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OK, so remember we left things
with this statement of the
8
00:00:29 --> 00:00:33
divergence theorem.
So, the divergence theorem
9
00:00:33 --> 00:00:36
gives us a way to compute the
flux of a vector field for a
10
00:00:36 --> 00:00:41
closed surface.
OK, it says if I have a closed
11
00:00:41 --> 00:00:47
surface, s,
bounding some region, D,
12
00:00:47 --> 00:00:54
and I have a vector field
defined in space,
13
00:00:54 --> 00:00:59
so that I can try to compute
the flux of my vector field
14
00:00:59 --> 00:01:04
through my surface.
Double integral of F.dS or
15
00:01:04 --> 00:01:08
F.ndS if you want,
and to set this up,
16
00:01:08 --> 00:01:11
of course, I need to use the
geometry of the surface
17
00:01:11 --> 00:01:13
depending on what the surface
is.
18
00:01:13 --> 00:01:17
We've seen various formulas for
how to set up the double
19
00:01:17 --> 00:01:20
integral.
But, we've also seen that if
20
00:01:20 --> 00:01:24
it's a closed surface,
and if a vector field is
21
00:01:24 --> 00:01:29
defined everywhere inside,
then we can actually reduce
22
00:01:29 --> 00:01:34
that to a calculation of the
triple integral of the
23
00:01:34 --> 00:01:38
divergence of F inside,
OK?
24
00:01:38 --> 00:01:40
So,
concretely, if I use
25
00:01:40 --> 00:01:43
coordinates,
let's say that the coordinates
26
00:01:43 --> 00:01:48
of my vector field are,
sorry, the components are P,
27
00:01:48 --> 00:01:52
Q, and R dot ndS,
then that will become the
28
00:01:52 --> 00:02:00
triple integral of,
well, so, divergence is P sub x
29
00:02:00 --> 00:02:07
plus Q sub y plus R sub z.
OK, so by the way,
30
00:02:07 --> 00:02:10
how to remember this formula
for divergence,
31
00:02:10 --> 00:02:14
and other formulas for other
things as well.
32
00:02:14 --> 00:02:22
Let me just tell you quickly
about the del notation.
33
00:02:22 --> 00:02:27
So,
this guy usually pronounced as
34
00:02:27 --> 00:02:29
del,
rather than as pointy triangle
35
00:02:29 --> 00:02:32
going downwards or something
like that,
36
00:02:32 --> 00:02:37
it's a symbolic notation for an
operator.
37
00:02:37 --> 00:02:42
So, you're probably going to
complain about putting these
38
00:02:42 --> 00:02:46
guys into a vector.
But, let's think of partial
39
00:02:46 --> 00:02:48
with respect to x,
with respect to y,
40
00:02:48 --> 00:02:51
and with respect to z as the
components of some formal
41
00:02:51 --> 00:02:53
vector.
Of course, it's not a real
42
00:02:53 --> 00:02:55
vector.
These are not like anything.
43
00:02:55 --> 00:03:02
These are just symbols.
But, so see for example,
44
00:03:02 --> 00:03:06
the gradient of function,
well, if you multiply this
45
00:03:06 --> 00:03:09
vector by scalar,
which is a function,
46
00:03:09 --> 00:03:14
then you will get partial,
partial x of f,
47
00:03:14 --> 00:03:20
partial, partial y of f,
partial, partial z, f,
48
00:03:20 --> 00:03:25
well, that's the gradient.
That seems to work.
49
00:03:25 --> 00:03:29
So now, the interesting thing
about divergence is I can think
50
00:03:29 --> 00:03:33
of divergence as del dot a
vector field.
51
00:03:33 --> 00:03:42
See, if I do the dot product
between this guy and my vector
52
00:03:42 --> 00:03:46
field P, Q, R,
well, it looks like I will
53
00:03:46 --> 00:03:51
indeed get partial,
partial x of P plus partial Q
54
00:03:51 --> 00:03:56
partial y plus partial R partial
z.
55
00:03:56 --> 00:04:06
That's the divergence.
and of course, similarly,
56
00:04:06 --> 00:04:08
when we have two variables
only, x and y,
57
00:04:08 --> 00:04:11
we could have thought of the
same notation,
58
00:04:11 --> 00:04:13
just with a two component
vector,
59
00:04:13 --> 00:04:16
partial, partial x,
partial, partial y.
60
00:04:16 --> 00:04:20
So, now, this is like of
slightly limited usefulness so
61
00:04:20 --> 00:04:22
far.
It's going to become very handy
62
00:04:22 --> 00:04:25
pretty soon because we are going
to see curl.
63
00:04:25 --> 00:04:28
And, the formula for curl in
the plane was kind of
64
00:04:28 --> 00:04:31
complicated.
But, if you thought about it in
65
00:04:31 --> 00:04:34
terms of this,
it was actually the determinant
66
00:04:34 --> 00:04:36
of del and f.
And now, in space,
67
00:04:36 --> 00:04:39
we are actually going to do del
cross f.
68
00:04:39 --> 00:04:40
But, I'm getting ahead of
things.
69
00:04:40 --> 00:04:44
So, let's not do anything with
that.
70
00:04:44 --> 00:04:52
Curl will be for next week.
Just getting you used to the
71
00:04:52 --> 00:04:54
notation, especially since you
might be using it in physics
72
00:04:54 --> 00:04:59
already.
So, it might be worth doing.
73
00:04:59 --> 00:05:03
OK, so the other thing I wanted
to say is, what does this
74
00:05:03 --> 00:05:06
theorem say physically?
How should I think of this
75
00:05:06 --> 00:05:09
statement?
So, I think I said that very
76
00:05:09 --> 00:05:13
quickly at the end of last time,
but not very carefully.
77
00:05:13 --> 00:05:22
So, what's the physical
interpretation of a divergence
78
00:05:22 --> 00:05:26
field?
So,
79
00:05:26 --> 00:05:30
I want to claim that the
divergence of a vector field
80
00:05:30 --> 00:05:35
corresponds to what I'm going to
call the source rate,
81
00:05:35 --> 00:05:52
which is somehow the amount of
flux generated per unit volume.
82
00:05:52 --> 00:05:56
So, to understand what that
means, let's think of what's
83
00:05:56 --> 00:06:00
called an incompressible fluid.
OK, so an incompressible fluid
84
00:06:00 --> 00:06:02
is something like water,
for example,
85
00:06:02 --> 00:06:06
where a fixed mass of it always
occupies the same amount of
86
00:06:06 --> 00:06:09
volume.
So, guesses are compressible.
87
00:06:09 --> 00:06:13
Liquids are incompressible,
basically.
88
00:06:13 --> 00:06:24
So, if you have an
incompressible fluid flow -- --
89
00:06:24 --> 00:06:34
well, so, again,
what that means is really,
90
00:06:34 --> 00:06:44
given mass occupies always a
fixed volume.
