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PROFESSOR STRANG: OK.
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So this is a fun lecture.
11
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This is the lecture where
understanding the gradient,
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00:00:28 --> 00:00:32
which we did last time, the
divergence, that we almost
13
00:00:32 --> 00:00:35
completed, those two pieces
now come together into
14
00:00:35 --> 00:00:37
Laplace's equation.
15
00:00:37 --> 00:00:42
Things work in a fantastic
way, so I enjoy this one.
16
00:00:42 --> 00:00:47
So the result will be that we
have the potential function,
17
00:00:47 --> 00:00:52
u, that we've spoken about,
and take its gradient.
18
00:00:52 --> 00:00:57
Now is coming a stream
function, S, that's connected
19
00:00:57 --> 00:00:58
with the divergence.
20
00:00:58 --> 00:01:01
When the divergence is zero,
there's something called
21
00:01:01 --> 00:01:03
a stream function, S.
22
00:01:03 --> 00:01:07
And these, the connections
between those two functions,
23
00:01:07 --> 00:01:11
u and S, are crucial.
24
00:01:11 --> 00:01:16
And connecting gradient to
divergence will take us
25
00:01:16 --> 00:01:18
to Laplace's equation.
26
00:01:18 --> 00:01:21
And then you'll see
the special, special
27
00:01:21 --> 00:01:25
role of x+iy in 2-D.
28
00:01:25 --> 00:01:27
So this is in two dimensions.
29
00:01:27 --> 00:01:28
OK.
30
00:01:28 --> 00:01:32
So let me begin
with divergence.
31
00:01:32 --> 00:01:36
This is the divergence
of w, of course.
32
00:01:36 --> 00:01:40
And solve it.
33
00:01:40 --> 00:01:42
Just as we wanted to solve
Kirchoff's current law, A
34
00:01:42 --> 00:01:47
transpose w equal zero, so now
in the continuous case, we want
35
00:01:47 --> 00:01:54
to find solutions to -- we want
to find divergence-free fields.
36
00:01:54 --> 00:01:57
Source-free fields,
you could say.
37
00:01:57 --> 00:01:59
That's about the best word we
have, divergence-free free,
38
00:01:59 --> 00:02:05
meaning there is no divergence.
39
00:02:05 --> 00:02:07
So what have we got here?
40
00:02:07 --> 00:02:12
We've got one equation
in two unknowns.
41
00:02:12 --> 00:02:14
Last time, with the gradient
business, we had two
42
00:02:14 --> 00:02:18
equations for u, and
only that one unknown.
43
00:02:18 --> 00:02:20
Now we've got one equation,
two unknowns, because we're
44
00:02:20 --> 00:02:23
looking at the transpose.
45
00:02:23 --> 00:02:25
So there should be a
lot of solutions.
46
00:02:25 --> 00:02:26
Right?
47
00:02:26 --> 00:02:30
If you give me a w_1,
then probably I'll be
48
00:02:30 --> 00:02:31
able to find a w_2.
49
00:02:31 --> 00:02:35
But there's a neat way to
describe the solutions
50
00:02:35 --> 00:02:38
to that equation, the
divergence-free fields.
51
00:02:38 --> 00:02:41
It's to introduce something
called a source function
52
00:02:41 --> 00:02:44
-- a stream function,
sorry, stream function.
53
00:02:44 --> 00:02:53
OK, and now the idea will be
that if I let -- let me try.
54
00:02:53 --> 00:02:57
I take any function S(x,y),
any function whatever.
55
00:02:57 --> 00:03:04
So now I try --
let w_1 be dS/dy.
56
00:03:04 --> 00:03:08
57
00:03:08 --> 00:03:14
Maybe I should say, maybe
I should go this way.
58
00:03:14 --> 00:03:16
I'm looking for solutions.
59
00:03:16 --> 00:03:16
OK.
60
00:03:16 --> 00:03:20
So I take any function,
S, I take its y
61
00:03:20 --> 00:03:21
derivative to be w_1.
62
00:03:23 --> 00:03:28
And you can tell me
what w_2 has to be.
63
00:03:28 --> 00:03:33
So if w_1 is dS/dy,
this will be what?
64
00:03:33 --> 00:03:38
This'll be the second -- the x
derivative of the y derivative.
65
00:03:38 --> 00:03:39
Right?
66
00:03:39 --> 00:03:43
If w_1 is the y derivative,
then when I take the x
67
00:03:43 --> 00:03:45
derivative, I've got
this cross derivative.
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00:03:45 --> 00:03:48
So what would be the
smart choice for w_2?
69
00:03:49 --> 00:03:56
So I'll say then, then w_2 will
be -- now I've sort of said
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00:03:56 --> 00:04:01
what w_1 is, what's the
w_2 that goes with it?
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00:04:01 --> 00:04:04
Well, it's whatever it
takes to cancel this.
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00:04:04 --> 00:04:08
I want this to be, in other
words I want it to get zero,
73
00:04:08 --> 00:04:12
so this should be the
second derivative S.
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00:04:12 --> 00:04:15
And what am I going
to put now? dydx.
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I'm using the same very crucial
fact that the cross derivative
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can be in either order.
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So what do I see for w_2?
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00:04:26 --> 00:04:28
Do you see what w_2 is?
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This is supposed to match that.
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00:04:30 --> 00:04:35
There's a minus
sign. w_2 is what?
81
00:04:35 --> 00:04:36
Minus dS/dx.
82
00:04:36 --> 00:04:39
83
00:04:39 --> 00:04:39
Right?
84
00:04:39 --> 00:04:40
Minus dS/dx.
85
00:04:42 --> 00:04:44
That's the minus dS/dx.
86
00:04:46 --> 00:04:50
If I take any function, S, I
let w_1 be its y derivative and
87
00:04:50 --> 00:04:55
w_2 be minus its x derivative,
then the divergence
88
00:04:55 --> 00:04:57
will be zero.
89
00:04:57 --> 00:05:00
Because I'll have the cross
derivative minus the cross
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00:05:00 --> 00:05:02
derivative, of course
that'll be zero.
91
00:05:02 --> 00:05:08
So these will be my w_1's, my
w's. w_1, w_2, coming from S.
92
00:05:08 --> 00:05:12
So I have what I expect,
with two unknowns, only
93
00:05:12 --> 00:05:15
one equation, I've got
lots of solutions.
94
00:05:15 --> 00:05:18
I create any function, S,
and that will be one.
95
00:05:18 --> 00:05:21
So that's a stream function.
96
00:05:21 --> 00:05:24
And it has a physical meaning,
so we get to see it.
97
00:05:24 --> 00:05:30
And it has a fantastic
connection to the potential.
98
00:05:30 --> 00:05:31
So up to now
99
00:05:31 --> 00:05:35
-- so this is now the moment
pieces come together.
100
00:05:35 --> 00:05:37
Up to now, we had the
divergence and the
101
00:05:37 --> 00:05:39
gradient separately.
102
00:05:39 --> 00:05:42
So up to now, what we had was,
we started with the potential
103
00:05:42 --> 00:05:48
u, and we went to v -- I called
it v for this application
104
00:05:48 --> 00:05:50
-- .
105
00:05:50 --> 00:05:54
106
00:05:54 --> 00:05:54
OK.
107
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And now, over on this side,
I had the divergence
108
00:05:57 --> 00:06:00
of w equals zero.
109
00:06:00 --> 00:06:05
And that lead me to w
being, what we just said,
110
00:06:05 --> 00:06:10
dS/dy, and minus dS/dx.
111
00:06:11 --> 00:06:15
OK, two separate pictures.
112
00:06:15 --> 00:06:20
Now we're going to connect the
framework, going to give the
113
00:06:20 --> 00:06:23
connection between v and w.
114
00:06:23 --> 00:06:25
And the connection will
be the easiest possible;
115
00:06:25 --> 00:06:26
they'll be equal.
116
00:06:26 --> 00:06:29
So the c in our framework
is the identity.
117
00:06:29 --> 00:06:33
I want to say, I want to
look at our framework
118
00:06:33 --> 00:06:36
when c is the identity.
119
00:06:36 --> 00:06:39
So v and w are the same.
