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Yesterday we learned about flux
and we have seen the first few
00:00:28.000 --> 00:00:33.000
examples of how to set up and
compute integrals for a flux of
00:00:33.000 --> 00:00:40.000
a vector field for a surface.
Remember the flux of a vector
00:00:40.000 --> 00:00:47.000
field F through the surface S is
defined by taking the double
00:00:47.000 --> 00:00:55.000
integral on the surface of F dot
n dS where n is the unit normal
00:00:55.000 --> 00:01:03.000
to the surface and dS is the
area element on the surface.
00:01:03.000 --> 00:01:06.000
As we have seen,
for various surfaces,
00:01:06.000 --> 00:01:09.000
we have various formulas
telling us what the normal
00:01:09.000 --> 00:01:12.000
vector is and what the area
element becomes.
00:01:12.000 --> 00:01:14.000
For example,
on spheres we typically
00:01:14.000 --> 00:01:18.000
integrate with respect to phi
and theta for latitude and
00:01:18.000 --> 00:01:21.000
longitude angles.
On a horizontal plane,
00:01:21.000 --> 00:01:25.000
we would just end up degrading
dx, dy and so on.
00:01:25.000 --> 00:01:29.000
At the end of lecture we saw a
formula.
00:01:29.000 --> 00:01:32.000
A lot of you asked me how we
got it.
00:01:32.000 --> 00:01:36.000
Well, we didn't get it yet.
We are going to try to explain
00:01:36.000 --> 00:01:39.000
where it comes from and why it
works.
00:01:39.000 --> 00:01:50.000
The case we want to look at is
if S is the graph of a function,
00:01:50.000 --> 00:02:01.000
it is given by z equals some
function in terms of x and y.
00:02:01.000 --> 00:02:08.000
Our surface is out here.
Z is a function of x and y.
00:02:08.000 --> 00:02:17.000
And x and y will range over
some domain in the x,
00:02:17.000 --> 00:02:28.000
y plane, namely the region that
is the shadow of the surface on
00:02:28.000 --> 00:02:34.000
the x, y plane.
I said that we will have a
00:02:34.000 --> 00:02:40.000
formula for n dS which will end
up being plus/minus minus f sub
00:02:40.000 --> 00:02:45.000
x, minus f sub y,
one dxdy,
00:02:45.000 --> 00:02:49.000
so that we will set up and
evaluate the integral in terms
00:02:49.000 --> 00:02:53.000
of x and y.
Every time we see z we will
00:02:53.000 --> 00:02:57.000
replace it by f of xy,
whatever the formula for f
00:02:57.000 --> 00:03:00.000
might be.
Actually, if we look at a very
00:03:00.000 --> 00:03:04.000
easy case where this is just a
horizontal plane,
00:03:04.000 --> 00:03:07.000
z equals constant,
the function is just a
00:03:07.000 --> 00:03:09.000
constant,
well, the partial derivatives
00:03:09.000 --> 00:03:14.000
become just zero.
You get dx, dy.
00:03:14.000 --> 00:03:17.000
That is what you would expect
for a horizontal plane just from
00:03:17.000 --> 00:03:19.000
common sense.
This is more interesting,
00:03:19.000 --> 00:03:23.000
of course, if a function is
more interesting.
00:03:23.000 --> 00:03:30.000
How do we get that?
Where does this come from?
00:03:30.000 --> 00:03:36.000
We need to figure out,
for a small piece of our
00:03:36.000 --> 00:03:43.000
surface, what will be n delta S.
Let's say that we take a small
00:03:43.000 --> 00:03:51.000
rectangle in here corresponding
to sides delta x and delta y and
00:03:51.000 --> 00:03:58.000
we look at the piece of surface
that is above that.
00:03:58.000 --> 00:04:03.000
Well, the question we have now
is what is the area of this
00:04:03.000 --> 00:04:08.000
little piece of surface and what
is its normal vector?
00:04:08.000 --> 00:04:10.000
Observe this little piece up
here.
00:04:10.000 --> 00:04:12.000
If it is small enough,
it will look like a
00:04:12.000 --> 00:04:14.000
parallelogram.
I mean it might be slightly
00:04:14.000 --> 00:04:17.000
curvy, but roughly it looks like
a parallelogram in space.
00:04:17.000 --> 00:04:22.000
And so we have seen how to find
the area of a parallelogram in
00:04:22.000 --> 00:04:25.000
space using cross-product.
If we can figure out what are
00:04:25.000 --> 00:04:29.000
the vectors for this side and
that side then taking that
00:04:29.000 --> 00:04:33.000
cross-product and taking the
magnitude of the cross-product
00:04:33.000 --> 00:04:36.000
will give us the area.
Moreover, the cross-product
00:04:36.000 --> 00:04:38.000
also gives us the normal
direction.
00:04:38.000 --> 00:04:40.000
In fact, the cross-product
gives us two in one.
00:04:40.000 --> 00:04:44.000
It gives us the normal
direction and the area element.
00:04:44.000 --> 00:04:48.000
And that is why I said that we
will have an easy formula for n
00:04:48.000 --> 00:04:52.000
dS while n and dS taken
separately are more complicated
00:04:52.000 --> 00:04:55.000
because you would have to
actually take the length of a
00:04:55.000 --> 00:05:03.000
direction of this guy.
Let's carry out this problem.
00:05:03.000 --> 00:05:13.000
Let's say I am going to look at
a small piece of the x,
00:05:13.000 --> 00:05:17.000
y plane.
Here I have delta x,
00:05:17.000 --> 00:05:22.000
here I have delta y,
and I am starting at some point
00:05:22.000 --> 00:05:28.000
(x, y).
Now, above that I will have a
00:05:28.000 --> 00:05:34.000
parallelogram on my surface.
This point here,
00:05:34.000 --> 00:05:36.000
the point where I start,
I know what it is.
00:05:36.000 --> 00:05:43.000
It is just (x, y).
And, well, z is f(x, y).
00:05:43.000 --> 00:05:44.000
Now what I want to find,
actually,
00:05:44.000 --> 00:05:50.000
is what are these two vectors,
let's call them U and V,
00:05:50.000 --> 00:05:54.000
that correspond to moving a bit
in the x direction or in the y
00:05:54.000 --> 00:05:58.000
direction?
And then U cross V will be,
00:05:58.000 --> 00:06:06.000
well, in terms of the magnitude
of this guy will just be the
00:06:06.000 --> 00:06:11.000
little piece of surface area,
delta S.
00:06:11.000 --> 00:06:15.000
And, in terms of direction,
it will be normal to the
00:06:15.000 --> 00:06:19.000
surface.
Actually, I will get just delta
00:06:19.000 --> 00:06:23.000
S times my normal vector.
Well, up to sign because,
00:06:23.000 --> 00:06:26.000
depending on whether I do U V
or V U, I might get the normal
00:06:26.000 --> 00:06:29.000
vector in the direction I want
or in the opposite direction.