91
00:06:44 --> 00:06:51
Then, well, let's say that we
have such a fluid with velocity
92
00:06:51 --> 00:06:57
given by our vector field.
OK, so we're thinking of F as
93
00:06:57 --> 00:07:03
the velocity and maybe something
containing water,
94
00:07:03 --> 00:07:08
a pipe, or something.
So, what does the divergence
95
00:07:08 --> 00:07:14
theorem say?
It says that if I take a region
96
00:07:14 --> 00:07:18
in space,
let's call it D,
97
00:07:18 --> 00:07:23
sorry, D is the inside,
and S is the surface around it,
98
00:07:23 --> 00:07:27
well, so if I sum the
divergence in D,
99
00:07:27 --> 00:07:35
well, I'm going to get the flux
going out through this surface,
100
00:07:35 --> 00:07:37
S.
I should have mentioned it
101
00:07:37 --> 00:07:39
earlier.
The convention in the
102
00:07:39 --> 00:07:43
divergence theorem is that we
orient the surface with a normal
103
00:07:43 --> 00:07:47
vector pointing always outwards.
OK, so now, we know what flux
104
00:07:47 --> 00:07:49
means.
Remember, we've been
105
00:07:49 --> 00:07:53
describing, flux means how much
fluid is passing through this
106
00:07:53 --> 00:08:00
surface.
So, that's the amount of fluid
107
00:08:00 --> 00:08:11
that's leaving the region,
D, per unit time.
108
00:08:11 --> 00:08:13
And, of course,
when I'm saying that,
109
00:08:13 --> 00:08:16
it means I'm counting
everything that's going out of D
110
00:08:16 --> 00:08:18
minus everything that's coming
into D.
111
00:08:18 --> 00:08:22
That's what the flux measures.
So, now, if there is stuff
112
00:08:22 --> 00:08:26
coming into D or going out of D,
well, it must come from
113
00:08:26 --> 00:08:28
somewhere.
So, one possibility would be
114
00:08:28 --> 00:08:32
that your fluid is actually
being compressed or expanded.
115
00:08:32 --> 00:08:34
But, I've said,
no, I'm looking at something
116
00:08:34 --> 00:08:37
like water that you cannot
squish into smaller volume.
117
00:08:37 --> 00:08:40
So, in that case,
the only explanation is that
118
00:08:40 --> 00:08:44
there is something it here that
actually is sucking up water or
119
00:08:44 --> 00:08:47
producing more water.
And so, integrating the
120
00:08:47 --> 00:08:52
divergence gives you the total
amount of sources minus the
121
00:08:52 --> 00:08:56
amount of syncs that are inside
this region.
122
00:08:56 --> 00:09:01
So, the divergence itself
measures basically the amount of
123
00:09:01 --> 00:09:06
sources or syncs per unit volume
in a given place.
124
00:09:06 --> 00:09:07
And now, if you think about it
that way,
125
00:09:07 --> 00:09:12
well,
it's basically the divergence
126
00:09:12 --> 00:09:17
theorem is just stating
something completely obvious
127
00:09:17 --> 00:09:23
about all the matter that is
leaving this region must come
128
00:09:23 --> 00:09:28
from somewhere.
So, that's basically how we
129
00:09:28 --> 00:09:30
think about it.
Now, of course,
130
00:09:30 --> 00:09:33
if you're doing 8.02,
then you might actually have
131
00:09:33 --> 00:09:35
seen the divergence theorem
already being used for things
132
00:09:35 --> 00:09:39
that are more like force fields,
say, electric fields and so on.
133
00:09:39 --> 00:09:42
Well, I'll try to say a few
things about that during the
134
00:09:42 --> 00:09:45
last week of classes.
But, then this kind of
135
00:09:45 --> 00:09:48
interpretation doesn't quite
work.
136
00:09:48 --> 00:09:51
OK, any questions,
generally speaking,
137
00:09:51 --> 00:09:56
before we move on to the proof
and other applications?
138
00:09:56 --> 00:10:05
Yes?
Oh, not the gradient.
139
00:10:05 --> 00:10:09
So, yeah, the divergence of F
measures the amount of sources
140
00:10:09 --> 00:10:11
or syncs in there.
Well, what makes it happen?
141
00:10:11 --> 00:10:13
If you want,
in a way, it's this theorem.
142
00:10:13 --> 00:10:16
Or, in another way,
if you think about it,
143
00:10:16 --> 00:10:20
try to look at your favorite
vector fields and compute their
144
00:10:20 --> 00:10:23
divergence.
And, if you take a vector field
145
00:10:23 --> 00:10:25
where maybe everything is
rotating,
146
00:10:25 --> 00:10:29
a flow that's just rotating
about some axis,
147
00:10:29 --> 00:10:31
then you'll find that its
divergence is zero.
148
00:10:31 --> 00:10:37
If you, sorry?
No, divergence is not equal to
149
00:10:37 --> 00:10:39
the gradient.
Sorry, there's a dot here that
150
00:10:39 --> 00:10:42
maybe is not very big,
but it's very important.
151
00:10:42 --> 00:10:44
OK, so you take the divergence
of a vector field.
152
00:10:44 --> 00:10:46
Well, you take the gradient of
a function.
153
00:10:46 --> 00:10:49
So, if the gradient of a
function is a vector,
154
00:10:49 --> 00:10:52
the divergence of a vector
field is a function.
155
00:10:52 --> 00:10:56
So, somehow these guys go back
and forth between.
156
00:10:56 --> 00:10:59
So, I should have said,
with new notations comes new
157
00:10:59 --> 00:11:04
responsibility.
I mean,
158
00:11:04 --> 00:11:07
now that we have this nice,
nifty notation that will let us
159
00:11:07 --> 00:11:10
do gradient divergence and later
curl in a unified way,
160
00:11:10 --> 00:11:12
if you choose this notation you
have to be really,
161
00:11:12 --> 00:11:17
really careful what you put
after it because otherwise it's
162
00:11:17 --> 00:11:21
easy to get completely confused.
OK, so divergence and gradients
163
00:11:21 --> 00:11:24
are completely different things.
The only thing they have in
164
00:11:24 --> 00:11:26
common is that both are what's
called a first order
165
00:11:26 --> 00:11:29
differential operator.
That means it involves the
166
00:11:29 --> 00:11:33
first partial derivatives of
whatever you put into it.
167
00:11:33 --> 00:11:35
But, one of them goes from
functions to vectors.
168
00:11:35 --> 00:11:38
That's gradient.
The other one goes from vectors
169
00:11:38 --> 00:11:41
to functions.
That's divergence.
170
00:11:41 --> 00:11:43
And, curl later will go from
vectors to vectors.
171
00:11:43 --> 00:11:57
But, that will be later.
Let's see, more questions?
172
00:11:57 --> 00:12:03
No?
OK, so let's see,
173
00:12:03 --> 00:12:12
so how are we going to actually
prove this theorem?
