JOEL LEWIS: Hi.
Welcome back to recitation.
In lecture, you've been learning about line integrals
and computing them around curves and closed curves and
in various different ways.
So here I have some problems on line integrals for you.
So in all cases I want C to be the circle of radius b.
So b is some constant, some positive constant.
It's the circle of radius b centered at the origin, and I
want to orient it counterclockwise.
And then what I'd like you to do is for each of the
following vector fields F, I'd like you to compute the line
integral around C of F dot dr.
So in the first case, where F is xi plus yj.
In the second, where F is g(xy) times (xi + yj).
So here g of xy is some scalar function.
But you don't know a formula for this function.
So your answer might be in terms of g, for example.
You can assume it's a continuous, differentiable
nice function.
And then the third one, F is -yi plus xj.
Now before you start, I want to give you a little
suggestion, which is often when we're given a line
integral like this, the first thing you want to do is jump
in and do a parameterization right away for the curve, and
then you get a normal single variable integral.
So what I'd like you to do for these problems is to think
about the setup and think about whether you can do this
without ever parameterizing C, so without ever substituting
in cosine and sine or whatever.
So for all three parts of this problem.
So if you can use some sort of geometric reasoning to save
yourself a little bit of work without ever going to the
parameterization.
So why don't you pause the video, spend some time, work
that out, come back, and we can work it out together.
Hopefully you had some luck working on these problems.
Let's get started.
So let's do the first problem first. Let's think about what
this vector field F looks like.
This first vector field.
So let me just draw a little picture over here.
So here's our circle of radius b.
And this vector field F given by xi plus yj, at every point
(x,y), the vector F is the same as the position vector of
that point.
So over here the vector's like that.
Over here, the vector's like that.
Up here, the vector is like that.
So these are just a few little values of F that
I've drawn in there.
And so down here, say, F is like that.
So in particular, so that's just sort of, you know, if you
wanted, you could draw in some more vectors, get a full
vector field picture.
So the thing to observe here is that a circle is a really
nice curve.
So the circle has the property that the position vector at a
point is orthogonal to the tangent vector to the circle.
At every point on the circle, the tangent vector to the
circle is perpendicular to the position vector.
So that means it's perpendicular to F, because F
is the same, in fact, but is parallel to
the position vector.
So in Part a, you have that F dot the tangent vector to your
curve is equal to zero at every point
on the entire curve.
All right?
So your field F dot your tangent vector is always zero.
So that means that the integral around C of F dot dr,
well, we know that dr is T ds.
So this is F dot T ds.
But that's just zero.
It's just an integral and the integrand is zero everywhere.
And whenever you take a definite integral of something
that's zero everywhere, you get zero.
So this is just zero right away.
We didn't have to parameterize the curve or anything.
We just had to look at this picture to sort of understand
that this kind of field, it's called a radial vector field,
where the vector F is always pointed directly outwards.
When you integrate a radial vector field around a circle
centered at the origin, you get zero, because the
contribution at every point is zero.
So that's Part a.
Part b is actually exactly the same.
If we look back at our formula over here in Part b, we have
that F is given by some function g(xy) times
(xi hat + yj hat).
Well, what is this g of xy doing?
It's just rescaling.
It's telling you every point you can scale that vector by
some amount.
So if we looked over at this picture, maybe over here you
would scale some of these vectors to be longer, and over
here they might be shorter, or you might switch them to be
negative, but you don't change the direction of any vector in
the field from Part a.
You just change their length.
So you still have a radial vector field.
And you still have the property that at every point
on our curve, the tangent vector to the curve is
orthogonal to the vector F. So the tangent vector is
orthogonal to F, so that means you again have F dot T is
equal to zero.
And so F dot dr is also equal to 0 ds, and so when you
integrate that, you just get zero.
So that's also what happens in Part b.
So Part b, I'm just going to write ditto.
The exact same reasoning applies in Part b as
applied in Part a.
And you also get zero as your integral without having to
parameterize, without having to do any tricky
calculations at all.
All right.
So let's now look at Part c.
I'm going to draw another little picture.
So in Part c, there's your curve.
At the point (x,y)--
so I'm going to draw some choices of F again.
So in Part c, at the point (x,y), your vector field F is
-y i hat plus x j hat.
Now if you draw that on the picture here, over there
that's that vector.
Over here, so at the point (0,1), say, that gives you the
vector {-1, 0}.
So that's horizontal to the left.
Here are some more.
There's one there, there's one there.
There's another one over here and so on.
In fact, what you'll notice is that this vector F is just
parallel to the tangent vector of the circle everywhere.
This field is a tangential field.
It's always pointing parallel to the curve.
OK?
It's perpendicular to the position vector.
It's in the same direction as the tangent
vector at every point.
So this is something that you've seen before, I think.
That this vector field is giving you a sort of nice
rotating motion.
You know, at every point it's circulating counterclockwise.
So what does that mean?
Well, again, it's not exactly the same as Part a and b, but
again we'll be able to compute this integral without
parameterizing.
Why?
Because F dot T in this case--
well, so, let's see.
What is the norm of F?
The magnitude of F is just the square root of (x squared plus
y squared).
So on our circle of radius b, that means the
magnitude of F is b.
And the magnitude of T, the unit tangent vector, is 1, and
they point in the same direction.
So when you have two vectors that point in the same
direction, their dot product is just the product of their
magnitudes.
So that means F dot T is equal to b.
This is a constant.
F dot T is equal to b.
So when you integrate around the circle, F dot dr, well,
this is equal to the integral around a circle of F dot the
tangent vector with respect to arc length.
But this integrand, F dot the tangent vector, is this
constant b.
So you're integrating over the curve b ds.
And when you integrate a constant ds, well, that just
gives you the total arc length.
So this is b times the total arc length.
And this is a circle of radius b.
So that's b times 2 pi b, which we could also write as 2
pi b squared.
So there you go.
So in this third case, you have a nice
tangential vector field.
So that means the integrand actually
works out to be constant.
Because the integrand is constant, we don't ever have
to parameterize the curve.
We can just use the fact that we already know its arc length
in order to compute this integral.
Again, we could do all of these integrals if we wanted
by parameterizing the circle, by x equals b cosine T, y
equals b sine T, and going through and writing this as an
integral from T equals 0 to 2 pi, and so on.
But these are examples of problems where it's helpful to
think about what's going on first, see if you can
understand the geometry of your situation.
And sometimes you'll have a problem
like this where you'll--
either in this class or elsewhere in your life--
where something that might seem complicated has a simple
geometric explanation.
And so when that does happen, it's nice when you can take
advantage of it.
Sometimes that won't happen and sometimes you'll have to
do the parameterization and the computation.
But in these cases we have these nice three examples
where with a radial vector field, you get that the
integrand is always zero, or with a tangential vector
field, you have that the integrand is constant.
All right.
So, I'll stop there.