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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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00:03:13 --> 00:03:20
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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00:03:31 --> 00:03:39
Such a field would be directed
towards the origin and its
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00:03:39 --> 00:03:47
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 --> 00:03:57
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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00:04:02 --> 00:04:09
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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00:04:46 --> 00:04:47
54
00:04:47 --> 00:04:49
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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00:04:52 --> 00:04:56
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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00:07:32 --> 00:07:41
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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00:07:54 --> 00:07:59
And that measured how much the
vector field was going across
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00:07:59 --> 00:08:01
the curve.
If you were thinking of a
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00:08:01 --> 00:08:05
velocity field,
that would measure how much
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00:08:05 --> 00:08:08
fluid is passing through the
curve in unit time.
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00:08:08 --> 00:08:10
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 --> 00:08:18
Because, if you have a curve in
space and say that you have wind
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00:08:18 --> 00:08:22
or something like that,
you cannot really ask how much
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00:08:22 --> 00:08:24
air is flowing through the
curve.
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00:08:24 --> 00:08:28
See, to have a flow through
something you need a surface.
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00:08:28 --> 00:08:32
If you have a net maybe then
you can ask how much stuff is
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00:08:32 --> 00:08:37
passing through that surface.
There is going to be a big
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00:08:37 --> 00:08:44
difference here.
In the three-dimensional space,
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00:08:44 --> 00:08:51
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 --> 00:08:59
anymore.
That means we will be
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00:08:59 --> 00:09:10
integrating, we will be summing
over all the pieces of a surface
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00:09:10 --> 00:09:13
in space.
Because a surface is a
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00:09:13 --> 00:09:15
two-dimensional object,
that will end up being a double
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00:09:15 --> 00:09:17
integral.
But, of course,
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00:09:17 --> 00:09:19
we will have to set it up
properly because the surface
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00:09:19 --> 00:09:22
that is in space,
and we will probably have x,
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00:09:22 --> 00:09:24
y and z to deal with at the
same time,
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00:09:24 --> 00:09:28
and we will have to somehow get
rid of one variable so that we
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00:09:28 --> 00:09:31
can set up and evaluate a double
integral.
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00:09:31 --> 00:09:35
So conceptually it is very
similar to line integrals.
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00:09:35 --> 00:09:39
In the line integral in the
plane, you had two variables
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00:09:39 --> 00:09:42
that you reduced to one by
figuring out what the curve was.
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00:09:42 --> 00:09:51
Here you have three variables
that you will reduce to two by
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00:09:51 --> 00:09:56
figuring out what the surface
is.
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Let me give you a definition of
flux in 3D.
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00:10:00 --> 00:10:15
Let's say that we have a vector
field and s, a surface in space.
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00:10:15 --> 00:10:17
Let me draw some kind of a
picture.
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00:10:17 --> 00:10:21
I have my surface and I have my
vector field F.
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00:10:21 --> 00:10:25
Well, at every point it changes
with a point.
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00:10:25 --> 00:10:28
Well, I want to figure out how
much my vector field is going
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00:10:28 --> 00:10:34
across that surface.
That means I want to figure out
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00:10:34 --> 00:10:40
the normal component of my
vector field,
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00:10:40 --> 00:10:45
so I will use,
as in the plane case,
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00:10:45 --> 00:10:53
the unit normal vector to s.
I take my point on the surface
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00:10:53 --> 00:10:59
and build a unit vector that is
standing on it perpendicularly.
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00:10:59 --> 00:11:05
Now, we have to decide which
way it is standing.
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00:11:05 --> 00:11:09
We can build our normal vector
to go this way or to go the
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00:11:09 --> 00:11:12
other way around.
There are two choices.
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00:11:12 --> 00:11:16
Basically, whenever you want to
set up a flux integral you have
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00:11:16 --> 00:11:19
to choose one side of the
surface.
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00:11:19 --> 00:11:23
And you will count positively
what flows toward that side and
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00:11:23 --> 00:11:26
negatively what flows towards
the other side.
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00:11:26 --> 00:11:40
There are two choices for n.
We need to choose a side of the
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00:11:40 --> 00:11:45
surface.
In the case of curves,
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00:11:45 --> 00:11:50
we made that choice by deciding
that because we were going along
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00:11:50 --> 00:11:54
some direction on the curve we
could choose one side by saying
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00:11:54 --> 00:11:57
let's rotate clockwise from the
tangent vector.
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00:11:57 --> 00:12:00
And, in a way,
what we were doing was really
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00:12:00 --> 00:12:04
it was a recipe to choose for us
one of the two sides.
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00:12:04 --> 00:12:09
Here we don't have a notion of
orienting the surface other than
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00:12:09 --> 00:12:14
by precisely choosing one of the
two possible normal vectors.
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00:12:14 --> 00:12:15
So, in fact,
this is called choosing an
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00:12:15 --> 00:12:18
orientation of a surface.
When you are saying you are
152
00:12:18 --> 00:12:22
orienting the surface that
really means you are deciding
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00:12:22 --> 00:12:31
which side is which.
Let's call that orientation.
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00:12:31 --> 00:12:35
Now, there is no set convention
that will work forever.
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00:12:35 --> 00:12:39
But the usually traditional
settings would be to take your
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00:12:39 --> 00:12:43
normal vector pointing maybe out
of the solid region because then
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00:12:43 --> 00:12:48
you will be looking at flux that
is coming out of that region of
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00:12:48 --> 00:12:51
space.
Or, if you have a surface that
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00:12:51 --> 00:12:55
is not like closed or anything
but maybe you will want the flux
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00:12:55 --> 00:12:59
going up through the region.
Or, there are various
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00:12:59 --> 00:13:02
conventions.
