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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

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over the second half of the
class very quickly.

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And so that was stuff about
double and triple integrals and

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vector calculus in the plane and
in space.

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As usual, what is on the final
is basically what was on the

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other tests, exactly the same
stuff.

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00:00:57,000 --> 00:01:03,000
Well, not the same problem,
unfortunately.

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00:01:03,000 --> 00:01:13,000
The first thing we learned
about was double integrals in

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00:01:13,000 --> 00:01:25,000
the plane and how to set up the
bounds and how to evaluate them.

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00:01:25,000 --> 00:01:30,000
Just to remind you quickly,
the important thing with

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00:01:30,000 --> 00:01:34,000
iterated integrals is when you
integrate a function f of x,

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00:01:34,000 --> 00:01:38,000
y,
say dy dx for example,

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00:01:38,000 --> 00:01:41,000
is that you have to draw a
picture of a region.

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00:01:41,000 --> 00:01:45,000
Unless it is completely obvious
you should really draw some

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00:01:45,000 --> 00:01:48,000
picture of the domain of
integration.

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00:01:48,000 --> 00:01:53,000
And once you have that picture
you can use it to find the

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00:01:53,000 --> 00:01:57,000
bounds.
Remember the general method is

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00:01:57,000 --> 00:02:03,000
that we first look at the inner
integral, here integral of f dy.

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00:02:03,000 --> 00:02:07,000
And in this inner integral the
outer variable here,

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00:02:07,000 --> 00:02:10,000
x, is fixed.
That means we are slicing our

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00:02:10,000 --> 00:02:13,000
region by a vertical line
corresponding to a fixed value

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00:02:13,000 --> 00:02:16,000
of x.
We fix a value of x.

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00:02:16,000 --> 00:02:20,000
And what we have to find out is
the bounds for y,

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00:02:20,000 --> 00:02:24,000
so the value of y at this
point, the value of y at that

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00:02:24,000 --> 00:02:28,000
point.
Let me call that y some bottom

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00:02:28,000 --> 00:02:34,000
of x, in general depends on x.
And this one will be y at that

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00:02:34,000 --> 00:02:43,000
top, and it also depends on x.
And then the bounds for y would

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00:02:43,000 --> 00:02:46,000
be this.
And then, when you look at the

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00:02:46,000 --> 00:02:48,000
outer bound, things are
different.

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00:02:48,000 --> 00:02:51,000
Because there you expect to
have just numbers,

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00:02:51,000 --> 00:02:53,000
no longer functions of
anything.

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00:02:53,000 --> 00:02:56,000
And what you do is look at the
shadow of your region.

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00:02:56,000 --> 00:03:00,000
We are doing it by shadow so
you just project to the x-axis.

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00:03:00,000 --> 00:03:04,000
If you project to the x-axis
your region will look like this.

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00:03:04,000 --> 00:03:08,000
Its shadow is going to be this
integral form,

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00:03:08,000 --> 00:03:12,000
some minimum value of x to some
maximum value of x.

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00:03:12,000 --> 00:03:24,000
And that will give us the
bounds for the outer integral.

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00:03:24,000 --> 00:03:27,000
And then, to evaluate,
we evaluate the usual way.

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00:03:27,000 --> 00:03:29,000
Speaking of evaluation,
what you need to know for the

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00:03:29,000 --> 00:03:31,000
final,
well, essentially the same kind

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00:03:31,000 --> 00:03:35,000
of evaluation techniques that we
were supposed to know for the

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00:03:35,000 --> 00:03:40,000
other tests.
That means the usual functions,

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00:03:40,000 --> 00:03:47,000
substitutions,
basic trig, stuff like that.

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00:03:47,000 --> 00:03:51,000
Well, I don't expect that you
would need integration by parts,

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00:03:51,000 --> 00:03:54,000
although I still hope that some
of you remember it from single

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00:03:54,000 --> 00:03:59,000
variable calculus.
If there is a need to integrate

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00:03:59,000 --> 00:04:06,000
some big power of cosine or sine
then the formula will be given

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00:04:06,000 --> 00:04:11,000
to you the way it is in the
notes.

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00:04:11,000 --> 00:04:17,000
And, of course,
we know also how to set up

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00:04:17,000 --> 00:04:23,000
these integrals in polar
coordinates.

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00:04:23,000 --> 00:04:28,000
And then the area element
becomes r dr d theta.

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00:04:28,000 --> 00:04:32,000
And because you integrate first
over r,

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00:04:32,000 --> 00:04:36,000
well, first of all you should
remember the polar coordinate

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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

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that what we do,
when we have our region,

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00:04:53,000 --> 00:04:56,000
is for a fixed value of theta
we look for the bounds for r,

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00:04:56,000 --> 00:04:59,000
just like before,
so the way we are slicing the

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region is now we are actually
shooting rays straight from the

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00:05:02,000 --> 00:05:04,000
origin.
And, in a given direction,

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00:05:04,000 --> 00:05:08,000
we are asking ourselves how far
does my region go?

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00:05:08,000 --> 00:05:12,000
You have to find a bound and
you have to find whatever the

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00:05:12,000 --> 00:05:15,000
value of r will be out here as a
function of theta.

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00:05:15,000 --> 00:05:19,000
And ways to do that can be
geometric or they can be by

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00:05:19,000 --> 00:05:22,000
starting from the x,
y equation of whatever curve

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00:05:22,000 --> 00:05:26,000
you have and then expressing it
in terms of r and theta and

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00:05:26,000 --> 00:05:28,000
solving for r.
For example,

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00:05:28,000 --> 00:05:33,000
just to illustrate it,
we have seen that one of our

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00:05:33,000 --> 00:05:39,000
classics has been the circle of
radius one centered at one,

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zero.
This guy, you have two

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00:05:42,000 --> 00:05:45,000
different ways of getting its
polar coordinate equation.

