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PROFESSOR STRANG: OK.
 

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So this is lecture 22, gradient
and divergence, headed

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for Laplace's equation.
 

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So the gradient will be our
operator A, the divergence, or

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minus the divergence, will be A
transpose, and then A transpose

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A will be the Laplacian.
 

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We get to Laplace's
equation Wednesday.

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Today I wanted to take
them separately.

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To understand the meaning
of gradient, the meaning

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of divergence, the
connection between them.

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I mentioned at the end of
last time that one is the

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transpose of the other,
or minus the transpose.

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I'll try to keep gradient on
this side, and if I could only

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transpose the blackboard, I
could do divergence -- I'll

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do divergence on that side.
 

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And I guess if I could get a
rotating blackboard, right in

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the middle I could do curl.
 

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That would be perfect.
 

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[LAUGHTER]
 

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OK.
 

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So some of this will
not be new to you.

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But maybe some of the insights
or the ways of looking

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at it could be new.
 

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This is the background
of vector calculus.

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So we have things
like vector fields.

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That means I have a vector,
, at each

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point, (x, y).
 

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So I could draw a little arrow
at every point to show the

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direction and magnitude
of that vector.

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I have a field of vectors.
 

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OK, so.
 

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From last time, our basic
setup is, the gradient is

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this first operator, A.
 

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The one we see at
the beginning.

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One change.
 

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Instead of calling the result
e, let me connect to velocity.

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I'll be thinking of u as a
potential, I'll use the word

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potential for u, and I'll use
v instead of e for the

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velocity component.
 

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So that's the A.
 

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And then on the other
side, I start with a w.

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Again, it's a vector field,
it's actually a momentum.

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Very often the step between
here and here -- well most

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often, the step between here
and here will be the identity.

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That's what gives
Laplace's equation.

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So you'll have to watch,
I'm sometimes confusing

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the v's with the w's.
 

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Because when I go to Laplace's
equation and c is the

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identity, they're the same.
 

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But I would like, today, to
try to keep this left side

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separate, the gradient, from
the right side, the divergence.

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And understand what they mean,
and how to work with them.

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And of course, a big connection
is the divergence theorem,

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or the Gauss-Green
connection, identity.

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We'll get to that.
 

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OK, gradient first.
 

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First, what does it mean?
 

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If I have a function u, what
does it is gradient tell me?

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And then the second is kind
of a backwards question.

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Suppose I have the v.
 

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Is it the gradient of some u?
 

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So one direction, from u to v,
and the second direction will

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be from v back to
u when possible.

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OK, meaning of the gradient.
 

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So the gradient is just the
obvious thing, the two

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derivatives in the
x and y direction.

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So of course the gradient gives
you the rate of change, the

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partial derivatives of u.
 

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But how do you see
that in a picture?

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Let me draw an important curve.
 

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I'll start with a very
simple u. u is x

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squared plus y squared.
 

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So that's my example.
 

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Example one.
 

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And what I've drawn is
an equipotential curve.

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Or isopotential might be a more
appropriate word these days,

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but we still say equipotential.
 

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It means that, along this
curve, u is a constant.

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00:04:47 --> 00:04:51
And for this particular
potential, this simple one to

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work with, x squared plus y
squared, the curve is a circle.

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So the curve would be a circle.
 

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OK, now what do I learn
by taking the gradient?

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00:05:08 --> 00:05:14
So the gradient of u, this is
my v, is the x derivative,

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which is 2x, and the y
derivative, which is 2y.

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OK.
 

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Let me take a typical point
on the curve and draw

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that gradient vector.
 

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00:05:29 --> 00:05:36
So this is the x, y plane, this
is the curve u equal constant.

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00:05:36 --> 00:05:40
When I have a curve u equal
constant and I draw the

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gradient of u, where
does it point?

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00:05:43 --> 00:05:48
This is the first and most
important, simple idea

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about the gradient vector.
 

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The gradient vector points --
does the gradient vector point,

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could it point any old way?
 

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No.
 

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The gradient vector is
perpendicular to the curve.

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00:06:04 --> 00:06:08
And we can see that, for this
simple example, that vector

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, that's a vector
radially outwards, right,

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if here's the origin.
 

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00:06:14 --> 00:06:18
And if, at this point, I don't
know its coordinates, whatever

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they are, maybe 2, 1 or
something, the gradient

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00:06:24 --> 00:06:25
vector would be .
 

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00:06:25 --> 00:06:28
It would be a multiple
-- here's the position

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vector, .
 

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The point is, the gradient
vector points out.

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Perpendicular to the curve.
 

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That's what the
gradient tells you.

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It tells you, in this situation
it's telling me which direction

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is perpendicular to the curve.
 

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Now how do I understand that?
 

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00:06:52 --> 00:06:54
How do I see that?
 

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00:06:54 --> 00:06:59
I think of this -- and
let me try to draw the

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00:06:59 --> 00:07:01
whole surface, u=x+y^2.
 

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What would that whole
surface look like?

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00:07:07 --> 00:07:12
Let me try to put that
picture -- this is a

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2-D picture right now.
 

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It's sort of a cross-section.
 

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Let me put this 2-D picture
into a 3-D picture to see.

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So the 3-D picture is a picture
of the whole surface. u going

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up, x and y going around.
 

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And what kind of a curve,
what kind of a surface do I

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00:07:31 --> 00:07:37
have for that function, x
squared plus y squared.

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Well, it grows, right?
 

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It's sort of, I use
the word bowl.

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And we've seen it before.
 

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It sort of goes up, right, and
it goes up faster and faster as

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I go out, because
of the squares.

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And now what I want to do
is, suppose I take the

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cross-section, suppose
I take this bowl.

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00:08:02 --> 00:08:07
Do you see a beautiful
bowl there?

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00:08:07 --> 00:08:13
And now, I cut through it,
horizontal cross-section,

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00:08:13 --> 00:08:15
at the height c.
 

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What do I get if I take this
surface in 3-D, and I cut

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through it by a plane, the
plane at that height, c.

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00:08:31 --> 00:08:37
So if the plane went through
there, that distance was c.

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It cuts us a cross-section
out of the bowl.

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00:08:40 --> 00:08:43
And what's the cross section?
 

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00:08:43 --> 00:08:44
It's the circle.
 

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00:08:44 --> 00:08:45
It's this one.
 

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00:08:45 --> 00:08:53
So out of the bowl, this plane
is cutting this cross-section.

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This is height u
equal constant.

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00:08:56 --> 00:09:00
So I'm really intersecting one
surface, the plane u equal

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constant, with the bowl.
 

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And the intersection of those
two is the equipotential,

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00:09:07 --> 00:09:09
the curve, u=c.
 

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Right?
 

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00:09:12 --> 00:09:14
You see the two pictures?
 

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00:09:14 --> 00:09:18
And now, what does the gradient
tell me in that picture?

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00:09:18 --> 00:09:22
What does the gradient of u
tell me in that picture?

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00:09:22 --> 00:09:26
Over here, it pointed
outwards, but now we're

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00:09:26 --> 00:09:28
going to kind of see why.
 

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So, it pointed outwards for
this nice function, u.

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I could have made a more
general function u, but let me

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just stay with this simple one.
 

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So suppose I'm climbing.
 

