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OK, so remember we left things
with this statement of the

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00:00:29,000 --> 00:00:33,000
divergence theorem.
So, the divergence theorem

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00:00:33,000 --> 00:00:36,000
gives us a way to compute the
flux of a vector field for a

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00:00:36,000 --> 00:00:41,000
closed surface.
OK, it says if I have a closed

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00:00:41,000 --> 00:00:47,000
surface, s,
bounding some region, D,

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00:00:47,000 --> 00:00:54,000
and I have a vector field
defined in space,

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00:00:54,000 --> 00:00:59,000
so that I can try to compute
the flux of my vector field

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00:00:59,000 --> 00:01:04,000
through my surface.
Double integral of F.dS or

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00:01:04,000 --> 00:01:08,000
F.ndS if you want,
and to set this up,

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00:01:08,000 --> 00:01:11,000
of course, I need to use the
geometry of the surface

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00:01:11,000 --> 00:01:13,000
depending on what the surface
is.

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00:01:13,000 --> 00:01:17,000
We've seen various formulas for
how to set up the double

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00:01:17,000 --> 00:01:20,000
integral.
But, we've also seen that if

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00:01:20,000 --> 00:01:24,000
it's a closed surface,
and if a vector field is

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00:01:24,000 --> 00:01:29,000
defined everywhere inside,
then we can actually reduce

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00:01:29,000 --> 00:01:34,000
that to a calculation of the
triple integral of the

23
00:01:34,000 --> 00:01:38,000
divergence of F inside,
OK?

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00:01:38,000 --> 00:01:40,000
So,
concretely, if I use

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00:01:40,000 --> 00:01:43,000
coordinates,
let's say that the coordinates

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00:01:43,000 --> 00:01:48,000
of my vector field are,
sorry, the components are P,

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00:01:48,000 --> 00:01:52,000
Q, and R dot ndS,
then that will become the

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00:01:52,000 --> 00:02:00,000
triple integral of,
well, so, divergence is P sub x

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00:02:00,000 --> 00:02:07,000
plus Q sub y plus R sub z.
OK, so by the way,

30
00:02:07,000 --> 00:02:10,000
how to remember this formula
for divergence,

31
00:02:10,000 --> 00:02:14,000
and other formulas for other
things as well.

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00:02:14,000 --> 00:02:22,000
Let me just tell you quickly
about the del notation.

33
00:02:22,000 --> 00:02:27,000
So,
this guy usually pronounced as

34
00:02:27,000 --> 00:02:29,000
del,
rather than as pointy triangle

35
00:02:29,000 --> 00:02:32,000
going downwards or something
like that,

36
00:02:32,000 --> 00:02:37,000
it's a symbolic notation for an
operator.

37
00:02:37,000 --> 00:02:42,000
So, you're probably going to
complain about putting these

38
00:02:42,000 --> 00:02:46,000
guys into a vector.
But, let's think of partial

39
00:02:46,000 --> 00:02:48,000
with respect to x,
with respect to y,

40
00:02:48,000 --> 00:02:51,000
and with respect to z as the
components of some formal

41
00:02:51,000 --> 00:02:53,000
vector.
Of course, it's not a real

42
00:02:53,000 --> 00:02:55,000
vector.
These are not like anything.

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00:02:55,000 --> 00:03:02,000
These are just symbols.
But, so see for example,

44
00:03:02,000 --> 00:03:06,000
the gradient of function,
well, if you multiply this

45
00:03:06,000 --> 00:03:09,000
vector by scalar,
which is a function,

46
00:03:09,000 --> 00:03:14,000
then you will get partial,
partial x of f,

47
00:03:14,000 --> 00:03:20,000
partial, partial y of f,
partial, partial z, f,

48
00:03:20,000 --> 00:03:25,000
well, that's the gradient.
That seems to work.

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00:03:25,000 --> 00:03:29,000
So now, the interesting thing
about divergence is I can think

50
00:03:29,000 --> 00:03:33,000
of divergence as del dot a
vector field.

51
00:03:33,000 --> 00:03:42,000
See, if I do the dot product
between this guy and my vector

52
00:03:42,000 --> 00:03:46,000
field P, Q, R,
well, it looks like I will

53
00:03:46,000 --> 00:03:51,000
indeed get partial,
partial x of P plus partial Q

54
00:03:51,000 --> 00:03:56,000
partial y plus partial R partial
z.

55
00:03:56,000 --> 00:04:06,000
That's the divergence.
and of course, similarly,

56
00:04:06,000 --> 00:04:08,000
when we have two variables
only, x and y,

57
00:04:08,000 --> 00:04:11,000
we could have thought of the
same notation,

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00:04:11,000 --> 00:04:13,000
just with a two component
vector,

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00:04:13,000 --> 00:04:16,000
partial, partial x,
partial, partial y.

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00:04:16,000 --> 00:04:20,000
So, now, this is like of
slightly limited usefulness so

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00:04:20,000 --> 00:04:22,000
far.
It's going to become very handy

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00:04:22,000 --> 00:04:25,000
pretty soon because we are going
to see curl.

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00:04:25,000 --> 00:04:28,000
And, the formula for curl in
the plane was kind of

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00:04:28,000 --> 00:04:31,000
complicated.
But, if you thought about it in

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00:04:31,000 --> 00:04:34,000
terms of this,
it was actually the determinant

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00:04:34,000 --> 00:04:36,000
of del and f.
And now, in space,

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00:04:36,000 --> 00:04:39,000
we are actually going to do del
cross f.

68
00:04:39,000 --> 00:04:40,000
But, I'm getting ahead of
things.

69
00:04:40,000 --> 00:04:44,000
So, let's not do anything with
that.

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00:04:44,000 --> 00:04:52,000
Curl will be for next week.
Just getting you used to the

71
00:04:52,000 --> 00:04:54,000
notation, especially since you
might be using it in physics

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00:04:54,000 --> 00:04:59,000
already.
So, it might be worth doing.

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00:04:59,000 --> 00:05:03,000
OK, so the other thing I wanted
to say is, what does this

74
00:05:03,000 --> 00:05:06,000
theorem say physically?
How should I think of this

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00:05:06,000 --> 00:05:09,000
statement?
So, I think I said that very

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00:05:09,000 --> 00:05:13,000
quickly at the end of last time,
but not very carefully.

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00:05:13,000 --> 00:05:22,000
So, what's the physical
interpretation of a divergence

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00:05:22,000 --> 00:05:26,000
field?
So,

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00:05:26,000 --> 00:05:30,000
I want to claim that the
divergence of a vector field

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00:05:30,000 --> 00:05:35,000
corresponds to what I'm going to
call the source rate,

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00:05:35,000 --> 00:05:52,000
which is somehow the amount of
flux generated per unit volume.

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00:05:52,000 --> 00:05:56,000
So, to understand what that
means, let's think of what's

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00:05:56,000 --> 00:06:00,000
called an incompressible fluid.
OK, so an incompressible fluid

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00:06:00,000 --> 00:06:02,000
is something like water,
for example,

85
00:06:02,000 --> 00:06:06,000
where a fixed mass of it always
occupies the same amount of

86
00:06:06,000 --> 00:06:09,000
volume.
So, guesses are compressible.

87
00:06:09,000 --> 00:06:13,000
Liquids are incompressible,
basically.

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00:06:13,000 --> 00:06:24,000
So, if you have an
incompressible fluid flow -- --

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00:06:24,000 --> 00:06:34,000
well, so, again,
what that means is really,

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00:06:34,000 --> 00:06:44,000
given mass occupies always a
fixed volume.

