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I guess last time on Friday we
went over the first half of the

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class very quickly.
And so today we are going to go

00:00:33.000 --> 00:00:38.000
over the second half of the
class very quickly.

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And so that was stuff about
double and triple integrals and

00:00:45.000 --> 00:00:51.000
vector calculus in the plane and
in space.

00:00:51.000 --> 00:00:55.000
As usual, what is on the final
is basically what was on the

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other tests, exactly the same
stuff.

00:00:57.000 --> 00:01:03.000
Well, not the same problem,
unfortunately.

00:01:03.000 --> 00:01:13.000
The first thing we learned
about was double integrals in

00:01:13.000 --> 00:01:25.000
the plane and how to set up the
bounds and how to evaluate them.

00:01:25.000 --> 00:01:30.000
Just to remind you quickly,
the important thing with

00:01:30.000 --> 00:01:34.000
iterated integrals is when you
integrate a function f of x,

00:01:34.000 --> 00:01:38.000
y,
say dy dx for example,

00:01:38.000 --> 00:01:41.000
is that you have to draw a
picture of a region.

00:01:41.000 --> 00:01:45.000
Unless it is completely obvious
you should really draw some

00:01:45.000 --> 00:01:48.000
picture of the domain of
integration.

00:01:48.000 --> 00:01:53.000
And once you have that picture
you can use it to find the

00:01:53.000 --> 00:01:57.000
bounds.
Remember the general method is

00:01:57.000 --> 00:02:03.000
that we first look at the inner
integral, here integral of f dy.

00:02:03.000 --> 00:02:07.000
And in this inner integral the
outer variable here,

00:02:07.000 --> 00:02:10.000
x, is fixed.
That means we are slicing our

00:02:10.000 --> 00:02:13.000
region by a vertical line
corresponding to a fixed value

00:02:13.000 --> 00:02:16.000
of x.
We fix a value of x.

00:02:16.000 --> 00:02:20.000
And what we have to find out is
the bounds for y,

00:02:20.000 --> 00:02:24.000
so the value of y at this
point, the value of y at that

00:02:24.000 --> 00:02:28.000
point.
Let me call that y some bottom

00:02:28.000 --> 00:02:34.000
of x, in general depends on x.
And this one will be y at that

00:02:34.000 --> 00:02:43.000
top, and it also depends on x.
And then the bounds for y would

00:02:43.000 --> 00:02:46.000
be this.
And then, when you look at the

00:02:46.000 --> 00:02:48.000
outer bound, things are
different.

00:02:48.000 --> 00:02:51.000
Because there you expect to
have just numbers,

00:02:51.000 --> 00:02:53.000
no longer functions of
anything.

00:02:53.000 --> 00:02:56.000
And what you do is look at the
shadow of your region.

00:02:56.000 --> 00:03:00.000
We are doing it by shadow so
you just project to the x-axis.

00:03:00.000 --> 00:03:04.000
If you project to the x-axis
your region will look like this.

00:03:04.000 --> 00:03:08.000
Its shadow is going to be this
integral form,

00:03:08.000 --> 00:03:12.000
some minimum value of x to some
maximum value of x.

00:03:12.000 --> 00:03:24.000
And that will give us the
bounds for the outer integral.

00:03:24.000 --> 00:03:27.000
And then, to evaluate,
we evaluate the usual way.

00:03:27.000 --> 00:03:29.000
Speaking of evaluation,
what you need to know for the

00:03:29.000 --> 00:03:31.000
final,
well, essentially the same kind

00:03:31.000 --> 00:03:35.000
of evaluation techniques that we
were supposed to know for the

00:03:35.000 --> 00:03:40.000
other tests.
That means the usual functions,

00:03:40.000 --> 00:03:47.000
substitutions,
basic trig, stuff like that.

00:03:47.000 --> 00:03:51.000
Well, I don't expect that you
would need integration by parts,

00:03:51.000 --> 00:03:54.000
although I still hope that some
of you remember it from single

00:03:54.000 --> 00:03:59.000
variable calculus.
If there is a need to integrate

00:03:59.000 --> 00:04:06.000
some big power of cosine or sine
then the formula will be given

00:04:06.000 --> 00:04:11.000
to you the way it is in the
notes.

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And, of course,
we know also how to set up

00:04:17.000 --> 00:04:23.000
these integrals in polar
coordinates.

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And then the area element
becomes r dr d theta.

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And because you integrate first
over r,

00:04:32.000 --> 00:04:36.000
well, first of all you should
remember the polar coordinate

00:04:36.000 --> 00:04:42.000
formulas,
namely x equals r cosine theta

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and y equals r sine theta.
And second you should remember

00:04:50.000 --> 00:04:53.000
that what we do,
when we have our region,

00:04:53.000 --> 00:04:56.000
is for a fixed value of theta
we look for the bounds for r,

00:04:56.000 --> 00:04:59.000
just like before,
so the way we are slicing the

00:04:59.000 --> 00:05:02.000
region is now we are actually
shooting rays straight from the

00:05:02.000 --> 00:05:04.000
origin.
And, in a given direction,

00:05:04.000 --> 00:05:08.000
we are asking ourselves how far
does my region go?

00:05:08.000 --> 00:05:12.000
You have to find a bound and
you have to find whatever the

00:05:12.000 --> 00:05:15.000
value of r will be out here as a
function of theta.

00:05:15.000 --> 00:05:19.000
And ways to do that can be
geometric or they can be by

00:05:19.000 --> 00:05:22.000
starting from the x,
y equation of whatever curve

00:05:22.000 --> 00:05:26.000
you have and then expressing it
in terms of r and theta and

00:05:26.000 --> 00:05:28.000
solving for r.
For example,

00:05:28.000 --> 00:05:33.000
just to illustrate it,
we have seen that one of our

00:05:33.000 --> 00:05:39.000
classics has been the circle of
radius one centered at one,

00:05:39.000 --> 00:05:42.000
zero.
This guy, you have two

00:05:42.000 --> 00:05:45.000
different ways of getting its
polar coordinate equation.

00:05:45.000 --> 00:05:49.000
One is to argue geometrically
that you have a right angle in

00:05:49.000 --> 00:05:51.000
here.
And this length is two,

00:05:51.000 --> 00:05:54.000
this angle is theta,
this length is r,

00:05:54.000 --> 00:05:59.000
so the polar equation is r
equals two cosine theta.

00:05:59.000 --> 00:06:01.000
The other way to do it,
if somehow you are missing the

00:06:01.000 --> 00:06:03.000
geometric trick,
is to start from the x,

00:06:03.000 --> 00:06:05.000
y equation.
What is the x,

00:06:05.000 --> 00:06:10.000
y equation of this guy?
Well, it is x minus one squared

00:06:10.000 --> 00:06:16.000
plus y squared equals one.
If you expand that you will get

00:06:16.000 --> 00:06:22.000
x squared minus two x plus one
plus y squared equals one.

