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Push
 

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PROFESSOR STRANG: So I'm ready
for anything, hope I am.

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Questions about any topic.
 

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Yes.
 

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AUDIENCE: [INAUDIBLE]
 

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PROFESSOR STRANG: I feel
this is like a White

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House press conference.
 

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I think there's always somebody
in the front row who gets to

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ask the first question, and
then gets to say thank you Mr.

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President at the end,
and then I'm off.

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Yes.
 

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I'm tempted by the way to ask
you, all are you going to vote

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next Tuesday and of course I'd
like to know who you vote for,

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and I'd like to give
you my advice.

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But I don't know
that that's proper.

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If anybody wants advice,
they can email.

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But please vote.
 

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Please vote.
 

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Yeah.
 

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Alright, question here and
then we'll have, well, yeah.

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AUDIENCE: [INAUDIBLE]
 

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PROFESSOR STRANG: Oh, OK,
so those were just posted.

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Like, I see.
 

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3.3 number two.
 

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AUDIENCE: [INAUDIBLE]
 

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PROFESSOR STRANG: OK, so this
was a case, yeah, right.

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Oh, OK.
 

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We could be wrong so this is
3.3 number two, asks you

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about the flow field.
 

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Which has no flow in the x
direction, the velocity in

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the x direction is zero.
 

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The velocity in the
y direction is x.

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00:02:06 --> 00:02:10
OK, so suppose we just take
that as a full field and try to

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understand, is it a gradient.
 

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I mean, so what are the
questions I would ask?

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Is it a gradient of anything?
 

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00:02:17 --> 00:02:21
Because we're now thinking
v and w are pretty

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much the same guy.
 

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00:02:23 --> 00:02:26
So is it a gradient, yes or no?
 

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00:02:26 --> 00:02:29
If so, what's the potential?
 

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00:02:29 --> 00:02:32
Is it divergence
free, yes or no?

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00:02:32 --> 00:02:34
If it is, what's the
stream function?

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And of course if the answer to
both tests was yes, then we

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would be talking about
Laplace's equation.

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I suspect for this example the
answer at least to one of the

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two questions will be no.
 

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So we won't have the two pieces
coming together into Laplace.

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00:02:58 --> 00:02:59
OK, so first of all.
 

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Is it a gradient.
 

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00:03:01 --> 00:03:09
What's the test for, so my
two questions are, is v

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the gradient of some u?
 

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And what's the test for that?
 

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You remember if, is the
gradient, and see if I

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can remember myself.
 

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If it is a gradient, then this
is du/dx, and this is du/dy,

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and the condition that v_1 and
v_2 would have to satisfy is

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that the y derivative of that
would have to equal the x

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derivative of that, because on
the right-hand side they are

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the same. u_xy is
the same as u_yx.

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So I would look at at, so
let me write that again.

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I need the dv_1/dy to equal
dv_2/dx, and is that

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true in this example?
 

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What's dv_1/dy?
 

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Zero.
 

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What's dv_2/dx?
 

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00:04:11 --> 00:04:14
 
 

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One.
 

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So the answer's no.
 

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OK.
 

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So test, failed.
 

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Alright, the second question is
does it possibly sit over in

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the divergence free world?
 

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Is the divergence of, now
I'll call it w, equal zero?

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So the answer was no to that
question but now I think

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the answer to this
question will be yes.

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00:04:43 --> 00:04:47
Because what's the
divergence of this thing?

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00:04:47 --> 00:04:51
It's the x derivative of that,
which is certainly zero, plus

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the y derivative of that,
which is certainly zero.

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So the answer is yes.
 

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So there's no potential
but there is a stream

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function, right?
 

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Because the stream function
comes in with this test.

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So let's remember what, just
from today's lecture, what was

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00:05:12 --> 00:05:21
the stream function from
dw_1/dx+dw_2/dy=0, that'll be

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satisfied if w_1 is the y
derivative of the stream

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function, and w_2 is
minus the x derivative.

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00:05:32 --> 00:05:39
Because then this matches the x
derivative of this plus the y

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00:05:39 --> 00:05:42
derivative of this, which is
the divergence on the

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right-hand side I
would get zero.

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So there's got to be
an S, and what is it?

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00:05:47 --> 00:05:49
Probably not hard to find.
 

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00:05:49 --> 00:05:50
Let's see.
 

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00:05:50 --> 00:05:55
Here w_1 is zero, so that tells
me s doesn't depend on y at

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00:05:55 --> 00:06:02
all. w_2 is x, so x is supposed
to be minus the x derivative of

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00:06:02 --> 00:06:05
the stream function, so what
is the stream function now?

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00:06:05 --> 00:06:10
Have I got room to put it here?
 

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00:06:10 --> 00:06:11
Just about.
 

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What will work?
 

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00:06:16 --> 00:06:19
So again, here's w_1.
 

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00:06:21 --> 00:06:26
The y derivative of S is zero.
w_2 tells me that the x

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00:06:26 --> 00:06:28
derivative of S is minus x.
 

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00:06:28 --> 00:06:31
So all I'm looking for is a
function that only depends on

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00:06:31 --> 00:06:36
x, has no dependence on y,
and its derivative

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00:06:36 --> 00:06:38
should be minus x.
 

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00:06:38 --> 00:06:41
So what's the function?
 

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00:06:41 --> 00:06:45
Minus 1/2 of x squared.
 

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00:06:45 --> 00:06:54
Yeah, so this gives me S equal
minus a half of x squared.

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00:06:54 --> 00:06:59
Alright, so you're saying
that, so there is a

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00:06:59 --> 00:07:02
stream function, right.
 

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And what does that
travel along?

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00:07:04 --> 00:07:09
That travels along steam, that
means that the flow buzzes

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00:07:09 --> 00:07:10
along stream lines.
 

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00:07:10 --> 00:07:12
And what are the stream lines?
 

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They're the lines
where s is constant.

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00:07:16 --> 00:07:19
Equipotentials were the lines
where u is constant, but here

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we don't have a u
in this problem.

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Stream lines are lines where
the S is constant, so minus 1/2

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00:07:26 --> 00:07:31
x squared is constant, what's
the picture look like?

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00:07:31 --> 00:07:39
Picture then, for that, well,
and you know what the flow is

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00:07:39 --> 00:07:42
doing at a typical point here.
 

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00:07:42 --> 00:07:44
Say, x=3, y=1.
 

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Let me draw the little arrow.
 

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With a big chalk Which
way is the flow going?

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00:07:55 --> 00:08:01
Well, the x component is
zero, the y component is x.

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So the flow is going
up there, right?

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00:08:05 --> 00:08:10
Here are the y components x.
 

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This whole line is all
traveling up with

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the same velocity.
 

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00:08:16 --> 00:08:19
If I drop a leaf there, it
buzzes up that straight line.

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00:08:19 --> 00:08:21
So that's the stream line.
 

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00:08:21 --> 00:08:28
And its velocity is, that's x
equal; if, say, the velocity is

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00:08:28 --> 00:08:33
three then this speed is three.
 

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00:08:33 --> 00:08:35
It's going up that line.
 

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00:08:35 --> 00:08:38
So that's a stream line.
 

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00:08:38 --> 00:08:41
And sure enough, on
that line minus 1/2 x

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00:08:41 --> 00:08:42
squared is a constant.
 

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00:08:42 --> 00:08:47
So you see we're not talking
parabolas here because our

146
00:08:47 --> 00:08:52
curve is not y equals minus
1/2 x squared, it's minus 1/2

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00:08:52 --> 00:08:54
x squared equal constant.
 

