1
00:00:01,140 --> 00:00:05,640
Today I'm speaking about
the first of the three great

2
00:00:05,640 --> 00:00:09,620
partial differential equations.

3
00:00:09,620 --> 00:00:14,710
So this one is called Laplace's
equation, named after Laplace.

4
00:00:14,710 --> 00:00:16,780
And you see partial derivatives.

5
00:00:16,780 --> 00:00:19,950
So we have-- I don't have time.

6
00:00:19,950 --> 00:00:24,160
This equation is
in steady state.

7
00:00:24,160 --> 00:00:27,850
I have x and y, I'm
in the xy plane.

8
00:00:27,850 --> 00:00:32,320
And I have second
derivatives in x and then y.

9
00:00:32,320 --> 00:00:35,730
So I'm looking for
solutions to that equation.

10
00:00:35,730 --> 00:00:40,370
And of course I'm given
some boundary values.

11
00:00:40,370 --> 00:00:41,820
So time is not here.

12
00:00:41,820 --> 00:00:43,880
The boundary
values, the boundary

13
00:00:43,880 --> 00:00:46,870
is in the xy plane,
maybe a circle.

14
00:00:46,870 --> 00:00:49,810
Think about a circle
in the xy plane.

15
00:00:49,810 --> 00:00:54,470
And on the circle, I
know the solution u.

16
00:00:54,470 --> 00:00:58,760
So the boundary values
around the circle are given.

17
00:00:58,760 --> 00:01:02,160
And I have to find
the temperature

18
00:01:02,160 --> 00:01:04,590
u inside the circle.

19
00:01:04,590 --> 00:01:07,560
So I know the temperature
on the boundary.

20
00:01:07,560 --> 00:01:12,670
I let it settle down and I want
to know the temperature inside.

21
00:01:12,670 --> 00:01:19,020
And the beauty is, it solves
that basic partial differential

22
00:01:19,020 --> 00:01:20,080
equation.

23
00:01:20,080 --> 00:01:23,240
So let's find some solutions.

24
00:01:23,240 --> 00:01:25,390
They might not match
the boundary values,

25
00:01:25,390 --> 00:01:27,300
but we can use them.

26
00:01:27,300 --> 00:01:31,320
So u equal constant certainly
solves the equation.

27
00:01:31,320 --> 00:01:35,410
U equal x, the second
derivatives will be 0.

28
00:01:35,410 --> 00:01:36,700
U equal y.

29
00:01:36,700 --> 00:01:40,430
Here is a better one, x
squared minus y squared.

30
00:01:40,430 --> 00:01:45,070
So the second derivative
in the x direction is 2.

31
00:01:45,070 --> 00:01:48,370
The second derivative in
the y direction is minus 2.

32
00:01:48,370 --> 00:01:51,710
So I have 2, minus 2,
it solves the equation.

33
00:01:51,710 --> 00:01:56,480
Or this one, the second
derivative in x is 0.

34
00:01:56,480 --> 00:02:01,580
Second derivative in y is 0,
those are simple solutions.

35
00:02:01,580 --> 00:02:04,230
But those are only
a few solutions

36
00:02:04,230 --> 00:02:08,770
and we need an infinite
sequence because we're going

37
00:02:08,770 --> 00:02:12,960
to match boundary conditions.

38
00:02:12,960 --> 00:02:15,240
So is there a pattern here?

39
00:02:15,240 --> 00:02:18,530
So this is degree 0, constant.

40
00:02:18,530 --> 00:02:21,180
These are degree 1, linear.

41
00:02:21,180 --> 00:02:23,450
These are degree 2, quadratic.

42
00:02:23,450 --> 00:02:27,580
So I hope for two cubic ones.

43
00:02:27,580 --> 00:02:31,170
And then I hope for
two fourth degree ones.

44
00:02:31,170 --> 00:02:34,450
And that's the pattern,
that's the pattern.

45
00:02:34,450 --> 00:02:41,180
Let me find-- let me
spot the cubic ones.

46
00:02:41,180 --> 00:02:45,760
X cubed, if I start with x
cubed, of course the second x

47
00:02:45,760 --> 00:02:50,320
derivative is probably 6x.

48
00:02:50,320 --> 00:02:55,230
So I need the second y
derivative to be minus 6x.