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00:06:44,000 --> 00:06:51,000
Then, well, let's say that we
have such a fluid with velocity

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00:06:51,000 --> 00:06:57,000
given by our vector field.
OK, so we're thinking of F as

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00:06:57,000 --> 00:07:03,000
the velocity and maybe something
containing water,

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00:07:03,000 --> 00:07:08,000
a pipe, or something.
So, what does the divergence

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00:07:08,000 --> 00:07:14,000
theorem say?
It says that if I take a region

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00:07:14,000 --> 00:07:18,000
in space,
let's call it D,

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00:07:18,000 --> 00:07:23,000
sorry, D is the inside,
and S is the surface around it,

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00:07:23,000 --> 00:07:27,000
well, so if I sum the
divergence in D,

99
00:07:27,000 --> 00:07:35,000
well, I'm going to get the flux
going out through this surface,

100
00:07:35,000 --> 00:07:37,000
S.
I should have mentioned it

101
00:07:37,000 --> 00:07:39,000
earlier.
The convention in the

102
00:07:39,000 --> 00:07:43,000
divergence theorem is that we
orient the surface with a normal

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00:07:43,000 --> 00:07:47,000
vector pointing always outwards.
OK, so now, we know what flux

104
00:07:47,000 --> 00:07:49,000
means.
Remember, we've been

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00:07:49,000 --> 00:07:53,000
describing, flux means how much
fluid is passing through this

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00:07:53,000 --> 00:08:00,000
surface.
So, that's the amount of fluid

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00:08:00,000 --> 00:08:11,000
that's leaving the region,
D, per unit time.

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00:08:11,000 --> 00:08:13,000
And, of course,
when I'm saying that,

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00:08:13,000 --> 00:08:16,000
it means I'm counting
everything that's going out of D

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00:08:16,000 --> 00:08:18,000
minus everything that's coming
into D.

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00:08:18,000 --> 00:08:22,000
That's what the flux measures.
So, now, if there is stuff

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00:08:22,000 --> 00:08:26,000
coming into D or going out of D,
well, it must come from

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00:08:26,000 --> 00:08:28,000
somewhere.
So, one possibility would be

114
00:08:28,000 --> 00:08:32,000
that your fluid is actually
being compressed or expanded.

115
00:08:32,000 --> 00:08:34,000
But, I've said,
no, I'm looking at something

116
00:08:34,000 --> 00:08:37,000
like water that you cannot
squish into smaller volume.

117
00:08:37,000 --> 00:08:40,000
So, in that case,
the only explanation is that

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00:08:40,000 --> 00:08:44,000
there is something it here that
actually is sucking up water or

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00:08:44,000 --> 00:08:47,000
producing more water.
And so, integrating the

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00:08:47,000 --> 00:08:52,000
divergence gives you the total
amount of sources minus the

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00:08:52,000 --> 00:08:56,000
amount of syncs that are inside
this region.

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00:08:56,000 --> 00:09:01,000
So, the divergence itself
measures basically the amount of

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00:09:01,000 --> 00:09:06,000
sources or syncs per unit volume
in a given place.

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00:09:06,000 --> 00:09:07,000
And now, if you think about it
that way,

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00:09:07,000 --> 00:09:12,000
well,
it's basically the divergence

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00:09:12,000 --> 00:09:17,000
theorem is just stating
something completely obvious

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00:09:17,000 --> 00:09:23,000
about all the matter that is
leaving this region must come

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00:09:23,000 --> 00:09:28,000
from somewhere.
So, that's basically how we

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00:09:28,000 --> 00:09:30,000
think about it.
Now, of course,

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00:09:30,000 --> 00:09:33,000
if you're doing 8.02,
then you might actually have

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00:09:33,000 --> 00:09:35,000
seen the divergence theorem
already being used for things

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00:09:35,000 --> 00:09:39,000
that are more like force fields,
say, electric fields and so on.

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00:09:39,000 --> 00:09:42,000
Well, I'll try to say a few
things about that during the

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00:09:42,000 --> 00:09:45,000
last week of classes.
But, then this kind of

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00:09:45,000 --> 00:09:48,000
interpretation doesn't quite
work.

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00:09:48,000 --> 00:09:51,000
OK, any questions,
generally speaking,

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00:09:51,000 --> 00:09:56,000
before we move on to the proof
and other applications?

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00:09:56,000 --> 00:10:05,000
Yes?
Oh, not the gradient.

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00:10:05,000 --> 00:10:09,000
So, yeah, the divergence of F
measures the amount of sources

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00:10:09,000 --> 00:10:11,000
or syncs in there.
Well, what makes it happen?

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00:10:11,000 --> 00:10:13,000
If you want,
in a way, it's this theorem.

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00:10:13,000 --> 00:10:16,000
Or, in another way,
if you think about it,

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00:10:16,000 --> 00:10:20,000
try to look at your favorite
vector fields and compute their

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00:10:20,000 --> 00:10:23,000
divergence.
And, if you take a vector field

145
00:10:23,000 --> 00:10:25,000
where maybe everything is
rotating,

146
00:10:25,000 --> 00:10:29,000
a flow that's just rotating
about some axis,

147
00:10:29,000 --> 00:10:31,000
then you'll find that its
divergence is zero.

148
00:10:31,000 --> 00:10:37,000
If you, sorry?
No, divergence is not equal to

149
00:10:37,000 --> 00:10:39,000
the gradient.
Sorry, there's a dot here that

150
00:10:39,000 --> 00:10:42,000
maybe is not very big,
but it's very important.

151
00:10:42,000 --> 00:10:44,000
OK, so you take the divergence
of a vector field.

152
00:10:44,000 --> 00:10:46,000
Well, you take the gradient of
a function.

153
00:10:46,000 --> 00:10:49,000
So, if the gradient of a
function is a vector,

154
00:10:49,000 --> 00:10:52,000
the divergence of a vector
field is a function.

155
00:10:52,000 --> 00:10:56,000
So, somehow these guys go back
and forth between.

156
00:10:56,000 --> 00:10:59,000
So, I should have said,
with new notations comes new

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00:10:59,000 --> 00:11:04,000
responsibility.
I mean,

158
00:11:04,000 --> 00:11:07,000
now that we have this nice,
nifty notation that will let us

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00:11:07,000 --> 00:11:10,000
do gradient divergence and later
curl in a unified way,

160
00:11:10,000 --> 00:11:12,000
if you choose this notation you
have to be really,

161
00:11:12,000 --> 00:11:17,000
really careful what you put
after it because otherwise it's

162
00:11:17,000 --> 00:11:21,000
easy to get completely confused.
OK, so divergence and gradients

163
00:11:21,000 --> 00:11:24,000
are completely different things.
The only thing they have in

164
00:11:24,000 --> 00:11:26,000
common is that both are what's
called a first order

165
00:11:26,000 --> 00:11:29,000
differential operator.
That means it involves the

166
00:11:29,000 --> 00:11:33,000
first partial derivatives of
whatever you put into it.

167
00:11:33,000 --> 00:11:35,000
But, one of them goes from
functions to vectors.

168
00:11:35,000 --> 00:11:38,000
That's gradient.
The other one goes from vectors

169
00:11:38,000 --> 00:11:41,000
to functions.
That's divergence.

170
00:11:41,000 --> 00:11:43,000
And, curl later will go from
vectors to vectors.

171
00:11:43,000 --> 00:11:57,000
But, that will be later.
Let's see, more questions?

172
00:11:57,000 --> 00:12:03,000
No?
OK, so let's see,

173
00:12:03,000 --> 00:12:12,000
so how are we going to actually
prove this theorem?

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00:12:12,000 --> 00:12:15,000
Well, if you remember how we
prove Green's theorem a while

175
00:12:15,000 --> 00:12:18,000
ago, the answer is we're going
to do it exactly the same way.