00:06:22.000 --> 00:06:27.000
The ones simplify.
X squared plus y squared

00:06:27.000 --> 00:06:34.000
becomes r squared minus two x
becomes r cosine theta equals

00:06:34.000 --> 00:06:36.000
zero.
That gives you,

00:06:36.000 --> 00:06:40.000
when you simplify by r,
r equals two cosine theta.

00:06:40.000 --> 00:06:49.000
Two ways to get the same polar
equation.

00:06:49.000 --> 00:06:53.000
I should say this is an
example, in case you were

00:06:53.000 --> 00:07:01.000
wondering what I was doing.
We have also actually seen how

00:07:01.000 --> 00:07:14.000
to change variables to more
complicated coordinate systems.

00:07:14.000 --> 00:07:16.000
Let's say u, v coordinates.
But, of course,

00:07:16.000 --> 00:07:20.000
you can call them whatever you
want.

00:07:20.000 --> 00:07:24.000
The main thing to remember is
that you have to look for the

00:07:24.000 --> 00:07:30.000
Jacobian which will give you the
conversion ratio between dx dy

00:07:30.000 --> 00:07:32.000
and du dv.
For example,

00:07:32.000 --> 00:07:36.000
if you know u and v as
functions of x and y then you

00:07:36.000 --> 00:07:42.000
will write du dv equals absolute
value of the Jacobian partial u,

00:07:42.000 --> 00:07:50.000
v over partial x, y times dx dy.
Or, if it is easier for you,

00:07:50.000 --> 00:07:52.000
you can do the Jacobian the
other way around.

00:07:52.000 --> 00:07:56.000
And this Jacobian,
remember, is the determinant by

00:07:56.000 --> 00:08:00.000
a two by two matrix that you
obtain by putting the partial

00:08:00.000 --> 00:08:03.000
derivatives of u and v with
respect to x and y.

00:08:03.000 --> 00:08:08.000
Then, when we have that,
we can change the integrant,

00:08:08.000 --> 00:08:13.000
f of x, y, into something
involving u and v possibly.

00:08:13.000 --> 00:08:15.000
And then we have to find the
bounds.

00:08:15.000 --> 00:08:19.000
And to find the bounds perhaps
the easiest is to draw a picture

00:08:19.000 --> 00:08:21.000
of a region in u,
v coordinates.

00:08:21.000 --> 00:08:24.000
Maybe you have some picture in
the x,

00:08:24.000 --> 00:08:28.000
y plane that might actually be
really hard to draw and maybe in

00:08:28.000 --> 00:08:32.000
terms of u and v the picture
will become much simpler.

00:08:32.000 --> 00:08:35.000
It might just become a
rectangle.

00:08:35.000 --> 00:08:37.000
Of course, if you see
immediately what the bounds are

00:08:37.000 --> 00:08:39.000
in terms of u and v,
and they turn out to be very

00:08:39.000 --> 00:08:41.000
easy,
then maybe you don't even have

00:08:41.000 --> 00:08:45.000
to draw this picture.
But if it is not completely

00:08:45.000 --> 00:08:49.000
obvious then that might be a
helpful way of figuring out what

00:08:49.000 --> 00:08:52.000
the bounds will be when you
switch from x,

00:08:52.000 --> 00:08:55.000
y to u, v.
We have seen some problems like

00:08:55.000 --> 00:09:00.000
that and there are more in the
notes in case you need more.

00:09:00.000 --> 00:09:07.000
Questions?
Yes?

00:09:07.000 --> 00:09:10.000
That is the second time you've
asked for something real quick

00:09:10.000 --> 00:09:13.000
in these review sessions.
You are in a hurry.

00:09:13.000 --> 00:09:23.000
Take your time.
Partial u, v over partial x,

00:09:23.000 --> 00:09:26.000
y is just going to be the
determinant of u sub x,

00:09:26.000 --> 00:09:29.000
u sub y, v sub x,
v sub y.

00:09:29.000 --> 00:09:37.000
That is the definition.
That is pretty direct.

00:09:37.000 --> 00:09:39.000
And, of course,
a general common sense thing

00:09:39.000 --> 00:09:43.000
that applies to actually all the
integrals that we are going to

00:09:43.000 --> 00:09:45.000
see, there are two things in an
integral.

00:09:45.000 --> 00:09:49.000
One is whatever you integral is
called the integrant.

00:09:49.000 --> 00:09:52.000
It could be a function here.
It is a vector field in some of

00:09:52.000 --> 00:09:55.000
the flux things and so on.
There is another thing which is

00:09:55.000 --> 00:09:57.000
the region over which you
integrate.

00:09:57.000 --> 00:10:02.000
And the two have strictly
nothing to do with each other.

00:10:02.000 --> 00:10:05.000
When you are given a piece of
data in the statement of a

00:10:05.000 --> 00:10:07.000
problem,
you have to figure out whether

00:10:07.000 --> 00:10:09.000
that is part of a function to be
integrated or whether that is

00:10:09.000 --> 00:10:11.000
part of the region of
integration.

00:10:11.000 --> 00:10:14.000
If it is the region of
integration then it will go into

00:10:14.000 --> 00:10:17.000
the bounds of the integral and
maybe in the choice of the

00:10:17.000 --> 00:10:19.000
coordinate system that you use
for integrating.

00:10:19.000 --> 00:10:23.000
While the function that you are
integrating goes before the dx

00:10:23.000 --> 00:10:26.000
dy and not into the bounds or
anything like that.

00:10:26.000 --> 00:10:32.000
I know it sounds kind of silly
but it is a good safety check.

00:10:32.000 --> 00:10:34.000
Ask yourselves,
when you have a piece of data,

00:10:34.000 --> 00:10:36.000
where in my formula should this
go.

00:10:36.000 --> 00:10:47.000
Yes?
I case you want the bounds for

00:10:47.000 --> 00:10:50.000
this region in polar
coordinates, indeed it would be

00:10:50.000 --> 00:10:53.000
double integral.
For a fixed theta,

00:10:53.000 --> 00:10:57.000
r goes from zero to whatever it
is on that curve.

00:10:57.000 --> 00:11:04.000
So it would be zero to two
cosine theta of whatever the

00:11:04.000 --> 00:11:10.000
function is r dr d theta.
And the bounds on theta would

00:11:10.000 --> 00:11:15.000
be from negative pi over two to
pi over two.

00:11:15.000 --> 00:11:20.000
We have seen that one several
times, so hopefully by now it is

00:11:20.000 --> 00:11:21.000
clearer.
OK.

00:11:21.000 --> 00:11:28.000
Let me move on a bit because we
have a lot of other kinds of

00:11:28.000 --> 00:11:32.000
integrals to see.
Other kinds of integrals we

00:11:32.000 --> 00:11:35.000
have seen are triple integrals.
And I am not doing things in

00:11:35.000 --> 00:11:39.000
the order that we did them in
the class just so you can see

00:11:39.000 --> 00:11:43.000
parallels between stuff in the
plane and in space.

