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We are going to continue to
look at stuff in space.

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We have been working with
triple integrals and seeing how

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to set them up in all sorts of
coordinate systems.

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And the next topic we will be
looking at are vector fields in

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space.
And so, in particular,

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we will be learning about flux
and work.

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So, just for a change,
we will be starting with flux

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first.
And we will do work,

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actually, after Thanksgiving.
Just to remind you,

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a vector field in space is just
the same thing as in the plane.

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At every point you have a
vector, and the components of

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this vector depend on the
coordinates x,

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y and z.
Let's say the components might

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be P, Q, R, or your favorite
three letters,

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00:01:42,000 --> 00:01:53,000
where each of these things is a
function of coordinates x,

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y, z.
You have seen that in the plane

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it is already pretty hard to
draw a vector field.

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Usually, in space,
we won't really try too hard.

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00:02:01,000 --> 00:02:04,000
But it is still useful to try
to have a general idea for what

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the vectors in there are doing,
whether they are all going in

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the same direction,
whether they may be all

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vertical or horizontal,
pointing away from the origin,

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towards it,
things like that.

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But, generally-speaking,
we won't really bother with

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00:02:19,000 --> 00:02:24,000
trying to draw a picture because
that is going to be quite hard.

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00:02:24,000 --> 00:02:30,000
Just to give you examples,
well, the same kinds of

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examples as the plane,
you can think of force fields.

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For example,
the gravitational attraction --

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-- of a solid mass,
let's call this mass big M,

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at the origin on a mass M at
point x, y, z.

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That would be given by a vector
field that points toward the

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origin and whose magnitude is
inversely proportional to the

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square of a distance from the
origin.

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Such a field would be directed
towards the origin and its

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magnitude would be of the order
of a constant over pho squared

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where pho is the distance from
the origin.

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00:03:53,000 --> 00:03:57,000
The picture,
if I really wanted to draw a

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picture, would be everywhere it
is a field that points towards

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the origin.
And if I am further away then

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it gets smaller.
And, of course,

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I am not going to try to draw
all these vectors in there.

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If I wanted to give a formula
for that -- A formula for that

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might be something of a form
minus c times x,

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y, z over pho cubed.
Let's see.

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Well, the direction of this
vector, this vector is

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proportional to negative x,
y, z.

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00:04:47,000 --> 00:04:49,000
is the vector that goes from
the origin to your point.

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The negative goes towards the
origin.

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Then the magnitude of this guy,
well, the magnitude of x,

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y, z is just the distance from
the origin rho.

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So the magnitude of this thing
is one over rho cubed times some

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constant factor.
That would be an example of a

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vector field that comes up in
physics.

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Well, other examples would be
electric fields.

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Actually, if you look at the
electric field generated by a

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charged particle at the origin,
it is given by exactly the same

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kind of formula,
and there are magnetic fields

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and so on.
Another example comes from

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velocity fields.
If you have a fluid flow,

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for example,
if you want to study wind

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patterns in the atmosphere.
Well, wind, most of the time,

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is kind of horizontal,
but maybe it depends on the

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altitude.
At high altitude you have jet

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streams, and the wind velocity
is not the same at all

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altitudes.
And, just to give you more

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examples, in math we have seen
that the gradient of a function

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of three variables gives you a
vector field.

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If you have a function u of x,
y, z then its gradient field

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has just components,
u sub x, u sub y and u sub z.

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And, of course,
the cases are not mutually

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exclusive.
For example,

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00:06:34,000 --> 00:06:38,000
the electric field or
gravitational field is given by

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the gradient of the
gravitational or electric

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potential.
So, these are not like

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different cases.
There is overlap.

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Anyway, hopefully,
you are kind of convinced that

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you should learn about vector
fields.

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What are we going to do with
them?

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Well, let's start with flux.
Remember not so long ago we

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looked at flux of a
two-dimensional field of a

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curve.
We had a curve in the plane and

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we had a vector field.
And we looked at the component

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of a vector field in the
direction that was normal to the

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curve.
We formed the flux integral

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that was a line integral F dot n
ds.

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And that measured how much the
vector field was going across

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the curve.
If you were thinking of a

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00:08:01,000 --> 00:08:05,000
velocity field,
that would measure how much

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00:08:05,000 --> 00:08:08,000
fluid is passing through the
curve in unit time.

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Now let's say that we were in
space.

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Well, we cannot really think of
flux as a line integral.

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00:08:14,000 --> 00:08:18,000
Because, if you have a curve in
space and say that you have wind

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00:08:18,000 --> 00:08:22,000
or something like that,
you cannot really ask how much

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00:08:22,000 --> 00:08:24,000
air is flowing through the
curve.

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00:08:24,000 --> 00:08:28,000
See, to have a flow through
something you need a surface.

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If you have a net maybe then
you can ask how much stuff is

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passing through that surface.
There is going to be a big

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difference here.
In the three-dimensional space,

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flux will be measured through a
surface.

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And so it will be a surface
integral, not a line integral

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00:08:54,000 --> 00:08:59,000
anymore.
That means we will be

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integrating, we will be summing
over all the pieces of a surface

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in space.
Because a surface is a

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two-dimensional object,
that will end up being a double

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00:09:15,000 --> 00:09:17,000
integral.
But, of course,

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we will have to set it up
properly because the surface

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00:09:19,000 --> 00:09:22,000
that is in space,
and we will probably have x,

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00:09:22,000 --> 00:09:24,000
y and z to deal with at the
same time,

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00:09:24,000 --> 00:09:28,000
and we will have to somehow get
rid of one variable so that we

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00:09:28,000 --> 00:09:31,000
can set up and evaluate a double
integral.

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00:09:31,000 --> 00:09:35,000
So conceptually it is very
similar to line integrals.

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00:09:35,000 --> 00:09:39,000
In the line integral in the
plane, you had two variables

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that you reduced to one by
figuring out what the curve was.

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00:09:42,000 --> 00:09:51,000
Here you have three variables
that you will reduce to two by

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00:09:51,000 --> 00:09:56,000
figuring out what the surface
is.

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00:09:56,000 --> 00:10:00,000
Let me give you a definition of
flux in 3D.

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00:10:00,000 --> 00:10:15,000
Let's say that we have a vector
field and s, a surface in space.

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00:10:15,000 --> 00:10:17,000
Let me draw some kind of a
picture.

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00:10:17,000 --> 00:10:21,000
I have my surface and I have my
vector field F.

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00:10:21,000 --> 00:10:25,000
Well, at every point it changes
with a point.

