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PROFESSOR STRANG: OK, it's
Laplace again today.
10
00:00:24 --> 00:00:26
Laplace's equation.
11
00:00:26 --> 00:00:32
And trying to describe, that's
a big area that's a lot of
12
00:00:32 --> 00:00:35
people have worked
on for centuries.
13
00:00:35 --> 00:00:40
And for the early centuries,
there were always
14
00:00:40 --> 00:00:42
analysis methods.
15
00:00:42 --> 00:00:45
And what we you got
started on last time.
16
00:00:45 --> 00:00:47
And we'll do a bit more.
17
00:00:47 --> 00:00:50
There's no way we could do
everything that people
18
00:00:50 --> 00:00:51
have worked on.
19
00:00:51 --> 00:00:55
Year and years, trying to
find ideas about solving.
20
00:00:55 --> 00:00:58
But we can get the idea.
21
00:00:58 --> 00:01:01
And this part, then is
in the section called
22
00:01:01 --> 00:01:03
Laplace's equation.
23
00:01:03 --> 00:01:08
And the exam Wednesday
would include some of
24
00:01:08 --> 00:01:12
these constructions.
25
00:01:12 --> 00:01:15
So this is what we did last
time, we identified a whole
26
00:01:15 --> 00:01:20
family of solutions to
Laplace's equation as
27
00:01:20 --> 00:01:23
polynomials in x and y.
28
00:01:23 --> 00:01:27
Of increasing degree n, and
then when we wrote them in
29
00:01:27 --> 00:01:30
polar form they were fantastic.
r^n*cos(n*theta) and
30
00:01:30 --> 00:01:30
r^n*sin(n*theta).
31
00:01:30 --> 00:01:34
32
00:01:34 --> 00:01:39
So my idea is just, we've got
them, now let's use them.
33
00:01:39 --> 00:01:42
So how to use these solutions.
34
00:01:42 --> 00:01:48
So because we can take
combinations of them, we
35
00:01:48 --> 00:01:51
can create series of
sines and cosines.
36
00:01:51 --> 00:01:53
So I'll do that first.
37
00:01:53 --> 00:01:54
Series.
38
00:01:54 --> 00:01:58
And then Green's function,
that's the name you remember
39
00:01:58 --> 00:02:00
we've seen it before.
40
00:02:00 --> 00:02:04
That's the solution when the
right side is a delta function.
41
00:02:04 --> 00:02:08
When we have Poisson's
equation with a delta there.
42
00:02:08 --> 00:02:10
So that an important one.
43
00:02:10 --> 00:02:17
And then well, a big part of
two dimensional and three
44
00:02:17 --> 00:02:21
dimensional problems is that
the region itself, not just the
45
00:02:21 --> 00:02:26
equation but the region itself,
can be all over the place.
46
00:02:26 --> 00:02:30
We'll solve when the region is
nice, like a circle or a
47
00:02:30 --> 00:02:36
square, and then there is a way
to, in principle, to
48
00:02:36 --> 00:02:39
get other regions.
49
00:02:39 --> 00:02:43
To change from a crazy region
to a circle or a square,
50
00:02:43 --> 00:02:45
and then solve it there.
51
00:02:45 --> 00:02:48
So that's called
conformal mapping.
52
00:02:48 --> 00:02:52
And I can't let the whole
course go without saying a
53
00:02:52 --> 00:02:53
word or two about that.
54
00:02:53 --> 00:03:01
But somehow among numerical
methods, it's conformal
55
00:03:01 --> 00:03:07
mapping, there are packages
that do conformal mapping.
56
00:03:07 --> 00:03:12
But they're not the central
way to solve these
57
00:03:12 --> 00:03:13
equations numerically.
58
00:03:13 --> 00:03:16
Finite differences,
finite elements are.
59
00:03:16 --> 00:03:19
And that's what's
coming next week.
60
00:03:19 --> 00:03:27
So this is the future, this
is the present, right there.
61
00:03:27 --> 00:03:31
So, can I just start
with an example or two?
62
00:03:31 --> 00:03:36
Like, how would you solve
Laplace's equation in a circle?
63
00:03:36 --> 00:03:39
So, in a circle.
64
00:03:39 --> 00:03:42
This is the idea here.
65
00:03:42 --> 00:03:44
I have Laplace's equation.
66
00:03:44 --> 00:03:47
OK, so I've got a whole
lot of solutions.
67
00:03:47 --> 00:03:51
And I've even got chalk
to write them down.
68
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OK, so here's my circle,
might as well make
69
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it the unit circle.
70
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Radius one.
71
00:04:01 --> 00:04:04
And inside here is
Laplace. u_xx+u_yy=0.
72
00:04:04 --> 00:04:09
73
00:04:09 --> 00:04:12
No sources inside.
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00:04:12 --> 00:04:14
So we have to have sources
from somewhere, and they
75
00:04:14 --> 00:04:16
will come from the boundary.
76
00:04:16 --> 00:04:21
So on the boundary, we
keep, let me think
77
00:04:21 --> 00:04:23
of u as temperature.
78
00:04:23 --> 00:04:29
So I set the temperature on the
boundary. u equals some u_0.
79
00:04:29 --> 00:04:30
Some known function.
80
00:04:30 --> 00:04:31
This is given.
81
00:04:31 --> 00:04:33
This is the boundary condition.
82
00:04:33 --> 00:04:36
This is the given
boundary condition.
83
00:04:36 --> 00:04:41
And it's a function of, I'm
going to use polar coordinates.
84
00:04:41 --> 00:04:44
Polar coordinates are
natural for a circle.
85
00:04:44 --> 00:04:52
So this is at r=1, so maybe I
should say on the boundary,
86
00:04:52 --> 00:04:59
which is r=1, and going around
the angle, theta is given.
87
00:04:59 --> 00:05:00
This is u_0(theta).
88
00:05:02 --> 00:05:08
That's my given
boundary conditions.
89
00:05:08 --> 00:05:12
This problem is named after
Dirichlet, because it's like
90
00:05:12 --> 00:05:17
giving fixed conditions and
now Neumann conditions.
91
00:05:17 --> 00:05:21
OK, so I'm just looking
for a combination.
92
00:05:21 --> 00:05:25
For a function that solves
Laplace's equation inside the
93
00:05:25 --> 00:05:30
circle, and takes on some
values around the boundary.
94
00:05:30 --> 00:05:36
And of course the boundary
values might be plus one
95
00:05:36 --> 00:05:39
on the top and minus
one on the bottom.
96
00:05:39 --> 00:05:44
Or the boundary condition might
vary around, it might, in
97
00:05:44 --> 00:05:46
variable then come around.
98
00:05:46 --> 00:05:50
But notice that this is
a periodic function.
99
00:05:50 --> 00:05:57
This is 2pi periodic, because
the problem's the same.
100
00:05:57 --> 00:06:01
If I increase theta by 2pi I've
come back to the same point.
101
00:06:01 --> 00:06:04
So it's got to have that value.
102
00:06:04 --> 00:06:12
OK, so well let me give you
a couple of examples first.
103
00:06:12 --> 00:06:18
Suppose u_0, suppose this
function - Example 1, easy.
104
00:06:18 --> 00:06:21
Suppose u_0 is sin(3theta).
105
00:06:21 --> 00:06:25
106
00:06:25 --> 00:06:31
So that means I've got a region
here, I'm prescribing its
107
00:06:31 --> 00:06:33
temperature on the boundary.
108
00:06:33 --> 00:06:36
And I want to say what
does it look like inside?
109
00:06:36 --> 00:06:40
And I'm prescribing right now
the sin(3theta), so there theta
110
00:06:40 --> 00:06:45
is zero, so it's zero, the
boundary condition's zero,
111
00:06:45 --> 00:06:50
there climbs to one, back to
zero, down to minus
112
00:06:50 --> 00:06:51
1, back to 0.