00:11:43.000 --> 00:11:47.000
When we do triple integrals in
space, well, it is the same kind

00:11:47.000 --> 00:11:50.000
of story, except now we have,
of course, more coordinate

00:11:50.000 --> 00:11:53.000
systems.
We have rectangular

00:11:53.000 --> 00:11:59.000
coordinates, we have cylindrical
coordinates and we have

00:11:59.000 --> 00:12:05.000
spherical coordinates.
And cylindrical coordinates

00:12:05.000 --> 00:12:09.000
only mean that we are,
instead of x, y and z,

00:12:09.000 --> 00:12:15.000
we are replacing x and y by the
polar coordinate in the x,

00:12:15.000 --> 00:12:17.000
y plane,
so the angle theta and the

00:12:17.000 --> 00:12:21.000
distance r.
So R is somehow the distance

00:12:21.000 --> 00:12:24.000
from the z-axis and z is the
height.

00:12:24.000 --> 00:12:28.000
Usually you don't have to
choose between rectangular and

00:12:28.000 --> 00:12:31.000
cylindrical until somewhat late
in the process,

00:12:31.000 --> 00:12:33.000
especially if you integrate
first of all z,

00:12:33.000 --> 00:12:36.000
because then the choice will
come up mostly when you try to

00:12:36.000 --> 00:12:39.000
figure out what are the bounds
for the shadow of your region.

00:12:39.000 --> 00:12:43.000
I mean the z part looks exactly
the same in rectangular and in

00:12:43.000 --> 00:12:45.000
cylindrical.
Spherical is,

00:12:45.000 --> 00:12:49.000
on the other hand,
a little bit more annoying

00:12:49.000 --> 00:12:52.000
because it looks quite
different.

00:12:52.000 --> 00:12:56.000
You should think of it as doing
polar coordinates not only in

00:12:56.000 --> 00:12:59.000
the horizontal direction but
also in the vertical direction

00:12:59.000 --> 00:13:02.000
at the same time.
You have this angle phi.

00:13:02.000 --> 00:13:06.000
That measures the angle down
from the positive z-axis.

00:13:06.000 --> 00:13:10.000
And you have rho which is the
distance from the origin.

00:13:10.000 --> 00:13:19.000
And if I project to the z-axis,
r becomes rho sine phi and z

00:13:19.000 --> 00:13:24.000
becomes rho cosine phi.
I hope that you all know these

00:13:24.000 --> 00:13:27.000
two formulas,
but if you ever have a small

00:13:27.000 --> 00:13:31.000
somehow memory lapse during the
final then you should consider

00:13:31.000 --> 00:13:35.000
drawing this kind of picture
because it will let you check

00:13:35.000 --> 00:13:42.000
very quickly which one is sine,
which one is cosine.

00:13:42.000 --> 00:13:44.000
Now, of course,
we have to have formulas for dv

00:13:44.000 --> 00:13:47.000
in all these coordinate systems.
Here, for example,

00:13:47.000 --> 00:13:52.000
that might be dz r dr d theta
or r dr d theta dz or anything

00:13:52.000 --> 00:13:58.000
like that.
Here it might be rho squared

00:13:58.000 --> 00:14:04.000
times phi times d rho d phi d
theta.

00:14:04.000 --> 00:14:07.000
And the general method for
setting up bounds is pretty much

00:14:07.000 --> 00:14:10.000
the same as in the plane,
just there is one more step.

00:14:10.000 --> 00:14:14.000
If you are doing rectangular or
cylindrical coordinates with z

00:14:14.000 --> 00:14:16.000
first, for example,
that is the most common.

00:14:16.000 --> 00:14:22.000
Well, if you do z first then
you have to actually start by

00:14:22.000 --> 00:14:28.000
figuring out for a given value
of x and y or r and theta what

00:14:28.000 --> 00:14:31.000
is the portion of a vertical
line above x,

00:14:31.000 --> 00:14:37.000
y that lies within my region?
That will go from z on the

00:14:37.000 --> 00:14:44.000
bottom of my solid which depends
on x and y to z at the top of my

00:14:44.000 --> 00:14:49.000
solid which also usually will
depend on x and y.

00:14:49.000 --> 00:14:52.000
And so that will give me the
bounds for dz.

00:14:52.000 --> 00:14:56.000
And then I will be left with
the shadow of my region in the

00:14:56.000 --> 00:15:02.000
x, y plane.
And that one I will set up like

00:15:02.000 --> 00:15:09.000
a double integral over there.
Strictly-speaking,

00:15:09.000 --> 00:15:12.000
if you are curious,
we could also change to weird

00:15:12.000 --> 00:15:15.000
coordinate systems using
Jacobian with three variables at

00:15:15.000 --> 00:15:18.000
the same time.
But we haven't seen that so it

00:15:18.000 --> 00:15:24.000
won't be on the final.
But it would work just the same

00:15:24.000 --> 00:15:30.000
way, just with more pictures to
do.

00:15:30.000 --> 00:15:33.000
And, in fact,
I just wanted to say this rho

00:15:33.000 --> 00:15:37.000
squared sine phi is actually the
Jacobian for the change of

00:15:37.000 --> 00:15:42.000
variables for rectangular to
spherical coordinates.

00:15:42.000 --> 00:15:48.000
OK.
Let's not think too much about

00:15:48.000 --> 00:15:51.000
that.
Applications.

00:15:51.000 --> 00:15:57.000
Well, we have seen how to use
double integrals to find the

00:15:57.000 --> 00:16:03.000
area of a volume of a piece of a
plane or a piece of space,

00:16:03.000 --> 00:16:08.000
and find also the mass.
Remember, area is just double

00:16:08.000 --> 00:16:12.000
integral of one dA,
volume is triple integral of

00:16:12.000 --> 00:16:14.000
one dV.
Sometimes if it's the volume

00:16:14.000 --> 00:16:16.000
between the x,
y plane and the graph of some

00:16:16.000 --> 00:16:19.000
function, you can just set it up
directly as a double integral.

00:16:19.000 --> 00:16:23.000
But there is no harm in doing
it as a triple integral if you

00:16:23.000 --> 00:16:27.000
feel better about that.
And mass will be double or

00:16:27.000 --> 00:16:31.000
triple integral,
depending on how many

00:16:31.000 --> 00:16:36.000
dimensions you have,
of whatever density function

00:16:36.000 --> 00:16:45.000
you have, dA or dV.
Then there is how to find the

00:16:45.000 --> 00:16:53.000
average value of some function.
Well, let me do the

00:16:53.000 --> 00:16:58.000
three-dimensional case.
You will just replace volume by

00:16:58.000 --> 00:17:02.000
area and dV by dA and so on,
if need be.