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00:08:54 --> 00:08:55
Yeah, that's what we want.
 

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00:08:55 --> 00:09:02
So, but now having got so
far, let me take x=1, say.

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00:09:02 --> 00:09:04
What's the flow like on that?
 

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00:09:04 --> 00:09:11
So there's a stream line
with S equal constant.

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And the velocity on that is
zero, so nothing is going

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00:09:15 --> 00:09:17
in that horizontally.
 

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00:09:17 --> 00:09:25
And now it's one, so the flow
is slower up this line.

155
00:09:25 --> 00:09:26
OK, slower flow.
 

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00:09:26 --> 00:09:30
This was faster flow.
 

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00:09:30 --> 00:09:34
And then the question that's in
that homework problem is, is

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00:09:34 --> 00:09:38
there any rotation
in this flow?

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00:09:38 --> 00:09:42
We think about rotation,
we have an image of

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00:09:42 --> 00:09:45
rotational flow.
 

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00:09:45 --> 00:09:49
And that could be the next
example, we could figure out.

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00:09:49 --> 00:09:54
A flow that goes around
in circles, right?

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00:09:54 --> 00:09:56
Those would be the
stream lines.

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00:09:56 --> 00:10:03
So this would be like pure
rotation, shall I call it.

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00:10:03 --> 00:10:05
But I don't have that there.
 

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00:10:05 --> 00:10:08
I just want to draw the
other picture, in which the

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00:10:08 --> 00:10:16
stream lines are circles.
 

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00:10:16 --> 00:10:19
To have another nice,
clean, beautiful example.

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00:10:19 --> 00:10:21
OK, but here we don't have.
 

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00:10:21 --> 00:10:24
Our streamlines are
straight lines, and

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00:10:24 --> 00:10:26
yet we have rotation.
 

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00:10:26 --> 00:10:30
That's the point here.
 

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00:10:30 --> 00:10:32
Why do I say we have rotation?
 

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00:10:32 --> 00:10:36
Because the test for rotation
was that original test of

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00:10:36 --> 00:10:43
looking at, which I just wrote
the answer to be no here.

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00:10:43 --> 00:10:47
So if it's not the gradient,
the reason is there's

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00:10:47 --> 00:10:48
some rotation.
 

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00:10:48 --> 00:10:51
Gradient fields don't
have any rotation.

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00:10:51 --> 00:11:00
The rotation is this thing that
comes out, yeah it's this

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00:11:00 --> 00:11:05
difference. dv_2/dx, it's the
difference between those

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00:11:05 --> 00:11:08
that tells us the rotation.
 

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00:11:08 --> 00:11:12
And that was not zero, right?
 

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00:11:12 --> 00:11:17
For this example dv_1/dy,
was zero, because v_1 is

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00:11:17 --> 00:11:23
zero. dv_2/dx was one,
because v_2 is x.

185
00:11:23 --> 00:11:25
So there's some rotation here.
 

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00:11:25 --> 00:11:29
And in other words
the test for being a

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00:11:29 --> 00:11:31
gradient is no rotation.
 

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00:11:31 --> 00:11:33
This fails that test.
 

189
00:11:33 --> 00:11:35
But how is it rotating?
 

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00:11:35 --> 00:11:41
How can it be rotating when
the all the flow is just

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00:11:41 --> 00:11:43
traveling vertically?
 

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00:11:43 --> 00:11:47
I guess I give you this
example because it

193
00:11:47 --> 00:11:49
meant something to me.
 

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00:11:49 --> 00:11:54
My image of rotation was this
simpleminded type of flow.

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00:11:54 --> 00:11:56
You know, like a phonograph
record or something.

196
00:11:56 --> 00:11:59
This would be called
a sheer flow.

197
00:11:59 --> 00:12:03
A very important type of flow.
 

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00:12:03 --> 00:12:10
And actually, you'll realize
that if x is negative then the

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00:12:10 --> 00:12:14
flow in the second component,
the velocity, is now negative.

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00:12:14 --> 00:12:17
So it would be the stream
line, the flow would be

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00:12:17 --> 00:12:19
going down this way.
 

202
00:12:19 --> 00:12:22
And this point
wouldn't move at all.

203
00:12:22 --> 00:12:24
This would be, well I don't
know if it's a stream line,

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00:12:24 --> 00:12:29
it's a stagnant stream
line, right? x=0.

205
00:12:30 --> 00:12:33
On that line, there's
no velocity. .

206
00:12:33 --> 00:12:37
So this is all just
staying there.

207
00:12:37 --> 00:12:40
These lines are moving, this
line moving faster, this line

208
00:12:40 --> 00:12:41
would be moving even faster.
 

209
00:12:41 --> 00:12:44
This line's going
the other way.

210
00:12:44 --> 00:12:46
Faster and faster
the other way.

211
00:12:46 --> 00:12:49
It's a important flow.
 

212
00:12:49 --> 00:12:52
You know, in earthquakes
and things like that.

213
00:12:52 --> 00:12:57
This happens, when one plate
shears with respect to another.

214
00:12:57 --> 00:12:59
So that's shearing.
 

215
00:12:59 --> 00:13:06
Shearing means that a
line that was, that line

216
00:13:06 --> 00:13:09
after a while is tilted.
 

217
00:13:09 --> 00:13:12
This is going faster than this.
 

218
00:13:12 --> 00:13:12
Yes.
 

219
00:13:12 --> 00:13:16
AUDIENCE: [INAUDIBLE]
 

220
00:13:16 --> 00:13:20
PROFESSOR STRANG: Right,
being a constant.

221
00:13:20 --> 00:13:21
AUDIENCE: [INAUDIBLE]
 

222
00:13:21 --> 00:13:24
That's true.
 

223
00:13:24 --> 00:13:28
Ah, well, OK.
 

224
00:13:28 --> 00:13:32
Let's see.
 

225
00:13:32 --> 00:13:33
Well, how do I fix that?
 

226
00:13:33 --> 00:13:39
AUDIENCE: [INAUDIBLE]
 

227
00:13:39 --> 00:13:41
PROFESSOR STRANG: Yes.
 

228
00:13:41 --> 00:13:42
That's a good question.
 

229
00:13:42 --> 00:13:49
Should I have allowed in my
stream function, I mean

230
00:13:49 --> 00:13:51
that's a stream function.
 

231
00:13:51 --> 00:13:54
Because it satisfies the
equations that stream functions

232
00:13:54 --> 00:13:57
are - I could have thrown
in a constant, yeah.

233
00:13:57 --> 00:14:00
So your pointing out a
difficulty makes me think

234
00:14:00 --> 00:14:02
I should have thrown
in a constant.

235
00:14:02 --> 00:14:07
So if I throw in constants when
I could get other lines, yeah.

236
00:14:07 --> 00:14:09
Thanks, that's a good point.
 

237
00:14:09 --> 00:14:13
I just want to see, do
you see rotation in this

238
00:14:13 --> 00:14:15
flow, in this shear flow?
 

239
00:14:15 --> 00:14:16
And I think you do.
 

240
00:14:16 --> 00:14:17
If you think about it.
 

241
00:14:17 --> 00:14:21
Suppose you put a little
leaf, or a little penny

242
00:14:21 --> 00:14:23
or something right there.
 

243
00:14:23 --> 00:14:27
OK, is it going to turn?
 

244
00:14:27 --> 00:14:31
It'll flow along, but as it
flows, is it going to turn?

245
00:14:31 --> 00:14:36
In other words, is there some
is there some difference

246
00:14:36 --> 00:14:41
in the speed on one side
compared to the other?