176
00:12:18,000 --> 00:12:22,000
So, if you don't remember,
then I'm going to explain.

177
00:12:22,000 --> 00:12:24,000
OK, so the first thing we need
to do is actually a

178
00:12:24,000 --> 00:12:28,000
simplification.
So, instead of proving the

179
00:12:28,000 --> 00:12:33,000
divergence theorem,
namely, the equality up there,

180
00:12:33,000 --> 00:12:38,000
I'm going to actually prove
something easier.

181
00:12:38,000 --> 00:12:44,000
I'm going to prove that the
flux of a vector field that has

182
00:12:44,000 --> 00:12:52,000
only a z component is actually
equal to the triple integral of,

183
00:12:52,000 --> 00:12:58,000
well, the divergence of this is
just R sub z dV.

184
00:12:58,000 --> 00:13:00,000
OK, now, how do I go back to
the general case?

185
00:13:00,000 --> 00:13:03,000
Well, I will just prove the
same thing for a vector field

186
00:13:03,000 --> 00:13:07,000
that has only an x component or
only a y component.

187
00:13:07,000 --> 00:13:10,000
And then, I will add these
things together.

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00:13:10,000 --> 00:13:12,000
So, if you think carefully
about what happens when you

189
00:13:12,000 --> 00:13:15,000
evaluate this,
you will have some formula for

190
00:13:15,000 --> 00:13:16,000
ndS,
and when you do the dot

191
00:13:16,000 --> 00:13:18,000
product,
you'll end up with the sum,

192
00:13:18,000 --> 00:13:21,000
P times something plus Q times
something plus R times

193
00:13:21,000 --> 00:13:22,000
something.
And basically,

194
00:13:22,000 --> 00:13:26,000
we are just dealing with the
last term, R times something,

195
00:13:26,000 --> 00:13:28,000
and showing that it's equal to
what it should be.

196
00:13:28,000 --> 00:13:30,000
And then, we the three such
terms together.

197
00:13:30,000 --> 00:13:44,000
We'll get the general case.
OK, so then we get the general

198
00:13:44,000 --> 00:14:01,000
case by summing one such
identity for each component.

199
00:14:01,000 --> 00:14:08,000
I should say three such
identities, one for each

200
00:14:08,000 --> 00:14:13,000
component, whatever.
Now, let's make a second

201
00:14:13,000 --> 00:14:17,000
simplification because I'm still
not feeling confident I can

202
00:14:17,000 --> 00:14:19,000
prove this right away for any
surface.

203
00:14:19,000 --> 00:14:23,000
I'm going to do it first or
what's called a vertically

204
00:14:23,000 --> 00:14:26,000
simple region.
OK, so vertically simple means

205
00:14:26,000 --> 00:14:30,000
it will be something which I can
setup an integral over the z

206
00:14:30,000 --> 00:14:36,000
variable first easily.
So, it's something that has a

207
00:14:36,000 --> 00:14:44,000
bottom face, and a top face,
and then some vertical sides.

208
00:14:44,000 --> 00:14:53,000
OK, so let's say first what
happens if the given region,

209
00:14:53,000 --> 00:15:02,000
D, is vertically simple.
So, vertically simple means it

210
00:15:02,000 --> 00:15:09,000
looks like this.
It has top.

211
00:15:09,000 --> 00:15:16,000
It has a bottom.
And, it has some vertical sides.

212
00:15:16,000 --> 00:15:20,000
So, if you want,
if I look at it from above,

213
00:15:20,000 --> 00:15:25,000
it projects to some region in
the xy plane.

214
00:15:25,000 --> 00:15:30,000
Let's call that R.
And, it lives between the top

215
00:15:30,000 --> 00:15:34,000
face and the bottom face.
Let's say the top face is z

216
00:15:34,000 --> 00:15:37,000
equals z2 of (x,
y).

217
00:15:37,000 --> 00:15:42,000
Let's say the bottom face is z
equals z1(x, y).

218
00:15:42,000 --> 00:15:44,000
OK, and I don't need to know
actual formulas.

219
00:15:44,000 --> 00:15:47,000
I'm just going to work with
these and prove things

220
00:15:47,000 --> 00:15:50,000
independently of what the
formulas will be for these

221
00:15:50,000 --> 00:15:52,000
functions.
OK, so anyway,

222
00:15:52,000 --> 00:15:56,000
a vertically simple region is
something that lives above a

223
00:15:56,000 --> 00:15:59,000
part of the xy plane,
and is between two graphs of

224
00:15:59,000 --> 00:16:03,000
two functions.
So, let's see what we can do in

225
00:16:03,000 --> 00:16:10,000
that case.
So, the right-hand side of this

226
00:16:10,000 --> 00:16:20,000
equality, so that's the triple
integral, let's start computing

227
00:16:20,000 --> 00:16:23,000
it.
OK, so of course we will not be

228
00:16:23,000 --> 00:16:26,000
able to get a number out of it
because we don't know,

229
00:16:26,000 --> 00:16:28,000
actually, formulas for
anything.

230
00:16:28,000 --> 00:16:32,000
But at least we can start
simplifying because the way this

231
00:16:32,000 --> 00:16:36,000
region looks like,
I should say this is D,

232
00:16:36,000 --> 00:16:40,000
tells me that I can start
setting up the triple integral

233
00:16:40,000 --> 00:16:45,000
at least in the order where I
integrate first over z.

234
00:16:45,000 --> 00:16:53,000
OK, so I can actually do it as
a triple integral with Rz dz

235
00:16:53,000 --> 00:16:57,000
dxdy or dydx,
doesn't matter.

236
00:16:57,000 --> 00:17:01,000
So, what are the bounds on z?
See, this is actually good

237
00:17:01,000 --> 00:17:04,000
practice to remember how we set
up triple integrals.

238
00:17:04,000 --> 00:17:06,000
So, remember,
when we did it first over z,

239
00:17:06,000 --> 00:17:09,000
we start by fixing a point,
x and y,

240
00:17:09,000 --> 00:17:12,000
and for that value of x and y,
we look at a small vertical

241
00:17:12,000 --> 00:17:16,000
slice and see from where to
where we have to go.

242
00:17:16,000 --> 00:17:21,000
Well, we start at z equals
whatever the value is at the

243
00:17:21,000 --> 00:17:28,000
bottom, so, z1 of x and y.
And, we go up to the top face,

244
00:17:28,000 --> 00:17:32,000
z2 of x and y.
Now, for x and y,

245
00:17:32,000 --> 00:17:37,000
I'm not going to actually set
up bounds because I've already

246
00:17:37,000 --> 00:17:41,000
called R the quantity that I'm
integrating.

247
00:17:41,000 --> 00:17:45,000
So let me change this to,
let's say, U or something like

248
00:17:45,000 --> 00:17:47,000
that.
If you already have an R,

249
00:17:47,000 --> 00:17:49,000
I mean, there's not much risk
for confusion,

250
00:17:49,000 --> 00:17:53,000
but still.
OK, so we're going to call U

251
00:17:53,000 --> 00:17:59,000
the shadow of my region instead.
So, now I want to integrate

252
00:17:59,000 --> 00:18:01,000
over all values of x and y that
are in the shadow of my region.

253
00:18:01,000 --> 00:18:04,000
That means it's a double
integral over this region,

254
00:18:04,000 --> 00:18:06,000
U, which I haven't described to
you.

255
00:18:06,000 --> 00:18:09,000
So, I can't actually set up
bounds for x and y.

256
00:18:09,000 --> 00:18:12,000
But, I'm going to just leave it
like this.

