1
00:00:00 --> 00:00:01
2
00:00:01 --> 00:00:02
The following content is
provided under a Creative
3
00:00:02 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to
5
00:00:06 --> 00:00:09
offer high-quality educational
resources for free.
6
00:00:09 --> 00:00:13
To make a donation, or to view
additional materials from
7
00:00:13 --> 00:00:16
hundreds of MIT courses, visit
MIT OpenCourseWare
8
00:00:16 --> 00:00:19
at ocw.mit.edu.
9
00:00:19 --> 00:00:21
PROFESSOR STRANG: OK.
10
00:00:21 --> 00:00:26
So this is lecture 22, gradient
and divergence, headed
11
00:00:26 --> 00:00:28
for Laplace's equation.
12
00:00:28 --> 00:00:33
So the gradient will be our
operator A, the divergence, or
13
00:00:33 --> 00:00:37
minus the divergence, will be A
transpose, and then A transpose
14
00:00:37 --> 00:00:40
A will be the Laplacian.
15
00:00:40 --> 00:00:43
We get to Laplace's
equation Wednesday.
16
00:00:43 --> 00:00:46
Today I wanted to take
them separately.
17
00:00:46 --> 00:00:50
To understand the meaning
of gradient, the meaning
18
00:00:50 --> 00:00:56
of divergence, the
connection between them.
19
00:00:56 --> 00:00:58
I mentioned at the end of
last time that one is the
20
00:00:58 --> 00:01:03
transpose of the other,
or minus the transpose.
21
00:01:03 --> 00:01:07
I'll try to keep gradient on
this side, and if I could only
22
00:01:07 --> 00:01:12
transpose the blackboard, I
could do divergence -- I'll
23
00:01:12 --> 00:01:15
do divergence on that side.
24
00:01:15 --> 00:01:18
And I guess if I could get a
rotating blackboard, right in
25
00:01:18 --> 00:01:21
the middle I could do curl.
26
00:01:21 --> 00:01:22
That would be perfect.
27
00:01:22 --> 00:01:23
[LAUGHTER]
28
00:01:23 --> 00:01:23
OK.
29
00:01:23 --> 00:01:27
So some of this will
not be new to you.
30
00:01:27 --> 00:01:30
But maybe some of the insights
or the ways of looking
31
00:01:30 --> 00:01:32
at it could be new.
32
00:01:32 --> 00:01:35
This is the background
of vector calculus.
33
00:01:35 --> 00:01:38
So we have things
like vector fields.
34
00:01:38 --> 00:01:43
That means I have a vector,
, at each
35
00:01:43 --> 00:01:44
point, (x, y).
36
00:01:44 --> 00:01:49
So I could draw a little arrow
at every point to show the
37
00:01:49 --> 00:01:53
direction and magnitude
of that vector.
38
00:01:53 --> 00:01:55
I have a field of vectors.
39
00:01:55 --> 00:01:58
OK, so.
40
00:01:58 --> 00:02:03
From last time, our basic
setup is, the gradient is
41
00:02:03 --> 00:02:06
this first operator, A.
42
00:02:06 --> 00:02:10
The one we see at
the beginning.
43
00:02:10 --> 00:02:13
One change.
44
00:02:13 --> 00:02:20
Instead of calling the result
e, let me connect to velocity.
45
00:02:20 --> 00:02:23
I'll be thinking of u as a
potential, I'll use the word
46
00:02:23 --> 00:02:28
potential for u, and I'll use
v instead of e for the
47
00:02:28 --> 00:02:29
velocity component.
48
00:02:29 --> 00:02:31
So that's the A.
49
00:02:31 --> 00:02:36
And then on the other
side, I start with a w.
50
00:02:36 --> 00:02:40
Again, it's a vector field,
it's actually a momentum.
51
00:02:40 --> 00:02:44
Very often the step between
here and here -- well most
52
00:02:44 --> 00:02:49
often, the step between here
and here will be the identity.
53
00:02:49 --> 00:02:51
That's what gives
Laplace's equation.
54
00:02:51 --> 00:02:56
So you'll have to watch,
I'm sometimes confusing
55
00:02:56 --> 00:02:57
the v's with the w's.
56
00:02:57 --> 00:03:01
Because when I go to Laplace's
equation and c is the
57
00:03:01 --> 00:03:03
identity, they're the same.
58
00:03:03 --> 00:03:07
But I would like, today, to
try to keep this left side
59
00:03:07 --> 00:03:12
separate, the gradient, from
the right side, the divergence.
60
00:03:12 --> 00:03:17
And understand what they mean,
and how to work with them.
61
00:03:17 --> 00:03:21
And of course, a big connection
is the divergence theorem,
62
00:03:21 --> 00:03:25
or the Gauss-Green
connection, identity.
63
00:03:25 --> 00:03:26
We'll get to that.
64
00:03:26 --> 00:03:29
OK, gradient first.
65
00:03:29 --> 00:03:31
First, what does it mean?
66
00:03:31 --> 00:03:35
If I have a function u, what
does it is gradient tell me?
67
00:03:35 --> 00:03:39
And then the second is kind
of a backwards question.
68
00:03:39 --> 00:03:41
Suppose I have the v.
69
00:03:41 --> 00:03:44
Is it the gradient of some u?
70
00:03:44 --> 00:03:49
So one direction, from u to v,
and the second direction will
71
00:03:49 --> 00:03:53
be from v back to
u when possible.
72
00:03:53 --> 00:03:56
OK, meaning of the gradient.
73
00:03:56 --> 00:04:00
So the gradient is just the
obvious thing, the two
74
00:04:00 --> 00:04:03
derivatives in the
x and y direction.
75
00:04:03 --> 00:04:08
So of course the gradient gives
you the rate of change, the
76
00:04:08 --> 00:04:10
partial derivatives of u.
77
00:04:10 --> 00:04:15
But how do you see
that in a picture?
78
00:04:15 --> 00:04:18
Let me draw an important curve.
79
00:04:18 --> 00:04:22
I'll start with a very
simple u. u is x
80
00:04:22 --> 00:04:23
squared plus y squared.
81
00:04:23 --> 00:04:27
So that's my example.
82
00:04:27 --> 00:04:28
Example one.
83
00:04:28 --> 00:04:32
And what I've drawn is
an equipotential curve.
84
00:04:32 --> 00:04:36
Or isopotential might be a more
appropriate word these days,
85
00:04:36 --> 00:04:39
but we still say equipotential.
86
00:04:39 --> 00:04:47
It means that, along this
curve, u is a constant.
87
00:04:47 --> 00:04:51
And for this particular
potential, this simple one to
88
00:04:51 --> 00:05:02
work with, x squared plus y
squared, the curve is a circle.
89
00:05:02 --> 00:05:04
So the curve would be a circle.
90
00:05:04 --> 00:05:08
OK, now what do I learn
by taking the gradient?
91
00:05:08 --> 00:05:14
So the gradient of u, this is
my v, is the x derivative,
92
00:05:14 --> 00:05:19
which is 2x, and the y
derivative, which is 2y.
93
00:05:20 --> 00:05:23
OK.
94
00:05:23 --> 00:05:28
Let me take a typical point
on the curve and draw
95
00:05:28 --> 00:05:29
that gradient vector.
96
00:05:29 --> 00:05:36
So this is the x, y plane, this
is the curve u equal constant.
97
00:05:36 --> 00:05:40
When I have a curve u equal
constant and I draw the
98
00:05:40 --> 00:05:43
gradient of u, where
does it point?
99
00:05:43 --> 00:05:48
This is the first and most
important, simple idea
100
00:05:48 --> 00:05:50
about the gradient vector.
101
00:05:50 --> 00:05:56
The gradient vector points --
does the gradient vector point,
102
00:05:56 --> 00:05:59
could it point any old way?
103
00:05:59 --> 00:06:00
No.