174
00:12:12 --> 00:12:15
Well, if you remember how we
prove Green's theorem a while
175
00:12:15 --> 00:12:18
ago, the answer is we're going
to do it exactly the same way.
176
00:12:18 --> 00:12:22
So, if you don't remember,
then I'm going to explain.
177
00:12:22 --> 00:12:24
OK, so the first thing we need
to do is actually a
178
00:12:24 --> 00:12:28
simplification.
So, instead of proving the
179
00:12:28 --> 00:12:33
divergence theorem,
namely, the equality up there,
180
00:12:33 --> 00:12:38
I'm going to actually prove
something easier.
181
00:12:38 --> 00:12:44
I'm going to prove that the
flux of a vector field that has
182
00:12:44 --> 00:12:52
only a z component is actually
equal to the triple integral of,
183
00:12:52 --> 00:12:58
well, the divergence of this is
just R sub z dV.
184
00:12:58 --> 00:13:00
OK, now, how do I go back to
the general case?
185
00:13:00 --> 00:13:03
Well, I will just prove the
same thing for a vector field
186
00:13:03 --> 00:13:07
that has only an x component or
only a y component.
187
00:13:07 --> 00:13:10
And then, I will add these
things together.
188
00:13:10 --> 00:13:12
So, if you think carefully
about what happens when you
189
00:13:12 --> 00:13:15
evaluate this,
you will have some formula for
190
00:13:15 --> 00:13:16
ndS,
and when you do the dot
191
00:13:16 --> 00:13:18
product,
you'll end up with the sum,
192
00:13:18 --> 00:13:21
P times something plus Q times
something plus R times
193
00:13:21 --> 00:13:22
something.
And basically,
194
00:13:22 --> 00:13:26
we are just dealing with the
last term, R times something,
195
00:13:26 --> 00:13:28
and showing that it's equal to
what it should be.
196
00:13:28 --> 00:13:30
And then, we the three such
terms together.
197
00:13:30 --> 00:13:44
We'll get the general case.
OK, so then we get the general
198
00:13:44 --> 00:14:01
case by summing one such
identity for each component.
199
00:14:01 --> 00:14:08
I should say three such
identities, one for each
200
00:14:08 --> 00:14:13
component, whatever.
Now, let's make a second
201
00:14:13 --> 00:14:17
simplification because I'm still
not feeling confident I can
202
00:14:17 --> 00:14:19
prove this right away for any
surface.
203
00:14:19 --> 00:14:23
I'm going to do it first or
what's called a vertically
204
00:14:23 --> 00:14:26
simple region.
OK, so vertically simple means
205
00:14:26 --> 00:14:30
it will be something which I can
setup an integral over the z
206
00:14:30 --> 00:14:36
variable first easily.
So, it's something that has a
207
00:14:36 --> 00:14:44
bottom face, and a top face,
and then some vertical sides.
208
00:14:44 --> 00:14:53
OK, so let's say first what
happens if the given region,
209
00:14:53 --> 00:15:02
D, is vertically simple.
So, vertically simple means it
210
00:15:02 --> 00:15:09
looks like this.
It has top.
211
00:15:09 --> 00:15:16
It has a bottom.
And, it has some vertical sides.
212
00:15:16 --> 00:15:20
So, if you want,
if I look at it from above,
213
00:15:20 --> 00:15:25
it projects to some region in
the xy plane.
214
00:15:25 --> 00:15:30
Let's call that R.
And, it lives between the top
215
00:15:30 --> 00:15:34
face and the bottom face.
Let's say the top face is z
216
00:15:34 --> 00:15:37
equals z2 of (x,
y).
217
00:15:37 --> 00:15:42
Let's say the bottom face is z
equals z1(x, y).
218
00:15:42 --> 00:15:44
OK, and I don't need to know
actual formulas.
219
00:15:44 --> 00:15:47
I'm just going to work with
these and prove things
220
00:15:47 --> 00:15:50
independently of what the
formulas will be for these
221
00:15:50 --> 00:15:52
functions.
OK, so anyway,
222
00:15:52 --> 00:15:56
a vertically simple region is
something that lives above a
223
00:15:56 --> 00:15:59
part of the xy plane,
and is between two graphs of
224
00:15:59 --> 00:16:03
two functions.
So, let's see what we can do in
225
00:16:03 --> 00:16:10
that case.
So, the right-hand side of this
226
00:16:10 --> 00:16:20
equality, so that's the triple
integral, let's start computing
227
00:16:20 --> 00:16:23
it.
OK, so of course we will not be
228
00:16:23 --> 00:16:26
able to get a number out of it
because we don't know,
229
00:16:26 --> 00:16:28
actually, formulas for
anything.
230
00:16:28 --> 00:16:32
But at least we can start
simplifying because the way this
231
00:16:32 --> 00:16:36
region looks like,
I should say this is D,
232
00:16:36 --> 00:16:40
tells me that I can start
setting up the triple integral
233
00:16:40 --> 00:16:45
at least in the order where I
integrate first over z.
234
00:16:45 --> 00:16:53
OK, so I can actually do it as
a triple integral with Rz dz
235
00:16:53 --> 00:16:57
dxdy or dydx,
doesn't matter.
236
00:16:57 --> 00:17:01
So, what are the bounds on z?
See, this is actually good
237
00:17:01 --> 00:17:04
practice to remember how we set
up triple integrals.
238
00:17:04 --> 00:17:06
So, remember,
when we did it first over z,
239
00:17:06 --> 00:17:09
we start by fixing a point,
x and y,
240
00:17:09 --> 00:17:12
and for that value of x and y,
we look at a small vertical
241
00:17:12 --> 00:17:16
slice and see from where to
where we have to go.
242
00:17:16 --> 00:17:21
Well, we start at z equals
whatever the value is at the
243
00:17:21 --> 00:17:28
bottom, so, z1 of x and y.
And, we go up to the top face,
244
00:17:28 --> 00:17:32
z2 of x and y.
Now, for x and y,
245
00:17:32 --> 00:17:37
I'm not going to actually set
up bounds because I've already
246
00:17:37 --> 00:17:41
called R the quantity that I'm
integrating.
247
00:17:41 --> 00:17:45
So let me change this to,
let's say, U or something like
248
00:17:45 --> 00:17:47
that.
If you already have an R,
249
00:17:47 --> 00:17:49
I mean, there's not much risk
for confusion,
250
00:17:49 --> 00:17:53
but still.
OK, so we're going to call U
251
00:17:53 --> 00:17:59
the shadow of my region instead.
So, now I want to integrate
252
00:17:59 --> 00:18:01
over all values of x and y that
are in the shadow of my region.
253
00:18:01 --> 00:18:04
That means it's a double
integral over this region,
254
00:18:04 --> 00:18:06
U, which I haven't described to
you.
255
00:18:06 --> 00:18:09
So, I can't actually set up
bounds for x and y.
256
00:18:09 --> 00:18:12
But, I'm going to just leave it
like this.