Concretely, on problem sets it
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00:13:02 --> 00:13:05
will either say which choice you
have to make or you get to
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00:13:05 --> 00:13:07
choose which one you want to
make.
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00:13:07 --> 00:13:10
And, of course,
if you choose the other one
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00:13:10 --> 00:13:12
then the sign becomes the
opposite.
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00:13:12 --> 00:13:17
Now, once we have made a choice
then we can define the flux
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00:13:17 --> 00:13:21
integral.
It will just be the double
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00:13:21 --> 00:13:26
integral over a surface of F dot
n dS.
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00:13:26 --> 00:13:33
Now I am using a big dS.
That stands for the surface
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00:13:33 --> 00:13:39
area element on this surface.
I am using dS rather than dA
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00:13:39 --> 00:13:43
because I still want to think of
dA as maybe the area in one of
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00:13:43 --> 00:13:47
the coordinate planes like the
one we had in double integrals.
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00:13:47 --> 00:13:51
You will see later where this
comes in.
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00:13:51 --> 00:13:54
But conceptually it is very
similar.
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00:13:54 --> 00:13:59
Concretely what this means is I
cut my surface into little
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00:13:59 --> 00:14:03
pieces.
Each of them has area delta S.
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00:14:03 --> 00:14:07
And, for each piece,
I take my vector field,
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00:14:07 --> 00:14:13
I take my normal vector,
I dot them and I multiply by
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00:14:13 --> 00:14:17
this surface area and sum all
these things together.
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00:14:17 --> 00:14:23
That is what a double integral
means.
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00:14:23 --> 00:14:25
In particular,
an easy case where you know you
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00:14:25 --> 00:14:28
can get away without computing
anything is, of course,
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00:14:28 --> 00:14:32
if your vector field is tangent
to the surface because then you
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00:14:32 --> 00:14:36
know that there is no flux.
Flux is going to be zero
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00:14:36 --> 00:14:38
because nothing passes through
the surface.
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00:14:38 --> 00:14:42
Otherwise, we have to figure
out how to compute these things.
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00:14:42 --> 00:14:50
That is what we are going to
learn now.
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00:14:50 --> 00:14:51
Well, maybe I should box this
formula.
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00:14:51 --> 00:14:57
I have noticed that some of you
seem to like it when I box the
190
00:14:57 --> 00:15:03
important formulas.
(APPLAUSE) By the way,
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00:15:03 --> 00:15:12
a piece of notation before I
move on, sometimes you will also
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00:15:12 --> 00:15:18
see the notation vector dS.
What is vector dS?
193
00:15:18 --> 00:15:24
Vector dS is this guy n dS put
together.
194
00:15:24 --> 00:15:30
Vector dS is a vector which
points perpendicular to the
195
00:15:30 --> 00:15:35
surface and whose length
corresponds to the surface
196
00:15:35 --> 00:15:37
element.
And the reason for having this
197
00:15:37 --> 00:15:41
shortcut notation,
well, it is not only laziness
198
00:15:41 --> 00:15:44
like saving one n,
but it is because this guy is
199
00:15:44 --> 00:15:49
very often easier to compute
than it is to set up n and dS
200
00:15:49 --> 00:15:52
separately.
Actually, if you remember in
201
00:15:52 --> 00:15:56
the plane, we have seen that
vector n little ds can be
202
00:15:56 --> 00:15:59
written directly as dy,
- dx.
203
00:15:59 --> 00:16:03
That was easier than finding n
and ds separately.
204
00:16:03 --> 00:16:15
And here the same is going to
be true in many cases.
205
00:16:15 --> 00:16:23
Well, any questions before we
do examples?
206
00:16:23 --> 00:16:24
No.
OK.
207
00:16:24 --> 00:16:38
Let's do examples.
The first example for today is
208
00:16:38 --> 00:16:51
we are going to look at the flux
of vector field xi yj xk through
209
00:16:51 --> 00:17:02
the sphere of radius a -- --
centered at the origin.
210
00:17:02 --> 00:17:18
What does the picture look like?
We have a sphere of radius a.
211
00:17:18 --> 00:17:22
I have my vector field.
Well, , see,
212
00:17:22 --> 00:17:25
that is a vector field that is
equal to the vector from the
213
00:17:25 --> 00:17:35
origin to the point where I am,
so it is pointing radially away
214
00:17:35 --> 00:17:42
from the origin.
My vector field is really
215
00:17:42 --> 00:17:49
sticking out everywhere away
from the origin.
216
00:17:49 --> 00:17:56
Now I have to find the normal
vector to the sphere if I want
217
00:17:56 --> 00:18:04
to set up double integral over
the sphere of F dot vector ds,
218
00:18:04 --> 00:18:09
or if you want F dot n dS.
What does the normal vector to
219
00:18:09 --> 00:18:12
the sphere look like?
Well, it depends,
220
00:18:12 --> 00:18:14
of course, whether I choose it
pointing out or in.
221
00:18:14 --> 00:18:18
Let's say I am choosing it
pointing out then it will be
222
00:18:18 --> 00:18:20
sticking straight out of a
sphere as well.
223
00:18:20 --> 00:18:27
Hopefully, you can see that if
I take a normal vector to the
224
00:18:27 --> 00:18:34
sphere it is actually pointing
radially out away from the
225
00:18:34 --> 00:18:38
origin.
In fact, our vector field and
226
00:18:38 --> 00:18:41
our normal vector are parallel
to each other.
227
00:18:41 --> 00:18:45
Let's think a bit more about
what a normal vector looks like.
228
00:18:45 --> 00:18:47
I said it is sticking straight
out.
229
00:18:47 --> 00:18:49
It is proportional to this
vector field.