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00:09:47 --> 00:09:52
This is like, I'm climbing out
of a volcano or something.

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Usually, I would say a
mountain, but the way I've

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drawn the thing, it's
not much of a mountain.

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00:09:57 --> 00:09:59
So OK, volcano.
 

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Climbing out of it.
 

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And I got up to this point,
which was this point.

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00:10:06 --> 00:10:09
Now, what does the
gradient tell me?

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00:10:09 --> 00:10:11
That when I'm climbing away,
I reach -- duh duh duh

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00:10:11 --> 00:10:13
duh -- I get up to here.
 

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What does the gradient tell me?
 

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00:10:15 --> 00:10:17
It tells me which way to go.
 

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00:10:17 --> 00:10:21
It tells me the steepest
direction upwards, and

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00:10:21 --> 00:10:23
it tells me how steep.
 

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00:10:23 --> 00:10:23
Right.
 

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00:10:23 --> 00:10:26
So it tells me those
two things, direction

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and magnitude.
 

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00:10:27 --> 00:10:34
Direction is, well for this
special function you know the

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00:10:34 --> 00:10:37
direction is, like, straight
outwards, straight upwards.

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00:10:37 --> 00:10:39
And how steep is it?
 

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00:10:39 --> 00:10:45
What's the steepness?
 

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00:10:45 --> 00:10:49
It's the size of, it's
the magnitude of

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00:10:49 --> 00:10:51
this vector, grad u.
 

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00:10:51 --> 00:11:02
Which is the-- this is not
-- how big is this vector?

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This is my vector, grad u.
 

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This is another shorthand
notation for gradient.

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00:11:09 --> 00:11:17
The size of that vector is
the square root of, what?

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00:11:17 --> 00:11:19
The length of a vector is
the square root of the

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00:11:19 --> 00:11:21
sum of the squares.
 

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00:11:21 --> 00:11:24
So I have the length of v is
the square root of 4x^2 + 4y^2.

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00:11:24 --> 00:11:31
 
 

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00:11:31 --> 00:11:33
OK.
 

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00:11:33 --> 00:11:37
Now, it gets steeper and
steeper, of course.

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00:11:37 --> 00:11:45
The cross sections here
would be all circles for

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this simple function.
 

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And the gradients would keep
pointing out, and we'd have a

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00:11:50 --> 00:11:58
function that comes from
a surface like that.

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00:11:58 --> 00:12:02
OK, this is what I wanted to
say about question one, the

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meaning of the gradient.
 

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00:12:06 --> 00:12:10
The thing to remember, if you
remember the two pictures,

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you've really got the idea of
what the gradient

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is telling you.
 

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For a function of one
variable, the derivative

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is telling you the slope.
 

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00:12:19 --> 00:12:23
Well, we've got slopes in the
x direction, slopes in the y

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direction, and the gradient
direction tells us

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00:12:27 --> 00:12:29
the steepest slope.
 

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And it tells us -- yeah,
tells us what that slope is.

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00:12:34 --> 00:12:38
OK, so much for the
meaning of the gradient.

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00:12:38 --> 00:12:40
Now, backwards.
 

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00:12:40 --> 00:12:42
Suppose I have v.
 

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Because these gradient fields
are extremely important.

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00:12:46 --> 00:12:50
I mean, they're
wonderful fields.

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00:12:50 --> 00:12:55
If you've got a v_1 and a v_2
that is the gradient of some

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00:12:55 --> 00:12:59
u, you want to know that.
 

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00:12:59 --> 00:13:02
I mean, that's good news.
 

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00:13:02 --> 00:13:03
Most fields will not.
 

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00:13:03 --> 00:13:10
So let's figure out, when is --
can I life that up now, and do

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00:13:10 --> 00:13:12
the other gradient thing now?
 

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00:13:12 --> 00:13:22
Or maybe I'll put the final
gradient blackboard here, and

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then make way for divergence.
 

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00:13:25 --> 00:13:29
OK, so I'm asking
now question two.

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00:13:29 --> 00:13:37
I'm given v_1 and v_2, and I
want this to be du/dx for some

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00:13:37 --> 00:13:39
u, and I want this to be du/dy.
 

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00:13:39 --> 00:13:42
 
 

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00:13:42 --> 00:13:45
So I've got two equations.
 

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00:13:45 --> 00:13:47
If I'm looking for u.
 

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00:13:47 --> 00:13:52
Remember, I'm now starting with
these, looking for the u.

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00:13:52 --> 00:13:53
OK.
 

232
00:13:53 --> 00:13:54
So, am I going to find Au?
 

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00:13:56 --> 00:14:01
Am I going to find a function,
u, whose x derivative is my v_1

234
00:14:01 --> 00:14:06
that was given, and the
y derivative is my v_2?

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00:14:07 --> 00:14:09
Well, chances are not good.
 

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Right?
 

237
00:14:10 --> 00:14:13
I've got two equations here,
but only one unknown.

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00:14:13 --> 00:14:17
So generally, it's like
a rectangular system.

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00:14:17 --> 00:14:21
Usually there's no solution,
but sometimes there is.

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So we have to find out, what's
the test for consistency for

241
00:14:29 --> 00:14:31
these to have a solution?
 

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00:14:31 --> 00:14:40
And this brings us right away
to the key identity from

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partial derivatives.
 

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OK.
 

245
00:14:43 --> 00:14:48
Can you see some condition that
has to-- how can I connect

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00:14:48 --> 00:14:52
these two equations, and
therefore connect v_1 and v_2?

247
00:14:53 --> 00:14:59
That'll be the test for weather
-- you could say, in matrix

248
00:14:59 --> 00:15:02
language, I'm asking whether
 is in the column

249
00:15:02 --> 00:15:06
space of the gradient.
 

250
00:15:06 --> 00:15:10
Is it something I can get
from this tall thin matrix?

251
00:15:10 --> 00:15:16
OK, they key idea is take
the y derivative of that

252
00:15:16 --> 00:15:22
equation, d second u/dydx.
 

253
00:15:24 --> 00:15:27
And take the x derivative
of this equation. dv_2/dy.

254
00:15:28 --> 00:15:36
dx is the second derivative of
u, respect to the x derivative

255
00:15:36 --> 00:15:38
of the y derivative.
 

256
00:15:38 --> 00:15:39
OK.
 

257
00:15:39 --> 00:15:45
So it's like I'm operating
on the equations.

258
00:15:45 --> 00:15:48
I'm doing what I'm allowed to
do, I'm doing the same thing to

259
00:15:48 --> 00:15:50
both sides of each equation.
 

260
00:15:50 --> 00:15:52
Now I'm going to eliminate.
 

261
00:15:52 --> 00:15:54
And what am I going to find?
 

262
00:15:54 --> 00:15:56
What's the key point here?
 

263
00:15:56 --> 00:16:00
The key, key point about
second derivatives is

264
00:16:00 --> 00:16:02
that that equals that.
 

265
00:16:02 --> 00:16:04
Right?
 

266
00:16:04 --> 00:16:06
Take any function.
 

267
00:16:06 --> 00:16:09
Do we want to practice
with a function, just

268
00:16:09 --> 00:16:10
to see it be true?
 

269
00:16:10 --> 00:16:14
Take, let u be x cubed y.
 