257
00:18:12 --> 00:18:16
OK,
now you see,
258
00:18:16 --> 00:18:19
if you look at how you would
start evaluating this,
259
00:18:19 --> 00:18:22
well, the inner integral
certainly is not scary because
260
00:18:22 --> 00:18:25
you're integrating the
derivative of R with respect to
261
00:18:25 --> 00:18:27
z,
integrating that with respect
262
00:18:27 --> 00:18:33
to z.
So, you should get R back.
263
00:18:33 --> 00:18:39
OK, so triple integral over D
of Rz dV becomes,
264
00:18:39 --> 00:18:42
well, we'll have a double
integral over U of,
265
00:18:42 --> 00:18:49
so, the inner integral becomes
R at the point on the top.
266
00:18:49 --> 00:18:53
So, that means,
remember, R is a function of x,
267
00:18:53 --> 00:18:56
y, and z.
And, in fact,
268
00:18:56 --> 00:19:03
I will plug into it the value
of z at the top,
269
00:19:03 --> 00:19:13
so, z of xy minus the value of
R at the point on the bottom,
270
00:19:13 --> 00:19:16
x, y, z1 of x,
y.
271
00:19:16 --> 00:19:26
OK, any questions about this?
No?
272
00:19:26 --> 00:19:29
Is it looking vaguely
believable?
273
00:19:29 --> 00:19:32
Yeah? OK.
So, now, let's compute the
274
00:19:32 --> 00:19:34
other side because here we are
stuck.
275
00:19:34 --> 00:19:36
We won't be able to do anything
else.
276
00:19:36 --> 00:19:39
So, let's look at the flux
integral.
277
00:19:39 --> 00:19:43
OK, we have to look at the flux
of this vector field through the
278
00:19:43 --> 00:19:46
entire surface,
S, which is the whole boundary
279
00:19:46 --> 00:19:51
of D.
So, that consists of a lot of
280
00:19:51 --> 00:19:56
pieces, namely the top,
bottom, and the sides.
281
00:19:56 --> 00:20:04
OK, so the other side -- So,
let me just remind you,
282
00:20:04 --> 00:20:12
S is bottom plus top plus side
of this vector field,
283
00:20:12 --> 00:20:19
dot ndS equals,
OK, so what do we have?
284
00:20:19 --> 00:20:21
So first, we have to look at
the bottom.
285
00:20:21 --> 00:20:23
No, let's start with the top
actually.
286
00:20:23 --> 00:20:35
Sorry.
OK, so let's start with the top.
287
00:20:35 --> 00:20:43
So, just remind you,
let's do all of them.
288
00:20:43 --> 00:20:50
So, let's look at the top first.
So, we need to set up the flux
289
00:20:50 --> 00:20:52
integral for a vector field dot
ndS.
290
00:20:52 --> 00:20:56
We need to know what ndS is.
Well, fortunately for us,
291
00:20:56 --> 00:20:59
we know that the top face is
going to be the graph of some
292
00:20:59 --> 00:21:03
function of x and y.
So, we've seen a formula for
293
00:21:03 --> 00:21:06
ndS in this kind of situation,
OK?
294
00:21:06 --> 00:21:11
We have seen that ndS,
sorry, so, just to remind you
295
00:21:11 --> 00:21:16
this is the graph of a function
z equals z2 of x,
296
00:21:16 --> 00:21:21
y.
So, we've seen ndS for that is
297
00:21:21 --> 00:21:30
negative partial derivative of
this function with respect to x,
298
00:21:30 --> 00:21:35
negative partial z2 with
respect to y,
299
00:21:35 --> 00:21:38
one, dxdy.
OK, and, well,
300
00:21:38 --> 00:21:44
we can't compute these guys,
but it's not a big deal because
301
00:21:44 --> 00:21:47
if we do the dot product with
302
00:21:47 --> 00:21:48
303
00:21:48 --> 00:21:51
dot ndS,
that will give us,
304
00:21:51 --> 00:21:53
well, if you dot this with
zero, zero, R,
305
00:21:53 --> 00:22:03
these terms go away.
You just have R dxdy.
306
00:22:03 --> 00:22:11
So, that means that the double
integral for flux through the
307
00:22:11 --> 00:22:19
top of R vector field dot ndS
becomes double integral of the
308
00:22:19 --> 00:22:23
top of R dxdy.
Now, how do we evaluate that,
309
00:22:23 --> 00:22:28
actually?
Well, so R is a function of x,
310
00:22:28 --> 00:22:29
y, z.
But we said,
311
00:22:29 --> 00:22:32
we have only two variables that
we're going to use.
312
00:22:32 --> 00:22:35
We're going to use x and y.
We're going to get rid of z.
313
00:22:35 --> 00:22:38
How do we get rid of z?
Well, if we are on the top
314
00:22:38 --> 00:22:41
surface, z is given by this
formula, z2 of x,
315
00:22:41 --> 00:22:45
y.
So, I plug z equals z2 of x,
316
00:22:45 --> 00:22:50
y into the formula for R,
whatever it may be.
317
00:22:50 --> 00:22:54
Then, I integrate dxdy.
And, what's the range for x and
318
00:22:54 --> 00:22:57
y?
Well, my surface sits exactly
319
00:22:57 --> 00:23:01
above this region U in the xy
plane.
320
00:23:01 --> 00:23:08
So, it's double integral over
U, OK?
321
00:23:08 --> 00:23:17
Any questions about how I set
up this flux integral?
322
00:23:17 --> 00:23:21
No?
OK, let me close the door,
323
00:23:21 --> 00:23:26
actually.
OK, so we've got one of the two
324
00:23:26 --> 00:23:31
terms that we had over there.
Let's try to get the others.
325
00:23:31 --> 00:23:44
326
00:23:44 --> 00:23:49
[LAUGHTER] No comment.
OK, so, we need to look,
327
00:23:49 --> 00:23:56
also, at the other parts of our
surface for the flux integral.
328
00:23:56 --> 00:24:00
So, the bottom,
well, it will work pretty much
329
00:24:00 --> 00:24:03
the same way,
right, because it's the graph
330
00:24:03 --> 00:24:06
of a function,
z equals z1 of x,
331
00:24:06 --> 00:24:10
y.
So, we should be able to get
332
00:24:10 --> 00:24:17
ndS using the same method,
negative partial with respect
333
00:24:17 --> 00:24:23
to x, negative partial with
respect to y,
334
00:24:23 --> 00:24:26
one dxdy.
Now, there's a small catch.
335
00:24:26 --> 00:24:30
OK, we have to think of it
carefully about orientations.
336
00:24:30 --> 00:24:34
So,
remember, when we set up the
337
00:24:34 --> 00:24:38
divergence theorem,
we need the normal vectors to
338
00:24:38 --> 00:24:42
point out of our region,
which means that on the top
339
00:24:42 --> 00:24:46
surface,
we want n pointing up.
340
00:24:46 --> 00:24:50
But, on the bottom face,
we want n pointing down.
