1
00:00:00,000 --> 00:00:00,030


2
00:00:00,030 --> 00:00:02,330
The following content is
provided under a Creative

3
00:00:02,330 --> 00:00:03,610
Commons license.

4
00:00:03,610 --> 00:00:05,950
Your support will help
MIT OpenCourseWare

5
00:00:05,950 --> 00:00:09,940
continue to offer high-quality
educational resources for free.

6
00:00:09,940 --> 00:00:12,530
To make a donation, or to
view additional materials

7
00:00:12,530 --> 00:00:16,830
from hundreds of MIT courses,
visit MIT OpenCourseWare

8
00:00:16,830 --> 00:00:19,320
at ocw.mit.edu.

9
00:00:19,320 --> 00:00:21,400
PROFESSOR STRANG: OK.

10
00:00:21,400 --> 00:00:25,750
So this is lecture 22,
gradient and divergence,

11
00:00:25,750 --> 00:00:28,400
headed for Laplace's equation.

12
00:00:28,400 --> 00:00:31,940
So the gradient will
be our operator A;

13
00:00:31,940 --> 00:00:34,480
the divergence, or
minus the divergence,

14
00:00:34,480 --> 00:00:38,570
will be A transpose,
and then A transpose A

15
00:00:38,570 --> 00:00:40,860
will be the Laplacian.

16
00:00:40,860 --> 00:00:43,980
We get to Laplace's
equation Wednesday.

17
00:00:43,980 --> 00:00:46,930
Today I wanted to
take them separately.

18
00:00:46,930 --> 00:00:50,080
To understand the
meaning of gradient,

19
00:00:50,080 --> 00:00:56,350
the meaning of divergence,
the connection between them.

20
00:00:56,350 --> 00:00:58,180
I mentioned at the
end of last time

21
00:00:58,180 --> 00:01:03,690
that one is the transpose of the
other, or minus the transpose.

22
00:01:03,690 --> 00:01:06,560
I'll try to keep
gradient on this side,

23
00:01:06,560 --> 00:01:09,300
and if I could only
transpose the blackboard,

24
00:01:09,300 --> 00:01:15,220
I could do divergence-- I'll
do divergence on that side.

25
00:01:15,220 --> 00:01:18,450
And I guess if I could
get a rotating blackboard,

26
00:01:18,450 --> 00:01:21,100
right in the middle
I could do curl.

27
00:01:21,100 --> 00:01:23,290
That would be
perfect. [LAUGHTER]

28
00:01:23,290 --> 00:01:23,790
OK.

29
00:01:23,790 --> 00:01:27,260
So some of this will
not be new to you.

30
00:01:27,260 --> 00:01:31,020
But maybe some of the insights
or the ways of looking at it

31
00:01:31,020 --> 00:01:32,840
could be new.

32
00:01:32,840 --> 00:01:35,960
This is the background
of vector calculus.

33
00:01:35,960 --> 00:01:38,710
So we have things
like vector fields.

34
00:01:38,710 --> 00:01:44,221
That means I have a vector
[v 1, v 2] at each point (x,

35
00:01:44,221 --> 00:01:44,720
y).

36
00:01:44,720 --> 00:01:48,170
So I could draw a little
arrow at every point

37
00:01:48,170 --> 00:01:53,550
to show the direction and
magnitude of that vector.

38
00:01:53,550 --> 00:01:55,320
I have a field of vectors.

39
00:01:55,320 --> 00:01:58,730
OK, so.

40
00:01:58,730 --> 00:02:02,310
From last time,
our basic setup is,

41
00:02:02,310 --> 00:02:09,070
the gradient is this
first operator, A. The one

42
00:02:09,070 --> 00:02:10,900
we see at the beginning.

43
00:02:10,900 --> 00:02:13,840
One change.

44
00:02:13,840 --> 00:02:20,360
Instead of calling the result
e, let me connect to velocity.

45
00:02:20,360 --> 00:02:23,010
I'll be thinking of
u as a potential,

46
00:02:23,010 --> 00:02:27,800
I'll use the word potential for
u, and I'll use v instead of e

47
00:02:27,800 --> 00:02:29,900
for the velocity component.

48
00:02:29,900 --> 00:02:31,810
So that's the A.

49
00:02:31,810 --> 00:02:36,300
And then on the other
side, I start with a w.

50
00:02:36,300 --> 00:02:40,590
Again, it's a vector field,
it's actually a momentum.

51
00:02:40,590 --> 00:02:45,820
Very often the step between
here and here, well most often,

52
00:02:45,820 --> 00:02:49,150
the step between here and
here will be the identity.

53
00:02:49,150 --> 00:02:51,110
That's what gives
Laplace's equation.

54
00:02:51,110 --> 00:02:56,260
So you'll have to watch,
I'm sometimes confusing

55
00:02:56,260 --> 00:02:57,720
the v's with the w's.

56
00:02:57,720 --> 00:03:02,290
Because when I go to Laplace's
equation and c is the identity,

57
00:03:02,290 --> 00:03:03,820
they're the same.

58
00:03:03,820 --> 00:03:07,780
But I would like, today, to
try to keep this left side

59
00:03:07,780 --> 00:03:12,450
separate, the gradient, from
the right side, the divergence.

60
00:03:12,450 --> 00:03:17,460
And understand what they mean,
and how to work with them.

61
00:03:17,460 --> 00:03:21,420
And of course, a big connection
is the divergence theorem,

62
00:03:21,420 --> 00:03:25,800
or the Gauss-Green
connection, identity.

63
00:03:25,800 --> 00:03:26,940
We'll get to that.

64
00:03:26,940 --> 00:03:29,920
OK, gradient first.

65
00:03:29,920 --> 00:03:31,090
First, what does it mean?

66
00:03:31,090 --> 00:03:35,790
If I have a function u, what
does its gradient tell me?

67
00:03:35,790 --> 00:03:39,030
And then the second is kind
of a backwards question.

68
00:03:39,030 --> 00:03:44,480
Suppose I have the v. Is
it the gradient of some u?

69
00:03:44,480 --> 00:03:49,740
So one direction from u to
v, and the second direction

70
00:03:49,740 --> 00:03:53,560
will be from v back
to u when possible.

71
00:03:53,560 --> 00:03:56,610
OK, meaning of the gradient.

72
00:03:56,610 --> 00:03:59,640
So the gradient is
just the obvious thing,

73
00:03:59,640 --> 00:04:03,000
the two derivatives in
the x and y direction.

74
00:04:03,000 --> 00:04:07,580
So of course the gradient
gives you the rate of change,

75
00:04:07,580 --> 00:04:10,730
the partial derivatives of u.

76
00:04:10,730 --> 00:04:15,550
But how do you see
that in a picture?

77
00:04:15,550 --> 00:04:18,470
Let me draw an important curve.

78
00:04:18,470 --> 00:04:23,510
I'll start with a very simple u.
u is x squared plus y squared.

79
00:04:23,510 --> 00:04:27,120
So that's my example.

80
00:04:27,120 --> 00:04:28,330
Example one.

81
00:04:28,330 --> 00:04:32,860
And what I've drawn is
an equipotential curve.

82
00:04:32,860 --> 00:04:36,980
Or isopotential might be a more
appropriate word these days,

83
00:04:36,980 --> 00:04:39,690
but we still say equipotential.

84
00:04:39,690 --> 00:04:47,540
It means that, along this
curve, u is a constant.

85
00:04:47,540 --> 00:04:50,150
And for this
particular potential,

86
00:04:50,150 --> 00:04:59,000
this simple one to work with,
x squared plus y squared,

87
00:04:59,000 --> 00:05:02,950
the curve is a circle.

88
00:05:02,950 --> 00:05:04,860
So the curve would be a circle.

89
00:05:04,860 --> 00:05:08,020
OK, now what do I learn
by taking the gradient?

90
00:05:08,020 --> 00:05:14,680
So the gradient of u -- this
is my v -- is the x derivative,

91
00:05:14,680 --> 00:05:20,130
which is 2x, and the y
derivative, which is 2y.

92
00:05:20,130 --> 00:05:23,630
OK.

93
00:05:23,630 --> 00:05:26,930
Let me take a typical
point on the curve

94
00:05:26,930 --> 00:05:29,800
and draw that gradient vector.

95
00:05:29,800 --> 00:05:36,990
So this is the xy-plane, this
is the curve u equal constant.

96
00:05:36,990 --> 00:05:39,840
When I have a curve
u equal constant

97
00:05:39,840 --> 00:05:43,850
and I draw the gradient
of u, where does it point?

98
00:05:43,850 --> 00:05:46,940
This is the first
and most important

99
00:05:46,940 --> 00:05:50,360
simple idea about
the gradient vector.

100
00:05:50,360 --> 00:05:56,610
The gradient vector points--
Does the gradient vector point,

101
00:05:56,610 --> 00:05:59,730
could it point any old way?

102
00:05:59,730 --> 00:06:00,780
No.