00:06:29.000 --> 00:06:36.000
But we will take care of that
later.
00:06:36.000 --> 00:06:41.000
Let's find U and V.
And, in case you have trouble
00:06:41.000 --> 00:06:47.000
with that small picture,
I have a better one here.
00:06:47.000 --> 00:06:53.000
Let's keep it just in case this
one gets really too cluttered.
00:06:53.000 --> 00:06:57.000
It really represents the same
thing.
00:06:57.000 --> 00:07:05.000
Let's try to figure out these
vectors U and V.
00:07:05.000 --> 00:07:13.000
Vector U starts at the point x,
y, f of x, y and it goes to --
00:07:13.000 --> 00:07:19.000
Whereas, its head,
well, I will have moved x by
00:07:19.000 --> 00:07:23.000
delta x.
So, x plus delta x and y
00:07:23.000 --> 00:07:25.000
doesn't change.
And, of course,
00:07:25.000 --> 00:07:31.000
the z coordinate has to change.
It becomes f of x plus delta x
00:07:31.000 --> 00:07:34.000
and y.
Now, how does f change if I
00:07:34.000 --> 00:07:37.000
change x a little bit?
Well, we have seen that it is
00:07:37.000 --> 00:07:41.000
given by the partial derivative
f sub x.
00:07:41.000 --> 00:07:47.000
This is approximately equal to
f of x, y plus delta x times f
00:07:47.000 --> 00:07:51.000
sub x at the given point x,
y.
00:07:51.000 --> 00:07:56.000
I am not going to add it
because the notation is already
00:07:56.000 --> 00:07:59.000
long enough.
That means my vector U,
00:07:59.000 --> 00:08:05.000
well, approximately because I
am using this linear
00:08:05.000 --> 00:08:08.000
approximation,
00:08:15.000
0, f sub x times delta x>.
Is that OK with everyone?
00:08:15.000 --> 00:08:19.000
Good.
Now, what about V?
00:08:19.000 --> 00:08:23.000
Well, V works the same way so I
am not going to do all the
00:08:23.000 --> 00:08:27.000
details.
When I move from here to here x
00:08:27.000 --> 00:08:30.000
doesn't change and y changes by
delta y.
00:08:30.000 --> 00:08:39.000
X component nothing happens.
Y component changes by delta y.
00:08:39.000 --> 00:08:42.000
What about the z component?
Well, f changes by f sub y
00:08:42.000 --> 00:08:51.000
times delta y.
That is how f changes if I
00:08:51.000 --> 00:08:59.000
increase y by delta y.
I have my two sides.
00:08:59.000 --> 00:09:03.000
Now I can take that
cross-product.
00:09:03.000 --> 00:09:08.000
Well, maybe I will first factor
something out.
00:09:08.000 --> 00:09:13.000
See, I can rewrite this as one,
zero, f sub x times delta x.
00:09:13.000 --> 00:09:20.000
And this one I will rewrite as
zero, one, f sub y delta y.
00:09:20.000 --> 00:09:26.000
And so now the cross-product,
n hat delta S up to sign is
00:09:26.000 --> 00:09:31.000
going to be U cross V.
We will have to do the
00:09:31.000 --> 00:09:35.000
cross-product,
and we will have a delta x,
00:09:35.000 --> 00:09:41.000
delta y coming out.
I am just saving myself the
00:09:41.000 --> 00:09:47.000
trouble of writing a lot of
delta x's and delta y's,
00:09:47.000 --> 00:09:55.000
but if you prefer you can just
do directly this cross-product.
00:09:55.000 --> 00:09:57.000
Let's compute this
cross-product.
00:09:57.000 --> 00:10:03.000
Well, the i component is zero
minus f sub x.
00:10:03.000 --> 00:10:07.000
The y component is going to be,
well, f sub y minus zero but
00:10:07.000 --> 00:10:12.000
with the minus sign in front of
everything, so negative f sub y.
00:10:12.000 --> 00:10:20.000
And the z component will be
just one times delta x delta y.
00:10:20.000 --> 00:10:25.000
Does that make sense?
Yes.
00:10:25.000 --> 00:10:30.000
Very good.
And so now we shrink this
00:10:30.000 --> 00:10:32.000
rectangle,
we shrink delta x and delta y
00:10:32.000 --> 00:10:35.000
to zero,
that is how we get this formula
00:10:35.000 --> 00:10:38.000
for n dS equals negative fx,
negative fy,
00:10:38.000 --> 00:10:43.000
one, dxdy.
Well, plus/minus because it is
00:10:43.000 --> 00:10:47.000
up to us to choose whether we
want to take the normal vector
00:10:47.000 --> 00:10:50.000
point up or down.
See, if you take this
00:10:50.000 --> 00:10:55.000
convention then the z component
of n dS is positive.
00:10:55.000 --> 00:10:59.000
That corresponds to normal
vector pointing up.
00:10:59.000 --> 00:11:02.000
If you take the opposite signs
then the z component will be
00:11:02.000 --> 00:11:06.000
negative.
That means your normal vector
00:11:06.000 --> 00:11:15.000
points down.
This one is with n pointing up.
00:11:15.000 --> 00:11:17.000
I mean when I say up,
of course it is still
00:11:17.000 --> 00:11:21.000
perpendicular to the surface.
If the surface really has a big
00:11:21.000 --> 00:11:26.000
slope then it is not really
going to go all that much up,
00:11:26.000 --> 00:11:31.000
but more up than down.
OK.
00:11:31.000 --> 00:11:40.000
That is how we get the formula.
Any questions?
00:11:40.000 --> 00:11:41.000
No.
OK.
00:11:41.000 --> 00:11:44.000
That is a really useful
formula.
00:11:44.000 --> 00:11:49.000
You don't really need to
remember all the details of how
00:11:49.000 --> 00:11:54.000
we got it, but please remember
that formula.
00:11:54.000 --> 00:12:06.000
Let's do an example, actually.
Let's say we want to find the
00:12:06.000 --> 00:12:08.000
flux of the vector field z times
k,
00:12:08.000 --> 00:12:12.000
so it is a vertical vector
field,
00:12:12.000 --> 00:12:27.000
through the portion of the
paraboloid z equals x^2 y^2 that
00:12:27.000 --> 00:12:36.000
lives above the unit disk.
What does that mean?
00:12:36.000 --> 00:12:39.000
z = x^2 y^2.
We have seen it many times.
00:12:39.000 --> 00:12:43.000
It is this parabola and is
pointing up.
00:12:43.000 --> 00:12:47.000
Above the unit disk means I
don't care about this infinite
00:12:47.000 --> 00:12:52.000
surface.
I will actually stop when I hit
00:12:52.000 --> 00:12:56.000
a radius of one away from the
z-axis.
00:12:56.000 --> 00:13:03.000
And so now I have my vector
field which is going to point
00:13:03.000 --> 00:13:09.000
overall up because,
well, it is z times k.