341
00:24:50 --> 00:24:52
So, in fact,
we shouldn't use this formula
342
00:24:52 --> 00:24:55
here because that one
corresponds to the other
343
00:24:55 --> 00:24:58
orientation.
Well, we could use it and then
344
00:24:58 --> 00:25:02
subtract, but I was just going
to say that ndS is actually the
345
00:25:02 --> 00:25:06
opposite of this.
So, I'm going to switch all my
346
00:25:06 --> 00:25:09
signs.
OK, that's the other side of
347
00:25:09 --> 00:25:13
the formula when you orient your
graph with n pointing downwards.
348
00:25:13 --> 00:25:18
Now, if I do things the same
way as before,
349
00:25:18 --> 00:25:24
I will get that <0,0,
R> dot ndS will be negative
350
00:25:24 --> 00:25:27
R dxdy.
And so,
351
00:25:27 --> 00:25:34
when I do the double integral
over the bottom,
352
00:25:34 --> 00:25:39
I'm going to get the double
integral over the bottom of
353
00:25:39 --> 00:25:42
negative R dxdy,
which, if I try to evaluate
354
00:25:42 --> 00:25:46
that,
well, I actually will have to
355
00:25:46 --> 00:25:48
integrate.
Sorry, first I'll have to
356
00:25:48 --> 00:25:53
substitute the value of z.
The value of z that I will want
357
00:25:53 --> 00:25:57
to plug into R will be given by,
now, z1 of x,
358
00:25:57 --> 00:26:00
y.
And, the bounds of integration
359
00:26:00 --> 00:26:04
will be given,
again, by the shadow of our
360
00:26:04 --> 00:26:07
surface, which is,
again, this guy,
361
00:26:07 --> 00:26:09
U.
OK, so we seem to be all set,
362
00:26:09 --> 00:26:12
except we are still missing one
part of our surface,
363
00:26:12 --> 00:26:14
S, because we also need to look
at the sides.
364
00:26:14 --> 00:26:20
Well, what about the sides?
Well, our vector field,
365
00:26:20 --> 00:26:23
,
is actually vertical.
366
00:26:23 --> 00:26:29
It's parallel to the z axis.
OK, so my vector field does
367
00:26:29 --> 00:26:35
something like this everywhere.
And, why that makes it very
368
00:26:35 --> 00:26:38
interesting on the top and
bottom, that means that on the
369
00:26:38 --> 00:26:40
sides, really not much is going
on.
370
00:26:40 --> 00:26:45
No matter is passing through
the vertical sides.
371
00:26:45 --> 00:26:57
So, the side -- The sides are
vertical.
372
00:26:57 --> 00:27:05
So, <0,0,
R> is tangent to the side,
373
00:27:05 --> 00:27:14
and therefore,
the flux through the sides is
374
00:27:14 --> 00:27:23
just going to be zero.
OK, no calculation needed.
375
00:27:23 --> 00:27:26
That's because, of course,
that's the reason why a
376
00:27:26 --> 00:27:31
simplified first things so that
my vector field would only have
377
00:27:31 --> 00:27:35
a z component,
well, not just that but that's
378
00:27:35 --> 00:27:39
the main place where it becomes
very useful.
379
00:27:39 --> 00:27:42
So, now, if I compare my double
integral and,
380
00:27:42 --> 00:27:45
sorry, my triple integral and
my flux integral,
381
00:27:45 --> 00:27:47
I get that they are,
indeed, the same.
382
00:27:47 --> 00:28:03
383
00:28:03 --> 00:28:05
Well, that's the statement of
the theorem we are trying to
384
00:28:05 --> 00:28:17
prove.
I shouldn't erase it, OK?
385
00:28:17 --> 00:28:22
[LAUGHTER]
So, just to recap,
386
00:28:22 --> 00:28:32
we've got a formula for the
triple integral of R sub z dV.
387
00:28:32 --> 00:28:36
It's up there at the very top.
And, we got formulas for the
388
00:28:36 --> 00:28:39
flux through the top and the
bottom, and the sides.
389
00:28:39 --> 00:28:41
And, when you add them
together,
390
00:28:41 --> 00:28:47
you get indeed the same
formula,
391
00:28:47 --> 00:29:03
top plus bottom -- -- plus
sides of,
392
00:29:03 --> 00:29:08
OK, and so we have, actually,
completed the proof for this
393
00:29:08 --> 00:29:11
part.
Now, well, that's only for a
394
00:29:11 --> 00:29:14
vertically simple region,
OK?
395
00:29:14 --> 00:29:24
So, if D is not vertically
simple, what do we do?
396
00:29:24 --> 00:29:39
Well, we cut it into vertically
simple pieces.
397
00:29:39 --> 00:29:44
OK so, concretely,
I wanted to integrate over a
398
00:29:44 --> 00:29:48
solid doughnut.
Then, that's not vertically
399
00:29:48 --> 00:29:52
simple because here in the
middle, I have not only does top
400
00:29:52 --> 00:29:56
in this bottom,
but I have this middle face.
401
00:29:56 --> 00:29:59
So, the way I would cut my
doughnut would be probably I
402
00:29:59 --> 00:30:03
would slice it not in the way
that you'd usually slice the
403
00:30:03 --> 00:30:06
doughnut or a bagel,
but at it's probably more
404
00:30:06 --> 00:30:09
spectacular if you think that
it's a bagel.
405
00:30:09 --> 00:30:15
Then, a mathematician's way of
slicing it is like this into
406
00:30:15 --> 00:30:17
five pieces, OK?
And, that's not very convenient
407
00:30:17 --> 00:30:20
for eating,
but that's convenient for
408
00:30:20 --> 00:30:24
integrating over it because now
each of these pieces has a
409
00:30:24 --> 00:30:26
well-defined top and bottom
face,
410
00:30:26 --> 00:30:32
and of course you've introduced
some vertical sides for two
411
00:30:32 --> 00:30:35
reasons.
One is that we've said the flux
412
00:30:35 --> 00:30:40
through them is zero anyway.
So, who cares?
413
00:30:40 --> 00:30:43
Why bother?
But, also, if you sum the flux
414
00:30:43 --> 00:30:47
through the surface of each
little piece,
415
00:30:47 --> 00:30:50
well, you will see that you
will be integrating twice over
416
00:30:50 --> 00:30:52
each of these vertical cuts.
Once, when you do this piece,
417
00:30:52 --> 00:30:56
you will be taking the flux
through this red guy with normal
418
00:30:56 --> 00:31:00
vector pointing to the right,
and once, when you take this
419
00:31:00 --> 00:31:03
middle little piece,
you will be taking the flux
420
00:31:03 --> 00:31:07
through that cut surface again,
but now with normal vector
421
00:31:07 --> 00:31:09
pointing the other way around.
So, in fact,
422
00:31:09 --> 00:31:12
you'll be summing the flux
through these guys twice with
423
00:31:12 --> 00:31:15
opposite orientations.
They cancel out.