00:17:02.000 --> 00:17:09.000
That would be one over volume
of the solid times the triple

00:17:09.000 --> 00:17:18.000
integral of f dV,
or if it's a weighted average,

00:17:18.000 --> 00:17:24.000
one over mass times the triple
integral of a function times

00:17:24.000 --> 00:17:33.000
density times dV.
If you don't have a density or

00:17:33.000 --> 00:17:44.000
if the density is constant then
that reduces to that one.

00:17:44.000 --> 00:17:46.000
In particular,
we have seen the notion of

00:17:46.000 --> 00:17:50.000
center of mass.
The center of mass is just

00:17:50.000 --> 00:17:54.000
given by taking the average
values of the coordinates,

00:17:54.000 --> 00:17:56.000
x bar, y bar,
z bar.

00:17:56.000 --> 00:18:00.000
It is just this formula but
taking x, y or z as the

00:18:00.000 --> 00:18:08.000
function.
There are moments of inertia.

00:18:08.000 --> 00:18:12.000
For example,
the moment of inertia about the

00:18:12.000 --> 00:18:17.000
z-axis is the triple integral of
x squared plus y squared density

00:18:17.000 --> 00:18:19.000
dV.
Or, if you have just a

00:18:19.000 --> 00:18:22.000
two-dimensional object,
it is the same formula,

00:18:22.000 --> 00:18:23.000
but, of course,
with dA.

00:18:23.000 --> 00:18:26.000
And then we call that the polar
moment of inertia because we

00:18:26.000 --> 00:18:30.000
thought of it as rotating the
plane about the origin,

00:18:30.000 --> 00:18:34.000
but the origin is just where
the z-axis hits the x,

00:18:34.000 --> 00:18:38.000
y plane so it is really the
same thing.

00:18:38.000 --> 00:18:42.000
And we have also seen
gravitational attraction in

00:18:42.000 --> 00:18:47.000
space, and I will let you look
at your notes for that.

00:18:47.000 --> 00:18:57.000
It is just one formula to
remember.

00:18:57.000 --> 00:19:02.000
Questions about iterated
integrals, things like that?

00:19:02.000 --> 00:19:18.000
Yes?
The formula that you should

00:19:18.000 --> 00:19:22.000
know for gravitational
attraction is that if yu have a

00:19:22.000 --> 00:19:28.000
point mass at the origin and you
have some solid centered on the

00:19:28.000 --> 00:19:33.000
z-axis that is attracting it
then the force will be given by

00:19:33.000 --> 00:19:38.000
G times the mass times the
triple integral of density times

00:19:38.000 --> 00:19:42.000
cosine phi over rho squared
times dV.

00:19:42.000 --> 00:19:45.000
And, of course,
you will actually do that in

00:19:45.000 --> 00:19:50.000
spherical coordinates because it
is easier that way.

00:19:50.000 --> 00:19:55.000
That is the formula I have in
mind.

00:19:55.000 --> 00:19:59.000
But, see, all these formulas
just give you examples of things

00:19:59.000 --> 00:20:01.000
to integrate.
And how to set up the bounds

00:20:01.000 --> 00:20:03.000
and so on does not depend on
what you are actually

00:20:03.000 --> 00:20:08.000
integrating.
It is done always using the

00:20:08.000 --> 00:20:22.000
same methods.
Let's move on to work and line

00:20:22.000 --> 00:20:28.000
integrals.
We have seen how to do that in

00:20:28.000 --> 00:20:35.000
the plane and in space.
And it looks very similar

00:20:35.000 --> 00:20:41.000
somehow.
Remember, you have to know how

00:20:41.000 --> 00:20:48.000
to set up and evaluate a line
integral of this form.

00:20:48.000 --> 00:20:49.000
Let me do it in the plane this
time.

00:20:49.000 --> 00:20:53.000
If you are in the plane you
have two components,

00:20:53.000 --> 00:20:58.000
and then this becomes the line
integral of M dx plus N dy.

00:20:58.000 --> 00:21:01.000
If you have a space curve then
you will have a third component

00:21:01.000 --> 00:21:05.000
here.
You will add that guy times dz.

00:21:05.000 --> 00:21:08.000
Now, how do we evaluate that?
Well, it is very different from

00:21:08.000 --> 00:21:11.000
there because here we are just
on a curve so there should be

00:21:11.000 --> 00:21:15.000
only one degree of freedom.
One variable should be enough

00:21:15.000 --> 00:21:21.000
to know where we are.
We will have to express x and y

00:21:21.000 --> 00:21:28.000
in terms of -- Well,
I should and z optionally if

00:21:28.000 --> 00:21:32.000
there is one,
in terms of a single

00:21:32.000 --> 00:21:36.000
parameters.
And that might be just one of

00:21:36.000 --> 00:21:38.000
the coordinates.
If you are told y equals z

00:21:38.000 --> 00:21:41.000
squared, that is easy.
You just substitute y equals x

00:21:41.000 --> 00:21:44.000
squared and dy equals two x dx
into everything,

00:21:44.000 --> 00:21:48.000
and you are left with an
integral over x.

00:21:48.000 --> 00:22:01.000
Maybe it will be something in
terms of time or in terms of an

00:22:01.000 --> 00:22:05.000
angle.
We express everything in terms

00:22:05.000 --> 00:22:09.000
of a single parameter,
and that will give us a usual

00:22:09.000 --> 00:22:14.000
single integral.
Any questions about that?

00:22:14.000 --> 00:22:23.000
Yes?
If you cannot parameterize the

00:22:23.000 --> 00:22:27.000
curve then it is really,
really hard to evaluate the

00:22:27.000 --> 00:22:31.000
line integral.
Well, you might be able to

00:22:31.000 --> 00:22:35.000
evaluate it numerically into a
computer, but that is the

00:22:35.000 --> 00:22:40.000
easiest way to describe a curve.
Indeed it could be that in the

00:22:40.000 --> 00:22:43.000
plane you have an equation in
terms of x and y given by some

00:22:43.000 --> 00:22:46.000
completed formulas defining some
curve.

00:22:46.000 --> 00:22:49.000
Then actually there are ways
you can use basically

00:22:49.000 --> 00:22:52.000
differentials and constrained
partials to figure out what the

00:22:52.000 --> 00:22:55.000
tangent vector to the curve is
and so on.

00:22:55.000 --> 00:22:57.000
But we haven't really seen how
to do that.

00:22:57.000 --> 00:23:00.000
That would be a really nice
topic for tying together the end

00:23:00.000 --> 00:23:02.000
of the second unit that we
discussed last time,

00:23:02.000 --> 00:23:04.000
constrained partials,
with this stuff.

00:23:04.000 --> 00:23:08.000
But that is not going to be
part of our topics.