257
00:18:12,000 --> 00:18:16,000
OK,
now you see,

258
00:18:16,000 --> 00:18:19,000
if you look at how you would
start evaluating this,

259
00:18:19,000 --> 00:18:22,000
well, the inner integral
certainly is not scary because

260
00:18:22,000 --> 00:18:25,000
you're integrating the
derivative of R with respect to

261
00:18:25,000 --> 00:18:27,000
z,
integrating that with respect

262
00:18:27,000 --> 00:18:33,000
to z.
So, you should get R back.

263
00:18:33,000 --> 00:18:39,000
OK, so triple integral over D
of Rz dV becomes,

264
00:18:39,000 --> 00:18:42,000
well, we'll have a double
integral over U of,

265
00:18:42,000 --> 00:18:49,000
so, the inner integral becomes
R at the point on the top.

266
00:18:49,000 --> 00:18:53,000
So, that means,
remember, R is a function of x,

267
00:18:53,000 --> 00:18:56,000
y, and z.
And, in fact,

268
00:18:56,000 --> 00:19:03,000
I will plug into it the value
of z at the top,

269
00:19:03,000 --> 00:19:13,000
so, z of xy minus the value of
R at the point on the bottom,

270
00:19:13,000 --> 00:19:16,000
x, y, z1 of x,
y.

271
00:19:16,000 --> 00:19:26,000
OK, any questions about this?
No?

272
00:19:26,000 --> 00:19:29,000
Is it looking vaguely
believable?

273
00:19:29,000 --> 00:19:32,000
Yeah? OK.
So, now, let's compute the

274
00:19:32,000 --> 00:19:34,000
other side because here we are
stuck.

275
00:19:34,000 --> 00:19:36,000
We won't be able to do anything
else.

276
00:19:36,000 --> 00:19:39,000
So, let's look at the flux
integral.

277
00:19:39,000 --> 00:19:43,000
OK, we have to look at the flux
of this vector field through the

278
00:19:43,000 --> 00:19:46,000
entire surface,
S, which is the whole boundary

279
00:19:46,000 --> 00:19:51,000
of D.
So, that consists of a lot of

280
00:19:51,000 --> 00:19:56,000
pieces, namely the top,
bottom, and the sides.

281
00:19:56,000 --> 00:20:04,000
OK, so the other side -- So,
let me just remind you,

282
00:20:04,000 --> 00:20:12,000
S is bottom plus top plus side
of this vector field,

283
00:20:12,000 --> 00:20:19,000
dot ndS equals,
OK, so what do we have?

284
00:20:19,000 --> 00:20:21,000
So first, we have to look at
the bottom.

285
00:20:21,000 --> 00:20:23,000
No, let's start with the top
actually.

286
00:20:23,000 --> 00:20:35,000
Sorry.
OK, so let's start with the top.

287
00:20:35,000 --> 00:20:43,000
So, just remind you,
let's do all of them.

288
00:20:43,000 --> 00:20:50,000
So, let's look at the top first.
So, we need to set up the flux

289
00:20:50,000 --> 00:20:52,000
integral for a vector field dot
ndS.

290
00:20:52,000 --> 00:20:56,000
We need to know what ndS is.
Well, fortunately for us,

291
00:20:56,000 --> 00:20:59,000
we know that the top face is
going to be the graph of some

292
00:20:59,000 --> 00:21:03,000
function of x and y.
So, we've seen a formula for

293
00:21:03,000 --> 00:21:06,000
ndS in this kind of situation,
OK?

294
00:21:06,000 --> 00:21:11,000
We have seen that ndS,
sorry, so, just to remind you

295
00:21:11,000 --> 00:21:16,000
this is the graph of a function
z equals z2 of x,

296
00:21:16,000 --> 00:21:21,000
y.
So, we've seen ndS for that is

297
00:21:21,000 --> 00:21:30,000
negative partial derivative of
this function with respect to x,

298
00:21:30,000 --> 00:21:35,000
negative partial z2 with
respect to y,

299
00:21:35,000 --> 00:21:38,000
one, dxdy.
OK, and, well,

300
00:21:38,000 --> 00:21:44,000
we can't compute these guys,
but it's not a big deal because

301
00:21:44,000 --> 00:21:47,000
if we do the dot product with

302
00:21:48,000 --> 00:21:51,000
dot ndS,
that will give us,

303
00:21:51,000 --> 00:21:53,000
well, if you dot this with
zero, zero, R,

304
00:21:53,000 --> 00:22:03,000
these terms go away.
You just have R dxdy.

305
00:22:03,000 --> 00:22:11,000
So, that means that the double
integral for flux through the

306
00:22:11,000 --> 00:22:19,000
top of R vector field dot ndS
becomes double integral of the

307
00:22:19,000 --> 00:22:23,000
top of R dxdy.
Now, how do we evaluate that,

308
00:22:23,000 --> 00:22:28,000
actually?
Well, so R is a function of x,

309
00:22:28,000 --> 00:22:29,000
y, z.
But we said,

310
00:22:29,000 --> 00:22:32,000
we have only two variables that
we're going to use.

311
00:22:32,000 --> 00:22:35,000
We're going to use x and y.
We're going to get rid of z.

312
00:22:35,000 --> 00:22:38,000
How do we get rid of z?
Well, if we are on the top

313
00:22:38,000 --> 00:22:41,000
surface, z is given by this
formula, z2 of x,

314
00:22:41,000 --> 00:22:45,000
y.
So, I plug z equals z2 of x,

315
00:22:45,000 --> 00:22:50,000
y into the formula for R,
whatever it may be.

316
00:22:50,000 --> 00:22:54,000
Then, I integrate dxdy.
And, what's the range for x and

317
00:22:54,000 --> 00:22:57,000
y?
Well, my surface sits exactly

318
00:22:57,000 --> 00:23:01,000
above this region U in the xy
plane.

319
00:23:01,000 --> 00:23:08,000
So, it's double integral over
U, OK?

320
00:23:08,000 --> 00:23:17,000
Any questions about how I set
up this flux integral?

321
00:23:17,000 --> 00:23:21,000
No?
OK, let me close the door,

322
00:23:21,000 --> 00:23:26,000
actually.
OK, so we've got one of the two

323
00:23:26,000 --> 00:23:31,000
terms that we had over there.
Let's try to get the others.

324
00:23:44,000 --> 00:23:49,000
[LAUGHTER] No comment.
OK, so, we need to look,

325
00:23:49,000 --> 00:23:56,000
also, at the other parts of our
surface for the flux integral.

326
00:23:56,000 --> 00:24:00,000
So, the bottom,
well, it will work pretty much

327
00:24:00,000 --> 00:24:03,000
the same way,
right, because it's the graph

328
00:24:03,000 --> 00:24:06,000
of a function,
z equals z1 of x,

329
00:24:06,000 --> 00:24:10,000
y.
So, we should be able to get

330
00:24:10,000 --> 00:24:17,000
ndS using the same method,
negative partial with respect

331
00:24:17,000 --> 00:24:23,000
to x, negative partial with
respect to y,

332
00:24:23,000 --> 00:24:26,000
one dxdy.
Now, there's a small catch.

333
00:24:26,000 --> 00:24:30,000
OK, we have to think of it
carefully about orientations.

334
00:24:30,000 --> 00:24:34,000
So,
remember, when we set up the

335
00:24:34,000 --> 00:24:38,000
divergence theorem,
we need the normal vectors to

336
00:24:38,000 --> 00:24:42,000
point out of our region,
which means that on the top

337
00:24:42,000 --> 00:24:46,000
surface,
we want n pointing up.

338
00:24:46,000 --> 00:24:50,000
But, on the bottom face,
we want n pointing down.