49
00:02:55,230 --> 00:03:01,780
And I think minus
3xy squared does it.

50
00:03:01,780 --> 00:03:06,950
The second derivative
in y is 2 times

51
00:03:06,950 --> 00:03:10,180
the minus 3x is
minus 6x, cancels

52
00:03:10,180 --> 00:03:15,760
the 6x from the second
derivative there, and it works.

53
00:03:15,760 --> 00:03:20,290
So that fits the pattern,
but what is the pattern?

54
00:03:20,290 --> 00:03:21,020
Here it is.

55
00:03:21,020 --> 00:03:21,700
It's fantastic.

56
00:03:25,050 --> 00:03:35,060
I get these crazy polynomials
from taking x plus iy

57
00:03:35,060 --> 00:03:37,740
to the different powers.

58
00:03:37,740 --> 00:03:46,400
Here to the first power, if n
is 1, and I just have x plus iy

59
00:03:46,400 --> 00:03:48,505
and I take the real
part, that's x.

60
00:03:48,505 --> 00:03:52,190
So I'll take the
real part of this.

61
00:03:52,190 --> 00:03:56,810
The real part of this when
n is 1, the real part is x.

62
00:03:56,810 --> 00:03:59,440
What about when n is 2?

63
00:03:59,440 --> 00:04:02,390
Can you square
that in your head?

64
00:04:02,390 --> 00:04:07,600
So we have x squared and we
have i squared y squared,

65
00:04:07,600 --> 00:04:09,470
i squared being minus 1.

66
00:04:09,470 --> 00:04:13,170
So I have x squared and
I have minus y squared.

67
00:04:13,170 --> 00:04:17,149
Look, the real
part of this when n

68
00:04:17,149 --> 00:04:21,649
is 2, the real part
of x plus iy squared,

69
00:04:21,649 --> 00:04:24,820
the real part is x
squared minus y squared.

70
00:04:24,820 --> 00:04:31,290
And the imaginary
part was the 2ixy.

71
00:04:31,290 --> 00:04:36,720
So the imaginary part that
multiplies i is the 2xy.

72
00:04:36,720 --> 00:04:40,250
This is our pattern when n is 2.

73
00:04:40,250 --> 00:04:46,040
And when n is 3, I take
x plus iy cubed, and that

74
00:04:46,040 --> 00:04:48,950
begins with x cubed like that.

75
00:04:48,950 --> 00:04:51,910
And then I think that
the other real part

76
00:04:51,910 --> 00:04:54,710
would be a minus 3xy squared.

77
00:04:54,710 --> 00:04:56,870
I think you should check that.

78
00:04:56,870 --> 00:04:59,970
And then there will
be an imaginary part.

79
00:04:59,970 --> 00:05:03,890
Well, I think I could figure out
the imaginary part as I think.

80
00:05:03,890 --> 00:05:13,540
Maybe something like
minus-- maybe it's

81
00:05:13,540 --> 00:05:21,036
3yx squared minus y cubed,
something like that.

82
00:05:21,036 --> 00:05:22,910
That would be the real
part and that would be

83
00:05:22,910 --> 00:05:25,130
the imaginary part when n is 3.

84
00:05:25,130 --> 00:05:28,140
And wonderfully,
wonderfully, it works

85
00:05:28,140 --> 00:05:34,280
for all powers, exponents n.

86
00:05:34,280 --> 00:05:39,300
So I have now sort of a pretty
big family of solutions.

87
00:05:39,300 --> 00:05:43,190
A list, a double list,
really, the real parts

88
00:05:43,190 --> 00:05:45,910
and the imaginary
parts for every n.

89
00:05:45,910 --> 00:05:53,300
So I can use those
to find the solution

90
00:05:53,300 --> 00:05:58,570
u, which I'm looking for, the
temperature inside the circle.

91
00:05:58,570 --> 00:06:03,860
Now of course, I have
a linear equation.

92
00:06:03,860 --> 00:06:08,430
So if I have several
solutions, I can combine them

93
00:06:08,430 --> 00:06:10,320
and I still have a solution.

94
00:06:10,320 --> 00:06:13,330
X plus 7y will be a solution.

95
00:06:13,330 --> 00:06:16,590
Plus 11x squared minus
y squared, no problem.

96
00:06:16,590 --> 00:06:19,760
Plus 56 times 2xy.