424
00:31:15 --> 00:31:18
Well, and again,
because of what you are doing
425
00:31:18 --> 00:31:20
actually, the integral was just
zero anyway.
426
00:31:20 --> 00:31:25
So, it didn't matter.
But, even if it hadn't
427
00:31:25 --> 00:31:30
simplified, that would do it for
us.
428
00:31:30 --> 00:31:32
OK, so that's how we do it with
the general region.
429
00:31:32 --> 00:31:34
And then, as I said at the
beginning,
430
00:31:34 --> 00:31:37
when we can do it for a vector
field that has only a z
431
00:31:37 --> 00:31:39
component,
well, we can also do it for a
432
00:31:39 --> 00:31:42
vector field that has only an x
or only a y component.
433
00:31:42 --> 00:31:45
And then, we sum together and
we get the general case.
434
00:31:45 --> 00:31:52
So, that's the end of the proof.
OK, so you see,
435
00:31:52 --> 00:31:55
the idea is really the same as
for Green's theorem.
436
00:31:55 --> 00:32:00
Yes?
Oh, there's only four pieces,
437
00:32:00 --> 00:32:05
thank you.
Yes, there's three kinds of
438
00:32:05 --> 00:32:13
mathematicians:
those who know how to count,
439
00:32:13 --> 00:32:30
and those who don't.
Well, OK.
440
00:32:30 --> 00:32:34
So, OK, now I hope that you can
see already the interest of this
441
00:32:34 --> 00:32:38
theorem for the divergence
theorem for computing flux
442
00:32:38 --> 00:32:42
integrals just for the sake of
computing flux integrals like
443
00:32:42 --> 00:32:46
might happen on the problem set
or on the next test.
444
00:32:46 --> 00:32:49
But let me tell you also why
it's important physically to
445
00:32:49 --> 00:32:54
understand equations that had
been observed empirically well
446
00:32:54 --> 00:32:57
before they were actually
understood in terms of how
447
00:32:57 --> 00:33:03
things go.
So, let's look at something
448
00:33:03 --> 00:33:10
called the diffusion equation.
So, let me explain to you what
449
00:33:10 --> 00:33:13
it does.
So, the diffusion equation is
450
00:33:13 --> 00:33:16
something that governs,
well, what's called diffusion.
451
00:33:16 --> 00:33:19
Diffusion is when you have a
fluid in which you are
452
00:33:19 --> 00:33:24
introducing some substance,
and you want to figure out how
453
00:33:24 --> 00:33:27
that thing is going to spread
out,
454
00:33:27 --> 00:33:30
the technical term is diffuse,
into the ambient fluid.
455
00:33:30 --> 00:33:36
So, for example,
that governs the motion of,
456
00:33:36 --> 00:33:43
say, smoke in the air,
or if you put dye in the
457
00:33:43 --> 00:33:49
solution or things like that.
That will tell you,
458
00:33:49 --> 00:33:53
say that you drop some ink into
a glass of water.
459
00:33:53 --> 00:33:57
Well, you can imagine that
obviously it will get diluted
460
00:33:57 --> 00:33:59
into there.
And, that equation will tell
461
00:33:59 --> 00:34:04
you how exactly over time this
thing is going to spread out and
462
00:34:04 --> 00:34:09
start filling the entire glass.
So, what's the equation?
463
00:34:09 --> 00:34:12
Well, we need,
first, to know what the unknown
464
00:34:12 --> 00:34:13
will be.
So, it's a partial differential
465
00:34:13 --> 00:34:16
equation, OK?
So the unknown is a function,
466
00:34:16 --> 00:34:20
and the equation will relate
the partial derivatives of that
467
00:34:20 --> 00:34:26
function to each other.
So, u, the unknown,
468
00:34:26 --> 00:34:36
will be the concentration at a
given point.
469
00:34:36 --> 00:34:38
And, of course,
that depends on the point where
470
00:34:38 --> 00:34:40
you are.
So, that depends on x,
471
00:34:40 --> 00:34:42
y, z, the location where you
are.
472
00:34:42 --> 00:34:45
But, since the goal is also to
understand how things spread
473
00:34:45 --> 00:34:47
over time, it should also depend
on time.
474
00:34:47 --> 00:34:51
Otherwise, we're not going to
get very far.
475
00:34:51 --> 00:34:53
And, in fact,
what the equation will give us
476
00:34:53 --> 00:34:55
is the derivative of u with
respect to t.
477
00:34:55 --> 00:34:59
It will tell us how the
concentration at a given point
478
00:34:59 --> 00:35:03
varies over time in terms of how
the concentration varied in
479
00:35:03 --> 00:35:06
space.
So, it will relate partial u
480
00:35:06 --> 00:35:10
partial t to partial derivatives
with respect to x,
481
00:35:10 --> 00:35:11
y, and z.
482
00:35:11 --> 00:35:42
483
00:35:42 --> 00:35:43
[APPLAUSE]
OK, [LAUGHTER]
484
00:35:43 --> 00:35:48
so what's the equation?
The equation is partial u
485
00:35:48 --> 00:35:55
partial t equals some constant.
Let me call it constant k times
486
00:35:55 --> 00:36:01
something I will call del
squared u, which is also called
487
00:36:01 --> 00:36:05
the Laplacian of u,
and what is that?
488
00:36:05 --> 00:36:09
Well,
that means,
489
00:36:09 --> 00:36:14
OK, so just to scare you,
del squared is this,
490
00:36:14 --> 00:36:20
which means it's the divergence
of gradient u that we've used
491
00:36:20 --> 00:36:25
this notation for gradient.
OK, so if you actually expand
492
00:36:25 --> 00:36:29
it in terms of variables,
that becomes partial u over
493
00:36:29 --> 00:36:35
partial x squared plus partial
squared u over partial y squared
494
00:36:35 --> 00:36:40
plus partial squared u over
partial z squared.
495
00:36:40 --> 00:36:48
OK, so the equation is this
equals that.
496
00:36:48 --> 00:36:51
OK, so that's a weird looking
equation.
497
00:36:51 --> 00:36:54
And, I'm going to have to
explain to you,
498
00:36:54 --> 00:36:57
where does it come from?
OK, but before I do that,
499
00:36:57 --> 00:37:02
well, let me point out actually
that the equation is not just
500
00:37:02 --> 00:37:10
the diffusion equation.
It's also known as the heat
501
00:37:10 --> 00:37:15
equation.
And, that's because exactly the
502
00:37:15 --> 00:37:21
same equation governs how
temperature changes over time
503
00:37:21 --> 00:37:25
when you have,
again, so, sorry I should have
504
00:37:25 --> 00:37:28
been actually more careful.
I should have said this is in
505
00:37:28 --> 00:37:31
air that's not moving,
OK?
506
00:37:31 --> 00:37:32
OK, and same thing with the
solution.
507
00:37:32 --> 00:37:35
If you drop some ink into your
glass of water,
508
00:37:35 --> 00:37:38
well, if you start stirring,
obviously it will mix much
509
00:37:38 --> 00:37:40
faster than if you don't do
anything.