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00:10:25,000 --> 00:10:28,000
Well, I want to figure out how
much my vector field is going

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00:10:28,000 --> 00:10:34,000
across that surface.
That means I want to figure out

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00:10:34,000 --> 00:10:40,000
the normal component of my
vector field,

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00:10:40,000 --> 00:10:45,000
so I will use,
as in the plane case,

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00:10:45,000 --> 00:10:53,000
the unit normal vector to s.
I take my point on the surface

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00:10:53,000 --> 00:10:59,000
and build a unit vector that is
standing on it perpendicularly.

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00:10:59,000 --> 00:11:05,000
Now, we have to decide which
way it is standing.

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00:11:05,000 --> 00:11:09,000
We can build our normal vector
to go this way or to go the

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00:11:09,000 --> 00:11:12,000
other way around.
There are two choices.

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00:11:12,000 --> 00:11:16,000
Basically, whenever you want to
set up a flux integral you have

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00:11:16,000 --> 00:11:19,000
to choose one side of the
surface.

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00:11:19,000 --> 00:11:23,000
And you will count positively
what flows toward that side and

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00:11:23,000 --> 00:11:26,000
negatively what flows towards
the other side.

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00:11:26,000 --> 00:11:40,000
There are two choices for n.
We need to choose a side of the

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00:11:40,000 --> 00:11:45,000
surface.
In the case of curves,

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00:11:45,000 --> 00:11:50,000
we made that choice by deciding
that because we were going along

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00:11:50,000 --> 00:11:54,000
some direction on the curve we
could choose one side by saying

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00:11:54,000 --> 00:11:57,000
let's rotate clockwise from the
tangent vector.

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00:11:57,000 --> 00:12:00,000
And, in a way,
what we were doing was really

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00:12:00,000 --> 00:12:04,000
it was a recipe to choose for us
one of the two sides.

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00:12:04,000 --> 00:12:09,000
Here we don't have a notion of
orienting the surface other than

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00:12:09,000 --> 00:12:14,000
by precisely choosing one of the
two possible normal vectors.

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00:12:14,000 --> 00:12:15,000
So, in fact,
this is called choosing an

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00:12:15,000 --> 00:12:18,000
orientation of a surface.
When you are saying you are

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00:12:18,000 --> 00:12:22,000
orienting the surface that
really means you are deciding

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which side is which.
Let's call that orientation.

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00:12:31,000 --> 00:12:35,000
Now, there is no set convention
that will work forever.

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00:12:35,000 --> 00:12:39,000
But the usually traditional
settings would be to take your

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normal vector pointing maybe out
of the solid region because then

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00:12:43,000 --> 00:12:48,000
you will be looking at flux that
is coming out of that region of

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00:12:48,000 --> 00:12:51,000
space.
Or, if you have a surface that

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is not like closed or anything
but maybe you will want the flux

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00:12:55,000 --> 00:12:59,000
going up through the region.
Or, there are various

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00:12:59,000 --> 00:13:02,000
conventions.
Concretely, on problem sets it

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00:13:02,000 --> 00:13:05,000
will either say which choice you
have to make or you get to

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00:13:05,000 --> 00:13:07,000
choose which one you want to
make.

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00:13:07,000 --> 00:13:10,000
And, of course,
if you choose the other one

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00:13:10,000 --> 00:13:12,000
then the sign becomes the
opposite.

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00:13:12,000 --> 00:13:17,000
Now, once we have made a choice
then we can define the flux

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00:13:17,000 --> 00:13:21,000
integral.
It will just be the double

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00:13:21,000 --> 00:13:26,000
integral over a surface of F dot
n dS.

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00:13:26,000 --> 00:13:33,000
Now I am using a big dS.
That stands for the surface

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00:13:33,000 --> 00:13:39,000
area element on this surface.
I am using dS rather than dA

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00:13:39,000 --> 00:13:43,000
because I still want to think of
dA as maybe the area in one of

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00:13:43,000 --> 00:13:47,000
the coordinate planes like the
one we had in double integrals.

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00:13:47,000 --> 00:13:51,000
You will see later where this
comes in.

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00:13:51,000 --> 00:13:54,000
But conceptually it is very
similar.

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00:13:54,000 --> 00:13:59,000
Concretely what this means is I
cut my surface into little

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00:13:59,000 --> 00:14:03,000
pieces.
Each of them has area delta S.

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00:14:03,000 --> 00:14:07,000
And, for each piece,
I take my vector field,

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00:14:07,000 --> 00:14:13,000
I take my normal vector,
I dot them and I multiply by

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00:14:13,000 --> 00:14:17,000
this surface area and sum all
these things together.

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00:14:17,000 --> 00:14:23,000
That is what a double integral
means.

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00:14:23,000 --> 00:14:25,000
In particular,
an easy case where you know you

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00:14:25,000 --> 00:14:28,000
can get away without computing
anything is, of course,

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00:14:28,000 --> 00:14:32,000
if your vector field is tangent
to the surface because then you

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00:14:32,000 --> 00:14:36,000
know that there is no flux.
Flux is going to be zero

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00:14:36,000 --> 00:14:38,000
because nothing passes through
the surface.

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00:14:38,000 --> 00:14:42,000
Otherwise, we have to figure
out how to compute these things.

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00:14:42,000 --> 00:14:50,000
That is what we are going to
learn now.

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00:14:50,000 --> 00:14:51,000
Well, maybe I should box this
formula.

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00:14:51,000 --> 00:14:57,000
I have noticed that some of you
seem to like it when I box the

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00:14:57,000 --> 00:15:03,000
important formulas.
(APPLAUSE) By the way,

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00:15:03,000 --> 00:15:12,000
a piece of notation before I
move on, sometimes you will also

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00:15:12,000 --> 00:15:18,000
see the notation vector dS.
What is vector dS?

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00:15:18,000 --> 00:15:24,000
Vector dS is this guy n dS put
together.

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00:15:24,000 --> 00:15:30,000
Vector dS is a vector which
points perpendicular to the

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00:15:30,000 --> 00:15:35,000
surface and whose length
corresponds to the surface

195
00:15:35,000 --> 00:15:37,000
element.
And the reason for having this

196
00:15:37,000 --> 00:15:41,000
shortcut notation,
well, it is not only laziness

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00:15:41,000 --> 00:15:44,000
like saving one n,
but it is because this guy is

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00:15:44,000 --> 00:15:49,000
very often easier to compute
than it is to set up n and dS

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00:15:49,000 --> 00:15:52,000
separately.
Actually, if you remember in

200
00:15:52,000 --> 00:15:56,000
the plane, we have seen that
vector n little ds can be

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00:15:56,000 --> 00:15:59,000
written directly as dy,
- dx.

202
00:15:59,000 --> 00:16:03,000
That was easier than finding n
and ds separately.

203
00:16:03,000 --> 00:16:15,000
And here the same is going to
be true in many cases.

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00:16:15,000 --> 00:16:23,000
Well, any questions before we
do examples?

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00:16:23,000 --> 00:16:24,000
No.
OK.