120
00:06:39 --> 00:06:41
In other words, what
equation do we get then?
121
00:06:41 --> 00:06:47
We started with u, we take
the gradient of u, that's v.
122
00:06:47 --> 00:06:50
We go over here, and we still
have the gradient of u,
123
00:06:50 --> 00:06:55
because this was the v
and now it's also the w.
124
00:06:55 --> 00:06:58
And then we take --
what do we take?
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00:06:58 --> 00:07:02
Minus the divergence
of the gradient of u,
126
00:07:02 --> 00:07:04
equals, let's say, f.
127
00:07:04 --> 00:07:07
Shall we say f, to make
something happen?
128
00:07:07 --> 00:07:12
Or zero -- I'll say zero.
129
00:07:12 --> 00:07:14
I'll say zero.
130
00:07:14 --> 00:07:17
Yeah, because zero is
the one that give us
131
00:07:17 --> 00:07:18
Laplace's equation.
132
00:07:18 --> 00:07:22
The official name would be --
that's Laplace's equation.
133
00:07:22 --> 00:07:25
Do you recognize it?
134
00:07:25 --> 00:07:27
We'd better write
it out properly.
135
00:07:27 --> 00:07:31
The divergence of the
gradient is the Laplacian.
136
00:07:31 --> 00:07:35
So this is Laplace.
137
00:07:35 --> 00:07:38
Famous equation.
138
00:07:38 --> 00:07:40
Steady state type of equation.
139
00:07:40 --> 00:07:42
It goes with steady
state problems.
140
00:07:42 --> 00:07:45
Let's just figure
out what this is.
141
00:07:45 --> 00:07:53
So w is, since these are the
same, w is now .
142
00:07:55 --> 00:07:56
It's the gradient.
143
00:07:56 --> 00:07:59
Now what happens, do you see
what happens when I take the
144
00:07:59 --> 00:08:07
divergence of the gradient,
the divergence of this w.
145
00:08:07 --> 00:08:09
I just want to write
that equation out.
146
00:08:09 --> 00:08:12
It's crucial.
147
00:08:12 --> 00:08:17
I need brilliant colors
and lights shining, now,
148
00:08:17 --> 00:08:19
for what goes in there.
149
00:08:19 --> 00:08:22
Because I want to take the
divergence of the gradient.
150
00:08:22 --> 00:08:23
So what does that mean?
151
00:08:23 --> 00:08:27
I take the x derivative of the
first component, plus the y
152
00:08:27 --> 00:08:29
derivative of the
second component.
153
00:08:29 --> 00:08:32
The minus sign is not going
to matter, with a zero
154
00:08:32 --> 00:08:34
on the right hand side.
155
00:08:34 --> 00:08:35
I could cancel the minus.
156
00:08:35 --> 00:08:38
So, but let me say it again.
157
00:08:38 --> 00:08:41
The divergence is, it
applies to a back vector,
158
00:08:41 --> 00:08:43
I've got a vector.
159
00:08:43 --> 00:08:47
I take the component of
the first -- of w_1.
160
00:08:47 --> 00:08:52
So that'll give me d
second u/dx squared.
161
00:08:52 --> 00:08:55
And I take the y derivative --
I should have said derivative
162
00:08:55 --> 00:08:59
-- y derivative of the
second component.
163
00:08:59 --> 00:09:05
And the y derivative of that
is d second u/dy squared.
164
00:09:05 --> 00:09:08
And I get zero.
165
00:09:08 --> 00:09:11
So that's Laplace's equation.
166
00:09:11 --> 00:09:15
You see, the whole idea is
Laplace's equation, in working
167
00:09:15 --> 00:09:20
with Laplace's equation, we
have three elements, here.
168
00:09:20 --> 00:09:25
The gradient comes in,
the divergence comes in,
169
00:09:25 --> 00:09:29
and equality comes in.
170
00:09:29 --> 00:09:33
Would you like to see a more
general Laplace's equation?
171
00:09:33 --> 00:09:35
Well, a more general
Poisson's equation.
172
00:09:35 --> 00:09:43
So underneath Laplace,
let me write Poisson.
173
00:09:43 --> 00:09:46
You got the pronunciation, the
brilliant French pronunciation
174
00:09:46 --> 00:09:49
there, of Poisson?
175
00:09:49 --> 00:09:50
That was my best.
176
00:09:50 --> 00:09:52
I can't improve on that one.
177
00:09:52 --> 00:09:55
So it comes in when there's
a right hand side.
178
00:09:55 --> 00:09:59
So the normal Poisson equation
is second derivative with
179
00:09:59 --> 00:10:03
respect to x, second derivative
with respect to y -- well,
180
00:10:03 --> 00:10:07
actually, it should
have a minus there.
181
00:10:07 --> 00:10:11
Really should be a minus there.
182
00:10:11 --> 00:10:13
You remember why the
minus is there.
183
00:10:13 --> 00:10:16
The minus is there to make
the whole thing positive.
184
00:10:16 --> 00:10:16
Right?
185
00:10:16 --> 00:10:19
That sounds crazy, that the
minus makes it positive.
186
00:10:19 --> 00:10:23
But these second derivatives
are negative definite, as
187
00:10:23 --> 00:10:26
always, and the minus makes
them positive definite.
188
00:10:26 --> 00:10:32
So I don't remember whether --
maybe I often include the minus
189
00:10:32 --> 00:10:34
over here -- equal f(x,y).
190
00:10:35 --> 00:10:36
So there's a source.
191
00:10:36 --> 00:10:38
Poisson has a source term.
192
00:10:38 --> 00:10:41
Laplace doesn't.
193
00:10:41 --> 00:10:44
And just while I'm talking
about this framework,
194
00:10:44 --> 00:10:49
if there was a c(x,y),
so this would be or.
195
00:10:49 --> 00:10:51
I'll put or.
196
00:10:51 --> 00:10:55
The more general one would be
the x derivative, because I'm
197
00:10:55 --> 00:11:04
taking the divergence, of a
c(x,y)du/dx, and the y
198
00:11:04 --> 00:11:08
derivative of a c -- so
you see the difference.
199
00:11:08 --> 00:11:13
I'm now allowing some
variable conductivity.
200
00:11:13 --> 00:11:18
Variable whatever,
variable material.
201
00:11:18 --> 00:11:22
du/dy equals f, again.
202
00:11:22 --> 00:11:24
So that would be the
more general one.
203
00:11:24 --> 00:11:27
I don't think we
plan to study that.
204
00:11:27 --> 00:11:31
Well, that's not the most
general, I could have more
205
00:11:31 --> 00:11:32
and more things there.
206
00:11:32 --> 00:11:36
But that shows you a
variable material.
207
00:11:36 --> 00:11:39
Yeah, that material
is variable.
208
00:11:39 --> 00:11:42
I would use the word --
what word would I use?
209
00:11:42 --> 00:11:44
It doesn't depend
on the direction.
210
00:11:44 --> 00:11:48
I'm using the same c(x,y)
in the x direction
211
00:11:48 --> 00:11:49
and the y direction.
212
00:11:49 --> 00:11:51
Therefore in all directions.
213
00:11:51 --> 00:11:53
And I didn't come
prepared with that word.
214
00:11:53 --> 00:11:56
What's the word for
when it doesn't depend
215
00:11:56 --> 00:11:58
on the direction?
216
00:11:58 --> 00:11:59
Isotropic!
217
00:11:59 --> 00:12:00
Thanks.
218
00:12:00 --> 00:12:01
Isotropic.
219
00:12:01 --> 00:12:03
So that's isotropic.
220
00:12:03 --> 00:12:07
So that's really our
framework there.
221
00:12:07 --> 00:12:09
At least for
isotropic materials.
222
00:12:09 --> 00:12:11
And then we could have
more general with an
223
00:12:11 --> 00:12:12
isotropic material.
224
00:12:12 --> 00:12:14
That would be fun.
225
00:12:14 --> 00:12:16
All of those things are fun.
226
00:12:16 --> 00:12:19
But Laplace's equation
is the most fun of all.
227
00:12:19 --> 00:12:24
So let me take f to be zero,
c to be one, get back up
228
00:12:24 --> 00:12:30
to Laplace's equation, and
begin to make connections.