104
00:06:00 --> 00:06:04
The gradient vector is
perpendicular to the curve.
105
00:06:04 --> 00:06:08
And we can see that, for this
simple example, that vector
106
00:06:08 --> 00:06:13
, that's a vector
radially outwards, right,
107
00:06:13 --> 00:06:14
if here's the origin.
108
00:06:14 --> 00:06:18
And if, at this point, I don't
know its coordinates, whatever
109
00:06:18 --> 00:06:24
they are, maybe 2, 1 or
something, the gradient
110
00:06:24 --> 00:06:25
vector would be .
111
00:06:25 --> 00:06:28
It would be a multiple
-- here's the position
112
00:06:28 --> 00:06:29
vector, .
113
00:06:29 --> 00:06:33
The point is, the gradient
vector points out.
114
00:06:33 --> 00:06:36
Perpendicular to the curve.
115
00:06:36 --> 00:06:38
That's what the
gradient tells you.
116
00:06:38 --> 00:06:48
It tells you, in this situation
it's telling me which direction
117
00:06:48 --> 00:06:50
is perpendicular to the curve.
118
00:06:50 --> 00:06:52
Now how do I understand that?
119
00:06:52 --> 00:06:54
How do I see that?
120
00:06:54 --> 00:06:59
I think of this -- and
let me try to draw the
121
00:06:59 --> 00:07:01
whole surface, u=x+y^2.
122
00:07:03 --> 00:07:07
What would that whole
surface look like?
123
00:07:07 --> 00:07:12
Let me try to put that
picture -- this is a
124
00:07:12 --> 00:07:14
2-D picture right now.
125
00:07:14 --> 00:07:16
It's sort of a cross-section.
126
00:07:16 --> 00:07:20
Let me put this 2-D picture
into a 3-D picture to see.
127
00:07:20 --> 00:07:24
So the 3-D picture is a picture
of the whole surface. u going
128
00:07:24 --> 00:07:27
up, x and y going around.
129
00:07:27 --> 00:07:31
And what kind of a curve,
what kind of a surface do I
130
00:07:31 --> 00:07:37
have for that function, x
squared plus y squared.
131
00:07:37 --> 00:07:40
Well, it grows, right?
132
00:07:40 --> 00:07:44
It's sort of, I use
the word bowl.
133
00:07:44 --> 00:07:48
And we've seen it before.
134
00:07:48 --> 00:07:53
It sort of goes up, right, and
it goes up faster and faster as
135
00:07:53 --> 00:07:56
I go out, because
of the squares.
136
00:07:56 --> 00:07:59
And now what I want to do
is, suppose I take the
137
00:07:59 --> 00:08:02
cross-section, suppose
I take this bowl.
138
00:08:02 --> 00:08:07
Do you see a beautiful
bowl there?
139
00:08:07 --> 00:08:13
And now, I cut through it,
horizontal cross-section,
140
00:08:13 --> 00:08:15
at the height c.
141
00:08:15 --> 00:08:21
What do I get if I take this
surface in 3-D, and I cut
142
00:08:21 --> 00:08:31
through it by a plane, the
plane at that height, c.
143
00:08:31 --> 00:08:37
So if the plane went through
there, that distance was c.
144
00:08:37 --> 00:08:40
It cuts us a cross-section
out of the bowl.
145
00:08:40 --> 00:08:43
And what's the cross section?
146
00:08:43 --> 00:08:44
It's the circle.
147
00:08:44 --> 00:08:45
It's this one.
148
00:08:45 --> 00:08:53
So out of the bowl, this plane
is cutting this cross-section.
149
00:08:53 --> 00:08:56
This is height u
equal constant.
150
00:08:56 --> 00:09:00
So I'm really intersecting one
surface, the plane u equal
151
00:09:00 --> 00:09:02
constant, with the bowl.
152
00:09:02 --> 00:09:07
And the intersection of those
two is the equipotential,
153
00:09:07 --> 00:09:09
the curve, u=c.
154
00:09:11 --> 00:09:12
Right?
155
00:09:12 --> 00:09:14
You see the two pictures?
156
00:09:14 --> 00:09:18
And now, what does the gradient
tell me in that picture?
157
00:09:18 --> 00:09:22
What does the gradient of u
tell me in that picture?
158
00:09:22 --> 00:09:26
Over here, it pointed
outwards, but now we're
159
00:09:26 --> 00:09:28
going to kind of see why.
160
00:09:28 --> 00:09:35
So, it pointed outwards for
this nice function, u.
161
00:09:35 --> 00:09:40
I could have made a more
general function u, but let me
162
00:09:40 --> 00:09:41
just stay with this simple one.
163
00:09:41 --> 00:09:47
So suppose I'm climbing.
164
00:09:47 --> 00:09:52
This is like, I'm climbing out
of a volcano or something.
165
00:09:52 --> 00:09:54
Usually, I would say a
mountain, but the way I've
166
00:09:54 --> 00:09:57
drawn the thing, it's
not much of a mountain.
167
00:09:57 --> 00:09:59
So OK, volcano.
168
00:09:59 --> 00:10:01
Climbing out of it.
169
00:10:01 --> 00:10:06
And I got up to this point,
which was this point.
170
00:10:06 --> 00:10:09
Now, what does the
gradient tell me?
171
00:10:09 --> 00:10:11
That when I'm climbing away,
I reach -- duh duh duh
172
00:10:11 --> 00:10:13
duh -- I get up to here.
173
00:10:13 --> 00:10:15
What does the gradient tell me?
174
00:10:15 --> 00:10:17
It tells me which way to go.
175
00:10:17 --> 00:10:21
It tells me the steepest
direction upwards, and
176
00:10:21 --> 00:10:23
it tells me how steep.
177
00:10:23 --> 00:10:23
Right.
178
00:10:23 --> 00:10:26
So it tells me those
two things, direction
179
00:10:26 --> 00:10:27
and magnitude.
180
00:10:27 --> 00:10:34
Direction is, well for this
special function you know the
181
00:10:34 --> 00:10:37
direction is, like, straight
outwards, straight upwards.
182
00:10:37 --> 00:10:39
And how steep is it?
183
00:10:39 --> 00:10:45
What's the steepness?
184
00:10:45 --> 00:10:49
It's the size of, it's
the magnitude of
185
00:10:49 --> 00:10:51
this vector, grad u.
186
00:10:51 --> 00:11:02
Which is the-- this is not
-- how big is this vector?
187
00:11:02 --> 00:11:04
This is my vector, grad u.
188
00:11:04 --> 00:11:09
This is another shorthand
notation for gradient.
189
00:11:09 --> 00:11:17
The size of that vector is
the square root of, what?
190
00:11:17 --> 00:11:19
The length of a vector is
the square root of the
191
00:11:19 --> 00:11:21
sum of the squares.
192
00:11:21 --> 00:11:24
So I have the length of v is
the square root of 4x^2 + 4y^2.
193
00:11:24 --> 00:11:31
194
00:11:31 --> 00:11:33
OK.
195
00:11:33 --> 00:11:37
Now, it gets steeper and
steeper, of course.
196
00:11:37 --> 00:11:45
The cross sections here
would be all circles for
197
00:11:45 --> 00:11:46
this simple function.
198
00:11:46 --> 00:11:50
And the gradients would keep
pointing out, and we'd have a
199
00:11:50 --> 00:11:58
function that comes from
a surface like that.
200
00:11:58 --> 00:12:02
OK, this is what I wanted to
say about question one, the
201
00:12:02 --> 00:12:06
meaning of the gradient.
202
00:12:06 --> 00:12:10
The thing to remember, if you
remember the two pictures,
203
00:12:10 --> 00:12:13
you've really got the idea of
what the gradient
204
00:12:13 --> 00:12:14
is telling you.
205
00:12:14 --> 00:12:17
For a function of one
variable, the derivative
206
00:12:17 --> 00:12:19
is telling you the slope.