206
00:16:24,000 --> 00:16:38,000
Let's do examples.
The first example for today is

207
00:16:38,000 --> 00:16:51,000
we are going to look at the flux
of vector field xi yj xk through

208
00:16:51,000 --> 00:17:02,000
the sphere of radius a -- --
centered at the origin.

209
00:17:02,000 --> 00:17:18,000
What does the picture look like?
We have a sphere of radius a.

210
00:17:18,000 --> 00:17:22,000
I have my vector field.
Well, , see,

211
00:17:22,000 --> 00:17:25,000
that is a vector field that is
equal to the vector from the

212
00:17:25,000 --> 00:17:35,000
origin to the point where I am,
so it is pointing radially away

213
00:17:35,000 --> 00:17:42,000
from the origin.
My vector field is really

214
00:17:42,000 --> 00:17:49,000
sticking out everywhere away
from the origin.

215
00:17:49,000 --> 00:17:56,000
Now I have to find the normal
vector to the sphere if I want

216
00:17:56,000 --> 00:18:04,000
to set up double integral over
the sphere of F dot vector ds,

217
00:18:04,000 --> 00:18:09,000
or if you want F dot n dS.
What does the normal vector to

218
00:18:09,000 --> 00:18:12,000
the sphere look like?
Well, it depends,

219
00:18:12,000 --> 00:18:14,000
of course, whether I choose it
pointing out or in.

220
00:18:14,000 --> 00:18:18,000
Let's say I am choosing it
pointing out then it will be

221
00:18:18,000 --> 00:18:20,000
sticking straight out of a
sphere as well.

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00:18:20,000 --> 00:18:27,000
Hopefully, you can see that if
I take a normal vector to the

223
00:18:27,000 --> 00:18:34,000
sphere it is actually pointing
radially out away from the

224
00:18:34,000 --> 00:18:38,000
origin.
In fact, our vector field and

225
00:18:38,000 --> 00:18:41,000
our normal vector are parallel
to each other.

226
00:18:41,000 --> 00:18:45,000
Let's think a bit more about
what a normal vector looks like.

227
00:18:45,000 --> 00:18:47,000
I said it is sticking straight
out.

228
00:18:47,000 --> 00:18:49,000
It is proportional to this
vector field.

229
00:18:49,000 --> 00:18:51,000
Maybe I should start by writing

230
00:18:52,000 --> 00:18:56,000
because that is the vector that
goes from the origin to my point

231
00:18:56,000 --> 00:18:59,000
so it points radially away from
the origin.

232
00:18:59,000 --> 00:19:00,000
Now there is a small problem
with that.

233
00:19:00,000 --> 00:19:04,000
It is not a unit vector.
So what is its length?

234
00:19:04,000 --> 00:19:08,000
Well, its length is square root
of x^2 y^2 z^2.

235
00:19:08,000 --> 00:19:13,000
But, if I am on the sphere,
then that length is just equal

236
00:19:13,000 --> 00:19:16,000
to a because distance from the
origin is a.

237
00:19:16,000 --> 00:19:23,000
In fact, I get my normal vector
by scaling this guy down by a

238
00:19:23,000 --> 00:19:27,000
factor of a.
And let me write it down just

239
00:19:27,000 --> 00:19:34,000
in case you are still unsure.
This is unit because square

240
00:19:34,000 --> 00:19:43,000
root of x^2 y^2 z^2 is equal to
a on the sphere.

241
00:19:43,000 --> 00:19:48,000
OK.
Any questions about this?

242
00:19:48,000 --> 00:19:52,000
No. It looks OK?
I see a lot of blank faces.

243
00:19:52,000 --> 00:19:58,000
That physics test must have
been hard.

244
00:19:58,000 --> 00:20:03,000
Yes?
I could have put a rho but I

245
00:20:03,000 --> 00:20:06,000
want to emphasize the fact that
here it is going to be a

246
00:20:06,000 --> 00:20:09,000
constant.
I mean rho has this connotation

247
00:20:09,000 --> 00:20:13,000
of being a variable that I will
need to then maybe integrate

248
00:20:13,000 --> 00:20:17,000
over or do something with.
Yes, it would be correct to put

249
00:20:17,000 --> 00:20:20,000
rho but I then later will want
to replace it by its actual

250
00:20:20,000 --> 00:20:24,000
value which is a number.
And the number is a.

251
00:20:24,000 --> 00:20:28,000
It is not going to actually
change from point to point.

252
00:20:28,000 --> 00:20:30,000
For example,
if this was the unit sphere

253
00:20:30,000 --> 00:20:32,000
then I would just put x,
y, z.

254
00:20:32,000 --> 00:20:41,000
I wouldn't divide by anything.
Now let's figure out F dot n.

255
00:20:41,000 --> 00:20:47,000
Let's do things one at a time.
Well, F and n are parallel to

256
00:20:47,000 --> 00:20:53,000
each other.
F dot n, the normal component

257
00:20:53,000 --> 00:21:00,000
of F, is actually equal to the
length of F.

258
00:21:00,000 --> 00:21:05,000
Well, times the length of n if
you want, but that is going to

259
00:21:05,000 --> 00:21:09,000
be a one since F and n are
parallel to each other.

260
00:21:09,000 --> 00:21:12,000
And what is the magnitude of F
if I am on the sphere?

261
00:21:12,000 --> 00:21:18,000
Well, the magnitude of F in
general is square root of x^2

262
00:21:18,000 --> 00:21:23,000
y^2 z^2 on the sphere that is
going be a.

263
00:21:23,000 --> 00:21:25,000
The other way to do it,
if you don't want to think

264
00:21:25,000 --> 00:21:28,000
geometrically like that,
is to just to do the dot

265
00:21:28,000 --> 00:21:31,000
product x, y,
z doted with x over a,

266
00:21:31,000 --> 00:21:35,000
y over a, z over a.
You will be x^2 y^2 z^2 divided

267
00:21:35,000 --> 00:21:40,000
by a.
That will simplify to a because

268
00:21:40,000 --> 00:21:45,000
we are on the sphere.
See, we are already using here

269
00:21:45,000 --> 00:21:47,000
the relation between x,
y and z.

270
00:21:47,000 --> 00:21:49,000
We are not letting x,
y and z be completely

271
00:21:49,000 --> 00:21:51,000
arbitrary.
But the slogan is everything

272
00:21:51,000 --> 00:21:54,000
happens on the surface where we
are doing the integral.

273
00:21:54,000 --> 00:21:56,000
We are not looking at anything
inside or outside.

274
00:21:56,000 --> 00:21:58,000
We are just on the surface.

275
00:22:34,000 --> 00:22:43,000
Now what do I do with that?
Well, I have turned my integral

276
00:22:43,000 --> 00:22:50,000
into the double integral of a
dS.