230
00:18:49 --> 00:18:51
Maybe I should start by writing
231
00:18:51 --> 00:18:52
232
00:18:52 --> 00:18:56
because that is the vector that
goes from the origin to my point
233
00:18:56 --> 00:18:59
so it points radially away from
the origin.
234
00:18:59 --> 00:19:00
Now there is a small problem
with that.
235
00:19:00 --> 00:19:04
It is not a unit vector.
So what is its length?
236
00:19:04 --> 00:19:08
Well, its length is square root
of x^2 y^2 z^2.
237
00:19:08 --> 00:19:13
But, if I am on the sphere,
then that length is just equal
238
00:19:13 --> 00:19:16
to a because distance from the
origin is a.
239
00:19:16 --> 00:19:23
In fact, I get my normal vector
by scaling this guy down by a
240
00:19:23 --> 00:19:27
factor of a.
And let me write it down just
241
00:19:27 --> 00:19:34
in case you are still unsure.
This is unit because square
242
00:19:34 --> 00:19:43
root of x^2 y^2 z^2 is equal to
a on the sphere.
243
00:19:43 --> 00:19:48
OK.
Any questions about this?
244
00:19:48 --> 00:19:52
No. It looks OK?
I see a lot of blank faces.
245
00:19:52 --> 00:19:58
That physics test must have
been hard.
246
00:19:58 --> 00:20:03
Yes?
I could have put a rho but I
247
00:20:03 --> 00:20:06
want to emphasize the fact that
here it is going to be a
248
00:20:06 --> 00:20:09
constant.
I mean rho has this connotation
249
00:20:09 --> 00:20:13
of being a variable that I will
need to then maybe integrate
250
00:20:13 --> 00:20:17
over or do something with.
Yes, it would be correct to put
251
00:20:17 --> 00:20:20
rho but I then later will want
to replace it by its actual
252
00:20:20 --> 00:20:24
value which is a number.
And the number is a.
253
00:20:24 --> 00:20:28
It is not going to actually
change from point to point.
254
00:20:28 --> 00:20:30
For example,
if this was the unit sphere
255
00:20:30 --> 00:20:32
then I would just put x,
y, z.
256
00:20:32 --> 00:20:41
I wouldn't divide by anything.
Now let's figure out F dot n.
257
00:20:41 --> 00:20:47
Let's do things one at a time.
Well, F and n are parallel to
258
00:20:47 --> 00:20:53
each other.
F dot n, the normal component
259
00:20:53 --> 00:21:00
of F, is actually equal to the
length of F.
260
00:21:00 --> 00:21:05
Well, times the length of n if
you want, but that is going to
261
00:21:05 --> 00:21:09
be a one since F and n are
parallel to each other.
262
00:21:09 --> 00:21:12
And what is the magnitude of F
if I am on the sphere?
263
00:21:12 --> 00:21:18
Well, the magnitude of F in
general is square root of x^2
264
00:21:18 --> 00:21:23
y^2 z^2 on the sphere that is
going be a.
265
00:21:23 --> 00:21:25
The other way to do it,
if you don't want to think
266
00:21:25 --> 00:21:28
geometrically like that,
is to just to do the dot
267
00:21:28 --> 00:21:31
product x, y,
z doted with x over a,
268
00:21:31 --> 00:21:35
y over a, z over a.
You will be x^2 y^2 z^2 divided
269
00:21:35 --> 00:21:40
by a.
That will simplify to a because
270
00:21:40 --> 00:21:45
we are on the sphere.
See, we are already using here
271
00:21:45 --> 00:21:47
the relation between x,
y and z.
272
00:21:47 --> 00:21:49
We are not letting x,
y and z be completely
273
00:21:49 --> 00:21:51
arbitrary.
But the slogan is everything
274
00:21:51 --> 00:21:54
happens on the surface where we
are doing the integral.
275
00:21:54 --> 00:21:56
We are not looking at anything
inside or outside.
276
00:21:56 --> 00:21:58
We are just on the surface.
277
00:21:58 --> 00:22:34
278
00:22:34 --> 00:22:43
Now what do I do with that?
Well, I have turned my integral
279
00:22:43 --> 00:22:50
into the double integral of a
dS.
280
00:22:50 --> 00:22:53
And a is just a constant,
so I am very lucky here.
281
00:22:53 --> 00:22:59
I can just say this will be a
times the double integral of dS.
282
00:22:59 --> 00:23:02
And, of course,
some day I will have to learn
283
00:23:02 --> 00:23:05
how to tackle that beast,
but for now I don't actually
284
00:23:05 --> 00:23:08
need to because the double
integral of dS just means I am
285
00:23:08 --> 00:23:11
summing the area of each little
piece of the sphere.
286
00:23:11 --> 00:23:16
I am just going to get the
total area of the sphere which I
287
00:23:16 --> 00:23:23
know to be 4pi a2.
This guy here is going to be
288
00:23:23 --> 00:23:29
the area of S.
I know that to be 4pi a^2.
289
00:23:29 --> 00:23:37
So I will get 4pi a^3.
That one was relatively
290
00:23:37 --> 00:23:44
painless.
That was too easy.
291
00:23:44 --> 00:23:49
Let's do a second example with
the same sphere.
292
00:23:49 --> 00:23:56
But now my vector field is
going to be just z times k.
293
00:23:56 --> 00:23:58
Well, let me give it a
different name.
294
00:23:58 --> 00:24:04
Let me call it H instead of f
or something like that just so
295
00:24:04 --> 00:24:08
that it is not called F anymore.
Well, the initial part of the
296
00:24:08 --> 00:24:11
setup is still the same.
The normal vector is still the
297
00:24:11 --> 00:24:13
same.
What changes is,
298
00:24:13 --> 00:24:16
of course, my vector field is
no longer sticking straight out
299
00:24:16 --> 00:24:18
so I cannot use this easy
geometric argument.