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00:05:45,000 --> 00:05:49,000
One is to argue geometrically
that you have a right angle in

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here.
And this length is two,

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00:05:51,000 --> 00:05:54,000
this angle is theta,
this length is r,

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00:05:54,000 --> 00:05:59,000
so the polar equation is r
equals two cosine theta.

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00:05:59,000 --> 00:06:01,000
The other way to do it,
if somehow you are missing the

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00:06:01,000 --> 00:06:03,000
geometric trick,
is to start from the x,

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00:06:03,000 --> 00:06:05,000
y equation.
What is the x,

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00:06:05,000 --> 00:06:10,000
y equation of this guy?
Well, it is x minus one squared

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00:06:10,000 --> 00:06:16,000
plus y squared equals one.
If you expand that you will get

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00:06:16,000 --> 00:06:22,000
x squared minus two x plus one
plus y squared equals one.

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00:06:22,000 --> 00:06:27,000
The ones simplify.
X squared plus y squared

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00:06:27,000 --> 00:06:34,000
becomes r squared minus two x
becomes r cosine theta equals

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00:06:34,000 --> 00:06:36,000
zero.
That gives you,

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00:06:36,000 --> 00:06:40,000
when you simplify by r,
r equals two cosine theta.

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00:06:40,000 --> 00:06:49,000
Two ways to get the same polar
equation.

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00:06:49,000 --> 00:06:53,000
I should say this is an
example, in case you were

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00:06:53,000 --> 00:07:01,000
wondering what I was doing.
We have also actually seen how

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00:07:01,000 --> 00:07:14,000
to change variables to more
complicated coordinate systems.

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00:07:14,000 --> 00:07:16,000
Let's say u, v coordinates.
But, of course,

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00:07:16,000 --> 00:07:20,000
you can call them whatever you
want.

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00:07:20,000 --> 00:07:24,000
The main thing to remember is
that you have to look for the

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00:07:24,000 --> 00:07:30,000
Jacobian which will give you the
conversion ratio between dx dy

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00:07:30,000 --> 00:07:32,000
and du dv.
For example,

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00:07:32,000 --> 00:07:36,000
if you know u and v as
functions of x and y then you

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00:07:36,000 --> 00:07:42,000
will write du dv equals absolute
value of the Jacobian partial u,

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00:07:42,000 --> 00:07:50,000
v over partial x, y times dx dy.
Or, if it is easier for you,

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00:07:50,000 --> 00:07:52,000
you can do the Jacobian the
other way around.

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00:07:52,000 --> 00:07:56,000
And this Jacobian,
remember, is the determinant by

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00:07:56,000 --> 00:08:00,000
a two by two matrix that you
obtain by putting the partial

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00:08:00,000 --> 00:08:03,000
derivatives of u and v with
respect to x and y.

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00:08:03,000 --> 00:08:08,000
Then, when we have that,
we can change the integrant,

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00:08:08,000 --> 00:08:13,000
f of x, y, into something
involving u and v possibly.

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00:08:13,000 --> 00:08:15,000
And then we have to find the
bounds.

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00:08:15,000 --> 00:08:19,000
And to find the bounds perhaps
the easiest is to draw a picture

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00:08:19,000 --> 00:08:21,000
of a region in u,
v coordinates.

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00:08:21,000 --> 00:08:24,000
Maybe you have some picture in
the x,

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00:08:24,000 --> 00:08:28,000
y plane that might actually be
really hard to draw and maybe in

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00:08:28,000 --> 00:08:32,000
terms of u and v the picture
will become much simpler.

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00:08:32,000 --> 00:08:35,000
It might just become a
rectangle.

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00:08:35,000 --> 00:08:37,000
Of course, if you see
immediately what the bounds are

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00:08:37,000 --> 00:08:39,000
in terms of u and v,
and they turn out to be very

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00:08:39,000 --> 00:08:41,000
easy,
then maybe you don't even have

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00:08:41,000 --> 00:08:45,000
to draw this picture.
But if it is not completely

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00:08:45,000 --> 00:08:49,000
obvious then that might be a
helpful way of figuring out what

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00:08:49,000 --> 00:08:52,000
the bounds will be when you
switch from x,

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00:08:52,000 --> 00:08:55,000
y to u, v.
We have seen some problems like

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00:08:55,000 --> 00:09:00,000
that and there are more in the
notes in case you need more.

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00:09:00,000 --> 00:09:07,000
Questions?
Yes?

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00:09:07,000 --> 00:09:10,000
That is the second time you've
asked for something real quick

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00:09:10,000 --> 00:09:13,000
in these review sessions.
You are in a hurry.

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00:09:13,000 --> 00:09:23,000
Take your time.
Partial u, v over partial x,

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00:09:23,000 --> 00:09:26,000
y is just going to be the
determinant of u sub x,

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00:09:26,000 --> 00:09:29,000
u sub y, v sub x,
v sub y.

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00:09:29,000 --> 00:09:37,000
That is the definition.
That is pretty direct.

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00:09:37,000 --> 00:09:39,000
And, of course,
a general common sense thing

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00:09:39,000 --> 00:09:43,000
that applies to actually all the
integrals that we are going to

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00:09:43,000 --> 00:09:45,000
see, there are two things in an
integral.

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00:09:45,000 --> 00:09:49,000
One is whatever you integral is
called the integrant.

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00:09:49,000 --> 00:09:52,000
It could be a function here.
It is a vector field in some of

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00:09:52,000 --> 00:09:55,000
the flux things and so on.
There is another thing which is

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00:09:55,000 --> 00:09:57,000
the region over which you
integrate.

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00:09:57,000 --> 00:10:02,000
And the two have strictly
nothing to do with each other.

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00:10:02,000 --> 00:10:05,000
When you are given a piece of
data in the statement of a

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00:10:05,000 --> 00:10:07,000
problem,
you have to figure out whether

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00:10:07,000 --> 00:10:09,000
that is part of a function to be
integrated or whether that is

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00:10:09,000 --> 00:10:11,000
part of the region of
integration.