277
00:22:50,000 --> 00:22:53,000
And a is just a constant,
so I am very lucky here.

278
00:22:53,000 --> 00:22:59,000
I can just say this will be a
times the double integral of dS.

279
00:22:59,000 --> 00:23:02,000
And, of course,
some day I will have to learn

280
00:23:02,000 --> 00:23:05,000
how to tackle that beast,
but for now I don't actually

281
00:23:05,000 --> 00:23:08,000
need to because the double
integral of dS just means I am

282
00:23:08,000 --> 00:23:11,000
summing the area of each little
piece of the sphere.

283
00:23:11,000 --> 00:23:16,000
I am just going to get the
total area of the sphere which I

284
00:23:16,000 --> 00:23:23,000
know to be 4pi a2.
This guy here is going to be

285
00:23:23,000 --> 00:23:29,000
the area of S.
I know that to be 4pi a^2.

286
00:23:29,000 --> 00:23:37,000
So I will get 4pi a^3.
That one was relatively

287
00:23:37,000 --> 00:23:44,000
painless.
That was too easy.

288
00:23:44,000 --> 00:23:49,000
Let's do a second example with
the same sphere.

289
00:23:49,000 --> 00:23:56,000
But now my vector field is
going to be just z times k.

290
00:23:56,000 --> 00:23:58,000
Well, let me give it a
different name.

291
00:23:58,000 --> 00:24:04,000
Let me call it H instead of f
or something like that just so

292
00:24:04,000 --> 00:24:08,000
that it is not called F anymore.
Well, the initial part of the

293
00:24:08,000 --> 00:24:11,000
setup is still the same.
The normal vector is still the

294
00:24:11,000 --> 00:24:13,000
same.
What changes is,

295
00:24:13,000 --> 00:24:16,000
of course, my vector field is
no longer sticking straight out

296
00:24:16,000 --> 00:24:18,000
so I cannot use this easy
geometric argument.

297
00:24:18,000 --> 00:24:22,000
It looks like I will have to
compute F dot n and then figure

298
00:24:22,000 --> 00:24:24,000
out how to integrate that with
dS.

299
00:24:24,000 --> 00:24:36,000
Let's do that.
We still have that n is <x,

300
00:24:36,000 --> 00:24:44,000
y, z>/a.
That tells us that H dot n will

301
00:24:44,000 --> 00:24:49,000
be
dot <x, y,

302
00:24:49,000 --> 00:24:57,000
z> / a.
It looks like I will be left

303
00:24:57,000 --> 00:25:10,000
with z^2 over a.
H dot n is z^2 over a.

304
00:25:10,000 --> 00:25:18,000
The double integral for flux
now becomes double integral on

305
00:25:18,000 --> 00:25:25,000
the sphere of z^2 over a dS.
Well, we can take out one over

306
00:25:25,000 --> 00:25:29,000
a, that is fine,
but it looks like we will have

307
00:25:29,000 --> 00:25:33,000
to integrate z^2 on the surface
of the sphere.

308
00:25:33,000 --> 00:25:37,000
How do we do that?
Well, we have to figure out

309
00:25:37,000 --> 00:25:41,000
what is dS in terms of our
favorite set of two variables

310
00:25:41,000 --> 00:25:45,000
that we will use to integrate.
Now, what is the best way to

311
00:25:45,000 --> 00:25:47,000
figure out where you are on the
sphere?

312
00:25:47,000 --> 00:25:51,000
Well, you could try to use
maybe theta and z.

313
00:25:51,000 --> 00:25:55,000
If you know how high you are
and where you are around,

314
00:25:55,000 --> 00:25:58,000
in principle you know where you
are on the sphere.

315
00:25:58,000 --> 00:26:02,000
But since spherical coordinates
we have actually learned about

316
00:26:02,000 --> 00:26:06,000
something much more interesting,
namely spherical coordinates.

317
00:26:06,000 --> 00:26:09,000
It looks like longitude /
latitude is the way to go when

318
00:26:09,000 --> 00:26:12,000
trying to figure out where you
are on a sphere.

319
00:26:12,000 --> 00:26:19,000
We are going to use phi and
theta.

320
00:26:19,000 --> 00:26:24,000
And, of course,
we have to figure out how to

321
00:26:24,000 --> 00:26:28,000
express dS in terms of d phi and
d theta.

322
00:26:28,000 --> 00:26:32,000
Well, if you were paying
really, really close attention

323
00:26:32,000 --> 00:26:36,000
last time, you will notice that
we have actually already done

324
00:26:36,000 --> 00:26:41,000
that.
Last time we saw that if I have

325
00:26:41,000 --> 00:26:48,000
a sphere of radius a and I take
a little piece of it that

326
00:26:48,000 --> 00:26:56,000
corresponds to small changes in
phi and theta then we said that

327
00:26:56,000 --> 00:27:01,000
-- Well,
we argued that this side here,

328
00:27:01,000 --> 00:27:08,000
the one that is going east-west
was a piece of the circle that

329
00:27:08,000 --> 00:27:14,000
has a radius a sin phi because
that is r,

330
00:27:14,000 --> 00:27:19,000
so that side is a sin phi delta
theta.

331
00:27:19,000 --> 00:27:22,000
And the side that goes
north-south is a piece of the

332
00:27:22,000 --> 00:27:26,000
circle of radius a corresponding
to angle delta phi,

333
00:27:26,000 --> 00:27:32,000
so it is a delta phi.
And so, just to get to the

334
00:27:32,000 --> 00:27:40,000
answer, we got dS equals a^2 sin
phi d phi d theta.

335
00:27:40,000 --> 00:27:45,000
When we set up a surface
integral on the surface of a

336
00:27:45,000 --> 00:27:48,000
sphere,
most likely we will be using

337
00:27:48,000 --> 00:27:52,000
phi and theta as our two
variables of integration and dS

338
00:27:52,000 --> 00:27:55,000
will become this.
Now, it is OK to think of them

339
00:27:55,000 --> 00:27:58,000
as spherical coordinates,
but I would like to encourage

340
00:27:58,000 --> 00:28:01,000
you not to think of them as
spherical coordinates.

341
00:28:01,000 --> 00:28:05,000
Spherical coordinates are a way
of describing points in space in

342
00:28:05,000 --> 00:28:09,000
terms of three variables.
Here it is more like we are

343
00:28:09,000 --> 00:28:12,000
parameterizing the sphere.
We are finding a parametric

344
00:28:12,000 --> 00:28:15,000
equation for the sphere using
two variables phi and theta

345
00:28:15,000 --> 00:28:18,000
which happen to be part of the
spherical coordinate system.

346
00:28:18,000 --> 00:28:22,000
But, see, there is no rho
involved in here.