00:13:09.000 --> 00:13:13.000
The more z is positive,
the more your vector field goes
00:13:13.000 --> 00:13:16.000
up.
Of course, if z were negative
00:13:16.000 --> 00:13:20.000
then it would point down,
but it will live above.
00:13:20.000 --> 00:13:25.000
Actually, a quick opinion poll.
What do you think the flux
00:13:25.000 --> 00:13:28.000
should be?
Should it be positive,
00:13:28.000 --> 00:13:32.000
zero, negative or we don't
know?
00:13:32.000 --> 00:13:36.000
I see some I don't know,
I see some negative and I see
00:13:36.000 --> 00:13:39.000
some positive.
Of course, I didn't tell you
00:13:39.000 --> 00:13:42.000
which way I am orienting my
paraboloid.
00:13:42.000 --> 00:13:44.000
So far both answers are correct.
The only one that is probably
00:13:44.000 --> 00:13:47.000
not correct is zero because,
no matter which way you choose
00:13:47.000 --> 00:13:49.000
to orient it you should get
something.
00:13:49.000 --> 00:13:51.000
It is not looking like it will
be zero.
00:13:51.000 --> 00:14:00.000
Let's say that I am going to do
it with the normal pointing
00:14:00.000 --> 00:14:06.000
upwards.
Second chance.
00:14:06.000 --> 00:14:11.000
I see some people changing back
and forth from one and two.
00:14:11.000 --> 00:14:14.000
Let's draw a picture.
Which one is pointing upwards?
00:14:14.000 --> 00:14:16.000
Well, let's look at the bottom
point.
00:14:16.000 --> 00:14:19.000
The normal vector pointing up,
here we know what it means.
00:14:19.000 --> 00:14:22.000
It is this guy.
If you continue to follow your
00:14:22.000 --> 00:14:26.000
normal vector,
see, they are actually pointing
00:14:26.000 --> 00:14:30.000
up and into the paraboloid.
And I claim that the answer
00:14:30.000 --> 00:14:34.000
should be positive because the
vector field is crossing our
00:14:34.000 --> 00:14:38.000
paraboliod going upwards,
going from the outside out and
00:14:38.000 --> 00:14:46.000
below to the inside and upside.
So, in the direction that we
00:14:46.000 --> 00:14:55.000
are counting positively.
We will see how it turns out
00:14:55.000 --> 00:15:03.000
when we do the calculation.
We have to compute the integral
00:15:03.000 --> 00:15:08.000
for flux.
Double integral over a surface
00:15:08.000 --> 00:15:15.000
of F dot n dS is going to be --
What are we going to do?
00:15:15.000 --> 00:15:21.000
Well, F we said is <0,0,
z>.
00:15:21.000 --> 00:15:25.000
What is n dS.
Well, let's use our brand new
00:15:25.000 --> 00:15:28.000
formula.
It says negative f sub x,
00:15:28.000 --> 00:15:31.000
negative f sub y,
one, dxdy.
00:15:31.000 --> 00:15:38.000
What does little f in here?
It is x^2 y^2.
00:15:38.000 --> 00:15:42.000
When we are using this formula,
we need to know what little x
00:15:42.000 --> 00:15:47.000
stands for.
It is whatever the formula is
00:15:47.000 --> 00:15:53.000
for z as a function of x and y.
We take x^2 y^2 and we take the
00:15:53.000 --> 00:15:57.000
partial derivatives with minus
signs.
00:15:57.000 --> 00:16:01.000
We get negative 2x,
negative 2y and one,
00:16:01.000 --> 00:16:03.000
dxdy.
Well, of course here it didn't
00:16:03.000 --> 00:16:05.000
really matter because we are
going to dot them with zero.
00:16:05.000 --> 00:16:11.000
Actually, even if we had made a
mistake we somehow wouldn't have
00:16:11.000 --> 00:16:15.000
had to pay the price.
But still.
00:16:15.000 --> 00:16:22.000
We will end up with double
integral on S of z dxdy.
00:16:22.000 --> 00:16:25.000
Now, what do we do with that?
Well, we have too many things.
00:16:25.000 --> 00:16:43.000
We have to get rid of z.
Let's use z equals x^2 y^2 once
00:16:43.000 --> 00:16:51.000
more.
That becomes double integral of
00:16:51.000 --> 00:16:54.000
x^2 y^2 dxdy.
And here, see,
00:16:54.000 --> 00:16:57.000
we are using the fact that we
are only looking at things that
00:16:57.000 --> 00:16:59.000
are on the surface.
It is not like in a triple
00:16:59.000 --> 00:17:01.000
integral.
You could never do that because
00:17:01.000 --> 00:17:04.000
z, x and y are independent.
Here they are related by the
00:17:04.000 --> 00:17:09.000
equation of a surface.
If I sound like I am ranting,
00:17:09.000 --> 00:17:14.000
but I know from experience this
is where one of the most sticky
00:17:14.000 --> 00:17:18.000
and tricky points is.
OK.
00:17:18.000 --> 00:17:20.000
How will we actually integrate
that?
00:17:20.000 --> 00:17:22.000
Well, now that we have just x
and y, we should figure out what
00:17:22.000 --> 00:17:25.000
is the range for x and y.
Well, the range for x and y is
00:17:25.000 --> 00:17:27.000
going to be the shadow of our
region.
00:17:27.000 --> 00:17:34.000
It is going to be this unit
disk.
00:17:34.000 --> 00:17:43.000
I can just do that for now.
And this is finally where I
00:17:43.000 --> 00:17:46.000
have left the world of surface
integrals to go back to a usual
00:17:46.000 --> 00:17:49.000
double integral.
And now I have to set it up.
00:17:49.000 --> 00:17:52.000
Well, I can do it this way with
dxdy, but it looks like there is
00:17:52.000 --> 00:17:55.000
a smarter thing to do.
I am going to use polar
00:17:55.000 --> 00:17:59.000
coordinates.
In fact, I am going to say this
00:17:59.000 --> 00:18:03.000
is double integral of r^2 times
r dr d theta.
00:18:03.000 --> 00:18:07.000
I am on the unit disk so r goes
zero to one, theta goes zero to
00:18:07.000 --> 00:18:12.000
2pi.
And, if you do the calculation,
00:18:12.000 --> 00:18:19.000
you will find that this is
going to be pi over two.
00:18:19.000 --> 00:18:26.000
Any questions about the example.
Yes?
00:18:26.000 --> 00:18:29.000
How did I get this negative 2x
and negative 2y?
00:18:29.000 --> 00:18:33.000
I want to use my formula for n
dS.
00:18:33.000 --> 00:18:36.000
My surface is given by the
graph of a function.
00:18:36.000 --> 00:18:40.000
It is the graph of a function
x^2 y^2.
00:18:40.000 --> 00:18:43.000
I will use this formula that is
up here.
00:18:43.000 --> 00:18:47.000
I will take the function x^2
y^2 and I will take its partial
00:18:47.000 --> 00:18:50.000
derivatives.