229
00:12:30 --> 00:12:32
Begin to make connections.
230
00:12:32 --> 00:12:34
OK.
231
00:12:34 --> 00:12:39
The beautiful connection
is the one right here.
232
00:12:39 --> 00:12:44
When v and u are the same,
when v and w are the same,
233
00:12:44 --> 00:12:47
then I just read off.
234
00:12:47 --> 00:12:52
If v is the same as w, then how
is the potential function,
235
00:12:52 --> 00:12:55
which is on the left side,
connected to the strain
236
00:12:55 --> 00:12:59
function, which is coming from,
any time div w is zero, I've
237
00:12:59 --> 00:13:02
got a stream function
in 2-D, here.
238
00:13:02 --> 00:13:08
The connection is just, those
two match. du/dx is the
239
00:13:08 --> 00:13:16
same as dS/dy, and du/dy is
the same as minus dS/dx.
240
00:13:17 --> 00:13:22
You see that the two sides of
our world are coming together.
241
00:13:22 --> 00:13:28
We're dealing with flows that
are both gradient flows, they
242
00:13:28 --> 00:13:34
come from a potential, you
could often say potential
243
00:13:34 --> 00:13:38
flows, I see the word
ideal flows coming in.
244
00:13:38 --> 00:13:41
These are very special flows.
245
00:13:41 --> 00:13:47
So aero couldn't work entirely
with these flows, of course.
246
00:13:47 --> 00:13:50
These are such ideal flows.
247
00:13:50 --> 00:13:53
Proper aerodynamics, you've
got shocks, you've got all
248
00:13:53 --> 00:13:54
sorts of stuff going on.
249
00:13:54 --> 00:13:59
But in a region where
everything's beautiful, then
250
00:13:59 --> 00:14:04
you get back to this, total
steady state, steady flow,
251
00:14:04 --> 00:14:06
steady potential flow.
252
00:14:06 --> 00:14:12
We've got those equations that
connect u on one side with the
253
00:14:12 --> 00:14:16
divergence business, the stream
function on the other side.
254
00:14:16 --> 00:14:19
So I want to focus on these.
255
00:14:19 --> 00:14:22
Actually, I mean, so
the heart of Laplace's
256
00:14:22 --> 00:14:24
quation is in there.
257
00:14:24 --> 00:14:26
OK.
258
00:14:26 --> 00:14:30
First, do you know the
names of those two guys?
259
00:14:30 --> 00:14:32
I shouldn't call them guys,
they're the greatest
260
00:14:32 --> 00:14:35
mathematicians ever.
261
00:14:35 --> 00:14:38
Maybe after Gauss.
262
00:14:38 --> 00:14:42
Do you know whose names
are associated with
263
00:14:42 --> 00:14:45
those two equations.
264
00:14:45 --> 00:14:47
Well, Lagrange was great.
265
00:14:47 --> 00:14:49
I'm not saying anything
about Lagrange.
266
00:14:49 --> 00:14:51
But he didn't do this.
267
00:14:51 --> 00:14:54
So two, one French
and one German.
268
00:14:54 --> 00:15:00
So the French guy's
name is Cauchy.
269
00:15:00 --> 00:15:05
And the German is -- so these
are Cauchy, and the German
270
00:15:05 --> 00:15:07
is a really fantastic guy.
271
00:15:07 --> 00:15:10
Anybody know his name?
272
00:15:10 --> 00:15:11
Cauchy-Riemann.
273
00:15:11 --> 00:15:11
Yeah.
274
00:15:11 --> 00:15:15
The other name is Riemann.
275
00:15:15 --> 00:15:20
Cauchy-Riemann equations,
that connect the two
276
00:15:20 --> 00:15:22
pieces, u and S.
277
00:15:22 --> 00:15:24
And they're the subject of
an enormous theory that
278
00:15:24 --> 00:15:27
we'll just touch on here.
279
00:15:27 --> 00:15:29
OK. and we'll see them
graphically, and we'll
280
00:15:29 --> 00:15:32
find solutions.
281
00:15:32 --> 00:15:35
So this is one way to
pose our problem.
282
00:15:35 --> 00:15:39
Notice something here.
283
00:15:39 --> 00:15:42
I think that if we have these
equations, a solution,
284
00:15:42 --> 00:15:45
u, should satisfy
Laplace's equation.
285
00:15:45 --> 00:15:49
Because this equality will
take us around the loop.
286
00:15:49 --> 00:15:53
So do you see that if I
could solve these two
287
00:15:53 --> 00:15:55
-- here's the point.
288
00:15:55 --> 00:15:59
I'm going to get Laplace's
equation solved by u.
289
00:15:59 --> 00:16:03
Laplace's equation will
also be solved by S.
290
00:16:03 --> 00:16:04
The string function.
291
00:16:04 --> 00:16:09
So what's going to happen
is, I get two solutions.
292
00:16:09 --> 00:16:11
A pair of solutions.
293
00:16:11 --> 00:16:17
Laplace's equation, solutions
to that come in pairs, u and S.
294
00:16:17 --> 00:16:19
So we get them two at a time.
295
00:16:19 --> 00:16:23
And they're connected in
this remarkable way.
296
00:16:23 --> 00:16:26
Let's see, can you see that
u will satisfy-- this is
297
00:16:26 --> 00:16:28
the key to everything.
298
00:16:28 --> 00:16:30
Does u satisfy
Laplace's equation?
299
00:16:30 --> 00:16:31
Sure.
300
00:16:31 --> 00:16:33
I take the x derivative of
that, and I add to the
301
00:16:33 --> 00:16:35
y derivative of that.
302
00:16:35 --> 00:16:36
Do you see, it works.
303
00:16:36 --> 00:16:40
The x derivative of this, plus
the y derivative of this is
304
00:16:40 --> 00:16:43
exactly the cancellation of
the cross that I wanted.
305
00:16:43 --> 00:16:51
So I do these, I combine
those into Laplace.
306
00:16:51 --> 00:16:56
Shall I just go through
that verbally again?
307
00:16:56 --> 00:16:59
I take the x derivative of
this, which gives me the
308
00:16:59 --> 00:17:01
cross derivative of S.
309
00:17:01 --> 00:17:05
This asked me to take the y
derivative, so I get the
310
00:17:05 --> 00:17:06
cross derivative again.
311
00:17:06 --> 00:17:08
With the minus sign,
they add to zero.
312
00:17:08 --> 00:17:14
I just want to point out
that also, S satisfies
313
00:17:14 --> 00:17:15
Laplace's equation.
314
00:17:15 --> 00:17:20
Can we do that one?
315
00:17:20 --> 00:17:24
I claim that also, the
stream function solves
316
00:17:24 --> 00:17:25
Laplace's equation.
317
00:17:25 --> 00:17:27
Because we want the x
derivative of dS/dx.
318
00:17:28 --> 00:17:31
So dS/dx is minus this.
319
00:17:31 --> 00:17:32
Do you see what's happening?
320
00:17:32 --> 00:17:37
When I take the x derivative of
dS/dx, I get minus the cross
321
00:17:37 --> 00:17:41
derivative of u for this guy.
322
00:17:41 --> 00:17:44
It's the y derivative of
this, which is plus the
323
00:17:44 --> 00:17:45
cross derivative of u.
324
00:17:45 --> 00:17:46
They cancel.
325
00:17:46 --> 00:17:50
So u and S are just together.
326
00:17:50 --> 00:17:53
Oh, let's find some solutions.
327
00:17:53 --> 00:17:55
They're great to find.
328
00:17:55 --> 00:17:57
And then draw them.
329
00:17:57 --> 00:18:03
OK, can I find some solutions
to Laplace's equation.
330
00:18:03 --> 00:18:05
And I'm going to
find them in pairs.
331
00:18:05 --> 00:18:10
So I'm going to have a
list of u's and their
332
00:18:10 --> 00:18:15
corresponding S's.
333
00:18:15 --> 00:18:18
OK.
334
00:18:18 --> 00:18:20
These are solutions to Laplace.
335
00:18:20 --> 00:18:23
These are, solve
Laplace's equation.