347
00:28:22,000 --> 00:28:26,000
I am not using any rho ever,
and I am not going to in this

348
00:28:26,000 --> 00:28:28,000
calculation.
I have two variable phi and

349
00:28:28,000 --> 00:28:37,000
theta.
That is it.

350
00:28:37,000 --> 00:28:40,000
It is basically in the same way
as when you parameterize a line

351
00:28:40,000 --> 00:28:45,000
integral in the circle,
we use theta as the parameter

352
00:28:45,000 --> 00:28:50,000
variable and never think about
r.

353
00:28:50,000 --> 00:28:52,000
That being said,
well, we are going to use phi

354
00:28:52,000 --> 00:28:54,000
and theta.
We know what dS is.

355
00:28:54,000 --> 00:28:58,000
We still need to figure out
what z is.

356
00:28:58,000 --> 00:29:01,000
There we want to think a tiny
bit about spherical coordinates

357
00:29:01,000 --> 00:29:08,000
again.
And we will know that z is just

358
00:29:08,000 --> 00:29:15,000
a cos phi.
In case you don't quite see it,

359
00:29:15,000 --> 00:29:25,000
let me draw a diagram.
Phi is the angle down from the

360
00:29:25,000 --> 00:29:31,000
positive z axes,
this distance is a,

361
00:29:31,000 --> 00:29:38,000
so this distance here is a cos
phi.

362
00:29:38,000 --> 00:29:44,000
Now I have everything I need to
compute my double integral.

363
00:29:44,000 --> 00:29:49,000
z^2 over a dS will become a
double integral.

364
00:29:49,000 --> 00:30:00,000
z^2 becomes a^2 cos^2 phi over
a times, ds becomes,

365
00:30:00,000 --> 00:30:07,000
a^2 sin phi d phi d theta.
Now I need to set up bounds.

366
00:30:07,000 --> 00:30:12,000
Well, what are the bounds?
Phi goes all the way from zero

367
00:30:12,000 --> 00:30:19,000
to pi because we go all the way
from the north pole to the south

368
00:30:19,000 --> 00:30:23,000
pole, and theta goes from zero
to 2pi.

369
00:30:23,000 --> 00:30:27,000
And, of course,
I can get rid of some a's in

370
00:30:27,000 --> 00:30:34,000
there and take them out.
Let's look at what number we

371
00:30:34,000 --> 00:30:37,000
get.
First of all,

372
00:30:37,000 --> 00:30:43,000
we can take out all those a's
and get a^3.

373
00:30:43,000 --> 00:30:50,000
Second, in the inner integral,
we are integrating cos^2 phi

374
00:30:50,000 --> 00:30:54,000
sin phi d phi.
I claim that integrates to cos3

375
00:30:54,000 --> 00:30:57,000
up to some factor,
and that factor should be

376
00:30:57,000 --> 00:31:02,000
negative one-third.
If you look at cos3 phi and you

377
00:31:02,000 --> 00:31:07,000
take its derivative,
you will get that guy with a

378
00:31:07,000 --> 00:31:12,000
negative three in front between
zero and pi.

379
00:31:12,000 --> 00:31:16,000
And, while integrating over
theta, we will just multiply

380
00:31:16,000 --> 00:31:24,000
things by 2pi.
Let me add the 2pi in front.

381
00:31:24,000 --> 00:31:27,000
Now, if I evaluate this guy
between zero and pi,

382
00:31:27,000 --> 00:31:32,000
well, at pi cos^3 is negative
one, at zero it is one,

383
00:31:32,000 --> 00:31:35,000
I will get two-thirds out of
this.

384
00:31:35,000 --> 00:31:39,000
I end up with four-thirds pi
a^3.

385
00:31:39,000 --> 00:31:46,000
Sorry I didn't write very much
because I am trying to save

386
00:31:46,000 --> 00:31:52,000
blackboard space.
Yes?

387
00:31:52,000 --> 00:31:55,000
That is a very natural question.
That looks a lot like somebody

388
00:31:55,000 --> 00:31:58,000
we know, like the volume of a
sphere.

389
00:31:58,000 --> 00:32:03,000
And ultimately it will be.
Wait until next class when we

390
00:32:03,000 --> 00:32:07,000
talk about the divergence
theorem.

391
00:32:07,000 --> 00:32:11,000
I mean the question was is this
related to the volume of a

392
00:32:11,000 --> 00:32:14,000
sphere, and ultimately it is,
but for now it is just some

393
00:32:14,000 --> 00:32:23,000
coincidence.
Yes?

394
00:32:23,000 --> 00:32:26,000
The question is there is a way
to do it M dx plus N dy plus

395
00:32:26,000 --> 00:32:28,000
stuff like that?
The answer is unfortunately no

396
00:32:28,000 --> 00:32:30,000
because it is not a line
integral.

397
00:32:30,000 --> 00:32:35,000
It is a surface integral,
so we need to have to variables

398
00:32:35,000 --> 00:32:38,000
in there.
In a way you would end up with

399
00:32:38,000 --> 00:32:41,000
things like some dx dy maybe and
so on.

400
00:32:41,000 --> 00:32:45,000
I mean it is not practical to
do it directly that way because

401
00:32:45,000 --> 00:32:49,000
you would have then to compute
Jacobians to switch from dx dy

402
00:32:49,000 --> 00:32:52,000
to something else.
We are going to see various

403
00:32:52,000 --> 00:32:54,000
ways of computing it.
Unfortunately,

404
00:32:54,000 --> 00:32:57,000
it is not quite as simple as
with line integrals.

405
00:32:57,000 --> 00:32:59,000
But it is not much harder.
It is the same spirit.

406
00:32:59,000 --> 00:33:04,000
We just use two variables and
set up everything in terms of

407
00:33:04,000 --> 00:33:12,000
these two variables.
Any other questions?

408
00:33:12,000 --> 00:33:14,000
No.
OK.

409
00:33:51,000 --> 00:33:54,000
By the way, just some food for
thought.

410
00:33:54,000 --> 00:34:01,000
Never mind.
Conclusion of looking at these

411
00:34:01,000 --> 00:34:05,000
two examples is that sometimes
we can use geometric.

412
00:34:05,000 --> 00:34:07,000
The first example,
we didn't actually have to

413
00:34:07,000 --> 00:34:11,000
compute an integral.
But most of the time we need to

414
00:34:11,000 --> 00:34:14,000
learn how to set up double
integrals.

415
00:34:14,000 --> 00:34:26,000
Use geometry or you need to set
up for double integral of a

416
00:34:26,000 --> 00:34:30,000
surface.
And so we are going to learn

417
00:34:30,000 --> 00:34:33,000
how to do that in general.
As I said, we need to have two

418
00:34:33,000 --> 00:34:37,000
parameters on the surface and
express everything in terms of

419
00:34:37,000 --> 00:34:43,000
these.
Let's look at various examples.

