WEBVTT
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JOEL LEWIS: Hi.
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Welcome back to recitation.
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I've got a nice exercise here
on Stokes' Theorem for you.
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Now this problem is
a little bit more
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sophisticated than
a lot of problems
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we've been doing in recitation.
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So it requires a little bit
more thought, and it also
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involves more mathematical
sophistication.
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So we're doing a clever kind
of proof here that I like.
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So hopefully you'll
like this one.
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It's a little bit in a different
style than some of the ones
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we've done.
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So I think I need to talk
about it a little bit before we
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get started.
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So let's let F be
the field [x, y, z].
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So this is our radial field that
we've seen a lot in recitation.
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So what I'd like
you to do is prove
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that this field is not the
curl of any field G. All right.
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So I'd like you to
show that there's
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no field G such that F is
equal to the curl of G.
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Now, rather than just
saying that to you
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and letting you run
off, I have a suggestion
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for an interesting way
you could go about this.
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And this interesting way is
going to use Stokes' Theorem.
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So what I'd like you to do
is a proof by contradiction.
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OK, so what you're
going to do is
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you're going to assume
that F is a curl.
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OK?
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So you're going to
assume that there
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is some G such that F is
curl G. And then you're
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going to use that to get
a ridiculous conclusion.
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So you're going to
start with that premise,
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and you're going to end
up with a contradiction.
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So these two arrows
colliding into each other
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is a symbol that mathematicians
use for a contradiction.
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So you're going to
start from this premise,
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and you're going to
reach a contradiction.
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And what that's going to
show is that your premise
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couldn't be right.
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Yeah?
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Because if you start
from a true premise,
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well then all your conclusions
should be true as well.
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So if you reach a
false conclusion,
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then you must have
had a false premise.
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So what you're going
to do is you're
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going to take a sphere of
radius b centered at the origin.
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And a curve C on the sphere.
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You know, a simple,
closed curve.
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So assuming that F is this curl
of G, what I'd like you to do
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is use Stokes'
Theorem to interpret
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the line integral of G dot dr
over C in two different ways.
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OK?
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And interpreting this line
integral in two different ways,
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you're going to reach
a contradiction,
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and that will show that
F really isn't a curl.
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So that's what I'd
like you to do.
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So why don't you
pause the video,
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go ahead and see if you can
work that out, come back,
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and we'll talk
about it together.
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I hope you enjoyed
working on this problem.
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Let's get started on it.
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So as I was saying before we
started, what we're going to do
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is we're looking for a
proof by contradiction.
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So as the problem says, we're
going to start with a sphere.
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And I'm going to
take this curve C--
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some simply connected closed
curve that's going to go around
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the back of the sphere, and
it's going to be oriented
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one way or the
other-- and it's going
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to divide this sphere
into two pieces.
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So there's the one cap
on one side of it, S_1.
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And then there's-- whatever the
other piece on the other side
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of it is, S_2.
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OK.
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And so what we're going to
do is we're going to think
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about, what is
this line integral?
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OK.
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So this is our curve
C here on the sphere.
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So the integral
over C of G dot dr.
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So this is what the problem
suggests we think about.
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So this is a line
integral of a field dot dr
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over the boundary of a surface.
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Well, actually, it's the
boundary of two surfaces.
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Right?
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C is the boundary of S1,
and C-- if we orient it
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the other way-- is the
boundary of S_2, when
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we orient them both outwards.
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OK, so what is this?
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So Stokes' Theorem
tells us something
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about this line integral.
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So let's first think about
this as the top cap-- that
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cap S1-- with boundary
C oriented so that they
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agree with each other.
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So the normal is
outwards on the sphere,
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and C is proceeding
in the direction
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that I've drawn the arrow here.
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Well, in that
circumstance, we have
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that the integral around C of
G dot dr, by Stokes' Theorem,
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is equal to the surface integral
over S_1 of curl of G dot
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n with respect to surface area.
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Right?
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So this is just Stokes' Theorem.
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Stokes' Theorem says
the line integral
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of G around the boundary
curve is equal to the surface
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integral of the curl
of G over the region,
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provided all of our
orientations are correct.
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OK.
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Well, we know though
what curl of G
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is, because by assumption,
F is equal to curl of G. OK,
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so this is equal to the surface
integral over S1 of F dot n dS.
