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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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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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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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trying to draw a picture because
that is going to be quite hard.
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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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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,
00:04:36.000 --> 00:04:41.000
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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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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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
00:07:51.000 --> 00:07:54.000
that was a line integral F dot n
ds.
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And that measured how much the
vector field was going across
00:07:59.000 --> 00:08:01.000
the curve.
If you were thinking of a
00:08:01.000 --> 00:08:05.000
velocity field,
that would measure how much
00:08:05.000 --> 00:08:08.000
fluid is passing through the
curve in unit time.
00:08:08.000 --> 00:08:10.000
Now let's say that we were in
space.
00:08:10.000 --> 00:08:14.000
Well, we cannot really think of
flux as a line integral.
00:08:14.000 --> 00:08:18.000
Because, if you have a curve in
space and say that you have wind
00:08:18.000 --> 00:08:22.000
or something like that,
you cannot really ask how much
00:08:22.000 --> 00:08:24.000
air is flowing through the
curve.
00:08:24.000 --> 00:08:28.000
See, to have a flow through
something you need a surface.
00:08:28.000 --> 00:08:32.000
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,
00:08:44.000 --> 00:08:51.000
flux will be measured through a
surface.
00:08:51.000 --> 00:08:54.000
And so it will be a surface
integral, not a line integral
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
00:09:10.000 --> 00:09:13.000
in space.
Because a surface is a
00:09:13.000 --> 00:09:15.000
two-dimensional object,
that will end up being a double
00:09:15.000 --> 00:09:17.000
integral.
But, of course,
00:09:17.000 --> 00:09:19.000
we will have to set it up
properly because the surface
00:09:19.000 --> 00:09:22.000
that is in space,
and we will probably have x,
00:09:22.000 --> 00:09:24.000
y and z to deal with at the
same time,
00:09:24.000 --> 00:09:28.000
and we will have to somehow get
rid of one variable so that we
00:09:28.000 --> 00:09:31.000
can set up and evaluate a double
integral.
00:09:31.000 --> 00:09:35.000
So conceptually it is very
similar to line integrals.
00:09:35.000 --> 00:09:39.000
In the line integral in the
plane, you had two variables
00:09:39.000 --> 00:09:42.000
that you reduced to one by
figuring out what the curve was.
00:09:42.000 --> 00:09:51.000
Here you have three variables
that you will reduce to two by
00:09:51.000 --> 00:09:56.000
figuring out what the surface
is.
00:09:56.000 --> 00:10:00.000
Let me give you a definition of
flux in 3D.
00:10:00.000 --> 00:10:15.000
Let's say that we have a vector
field and s, a surface in space.
00:10:15.000 --> 00:10:17.000
Let me draw some kind of a
picture.
00:10:17.000 --> 00:10:21.000
I have my surface and I have my
vector field F.
00:10:21.000 --> 00:10:25.000
Well, at every point it changes
with a point.
00:10:25.000 --> 00:10:28.000
Well, I want to figure out how
much my vector field is going
00:10:28.000 --> 00:10:34.000
across that surface.
That means I want to figure out
00:10:34.000 --> 00:10:40.000
the normal component of my
vector field,
00:10:40.000 --> 00:10:45.000
so I will use,
as in the plane case,
00:10:45.000 --> 00:10:53.000
the unit normal vector to s.
I take my point on the surface
00:10:53.000 --> 00:10:59.000
and build a unit vector that is
standing on it perpendicularly.
00:10:59.000 --> 00:11:05.000
Now, we have to decide which
way it is standing.
00:11:05.000 --> 00:11:09.000
We can build our normal vector
to go this way or to go the
00:11:09.000 --> 00:11:12.000
other way around.
There are two choices.
00:11:12.000 --> 00:11:16.000
Basically, whenever you want to
set up a flux integral you have
00:11:16.000 --> 00:11:19.000
to choose one side of the
surface.
00:11:19.000 --> 00:11:23.000
And you will count positively
what flows toward that side and
00:11:23.000 --> 00:11:26.000
negatively what flows towards
the other side.
00:11:26.000 --> 00:11:40.000
There are two choices for n.
We need to choose a side of the
00:11:40.000 --> 00:11:45.000
surface.
In the case of curves,
00:11:45.000 --> 00:11:50.000
we made that choice by deciding
that because we were going along
00:11:50.000 --> 00:11:54.000
some direction on the curve we
could choose one side by saying
00:11:54.000 --> 00:11:57.000
let's rotate clockwise from the
tangent vector.
00:11:57.000 --> 00:12:00.000
And, in a way,
what we were doing was really
00:12:00.000 --> 00:12:04.000
it was a recipe to choose for us
one of the two sides.
00:12:04.000 --> 00:12:09.000
Here we don't have a notion of
orienting the surface other than
00:12:09.000 --> 00:12:14.000
by precisely choosing one of the
two possible normal vectors.
00:12:14.000 --> 00:12:15.000
So, in fact,
this is called choosing an
00:12:15.000 --> 00:12:18.000
orientation of a surface.
When you are saying you are
00:12:18.000 --> 00:12:22.000
orienting the surface that
really means you are deciding
00:12:22.000 --> 00:12:31.000
which side is which.
Let's call that orientation.
00:12:31.000 --> 00:12:35.000
Now, there is no set convention
that will work forever.
00:12:35.000 --> 00:12:39.000
But the usually traditional
settings would be to take your
00:12:39.000 --> 00:12:43.000
normal vector pointing maybe out
of the solid region because then
00:12:43.000 --> 00:12:48.000
you will be looking at flux that
is coming out of that region of
00:12:48.000 --> 00:12:51.000
space.
Or, if you have a surface that
00:12:51.000 --> 00:12:55.000
is not like closed or anything
but maybe you will want the flux
00:12:55.000 --> 00:12:59.000
going up through the region.
Or, there are various
00:12:59.000 --> 00:13:02.000
conventions.
Concretely, on problem sets it
00:13:02.000 --> 00:13:05.000
will either say which choice you
have to make or you get to
00:13:05.000 --> 00:13:07.000
choose which one you want to
make.