If I take the partial of f,
00:18:50.000 --> 00:18:55.000
so x^2 y^2 with respect to x,
I get 2x, so I put negative 2x.
00:18:55.000 --> 00:18:57.000
And then the same thing,
negative 2y,
00:18:57.000 --> 00:19:01.000
one, dxdy.
Yes?
00:19:01.000 --> 00:19:04.000
Which k hat?
Oh, you mean the vector field.
00:19:04.000 --> 00:19:06.000
It is a different part of the
story.
00:19:06.000 --> 00:19:10.000
Whenever you do a surface
integral for flux you have two
00:19:10.000 --> 00:19:13.000
parts of the story.
One is the vector field whose
00:19:13.000 --> 00:19:17.000
flux you are taking.
The other one is the surface
00:19:17.000 --> 00:19:21.000
for which you will be taking
flux.
00:19:21.000 --> 00:19:26.000
The vector field only comes as
this f in the notation,
00:19:26.000 --> 00:19:28.000
and everything else,
the bounds in the double
00:19:28.000 --> 00:19:32.000
integral and the n dS,
all come from the surface that
00:19:32.000 --> 00:19:36.000
we are looking at.
Basically, in all of this
00:19:36.000 --> 00:19:40.000
calculation, this is coming from
f equals zk.
00:19:40.000 --> 00:19:47.000
Everything else comes from the
information paraboloid z = x^2
00:19:47.000 --> 00:19:51.000
y^2 above the unit disk.
In particular,
00:19:51.000 --> 00:19:55.000
if we wanted to now find the
flux of any other vector field
00:19:55.000 --> 00:19:58.000
for the same paraboloid,
well, all we would have to do
00:19:58.000 --> 00:20:02.000
is just replace this guy by
whatever the new vector field
00:20:02.000 --> 00:20:05.000
is.
We have learned how to set up
00:20:05.000 --> 00:20:09.000
flux integrals for this
paraboloid.
00:20:09.000 --> 00:20:10.000
Not that you should remember
this one by heart.
00:20:10.000 --> 00:20:13.000
I mean there are many
paraboloids in life and other
00:20:13.000 --> 00:20:17.000
surfaces, too.
It is better to remember the
00:20:17.000 --> 00:20:22.000
general method.
Any other questions?
00:20:22.000 --> 00:20:25.000
No.
OK.
00:20:25.000 --> 00:20:31.000
Let's see more ways of taking
flux integrals.
00:20:31.000 --> 00:20:33.000
But, just to reassure you,
at this point we have seen the
00:20:33.000 --> 00:20:36.000
most important ones.
90% of the problems that we
00:20:36.000 --> 00:20:41.000
will be looking at we can do
with what we have seen so far in
00:20:41.000 --> 00:20:49.000
less time and this formula.
Let's look a little bit at a
00:20:49.000 --> 00:20:56.000
more general situation.
Let's say that my surface is so
00:20:56.000 --> 00:21:00.000
complicated that I cannot
actually express z as a function
00:21:00.000 --> 00:21:04.000
of x and y, but let's say that I
know how to parametize it.
00:21:04.000 --> 00:21:06.000
I have a parametric equation
for my surface.
00:21:06.000 --> 00:21:11.000
That means I can express x,
y and z in terms of any two
00:21:11.000 --> 00:21:16.000
parameter variables that might
be relevant for me.
00:21:16.000 --> 00:21:19.000
If you want,
this one here is a special case
00:21:19.000 --> 00:21:23.000
where you can parameterize
things in terms of x and y as
00:21:23.000 --> 00:21:26.000
your two variables.
How would you do it in the
00:21:26.000 --> 00:21:29.000
fully general case?
In a way, that will answer your
00:21:29.000 --> 00:21:30.000
question that,
I think one of you,
00:21:30.000 --> 00:21:34.000
I forgot, asked yesterday how
would I do it in general?
00:21:34.000 --> 00:21:36.000
Is there a formula like M dx
plus N dy?
00:21:36.000 --> 00:21:39.000
Well, that is going to be the
general formula.
00:21:39.000 --> 00:21:42.000
And you will see that it is a
little bit too complicated,
00:21:42.000 --> 00:21:46.000
so the really useful ones are
actually the special ones.
00:21:46.000 --> 00:21:56.000
Let's say that we are given a
parametric description -- -- of
00:21:56.000 --> 00:22:01.000
a surface S.
That means we can describe S by
00:22:01.000 --> 00:22:05.000
formulas saying x is some
function of two parameter
00:22:05.000 --> 00:22:08.000
variables.
I am going to call them u and v.
00:22:08.000 --> 00:22:10.000
I hope you don't mind.
You can call them t1 and t2.
00:22:10.000 --> 00:22:18.000
You can call them whatever you
want.
00:22:18.000 --> 00:22:21.000
One of the basic properties of
a surface is because I have only
00:22:21.000 --> 00:22:23.000
two independent directions to
move on.
00:22:23.000 --> 00:22:26.000
I should be able to express x,
y and z in terms of two
00:22:26.000 --> 00:22:29.000
variables.
Now, let's say that I know how
00:22:29.000 --> 00:22:32.000
to do that.
Or, maybe I should instead
00:22:32.000 --> 00:22:35.000
think of it in terms of a
position vector if it helps you.
00:22:35.000 --> 00:22:40.000
That is just a vector with
components 00:22:46.000
y, z> is given as a function
of u and v.
00:22:46.000 --> 00:22:50.000
It works like a parametric
curve but with two parameters.
00:22:50.000 --> 00:22:54.000
Now, how would we actually set
up a flux integral on such a
00:22:54.000 --> 00:22:57.000
surface.
Well, because we are locating
00:22:57.000 --> 00:23:01.000
ourselves in terms of u and v,
we will end up with an integral
00:23:01.000 --> 00:23:06.000
du dv.
We need to figure out how to
00:23:06.000 --> 00:23:11.000
express n dS in terms of du and
dv.
00:23:11.000 --> 00:23:19.000
N dS should be something du dv.
How do we do that?
00:23:19.000 --> 00:23:25.000
Well, we can use the same
method that we have actually
00:23:25.000 --> 00:23:28.000
used over here.
Because, if you think for a
00:23:28.000 --> 00:23:30.000
second, here we used,
of course, a rectangle in the
00:23:30.000 --> 00:23:34.000
x, y plane and we lifted it to a
parallelogram and so on.
00:23:34.000 --> 00:23:37.000
But more generally you can
think what happens if I change u
00:23:37.000 --> 00:23:41.000
by delta u keeping v constant or
the other way around?
00:23:41.000 --> 00:23:45.000
You will get some sort of mesh
grid on your surface and you
00:23:45.000 --> 00:23:48.000
will look at a little
parallelogram that is an
00:23:48.000 --> 00:23:52.000
elementary piece of that mesh
and figure out what is its area
00:23:52.000 --> 00:23:57.000
and normal vector.