103
00:06:00,780 --> 00:06:04,370
The gradient vector is
perpendicular to the curve.

104
00:06:04,370 --> 00:06:08,030
And we can see that,
for this simple example,

105
00:06:08,030 --> 00:06:12,650
that vector [2x, 2y], that's
a vector radially outwards,

106
00:06:12,650 --> 00:06:13,310
right?

107
00:06:13,310 --> 00:06:16,490
If here's the origin,
and if, at this point --

108
00:06:16,490 --> 00:06:20,165
I don't know its coordinates,
whatever they are, maybe (2,

109
00:06:20,165 --> 00:06:25,750
1) or something -- the gradient
vector would be [4, 2].

110
00:06:25,750 --> 00:06:29,950
It would be a multiple-- Here's
the position vector, [x, y].

111
00:06:29,950 --> 00:06:33,660
The point is, the gradient
vector points out.

112
00:06:33,660 --> 00:06:36,600
Perpendicular to the curve.

113
00:06:36,600 --> 00:06:38,900
That's what the
gradient tells you.

114
00:06:38,900 --> 00:06:44,720
It tells you, in
this situation it's

115
00:06:44,720 --> 00:06:50,170
telling me which direction is
perpendicular to the curve.

116
00:06:50,170 --> 00:06:52,290
Now how do I understand that?

117
00:06:52,290 --> 00:06:54,230
How do I see that?

118
00:06:54,230 --> 00:07:01,120
I think of this-- Let me try
to draw the whole surface,

119
00:07:01,120 --> 00:07:03,580
u equal x plus y squared.

120
00:07:03,580 --> 00:07:07,070
What would that whole
surface look like?

121
00:07:07,070 --> 00:07:13,780
Let me try to put that picture--
This is a 2-D picture right

122
00:07:13,780 --> 00:07:14,290
now.

123
00:07:14,290 --> 00:07:16,440
It's sort of a cross-section.

124
00:07:16,440 --> 00:07:20,810
Let me put this 2-D picture
into a 3-D picture to see.

125
00:07:20,810 --> 00:07:24,290
So the 3-D picture is a
picture of the whole surface. u

126
00:07:24,290 --> 00:07:27,840
going up, x and y going around.

127
00:07:27,840 --> 00:07:31,510
And what kind of a curve,
what kind of a surface

128
00:07:31,510 --> 00:07:37,160
do I have for that function,
x squared plus y squared?

129
00:07:37,160 --> 00:07:40,360
Well, it grows, right?

130
00:07:40,360 --> 00:07:44,960
It's sort of, I
use the word bowl.

131
00:07:44,960 --> 00:07:48,430
And we've seen it before.

132
00:07:48,430 --> 00:07:51,990
It sort of goes
up, right, and it

133
00:07:51,990 --> 00:07:56,060
goes up faster and faster as I
go out, because of the squares.

134
00:07:56,060 --> 00:07:58,610
And now what I want
to do is, suppose

135
00:07:58,610 --> 00:08:02,360
I take the cross-section,
suppose I take this bowl.

136
00:08:02,360 --> 00:08:07,840
Do you see a
beautiful bowl there?

137
00:08:07,840 --> 00:08:13,370
And now, I cut through it,
horizontal cross-section,

138
00:08:13,370 --> 00:08:15,690
at the height c.

139
00:08:15,690 --> 00:08:20,750
What do I get if I take
this surface in 3-D,

140
00:08:20,750 --> 00:08:31,090
and I cut through it by a plane,
the plane at that height, c.

141
00:08:31,090 --> 00:08:37,270
So if the plane went through
there, that distance was c.

142
00:08:37,270 --> 00:08:40,610
It cuts out a cross-section
out of the bowl.

143
00:08:40,610 --> 00:08:43,040
And what's the cross-section?

144
00:08:43,040 --> 00:08:44,020
It's the circle.

145
00:08:44,020 --> 00:08:45,690
It's this one.

146
00:08:45,690 --> 00:08:53,030
So out of the bowl, this plane
is cutting this cross-section.

147
00:08:53,030 --> 00:08:56,670
This is height u equal constant.

148
00:08:56,670 --> 00:09:00,030
So I'm really intersecting
one surface, the plane u

149
00:09:00,030 --> 00:09:02,650
equal constant, with the bowl.

150
00:09:02,650 --> 00:09:07,400
And the intersection of those
two is the equipotential,

151
00:09:07,400 --> 00:09:11,570
the curve u=c.

152
00:09:11,570 --> 00:09:12,530
Right?

153
00:09:12,530 --> 00:09:14,610
You see the two pictures?

154
00:09:14,610 --> 00:09:18,740
And now, what does the gradient
tell me in that picture?

155
00:09:18,740 --> 00:09:22,170
What does the gradient of
u tell me in that picture?

156
00:09:22,170 --> 00:09:26,240
Over here, it pointed
outwards, but now we're

157
00:09:26,240 --> 00:09:28,690
going to kind of see why.

158
00:09:28,690 --> 00:09:35,330
So, it pointed outwards
for this nice function u.

159
00:09:35,330 --> 00:09:39,420
I could have made a
more general function u,

160
00:09:39,420 --> 00:09:41,940
but let me just stay
with this simple one.

161
00:09:41,940 --> 00:09:47,380
So suppose I'm climbing.

162
00:09:47,380 --> 00:09:52,490
This is like, I'm climbing
out of a volcano or something.

163
00:09:52,490 --> 00:09:54,740
Usually, I would say a
mountain, but the way

164
00:09:54,740 --> 00:09:57,280
I've drawn the thing, it's
not much of a mountain.

165
00:09:57,280 --> 00:09:59,120
So OK, volcano.

166
00:09:59,120 --> 00:10:01,050
Climbing out of it.

167
00:10:01,050 --> 00:10:06,370
And I got up to this point,
which was this point.

168
00:10:06,370 --> 00:10:09,040
Now, what does the
gradient tell me?

169
00:10:09,040 --> 00:10:11,750
That when I'm climbing away,
I reach -- duh duh duh duh --

170
00:10:11,750 --> 00:10:13,000
I get up to here.

171
00:10:13,000 --> 00:10:15,780
What does the gradient tell me?

172
00:10:15,780 --> 00:10:17,690
It tells me which way to go.

173
00:10:17,690 --> 00:10:20,980
It tells me the steepest
direction upwards,

174
00:10:20,980 --> 00:10:23,040
and it tells me how steep.

175
00:10:23,040 --> 00:10:23,980
Right.

176
00:10:23,980 --> 00:10:27,440
So it tells me those two
things, direction and magnitude.

177
00:10:27,440 --> 00:10:33,260
Direction is, well for
this special function

178
00:10:33,260 --> 00:10:36,700
you know the direction is,
like, straight outwards,

179
00:10:36,700 --> 00:10:37,540
straight upwards.

180
00:10:37,540 --> 00:10:39,510
And how steep is it?

181
00:10:39,510 --> 00:10:45,230
What's the steepness?

182
00:10:45,230 --> 00:10:50,050
It's the size of, it's the
magnitude of this vector,

183
00:10:50,050 --> 00:10:51,480
grad u.

184
00:10:51,480 --> 00:11:02,270
Which is the-- This is not--
How big is this vector?

185
00:11:02,270 --> 00:11:04,770
This is my vector grad u.

186
00:11:04,770 --> 00:11:09,260
This is another shorthand
notation for gradient.

187
00:11:09,260 --> 00:11:17,330
The size of that vector is
the square root of what?

188
00:11:17,330 --> 00:11:19,730
The length of a vector
is the square root

189
00:11:19,730 --> 00:11:21,050
of the sum of the squares.

190
00:11:21,050 --> 00:11:25,550
So I have the length of v is
the square root of 4 x squared

191
00:11:25,550 --> 00:11:31,070
plus 4 y squared.

192
00:11:31,070 --> 00:11:33,970
OK.

193
00:11:33,970 --> 00:11:37,750
Now, it gets steeper
and steeper, of course.

194
00:11:37,750 --> 00:11:45,350
The cross-sections here
would be all circles

195
00:11:45,350 --> 00:11:46,910
for this simple function.

196
00:11:46,910 --> 00:11:49,420
And the gradients would
keep pointing out,

197
00:11:49,420 --> 00:11:58,230
and we'd have a function that
comes from a surface like that.

198
00:11:58,230 --> 00:12:02,790
OK, this is what I wanted to say
about question one, the meaning

199
00:12:02,790 --> 00:12:06,110
of the gradient.

200
00:12:06,110 --> 00:12:10,960
The thing to remember, if you
remember the two pictures,

201
00:12:10,960 --> 00:12:12,450
you've really got
the idea of what

202
00:12:12,450 --> 00:12:14,940
the gradient is telling you.

203
00:12:14,940 --> 00:12:17,220
For a function of one
variable, the derivative

204
00:12:17,220 --> 00:12:19,870
is telling you the slope.