00:13:07.000 --> 00:13:10.000
And, of course,
if you choose the other one
00:13:10.000 --> 00:13:12.000
then the sign becomes the
opposite.
00:13:12.000 --> 00:13:17.000
Now, once we have made a choice
then we can define the flux
00:13:17.000 --> 00:13:21.000
integral.
It will just be the double
00:13:21.000 --> 00:13:26.000
integral over a surface of F dot
n dS.
00:13:26.000 --> 00:13:33.000
Now I am using a big dS.
That stands for the surface
00:13:33.000 --> 00:13:39.000
area element on this surface.
I am using dS rather than dA
00:13:39.000 --> 00:13:43.000
because I still want to think of
dA as maybe the area in one of
00:13:43.000 --> 00:13:47.000
the coordinate planes like the
one we had in double integrals.
00:13:47.000 --> 00:13:51.000
You will see later where this
comes in.
00:13:51.000 --> 00:13:54.000
But conceptually it is very
similar.
00:13:54.000 --> 00:13:59.000
Concretely what this means is I
cut my surface into little
00:13:59.000 --> 00:14:03.000
pieces.
Each of them has area delta S.
00:14:03.000 --> 00:14:07.000
And, for each piece,
I take my vector field,
00:14:07.000 --> 00:14:13.000
I take my normal vector,
I dot them and I multiply by
00:14:13.000 --> 00:14:17.000
this surface area and sum all
these things together.
00:14:17.000 --> 00:14:23.000
That is what a double integral
means.
00:14:23.000 --> 00:14:25.000
In particular,
an easy case where you know you
00:14:25.000 --> 00:14:28.000
can get away without computing
anything is, of course,
00:14:28.000 --> 00:14:32.000
if your vector field is tangent
to the surface because then you
00:14:32.000 --> 00:14:36.000
know that there is no flux.
Flux is going to be zero
00:14:36.000 --> 00:14:38.000
because nothing passes through
the surface.
00:14:38.000 --> 00:14:42.000
Otherwise, we have to figure
out how to compute these things.
00:14:42.000 --> 00:14:50.000
That is what we are going to
learn now.
00:14:50.000 --> 00:14:51.000
Well, maybe I should box this
formula.
00:14:51.000 --> 00:14:57.000
I have noticed that some of you
seem to like it when I box the
00:14:57.000 --> 00:15:03.000
important formulas.
(APPLAUSE) By the way,
00:15:03.000 --> 00:15:12.000
a piece of notation before I
move on, sometimes you will also
00:15:12.000 --> 00:15:18.000
see the notation vector dS.
What is vector dS?
00:15:18.000 --> 00:15:24.000
Vector dS is this guy n dS put
together.
00:15:24.000 --> 00:15:30.000
Vector dS is a vector which
points perpendicular to the
00:15:30.000 --> 00:15:35.000
surface and whose length
corresponds to the surface
00:15:35.000 --> 00:15:37.000
element.
And the reason for having this
00:15:37.000 --> 00:15:41.000
shortcut notation,
well, it is not only laziness
00:15:41.000 --> 00:15:44.000
like saving one n,
but it is because this guy is
00:15:44.000 --> 00:15:49.000
very often easier to compute
than it is to set up n and dS
00:15:49.000 --> 00:15:52.000
separately.
Actually, if you remember in
00:15:52.000 --> 00:15:56.000
the plane, we have seen that
vector n little ds can be
00:15:56.000 --> 00:15:59.000
written directly as dy,
- dx.
00:15:59.000 --> 00:16:03.000
That was easier than finding n
and ds separately.
00:16:03.000 --> 00:16:15.000
And here the same is going to
be true in many cases.
00:16:15.000 --> 00:16:23.000
Well, any questions before we
do examples?
00:16:23.000 --> 00:16:24.000
No.
OK.
00:16:24.000 --> 00:16:38.000
Let's do examples.
The first example for today is
00:16:38.000 --> 00:16:51.000
we are going to look at the flux
of vector field xi yj xk through
00:16:51.000 --> 00:17:02.000
the sphere of radius a -- --
centered at the origin.
00:17:02.000 --> 00:17:18.000
What does the picture look like?
We have a sphere of radius a.
00:17:18.000 --> 00:17:22.000
I have my vector field.
Well, , see,
00:17:22.000 --> 00:17:25.000
that is a vector field that is
equal to the vector from the
00:17:25.000 --> 00:17:35.000
origin to the point where I am,
so it is pointing radially away
00:17:35.000 --> 00:17:42.000
from the origin.
My vector field is really
00:17:42.000 --> 00:17:49.000
sticking out everywhere away
from the origin.
00:17:49.000 --> 00:17:56.000
Now I have to find the normal
vector to the sphere if I want
00:17:56.000 --> 00:18:04.000
to set up double integral over
the sphere of F dot vector ds,
00:18:04.000 --> 00:18:09.000
or if you want F dot n dS.
What does the normal vector to
00:18:09.000 --> 00:18:12.000
the sphere look like?
Well, it depends,
00:18:12.000 --> 00:18:14.000
of course, whether I choose it
pointing out or in.
00:18:14.000 --> 00:18:18.000
Let's say I am choosing it
pointing out then it will be
00:18:18.000 --> 00:18:20.000
sticking straight out of a
sphere as well.
00:18:20.000 --> 00:18:27.000
Hopefully, you can see that if
I take a normal vector to the
00:18:27.000 --> 00:18:34.000
sphere it is actually pointing
radially out away from the
00:18:34.000 --> 00:18:38.000
origin.
In fact, our vector field and
00:18:38.000 --> 00:18:41.000
our normal vector are parallel
to each other.
00:18:41.000 --> 00:18:45.000
Let's think a bit more about
what a normal vector looks like.
00:18:45.000 --> 00:18:47.000
I said it is sticking straight
out.
00:18:47.000 --> 00:18:49.000
It is proportional to this
vector field.