339
00:24:50,000 --> 00:24:52,000
So, in fact,
we shouldn't use this formula

340
00:24:52,000 --> 00:24:55,000
here because that one
corresponds to the other

341
00:24:55,000 --> 00:24:58,000
orientation.
Well, we could use it and then

342
00:24:58,000 --> 00:25:02,000
subtract, but I was just going
to say that ndS is actually the

343
00:25:02,000 --> 00:25:06,000
opposite of this.
So, I'm going to switch all my

344
00:25:06,000 --> 00:25:09,000
signs.
OK, that's the other side of

345
00:25:09,000 --> 00:25:13,000
the formula when you orient your
graph with n pointing downwards.

346
00:25:13,000 --> 00:25:18,000
Now, if I do things the same
way as before,

347
00:25:18,000 --> 00:25:24,000
I will get that <0,0,
R> dot ndS will be negative

348
00:25:24,000 --> 00:25:27,000
R dxdy.
And so,

349
00:25:27,000 --> 00:25:34,000
when I do the double integral
over the bottom,

350
00:25:34,000 --> 00:25:39,000
I'm going to get the double
integral over the bottom of

351
00:25:39,000 --> 00:25:42,000
negative R dxdy,
which, if I try to evaluate

352
00:25:42,000 --> 00:25:46,000
that,
well, I actually will have to

353
00:25:46,000 --> 00:25:48,000
integrate.
Sorry, first I'll have to

354
00:25:48,000 --> 00:25:53,000
substitute the value of z.
The value of z that I will want

355
00:25:53,000 --> 00:25:57,000
to plug into R will be given by,
now, z1 of x,

356
00:25:57,000 --> 00:26:00,000
y.
And, the bounds of integration

357
00:26:00,000 --> 00:26:04,000
will be given,
again, by the shadow of our

358
00:26:04,000 --> 00:26:07,000
surface, which is,
again, this guy,

359
00:26:07,000 --> 00:26:09,000
U.
OK, so we seem to be all set,

360
00:26:09,000 --> 00:26:12,000
except we are still missing one
part of our surface,

361
00:26:12,000 --> 00:26:14,000
S, because we also need to look
at the sides.

362
00:26:14,000 --> 00:26:20,000
Well, what about the sides?
Well, our vector field,

363
00:26:20,000 --> 00:26:23,000
,
is actually vertical.

364
00:26:23,000 --> 00:26:29,000
It's parallel to the z axis.
OK, so my vector field does

365
00:26:29,000 --> 00:26:35,000
something like this everywhere.
And, why that makes it very

366
00:26:35,000 --> 00:26:38,000
interesting on the top and
bottom, that means that on the

367
00:26:38,000 --> 00:26:40,000
sides, really not much is going
on.

368
00:26:40,000 --> 00:26:45,000
No matter is passing through
the vertical sides.

369
00:26:45,000 --> 00:26:57,000
So, the side -- The sides are
vertical.

370
00:26:57,000 --> 00:27:05,000
So, <0,0,
R> is tangent to the side,

371
00:27:05,000 --> 00:27:14,000
and therefore,
the flux through the sides is

372
00:27:14,000 --> 00:27:23,000
just going to be zero.
OK, no calculation needed.

373
00:27:23,000 --> 00:27:26,000
That's because, of course,
that's the reason why a

374
00:27:26,000 --> 00:27:31,000
simplified first things so that
my vector field would only have

375
00:27:31,000 --> 00:27:35,000
a z component,
well, not just that but that's

376
00:27:35,000 --> 00:27:39,000
the main place where it becomes
very useful.

377
00:27:39,000 --> 00:27:42,000
So, now, if I compare my double
integral and,

378
00:27:42,000 --> 00:27:45,000
sorry, my triple integral and
my flux integral,

379
00:27:45,000 --> 00:27:47,000
I get that they are,
indeed, the same.

380
00:28:03,000 --> 00:28:05,000
Well, that's the statement of
the theorem we are trying to

381
00:28:05,000 --> 00:28:17,000
prove.
I shouldn't erase it, OK?

382
00:28:17,000 --> 00:28:22,000
[LAUGHTER]
So, just to recap,

383
00:28:22,000 --> 00:28:32,000
we've got a formula for the
triple integral of R sub z dV.

384
00:28:32,000 --> 00:28:36,000
It's up there at the very top.
And, we got formulas for the

385
00:28:36,000 --> 00:28:39,000
flux through the top and the
bottom, and the sides.

386
00:28:39,000 --> 00:28:41,000
And, when you add them
together,

387
00:28:41,000 --> 00:28:47,000
you get indeed the same
formula,

388
00:28:47,000 --> 00:29:03,000
top plus bottom -- -- plus
sides of,

389
00:29:03,000 --> 00:29:08,000
OK, and so we have, actually,
completed the proof for this

390
00:29:08,000 --> 00:29:11,000
part.
Now, well, that's only for a

391
00:29:11,000 --> 00:29:14,000
vertically simple region,
OK?

392
00:29:14,000 --> 00:29:24,000
So, if D is not vertically
simple, what do we do?

393
00:29:24,000 --> 00:29:39,000
Well, we cut it into vertically
simple pieces.

394
00:29:39,000 --> 00:29:44,000
OK so, concretely,
I wanted to integrate over a

395
00:29:44,000 --> 00:29:48,000
solid doughnut.
Then, that's not vertically

396
00:29:48,000 --> 00:29:52,000
simple because here in the
middle, I have not only does top

397
00:29:52,000 --> 00:29:56,000
in this bottom,
but I have this middle face.

398
00:29:56,000 --> 00:29:59,000
So, the way I would cut my
doughnut would be probably I

399
00:29:59,000 --> 00:30:03,000
would slice it not in the way
that you'd usually slice the

400
00:30:03,000 --> 00:30:06,000
doughnut or a bagel,
but at it's probably more

401
00:30:06,000 --> 00:30:09,000
spectacular if you think that
it's a bagel.

402
00:30:09,000 --> 00:30:15,000
Then, a mathematician's way of
slicing it is like this into

403
00:30:15,000 --> 00:30:17,000
five pieces, OK?
And, that's not very convenient

404
00:30:17,000 --> 00:30:20,000
for eating,
but that's convenient for

405
00:30:20,000 --> 00:30:24,000
integrating over it because now
each of these pieces has a

406
00:30:24,000 --> 00:30:26,000
well-defined top and bottom
face,

407
00:30:26,000 --> 00:30:32,000
and of course you've introduced
some vertical sides for two

408
00:30:32,000 --> 00:30:35,000
reasons.
One is that we've said the flux

409
00:30:35,000 --> 00:30:40,000
through them is zero anyway.
So, who cares?

410
00:30:40,000 --> 00:30:43,000
Why bother?
But, also, if you sum the flux

411
00:30:43,000 --> 00:30:47,000
through the surface of each
little piece,

412
00:30:47,000 --> 00:30:50,000
well, you will see that you
will be integrating twice over

413
00:30:50,000 --> 00:30:52,000
each of these vertical cuts.
Once, when you do this piece,

414
00:30:52,000 --> 00:30:56,000
you will be taking the flux
through this red guy with normal

415
00:30:56,000 --> 00:31:00,000
vector pointing to the right,
and once, when you take this

416
00:31:00,000 --> 00:31:03,000
middle little piece,
you will be taking the flux

417
00:31:03,000 --> 00:31:07,000
through that cut surface again,
but now with normal vector

418
00:31:07,000 --> 00:31:09,000
pointing the other way around.
So, in fact,

419
00:31:09,000 --> 00:31:12,000
you'll be summing the flux
through these guys twice with

420
00:31:12,000 --> 00:31:15,000
opposite orientations.
They cancel out.

421
00:31:15,000 --> 00:31:18,000
Well, and again,
because of what you are doing

422
00:31:18,000 --> 00:31:20,000
actually, the integral was just
zero anyway.