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00:10:11,000 --> 00:10:14,000
If it is the region of
integration then it will go into

146
00:10:14,000 --> 00:10:17,000
the bounds of the integral and
maybe in the choice of the

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00:10:17,000 --> 00:10:19,000
coordinate system that you use
for integrating.

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00:10:19,000 --> 00:10:23,000
While the function that you are
integrating goes before the dx

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00:10:23,000 --> 00:10:26,000
dy and not into the bounds or
anything like that.

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00:10:26,000 --> 00:10:32,000
I know it sounds kind of silly
but it is a good safety check.

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00:10:32,000 --> 00:10:34,000
Ask yourselves,
when you have a piece of data,

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00:10:34,000 --> 00:10:36,000
where in my formula should this
go.

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00:10:36,000 --> 00:10:47,000
Yes?
I case you want the bounds for

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00:10:47,000 --> 00:10:50,000
this region in polar
coordinates, indeed it would be

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double integral.
For a fixed theta,

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00:10:53,000 --> 00:10:57,000
r goes from zero to whatever it
is on that curve.

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00:10:57,000 --> 00:11:04,000
So it would be zero to two
cosine theta of whatever the

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00:11:04,000 --> 00:11:10,000
function is r dr d theta.
And the bounds on theta would

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00:11:10,000 --> 00:11:15,000
be from negative pi over two to
pi over two.

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00:11:15,000 --> 00:11:20,000
We have seen that one several
times, so hopefully by now it is

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00:11:20,000 --> 00:11:21,000
clearer.
OK.

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00:11:21,000 --> 00:11:28,000
Let me move on a bit because we
have a lot of other kinds of

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00:11:28,000 --> 00:11:32,000
integrals to see.
Other kinds of integrals we

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00:11:32,000 --> 00:11:35,000
have seen are triple integrals.
And I am not doing things in

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00:11:35,000 --> 00:11:39,000
the order that we did them in
the class just so you can see

166
00:11:39,000 --> 00:11:43,000
parallels between stuff in the
plane and in space.

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00:11:43,000 --> 00:11:47,000
When we do triple integrals in
space, well, it is the same kind

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00:11:47,000 --> 00:11:50,000
of story, except now we have,
of course, more coordinate

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00:11:50,000 --> 00:11:53,000
systems.
We have rectangular

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00:11:53,000 --> 00:11:59,000
coordinates, we have cylindrical
coordinates and we have

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00:11:59,000 --> 00:12:05,000
spherical coordinates.
And cylindrical coordinates

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00:12:05,000 --> 00:12:09,000
only mean that we are,
instead of x, y and z,

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00:12:09,000 --> 00:12:15,000
we are replacing x and y by the
polar coordinate in the x,

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00:12:15,000 --> 00:12:17,000
y plane,
so the angle theta and the

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00:12:17,000 --> 00:12:21,000
distance r.
So R is somehow the distance

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00:12:21,000 --> 00:12:24,000
from the z-axis and z is the
height.

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00:12:24,000 --> 00:12:28,000
Usually you don't have to
choose between rectangular and

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00:12:28,000 --> 00:12:31,000
cylindrical until somewhat late
in the process,

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00:12:31,000 --> 00:12:33,000
especially if you integrate
first of all z,

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00:12:33,000 --> 00:12:36,000
because then the choice will
come up mostly when you try to

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00:12:36,000 --> 00:12:39,000
figure out what are the bounds
for the shadow of your region.

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00:12:39,000 --> 00:12:43,000
I mean the z part looks exactly
the same in rectangular and in

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00:12:43,000 --> 00:12:45,000
cylindrical.
Spherical is,

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00:12:45,000 --> 00:12:49,000
on the other hand,
a little bit more annoying

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00:12:49,000 --> 00:12:52,000
because it looks quite
different.

186
00:12:52,000 --> 00:12:56,000
You should think of it as doing
polar coordinates not only in

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the horizontal direction but
also in the vertical direction

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00:12:59,000 --> 00:13:02,000
at the same time.
You have this angle phi.

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00:13:02,000 --> 00:13:06,000
That measures the angle down
from the positive z-axis.

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00:13:06,000 --> 00:13:10,000
And you have rho which is the
distance from the origin.

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00:13:10,000 --> 00:13:19,000
And if I project to the z-axis,
r becomes rho sine phi and z

192
00:13:19,000 --> 00:13:24,000
becomes rho cosine phi.
I hope that you all know these

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00:13:24,000 --> 00:13:27,000
two formulas,
but if you ever have a small

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00:13:27,000 --> 00:13:31,000
somehow memory lapse during the
final then you should consider

195
00:13:31,000 --> 00:13:35,000
drawing this kind of picture
because it will let you check

196
00:13:35,000 --> 00:13:42,000
very quickly which one is sine,
which one is cosine.

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00:13:42,000 --> 00:13:44,000
Now, of course,
we have to have formulas for dv

198
00:13:44,000 --> 00:13:47,000
in all these coordinate systems.
Here, for example,

199
00:13:47,000 --> 00:13:52,000
that might be dz r dr d theta
or r dr d theta dz or anything

200
00:13:52,000 --> 00:13:58,000
like that.
Here it might be rho squared

201
00:13:58,000 --> 00:14:04,000
times phi times d rho d phi d
theta.

202
00:14:04,000 --> 00:14:07,000
And the general method for
setting up bounds is pretty much

203
00:14:07,000 --> 00:14:10,000
the same as in the plane,
just there is one more step.

204
00:14:10,000 --> 00:14:14,000
If you are doing rectangular or
cylindrical coordinates with z

205
00:14:14,000 --> 00:14:16,000
first, for example,
that is the most common.

206
00:14:16,000 --> 00:14:22,000
Well, if you do z first then
you have to actually start by

207
00:14:22,000 --> 00:14:28,000
figuring out for a given value
of x and y or r and theta what

208
00:14:28,000 --> 00:14:31,000
is the portion of a vertical
line above x,

209
00:14:31,000 --> 00:14:37,000
y that lies within my region?
That will go from z on the

210
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

211
00:14:44,000 --> 00:14:49,000
solid which also usually will
depend on x and y.