207
00:12:19 --> 00:12:23
Well, we've got slopes in the
x direction, slopes in the y
208
00:12:23 --> 00:12:27
direction, and the gradient
direction tells us
209
00:12:27 --> 00:12:29
the steepest slope.
210
00:12:29 --> 00:12:34
And it tells us -- yeah,
tells us what that slope is.
211
00:12:34 --> 00:12:38
OK, so much for the
meaning of the gradient.
212
00:12:38 --> 00:12:40
Now, backwards.
213
00:12:40 --> 00:12:42
Suppose I have v.
214
00:12:42 --> 00:12:46
Because these gradient fields
are extremely important.
215
00:12:46 --> 00:12:50
I mean, they're
wonderful fields.
216
00:12:50 --> 00:12:55
If you've got a v_1 and a v_2
that is the gradient of some
217
00:12:55 --> 00:12:59
u, you want to know that.
218
00:12:59 --> 00:13:02
I mean, that's good news.
219
00:13:02 --> 00:13:03
Most fields will not.
220
00:13:03 --> 00:13:10
So let's figure out, when is --
can I life that up now, and do
221
00:13:10 --> 00:13:12
the other gradient thing now?
222
00:13:12 --> 00:13:22
Or maybe I'll put the final
gradient blackboard here, and
223
00:13:22 --> 00:13:25
then make way for divergence.
224
00:13:25 --> 00:13:29
OK, so I'm asking
now question two.
225
00:13:29 --> 00:13:37
I'm given v_1 and v_2, and I
want this to be du/dx for some
226
00:13:37 --> 00:13:39
u, and I want this to be du/dy.
227
00:13:39 --> 00:13:42
228
00:13:42 --> 00:13:45
So I've got two equations.
229
00:13:45 --> 00:13:47
If I'm looking for u.
230
00:13:47 --> 00:13:52
Remember, I'm now starting with
these, looking for the u.
231
00:13:52 --> 00:13:53
OK.
232
00:13:53 --> 00:13:54
So, am I going to find Au?
233
00:13:56 --> 00:14:01
Am I going to find a function,
u, whose x derivative is my v_1
234
00:14:01 --> 00:14:06
that was given, and the
y derivative is my v_2?
235
00:14:07 --> 00:14:09
Well, chances are not good.
236
00:14:09 --> 00:14:10
Right?
237
00:14:10 --> 00:14:13
I've got two equations here,
but only one unknown.
238
00:14:13 --> 00:14:17
So generally, it's like
a rectangular system.
239
00:14:17 --> 00:14:21
Usually there's no solution,
but sometimes there is.
240
00:14:21 --> 00:14:29
So we have to find out, what's
the test for consistency for
241
00:14:29 --> 00:14:31
these to have a solution?
242
00:14:31 --> 00:14:40
And this brings us right away
to the key identity from
243
00:14:40 --> 00:14:42
partial derivatives.
244
00:14:42 --> 00:14:43
OK.
245
00:14:43 --> 00:14:48
Can you see some condition that
has to-- how can I connect
246
00:14:48 --> 00:14:52
these two equations, and
therefore connect v_1 and v_2?
247
00:14:53 --> 00:14:59
That'll be the test for weather
-- you could say, in matrix
248
00:14:59 --> 00:15:02
language, I'm asking whether
is in the column
249
00:15:02 --> 00:15:06
space of the gradient.
250
00:15:06 --> 00:15:10
Is it something I can get
from this tall thin matrix?
251
00:15:10 --> 00:15:16
OK, they key idea is take
the y derivative of that
252
00:15:16 --> 00:15:22
equation, d second u/dydx.
253
00:15:24 --> 00:15:27
And take the x derivative
of this equation. dv_2/dy.
254
00:15:28 --> 00:15:36
dx is the second derivative of
u, respect to the x derivative
255
00:15:36 --> 00:15:38
of the y derivative.
256
00:15:38 --> 00:15:39
OK.
257
00:15:39 --> 00:15:45
So it's like I'm operating
on the equations.
258
00:15:45 --> 00:15:48
I'm doing what I'm allowed to
do, I'm doing the same thing to
259
00:15:48 --> 00:15:50
both sides of each equation.
260
00:15:50 --> 00:15:52
Now I'm going to eliminate.
261
00:15:52 --> 00:15:54
And what am I going to find?
262
00:15:54 --> 00:15:56
What's the key point here?
263
00:15:56 --> 00:16:00
The key, key point about
second derivatives is
264
00:16:00 --> 00:16:02
that that equals that.
265
00:16:02 --> 00:16:04
Right?
266
00:16:04 --> 00:16:06
Take any function.
267
00:16:06 --> 00:16:09
Do we want to practice
with a function, just
268
00:16:09 --> 00:16:10
to see it be true?
269
00:16:10 --> 00:16:14
Take, let u be x cubed y.
270
00:16:14 --> 00:16:16
Just for the heck of it.
271
00:16:16 --> 00:16:18
OK, I don't know what --
is that going to produce
272
00:16:18 --> 00:16:19
anything interesting?
273
00:16:19 --> 00:16:22
Then maybe I'd better
make it y squared.
274
00:16:22 --> 00:16:25
I don't know why I'm doing
this, it just is like an
275
00:16:25 --> 00:16:27
example, to make it believable.
276
00:16:27 --> 00:16:34
OK, so du/dx, u_x is -- and
I'll often use u_x as a sort
277
00:16:34 --> 00:16:36
of shorthand -- will be what?
278
00:16:36 --> 00:16:39
3x squared y squared.
279
00:16:39 --> 00:16:44
And u_y, taking the y
derivative is just, x is
280
00:16:44 --> 00:16:47
constant, 2x cubed y.
281
00:16:47 --> 00:16:53
And now let me do u_xy,
the y derivative of this.
282
00:16:53 --> 00:16:56
Which is what?
283
00:16:56 --> 00:16:59
It's been a long weekend,
but hey, we can do these.
284
00:16:59 --> 00:17:05
The y derivative of that is
going to be 6x squared y.
285
00:17:05 --> 00:17:11
And the x derivative of this is
going to be, so I should say y
286
00:17:11 --> 00:17:16
-- I've got the y derivative
and now I should take the
287
00:17:16 --> 00:17:17
x derivative of that.
288
00:17:17 --> 00:17:19
And what do I get?
289
00:17:19 --> 00:17:25
The x derivative of
that is 6x squared y.
290
00:17:25 --> 00:17:25
And look!
291
00:17:25 --> 00:17:27
They're the same.
292
00:17:27 --> 00:17:28
Hooray.
293
00:17:28 --> 00:17:29
OK.
294
00:17:29 --> 00:17:36
So if the function is smooth
and has these derivatives,
295
00:17:36 --> 00:17:39
they'll come out the same.
296
00:17:39 --> 00:17:42
And therefore, so
what do I conclude?
297
00:17:42 --> 00:17:49
What's the test on v_1 and
v_2 that it must pass
298
00:17:49 --> 00:17:52
to be a gradient field?
299
00:17:52 --> 00:17:56
For there to be a function, u,
that solves these equations.
300
00:17:56 --> 00:18:01
These are solvable
only when what?
301
00:18:01 --> 00:18:03
Well, if these are the same,
these have to be the same.
302
00:18:03 --> 00:18:16
So it's solvable only when
-- I need dv_2/dx-dv_1/dy
303
00:18:16 --> 00:18:19
to be what?
304
00:18:19 --> 00:18:20
Zero.
305
00:18:20 --> 00:18:21
OK.
306
00:18:21 --> 00:18:23
That's the conclusion.
307
00:18:23 --> 00:18:28
Those have to be the same.
dv_2/dx-dv_1/dy has to be zero.
308
00:18:28 --> 00:18:31
OK.
309
00:18:31 --> 00:18:35
Because those are the same.