00:18:49.000 --> 00:18:51.000
Maybe I should start by writing
00:18:52.000 --> 00:18:56.000
because that is the vector that
goes from the origin to my point
00:18:56.000 --> 00:18:59.000
so it points radially away from
the origin.
00:18:59.000 --> 00:19:00.000
Now there is a small problem
with that.
00:19:00.000 --> 00:19:04.000
It is not a unit vector.
So what is its length?
00:19:04.000 --> 00:19:08.000
Well, its length is square root
of x^2 y^2 z^2.
00:19:08.000 --> 00:19:13.000
But, if I am on the sphere,
then that length is just equal
00:19:13.000 --> 00:19:16.000
to a because distance from the
origin is a.
00:19:16.000 --> 00:19:23.000
In fact, I get my normal vector
by scaling this guy down by a
00:19:23.000 --> 00:19:27.000
factor of a.
And let me write it down just
00:19:27.000 --> 00:19:34.000
in case you are still unsure.
This is unit because square
00:19:34.000 --> 00:19:43.000
root of x^2 y^2 z^2 is equal to
a on the sphere.
00:19:43.000 --> 00:19:48.000
OK.
Any questions about this?
00:19:48.000 --> 00:19:52.000
No. It looks OK?
I see a lot of blank faces.
00:19:52.000 --> 00:19:58.000
That physics test must have
been hard.
00:19:58.000 --> 00:20:03.000
Yes?
I could have put a rho but I
00:20:03.000 --> 00:20:06.000
want to emphasize the fact that
here it is going to be a
00:20:06.000 --> 00:20:09.000
constant.
I mean rho has this connotation
00:20:09.000 --> 00:20:13.000
of being a variable that I will
need to then maybe integrate
00:20:13.000 --> 00:20:17.000
over or do something with.
Yes, it would be correct to put
00:20:17.000 --> 00:20:20.000
rho but I then later will want
to replace it by its actual
00:20:20.000 --> 00:20:24.000
value which is a number.
And the number is a.
00:20:24.000 --> 00:20:28.000
It is not going to actually
change from point to point.
00:20:28.000 --> 00:20:30.000
For example,
if this was the unit sphere
00:20:30.000 --> 00:20:32.000
then I would just put x,
y, z.
00:20:32.000 --> 00:20:41.000
I wouldn't divide by anything.
Now let's figure out F dot n.
00:20:41.000 --> 00:20:47.000
Let's do things one at a time.
Well, F and n are parallel to
00:20:47.000 --> 00:20:53.000
each other.
F dot n, the normal component
00:20:53.000 --> 00:21:00.000
of F, is actually equal to the
length of F.
00:21:00.000 --> 00:21:05.000
Well, times the length of n if
you want, but that is going to
00:21:05.000 --> 00:21:09.000
be a one since F and n are
parallel to each other.
00:21:09.000 --> 00:21:12.000
And what is the magnitude of F
if I am on the sphere?
00:21:12.000 --> 00:21:18.000
Well, the magnitude of F in
general is square root of x^2
00:21:18.000 --> 00:21:23.000
y^2 z^2 on the sphere that is
going be a.
00:21:23.000 --> 00:21:25.000
The other way to do it,
if you don't want to think
00:21:25.000 --> 00:21:28.000
geometrically like that,
is to just to do the dot
00:21:28.000 --> 00:21:31.000
product x, y,
z doted with x over a,
00:21:31.000 --> 00:21:35.000
y over a, z over a.
You will be x^2 y^2 z^2 divided
00:21:35.000 --> 00:21:40.000
by a.
That will simplify to a because
00:21:40.000 --> 00:21:45.000
we are on the sphere.
See, we are already using here
00:21:45.000 --> 00:21:47.000
the relation between x,
y and z.
00:21:47.000 --> 00:21:49.000
We are not letting x,
y and z be completely
00:21:49.000 --> 00:21:51.000
arbitrary.
But the slogan is everything
00:21:51.000 --> 00:21:54.000
happens on the surface where we
are doing the integral.
00:21:54.000 --> 00:21:56.000
We are not looking at anything
inside or outside.
00:21:56.000 --> 00:21:58.000
We are just on the surface.
00:22:34.000 --> 00:22:43.000
Now what do I do with that?
Well, I have turned my integral
00:22:43.000 --> 00:22:50.000
into the double integral of a
dS.
00:22:50.000 --> 00:22:53.000
And a is just a constant,
so I am very lucky here.
00:22:53.000 --> 00:22:59.000
I can just say this will be a
times the double integral of dS.
00:22:59.000 --> 00:23:02.000
And, of course,
some day I will have to learn
00:23:02.000 --> 00:23:05.000
how to tackle that beast,
but for now I don't actually
00:23:05.000 --> 00:23:08.000
need to because the double
integral of dS just means I am
00:23:08.000 --> 00:23:11.000
summing the area of each little
piece of the sphere.
00:23:11.000 --> 00:23:16.000
I am just going to get the
total area of the sphere which I
00:23:16.000 --> 00:23:23.000
know to be 4pi a2.
This guy here is going to be
00:23:23.000 --> 00:23:29.000
the area of S.
I know that to be 4pi a^2.
00:23:29.000 --> 00:23:37.000
So I will get 4pi a^3.
That one was relatively
00:23:37.000 --> 00:23:44.000
painless.
That was too easy.
00:23:44.000 --> 00:23:49.000
Let's do a second example with
the same sphere.
00:23:49.000 --> 00:23:56.000
But now my vector field is
going to be just z times k.
00:23:56.000 --> 00:23:58.000
Well, let me give it a
different name.
00:23:58.000 --> 00:24:04.000
Let me call it H instead of f
or something like that just so
00:24:04.000 --> 00:24:08.000
that it is not called F anymore.
Well, the initial part of the
00:24:08.000 --> 00:24:11.000
setup is still the same.
The normal vector is still the
00:24:11.000 --> 00:24:13.000
same.
What changes is,
00:24:13.000 --> 00:24:16.000
of course, my vector field is
no longer sticking straight out
00:24:16.000 --> 00:24:18.000
so I cannot use this easy
geometric argument.