300
00:24:18 --> 00:24:22
It looks like I will have to
compute F dot n and then figure
301
00:24:22 --> 00:24:24
out how to integrate that with
dS.
302
00:24:24 --> 00:24:36
Let's do that.
We still have that n is 00:24:44
y, z>/a.
That tells us that H dot n will
304
00:24:44 --> 00:24:49
be
dot 00:24:57
z> / a.
It looks like I will be left
306
00:24:57 --> 00:25:10
with z^2 over a.
H dot n is z^2 over a.
307
00:25:10 --> 00:25:18
The double integral for flux
now becomes double integral on
308
00:25:18 --> 00:25:25
the sphere of z^2 over a dS.
Well, we can take out one over
309
00:25:25 --> 00:25:29
a, that is fine,
but it looks like we will have
310
00:25:29 --> 00:25:33
to integrate z^2 on the surface
of the sphere.
311
00:25:33 --> 00:25:37
How do we do that?
Well, we have to figure out
312
00:25:37 --> 00:25:41
what is dS in terms of our
favorite set of two variables
313
00:25:41 --> 00:25:45
that we will use to integrate.
Now, what is the best way to
314
00:25:45 --> 00:25:47
figure out where you are on the
sphere?
315
00:25:47 --> 00:25:51
Well, you could try to use
maybe theta and z.
316
00:25:51 --> 00:25:55
If you know how high you are
and where you are around,
317
00:25:55 --> 00:25:58
in principle you know where you
are on the sphere.
318
00:25:58 --> 00:26:02
But since spherical coordinates
we have actually learned about
319
00:26:02 --> 00:26:06
something much more interesting,
namely spherical coordinates.
320
00:26:06 --> 00:26:09
It looks like longitude /
latitude is the way to go when
321
00:26:09 --> 00:26:12
trying to figure out where you
are on a sphere.
322
00:26:12 --> 00:26:19
We are going to use phi and
theta.
323
00:26:19 --> 00:26:24
And, of course,
we have to figure out how to
324
00:26:24 --> 00:26:28
express dS in terms of d phi and
d theta.
325
00:26:28 --> 00:26:32
Well, if you were paying
really, really close attention
326
00:26:32 --> 00:26:36
last time, you will notice that
we have actually already done
327
00:26:36 --> 00:26:41
that.
Last time we saw that if I have
328
00:26:41 --> 00:26:48
a sphere of radius a and I take
a little piece of it that
329
00:26:48 --> 00:26:56
corresponds to small changes in
phi and theta then we said that
330
00:26:56 --> 00:27:01
-- Well,
we argued that this side here,
331
00:27:01 --> 00:27:08
the one that is going east-west
was a piece of the circle that
332
00:27:08 --> 00:27:14
has a radius a sin phi because
that is r,
333
00:27:14 --> 00:27:19
so that side is a sin phi delta
theta.
334
00:27:19 --> 00:27:22
And the side that goes
north-south is a piece of the
335
00:27:22 --> 00:27:26
circle of radius a corresponding
to angle delta phi,
336
00:27:26 --> 00:27:32
so it is a delta phi.
And so, just to get to the
337
00:27:32 --> 00:27:40
answer, we got dS equals a^2 sin
phi d phi d theta.
338
00:27:40 --> 00:27:45
When we set up a surface
integral on the surface of a
339
00:27:45 --> 00:27:48
sphere,
most likely we will be using
340
00:27:48 --> 00:27:52
phi and theta as our two
variables of integration and dS
341
00:27:52 --> 00:27:55
will become this.
Now, it is OK to think of them
342
00:27:55 --> 00:27:58
as spherical coordinates,
but I would like to encourage
343
00:27:58 --> 00:28:01
you not to think of them as
spherical coordinates.
344
00:28:01 --> 00:28:05
Spherical coordinates are a way
of describing points in space in
345
00:28:05 --> 00:28:09
terms of three variables.
Here it is more like we are
346
00:28:09 --> 00:28:12
parameterizing the sphere.
We are finding a parametric
347
00:28:12 --> 00:28:15
equation for the sphere using
two variables phi and theta
348
00:28:15 --> 00:28:18
which happen to be part of the
spherical coordinate system.
349
00:28:18 --> 00:28:22
But, see, there is no rho
involved in here.
350
00:28:22 --> 00:28:26
I am not using any rho ever,
and I am not going to in this
351
00:28:26 --> 00:28:28
calculation.
I have two variable phi and
352
00:28:28 --> 00:28:37
theta.
That is it.
353
00:28:37 --> 00:28:40
It is basically in the same way
as when you parameterize a line
354
00:28:40 --> 00:28:45
integral in the circle,
we use theta as the parameter
355
00:28:45 --> 00:28:50
variable and never think about
r.
356
00:28:50 --> 00:28:52
That being said,
well, we are going to use phi
357
00:28:52 --> 00:28:54
and theta.
We know what dS is.
358
00:28:54 --> 00:28:58
We still need to figure out
what z is.
359
00:28:58 --> 00:29:01
There we want to think a tiny
bit about spherical coordinates
360
00:29:01 --> 00:29:08
again.
And we will know that z is just
361
00:29:08 --> 00:29:15
a cos phi.
In case you don't quite see it,
362
00:29:15 --> 00:29:25
let me draw a diagram.
Phi is the angle down from the
363
00:29:25 --> 00:29:31
positive z axes,
this distance is a,
364
00:29:31 --> 00:29:38
so this distance here is a cos
phi.
365
00:29:38 --> 00:29:44
Now I have everything I need to
compute my double integral.
366
00:29:44 --> 00:29:49
z^2 over a dS will become a
double integral.
367
00:29:49 --> 00:30:00
z^2 becomes a^2 cos^2 phi over
a times, ds becomes,
368
00:30:00 --> 00:30:07
a^2 sin phi d phi d theta.