205
00:12:19,870 --> 00:12:23,680
Well, we've got slopes in the
x direction, slopes in the y

206
00:12:23,680 --> 00:12:26,740
direction, and the
gradient direction

207
00:12:26,740 --> 00:12:29,300
tells us the steepest slope.

208
00:12:29,300 --> 00:12:34,300
And it tells us, yeah,
tells us what that slope is.

209
00:12:34,300 --> 00:12:38,020
OK, so much for the
meaning of the gradient.

210
00:12:38,020 --> 00:12:40,040
Now, backwards.

211
00:12:40,040 --> 00:12:44,490
Suppose I have v.
Because these gradient

212
00:12:44,490 --> 00:12:46,560
fields are extremely important.

213
00:12:46,560 --> 00:12:50,360
I mean, they're
wonderful fields.

214
00:12:50,360 --> 00:12:57,760
If you've got a v_1 and a v_2
that is the gradient of some u,

215
00:12:57,760 --> 00:12:59,200
you want to know that.

216
00:12:59,200 --> 00:13:02,080
I mean, that's good news.

217
00:13:02,080 --> 00:13:03,630
Most fields will not.

218
00:13:03,630 --> 00:13:10,630
So let's figure out, when is--
Can I lift that up now, and do

219
00:13:10,630 --> 00:13:12,100
the other gradient thing now?

220
00:13:12,100 --> 00:13:22,060
Or maybe I'll put the final
gradient blackboard here,

221
00:13:22,060 --> 00:13:25,760
and then make way
for divergence.

222
00:13:25,760 --> 00:13:29,260
OK, so I'm asking
now question two.

223
00:13:29,260 --> 00:13:34,250
I'm given v_1 and
v_2, and I want

224
00:13:34,250 --> 00:13:39,110
this to be du/dx for some u,
and I want this to be du/dy.

225
00:13:39,110 --> 00:13:42,920


226
00:13:42,920 --> 00:13:45,640
So I've got two equations.

227
00:13:45,640 --> 00:13:47,930
If I'm looking for u.

228
00:13:47,930 --> 00:13:52,230
Remember, I'm now starting
with these, looking for the u.

229
00:13:52,230 --> 00:13:53,230
OK.

230
00:13:53,230 --> 00:13:56,130
So, am I going to find Au?

231
00:13:56,130 --> 00:14:00,870
Am I going to find a function
u, whose x derivative

232
00:14:00,870 --> 00:14:07,180
is my v_1 that was given, and
the y derivative is my v_2?

233
00:14:07,180 --> 00:14:09,600
Well, chances are not good.

234
00:14:09,600 --> 00:14:10,100
Right?

235
00:14:10,100 --> 00:14:13,740
I've got two equations
here, but only one unknown.

236
00:14:13,740 --> 00:14:17,430
So generally, it's like
a rectangular system.

237
00:14:17,430 --> 00:14:21,170
Usually there's no solution,
but sometimes there is.

238
00:14:21,170 --> 00:14:28,200
So we have to find out, what's
the test for consistency

239
00:14:28,200 --> 00:14:31,050
for these to have a solution?

240
00:14:31,050 --> 00:14:39,330
And this brings us right
away to the key identity

241
00:14:39,330 --> 00:14:42,300
from partial derivatives.

242
00:14:42,300 --> 00:14:43,940
OK.

243
00:14:43,940 --> 00:14:46,460
Can you see some
condition that has

244
00:14:46,460 --> 00:14:50,310
to-- How can I connect
these two equations,

245
00:14:50,310 --> 00:14:53,930
and therefore
connect v_1 and v_2?

246
00:14:53,930 --> 00:14:58,360
That'll be the test for
weather-- You could say,

247
00:14:58,360 --> 00:15:01,930
in matrix language, I'm
asking whether [v 1, v 2] is

248
00:15:01,930 --> 00:15:06,320
in the column space
of the gradient.

249
00:15:06,320 --> 00:15:10,850
Is it something I can get
from this tall thin matrix?

250
00:15:10,850 --> 00:15:21,660
OK, they key idea is take the y
derivative of that equation, d

251
00:15:21,660 --> 00:15:24,440
second u/dydx.

252
00:15:24,440 --> 00:15:30,400
And take the x derivative of
this equation. dv_2 dy-- dx,

253
00:15:30,400 --> 00:15:35,100
is the second derivative
of u, with respect

254
00:15:35,100 --> 00:15:38,210
to-- the x derivative
of the y derivative.

255
00:15:38,210 --> 00:15:39,540
OK.

256
00:15:39,540 --> 00:15:45,640
So it's like I'm operating
on the equations.

257
00:15:45,640 --> 00:15:47,260
I'm doing what
I'm allowed to do,

258
00:15:47,260 --> 00:15:50,590
I'm doing the same thing to
both sides of each equation.

259
00:15:50,590 --> 00:15:52,060
Now I'm going to eliminate.

260
00:15:52,060 --> 00:15:54,020
And what am I going to find?

261
00:15:54,020 --> 00:15:56,430
What's the key point here?

262
00:15:56,430 --> 00:15:59,710
The key, key point
about second derivatives

263
00:15:59,710 --> 00:16:02,340
is that that equals that.

264
00:16:02,340 --> 00:16:04,310
Right?

265
00:16:04,310 --> 00:16:06,690
Take any function.

266
00:16:06,690 --> 00:16:09,130
Do we want to practice
with a function,

267
00:16:09,130 --> 00:16:10,990
just to see it be true?

268
00:16:10,990 --> 00:16:14,360
Take, let u be x cubed y.

269
00:16:14,360 --> 00:16:16,300
Just for the heck of it.

270
00:16:16,300 --> 00:16:18,500
OK, I don't know what--
Is that going to produce

271
00:16:18,500 --> 00:16:19,440
anything interesting?

272
00:16:19,440 --> 00:16:22,760
Then maybe I'd better
make it y squared.

273
00:16:22,760 --> 00:16:26,240
I don't know why I'm doing this,
it just is like an example,

274
00:16:26,240 --> 00:16:27,770
to make it believable.

275
00:16:27,770 --> 00:16:33,380
OK, so du/dx, u_x, is --
and I'll often use u sub x

276
00:16:33,380 --> 00:16:36,070
as a sort of shorthand
-- will be what?

277
00:16:36,070 --> 00:16:39,390
3x squared y squared.

278
00:16:39,390 --> 00:16:43,830
And u_y, taking the
y derivative is just,

279
00:16:43,830 --> 00:16:47,270
x is constant, 2x cubed y.

280
00:16:47,270 --> 00:16:53,560
And now let me do u_xy,
the y derivative of this.

281
00:16:53,560 --> 00:16:56,500
Which is what?

282
00:16:56,500 --> 00:16:59,490
It's been a long weekend,
but hey, we can do these.

283
00:16:59,490 --> 00:17:05,290
The y derivative of that is
going to be 6x squared y.

284
00:17:05,290 --> 00:17:09,100
And the x derivative
of this is going to be,

285
00:17:09,100 --> 00:17:15,210
so I should say y-- I've
got the y derivative

286
00:17:15,210 --> 00:17:17,500
and now I should take
the x derivative of that.

287
00:17:17,500 --> 00:17:19,230
And what do I get?

288
00:17:19,230 --> 00:17:25,270
The x derivative of
that is 6x squared y.

289
00:17:25,270 --> 00:17:25,830
And look!

290
00:17:25,830 --> 00:17:27,680
They're the same.

291
00:17:27,680 --> 00:17:28,380
Hooray.

292
00:17:28,380 --> 00:17:29,010
OK.

293
00:17:29,010 --> 00:17:36,160
So if the function is smooth
and has these derivatives,

294
00:17:36,160 --> 00:17:39,140
they'll come out the same.

295
00:17:39,140 --> 00:17:42,620
And therefore, so
what do I conclude?

296
00:17:42,620 --> 00:17:48,310
What's the test on v_1
and v_2 that it must

297
00:17:48,310 --> 00:17:52,930
pass to be a gradient field?

298
00:17:52,930 --> 00:17:56,820
For there to be a function u,
that solves these equations.

299
00:17:56,820 --> 00:18:01,000
These are solvable
only when what?

300
00:18:01,000 --> 00:18:03,900
Well, if these are the same,
these have to be the same.

301
00:18:03,900 --> 00:18:11,552
So it's solvable only
when-- I need dv_2/dx -

302
00:18:11,552 --> 00:18:19,140
dv_1/dy to be what?

303
00:18:19,140 --> 00:18:20,810
Zero.

304
00:18:20,810 --> 00:18:21,670
OK.

305
00:18:21,670 --> 00:18:23,080
That's the conclusion.

306
00:18:23,080 --> 00:18:25,390
Those have to be the same.

307
00:18:25,390 --> 00:18:28,460
dv_2/dx - dv_1/dy
has to be zero.

308
00:18:28,460 --> 00:18:31,990
OK.

309
00:18:31,990 --> 00:18:35,420
Because those are the same.