336
00:18:23 --> 00:18:26
So we've got Laplace's
equation in our minds.
337
00:18:26 --> 00:18:29
Actually, furthermore, they
solve Cauchy-Riemann.
338
00:18:29 --> 00:18:31
Because they're going
to be connected by our
339
00:18:31 --> 00:18:34
Cauchy-Riemann equation.
340
00:18:34 --> 00:18:39
So, solve Laplace
and Cauchy-Riemann.
341
00:18:39 --> 00:18:40
OK.
342
00:18:40 --> 00:18:41
Suppose I take u(x,y)=x.
343
00:18:41 --> 00:18:44
344
00:18:44 --> 00:18:48
I'm going to start with
an easy solution.
345
00:18:48 --> 00:18:50
That certainly solves
Laplace's equation.
346
00:18:50 --> 00:18:53
You've got Laplace's
equation in mind?
347
00:18:53 --> 00:18:55
Let me write it up here
again. u_xx+u_yy=0.
348
00:18:55 --> 00:18:59
349
00:18:59 --> 00:19:03
I take the chance to write it
again, to do it in this little
350
00:19:03 --> 00:19:08
bit shorter notation, just
subscripts instead of partials.
351
00:19:08 --> 00:19:10
And also, S_xx+S_yy=0.
352
00:19:10 --> 00:19:13
353
00:19:13 --> 00:19:19
But most of all, the
Cauchy-Riemann that
354
00:19:19 --> 00:19:21
connects the two.
355
00:19:21 --> 00:19:25
Well, does u=x solve
Laplace's equation?
356
00:19:25 --> 00:19:27
Of course it does.
357
00:19:27 --> 00:19:31
The second x derivative, if
the function is x, is zero.
358
00:19:31 --> 00:19:34
And the second y derivative
is very, very zero.
359
00:19:34 --> 00:19:35
[LAUGHTER]
360
00:19:35 --> 00:19:37
So what's S?
361
00:19:37 --> 00:19:40
What's the S that goes with it?
362
00:19:40 --> 00:19:43
It'll be simple, too.
363
00:19:43 --> 00:19:47
So the S at u is x, right?
364
00:19:47 --> 00:19:49
I'm starting with this x.
365
00:19:49 --> 00:19:52
So du/dx is one.
366
00:19:52 --> 00:19:55
So what do you figure s is?
367
00:19:55 --> 00:20:00
If du/dx -- see, u is just x
itself, so du/dx is only a one.
368
00:20:00 --> 00:20:03
That derivative was easy.
369
00:20:03 --> 00:20:08
Then dS/dy is supposed to be
one, and dS/dx is supposed
370
00:20:08 --> 00:20:09
to be zero, I guess.
371
00:20:09 --> 00:20:13
Do you see what S is? y.
372
00:20:13 --> 00:20:14
S is y.
373
00:20:14 --> 00:20:16
S is y.
374
00:20:16 --> 00:20:19
Of course, that solves
Laplace's equation, too, and
375
00:20:19 --> 00:20:20
it solves Cauchy-Riemann.
376
00:20:20 --> 00:20:24
The x derivative of this is
the y derivative of that.
377
00:20:24 --> 00:20:25
One equal one.
378
00:20:25 --> 00:20:28
And the y derivative of that
is minus the x derivative
379
00:20:28 --> 00:20:30
of that, zero equal zero.
380
00:20:30 --> 00:20:33
OK, so that's an easy one.
381
00:20:33 --> 00:20:35
I'm going to go up a level.
382
00:20:35 --> 00:20:37
I want to take a second degree.
383
00:20:37 --> 00:20:41
So my next guy in the list will
be -- a pair, it's a list
384
00:20:41 --> 00:20:48
of pairs -- will be x
squared minus y squared.
385
00:20:48 --> 00:20:51
First of all, it better not
be in that list unless it
386
00:20:51 --> 00:20:53
solves Laplace's equation.
387
00:20:53 --> 00:20:56
And then if it is,
we'll find an S.
388
00:20:56 --> 00:20:59
So plug it in mentally,
can you plug this into
389
00:20:59 --> 00:21:01
Laplace's equation?
390
00:21:01 --> 00:21:05
What's the second x
derivative of this function?
391
00:21:05 --> 00:21:06
Two.
392
00:21:06 --> 00:21:10
Right? xx brings down a two.
393
00:21:10 --> 00:21:13
What's the second y derivative?
394
00:21:13 --> 00:21:15
Minus two, from this term.
395
00:21:15 --> 00:21:17
And then put them into
Laplace's equation,
396
00:21:17 --> 00:21:19
two minus two.
397
00:21:19 --> 00:21:19
Correct.
398
00:21:19 --> 00:21:20
Zero.
399
00:21:20 --> 00:21:21
All right.
400
00:21:21 --> 00:21:24
Now I'm looking for the
S that goes with it.
401
00:21:24 --> 00:21:26
The other one in the pair.
402
00:21:26 --> 00:21:30
OK, maybe I'd better think
through what -- so this is
403
00:21:30 --> 00:21:33
supposed to give
me the S, du/dx.
404
00:21:33 --> 00:21:37
405
00:21:37 --> 00:21:44
Let me copy Cauchy-Riemann
here, so we can just focus
406
00:21:44 --> 00:21:46
entirely on that board.
407
00:21:46 --> 00:21:52
So du/dx, this is 2x in my
example. du/dy is minus 2y.
408
00:21:54 --> 00:21:58
So what am I learning?
dS/dy should be 2x.
409
00:21:59 --> 00:22:00
dS/dx should be 2y.
410
00:22:02 --> 00:22:04
Because our minus
signs both there.
411
00:22:04 --> 00:22:05
What's S?
412
00:22:05 --> 00:22:08
Do you see S?
413
00:22:08 --> 00:22:15
The y derivative is 2x, the x
derivative is 2y, and that
414
00:22:15 --> 00:22:20
stream function is 2xy.
415
00:22:21 --> 00:22:22
Right?
416
00:22:22 --> 00:22:23
2xy.
417
00:22:24 --> 00:22:29
Because the x derivative
of this is 2y, and the y
418
00:22:29 --> 00:22:30
derivative of a this is 2x.
419
00:22:31 --> 00:22:34
And we saw 2xy last time also.
420
00:22:34 --> 00:22:38
And of course, it solves
Laplace's equation easily.
421
00:22:38 --> 00:22:43
Plug that in, the second
derivative is zero.
422
00:22:43 --> 00:22:45
The second x derivative is
zero, second y derivative
423
00:22:45 --> 00:22:47
is zero, everything.
424
00:22:47 --> 00:22:50
So that's a pair.
425
00:22:50 --> 00:22:55
This is a nice pair.
426
00:22:55 --> 00:22:59
You want to shoot
for third degree?
427
00:22:59 --> 00:23:04
We could maybe figure out third
degree, just by jiggling it.
428
00:23:04 --> 00:23:07
After that, we're going
to need an idea to get
429
00:23:07 --> 00:23:08
up to fourth degree.
430
00:23:08 --> 00:23:11
Let me try third degree.
431
00:23:11 --> 00:23:13
Cubics, now.
432
00:23:13 --> 00:23:16
So I'm looking, first I just
want to get somebody here.
433
00:23:16 --> 00:23:19
So it's some x cubed.
434
00:23:19 --> 00:23:22
And then I'm going to need some
more stuff, because x cubed by
435
00:23:22 --> 00:23:26
itself certainly won't work.
436
00:23:26 --> 00:23:28
I need something more.
437
00:23:28 --> 00:23:31
And I want to plug it into
Laplace's equation and figure
438
00:23:31 --> 00:23:33
out what should it be?
439
00:23:33 --> 00:23:37
OK, so when I plug this
into Laplace's equation,
440
00:23:37 --> 00:23:39
what do I get?
441
00:23:39 --> 00:23:45
Let me do Laplace's equation
over here, to try to get
442
00:23:45 --> 00:23:51
the u of degree three.
443
00:23:51 --> 00:23:56
So what's u_xx, so far?
444
00:23:56 --> 00:23:57
6x, right?