00:24:18.000 --> 00:24:22.000
It looks like I will have to
compute F dot n and then figure
00:24:22.000 --> 00:24:24.000
out how to integrate that with
dS.
00:24:24.000 --> 00:24:36.000
Let's do that.
We still have that n is 00:24:44.000
y, z>/a.
That tells us that H dot n will
00:24:44.000 --> 00:24:49.000
be
dot 00:24:57.000
z> / a.
It looks like I will be left
00:24:57.000 --> 00:25:10.000
with z^2 over a.
H dot n is z^2 over a.
00:25:10.000 --> 00:25:18.000
The double integral for flux
now becomes double integral on
00:25:18.000 --> 00:25:25.000
the sphere of z^2 over a dS.
Well, we can take out one over
00:25:25.000 --> 00:25:29.000
a, that is fine,
but it looks like we will have
00:25:29.000 --> 00:25:33.000
to integrate z^2 on the surface
of the sphere.
00:25:33.000 --> 00:25:37.000
How do we do that?
Well, we have to figure out
00:25:37.000 --> 00:25:41.000
what is dS in terms of our
favorite set of two variables
00:25:41.000 --> 00:25:45.000
that we will use to integrate.
Now, what is the best way to
00:25:45.000 --> 00:25:47.000
figure out where you are on the
sphere?
00:25:47.000 --> 00:25:51.000
Well, you could try to use
maybe theta and z.
00:25:51.000 --> 00:25:55.000
If you know how high you are
and where you are around,
00:25:55.000 --> 00:25:58.000
in principle you know where you
are on the sphere.
00:25:58.000 --> 00:26:02.000
But since spherical coordinates
we have actually learned about
00:26:02.000 --> 00:26:06.000
something much more interesting,
namely spherical coordinates.
00:26:06.000 --> 00:26:09.000
It looks like longitude /
latitude is the way to go when
00:26:09.000 --> 00:26:12.000
trying to figure out where you
are on a sphere.
00:26:12.000 --> 00:26:19.000
We are going to use phi and
theta.
00:26:19.000 --> 00:26:24.000
And, of course,
we have to figure out how to
00:26:24.000 --> 00:26:28.000
express dS in terms of d phi and
d theta.
00:26:28.000 --> 00:26:32.000
Well, if you were paying
really, really close attention
00:26:32.000 --> 00:26:36.000
last time, you will notice that
we have actually already done
00:26:36.000 --> 00:26:41.000
that.
Last time we saw that if I have
00:26:41.000 --> 00:26:48.000
a sphere of radius a and I take
a little piece of it that
00:26:48.000 --> 00:26:56.000
corresponds to small changes in
phi and theta then we said that
00:26:56.000 --> 00:27:01.000
-- Well,
we argued that this side here,
00:27:01.000 --> 00:27:08.000
the one that is going east-west
was a piece of the circle that
00:27:08.000 --> 00:27:14.000
has a radius a sin phi because
that is r,
00:27:14.000 --> 00:27:19.000
so that side is a sin phi delta
theta.
00:27:19.000 --> 00:27:22.000
And the side that goes
north-south is a piece of the
00:27:22.000 --> 00:27:26.000
circle of radius a corresponding
to angle delta phi,
00:27:26.000 --> 00:27:32.000
so it is a delta phi.
And so, just to get to the
00:27:32.000 --> 00:27:40.000
answer, we got dS equals a^2 sin
phi d phi d theta.
00:27:40.000 --> 00:27:45.000
When we set up a surface
integral on the surface of a
00:27:45.000 --> 00:27:48.000
sphere,
most likely we will be using
00:27:48.000 --> 00:27:52.000
phi and theta as our two
variables of integration and dS
00:27:52.000 --> 00:27:55.000
will become this.
Now, it is OK to think of them
00:27:55.000 --> 00:27:58.000
as spherical coordinates,
but I would like to encourage
00:27:58.000 --> 00:28:01.000
you not to think of them as
spherical coordinates.
00:28:01.000 --> 00:28:05.000
Spherical coordinates are a way
of describing points in space in
00:28:05.000 --> 00:28:09.000
terms of three variables.
Here it is more like we are
00:28:09.000 --> 00:28:12.000
parameterizing the sphere.
We are finding a parametric
00:28:12.000 --> 00:28:15.000
equation for the sphere using
two variables phi and theta
00:28:15.000 --> 00:28:18.000
which happen to be part of the
spherical coordinate system.
00:28:18.000 --> 00:28:22.000
But, see, there is no rho
involved in here.
00:28:22.000 --> 00:28:26.000
I am not using any rho ever,
and I am not going to in this
00:28:26.000 --> 00:28:28.000
calculation.
I have two variable phi and
00:28:28.000 --> 00:28:37.000
theta.
That is it.
00:28:37.000 --> 00:28:40.000
It is basically in the same way
as when you parameterize a line
00:28:40.000 --> 00:28:45.000
integral in the circle,
we use theta as the parameter
00:28:45.000 --> 00:28:50.000
variable and never think about
r.
00:28:50.000 --> 00:28:52.000
That being said,
well, we are going to use phi
00:28:52.000 --> 00:28:54.000
and theta.
We know what dS is.
00:28:54.000 --> 00:28:58.000
We still need to figure out
what z is.
00:28:58.000 --> 00:29:01.000
There we want to think a tiny
bit about spherical coordinates
00:29:01.000 --> 00:29:08.000
again.
And we will know that z is just
00:29:08.000 --> 00:29:15.000
a cos phi.
In case you don't quite see it,
00:29:15.000 --> 00:29:25.000
let me draw a diagram.
Phi is the angle down from the
00:29:25.000 --> 00:29:31.000
positive z axes,
this distance is a,
00:29:31.000 --> 00:29:38.000
so this distance here is a cos
phi.
00:29:38.000 --> 00:29:44.000
Now I have everything I need to
compute my double integral.
00:29:44.000 --> 00:29:49.000
z^2 over a dS will become a
double integral.
00:29:49.000 --> 00:30:00.000
z^2 becomes a^2 cos^2 phi over
a times, ds becomes,
00:30:00.000 --> 00:30:07.000
a^2 sin phi d phi d theta.