310
00:18:35,420 --> 00:18:40,520
I guess I came to this early
because it's the key identity

311
00:18:40,520 --> 00:18:43,140
of vector calculus.

312
00:18:43,140 --> 00:18:46,140
Well, the key identity
behind vector calculus

313
00:18:46,140 --> 00:18:51,510
is this fact about
the derivatives.

314
00:18:51,510 --> 00:18:58,790
Can I just throw out a question
that you might think about?

315
00:18:58,790 --> 00:19:02,510
We already have seen, so often
now, the second derivative,

316
00:19:02,510 --> 00:19:05,570
like v_xx.

317
00:19:05,570 --> 00:19:07,090
Or u_xx, suppose.

318
00:19:07,090 --> 00:19:09,940
So here's a little
question to think about.

319
00:19:09,940 --> 00:19:13,710
So think.

320
00:19:13,710 --> 00:19:17,030
OK.

321
00:19:17,030 --> 00:19:21,170
I just want to bring
finite differences

322
00:19:21,170 --> 00:19:26,930
in for a moment. u_xx
we've got a handle on.

323
00:19:26,930 --> 00:19:33,150
We know that that's like u at
x+h -- and if there was a y,

324
00:19:33,150 --> 00:19:40,970
put in the y -- minus 2u at x
-- and I can put in a y there --

325
00:19:40,970 --> 00:19:45,210
plus u at x-h, y.

326
00:19:45,210 --> 00:19:47,900
We've seen that.

327
00:19:47,900 --> 00:19:52,140
That'll be the x derivative,
second x derivative-- sorry,

328
00:19:52,140 --> 00:19:56,530
second x difference.

329
00:19:56,530 --> 00:20:00,280
Since we're taking x
derivatives and x differences,

330
00:20:00,280 --> 00:20:03,350
it's x that moves,
and y doesn't move.

331
00:20:03,350 --> 00:20:06,640
Just the central idea
of partial derivatives.

332
00:20:06,640 --> 00:20:11,000
u_yy will be similar
with y moving.

333
00:20:11,000 --> 00:20:14,470
And my question to you is -- and
we could have asked it way way

334
00:20:14,470 --> 00:20:17,690
back -- what's a finite
difference approximation

335
00:20:17,690 --> 00:20:23,190
to u_xy, to the
cross derivative?

336
00:20:23,190 --> 00:20:27,630
That equals what?

337
00:20:27,630 --> 00:20:30,940
I want to go to
finite differences.

338
00:20:30,940 --> 00:20:39,260
OK, what finite differences?

339
00:20:39,260 --> 00:20:40,790
Which finite differences?

340
00:20:40,790 --> 00:20:43,440
Maybe that's a question
to think about.

341
00:20:43,440 --> 00:20:49,820
If I can remember,
I'll include it just

342
00:20:49,820 --> 00:20:55,130
as a small homework question for
the homework on this material.

343
00:20:55,130 --> 00:20:57,570
So that's looking
ahead, really, today.

344
00:20:57,570 --> 00:21:01,440
We're not making
things discrete.

345
00:21:01,440 --> 00:21:05,780
We're in continuous x and y.

346
00:21:05,780 --> 00:21:07,090
We have vector fields.

347
00:21:07,090 --> 00:21:20,440
And we now know the test
for a gradient field.

348
00:21:20,440 --> 00:21:24,190
I'm tempted to use
the word curl here.

349
00:21:24,190 --> 00:21:28,100
I'm tempted to
use the word curl.

350
00:21:28,100 --> 00:21:32,275
I want to connect that test --
may I use the word curl without

351
00:21:32,275 --> 00:21:39,170
-- and I'll say why I'm not
going to do everything properly

352
00:21:39,170 --> 00:21:40,710
with curl right away.

353
00:21:40,710 --> 00:21:45,200
I would describe this as curl.

354
00:21:45,200 --> 00:21:51,820
The test is, the curl
of v has to be zero.

355
00:21:51,820 --> 00:22:01,080
So for me, that's the curl of v.

356
00:22:01,080 --> 00:22:03,790
You're going to
say, wait a minute.

357
00:22:03,790 --> 00:22:07,040
I learned about curl, and that
doesn't look like the curl

358
00:22:07,040 --> 00:22:10,650
to me.

359
00:22:10,650 --> 00:22:14,390
So I'll say wait another minute.

360
00:22:14,390 --> 00:22:16,160
It's not that far off.

361
00:22:16,160 --> 00:22:18,090
OK, what's your objection?

362
00:22:18,090 --> 00:22:22,230
Your objection is that the curl
is in three-dimensional space.

363
00:22:22,230 --> 00:22:23,490
Right?

364
00:22:23,490 --> 00:22:25,580
When you saw curl,
and of course it

365
00:22:25,580 --> 00:22:29,580
comes in this
section of the book,

366
00:22:29,580 --> 00:22:33,850
we had functions of x, y, z.

367
00:22:33,850 --> 00:22:39,300
And the curl had
three components.

368
00:22:39,300 --> 00:22:42,035
And those three components--
Do you remember curl?

369
00:22:42,035 --> 00:22:44,160
I mean, if you remember
curl, you're a good person.

370
00:22:44,160 --> 00:22:48,800
Because it's got
this-- You sort of

371
00:22:48,800 --> 00:22:54,580
remember that it has
things like this.

372
00:22:54,580 --> 00:22:55,110
Right?

373
00:22:55,110 --> 00:23:00,100
Sort of, differences of
derivatives, and the indices

374
00:23:00,100 --> 00:23:01,250
follow a certain pattern.

375
00:23:01,250 --> 00:23:04,300
I'm saying that this
is the natural--

376
00:23:04,300 --> 00:23:08,280
this is the curl in a plane.

377
00:23:08,280 --> 00:23:13,350
What do I mean by
the curl in a plane?

378
00:23:13,350 --> 00:23:23,211
So in 3-D, v has components
of v_1, v_2, and v_3.

379
00:23:23,211 --> 00:23:23,710
Right?

380
00:23:23,710 --> 00:23:27,830
That depend on x, y, z.

381
00:23:27,830 --> 00:23:33,540
In the plane, in 3-D, I
have a vector field v, which

382
00:23:33,540 --> 00:23:38,260
has components v_1, v_2, v_3.

383
00:23:38,260 --> 00:23:39,820
All depending on x, y, z.

384
00:23:39,820 --> 00:23:45,550
That's the general 3-D picture,
where you usually see the curl.

385
00:23:45,550 --> 00:23:48,350
Now, in the plane,
what's happening?

386
00:23:48,350 --> 00:23:52,930
In the plane, we
have two components.

387
00:23:52,930 --> 00:23:56,290
So what's happening?

388
00:23:56,290 --> 00:23:58,460
Think of this.

389
00:23:58,460 --> 00:24:02,520
So in 3-D, our velocity
field is like, the flow

390
00:24:02,520 --> 00:24:03,920
is going all over the place.

391
00:24:03,920 --> 00:24:04,530
Right?

392
00:24:04,530 --> 00:24:06,910
It's a three-dimensional flow.

393
00:24:06,910 --> 00:24:11,750
But now suppose my flow
stays in the plane.

394
00:24:11,750 --> 00:24:18,200
So in 2-D-- so now I have
to put 2-D up above here.

395
00:24:18,200 --> 00:24:26,280
So in 2-D, a plane field is
what I'm working with today.

396
00:24:26,280 --> 00:24:29,080
My v is some v_1(x,y).

397
00:24:29,080 --> 00:24:31,910


398
00:24:31,910 --> 00:24:37,870
No z, no dependence
on z. v_2(x,y),

399
00:24:37,870 --> 00:24:41,290
the y direction of the
velocity doesn't depend on z,

400
00:24:41,290 --> 00:24:44,300
because this is a plane
field, same on every plane.

401
00:24:44,300 --> 00:24:46,730
And the third component
of this plane field,

402
00:24:46,730 --> 00:24:52,060
the velocity perpendicular
in the z direction, is zero.

403
00:24:52,060 --> 00:24:53,140
OK.

404
00:24:53,140 --> 00:24:53,960
Zero.

405
00:24:53,960 --> 00:24:57,850
So what I want to
say is that if I

406
00:24:57,850 --> 00:25:03,790
look, if I specialize to plane
fields, to fields like these,

407
00:25:03,790 --> 00:25:08,970
then the only component of the
curl that survives is this one.

408
00:25:08,970 --> 00:25:11,680
See, the other components of
the curl, which I'm not even

409
00:25:11,680 --> 00:25:14,860
writing down-- the other
components of a curl

410
00:25:14,860 --> 00:25:19,130
have derivatives of
v_3, but v_3 is zero.

411
00:25:19,130 --> 00:25:22,510
And they also have derivatives
of these guys with respect

412
00:25:22,510 --> 00:25:24,310
to z.

413
00:25:24,310 --> 00:25:26,270
But they don't depend on z.

414
00:25:26,270 --> 00:25:28,610
So that's why all the
other pieces of the curl,

415
00:25:28,610 --> 00:25:31,820
like, are automatically
zero for a plane field.