270
00:16:14 --> 00:16:16
Just for the heck of it.
 

271
00:16:16 --> 00:16:18
OK, I don't know what --
is that going to produce

272
00:16:18 --> 00:16:19
anything interesting?
 

273
00:16:19 --> 00:16:22
Then maybe I'd better
make it y squared.

274
00:16:22 --> 00:16:25
I don't know why I'm doing
this, it just is like an

275
00:16:25 --> 00:16:27
example, to make it believable.
 

276
00:16:27 --> 00:16:34
OK, so du/dx, u_x is -- and
I'll often use u_x as a sort

277
00:16:34 --> 00:16:36
of shorthand -- will be what?
 

278
00:16:36 --> 00:16:39
3x squared y squared.
 

279
00:16:39 --> 00:16:44
And u_y, taking the y
derivative is just, x is

280
00:16:44 --> 00:16:47
constant, 2x cubed y.
 

281
00:16:47 --> 00:16:53
And now let me do u_xy,
the y derivative of this.

282
00:16:53 --> 00:16:56
Which is what?
 

283
00:16:56 --> 00:16:59
It's been a long weekend,
but hey, we can do these.

284
00:16:59 --> 00:17:05
The y derivative of that is
going to be 6x squared y.

285
00:17:05 --> 00:17:11
And the x derivative of this is
going to be, so I should say y

286
00:17:11 --> 00:17:16
-- I've got the y derivative
and now I should take the

287
00:17:16 --> 00:17:17
x derivative of that.
 

288
00:17:17 --> 00:17:19
And what do I get?
 

289
00:17:19 --> 00:17:25
The x derivative of
that is 6x squared y.

290
00:17:25 --> 00:17:25
And look!
 

291
00:17:25 --> 00:17:27
They're the same.
 

292
00:17:27 --> 00:17:28
Hooray.
 

293
00:17:28 --> 00:17:29
OK.
 

294
00:17:29 --> 00:17:36
So if the function is smooth
and has these derivatives,

295
00:17:36 --> 00:17:39
they'll come out the same.
 

296
00:17:39 --> 00:17:42
And therefore, so
what do I conclude?

297
00:17:42 --> 00:17:49
What's the test on v_1 and
v_2 that it must pass

298
00:17:49 --> 00:17:52
to be a gradient field?
 

299
00:17:52 --> 00:17:56
For there to be a function, u,
that solves these equations.

300
00:17:56 --> 00:18:01
These are solvable
only when what?

301
00:18:01 --> 00:18:03
Well, if these are the same,
these have to be the same.

302
00:18:03 --> 00:18:16
So it's solvable only when
-- I need dv_2/dx-dv_1/dy

303
00:18:16 --> 00:18:19
to be what?
 

304
00:18:19 --> 00:18:20
Zero.
 

305
00:18:20 --> 00:18:21
OK.
 

306
00:18:21 --> 00:18:23
That's the conclusion.
 

307
00:18:23 --> 00:18:28
Those have to be the same.
dv_2/dx-dv_1/dy has to be zero.

308
00:18:28 --> 00:18:31
OK.
 

309
00:18:31 --> 00:18:35
Because those are the same.
 

310
00:18:35 --> 00:18:40
I guess I came to this early
because it's the key identity

311
00:18:40 --> 00:18:43
of vector calculus.
 

312
00:18:43 --> 00:18:46
Well, the key identity behind
vector calculus is this

313
00:18:46 --> 00:18:51
fact about the derivatives.
 

314
00:18:51 --> 00:18:58
Can I just throw out a question
that you might think about?

315
00:18:58 --> 00:19:01
We already have seen, so
often now, the second

316
00:19:01 --> 00:19:04
derivative, like v_xx.
 

317
00:19:05 --> 00:19:07
Or u_xx, suppose.
 

318
00:19:07 --> 00:19:09
So here's a little
question to think about.

319
00:19:09 --> 00:19:13
So think.
 

320
00:19:13 --> 00:19:17
OK.
 

321
00:19:17 --> 00:19:23
I just want to bring finite
differences in for a moment.

322
00:19:23 --> 00:19:26
u_xx we've got a handle on.
 

323
00:19:26 --> 00:19:29
We know that that's
like u(x+h).

324
00:19:31 --> 00:19:36
And if there was a y, put
in the y, minus 2u at x.

325
00:19:36 --> 00:19:42
And I can put in a y
there, plus u(x-h,y).

326
00:19:42 --> 00:19:45
 
 

327
00:19:45 --> 00:19:47
We've seen that.
 

328
00:19:47 --> 00:19:52
That'll be the x derivative,
second x derivative -- sorry!

329
00:19:52 --> 00:19:56
Second x difference.
 

330
00:19:56 --> 00:20:00
Since we're taking x
derivatives and x differences,

331
00:20:00 --> 00:20:03
it's x that moves,
and y doesn't move.

332
00:20:03 --> 00:20:08
Just the central idea of
partial derivatives. u_yy will

333
00:20:08 --> 00:20:11
be similar with y moving.
 

334
00:20:11 --> 00:20:14
And my question to you is --
and we could have asked it way

335
00:20:14 --> 00:20:17
way back -- what's a finite
difference approximation

336
00:20:17 --> 00:20:23
to u(x,y), to the
cross derivative?

337
00:20:23 --> 00:20:27
That equals what?
 

338
00:20:27 --> 00:20:30
I want to go to
finite differences.

339
00:20:30 --> 00:20:39
OK, what finite differences?
 

340
00:20:39 --> 00:20:40
Which finite differences?
 

341
00:20:40 --> 00:20:43
Maybe that's a question
to think about.

342
00:20:43 --> 00:20:50
If I can remember, I'll include
it just as a small homework

343
00:20:50 --> 00:20:55
question for the homework
on this material.

344
00:20:55 --> 00:20:57
So that's looking
ahead, really, today.

345
00:20:57 --> 00:21:01
We're not making
things discrete.

346
00:21:01 --> 00:21:05
We're in continuous x and y.
 

347
00:21:05 --> 00:21:07
We have vector fields.
 

348
00:21:07 --> 00:21:20
And we now know the test
for a gradient field.

349
00:21:20 --> 00:21:24
I'm tempted to use
the word curl here.

350
00:21:24 --> 00:21:28
I'm tempted to use
the word curl.

351
00:21:28 --> 00:21:32
I want to connect that test --
may I use the word curl without

352
00:21:32 --> 00:21:39
-- and I'll say why I'm not
going to do everything properly

353
00:21:39 --> 00:21:40
with curl right away.
 

354
00:21:40 --> 00:21:45
I would describe this as curl.
 

355
00:21:45 --> 00:21:51
The test is, the curl
of v has to be zero.

356
00:21:51 --> 00:22:01
So for me, that's
the curl of v.

357
00:22:01 --> 00:22:03
You're going to say,
wait a minute.

358
00:22:03 --> 00:22:06
I learned about curl,
and that doesn't look

359
00:22:06 --> 00:22:10
like the curl to me.
 

360
00:22:10 --> 00:22:14
So I'll say wait
another minute.

361
00:22:14 --> 00:22:16
It's not that far off.
 

362
00:22:16 --> 00:22:18
OK, what's your objection?
 