445
00:23:57 --> 00:24:00
Bring down, we've got 3x
squared, then we get 6x,
446
00:24:00 --> 00:24:01
x so I've got a 6x.
447
00:24:03 --> 00:24:07
And I'm looking for --
so u_yy should be minus
448
00:24:07 --> 00:24:10
6x to cancel that.
449
00:24:10 --> 00:24:13
So what do I want there?
450
00:24:13 --> 00:24:15
What do I need?
451
00:24:15 --> 00:24:19
I need a minus,
I'm sure of that.
452
00:24:19 --> 00:24:25
So the second y derivative
should be this 6x deal.
453
00:24:25 --> 00:24:29
What do I want?
454
00:24:29 --> 00:24:30
3xy squared?
455
00:24:30 --> 00:24:31
That sounds good.
456
00:24:31 --> 00:24:34
Let me write it down
and see if it is good.
457
00:24:34 --> 00:24:34
OK.
458
00:24:34 --> 00:24:37
The second y derivative.
459
00:24:37 --> 00:24:38
Yes.
460
00:24:38 --> 00:24:41
We'll bring down a two, and
then the y's will disappear
461
00:24:41 --> 00:24:43
and I'll have the minus 6x.
462
00:24:43 --> 00:24:44
Golden.
463
00:24:44 --> 00:24:46
OK, that's great.
464
00:24:46 --> 00:24:48
That's great.
465
00:24:48 --> 00:24:49
Is that correct?
466
00:24:49 --> 00:24:51
I mean, it's great,
but is it right?
467
00:24:51 --> 00:24:52
Yes.
468
00:24:52 --> 00:24:52
Yes.
469
00:24:52 --> 00:24:54
OK.
470
00:24:54 --> 00:25:01
Now, you have faith that
there's another one?
471
00:25:01 --> 00:25:04
Well, yeah, there
is another one.
472
00:25:04 --> 00:25:06
Cauchy-Riemann
never let us down.
473
00:25:06 --> 00:25:10
There will be an S that'll go
with that, that'll solve a
474
00:25:10 --> 00:25:13
Cauchy-Riemann equation,
and it'll look like it.
475
00:25:13 --> 00:25:22
And I think, I think -- and I
just sort of reverse x and y to
476
00:25:22 --> 00:25:25
get another one, because if I
exchange x and y, I'm still
477
00:25:25 --> 00:25:27
OK with Laplace's equation.
478
00:25:27 --> 00:25:35
I think something like 3yx
squared minus y cubed.
479
00:25:35 --> 00:25:39
If that worked, then this
one should work, too.
480
00:25:39 --> 00:25:44
Because I just switched
x and y, and I think
481
00:25:44 --> 00:25:45
it'll work right here.
482
00:25:45 --> 00:25:49
The second x derivative will
be 6y, and the second y
483
00:25:49 --> 00:25:51
derivative will be minus 6y.
484
00:25:52 --> 00:25:54
I think that's good.
485
00:25:54 --> 00:25:58
OK.
486
00:25:58 --> 00:26:01
Let's put this list on
hold for a moment.
487
00:26:01 --> 00:26:03
Something's, obviously, there's
some pattern here that
488
00:26:03 --> 00:26:06
we've got to locate.
489
00:26:06 --> 00:26:10
Can I put it on hold for a
moment and take, for example,
490
00:26:10 --> 00:26:11
the graph of this one.
491
00:26:11 --> 00:26:15
And I want to draw
the pictures.
492
00:26:15 --> 00:26:20
Before I get a complete list,
I want to draw the pictures
493
00:26:20 --> 00:26:22
of these functions.
494
00:26:22 --> 00:26:24
And what do I mean by pictures?
495
00:26:24 --> 00:26:31
I mean draw the -- so now I'm
taking the u to be x squared
496
00:26:31 --> 00:26:34
minus y squared, and
the S to be 2xy.
497
00:26:36 --> 00:26:41
And I want to draw those, I
want to draw the vector field
498
00:26:41 --> 00:26:43
of the gradients of those guys.
499
00:26:43 --> 00:26:48
OK, so this is the u -- I
want to draw -- this is
500
00:26:48 --> 00:26:51
the potential, x squared
minus y squared.
501
00:26:51 --> 00:26:54
So what are they
equipotential curves?
502
00:26:54 --> 00:26:58
So this is now, I'm
drawing the flow.
503
00:26:58 --> 00:27:01
And to draw the flow, I draw
the curves on which the
504
00:27:01 --> 00:27:05
potential is a constant. x
squared minus y squared
505
00:27:05 --> 00:27:06
equal constant.
506
00:27:06 --> 00:27:08
So what kind of a curve
is x squared minus y
507
00:27:08 --> 00:27:10
squared equal constant?
508
00:27:10 --> 00:27:12
It's a hyperbola.
509
00:27:12 --> 00:27:15
It's a hyperbola.
510
00:27:15 --> 00:27:17
So x squared minus y
squared equal, let's
511
00:27:17 --> 00:27:21
take one, for example.
512
00:27:21 --> 00:27:22
As one constant.
513
00:27:22 --> 00:27:27
So x=1 will be on the
curve, when y is zero.
514
00:27:27 --> 00:27:30
And then what else
will be on the curve?
515
00:27:30 --> 00:27:37
If x is a little bigger, like
two -- so here's (1, 0).
516
00:27:37 --> 00:27:41
That point is certainly, x
squared minus y squared
517
00:27:41 --> 00:27:43
is one for that.
518
00:27:43 --> 00:27:44
Suppose I go out to x=2.
519
00:27:45 --> 00:27:50
What should y be then,
to make this right?
520
00:27:50 --> 00:27:52
Where's the curve going?
521
00:27:52 --> 00:27:56
When x is two, what is y?
522
00:27:56 --> 00:27:59
Square root of three,
plus or minus.
523
00:27:59 --> 00:28:01
So square root of three
is something like
524
00:28:01 --> 00:28:04
this, up or down.
525
00:28:04 --> 00:28:09
It's a curve, like so.
526
00:28:09 --> 00:28:14
It's a hyperbola.
527
00:28:14 --> 00:28:20
And now, if I change c to
four, let's say, then it'll
528
00:28:20 --> 00:28:26
go through (2, 0), and
it'll go up this way.
529
00:28:26 --> 00:28:32
And if I change x to something
very small, it'll still --
530
00:28:32 --> 00:28:39
oh, there's a Greek
word, asymptotes.
531
00:28:39 --> 00:28:39
Oh, yeah.
532
00:28:39 --> 00:28:43
What do I get if -- the
asymptote is when this is zero.
533
00:28:43 --> 00:28:43
Yeah.
534
00:28:43 --> 00:28:47
When that is zero,
what's my curve?
535
00:28:47 --> 00:28:54
What are x and y?
536
00:28:54 --> 00:28:57
They'll be the same. x
will be y, or minus y.
537
00:28:57 --> 00:29:02
I'll be on that straight line,
or on this straight line.
538
00:29:02 --> 00:29:07
So all these other hyperbolas
are kind of asymptotic,
539
00:29:07 --> 00:29:10
whatever the word is.
540
00:29:10 --> 00:29:11
Right, do you see them?
541
00:29:11 --> 00:29:15
As a bunch of hyperbolas?
542
00:29:15 --> 00:29:23
And actually, more
hyperbolas -- well, yeah.
543
00:29:23 --> 00:29:28
Let's see, back when I had a
one there, I took x and y
544
00:29:28 --> 00:29:32
to be positive, and I
got that hyperbola.
545
00:29:32 --> 00:29:36
But since I'm squaring
them there, also these
546
00:29:36 --> 00:29:39
hyperbolas are here.
547
00:29:39 --> 00:29:46
So these same guys are on
that side of the picture.
548
00:29:46 --> 00:29:49
Those are the equipotentials.
549
00:29:49 --> 00:29:55
So these are the
equipotentials.
550
00:29:55 --> 00:29:57
OK.
551
00:29:57 --> 00:30:03
Now, let me draw S.
552
00:30:03 --> 00:30:08
So those will be the equi
-- no, I don't want to
553
00:30:08 --> 00:30:09
say equistream functions.
554
00:30:09 --> 00:30:12
That's awkward.