416
00:25:31,820 --> 00:25:34,040
So that the only
component that's

417
00:25:34,040 --> 00:25:38,040
significant-- The test curl
v equals zero boils down

418
00:25:38,040 --> 00:25:41,990
to a test not on three
things, but just on one.

419
00:25:41,990 --> 00:25:44,230
And that's the one.

420
00:25:44,230 --> 00:25:50,365
So because I want to
stay mostly with plane

421
00:25:50,365 --> 00:25:52,650
fields and
two-dimensional problems,

422
00:25:52,650 --> 00:25:57,490
I just had to comment that, if
the curl was to get in here,

423
00:25:57,490 --> 00:25:59,790
it would fit fine.

424
00:25:59,790 --> 00:26:03,770
And if I restrict the curl to
the fields I'm working with,

425
00:26:03,770 --> 00:26:07,190
plane fields, then there's
only one component I'll

426
00:26:07,190 --> 00:26:10,180
have to think about,
it has to be zero

427
00:26:10,180 --> 00:26:11,950
to have a gradient field.

428
00:26:11,950 --> 00:26:15,830
OK, now I guess I should
just do an example or two.

429
00:26:15,830 --> 00:26:18,170
Can I give you a
v_1 and v_2, and you

430
00:26:18,170 --> 00:26:22,870
tell me, is it a gradient field
or is it not a gradient field.

431
00:26:22,870 --> 00:26:27,330
Let me give you a different,
let me just change these guys.

432
00:26:27,330 --> 00:26:33,270
Suppose I change
that to y and x.

433
00:26:33,270 --> 00:26:39,730
So there is a v, a different
v. That's a vector field.

434
00:26:39,730 --> 00:26:44,400
At every point x, y,
I've got a little vector.

435
00:26:44,400 --> 00:26:47,160
I could try, even, to draw them.

436
00:26:47,160 --> 00:26:50,970
And I'm going to ask you,
is it the gradient of any u.

437
00:26:50,970 --> 00:26:53,620
And if it is, what's that u?

438
00:26:53,620 --> 00:26:57,120
So let me show you what
I mean by a vector field.

439
00:26:57,120 --> 00:27:03,710
I mean, at a typical point
like x=1, y=0, the vector--

440
00:27:03,710 --> 00:27:07,920
Let's see, if x is
one and y is zero,

441
00:27:07,920 --> 00:27:19,300
then what's the gradient at
that point? [0, 2], am I right?

442
00:27:19,300 --> 00:27:22,590
I won't draw it too
big, or you won't

443
00:27:22,590 --> 00:27:24,100
be able to see a darn thing.

444
00:27:24,100 --> 00:27:31,710
OK, what about at
the point (1, 1)?

445
00:27:31,710 --> 00:27:37,590
Which way is my vector
field going? [2, 2].

446
00:27:37,590 --> 00:27:41,960
So what's that look like?

447
00:27:41,960 --> 00:27:45,050
Plotting the vector field
v at a bunch of points.

448
00:27:45,050 --> 00:27:51,790
So you get like a
map of little arrows.

449
00:27:51,790 --> 00:27:53,260
So here it would go that way.

450
00:27:53,260 --> 00:27:55,430
Is that right?

451
00:27:55,430 --> 00:27:56,840
Huh.

452
00:27:56,840 --> 00:28:00,350
I wasn't expecting
that, to tell the truth.

453
00:28:00,350 --> 00:28:05,330
Let's see, so can I get
in some more points?

454
00:28:05,330 --> 00:28:06,650
Let's see.

455
00:28:06,650 --> 00:28:09,110
What if I have there.

456
00:28:09,110 --> 00:28:12,000
What's-- At (1/2, 0).

457
00:28:12,000 --> 00:28:20,770
Then, v is [0, 1].

458
00:28:20,770 --> 00:28:24,160
Where is this flow going?

459
00:28:24,160 --> 00:28:28,680
See, if the point is
along this line, where

460
00:28:28,680 --> 00:28:34,230
y is equal to x, then the flow
was going out along this line.

461
00:28:34,230 --> 00:28:36,070
Can you give me some
other point here,

462
00:28:36,070 --> 00:28:38,420
just so we get some
handle on this?

463
00:28:38,420 --> 00:28:42,090
There's the point
(2, 1), let's say.

464
00:28:42,090 --> 00:28:43,860
Let me put it over a little bit.

465
00:28:43,860 --> 00:28:46,460
How about the point (2, 1)?

466
00:28:46,460 --> 00:28:48,920
What's the vector if
I just want to draw --

467
00:28:48,920 --> 00:28:50,500
I'm just drawing here.

468
00:28:50,500 --> 00:28:55,290
And of course, code would do
it much better than I'm doing.

469
00:28:55,290 --> 00:28:59,580
(2, 1), that gives
me [2, 4], right?

470
00:28:59,580 --> 00:29:04,870
So [2, 4] is over two
and up, it's like this.

471
00:29:04,870 --> 00:29:10,630
I think -- but I
don't swear to it --

472
00:29:10,630 --> 00:29:12,450
that if I connect
all this-- See,

473
00:29:12,450 --> 00:29:14,900
now you have to take
a big leap of faith.

474
00:29:14,900 --> 00:29:19,670
Imagine, like, at every point
we've got these little arrows,

475
00:29:19,670 --> 00:29:24,340
and I want to connect them up.

476
00:29:24,340 --> 00:29:26,510
Let me do something.

477
00:29:26,510 --> 00:29:29,970
I'll do that, but let me
come back to the question

478
00:29:29,970 --> 00:29:32,130
I should have asked you first.

479
00:29:32,130 --> 00:29:35,420
Is this a gradient field?

480
00:29:35,420 --> 00:29:39,730
Does it satisfy the curl zero
condition that we put in a box

481
00:29:39,730 --> 00:29:40,820
here?

482
00:29:40,820 --> 00:29:45,930
Does that satisfy dv-- There
is v_2 and there's v_1.

483
00:29:45,930 --> 00:29:54,210
Is dv_2/dx, whatever that test
was, minus dv_1/dy equal zero?

484
00:29:54,210 --> 00:29:57,180
Yes, no?

485
00:29:57,180 --> 00:30:05,010
Yes, right? dv_2/dx is
two, and dv_1/dy is two.

486
00:30:05,010 --> 00:30:09,070
So what's the conclusion then?

487
00:30:09,070 --> 00:30:12,900
It satisfied my
little test, so this

488
00:30:12,900 --> 00:30:16,590
must be the gradient of some u.

489
00:30:16,590 --> 00:30:18,720
Right?

490
00:30:18,720 --> 00:30:20,590
That's the question we have.

491
00:30:20,590 --> 00:30:23,960
Which vector fields-- And we
found that this is one of them,

492
00:30:23,960 --> 00:30:25,300
it passes the test.

493
00:30:25,300 --> 00:30:26,790
It's the gradient of some u.

494
00:30:26,790 --> 00:30:28,920
What's the u?

495
00:30:28,920 --> 00:30:33,790
What's the function u,
which is supposed to exist,

496
00:30:33,790 --> 00:30:37,630
whose gradient is that.

497
00:30:37,630 --> 00:30:38,800
2xy.

498
00:30:38,800 --> 00:30:40,610
Did everybody spot that one?

499
00:30:40,610 --> 00:30:41,120
2xy.

500
00:30:41,120 --> 00:30:43,970


501
00:30:43,970 --> 00:30:48,060
Because the x
derivative has to be 2y.

502
00:30:48,060 --> 00:30:51,080
So I just integrate
with respect to x.

503
00:30:51,080 --> 00:30:53,050
This is the x derivative.

504
00:30:53,050 --> 00:30:54,080
It's 2y.

505
00:30:54,080 --> 00:30:57,640
Take the integral with
respect to x, and I get 2xy.

506
00:30:57,640 --> 00:31:05,250
And there could be some term
that depended only on y.

507
00:31:05,250 --> 00:31:11,530
Anyway, this works.

508
00:31:11,530 --> 00:31:21,900
I think that this, maybe
gives me somehow-- Oh, yeah.

509
00:31:21,900 --> 00:31:24,810
What's the
equipotential curve now?

510
00:31:24,810 --> 00:31:28,690
Oh, yeah, this picture's
going to come together.

511
00:31:28,690 --> 00:31:34,100
What is the equipotential
curve for this potential?

512
00:31:34,100 --> 00:31:38,370
It was a circle for the first
guy, but circles are out now.

513
00:31:38,370 --> 00:31:42,760
I changed v. I've got a
new potential function.

514
00:31:42,760 --> 00:31:50,310
And now I want to draw, in
this graph, the equipotentials.

515
00:31:50,310 --> 00:31:53,430
Suppose u is one.

516
00:31:53,430 --> 00:32:00,180
Suppose I draw the curve
2xy=1 in that picture.

517
00:32:00,180 --> 00:32:02,830
What kind of a curve is it?

