WEBVTT
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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.
00:09:38.840 --> 00:09:40.390
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,
00:09:44.940 --> 00:09:47.250
and a t sine t times a cosine t.
00:09:47.250 --> 00:09:49.600
And so these two add
up to 0, and so I only
00:09:49.600 --> 00:09:51.080
have to integrate one thing.
00:09:51.080 --> 00:09:56.710
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.
00:10:08.720 --> 00:10:10.980
And sine of pi I believe is 0.
00:10:10.980 --> 00:10:13.830
And sine of 0 I believe is 0.
00:10:13.830 --> 00:10:17.050
And so I get 0
minus 0, so I get 0.
00:10:17.050 --> 00:10:19.480
So they're actually--
when I compute
00:10:19.480 --> 00:10:22.050
the line integral of F
dot dr over that helix,
00:10:22.050 --> 00:10:23.747
I actually get 0.
00:10:23.747 --> 00:10:25.330
So let me just remind
you, real quick,
00:10:25.330 --> 00:10:28.010
what the point of the
problem was and what we did.
00:10:28.010 --> 00:10:33.490
We had-- at the very beginning,
we had a vector field,
00:10:33.490 --> 00:10:35.872
we had a curve, and
essentially all we were doing
00:10:35.872 --> 00:10:38.330
is a problem we've done in two
dimensions many times, which
00:10:38.330 --> 00:10:40.470
is compute a line
integral along a curve.
00:10:40.470 --> 00:10:42.330
And so we just
added a dimension.
00:10:42.330 --> 00:10:43.910
The problem is exactly the same.
00:10:43.910 --> 00:10:48.750
Instead of now just dx and dy,
now we have a dx, dy, and a dz.
00:10:48.750 --> 00:10:51.400
We have one extra
direction you're moving.
00:10:51.400 --> 00:10:53.302
But that's all that's different.
00:10:53.302 --> 00:10:55.260
So the first thing we
did was sketch the curve.
00:10:55.260 --> 00:10:57.343
Then we computed the line
integral, as I'm saying,
00:10:57.343 --> 00:11:00.190
by exactly the same methods
that we did in two dimensions.
00:11:00.190 --> 00:11:02.010
So everything, really,
should remind you
00:11:02.010 --> 00:11:04.020
of what you've done
previously, we now just
00:11:04.020 --> 00:11:06.720
have a third component
we have to deal with.
00:11:06.720 --> 00:11:10.530
And that was, in our case, in
this problem, the R*dz part.
00:11:10.530 --> 00:11:12.506
So I think that's
where I'll stop.