Well, that will again be given
00:23:57.000 --> 00:24:01.000
by the cross-product of the two
sides.
00:24:01.000 --> 00:24:07.000
Let's think a little bit about
what happens when I move a
00:24:07.000 --> 00:24:12.000
little bit on my surface.
I am taking this grid on my
00:24:12.000 --> 00:24:16.000
surface given by the u and v
directions.
00:24:16.000 --> 00:24:25.000
And, if I take a piece of that
corresponding to small changes
00:24:25.000 --> 00:24:33.000
delta u and delta v,
what is going to be going on
00:24:33.000 --> 00:24:36.000
here?
Well, I have to deal with two
00:24:36.000 --> 00:24:39.000
vectors, one corresponding to
changing u, the other one
00:24:39.000 --> 00:24:42.000
corresponding to changing v.
If I change u,
00:24:42.000 --> 00:24:46.000
how does my point change?
Well, it is given by the
00:24:46.000 --> 00:24:49.000
derivative of this with respect
to u.
00:24:49.000 --> 00:24:57.000
This vector here I will call,
so the sides are given by,
00:24:57.000 --> 00:25:05.000
let me say, partial r over
partial u times delta u.
00:25:05.000 --> 00:25:09.000
If you prefer,
maybe I should write it as
00:25:09.000 --> 00:25:13.000
partial x over partial u times
delta u.
00:25:13.000 --> 00:25:17.000
Well, it is just too boring to
write.
00:25:17.000 --> 00:25:21.000
And so on.
It means if I change u a little
00:25:21.000 --> 00:25:24.000
bit, keeping v constant,
then how x changes is,
00:25:24.000 --> 00:25:26.000
given by partial x over partial
u times delta u,
00:25:26.000 --> 00:25:28.000
same thing with y,
same thing with z,
00:25:28.000 --> 00:25:33.000
and I am just using vector
notation to do it this way.
00:25:33.000 --> 00:25:41.000
That is the analog of when I
said delta r for line integrals
00:25:41.000 --> 00:25:47.000
along a curve,
vector delta r is the velocity
00:25:47.000 --> 00:25:58.000
vector dr dt times delta t.
Now, if I look at the other
00:25:58.000 --> 00:26:07.000
side -- Let me start again.
I ran out of space.
00:26:07.000 --> 00:26:12.000
One side is partial r over
partial u times delta u.
00:26:12.000 --> 00:26:17.000
And the other one would be
partial r over partial v times
00:26:17.000 --> 00:26:19.000
delta v.
Because that is how the
00:26:19.000 --> 00:26:24.000
position of your point changes
if you just change u or v and
00:26:24.000 --> 00:26:31.000
not the other one.
To find the surface element
00:26:31.000 --> 00:26:40.000
together with a normal vector,
I would just take the
00:26:40.000 --> 00:26:46.000
cross-product between these
guys.
00:26:46.000 --> 00:26:50.000
If you prefer,
that is the cross-product of
00:26:50.000 --> 00:26:56.000
partial r over partial u with
partial r over partial v,
00:26:56.000 --> 00:27:02.000
delta u delta v.
And so n dS is this
00:27:02.000 --> 00:27:11.000
cross-product times du dv up to
sign.
00:27:11.000 --> 00:27:27.000
It depends on which choice I
make for my normal vector,
00:27:27.000 --> 00:27:32.000
of course.
That, of course,
00:27:32.000 --> 00:27:35.000
is a slightly confusing
equation to think of.
00:27:35.000 --> 00:27:37.000
A good exercise,
if you want to really
00:27:37.000 --> 00:27:40.000
understand what is going on,
try this in two good examples
00:27:40.000 --> 00:27:43.000
to look at.
One good example to look at is
00:27:43.000 --> 00:27:45.000
the previous one.
What is it?
00:27:45.000 --> 00:27:47.000
It is when u and v are just x
and y.
00:27:47.000 --> 00:27:51.000
The parametric equations are
just x equals x,
00:27:51.000 --> 00:27:54.000
y equals y and z is f of x,
y.
00:27:54.000 --> 00:27:59.000
You should end up with the same
formula that we had over there.
00:27:59.000 --> 00:28:03.000
And you should see why because
both of them are given by a
00:28:03.000 --> 00:28:06.000
cross-product.
The other case you can look at
00:28:06.000 --> 00:28:08.000
just to convince yourselves even
further.
00:28:08.000 --> 00:28:12.000
We don't need to do that
because we have seen the formula
00:28:12.000 --> 00:28:17.000
before, but in the case of a
sphere we have seen the formula
00:28:17.000 --> 00:28:22.000
for n and for dS separately.
We know what n dS are in terms
00:28:22.000 --> 00:28:26.000
of d phi, d theta.
Well, you could parametize a
00:28:26.000 --> 00:28:28.000
sphere in terms of phi and
theta.
00:28:28.000 --> 00:28:33.000
Namely, the formulas would be x
equals a sine phi cosine theta,
00:28:33.000 --> 00:28:38.000
y equals a sign phi sine theta,
z equals a cosine phi.
00:28:38.000 --> 00:28:42.000
The formulas for circle
coordinates setting Ro equals a
00:28:42.000 --> 00:28:44.000
.
That is a parametric equation
00:28:44.000 --> 00:28:47.000
for the sphere.
And then, if you try to use
00:28:47.000 --> 00:28:50.000
this formula here,
you should end up with the same
00:28:50.000 --> 00:28:52.000
things we have already seen for
n dS,
00:28:52.000 --> 00:28:57.000
just with a lot more pain to
actually get there because
00:28:57.000 --> 00:29:00.000
cross-product is going to be a
bit complicated.
00:29:00.000 --> 00:29:03.000
But we are seeing all of these
formulas all fitting together.
00:29:03.000 --> 00:29:05.000
Somehow it is always the same
question.
00:29:05.000 --> 00:29:10.000
We just have different angles
of attack on this general
00:29:10.000 --> 00:29:18.000
problem.
Questions?
00:29:18.000 --> 00:29:20.000
No.
OK.
00:29:20.000 --> 00:29:30.000
Let's look at yet another last
way of finding n dS.
00:29:30.000 --> 00:29:36.000
And then I promise we will
switch to something else because
00:29:36.000 --> 00:29:42.000
I can feel that you are getting
a bit overwhelmed for all these
00:29:42.000 --> 00:29:47.000
formulas for n dS.
What happens very often is we
00:29:47.000 --> 00:29:51.000
don't actually know how to
parametize our surface.
00:29:51.000 --> 00:29:54.000
Maybe we don't know how to
solve for z as a function of x
00:29:54.000 --> 00:29:58.000
and y, but our surface is given
by some equation.
00:29:58.000 --> 00:30:05.000
And so what that means is
actually maybe what we know is
00:30:05.000 --> 00:30:12.000
not really these kinds of
formulas, but maybe we know a
00:30:12.000 --> 00:30:17.000
normal vector.