Now I need to set up bounds.
369
00:30:07 --> 00:30:12
Well, what are the bounds?
Phi goes all the way from zero
370
00:30:12 --> 00:30:19
to pi because we go all the way
from the north pole to the south
371
00:30:19 --> 00:30:23
pole, and theta goes from zero
to 2pi.
372
00:30:23 --> 00:30:27
And, of course,
I can get rid of some a's in
373
00:30:27 --> 00:30:34
there and take them out.
Let's look at what number we
374
00:30:34 --> 00:30:37
get.
First of all,
375
00:30:37 --> 00:30:43
we can take out all those a's
and get a^3.
376
00:30:43 --> 00:30:50
Second, in the inner integral,
we are integrating cos^2 phi
377
00:30:50 --> 00:30:54
sin phi d phi.
I claim that integrates to cos3
378
00:30:54 --> 00:30:57
up to some factor,
and that factor should be
379
00:30:57 --> 00:31:02
negative one-third.
If you look at cos3 phi and you
380
00:31:02 --> 00:31:07
take its derivative,
you will get that guy with a
381
00:31:07 --> 00:31:12
negative three in front between
zero and pi.
382
00:31:12 --> 00:31:16
And, while integrating over
theta, we will just multiply
383
00:31:16 --> 00:31:24
things by 2pi.
Let me add the 2pi in front.
384
00:31:24 --> 00:31:27
Now, if I evaluate this guy
between zero and pi,
385
00:31:27 --> 00:31:32
well, at pi cos^3 is negative
one, at zero it is one,
386
00:31:32 --> 00:31:35
I will get two-thirds out of
this.
387
00:31:35 --> 00:31:39
I end up with four-thirds pi
a^3.
388
00:31:39 --> 00:31:46
Sorry I didn't write very much
because I am trying to save
389
00:31:46 --> 00:31:52
blackboard space.
Yes?
390
00:31:52 --> 00:31:55
That is a very natural question.
That looks a lot like somebody
391
00:31:55 --> 00:31:58
we know, like the volume of a
sphere.
392
00:31:58 --> 00:32:03
And ultimately it will be.
Wait until next class when we
393
00:32:03 --> 00:32:07
talk about the divergence
theorem.
394
00:32:07 --> 00:32:11
I mean the question was is this
related to the volume of a
395
00:32:11 --> 00:32:14
sphere, and ultimately it is,
but for now it is just some
396
00:32:14 --> 00:32:23
coincidence.
Yes?
397
00:32:23 --> 00:32:26
The question is there is a way
to do it M dx plus N dy plus
398
00:32:26 --> 00:32:28
stuff like that?
The answer is unfortunately no
399
00:32:28 --> 00:32:30
because it is not a line
integral.
400
00:32:30 --> 00:32:35
It is a surface integral,
so we need to have to variables
401
00:32:35 --> 00:32:38
in there.
In a way you would end up with
402
00:32:38 --> 00:32:41
things like some dx dy maybe and
so on.
403
00:32:41 --> 00:32:45
I mean it is not practical to
do it directly that way because
404
00:32:45 --> 00:32:49
you would have then to compute
Jacobians to switch from dx dy
405
00:32:49 --> 00:32:52
to something else.
We are going to see various
406
00:32:52 --> 00:32:54
ways of computing it.
Unfortunately,
407
00:32:54 --> 00:32:57
it is not quite as simple as
with line integrals.
408
00:32:57 --> 00:32:59
But it is not much harder.
It is the same spirit.
409
00:32:59 --> 00:33:04
We just use two variables and
set up everything in terms of
410
00:33:04 --> 00:33:12
these two variables.
Any other questions?
411
00:33:12 --> 00:33:14
No.
OK.
412
00:33:14 --> 00:33:51
413
00:33:51 --> 00:33:54
By the way, just some food for
thought.
414
00:33:54 --> 00:34:01
Never mind.
Conclusion of looking at these
415
00:34:01 --> 00:34:05
two examples is that sometimes
we can use geometric.
416
00:34:05 --> 00:34:07
The first example,
we didn't actually have to
417
00:34:07 --> 00:34:11
compute an integral.
But most of the time we need to
418
00:34:11 --> 00:34:14
learn how to set up double
integrals.
419
00:34:14 --> 00:34:26
Use geometry or you need to set
up for double integral of a
420
00:34:26 --> 00:34:30
surface.
And so we are going to learn
421
00:34:30 --> 00:34:33
how to do that in general.
As I said, we need to have two
422
00:34:33 --> 00:34:37
parameters on the surface and
express everything in terms of
423
00:34:37 --> 00:34:43
these.
Let's look at various examples.
424
00:34:43 --> 00:34:46
We are going to see various
situations where we can do
425
00:34:46 --> 00:34:49
things.
Well, let's start with an easy
426
00:34:49 --> 00:34:53
one.
Let's call that number zero.
427
00:34:53 --> 00:35:02
Say that my surface S is a
horizontal plane,
428
00:35:02 --> 00:35:07
say z equals a.
When I say a horizontal plane,
429
00:35:07 --> 00:35:09
it doesn't have to be the
entire horizontal plane.
430
00:35:09 --> 00:35:14
It could be a small piece of it.
It could even be,
431
00:35:14 --> 00:35:16
to trick you,
maybe an ellipse in there or a
432
00:35:16 --> 00:35:19
triangle in there or something
like that.
433
00:35:19 --> 00:35:23
What you have to recognize is
my surface is a piece of just a
434
00:35:23 --> 00:35:27
flat plane, so I shouldn't worry
too much about what part of a
435
00:35:27 --> 00:35:30
plane it is.