247
00:14:41 --> 00:14:45
I mean, it's what makes a
curveball curve, right?

248
00:14:45 --> 00:14:50
When the pitcher throws the
ball, he imparts a spin to it,

249
00:14:50 --> 00:14:53
and that gives a different
pressure on the two sides of

250
00:14:53 --> 00:14:56
the ball, and the ball moves.
 

251
00:14:56 --> 00:14:59
I think that's going
to happen here.

252
00:14:59 --> 00:15:02
Maybe you see it, and
I'm just talking.

253
00:15:02 --> 00:15:05
I mean, this side is going
faster than this side.

254
00:15:05 --> 00:15:11
So the net result is that even
though the thing is traveling

255
00:15:11 --> 00:15:15
up and down, it's turning.
 

256
00:15:15 --> 00:15:18
It's turning because the
right-hand side is going faster

257
00:15:18 --> 00:15:19
than the left-hand side.
 

258
00:15:19 --> 00:15:22
So it does have a rotation.
 

259
00:15:22 --> 00:15:28
This quantity, this difference
between dv_1/dy and dv_2/dx,

260
00:15:28 --> 00:15:33
which is the component of the
curl, maybe the sign should be

261
00:15:33 --> 00:15:36
the opposite, maybe I think
it should be minus this

262
00:15:36 --> 00:15:37
plus this or something.
 

263
00:15:37 --> 00:15:40
Point is that it's not zero.
 

264
00:15:40 --> 00:15:43
So there is curl,
there is rotation.

265
00:15:43 --> 00:15:44
OK.
 

266
00:15:44 --> 00:15:48
I was going to ask about
this picture, too.

267
00:15:48 --> 00:15:50
And then I'll open
to more examples.

268
00:15:50 --> 00:15:53
I just feel examples are good.
 

269
00:15:53 --> 00:15:57
Simple velocity
fields, like 0x.

270
00:15:57 --> 00:16:00
 
 

271
00:16:00 --> 00:16:04
Just to think through,
OK, what does that mean?

272
00:16:04 --> 00:16:06
Is it curl free?
 

273
00:16:06 --> 00:16:09
Another way of saying is it
a gradient field would be

274
00:16:09 --> 00:16:11
to say is it curl free?
 

275
00:16:11 --> 00:16:15
Irrotational is the
right word here.

276
00:16:15 --> 00:16:18
Test one, is it
irrotational, answer no.

277
00:16:18 --> 00:16:23
Is it divergence free, is
it source free, the answer

278
00:16:23 --> 00:16:25
was yes, for this example.
 

279
00:16:25 --> 00:16:29
If we pick another example I
could reverse those, or another

280
00:16:29 --> 00:16:33
example I can probably come
up with an example here.

281
00:16:33 --> 00:16:39
Let's see, what if I wanted the
stream lines to be circles,

282
00:16:39 --> 00:16:46
what would be a good velocity
field that goes in circles?

283
00:16:46 --> 00:16:47
Let's see.
 

284
00:16:47 --> 00:16:52
At a typical point, if I want
the velocity to be going that

285
00:16:52 --> 00:16:58
way, here's the vector, the
position vector, the radial

286
00:16:58 --> 00:17:02
victor that goes, so what are
the components of this vector?

287
00:17:02 --> 00:17:04
Just, .
 

288
00:17:04 --> 00:17:12
So now if I want the velocity
field to go other way, what

289
00:17:12 --> 00:17:18
would be a good thing
with rotation?

290
00:17:18 --> 00:17:20
 would sound
good. v=.

291
00:17:20 --> 00:17:24
 
 

292
00:17:24 --> 00:17:29
So are we expecting this to
be a gradient of anything?

293
00:17:29 --> 00:17:32
I'm not.
 

294
00:17:32 --> 00:17:35
We've built in rotation here.
 

295
00:17:35 --> 00:17:40
I'm expecting the curl of this
thing, this quantity, I think I

296
00:17:40 --> 00:17:47
take the x derivative, I look
at the y derivative of this and

297
00:17:47 --> 00:17:49
compare it with the x
derivative of that.

298
00:17:49 --> 00:17:52
And they're not the same; in
fact one is minus one and

299
00:17:52 --> 00:17:54
the other's plus one.
 

300
00:17:54 --> 00:17:56
So I've got rotation here.
 

301
00:17:56 --> 00:18:03
I've got sort of two, is
the component of the curl.

302
00:18:03 --> 00:18:12
So let's just write it down.
dv_2/dx, this vorticity that

303
00:18:12 --> 00:18:17
measures the turning speed is
one from dv_2/dx,

304
00:18:17 --> 00:18:20
minus one is two.
 

305
00:18:20 --> 00:18:23
So it's not a gradient
of anything.

306
00:18:23 --> 00:18:28
If the x derivative of u is
minus y, then the y derivative

307
00:18:28 --> 00:18:30
can't be plus x, no good.
 

308
00:18:30 --> 00:18:38
OK, what about, is
it divergence free?

309
00:18:38 --> 00:18:41
Do I need a source to
keep this flow going?

310
00:18:41 --> 00:18:42
Well, what's the test?
 

311
00:18:42 --> 00:18:47
In other words, is there a
stream function for this guy?

312
00:18:47 --> 00:18:50
I think probably there is.
 

313
00:18:50 --> 00:18:54
What's the test to know if
there is a stream function?

314
00:18:54 --> 00:18:57
I take the divergence, I'm
over on the right-hand

315
00:18:57 --> 00:18:59
side of my picture now.
 

316
00:18:59 --> 00:19:01
I take the divergence.
 

317
00:19:01 --> 00:19:08
Divergence of this v is the x
derivative of that plus the y

318
00:19:08 --> 00:19:11
derivative of that, good, zero.
 

319
00:19:11 --> 00:19:13
So there is a stream function.
 

320
00:19:13 --> 00:19:15
And what is it?
 

321
00:19:15 --> 00:19:20
Well, I'm pretty sure that
these stream lines are circles,

322
00:19:20 --> 00:19:24
I think the stream function is
going to be x squared

323
00:19:24 --> 00:19:27
plus y squared.
 

324
00:19:27 --> 00:19:28
Yep.
 

325
00:19:28 --> 00:19:36
Then, am I right that the y
derivative of that will be 2y.

326
00:19:38 --> 00:19:41
That's not looking too good.
 

327
00:19:41 --> 00:19:45
What do I want here?
 

328
00:19:45 --> 00:19:49
Here's my v, which
is the same as w.

329
00:19:49 --> 00:19:54
And what I'm looking for is
to get these guys correct.

330
00:19:54 --> 00:19:57
So and I should be
able to do it.

331
00:19:57 --> 00:19:59
And what would s be?
 

332
00:19:59 --> 00:20:02
I haven't got s quite right.
 

333
00:20:02 --> 00:20:04
I think if I multiply
by negative 1/2, that

334
00:20:04 --> 00:20:06
might have done it.
 

335
00:20:06 --> 00:20:13
Yeah, because now the y
derivative is now minus y.

336
00:20:13 --> 00:20:15
Great.
 

337
00:20:15 --> 00:20:21
And the x derivative of S is
minus x, and then I should take

338
00:20:21 --> 00:20:24
a minus that, so I should want
a plus x, which is

339
00:20:24 --> 00:20:25
what I've got.
 

340
00:20:25 --> 00:20:27
So those are the stream lines.
 

341
00:20:27 --> 00:20:30
Circles.
 