510
00:37:40 --> 00:37:43
OK, so that's the case where we
don't actually,
511
00:37:43 --> 00:37:47
the fluid is not moving.
And, the heat equation now does
512
00:37:47 --> 00:37:51
the same, but for temperature in
a fluid that's at rest,
513
00:37:51 --> 00:37:55
that's not moving.
It tells you how the heat goes
514
00:37:55 --> 00:37:58
from the warmest parts to the
coldest parts,
515
00:37:58 --> 00:38:03
and eventually temperatures
should somehow even out.
516
00:38:03 --> 00:38:08
So, in the heat equation,
that would just be,
517
00:38:08 --> 00:38:15
this u would just measure the
temperature for temperature of
518
00:38:15 --> 00:38:19
your fluid at a given point.
Actually, it doesn't have to be
519
00:38:19 --> 00:38:23
a fluid.
It could be a solid for that
520
00:38:23 --> 00:38:26
heat equation.
So, for example,
521
00:38:26 --> 00:38:31
say that you have a big box
made of metal or something,
522
00:38:31 --> 00:38:34
and you do some heating at one
side.
523
00:38:34 --> 00:38:38
You want to know how quickly is
the other side going to get hot?
524
00:38:38 --> 00:38:40
Well, you can use the equation
to figure out,
525
00:38:40 --> 00:38:44
you know, say that you have a
metal bar, and you keep one side
526
00:38:44 --> 00:38:46
at 400� because it's in your
oven.
527
00:38:46 --> 00:38:52
How quickly will the other side
get warm?
528
00:38:52 --> 00:38:57
OK, so it's the same equation
for both phenomena even though
529
00:38:57 --> 00:39:00
they are, of course,
different phenomena.
530
00:39:00 --> 00:39:02
Well, the physical reason why
they're the same is actually
531
00:39:02 --> 00:39:05
that phenomena are different,
but not all that much.
532
00:39:05 --> 00:39:07
They involve,
actually, how the atoms and
533
00:39:07 --> 00:39:11
molecules are actually moving,
and hitting each other inside
534
00:39:11 --> 00:39:14
this medium.
OK, so anyway,
535
00:39:14 --> 00:39:17
what's the explanation for
this?
536
00:39:17 --> 00:39:20
So, to understand the
explanation, and given what
537
00:39:20 --> 00:39:22
we've been doing,
we should have a vector field
538
00:39:22 --> 00:39:26
in there.
So, I'm going to think of the
539
00:39:26 --> 00:39:30
flow of, well,
let's imagine that we are doing
540
00:39:30 --> 00:39:35
motion of smoke in air.
So, that's the flow of the
541
00:39:35 --> 00:39:39
smoke: that means at every
point, it's a vector whose
542
00:39:39 --> 00:39:43
direction tells me in which
direction the smoke is actually
543
00:39:43 --> 00:39:47
moving.
And, its magnitude tells me how
544
00:39:47 --> 00:39:52
fast it's moving,
and also what amount of smoke
545
00:39:52 --> 00:39:56
is moving.
So, there's two things to
546
00:39:56 --> 00:40:01
understand.
One is how the disparities in
547
00:40:01 --> 00:40:06
the concentration between
different points causes the flow
548
00:40:06 --> 00:40:10
to be there,
how smoke will flow from the
549
00:40:10 --> 00:40:14
regions where there's more smoke
to the regions where there's
550
00:40:14 --> 00:40:17
less smoke.
And, that's actually physics.
551
00:40:17 --> 00:40:24
But, in a way,
it's also common sense.
552
00:40:24 --> 00:40:40
So, physics and common sense
tell us that the smoke will flow
553
00:40:40 --> 00:40:56
from high concentration towards
low concentration regions.
554
00:40:56 --> 00:41:01
OK, so if we think of this
function, U,
555
00:41:01 --> 00:41:04
that measures concentration,
that means we are always going
556
00:41:04 --> 00:41:07
to go in the direction where the
concentration decreases the
557
00:41:07 --> 00:41:09
fastest.
Well, what's that?
558
00:41:09 --> 00:41:25
That's negative the gradient.
So, F is directed along minus
559
00:41:25 --> 00:41:32
gradient u.
Now, how big is F going to be?
560
00:41:32 --> 00:41:35
Well, they are,
you have to come up with some
561
00:41:35 --> 00:41:39
intuition for how the two are
related.
562
00:41:39 --> 00:41:42
And, the easiest relation I can
think of is that they might be
563
00:41:42 --> 00:41:44
just proportional.
So, the steeper the differences
564
00:41:44 --> 00:41:47
in concentration,
the faster the flow will be,
565
00:41:47 --> 00:41:50
or the more there will be flow.
And, if you try to think about
566
00:41:50 --> 00:41:53
it as molecules moving in random
directions, you will see it
567
00:41:53 --> 00:41:56
makes actually complete sense.
Anyway, it should be part of
568
00:41:56 --> 00:42:00
your physics class,
not part of what I'm telling
569
00:42:00 --> 00:42:04
you.
So, I'm just going to accept
570
00:42:04 --> 00:42:12
that the flow is just
proportional to the gradient of
571
00:42:12 --> 00:42:13
u.
So, if you want,
572
00:42:13 --> 00:42:16
the differences between
concentrations of different
573
00:42:16 --> 00:42:18
points are very small,
then the flow will be very
574
00:42:18 --> 00:42:22
gentle.
And, if on the other hand you
575
00:42:22 --> 00:42:26
have huge disparities,
then things will try to even
576
00:42:26 --> 00:42:31
out faster.
OK, so that's the first part.
577
00:42:31 --> 00:42:35
Now, we need to understand the
second part, which is once we
578
00:42:35 --> 00:42:38
know how flow is going,
how does that affect the
579
00:42:38 --> 00:42:40
concentration?
If smoke is going that way,
580
00:42:40 --> 00:42:43
then it means the concentration
here should be decreasing.
581
00:42:43 --> 00:42:45
And there, it should be
increasing.
582
00:42:45 --> 00:42:58
So, that's the relation between
F and partial u partial t.
583
00:42:58 --> 00:43:07
At that part is actually math,
namely, the divergence theorem.
584
00:43:07 --> 00:43:19
So, let's try to understand
that part more carefully.
585
00:43:19 --> 00:43:25
So, let's take a small piece of
a small region in space,
586
00:43:25 --> 00:43:28
D, bounded by a surface,
S.
587
00:43:28 --> 00:43:33
So, I want to figure out what's
going on in here.
588
00:43:33 --> 00:43:42
So, let's look at the flux out
of D through S.
589
00:43:42 --> 00:43:49
Well, we said that this flux
would be given by double
590
00:43:49 --> 00:43:58
integral on S of F dot n dS.
So, this flux measures how much
591
00:43:58 --> 00:44:05
smoke is passing through S per
unit time.