212
00:14:49,000 --> 00:14:52,000
And so that will give me the
bounds for dz.

213
00:14:52,000 --> 00:14:56,000
And then I will be left with
the shadow of my region in the

214
00:14:56,000 --> 00:15:02,000
x, y plane.
And that one I will set up like

215
00:15:02,000 --> 00:15:09,000
a double integral over there.
Strictly-speaking,

216
00:15:09,000 --> 00:15:12,000
if you are curious,
we could also change to weird

217
00:15:12,000 --> 00:15:15,000
coordinate systems using
Jacobian with three variables at

218
00:15:15,000 --> 00:15:18,000
the same time.
But we haven't seen that so it

219
00:15:18,000 --> 00:15:24,000
won't be on the final.
But it would work just the same

220
00:15:24,000 --> 00:15:30,000
way, just with more pictures to
do.

221
00:15:30,000 --> 00:15:33,000
And, in fact,
I just wanted to say this rho

222
00:15:33,000 --> 00:15:37,000
squared sine phi is actually the
Jacobian for the change of

223
00:15:37,000 --> 00:15:42,000
variables for rectangular to
spherical coordinates.

224
00:15:42,000 --> 00:15:48,000
OK.
Let's not think too much about

225
00:15:48,000 --> 00:15:51,000
that.
Applications.

226
00:15:51,000 --> 00:15:57,000
Well, we have seen how to use
double integrals to find the

227
00:15:57,000 --> 00:16:03,000
area of a volume of a piece of a
plane or a piece of space,

228
00:16:03,000 --> 00:16:08,000
and find also the mass.
Remember, area is just double

229
00:16:08,000 --> 00:16:12,000
integral of one dA,
volume is triple integral of

230
00:16:12,000 --> 00:16:14,000
one dV.
Sometimes if it's the volume

231
00:16:14,000 --> 00:16:16,000
between the x,
y plane and the graph of some

232
00:16:16,000 --> 00:16:19,000
function, you can just set it up
directly as a double integral.

233
00:16:19,000 --> 00:16:23,000
But there is no harm in doing
it as a triple integral if you

234
00:16:23,000 --> 00:16:27,000
feel better about that.
And mass will be double or

235
00:16:27,000 --> 00:16:31,000
triple integral,
depending on how many

236
00:16:31,000 --> 00:16:36,000
dimensions you have,
of whatever density function

237
00:16:36,000 --> 00:16:45,000
you have, dA or dV.
Then there is how to find the

238
00:16:45,000 --> 00:16:53,000
average value of some function.
Well, let me do the

239
00:16:53,000 --> 00:16:58,000
three-dimensional case.
You will just replace volume by

240
00:16:58,000 --> 00:17:02,000
area and dV by dA and so on,
if need be.

241
00:17:02,000 --> 00:17:09,000
That would be one over volume
of the solid times the triple

242
00:17:09,000 --> 00:17:18,000
integral of f dV,
or if it's a weighted average,

243
00:17:18,000 --> 00:17:24,000
one over mass times the triple
integral of a function times

244
00:17:24,000 --> 00:17:33,000
density times dV.
If you don't have a density or

245
00:17:33,000 --> 00:17:44,000
if the density is constant then
that reduces to that one.

246
00:17:44,000 --> 00:17:46,000
In particular,
we have seen the notion of

247
00:17:46,000 --> 00:17:50,000
center of mass.
The center of mass is just

248
00:17:50,000 --> 00:17:54,000
given by taking the average
values of the coordinates,

249
00:17:54,000 --> 00:17:56,000
x bar, y bar,
z bar.

250
00:17:56,000 --> 00:18:00,000
It is just this formula but
taking x, y or z as the

251
00:18:00,000 --> 00:18:08,000
function.
There are moments of inertia.

252
00:18:08,000 --> 00:18:12,000
For example,
the moment of inertia about the

253
00:18:12,000 --> 00:18:17,000
z-axis is the triple integral of
x squared plus y squared density

254
00:18:17,000 --> 00:18:19,000
dV.
Or, if you have just a

255
00:18:19,000 --> 00:18:22,000
two-dimensional object,
it is the same formula,

256
00:18:22,000 --> 00:18:23,000
but, of course,
with dA.

257
00:18:23,000 --> 00:18:26,000
And then we call that the polar
moment of inertia because we

258
00:18:26,000 --> 00:18:30,000
thought of it as rotating the
plane about the origin,

259
00:18:30,000 --> 00:18:34,000
but the origin is just where
the z-axis hits the x,

260
00:18:34,000 --> 00:18:38,000
y plane so it is really the
same thing.

261
00:18:38,000 --> 00:18:42,000
And we have also seen
gravitational attraction in

262
00:18:42,000 --> 00:18:47,000
space, and I will let you look
at your notes for that.

263
00:18:47,000 --> 00:18:57,000
It is just one formula to
remember.

264
00:18:57,000 --> 00:19:02,000
Questions about iterated
integrals, things like that?

265
00:19:02,000 --> 00:19:18,000
Yes?
The formula that you should

266
00:19:18,000 --> 00:19:22,000
know for gravitational
attraction is that if yu have a

267
00:19:22,000 --> 00:19:28,000
point mass at the origin and you
have some solid centered on the

268
00:19:28,000 --> 00:19:33,000
z-axis that is attracting it
then the force will be given by

269
00:19:33,000 --> 00:19:38,000
G times the mass times the
triple integral of density times

270
00:19:38,000 --> 00:19:42,000
cosine phi over rho squared
times dV.

271
00:19:42,000 --> 00:19:45,000
And, of course,
you will actually do that in

272
00:19:45,000 --> 00:19:50,000
spherical coordinates because it
is easier that way.