113
00:06:51 --> 00:06:57
3 times, and again comes
back to 0 again there.
114
00:06:57 --> 00:07:02
So, I'm looking for a solution
to Laplace's equation and
115
00:07:02 --> 00:07:05
I've got a pretty good list.
116
00:07:05 --> 00:07:08
That will match u_0
when r is one.
117
00:07:08 --> 00:07:11
So that's the
boundary, r is one.
118
00:07:11 --> 00:07:13
So you can tell me what it is.
119
00:07:13 --> 00:07:15
So you can solve this
problem, right away.
120
00:07:15 --> 00:07:26
The answer is u(r,theta)
is, what function will?
121
00:07:26 --> 00:07:29
Remember I've got my
eye on that list.
122
00:07:29 --> 00:07:31
You too, right?
123
00:07:31 --> 00:07:34
I'm just trying to get one
that when r is one it
124
00:07:34 --> 00:07:35
will match sin(3theta).
125
00:07:37 --> 00:07:40
What's the good guy?
126
00:07:40 --> 00:07:43
It'll be on that list.
127
00:07:43 --> 00:07:48
Which of those, by itself,
here I don't need a series
128
00:07:48 --> 00:07:53
because I've got such
a neat u_0 function.
129
00:07:53 --> 00:07:57
I'll get it right with
one answer, and what
130
00:07:57 --> 00:08:01
is that answer?
131
00:08:01 --> 00:08:02
I look there.
132
00:08:02 --> 00:08:08
I say what do I do, so that at
r=1, I'll match sin(3theta).
133
00:08:09 --> 00:08:11
I'll use r cubed sin(3theta).
134
00:08:12 --> 00:08:15
So the good winner will
be r cubed sin(3theta).
135
00:08:15 --> 00:08:17
136
00:08:17 --> 00:08:19
That solves Laplace's equation.
137
00:08:19 --> 00:08:24
We checked it out, it's the
imaginary part of x+iy cubed.
138
00:08:24 --> 00:08:28
We could write it in x and
y coordinates if we wanted
139
00:08:28 --> 00:08:31
but we don't want to.
140
00:08:31 --> 00:08:36
And it matches when r is one,
it gives us sin(3theta).
141
00:08:36 --> 00:08:42
That it.
142
00:08:42 --> 00:08:46
And of course I
could take any one.
143
00:08:46 --> 00:08:49
Now suppose I'm trying to
match something that's not
144
00:08:49 --> 00:08:50
as simple as sin(3theta).
145
00:08:50 --> 00:08:53
146
00:08:53 --> 00:08:57
In that case, I may have
to use all of them.
147
00:08:57 --> 00:09:01
I mean, it's very, very,
fluky that one term
148
00:09:01 --> 00:09:02
is going to do it.
149
00:09:02 --> 00:09:06
Usually, so my main examples
would be I'll have to
150
00:09:06 --> 00:09:08
match all of them.
151
00:09:08 --> 00:09:09
So what do I do?
152
00:09:09 --> 00:09:19
At r=1, so my general solution
is a combination of these guys
153
00:09:19 --> 00:09:21
I worked so hard to get.
154
00:09:21 --> 00:09:23
The solution is of this form.
155
00:09:23 --> 00:09:27
It's some a_0, the constant.
156
00:09:27 --> 00:09:33
And then a_1*rcos(theta),
and b_1*rsin(theta).
157
00:09:33 --> 00:09:36
158
00:09:36 --> 00:09:41
And a_2*r squared cos(2theta).
159
00:09:42 --> 00:09:45
And so on.
160
00:09:45 --> 00:09:50
I'm just taking any combination
of, I'm using the a's as the
161
00:09:50 --> 00:09:54
coefficients for
the cosine guys.
162
00:09:54 --> 00:09:59
And the b's, b_1, b_2, b_3
would be the coefficients
163
00:09:59 --> 00:10:01
for the sine one.
164
00:10:01 --> 00:10:04
OK, that's my general solution.
165
00:10:04 --> 00:10:06
That solves Laplace's equation.
166
00:10:06 --> 00:10:09
Every term did, so every
combination will.
167
00:10:09 --> 00:10:10
Now, set r=1.
168
00:10:12 --> 00:10:18
To match that r=1 and
match the boundary.
169
00:10:18 --> 00:10:24
And match u_0(theta), the
required temperature around
170
00:10:24 --> 00:10:26
the on the boundary.
171
00:10:26 --> 00:10:30
The boundary being
where r is one.
172
00:10:30 --> 00:10:31
So this is set r=1.
173
00:10:32 --> 00:10:39
So then u_0(theta), this
given thing, has to match
174
00:10:39 --> 00:10:41
this, when r is one.
175
00:10:41 --> 00:10:51
So it's a_0 plus a_1, now what
do I write here? r is one,
176
00:10:51 --> 00:10:52
so it's just cos(theta).
177
00:10:52 --> 00:10:54
178
00:10:54 --> 00:10:59
Now, b_1, r is one, so I
just have sin(theta).
179
00:11:00 --> 00:11:06
And I have an a_2*cos(2theta),
and a b_2*sin(2theta),
180
00:11:06 --> 00:11:12
and so on.
181
00:11:12 --> 00:11:16
Here, just let me put
it together now.
182
00:11:16 --> 00:11:21
I'm given any temperature
distribution around
183
00:11:21 --> 00:11:24
the boundary.
184
00:11:24 --> 00:11:30
It's in equilibrium, the
temperature, where if the
185
00:11:30 --> 00:11:37
temperature's high near that
point and low over here the
186
00:11:37 --> 00:11:40
temperature inside will
gradually go from that high
187
00:11:40 --> 00:11:45
point, dot dot dot dot,
to the lower one.
188
00:11:45 --> 00:11:47
By matching on the boundary.
189
00:11:47 --> 00:11:49
And this is the match
on the boundary.
190
00:11:49 --> 00:11:59
Now, this is really a
lead in to the last
191
00:11:59 --> 00:12:02
part of this course.
192
00:12:02 --> 00:12:07
So whose name is associated
with a series like that?
193
00:12:07 --> 00:12:08
Fourier.
194
00:12:08 --> 00:12:12
You recognize that as what's
called a Fourier series.
195
00:12:12 --> 00:12:17
So the idea is, I'm given
these boundary values.
196
00:12:17 --> 00:12:22
I find their expansion in
sines and cosines, and that's
197
00:12:22 --> 00:12:25
what we'll do in November.
198
00:12:25 --> 00:12:29
And then I've got it.
199
00:12:29 --> 00:12:32
Then I know the
a's and the b's.
200
00:12:32 --> 00:12:36
And then basically I just put
in the r's. r and r squareds
201
00:12:36 --> 00:12:38
and r cubeds and so on.
202
00:12:38 --> 00:12:43
So then I've got
the answer inside.
203
00:12:43 --> 00:12:48
In principle it's so easy.
204
00:12:48 --> 00:12:51
So, why is it easy, though?
205
00:12:51 --> 00:12:54
First, it's easy because it's
a circle we're working in.
206
00:12:54 --> 00:12:59
If I was in an ellipse or a
strange shape, forget it.
207
00:12:59 --> 00:13:03
I mean, so this is
quite special.
208
00:13:03 --> 00:13:09
And secondly, it's easy because
these functions are so nice.
209
00:13:09 --> 00:13:14
Fourier works with the
best functions ever.
210
00:13:14 --> 00:13:16
These sines and cosines.
211
00:13:16 --> 00:13:20
So I'll find a way to
find those coefficients,
212
00:13:20 --> 00:13:24
the a's and the b's.