310
00:18:35 --> 00:18:40
I guess I came to this early
because it's the key identity
311
00:18:40 --> 00:18:43
of vector calculus.
312
00:18:43 --> 00:18:46
Well, the key identity behind
vector calculus is this
313
00:18:46 --> 00:18:51
fact about the derivatives.
314
00:18:51 --> 00:18:58
Can I just throw out a question
that you might think about?
315
00:18:58 --> 00:19:01
We already have seen, so
often now, the second
316
00:19:01 --> 00:19:04
derivative, like v_xx.
317
00:19:05 --> 00:19:07
Or u_xx, suppose.
318
00:19:07 --> 00:19:09
So here's a little
question to think about.
319
00:19:09 --> 00:19:13
So think.
320
00:19:13 --> 00:19:17
OK.
321
00:19:17 --> 00:19:23
I just want to bring finite
differences in for a moment.
322
00:19:23 --> 00:19:26
u_xx we've got a handle on.
323
00:19:26 --> 00:19:29
We know that that's
like u(x+h).
324
00:19:31 --> 00:19:36
And if there was a y, put
in the y, minus 2u at x.
325
00:19:36 --> 00:19:42
And I can put in a y
there, plus u(x-h,y).
326
00:19:42 --> 00:19:45
327
00:19:45 --> 00:19:47
We've seen that.
328
00:19:47 --> 00:19:52
That'll be the x derivative,
second x derivative -- sorry!
329
00:19:52 --> 00:19:56
Second x difference.
330
00:19:56 --> 00:20:00
Since we're taking x
derivatives and x differences,
331
00:20:00 --> 00:20:03
it's x that moves,
and y doesn't move.
332
00:20:03 --> 00:20:08
Just the central idea of
partial derivatives. u_yy will
333
00:20:08 --> 00:20:11
be similar with y moving.
334
00:20:11 --> 00:20:14
And my question to you is --
and we could have asked it way
335
00:20:14 --> 00:20:17
way back -- what's a finite
difference approximation
336
00:20:17 --> 00:20:23
to u(x,y), to the
cross derivative?
337
00:20:23 --> 00:20:27
That equals what?
338
00:20:27 --> 00:20:30
I want to go to
finite differences.
339
00:20:30 --> 00:20:39
OK, what finite differences?
340
00:20:39 --> 00:20:40
Which finite differences?
341
00:20:40 --> 00:20:43
Maybe that's a question
to think about.
342
00:20:43 --> 00:20:50
If I can remember, I'll include
it just as a small homework
343
00:20:50 --> 00:20:55
question for the homework
on this material.
344
00:20:55 --> 00:20:57
So that's looking
ahead, really, today.
345
00:20:57 --> 00:21:01
We're not making
things discrete.
346
00:21:01 --> 00:21:05
We're in continuous x and y.
347
00:21:05 --> 00:21:07
We have vector fields.
348
00:21:07 --> 00:21:20
And we now know the test
for a gradient field.
349
00:21:20 --> 00:21:24
I'm tempted to use
the word curl here.
350
00:21:24 --> 00:21:28
I'm tempted to use
the word curl.
351
00:21:28 --> 00:21:32
I want to connect that test --
may I use the word curl without
352
00:21:32 --> 00:21:39
-- and I'll say why I'm not
going to do everything properly
353
00:21:39 --> 00:21:40
with curl right away.
354
00:21:40 --> 00:21:45
I would describe this as curl.
355
00:21:45 --> 00:21:51
The test is, the curl
of v has to be zero.
356
00:21:51 --> 00:22:01
So for me, that's
the curl of v.
357
00:22:01 --> 00:22:03
You're going to say,
wait a minute.
358
00:22:03 --> 00:22:06
I learned about curl,
and that doesn't look
359
00:22:06 --> 00:22:10
like the curl to me.
360
00:22:10 --> 00:22:14
So I'll say wait
another minute.
361
00:22:14 --> 00:22:16
It's not that far off.
362
00:22:16 --> 00:22:18
OK, what's your objection?
363
00:22:18 --> 00:22:22
Your objection is that the curl
is in three dimensional space.
364
00:22:22 --> 00:22:23
Right?
365
00:22:23 --> 00:22:26
When you saw curl, and of
course it comes in this
366
00:22:26 --> 00:22:33
section of the book, we
had functions of x, y, z.
367
00:22:33 --> 00:22:39
And the curl had
three components.
368
00:22:39 --> 00:22:42
And those three components
-- do you remember curl?
369
00:22:42 --> 00:22:44
I mean, if you remember
curl, you're a good person.
370
00:22:44 --> 00:22:49
Because it's got this -- you
sort of remember that it
371
00:22:49 --> 00:22:54
has things like this.
372
00:22:54 --> 00:22:55
Right?
373
00:22:55 --> 00:23:00
Sort of, differences of
derivatives, and the indices
374
00:23:00 --> 00:23:01
follow a certain pattern.
375
00:23:01 --> 00:23:04
I'm saying that this is
the natural -- this is
376
00:23:04 --> 00:23:08
the curl in a plane.
377
00:23:08 --> 00:23:13
What do I mean by the
curl in a plane?
378
00:23:13 --> 00:23:22
So in 3-D, v has components
of v_1, v_2, and v_3.
379
00:23:23 --> 00:23:23
Right?
380
00:23:23 --> 00:23:27
That depend on x, y, z.
381
00:23:27 --> 00:23:33
In the plane, in 3-D, I have
a vector field, v, which has
382
00:23:33 --> 00:23:38
components v_1, v_2, v_3.
383
00:23:38 --> 00:23:39
All depending on x, y, z.
384
00:23:39 --> 00:23:45
That's the general 3-D picture,
where you usually see the curl.
385
00:23:45 --> 00:23:48
Now, in the plane,
what's happening?
386
00:23:48 --> 00:23:52
In the plane, we have
two components.
387
00:23:52 --> 00:23:56
So what's happening?
388
00:23:56 --> 00:23:58
Think of this.
389
00:23:58 --> 00:24:02
So in 3-D, our velocity
field is like, the flow is
390
00:24:02 --> 00:24:03
going all over the place.
391
00:24:03 --> 00:24:04
Right?
392
00:24:04 --> 00:24:06
It's a three dimensional flow.
393
00:24:06 --> 00:24:11
But now suppose my flow
stays in the plane.
394
00:24:11 --> 00:24:18
So in 2-D -- so now I have
to put 2-D up above here.
395
00:24:18 --> 00:24:26
So in 2-D, a plane field is
what I'm working with today.
396
00:24:26 --> 00:24:29
My v is some v_1(x,y).
397
00:24:29 --> 00:24:31
398
00:24:31 --> 00:24:39
No z, no dependence on z.
v_2(x,y), the y direction of
399
00:24:39 --> 00:24:42
the velocity doesn't depend on
z, because this is a plane
400
00:24:42 --> 00:24:44
field, same on every plane.
401
00:24:44 --> 00:24:47
And the third component of this
plane field, the velocity
402
00:24:47 --> 00:24:52
perpendicular in the z
direction, is zero.
403
00:24:52 --> 00:24:53
OK. one.
404
00:24:53 --> 00:25:01
So what I want to say is that
if I look, if I specialize too
405
00:25:01 --> 00:25:05
plane fields, to fields
like these, then the only
406
00:25:05 --> 00:25:08
component of the curl that
survives is this one.
407
00:25:08 --> 00:25:11
See, the other components of
the curl -- which I'm not even
408
00:25:11 --> 00:25:15
writing down -- the other
components of a curl have
409
00:25:15 --> 00:25:19
derivatives of v_3,
but v_3 is zero.
410
00:25:19 --> 00:25:21
And they also have derivatives
of these guys with
411
00:25:21 --> 00:25:24
respect to z.
412
00:25:24 --> 00:25:26
But they don't depend on z.