592
00:44:05 --> 00:44:14
That's the amount of smoke
through S per unit time.
593
00:44:14 --> 00:44:19
But now, how can I compute that
differently?
594
00:44:19 --> 00:44:23
Well, I can compute it just by
looking at the total amount of
595
00:44:23 --> 00:44:26
smoke in this region,
and seeing how much it changes.
596
00:44:26 --> 00:44:29
If I'm gaining or losing smoke,
it means it must be going up
597
00:44:29 --> 00:44:32
there.
Well, unless I have a smoker in
598
00:44:32 --> 00:44:35
here, but that's not part of the
data.
599
00:44:35 --> 00:44:41
So,
that should be, sorry,
600
00:44:41 --> 00:44:44
that's the same thing as the
derivative with respect to t of
601
00:44:44 --> 00:44:47
the total amount of smoke in the
region,
602
00:44:47 --> 00:44:50
which is given by the triple
integral of u.
603
00:44:50 --> 00:44:52
If I integrate the
concentration of smoke,
604
00:44:52 --> 00:44:56
which means the amount of smoke
per unit volume over d,
605
00:44:56 --> 00:44:59
I will get the total amount of
smoke in d,
606
00:44:59 --> 00:45:02
except,
well,
607
00:45:02 --> 00:45:05
let's see.
This flux is counted positively
608
00:45:05 --> 00:45:07
if we go out of d.
So, actually,
609
00:45:07 --> 00:45:12
it's minus the derivative.
This is the amount of smoke
610
00:45:12 --> 00:45:16
that we are losing per unit
time.
611
00:45:16 --> 00:45:33
OK, so now we are almost there.
Well, let me actually --
612
00:45:33 --> 00:45:42
Because we know yet another way
to compute this guy using the
613
00:45:42 --> 00:45:48
divergence theorem.
Right, so this part here is
614
00:45:48 --> 00:45:53
just common sense and thinking
about what it means.
615
00:45:53 --> 00:46:00
The divergence theorem tells me
this is also equal to the triple
616
00:46:00 --> 00:46:06
integral, d, of div f dV.
So, what I got is that the
617
00:46:06 --> 00:46:15
triple integral over d of div F
dV equals this derivative.
618
00:46:15 --> 00:46:18
Well, let's think a bit about
this derivative so,
619
00:46:18 --> 00:46:20
see, you are integrating
function over x,
620
00:46:20 --> 00:46:22
y, and z.
And then, you are
621
00:46:22 --> 00:46:24
differentiating with respect to
t.
622
00:46:24 --> 00:46:28
I claim that you can actually
switch the order in which you do
623
00:46:28 --> 00:46:30
things.
So, when we think about it,
624
00:46:30 --> 00:46:33
is, here, you are taking the
total amount of smoke and then
625
00:46:33 --> 00:46:37
see how that changes over time.
So, you're taking the
626
00:46:37 --> 00:46:40
derivative of the sum of all the
small amounts of smoke
627
00:46:40 --> 00:46:42
everywhere.
Well, that will be the sum of
628
00:46:42 --> 00:46:47
the derivatives of the amounts
of smoke inside each little box.
629
00:46:47 --> 00:46:55
So, we can actually move the
derivatives into the integral.
630
00:46:55 --> 00:47:00
So, let's see,
I said minus d dt of triple
631
00:47:00 --> 00:47:07
integral over d udV.
And, now I'm saying this is the
632
00:47:07 --> 00:47:14
same as the triple integral in d
of du dt dv.
633
00:47:14 --> 00:47:19
And the reason why this is
going to work is you should
634
00:47:19 --> 00:47:24
think of it as d dt of a sum of
u of some values.
635
00:47:24 --> 00:47:30
You plug in the values of your
points at that given time times
636
00:47:30 --> 00:47:32
the small volume.
You sum them,
637
00:47:32 --> 00:47:33
and then you take the
derivative.
638
00:47:33 --> 00:47:42
And now, you see,
the derivative of this sum is
639
00:47:42 --> 00:47:49
the sum of the derivatives.
yi, zi, t, so,
640
00:47:49 --> 00:47:53
if you think about what the
integral measures,
641
00:47:53 --> 00:47:58
that's actually pretty easy.
And it's because this variable
642
00:47:58 --> 00:48:01
here is not the same as the
variables on which we are
643
00:48:01 --> 00:48:03
integrating.
That's why we can do it.
644
00:48:03 --> 00:48:13
OK, so now, if we have this for
any region, d.
645
00:48:13 --> 00:48:18
So, we have a function of x,
y, z, t, and we have another
646
00:48:18 --> 00:48:21
function here.
And whenever we integrate them
647
00:48:21 --> 00:48:23
anywhere, we get the same
answer.
648
00:48:23 --> 00:48:26
Well, that must mean they're
the same.
649
00:48:26 --> 00:48:29
Just think of what happens if
you just take d to be a tiny
650
00:48:29 --> 00:48:31
little box.
You will get roughly the value
651
00:48:31 --> 00:48:33
of div f at that point times the
volume of the box.
652
00:48:33 --> 00:48:36
Or, you will get roughly the
value of du dt at that point
653
00:48:36 --> 00:48:41
times the value of a little box.
So, the values must be the same.
654
00:48:41 --> 00:48:46
Well, let me actually explain
that a tiny bit better.
655
00:48:46 --> 00:48:50
So, what I get is that one
over, let me divide by the
656
00:48:50 --> 00:49:00
volume of D, sorry.
I promise, I'm done in a minute.
657
00:49:00 --> 00:49:08
Is the same thing as one over
volume D of negative du dt,
658
00:49:08 --> 00:49:10
dV.
So, that means the average
659
00:49:10 --> 00:49:12
value,
OK, maybe that's the best way
660
00:49:12 --> 00:49:17
of telling it,
the average of div f in D is
661
00:49:17 --> 00:49:27
equal to the average of minus
partial u partial t in D.
662
00:49:27 --> 00:49:30
And, that's true for any
region, D, not just for some
663
00:49:30 --> 00:49:33
regions, but for,
really, any region I can think
664
00:49:33 --> 00:49:37
of.
So, the outcome is that
665
00:49:37 --> 00:49:43
actually the divergence of f is
equal to minus du dt.
666
00:49:43 --> 00:49:47
And, that's another way to
think about what divergence
667
00:49:47 --> 00:49:48
means.
The divergence,
668
00:49:48 --> 00:49:50
we said, is how much stuff is
actually expanding,
669
00:49:50 --> 00:49:54
flowing out.
That's how much we're losing.
670
00:49:54 --> 00:49:58
And so, now,
if you combine this with that,
671
00:49:58 --> 00:50:02
you will get that du dt is
minus divergence f,
672
00:50:02 --> 00:50:08
which is plus K del squared u.
OK, so you combine this guy
673
00:50:08 --> 00:50:10
with that guy,
and you get the diffusion
674
00:50:10 --> 00:50:13
equation.
675
00:50:13 --> 00:50:18