518
00:32:02,830 --> 00:32:04,870
Do you recognize this?

519
00:32:04,870 --> 00:32:06,550
The Greeks would.

520
00:32:06,550 --> 00:32:08,790
Recognize 2xy=1.

521
00:32:08,790 --> 00:32:11,690
Or you could say y=1/(2x).

522
00:32:11,690 --> 00:32:14,890
That gives you a quick
handle on the curve.

523
00:32:14,890 --> 00:32:19,420
It comes down like that.

524
00:32:19,420 --> 00:32:20,740
Right?

525
00:32:20,740 --> 00:32:25,290
And what's the Greek
name for that curve?

526
00:32:25,290 --> 00:32:28,930
Oh, come on.

527
00:32:28,930 --> 00:32:30,960
It's a hyperbola.

528
00:32:30,960 --> 00:32:32,370
It's a hyperbola.

529
00:32:32,370 --> 00:32:35,190
Hyperbolas-- You
remember the Greeks,

530
00:32:35,190 --> 00:32:37,290
they had these conic sections.

531
00:32:37,290 --> 00:32:41,340
They had ellipses, they had
parabolas, the marginal case,

532
00:32:41,340 --> 00:32:43,340
and then they had hyperbolas.

533
00:32:43,340 --> 00:32:47,240
And they all come from
second degree things.

534
00:32:47,240 --> 00:32:54,740
If I have a x squared and 2bxy
and c y squared equal one,

535
00:32:54,740 --> 00:32:57,530
that's one of those curves.

536
00:32:57,530 --> 00:33:01,345
And if it was x squared
plus y squared equal one,

537
00:33:01,345 --> 00:33:02,750
it was a circle.

538
00:33:02,750 --> 00:33:05,410
If it was x squared plus
7y squared equal one,

539
00:33:05,410 --> 00:33:11,030
it would be an ellipse.

540
00:33:11,030 --> 00:33:15,430
The positive definite-- It all
comes down to linear algebra,

541
00:33:15,430 --> 00:33:18,040
of course.

542
00:33:18,040 --> 00:33:21,870
If that little matrix
is positive definite,

543
00:33:21,870 --> 00:33:24,650
so that means a
and c are positive,

544
00:33:24,650 --> 00:33:26,680
and ac is bigger
than b squared, you

545
00:33:26,680 --> 00:33:29,950
know the test for
positive definite.

546
00:33:29,950 --> 00:33:32,290
What kind of curve
do the Greeks have?

547
00:33:32,290 --> 00:33:35,570
What kind of equipotential--
What kind of a curve have we

548
00:33:35,570 --> 00:33:36,730
got here?

549
00:33:36,730 --> 00:33:38,750
An ellipse.

550
00:33:38,750 --> 00:33:42,500
If this curve, if
that little matrix

551
00:33:42,500 --> 00:33:48,710
is indefinite, as,
for example, here.

552
00:33:48,710 --> 00:33:53,590
So with this one, what would be
the matrix, what's the matrix

553
00:33:53,590 --> 00:33:58,210
that goes with 2xy, if
I match this with this.

554
00:33:58,210 --> 00:33:59,300
That's the matrix.

555
00:33:59,300 --> 00:34:02,170
There's no a squareds,
there's no x squareds,

556
00:34:02,170 --> 00:34:04,350
there's no y squareds.

557
00:34:04,350 --> 00:34:08,700
And there are 2xy's, I
think the matrix is that.

558
00:34:08,700 --> 00:34:11,690
So it's this nice
symmetric matrix.

559
00:34:11,690 --> 00:34:14,760
Is that a positive
definite matrix?

560
00:34:14,760 --> 00:34:16,570
Certainly not.

561
00:34:16,570 --> 00:34:17,590
It's indefinite.

562
00:34:17,590 --> 00:34:20,380
It's eigenvalues are
plus one and minus one,

563
00:34:20,380 --> 00:34:22,550
adding to the trace zero.

564
00:34:22,550 --> 00:34:29,540
So indefinite matrices
correspond to hyperbolas.

565
00:34:29,540 --> 00:34:32,970
And later on,
definite matrices will

566
00:34:32,970 --> 00:34:36,170
correspond to elliptic partial
differential equations.

567
00:34:36,170 --> 00:34:39,330
And indefinite
matrices-- Like Laplace.

568
00:34:39,330 --> 00:34:41,950
And indefinite matrices
will correspond

569
00:34:41,950 --> 00:34:45,780
to hyperbolic partial
differential equations,

570
00:34:45,780 --> 00:34:48,180
like the wave equation.

571
00:34:48,180 --> 00:34:50,930
What's the-- Now we're here.

572
00:34:50,930 --> 00:34:54,840
I didn't expect to get here.

573
00:34:54,840 --> 00:34:57,740
What's the marginal case?

574
00:34:57,740 --> 00:35:01,550
What's the marginal case
between positive definite

575
00:35:01,550 --> 00:35:04,100
and indefinite is...?

576
00:35:04,100 --> 00:35:05,130
Semidefinite.

577
00:35:05,130 --> 00:35:05,860
Great.

578
00:35:05,860 --> 00:35:07,610
And what kind of a
curve do you think

579
00:35:07,610 --> 00:35:12,670
comes when this little
matrix is semidefinite.

580
00:35:12,670 --> 00:35:18,010
It's the one in between
ellipses and hyperbolas.

581
00:35:18,010 --> 00:35:23,551
The marginal guy is a parabola.

582
00:35:23,551 --> 00:35:24,050
Right.

583
00:35:24,050 --> 00:35:27,480
So semidefinite would
correspond to a parabola.

584
00:35:27,480 --> 00:35:28,010
Right.

585
00:35:28,010 --> 00:35:28,690
OK.

586
00:35:28,690 --> 00:35:29,190
Good.

587
00:35:29,190 --> 00:35:35,210
Anyway, all I was going
to say is, this u=2xy,

588
00:35:35,210 --> 00:35:37,040
that's our potential.

589
00:35:37,040 --> 00:35:43,530
If I draw equipotential
curves, they're hyperbolas.

590
00:35:43,530 --> 00:35:47,880
And now, what's the point about
these little arrows that I got

591
00:35:47,880 --> 00:35:53,790
started on.

592
00:35:53,790 --> 00:35:57,850
What was the very first point
about the answer to the meaning

593
00:35:57,850 --> 00:36:00,600
of the gradient was what?

594
00:36:00,600 --> 00:36:07,730
These are the gradients of u,
so those arrows point where?

595
00:36:07,730 --> 00:36:10,990
Perpendicular to the hyperbolas.

596
00:36:10,990 --> 00:36:12,950
Perpendicular to the hyperbola.

597
00:36:12,950 --> 00:36:19,300
We're trying to see the geometry
-- its beautiful geometry --

598
00:36:19,300 --> 00:36:21,760
behind the gradient.

599
00:36:21,760 --> 00:36:29,630
So if v is a gradient,
then it comes from some u.

600
00:36:29,630 --> 00:36:33,410
I can plot the u equal
constant, the equipotential,

601
00:36:33,410 --> 00:36:36,680
and then the gradients
will be perpendicular.

602
00:36:36,680 --> 00:36:42,130
So they really are
a little-- OK, good.

603
00:36:42,130 --> 00:36:45,870
OK, those are pieces
of information

604
00:36:45,870 --> 00:36:51,560
that you have, but
always need saying again.

605
00:36:51,560 --> 00:36:54,930
And to get the picture
in your mind-- I suppose,

606
00:36:54,930 --> 00:37:01,660
finally, I should choose a
v which is not a gradient.

607
00:37:01,660 --> 00:37:04,030
Just to finish.

608
00:37:04,030 --> 00:37:06,450
How shall I adjust that v?

609
00:37:06,450 --> 00:37:07,980
This v was a gradient.

610
00:37:07,980 --> 00:37:10,260
Can you just change
it a little bit --

611
00:37:10,260 --> 00:37:15,650
practically anything you
do will screw it up --

612
00:37:15,650 --> 00:37:18,030
to make it not a gradient?

613
00:37:18,030 --> 00:37:23,150
So I just changed this two,
what shall I change the two to?

614
00:37:23,150 --> 00:37:26,710
To three.

615
00:37:26,710 --> 00:37:28,830
That would totally foul it up.

616
00:37:28,830 --> 00:37:32,120
So that vector field, which
I could draw little pictures

617
00:37:32,120 --> 00:37:36,570
of, but there would be no
u that it's coming from.

618
00:37:36,570 --> 00:37:37,800
There would be no u.

619
00:37:37,800 --> 00:37:40,080
These little arrows would
not line up perpendicular

620
00:37:40,080 --> 00:37:45,630
to some beautiful curves.

621
00:37:45,630 --> 00:37:47,190
I don't get a u from that.

622
00:37:47,190 --> 00:37:50,870
Because the y derivative
of that is three,

623
00:37:50,870 --> 00:37:53,630
and it doesn't equal the
x derivative of that.