00:23:08.000 --> 00:23:11.000
Basically, all the curves we
have seen in this class,

00:23:11.000 --> 00:23:14.000
there is a way to express the
position of a point in terms of

00:23:14.000 --> 00:23:23.000
a parameter.
We haven't seen any curves that

00:23:23.000 --> 00:23:34.000
are so complicated that you
cannot do that.

00:23:34.000 --> 00:23:37.000
The other thing we have seen is
that there are some special

00:23:37.000 --> 00:23:40.000
cases of vector fields where we
don't actually have to compute

00:23:40.000 --> 00:23:43.000
this thing because maybe we know
that it is the gradient of some

00:23:43.000 --> 00:23:48.000
potential function.
And then we have a fundamental

00:23:48.000 --> 00:23:55.000
theorem that gives us a way to
compute this without computing

00:23:55.000 --> 00:24:04.000
it.
We've seen about gradient

00:24:04.000 --> 00:24:16.000
fields and path independence.
The thing to check is whether

00:24:16.000 --> 00:24:18.000
the curl of our vector field is
zero.

00:24:18.000 --> 00:24:27.000
And remember in the plane that
is one condition,

00:24:27.000 --> 00:24:33.000
Nx equals My.
In 3D in space that is actually

00:24:33.000 --> 00:24:39.000
three conditions because you
have to check all the mixed

00:24:39.000 --> 00:24:44.000
partials of the various
components.

00:24:44.000 --> 00:24:47.000
If the curl of f is zero that
tells us we are likely to have a

00:24:47.000 --> 00:24:52.000
gradient field.
Strictly-speaking,

00:24:52.000 --> 00:25:06.000
I should mention and F is
defined in a simply-connected

00:25:06.000 --> 00:25:16.000
region.
Then F is a gradient field.

00:25:16.000 --> 00:25:25.000
That means that we can find a
potential function.

00:25:25.000 --> 00:25:30.000
You can write F as the gradient
of little f for some potential

00:25:30.000 --> 00:25:39.000
function little f.
And we have seen how to find

00:25:39.000 --> 00:25:46.000
the potential.
In fact, we have seen two

00:25:46.000 --> 00:25:50.000
methods for that.
And we have seen them twice.

00:25:50.000 --> 00:25:53.000
We have seen them once for
functions of two variables,

00:25:53.000 --> 00:25:55.000
once for functions of three
variables.

00:25:55.000 --> 00:25:58.000
They look very much the same.
I encourage you to compare your

00:25:58.000 --> 00:26:03.000
notes for the two side by side
to see where they differ.

00:26:03.000 --> 00:26:05.000
Where they differ,
roughly-speaking,

00:26:05.000 --> 00:26:09.000
well, I never know if it is the
first or the second,

00:26:09.000 --> 00:26:14.000
but one of the two methods was
to compute a line integral.

00:26:14.000 --> 00:26:18.000
In the plane,
what we did is we set up and

00:26:18.000 --> 00:26:23.000
evaluated a line integral along
our favorite path from the

00:26:23.000 --> 00:26:27.000
origin to a point with
coordinates say x1,

00:26:27.000 --> 00:26:32.000
y1.
And then we had to evaluate the

00:26:32.000 --> 00:26:38.000
line integral for the work done
along this path.

00:26:38.000 --> 00:26:43.000
And that will give us the value
of potential at that point.

00:26:43.000 --> 00:26:46.000
If we are doing it with three
variables, that method remains

00:26:46.000 --> 00:26:49.000
very similar.
The only difference is now we

00:26:49.000 --> 00:26:52.000
have to go also up in space to
some point x1,

00:26:52.000 --> 00:26:55.000
y1, z1.
And so we actually sum three

00:26:55.000 --> 00:26:59.000
pieces together.
But on each piece it is the

00:26:59.000 --> 00:27:01.000
same story, only one variable
changes.

00:27:01.000 --> 00:27:06.000
Here it is only x that changes,
it is only y that changes,

00:27:06.000 --> 00:27:11.000
and on the third one only z
would be changing.

00:27:11.000 --> 00:27:18.000
That is one possibility.
And the other possibility for

00:27:18.000 --> 00:27:23.000
finding the potential is that we
start with the condition that

00:27:23.000 --> 00:27:28.000
the first component of our
vector field should be equal to

00:27:28.000 --> 00:27:32.000
f sub x for the unknown
potential function.

00:27:32.000 --> 00:27:36.000
What we do is integrate with
respect to x,

00:27:36.000 --> 00:27:40.000
and we will get our potential
function up to an integration

00:27:40.000 --> 00:27:44.000
constant.
And that integration constant

00:27:44.000 --> 00:27:48.000
typically depends on the
remaining variables that might

00:27:48.000 --> 00:27:53.000
be y or equal in space y and z.
And then what we have to do is

00:27:53.000 --> 00:27:56.000
take the partial of this with
respect to y and compare it to

00:27:56.000 --> 00:28:00.000
what we want it to be,
namely the y component of a

00:28:00.000 --> 00:28:02.000
vector field,
and match them to get some

00:28:02.000 --> 00:28:08.000
information about this guy.
And if we have three variables

00:28:08.000 --> 00:28:13.000
then there is a third step
because there you will still

00:28:13.000 --> 00:28:18.000
have an unknown function of z
that you need to get by

00:28:18.000 --> 00:28:23.000
comparing the partials with
respect to z.

00:28:23.000 --> 00:28:30.000
I see a lot of very quiet faces
somehow.

00:28:30.000 --> 00:28:36.000
Well, hopefully that is because
you know that stuff.

00:28:36.000 --> 00:28:42.000
If it is because you are
hopelessly confused then please

00:28:42.000 --> 00:28:49.000
review a lot before the final,
but I really hope that is not

00:28:49.000 --> 00:28:52.000
the case.
And so,

00:29:22.000 --> 00:29:26.000
in particular,
what we have seen is once we

00:29:26.000 --> 00:29:32.000
have the potential then we can
use the fundamental theorem of

00:29:32.000 --> 00:29:38.000
calculus to tell us that if we
have a line integral to compute

00:29:38.000 --> 00:29:44.000
for work along a curve that goes
from some point P zero to some

00:29:44.000 --> 00:29:49.000
point P one then the line
integral for the work done by

00:29:49.000 --> 00:29:54.000
gradient F is actually going
just to be the change in value

00:29:54.000 --> 00:29:58.000
of a potential.
And, in particular,

00:29:58.000 --> 00:30:02.000
that does not depend on how we
got from P zero to P one.

00:30:02.000 --> 00:30:10.000
That is why we say that we have
path independence.

00:30:10.000 --> 00:30:22.000
Next topic is flux in plane and
space.

00:30:22.000 --> 00:30:26.000
Flux looks quite different in
the plane and in space because,

00:30:26.000 --> 00:30:29.000
in the plane,
it is just another kind of line

00:30:29.000 --> 00:30:32.000
integral,
while in space it is a surface

00:30:32.000 --> 00:30:34.000
integral.
If you were in four-dimensional

00:30:34.000 --> 00:30:35.000
space it would be a triple
integral.

