JOEL LEWIS: Hi.
Welcome back to recitation.
I've got a nice exercise here on Stokes' Theorem for you.
Now this problem is a little bit more
sophisticated than a lot of problems
we've been doing in recitation.
So it requires a little bit more thought, and it also
involves more mathematical sophistication.
So we're doing a clever kind of proof here that I like.
So hopefully you'll like this one.
It's a little bit in a different style than some of the ones
we've done.
So I think I need to talk about it a little bit before we
get started.
So let's let F be the field [x, y, z].
So this is our radial field that we've seen a lot in recitation.
So what I'd like you to do is prove
that this field is not the curl of any field G. All right.
So I'd like you to show that there's
no field G such that F is equal to the curl of G.
Now, rather than just saying that to you
and letting you run off, I have a suggestion
for an interesting way you could go about this.
And this interesting way is going to use Stokes' Theorem.
So what I'd like you to do is a proof by contradiction.
OK, so what you're going to do is
you're going to assume that F is a curl.
OK?
So you're going to assume that there
is some G such that F is curl G. And then you're
going to use that to get a ridiculous conclusion.
So you're going to start with that premise,
and you're going to end up with a contradiction.
So these two arrows colliding into each other
is a symbol that mathematicians use for a contradiction.
So you're going to start from this premise,
and you're going to reach a contradiction.
And what that's going to show is that your premise
couldn't be right.
Yeah?
Because if you start from a true premise,
well then all your conclusions should be true as well.
So if you reach a false conclusion,
then you must have had a false premise.
So what you're going to do is you're
going to take a sphere of radius b centered at the origin.
And a curve C on the sphere.
You know, a simple, closed curve.
So assuming that F is this curl of G, what I'd like you to do
is use Stokes' Theorem to interpret
the line integral of G dot dr over C in two different ways.
OK?
And interpreting this line integral in two different ways,
you're going to reach a contradiction,
and that will show that F really isn't a curl.
So that's what I'd like you to do.
So why don't you pause the video,
go ahead and see if you can work that out, come back,
and we'll talk about it together.
I hope you enjoyed working on this problem.
Let's get started on it.
So as I was saying before we started, what we're going to do
is we're looking for a proof by contradiction.
So as the problem says, we're going to start with a sphere.
And I'm going to take this curve C--
some simply connected closed curve that's going to go around
the back of the sphere, and it's going to be oriented
one way or the other-- and it's going
to divide this sphere into two pieces.
So there's the one cap on one side of it, S_1.
And then there's-- whatever the other piece on the other side
of it is, S_2.
OK.
And so what we're going to do is we're going to think
about, what is this line integral?
OK.
So this is our curve C here on the sphere.
So the integral over C of G dot dr.
So this is what the problem suggests we think about.
So this is a line integral of a field dot dr
over the boundary of a surface.
Well, actually, it's the boundary of two surfaces.
Right?
C is the boundary of S1, and C-- if we orient it
the other way-- is the boundary of S_2, when
we orient them both outwards.
OK, so what is this?
So Stokes' Theorem tells us something
about this line integral.
So let's first think about this as the top cap-- that
cap S1-- with boundary C oriented so that they
agree with each other.
So the normal is outwards on the sphere,
and C is proceeding in the direction
that I've drawn the arrow here.
Well, in that circumstance, we have
that the integral around C of G dot dr, by Stokes' Theorem,
is equal to the surface integral over S_1 of curl of G dot
n with respect to surface area.
Right?
So this is just Stokes' Theorem.
Stokes' Theorem says the line integral
of G around the boundary curve is equal to the surface
integral of the curl of G over the region,
provided all of our orientations are correct.
OK.
Well, we know though what curl of G
is, because by assumption, F is equal to curl of G. OK,
so this is equal to the surface integral over S1 of F dot n dS.
So in the first step, we use Stokes' Theorem.
In the second step, we use our assumption
that curl G is equal to F. Well, now what?
But we know what F is.
Right?
F is this radial field [x, y, z].
So F and n are pointing in the same direction.
They're parallel to each other. n is a unit vector,
so this is just the length of F. This F dot
n is just the length of F. And since we're
on a sphere of radius b, this is just b.
OK, so the integrand is just b.
So this is the integral over S_1 of b
dS, which is b times the area of S_1.
OK.
One thing I'd like you to notice is that in particular, this
is a positive number.
b is positive and the area is positive.
OK.
So that's our first interpretation.
So we took our field G that we suppose exists,
and we integrate it around this curve C,
and we apply Stokes' Theorem, and then
the fact that we know what F is means
that we know what F dot n is, and so that makes our surface
integral very easy to compute.
And it turns out to be b times the area of S, which,
I just happened to notice, is a positive number.
OK.
Well, now we can do the same trick
on the other half of the sphere.
Right?
So we just did the top cap here.
We did S_1.
So now we have the bottom cap, or whatever.
All the rest of the sphere, S_2.
OK.
So we can also get that the integral over C of G dot dr,
we can interpret it in terms of Stokes' Theorem.
But notice then that C-- we still
want to use the same normal.
We like outwards pointing normals.
So we're going to have to orient C the other way in order
to make Stokes' Theorem make sense.
So let's walk over here where we have some empty board space.
So we want to orient C the other way.
So in other words, we're going to take the negative
of this line integral.
So it's minus G dot dr. And if we
apply Stokes' Theorem to this line integral-- so this
is the same line integral, but with the opposite orientation
on C and so with the opposite sign-- by Stokes' Theorem,
this is equal to the integral over S_2 of the curl of G
dot n dS.
And so the way I've set this up, this is still
my outward-pointing normal.
OK.
But again, we can use our assumption
and we can get curl of G is equal to F,
because we're assuming that G has this property.
So this is equal to the integral over S_2
of F dot n with respect to surface area.
And again, n is the outward-pointing normal,
and F is parallel to it.
So this dot product is just the length
of F. The outward pointing unit normal.
So this is just the length of F, which is b.
So this is equal to b times the area of S_2,
which is also positive.
So what have we just shown?
Well, we started from the assumption
that there exists a G such that F is the curl of G.
And starting from that assumption--
let's look-- we showed that the line integral around C of G
dot dr is equal to some positive number.
And we also showed, over here, that the negative
of the line integral of G around C
is equal to some positive number.
Well, this is clearly absurd.
That can't be true.
So starting from our assumption that F was the curl of G--
that there is a G such the F is the curl of G--
we reached an absurd conclusion.
We reached a conclusion that the same number
is both positive and negative.
But that can't happen.
So that means our premise had to be false.
OK.
So this is a contradiction.
So our assumption is false.
And our assumption was-- that we used
to get this whole thing started-- that F was
the curl of some G. All right.
So what have we shown?
So we used a nice argument here with Stokes' Theorem
in order to show that certain fields aren't
the curl of other fields.
So Stokes' Theorem limits the kind of fields that
can be curls of other fields.
Now perhaps, you may have thought of other theorems
that you can use that also limit what sorts of fields
can be curls.
And so there are other ways to reach this true conclusion
that our field F-- whose components are x, y, and z--
is not the curl of any field.
This isn't the only way to reach that conclusion.
But this is a nice way that shows that Stokes' Theorem puts
some limitations on what fields can behave like if they're
going to be curls.
I'll stop there.