342
00:20:30 --> 00:20:35
So I have circle, the flow is
going around in a circle.

343
00:20:35 --> 00:20:40
I don't have to, I don't need
any source to keep it going.

344
00:20:40 --> 00:20:48
But it's not a gradient.
 

345
00:20:48 --> 00:20:57
So this is like a sample test,
to take a simple flow field,

346
00:20:57 --> 00:21:01
apply the two tests, and I
guess we should complete

347
00:21:01 --> 00:21:05
with an example that
passes both tests.

348
00:21:05 --> 00:21:07
Right?
 

349
00:21:07 --> 00:21:11
Let me open to any other
question, and then we could

350
00:21:11 --> 00:21:16
cook up an example that passes
both tests before we stop.

351
00:21:16 --> 00:21:17
I'll stop talking
first, though.

352
00:21:17 --> 00:21:23
Just listen for a
question on any topic.

353
00:21:23 --> 00:21:26
Or is it useful just to take
fields like this and go

354
00:21:26 --> 00:21:27
through those steps?
 

355
00:21:27 --> 00:21:28
It probably is.
 

356
00:21:28 --> 00:21:32
It's certainly good for me.
 

357
00:21:32 --> 00:21:36
OK, what's a field that will
satisfy everybody, that will

358
00:21:36 --> 00:21:43
be a gradient field and also
divergence free, so that

359
00:21:43 --> 00:21:48
we'll have solutions
to Laplace's equation.

360
00:21:48 --> 00:21:49
Let's see.
 

361
00:21:49 --> 00:21:59
Well we had some solutions to
Laplace's equation there.

362
00:21:59 --> 00:22:03
You know if I make it
linear it's real easy.

363
00:22:03 --> 00:22:11
If I make it quadratic - huh.
 

364
00:22:11 --> 00:22:15
Can I anticipate a little
what's coming Friday?

365
00:22:15 --> 00:22:20
I so recommend to come to
Friday's lecture, but

366
00:22:20 --> 00:22:21
so what's coming?
 

367
00:22:21 --> 00:22:23
What did we do today?
 

368
00:22:23 --> 00:22:27
We discovered that we got
solutions to Laplace's equation

369
00:22:27 --> 00:22:31
from all, by real and imaginary
parts of all these guys.

370
00:22:31 --> 00:22:33
Those were terrific.
 

371
00:22:33 --> 00:22:38
And then we could take
combinations of those.

372
00:22:38 --> 00:22:40
So here's what's coming Friday.
 

373
00:22:40 --> 00:22:46
When I take combinations of
these guys I get some function

374
00:22:46 --> 00:22:53
of this magic complex, of
this magic combination x+iy.

375
00:22:54 --> 00:22:56
Some function, any function.
 

376
00:22:56 --> 00:22:59
Any reasonable function, and
we'll say what reasonable

377
00:22:59 --> 00:23:04
means, of x+iy, its real part
and its imaginary part

378
00:23:04 --> 00:23:06
are going to be great.
 

379
00:23:06 --> 00:23:10
This is like the center of a
big, big, part of mathematics.

380
00:23:10 --> 00:23:11
Functions of x+iy.
 

381
00:23:11 --> 00:23:14
 
 

382
00:23:14 --> 00:23:20
And by nice I mean that
these series converge.

383
00:23:20 --> 00:23:22
So that we have really
a nice function.

384
00:23:22 --> 00:23:25
Let me take the first function
that comes to mind. e^(x+iy).

385
00:23:25 --> 00:23:28
 
 

386
00:23:28 --> 00:23:31
So let me take this
to be e^(x+iy).

387
00:23:31 --> 00:23:34
 
 

388
00:23:34 --> 00:23:37
OK.
 

389
00:23:37 --> 00:23:38
Right.
 

390
00:23:38 --> 00:23:41
So you remember, I'm aiming to
get solutions to Laplace's

391
00:23:41 --> 00:23:44
equation, because that will
give me automatically the two

392
00:23:44 --> 00:23:49
pieces both working, so I claim
that the real part of that, and

393
00:23:49 --> 00:23:54
the imaginary part, those are
my twins, u and S, both solve.

394
00:23:54 --> 00:23:59
So u is going to be the real
part of this function.

395
00:23:59 --> 00:24:02
And S is going to be the
imaginary part of it.

396
00:24:02 --> 00:24:06
And I claim that those will
both solve Laplace's equation.

397
00:24:06 --> 00:24:09
We can plug it in and
see that it does.

398
00:24:09 --> 00:24:13
And that they will have,
they're twinned by the

399
00:24:13 --> 00:24:15
Cauchy-Riemann equations.
 

400
00:24:15 --> 00:24:20
So how am I going to simplify
that, so that I can identify

401
00:24:20 --> 00:24:23
what's the real part of that
thing and what's the

402
00:24:23 --> 00:24:25
imaginary part?
 

403
00:24:25 --> 00:24:30
This is actually, that's
a good question.

404
00:24:30 --> 00:24:36
I don't know how much you've
run into i, in the past.

405
00:24:36 --> 00:24:40
Are you happy with
something like that?

406
00:24:40 --> 00:24:42
How could you find the real
part of it, how could

407
00:24:42 --> 00:24:44
you simplify it?
 

408
00:24:44 --> 00:24:51
How else could I write
e to the something?

409
00:24:51 --> 00:24:52
Exactly.
 

410
00:24:52 --> 00:24:56
Think of it as the product of
two, so the key fact about

411
00:24:56 --> 00:25:00
exponentials is that that's
the same as e^x times e^(iy).

412
00:25:00 --> 00:25:03
 
 

413
00:25:03 --> 00:25:07
The exponents add, so that's
the first thing always to

414
00:25:07 --> 00:25:09
think about as a possibility.
 

415
00:25:09 --> 00:25:10
Now, what am I going to do?
 

416
00:25:10 --> 00:25:13
I still want to
get a real part.

417
00:25:13 --> 00:25:18
This is clearly all real,
right? e^x is real.

418
00:25:18 --> 00:25:22
So it's this part that's going
to give me the two pieces.

419
00:25:22 --> 00:25:26
So this is going to be e^x time
now, what do I put for e^(iy)?

420
00:25:28 --> 00:25:32
cos(y), good.
 

421
00:25:32 --> 00:25:35
Plus i*sin(y), good.
 

422
00:25:35 --> 00:25:38
And now I can read
off, no problem.

423
00:25:38 --> 00:25:43
What is this real part that I
was looking for? e^x*cos(y).

424
00:25:43 --> 00:25:48
 
 

425
00:25:48 --> 00:25:52
And the imaginary part is
just what's multiplying the

426
00:25:52 --> 00:25:53
i, it's the e^x*sin(y).
 

427
00:25:53 --> 00:25:57
 
 

428
00:25:57 --> 00:26:02
OK, so what's my claim?
 

429
00:26:02 --> 00:26:05
I claim that that function
solves Laplace's equation.

430
00:26:05 --> 00:26:08
And this one too.
 

431
00:26:08 --> 00:26:10
And that they're twinned.
 

432
00:26:10 --> 00:26:14
And that they give stream lines
and equipotentials that meet at

433
00:26:14 --> 00:26:22
right angles, it's
another pair.

434
00:26:22 --> 00:26:25
Plug that into
Laplace's equation.

435
00:26:25 --> 00:26:33
So let me do u_xx+u_yy,
just to satisfy that it is

436
00:26:33 --> 00:26:35
going to come out zero.
 

437
00:26:35 --> 00:26:39
So what's the xx derivative,
the second x derivative

438
00:26:39 --> 00:26:41
of that function?
 