Now I need to set up bounds.
00:30:07.000 --> 00:30:12.000
Well, what are the bounds?
Phi goes all the way from zero
00:30:12.000 --> 00:30:19.000
to pi because we go all the way
from the north pole to the south
00:30:19.000 --> 00:30:23.000
pole, and theta goes from zero
to 2pi.
00:30:23.000 --> 00:30:27.000
And, of course,
I can get rid of some a's in
00:30:27.000 --> 00:30:34.000
there and take them out.
Let's look at what number we
00:30:34.000 --> 00:30:37.000
get.
First of all,
00:30:37.000 --> 00:30:43.000
we can take out all those a's
and get a^3.
00:30:43.000 --> 00:30:50.000
Second, in the inner integral,
we are integrating cos^2 phi
00:30:50.000 --> 00:30:54.000
sin phi d phi.
I claim that integrates to cos3
00:30:54.000 --> 00:30:57.000
up to some factor,
and that factor should be
00:30:57.000 --> 00:31:02.000
negative one-third.
If you look at cos3 phi and you
00:31:02.000 --> 00:31:07.000
take its derivative,
you will get that guy with a
00:31:07.000 --> 00:31:12.000
negative three in front between
zero and pi.
00:31:12.000 --> 00:31:16.000
And, while integrating over
theta, we will just multiply
00:31:16.000 --> 00:31:24.000
things by 2pi.
Let me add the 2pi in front.
00:31:24.000 --> 00:31:27.000
Now, if I evaluate this guy
between zero and pi,
00:31:27.000 --> 00:31:32.000
well, at pi cos^3 is negative
one, at zero it is one,
00:31:32.000 --> 00:31:35.000
I will get two-thirds out of
this.
00:31:35.000 --> 00:31:39.000
I end up with four-thirds pi
a^3.
00:31:39.000 --> 00:31:46.000
Sorry I didn't write very much
because I am trying to save
00:31:46.000 --> 00:31:52.000
blackboard space.
Yes?
00:31:52.000 --> 00:31:55.000
That is a very natural question.
That looks a lot like somebody
00:31:55.000 --> 00:31:58.000
we know, like the volume of a
sphere.
00:31:58.000 --> 00:32:03.000
And ultimately it will be.
Wait until next class when we
00:32:03.000 --> 00:32:07.000
talk about the divergence
theorem.
00:32:07.000 --> 00:32:11.000
I mean the question was is this
related to the volume of a
00:32:11.000 --> 00:32:14.000
sphere, and ultimately it is,
but for now it is just some
00:32:14.000 --> 00:32:23.000
coincidence.
Yes?
00:32:23.000 --> 00:32:26.000
The question is there is a way
to do it M dx plus N dy plus
00:32:26.000 --> 00:32:28.000
stuff like that?
The answer is unfortunately no
00:32:28.000 --> 00:32:30.000
because it is not a line
integral.
00:32:30.000 --> 00:32:35.000
It is a surface integral,
so we need to have to variables
00:32:35.000 --> 00:32:38.000
in there.
In a way you would end up with
00:32:38.000 --> 00:32:41.000
things like some dx dy maybe and
so on.
00:32:41.000 --> 00:32:45.000
I mean it is not practical to
do it directly that way because
00:32:45.000 --> 00:32:49.000
you would have then to compute
Jacobians to switch from dx dy
00:32:49.000 --> 00:32:52.000
to something else.
We are going to see various
00:32:52.000 --> 00:32:54.000
ways of computing it.
Unfortunately,
00:32:54.000 --> 00:32:57.000
it is not quite as simple as
with line integrals.
00:32:57.000 --> 00:32:59.000
But it is not much harder.
It is the same spirit.
00:32:59.000 --> 00:33:04.000
We just use two variables and
set up everything in terms of
00:33:04.000 --> 00:33:12.000
these two variables.
Any other questions?
00:33:12.000 --> 00:33:14.000
No.
OK.
00:33:51.000 --> 00:33:54.000
By the way, just some food for
thought.
00:33:54.000 --> 00:34:01.000
Never mind.
Conclusion of looking at these
00:34:01.000 --> 00:34:05.000
two examples is that sometimes
we can use geometric.
00:34:05.000 --> 00:34:07.000
The first example,
we didn't actually have to
00:34:07.000 --> 00:34:11.000
compute an integral.
But most of the time we need to
00:34:11.000 --> 00:34:14.000
learn how to set up double
integrals.
00:34:14.000 --> 00:34:26.000
Use geometry or you need to set
up for double integral of a
00:34:26.000 --> 00:34:30.000
surface.
And so we are going to learn
00:34:30.000 --> 00:34:33.000
how to do that in general.
As I said, we need to have two
00:34:33.000 --> 00:34:37.000
parameters on the surface and
express everything in terms of
00:34:37.000 --> 00:34:43.000
these.
Let's look at various examples.
00:34:43.000 --> 00:34:46.000
We are going to see various
situations where we can do
00:34:46.000 --> 00:34:49.000
things.
Well, let's start with an easy
00:34:49.000 --> 00:34:53.000
one.
Let's call that number zero.
00:34:53.000 --> 00:35:02.000
Say that my surface S is a
horizontal plane,
00:35:02.000 --> 00:35:07.000
say z equals a.
When I say a horizontal plane,
00:35:07.000 --> 00:35:09.000
it doesn't have to be the
entire horizontal plane.
00:35:09.000 --> 00:35:14.000
It could be a small piece of it.
It could even be,
00:35:14.000 --> 00:35:16.000
to trick you,
maybe an ellipse in there or a
00:35:16.000 --> 00:35:19.000
triangle in there or something
like that.
00:35:19.000 --> 00:35:23.000
What you have to recognize is
my surface is a piece of just a
00:35:23.000 --> 00:35:27.000
flat plane, so I shouldn't worry
too much about what part of a
00:35:27.000 --> 00:35:30.000
plane it is.
Well, it will become important
00:35:30.000 --> 00:35:32.000
when I set up bounds for
integration.