420
00:34:43,000 --> 00:34:46,000
We are going to see various
situations where we can do

421
00:34:46,000 --> 00:34:49,000
things.
Well, let's start with an easy

422
00:34:49,000 --> 00:34:53,000
one.
Let's call that number zero.

423
00:34:53,000 --> 00:35:02,000
Say that my surface S is a
horizontal plane,

424
00:35:02,000 --> 00:35:07,000
say z equals a.
When I say a horizontal plane,

425
00:35:07,000 --> 00:35:09,000
it doesn't have to be the
entire horizontal plane.

426
00:35:09,000 --> 00:35:14,000
It could be a small piece of it.
It could even be,

427
00:35:14,000 --> 00:35:16,000
to trick you,
maybe an ellipse in there or a

428
00:35:16,000 --> 00:35:19,000
triangle in there or something
like that.

429
00:35:19,000 --> 00:35:23,000
What you have to recognize is
my surface is a piece of just a

430
00:35:23,000 --> 00:35:27,000
flat plane, so I shouldn't worry
too much about what part of a

431
00:35:27,000 --> 00:35:30,000
plane it is.
Well, it will become important

432
00:35:30,000 --> 00:35:32,000
when I set up bounds for
integration.

433
00:35:32,000 --> 00:35:36,000
But, when it comes to looking
for the normal vector,

434
00:35:36,000 --> 00:35:40,000
be rest assured that the normal
vector to a horizontal plane is

435
00:35:40,000 --> 00:35:44,000
just vertical.
It is going to be either k or

436
00:35:44,000 --> 00:35:49,000
negative k depending on whether
I have chosen to orient it

437
00:35:49,000 --> 00:35:54,000
pointing up or down.
And which one I choose might

438
00:35:54,000 --> 00:35:57,000
depend on what I am going to try
to do.

439
00:35:57,000 --> 00:36:02,000
The normal vector is just
sticking straight up or straight

440
00:36:02,000 --> 00:36:05,000
down.
Now, what about dS?

441
00:36:05,000 --> 00:36:11,000
Well, it is just going to be
the area element in a horizontal

442
00:36:11,000 --> 00:36:14,000
plane.
It just looks like it should be

443
00:36:14,000 --> 00:36:16,000
dx dy.
I mean if I am moving on a

444
00:36:16,000 --> 00:36:18,000
horizontal plane,
to know where I am,

445
00:36:18,000 --> 00:36:26,000
I should know x and y.
So dS will be dx dy.

446
00:36:26,000 --> 00:36:31,000
If I play the game that way,
I have my vector field F.

447
00:36:31,000 --> 00:36:34,000
I do F dot n.
That just gives me the z

448
00:36:34,000 --> 00:36:37,000
component which might involve x,
y and z.

449
00:36:37,000 --> 00:36:40,000
x and y I am very happy with.
They will stay as my variables.

450
00:36:40,000 --> 00:36:43,000
Whenever I see z,
well, I want to get rid of it.

451
00:36:43,000 --> 00:36:46,000
That is easy because z is just
equal to a.

452
00:36:46,000 --> 00:36:50,000
I just plug that value and I am
left with only x and y,

453
00:36:50,000 --> 00:36:53,000
and I am integrating that dx
dy.

454
00:36:53,000 --> 00:36:58,000
It is actually ending up being
just a usual double integral in

455
00:36:58,000 --> 00:37:00,000
x, y coordinates.
And, of course,

456
00:37:00,000 --> 00:37:02,000
once it is set up anything is
fair game.

457
00:37:02,000 --> 00:37:05,000
I might want to switch to polar
coordinates or something like

458
00:37:05,000 --> 00:37:09,000
that.
Or, I can set it up dx dy or dy

459
00:37:09,000 --> 00:37:12,000
dx.
All the usual stuff applies.

460
00:37:12,000 --> 00:37:17,000
But, for the initial setup,
we are just going to use these

461
00:37:17,000 --> 00:37:21,000
and express everything in terms
of x and y.

462
00:37:21,000 --> 00:37:27,000
A small variation on that.
Let's say that we take vertical

463
00:37:27,000 --> 00:37:35,000
planes that are parallel to
maybe the blackboard plane,

464
00:37:35,000 --> 00:37:42,000
so parallel to the yz plane.
That might be something like x

465
00:37:42,000 --> 00:37:47,000
equals some constant.
Well, what would I do then?

466
00:37:47,000 --> 00:37:52,000
It could be pretty much the
same.

467
00:37:52,000 --> 00:37:55,000
The normal vector for this guy
would be sticking straight out

468
00:37:55,000 --> 00:37:59,000
towards me or away from me.
Let's say I am having it come

469
00:37:59,000 --> 00:38:03,000
to the front.
The normal vector would be

470
00:38:03,000 --> 00:38:07,000
plus/minus i.
And the variables that I would

471
00:38:07,000 --> 00:38:11,000
be using, to find out my
position on this guy,

472
00:38:11,000 --> 00:38:14,000
would be y and z.
In terms of those,

473
00:38:14,000 --> 00:38:19,000
the surface element is just dy
dz.

474
00:38:19,000 --> 00:38:25,000
Similarly for planes parallel
to the xz plane.

475
00:38:25,000 --> 00:38:33,000
You can figure that one out.
These are somehow the easiest

476
00:38:33,000 --> 00:38:39,000
ones, because those we already
know how to compute without too

477
00:38:39,000 --> 00:38:41,000
much trouble.
What if it is a more

478
00:38:41,000 --> 00:38:43,000
complicated plane?
We will come back to that next

479
00:38:43,000 --> 00:38:49,000
time.
Let's explore some other

480
00:38:49,000 --> 00:38:56,000
situations first.
Number one on the list.

481
00:38:56,000 --> 00:39:03,000
Let's say that I gave you a
sphere of radius a centered at

482
00:39:03,000 --> 00:39:09,000
the origin, or maybe just half
of that sphere or some portion

483
00:39:09,000 --> 00:39:12,000
of it.
Well, we have already seen how

484
00:39:12,000 --> 00:39:15,000
to do things.
Namely, we will be saying the

485
00:39:15,000 --> 00:39:18,000
normal vector is x,
y, z over a,

486
00:39:18,000 --> 00:39:23,000
plus or minus depending on
whether we want it pointing in

487
00:39:23,000 --> 00:39:29,000
or out.
And dS will be a^2 sin phi d

488
00:39:29,000 --> 00:39:32,000
phi d theta.
In fact, we will express

489
00:39:32,000 --> 00:39:35,000
everything in terms of phi and
theta.

490
00:39:35,000 --> 00:39:37,000
If I wanted to I could tell you
what the formulas are for x,

491
00:39:37,000 --> 00:39:40,000
y, z in terms of phi and theta.
You know them.