555
00:30:12 --> 00:30:17
So now, I want to draw S equal
constant, like one or whatever.
556
00:30:17 --> 00:30:21
So I've drawn this with a whole
lot of constants, and now I
557
00:30:21 --> 00:30:24
want to draw the other guys.
558
00:30:24 --> 00:30:27
What do those curves look like?
559
00:30:27 --> 00:30:30
And what's their name?
560
00:30:30 --> 00:30:34
A curve on which the stream
function is a constant has
561
00:30:34 --> 00:30:37
a nice name streamline.
562
00:30:37 --> 00:30:40
So now I'm going to
draw the streamlines.
563
00:30:40 --> 00:30:45
And what are the streamlines?
564
00:30:45 --> 00:30:48
The streamlines will be the
curves that the actual
565
00:30:48 --> 00:30:50
material flows.
566
00:30:50 --> 00:30:56
If you drop a leaf into this
flow, and you watch it, it'll
567
00:30:56 --> 00:31:00
flow along a streamline.
568
00:31:00 --> 00:31:02
And we can draw those lines.
569
00:31:02 --> 00:31:03
So what are those?
570
00:31:03 --> 00:31:06
2xy=1, or the equation y=1/2x.
571
00:31:06 --> 00:31:09
572
00:31:09 --> 00:31:11
That's also a hyperbola, right?
573
00:31:11 --> 00:31:12
That's also a hyperbola.
574
00:31:12 --> 00:31:18
This is a fantastic picture,
in which we have two
575
00:31:18 --> 00:31:21
sets of hyperbolas.
576
00:31:21 --> 00:31:25
We're second degree, that's why
we're getting two hyperbolas.
577
00:31:25 --> 00:31:29
I'm not going to tackle drawing
-- MATLAB could do it --
578
00:31:29 --> 00:31:33
drawing the equipotentials and
the streamlines for this guy.
579
00:31:33 --> 00:31:37
Oh, but I'm willing
to tackle this one.
580
00:31:37 --> 00:31:42
What are the equipotentials and
the streamlines for the easiest
581
00:31:42 --> 00:31:46
one in the list, there?
582
00:31:46 --> 00:31:47
Can I just draw that one?
583
00:31:47 --> 00:31:50
Because it makes a point
very clear, that we'll
584
00:31:50 --> 00:31:53
see when we draw these.
585
00:31:53 --> 00:31:59
Okay, so what's the picture,
the corresponding picture?
586
00:31:59 --> 00:32:01
Here is the xy plane again.
587
00:32:01 --> 00:32:07
What are the equipotentials
for this pair?
588
00:32:07 --> 00:32:09
And the streamlines.
589
00:32:09 --> 00:32:15
The equipotentials are x equal
constant, what are those?
590
00:32:15 --> 00:32:17
Those are lines,
vertical lines.
591
00:32:17 --> 00:32:21
So the equipotentials are
just vertical lines.
592
00:32:21 --> 00:32:26
Equipotentials, x
equal a constant.
593
00:32:26 --> 00:32:30
And what are the streamlines?
594
00:32:30 --> 00:32:32
Horizontal lines.
595
00:32:32 --> 00:32:34
Streamlines go this way.
596
00:32:34 --> 00:32:37
And what's the great
point about these?
597
00:32:37 --> 00:32:40
These are the streamlines,
S equal constant.
598
00:32:40 --> 00:32:45
And of course, what
do you notice here?
599
00:32:45 --> 00:32:47
They're perpendicular.
600
00:32:47 --> 00:32:50
The streamlines are
perpendicular to the
601
00:32:50 --> 00:32:52
equipotentials.
602
00:32:52 --> 00:32:53
And why?
603
00:32:53 --> 00:32:57
It's because -- you remember
we talked about, what
604
00:32:57 --> 00:32:59
does the gradient mean?
605
00:32:59 --> 00:33:01
Which way does the
gradient point?
606
00:33:01 --> 00:33:04
It points perpendicular
to these equipotentials.
607
00:33:04 --> 00:33:09
And in this case, all these
equipotentials are parallel,
608
00:33:09 --> 00:33:12
and the perpendicular lines
are the streamlines, and
609
00:33:12 --> 00:33:15
they're all parallel.
610
00:33:15 --> 00:33:20
Now over here, we haven't got
straight lines. but we still
611
00:33:20 --> 00:33:23
have the beautiful figure.
612
00:33:23 --> 00:33:24
Now I'm ready to tackle it.
613
00:33:24 --> 00:33:26
I'll draw these curves.
614
00:33:26 --> 00:33:29
They're hyperbolas,
like y=1/2x.
615
00:33:31 --> 00:33:33
If I just make that y=1/2x.
616
00:33:34 --> 00:33:36
That's a line that
comes down this way.
617
00:33:36 --> 00:33:42
Let me try to draw it. i'll
use dashed lines, of course.
618
00:33:42 --> 00:33:46
As x gets bigger,
y gets smaller.
619
00:33:46 --> 00:33:52
But y never makes it to zero,
because if y was zero,
620
00:33:52 --> 00:33:55
no x would work.
621
00:33:55 --> 00:34:00
But if I change that one to
to a four, I've got a bigger
622
00:34:00 --> 00:34:03
-- this is coming out here.
623
00:34:03 --> 00:34:09
If I change that to something
very small, I'll get
624
00:34:09 --> 00:34:12
one that's coming --
625
00:34:12 --> 00:34:17
Do you see how the
picture works?
626
00:34:17 --> 00:34:19
These are all right angles.
627
00:34:19 --> 00:34:20
That's the great thing.
628
00:34:20 --> 00:34:23
Right angles.
629
00:34:23 --> 00:34:27
90 degree angles.
630
00:34:27 --> 00:34:31
Between the streamlines
and the equipotentials.
631
00:34:31 --> 00:34:34
You may have seen this
before, and now I just
632
00:34:34 --> 00:34:36
want to ask you why.
633
00:34:36 --> 00:34:39
Why are those 90 degrees?
634
00:34:39 --> 00:34:45
We kind of see it physically,
that the gradient, the flow is
635
00:34:45 --> 00:34:47
in the gradient direction.
636
00:34:47 --> 00:34:51
And we know that gradients
are always perpendicular
637
00:34:51 --> 00:34:54
to equipotential lines.
638
00:34:54 --> 00:34:58
The gradient of any function
is perpendicular to
639
00:34:58 --> 00:35:00
the level curves.
640
00:35:00 --> 00:35:02
That's all we're seeing here.
641
00:35:02 --> 00:35:08
But we're seeing these two
fantastic families of curves.
642
00:35:08 --> 00:35:11
Equipotentials perpendicular
to the streamline.
643
00:35:11 --> 00:35:14
And the reason they're
perpendicular is
644
00:35:14 --> 00:35:15
Cauchy-Riemann.
645
00:35:15 --> 00:35:19
Cauchy-Riemann is telling
us they're perpendicular.
646
00:35:19 --> 00:35:24
Because the gradient --
yeah, you see that they're
647
00:35:24 --> 00:35:28
perpendicular, and this may
be overkill to try
648
00:35:28 --> 00:35:32
to give a proof.
649
00:35:32 --> 00:35:39
The gradient of u -- what I
want to say is, the gradient
650
00:35:39 --> 00:35:43
of u is a 90 degree rotation
of the gradient of S.
651
00:35:43 --> 00:35:44
Let me put it that way.
652
00:35:44 --> 00:35:50
The gradient of S -- the
gradient of u, rotate 90
653
00:35:50 --> 00:35:58
degrees, rotate gradient
of u by 90 degrees, pi/2,
654
00:35:58 --> 00:36:04
and you get grad S.
655
00:36:04 --> 00:36:07
That's what these
equations say.
656
00:36:07 --> 00:36:12
That if I take the gradient
of u -- yeah, let
657
00:36:12 --> 00:36:14
me try to do that.
658
00:36:14 --> 00:36:16
So I take the gradient of u.
659
00:36:16 --> 00:36:17
That's .
660
00:36:19 --> 00:36:23
So if I have a vector in
2-D, .
661
00:36:25 --> 00:36:30
And I want to rotate a
vector by 90 degrees.
