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JOEL LEWIS: Hi.

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Welcome back to recitation.

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In lecture, you've been learning
about line integrals

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and computing them around curves
and closed curves and

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in various different ways.

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So here I have some problems
on line integrals for you.

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So in all cases I want C to
be the circle of radius b.

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So b is some constant, some
positive constant.

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It's the circle of radius b
centered at the origin, and I

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want to orient it
counterclockwise.

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And then what I'd like you
to do is for each of the

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following vector fields F, I'd
like you to compute the line

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integral around C of F dot dr.

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So in the first case, where
F is xi plus yj.

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In the second, where F is
g(xy) times (xi + yj).

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So here g of xy is some
scalar function.

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But you don't know a formula
for this function.

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So your answer might be in
terms of g, for example.

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You can assume it's a
continuous, differentiable

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nice function.

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And then the third one,
F is -yi plus xj.

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Now before you start, I want
to give you a little

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suggestion, which is often
when we're given a line

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integral like this, the first
thing you want to do is jump

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in and do a parameterization
right away for the curve, and

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then you get a normal single
variable integral.

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So what I'd like you to do for
these problems is to think

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about the setup and think about
whether you can do this

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without ever parameterizing C,
so without ever substituting

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in cosine and sine
or whatever.

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So for all three parts
of this problem.

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So if you can use some sort of
geometric reasoning to save

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yourself a little bit of work
without ever going to the

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parameterization.

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So why don't you pause the
video, spend some time, work

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that out, come back, and we
can work it out together.

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Hopefully you had some luck
working on these problems.

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Let's get started.

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So let's do the first problem
first. Let's think about what

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this vector field
F looks like.

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This first vector field.

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So let me just draw a little
picture over here.

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So here's our circle
of radius b.

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And this vector field F given by
xi plus yj, at every point

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(x,y), the vector F is the same
as the position vector of

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that point.

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So over here the vector's
like that.

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Over here, the vector's
like that.

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Up here, the vector
is like that.

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So these are just a few little
values of F that

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I've drawn in there.

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And so down here, say,
F is like that.

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So in particular, so that's just
sort of, you know, if you

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wanted, you could draw in some
more vectors, get a full

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vector field picture.

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So the thing to observe here is
that a circle is a really

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nice curve.

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So the circle has the property
that the position vector at a

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point is orthogonal to the
tangent vector to the circle.

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At every point on the circle,
the tangent vector to the

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circle is perpendicular to
the position vector.

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So that means it's perpendicular
to F, because F

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is the same, in fact,
but is parallel to

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the position vector.

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So in Part a, you have that F
dot the tangent vector to your

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curve is equal to zero
at every point

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on the entire curve.

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All right?

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So your field F dot your tangent
vector is always zero.

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So that means that the integral
around C of F dot dr,

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well, we know that dr is T ds.

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So this is F dot T ds.

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But that's just zero.

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It's just an integral and the
integrand is zero everywhere.

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And whenever you take a definite
integral of something

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that's zero everywhere,
you get zero.

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So this is just zero
right away.

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We didn't have to parameterize
the curve or anything.

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We just had to look at this
picture to sort of understand

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that this kind of field, it's
called a radial vector field,

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where the vector F is always
pointed directly outwards.

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When you integrate a radial
vector field around a circle

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centered at the origin, you
get zero, because the

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contribution at every
point is zero.

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So that's Part a.

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Part b is actually
exactly the same.

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If we look back at our formula
over here in Part b, we have

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that F is given by some
function g(xy) times

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(xi hat + yj hat).

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Well, what is this
g of xy doing?

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It's just rescaling.

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It's telling you every point you
can scale that vector by

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some amount.

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So if we looked over at this
picture, maybe over here you

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would scale some of these
vectors to be longer, and over

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here they might be shorter, or
you might switch them to be

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negative, but you don't change
the direction of any vector in

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the field from Part a.

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You just change their length.