363
00:22:18 --> 00:22:22
Your objection is that the curl
is in three dimensional space.

364
00:22:22 --> 00:22:23
Right?
 

365
00:22:23 --> 00:22:26
When you saw curl, and of
course it comes in this

366
00:22:26 --> 00:22:33
section of the book, we
had functions of x, y, z.

367
00:22:33 --> 00:22:39
And the curl had
three components.

368
00:22:39 --> 00:22:42
And those three components
-- do you remember curl?

369
00:22:42 --> 00:22:44
I mean, if you remember
curl, you're a good person.

370
00:22:44 --> 00:22:49
Because it's got this -- you
sort of remember that it

371
00:22:49 --> 00:22:54
has things like this.
 

372
00:22:54 --> 00:22:55
Right?
 

373
00:22:55 --> 00:23:00
Sort of, differences of
derivatives, and the indices

374
00:23:00 --> 00:23:01
follow a certain pattern.
 

375
00:23:01 --> 00:23:04
I'm saying that this is
the natural -- this is

376
00:23:04 --> 00:23:08
the curl in a plane.
 

377
00:23:08 --> 00:23:13
What do I mean by the
curl in a plane?

378
00:23:13 --> 00:23:22
So in 3-D, v has components
of v_1, v_2, and v_3.

379
00:23:23 --> 00:23:23
Right?
 

380
00:23:23 --> 00:23:27
That depend on x, y, z.
 

381
00:23:27 --> 00:23:33
In the plane, in 3-D, I have
a vector field, v, which has

382
00:23:33 --> 00:23:38
components v_1, v_2, v_3.
 

383
00:23:38 --> 00:23:39
All depending on x, y, z.
 

384
00:23:39 --> 00:23:45
That's the general 3-D picture,
where you usually see the curl.

385
00:23:45 --> 00:23:48
Now, in the plane,
what's happening?

386
00:23:48 --> 00:23:52
In the plane, we have
two components.

387
00:23:52 --> 00:23:56
So what's happening?
 

388
00:23:56 --> 00:23:58
Think of this.
 

389
00:23:58 --> 00:24:02
So in 3-D, our velocity
field is like, the flow is

390
00:24:02 --> 00:24:03
going all over the place.
 

391
00:24:03 --> 00:24:04
Right?
 

392
00:24:04 --> 00:24:06
It's a three dimensional flow.
 

393
00:24:06 --> 00:24:11
But now suppose my flow
stays in the plane.

394
00:24:11 --> 00:24:18
So in 2-D -- so now I have
to put 2-D up above here.

395
00:24:18 --> 00:24:26
So in 2-D, a plane field is
what I'm working with today.

396
00:24:26 --> 00:24:29
My v is some v_1(x,y).
 

397
00:24:29 --> 00:24:31
 
 

398
00:24:31 --> 00:24:39
No z, no dependence on z.
v_2(x,y), the y direction of

399
00:24:39 --> 00:24:42
the velocity doesn't depend on
z, because this is a plane

400
00:24:42 --> 00:24:44
field, same on every plane.
 

401
00:24:44 --> 00:24:47
And the third component of this
plane field, the velocity

402
00:24:47 --> 00:24:52
perpendicular in the z
direction, is zero.

403
00:24:52 --> 00:24:53
OK. one.
 

404
00:24:53 --> 00:25:01
So what I want to say is that
if I look, if I specialize too

405
00:25:01 --> 00:25:05
plane fields, to fields
like these, then the only

406
00:25:05 --> 00:25:08
component of the curl that
survives is this one.

407
00:25:08 --> 00:25:11
See, the other components of
the curl -- which I'm not even

408
00:25:11 --> 00:25:15
writing down -- the other
components of a curl have

409
00:25:15 --> 00:25:19
derivatives of v_3,
but v_3 is zero.

410
00:25:19 --> 00:25:21
And they also have derivatives
of these guys with

411
00:25:21 --> 00:25:24
respect to z.
 

412
00:25:24 --> 00:25:26
But they don't depend on z.
 

413
00:25:26 --> 00:25:28
So that's why all the other
pieces of the curl, like,

414
00:25:28 --> 00:25:31
are automatically zero
for a plane field.

415
00:25:31 --> 00:25:36
So that the only component
that's significant, the test

416
00:25:36 --> 00:25:40
curl v equals zero, boils
down to a test not on three

417
00:25:40 --> 00:25:41
things, but just on one.
 

418
00:25:41 --> 00:25:44
And that's the one.
 

419
00:25:44 --> 00:25:51
So because I want to stay
mostly with plane fields and

420
00:25:51 --> 00:25:54
two dimensional problems, I
just had to comment that,

421
00:25:54 --> 00:25:59
if the curl was to get in
here, it would fit fine.

422
00:25:59 --> 00:26:03
And if I restrict the curl to
the fields I'm working with,

423
00:26:03 --> 00:26:07
plane fields, then there's only
one component I'll have to

424
00:26:07 --> 00:26:11
think about, it has to be zero
to have a gradient field.

425
00:26:11 --> 00:26:15
OK, now I guess I should
just do an example or two.

426
00:26:15 --> 00:26:19
Can I give you a v_1 and v_2,
and you tell me, is it a

427
00:26:19 --> 00:26:22
gradient field or is it
not a gradient field.

428
00:26:22 --> 00:26:27
Let me give you a different,
let me just change these guys.

429
00:26:27 --> 00:26:33
Suppose I change
that to y and x.

430
00:26:33 --> 00:26:38
So there is a v, a different v.
 

431
00:26:38 --> 00:26:39
That's a vector field.
 

432
00:26:39 --> 00:26:44
At every point x, y, I've
got a little vector.

433
00:26:44 --> 00:26:47
I could try, even,
to draw them.

434
00:26:47 --> 00:26:50
And I'm going to ask you, is
it the gradient of any u.

435
00:26:50 --> 00:26:53
And if it is, what's that u?
 

436
00:26:53 --> 00:26:57
So let me show you what I
mean by a vector field.

437
00:26:57 --> 00:27:04
I mean, at a typical point like
x=1, y=0, the vector -- let's

438
00:27:04 --> 00:27:11
see, if x is one and y is zero,
then what's the gradient at

439
00:27:11 --> 00:27:19
that point? , am I right?
 

440
00:27:19 --> 00:27:22
I won't draw it too big,
or you won't be able

441
00:27:22 --> 00:27:24
to see a darn thing.
 

442
00:27:24 --> 00:27:31
OK, what about at
the point (1, 1)?

443
00:27:31 --> 00:27:37
Which way is my vector
field going? .

444
00:27:37 --> 00:27:41
So what's that look like?
 

445
00:27:41 --> 00:27:45
Plotting the vector field,
v, at a bunch of points.

446
00:27:45 --> 00:27:51
So you get like a map
of little arrows.

447
00:27:51 --> 00:27:53
So here it would go that way.
 

448
00:27:53 --> 00:27:55
Is that right?
 

449
00:27:55 --> 00:27:56
Huh.
 

450
00:27:56 --> 00:28:00
I wasn't expecting that,
to tell the truth.

451
00:28:00 --> 00:28:05
Let's see, so can I
get some more points?

452
00:28:05 --> 00:28:06
Let's see.
 

453
00:28:06 --> 00:28:09
What if I have, there.
 