Well, it will become important
436
00:35:30 --> 00:35:32
when I set up bounds for
integration.
437
00:35:32 --> 00:35:36
But, when it comes to looking
for the normal vector,
438
00:35:36 --> 00:35:40
be rest assured that the normal
vector to a horizontal plane is
439
00:35:40 --> 00:35:44
just vertical.
It is going to be either k or
440
00:35:44 --> 00:35:49
negative k depending on whether
I have chosen to orient it
441
00:35:49 --> 00:35:54
pointing up or down.
And which one I choose might
442
00:35:54 --> 00:35:57
depend on what I am going to try
to do.
443
00:35:57 --> 00:36:02
The normal vector is just
sticking straight up or straight
444
00:36:02 --> 00:36:05
down.
Now, what about dS?
445
00:36:05 --> 00:36:11
Well, it is just going to be
the area element in a horizontal
446
00:36:11 --> 00:36:14
plane.
It just looks like it should be
447
00:36:14 --> 00:36:16
dx dy.
I mean if I am moving on a
448
00:36:16 --> 00:36:18
horizontal plane,
to know where I am,
449
00:36:18 --> 00:36:26
I should know x and y.
So dS will be dx dy.
450
00:36:26 --> 00:36:31
If I play the game that way,
I have my vector field F.
451
00:36:31 --> 00:36:34
I do F dot n.
That just gives me the z
452
00:36:34 --> 00:36:37
component which might involve x,
y and z.
453
00:36:37 --> 00:36:40
x and y I am very happy with.
They will stay as my variables.
454
00:36:40 --> 00:36:43
Whenever I see z,
well, I want to get rid of it.
455
00:36:43 --> 00:36:46
That is easy because z is just
equal to a.
456
00:36:46 --> 00:36:50
I just plug that value and I am
left with only x and y,
457
00:36:50 --> 00:36:53
and I am integrating that dx
dy.
458
00:36:53 --> 00:36:58
It is actually ending up being
just a usual double integral in
459
00:36:58 --> 00:37:00
x, y coordinates.
And, of course,
460
00:37:00 --> 00:37:02
once it is set up anything is
fair game.
461
00:37:02 --> 00:37:05
I might want to switch to polar
coordinates or something like
462
00:37:05 --> 00:37:09
that.
Or, I can set it up dx dy or dy
463
00:37:09 --> 00:37:12
dx.
All the usual stuff applies.
464
00:37:12 --> 00:37:17
But, for the initial setup,
we are just going to use these
465
00:37:17 --> 00:37:21
and express everything in terms
of x and y.
466
00:37:21 --> 00:37:27
A small variation on that.
Let's say that we take vertical
467
00:37:27 --> 00:37:35
planes that are parallel to
maybe the blackboard plane,
468
00:37:35 --> 00:37:42
so parallel to the yz plane.
That might be something like x
469
00:37:42 --> 00:37:47
equals some constant.
Well, what would I do then?
470
00:37:47 --> 00:37:52
It could be pretty much the
same.
471
00:37:52 --> 00:37:55
The normal vector for this guy
would be sticking straight out
472
00:37:55 --> 00:37:59
towards me or away from me.
Let's say I am having it come
473
00:37:59 --> 00:38:03
to the front.
The normal vector would be
474
00:38:03 --> 00:38:07
plus/minus i.
And the variables that I would
475
00:38:07 --> 00:38:11
be using, to find out my
position on this guy,
476
00:38:11 --> 00:38:14
would be y and z.
In terms of those,
477
00:38:14 --> 00:38:19
the surface element is just dy
dz.
478
00:38:19 --> 00:38:25
Similarly for planes parallel
to the xz plane.
479
00:38:25 --> 00:38:33
You can figure that one out.
These are somehow the easiest
480
00:38:33 --> 00:38:39
ones, because those we already
know how to compute without too
481
00:38:39 --> 00:38:41
much trouble.
What if it is a more
482
00:38:41 --> 00:38:43
complicated plane?
We will come back to that next
483
00:38:43 --> 00:38:49
time.
Let's explore some other
484
00:38:49 --> 00:38:56
situations first.
Number one on the list.
485
00:38:56 --> 00:39:03
Let's say that I gave you a
sphere of radius a centered at
486
00:39:03 --> 00:39:09
the origin, or maybe just half
of that sphere or some portion
487
00:39:09 --> 00:39:12
of it.
Well, we have already seen how
488
00:39:12 --> 00:39:15
to do things.
Namely, we will be saying the
489
00:39:15 --> 00:39:18
normal vector is x,
y, z over a,
490
00:39:18 --> 00:39:23
plus or minus depending on
whether we want it pointing in
491
00:39:23 --> 00:39:29
or out.
And dS will be a^2 sin phi d
492
00:39:29 --> 00:39:32
phi d theta.
In fact, we will express
493
00:39:32 --> 00:39:35
everything in terms of phi and
theta.
494
00:39:35 --> 00:39:37
If I wanted to I could tell you
what the formulas are for x,
495
00:39:37 --> 00:39:40
y, z in terms of phi and theta.
You know them.
496
00:39:40 --> 00:39:44
But it is actually better to
wait a little bit.
497
00:39:44 --> 00:39:48
It is better to do F dot n,
because F is also going to have
498
00:39:48 --> 00:39:49
a bunch of x's,
y's and z's.
499
00:39:49 --> 00:39:53
And if there is any kind of
symmetry to the problem then you
500
00:39:53 --> 00:39:57
might end up with things like
x^2 y^2 z^2 or things that have
501
00:39:57 --> 00:40:01
more symmetry that are easier to
express in terms of phi and
502
00:40:01 --> 00:40:05
theta.
The advice would be first do
503
00:40:05 --> 00:40:10
the dot product with F,
and then see what you get and
504
00:40:10 --> 00:40:17
then turn it into phi and theta.