413
00:25:26 --> 00:25:28
So that's why all the other
pieces of the curl, like,
414
00:25:28 --> 00:25:31
are automatically zero
for a plane field.
415
00:25:31 --> 00:25:36
So that the only component
that's significant, the test
416
00:25:36 --> 00:25:40
curl v equals zero, boils
down to a test not on three
417
00:25:40 --> 00:25:41
things, but just on one.
418
00:25:41 --> 00:25:44
And that's the one.
419
00:25:44 --> 00:25:51
So because I want to stay
mostly with plane fields and
420
00:25:51 --> 00:25:54
two dimensional problems, I
just had to comment that,
421
00:25:54 --> 00:25:59
if the curl was to get in
here, it would fit fine.
422
00:25:59 --> 00:26:03
And if I restrict the curl to
the fields I'm working with,
423
00:26:03 --> 00:26:07
plane fields, then there's only
one component I'll have to
424
00:26:07 --> 00:26:11
think about, it has to be zero
to have a gradient field.
425
00:26:11 --> 00:26:15
OK, now I guess I should
just do an example or two.
426
00:26:15 --> 00:26:19
Can I give you a v_1 and v_2,
and you tell me, is it a
427
00:26:19 --> 00:26:22
gradient field or is it
not a gradient field.
428
00:26:22 --> 00:26:27
Let me give you a different,
let me just change these guys.
429
00:26:27 --> 00:26:33
Suppose I change
that to y and x.
430
00:26:33 --> 00:26:38
So there is a v, a different v.
431
00:26:38 --> 00:26:39
That's a vector field.
432
00:26:39 --> 00:26:44
At every point x, y, I've
got a little vector.
433
00:26:44 --> 00:26:47
I could try, even,
to draw them.
434
00:26:47 --> 00:26:50
And I'm going to ask you, is
it the gradient of any u.
435
00:26:50 --> 00:26:53
And if it is, what's that u?
436
00:26:53 --> 00:26:57
So let me show you what I
mean by a vector field.
437
00:26:57 --> 00:27:04
I mean, at a typical point like
x=1, y=0, the vector -- let's
438
00:27:04 --> 00:27:11
see, if x is one and y is zero,
then what's the gradient at
439
00:27:11 --> 00:27:19
that point? , am I right?
440
00:27:19 --> 00:27:22
I won't draw it too big,
or you won't be able
441
00:27:22 --> 00:27:24
to see a darn thing.
442
00:27:24 --> 00:27:31
OK, what about at
the point (1, 1)?
443
00:27:31 --> 00:27:37
Which way is my vector
field going? .
444
00:27:37 --> 00:27:41
So what's that look like?
445
00:27:41 --> 00:27:45
Plotting the vector field,
v, at a bunch of points.
446
00:27:45 --> 00:27:51
So you get like a map
of little arrows.
447
00:27:51 --> 00:27:53
So here it would go that way.
448
00:27:53 --> 00:27:55
Is that right?
449
00:27:55 --> 00:27:56
Huh.
450
00:27:56 --> 00:28:00
I wasn't expecting that,
to tell the truth.
451
00:28:00 --> 00:28:05
Let's see, so can I
get some more points?
452
00:28:05 --> 00:28:06
Let's see.
453
00:28:06 --> 00:28:09
What if I have, there.
454
00:28:09 --> 00:28:12
What's -- at (1/2, 0).
455
00:28:12 --> 00:28:20
Then, v is .
456
00:28:20 --> 00:28:24
Where is this flow going?
457
00:28:24 --> 00:28:30
See, if the point is along this
line, where y is equal to x,
458
00:28:30 --> 00:28:34
then the flow was going
out along this line.
459
00:28:34 --> 00:28:36
Can you give me some other
point here, just so we
460
00:28:36 --> 00:28:38
get some handle on this?
461
00:28:38 --> 00:28:42
There's the point
(2, 1), let's say.
462
00:28:42 --> 00:28:43
Let me put it over
a little bit.
463
00:28:43 --> 00:28:46
How about the point (2, 1).
464
00:28:46 --> 00:28:48
What's the vector if I
just want to draw --
465
00:28:48 --> 00:28:50
I'm just drawing here.
466
00:28:50 --> 00:28:55
And of course, code would do it
much better than I'm doing. (2,
467
00:28:55 --> 00:28:59
1), that gives me
, right?
468
00:28:59 --> 00:29:04
So at (2, 4) is over two
and up, it's like this.
469
00:29:04 --> 00:29:11
I think -- but I don't swear to
it -- that if I connect all
470
00:29:11 --> 00:29:14
this -- see, now you have to
take a big leap of faith.
471
00:29:14 --> 00:29:19
Imagine, like at every point
we've got these little arrows,
472
00:29:19 --> 00:29:24
and I want to connect them up.
473
00:29:24 --> 00:29:26
Let me do something.
474
00:29:26 --> 00:29:30
I'll do that, but let me come
back to the question I should
475
00:29:30 --> 00:29:32
have asked you first.
476
00:29:32 --> 00:29:35
Is this a gradient field?
477
00:29:35 --> 00:29:38
Does it satisfy the curl
zero condition that
478
00:29:38 --> 00:29:40
we put in a box here?
479
00:29:40 --> 00:29:45
Does that satisfy dv -- there
is v_2 and there's v_1.
480
00:29:45 --> 00:29:54
Is dv_2/dx, whatever that test
was, minus dv_1/dy equal zero?
481
00:29:54 --> 00:29:57
Yes, no?
482
00:29:57 --> 00:30:05
Yes, right? dv_2/dx is
two, and dv_1/dy is two.
483
00:30:05 --> 00:30:09
So what's the conclusion then?
484
00:30:09 --> 00:30:13
It satisfied my little
test, so this must be
485
00:30:13 --> 00:30:16
the gradient of some u.
486
00:30:16 --> 00:30:18
Right?
487
00:30:18 --> 00:30:20
That's the question we have.
488
00:30:20 --> 00:30:23
Which vector fields -- and we
found that this is one of
489
00:30:23 --> 00:30:25
them, it passes the test.
490
00:30:25 --> 00:30:26
It's the gradient of some u.
491
00:30:26 --> 00:30:28
What's the u?
492
00:30:28 --> 00:30:33
What's the function, u,
which is supposed to exist,
493
00:30:33 --> 00:30:37
whose gradient is that.
494
00:30:37 --> 00:30:38
2xy.
495
00:30:38 --> 00:30:40
Did everybody spot that one?
496
00:30:40 --> 00:30:41
2xy.
497
00:30:41 --> 00:30:43
498
00:30:43 --> 00:30:46
Because the x derivative
has to be 2y.
499
00:30:48 --> 00:30:51
So I just integrate
with respect to x.
500
00:30:51 --> 00:30:53
This is the x derivative.
501
00:30:53 --> 00:30:53
It's 2y.
502
00:30:54 --> 00:30:56
Take the integral with
respect to x, and I get 2xy.
503
00:30:57 --> 00:31:05
And there could be some term
that depended only on y.
504
00:31:05 --> 00:31:11
Anyway, this works.
505
00:31:11 --> 00:31:21
I think that this, maybe gives
me somehow -- oh, yeah.
506
00:31:21 --> 00:31:24
What's the equipotential
curve now?
507
00:31:24 --> 00:31:28
Oh, yeah, this picture's
going to come together.
508
00:31:28 --> 00:31:34
What is the equipotential
curve for this potential?
509
00:31:34 --> 00:31:38
It was a circle for the first
guy, but circles are out now.
510
00:31:38 --> 00:31:40
I changed v.
511
00:31:40 --> 00:31:42
I've got a new
potential function.
512
00:31:42 --> 00:31:50
And now I want to draw, in this
graph, the equipotentials.
513
00:31:50 --> 00:31:53
Suppose u is one.