00:30:35.000 --> 00:30:43.000
Generally, you do flux for
something that is somehow a wall

00:30:43.000 --> 00:30:50.000
that separates regions of space
from each other.

00:30:50.000 --> 00:30:56.000
In the plane,
the way we do it is we have a

00:30:56.000 --> 00:31:02.000
curve C and we look at its
tangent vector,

00:31:02.000 --> 00:31:09.000
let's call that T,
and we rotate it by 90 degrees

00:31:09.000 --> 00:31:13.000
clockwise.
That is our convention to get a

00:31:13.000 --> 00:31:18.000
unit normal vector that points
to the right of the curve as we

00:31:18.000 --> 00:31:21.000
move along the curve.
That is our convention for

00:31:21.000 --> 00:31:24.000
orienting curves.
And we are always going to be

00:31:24.000 --> 00:31:33.000
using that one.
N equals T rotated 90 degrees

00:31:33.000 --> 00:31:37.000
clockwise.
In particular,

00:31:37.000 --> 00:31:44.000
that means that n ds,
which will be what we integrate

00:31:44.000 --> 00:31:52.000
against when we try to compute
flux, will just end up being dy,

00:31:52.000 --> 00:31:58.000
negative dx.
Concretely, when we have to

00:31:58.000 --> 00:32:03.000
evaluate a line integral of F
dot n ds,

00:32:03.000 --> 00:32:07.000
geometrically we could try to
take the dot product of our

00:32:07.000 --> 00:32:11.000
field with the normal vector and
then sum the length element

00:32:11.000 --> 00:32:14.000
along the curve.
And, in some cases,

00:32:14.000 --> 00:32:16.000
for example,
if you know that the vector

00:32:16.000 --> 00:32:19.000
field is tangent to the curve or
if a dot product is constant or

00:32:19.000 --> 00:32:22.000
things like that then that might
actually give you a very easy

00:32:22.000 --> 00:32:24.000
answer.
But, in general,

00:32:24.000 --> 00:32:28.000
the most efficient way to do it
will be to say that if your

00:32:28.000 --> 00:32:31.000
vector field has components,
I don't know,

00:32:31.000 --> 00:32:36.000
let's call them P and Q,
then that will be just the line

00:32:36.000 --> 00:32:39.000
integral of PQ dot dy,
negative dx,

00:32:39.000 --> 00:32:44.000
which means negative Q dx plus
P dy.

00:32:44.000 --> 00:32:49.000
And, from that point onward,
you evaluate it exactly the

00:32:49.000 --> 00:32:54.000
same way as you would for a work
integral.

00:32:54.000 --> 00:32:56.000
But, of course,
the geometric meaning is very

00:32:56.000 --> 00:32:57.000
different.
It is the same meaning that we

00:32:57.000 --> 00:33:08.000
have always seen for flux.
It measures how much a vector

00:33:08.000 --> 00:33:22.000
field goes across the curve.
Now, if we are in space then

00:33:22.000 --> 00:33:30.000
you take flux for a surface,
not for a curve.

00:33:30.000 --> 00:33:33.000
And the way it will work is
that you have to choose an

00:33:33.000 --> 00:33:37.000
orientation of a surface,
which just means choosing one

00:33:37.000 --> 00:33:40.000
of the two possible unit normal
vectors.

00:33:40.000 --> 00:33:51.000
And then you will do a surface
integral for F dot n dS.

00:33:51.000 --> 00:34:00.000
That is the surface i element.
The setup for this surface

00:34:00.000 --> 00:34:07.000
integral is that first we have
to express n and dS in some way.

00:34:07.000 --> 00:34:14.000
One possibility is that we can
express the normal vector n dS

00:34:14.000 --> 00:34:18.000
geometrically.
That is, for example,

00:34:18.000 --> 00:34:23.000
what we do when we look at,
say, a horizontal plane or a

00:34:23.000 --> 00:34:27.000
vertical plane or a sphere or a
cylinder.

00:34:27.000 --> 00:34:31.000
Then we have some geometric
idea of why the normal vector is

00:34:31.000 --> 00:34:34.000
what it is and we have some
formula for dS.

00:34:34.000 --> 00:34:39.000
Or, we can use one of the
standard formulas.

00:34:39.000 --> 00:34:43.000
Basically, we have seen two
formulas that work in fairly

00:34:43.000 --> 00:34:50.000
generate situations.
One of them says -- If S is

00:34:50.000 --> 00:35:00.000
given by an equation z equals
some function of x,

00:35:00.000 --> 00:35:06.000
y then you can just say n dS
equals minus f sub x,

00:35:06.000 --> 00:35:09.000
minus f sub y,
one, dx dy.

00:35:09.000 --> 00:35:14.000
And I need to rewrite that
because I am running out of

00:35:14.000 --> 00:35:16.000
space.
But, while I erase,

00:35:16.000 --> 00:35:20.000
I would like to point out the
most important there in here.

00:35:20.000 --> 00:35:24.000
When I say n dS equals blah,
blah, blah times dx dy,

00:35:24.000 --> 00:35:28.000
dx dy is not the same thing as
dS at all.

00:35:28.000 --> 00:35:31.000
If you make that mistake you
are going to get into trouble

00:35:31.000 --> 00:35:34.000
the next time that you try to
buy real estate in a region

00:35:34.000 --> 00:35:37.000
which hills or cliffs or things
like that.

00:35:37.000 --> 00:35:40.000
dS is the area on the slanted
surface.

00:35:40.000 --> 00:35:46.000
dx dy is the area on the map
that shows the x,

00:35:46.000 --> 00:35:49.000
y plane.
And these are not the same

00:35:49.000 --> 00:35:51.000
thing.
In particular,

00:35:51.000 --> 00:36:00.000
you cannot just take one piece
of it and not the other piece.

00:36:00.000 --> 00:36:06.000
Let me give you formulas for n
and for dS separately just to

00:36:06.000 --> 00:36:14.000
convince you.
That way, if you feel that you

00:36:14.000 --> 00:36:22.000
need them, then you will have
them.

00:36:22.000 --> 00:36:26.000
N is minus f sub x,
minus f sub y,

00:36:26.000 --> 00:36:31.000
one, but scaled down to unit
length.

00:36:31.000 --> 00:36:35.000
This is not a unit vector.
It is actually divided by the

00:36:35.000 --> 00:36:38.000
length of this guy which is fx
squared plus fy squared plus

00:36:38.000 --> 00:36:46.000
one.
And dS is that same vector

00:36:46.000 --> 00:36:52.000
times dx dy.
And so the square roots cancel

00:36:52.000 --> 00:36:56.000
out when you multiply them
together.