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So in the first step,
we use Stokes' Theorem.
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In the second step,
we use our assumption
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that curl G is equal
to F. Well, now what?
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But we know what F is.
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Right?
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F is this radial
field [x, y, z].
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So F and n are pointing
in the same direction.
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They're parallel to each
other. n is a unit vector,
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so this is just the
length of F. This F dot
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n is just the length
of F. And since we're
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on a sphere of radius
b, this is just b.
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OK, so the integrand is just b.
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So this is the
integral over S_1 of b
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dS, which is b times
the area of S_1.
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OK.
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One thing I'd like you to notice
is that in particular, this
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is a positive number.
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b is positive and
the area is positive.
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OK.
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So that's our first
interpretation.
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So we took our field G
that we suppose exists,
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and we integrate it
around this curve C,
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and we apply Stokes'
Theorem, and then
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the fact that we
know what F is means
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that we know what F dot n is,
and so that makes our surface
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integral very easy to compute.
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And it turns out to be b
times the area of S, which,
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I just happened to notice,
is a positive number.
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OK.
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Well, now we can
do the same trick
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on the other half of the sphere.
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Right?
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So we just did the top cap here.
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We did S_1.
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So now we have the
bottom cap, or whatever.
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All the rest of the sphere, S_2.
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OK.
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So we can also get that the
integral over C of G dot dr,
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we can interpret it in
terms of Stokes' Theorem.
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But notice then
that C-- we still
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want to use the same normal.
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We like outwards
pointing normals.
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So we're going to have to
orient C the other way in order
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to make Stokes'
Theorem make sense.
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So let's walk over here where
we have some empty board space.
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So we want to orient
C the other way.
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So in other words, we're
going to take the negative
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of this line integral.
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So it's minus G
dot dr. And if we
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apply Stokes' Theorem to
this line integral-- so this
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is the same line integral, but
with the opposite orientation
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on C and so with the opposite
sign-- by Stokes' Theorem,
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this is equal to the integral
over S_2 of the curl of G
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dot n dS.
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And so the way I've set
this up, this is still
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my outward-pointing normal.
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OK.
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But again, we can
use our assumption
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and we can get curl
of G is equal to F,
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because we're assuming
that G has this property.
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So this is equal to
the integral over S_2
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of F dot n with respect
to surface area.
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And again, n is the
outward-pointing normal,
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and F is parallel to it.
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So this dot product
is just the length
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of F. The outward
pointing unit normal.
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So this is just the
length of F, which is b.
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So this is equal to b
times the area of S_2,
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which is also positive.
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So what have we just shown?
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Well, we started
from the assumption
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that there exists a G such
that F is the curl of G.
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And starting from
that assumption--
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let's look-- we showed that
the line integral around C of G
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dot dr is equal to
some positive number.
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And we also showed, over
here, that the negative
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of the line integral
of G around C
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is equal to some
positive number.
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Well, this is clearly absurd.
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That can't be true.
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So starting from our assumption
that F was the curl of G--
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that there is a G such
the F is the curl of G--
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we reached an absurd conclusion.
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We reached a conclusion
that the same number
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is both positive and negative.
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But that can't happen.
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So that means our
premise had to be false.
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OK.
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So this is a contradiction.
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So our assumption is false.
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And our assumption
was-- that we used
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to get this whole thing
started-- that F was
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the curl of some G. All right.
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So what have we shown?
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So we used a nice argument
here with Stokes' Theorem
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in order to show that
certain fields aren't
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the curl of other fields.
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So Stokes' Theorem limits
the kind of fields that
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can be curls of other fields.
00:10:44.630 --> 00:10:48.060
Now perhaps, you may have
thought of other theorems
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that you can use that also
limit what sorts of fields
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can be curls.
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And so there are other ways
to reach this true conclusion
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that our field F-- whose
components are x, y, and z--
00:10:58.250 --> 00:11:00.310
is not the curl of any field.
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This isn't the only way
to reach that conclusion.
00:11:02.700 --> 00:11:06.000
But this is a nice way that
shows that Stokes' Theorem puts
00:11:06.000 --> 00:11:08.970
some limitations on what fields
can behave like if they're
00:11:08.970 --> 00:11:10.380
going to be curls.
00:11:10.380 --> 00:11:11.972
I'll stop there.