213
00:13:24 --> 00:13:27
Even though there are lots
of them, I'll be able to
214
00:13:27 --> 00:13:31
pick them off one at the
time, the a's and b's.
215
00:13:31 --> 00:13:35
Once I know the a's and
b's, I know the answer.
216
00:13:35 --> 00:13:38
So do you see this is
in principle a great
217
00:13:38 --> 00:13:39
way to solve it?
218
00:13:39 --> 00:13:43
In fact, it's the way we used
over here, when my u_0 was
219
00:13:43 --> 00:13:48
sin(3theta) then the only term
in its Fourier series
220
00:13:48 --> 00:13:49
was one sin(3theta).
221
00:13:50 --> 00:13:54
And then the solution was
one r cubed sin(3theta).
222
00:13:55 --> 00:13:59
So you can learn
things from this.
223
00:13:59 --> 00:14:03
For example, oh,
what can you learn?
224
00:14:03 --> 00:14:09
One thing I noticed, an
important feature of Laplace's
225
00:14:09 --> 00:14:15
equation is that this solution
inside the circle
226
00:14:15 --> 00:14:22
gets very smooth.
227
00:14:22 --> 00:14:25
The boundary conditions could
be like a delta function.
228
00:14:25 --> 00:14:29
I could say that on the
boundary, the temperature
229
00:14:29 --> 00:14:35
is zero everywhere except
at that point it spikes.
230
00:14:35 --> 00:14:36
So I could take u_0.
231
00:14:36 --> 00:14:43
So example 2, and I won't do it
in full, would be u_0 on the
232
00:14:43 --> 00:14:46
boundary equal a
delta function.
233
00:14:46 --> 00:14:49
A spike at that one point.
234
00:14:49 --> 00:14:55
So all the heat is coming from
the source at that one point.
235
00:14:55 --> 00:14:59
Like I've got a
fire going there.
236
00:14:59 --> 00:15:05
Keeping the rest of the
boundary frozen, the heat's
237
00:15:05 --> 00:15:07
kind of going to come inside.
238
00:15:07 --> 00:15:09
So then how would I proceed?
239
00:15:09 --> 00:15:16
Well, if I have this boundary
value as a delta function, I
240
00:15:16 --> 00:15:21
look for its Fourier series,
and it's a very important,
241
00:15:21 --> 00:15:25
beautiful, Fourier series
for a delta function.
242
00:15:25 --> 00:15:27
Would you want to know it?
243
00:15:27 --> 00:15:30
I mean, we'll know it
well in November.
244
00:15:30 --> 00:15:32
Would you want to
know it in October?
245
00:15:32 --> 00:15:37
This is Halloween,
I guess, so delta.
246
00:15:37 --> 00:15:39
I'll tell you what it is.
247
00:15:39 --> 00:15:42
Since you insist.
248
00:15:42 --> 00:15:48
Delta theta, I think,
will, I think there's
249
00:15:48 --> 00:15:53
a 1/2pi or something.
250
00:15:53 --> 00:15:55
Ah, shoot.
251
00:15:55 --> 00:15:57
We'll get it exactly right.
252
00:15:57 --> 00:16:06
It's something like 1 and 2
cos(theta) and 2cos(2theta),
253
00:16:06 --> 00:16:07
I'm not sure about the 2pi.
254
00:16:08 --> 00:16:14
2cos(2theta), and
2cos(3theta), and so on.
255
00:16:14 --> 00:16:18
We'll know it well
when we get there.
256
00:16:18 --> 00:16:22
What I notice about this delta
function, of course you're
257
00:16:22 --> 00:16:25
going to expect the delta
function being somehow
258
00:16:25 --> 00:16:26
a little bit strange.
259
00:16:26 --> 00:16:32
At theta=0, what does
that series add up to?
260
00:16:32 --> 00:16:35
Just so you begin to get a
hang of Fourier series.
261
00:16:35 --> 00:16:40
At theta=0, what does
that series look like?
262
00:16:40 --> 00:16:44
Well, all these
cosine thetas are?
263
00:16:44 --> 00:16:45
One.
264
00:16:45 --> 00:16:48
So this series at
theta=0 is 1+2+2+2+2.
265
00:16:50 --> 00:16:51
It's infinite.
266
00:16:51 --> 00:16:54
And that's what we want.
267
00:16:54 --> 00:16:57
The delta function is
infinite at theta=0.
268
00:16:59 --> 00:17:02
And it's periodic, of course,
so that if I go around to
269
00:17:02 --> 00:17:07
theta=2pi I'll come
back to zero again.
270
00:17:07 --> 00:17:12
At theta=pi, you could sort of
see, well, yeah, theta=pi is
271
00:17:12 --> 00:17:14
a sort of interesting point.
272
00:17:14 --> 00:17:18
At theta is pi,
what's the cosine?
273
00:17:18 --> 00:17:20
Is negative one, right?
274
00:17:20 --> 00:17:23
But then the cos(2pi)
will be plus one.
275
00:17:23 --> 00:17:28
So at theta=pi, I think I'm
getting a one minus a two plus
276
00:17:28 --> 00:17:31
a two, minus a two, plus a two.
277
00:17:31 --> 00:17:35
You see, it's doing its best to
cancel itself out and give me
278
00:17:35 --> 00:17:40
the zero that I want, the
theta=pi over on the left
279
00:17:40 --> 00:17:42
side of the circle.
280
00:17:42 --> 00:17:49
Anyway, so that's an
extreme example.
281
00:17:49 --> 00:17:52
But now, what's the
temperature inside?
282
00:17:52 --> 00:17:54
Can you just follow
the same rule?
283
00:17:54 --> 00:17:56
What will be the
temperature inside?
284
00:17:56 --> 00:18:02
If that's the delta function,
if that's the right series,
285
00:18:02 --> 00:18:10
whatever, it may be a 4pi,
I'm not sure, for that.
286
00:18:10 --> 00:18:13
Now, you can tell me what's
the solution, what's the
287
00:18:13 --> 00:18:17
temperature distribution inside
a circle when one point on
288
00:18:17 --> 00:18:26
the boundary has a heat
source, a delta function.
289
00:18:26 --> 00:18:27
What do I do?
290
00:18:27 --> 00:18:30
How do I match this
with this guy?
291
00:18:30 --> 00:18:33
I just put in the r's, right?
292
00:18:33 --> 00:18:37
If this is what it's supposed
to match when r is one, then
293
00:18:37 --> 00:18:44
when r is, so maybe I'll
put it under here.
294
00:18:44 --> 00:18:54
So the u(r,theta), from
the delta guy, is
295
00:18:54 --> 00:18:56
just put in the r's.
296
00:18:56 --> 00:19:06
1+2rcos(theta), and 2r squared
cos(2theta), and so on.
297
00:19:06 --> 00:19:17
OK, and eventually 2r to
cos(100theta), and more.
298
00:19:17 --> 00:19:24
OK, I write this out, you could
say why did he write this down?
299
00:19:24 --> 00:19:29
I wanted to make this point
that the important feature of
300
00:19:29 --> 00:19:34
the solution to Laplace's
equation is how smooth it gets
301
00:19:34 --> 00:19:37
when you go inside the region.
302
00:19:37 --> 00:19:38
And why is that?
303
00:19:38 --> 00:19:45
Because at r=1/2, this term
is practically gone, right?
304
00:19:45 --> 00:19:49
If I go halfway into the
circle, this term is
305
00:19:49 --> 00:19:50
practically gone.
306
00:19:50 --> 00:19:53
1/2 to the hundredth power.
307
00:19:53 --> 00:19:55
And if I go to the center
of the circle, it's
308
00:19:55 --> 00:19:56
completely gone.