00:35:32.000 --> 00:35:36.000
But, when it comes to looking
for the normal vector,
00:35:36.000 --> 00:35:40.000
be rest assured that the normal
vector to a horizontal plane is
00:35:40.000 --> 00:35:44.000
just vertical.
It is going to be either k or
00:35:44.000 --> 00:35:49.000
negative k depending on whether
I have chosen to orient it
00:35:49.000 --> 00:35:54.000
pointing up or down.
And which one I choose might
00:35:54.000 --> 00:35:57.000
depend on what I am going to try
to do.
00:35:57.000 --> 00:36:02.000
The normal vector is just
sticking straight up or straight
00:36:02.000 --> 00:36:05.000
down.
Now, what about dS?
00:36:05.000 --> 00:36:11.000
Well, it is just going to be
the area element in a horizontal
00:36:11.000 --> 00:36:14.000
plane.
It just looks like it should be
00:36:14.000 --> 00:36:16.000
dx dy.
I mean if I am moving on a
00:36:16.000 --> 00:36:18.000
horizontal plane,
to know where I am,
00:36:18.000 --> 00:36:26.000
I should know x and y.
So dS will be dx dy.
00:36:26.000 --> 00:36:31.000
If I play the game that way,
I have my vector field F.
00:36:31.000 --> 00:36:34.000
I do F dot n.
That just gives me the z
00:36:34.000 --> 00:36:37.000
component which might involve x,
y and z.
00:36:37.000 --> 00:36:40.000
x and y I am very happy with.
They will stay as my variables.
00:36:40.000 --> 00:36:43.000
Whenever I see z,
well, I want to get rid of it.
00:36:43.000 --> 00:36:46.000
That is easy because z is just
equal to a.
00:36:46.000 --> 00:36:50.000
I just plug that value and I am
left with only x and y,
00:36:50.000 --> 00:36:53.000
and I am integrating that dx
dy.
00:36:53.000 --> 00:36:58.000
It is actually ending up being
just a usual double integral in
00:36:58.000 --> 00:37:00.000
x, y coordinates.
And, of course,
00:37:00.000 --> 00:37:02.000
once it is set up anything is
fair game.
00:37:02.000 --> 00:37:05.000
I might want to switch to polar
coordinates or something like
00:37:05.000 --> 00:37:09.000
that.
Or, I can set it up dx dy or dy
00:37:09.000 --> 00:37:12.000
dx.
All the usual stuff applies.
00:37:12.000 --> 00:37:17.000
But, for the initial setup,
we are just going to use these
00:37:17.000 --> 00:37:21.000
and express everything in terms
of x and y.
00:37:21.000 --> 00:37:27.000
A small variation on that.
Let's say that we take vertical
00:37:27.000 --> 00:37:35.000
planes that are parallel to
maybe the blackboard plane,
00:37:35.000 --> 00:37:42.000
so parallel to the yz plane.
That might be something like x
00:37:42.000 --> 00:37:47.000
equals some constant.
Well, what would I do then?
00:37:47.000 --> 00:37:52.000
It could be pretty much the
same.
00:37:52.000 --> 00:37:55.000
The normal vector for this guy
would be sticking straight out
00:37:55.000 --> 00:37:59.000
towards me or away from me.
Let's say I am having it come
00:37:59.000 --> 00:38:03.000
to the front.
The normal vector would be
00:38:03.000 --> 00:38:07.000
plus/minus i.
And the variables that I would
00:38:07.000 --> 00:38:11.000
be using, to find out my
position on this guy,
00:38:11.000 --> 00:38:14.000
would be y and z.
In terms of those,
00:38:14.000 --> 00:38:19.000
the surface element is just dy
dz.
00:38:19.000 --> 00:38:25.000
Similarly for planes parallel
to the xz plane.
00:38:25.000 --> 00:38:33.000
You can figure that one out.
These are somehow the easiest
00:38:33.000 --> 00:38:39.000
ones, because those we already
know how to compute without too
00:38:39.000 --> 00:38:41.000
much trouble.
What if it is a more
00:38:41.000 --> 00:38:43.000
complicated plane?
We will come back to that next
00:38:43.000 --> 00:38:49.000
time.
Let's explore some other
00:38:49.000 --> 00:38:56.000
situations first.
Number one on the list.
00:38:56.000 --> 00:39:03.000
Let's say that I gave you a
sphere of radius a centered at
00:39:03.000 --> 00:39:09.000
the origin, or maybe just half
of that sphere or some portion
00:39:09.000 --> 00:39:12.000
of it.
Well, we have already seen how
00:39:12.000 --> 00:39:15.000
to do things.
Namely, we will be saying the
00:39:15.000 --> 00:39:18.000
normal vector is x,
y, z over a,
00:39:18.000 --> 00:39:23.000
plus or minus depending on
whether we want it pointing in
00:39:23.000 --> 00:39:29.000
or out.
And dS will be a^2 sin phi d
00:39:29.000 --> 00:39:32.000
phi d theta.
In fact, we will express
00:39:32.000 --> 00:39:35.000
everything in terms of phi and
theta.
00:39:35.000 --> 00:39:37.000
If I wanted to I could tell you
what the formulas are for x,
00:39:37.000 --> 00:39:40.000
y, z in terms of phi and theta.
You know them.
00:39:40.000 --> 00:39:44.000
But it is actually better to
wait a little bit.
00:39:44.000 --> 00:39:48.000
It is better to do F dot n,
because F is also going to have
00:39:48.000 --> 00:39:49.000
a bunch of x's,
y's and z's.
00:39:49.000 --> 00:39:53.000
And if there is any kind of
symmetry to the problem then you
00:39:53.000 --> 00:39:57.000
might end up with things like
x^2 y^2 z^2 or things that have
00:39:57.000 --> 00:40:01.000
more symmetry that are easier to
express in terms of phi and
00:40:01.000 --> 00:40:05.000
theta.
The advice would be first do
00:40:05.000 --> 00:40:10.000
the dot product with F,
and then see what you get and
00:40:10.000 --> 00:40:17.000
then turn it into phi and theta.