273
00:19:50,000 --> 00:19:55,000
That is the formula I have in
mind.

274
00:19:55,000 --> 00:19:59,000
But, see, all these formulas
just give you examples of things

275
00:19:59,000 --> 00:20:01,000
to integrate.
And how to set up the bounds

276
00:20:01,000 --> 00:20:03,000
and so on does not depend on
what you are actually

277
00:20:03,000 --> 00:20:08,000
integrating.
It is done always using the

278
00:20:08,000 --> 00:20:22,000
same methods.
Let's move on to work and line

279
00:20:22,000 --> 00:20:28,000
integrals.
We have seen how to do that in

280
00:20:28,000 --> 00:20:35,000
the plane and in space.
And it looks very similar

281
00:20:35,000 --> 00:20:41,000
somehow.
Remember, you have to know how

282
00:20:41,000 --> 00:20:48,000
to set up and evaluate a line
integral of this form.

283
00:20:48,000 --> 00:20:49,000
Let me do it in the plane this
time.

284
00:20:49,000 --> 00:20:53,000
If you are in the plane you
have two components,

285
00:20:53,000 --> 00:20:58,000
and then this becomes the line
integral of M dx plus N dy.

286
00:20:58,000 --> 00:21:01,000
If you have a space curve then
you will have a third component

287
00:21:01,000 --> 00:21:05,000
here.
You will add that guy times dz.

288
00:21:05,000 --> 00:21:08,000
Now, how do we evaluate that?
Well, it is very different from

289
00:21:08,000 --> 00:21:11,000
there because here we are just
on a curve so there should be

290
00:21:11,000 --> 00:21:15,000
only one degree of freedom.
One variable should be enough

291
00:21:15,000 --> 00:21:21,000
to know where we are.
We will have to express x and y

292
00:21:21,000 --> 00:21:28,000
in terms of -- Well,
I should and z optionally if

293
00:21:28,000 --> 00:21:32,000
there is one,
in terms of a single

294
00:21:32,000 --> 00:21:36,000
parameters.
And that might be just one of

295
00:21:36,000 --> 00:21:38,000
the coordinates.
If you are told y equals z

296
00:21:38,000 --> 00:21:41,000
squared, that is easy.
You just substitute y equals x

297
00:21:41,000 --> 00:21:44,000
squared and dy equals two x dx
into everything,

298
00:21:44,000 --> 00:21:48,000
and you are left with an
integral over x.

299
00:21:48,000 --> 00:22:01,000
Maybe it will be something in
terms of time or in terms of an

300
00:22:01,000 --> 00:22:05,000
angle.
We express everything in terms

301
00:22:05,000 --> 00:22:09,000
of a single parameter,
and that will give us a usual

302
00:22:09,000 --> 00:22:14,000
single integral.
Any questions about that?

303
00:22:14,000 --> 00:22:23,000
Yes?
If you cannot parameterize the

304
00:22:23,000 --> 00:22:27,000
curve then it is really,
really hard to evaluate the

305
00:22:27,000 --> 00:22:31,000
line integral.
Well, you might be able to

306
00:22:31,000 --> 00:22:35,000
evaluate it numerically into a
computer, but that is the

307
00:22:35,000 --> 00:22:40,000
easiest way to describe a curve.
Indeed it could be that in the

308
00:22:40,000 --> 00:22:43,000
plane you have an equation in
terms of x and y given by some

309
00:22:43,000 --> 00:22:46,000
completed formulas defining some
curve.

310
00:22:46,000 --> 00:22:49,000
Then actually there are ways
you can use basically

311
00:22:49,000 --> 00:22:52,000
differentials and constrained
partials to figure out what the

312
00:22:52,000 --> 00:22:55,000
tangent vector to the curve is
and so on.

313
00:22:55,000 --> 00:22:57,000
But we haven't really seen how
to do that.

314
00:22:57,000 --> 00:23:00,000
That would be a really nice
topic for tying together the end

315
00:23:00,000 --> 00:23:02,000
of the second unit that we
discussed last time,

316
00:23:02,000 --> 00:23:04,000
constrained partials,
with this stuff.

317
00:23:04,000 --> 00:23:08,000
But that is not going to be
part of our topics.

318
00:23:08,000 --> 00:23:11,000
Basically, all the curves we
have seen in this class,

319
00:23:11,000 --> 00:23:14,000
there is a way to express the
position of a point in terms of

320
00:23:14,000 --> 00:23:23,000
a parameter.
We haven't seen any curves that

321
00:23:23,000 --> 00:23:34,000
are so complicated that you
cannot do that.

322
00:23:34,000 --> 00:23:37,000
The other thing we have seen is
that there are some special

323
00:23:37,000 --> 00:23:40,000
cases of vector fields where we
don't actually have to compute

324
00:23:40,000 --> 00:23:43,000
this thing because maybe we know
that it is the gradient of some

325
00:23:43,000 --> 00:23:48,000
potential function.
And then we have a fundamental

326
00:23:48,000 --> 00:23:55,000
theorem that gives us a way to
compute this without computing

327
00:23:55,000 --> 00:24:04,000
it.
We've seen about gradient

328
00:24:04,000 --> 00:24:16,000
fields and path independence.
The thing to check is whether

329
00:24:16,000 --> 00:24:18,000
the curl of our vector field is
zero.

330
00:24:18,000 --> 00:24:27,000
And remember in the plane that
is one condition,

331
00:24:27,000 --> 00:24:33,000
Nx equals My.
In 3D in space that is actually

332
00:24:33,000 --> 00:24:39,000
three conditions because you
have to check all the mixed

333
00:24:39,000 --> 00:24:44,000
partials of the various
components.

334
00:24:44,000 --> 00:24:47,000
If the curl of f is zero that
tells us we are likely to have a

335
00:24:47,000 --> 00:24:52,000
gradient field.
Strictly-speaking,

336
00:24:52,000 --> 00:25:06,000
I should mention and F is
defined in a simply-connected

337
00:25:06,000 --> 00:25:16,000
region.
Then F is a gradient field.