423
00:31:20,000 --> 00:31:25,000
So, it didn't matter.
But, even if it hadn't

424
00:31:25,000 --> 00:31:30,000
simplified, that would do it for
us.

425
00:31:30,000 --> 00:31:32,000
OK, so that's how we do it with
the general region.

426
00:31:32,000 --> 00:31:34,000
And then, as I said at the
beginning,

427
00:31:34,000 --> 00:31:37,000
when we can do it for a vector
field that has only a z

428
00:31:37,000 --> 00:31:39,000
component,
well, we can also do it for a

429
00:31:39,000 --> 00:31:42,000
vector field that has only an x
or only a y component.

430
00:31:42,000 --> 00:31:45,000
And then, we sum together and
we get the general case.

431
00:31:45,000 --> 00:31:52,000
So, that's the end of the proof.
OK, so you see,

432
00:31:52,000 --> 00:31:55,000
the idea is really the same as
for Green's theorem.

433
00:31:55,000 --> 00:32:00,000
Yes?
Oh, there's only four pieces,

434
00:32:00,000 --> 00:32:05,000
thank you.
Yes, there's three kinds of

435
00:32:05,000 --> 00:32:13,000
mathematicians:
those who know how to count,

436
00:32:13,000 --> 00:32:30,000
and those who don't.
Well, OK.

437
00:32:30,000 --> 00:32:34,000
So, OK, now I hope that you can
see already the interest of this

438
00:32:34,000 --> 00:32:38,000
theorem for the divergence
theorem for computing flux

439
00:32:38,000 --> 00:32:42,000
integrals just for the sake of
computing flux integrals like

440
00:32:42,000 --> 00:32:46,000
might happen on the problem set
or on the next test.

441
00:32:46,000 --> 00:32:49,000
But let me tell you also why
it's important physically to

442
00:32:49,000 --> 00:32:54,000
understand equations that had
been observed empirically well

443
00:32:54,000 --> 00:32:57,000
before they were actually
understood in terms of how

444
00:32:57,000 --> 00:33:03,000
things go.
So, let's look at something

445
00:33:03,000 --> 00:33:10,000
called the diffusion equation.
So, let me explain to you what

446
00:33:10,000 --> 00:33:13,000
it does.
So, the diffusion equation is

447
00:33:13,000 --> 00:33:16,000
something that governs,
well, what's called diffusion.

448
00:33:16,000 --> 00:33:19,000
Diffusion is when you have a
fluid in which you are

449
00:33:19,000 --> 00:33:24,000
introducing some substance,
and you want to figure out how

450
00:33:24,000 --> 00:33:27,000
that thing is going to spread
out,

451
00:33:27,000 --> 00:33:30,000
the technical term is diffuse,
into the ambient fluid.

452
00:33:30,000 --> 00:33:36,000
So, for example,
that governs the motion of,

453
00:33:36,000 --> 00:33:43,000
say, smoke in the air,
or if you put dye in the

454
00:33:43,000 --> 00:33:49,000
solution or things like that.
That will tell you,

455
00:33:49,000 --> 00:33:53,000
say that you drop some ink into
a glass of water.

456
00:33:53,000 --> 00:33:57,000
Well, you can imagine that
obviously it will get diluted

457
00:33:57,000 --> 00:33:59,000
into there.
And, that equation will tell

458
00:33:59,000 --> 00:34:04,000
you how exactly over time this
thing is going to spread out and

459
00:34:04,000 --> 00:34:09,000
start filling the entire glass.
So, what's the equation?

460
00:34:09,000 --> 00:34:12,000
Well, we need,
first, to know what the unknown

461
00:34:12,000 --> 00:34:13,000
will be.
So, it's a partial differential

462
00:34:13,000 --> 00:34:16,000
equation, OK?
So the unknown is a function,

463
00:34:16,000 --> 00:34:20,000
and the equation will relate
the partial derivatives of that

464
00:34:20,000 --> 00:34:26,000
function to each other.
So, u, the unknown,

465
00:34:26,000 --> 00:34:36,000
will be the concentration at a
given point.

466
00:34:36,000 --> 00:34:38,000
And, of course,
that depends on the point where

467
00:34:38,000 --> 00:34:40,000
you are.
So, that depends on x,

468
00:34:40,000 --> 00:34:42,000
y, z, the location where you
are.

469
00:34:42,000 --> 00:34:45,000
But, since the goal is also to
understand how things spread

470
00:34:45,000 --> 00:34:47,000
over time, it should also depend
on time.

471
00:34:47,000 --> 00:34:51,000
Otherwise, we're not going to
get very far.

472
00:34:51,000 --> 00:34:53,000
And, in fact,
what the equation will give us

473
00:34:53,000 --> 00:34:55,000
is the derivative of u with
respect to t.

474
00:34:55,000 --> 00:34:59,000
It will tell us how the
concentration at a given point

475
00:34:59,000 --> 00:35:03,000
varies over time in terms of how
the concentration varied in

476
00:35:03,000 --> 00:35:06,000
space.
So, it will relate partial u

477
00:35:06,000 --> 00:35:10,000
partial t to partial derivatives
with respect to x,

478
00:35:10,000 --> 00:35:11,000
y, and z.

479
00:35:42,000 --> 00:35:43,000
[APPLAUSE]
OK, [LAUGHTER]

480
00:35:43,000 --> 00:35:48,000
so what's the equation?
The equation is partial u

481
00:35:48,000 --> 00:35:55,000
partial t equals some constant.
Let me call it constant k times

482
00:35:55,000 --> 00:36:01,000
something I will call del
squared u, which is also called

483
00:36:01,000 --> 00:36:05,000
the Laplacian of u,
and what is that?

484
00:36:05,000 --> 00:36:09,000
Well,
that means,

485
00:36:09,000 --> 00:36:14,000
OK, so just to scare you,
del squared is this,

486
00:36:14,000 --> 00:36:20,000
which means it's the divergence
of gradient u that we've used

487
00:36:20,000 --> 00:36:25,000
this notation for gradient.
OK, so if you actually expand

488
00:36:25,000 --> 00:36:29,000
it in terms of variables,
that becomes partial u over

489
00:36:29,000 --> 00:36:35,000
partial x squared plus partial
squared u over partial y squared

490
00:36:35,000 --> 00:36:40,000
plus partial squared u over
partial z squared.

491
00:36:40,000 --> 00:36:48,000
OK, so the equation is this
equals that.

492
00:36:48,000 --> 00:36:51,000
OK, so that's a weird looking
equation.

493
00:36:51,000 --> 00:36:54,000
And, I'm going to have to
explain to you,

494
00:36:54,000 --> 00:36:57,000
where does it come from?
OK, but before I do that,

495
00:36:57,000 --> 00:37:02,000
well, let me point out actually
that the equation is not just

496
00:37:02,000 --> 00:37:10,000
the diffusion equation.
It's also known as the heat

497
00:37:10,000 --> 00:37:15,000
equation.
And, that's because exactly the

498
00:37:15,000 --> 00:37:21,000
same equation governs how
temperature changes over time

499
00:37:21,000 --> 00:37:25,000
when you have,
again, so, sorry I should have

500
00:37:25,000 --> 00:37:28,000
been actually more careful.
I should have said this is in

501
00:37:28,000 --> 00:37:31,000
air that's not moving,
OK?

502
00:37:31,000 --> 00:37:32,000
OK, and same thing with the
solution.

503
00:37:32,000 --> 00:37:35,000
If you drop some ink into your
glass of water,

504
00:37:35,000 --> 00:37:38,000
well, if you start stirring,
obviously it will mix much

505
00:37:38,000 --> 00:37:40,000
faster than if you don't do
anything.