97
00:06:19,760 --> 00:06:21,410
Those are all solutions.

98
00:06:21,410 --> 00:06:27,670
So I'm going to find a
solution, my final solution

99
00:06:27,670 --> 00:06:31,694
u will be a combination of this,
this, this, this, this, this,

100
00:06:31,694 --> 00:06:35,520
this, and all the
others for higher n.

101
00:06:35,520 --> 00:06:37,580
That's going to be my solution.

102
00:06:37,580 --> 00:06:40,990
And I will need that
infinite family.

103
00:06:40,990 --> 00:06:42,870
See, partial
differential equations,

104
00:06:42,870 --> 00:06:48,390
we move up to infinite family
of solutions instead of just

105
00:06:48,390 --> 00:06:51,580
a couple of null solutions.

106
00:06:51,580 --> 00:06:53,810
So let me take an example.

107
00:06:53,810 --> 00:06:54,785
Let me take an example.

108
00:07:01,410 --> 00:07:03,320
We're taking the
region to be a circle.

109
00:07:09,560 --> 00:07:16,050
So in that circle, I'm looking
for the solution u of x and y.

110
00:07:16,050 --> 00:07:19,530
And actually in a circle,
it's pretty natural

111
00:07:19,530 --> 00:07:22,140
to use polar coordinates.

112
00:07:22,140 --> 00:07:27,210
Instead of x and y
inside a circle that's

113
00:07:27,210 --> 00:07:31,680
inconvenient in the
xy plane, its equation

114
00:07:31,680 --> 00:07:35,870
involves x equals square root of
1 minus y squared or something,

115
00:07:35,870 --> 00:07:43,480
I'll switch to polar
coordinates r and theta.

116
00:07:43,480 --> 00:07:47,090
Well, you might say
you remember we had

117
00:07:47,090 --> 00:07:51,290
these nice family of solutions.

118
00:07:51,290 --> 00:07:54,310
Is it still good in
polar coordinates?

119
00:07:54,310 --> 00:07:56,980
Well the fact is,
it's even better.

120
00:07:56,980 --> 00:08:01,080
So the solution of u
will be the real part

121
00:08:01,080 --> 00:08:03,040
and the imaginary part.

122
00:08:03,040 --> 00:08:10,550
Now what is x plus
iy in r and theta?

123
00:08:10,550 --> 00:08:21,690
Well, we all know x is r Cos
theta plus ir sine theta.

124
00:08:21,690 --> 00:08:27,420
And that's r times Cos
theta plus i sine theta,

125
00:08:27,420 --> 00:08:34,510
the one unforgettable
complex Euler's formula, e

126
00:08:34,510 --> 00:08:35,429
to the I theta.

127
00:08:39,429 --> 00:08:42,390
Now, I need its nth power.

128
00:08:42,390 --> 00:08:45,560
The nth power of
this is wonderful.

129
00:08:45,560 --> 00:08:48,140
The real part and imaginary
part of the nth power

130
00:08:48,140 --> 00:08:53,320
is r to the nth e
to the in theta.

131
00:08:53,320 --> 00:08:57,800
That's my x plus iy to the nth.

132
00:08:57,800 --> 00:09:00,510
Much nicer in polar
coordinates, because I

133
00:09:00,510 --> 00:09:04,190
can take the real part and
the imaginary part right away.

134
00:09:04,190 --> 00:09:12,100
It's r to the nth Cos n theta
and r to the nth sine n theta.

135
00:09:12,100 --> 00:09:17,690
These are my solutions,
my long list of solutions,

136
00:09:17,690 --> 00:09:20,100
to Laplace's equation.

137
00:09:20,100 --> 00:09:23,630
And it's some
combination of those,

138
00:09:23,630 --> 00:09:27,150
my final thing is going to
be some combination of those,

139
00:09:27,150 --> 00:09:28,820
some combination.

140
00:09:28,820 --> 00:09:31,730
Maybe coefficients a sub n.

141
00:09:34,330 --> 00:09:41,820
I can use these and
I can use these.

142
00:09:41,820 --> 00:09:49,550
So maybe b sub n r to
the nth sine n theta.

143
00:09:49,550 --> 00:09:52,780
You may wonder what I'm
doing, but what I'm achieved,

144
00:09:52,780 --> 00:10:00,540
it's done now, is to
find the general solution

145
00:10:00,540 --> 00:10:02,800
of Laplace's equation.