514
00:31:53 --> 00:32:00
Suppose I draw the curve
2xy=1 in that picture.
515
00:32:00 --> 00:32:02
What kind of a curve is it?
516
00:32:02 --> 00:32:04
Do you recognize this?
517
00:32:04 --> 00:32:06
The Greeks would.
518
00:32:06 --> 00:32:07
Recognize 2xy=1.
519
00:32:08 --> 00:32:10
Or you could say y=1/2x.
520
00:32:11 --> 00:32:14
That gives you a quick
handle on the curve.
521
00:32:14 --> 00:32:19
It comes down like that.
522
00:32:19 --> 00:32:20
Right?
523
00:32:20 --> 00:32:25
And what's the Greek
name for that curve?
524
00:32:25 --> 00:32:28
Oh, come on.
525
00:32:28 --> 00:32:30
It's a hyperbola.
526
00:32:30 --> 00:32:32
It's a hyperbola.
527
00:32:32 --> 00:32:35
Hyperbolas -- you remember
the Greeks, they had
528
00:32:35 --> 00:32:37
these conic sections.
529
00:32:37 --> 00:32:41
They had ellipses, they had
parabolas, the marginal case,
530
00:32:41 --> 00:32:43
and then they had hyperbolas.
531
00:32:43 --> 00:32:47
And they all come from
second degree things.
532
00:32:47 --> 00:32:54
If I have a x squared and 2bxy
and cy squared equal one,
533
00:32:54 --> 00:32:57
that's one of those curves.
534
00:32:57 --> 00:33:01
And if it was x squared
plus y squared equal
535
00:33:01 --> 00:33:02
one, it was a circle.
536
00:33:02 --> 00:33:05
If it was x squared plus
7y squared equal one,
537
00:33:05 --> 00:33:11
it would be an ellipse.
538
00:33:11 --> 00:33:14
The positive definite -- it
all comes down to linear
539
00:33:14 --> 00:33:18
algebra, of course.
540
00:33:18 --> 00:33:22
If that little matrix is
positive definite, so that
541
00:33:22 --> 00:33:26
means a and c are positive, and
a c is bigger than b squared,
542
00:33:26 --> 00:33:29
you know the test for
positive definite.
543
00:33:29 --> 00:33:32
What kind of curve
do the Greeks have?
544
00:33:32 --> 00:33:35
What kind of equipotential
-- what kind of a curve
545
00:33:35 --> 00:33:36
have we got here?
546
00:33:36 --> 00:33:38
An ellipse.
547
00:33:38 --> 00:33:45
If this curve, if that little
matrix is indefinite,
548
00:33:45 --> 00:33:48
as, for example, here.
549
00:33:48 --> 00:33:53
So with this one, what would be
the matrix, what's the matrix
550
00:33:53 --> 00:33:58
that goes with 2xy, if I
match this with this.
551
00:33:58 --> 00:33:59
That's the matrix.
552
00:33:59 --> 00:34:02
There's no a squareds, there's
no x squareds, there's
553
00:34:02 --> 00:34:04
no y squareds.
554
00:34:04 --> 00:34:08
And there are 2xy's, I
think the matrix is that.
555
00:34:08 --> 00:34:11
So it's this nice
symmetric matrix.
556
00:34:11 --> 00:34:14
Is that a positive
definite matrix?
557
00:34:14 --> 00:34:16
Certainly not.
558
00:34:16 --> 00:34:17
It's indefinite.
559
00:34:17 --> 00:34:20
It's Eigenvalues are plus
one and minus one, adding
560
00:34:20 --> 00:34:22
to the trace zero.
561
00:34:22 --> 00:34:29
So indefinite matrices
correspond to hyperbolas.
562
00:34:29 --> 00:34:34
And later on, definite matrices
will correspond to elliptic
563
00:34:34 --> 00:34:36
partial differential equations.
564
00:34:36 --> 00:34:40
And indefinite matrices -- like
Laplace -- and indefinite
565
00:34:40 --> 00:34:45
matrices will correspond to
hyperbolic partial differential
566
00:34:45 --> 00:34:48
equations, like the
wave equation.
567
00:34:48 --> 00:34:50
What's the -- now we're here.
568
00:34:50 --> 00:34:57
I didn't expect to get here
-- what's the marginal case?
569
00:34:57 --> 00:35:02
What's the marginal case
between positive definite and
570
00:35:02 --> 00:35:05
indefinite is -- semidefinite.
571
00:35:05 --> 00:35:05
Great.
572
00:35:05 --> 00:35:08
And what kind of a curve do you
think comes when this little
573
00:35:08 --> 00:35:12
matrix is semidefinite.
574
00:35:12 --> 00:35:18
It's the one in between
ellipses and hyperbolas.
575
00:35:18 --> 00:35:23
The marginal guy is a parabola.
576
00:35:23 --> 00:35:24
Right.
577
00:35:24 --> 00:35:27
So semidefinite would
correspond to a parabola.
578
00:35:27 --> 00:35:28
Right.
579
00:35:28 --> 00:35:28
OK.
580
00:35:28 --> 00:35:29
Good.
581
00:35:29 --> 00:35:35
Anyway, all I was going
to say is, this u=2xy,
582
00:35:35 --> 00:35:37
that's our potential.
583
00:35:37 --> 00:35:43
If I draw equipotential
curves, they're hyperbolas.
584
00:35:43 --> 00:35:47
And now, what's the point
about these little arrows
585
00:35:47 --> 00:35:53
that I got started on.
586
00:35:53 --> 00:35:57
What was the very first point
about the answer to the meaning
587
00:35:57 --> 00:36:00
of the gradient was what?
588
00:36:00 --> 00:36:07
These are the gradients of u,
so those arrows point where?
589
00:36:07 --> 00:36:10
Perpendicular to
the hyperbolas.
590
00:36:10 --> 00:36:12
Perpendicular to the hyperbola.
591
00:36:12 --> 00:36:18
We're trying to see the
geometry -- it's beautiful
592
00:36:18 --> 00:36:21
geometry -- behind
the gradient.
593
00:36:21 --> 00:36:29
So if v is a gradient, then
it comes from some u.
594
00:36:29 --> 00:36:33
I can plot the u equal
constant, the equipotential,
595
00:36:33 --> 00:36:36
and then the gradients
will be perpendicular.
596
00:36:36 --> 00:36:42
So they really are a
little -- OK, good.
597
00:36:42 --> 00:36:49
OK, those are pieces of
information that you have, but
598
00:36:49 --> 00:36:51
always need saying again.
599
00:36:51 --> 00:36:55
And to get the picture in your
mind -- I suppose, finally, I
600
00:36:55 --> 00:37:01
should choose a v which
is not a gradient.
601
00:37:01 --> 00:37:04
Just to finish.
602
00:37:04 --> 00:37:06
How shall I adjust that v?
603
00:37:06 --> 00:37:07
This v was a gradient.
604
00:37:07 --> 00:37:11
Can you just change it a little
bit -- practically anything you
605
00:37:11 --> 00:37:18
do will screw it up -- to
make it not a gradient?
606
00:37:18 --> 00:37:23
So I just changed this two,
what shall I change the two to?
607
00:37:23 --> 00:37:26
To three.
608
00:37:26 --> 00:37:28
That would totally foul it up.
609
00:37:28 --> 00:37:32
So that vector field, which I
could draw little pictures of,
610
00:37:32 --> 00:37:36
but there would be no u
that it's coming from.
611
00:37:36 --> 00:37:37
There would be no u.
612
00:37:37 --> 00:37:40
These little arrows would not
line up perpendicular to
613
00:37:40 --> 00:37:45
some beautiful curves.
614
00:37:45 --> 00:37:47
I don't get a u from that.
615
00:37:47 --> 00:37:51
Because the y derivative of
that is three, and it doesn't
616
00:37:51 --> 00:37:53
equal the x derivative of that.