506
00:37:40,000 --> 00:37:43,000
OK, so that's the case where we
don't actually,

507
00:37:43,000 --> 00:37:47,000
the fluid is not moving.
And, the heat equation now does

508
00:37:47,000 --> 00:37:51,000
the same, but for temperature in
a fluid that's at rest,

509
00:37:51,000 --> 00:37:55,000
that's not moving.
It tells you how the heat goes

510
00:37:55,000 --> 00:37:58,000
from the warmest parts to the
coldest parts,

511
00:37:58,000 --> 00:38:03,000
and eventually temperatures
should somehow even out.

512
00:38:03,000 --> 00:38:08,000
So, in the heat equation,
that would just be,

513
00:38:08,000 --> 00:38:15,000
this u would just measure the
temperature for temperature of

514
00:38:15,000 --> 00:38:19,000
your fluid at a given point.
Actually, it doesn't have to be

515
00:38:19,000 --> 00:38:23,000
a fluid.
It could be a solid for that

516
00:38:23,000 --> 00:38:26,000
heat equation.
So, for example,

517
00:38:26,000 --> 00:38:31,000
say that you have a big box
made of metal or something,

518
00:38:31,000 --> 00:38:34,000
and you do some heating at one
side.

519
00:38:34,000 --> 00:38:38,000
You want to know how quickly is
the other side going to get hot?

520
00:38:38,000 --> 00:38:40,000
Well, you can use the equation
to figure out,

521
00:38:40,000 --> 00:38:44,000
you know, say that you have a
metal bar, and you keep one side

522
00:38:44,000 --> 00:38:46,000
at 400� because it's in your
oven.

523
00:38:46,000 --> 00:38:52,000
How quickly will the other side
get warm?

524
00:38:52,000 --> 00:38:57,000
OK, so it's the same equation
for both phenomena even though

525
00:38:57,000 --> 00:39:00,000
they are, of course,
different phenomena.

526
00:39:00,000 --> 00:39:02,000
Well, the physical reason why
they're the same is actually

527
00:39:02,000 --> 00:39:05,000
that phenomena are different,
but not all that much.

528
00:39:05,000 --> 00:39:07,000
They involve,
actually, how the atoms and

529
00:39:07,000 --> 00:39:11,000
molecules are actually moving,
and hitting each other inside

530
00:39:11,000 --> 00:39:14,000
this medium.
OK, so anyway,

531
00:39:14,000 --> 00:39:17,000
what's the explanation for
this?

532
00:39:17,000 --> 00:39:20,000
So, to understand the
explanation, and given what

533
00:39:20,000 --> 00:39:22,000
we've been doing,
we should have a vector field

534
00:39:22,000 --> 00:39:26,000
in there.
So, I'm going to think of the

535
00:39:26,000 --> 00:39:30,000
flow of, well,
let's imagine that we are doing

536
00:39:30,000 --> 00:39:35,000
motion of smoke in air.
So, that's the flow of the

537
00:39:35,000 --> 00:39:39,000
smoke: that means at every
point, it's a vector whose

538
00:39:39,000 --> 00:39:43,000
direction tells me in which
direction the smoke is actually

539
00:39:43,000 --> 00:39:47,000
moving.
And, its magnitude tells me how

540
00:39:47,000 --> 00:39:52,000
fast it's moving,
and also what amount of smoke

541
00:39:52,000 --> 00:39:56,000
is moving.
So, there's two things to

542
00:39:56,000 --> 00:40:01,000
understand.
One is how the disparities in

543
00:40:01,000 --> 00:40:06,000
the concentration between
different points causes the flow

544
00:40:06,000 --> 00:40:10,000
to be there,
how smoke will flow from the

545
00:40:10,000 --> 00:40:14,000
regions where there's more smoke
to the regions where there's

546
00:40:14,000 --> 00:40:17,000
less smoke.
And, that's actually physics.

547
00:40:17,000 --> 00:40:24,000
But, in a way,
it's also common sense.

548
00:40:24,000 --> 00:40:40,000
So, physics and common sense
tell us that the smoke will flow

549
00:40:40,000 --> 00:40:56,000
from high concentration towards
low concentration regions.

550
00:40:56,000 --> 00:41:01,000
OK, so if we think of this
function, U,

551
00:41:01,000 --> 00:41:04,000
that measures concentration,
that means we are always going

552
00:41:04,000 --> 00:41:07,000
to go in the direction where the
concentration decreases the

553
00:41:07,000 --> 00:41:09,000
fastest.
Well, what's that?

554
00:41:09,000 --> 00:41:25,000
That's negative the gradient.
So, F is directed along minus

555
00:41:25,000 --> 00:41:32,000
gradient u.
Now, how big is F going to be?

556
00:41:32,000 --> 00:41:35,000
Well, they are,
you have to come up with some

557
00:41:35,000 --> 00:41:39,000
intuition for how the two are
related.

558
00:41:39,000 --> 00:41:42,000
And, the easiest relation I can
think of is that they might be

559
00:41:42,000 --> 00:41:44,000
just proportional.
So, the steeper the differences

560
00:41:44,000 --> 00:41:47,000
in concentration,
the faster the flow will be,

561
00:41:47,000 --> 00:41:50,000
or the more there will be flow.
And, if you try to think about

562
00:41:50,000 --> 00:41:53,000
it as molecules moving in random
directions, you will see it

563
00:41:53,000 --> 00:41:56,000
makes actually complete sense.
Anyway, it should be part of

564
00:41:56,000 --> 00:42:00,000
your physics class,
not part of what I'm telling

565
00:42:00,000 --> 00:42:04,000
you.
So, I'm just going to accept

566
00:42:04,000 --> 00:42:12,000
that the flow is just
proportional to the gradient of

567
00:42:12,000 --> 00:42:13,000
u.
So, if you want,

568
00:42:13,000 --> 00:42:16,000
the differences between
concentrations of different

569
00:42:16,000 --> 00:42:18,000
points are very small,
then the flow will be very

570
00:42:18,000 --> 00:42:22,000
gentle.
And, if on the other hand you

571
00:42:22,000 --> 00:42:26,000
have huge disparities,
then things will try to even

572
00:42:26,000 --> 00:42:31,000
out faster.
OK, so that's the first part.

573
00:42:31,000 --> 00:42:35,000
Now, we need to understand the
second part, which is once we

574
00:42:35,000 --> 00:42:38,000
know how flow is going,
how does that affect the

575
00:42:38,000 --> 00:42:40,000
concentration?
If smoke is going that way,

576
00:42:40,000 --> 00:42:43,000
then it means the concentration
here should be decreasing.

577
00:42:43,000 --> 00:42:45,000
And there, it should be
increasing.

578
00:42:45,000 --> 00:42:58,000
So, that's the relation between
F and partial u partial t.

579
00:42:58,000 --> 00:43:07,000
At that part is actually math,
namely, the divergence theorem.

580
00:43:07,000 --> 00:43:19,000
So, let's try to understand
that part more carefully.

581
00:43:19,000 --> 00:43:25,000
So, let's take a small piece of
a small region in space,

582
00:43:25,000 --> 00:43:28,000
D, bounded by a surface,
S.

583
00:43:28,000 --> 00:43:33,000
So, I want to figure out what's
going on in here.

584
00:43:33,000 --> 00:43:42,000
So, let's look at the flux out
of D through S.

585
00:43:42,000 --> 00:43:49,000
Well, we said that this flux
would be given by double

586
00:43:49,000 --> 00:43:58,000
integral on S of F dot n dS.
So, this flux measures how much

587
00:43:58,000 --> 00:44:05,000
smoke is passing through S per
unit time.