492
00:39:40,000 --> 00:39:44,000
But it is actually better to
wait a little bit.

493
00:39:44,000 --> 00:39:48,000
It is better to do F dot n,
because F is also going to have

494
00:39:48,000 --> 00:39:49,000
a bunch of x's,
y's and z's.

495
00:39:49,000 --> 00:39:53,000
And if there is any kind of
symmetry to the problem then you

496
00:39:53,000 --> 00:39:57,000
might end up with things like
x^2 y^2 z^2 or things that have

497
00:39:57,000 --> 00:40:01,000
more symmetry that are easier to
express in terms of phi and

498
00:40:01,000 --> 00:40:05,000
theta.
The advice would be first do

499
00:40:05,000 --> 00:40:10,000
the dot product with F,
and then see what you get and

500
00:40:10,000 --> 00:40:17,000
then turn it into phi and theta.
That is one we have seen.

501
00:40:17,000 --> 00:40:20,000
Let's say that I have -- It is
a close cousin.

502
00:40:20,000 --> 00:40:30,000
Let's say I have a cylinder of
radius a centered on the z-axis.

503
00:40:30,000 --> 00:40:37,000
What does that look like?
And, again, when I say

504
00:40:37,000 --> 00:40:40,000
cylinder, it could be a piece of
cylinder.

505
00:40:40,000 --> 00:40:44,000
First of all,
what does the normal vector to

506
00:40:44,000 --> 00:40:47,000
a cylinder look like?
Well, it is sticking straight

507
00:40:47,000 --> 00:40:50,000
out, but sticking straight out
in a slightly different way from

508
00:40:50,000 --> 00:40:52,000
what happens with a sphere.
See, the sides of a cylinder

509
00:40:52,000 --> 00:40:54,000
are vertical.
If you imagine that you have

510
00:40:54,000 --> 00:40:56,000
this big cylindrical type in
front of you,

511
00:40:56,000 --> 00:40:59,000
hopefully you can see that a
normal vector is going to always

512
00:40:59,000 --> 00:41:02,000
be horizontal.
It is sticking straight out in

513
00:41:02,000 --> 00:41:07,000
the horizontal directions.
It doesn't have any z component.

514
00:41:07,000 --> 00:41:13,000
I claim the normal vector for
the cylinder,

515
00:41:13,000 --> 00:41:21,000
if you have a point here at (x,
y, z), it would be pointing

516
00:41:21,000 --> 00:41:27,000
straight out away from the
central axis.

517
00:41:27,000 --> 00:41:31,000
My normal vector,
well, if I am taking it two

518
00:41:31,000 --> 00:41:34,000
points outwards,
will be going straight away

519
00:41:34,000 --> 00:41:38,000
from the central axis.
If I look at it from above,

520
00:41:38,000 --> 00:41:42,000
maybe it is easier if I look at
it from above,

521
00:41:42,000 --> 00:41:45,000
look at x, y,
then my cylinder looks like a

522
00:41:45,000 --> 00:41:49,000
circle and the normal vector
just points straight out.

523
00:41:49,000 --> 00:41:53,000
It is the same situation as
when we had a circle in the 2D

524
00:41:53,000 --> 00:41:57,000
case.
The normal vector for that is

525
00:41:57,000 --> 00:42:02,000
just going to be x,
y and 0 in the z component.

526
00:42:02,000 --> 00:42:05,000
Well, plus/minus,
depending on whether you want

527
00:42:05,000 --> 00:42:06,000
it sticking in or out.

528
00:42:41,000 --> 00:42:47,000
We said in our cylinder normal
vector is plus or minus x,

529
00:42:47,000 --> 00:42:54,000
y, zero over a.
What about the surface element?

530
00:42:54,000 --> 00:42:57,000
Before we ask that,
maybe we should first figure

531
00:42:57,000 --> 00:43:00,000
out what coordinates are we
going to use to locate ourselves

532
00:43:00,000 --> 00:43:02,000
in a cylinder.
Well, yes,

533
00:43:02,000 --> 00:43:05,000
we probably want to use part of
a cylindrical coordinate,

534
00:43:05,000 --> 00:43:08,000
except for,
well, we don't want r because r

535
00:43:08,000 --> 00:43:11,000
doesn't change,
it is not a variable here.

536
00:43:11,000 --> 00:43:15,000
Indeed, you probably want to
use z to tell how high you are

537
00:43:15,000 --> 00:43:18,000
and theta to tell you where you
are around.

538
00:43:18,000 --> 00:43:27,000
dS should be in terms of dz d
theta.

539
00:43:27,000 --> 00:43:33,000
Now, what is the constant?
Well, let's look at a small

540
00:43:33,000 --> 00:43:39,000
piece of our cylinder
corresponding to a small angle

541
00:43:39,000 --> 00:43:44,000
delta theta and a small height
delta z.

542
00:43:44,000 --> 00:43:47,000
Well, the height,
as I said, is going to be delta

543
00:43:47,000 --> 00:43:50,000
z.
What about the width?

544
00:43:50,000 --> 00:43:55,000
It is going to be a piece of a
circle of radius a corresponding

545
00:43:55,000 --> 00:43:59,000
to the angle delta theta,
so this side will be a delta

546
00:43:59,000 --> 00:44:05,000
theta.
Delta S is a delta theta delta

547
00:44:05,000 --> 00:44:08,000
z.
DS is just a dz d theta or d

548
00:44:08,000 --> 00:44:13,000
theta dz.
It doesn't matter which way you

549
00:44:13,000 --> 00:44:16,000
do it.
And so when we set up the flux

550
00:44:16,000 --> 00:44:21,000
integral, we will take first the
dot product of f with this

551
00:44:21,000 --> 00:44:25,000
normal vector.
Then we will stick in this dS.

552
00:44:25,000 --> 00:44:28,000
And then, of course,
we will get rid of any x and y

553
00:44:28,000 --> 00:44:31,000
that are left by expressing them
in terms of theta.

554
00:44:31,000 --> 00:44:37,000
Maybe x becomes a cos theta,
y becomes a sin theta.

555
00:44:37,000 --> 00:44:41,000
These various formulas,
you should try to remember them

556
00:44:41,000 --> 00:44:45,000
because they are really useful,
for the sphere,

557
00:44:45,000 --> 00:44:48,000
for the cylinder.
And, hopefully,

558
00:44:48,000 --> 00:44:52,000
those for the planes you kind
of know already intuitively.

559
00:44:52,000 --> 00:44:59,000
What about marginals or faces?
Not everything in life is made

560
00:44:59,000 --> 00:45:08,000
out of cylinders and spheres.
I mean it is a good try.

561
00:45:08,000 --> 00:45:11,000
Let's look at a marginal kind
of surface.