309
00:19:56 --> 00:19:59
In fact, what's the value at
the center of the circle?
310
00:19:59 --> 00:20:07
What's the temperature
at the center?
311
00:20:07 --> 00:20:07
1/2pi.
312
00:20:08 --> 00:20:11
This is the only term
that's remaining.
313
00:20:11 --> 00:20:19
And it's the average,
around the circle.
314
00:20:19 --> 00:20:23
That makes physical
sense, I guess.
315
00:20:23 --> 00:20:27
Since the whole thing's
completely isotropic, we've
316
00:20:27 --> 00:20:34
got a perfect circle.
317
00:20:34 --> 00:20:37
The value at the center of
the circle is always the
318
00:20:37 --> 00:20:39
average going around.
319
00:20:39 --> 00:20:43
The constant term in the
Fourier series, this guy.
320
00:20:43 --> 00:20:46
We'll get to know
that one very well.
321
00:20:46 --> 00:20:50
That's the average.
322
00:20:50 --> 00:20:55
You're just seeing a little bit
of Fourier series early, here.
323
00:20:55 --> 00:21:02
But my point is that you could
have high oscillation around
324
00:21:02 --> 00:21:07
the boundary, that damps out
because of these powers of r.
325
00:21:07 --> 00:21:13
And inside the circle it's
only the low order terms
326
00:21:13 --> 00:21:22
that begin to take over.
327
00:21:22 --> 00:21:25
This is the kind of trick you
have, or not trick but the kind
328
00:21:25 --> 00:21:30
of method that you can use for
solving Laplace's equation
329
00:21:30 --> 00:21:34
by an infinite series.
330
00:21:34 --> 00:21:40
Of course, a person who wants a
number can complain that, wait
331
00:21:40 --> 00:21:43
a minute, how do I use
that infinite series?
332
00:21:43 --> 00:21:48
Well, of course, if you wanted
to know the temperature at a
333
00:21:48 --> 00:21:51
particular point you'd have to
plug in that value of r, that
334
00:21:51 --> 00:21:55
value of theta, add up the
terms until you hope that they
335
00:21:55 --> 00:22:00
become so small that
you can ignore them.
336
00:22:00 --> 00:22:03
So infinite series is
one form of a solution.
337
00:22:03 --> 00:22:12
And somehow these are examples,
I should use the words
338
00:22:12 --> 00:22:14
separation of variables.
339
00:22:14 --> 00:22:21
Separation of variables
is the golden idea in
340
00:22:21 --> 00:22:22
this analysis stuff.
341
00:22:22 --> 00:22:26
Separation of variables means
I got the r part separated
342
00:22:26 --> 00:22:29
from the theta part.
343
00:22:29 --> 00:22:33
And that worked great,
worked well for a circle.
344
00:22:33 --> 00:22:39
Let's see, maybe for a square I
could try to separate x from y.
345
00:22:39 --> 00:22:46
Maybe there's a homework
problem, a solution that
346
00:22:46 --> 00:22:51
separates x from y, I
think is something like.
347
00:22:51 --> 00:22:53
So this would be
another family.
348
00:22:53 --> 00:23:01
Good for squares, something
like sin(kx) cinch times, so
349
00:23:01 --> 00:23:08
this separation is something
in x times something in y.
350
00:23:08 --> 00:23:10
Again I'm just
mentioning things.
351
00:23:10 --> 00:23:15
I think that that solves
Laplace's equation because if I
352
00:23:15 --> 00:23:19
take two x derivatives, that'll
bring down k squared, but
353
00:23:19 --> 00:23:22
it'll flip the sine, right?
354
00:23:22 --> 00:23:25
These two derivatives of
the sine will be a minus.
355
00:23:25 --> 00:23:30
And if I take I
need a ky there.
356
00:23:30 --> 00:23:35
And if I took two derivatives
of this hyperbolic sine, you
357
00:23:35 --> 00:23:36
remember that's the
e^(ky) and e^(-ky).
358
00:23:36 --> 00:23:40
359
00:23:40 --> 00:23:43
The two derivatives of
that will bring out a k
360
00:23:43 --> 00:23:45
squared with a plus sign.
361
00:23:45 --> 00:23:50
So two x derivatives bring out
the minus k squared, two y
362
00:23:50 --> 00:23:53
derivatives bring out a plus k
squared and together that
363
00:23:53 --> 00:23:56
solves Laplace's equation.
364
00:23:56 --> 00:23:58
We'll check that in
our homework problem.
365
00:23:58 --> 00:24:02
So there would be an
example, good for a square.
366
00:24:02 --> 00:24:13
So, there's hope to do an exact
solution in a special region.
367
00:24:13 --> 00:24:19
Now, what's this
Green's function idea?
368
00:24:19 --> 00:24:26
OK, that's now this
is another thing.
369
00:24:26 --> 00:24:34
So last time we appreciated
that this combination
370
00:24:34 --> 00:24:37
x+iy was magic.
371
00:24:37 --> 00:24:43
The idea was that we could take
any function of x+iy, and it
372
00:24:43 --> 00:24:47
solves Laplace's equation.
373
00:24:47 --> 00:24:52
Can we just see, sort of
very crudely why that is?
374
00:24:52 --> 00:25:00
We saw the pattern, we
saw x+iy to the nth.
375
00:25:00 --> 00:25:04
Sort of, we went as far as
n=3, checked it all out.
376
00:25:04 --> 00:25:08
But now, really if I want to be
able to, why does that solve
377
00:25:08 --> 00:25:13
Laplace's equation for any n?
378
00:25:13 --> 00:25:16
Should I just plug that
into Laplace's equation?
379
00:25:16 --> 00:25:24
What happens if I take the two
x derivatives of this thing?
380
00:25:24 --> 00:25:28
So this going to be a
typical function of x+iy,
381
00:25:28 --> 00:25:29
typically nice one.
382
00:25:29 --> 00:25:33
If I take two x derivatives, I
want to plug it in and see that
383
00:25:33 --> 00:25:36
it really does solve
Laplace's equation.
384
00:25:36 --> 00:25:40
So two x derivatives of
that will give me what?
385
00:25:40 --> 00:25:44
The first x derivative will
bring down an n times
386
00:25:44 --> 00:25:45
this thing to the n-1.
387
00:25:47 --> 00:25:51
And then the next x derivative
will bring down an n-1 times
388
00:25:51 --> 00:25:53
this thing to the n-2.
389
00:25:55 --> 00:25:57
So that'll be the u_xx.
390
00:25:59 --> 00:26:00
And what about u_yy?
391
00:26:00 --> 00:26:05
392
00:26:05 --> 00:26:08
This is my u.
393
00:26:08 --> 00:26:14
I'm sort of just checking that
yes, this scene again, see
394
00:26:14 --> 00:26:17
if it still works Friday
what worked Wednesday.
395
00:26:17 --> 00:26:23
That this x+iy is magic and
functions of it like powers,
396
00:26:23 --> 00:26:27
exponentials, logarithims,
whatever, all solve
397
00:26:27 --> 00:26:28
Laplace's equation.
398
00:26:28 --> 00:26:33
OK, so we did u_xx,
and we got easy.
399
00:26:33 --> 00:26:34
Now, what happens with u_yy?
400
00:26:35 --> 00:26:36
Do you see the point?
401
00:26:36 --> 00:26:39
AUDIENCE: [INAUDIBLE]
402
00:26:39 --> 00:26:41
PROFESSOR STRANG:
Sorry opposite sign.
403
00:26:41 --> 00:26:44
And why does the sign
come out opposite?
404
00:26:44 --> 00:26:46
Because of that guy.