588
00:44:05,000 --> 00:44:14,000
That's the amount of smoke
through S per unit time.

589
00:44:14,000 --> 00:44:19,000
But now, how can I compute that
differently?

590
00:44:19,000 --> 00:44:23,000
Well, I can compute it just by
looking at the total amount of

591
00:44:23,000 --> 00:44:26,000
smoke in this region,
and seeing how much it changes.

592
00:44:26,000 --> 00:44:29,000
If I'm gaining or losing smoke,
it means it must be going up

593
00:44:29,000 --> 00:44:32,000
there.
Well, unless I have a smoker in

594
00:44:32,000 --> 00:44:35,000
here, but that's not part of the
data.

595
00:44:35,000 --> 00:44:41,000
So,
that should be, sorry,

596
00:44:41,000 --> 00:44:44,000
that's the same thing as the
derivative with respect to t of

597
00:44:44,000 --> 00:44:47,000
the total amount of smoke in the
region,

598
00:44:47,000 --> 00:44:50,000
which is given by the triple
integral of u.

599
00:44:50,000 --> 00:44:52,000
If I integrate the
concentration of smoke,

600
00:44:52,000 --> 00:44:56,000
which means the amount of smoke
per unit volume over d,

601
00:44:56,000 --> 00:44:59,000
I will get the total amount of
smoke in d,

602
00:44:59,000 --> 00:45:02,000
except,
well,

603
00:45:02,000 --> 00:45:05,000
let's see.
This flux is counted positively

604
00:45:05,000 --> 00:45:07,000
if we go out of d.
So, actually,

605
00:45:07,000 --> 00:45:12,000
it's minus the derivative.
This is the amount of smoke

606
00:45:12,000 --> 00:45:16,000
that we are losing per unit
time.

607
00:45:16,000 --> 00:45:33,000
OK, so now we are almost there.
Well, let me actually --

608
00:45:33,000 --> 00:45:42,000
Because we know yet another way
to compute this guy using the

609
00:45:42,000 --> 00:45:48,000
divergence theorem.
Right, so this part here is

610
00:45:48,000 --> 00:45:53,000
just common sense and thinking
about what it means.

611
00:45:53,000 --> 00:46:00,000
The divergence theorem tells me
this is also equal to the triple

612
00:46:00,000 --> 00:46:06,000
integral, d, of div f dV.
So, what I got is that the

613
00:46:06,000 --> 00:46:15,000
triple integral over d of div F
dV equals this derivative.

614
00:46:15,000 --> 00:46:18,000
Well, let's think a bit about
this derivative so,

615
00:46:18,000 --> 00:46:20,000
see, you are integrating
function over x,

616
00:46:20,000 --> 00:46:22,000
y, and z.
And then, you are

617
00:46:22,000 --> 00:46:24,000
differentiating with respect to
t.

618
00:46:24,000 --> 00:46:28,000
I claim that you can actually
switch the order in which you do

619
00:46:28,000 --> 00:46:30,000
things.
So, when we think about it,

620
00:46:30,000 --> 00:46:33,000
is, here, you are taking the
total amount of smoke and then

621
00:46:33,000 --> 00:46:37,000
see how that changes over time.
So, you're taking the

622
00:46:37,000 --> 00:46:40,000
derivative of the sum of all the
small amounts of smoke

623
00:46:40,000 --> 00:46:42,000
everywhere.
Well, that will be the sum of

624
00:46:42,000 --> 00:46:47,000
the derivatives of the amounts
of smoke inside each little box.

625
00:46:47,000 --> 00:46:55,000
So, we can actually move the
derivatives into the integral.

626
00:46:55,000 --> 00:47:00,000
So, let's see,
I said minus d dt of triple

627
00:47:00,000 --> 00:47:07,000
integral over d udV.
And, now I'm saying this is the

628
00:47:07,000 --> 00:47:14,000
same as the triple integral in d
of du dt dv.

629
00:47:14,000 --> 00:47:19,000
And the reason why this is
going to work is you should

630
00:47:19,000 --> 00:47:24,000
think of it as d dt of a sum of
u of some values.

631
00:47:24,000 --> 00:47:30,000
You plug in the values of your
points at that given time times

632
00:47:30,000 --> 00:47:32,000
the small volume.
You sum them,

633
00:47:32,000 --> 00:47:33,000
and then you take the
derivative.

634
00:47:33,000 --> 00:47:42,000
And now, you see,
the derivative of this sum is

635
00:47:42,000 --> 00:47:49,000
the sum of the derivatives.
yi, zi, t, so,

636
00:47:49,000 --> 00:47:53,000
if you think about what the
integral measures,

637
00:47:53,000 --> 00:47:58,000
that's actually pretty easy.
And it's because this variable

638
00:47:58,000 --> 00:48:01,000
here is not the same as the
variables on which we are

639
00:48:01,000 --> 00:48:03,000
integrating.
That's why we can do it.

640
00:48:03,000 --> 00:48:13,000
OK, so now, if we have this for
any region, d.

641
00:48:13,000 --> 00:48:18,000
So, we have a function of x,
y, z, t, and we have another

642
00:48:18,000 --> 00:48:21,000
function here.
And whenever we integrate them

643
00:48:21,000 --> 00:48:23,000
anywhere, we get the same
answer.

644
00:48:23,000 --> 00:48:26,000
Well, that must mean they're
the same.

645
00:48:26,000 --> 00:48:29,000
Just think of what happens if
you just take d to be a tiny

646
00:48:29,000 --> 00:48:31,000
little box.
You will get roughly the value

647
00:48:31,000 --> 00:48:33,000
of div f at that point times the
volume of the box.

648
00:48:33,000 --> 00:48:36,000
Or, you will get roughly the
value of du dt at that point

649
00:48:36,000 --> 00:48:41,000
times the value of a little box.
So, the values must be the same.

650
00:48:41,000 --> 00:48:46,000
Well, let me actually explain
that a tiny bit better.

651
00:48:46,000 --> 00:48:50,000
So, what I get is that one
over, let me divide by the

652
00:48:50,000 --> 00:49:00,000
volume of D, sorry.
I promise, I'm done in a minute.

653
00:49:00,000 --> 00:49:08,000
Is the same thing as one over
volume D of negative du dt,

654
00:49:08,000 --> 00:49:10,000
dV.
So, that means the average

655
00:49:10,000 --> 00:49:12,000
value,
OK, maybe that's the best way

656
00:49:12,000 --> 00:49:17,000
of telling it,
the average of div f in D is

657
00:49:17,000 --> 00:49:27,000
equal to the average of minus
partial u partial t in D.

658
00:49:27,000 --> 00:49:30,000
And, that's true for any
region, D, not just for some

659
00:49:30,000 --> 00:49:33,000
regions, but for,
really, any region I can think

660
00:49:33,000 --> 00:49:37,000
of.
So, the outcome is that

661
00:49:37,000 --> 00:49:43,000
actually the divergence of f is
equal to minus du dt.

662
00:49:43,000 --> 00:49:47,000
And, that's another way to
think about what divergence

663
00:49:47,000 --> 00:49:48,000
means.
The divergence,

664
00:49:48,000 --> 00:49:50,000
we said, is how much stuff is
actually expanding,

665
00:49:50,000 --> 00:49:54,000
flowing out.
That's how much we're losing.

666
00:49:54,000 --> 00:49:58,000
And so, now,
if you combine this with that,

667
00:49:58,000 --> 00:50:02,000
you will get that du dt is
minus divergence f,

668
00:50:02,000 --> 00:50:08,000
which is plus K del squared u.
OK, so you combine this guy

669
00:50:08,000 --> 00:50:10,000
with that guy,
and you get the diffusion

670
00:50:10,000 --> 00:50:13,000
equation.