And I am going to call this one
00:30:17.000 --> 00:30:22.000
capital N because I don't even
need it to be a unit vector.
00:30:22.000 --> 00:30:27.000
You will see.
It can be a normal vector of
00:30:27.000 --> 00:30:33.000
any length you want to the
surfaces.
00:30:33.000 --> 00:30:35.000
Why would we ever know a normal
vector?
00:30:35.000 --> 00:30:39.000
Well, for example,
if our surface is a plane,
00:30:39.000 --> 00:30:43.000
a slanted plane given by some
equation, ax by cz = d.
00:30:43.000 --> 00:30:44.000
Well, you know the normal
vector.
00:30:44.000 --> 00:30:48.000
It is .
Of course, you could solve for
00:30:48.000 --> 00:30:52.000
z and then go back to that case,
which is why I said that one is
00:30:52.000 --> 00:30:55.000
very useful.
But you can also just stay with
00:30:55.000 --> 00:30:58.000
a normal vector.
Why else would you know a
00:30:58.000 --> 00:31:01.000
normal vector?
Well, let's say that you know
00:31:01.000 --> 00:31:05.000
an equation that is of a form g
of x, y, z equals zero.
00:31:05.000 --> 00:31:08.000
Well, then you know that the
gradient of g is perpendicular
00:31:08.000 --> 00:31:14.000
to the level surface.
Let me just give you two
00:31:14.000 --> 00:31:19.000
examples.
If you have a plane,
00:31:19.000 --> 00:31:24.000
ax by cz = d,
then the normal vector would
00:31:24.000 --> 00:31:28.000
just be .
00:31:28.000 --> 00:31:34.000
If you have a surface S given
by an equation,
00:31:34.000 --> 00:31:40.000
g(x, y, z) = 0,
then you can take a normal
00:31:40.000 --> 00:31:46.000
vector to be the gradient of g.
We have seen that the gradient
00:31:46.000 --> 00:31:49.000
is perpendicular to the level
surface.
00:31:49.000 --> 00:31:51.000
Now, of course,
we don't necessarily have to
00:31:51.000 --> 00:31:55.000
follow what is going to come.
Because, if we could solve for
00:31:55.000 --> 00:31:59.000
z, then we might be better off
doing what we did over there.
00:31:59.000 --> 00:32:02.000
But let's say that we want to
do it this.
00:32:02.000 --> 00:32:06.000
What can we do?
Well, I am going to give you
00:32:06.000 --> 00:32:09.000
another way to think
geometrically about n dS.
00:32:40.000 --> 00:32:43.000
Let's start by thinking about
the slanted plane.
00:32:43.000 --> 00:32:47.000
Let's say that my surface is
just a slanted plane.
00:32:47.000 --> 00:32:52.000
My normal vector would be maybe
somewhere here.
00:32:52.000 --> 00:32:55.000
And let's say that I am going
to try -- I need to get some
00:32:55.000 --> 00:32:57.000
handle on how to set up my
integrals,
00:32:57.000 --> 00:33:00.000
so maybe I am going to express
things in terms of x and y.
00:33:00.000 --> 00:33:05.000
I have my coordinates,
and I will try to use x and y.
00:33:05.000 --> 00:33:11.000
Then I would like to relate
delta S or dS to the area in the
00:33:11.000 --> 00:33:14.000
x y plane.
That means I want maybe to look
00:33:14.000 --> 00:33:19.000
at the projection of this guy
onto a horizontal plane.
00:33:19.000 --> 00:33:31.000
Let's squish it horizontally.
Then you have here another area.
00:33:31.000 --> 00:33:35.000
The guy on the slanted plane,
let's call that delta S.
00:33:35.000 --> 00:33:38.000
And let's call this guy down
here delta A.
00:33:38.000 --> 00:33:42.000
And delta A would become
ultimately maybe delta x,
00:33:42.000 --> 00:33:47.000
delta y or something like that.
The question is how do we find
00:33:47.000 --> 00:33:51.000
the conversion rate between
these two areas?
00:33:51.000 --> 00:33:53.000
I mean they are not the same.
Visually, I hope it is clear to
00:33:53.000 --> 00:33:56.000
you that if my plane is actually
horizontal then,
00:33:56.000 --> 00:34:00.000
of course, they are the same.
But the more slanted it becomes
00:34:00.000 --> 00:34:04.000
the more delta A becomes smaller
than delta S.
00:34:04.000 --> 00:34:09.000
If you buy land and it is on
the side of a cliff,
00:34:09.000 --> 00:34:12.000
well, whether you look at it on
a map or whether you look at it
00:34:12.000 --> 00:34:15.000
on the actual cliff,
the area is going to be very
00:34:15.000 --> 00:34:18.000
different.
I am not sure if that is a wise
00:34:18.000 --> 00:34:22.000
thing to do if you want to build
a house there,
00:34:22.000 --> 00:34:26.000
but I bet you can get really
cheap land.
00:34:26.000 --> 00:34:31.000
Anyway, delta S versus delta A
depends on how slanted things
00:34:31.000 --> 00:34:34.000
are.
And let's try to make that more
00:34:34.000 --> 00:34:41.000
precise by looking at the angel
that our plane makes with the
00:34:41.000 --> 00:34:47.000
horizontal direction.
Let's call this angle alpha,
00:34:47.000 --> 00:34:53.000
the angle that our plane makes
with the horizontal direction.
00:34:53.000 --> 00:34:57.000
See, it is all coming together.
The first unit about
00:34:57.000 --> 00:35:03.000
cross-products,
normal vectors and so on is
00:35:03.000 --> 00:35:09.000
actually useful now.
I claim that the surface
00:35:09.000 --> 00:35:16.000
element is related to the area
in the plane by delta A equals
00:35:16.000 --> 00:35:19.000
delta S times the cosine of
alpha.
00:35:19.000 --> 00:35:24.000
Why is that?
Well, let's look at this small
00:35:24.000 --> 00:35:27.000
rectangle with one horizontal
side and one slanted side.
00:35:27.000 --> 00:35:33.000
When you project this side does
not change, but this side gets
00:35:33.000 --> 00:35:37.000
shortened by a factor of cosine
alpha.
00:35:37.000 --> 00:35:41.000
Whatever this length was,
this length here is that one
00:35:41.000 --> 00:35:44.000
times cosine alpha.
That is why the area gets
00:35:44.000 --> 00:35:48.000
shrunk by cosine alpha.
In one direction nothing
00:35:48.000 --> 00:35:52.000
happens.
In the other direction you
00:35:52.000 --> 00:35:59.000
squish by cosine alpha.
What that means is that,
00:35:59.000 --> 00:36:05.000
well, we will have to deal with
this.
00:36:05.000 --> 00:36:08.000
And, of course,
the one we will care about
00:36:08.000 --> 00:36:11.000
actually is delta S expressed in
terms of delta A.
00:36:11.000 --> 00:36:13.000
But what are we going to do
with this cosine?