That is one we have seen.
505
00:40:17 --> 00:40:20
Let's say that I have -- It is
a close cousin.
506
00:40:20 --> 00:40:30
Let's say I have a cylinder of
radius a centered on the z-axis.
507
00:40:30 --> 00:40:37
What does that look like?
And, again, when I say
508
00:40:37 --> 00:40:40
cylinder, it could be a piece of
cylinder.
509
00:40:40 --> 00:40:44
First of all,
what does the normal vector to
510
00:40:44 --> 00:40:47
a cylinder look like?
Well, it is sticking straight
511
00:40:47 --> 00:40:50
out, but sticking straight out
in a slightly different way from
512
00:40:50 --> 00:40:52
what happens with a sphere.
See, the sides of a cylinder
513
00:40:52 --> 00:40:54
are vertical.
If you imagine that you have
514
00:40:54 --> 00:40:56
this big cylindrical type in
front of you,
515
00:40:56 --> 00:40:59
hopefully you can see that a
normal vector is going to always
516
00:40:59 --> 00:41:02
be horizontal.
It is sticking straight out in
517
00:41:02 --> 00:41:07
the horizontal directions.
It doesn't have any z component.
518
00:41:07 --> 00:41:13
I claim the normal vector for
the cylinder,
519
00:41:13 --> 00:41:21
if you have a point here at (x,
y, z), it would be pointing
520
00:41:21 --> 00:41:27
straight out away from the
central axis.
521
00:41:27 --> 00:41:31
My normal vector,
well, if I am taking it two
522
00:41:31 --> 00:41:34
points outwards,
will be going straight away
523
00:41:34 --> 00:41:38
from the central axis.
If I look at it from above,
524
00:41:38 --> 00:41:42
maybe it is easier if I look at
it from above,
525
00:41:42 --> 00:41:45
look at x, y,
then my cylinder looks like a
526
00:41:45 --> 00:41:49
circle and the normal vector
just points straight out.
527
00:41:49 --> 00:41:53
It is the same situation as
when we had a circle in the 2D
528
00:41:53 --> 00:41:57
case.
The normal vector for that is
529
00:41:57 --> 00:42:02
just going to be x,
y and 0 in the z component.
530
00:42:02 --> 00:42:05
Well, plus/minus,
depending on whether you want
531
00:42:05 --> 00:42:06
it sticking in or out.
532
00:42:06 --> 00:42:41
533
00:42:41 --> 00:42:47
We said in our cylinder normal
vector is plus or minus x,
534
00:42:47 --> 00:42:54
y, zero over a.
What about the surface element?
535
00:42:54 --> 00:42:57
Before we ask that,
maybe we should first figure
536
00:42:57 --> 00:43:00
out what coordinates are we
going to use to locate ourselves
537
00:43:00 --> 00:43:02
in a cylinder.
Well, yes,
538
00:43:02 --> 00:43:05
we probably want to use part of
a cylindrical coordinate,
539
00:43:05 --> 00:43:08
except for,
well, we don't want r because r
540
00:43:08 --> 00:43:11
doesn't change,
it is not a variable here.
541
00:43:11 --> 00:43:15
Indeed, you probably want to
use z to tell how high you are
542
00:43:15 --> 00:43:18
and theta to tell you where you
are around.
543
00:43:18 --> 00:43:27
dS should be in terms of dz d
theta.
544
00:43:27 --> 00:43:33
Now, what is the constant?
Well, let's look at a small
545
00:43:33 --> 00:43:39
piece of our cylinder
corresponding to a small angle
546
00:43:39 --> 00:43:44
delta theta and a small height
delta z.
547
00:43:44 --> 00:43:47
Well, the height,
as I said, is going to be delta
548
00:43:47 --> 00:43:50
z.
What about the width?
549
00:43:50 --> 00:43:55
It is going to be a piece of a
circle of radius a corresponding
550
00:43:55 --> 00:43:59
to the angle delta theta,
so this side will be a delta
551
00:43:59 --> 00:44:05
theta.
Delta S is a delta theta delta
552
00:44:05 --> 00:44:08
z.
DS is just a dz d theta or d
553
00:44:08 --> 00:44:13
theta dz.
It doesn't matter which way you
554
00:44:13 --> 00:44:16
do it.
And so when we set up the flux
555
00:44:16 --> 00:44:21
integral, we will take first the
dot product of f with this
556
00:44:21 --> 00:44:25
normal vector.
Then we will stick in this dS.
557
00:44:25 --> 00:44:28
And then, of course,
we will get rid of any x and y
558
00:44:28 --> 00:44:31
that are left by expressing them
in terms of theta.
559
00:44:31 --> 00:44:37
Maybe x becomes a cos theta,
y becomes a sin theta.
560
00:44:37 --> 00:44:41
These various formulas,
you should try to remember them
561
00:44:41 --> 00:44:45
because they are really useful,
for the sphere,
562
00:44:45 --> 00:44:48
for the cylinder.
And, hopefully,
563
00:44:48 --> 00:44:52
those for the planes you kind
of know already intuitively.
564
00:44:52 --> 00:44:59
What about marginals or faces?
Not everything in life is made
565
00:44:59 --> 00:45:08
out of cylinders and spheres.
I mean it is a good try.
566
00:45:08 --> 00:45:11
Let's look at a marginal kind
of surface.
567
00:45:11 --> 00:45:19
Let's say I give you a graph of
a function z equals f of x,
568
00:45:19 --> 00:45:21
y.
This guy has nothing to do with
569
00:45:21 --> 00:45:22
the integrand.
It is not what we are
570
00:45:22 --> 00:45:24
integrating.
We are just integrating a
571
00:45:24 --> 00:45:26
vector field that has nothing to
do with that.