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So you still have a radial
vector field.

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And you still have the property
that at every point

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on our curve, the tangent
vector to the curve is

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orthogonal to the vector F.
So the tangent vector is

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orthogonal to F, so that means
you again have F dot T is

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equal to zero.

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And so F dot dr is also equal
to 0 ds, and so when you

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integrate that, you
just get zero.

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So that's also what
happens in Part b.

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So Part b, I'm just going
to write ditto.

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The exact same reasoning
applies in Part b as

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applied in Part a.

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And you also get zero as your
integral without having to

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parameterize, without having
to do any tricky

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calculations at all.

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All right.

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So let's now look at Part c.

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I'm going to draw another
little picture.

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So in Part c, there's
your curve.

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At the point (x,y)--

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so I'm going to draw some
choices of F again.

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So in Part c, at the point
(x,y), your vector field F is

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-y i hat plus x j hat.

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Now if you draw that on the
picture here, over there

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that's that vector.

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Over here, so at the point
(0,1), say, that gives you the

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vector {-1, 0}.

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So that's horizontal
to the left.

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Here are some more.

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There's one there, there's
one there.

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There's another one over
here and so on.

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In fact, what you'll notice is
that this vector F is just

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parallel to the tangent vector
of the circle everywhere.

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This field is a tangential
field.

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It's always pointing parallel
to the curve.

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OK?

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It's perpendicular to
the position vector.

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It's in the same direction
as the tangent

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vector at every point.

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So this is something that you've
seen before, I think.

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That this vector field is giving
you a sort of nice

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rotating motion.

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You know, at every point it's
circulating counterclockwise.

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So what does that mean?

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Well, again, it's not exactly
the same as Part a and b, but

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again we'll be able to compute
this integral without

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parameterizing.

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Why?

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Because F dot T in this case--

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well, so, let's see.

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What is the norm of F?

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The magnitude of F is just the
square root of (x squared plus

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y squared).

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So on our circle of radius
b, that means the

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magnitude of F is b.

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And the magnitude of T, the unit
tangent vector, is 1, and

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they point in the
same direction.

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So when you have two vectors
that point in the same

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direction, their dot product is
just the product of their

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magnitudes.

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So that means F dot
T is equal to b.

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This is a constant.

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F dot T is equal to b.

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So when you integrate around
the circle, F dot dr, well,

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this is equal to the integral
around a circle of F dot the

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tangent vector with respect
to arc length.

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But this integrand, F dot the
tangent vector, is this

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constant b.

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So you're integrating
over the curve b ds.

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And when you integrate a
constant ds, well, that just

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gives you the total
arc length.

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So this is b times the
total arc length.

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And this is a circle
of radius b.

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So that's b times 2 pi b, which
we could also write as 2

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pi b squared.

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So there you go.

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So in this third case,
you have a nice

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tangential vector field.

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So that means the integrand
actually

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works out to be constant.

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Because the integrand is
constant, we don't ever have

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to parameterize the curve.

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We can just use the fact that we
already know its arc length

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in order to compute
this integral.

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Again, we could do all of these
integrals if we wanted

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by parameterizing the circle,
by x equals b cosine T, y

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equals b sine T, and going
through and writing this as an

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integral from T equals
0 to 2 pi, and so on.

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But these are examples of
problems where it's helpful to

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think about what's going on
first, see if you can

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understand the geometry
of your situation.

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And sometimes you'll
have a problem

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like this where you'll--

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either in this class or
elsewhere in your life--

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where something that might seem
complicated has a simple

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geometric explanation.

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And so when that does happen,
it's nice when you can take

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advantage of it.

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Sometimes that won't happen and
sometimes you'll have to

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do the parameterization
and the computation.

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But in these cases we have
these nice three examples

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where with a radial vector
field, you get that the

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integrand is always zero, or
with a tangential vector

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field, you have that the
integrand is constant.

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All right.

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So, I'll stop there.

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