454
00:28:09 --> 00:28:12
What's -- at (1/2, 0).
 

455
00:28:12 --> 00:28:20
Then, v is .
 

456
00:28:20 --> 00:28:24
Where is this flow going?
 

457
00:28:24 --> 00:28:30
See, if the point is along this
line, where y is equal to x,

458
00:28:30 --> 00:28:34
then the flow was going
out along this line.

459
00:28:34 --> 00:28:36
Can you give me some other
point here, just so we

460
00:28:36 --> 00:28:38
get some handle on this?
 

461
00:28:38 --> 00:28:42
There's the point
(2, 1), let's say.

462
00:28:42 --> 00:28:43
Let me put it over
a little bit.

463
00:28:43 --> 00:28:46
How about the point (2, 1).
 

464
00:28:46 --> 00:28:48
What's the vector if I
just want to draw --

465
00:28:48 --> 00:28:50
I'm just drawing here.
 

466
00:28:50 --> 00:28:55
And of course, code would do it
much better than I'm doing. (2,

467
00:28:55 --> 00:28:59
1), that gives me
, right?

468
00:28:59 --> 00:29:04
So at (2, 4) is over two
and up, it's like this.

469
00:29:04 --> 00:29:11
I think -- but I don't swear to
it -- that if I connect all

470
00:29:11 --> 00:29:14
this -- see, now you have to
take a big leap of faith.

471
00:29:14 --> 00:29:19
Imagine, like at every point
we've got these little arrows,

472
00:29:19 --> 00:29:24
and I want to connect them up.
 

473
00:29:24 --> 00:29:26
Let me do something.
 

474
00:29:26 --> 00:29:30
I'll do that, but let me come
back to the question I should

475
00:29:30 --> 00:29:32
have asked you first.
 

476
00:29:32 --> 00:29:35
Is this a gradient field?
 

477
00:29:35 --> 00:29:38
Does it satisfy the curl
zero condition that

478
00:29:38 --> 00:29:40
we put in a box here?
 

479
00:29:40 --> 00:29:45
Does that satisfy dv -- there
is v_2 and there's v_1.

480
00:29:45 --> 00:29:54
Is dv_2/dx, whatever that test
was, minus dv_1/dy equal zero?

481
00:29:54 --> 00:29:57
Yes, no?
 

482
00:29:57 --> 00:30:05
Yes, right? dv_2/dx is
two, and dv_1/dy is two.

483
00:30:05 --> 00:30:09
So what's the conclusion then?
 

484
00:30:09 --> 00:30:13
It satisfied my little
test, so this must be

485
00:30:13 --> 00:30:16
the gradient of some u.
 

486
00:30:16 --> 00:30:18
Right?
 

487
00:30:18 --> 00:30:20
That's the question we have.
 

488
00:30:20 --> 00:30:23
Which vector fields -- and we
found that this is one of

489
00:30:23 --> 00:30:25
them, it passes the test.
 

490
00:30:25 --> 00:30:26
It's the gradient of some u.
 

491
00:30:26 --> 00:30:28
What's the u?
 

492
00:30:28 --> 00:30:33
What's the function, u,
which is supposed to exist,

493
00:30:33 --> 00:30:37
whose gradient is that.
 

494
00:30:37 --> 00:30:38
2xy.
 

495
00:30:38 --> 00:30:40
Did everybody spot that one?
 

496
00:30:40 --> 00:30:41
2xy.
 

497
00:30:41 --> 00:30:43
 
 

498
00:30:43 --> 00:30:46
Because the x derivative
has to be 2y.

499
00:30:48 --> 00:30:51
So I just integrate
with respect to x.

500
00:30:51 --> 00:30:53
This is the x derivative.
 

501
00:30:53 --> 00:30:53
It's 2y.
 

502
00:30:54 --> 00:30:56
Take the integral with
respect to x, and I get 2xy.

503
00:30:57 --> 00:31:05
And there could be some term
that depended only on y.

504
00:31:05 --> 00:31:11
Anyway, this works.
 

505
00:31:11 --> 00:31:21
I think that this, maybe gives
me somehow -- oh, yeah.

506
00:31:21 --> 00:31:24
What's the equipotential
curve now?

507
00:31:24 --> 00:31:28
Oh, yeah, this picture's
going to come together.

508
00:31:28 --> 00:31:34
What is the equipotential
curve for this potential?

509
00:31:34 --> 00:31:38
It was a circle for the first
guy, but circles are out now.

510
00:31:38 --> 00:31:40
I changed v.
 

511
00:31:40 --> 00:31:42
I've got a new
potential function.

512
00:31:42 --> 00:31:50
And now I want to draw, in this
graph, the equipotentials.

513
00:31:50 --> 00:31:53
Suppose u is one.
 

514
00:31:53 --> 00:32:00
Suppose I draw the curve
2xy=1 in that picture.

515
00:32:00 --> 00:32:02
What kind of a curve is it?
 

516
00:32:02 --> 00:32:04
Do you recognize this?
 

517
00:32:04 --> 00:32:06
The Greeks would.
 

518
00:32:06 --> 00:32:07
Recognize 2xy=1.
 

519
00:32:08 --> 00:32:10
Or you could say y=1/2x.
 

520
00:32:11 --> 00:32:14
That gives you a quick
handle on the curve.

521
00:32:14 --> 00:32:19
It comes down like that.
 

522
00:32:19 --> 00:32:20
Right?
 

523
00:32:20 --> 00:32:25
And what's the Greek
name for that curve?

524
00:32:25 --> 00:32:28
Oh, come on.
 

525
00:32:28 --> 00:32:30
It's a hyperbola.
 

526
00:32:30 --> 00:32:32
It's a hyperbola.
 

527
00:32:32 --> 00:32:35
Hyperbolas -- you remember
the Greeks, they had

528
00:32:35 --> 00:32:37
these conic sections.
 

529
00:32:37 --> 00:32:41
They had ellipses, they had
parabolas, the marginal case,

530
00:32:41 --> 00:32:43
and then they had hyperbolas.
 

531
00:32:43 --> 00:32:47
And they all come from
second degree things.

532
00:32:47 --> 00:32:54
If I have a x squared and 2bxy
and cy squared equal one,

533
00:32:54 --> 00:32:57
that's one of those curves.
 

534
00:32:57 --> 00:33:01
And if it was x squared
plus y squared equal

535
00:33:01 --> 00:33:02
one, it was a circle.
 

536
00:33:02 --> 00:33:05
If it was x squared plus
7y squared equal one,

537
00:33:05 --> 00:33:11
it would be an ellipse.
 

538
00:33:11 --> 00:33:14
The positive definite -- it
all comes down to linear

539
00:33:14 --> 00:33:18
algebra, of course.
 

540
00:33:18 --> 00:33:22
If that little matrix is
positive definite, so that

541
00:33:22 --> 00:33:26
means a and c are positive, and
a c is bigger than b squared,

542
00:33:26 --> 00:33:29
you know the test for
positive definite.

543
00:33:29 --> 00:33:32
What kind of curve
do the Greeks have?

544
00:33:32 --> 00:33:35
What kind of equipotential
-- what kind of a curve

545
00:33:35 --> 00:33:36
have we got here?
 