405
00:26:46 --> 00:26:49
Yeah, it's the chain rule,
right the derivative of this
406
00:26:49 --> 00:26:53
with respect to y will give
me an n times this thing
407
00:26:53 --> 00:26:55
to one lower power.
408
00:26:55 --> 00:26:58
Times the derivative
of what's inside.
409
00:26:58 --> 00:27:01
And the derivative of
what's inside is an i.
410
00:27:01 --> 00:27:05
And then the second derivative
will bring down an n-1, this
411
00:27:05 --> 00:27:12
guy will be down to n-2,
another i will come out and
412
00:27:12 --> 00:27:14
just what you want, right?
413
00:27:14 --> 00:27:17
Because the i squared is
minus one, those cancel.
414
00:27:17 --> 00:27:21
When those are equal
opposite signs.
415
00:27:21 --> 00:27:29
And we get u_xx+u_yy
equaling 0.
416
00:27:29 --> 00:27:31
So that works.
417
00:27:31 --> 00:27:33
And, actually, the same
idea would work for
418
00:27:33 --> 00:27:35
any function of x+iy.
419
00:27:36 --> 00:27:40
The two x derivatives
just give f''.
420
00:27:42 --> 00:27:46
Two y derivatives will give
f'' but the chain rule will
421
00:27:46 --> 00:27:49
bring out i both times
and we've got it.
422
00:27:49 --> 00:28:00
OK, I think we just need
another couple of examples.
423
00:28:00 --> 00:28:04
And this of course could
be in polar coordinates,
424
00:28:04 --> 00:28:05
f(r*e^(i*theta)).
425
00:28:05 --> 00:28:08
426
00:28:08 --> 00:28:11
That's just, everybody
recognizes re^(i*theta)
427
00:28:11 --> 00:28:12
is the same as x+iy?
428
00:28:13 --> 00:28:18
Better just be sure we've got
that. x is some point here in
429
00:28:18 --> 00:28:23
the complex plane. iy
takes us up to here.
430
00:28:23 --> 00:28:24
So there's x+iy.
431
00:28:26 --> 00:28:31
That's x+iy there, but
it's also, so let me
432
00:28:31 --> 00:28:32
put those in better.
433
00:28:32 --> 00:28:37
So there's x and there's y.
434
00:28:37 --> 00:28:39
Everybody knows this
picture, right?
435
00:28:39 --> 00:28:42
This x and this y, now if
I want to go to polar
436
00:28:42 --> 00:28:48
coordinates, that angle is
theta, this x is r*cos(theta),
437
00:28:48 --> 00:28:54
this y is r*sin(theta), and
this guy is re^(i*theta).
438
00:28:54 --> 00:29:00
439
00:29:00 --> 00:29:04
cos(theta)+i*sin(theta) is
the same as re^i*theta).
440
00:29:05 --> 00:29:07
That's utterly fundamental.
441
00:29:07 --> 00:29:13
Everybody's responsible for
that picture of putting the
442
00:29:13 --> 00:29:18
complex numbers into their
beautiful polar form.
443
00:29:18 --> 00:29:25
That's what made our r to the
nth cos(n*theta) all so simple.
444
00:29:25 --> 00:29:30
Now, what was I aiming to do?
445
00:29:30 --> 00:29:33
Give a particular f.
446
00:29:33 --> 00:29:38
Now I want to give a particular
function f, or maybe
447
00:29:38 --> 00:29:40
a couple of choices.
448
00:29:40 --> 00:29:44
A couple of functions f, and
see that they're real parts
449
00:29:44 --> 00:29:50
and their imaginary parts
solve Laplace's equation.
450
00:29:50 --> 00:30:00
Let me take first a one
that works completely.
451
00:30:00 --> 00:30:05
Take the real part and
the imaginary part.
452
00:30:05 --> 00:30:06
Let me take e^(x+iy).
453
00:30:06 --> 00:30:10
454
00:30:10 --> 00:30:15
It's a function of x+iy,
extremely nice function of
455
00:30:15 --> 00:30:20
x+iy, and we can figure out its
real and imaginary parts, and
456
00:30:20 --> 00:30:25
we get two solutions to
Laplace's equation.
457
00:30:25 --> 00:30:29
The good way is to write this
thing as e^x times e^(iy).
458
00:30:31 --> 00:30:35
And again we'll write it as
e^x times cos(y)+i*sin(y).
459
00:30:35 --> 00:30:39
460
00:30:39 --> 00:30:43
So now I can see that the
real part, I can see
461
00:30:43 --> 00:30:44
what the real part is.
462
00:30:44 --> 00:30:46
And I can see what the
imaginary part is.
463
00:30:46 --> 00:30:49
The real part will
be, that's real.
464
00:30:49 --> 00:30:52
And that's real. so this
will so give me e^x*cos(y).
465
00:30:54 --> 00:30:56
And the imaginary part
will be e^x*sin(y).
466
00:30:59 --> 00:31:03
You see it.
467
00:31:03 --> 00:31:08
And those will solve
Laplace's equation.
468
00:31:08 --> 00:31:13
Can I give a name to this
whole field of analysis?
469
00:31:13 --> 00:31:20
This e^z is an analytic, I
should just use that word,
470
00:31:20 --> 00:31:25
an analytic function.
471
00:31:25 --> 00:31:27
And these guys, the real
and imaginary parts, are
472
00:31:27 --> 00:31:34
two harmonic functions.
473
00:31:34 --> 00:31:36
Maybe it's not so important
to know the word
474
00:31:36 --> 00:31:38
harmonic function.
475
00:31:38 --> 00:31:41
But analytic function,
yeah I would say that's
476
00:31:41 --> 00:31:44
an important word.
477
00:31:44 --> 00:31:48
Actually, what does it mean?
478
00:31:48 --> 00:31:52
It's a function of z.
479
00:31:52 --> 00:31:57
So we're in the complex
plane here now.
480
00:31:57 --> 00:32:04
It's a function of z, e^z, and
it can be written as a power
481
00:32:04 --> 00:32:11
series, of course, one plus z
plus 1/2 factorial z squared
482
00:32:11 --> 00:32:13
and all those guys.
483
00:32:13 --> 00:32:15
So it has a power series.
484
00:32:15 --> 00:32:19
That makes it a combination
of our special one.
485
00:32:19 --> 00:32:25
The great thing about that
series is it converges.
486
00:32:25 --> 00:32:31
So an analytic function, an
analytic function is the sum of
487
00:32:31 --> 00:32:34
a power series that converges.
488
00:32:34 --> 00:32:35
And this one does.
489
00:32:35 --> 00:32:37
So there's an example.
490
00:32:37 --> 00:32:41
Yeah, so the whole theory
of analytic functions is
491
00:32:41 --> 00:32:44
actually, that's Chapter
5 of the textbook.
492
00:32:44 --> 00:32:53
And we won't get beyond this
point, I think, in one semester
493
00:32:53 --> 00:32:57
with analytic functions.
494
00:32:57 --> 00:32:59
So what am I saying, though?
495
00:32:59 --> 00:33:02
I'm saying that the theory of
analytic functions is closely
496
00:33:02 --> 00:33:05
tied to Laplace's equation.
497
00:33:05 --> 00:33:08
Because the real and the
imaginary parts give me this
498
00:33:08 --> 00:33:13
pair u and S that satisfy,
they each satisfy
499
00:33:13 --> 00:33:14
Laplace's equation.
500
00:33:14 --> 00:33:19
And they're connected by the
Cauchy-Riemann equations.
501
00:33:19 --> 00:33:25
Boy, it's a lot of mathematics
coming real fast here.
502
00:33:25 --> 00:33:29
Now I'd like to take
one more example.
503
00:33:29 --> 00:33:33
Instead of the exponential,
can we take the logarithm.