572
00:45:26 --> 00:45:31
This is how I want to describe
the surface on which I will be
573
00:45:31 --> 00:45:37
integrating.
My surface is given by z as a
574
00:45:37 --> 00:45:40
function of x,
y.
575
00:45:40 --> 00:45:48
Well, I would need to tell you
what n is and what dS is.
576
00:45:48 --> 00:45:51
That is going to be slightly
annoying.
577
00:45:51 --> 00:45:54
I mean, I don't want to tell
them separately because you see
578
00:45:54 --> 00:45:58
they are pretty hard.
Instead, I am going to tell you
579
00:45:58 --> 00:46:01
that in this case,
well, let's see.
580
00:46:01 --> 00:46:07
What variables do we want?
I am going to tell you a
581
00:46:07 --> 00:46:11
formula for n dS.
What variables do we want to
582
00:46:11 --> 00:46:15
express this in terms of?
Well, most likely x and y
583
00:46:15 --> 00:46:19
because we know how to express z
in terms of x and y.
584
00:46:19 --> 00:46:26
This is an invitation to get
rid of any z that might be left
585
00:46:26 --> 00:46:30
and set everything up in terms
of dx dy.
586
00:46:30 --> 00:46:31
The formula that we are going
to see,
587
00:46:31 --> 00:46:36
I think we are going to see the
details of why it works
588
00:46:36 --> 00:46:40
tomorrow,
is that you can take negative
589
00:46:40 --> 00:46:44
partial f partial x,
negative partial f partial y,
590
00:46:44 --> 00:46:48
one,
dx dy.
591
00:46:48 --> 00:46:52
Plus/minus depending on which
way you want it to go.
592
00:46:52 --> 00:46:57
If you really want to know what
dS is, well, dS is the magnitude
593
00:46:57 --> 00:47:01
of this vector times dx dy.
There will be a square root and
594
00:47:01 --> 00:47:04
some squares and some stuff.
What is the normal vector?
595
00:47:04 --> 00:47:10
Well, you take this vector and
you scale it down to unit
596
00:47:10 --> 00:47:14
length.
Just to emphasize it,
597
00:47:14 --> 00:47:24
this guy here is not n and this
guy here is not dS.
598
00:47:24 --> 00:47:27
Each of them is more
complicated than that,
599
00:47:27 --> 00:47:31
but the combination somehow
simplifies nicely.
600
00:47:31 --> 00:47:36
And that is good news for us.
Now, concretely,
601
00:47:36 --> 00:47:40
one way you can think about it
is this tells you how to reduce
602
00:47:40 --> 00:47:42
things to an integral of x and
y.
603
00:47:42 --> 00:47:44
And, of course,
you will have to figure out
604
00:47:44 --> 00:47:47
what are the bounds on x and y.
That means you will need to
605
00:47:47 --> 00:47:51
know what does the shadow of
your surface look like in the x,
606
00:47:51 --> 00:47:59
y plane.
To set up bounds on whatever
607
00:47:59 --> 00:48:06
you will get dx dy,
well, of course you can switch
608
00:48:06 --> 00:48:08
to dy dx or anything you would
like,
609
00:48:08 --> 00:48:19
but you will need to look at
the shadow of S in the x y
610
00:48:19 --> 00:48:22
plane.
But only do that after you
611
00:48:22 --> 00:48:27
gotten rid of all the z.
When you no longer have z then
612
00:48:27 --> 00:48:33
you can figure out what the
bounds are for x and y.
613
00:48:33 --> 00:48:40
Any questions about that?
Yes?
614
00:48:40 --> 00:48:42
For the cylinder.
OK.
615
00:48:42 --> 00:48:44
Let me re-explain quickly how I
got a normal vector for the
616
00:48:44 --> 00:48:47
cylinder.
If you know what a cylinder
617
00:48:47 --> 00:48:50
looks like, you probably can see
that the normal vector sticks
618
00:48:50 --> 00:48:56
straight out of it horizontally.
That means the z component of n
619
00:48:56 --> 00:48:59
is going to be zero.
And then the x,
620
00:48:59 --> 00:49:02
y components you get by looking
at it from above.
621
00:49:02 --> 00:49:12
One last thing I want to say.
What about the geometric
622
00:49:12 --> 00:49:15
interpretation and how to prove
it?
623
00:49:15 --> 00:49:27
Well, if your vector field F is
a velocity field then the flux
624
00:49:27 --> 00:49:37
is the amount of matter that
crosses the surface that passes
625
00:49:37 --> 00:49:44
through S per unit time.
And the way that you would
626
00:49:44 --> 00:49:47
prove it would be similar to the
picture that I drew when we did
627
00:49:47 --> 00:49:50
it in the plane.
Namely, you would consider a
628
00:49:50 --> 00:49:53
small element of a surface delta
S.
629
00:49:53 --> 00:49:55
And you would try to figure out
what is the stuff that flows
630
00:49:55 --> 00:49:59
through it in a second.
Well, it is the stuff that
631
00:49:59 --> 00:50:05
lives in a small box whose base
is that piece of surface and
632
00:50:05 --> 00:50:10
whose other side is given by the
vector field.
633
00:50:10 --> 00:50:15
And then the volume of that is
given by base times height,
634
00:50:15 --> 00:50:20
and the height is F dot n.
It is the same argument as what
635
00:50:20 --> 00:50:22
we saw in the plane.
OK.
636
00:50:22 --> 00:50:24
Next time we will see more
formulas.
637
00:50:24 --> 00:50:28
We will first see how to prove
this, more ways to do it,
638
00:50:28 --> 00:50:31
more examples.
And then we will get to the
639
00:50:31 --> 00:50:34
divergence theorem.
640
00:50:34 --> 00:50:39