546
00:33:36 --> 00:33:38
An ellipse.
 

547
00:33:38 --> 00:33:45
If this curve, if that little
matrix is indefinite,

548
00:33:45 --> 00:33:48
as, for example, here.
 

549
00:33:48 --> 00:33:53
So with this one, what would be
the matrix, what's the matrix

550
00:33:53 --> 00:33:58
that goes with 2xy, if I
match this with this.

551
00:33:58 --> 00:33:59
That's the matrix.
 

552
00:33:59 --> 00:34:02
There's no a squareds, there's
no x squareds, there's

553
00:34:02 --> 00:34:04
no y squareds.
 

554
00:34:04 --> 00:34:08
And there are 2xy's, I
think the matrix is that.

555
00:34:08 --> 00:34:11
So it's this nice
symmetric matrix.

556
00:34:11 --> 00:34:14
Is that a positive
definite matrix?

557
00:34:14 --> 00:34:16
Certainly not.
 

558
00:34:16 --> 00:34:17
It's indefinite.
 

559
00:34:17 --> 00:34:20
It's Eigenvalues are plus
one and minus one, adding

560
00:34:20 --> 00:34:22
to the trace zero.
 

561
00:34:22 --> 00:34:29
So indefinite matrices
correspond to hyperbolas.

562
00:34:29 --> 00:34:34
And later on, definite matrices
will correspond to elliptic

563
00:34:34 --> 00:34:36
partial differential equations.
 

564
00:34:36 --> 00:34:40
And indefinite matrices -- like
Laplace -- and indefinite

565
00:34:40 --> 00:34:45
matrices will correspond to
hyperbolic partial differential

566
00:34:45 --> 00:34:48
equations, like the
wave equation.

567
00:34:48 --> 00:34:50
What's the -- now we're here.
 

568
00:34:50 --> 00:34:57
I didn't expect to get here
-- what's the marginal case?

569
00:34:57 --> 00:35:02
What's the marginal case
between positive definite and

570
00:35:02 --> 00:35:05
indefinite is -- semidefinite.
 

571
00:35:05 --> 00:35:05
Great.
 

572
00:35:05 --> 00:35:08
And what kind of a curve do you
think comes when this little

573
00:35:08 --> 00:35:12
matrix is semidefinite.
 

574
00:35:12 --> 00:35:18
It's the one in between
ellipses and hyperbolas.

575
00:35:18 --> 00:35:23
The marginal guy is a parabola.
 

576
00:35:23 --> 00:35:24
Right.
 

577
00:35:24 --> 00:35:27
So semidefinite would
correspond to a parabola.

578
00:35:27 --> 00:35:28
Right.
 

579
00:35:28 --> 00:35:28
OK.
 

580
00:35:28 --> 00:35:29
Good.
 

581
00:35:29 --> 00:35:35
Anyway, all I was going
to say is, this u=2xy,

582
00:35:35 --> 00:35:37
that's our potential.
 

583
00:35:37 --> 00:35:43
If I draw equipotential
curves, they're hyperbolas.

584
00:35:43 --> 00:35:47
And now, what's the point
about these little arrows

585
00:35:47 --> 00:35:53
that I got started on.
 

586
00:35:53 --> 00:35:57
What was the very first point
about the answer to the meaning

587
00:35:57 --> 00:36:00
of the gradient was what?
 

588
00:36:00 --> 00:36:07
These are the gradients of u,
so those arrows point where?

589
00:36:07 --> 00:36:10
Perpendicular to
the hyperbolas.

590
00:36:10 --> 00:36:12
Perpendicular to the hyperbola.
 

591
00:36:12 --> 00:36:18
We're trying to see the
geometry -- it's beautiful

592
00:36:18 --> 00:36:21
geometry -- behind
the gradient.

593
00:36:21 --> 00:36:29
So if v is a gradient, then
it comes from some u.

594
00:36:29 --> 00:36:33
I can plot the u equal
constant, the equipotential,

595
00:36:33 --> 00:36:36
and then the gradients
will be perpendicular.

596
00:36:36 --> 00:36:42
So they really are a
little -- OK, good.

597
00:36:42 --> 00:36:49
OK, those are pieces of
information that you have, but

598
00:36:49 --> 00:36:51
always need saying again.
 

599
00:36:51 --> 00:36:55
And to get the picture in your
mind -- I suppose, finally, I

600
00:36:55 --> 00:37:01
should choose a v which
is not a gradient.

601
00:37:01 --> 00:37:04
Just to finish.
 

602
00:37:04 --> 00:37:06
How shall I adjust that v?
 

603
00:37:06 --> 00:37:07
This v was a gradient.
 

604
00:37:07 --> 00:37:11
Can you just change it a little
bit -- practically anything you

605
00:37:11 --> 00:37:18
do will screw it up -- to
make it not a gradient?

606
00:37:18 --> 00:37:23
So I just changed this two,
what shall I change the two to?

607
00:37:23 --> 00:37:26
To three.
 

608
00:37:26 --> 00:37:28
That would totally foul it up.
 

609
00:37:28 --> 00:37:32
So that vector field, which I
could draw little pictures of,

610
00:37:32 --> 00:37:36
but there would be no u
that it's coming from.

611
00:37:36 --> 00:37:37
There would be no u.
 

612
00:37:37 --> 00:37:40
These little arrows would not
line up perpendicular to

613
00:37:40 --> 00:37:45
some beautiful curves.
 

614
00:37:45 --> 00:37:47
I don't get a u from that.
 

615
00:37:47 --> 00:37:51
Because the y derivative of
that is three, and it doesn't

616
00:37:51 --> 00:37:53
equal the x derivative of that.
 

617
00:37:53 --> 00:37:54
So that's a no good one.
 

618
00:37:54 --> 00:37:57
Let's go back to the good one.
 

619
00:37:57 --> 00:37:58
OK.
 

620
00:37:58 --> 00:38:00
OK, good.
 

621
00:38:00 --> 00:38:03
Is that OK for gradients?
 

622
00:38:03 --> 00:38:05
We got the meaning
of gradients.

623
00:38:05 --> 00:38:09
They point perpendicular
to equipotentials.

624
00:38:09 --> 00:38:12
They tell how steeply
those -- they tell the

625
00:38:12 --> 00:38:15
separation between the
equipotentials, right.

626
00:38:15 --> 00:38:18
It's like, if you're a mountain
climber, you're looking at your

627
00:38:18 --> 00:38:23
map, your contour map, and
that's all I'm drawing here.

628
00:38:23 --> 00:38:26
I'm drawing a contour map that
every guy who goes climbing in

629
00:38:26 --> 00:38:29
New Hampshire is going to have.
 

630
00:38:29 --> 00:38:33
And it shows little circles,
those are level heights, right?

631
00:38:33 --> 00:38:36
Those are level contours.
 

632
00:38:36 --> 00:38:42
And if you want to climb as
fast as possible, you go

633
00:38:42 --> 00:38:44
perpendicular to
those contours.

634
00:38:44 --> 00:38:47
And the distance between
contours tells you

635
00:38:47 --> 00:38:48
how steep it is.
 

636
00:38:48 --> 00:38:52
So it's all nice geometry.
 

637
00:38:52 --> 00:38:53
OK.
 