146
00:10:02,800 --> 00:10:04,780
Instead of two
constants that we had

147
00:10:04,780 --> 00:10:09,180
for an ordinary differential
equation, a C1 and a C2,

148
00:10:09,180 --> 00:10:14,740
here I have these guys
go from up to infinity.

149
00:10:14,740 --> 00:10:16,470
N goes up to infinity.

150
00:10:16,470 --> 00:10:20,010
So I have many solutions.

151
00:10:20,010 --> 00:10:24,030
And any combination working,
so that's the general solution.

152
00:10:24,030 --> 00:10:25,790
That's the general solution.

153
00:10:25,790 --> 00:10:29,310
And I would have to
match that-- now here's

154
00:10:29,310 --> 00:10:33,020
the final step and not
simple, not always simple--

155
00:10:33,020 --> 00:10:36,590
I have to match this to
the boundary conditions.

156
00:10:36,590 --> 00:10:40,090
That's what will tell me
the constants, of course.

157
00:10:40,090 --> 00:10:45,790
As usual, c1 and c2 came from
the matching the conditions.

158
00:10:45,790 --> 00:10:50,140
Now I don't have just c1 and
c2, I have this infinite family

159
00:10:50,140 --> 00:10:53,560
of a's, infinite family of b's.

160
00:10:53,560 --> 00:10:57,980
And I have a lot more to
match because on the boundary,

161
00:10:57,980 --> 00:11:04,810
here I have to match
u0, which is given.

162
00:11:04,810 --> 00:11:08,100
So I might be
given, suppose I was

163
00:11:08,100 --> 00:11:13,370
given u0 equal to the
temperature was equal 1

164
00:11:13,370 --> 00:11:14,990
on the top half.

165
00:11:14,990 --> 00:11:20,410
And on the bottom half, say
the temperature is minus 1.

166
00:11:23,300 --> 00:11:25,740
That's a typical problem.

167
00:11:25,740 --> 00:11:28,720
I have a circular region.

168
00:11:31,810 --> 00:11:34,910
The top half is held
at one temperature,

169
00:11:34,910 --> 00:11:38,570
the lower half is held at
a different temperature.

170
00:11:38,570 --> 00:11:40,160
I reach equilibrium.

171
00:11:40,160 --> 00:11:42,700
Everybody knows that
along that line,

172
00:11:42,700 --> 00:11:47,150
probably the temperature
would be 0 by symmetry.

173
00:11:47,150 --> 00:11:52,550
But once the temperature
there halfway up, not so easy,

174
00:11:52,550 --> 00:11:54,420
or anywhere in there.

175
00:11:54,420 --> 00:12:01,400
Well, the answer is u in
the middle, u of r and theta

176
00:12:01,400 --> 00:12:08,350
inside is given by that formula.

177
00:12:08,350 --> 00:12:11,610
And again, the ANs
and the BNs come

178
00:12:11,610 --> 00:12:17,090
by matching the-- getting the
right answer on the boundary.

179
00:12:17,090 --> 00:12:24,430
Well, there's a big theory
there how do I match these?

180
00:12:24,430 --> 00:12:26,680
That's called a Fourier series.

181
00:12:26,680 --> 00:12:28,670
That's called a Fourier series.

182
00:12:28,670 --> 00:12:33,080
So I'm finding the coefficients
for a Fourier series, the A's

183
00:12:33,080 --> 00:12:39,040
and B's, that match a
function around the boundary.

184
00:12:39,040 --> 00:12:42,150
And I could match any
function, and Fourier series

185
00:12:42,150 --> 00:12:47,230
is another entirely
separate video.

186
00:12:47,230 --> 00:12:52,830
We've done the job with
Laplace's equation in a circle.

187
00:12:52,830 --> 00:12:56,910
We've reduced the problem
to a Fourier series problem.

188
00:12:56,910 --> 00:13:00,290
We have found the
general solution.

189
00:13:00,290 --> 00:13:04,490
And then to match it to
a specific given boundary

190
00:13:04,490 --> 00:13:07,730
value, that's a
Fourier series problem.

191
00:13:07,730 --> 00:13:11,130
So I'll have to put that off
to the Fourier series video.

192
00:13:11,130 --> 00:13:12,840
Thank you.