624
00:37:53,630 --> 00:37:54,950
So that's a no-good one.

625
00:37:54,950 --> 00:37:57,250
Let's go back to the good one.

626
00:37:57,250 --> 00:37:58,470
OK.

627
00:37:58,470 --> 00:38:00,400
OK, good.

628
00:38:00,400 --> 00:38:03,450
Is that OK for gradients?

629
00:38:03,450 --> 00:38:05,660
We got the meaning of gradients.

630
00:38:05,660 --> 00:38:09,620
They point perpendicular
to equipotentials.

631
00:38:09,620 --> 00:38:12,170
They tell how
steeply those-- They

632
00:38:12,170 --> 00:38:15,470
tell the separation between
the equipotentials, right.

633
00:38:15,470 --> 00:38:17,680
It's like, if you're
a mountain climber,

634
00:38:17,680 --> 00:38:21,190
you're looking at your
map, your contour map,

635
00:38:21,190 --> 00:38:23,020
and that's all I'm drawing here.

636
00:38:23,020 --> 00:38:26,160
I'm drawing a contour
map that every guy who

637
00:38:26,160 --> 00:38:29,030
goes climbing in New
Hampshire is going to have.

638
00:38:29,030 --> 00:38:33,880
And it shows little circles,
those are level heights, right?

639
00:38:33,880 --> 00:38:36,600
Those are level contours.

640
00:38:36,600 --> 00:38:41,700
And if you want to climb
as fast as possible,

641
00:38:41,700 --> 00:38:44,070
you go perpendicular
to those contours.

642
00:38:44,070 --> 00:38:46,290
And the distance
between contours

643
00:38:46,290 --> 00:38:48,950
tells you how steep it is.

644
00:38:48,950 --> 00:38:52,370
So it's all nice geometry.

645
00:38:52,370 --> 00:38:53,040
OK.

646
00:38:53,040 --> 00:38:57,900
I've got to get to
divergence here.

647
00:38:57,900 --> 00:39:02,390
Divergence.

648
00:39:02,390 --> 00:39:04,370
I should have said
though-- Damn.

649
00:39:04,370 --> 00:39:08,990
There's more to say
about gradients.

650
00:39:08,990 --> 00:39:17,170
That question of whether
[v 1, v 2], the vector field,

651
00:39:17,170 --> 00:39:18,510
is this question.

652
00:39:18,510 --> 00:39:21,970
The question of is it
the gradient of some u.

653
00:39:21,970 --> 00:39:24,300
So we now have a test.

654
00:39:24,300 --> 00:39:28,070
We now have a test.

655
00:39:28,070 --> 00:39:32,840
This is our test, right?

656
00:39:32,840 --> 00:39:34,320
That's our test.

657
00:39:34,320 --> 00:39:41,190
But I have to connect it
with Kirchhoff's voltage law.

658
00:39:41,190 --> 00:39:43,790
Do you remember, we
haven't talked so much

659
00:39:43,790 --> 00:39:45,680
about Kirchhoff's
voltage law, but I'm

660
00:39:45,680 --> 00:39:47,610
connecting it with
the discrete case,

661
00:39:47,610 --> 00:39:51,900
to add in a little more insight.

662
00:39:51,900 --> 00:39:56,170
What did Kirchhoff's
voltage law say?

663
00:39:56,170 --> 00:40:01,860
In that case, A was
a difference matrix.

664
00:40:01,860 --> 00:40:05,070
It was the incidence
matrix for our graph.

665
00:40:05,070 --> 00:40:12,550
And the question was-- I
have to take two moments

666
00:40:12,550 --> 00:40:14,270
to think about that.

667
00:40:14,270 --> 00:40:20,090
So Kirchhoff's voltage
law, for a graph.

668
00:40:20,090 --> 00:40:22,380
A is an incidence matrix.

669
00:40:22,380 --> 00:40:30,020
You know, the minus one, one
guys, one for every edge?

670
00:40:30,020 --> 00:40:38,130
And let me call
it v again, or e.

671
00:40:38,130 --> 00:40:40,630
I called it e at that time.

672
00:40:40,630 --> 00:40:44,820
Let's just look at an x.

673
00:40:44,820 --> 00:40:45,320
Right.

674
00:40:45,320 --> 00:40:58,920
A is this long thin matrix,
times-- sorry, u's. u's.

675
00:40:58,920 --> 00:41:04,800
Let me say it all at once.

676
00:41:04,800 --> 00:41:07,950
Which vectors have the form Au?

677
00:41:07,950 --> 00:41:13,500
Which vectors are combinations
of the columns of A?

678
00:41:13,500 --> 00:41:18,980
The test is,
Kirchhoff's voltage law,

679
00:41:18,980 --> 00:41:23,220
that if I go around any
loop in the graph --

680
00:41:23,220 --> 00:41:31,060
so if I have a u_1 here, u_2
here, u_5 here, and u_7 here --

681
00:41:31,060 --> 00:41:35,900
then Au will produce
u_1-u_7 on that edge.

682
00:41:35,900 --> 00:41:39,030
It'll produce a
u_2-u_1 on that edge.

683
00:41:39,030 --> 00:41:42,210
It'll produce a
u_5-u_2 on that edge,

684
00:41:42,210 --> 00:41:45,590
if the edges are
all going that way.

685
00:41:45,590 --> 00:41:49,530
And it'll produce a
u_7-u_5 on that edge.

686
00:41:49,530 --> 00:41:56,460
So I've got four components
of Au, four differences.

687
00:41:56,460 --> 00:41:59,140
And what does Kirchhoff's
voltage law tell me

688
00:41:59,140 --> 00:42:00,860
about those four differences?

689
00:42:00,860 --> 00:42:04,730
Which I can certainly
see directly.

690
00:42:04,730 --> 00:42:12,390
Those four differences, u_1-u_7,
u_2-u_1, u_5-u_2 and u_7-u_5.

691
00:42:12,390 --> 00:42:16,930
What's the obvious fact
about those four guys?

692
00:42:16,930 --> 00:42:21,530
They add to zero.

693
00:42:21,530 --> 00:42:27,560
The total drop around
a loop is zero.

694
00:42:27,560 --> 00:42:29,980
You see, if I cancel
those, if I add

695
00:42:29,980 --> 00:42:33,880
them, the u_1's cancel, the
u_2's cancel, the u_5's cancel,

696
00:42:33,880 --> 00:42:34,980
the u_7's cancel.

697
00:42:34,980 --> 00:42:35,951
We know this.

698
00:42:35,951 --> 00:42:36,450
OK.

699
00:42:36,450 --> 00:42:38,830
So that's Kirchhoff's
voltage law.

700
00:42:38,830 --> 00:42:46,150
It's got to have
a continuous form.

701
00:42:46,150 --> 00:42:53,910
This tells me, this is
the test on v at a point.

702
00:42:53,910 --> 00:42:57,690
What's the test on
v around a loop?

703
00:42:57,690 --> 00:43:01,110
I just want to connect
that-- I have to connect that

704
00:43:01,110 --> 00:43:06,900
to a second test.

705
00:43:06,900 --> 00:43:10,750
I'll just mention it, and
you'll find it in the book.

706
00:43:10,750 --> 00:43:12,990
That's the pointwise test.

707
00:43:12,990 --> 00:43:14,760
That was the easy test.

708
00:43:14,760 --> 00:43:17,880
We applied it to this and
we got the answer yes.

709
00:43:17,880 --> 00:43:21,160
If it was 3y, 2x, we
got the answer no.

710
00:43:21,160 --> 00:43:26,070
Now let me give you
a test that looks

711
00:43:26,070 --> 00:43:28,160
like Kirchhoff's voltage law.

712
00:43:28,160 --> 00:43:32,170
So I'm going to integrate
around a closed loop.

713
00:43:32,170 --> 00:43:33,480
What am I going to integrate?

714
00:43:33,480 --> 00:43:38,970
I think I integrate v--
Oh boy, I'd better look.

715
00:43:38,970 --> 00:43:41,251
It's easy to get these wrong.

716
00:43:41,251 --> 00:43:41,750
Yeah.

717
00:43:41,750 --> 00:43:45,080
So I would call
this the vorticity.

718
00:43:45,080 --> 00:43:47,110
And then I would
say the vorticity

719
00:43:47,110 --> 00:43:49,290
is zero for a gradient field.

720
00:43:49,290 --> 00:43:53,110
Now my integral guy is
going to be the circulation.

721
00:43:53,110 --> 00:43:53,960
Oh yeah.

722
00:43:53,960 --> 00:43:55,500
Because I'm following the path.

723
00:43:55,500 --> 00:44:03,550
So it's just v_1*dx+v_2*dy
should be zero.

724
00:44:03,550 --> 00:44:08,020
Around every closed loop --
that's idea of this thing,

725
00:44:08,020 --> 00:44:11,840
that it tells me the integral
goes around a closed loop --

726
00:44:11,840 --> 00:44:23,370
if I follow the velocity field,
the total circulation is zero.