338
00:25:16,000 --> 00:25:25,000
That means that we can find a
potential function.

339
00:25:25,000 --> 00:25:30,000
You can write F as the gradient
of little f for some potential

340
00:25:30,000 --> 00:25:39,000
function little f.
And we have seen how to find

341
00:25:39,000 --> 00:25:46,000
the potential.
In fact, we have seen two

342
00:25:46,000 --> 00:25:50,000
methods for that.
And we have seen them twice.

343
00:25:50,000 --> 00:25:53,000
We have seen them once for
functions of two variables,

344
00:25:53,000 --> 00:25:55,000
once for functions of three
variables.

345
00:25:55,000 --> 00:25:58,000
They look very much the same.
I encourage you to compare your

346
00:25:58,000 --> 00:26:03,000
notes for the two side by side
to see where they differ.

347
00:26:03,000 --> 00:26:05,000
Where they differ,
roughly-speaking,

348
00:26:05,000 --> 00:26:09,000
well, I never know if it is the
first or the second,

349
00:26:09,000 --> 00:26:14,000
but one of the two methods was
to compute a line integral.

350
00:26:14,000 --> 00:26:18,000
In the plane,
what we did is we set up and

351
00:26:18,000 --> 00:26:23,000
evaluated a line integral along
our favorite path from the

352
00:26:23,000 --> 00:26:27,000
origin to a point with
coordinates say x1,

353
00:26:27,000 --> 00:26:32,000
y1.
And then we had to evaluate the

354
00:26:32,000 --> 00:26:38,000
line integral for the work done
along this path.

355
00:26:38,000 --> 00:26:43,000
And that will give us the value
of potential at that point.

356
00:26:43,000 --> 00:26:46,000
If we are doing it with three
variables, that method remains

357
00:26:46,000 --> 00:26:49,000
very similar.
The only difference is now we

358
00:26:49,000 --> 00:26:52,000
have to go also up in space to
some point x1,

359
00:26:52,000 --> 00:26:55,000
y1, z1.
And so we actually sum three

360
00:26:55,000 --> 00:26:59,000
pieces together.
But on each piece it is the

361
00:26:59,000 --> 00:27:01,000
same story, only one variable
changes.

362
00:27:01,000 --> 00:27:06,000
Here it is only x that changes,
it is only y that changes,

363
00:27:06,000 --> 00:27:11,000
and on the third one only z
would be changing.

364
00:27:11,000 --> 00:27:18,000
That is one possibility.
And the other possibility for

365
00:27:18,000 --> 00:27:23,000
finding the potential is that we
start with the condition that

366
00:27:23,000 --> 00:27:28,000
the first component of our
vector field should be equal to

367
00:27:28,000 --> 00:27:32,000
f sub x for the unknown
potential function.

368
00:27:32,000 --> 00:27:36,000
What we do is integrate with
respect to x,

369
00:27:36,000 --> 00:27:40,000
and we will get our potential
function up to an integration

370
00:27:40,000 --> 00:27:44,000
constant.
And that integration constant

371
00:27:44,000 --> 00:27:48,000
typically depends on the
remaining variables that might

372
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

373
00:27:53,000 --> 00:27:56,000
take the partial of this with
respect to y and compare it to

374
00:27:56,000 --> 00:28:00,000
what we want it to be,
namely the y component of a

375
00:28:00,000 --> 00:28:02,000
vector field,
and match them to get some

376
00:28:02,000 --> 00:28:08,000
information about this guy.
And if we have three variables

377
00:28:08,000 --> 00:28:13,000
then there is a third step
because there you will still

378
00:28:13,000 --> 00:28:18,000
have an unknown function of z
that you need to get by

379
00:28:18,000 --> 00:28:23,000
comparing the partials with
respect to z.

380
00:28:23,000 --> 00:28:30,000
I see a lot of very quiet faces
somehow.

381
00:28:30,000 --> 00:28:36,000
Well, hopefully that is because
you know that stuff.

382
00:28:36,000 --> 00:28:42,000
If it is because you are
hopelessly confused then please

383
00:28:42,000 --> 00:28:49,000
review a lot before the final,
but I really hope that is not

384
00:28:49,000 --> 00:28:52,000
the case.
And so,

385
00:29:22,000 --> 00:29:26,000
in particular,
what we have seen is once we

386
00:29:26,000 --> 00:29:32,000
have the potential then we can
use the fundamental theorem of

387
00:29:32,000 --> 00:29:38,000
calculus to tell us that if we
have a line integral to compute

388
00:29:38,000 --> 00:29:44,000
for work along a curve that goes
from some point P zero to some

389
00:29:44,000 --> 00:29:49,000
point P one then the line
integral for the work done by

390
00:29:49,000 --> 00:29:54,000
gradient F is actually going
just to be the change in value

391
00:29:54,000 --> 00:29:58,000
of a potential.
And, in particular,

392
00:29:58,000 --> 00:30:02,000
that does not depend on how we
got from P zero to P one.

393
00:30:02,000 --> 00:30:10,000
That is why we say that we have
path independence.

394
00:30:10,000 --> 00:30:22,000
Next topic is flux in plane and
space.

395
00:30:22,000 --> 00:30:26,000
Flux looks quite different in
the plane and in space because,

396
00:30:26,000 --> 00:30:29,000
in the plane,
it is just another kind of line

397
00:30:29,000 --> 00:30:32,000
integral,
while in space it is a surface

398
00:30:32,000 --> 00:30:34,000
integral.
If you were in four-dimensional

399
00:30:34,000 --> 00:30:35,000
space it would be a triple
integral.

400
00:30:35,000 --> 00:30:43,000
Generally, you do flux for
something that is somehow a wall

401
00:30:43,000 --> 00:30:50,000
that separates regions of space
from each other.