638
00:38:53 --> 00:38:57
I've got to get to
divergence here.

639
00:38:57 --> 00:39:02
Divergence.
 

640
00:39:02 --> 00:39:04
I should have said
though, -- damn.

641
00:39:04 --> 00:39:08
There's more to say
about gradients.

642
00:39:08 --> 00:39:17
That question of whether <v_1,
v_2>, the vector field,

643
00:39:17 --> 00:39:18
is this question.
 

644
00:39:18 --> 00:39:21
The question of is it
the gradient of some u.

645
00:39:21 --> 00:39:24
So we now have a test.
 

646
00:39:24 --> 00:39:28
We now have a test.
 

647
00:39:28 --> 00:39:32
This is our test, right?
 

648
00:39:32 --> 00:39:34
That's our test.
 

649
00:39:34 --> 00:39:41
But I have to connect it with
Kirchhoff's voltage law.

650
00:39:41 --> 00:39:44
Do you remember, we haven't
talked so much about

651
00:39:44 --> 00:39:46
Kirchhoff's voltage law, but
I'm connecting it with the

652
00:39:46 --> 00:39:51
discrete case, to add in a
little more insight.

653
00:39:51 --> 00:39:56
What did Kirchhoff's
voltage law say?

654
00:39:56 --> 00:40:01
In that case, a was a
difference matrix.

655
00:40:01 --> 00:40:05
It was the incidence
matrix for our graph.

656
00:40:05 --> 00:40:12
And the question was -- I
have to take two moments

657
00:40:12 --> 00:40:14
to think about that.
 

658
00:40:14 --> 00:40:18
So Kirchhoff's voltage law.
 

659
00:40:18 --> 00:40:22
For a graph. a is an
incidence matrix.

660
00:40:22 --> 00:40:30
You know, the minus one, one
guys, one for every edge?

661
00:40:30 --> 00:40:38
And let me call it
v again, or e.

662
00:40:38 --> 00:40:40
I called it e at that time.
 

663
00:40:40 --> 00:40:44
Let's just look at an x.
 

664
00:40:44 --> 00:40:45
Right.
 

665
00:40:45 --> 00:40:58
A is this long thin matrix,
times -- sorry, u's. u's.

666
00:40:58 --> 00:41:04
Let me say it all at once.
 

667
00:41:04 --> 00:41:06
Which vectors have the form Au?
 

668
00:41:07 --> 00:41:13
Which vectors are combinations
of the columns of A?

669
00:41:13 --> 00:41:19
The test is, Kirchhoff's
voltage law, that if I go

670
00:41:19 --> 00:41:25
around any loop in the graph --
so if I have a u_1 here, u_2

671
00:41:25 --> 00:41:33
here, u_5 here, and u_7 here --
then a u will produce

672
00:41:33 --> 00:41:35
u_1-u_7 on that edge.
 

673
00:41:35 --> 00:41:39
It'll produce a
u_2-u_1 on that edge.

674
00:41:39 --> 00:41:42
It'll produce a u_5-u_2 on
that edge, if the edges

675
00:41:42 --> 00:41:45
are all going that way.
 

676
00:41:45 --> 00:41:49
And it'll produce a
u_7-u_5 on that edge.

677
00:41:49 --> 00:41:56
So I've got four components
of Au, four differences.

678
00:41:56 --> 00:41:59
And what does Kirchhoff's
voltage law tell me about

679
00:41:59 --> 00:42:00
those four differences?
 

680
00:42:00 --> 00:42:04
Which I can certainly
see directly.

681
00:42:04 --> 00:42:09
Those four differences,
u_1-u_7, u_2-u_1,

682
00:42:09 --> 00:42:11
u_5-u_2 and u_7-u_5.
 

683
00:42:12 --> 00:42:16
What's the obvious fact
about those four guys?

684
00:42:16 --> 00:42:21
They add to zero.
 

685
00:42:21 --> 00:42:27
The total drop around
a loop is zero.

686
00:42:27 --> 00:42:31
You see, if I cancel those, if
I add them, the u_1's cancel,

687
00:42:31 --> 00:42:34
the u_2's cancel, the u_5's
cancel, the u_7's cancel.

688
00:42:34 --> 00:42:36
We know this.
 

689
00:42:36 --> 00:42:36
OK.
 

690
00:42:36 --> 00:42:38
So that's Kirchhoff's
voltage law.

691
00:42:38 --> 00:42:46
It's got to have a
continuous form.

692
00:42:46 --> 00:42:53
This tells me, this is the
test on v at a point.

693
00:42:53 --> 00:42:57
What's the test on
v around a loop?

694
00:42:57 --> 00:43:00
I just want to connect
that -- I have to connect

695
00:43:00 --> 00:43:06
that to a second test.
 

696
00:43:06 --> 00:43:10
I'll just mention it, and
you'll find it in the book.

697
00:43:10 --> 00:43:12
That's the pointwise test.
 

698
00:43:12 --> 00:43:14
That was the easy test.
 

699
00:43:14 --> 00:43:17
We applied it to this and
we got the answer yes.

700
00:43:17 --> 00:43:21
If it was 3y, 2x, we
got the answer no.

701
00:43:21 --> 00:43:26
Now let me give you a
test that looks like

702
00:43:26 --> 00:43:28
Kirchhoff's voltage law.
 

703
00:43:28 --> 00:43:32
So I'm going to integrate
around a closed loop.

704
00:43:32 --> 00:43:33
What am I going to integrate?
 

705
00:43:33 --> 00:43:38
I think I integrate v --
oh boy, I'd better look.

706
00:43:38 --> 00:43:41
It's easy to get these wrong.
 

707
00:43:41 --> 00:43:41
Yeah.
 

708
00:43:41 --> 00:43:45
So I would call this
the vorticity.

709
00:43:45 --> 00:43:47
And then I would say the
vorticity is zero for

710
00:43:47 --> 00:43:49
a gradient field.
 

711
00:43:49 --> 00:43:53
Now my integral guy is going
to be the circulation.

712
00:43:53 --> 00:43:53
Oh yeah.
 

713
00:43:53 --> 00:43:55
Because I'm following the path.
 

714
00:43:55 --> 00:44:03
So it's just v_1*dx+v_2*dy
should be zero.

715
00:44:03 --> 00:44:08
Around every closed loop --
that's idea of this thing, that

716
00:44:08 --> 00:44:12
it tells me the integral goes
around a closed loop -- if I

717
00:44:12 --> 00:44:23
follow the velocity field, the
total circulation is zero.

718
00:44:23 --> 00:44:28
I put this up here as a fact
in vector calculus that's

719
00:44:28 --> 00:44:29
connected to that.
 

720
00:44:29 --> 00:44:32
These, one is zero and
the other is zero.

721
00:44:32 --> 00:44:36
There's a Stokes' theorem that
tells me that this integral is

722
00:44:36 --> 00:44:38
found from a double
integral of this.

723
00:44:38 --> 00:44:41
So if one is zero,
the other is zero.

724
00:44:41 --> 00:44:46
I'm just saying, here is
the natural analog of

725
00:44:46 --> 00:44:48
Kirchhoff's voltage law.
 

726
00:44:48 --> 00:44:49
OK.
 

