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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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00:00:27 --> 00:00:29
AUDIENCE: [INAUDIBLE]
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00:00:29 --> 00:00:30
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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00:00:50 --> 00:00:54
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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00:00:57 --> 00:01:00
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.
26
00:01:08 --> 00:01:09
Please vote.
27
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Yeah.
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00:01:10 --> 00:01:13
Alright, question here and
then we'll have, well, yeah.
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00:01:13 --> 00:01:19
AUDIENCE: [INAUDIBLE]
30
00:01:19 --> 00:01:21
PROFESSOR STRANG: Oh, OK,
so those were just posted.
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Like, I see.
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00:01:24 --> 00:01:29
3.3 number two.
33
00:01:29 --> 00:01:33
AUDIENCE: [INAUDIBLE]
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PROFESSOR STRANG: OK, so this
was a case, yeah, right.
35
00:01:40 --> 00:01:42
Oh, OK.
36
00:01:42 --> 00:01:51
We could be wrong so this is
3.3 number two, asks you
37
00:01:51 --> 00:01:55
about the flow field.
38
00:01:55 --> 00:02:01
Which has no flow in the x
direction, the velocity in
39
00:02:01 --> 00:02:03
the x direction is zero.
40
00:02:03 --> 00:02:06
The velocity in the
y direction is x.
41
00:02:06 --> 00:02:10
OK, so suppose we just take
that as a full field and try to
42
00:02:10 --> 00:02:13
understand, is it a gradient.
43
00:02:13 --> 00:02:16
I mean, so what are the
questions I would ask?
44
00:02:16 --> 00:02:17
Is it a gradient of anything?
45
00:02:17 --> 00:02:21
Because we're now thinking
v and w are pretty
46
00:02:21 --> 00:02:23
much the same guy.
47
00:02:23 --> 00:02:26
So is it a gradient, yes or no?
48
00:02:26 --> 00:02:29
If so, what's the potential?
49
00:02:29 --> 00:02:32
Is it divergence
free, yes or no?
50
00:02:32 --> 00:02:34
If it is, what's the
stream function?
51
00:02:34 --> 00:02:40
And of course if the answer to
both tests was yes, then we
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00:02:40 --> 00:02:44
would be talking about
Laplace's equation.
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00:02:44 --> 00:02:50
I suspect for this example the
answer at least to one of the
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00:02:50 --> 00:02:52
two questions will be no.
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00:02:52 --> 00:02:58
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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00:02:59 --> 00:03:01
Is it a gradient.
58
00:03:01 --> 00:03:09
What's the test for, so my
two questions are, is v
59
00:03:09 --> 00:03:13
the gradient of some u?
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00:03:13 --> 00:03:17
And what's the test for that?
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00:03:17 --> 00:03:20
You remember if, is the
gradient, and see if I
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00:03:20 --> 00:03:21
can remember myself.
63
00:03:21 --> 00:03:28
If it is a gradient, then this
is du/dx, and this is du/dy,
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00:03:28 --> 00:03:33
and the condition that v_1 and
v_2 would have to satisfy is
65
00:03:33 --> 00:03:37
that the y derivative of that
would have to equal the x
66
00:03:37 --> 00:03:41
derivative of that, because on
the right-hand side they are
67
00:03:41 --> 00:03:44
the same. u_xy is
the same as u_yx.
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00:03:45 --> 00:03:53
So I would look at at, so
let me write that again.
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00:03:53 --> 00:04:02
I need the dv_1/dy to equal
dv_2/dx, and is that
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00:04:02 --> 00:04:06
true in this example?
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00:04:06 --> 00:04:07
What's dv_1/dy?
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00:04:09 --> 00:04:10
Zero.
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00:04:10 --> 00:04:11
What's dv_2/dx?
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00:04:11 --> 00:04:14
75
00:04:14 --> 00:04:15
One.
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00:04:15 --> 00:04:17
So the answer's no.
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OK.
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So test, failed.
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00:04:22 --> 00:04:29
Alright, the second question is
does it possibly sit over in
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the divergence free world?