402
00:30:50,000 --> 00:30:56,000
In the plane,
the way we do it is we have a

403
00:30:56,000 --> 00:31:02,000
curve C and we look at its
tangent vector,

404
00:31:02,000 --> 00:31:09,000
let's call that T,
and we rotate it by 90 degrees

405
00:31:09,000 --> 00:31:13,000
clockwise.
That is our convention to get a

406
00:31:13,000 --> 00:31:18,000
unit normal vector that points
to the right of the curve as we

407
00:31:18,000 --> 00:31:21,000
move along the curve.
That is our convention for

408
00:31:21,000 --> 00:31:24,000
orienting curves.
And we are always going to be

409
00:31:24,000 --> 00:31:33,000
using that one.
N equals T rotated 90 degrees

410
00:31:33,000 --> 00:31:37,000
clockwise.
In particular,

411
00:31:37,000 --> 00:31:44,000
that means that n ds,
which will be what we integrate

412
00:31:44,000 --> 00:31:52,000
against when we try to compute
flux, will just end up being dy,

413
00:31:52,000 --> 00:31:58,000
negative dx.
Concretely, when we have to

414
00:31:58,000 --> 00:32:03,000
evaluate a line integral of F
dot n ds,

415
00:32:03,000 --> 00:32:07,000
geometrically we could try to
take the dot product of our

416
00:32:07,000 --> 00:32:11,000
field with the normal vector and
then sum the length element

417
00:32:11,000 --> 00:32:14,000
along the curve.
And, in some cases,

418
00:32:14,000 --> 00:32:16,000
for example,
if you know that the vector

419
00:32:16,000 --> 00:32:19,000
field is tangent to the curve or
if a dot product is constant or

420
00:32:19,000 --> 00:32:22,000
things like that then that might
actually give you a very easy

421
00:32:22,000 --> 00:32:24,000
answer.
But, in general,

422
00:32:24,000 --> 00:32:28,000
the most efficient way to do it
will be to say that if your

423
00:32:28,000 --> 00:32:31,000
vector field has components,
I don't know,

424
00:32:31,000 --> 00:32:36,000
let's call them P and Q,
then that will be just the line

425
00:32:36,000 --> 00:32:39,000
integral of PQ dot dy,
negative dx,

426
00:32:39,000 --> 00:32:44,000
which means negative Q dx plus
P dy.

427
00:32:44,000 --> 00:32:49,000
And, from that point onward,
you evaluate it exactly the

428
00:32:49,000 --> 00:32:54,000
same way as you would for a work
integral.

429
00:32:54,000 --> 00:32:56,000
But, of course,
the geometric meaning is very

430
00:32:56,000 --> 00:32:57,000
different.
It is the same meaning that we

431
00:32:57,000 --> 00:33:08,000
have always seen for flux.
It measures how much a vector

432
00:33:08,000 --> 00:33:22,000
field goes across the curve.
Now, if we are in space then

433
00:33:22,000 --> 00:33:30,000
you take flux for a surface,
not for a curve.

434
00:33:30,000 --> 00:33:33,000
And the way it will work is
that you have to choose an

435
00:33:33,000 --> 00:33:37,000
orientation of a surface,
which just means choosing one

436
00:33:37,000 --> 00:33:40,000
of the two possible unit normal
vectors.

437
00:33:40,000 --> 00:33:51,000
And then you will do a surface
integral for F dot n dS.

438
00:33:51,000 --> 00:34:00,000
That is the surface i element.
The setup for this surface

439
00:34:00,000 --> 00:34:07,000
integral is that first we have
to express n and dS in some way.

440
00:34:07,000 --> 00:34:14,000
One possibility is that we can
express the normal vector n dS

441
00:34:14,000 --> 00:34:18,000
geometrically.
That is, for example,

442
00:34:18,000 --> 00:34:23,000
what we do when we look at,
say, a horizontal plane or a

443
00:34:23,000 --> 00:34:27,000
vertical plane or a sphere or a
cylinder.

444
00:34:27,000 --> 00:34:31,000
Then we have some geometric
idea of why the normal vector is

445
00:34:31,000 --> 00:34:34,000
what it is and we have some
formula for dS.

446
00:34:34,000 --> 00:34:39,000
Or, we can use one of the
standard formulas.

447
00:34:39,000 --> 00:34:43,000
Basically, we have seen two
formulas that work in fairly

448
00:34:43,000 --> 00:34:50,000
generate situations.
One of them says -- If S is

449
00:34:50,000 --> 00:35:00,000
given by an equation z equals
some function of x,

450
00:35:00,000 --> 00:35:06,000
y then you can just say n dS
equals minus f sub x,

451
00:35:06,000 --> 00:35:09,000
minus f sub y,
one, dx dy.

452
00:35:09,000 --> 00:35:14,000
And I need to rewrite that
because I am running out of

453
00:35:14,000 --> 00:35:16,000
space.
But, while I erase,

454
00:35:16,000 --> 00:35:20,000
I would like to point out the
most important there in here.

455
00:35:20,000 --> 00:35:24,000
When I say n dS equals blah,
blah, blah times dx dy,

456
00:35:24,000 --> 00:35:28,000
dx dy is not the same thing as
dS at all.

457
00:35:28,000 --> 00:35:31,000
If you make that mistake you
are going to get into trouble

458
00:35:31,000 --> 00:35:34,000
the next time that you try to
buy real estate in a region

459
00:35:34,000 --> 00:35:37,000
which hills or cliffs or things
like that.

460
00:35:37,000 --> 00:35:40,000
dS is the area on the slanted
surface.

461
00:35:40,000 --> 00:35:46,000
dx dy is the area on the map
that shows the x,

462
00:35:46,000 --> 00:35:49,000
y plane.
And these are not the same

463
00:35:49,000 --> 00:35:51,000
thing.
In particular,

464
00:35:51,000 --> 00:36:00,000
you cannot just take one piece
of it and not the other piece.

