

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So let me just remind
you, real quick,

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what the point of the
problem was and what we did.

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We had-- at the very beginning,
we had a vector field,

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we had a curve, and
essentially all we were doing

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is a problem we've done in two
dimensions many times, which

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is compute a line
integral along a curve.

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And so we just
added a dimension.

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The problem is exactly the same.

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Instead of now just dx and dy,
now we have a dx, dy, and a dz.

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We have one extra
direction you're moving.

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But that's all that's different.

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So the first thing we
did was sketch the curve.

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Then we computed the line
integral, as I'm saying,

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by exactly the same methods
that we did in two dimensions.

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So everything, really,
should remind you

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of what you've done
previously, we now just

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have a third component
we have to deal with.

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And that was, in our case, in
this problem, the R*dz part.

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So I think that's
where I'll stop.

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