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PROFESSOR STRANG: OK, it's
Laplace again today.

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00:00:24 --> 00:00:26
Laplace's equation.
 

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And trying to describe, that's
a big area that's a lot of

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people have worked
on for centuries.

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00:00:35 --> 00:00:40
And for the early centuries,
there were always

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analysis methods.
 

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And what we you got
started on last time.

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And we'll do a bit more.
 

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There's no way we could do
everything that people

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have worked on.
 

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00:00:51 --> 00:00:55
Year and years, trying to
find ideas about solving.

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00:00:55 --> 00:00:58
But we can get the idea.
 

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And this part, then is
in the section called

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Laplace's equation.
 

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And the exam Wednesday
would include some of

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these constructions.
 

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So this is what we did last
time, we identified a whole

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family of solutions to
Laplace's equation as

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polynomials in x and y.
 

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00:01:23 --> 00:01:27
Of increasing degree n, and
then when we wrote them in

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polar form they were fantastic.
r^n*cos(n*theta) and

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00:01:30 --> 00:01:30
r^n*sin(n*theta).
 

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00:01:30 --> 00:01:34
 
 

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00:01:34 --> 00:01:39
So my idea is just, we've got
them, now let's use them.

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So how to use these solutions.
 

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So because we can take
combinations of them, we

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can create series of
sines and cosines.

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00:01:51 --> 00:01:53
So I'll do that first.
 

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Series.
 

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00:01:54 --> 00:01:58
And then Green's function,
that's the name you remember

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00:01:58 --> 00:02:00
we've seen it before.
 

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00:02:00 --> 00:02:04
That's the solution when the
right side is a delta function.

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00:02:04 --> 00:02:08
When we have Poisson's
equation with a delta there.

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So that an important one.
 

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And then well, a big part of
two dimensional and three

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dimensional problems is that
the region itself, not just the

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equation but the region itself,
can be all over the place.

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We'll solve when the region is
nice, like a circle or a

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square, and then there is a way
to, in principle, to

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get other regions.
 

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To change from a crazy region
to a circle or a square,

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and then solve it there.
 

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So that's called
conformal mapping.

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And I can't let the whole
course go without saying a

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word or two about that.
 

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But somehow among numerical
methods, it's conformal

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mapping, there are packages
that do conformal mapping.

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But they're not the central
way to solve these

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equations numerically.
 

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Finite differences,
finite elements are.

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00:03:16 --> 00:03:19
And that's what's
coming next week.

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So this is the future, this
is the present, right there.

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So, can I just start
with an example or two?

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Like, how would you solve
Laplace's equation in a circle?

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So, in a circle.
 

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This is the idea here.
 

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I have Laplace's equation.
 

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OK, so I've got a whole
lot of solutions.

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And I've even got chalk
to write them down.

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OK, so here's my circle,
might as well make

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it the unit circle.
 

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Radius one.
 

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And inside here is
Laplace. u_xx+u_yy=0.

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No sources inside.
 

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So we have to have sources
from somewhere, and they

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will come from the boundary.
 

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So on the boundary, we
keep, let me think

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of u as temperature.
 

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So I set the temperature on the
boundary. u equals some u_0.

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Some known function.
 

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This is given.
 

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This is the boundary condition.
 

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This is the given
boundary condition.

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And it's a function of, I'm
going to use polar coordinates.

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00:04:41 --> 00:04:44
Polar coordinates are
natural for a circle.

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So this is at r=1, so maybe I
should say on the boundary,

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00:04:52 --> 00:04:59
which is r=1, and going around
the angle, theta is given.

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00:04:59 --> 00:05:00
This is u_0(theta).
 

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That's my given
boundary conditions.

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This problem is named after
Dirichlet, because it's like

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giving fixed conditions and
now Neumann conditions.

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OK, so I'm just looking
for a combination.

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00:05:21 --> 00:05:25
For a function that solves
Laplace's equation inside the

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circle, and takes on some
values around the boundary.

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00:05:30 --> 00:05:36
And of course the boundary
values might be plus one

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on the top and minus
one on the bottom.

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00:05:39 --> 00:05:44
Or the boundary condition might
vary around, it might, in

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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).
 

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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

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00:06:31 --> 00:06:33
temperature on the boundary.
 

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00:06:33 --> 00:06:36
And I want to say what
does it look like inside?

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00:06:36 --> 00:06:40
And I'm prescribing right now
the sin(3theta), so there theta

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00:06:40 --> 00:06:45
is zero, so it's zero, the
boundary condition's zero,

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00:06:45 --> 00:06:50
there climbs to one, back to
zero, down to minus

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00:06:50 --> 00:06:51
1, back to 0.
 

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00:06:51 --> 00:06:57
3 times, and again comes
back to 0 again there.

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00:06:57 --> 00:07:02
So, I'm looking for a solution
to Laplace's equation and

115
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I've got a pretty good list.
 

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That will match u_0
when r is one.

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00:07:08 --> 00:07:11
So that's the
boundary, r is one.

118
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So you can tell me what it is.
 

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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?

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00:07:26 --> 00:07:29
Remember I've got my
eye on that list.

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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

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will match sin(3theta).
 

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00:07:37 --> 00:07:40
What's the good guy?
 

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It'll be on that list.
 

127
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Which of those, by itself,
here I don't need a series

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because I've got such
a neat u_0 function.

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I'll get it right with
one answer, and what

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00:07:57 --> 00:08:01
is that answer?
 

131
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I look there.
 

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00:08:02 --> 00:08:08
I say what do I do, so that at
r=1, I'll match sin(3theta).

133
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I'll use r cubed sin(3theta).
 

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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.
 

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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.
 

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00:08:42 --> 00:08:46
And of course I
could take any one.

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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).
 

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00:08:50 --> 00:08:53
 
 

146
00:08:53 --> 00:08:57
In that case, I may have
to use all of them.

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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

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00:09:06 --> 00:09:08
match all of them.
 

151
00:09:08 --> 00:09:09
So what do I do?
 

152
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At r=1, so my general solution
is a combination of these guys

153
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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).

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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

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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.

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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