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, S1.
And then there's--
whatever the other piece on the other side of it--
S2.
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 two surfaces.
Right?
C is the boundary of S1, and C-- if we orient
it the other way--
is the boundary of S2, 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 S1 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 S1 of b dS, which is b times
the area of S1.
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 take 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 S1.
So now we have the bottom cap, or whatever.
All the rest of the sphere, S2.
OK.
So we can also get the integral over C of G dot dr.
We can interpret it in terms of Stokes' Theorem.
But notice then that in 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 S2
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 S2 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 S2,
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 that we used to get this whole thing
started was 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.