727
00:44:49 --> 00:44:52
I had to say something about
voltage law, because for the

728
00:44:52 --> 00:44:59
divergence, which I'm now
going to get to -- whatever.

729
00:44:59 --> 00:45:03
Let me ask about
divergence of w=0.

730
00:45:05 --> 00:45:10
What does that mean?
 

731
00:45:10 --> 00:45:14
That's going to be the
equivalent of who's law.

732
00:45:14 --> 00:45:16
Please tell me.
 

733
00:45:16 --> 00:45:20
Which law is going to be the
equivalent of -- divergence

734
00:45:20 --> 00:45:24
of w=0 is going to mean
there's no source.

735
00:45:24 --> 00:45:27
Whatever goes in, comes out.
 

736
00:45:27 --> 00:45:30
Whose law is that?
 

737
00:45:30 --> 00:45:31
That's Kirchhoff again.
 

738
00:45:31 --> 00:45:33
Well, yeah, other
people in physics.

739
00:45:33 --> 00:45:33
Right.
 

740
00:45:33 --> 00:45:38
But in our little world, it's
the other Kirchhoff law.

741
00:45:38 --> 00:45:41
It's Kirchhoff's current law.
 

742
00:45:41 --> 00:45:46
It's the one, it's the
A transpose Right?

743
00:45:46 --> 00:45:49
This is what we're thinking of
as A transpose w equal zero.

744
00:45:49 --> 00:45:55
Kirchhoff's current
law, in equals out.

745
00:45:55 --> 00:46:00
How will I translate that in
equal out for functions?

746
00:46:00 --> 00:46:07
Now I don't have -- on a graph,
I just had the total flow,

747
00:46:07 --> 00:46:09
the net flow at every node.
 

748
00:46:09 --> 00:46:12
Notice the divergence
is at every node.

749
00:46:12 --> 00:46:17
The circulation was
around every loop.

750
00:46:17 --> 00:46:18
OK.
 

751
00:46:18 --> 00:46:21
So in equals out was just
the sum of four things.

752
00:46:21 --> 00:46:27
OK, here I'm going to have in
equal out -- how am I going

753
00:46:27 --> 00:46:31
to express in equal out?
 

754
00:46:31 --> 00:46:32
Divergence of w equals 0.
 

755
00:46:33 --> 00:46:37
Yeah, what I need is the
divergence theorem.

756
00:46:37 --> 00:46:41
Let's just face it,
we've got to have that.

757
00:46:41 --> 00:46:45
So I have a region here.
 

758
00:46:45 --> 00:46:51
I have a w everywhere,
w, .

759
00:46:52 --> 00:46:54
Then the divergence theorem.
 

760
00:46:54 --> 00:46:58
This is the great identity,
which of course has

761
00:46:58 --> 00:47:01
a discrete form.
 

762
00:47:01 --> 00:47:01
OK.
 

763
00:47:01 --> 00:47:05
The divergence theorem says
that if I integrate over the

764
00:47:05 --> 00:47:13
region, over this region, R,
the divergence, that's

765
00:47:13 --> 00:47:14
(dw_1/dx+dw_2/dy)dxdy.
 

766
00:47:14 --> 00:47:23
 
 

767
00:47:23 --> 00:47:27
So that's like telling me the
source, I'm integrating over

768
00:47:27 --> 00:47:30
the source at every point.
 

769
00:47:30 --> 00:47:38
At every point here, this
measures in minus out.

770
00:47:38 --> 00:47:44
But now, when I put the whole
thing together by integrating,

771
00:47:44 --> 00:47:47
what's the right hand
side of this equation?

772
00:47:47 --> 00:47:49
Do you know the
divergence theorem?

773
00:47:49 --> 00:47:55
And let's remember it
and see why it's so.

774
00:47:55 --> 00:48:01
What I'm doing is in equals out
for the whole region at once.

775
00:48:01 --> 00:48:02
Right?
 

776
00:48:02 --> 00:48:05
When I this is like in
equal out at a point.

777
00:48:05 --> 00:48:08
But now I'm putting all
the points together.

778
00:48:08 --> 00:48:15
So the only way out will be
out through the boundary.

779
00:48:15 --> 00:48:19
And so I'll need to say
how much flows out.

780
00:48:19 --> 00:48:25
This is the total source, the
total in equal out inside.

781
00:48:25 --> 00:48:28
The only way to get out
is through the boundary.

782
00:48:28 --> 00:48:35
So this is the integral
around the boundary of --

783
00:48:35 --> 00:48:39
so what's the flow out?
 

784
00:48:39 --> 00:48:45
It's, yeah, it's somehow --
think now what should go there.

785
00:48:45 --> 00:48:49
This is flux I'm talking about.
 

786
00:48:49 --> 00:48:53
Flux is short word for
the total flow out.

787
00:48:53 --> 00:48:57
OK.
 

788
00:48:57 --> 00:49:02
So now I've got to
get this right.

789
00:49:02 --> 00:49:10
In vector notation, it would
be -- w tells me the flow.

790
00:49:10 --> 00:49:16
But flow outwards, see,
suppose w points that way.

791
00:49:16 --> 00:49:20
Then the actual flow
out is not all that.

792
00:49:20 --> 00:49:22
Because a lot of that is
just going sideways.

793
00:49:22 --> 00:49:24
It's this part.
 

794
00:49:24 --> 00:49:26
It's the flow perpendicular
to the boundary.

795
00:49:26 --> 00:49:33
So it's w dot n, the
normal component of flow.

796
00:49:33 --> 00:49:42
And I integrate that
around the boundary.

797
00:49:42 --> 00:49:46
There you have a
key, key theorem.

798
00:49:46 --> 00:49:46
In 2-D.
 

799
00:49:47 --> 00:49:51
And it's an equation
for the flux.

800
00:49:51 --> 00:49:53
It's like the fundamental
theorem of calculus, but now

801
00:49:53 --> 00:49:58
we're in two dimensions.
 

802
00:49:58 --> 00:50:00
And this is what it looks like.
 

803
00:50:00 --> 00:50:04
OK, so I'm obviously not
going to finish with the

804
00:50:04 --> 00:50:07
divergence theorem today.
 

805
00:50:07 --> 00:50:10
So what's the conclusion?
 

806
00:50:10 --> 00:50:15
If the divergence is
zero, then what?

807
00:50:15 --> 00:50:21
If the divergence is zero,
if this is zero at every

808
00:50:21 --> 00:50:35
point, then this is
zero across every loop.

809
00:50:35 --> 00:50:38
Can I call this thing a loop?
 

810
00:50:38 --> 00:50:42
That closed loop.
 

811
00:50:42 --> 00:50:49
That's the conclusion
that we want to reach.

812
00:50:49 --> 00:50:52
So this is the
divergence theorem.

813
00:50:52 --> 00:51:00
The text gives a proof, not
to repeat in class, but it's

814
00:51:00 --> 00:51:03
a crucial formula to know.
 

815
00:51:03 --> 00:51:08
That the integral of the
divergence is the flux.

816
00:51:08 --> 00:51:09
OK.
 

817
00:51:09 --> 00:51:11
Let's come back to that
Wednesday, and I'll have

818
00:51:11 --> 00:51:13
lots of homework for you.
 

819
00:51:13 --> 00:51:18
Thanks for turning
in these today.