00:36:13.000 --> 00:36:15.000
It is not very convenient to
have a cosine left in here.
00:36:15.000 --> 00:36:19.000
Remember, the angle between two
planes is the same thing as the
00:36:19.000 --> 00:36:21.000
angle between the normal
vectors.
00:36:21.000 --> 00:36:23.000
If you want to see this angle
alpha elsewhere,
00:36:23.000 --> 00:36:27.000
what you can do is you can just
take the vertical direction.
00:36:27.000 --> 00:36:38.000
Let's take k.
Then here we have our angle
00:36:38.000 --> 00:36:43.000
alpha again.
In particular,
00:36:43.000 --> 00:36:46.000
cosine of alpha,
I can get, well,
00:36:46.000 --> 00:36:50.000
we know how to find the angle
between two vectors.
00:36:50.000 --> 00:36:57.000
If we have our normal vector N,
we will do N dot k,
00:36:57.000 --> 00:37:02.000
and we will divide by length N,
length k.
00:37:02.000 --> 00:37:06.000
Well, length k is one.
That is one easy guy.
00:37:06.000 --> 00:37:12.000
That is how we find the angle.
Now I am going to say,
00:37:12.000 --> 00:37:21.000
well, delta S is going to be
one over cosine alpha delta A.
00:37:21.000 --> 00:37:36.000
And I can rewrite that as
length of N divided by N dot k
00:37:36.000 --> 00:37:42.000
times delta A.
Now, let's multiply that by the
00:37:42.000 --> 00:37:48.000
unit normal vector.
Because what we are about is
00:37:48.000 --> 00:37:52.000
not so much dS but actually n
dS.
00:37:52.000 --> 00:38:04.000
N delta S will be,
I am just going to multiply by
00:38:04.000 --> 00:38:07.000
N.
Well, let's think for a second.
00:38:07.000 --> 00:38:11.000
What happens if I take a unit
normal N and I multiply it by
00:38:11.000 --> 00:38:14.000
the length of my other normal
big N?
00:38:14.000 --> 00:38:19.000
Well, I get big N again.
This is a normal vector of the
00:38:19.000 --> 00:38:22.000
same length as N,
well, up to sign.
00:38:22.000 --> 00:38:27.000
The only thing I don't know is
whether this guy will be going
00:38:27.000 --> 00:38:32.000
in the same direction as big N
or in the opposite direction.
00:38:32.000 --> 00:38:35.000
Say that, for example,
my capital N has,
00:38:35.000 --> 00:38:39.000
I don't know,
length three for example.
00:38:39.000 --> 00:38:43.000
Then the normal unit vector
might be this guy,
00:38:43.000 --> 00:38:47.000
in which case indeed three
times little n will be big n.
00:38:47.000 --> 00:38:52.000
Or it might be this one in
which case three times little n
00:38:52.000 --> 00:38:58.000
will be negative big N.
But up to sign it is N.
00:38:58.000 --> 00:39:02.000
And then I will have N over N
dot k delta A.
00:39:02.000 --> 00:39:07.000
And so the final formula,
the one that we care about in
00:39:07.000 --> 00:39:11.000
case you don't really like my
explanations of how we get
00:39:11.000 --> 00:39:19.000
there,
is that N dS is plus or minus N
00:39:19.000 --> 00:39:27.000
over N dot k dx dy.
That one is actually kind of
00:39:27.000 --> 00:39:31.000
useful so let's box it.
Now,
00:39:31.000 --> 00:39:35.000
just in case you are wondering,
of course, if you didn't want
00:39:35.000 --> 00:39:36.000
to project to x,
y,
00:39:36.000 --> 00:39:39.000
you would have maybe preferred
to project to say the plane of a
00:39:39.000 --> 00:39:40.000
blackboard, y,
z,
00:39:40.000 --> 00:39:45.000
well, you can do the same thing.
To express n dS in terms of dy
00:39:45.000 --> 00:39:49.000
dz you do the same argument.
Simply, the only thing that
00:39:49.000 --> 00:39:51.000
changes, instead of using the
vertical vector k,
00:39:51.000 --> 00:39:56.000
you use the normal vector i.
So you would be doing N over N
00:39:56.000 --> 00:39:58.000
dot i dy dz.
The same thing.
00:39:58.000 --> 00:40:04.000
So just keep an open mind that
this also works with other
00:40:04.000 --> 00:40:09.000
variables.
Anyway, that is how you can
00:40:09.000 --> 00:40:17.000
basically project the vectors of
this area element onto the x,
00:40:17.000 --> 00:40:23.000
y plane in a way.
Let's look at the special case
00:40:23.000 --> 00:40:29.000
just to see how this fits with
stuff we have seen before.
00:40:29.000 --> 00:40:37.000
Let's do a special example
where our surface is given by
00:40:37.000 --> 00:40:44.000
the equation z minus f of x,
y equals zero.
00:40:44.000 --> 00:40:46.000
That is a strange way to write
the equation.
00:40:46.000 --> 00:40:50.000
z equals f of x, y.
That we saw before.
00:40:50.000 --> 00:40:53.000
But now it looks like some
function of x,
00:40:53.000 --> 00:40:57.000
y, z equals zero.
Let's try to use this new
00:40:57.000 --> 00:41:05.000
method.
Let's call this guy g(x, y, z).
00:41:05.000 --> 00:41:07.000
Well, now let's look at the
normal vector.
00:41:07.000 --> 00:41:10.000
The normal vector would be the
gradient of g,
00:41:10.000 --> 00:41:13.000
you see.
What is the gradient of this
00:41:13.000 --> 00:41:17.000
function?
The gradient of g -- Well,
00:41:17.000 --> 00:41:22.000
partial g, partial x,
that is just negative partial
00:41:22.000 --> 00:41:25.000
f, partial x.
The y component,
00:41:25.000 --> 00:41:32.000
partial g, partial y is going
to be negative f sub y,
00:41:32.000 --> 00:41:42.000
and g sub z is just one.
Now, if you take N over N dot k
00:41:42.000 --> 00:41:46.000
dx dy,
well, it looks like it is going
00:41:46.000 --> 00:41:50.000
to be negative f sub x,
negative f sub y,
00:41:50.000 --> 00:41:53.000
one divided by -- Well,
what is N dot k?
00:41:53.000 --> 00:41:57.000
If you dot that with k you will
get just one,
00:41:57.000 --> 00:42:01.000
so I am not going to write it,
dx dy.
00:42:01.000 --> 00:42:05.000
See, that is again our favorite
formula.
00:42:05.000 --> 00:42:12.000
This one is actually more
general because you don't need
00:42:12.000 --> 00:42:18.000
to solve for z,
but if you cannot solve for z
00:42:18.000 --> 00:42:28.000
then it is the same as before.
I think that is enough formulas
00:42:28.000 --> 00:42:34.000
for n dS.