00:36:56.000 --> 00:37:00.000
But it would be completely
wrong to just say I will replace

00:37:00.000 --> 00:37:03.000
n dS by minus f sub x,
minus f sub y and one.

00:37:03.000 --> 00:37:07.000
Then I end up again with the dS
and I do something else with dS.

00:37:07.000 --> 00:37:16.000
That is a pretty bad conceptual
mistake because it gives you the

00:37:16.000 --> 00:37:20.000
wrong answer.
Another option more general

00:37:20.000 --> 00:37:24.000
than that.
If we have not seen how to

00:37:24.000 --> 00:37:32.000
solve for z, how to express z as
a function of x and y,

00:37:32.000 --> 00:37:40.000
well, maybe we still know some
normal vector to the surface.

00:37:40.000 --> 00:37:49.000
Then there is another formula
for n dS which is up to sine,

00:37:49.000 --> 00:37:55.000
N divided by N dot k dx dy.
And, see, that projection

00:37:55.000 --> 00:37:58.000
formula works also if you have
to project to another coordinate

00:37:58.000 --> 00:37:59.000
plane.
For example,

00:37:59.000 --> 00:38:02.000
if you want to project to the
x, z coordinate plane,

00:38:02.000 --> 00:38:10.000
the relation between n dS and
dx dz is given by N over N dot

00:38:10.000 --> 00:38:13.000
j,
because j is the direction

00:38:13.000 --> 00:38:19.000
perpendicular to the xz plane.
But this one is more useful.

00:38:19.000 --> 00:38:24.000
What is a good example of that?
If you have a slanted plane

00:38:24.000 --> 00:38:28.000
given to you,
you can easily find its normal

00:38:28.000 --> 00:38:31.000
vector.
That is just given by the

00:38:31.000 --> 00:38:33.000
coefficients of x,
y, z in the equation.

00:38:33.000 --> 00:38:38.000
Another situation where that
might happen is if your surface

00:38:38.000 --> 00:38:41.000
is given by an equation of a
form of g of x,

00:38:41.000 --> 00:38:44.000
y, z equals zero.
If that is the case then you

00:38:44.000 --> 00:38:47.000
know this is a level set of g.
And we know how to find a

00:38:47.000 --> 00:38:50.000
normal vector to the level set,
namely the gradient vector is

00:38:50.000 --> 00:38:52.000
always perpendicular to the
level set.

00:38:52.000 --> 00:38:59.000
You would take the gradient of
g to be your big N.

00:38:59.000 --> 00:39:06.000
OK.
Now, these are basically all

00:39:06.000 --> 00:39:10.000
the integrals we have seen how
to set up.

00:39:10.000 --> 00:39:15.000
Now we have a bunch of theorems
relating them.

00:39:15.000 --> 00:39:23.000
Let me think about how I am
going to organize that.

00:39:23.000 --> 00:39:32.000
Let me try like this.
This part of the board will be

00:39:32.000 --> 00:39:35.000
work,
this part of the board will be

00:39:35.000 --> 00:39:40.000
about flux and the left part of
the board will be about things

00:39:40.000 --> 00:39:46.000
in the plane and the right one
will be about things in space.

00:39:46.000 --> 00:39:53.000
What have we seen?
Well, we have seen Green's

00:39:53.000 --> 00:39:57.000
theorem for work.
That doesn't work so well

00:39:57.000 --> 00:39:59.000
because that is too small,
so I am going to actually use

00:39:59.000 --> 00:40:01.000
more blackboards to do that.

00:40:26.000 --> 00:40:32.000
This side will be space,
this side will be the plane and

00:40:32.000 --> 00:40:37.000
we are going to start with
theorems about work.

00:40:37.000 --> 00:40:41.000
And we will see theorems about
flux pretty soon.

00:40:41.000 --> 00:40:47.000
We have two theorems about work.
In the plane that is called

00:40:47.000 --> 00:40:51.000
Green's theorem.
In space that is called Stokes'

00:40:51.000 --> 00:40:56.000
theorem.
Green's theorem says if I have

00:40:56.000 --> 00:41:03.000
a closed curve in the plane
going counterclockwise enclosing

00:41:03.000 --> 00:41:11.000
entirely some region R then the
line integral along C for the

00:41:11.000 --> 00:41:20.000
work of F is equal to the double
integral of a region inside of

00:41:20.000 --> 00:41:27.000
the curl of F dA.
Concretely, if my components of

00:41:27.000 --> 00:41:35.000
F are called M and N that is the
line integral of M dx plus N dy

00:41:35.000 --> 00:41:43.000
is equal to the double integral
of R of N sub x minus M sub y

00:41:43.000 --> 00:41:46.000
dA.
This side here is a usual line

00:41:46.000 --> 00:41:49.000
integral.
This side here is a usual

00:41:49.000 --> 00:41:53.000
double integral in the plane.
And somehow their values end up

00:41:53.000 --> 00:41:58.000
being magically related.
Well, not quite magically.

00:41:58.000 --> 00:42:03.000
We actually have seen how to
prove it.

00:42:03.000 --> 00:42:10.000
And now the analog of that in
space is Stokes' theorem.

00:42:10.000 --> 00:42:15.000
Stokes says if I have a closed
curve in space,

00:42:15.000 --> 00:42:20.000
now I have to decide what kind
of thing it bounds.

00:42:20.000 --> 00:42:23.000
And the answer is it will have
to bound some surface,

00:42:23.000 --> 00:42:28.000
but I have a choice of surface.
I choose my favorite surface

00:42:28.000 --> 00:42:31.000
bounded by C.
I guess I will just draw it

00:42:31.000 --> 00:42:33.000
like that.
And I have to choose a

00:42:33.000 --> 00:42:36.000
compatible orientation.
Remember, we have seen this

00:42:36.000 --> 00:42:39.000
right hand rule for choosing how
to orient the surface.

00:42:39.000 --> 00:42:43.000
I believe, in this case,
if I take C like that then the

00:42:43.000 --> 00:42:49.000
normal vector has to go up.
And then it tells me how to

00:42:49.000 --> 00:42:53.000
compute the work done by F along
C.

00:42:53.000 --> 00:43:01.000
Namely, that becomes the double
integral over that surface S of

00:43:01.000 --> 00:43:08.000
curl F, which I will write as
dell cross F dot n dS.

00:43:08.000 --> 00:43:10.000
This line integral is a usual
line integral,

00:43:10.000 --> 00:43:14.000
but if for some reason we don't
want to compute it directly we

00:43:14.000 --> 00:43:18.000
can actually replace it by a
surface integral over any

00:43:18.000 --> 00:43:22.000
surface bounded by the curve.
It might be that a problem will

00:43:22.000 --> 00:43:23.000
tell you which surface you have
to consider.