504
00:33:33 --> 00:33:39
I want to take the log of x+iy,
and I want you to split it into
505
00:33:39 --> 00:33:42
its real and imaginary parts,
and get the u and the
506
00:33:42 --> 00:33:44
S that go with that.
507
00:33:44 --> 00:33:48
So this was like the
nicest possible.
508
00:33:48 --> 00:33:53
We got a series of, e^z is
good for every z, the series
509
00:33:53 --> 00:33:55
converges, fantastic.
510
00:33:55 --> 00:33:58
It's an analytic
function everywhere.
511
00:33:58 --> 00:34:01
Best possible.
512
00:34:01 --> 00:34:06
Now we go to one that's not
best possible but nevertheless
513
00:34:06 --> 00:34:08
highly valuable.
514
00:34:08 --> 00:34:11
OK, so e^z, I've done.
515
00:34:11 --> 00:34:17
Let me erase e^z, take log z.
516
00:34:17 --> 00:34:23
OK, so now I'm not
doing e^z any more.
517
00:34:23 --> 00:34:28
And I want to find
the logarithm, OK.
518
00:34:28 --> 00:34:30
So, what's the deal
with the logarithm?
519
00:34:30 --> 00:34:32
Real and imaginary parts.
520
00:34:32 --> 00:34:36
Now I'm going to take
the log of x+iy.
521
00:34:36 --> 00:34:42
522
00:34:42 --> 00:34:50
That is a function of x+iy,
except at one point it
523
00:34:50 --> 00:34:52
has a problem, right?
524
00:34:52 --> 00:35:01
There's a point where this is
not going to be analytic, and
525
00:35:01 --> 00:35:06
there's going to be a special
point in the flow which
526
00:35:06 --> 00:35:07
is singular somehow.
527
00:35:07 --> 00:35:13
But away from that point, we
have a nice-looking function,
528
00:35:13 --> 00:35:17
the logarithm of x+iy, and now
I'd like to get its real
529
00:35:17 --> 00:35:19
and imaginary parts.
530
00:35:19 --> 00:35:22
I'd like to know
the u and the S.
531
00:35:22 --> 00:35:24
But nobody in their right
mind wants to take the
532
00:35:24 --> 00:35:26
logarithm of a sum, right?
533
00:35:26 --> 00:35:32
That's a very foolish thing to
try to do, the log of a sum.
534
00:35:32 --> 00:35:35
What's the good way to
get somewhere with this?
535
00:35:35 --> 00:35:39
Real and imaginary part.
536
00:35:39 --> 00:35:44
I can take the log
of a product.
537
00:35:44 --> 00:35:48
So the polar is
way better again.
538
00:35:48 --> 00:35:53
I want to write this as
a log of re^i, I want
539
00:35:53 --> 00:35:55
to write it that way.
540
00:35:55 --> 00:36:00
And now what's the
log of a product?
541
00:36:00 --> 00:36:03
The sum of the two pieces.
542
00:36:03 --> 00:36:13
So I have log r, and the log
of e^(i*theta), which is?
543
00:36:13 --> 00:36:13
Which is i*theta.
544
00:36:14 --> 00:36:17
Boy, look, this is fantastic.
545
00:36:17 --> 00:36:21
Fantastic except it's zero.
546
00:36:21 --> 00:36:27
I mean, it's fantastic but it's
got a big problem at zero.
547
00:36:27 --> 00:36:30
But it's an extremely
important example.
548
00:36:30 --> 00:36:33
So what's the real part?
549
00:36:33 --> 00:36:36
It's sitting there.
550
00:36:36 --> 00:36:38
This is my u.
551
00:36:38 --> 00:36:41
This is my u(r,theta),
my u(x,y), whatever you
552
00:36:41 --> 00:36:45
want is the log of r.
553
00:36:45 --> 00:36:51
The log of the square root of
x squared plus y squared.
554
00:36:51 --> 00:36:56
I claim that again by this
magic combination, this log,
555
00:36:56 --> 00:37:00
this r is the square root of
x squared plus y squared.
556
00:37:00 --> 00:37:02
I claim if you substitute
that into Laplace's
557
00:37:02 --> 00:37:05
equation you get zero.
558
00:37:05 --> 00:37:07
It works.
559
00:37:07 --> 00:37:12
And what's the
imaginary part, the S?
560
00:37:12 --> 00:37:14
The twin?
561
00:37:14 --> 00:37:18
Is the imaginary part,
which is theta.
562
00:37:18 --> 00:37:23
Oh, what is theta in x, if
I wanted it in x and y?
563
00:37:23 --> 00:37:25
What would theta be?
564
00:37:25 --> 00:37:32
It's the arctan, it's the angle
whose tangent is something.
565
00:37:32 --> 00:37:39
y/x, so if I really want it in
rectangular xy stuff, it's the
566
00:37:39 --> 00:37:41
angle whose tangent is y/x.
567
00:37:42 --> 00:37:45
And again, if you remember in
calculus how to take
568
00:37:45 --> 00:37:48
derivatives of this thing and
you plug it into Laplace's
569
00:37:48 --> 00:37:50
equation you get zero.
570
00:37:50 --> 00:37:53
It works.
571
00:37:53 --> 00:37:57
So that's a great
solution except where?
572
00:37:57 --> 00:37:58
At zero.
573
00:37:58 --> 00:38:00
Except at zero.
574
00:38:00 --> 00:38:11
And this doesn't tell us
what's happening at zero.
575
00:38:11 --> 00:38:13
It's an excellent solution.
576
00:38:13 --> 00:38:18
What's the picture?
577
00:38:18 --> 00:38:25
So by Wednesday's exam I'm
not expecting you to be
578
00:38:25 --> 00:38:30
an expert on the theory
of analytic functions.
579
00:38:30 --> 00:38:35
I don't expect you to know
any conformal mappings.
580
00:38:35 --> 00:38:38
By Wednesday, God, that's.
581
00:38:38 --> 00:38:44
But, I do expect you to have
these pictures in mind.
582
00:38:44 --> 00:38:48
So when I draw those
axes, what picture is it
583
00:38:48 --> 00:38:50
that I'm planning on?
584
00:38:50 --> 00:38:53
I'm planning on the
equipotentials u equal
585
00:38:53 --> 00:39:02
constant, and the, who
are the other guys?
586
00:39:02 --> 00:39:04
The stream lines.
587
00:39:04 --> 00:39:07
The places where the
stream functions.
588
00:39:07 --> 00:39:10
So here is the
potential function.
589
00:39:10 --> 00:39:14
So what are the
equipotential curves?
590
00:39:14 --> 00:39:16
For that guy.
591
00:39:16 --> 00:39:18
Circles.
592
00:39:18 --> 00:39:24
This is a constant when r is a
constant, so the equipotential
593
00:39:24 --> 00:39:28
functions would be circles.
594
00:39:28 --> 00:39:30
I don't want to draw
that circle with
595
00:39:30 --> 00:39:31
radius zero, though.
596
00:39:31 --> 00:39:34
I'm nervous about that one.
597
00:39:34 --> 00:39:37
But all the others are great.
598
00:39:37 --> 00:39:40
And what are the
stream lines, now?
599
00:39:40 --> 00:39:47
The stream lines are, well,
what will the stream lines be?
600
00:39:47 --> 00:39:51
If I've drawn one family, you
can tell me the other family.
601
00:39:51 --> 00:39:54
The stream lines will be?
602
00:39:54 --> 00:39:57
Radial lines.
603
00:39:57 --> 00:39:59
Because they're going to
be perpendicular to this.
604
00:39:59 --> 00:40:06
And so what do I get, this is
the stream function, theta.
605
00:40:06 --> 00:40:08
So what's a stream line?