662
00:36:30 --> 00:36:32
What's the result?
663
00:36:32 --> 00:36:38
Suppose I have a vector ,
and I'm looking to rotate it.
664
00:36:38 --> 00:36:40
What's the vector
that goes that way?
665
00:36:40 --> 00:36:47
If that vector is ,
that vector should be what?
666
00:36:47 --> 00:36:50
You may not have done this,
but it's worth just noticing,
667
00:36:50 --> 00:36:52
and you won't forget it.
668
00:36:52 --> 00:36:55
It's got to have a zero
dot product, right?
669
00:36:55 --> 00:36:58
So I'm looking for
a vector, here.
670
00:36:58 --> 00:37:01
This went out a and up b.
671
00:37:01 --> 00:37:03
I'm looking for a vector
there that's perpendicular
672
00:37:03 --> 00:37:05
to this vector.
673
00:37:05 --> 00:37:08
So what should it do?
674
00:37:08 --> 00:37:11
What am I going to put there?
675
00:37:11 --> 00:37:12
A b.
676
00:37:12 --> 00:37:16
And what am I going
to put -- oh no.
677
00:37:16 --> 00:37:17
It went backwards, sorry.
678
00:37:17 --> 00:37:21
When I put there, I should
have put -- minus b.
679
00:37:21 --> 00:37:23
And what goes there? a.
680
00:37:23 --> 00:37:25
Yeah.
681
00:37:25 --> 00:37:28
That's the perpendicular one.
682
00:37:28 --> 00:37:29
Right?
683
00:37:29 --> 00:37:31
That's the 90 degree rotation.
684
00:37:31 --> 00:37:34
And that's what Cauchy-Riemann
is doing for us.
685
00:37:34 --> 00:37:38
I take the gradient,
I take this vector.
686
00:37:38 --> 00:37:42
Now I rotate by 90 degrees,
means I reverse these, reverse
687
00:37:42 --> 00:37:46
the sign, and I've got
the gradient of S.
688
00:37:46 --> 00:37:48
Or minus the gradient of S.
689
00:37:48 --> 00:37:51
I won't say whether the
rotation is plus 90 degrees
690
00:37:51 --> 00:37:55
or minus 90 degrees.
691
00:37:55 --> 00:38:05
This can remain an exercise to
see it slowly and clearly.
692
00:38:05 --> 00:38:07
I'm happy if you see it
in the picture there.
693
00:38:07 --> 00:38:12
And of course, you saw it
in this picture, here.
694
00:38:12 --> 00:38:16
So is this is a moment, then,
to take a little pause.
695
00:38:16 --> 00:38:21
Because we've got ten minutes
for the great event.
696
00:38:21 --> 00:38:23
We've got the general idea.
697
00:38:23 --> 00:38:26
The u equal constants and
the S equal constant.
698
00:38:26 --> 00:38:29
These two families of
perpendicular curves, one
699
00:38:29 --> 00:38:34
telling us where level sets
for the potential, the
700
00:38:34 --> 00:38:37
other telling us the
direction of the flow.
701
00:38:37 --> 00:38:39
The flow goes perpendicular
to the level sets.
702
00:38:39 --> 00:38:43
It's just wonderful.
703
00:38:43 --> 00:38:50
And now I would like to get the
pattern that's going on here
704
00:38:50 --> 00:38:52
and complete that list.
705
00:38:52 --> 00:38:57
OK, well that pattern, like,
comes out of the blue,
706
00:38:57 --> 00:39:01
I have to admit.
707
00:39:01 --> 00:39:04
You might sort of recognize it,
that something -- here is,
708
00:39:04 --> 00:39:06
obviously, stuff to
the first power.
709
00:39:06 --> 00:39:09
Here we've squared
something to get here.
710
00:39:09 --> 00:39:11
Here we've cubed something.
711
00:39:11 --> 00:39:14
You sort of recognize these
numbers, 1 3 3 1, or one,
712
00:39:14 --> 00:39:23
minus three, whatever.
713
00:39:23 --> 00:39:25
We're taking powers
of something.
714
00:39:25 --> 00:39:31
And that something is what
comes out of the blue.
715
00:39:31 --> 00:39:35
It's this quantity, x+iy.
716
00:39:35 --> 00:39:39
717
00:39:39 --> 00:39:41
Complex variables.
718
00:39:41 --> 00:39:44
Everything here was real
until this moment.
719
00:39:44 --> 00:39:48
And then I'm saying that the
complex number, i, the
720
00:39:48 --> 00:39:52
imaginary number, i, the
square root of minus one.
721
00:39:52 --> 00:39:57
Which of course, it's not a
real number, but it has the
722
00:39:57 --> 00:40:00
property, whenever we see i
squared, we write minus one.
723
00:40:00 --> 00:40:03
So then, we know how
to deal with it.
724
00:40:03 --> 00:40:04
OK.
725
00:40:04 --> 00:40:04
Sort of.
726
00:40:04 --> 00:40:09
Anyway, so that complex
number I'll call z.
727
00:40:09 --> 00:40:14
And here's what I think.
728
00:40:14 --> 00:40:19
I think that these two
pieces are the real and
729
00:40:19 --> 00:40:24
the imaginary parts of z.
730
00:40:24 --> 00:40:29
Now, these two pieces are
the real and the imaginary
731
00:40:29 --> 00:40:34
parts of z squared.
732
00:40:34 --> 00:40:38
These two pieces will be
the real part and the
733
00:40:38 --> 00:40:42
imaginary part of z cubed.
734
00:40:42 --> 00:40:46
And if we check that, and then
we begin to see, why should
735
00:40:46 --> 00:40:49
these satisfy Laplace's
equation, we'll have
736
00:40:49 --> 00:40:50
the whole pattern.
737
00:40:50 --> 00:40:54
It'll just be x+iy, fourth
power, fifth power,
738
00:40:54 --> 00:40:55
sixth power.
739
00:40:55 --> 00:40:58
We can make a complete,
infinite list of
740
00:40:58 --> 00:41:02
pairs of solutions to
Laplace's equation.
741
00:41:02 --> 00:41:09
So let me just check what I
said about the squares first.
742
00:41:09 --> 00:41:11
How do you do x+iy squared.
743
00:41:11 --> 00:41:14
Because that's what I
believe we're seeing
744
00:41:14 --> 00:41:16
in the quadratic list.
745
00:41:16 --> 00:41:23
So I claim that if I take x+iy
squared, just do it normally, I
746
00:41:23 --> 00:41:29
get x squared, and I get
2ixy's, and I get i
747
00:41:29 --> 00:41:32
squared, y squared.
748
00:41:32 --> 00:41:34
Right?
749
00:41:34 --> 00:41:39
I just squared it, following
normal algebra rules.
750
00:41:39 --> 00:41:41
Now what's the real part?
751
00:41:41 --> 00:41:46
What's the real part of
this x+iy squared? x
752
00:41:46 --> 00:41:49
squared, this is real.
753
00:41:49 --> 00:41:52
And this is real, because i
squared is minus one, this
754
00:41:52 --> 00:41:55
says minus y squared,
that's our guy.
755
00:41:55 --> 00:41:59
And the imaginary part is 2xy.
756
00:42:01 --> 00:42:04
The imaginary part is the
part that multiplies i.
757
00:42:04 --> 00:42:12
So by some magic -- next time
is the fun of exploring this
758
00:42:12 --> 00:42:17
magic -- we get solutions
to a real equation.
759
00:42:17 --> 00:42:21
We get two solutions to a real
equation, by working with this
760
00:42:21 --> 00:42:26
complex thing, x+iy, and taking
things like -- now, if I go to
761
00:42:26 --> 00:42:32
x+iy cubed, well, let me
just say it'll work.
762
00:42:32 --> 00:42:39
We'll have -- it's like, if
you cube something, you're
763
00:42:39 --> 00:42:41
going to see 1 3 3 1.
764
00:42:41 --> 00:42:47
You'll have an x cubed,
and a 3x squared iy,
765
00:42:47 --> 00:42:57
and a 3xiy twice, and
a one of the iy cubed.
766
00:42:57 --> 00:42:59
And then you look to see
what part is real, and
767
00:42:59 --> 00:43:01
you say, x cubed is real.