00:43:23.000 --> 00:43:28.000
It might be that you will be
left to choose the simplest

00:43:28.000 --> 00:43:33.000
possible surface you can think
of that is somehow having this

00:43:33.000 --> 00:43:36.000
curve as its boundary.
And so now, remember,

00:43:36.000 --> 00:43:40.000
curl of a vector field in space
is going to be another vector

00:43:40.000 --> 00:43:42.000
expression.
It has three components.

00:43:42.000 --> 00:43:45.000
And the way you compute it is
not by remembering the actual

00:43:45.000 --> 00:43:48.000
formula,
which is really complicated by,

00:43:48.000 --> 00:43:54.000
but by instead computing the
cross-product between dell and

00:43:54.000 --> 00:44:01.000
F.
You set up the cross-product.

00:44:01.000 --> 00:44:03.000
And, of course,
it is a highly symbolic

00:44:03.000 --> 00:44:07.000
cross-product.
I mean it is not an

00:44:07.000 --> 00:44:16.000
cross-product of actual vectors
but it works the same way.

00:44:16.000 --> 00:44:20.000
Both of these formulas
basically relate work on a curve

00:44:20.000 --> 00:44:24.000
with what happens to the curl on
the surface that is enclosed by

00:44:24.000 --> 00:44:27.000
this curve,
that is bounded by this curve.

00:44:27.000 --> 00:44:30.000
And in this one you have less
freedom of choice because you

00:44:30.000 --> 00:44:34.000
don't have somehow a z direction
in which you could move your

00:44:34.000 --> 00:44:36.000
surface.
There is only possible choice

00:44:36.000 --> 00:44:39.000
of surface.
There is only one thing that is

00:44:39.000 --> 00:44:42.000
enclosed by this curve in the
plane.

00:44:42.000 --> 00:44:45.000
In both cases,
these things tell you that you

00:44:45.000 --> 00:44:48.000
can think of curl as measuring
how much the field fails to be

00:44:48.000 --> 00:44:51.000
conservative.
See, if your field was

00:44:51.000 --> 00:44:55.000
conservative -- If a curl was
zero then the right-hand side

00:44:55.000 --> 00:44:57.000
would just be zero.
And that would be fortunate

00:44:57.000 --> 00:45:01.000
because if a curl is zero then
your field is less conservative.

00:45:01.000 --> 00:45:02.000
That means it comes from a
potential.

00:45:02.000 --> 00:45:05.000
That means when you go along a
closed curve,

00:45:05.000 --> 00:45:09.000
well, the change of value of a
potential should be zero.

00:45:09.000 --> 00:45:13.000
Another way to say it is path
independence tells you no work.

00:45:13.000 --> 00:45:16.000
And, of course,
if you have a vector field that

00:45:16.000 --> 00:45:20.000
is not a gradient field then the
curl is not necessarily zero and

00:45:20.000 --> 00:45:23.000
then you get a more interesting
answer.

00:45:23.000 --> 00:45:27.000
Finally, let's move onto the
theorems about flux.

00:45:27.000 --> 00:45:43.000
That is Green for flux and that
is the divergence theorem.

00:45:43.000 --> 00:45:54.000
Flux theorems.
Here I say that will be

00:45:54.000 --> 00:46:01.000
divergence.
And here it will be Green again.

00:46:01.000 --> 00:46:07.000
Green's theorem for flux says I
have a closed curve that goes

00:46:07.000 --> 00:46:10.000
counterclockwise around some
region.

00:46:10.000 --> 00:46:13.000
In particular,
counterclockwise means that the

00:46:13.000 --> 00:46:16.000
normal vector will be going out
of the region.

00:46:16.000 --> 00:46:20.000
And then it tells us that the
flux out of the region,

00:46:20.000 --> 00:46:24.000
through the curve C,
so that will be the line

00:46:24.000 --> 00:46:30.000
integral of F dot n ds is equal
to the integral of a region

00:46:30.000 --> 00:46:37.000
inside of div F dA.
And remember the divergence of

00:46:37.000 --> 00:46:42.000
M, N is just Mx plus Ny.
This one here,

00:46:42.000 --> 00:46:46.000
the divergence theorem,
tells you something similar but

00:46:46.000 --> 00:46:49.000
now for a region of space
bounded by a closed surface.

00:46:49.000 --> 00:46:55.000
So if you have some region of
space and you call its boundary

00:46:55.000 --> 00:47:00.000
surface S and you let n be the
normal vector that goes out of

00:47:00.000 --> 00:47:07.000
the region R.
You orient S outwards.

00:47:07.000 --> 00:47:15.000
Then the flux out of the region
through S is going to be the

00:47:15.000 --> 00:47:23.000
same as the triple integral over
the region of divergence F dV.

00:47:23.000 --> 00:47:31.000
Remember, the divergence of a
vector field with components P,

00:47:31.000 --> 00:47:36.000
Q, R is Px plus Qy plus Rz.
What do these two theorems say?

00:47:36.000 --> 00:47:39.000
Well, they say essentially the
same thing.

00:47:39.000 --> 00:47:43.000
They say the total flux out of
a region is equal to the

00:47:43.000 --> 00:47:46.000
integral of divergence over
whatever is inside.

00:47:46.000 --> 00:47:49.000
And the reason for that is,
again, we have seen for a

00:47:49.000 --> 00:47:52.000
velocity field that divergence
measures how much things are

00:47:52.000 --> 00:47:55.000
expanding or how much stuff is
being created.

00:47:55.000 --> 00:47:59.000
It tells you the amount of
sources per unit portion of the

00:47:59.000 --> 00:48:01.000
region.
When you sum that over

00:48:01.000 --> 00:48:03.000
everything,
you get the amount of fluid

00:48:03.000 --> 00:48:05.000
that is being,
you know, the total amount of

00:48:05.000 --> 00:48:08.000
sources inside here,
and that tells us how much

00:48:08.000 --> 00:48:10.000
stuff has to go out per unit
time.

00:48:10.000 --> 00:48:12.000
That is basically the
interpretation.

00:48:12.000 --> 00:48:16.000
In a way, I would be tempted to
say that this table of four

00:48:16.000 --> 00:48:20.000
theorems is somehow the crucial
point of 18.02.

00:48:20.000 --> 00:48:23.000
And you would do well to
remember them.

00:48:23.000 --> 00:48:26.000
However, I would like also to
point out that these theorems

00:48:26.000 --> 00:48:28.000
are completely useless if you
don't know how to compute any of

00:48:28.000 --> 00:48:31.000
the integrals that are in there.
So all the stuff that was

00:48:31.000 --> 00:48:35.000
around there before is actually
somehow more fundamental.

00:48:35.000 --> 00:48:38.000
And if you don't know how to
compute the double or triple

00:48:38.000 --> 00:48:41.000
integrals then this is of little
use to you.

00:48:41.000 --> 00:48:48.000
That is the end.
I guess I have to wish you

00:48:48.000 --> 00:48:52.000
happy holidays.