439
00:26:41 --> 00:26:44
Take its derivative with
respect to x, and then do it

440
00:26:44 --> 00:26:46
again, and what do you have?
 

441
00:26:46 --> 00:26:53
Same. didn't change. e^x is
just, and now what about

442
00:26:53 --> 00:26:57
the second y derivative?
 

443
00:26:57 --> 00:26:59
So now e^x is just a constant.
 

444
00:26:59 --> 00:27:01
What's the second
derivative of cos(y)?

445
00:27:01 --> 00:27:03
 
 

446
00:27:03 --> 00:27:04
Negative cos(y).
 

447
00:27:05 --> 00:27:05
Right.
 

448
00:27:05 --> 00:27:08
Because the first derivative
is negative sine, the second

449
00:27:08 --> 00:27:10
derivative is negative cosine.
 

450
00:27:10 --> 00:27:14
So the second derivative is
e^x, it didn't change, times

451
00:27:14 --> 00:27:17
cos(y) with a minus sine.
 

452
00:27:17 --> 00:27:23
And you see what,
did I write sine?

453
00:27:23 --> 00:27:25
I meant to write cosine.
 

454
00:27:25 --> 00:27:27
Cancel that from the tape.
 

455
00:27:27 --> 00:27:29
OK, right.
 

456
00:27:29 --> 00:27:29
Yeah.
 

457
00:27:29 --> 00:27:32
So the second x
derivative was just e^x,

458
00:27:32 --> 00:27:34
e^x*cos(y) didn't move.
 

459
00:27:34 --> 00:27:35
Sorry.
 

460
00:27:35 --> 00:27:37
That was frightening.
 

461
00:27:37 --> 00:27:41
OK, and then now here's
the second y derivative.

462
00:27:41 --> 00:27:44
In other words, it gives zero.
 

463
00:27:44 --> 00:27:48
Gives zero, and this
one would too.

464
00:27:48 --> 00:27:54
Now, I don't really have
an idea of what the

465
00:27:54 --> 00:28:01
picture is like.
 

466
00:28:01 --> 00:28:03
But it's important.
 

467
00:28:03 --> 00:28:08
We've got a flow field here,
and it's from e^z, e^(x+iy)

468
00:28:08 --> 00:28:12
exponential has gotta be
an important function.

469
00:28:12 --> 00:28:15
So it's got to be
somehow interesting.

470
00:28:15 --> 00:28:20
What do you think, so what
would the, what would the

471
00:28:20 --> 00:28:23
equipotential lines looks like?
 

472
00:28:23 --> 00:28:29
Oh, boy. e to the x cos
y equal a constant.

473
00:28:29 --> 00:28:33
My gosh. e to the x cos y.
 

474
00:28:33 --> 00:28:37
So let's see.
 

475
00:28:37 --> 00:28:42
I don't know how to draw this
picture, but one thing I know

476
00:28:42 --> 00:28:48
is that if I changed y by 2pi,
I would get another copy

477
00:28:48 --> 00:28:50
of this curve, right?
 

478
00:28:50 --> 00:28:54
If I changed y by every time
you see cosine over sine, you

479
00:28:54 --> 00:28:55
think hey, that's periodic.
 

480
00:28:55 --> 00:28:57
If I change it by 2pi.
 

481
00:28:57 --> 00:29:08
So I'm thinking that y between
zero and 2 pi, so here's y=0.

482
00:29:10 --> 00:29:17
And y=2pi, I'm thinking that
my flow probably somehow

483
00:29:17 --> 00:29:20
stays in a strip.
 

484
00:29:20 --> 00:29:21
Like that.
 

485
00:29:21 --> 00:29:25
And then the whole thing
just repeats, and

486
00:29:25 --> 00:29:26
repeats, and repeats.
 

487
00:29:26 --> 00:29:31
So I'm thinking really this is
flow in an infinite strip.

488
00:29:31 --> 00:29:33
Infinite height or
something like that.

489
00:29:33 --> 00:29:38
You can imagine that there
could be applications.

490
00:29:38 --> 00:29:40
But I still haven't
drawn the curve.

491
00:29:40 --> 00:29:46
I just think, let's see, what
would it look like when y is a

492
00:29:46 --> 00:29:50
little - suppose I'm trying
to draw the picture of

493
00:29:50 --> 00:29:55
e^x*cos(y)=1, whatever.
 

494
00:29:55 --> 00:30:00
OK, I'll just attempt
to draw that curve.

495
00:30:00 --> 00:30:10
Just, so if y was a little bit
off of zero, the cosine would

496
00:30:10 --> 00:30:14
be, yeah, how's it going to go?
 

497
00:30:14 --> 00:30:25
If y is just a little
off zero, tell me any

498
00:30:25 --> 00:30:30
points on this curve?
 

499
00:30:30 --> 00:30:36
I can see that e^x is
going to be a big number.

500
00:30:36 --> 00:30:39
Is (0,0) on the curve?
 

501
00:30:39 --> 00:30:40
Good.
 

502
00:30:40 --> 00:30:44
Got one point.
 

503
00:30:44 --> 00:30:47
Alright.
 

504
00:30:47 --> 00:30:52
Now, suppose y is a little
bit more than zero.

505
00:30:52 --> 00:30:55
So suppose y goes
up a little bit.

506
00:30:55 --> 00:31:00
Then what? (1,0) or something?
 

507
00:31:00 --> 00:31:00
Yeah.
 

508
00:31:00 --> 00:31:03
I suppose (1,0)?
 

509
00:31:04 --> 00:31:08
No, no.
 

510
00:31:08 --> 00:31:16
So if y goes up a little, then
x would go out a little bit.

511
00:31:16 --> 00:31:18
So what's happening?
 

512
00:31:18 --> 00:31:25
So cos(y), so the cos(y)
is going to drop from

513
00:31:25 --> 00:31:28
one to zero, right?
 

514
00:31:28 --> 00:31:30
To start with.
 

515
00:31:30 --> 00:31:33
Then, if this cos(y) is
dropping from one to zero then

516
00:31:33 --> 00:31:39
this e^x has got to climb up,
to to keep the product one.

517
00:31:39 --> 00:31:40
So I'll move out.
 

518
00:31:40 --> 00:31:42
So somehow it'll move out.
 

519
00:31:42 --> 00:32:00
I I think maybe when y reaches
pi/2, then the cosine

520
00:32:00 --> 00:32:06
has got down to zero.
 

521
00:32:06 --> 00:32:09
We could work on
this for a while.

522
00:32:09 --> 00:32:11
Or we could let MATLAB draw it.
 

523
00:32:11 --> 00:32:17
But I think that we would see
these, and I could do better.

524
00:32:17 --> 00:32:21
I'm feeling pretty humiliated
to not have a better

525
00:32:21 --> 00:32:22
picture here.
 

526
00:32:22 --> 00:32:25
Suppose y is a little
less than zero?

527
00:32:25 --> 00:32:26
Do we get anything
interesting there?

528
00:32:26 --> 00:32:30
Oh well, the cosine
is an even function.

529
00:32:30 --> 00:32:34
So I think the thing might,
is it just going to

530
00:32:34 --> 00:32:38
turn around like that?
 

531
00:32:38 --> 00:32:44
So that y and minus y for a
certain x, the y value and the

532
00:32:44 --> 00:32:48
minus y will both be on the
curve because the cosine

533
00:32:48 --> 00:32:52
doesn't know whether it's the
cosine of of a plus or a minus.