After spending a lot of time
00:42:34.000 --> 00:42:41.000
telling you how to compute
surface integrals,
00:42:41.000 --> 00:42:51.000
now I am going to try to tell
you how to avoid computing them.
00:42:51.000 --> 00:43:05.000
And that is called the
divergence theorem.
00:43:05.000 --> 00:43:09.000
And we will see the proof and
everything and applications on
00:43:09.000 --> 00:43:12.000
Tuesday, but I want to at least
the theorem and see how it works
00:43:12.000 --> 00:43:15.000
in one example.
It is also known as the
00:43:15.000 --> 00:43:19.000
Gauss-Green theorem or just the
Gauss theorem,
00:43:19.000 --> 00:43:24.000
depending in who you talk to.
The Green here is the same
00:43:24.000 --> 00:43:28.000
Green as in Green's theorem,
because somehow that is a space
00:43:28.000 --> 00:43:31.000
version of Green's theorem.
What does it say?
00:43:31.000 --> 00:43:43.000
It is 3D analog of Green for
flux.
00:43:43.000 --> 00:43:47.000
What it says is if S is a
closed surface -- Remember,
00:43:47.000 --> 00:43:52.000
it is the same as with Green's
theorem, we need to have
00:43:52.000 --> 00:43:56.000
something that is completely
enclosed.
00:43:56.000 --> 00:44:00.000
You have a surface and there is
somehow no gaps in it.
00:44:00.000 --> 00:44:06.000
There is no boundary to it.
It is really completely
00:44:06.000 --> 00:44:17.000
enclosing a region in space that
I will call D.
00:44:17.000 --> 00:44:19.000
And I need to choose my
orientation.
00:44:19.000 --> 00:44:29.000
The orientation that will work
for this theorem is choosing the
00:44:29.000 --> 00:44:43.000
normal vector to point outwards.
N needs to be outwards.
00:44:43.000 --> 00:44:46.000
That is one part of the puzzle.
The other part is a vector
00:44:46.000 --> 00:44:51.000
field.
I need to have a vector field
00:44:51.000 --> 00:44:58.000
that is defined and
differentiable -- -- everywhere
00:44:58.000 --> 00:45:04.000
in D, so same instructions as
usual.
00:45:04.000 --> 00:45:12.000
Then I don't have actually to
compute the flux integral.
00:45:12.000 --> 00:45:17.000
Double integral of f dot n dS
of a closed surface S.
00:45:17.000 --> 00:45:19.000
I am going to put a circle just
to remind you it is has got to
00:45:19.000 --> 00:45:22.000
be a closed surface.
It is just a notation to remind
00:45:22.000 --> 00:45:26.000
us closed surface.
I can replace that by the
00:45:26.000 --> 00:45:32.000
triple integral of a region
inside of divergence of F dV.
00:45:32.000 --> 00:45:36.000
Now, I need to tell you what
the divergence of a 3D vector
00:45:36.000 --> 00:45:39.000
field is.
Well, you will see that it is
00:45:39.000 --> 00:45:41.000
not much harder than in the 2D
case.
00:45:41.000 --> 00:45:58.000
What you do is just -- Say that
your vector field has components
00:45:58.000 --> 00:46:06.000
P, Q and R.
Then you will take P sub x Q
00:46:06.000 --> 00:46:10.000
sub y R sub z.
That is the definition.
00:46:10.000 --> 00:46:13.000
It is pretty easy to remember.
You take the x component
00:46:13.000 --> 00:46:18.000
partial respect to S plus
partial respect to y over y
00:46:18.000 --> 00:46:23.000
component plus partial respect
to z of the z component.
00:46:23.000 --> 00:46:33.000
For example,
last time we saw that the flux
00:46:33.000 --> 00:46:42.000
of the vector field zk through a
sphere of radius a was
00:46:42.000 --> 00:46:53.000
four-thirds pi a cubed by
computing the surface integral.
00:46:53.000 --> 00:46:56.000
Well, if we do it more
efficiently now by Green's
00:46:56.000 --> 00:46:59.000
theorem, we are going to use
Green's theorem for this sphere
00:46:59.000 --> 00:47:02.000
because we are doing the whole
sphere.
00:47:02.000 --> 00:47:04.000
It is fine.
It is a closed surface.
00:47:04.000 --> 00:47:05.000
We couldn't do it for,
say, the hemisphere or
00:47:05.000 --> 00:47:09.000
something like that.
Well, for a hemisphere we would
00:47:09.000 --> 00:47:15.000
need to add maybe the flat face
of a bottom or something like
00:47:15.000 --> 00:47:20.000
that.
Green's theorem says that our
00:47:20.000 --> 00:47:28.000
flux integral can actually be
replaced by the triple integral
00:47:28.000 --> 00:47:36.000
over the solid bowl of radius a
of the divergence of zk dV.
00:47:36.000 --> 00:47:40.000
But now what is the divergence
of this field?
00:47:40.000 --> 00:47:48.000
Well, you have zero,
zero, z so you get zero plus
00:47:48.000 --> 00:47:52.000
zero plus one.
It looks like it will be one.
00:47:52.000 --> 00:47:59.000
If you do the triple integral
of 1dV, you will get just the
00:47:59.000 --> 00:48:06.000
volume -- -- of the region
inside, which is four-thirds by
00:48:06.000 --> 00:48:09.000
a cubed.
And so it was no accident.
00:48:09.000 --> 00:48:13.000
In fact, before that we looked
at also xi yj zk and we found
00:48:13.000 --> 00:48:17.000
three times the volume.
That is because the divergence
00:48:17.000 --> 00:48:19.000
of that field was actually
three.
00:48:19.000 --> 00:48:22.000
Very quickly,
let me just say what this means
00:48:22.000 --> 00:48:24.000
physically.
Physically, see,
00:48:24.000 --> 00:48:29.000
this guy on the left is the
total amount of stuff that goes
00:48:29.000 --> 00:48:34.000
out of the region per unit time.
I want to figure out how much
00:48:34.000 --> 00:48:37.000
stuff comes out of there.
What does the divergence mean?
00:48:37.000 --> 00:48:41.000
The divergence means it
measures how much the flow is
00:48:41.000 --> 00:48:43.000
expanding things.
It measures how much,
00:48:43.000 --> 00:48:46.000
I said that probably when we
were trying to understand 2D
00:48:46.000 --> 00:48:49.000
divergence.
It measures the amount of
00:48:49.000 --> 00:48:54.000
sources or sinks that you have
inside your fluid.
00:48:54.000 --> 00:48:57.000
Now it becomes commonsense.
If you take a region of space,
00:48:57.000 --> 00:49:01.000
the total amount of water that
flows out of it is the total
00:49:01.000 --> 00:49:05.000
amount of sources that you have
in there minus the sinks.
00:49:05.000 --> 00:49:08.000
I mean, in spite of this
commonsense explanation,
00:49:08.000 --> 00:49:10.000
we are going to see how to
prove this.
00:49:10.000 --> 00:49:14.000
And we will see how it works
and what it says.