768
00:43:01 --> 00:43:07
And i squared is the minus,
so I have minus 3xy squared.
769
00:43:07 --> 00:43:08
Golden.
770
00:43:08 --> 00:43:12
And you look for what part is
imaginary, and you see the 3x
771
00:43:12 --> 00:43:15
squared y, and you see the
i cubed -- so what's i
772
00:43:15 --> 00:43:19
cubed? -- is minus i.
773
00:43:19 --> 00:43:24
So that's an imaginary term,
with the minus we wanted.
774
00:43:24 --> 00:43:26
So now we've got
the whole thing.
775
00:43:26 --> 00:43:31
We've got solutions to
Laplace's equation, coming
776
00:43:31 --> 00:43:37
from all the powers.
777
00:43:37 --> 00:43:41
This is now the
moment to celebrate.
778
00:43:41 --> 00:43:45
Because we've got a giant
family of solutions to
779
00:43:45 --> 00:43:50
Laplace's equation.
780
00:43:50 --> 00:43:51
We've got the real parts.
781
00:43:51 --> 00:43:57
So u is the real part
of x+iy to any power.
782
00:43:57 --> 00:44:06
And S will be the
imaginary part.
783
00:44:06 --> 00:44:10
And I claim that, just as we've
held for n equal one, two,
784
00:44:10 --> 00:44:15
three, for every n, these
will be solutions to
785
00:44:15 --> 00:44:18
Laplace's equation.
786
00:44:18 --> 00:44:21
Not only that, they'll be
connected by Cauchy-Riemann, so
787
00:44:21 --> 00:44:31
they'll be the potential and
the stream function for a flow.
788
00:44:31 --> 00:44:34
And we've got lots of them.
789
00:44:34 --> 00:44:39
And, why don't we get even
more, because we have a
790
00:44:39 --> 00:44:41
linear equation here.
791
00:44:41 --> 00:44:44
So what are we allowed to do,
if we have solutions to a
792
00:44:44 --> 00:44:46
linear equation, what are we
allowed to do with those
793
00:44:46 --> 00:44:49
solutions to get
more solutions?
794
00:44:49 --> 00:44:51
Combine them.
795
00:44:51 --> 00:44:56
I can take any combination
of these guys.
796
00:44:56 --> 00:45:00
I can take "or." "And," "or," I
don't know which, should I put
797
00:45:00 --> 00:45:08
"or," "and?" The real part of
any combination of these.
798
00:45:08 --> 00:45:13
So I could take a combination
of, I can take coefficients
799
00:45:13 --> 00:45:22
c_k, or c_n, maybe I should
say c_n(x+iy) to the nth.
800
00:45:22 --> 00:45:26
So if I take a combination of
solutions, with coefficient
801
00:45:26 --> 00:45:29
c_n, I still have solutions.
802
00:45:29 --> 00:45:38
And what will be the twin
solution, or the stream
803
00:45:38 --> 00:45:40
function that goes with this u?
804
00:45:40 --> 00:45:44
So this is another u,
this is virtually a
805
00:45:44 --> 00:45:48
complete family of u.
806
00:45:48 --> 00:45:51
Because we have all these
coefficients to choose.
807
00:45:51 --> 00:45:52
And what will be the S?
808
00:45:52 --> 00:45:55
So what's the
corresponding S, then?
809
00:45:55 --> 00:46:02
It's the imaginary part of
-- what? -- the same thing.
810
00:46:02 --> 00:46:10
We're just taking that
same combination of
811
00:46:10 --> 00:46:11
the special ones.
812
00:46:11 --> 00:46:13
So the special ones
were (x+iy)^n.
813
00:46:15 --> 00:46:17
And the combinations are those.
814
00:46:17 --> 00:46:19
Yeah.
815
00:46:19 --> 00:46:21
These are fantastic solutions.
816
00:46:21 --> 00:46:25
And now, since I'm blessed
with two more minutes, I get
817
00:46:25 --> 00:46:29
to make them much easier.
818
00:46:29 --> 00:46:33
Because already when I got
up to x cubed and y cubed,
819
00:46:33 --> 00:46:35
they're looking messy.
820
00:46:35 --> 00:46:37
You say, okay, great to have
all these solutions, but how
821
00:46:37 --> 00:46:40
am I going to use them?
822
00:46:40 --> 00:46:45
They're getting more and more
complicated as n increases.
823
00:46:45 --> 00:46:49
But switch to polar
coordinates.
824
00:46:49 --> 00:46:54
Make the same list in
polar coordinates.
825
00:46:54 --> 00:46:57
So again, I'm just going to
list the same guys in polar
826
00:46:57 --> 00:47:02
coordinates r, theta.
827
00:47:02 --> 00:47:04
Then you'll see the
pattern, then the pattern
828
00:47:04 --> 00:47:05
really jumps out.
829
00:47:05 --> 00:47:10
So what is x in
polar coordinates?
830
00:47:10 --> 00:47:15
If I switch from xy rectangular
coordinates, to r, theta, polar
831
00:47:15 --> 00:47:18
coordinates, x is r*cos(theta).
832
00:47:20 --> 00:47:23
And y is r*sin(theta).
833
00:47:25 --> 00:47:30
Well, so far it doesn't look
any easier. x and y look fine.
834
00:47:30 --> 00:47:36
But let me go to
the second one.
835
00:47:36 --> 00:47:41
So this is now r squared
cos squared theta, right?
836
00:47:41 --> 00:47:48
Minus, this is r squared
sine squared theta.
837
00:47:48 --> 00:47:51
Trigonometry comes in.
838
00:47:51 --> 00:47:54
I have r squared cos squared
theta, minus r squared sine
839
00:47:54 --> 00:47:58
squared theta, I want
to simplify that.
840
00:47:58 --> 00:48:03
So it's got an r squared, and
it's got a cos squared theta
841
00:48:03 --> 00:48:04
minus sine squared theta.
842
00:48:04 --> 00:48:07
Anybody remember
that? cos(2theta)!
843
00:48:07 --> 00:48:12
844
00:48:12 --> 00:48:14
And this one.
845
00:48:14 --> 00:48:17
Well, what do you
think is coming?
846
00:48:17 --> 00:48:19
What do you think
is coming here?
847
00:48:19 --> 00:48:22
I have 2r*cos(theta)
times r*sin(theta).
848
00:48:22 --> 00:48:27
I have two, I've got an r
squared, again, times
849
00:48:27 --> 00:48:29
2cos(theta)*sin(theta).
850
00:48:30 --> 00:48:31
So what's
2cos(theta)*sin(theta)?
851
00:48:33 --> 00:48:33
sin(2theta).
852
00:48:34 --> 00:48:36
You get it. sin(2theta).
853
00:48:36 --> 00:48:41
And you know what's
coming next, right?
854
00:48:41 --> 00:48:44
You know that now, the whole
family in polar coordinates
855
00:48:44 --> 00:48:50
-- what's the nth power?
856
00:48:50 --> 00:48:53
We can now write down the
nth one in this list.
857
00:48:53 --> 00:49:04
It is r^n -- for the nth pair,
r^n times cos(n*theta),
858
00:49:04 --> 00:49:09
and r^n sin(n*theta).
859
00:49:09 --> 00:49:18
So those are the twins, the
u and the S that we get if
860
00:49:18 --> 00:49:21
we use polar coordinates.
861
00:49:21 --> 00:49:23
So that's just terrific.
862
00:49:23 --> 00:49:25
Now we have a whole
lot of solutions to
863
00:49:25 --> 00:49:27
Laplace's equation.
864
00:49:27 --> 00:49:31
And we have, don't forget,
all combinations of them.
865
00:49:31 --> 00:49:33
And we're really ready to go.
866
00:49:33 --> 00:49:40
So Friday we'll be solving
Laplace's equation in the
867
00:49:40 --> 00:49:45
cases that we can do it, by
pencil and paper, by chalk.
868
00:49:45 --> 00:49:48
And then, after that, comes
solving Laplace's equation
869
00:49:48 --> 00:49:52
by finite differences
and finite elements.
870
00:49:52 --> 00:49:52