534
00:32:52 --> 00:32:58
Yeah, I think we would
get curves of that sort.

535
00:32:58 --> 00:33:04
And then the other curves, S
equal constant, the stream

536
00:33:04 --> 00:33:08
lines will somehow
go vertically.

537
00:33:08 --> 00:33:16
Maybe I'll just not use
the whole time to work on

538
00:33:16 --> 00:33:18
that particular curve.
 

539
00:33:18 --> 00:33:20
We'd have to prepare it.
 

540
00:33:20 --> 00:33:25
The point is, you see how
incredibly easily we produce

541
00:33:25 --> 00:33:29
solutions to Laplace's equation
that you wouldn't have thought

542
00:33:29 --> 00:33:31
of, and I wouldn't
have thought of.

543
00:33:31 --> 00:33:34
So that would be one way
to produce solutions.

544
00:33:34 --> 00:33:39
I might even repeat this one in
class Friday, or I might not.

545
00:33:39 --> 00:33:43
Let me suggest another couple
of possibilities that

546
00:33:43 --> 00:33:45
I will do in class.
 

547
00:33:45 --> 00:33:49
Can I just give you a couple
of other functions, f.

548
00:33:49 --> 00:33:54
In fact, I'll just erase that
one and put in some other ones.

549
00:33:54 --> 00:33:57
Suppose I took the
function 1/(x_iy).

550
00:33:57 --> 00:34:07
 
 

551
00:34:07 --> 00:34:09
So that's a function of this
magic combination, x+iy.

552
00:34:11 --> 00:34:15
What's its real part and
what's its imaginary part?

553
00:34:15 --> 00:34:19
Do you know how to split that
guy into real and imaginary?

554
00:34:19 --> 00:34:21
There's a little trick,
if you remember from

555
00:34:21 --> 00:34:24
learning complex numbers.
 

556
00:34:24 --> 00:34:26
Do you remember the trick?
 

557
00:34:26 --> 00:34:31
The problem is that this
thing is down in the

558
00:34:31 --> 00:34:33
denominator, right?
 

559
00:34:33 --> 00:34:34
We don't want it there.
 

560
00:34:34 --> 00:34:38
Because we can't split the real
and imaginary parts down there.

561
00:34:38 --> 00:34:43
So I would like to rewrite it
in a way that gets something

562
00:34:43 --> 00:34:47
real down in the denominator,
moves all the i stuff up in

563
00:34:47 --> 00:34:50
the numerator where
I can separate it.

564
00:34:50 --> 00:34:52
How do I do it?
 

565
00:34:52 --> 00:34:58
Multiply both sides by, both
top and bottom, by x-iy.

566
00:34:59 --> 00:35:03
Good.
 

567
00:35:03 --> 00:35:05
So what does that
put down here now?

568
00:35:05 --> 00:35:07
That's a number times its
conjugate and that's going

569
00:35:07 --> 00:35:11
to produce x squared.
 

570
00:35:11 --> 00:35:13
Minus or plus?
 

571
00:35:13 --> 00:35:16
Plus y squared, right.
 

572
00:35:16 --> 00:35:19
The number times its conjugate
is the length squared.

573
00:35:19 --> 00:35:27
And now we just have x-iy, and
now it's obvious what the u is.

574
00:35:27 --> 00:35:33
This is real now, so the u is
just x over x squared plus y

575
00:35:33 --> 00:35:39
squared, and the S is the
minus y over x squared

576
00:35:39 --> 00:35:41
plus y squared.
 

577
00:35:41 --> 00:35:44
That's a very interesting flow.
 

578
00:35:44 --> 00:35:47
That's an interesting
flow, and we could do

579
00:35:47 --> 00:35:48
its picture and so on.
 

580
00:35:48 --> 00:35:51
And in fact it would be a
nicer picture than the

581
00:35:51 --> 00:35:57
one we stopped on.
 

582
00:35:57 --> 00:36:03
What should I notice
about this flow?

583
00:36:03 --> 00:36:08
Of course, the flow is
automatically irrotational,

584
00:36:08 --> 00:36:13
the curl is zero because
there is a potential.

585
00:36:13 --> 00:36:18
A gradient of a potential, the
gradient of a potential is

586
00:36:18 --> 00:36:22
going to be free of rotation.
 

587
00:36:22 --> 00:36:30
And there will be stream
lines, all those good things.

588
00:36:30 --> 00:36:34
There's one bad point
about the flow, though.

589
00:36:34 --> 00:36:36
Which is where?
 

590
00:36:36 --> 00:36:36
At (0,0).
 

591
00:36:37 --> 00:36:42
The whole thing falls apart, at
the origin this falls apart.

592
00:36:42 --> 00:36:46
So this is a great flow
except at the origin,

593
00:36:46 --> 00:36:49
it's very problematic.
 

594
00:36:49 --> 00:36:51
It's singular at the origin.
 

595
00:36:51 --> 00:36:55
So if we drew the pictures
we would see you

596
00:36:55 --> 00:36:56
something strange.
 

597
00:36:56 --> 00:36:59
This is going to
zero at the origin.

598
00:36:59 --> 00:37:03
So, yeah, we have trouble at
the origin but an important

599
00:37:03 --> 00:37:05
flow otherwise, yep.
 

600
00:37:05 --> 00:37:07
And I'll just mention the
third but I won't do

601
00:37:07 --> 00:37:08
anything with it.
 

602
00:37:08 --> 00:37:13
Because it's such a neat
one that I have to

603
00:37:13 --> 00:37:15
save it for Friday.
 

604
00:37:15 --> 00:37:19
The other natural function to
think of is the logarithm.

605
00:37:19 --> 00:37:20
The logarithm of x+iy.
 

606
00:37:22 --> 00:37:28
Split that into u and S.
 

607
00:37:28 --> 00:37:31
What kind of a thing
do we have here?

608
00:37:31 --> 00:37:32
What kind of singularity?
 

609
00:37:32 --> 00:37:38
Yeah, let me just do two
moments on this example, and

610
00:37:38 --> 00:37:40
then leave it for Friday
because the whole

611
00:37:40 --> 00:37:44
class has to see it.
 

612
00:37:44 --> 00:37:47
Is there a singularity
for this guy?

613
00:37:47 --> 00:37:56
Is there a point (x,y) where
the logarithm is not great?

614
00:37:56 --> 00:37:59
At the origin, again.
 

615
00:37:59 --> 00:38:01
We'll again have a
singularity at the origin.

616
00:38:01 --> 00:38:05
Something strange is
happening at the origin.

617
00:38:05 --> 00:38:08
And what we'll find is there's
a delta function there.

618
00:38:08 --> 00:38:14
We're feeding in, we have a
source right at the origin and

619
00:38:14 --> 00:38:20
then it's flowing out on,
I think on radial lines.

620
00:38:20 --> 00:38:23
I think the stream lines go
out from the origin and

621
00:38:23 --> 00:38:26
the equipotentials go
around the origin.

622
00:38:26 --> 00:38:28
Yeah, it's a great example.
 

623
00:38:28 --> 00:38:31
So that another one to come.
 

624
00:38:31 --> 00:38:38
OK, so examples like these are,
I mean generations of thinking

625
00:38:38 --> 00:38:44
went into solutions of
Laplace's equation.

626
00:38:44 --> 00:38:50
And 2-D particularly where we
have this special combination.

627
00:38:50 --> 00:38:53
I wish we had such a
combination in 3-D

628
00:38:53 --> 00:38:55
but we simply don't.
 