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00:04:31 --> 00:04:36
Is the divergence of, now
I'll call it w, equal zero?
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00:04:36 --> 00:04:40
So the answer was no to that
question but now I think
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00:04:40 --> 00:04:43
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?
85
00:04:47 --> 00:04:51
It's the x derivative of that,
which is certainly zero, plus
86
00:04:51 --> 00:04:55
the y derivative of that,
which is certainly zero.
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00:04:55 --> 00:04:58
So the answer is yes.
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00:04:58 --> 00:05:03
So there's no potential
but there is a stream
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function, right?
90
00:05:04 --> 00:05:08
Because the stream function
comes in with this test.
91
00:05:08 --> 00:05:12
So let's remember what, just
from today's lecture, what was
92
00:05:12 --> 00:05:21
the stream function from
dw_1/dx+dw_2/dy=0, that'll be
93
00:05:21 --> 00:05:27
satisfied if w_1 is the y
derivative of the stream
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00:05:27 --> 00:05:32
function, and w_2 is
minus the x derivative.
95
00:05:32 --> 00:05:39
Because then this matches the x
derivative of this plus the y
96
00:05:39 --> 00:05:42
derivative of this, which is
the divergence on the
97
00:05:42 --> 00:05:44
right-hand side I
would get zero.
98
00:05:44 --> 00:05:47
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.
100
00:05:49 --> 00:05:50
Let's see.
101
00:05:50 --> 00:05:55
Here w_1 is zero, so that tells
me s doesn't depend on y at
102
00:05:55 --> 00:06:02
all. w_2 is x, so x is supposed
to be minus the x derivative of
103
00:06:02 --> 00:06:05
the stream function, so what
is the stream function now?
104
00:06:05 --> 00:06:10
Have I got room to put it here?
105
00:06:10 --> 00:06:11
Just about.
106
00:06:11 --> 00:06:16
What will work?
107
00:06:16 --> 00:06:19
So again, here's w_1.
108
00:06:21 --> 00:06:26
The y derivative of S is zero.
w_2 tells me that the x
109
00:06:26 --> 00:06:28
derivative of S is minus x.
110
00:06:28 --> 00:06:31
So all I'm looking for is a
function that only depends on
111
00:06:31 --> 00:06:36
x, has no dependence on y,
and its derivative
112
00:06:36 --> 00:06:38
should be minus x.
113
00:06:38 --> 00:06:41
So what's the function?
114
00:06:41 --> 00:06:45
Minus 1/2 of x squared.
115
00:06:45 --> 00:06:54
Yeah, so this gives me S equal
minus a half of x squared.
116
00:06:54 --> 00:06:59
Alright, so you're saying
that, so there is a
117
00:06:59 --> 00:07:02
stream function, right.
118
00:07:02 --> 00:07:04
And what does that
travel along?
119
00:07:04 --> 00:07:09
That travels along steam, that
means that the flow buzzes
120
00:07:09 --> 00:07:10
along stream lines.
121
00:07:10 --> 00:07:12
And what are the stream lines?
122
00:07:12 --> 00:07:16
They're the lines
where s is constant.
123
00:07:16 --> 00:07:19
Equipotentials were the lines
where u is constant, but here
124
00:07:19 --> 00:07:22
we don't have a u
in this problem.
125
00:07:22 --> 00:07:26
Stream lines are lines where
the S is constant, so minus 1/2
126
00:07:26 --> 00:07:31
x squared is constant, what's
the picture look like?
127
00:07:31 --> 00:07:39
Picture then, for that, well,
and you know what the flow is
128
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.
130
00:07:46 --> 00:07:49
Let me draw the little arrow.
131
00:07:49 --> 00:07:55
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.
133
00:08:01 --> 00:08:05
So the flow is going
up there, right?
134
00:08:05 --> 00:08:10
Here are the y components x.
135
00:08:10 --> 00:08:13
This whole line is all
traveling up with
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00:08:13 --> 00:08:16
the same velocity.
137
00:08:16 --> 00:08:19
If I drop a leaf there, it
buzzes up that straight line.