606
00:40:08 --> 00:40:10
The stream function
should be a constant.
607
00:40:10 --> 00:40:12
Theta's a constant.
608
00:40:12 --> 00:40:15
That means I'm
going out on rays.
609
00:40:15 --> 00:40:20
Those are all streamlined.
610
00:40:20 --> 00:40:23
Again, everything fantastic.
611
00:40:23 --> 00:40:28
If you look in a little region
here you see just a beautiful
612
00:40:28 --> 00:40:34
picture of of equipotentials
and stream lines crossing
613
00:40:34 --> 00:40:35
them at right angles.
614
00:40:35 --> 00:40:37
Everything great.
615
00:40:37 --> 00:40:43
Just that point is
obviously a problem.
616
00:40:43 --> 00:40:51
Now, and I'm suspecting that
there's a source here.
617
00:40:51 --> 00:41:00
I think this flow, which is
given by these guys, comes
618
00:41:00 --> 00:41:06
from some kind of a delta
function right there.
619
00:41:06 --> 00:41:12
And the flow goes outwards.
620
00:41:12 --> 00:41:16
So I know u, I know v is
the gradient of u, right?
621
00:41:16 --> 00:41:19
I could take the x and
y derivatives, I'd
622
00:41:19 --> 00:41:22
know the velocity.
623
00:41:22 --> 00:41:26
I know the stream function,
the divergence would be zero.
624
00:41:26 --> 00:41:32
Everything great,
except at the origin.
625
00:41:32 --> 00:41:35
I think we've got some
action at the origin.
626
00:41:35 --> 00:41:43
Because, here's the
way to test it.
627
00:41:43 --> 00:41:49
I want to see what's
happening at the origin.
628
00:41:49 --> 00:41:52
And I'm going to use the
divergence theorem.
629
00:41:52 --> 00:41:52
Yeah.
630
00:41:52 --> 00:41:52
Yeah.
631
00:41:52 --> 00:41:55
I'm going to use the
divergence theorem.
632
00:41:55 --> 00:42:00
So the divergence theorem says,
what is the divergence theorem?
633
00:42:00 --> 00:42:09
So this is the key thing that
connects double integrals.
634
00:42:09 --> 00:42:13
Let me take a circle
of radius r.
635
00:42:13 --> 00:42:19
So that's the circle of radius
r. r could be big, or little.
636
00:42:19 --> 00:42:24
So I integrate over the
circle of radius r.
637
00:42:24 --> 00:42:30
So what's the deal?
v is the same as w.
638
00:42:30 --> 00:42:32
What does the divergence
theorem tell me?
639
00:42:32 --> 00:42:36
It tells me that if I
integrate, what do I integrate,
640
00:42:36 --> 00:42:43
the divergence of w?
dx/dy, or r*dr*d theta.
641
00:42:43 --> 00:42:45
642
00:42:45 --> 00:42:52
Then I get the flux.
643
00:42:52 --> 00:42:56
So this is a key identity.
644
00:42:56 --> 00:42:59
Fundamentally, more than
just the key identity,
645
00:42:59 --> 00:43:01
it's central here.
646
00:43:01 --> 00:43:07
The total flow out of
the region must make it
647
00:43:07 --> 00:43:09
through the boundary.
648
00:43:09 --> 00:43:12
So I integrate this boundary
and this boundary is a circle
649
00:43:12 --> 00:43:19
of radius r, and what do I
integrate along that circle?
650
00:43:19 --> 00:43:26
What's the other side of the
divergence theorem? w dot n. w
651
00:43:26 --> 00:43:30
dot n, around the boundary.
652
00:43:30 --> 00:43:37
And remember, I have this
nice, my curve here
653
00:43:37 --> 00:43:42
is this nice circle.
654
00:43:42 --> 00:43:44
So I'm going to integrate
around that circle.
655
00:43:44 --> 00:43:52
First, of all what is n?
656
00:43:52 --> 00:43:56
By definition, n is
the norm that points
657
00:43:56 --> 00:43:58
outward, straight out.
658
00:43:58 --> 00:44:02
So it's actually
going out that way.
659
00:44:02 --> 00:44:05
At every point it's
pointing straight out.
660
00:44:05 --> 00:44:09
And dS, yeah, I think we can
figure out exactly what
661
00:44:09 --> 00:44:18
that right-hand side is.
662
00:44:18 --> 00:44:23
How do I get that
right-hand side?
663
00:44:23 --> 00:44:29
I'm looking for w, and
then I have to integrate.
664
00:44:29 --> 00:44:35
OK, here is my u.
665
00:44:35 --> 00:44:42
My u is log r.
666
00:44:42 --> 00:44:45
So what's the
gradient of log r?
667
00:44:45 --> 00:44:47
It points outwards.
668
00:44:47 --> 00:44:49
And how large is
the derivative?
669
00:44:49 --> 00:44:53
So the derivative of
this log r is 1/r.
670
00:44:55 --> 00:45:03
I think that this comes down
to, this is the integral.
671
00:45:03 --> 00:45:05
Around the circle.
672
00:45:05 --> 00:45:07
I think that this thing is 1/r.
673
00:45:07 --> 00:45:13
674
00:45:13 --> 00:45:17
I went pretty quickly there, so
I'll ask you to look in the
675
00:45:17 --> 00:45:21
book because this is such an
important example it's done
676
00:45:21 --> 00:45:26
there in more detail.
677
00:45:26 --> 00:45:30
So I'm claiming that the
derivative is 1/r, and that
678
00:45:30 --> 00:45:32
it points directly out.
679
00:45:32 --> 00:45:35
So the gradient points out.
680
00:45:35 --> 00:45:39
The normal points out, so
that I just get exactly 1/r.
681
00:45:40 --> 00:45:41
Now, what is dS?
682
00:45:41 --> 00:45:45
683
00:45:45 --> 00:45:49
For integrating around the
circle what's a little tiny
684
00:45:49 --> 00:45:54
piece of r on a circle?
685
00:45:54 --> 00:45:57
Of radius r? r d theta.
686
00:45:57 --> 00:46:03
Good man. r d theta.
687
00:46:03 --> 00:46:09
Now that's an integral
I can do, right?
688
00:46:09 --> 00:46:11
And what do I get?
689
00:46:11 --> 00:46:16
2 pi. r cancels r, I'm
integrating d theta
690
00:46:16 --> 00:46:18
around from zero to 2pi.
691
00:46:19 --> 00:46:20
The answer is 2pi.
692
00:46:20 --> 00:46:24
693
00:46:24 --> 00:46:27
So what do I learn from that?
694
00:46:27 --> 00:46:32
I learn that somehow this
source in the inside
695
00:46:32 --> 00:46:34
has strength 2pi.
696
00:46:35 --> 00:46:42
What's sitting in there is
2pi times a delta function.
697
00:46:42 --> 00:46:49
This is the solution to
Laplace's equation except at
698
00:46:49 --> 00:46:53
that source term, so I really
should say Poisson's equation.
699
00:46:53 --> 00:46:58
This has turned out to be the
solution to Poisson with a
700
00:46:58 --> 00:47:03
delta, or with 2pi
times a delta.
701
00:47:03 --> 00:47:08
We have just solved this
important equation.
702
00:47:08 --> 00:47:12
Poisson's equation
with a point source.
703
00:47:12 --> 00:47:16
And, of course, that's
important because when you can
704
00:47:16 --> 00:47:19
solve with a point source, you
can put together all
705
00:47:19 --> 00:47:22
sorts of sources.
706
00:47:22 --> 00:47:24
And this is called the
Green's function.
707
00:47:24 --> 00:47:28
The Green's function is
the solution when the
708
00:47:28 --> 00:47:29
source is a delta.