629
00:38:55 --> 00:38:58
We can discuss Laplace's
equation in 3-D of

630
00:38:58 --> 00:38:59
course, very important.
 

631
00:38:59 --> 00:39:03
But I mean wave equation, this
fact that we're talking to each

632
00:39:03 --> 00:39:16
other is got the Laplacian in
3-D but there's no x+iy magic.

633
00:39:16 --> 00:39:21
OK, that's some u's and
s's and v's and w's.

634
00:39:21 --> 00:39:25
What else is on your mind?
 

635
00:39:25 --> 00:39:30
Questions?
 

636
00:39:30 --> 00:39:33
I could ask this question,
or here's something I

637
00:39:33 --> 00:39:37
did not do in class.
 

638
00:39:37 --> 00:39:40
I think I wrote down the
divergence theorem.

639
00:39:40 --> 00:39:42
So can we start by doing that?
 

640
00:39:42 --> 00:39:45
Let me write down the
divergence theorem,

641
00:39:45 --> 00:39:47
with your help.
 

642
00:39:47 --> 00:39:55
And then use it.
 

643
00:39:55 --> 00:39:58
So what does the divergence
theorem, we're in 2-D.

644
00:39:58 --> 00:40:03
So this is 2-D, just
the similar theorem.

645
00:40:03 --> 00:40:08
So what does the theorem say,
that if I take the divergence

646
00:40:08 --> 00:40:19
of some w, some vector field,
then if I integrate that over

647
00:40:19 --> 00:40:23
some region, so I have some
region here and at every

648
00:40:23 --> 00:40:26
point there's a flow, w.
 

649
00:40:26 --> 00:40:33
And I look at the divergence of
w and I integrate that, dx/dy,

650
00:40:33 --> 00:40:40
so that's a double integral
over a region, I will get,

651
00:40:40 --> 00:40:44
what's the right-hand side?
 

652
00:40:44 --> 00:40:48
What is it divergence measure?
 

653
00:40:48 --> 00:40:52
So I'm really asking like
just memory, what is the

654
00:40:52 --> 00:40:54
divergence, it's an identity.
 

655
00:40:54 --> 00:41:00
It's integration by parts
in some way, as we'll see.

656
00:41:00 --> 00:41:06
But what do you remember for
the divergence theorem?

657
00:41:06 --> 00:41:09
You get what?
 

658
00:41:09 --> 00:41:14
It measures how much
flux out, right?

659
00:41:14 --> 00:41:17
So when we measure the flux out
by integrating around the

660
00:41:17 --> 00:41:21
boundary, how much is getting
through the boundary?

661
00:41:21 --> 00:41:24
And what's the flow
through the boundary?

662
00:41:24 --> 00:41:28
I take w, but that's a vector.
 

663
00:41:28 --> 00:41:32
And I'm looking for
what component of w?

664
00:41:32 --> 00:41:36
The normal component, the
component of w, w dot n, the

665
00:41:36 --> 00:41:42
component of w that's headed
out. n is defined to be

666
00:41:42 --> 00:41:45
whatever the boundary is, here
I've made it look like a

667
00:41:45 --> 00:41:47
circle but I shouldn't have.
 

668
00:41:47 --> 00:41:51
Let me make it a little
wobblier or something.

669
00:41:51 --> 00:41:57
So the normal component at any,
there, look at that point.

670
00:41:57 --> 00:42:02
The normal direction through
the boundary, down in that

671
00:42:02 --> 00:42:05
crazy point, is this way.
 

672
00:42:05 --> 00:42:08
So the normal is
going this way here.

673
00:42:08 --> 00:42:10
Here, it's going over this way.
 

674
00:42:10 --> 00:42:13
It's perpendicular to
the boundary, OK?

675
00:42:13 --> 00:42:17
And then we integrate
around the boundary.

676
00:42:17 --> 00:42:20
Alright.
 

677
00:42:20 --> 00:42:27
So that's the identity of that,
that's the divergence theorem.

678
00:42:27 --> 00:42:32
Now, let's see.
 

679
00:42:32 --> 00:42:33
Could you, yeah.
 

680
00:42:33 --> 00:42:38
So we have a minute.
 

681
00:42:38 --> 00:42:42
You want to take a particular
w and see if this

682
00:42:42 --> 00:42:45
would be correct?
 

683
00:42:45 --> 00:42:50
How about w=0x, our
first example?

684
00:42:50 --> 00:42:51
Suppose I tried w=0x.
 

685
00:42:52 --> 00:42:56
I just want to see if the
divergence, what the flux

686
00:42:56 --> 00:42:58
is through the boundary.
 

687
00:42:58 --> 00:43:04
What what region shall I take
for the, so the divergence

688
00:43:04 --> 00:43:05
theorem has two inputs.
 

689
00:43:05 --> 00:43:07
It has a flow field.
 

690
00:43:07 --> 00:43:11
And let me take w to be
0x, just so it's a shear.

691
00:43:11 --> 00:43:14
And a region.
 

692
00:43:14 --> 00:43:18
And of course the integral
might not be that much fun

693
00:43:18 --> 00:43:21
to do, unless we make
the region nice.

694
00:43:21 --> 00:43:29
What do you take as a nice
region for - actually, it

695
00:43:29 --> 00:43:32
doesn't matter what
the region is.

696
00:43:32 --> 00:43:33
Take any old region.
 

697
00:43:33 --> 00:43:36
For the moment.
 

698
00:43:36 --> 00:43:38
What's the answer?
 

699
00:43:38 --> 00:43:41
For this particular flow, w=0x?
 

700
00:43:41 --> 00:43:45
 
 

701
00:43:45 --> 00:43:48
Zero.
 

702
00:43:48 --> 00:43:49
That's the cool part.
 

703
00:43:49 --> 00:43:54
If the answer's zero
then work is suspended.

704
00:43:54 --> 00:43:56
And why is it zero?
 

705
00:43:56 --> 00:44:01
Because the divergence of this
particular w, the x derivative

706
00:44:01 --> 00:44:04
of that plus the y
derivative of that is zero.

707
00:44:04 --> 00:44:05
This has divergence.
 

708
00:44:05 --> 00:44:07
Everywhere zero.
 

709
00:44:07 --> 00:44:12
So integrating is no problem
at all, so that would be

710
00:44:12 --> 00:44:15
zero, for this flow field.
 

711
00:44:15 --> 00:44:17
For this divergence free field.
 

712
00:44:17 --> 00:44:24
Zero for that because
div w is zero.

713
00:44:24 --> 00:44:26
But is that correct?
 

714
00:44:26 --> 00:44:30
What does that tell me,
these flows are - we saw

715
00:44:30 --> 00:44:33
what the flow is like.
 

716
00:44:33 --> 00:44:35
Say there's the origin.
 

717
00:44:35 --> 00:44:40
It doesn't have to be a circle,
it looks like a circle.

718
00:44:40 --> 00:44:42
Do you see why the
flux is zero?

719
00:44:42 --> 00:44:46
There is flow through
the boundary, right?

720
00:44:46 --> 00:44:52
Flow is going buzz, buzz,
buzz up this line and out.

721
00:44:52 --> 00:44:54
And it's coming in here.
 

722
00:44:54 --> 00:44:57
So there that's what
we discovered.

723
00:44:57 --> 00:45:04
This 0x is vertical flow.
 

724
00:45:04 --> 00:45:07
It hasn't gotten any
horizontal component.

725
00:45:07 --> 00:45:10
It's got a vertical
component, it's going out.

726
00:45:10 --> 00:45:13
And would we want to do
this right-hand side?