727
00:44:23,370 --> 00:44:27,740
I put this up here as a
fact in vector calculus

728
00:44:27,740 --> 00:44:29,230
that's connected to that.

729
00:44:29,230 --> 00:44:32,050
These, one is zero
when the other is zero.

730
00:44:32,050 --> 00:44:34,160
There's a Stokes'
theorem that tells me

731
00:44:34,160 --> 00:44:38,480
that this integral is found
from a double integral of this.

732
00:44:38,480 --> 00:44:41,520
So if one is zero,
the other is zero.

733
00:44:41,520 --> 00:44:45,910
I'm just saying, here
is the natural analog

734
00:44:45,910 --> 00:44:48,110
of Kirchhoff's voltage law.

735
00:44:48,110 --> 00:44:49,310
OK.

736
00:44:49,310 --> 00:44:51,600
I had to say something
about voltage law,

737
00:44:51,600 --> 00:44:54,440
because for the
divergence, which I'm now

738
00:44:54,440 --> 00:44:59,910
going to get to-- Whatever.

739
00:44:59,910 --> 00:45:05,950
Let me ask about
divergence of w equal zero.

740
00:45:05,950 --> 00:45:10,030
What does that mean?

741
00:45:10,030 --> 00:45:14,270
That's going to be the
equivalent of whose law?

742
00:45:14,270 --> 00:45:16,790
Please tell me.

743
00:45:16,790 --> 00:45:20,200
Which law is going to be the
equivalent of-- Divergence

744
00:45:20,200 --> 00:45:24,460
of w equal zero is going
to mean there's no source.

745
00:45:24,460 --> 00:45:27,970
Whatever goes in, comes out.

746
00:45:27,970 --> 00:45:30,500
Whose law is that?

747
00:45:30,500 --> 00:45:31,470
That's Kirchhoff again.

748
00:45:31,470 --> 00:45:33,350
Well, yeah, other
people in physics.

749
00:45:33,350 --> 00:45:33,850
Right.

750
00:45:33,850 --> 00:45:38,660
But in our little world,
it's the other Kirchhoff law.

751
00:45:38,660 --> 00:45:41,750
It's Kirchhoff's current law.

752
00:45:41,750 --> 00:45:46,740
It's the one, it's the
A transpose, right?

753
00:45:46,740 --> 00:45:49,620
This is what we're thinking of
as A transpose w equal zero.

754
00:45:49,620 --> 00:45:55,670
Kirchhoff's current
law, in equals out.

755
00:45:55,670 --> 00:46:00,190
How will I translate that
in equal out for functions?

756
00:46:00,190 --> 00:46:05,240
Now I don't have--
On a graph, I just

757
00:46:05,240 --> 00:46:09,530
had the total flow, the
net flow at every node.

758
00:46:09,530 --> 00:46:12,990
Notice the divergence
is at every node.

759
00:46:12,990 --> 00:46:17,640
The circulation was
around every loop.

760
00:46:17,640 --> 00:46:18,370
OK.

761
00:46:18,370 --> 00:46:21,720
So in equals out was just
the sum of four things.

762
00:46:21,720 --> 00:46:25,660
OK, here I'm going
to have in equal

763
00:46:25,660 --> 00:46:31,250
out-- How am I going to
express in equal out?

764
00:46:31,250 --> 00:46:33,220
Divergence of w equals 0.

765
00:46:33,220 --> 00:46:37,380
Yeah, what I need is
the divergence theorem.

766
00:46:37,380 --> 00:46:41,830
Let's just face it,
we've got to have that.

767
00:46:41,830 --> 00:46:45,240
So I have a region here.

768
00:46:45,240 --> 00:46:52,690
I have a w everywhere,
w, [w 1, w 2].

769
00:46:52,690 --> 00:46:54,060
Then the divergence theorem.

770
00:46:54,060 --> 00:46:58,150
This is the great
identity, which

771
00:46:58,150 --> 00:47:00,851
of course has a discrete form.

772
00:47:00,851 --> 00:47:01,350
OK.

773
00:47:01,350 --> 00:47:06,880
The divergence theorem says that
if I integrate over the region,

774
00:47:06,880 --> 00:47:12,280
over this region
R, the divergence,

775
00:47:12,280 --> 00:47:17,120
that's (dw_1/dx+dw_2/dy), dxdy.

776
00:47:17,120 --> 00:47:23,780


777
00:47:23,780 --> 00:47:26,545
So that's like
telling me the source,

778
00:47:26,545 --> 00:47:30,840
I'm integrating over the
source at every point.

779
00:47:30,840 --> 00:47:38,460
At every point here, this
measures in minus out.

780
00:47:38,460 --> 00:47:44,480
But now, when I put the whole
thing together by integrating,

781
00:47:44,480 --> 00:47:47,040
what's the right-hand
side of this equation?

782
00:47:47,040 --> 00:47:49,800
Do you know the
divergence theorem?

783
00:47:49,800 --> 00:47:55,390
And let's remember it
and see why it's so.

784
00:47:55,390 --> 00:48:01,660
What I'm doing is in equals out
for the whole region at once.

785
00:48:01,660 --> 00:48:02,160
Right?

786
00:48:02,160 --> 00:48:05,830
When I-- This is like
in equal out at a point.

787
00:48:05,830 --> 00:48:08,190
But now I'm putting all
the points together.

788
00:48:08,190 --> 00:48:15,090
So the only way out will be
out through the boundary.

789
00:48:15,090 --> 00:48:19,450
And so I'll need to
say how much flows out.

790
00:48:19,450 --> 00:48:25,180
This is the total source, the
total in equal out inside.

791
00:48:25,180 --> 00:48:28,370
The only way to get out
is through the boundary.

792
00:48:28,370 --> 00:48:35,650
So this is the integral
around the boundary of-- So

793
00:48:35,650 --> 00:48:39,730
what's the flow out?

794
00:48:39,730 --> 00:48:45,610
It's, yeah, it's somehow--
Think now, what should go there?

795
00:48:45,610 --> 00:48:49,850
This is flux I'm talking about.

796
00:48:49,850 --> 00:48:53,890
Flux is short word for
the total flow out.

797
00:48:53,890 --> 00:48:57,260
OK.

798
00:48:57,260 --> 00:49:02,830
So now I've got
to get this right.

799
00:49:02,830 --> 00:49:10,510
In vector notation, it would
be-- w tells me the flow.

800
00:49:10,510 --> 00:49:16,210
But flow outwards, see,
suppose w points that way.

801
00:49:16,210 --> 00:49:20,080
Then the actual flow
out is not all that.

802
00:49:20,080 --> 00:49:22,690
Because a lot of that
is just going sideways.

803
00:49:22,690 --> 00:49:24,230
It's this part.

804
00:49:24,230 --> 00:49:26,810
It's the flow perpendicular
to the boundary.

805
00:49:26,810 --> 00:49:33,130
So it's w dot n, the
normal component of flow.

806
00:49:33,130 --> 00:49:42,070
And I integrate that
around the boundary.

807
00:49:42,070 --> 00:49:46,190
There you have a
key, key theorem.

808
00:49:46,190 --> 00:49:51,080
In 2-D. And it's an
equation for the flux.

809
00:49:51,080 --> 00:49:53,370
It's like the fundamental
theorem of calculus,

810
00:49:53,370 --> 00:49:58,730
but now we're in two dimensions.

811
00:49:58,730 --> 00:50:00,260
And this is what it looks like.

812
00:50:00,260 --> 00:50:05,090
OK, so I'm obviously not going
to finish with the divergence

813
00:50:05,090 --> 00:50:07,750
theorem today.

814
00:50:07,750 --> 00:50:10,090
So what's the conclusion?

815
00:50:10,090 --> 00:50:15,200
If the divergence
is zero, then what?

816
00:50:15,200 --> 00:50:25,500
If the divergence is zero, if
this is zero at every point,

817
00:50:25,500 --> 00:50:35,390
then this is zero
across every loop.

818
00:50:35,390 --> 00:50:38,990
Can I call this thing a loop?

819
00:50:38,990 --> 00:50:42,500
That closed loop.

820
00:50:42,500 --> 00:50:49,860
That's the conclusion
that we want to reach.

821
00:50:49,860 --> 00:50:52,700
So this is the
divergence theorem.

822
00:50:52,700 --> 00:50:59,660
The text gives a proof,
not to repeat in class,

823
00:50:59,660 --> 00:51:03,680
but it's a crucial
formula to know.

824
00:51:03,680 --> 00:51:08,700
That the integral of the
divergence is the flux.

825
00:51:08,700 --> 00:51:09,450
OK.

826
00:51:09,450 --> 00:51:11,040
Let's come back
to that Wednesday,

827
00:51:11,040 --> 00:51:13,160
and I'll have lots
of homework for you.

828
00:51:13,160 --> 00:51:17,577
Thanks for turning
in these today.

829
00:51:17,577 --> 00:51:18,077