562
00:45:11,000 --> 00:45:19,000
Let's say I give you a graph of
a function z equals f of x,

563
00:45:19,000 --> 00:45:21,000
y.
This guy has nothing to do with

564
00:45:21,000 --> 00:45:22,000
the integrand.
It is not what we are

565
00:45:22,000 --> 00:45:24,000
integrating.
We are just integrating a

566
00:45:24,000 --> 00:45:26,000
vector field that has nothing to
do with that.

567
00:45:26,000 --> 00:45:31,000
This is how I want to describe
the surface on which I will be

568
00:45:31,000 --> 00:45:37,000
integrating.
My surface is given by z as a

569
00:45:37,000 --> 00:45:40,000
function of x,
y.

570
00:45:40,000 --> 00:45:48,000
Well, I would need to tell you
what n is and what dS is.

571
00:45:48,000 --> 00:45:51,000
That is going to be slightly
annoying.

572
00:45:51,000 --> 00:45:54,000
I mean, I don't want to tell
them separately because you see

573
00:45:54,000 --> 00:45:58,000
they are pretty hard.
Instead, I am going to tell you

574
00:45:58,000 --> 00:46:01,000
that in this case,
well, let's see.

575
00:46:01,000 --> 00:46:07,000
What variables do we want?
I am going to tell you a

576
00:46:07,000 --> 00:46:11,000
formula for n dS.
What variables do we want to

577
00:46:11,000 --> 00:46:15,000
express this in terms of?
Well, most likely x and y

578
00:46:15,000 --> 00:46:19,000
because we know how to express z
in terms of x and y.

579
00:46:19,000 --> 00:46:26,000
This is an invitation to get
rid of any z that might be left

580
00:46:26,000 --> 00:46:30,000
and set everything up in terms
of dx dy.

581
00:46:30,000 --> 00:46:31,000
The formula that we are going
to see,

582
00:46:31,000 --> 00:46:36,000
I think we are going to see the
details of why it works

583
00:46:36,000 --> 00:46:40,000
tomorrow,
is that you can take negative

584
00:46:40,000 --> 00:46:44,000
partial f partial x,
negative partial f partial y,

585
00:46:44,000 --> 00:46:48,000
one,
dx dy.

586
00:46:48,000 --> 00:46:52,000
Plus/minus depending on which
way you want it to go.

587
00:46:52,000 --> 00:46:57,000
If you really want to know what
dS is, well, dS is the magnitude

588
00:46:57,000 --> 00:47:01,000
of this vector times dx dy.
There will be a square root and

589
00:47:01,000 --> 00:47:04,000
some squares and some stuff.
What is the normal vector?

590
00:47:04,000 --> 00:47:10,000
Well, you take this vector and
you scale it down to unit

591
00:47:10,000 --> 00:47:14,000
length.
Just to emphasize it,

592
00:47:14,000 --> 00:47:24,000
this guy here is not n and this
guy here is not dS.

593
00:47:24,000 --> 00:47:27,000
Each of them is more
complicated than that,

594
00:47:27,000 --> 00:47:31,000
but the combination somehow
simplifies nicely.

595
00:47:31,000 --> 00:47:36,000
And that is good news for us.
Now, concretely,

596
00:47:36,000 --> 00:47:40,000
one way you can think about it
is this tells you how to reduce

597
00:47:40,000 --> 00:47:42,000
things to an integral of x and
y.

598
00:47:42,000 --> 00:47:44,000
And, of course,
you will have to figure out

599
00:47:44,000 --> 00:47:47,000
what are the bounds on x and y.
That means you will need to

600
00:47:47,000 --> 00:47:51,000
know what does the shadow of
your surface look like in the x,

601
00:47:51,000 --> 00:47:59,000
y plane.
To set up bounds on whatever

602
00:47:59,000 --> 00:48:06,000
you will get dx dy,
well, of course you can switch

603
00:48:06,000 --> 00:48:08,000
to dy dx or anything you would
like,

604
00:48:08,000 --> 00:48:19,000
but you will need to look at
the shadow of S in the x y

605
00:48:19,000 --> 00:48:22,000
plane.
But only do that after you

606
00:48:22,000 --> 00:48:27,000
gotten rid of all the z.
When you no longer have z then

607
00:48:27,000 --> 00:48:33,000
you can figure out what the
bounds are for x and y.

608
00:48:33,000 --> 00:48:40,000
Any questions about that?
Yes?

609
00:48:40,000 --> 00:48:42,000
For the cylinder.
OK.

610
00:48:42,000 --> 00:48:44,000
Let me re-explain quickly how I
got a normal vector for the

611
00:48:44,000 --> 00:48:47,000
cylinder.
If you know what a cylinder

612
00:48:47,000 --> 00:48:50,000
looks like, you probably can see
that the normal vector sticks

613
00:48:50,000 --> 00:48:56,000
straight out of it horizontally.
That means the z component of n

614
00:48:56,000 --> 00:48:59,000
is going to be zero.
And then the x,

615
00:48:59,000 --> 00:49:02,000
y components you get by looking
at it from above.

616
00:49:02,000 --> 00:49:12,000
One last thing I want to say.
What about the geometric

617
00:49:12,000 --> 00:49:15,000
interpretation and how to prove
it?

618
00:49:15,000 --> 00:49:27,000
Well, if your vector field F is
a velocity field then the flux

619
00:49:27,000 --> 00:49:37,000
is the amount of matter that
crosses the surface that passes

620
00:49:37,000 --> 00:49:44,000
through S per unit time.
And the way that you would

621
00:49:44,000 --> 00:49:47,000
prove it would be similar to the
picture that I drew when we did

622
00:49:47,000 --> 00:49:50,000
it in the plane.
Namely, you would consider a

623
00:49:50,000 --> 00:49:53,000
small element of a surface delta
S.

624
00:49:53,000 --> 00:49:55,000
And you would try to figure out
what is the stuff that flows

625
00:49:55,000 --> 00:49:59,000
through it in a second.
Well, it is the stuff that

626
00:49:59,000 --> 00:50:05,000
lives in a small box whose base
is that piece of surface and

627
00:50:05,000 --> 00:50:10,000
whose other side is given by the
vector field.

628
00:50:10,000 --> 00:50:15,000
And then the volume of that is
given by base times height,

629
00:50:15,000 --> 00:50:20,000
and the height is F dot n.
It is the same argument as what

630
00:50:20,000 --> 00:50:22,000
we saw in the plane.
OK.

631
00:50:22,000 --> 00:50:24,000
Next time we will see more
formulas.

632
00:50:24,000 --> 00:50:28,000
We will first see how to prove
this, more ways to do it,

633
00:50:28,000 --> 00:50:31,000
more examples.
And then we will get to the

634
00:50:31,000 --> 00:50:34,000
divergence theorem.