727
00:45:13 --> 00:45:16
I don't think so, right.
 

728
00:45:16 --> 00:45:19
This right-hand side is asking
me what, I have to find the

729
00:45:19 --> 00:45:23
normal direction on this
whatever curve that is.

730
00:45:23 --> 00:45:28
I have to take its dot
product with the flow 0x,

731
00:45:28 --> 00:45:33
so this is some quantity.
 

732
00:45:33 --> 00:45:36
And then I have to do this dS
which I haven't even mentioned,

733
00:45:36 --> 00:45:38
dS is arc length around.
 

734
00:45:38 --> 00:45:40
I'm integrating
around these pieces.

735
00:45:40 --> 00:45:48
But yet somehow we have some
idea from that picture that

736
00:45:48 --> 00:45:52
the total flux is zero.
 

737
00:45:52 --> 00:45:57
How would you say it in words,
if I say here's the flow field.

738
00:45:57 --> 00:46:01
There's a region, funny shape.
 

739
00:46:01 --> 00:46:03
The flux is zero
through that boundary.

740
00:46:03 --> 00:46:07
And if I asked you why,
what would you say?

741
00:46:07 --> 00:46:11
I mean, a math answer would be
use the divergence theorem.

742
00:46:11 --> 00:46:19
But why from this picture does
it look like we have zero flux?

743
00:46:19 --> 00:46:21
What comes in goes out, yeah.
 

744
00:46:21 --> 00:46:25
What's coming in the bottom
here is going out the top.

745
00:46:25 --> 00:46:29
That's basically it.
 

746
00:46:29 --> 00:46:34
So we would get
zero for that one.

747
00:46:34 --> 00:46:38
So I think the homework, the
suggested homework maybe

748
00:46:38 --> 00:46:42
includes an example where
the divergence isn't zero.

749
00:46:42 --> 00:46:44
And then you actually have
to do these integrals.

750
00:46:44 --> 00:46:50
Just as practice for what
do those integrals mean.

751
00:46:50 --> 00:46:55
Maybe I won't go through one
now, but that's good practice.

752
00:46:55 --> 00:47:04
Take some simple w, but one
with a non-zero divergence and

753
00:47:04 --> 00:47:08
then see if you can do either
or both of the integrals that

754
00:47:08 --> 00:47:11
are supposed to come out equal.
 

755
00:47:11 --> 00:47:16
That's a good one Now, there's
one thing I could - any

756
00:47:16 --> 00:47:18
questions, or discussion?
 

757
00:47:18 --> 00:47:25
You guys are seeing these
examples come up; it's the only

758
00:47:25 --> 00:47:28
way I would know to learn
this subject is take

759
00:47:28 --> 00:47:30
simple v's and w's.
 

760
00:47:30 --> 00:47:36
And see what you
can do with them.

761
00:47:36 --> 00:47:39
We've got the general
principles, but then apply

762
00:47:39 --> 00:47:42
them to specific flows.
 

763
00:47:42 --> 00:47:44
AUDIENCE: [INAUDIBLE]
 

764
00:47:44 --> 00:47:44
PROFESSOR STRANG: Yes, thanks
 

765
00:47:44 --> 00:47:52
AUDIENCE: [INAUDIBLE]
 

766
00:47:52 --> 00:47:53
PROFESSOR STRANG: This theorem?
 

767
00:47:53 --> 00:47:58
AUDIENCE: [INAUDIBLE]
 

768
00:47:58 --> 00:48:00
PROFESSOR STRANG: Yeah
if it was, well let's

769
00:48:00 --> 00:48:02
draw a funny shape.
 

770
00:48:02 --> 00:48:03
See what we think.
 

771
00:48:03 --> 00:48:11
I mean, with this flow, right?
 

772
00:48:11 --> 00:48:16
Let me just say, if the flow
has some difficult divergence

773
00:48:16 --> 00:48:19
and the region is some mess,
nobody's going to

774
00:48:19 --> 00:48:20
be able to do it.
 

775
00:48:20 --> 00:48:21
I mean, yeah.
 

776
00:48:21 --> 00:48:24
So don't think that
these can all be done.

777
00:48:24 --> 00:48:29
The equality, it's like
integrals in calculus.

778
00:48:29 --> 00:48:31
No problem to think of
integrations that are just

779
00:48:31 --> 00:48:33
beyond human capacity.
 

780
00:48:33 --> 00:48:36
But the formulas still hold.
 

781
00:48:36 --> 00:48:43
Suppose my region was
even, like this?

782
00:48:43 --> 00:48:48
Would that still be, do we
still see flow in equals flow

783
00:48:48 --> 00:48:50
out for this particular flow?
 

784
00:48:50 --> 00:48:53
I think, yeah, the
flow's going this way.

785
00:48:53 --> 00:48:56
So it's coming in here,
it's going out again.

786
00:48:56 --> 00:48:57
That contributes.
 

787
00:48:57 --> 00:49:00
Back in again here,
and out again here.

788
00:49:00 --> 00:49:09
Yeah, I think our instinct
would be correct there, yeah.

789
00:49:09 --> 00:49:11
All sorts of examples.
 

790
00:49:11 --> 00:49:16
I was going to, well, I'll
maybe do it in class.

791
00:49:16 --> 00:49:21
This divergence theorem is the
fundamental theorem of 2-D

792
00:49:21 --> 00:49:22
calculus, you could say.
 

793
00:49:22 --> 00:49:27
Or one of them.
 

794
00:49:27 --> 00:49:42
And to write these things and
see what they lead to, yeah.

795
00:49:42 --> 00:49:44
I'll tell you what
I was going to do.

796
00:49:44 --> 00:49:52
I was going to apply this to
the vector field, u times w.

797
00:49:52 --> 00:49:56
So u is a scalar, w is a
vector, and therefore

798
00:49:56 --> 00:49:58
uw is a vector.
 

799
00:49:58 --> 00:50:06
It's got two components, uw_1,
are you willing to do that one?

800
00:50:06 --> 00:50:16
So I apply this not to w
itself, but to u times w.

801
00:50:16 --> 00:50:20
Which has two components,
uw_1 and uw_2.

802
00:50:22 --> 00:50:24
OK, so I should take
the divergence of uw.

803
00:50:25 --> 00:50:31
I mean, that's a vector
field, uw, this guy.

804
00:50:31 --> 00:50:35
And I'll get the uw dot n.
 

805
00:50:35 --> 00:50:41
And it just turns out that this
is the right way to do it.

806
00:50:41 --> 00:50:45
To see the fact that
gradient and divergence are

807
00:50:45 --> 00:50:47
transposes of each other.
 

808
00:50:47 --> 00:50:48
Yeah, yeah.
 

809
00:50:48 --> 00:50:51
I maybe I won't do that
calculation now, I'll just say

810
00:50:51 --> 00:50:56
that if you take the divergence
theorem and you apply it to

811
00:50:56 --> 00:51:04
this guy, , and
write out what it means, you

812
00:51:04 --> 00:51:10
get a very interesting formula.
 

813
00:51:10 --> 00:51:12
I'll just leave that there.
 

814
00:51:12 --> 00:51:14
So I'm ready for a final
question if there is one,

815
00:51:14 --> 00:51:21
or otherwise keep going
Friday with these Laplace

816
00:51:21 --> 00:51:23
equation solutions.
 

817
00:51:23 --> 00:51:25
Play with some vector fields.
 

818
00:51:25 --> 00:51:29
That's my best advice.
 

819
00:51:29 --> 00:51:34
And I'll see you Friday.