138
00:08:19 --> 00:08:21
So that's the stream line.
139
00:08:21 --> 00:08:28
And its velocity is, that's x
equal; if, say, the velocity is
140
00:08:28 --> 00:08:33
three then this speed is three.
141
00:08:33 --> 00:08:35
It's going up that line.
142
00:08:35 --> 00:08:38
So that's a stream line.
143
00:08:38 --> 00:08:41
And sure enough, on
that line minus 1/2 x
144
00:08:41 --> 00:08:42
squared is a constant.
145
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
147
00:08:52 --> 00:08:54
x squared equal constant.
148
00:08:54 --> 00:08:55
Yeah, that's what we want.
149
00:08:55 --> 00:09:02
So, but now having got so
far, let me take x=1, say.
150
00:09:02 --> 00:09:04
What's the flow like on that?
151
00:09:04 --> 00:09:11
So there's a stream line
with S equal constant.
152
00:09:11 --> 00:09:15
And the velocity on that is
zero, so nothing is going
153
00:09:15 --> 00:09:17
in that horizontally.
154
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.
156
00:09:26 --> 00:09:30
This was faster flow.
157
00:09:30 --> 00:09:34
And then the question that's in
that homework problem is, is
158
00:09:34 --> 00:09:38
there any rotation
in this flow?
159
00:09:38 --> 00:09:42
We think about rotation,
we have an image of
160
00:09:42 --> 00:09:45
rotational flow.
161
00:09:45 --> 00:09:49
And that could be the next
example, we could figure out.
162
00:09:49 --> 00:09:54
A flow that goes around
in circles, right?
163
00:09:54 --> 00:09:56
Those would be the
stream lines.
164
00:09:56 --> 00:10:03
So this would be like pure
rotation, shall I call it.
165
00:10:03 --> 00:10:05
But I don't have that there.
166
00:10:05 --> 00:10:08
I just want to draw the
other picture, in which the
167
00:10:08 --> 00:10:16
stream lines are circles.
168
00:10:16 --> 00:10:19
To have another nice,
clean, beautiful example.
169
00:10:19 --> 00:10:21
OK, but here we don't have.
170
00:10:21 --> 00:10:24
Our streamlines are
straight lines, and
171
00:10:24 --> 00:10:26
yet we have rotation.
172
00:10:26 --> 00:10:30
That's the point here.
173
00:10:30 --> 00:10:32
Why do I say we have rotation?
174
00:10:32 --> 00:10:36
Because the test for rotation
was that original test of
175
00:10:36 --> 00:10:43
looking at, which I just wrote
the answer to be no here.
176
00:10:43 --> 00:10:47
So if it's not the gradient,
the reason is there's
177
00:10:47 --> 00:10:48
some rotation.
178
00:10:48 --> 00:10:51
Gradient fields don't
have any rotation.
179
00:10:51 --> 00:11:00
The rotation is this thing that
comes out, yeah it's this
180
00:11:00 --> 00:11:05
difference. dv_2/dx, it's the
difference between those
181
00:11:05 --> 00:11:08
that tells us the rotation.
182
00:11:08 --> 00:11:12
And that was not zero, right?
183
00:11:12 --> 00:11:17
For this example dv_1/dy,
was zero, because v_1 is
184
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.
186
00:11:25 --> 00:11:29
And in other words
the test for being a
187
00:11:29 --> 00:11:31
gradient is no rotation.
188
00:11:31 --> 00:11:33
This fails that test.
189
00:11:33 --> 00:11:35
But how is it rotating?
190
00:11:35 --> 00:11:41
How can it be rotating when
the all the flow is just
191
00:11:41 --> 00:11:43
traveling vertically?
192
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.
194
00:11:49 --> 00:11:54
My image of rotation was this
simpleminded type of flow.
195
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.
198
00:12:03 --> 00:12:10
And actually, you'll realize
that if x is negative then the
199
00:12:10 --> 00:12:14
flow in the second component,
the velocity, is now negative.
200
00:12:14 --> 00:12:17
So it would be the stream
line, the flow would be
201
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,
204
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.