617
00:37:53 --> 00:37:54
So that's a no good one.
618
00:37:54 --> 00:37:57
Let's go back to the good one.
619
00:37:57 --> 00:37:58
OK.
620
00:37:58 --> 00:38:00
OK, good.
621
00:38:00 --> 00:38:03
Is that OK for gradients?
622
00:38:03 --> 00:38:05
We got the meaning
of gradients.
623
00:38:05 --> 00:38:09
They point perpendicular
to equipotentials.
624
00:38:09 --> 00:38:12
They tell how steeply
those -- they tell the
625
00:38:12 --> 00:38:15
separation between the
equipotentials, right.
626
00:38:15 --> 00:38:18
It's like, if you're a mountain
climber, you're looking at your
627
00:38:18 --> 00:38:23
map, your contour map, and
that's all I'm drawing here.
628
00:38:23 --> 00:38:26
I'm drawing a contour map that
every guy who goes climbing in
629
00:38:26 --> 00:38:29
New Hampshire is going to have.
630
00:38:29 --> 00:38:33
And it shows little circles,
those are level heights, right?
631
00:38:33 --> 00:38:36
Those are level contours.
632
00:38:36 --> 00:38:42
And if you want to climb as
fast as possible, you go
633
00:38:42 --> 00:38:44
perpendicular to
those contours.
634
00:38:44 --> 00:38:47
And the distance between
contours tells you
635
00:38:47 --> 00:38:48
how steep it is.
636
00:38:48 --> 00:38:52
So it's all nice geometry.
637
00:38:52 --> 00:38:53
OK.
638
00:38:53 --> 00:38:57
I've got to get to
divergence here.
639
00:38:57 --> 00:39:02
Divergence.
640
00:39:02 --> 00:39:04
I should have said
though, -- damn.
641
00:39:04 --> 00:39:08
There's more to say
about gradients.
642
00:39:08 --> 00:39:17
That question of whether , the vector field,
643
00:39:17 --> 00:39:18
is this question.
644
00:39:18 --> 00:39:21
The question of is it
the gradient of some u.
645
00:39:21 --> 00:39:24
So we now have a test.
646
00:39:24 --> 00:39:28
We now have a test.
647
00:39:28 --> 00:39:32
This is our test, right?
648
00:39:32 --> 00:39:34
That's our test.
649
00:39:34 --> 00:39:41
But I have to connect it with
Kirchhoff's voltage law.
650
00:39:41 --> 00:39:44
Do you remember, we haven't
talked so much about
651
00:39:44 --> 00:39:46
Kirchhoff's voltage law, but
I'm connecting it with the
652
00:39:46 --> 00:39:51
discrete case, to add in a
little more insight.
653
00:39:51 --> 00:39:56
What did Kirchhoff's
voltage law say?
654
00:39:56 --> 00:40:01
In that case, a was a
difference matrix.
655
00:40:01 --> 00:40:05
It was the incidence
matrix for our graph.
656
00:40:05 --> 00:40:12
And the question was -- I
have to take two moments
657
00:40:12 --> 00:40:14
to think about that.
658
00:40:14 --> 00:40:18
So Kirchhoff's voltage law.
659
00:40:18 --> 00:40:22
For a graph. a is an
incidence matrix.
660
00:40:22 --> 00:40:30
You know, the minus one, one
guys, one for every edge?
661
00:40:30 --> 00:40:38
And let me call it
v again, or e.
662
00:40:38 --> 00:40:40
I called it e at that time.
663
00:40:40 --> 00:40:44
Let's just look at an x.
664
00:40:44 --> 00:40:45
Right.
665
00:40:45 --> 00:40:58
A is this long thin matrix,
times -- sorry, u's. u's.
666
00:40:58 --> 00:41:04
Let me say it all at once.
667
00:41:04 --> 00:41:06
Which vectors have the form Au?
668
00:41:07 --> 00:41:13
Which vectors are combinations
of the columns of A?
669
00:41:13 --> 00:41:19
The test is, Kirchhoff's
voltage law, that if I go
670
00:41:19 --> 00:41:25
around any loop in the graph --
so if I have a u_1 here, u_2
671
00:41:25 --> 00:41:33
here, u_5 here, and u_7 here --
then a u will produce
672
00:41:33 --> 00:41:35
u_1-u_7 on that edge.
673
00:41:35 --> 00:41:39
It'll produce a
u_2-u_1 on that edge.
674
00:41:39 --> 00:41:42
It'll produce a u_5-u_2 on
that edge, if the edges
675
00:41:42 --> 00:41:45
are all going that way.
676
00:41:45 --> 00:41:49
And it'll produce a
u_7-u_5 on that edge.
677
00:41:49 --> 00:41:56
So I've got four components
of Au, four differences.
678
00:41:56 --> 00:41:59
And what does Kirchhoff's
voltage law tell me about
679
00:41:59 --> 00:42:00
those four differences?
680
00:42:00 --> 00:42:04
Which I can certainly
see directly.
681
00:42:04 --> 00:42:09
Those four differences,
u_1-u_7, u_2-u_1,
682
00:42:09 --> 00:42:11
u_5-u_2 and u_7-u_5.
683
00:42:12 --> 00:42:16
What's the obvious fact
about those four guys?
684
00:42:16 --> 00:42:21
They add to zero.
685
00:42:21 --> 00:42:27
The total drop around
a loop is zero.
686
00:42:27 --> 00:42:31
You see, if I cancel those, if
I add them, the u_1's cancel,
687
00:42:31 --> 00:42:34
the u_2's cancel, the u_5's
cancel, the u_7's cancel.
688
00:42:34 --> 00:42:36
We know this.
689
00:42:36 --> 00:42:36
OK.
690
00:42:36 --> 00:42:38
So that's Kirchhoff's
voltage law.
691
00:42:38 --> 00:42:46
It's got to have a
continuous form.
692
00:42:46 --> 00:42:53
This tells me, this is the
test on v at a point.
693
00:42:53 --> 00:42:57
What's the test on
v around a loop?
694
00:42:57 --> 00:43:00
I just want to connect
that -- I have to connect
695
00:43:00 --> 00:43:06
that to a second test.
696
00:43:06 --> 00:43:10
I'll just mention it, and
you'll find it in the book.
697
00:43:10 --> 00:43:12
That's the pointwise test.
698
00:43:12 --> 00:43:14
That was the easy test.
699
00:43:14 --> 00:43:17
We applied it to this and
we got the answer yes.
700
00:43:17 --> 00:43:21
If it was 3y, 2x, we
got the answer no.
701
00:43:21 --> 00:43:26
Now let me give you a
test that looks like
702
00:43:26 --> 00:43:28
Kirchhoff's voltage law.
703
00:43:28 --> 00:43:32
So I'm going to integrate
around a closed loop.
704
00:43:32 --> 00:43:33
What am I going to integrate?
705
00:43:33 --> 00:43:38
I think I integrate v --
oh boy, I'd better look.
706
00:43:38 --> 00:43:41
It's easy to get these wrong.
707
00:43:41 --> 00:43:41
Yeah.
708
00:43:41 --> 00:43:45
So I would call this
the vorticity.
709
00:43:45 --> 00:43:47
And then I would say the
vorticity is zero for
710
00:43:47 --> 00:43:49
a gradient field.
711
00:43:49 --> 00:43:53
Now my integral guy is going
to be the circulation.
712
00:43:53 --> 00:43:53
Oh yeah.
713
00:43:53 --> 00:43:55
Because I'm following the path.
714
00:43:55 --> 00:44:03
So it's just v_1*dx+v_2*dy
should be zero.
715
00:44:03 --> 00:44:08
Around every closed loop --
that's idea of this thing, that
716
00:44:08 --> 00:44:12
it tells me the integral goes
around a closed loop -- if I
717
00:44:12 --> 00:44:23
follow the velocity field, the
total circulation is zero.