709
00:47:29 --> 00:47:33
So if I divide by 2pi,
now I've got it.
710
00:47:33 --> 00:47:37
I divide this by 2pi and there
is the Green's function.
711
00:47:37 --> 00:47:43
I have to put that
in bold letters.
712
00:47:43 --> 00:47:48
Green's function.
713
00:47:48 --> 00:47:52
It's the solution to the
equation when the source is
714
00:47:52 --> 00:47:59
a delta and the answer is u
is the log of r over 2pi.
715
00:47:59 --> 00:48:05
So that's the Green's
function in 2-D.
716
00:48:05 --> 00:48:08
Physicists, you know, they
live and die with these
717
00:48:08 --> 00:48:10
Green's function.
718
00:48:10 --> 00:48:12
Live, let's say, with
Green's function.
719
00:48:12 --> 00:48:18
And they would want to know
the Green's function in 3-D.
720
00:48:18 --> 00:48:21
So the Green's function
in three dimensions also
721
00:48:21 --> 00:48:23
turns out beautifully.
722
00:48:23 --> 00:48:28
This is in, they would
say, in free space.
723
00:48:28 --> 00:48:32
This is the Green's function
when there's no other charges.
724
00:48:32 --> 00:48:35
Nothing is happening, except
for the charge right
725
00:48:35 --> 00:48:36
at the center.
726
00:48:36 --> 00:48:42
And if I'm in two dimensions
the Green's function
727
00:48:42 --> 00:48:44
is this log r.
728
00:48:44 --> 00:48:49
So it grows more slowly.
729
00:48:49 --> 00:48:51
It behaves like log r.
730
00:48:51 --> 00:48:54
And in 3-D I think the
answer is 1/4pi*r.
731
00:48:56 --> 00:49:02
It's just amazing that those
Green's functions, when
732
00:49:02 --> 00:49:09
the right side is a delta,
have such nice formulas.
733
00:49:09 --> 00:49:16
OK, let me take
one moment here.
734
00:49:16 --> 00:49:22
I'll tell you what conformal
mapping is about.
735
00:49:22 --> 00:49:26
But what's your take-home
from this lecture?
736
00:49:26 --> 00:49:33
Your take-home is two methods
that we can really use to get
737
00:49:33 --> 00:49:35
a formula for the answer.
738
00:49:35 --> 00:49:43
One method was for Laplace's
equation in a circle.
739
00:49:43 --> 00:49:47
Get the boundary conditions in
a series of sines and cosines,
740
00:49:47 --> 00:49:52
and then just put in
the r's that we need.
741
00:49:52 --> 00:49:56
That's a simple, simple method.
742
00:49:56 --> 00:50:00
Provided we can get started
with the Fourier series.
743
00:50:00 --> 00:50:06
The second method is, look at
functions of x+iy, and try to
744
00:50:06 --> 00:50:09
pick one that matches
your problem.
745
00:50:09 --> 00:50:13
And if your problem has
a point source, at the
746
00:50:13 --> 00:50:17
origin we found that one.
747
00:50:17 --> 00:50:22
So the literature for hundreds
of years is aimed at
748
00:50:22 --> 00:50:24
solving other problems.
749
00:50:24 --> 00:50:27
If the point source is
somewhere else, what happens?
750
00:50:27 --> 00:50:28
That's not hard.
751
00:50:28 --> 00:50:32
If it's not a point source but
some other kind of source, or
752
00:50:32 --> 00:50:37
if the region is not a circle.
753
00:50:37 --> 00:50:43
Can I say in one final sentence
just what to do, this conformal
754
00:50:43 --> 00:50:51
mapping idea, when the
region is not a circle.
755
00:50:51 --> 00:50:54
Well, I can say it in one
word, make it a circle.
756
00:50:54 --> 00:50:57
I mean, that's what Riemann
said you could do it.
757
00:50:57 --> 00:51:02
You could think of a function,
so Riemann said that there's
758
00:51:02 --> 00:51:07
always some function of x+iy,
let me call this Riemann's
759
00:51:07 --> 00:51:08
function capital F(x,y).
760
00:51:10 --> 00:51:13
So this is now the idea
of conformal mapping.
761
00:51:13 --> 00:51:16
Change variables.
762
00:51:16 --> 00:51:19
Conformal mapping is a
change of variables.
763
00:51:19 --> 00:51:24
He picked some function and
let its real part be and let
764
00:51:24 --> 00:51:26
its imaginary part be Y.
765
00:51:26 --> 00:51:28
Capital Y.
766
00:51:28 --> 00:51:32
OK, this is totally
ridiculous to put conformal
767
00:51:32 --> 00:51:34
mapping in 30 seconds.
768
00:51:34 --> 00:51:42
But, never mind,
let's just do it.
769
00:51:42 --> 00:51:46
The book describes conformal
mappings and classical applied
770
00:51:46 --> 00:51:51
math courses do much more
with conformal mapping.
771
00:51:51 --> 00:51:54
But the truth is,
computationally they're
772
00:51:54 --> 00:51:59
not anything like as
much used as these.
773
00:51:59 --> 00:52:00
So what's the idea?
774
00:52:00 --> 00:52:07
The idea is to find a neat
function of x+iy, so that
775
00:52:07 --> 00:52:11
your crazy boundary
becomes a circle.
776
00:52:11 --> 00:52:15
In the capital X,
capital Y variable.
777
00:52:15 --> 00:52:19
So you're mapping the region,
ellipse, whatever it looks
778
00:52:19 --> 00:52:24
like, by changing from little
x, little y, where it was an
779
00:52:24 --> 00:52:29
ellipse, to capital X, capital
Y, where it's a circle.
780
00:52:29 --> 00:52:32
And the point is
Laplace's equation stays
781
00:52:32 --> 00:52:33
Laplace's equation.
782
00:52:33 --> 00:52:37
That change of variables does
not mess up Laplace's equation.
783
00:52:37 --> 00:52:40
So that then you've
got it in a circle.
784
00:52:40 --> 00:52:44
You solve it in a
circle, for these guys.
785
00:52:44 --> 00:52:46
And then you go back.
786
00:52:46 --> 00:52:51
In a word, you're able to solve
Laplace's equation in this
787
00:52:51 --> 00:52:56
crazy region because you
never leave the magic x+iy.
788
00:52:57 --> 00:53:02
You find a combination with
that magic x+iy that makes
789
00:53:02 --> 00:53:04
your region into a circle.
790
00:53:04 --> 00:53:08
In the circle we now know
how to use capital X+iY.
791
00:53:08 --> 00:53:12
792
00:53:12 --> 00:53:16
You're staying with that magic
combination and getting the
793
00:53:16 --> 00:53:18
region to be what you like.
794
00:53:18 --> 00:53:21
So people know a lot of
these conformal mappings.
795
00:53:21 --> 00:53:28
A famous one is the Joukowski
one, that takes something that
796
00:53:28 --> 00:53:33
looks very like an airfoil, and
you can get a circle out of it.
797
00:53:33 --> 00:53:37
So I'll put down
Joukowski's name.
798
00:53:37 --> 00:53:48
So that's one that I trust
Course 16 still finds valuable.
799
00:53:48 --> 00:53:56
It's a transformation that
takes certain shapes and they
800
00:53:56 --> 00:54:00
include shapes that look like
airfoils, and produce circles.
801
00:54:00 --> 00:54:07
OK, so sorry about such
a quick presentation of
802
00:54:07 --> 00:54:10
such a basic subject.
803
00:54:10 --> 00:54:15
Conformal mapping, not on any
exam, that'd be impossible.
804
00:54:15 --> 00:54:18
It's really this stuff
that you're number
805
00:54:18 --> 00:54:20
one responsible for.
806
00:54:20 --> 00:54:20