That is one we have seen.
00:40:17.000 --> 00:40:20.000
Let's say that I have -- It is
a close cousin.
00:40:20.000 --> 00:40:30.000
Let's say I have a cylinder of
radius a centered on the z-axis.
00:40:30.000 --> 00:40:37.000
What does that look like?
And, again, when I say
00:40:37.000 --> 00:40:40.000
cylinder, it could be a piece of
cylinder.
00:40:40.000 --> 00:40:44.000
First of all,
what does the normal vector to
00:40:44.000 --> 00:40:47.000
a cylinder look like?
Well, it is sticking straight
00:40:47.000 --> 00:40:50.000
out, but sticking straight out
in a slightly different way from
00:40:50.000 --> 00:40:52.000
what happens with a sphere.
See, the sides of a cylinder
00:40:52.000 --> 00:40:54.000
are vertical.
If you imagine that you have
00:40:54.000 --> 00:40:56.000
this big cylindrical type in
front of you,
00:40:56.000 --> 00:40:59.000
hopefully you can see that a
normal vector is going to always
00:40:59.000 --> 00:41:02.000
be horizontal.
It is sticking straight out in
00:41:02.000 --> 00:41:07.000
the horizontal directions.
It doesn't have any z component.
00:41:07.000 --> 00:41:13.000
I claim the normal vector for
the cylinder,
00:41:13.000 --> 00:41:21.000
if you have a point here at (x,
y, z), it would be pointing
00:41:21.000 --> 00:41:27.000
straight out away from the
central axis.
00:41:27.000 --> 00:41:31.000
My normal vector,
well, if I am taking it two
00:41:31.000 --> 00:41:34.000
points outwards,
will be going straight away
00:41:34.000 --> 00:41:38.000
from the central axis.
If I look at it from above,
00:41:38.000 --> 00:41:42.000
maybe it is easier if I look at
it from above,
00:41:42.000 --> 00:41:45.000
look at x, y,
then my cylinder looks like a
00:41:45.000 --> 00:41:49.000
circle and the normal vector
just points straight out.
00:41:49.000 --> 00:41:53.000
It is the same situation as
when we had a circle in the 2D
00:41:53.000 --> 00:41:57.000
case.
The normal vector for that is
00:41:57.000 --> 00:42:02.000
just going to be x,
y and 0 in the z component.
00:42:02.000 --> 00:42:05.000
Well, plus/minus,
depending on whether you want
00:42:05.000 --> 00:42:06.000
it sticking in or out.
00:42:41.000 --> 00:42:47.000
We said in our cylinder normal
vector is plus or minus x,
00:42:47.000 --> 00:42:54.000
y, zero over a.
What about the surface element?
00:42:54.000 --> 00:42:57.000
Before we ask that,
maybe we should first figure
00:42:57.000 --> 00:43:00.000
out what coordinates are we
going to use to locate ourselves
00:43:00.000 --> 00:43:02.000
in a cylinder.
Well, yes,
00:43:02.000 --> 00:43:05.000
we probably want to use part of
a cylindrical coordinate,
00:43:05.000 --> 00:43:08.000
except for,
well, we don't want r because r
00:43:08.000 --> 00:43:11.000
doesn't change,
it is not a variable here.
00:43:11.000 --> 00:43:15.000
Indeed, you probably want to
use z to tell how high you are
00:43:15.000 --> 00:43:18.000
and theta to tell you where you
are around.
00:43:18.000 --> 00:43:27.000
dS should be in terms of dz d
theta.
00:43:27.000 --> 00:43:33.000
Now, what is the constant?
Well, let's look at a small
00:43:33.000 --> 00:43:39.000
piece of our cylinder
corresponding to a small angle
00:43:39.000 --> 00:43:44.000
delta theta and a small height
delta z.
00:43:44.000 --> 00:43:47.000
Well, the height,
as I said, is going to be delta
00:43:47.000 --> 00:43:50.000
z.
What about the width?
00:43:50.000 --> 00:43:55.000
It is going to be a piece of a
circle of radius a corresponding
00:43:55.000 --> 00:43:59.000
to the angle delta theta,
so this side will be a delta
00:43:59.000 --> 00:44:05.000
theta.
Delta S is a delta theta delta
00:44:05.000 --> 00:44:08.000
z.
DS is just a dz d theta or d
00:44:08.000 --> 00:44:13.000
theta dz.
It doesn't matter which way you
00:44:13.000 --> 00:44:16.000
do it.
And so when we set up the flux
00:44:16.000 --> 00:44:21.000
integral, we will take first the
dot product of f with this
00:44:21.000 --> 00:44:25.000
normal vector.
Then we will stick in this dS.
00:44:25.000 --> 00:44:28.000
And then, of course,
we will get rid of any x and y
00:44:28.000 --> 00:44:31.000
that are left by expressing them
in terms of theta.
00:44:31.000 --> 00:44:37.000
Maybe x becomes a cos theta,
y becomes a sin theta.
00:44:37.000 --> 00:44:41.000
These various formulas,
you should try to remember them
00:44:41.000 --> 00:44:45.000
because they are really useful,
for the sphere,
00:44:45.000 --> 00:44:48.000
for the cylinder.
And, hopefully,
00:44:48.000 --> 00:44:52.000
those for the planes you kind
of know already intuitively.
00:44:52.000 --> 00:44:59.000
What about marginals or faces?
Not everything in life is made
00:44:59.000 --> 00:45:08.000
out of cylinders and spheres.
I mean it is a good try.
00:45:08.000 --> 00:45:11.000
Let's look at a marginal kind
of surface.
00:45:11.000 --> 00:45:19.000
Let's say I give you a graph of
a function z equals f of x,
00:45:19.000 --> 00:45:21.000
y.
This guy has nothing to do with
00:45:21.000 --> 00:45:22.000
the integrand.
It is not what we are
00:45:22.000 --> 00:45:24.000
integrating.
We are just integrating a
00:45:24.000 --> 00:45:26.000
vector field that has nothing to
do with that.