718
00:44:23 --> 00:44:28
I put this up here as a fact
in vector calculus that's
719
00:44:28 --> 00:44:29
connected to that.
720
00:44:29 --> 00:44:32
These, one is zero and
the other is zero.
721
00:44:32 --> 00:44:36
There's a Stokes' theorem that
tells me that this integral is
722
00:44:36 --> 00:44:38
found from a double
integral of this.
723
00:44:38 --> 00:44:41
So if one is zero,
the other is zero.
724
00:44:41 --> 00:44:46
I'm just saying, here is
the natural analog of
725
00:44:46 --> 00:44:48
Kirchhoff's voltage law.
726
00:44:48 --> 00:44:49
OK.
727
00:44:49 --> 00:44:52
I had to say something about
voltage law, because for the
728
00:44:52 --> 00:44:59
divergence, which I'm now
going to get to -- whatever.
729
00:44:59 --> 00:45:03
Let me ask about
divergence of w=0.
730
00:45:05 --> 00:45:10
What does that mean?
731
00:45:10 --> 00:45:14
That's going to be the
equivalent of who's law.
732
00:45:14 --> 00:45:16
Please tell me.
733
00:45:16 --> 00:45:20
Which law is going to be the
equivalent of -- divergence
734
00:45:20 --> 00:45:24
of w=0 is going to mean
there's no source.
735
00:45:24 --> 00:45:27
Whatever goes in, comes out.
736
00:45:27 --> 00:45:30
Whose law is that?
737
00:45:30 --> 00:45:31
That's Kirchhoff again.
738
00:45:31 --> 00:45:33
Well, yeah, other
people in physics.
739
00:45:33 --> 00:45:33
Right.
740
00:45:33 --> 00:45:38
But in our little world, it's
the other Kirchhoff law.
741
00:45:38 --> 00:45:41
It's Kirchhoff's current law.
742
00:45:41 --> 00:45:46
It's the one, it's the
A transpose Right?
743
00:45:46 --> 00:45:49
This is what we're thinking of
as A transpose w equal zero.
744
00:45:49 --> 00:45:55
Kirchhoff's current
law, in equals out.
745
00:45:55 --> 00:46:00
How will I translate that in
equal out for functions?
746
00:46:00 --> 00:46:07
Now I don't have -- on a graph,
I just had the total flow,
747
00:46:07 --> 00:46:09
the net flow at every node.
748
00:46:09 --> 00:46:12
Notice the divergence
is at every node.
749
00:46:12 --> 00:46:17
The circulation was
around every loop.
750
00:46:17 --> 00:46:18
OK.
751
00:46:18 --> 00:46:21
So in equals out was just
the sum of four things.
752
00:46:21 --> 00:46:27
OK, here I'm going to have in
equal out -- how am I going
753
00:46:27 --> 00:46:31
to express in equal out?
754
00:46:31 --> 00:46:32
Divergence of w equals 0.
755
00:46:33 --> 00:46:37
Yeah, what I need is the
divergence theorem.
756
00:46:37 --> 00:46:41
Let's just face it,
we've got to have that.
757
00:46:41 --> 00:46:45
So I have a region here.
758
00:46:45 --> 00:46:51
I have a w everywhere,
w, .
759
00:46:52 --> 00:46:54
Then the divergence theorem.
760
00:46:54 --> 00:46:58
This is the great identity,
which of course has
761
00:46:58 --> 00:47:01
a discrete form.
762
00:47:01 --> 00:47:01
OK.
763
00:47:01 --> 00:47:05
The divergence theorem says
that if I integrate over the
764
00:47:05 --> 00:47:13
region, over this region, R,
the divergence, that's
765
00:47:13 --> 00:47:14
(dw_1/dx+dw_2/dy)dxdy.
766
00:47:14 --> 00:47:23
767
00:47:23 --> 00:47:27
So that's like telling me the
source, I'm integrating over
768
00:47:27 --> 00:47:30
the source at every point.
769
00:47:30 --> 00:47:38
At every point here, this
measures in minus out.
770
00:47:38 --> 00:47:44
But now, when I put the whole
thing together by integrating,
771
00:47:44 --> 00:47:47
what's the right hand
side of this equation?
772
00:47:47 --> 00:47:49
Do you know the
divergence theorem?
773
00:47:49 --> 00:47:55
And let's remember it
and see why it's so.
774
00:47:55 --> 00:48:01
What I'm doing is in equals out
for the whole region at once.
775
00:48:01 --> 00:48:02
Right?
776
00:48:02 --> 00:48:05
When I this is like in
equal out at a point.
777
00:48:05 --> 00:48:08
But now I'm putting all
the points together.
778
00:48:08 --> 00:48:15
So the only way out will be
out through the boundary.
779
00:48:15 --> 00:48:19
And so I'll need to say
how much flows out.
780
00:48:19 --> 00:48:25
This is the total source, the
total in equal out inside.
781
00:48:25 --> 00:48:28
The only way to get out
is through the boundary.
782
00:48:28 --> 00:48:35
So this is the integral
around the boundary of --
783
00:48:35 --> 00:48:39
so what's the flow out?
784
00:48:39 --> 00:48:45
It's, yeah, it's somehow --
think now what should go there.
785
00:48:45 --> 00:48:49
This is flux I'm talking about.
786
00:48:49 --> 00:48:53
Flux is short word for
the total flow out.
787
00:48:53 --> 00:48:57
OK.
788
00:48:57 --> 00:49:02
So now I've got to
get this right.
789
00:49:02 --> 00:49:10
In vector notation, it would
be -- w tells me the flow.
790
00:49:10 --> 00:49:16
But flow outwards, see,
suppose w points that way.
791
00:49:16 --> 00:49:20
Then the actual flow
out is not all that.
792
00:49:20 --> 00:49:22
Because a lot of that is
just going sideways.
793
00:49:22 --> 00:49:24
It's this part.
794
00:49:24 --> 00:49:26
It's the flow perpendicular
to the boundary.
795
00:49:26 --> 00:49:33
So it's w dot n, the
normal component of flow.
796
00:49:33 --> 00:49:42
And I integrate that
around the boundary.
797
00:49:42 --> 00:49:46
There you have a
key, key theorem.
798
00:49:46 --> 00:49:46
In 2-D.
799
00:49:47 --> 00:49:51
And it's an equation
for the flux.
800
00:49:51 --> 00:49:53
It's like the fundamental
theorem of calculus, but now
801
00:49:53 --> 00:49:58
we're in two dimensions.
802
00:49:58 --> 00:50:00
And this is what it looks like.
803
00:50:00 --> 00:50:04
OK, so I'm obviously not
going to finish with the
804
00:50:04 --> 00:50:07
divergence theorem today.
805
00:50:07 --> 00:50:10
So what's the conclusion?
806
00:50:10 --> 00:50:15
If the divergence is
zero, then what?
807
00:50:15 --> 00:50:21
If the divergence is zero,
if this is zero at every
808
00:50:21 --> 00:50:35
point, then this is
zero across every loop.
809
00:50:35 --> 00:50:38
Can I call this thing a loop?
810
00:50:38 --> 00:50:42
That closed loop.
811
00:50:42 --> 00:50:49
That's the conclusion
that we want to reach.
812
00:50:49 --> 00:50:52
So this is the
divergence theorem.
813
00:50:52 --> 00:51:00
The text gives a proof, not
to repeat in class, but it's
814
00:51:00 --> 00:51:03
a crucial formula to know.
815
00:51:03 --> 00:51:08
That the integral of the
divergence is the flux.
816
00:51:08 --> 00:51:09
OK.
817
00:51:09 --> 00:51:11
Let's come back to that
Wednesday, and I'll have
818
00:51:11 --> 00:51:13
lots of homework for you.
819
00:51:13 --> 00:51:18
Thanks for turning
in these today.