00:45:26.000 --> 00:45:31.000
This is how I want to describe
the surface on which I will be
00:45:31.000 --> 00:45:37.000
integrating.
My surface is given by z as a
00:45:37.000 --> 00:45:40.000
function of x,
y.
00:45:40.000 --> 00:45:48.000
Well, I would need to tell you
what n is and what dS is.
00:45:48.000 --> 00:45:51.000
That is going to be slightly
annoying.
00:45:51.000 --> 00:45:54.000
I mean, I don't want to tell
them separately because you see
00:45:54.000 --> 00:45:58.000
they are pretty hard.
Instead, I am going to tell you
00:45:58.000 --> 00:46:01.000
that in this case,
well, let's see.
00:46:01.000 --> 00:46:07.000
What variables do we want?
I am going to tell you a
00:46:07.000 --> 00:46:11.000
formula for n dS.
What variables do we want to
00:46:11.000 --> 00:46:15.000
express this in terms of?
Well, most likely x and y
00:46:15.000 --> 00:46:19.000
because we know how to express z
in terms of x and y.
00:46:19.000 --> 00:46:26.000
This is an invitation to get
rid of any z that might be left
00:46:26.000 --> 00:46:30.000
and set everything up in terms
of dx dy.
00:46:30.000 --> 00:46:31.000
The formula that we are going
to see,
00:46:31.000 --> 00:46:36.000
I think we are going to see the
details of why it works
00:46:36.000 --> 00:46:40.000
tomorrow,
is that you can take negative
00:46:40.000 --> 00:46:44.000
partial f partial x,
negative partial f partial y,
00:46:44.000 --> 00:46:48.000
one,
dx dy.
00:46:48.000 --> 00:46:52.000
Plus/minus depending on which
way you want it to go.
00:46:52.000 --> 00:46:57.000
If you really want to know what
dS is, well, dS is the magnitude
00:46:57.000 --> 00:47:01.000
of this vector times dx dy.
There will be a square root and
00:47:01.000 --> 00:47:04.000
some squares and some stuff.
What is the normal vector?
00:47:04.000 --> 00:47:10.000
Well, you take this vector and
you scale it down to unit
00:47:10.000 --> 00:47:14.000
length.
Just to emphasize it,
00:47:14.000 --> 00:47:24.000
this guy here is not n and this
guy here is not dS.
00:47:24.000 --> 00:47:27.000
Each of them is more
complicated than that,
00:47:27.000 --> 00:47:31.000
but the combination somehow
simplifies nicely.
00:47:31.000 --> 00:47:36.000
And that is good news for us.
Now, concretely,
00:47:36.000 --> 00:47:40.000
one way you can think about it
is this tells you how to reduce
00:47:40.000 --> 00:47:42.000
things to an integral of x and
y.
00:47:42.000 --> 00:47:44.000
And, of course,
you will have to figure out
00:47:44.000 --> 00:47:47.000
what are the bounds on x and y.
That means you will need to
00:47:47.000 --> 00:47:51.000
know what does the shadow of
your surface look like in the x,
00:47:51.000 --> 00:47:59.000
y plane.
To set up bounds on whatever
00:47:59.000 --> 00:48:06.000
you will get dx dy,
well, of course you can switch
00:48:06.000 --> 00:48:08.000
to dy dx or anything you would
like,
00:48:08.000 --> 00:48:19.000
but you will need to look at
the shadow of S in the x y
00:48:19.000 --> 00:48:22.000
plane.
But only do that after you
00:48:22.000 --> 00:48:27.000
gotten rid of all the z.
When you no longer have z then
00:48:27.000 --> 00:48:33.000
you can figure out what the
bounds are for x and y.
00:48:33.000 --> 00:48:40.000
Any questions about that?
Yes?
00:48:40.000 --> 00:48:42.000
For the cylinder.
OK.
00:48:42.000 --> 00:48:44.000
Let me re-explain quickly how I
got a normal vector for the
00:48:44.000 --> 00:48:47.000
cylinder.
If you know what a cylinder
00:48:47.000 --> 00:48:50.000
looks like, you probably can see
that the normal vector sticks
00:48:50.000 --> 00:48:56.000
straight out of it horizontally.
That means the z component of n
00:48:56.000 --> 00:48:59.000
is going to be zero.
And then the x,
00:48:59.000 --> 00:49:02.000
y components you get by looking
at it from above.
00:49:02.000 --> 00:49:12.000
One last thing I want to say.
What about the geometric
00:49:12.000 --> 00:49:15.000
interpretation and how to prove
it?
00:49:15.000 --> 00:49:27.000
Well, if your vector field F is
a velocity field then the flux
00:49:27.000 --> 00:49:37.000
is the amount of matter that
crosses the surface that passes
00:49:37.000 --> 00:49:44.000
through S per unit time.
And the way that you would
00:49:44.000 --> 00:49:47.000
prove it would be similar to the
picture that I drew when we did
00:49:47.000 --> 00:49:50.000
it in the plane.
Namely, you would consider a
00:49:50.000 --> 00:49:53.000
small element of a surface delta
S.
00:49:53.000 --> 00:49:55.000
And you would try to figure out
what is the stuff that flows
00:49:55.000 --> 00:49:59.000
through it in a second.
Well, it is the stuff that
00:49:59.000 --> 00:50:05.000
lives in a small box whose base
is that piece of surface and
00:50:05.000 --> 00:50:10.000
whose other side is given by the
vector field.
00:50:10.000 --> 00:50:15.000
And then the volume of that is
given by base times height,
00:50:15.000 --> 00:50:20.000
and the height is F dot n.
It is the same argument as what
00:50:20.000 --> 00:50:22.000
we saw in the plane.
OK.
00:50:22.000 --> 00:50:24.000
Next time we will see more
formulas.
00:50:24.000 --> 00:50:28.000
We will first see how to prove
this, more ways to do it,
00:50:28.000 --> 00:50:31.000
more examples.
And then we will get to the
00:50:31.000 --> 00:50:34.000
divergence theorem.