465
00:36:00,000 --> 00:36:06,000
Let me give you formulas for n
and for dS separately just to

466
00:36:06,000 --> 00:36:14,000
convince you.
That way, if you feel that you

467
00:36:14,000 --> 00:36:22,000
need them, then you will have
them.

468
00:36:22,000 --> 00:36:26,000
N is minus f sub x,
minus f sub y,

469
00:36:26,000 --> 00:36:31,000
one, but scaled down to unit
length.

470
00:36:31,000 --> 00:36:35,000
This is not a unit vector.
It is actually divided by the

471
00:36:35,000 --> 00:36:38,000
length of this guy which is fx
squared plus fy squared plus

472
00:36:38,000 --> 00:36:46,000
one.
And dS is that same vector

473
00:36:46,000 --> 00:36:52,000
times dx dy.
And so the square roots cancel

474
00:36:52,000 --> 00:36:56,000
out when you multiply them
together.

475
00:36:56,000 --> 00:37:00,000
But it would be completely
wrong to just say I will replace

476
00:37:00,000 --> 00:37:03,000
n dS by minus f sub x,
minus f sub y and one.

477
00:37:03,000 --> 00:37:07,000
Then I end up again with the dS
and I do something else with dS.

478
00:37:07,000 --> 00:37:16,000
That is a pretty bad conceptual
mistake because it gives you the

479
00:37:16,000 --> 00:37:20,000
wrong answer.
Another option more general

480
00:37:20,000 --> 00:37:24,000
than that.
If we have not seen how to

481
00:37:24,000 --> 00:37:32,000
solve for z, how to express z as
a function of x and y,

482
00:37:32,000 --> 00:37:40,000
well, maybe we still know some
normal vector to the surface.

483
00:37:40,000 --> 00:37:49,000
Then there is another formula
for n dS which is up to sine,

484
00:37:49,000 --> 00:37:55,000
N divided by N dot k dx dy.
And, see, that projection

485
00:37:55,000 --> 00:37:58,000
formula works also if you have
to project to another coordinate

486
00:37:58,000 --> 00:37:59,000
plane.
For example,

487
00:37:59,000 --> 00:38:02,000
if you want to project to the
x, z coordinate plane,

488
00:38:02,000 --> 00:38:10,000
the relation between n dS and
dx dz is given by N over N dot

489
00:38:10,000 --> 00:38:13,000
j,
because j is the direction

490
00:38:13,000 --> 00:38:19,000
perpendicular to the xz plane.
But this one is more useful.

491
00:38:19,000 --> 00:38:24,000
What is a good example of that?
If you have a slanted plane

492
00:38:24,000 --> 00:38:28,000
given to you,
you can easily find its normal

493
00:38:28,000 --> 00:38:31,000
vector.
That is just given by the

494
00:38:31,000 --> 00:38:33,000
coefficients of x,
y, z in the equation.

495
00:38:33,000 --> 00:38:38,000
Another situation where that
might happen is if your surface

496
00:38:38,000 --> 00:38:41,000
is given by an equation of a
form of g of x,

497
00:38:41,000 --> 00:38:44,000
y, z equals zero.
If that is the case then you

498
00:38:44,000 --> 00:38:47,000
know this is a level set of g.
And we know how to find a

499
00:38:47,000 --> 00:38:50,000
normal vector to the level set,
namely the gradient vector is

500
00:38:50,000 --> 00:38:52,000
always perpendicular to the
level set.

501
00:38:52,000 --> 00:38:59,000
You would take the gradient of
g to be your big N.

502
00:38:59,000 --> 00:39:06,000
OK.
Now, these are basically all

503
00:39:06,000 --> 00:39:10,000
the integrals we have seen how
to set up.

504
00:39:10,000 --> 00:39:15,000
Now we have a bunch of theorems
relating them.

505
00:39:15,000 --> 00:39:23,000
Let me think about how I am
going to organize that.

506
00:39:23,000 --> 00:39:32,000
Let me try like this.
This part of the board will be

507
00:39:32,000 --> 00:39:35,000
work,
this part of the board will be

508
00:39:35,000 --> 00:39:40,000
about flux and the left part of
the board will be about things

509
00:39:40,000 --> 00:39:46,000
in the plane and the right one
will be about things in space.

510
00:39:46,000 --> 00:39:53,000
What have we seen?
Well, we have seen Green's

511
00:39:53,000 --> 00:39:57,000
theorem for work.
That doesn't work so well

512
00:39:57,000 --> 00:39:59,000
because that is too small,
so I am going to actually use

513
00:39:59,000 --> 00:40:01,000
more blackboards to do that.

514
00:40:26,000 --> 00:40:32,000
This side will be space,
this side will be the plane and

515
00:40:32,000 --> 00:40:37,000
we are going to start with
theorems about work.

516
00:40:37,000 --> 00:40:41,000
And we will see theorems about
flux pretty soon.

517
00:40:41,000 --> 00:40:47,000
We have two theorems about work.
In the plane that is called

518
00:40:47,000 --> 00:40:51,000
Green's theorem.
In space that is called Stokes'

519
00:40:51,000 --> 00:40:56,000
theorem.
Green's theorem says if I have

520
00:40:56,000 --> 00:41:03,000
a closed curve in the plane
going counterclockwise enclosing

521
00:41:03,000 --> 00:41:11,000
entirely some region R then the
line integral along C for the

522
00:41:11,000 --> 00:41:20,000
work of F is equal to the double
integral of a region inside of

523
00:41:20,000 --> 00:41:27,000
the curl of F dA.
Concretely, if my components of

524
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

525
00:41:35,000 --> 00:41:43,000
is equal to the double integral
of R of N sub x minus M sub y

526
00:41:43,000 --> 00:41:46,000
dA.
This side here is a usual line

527
00:41:46,000 --> 00:41:49,000
integral.
This side here is a usual

528
00:41:49,000 --> 00:41:53,000
double integral in the plane.
And somehow their values end up

529
00:41:53,000 --> 00:41:58,000
being magically related.
Well, not quite magically.

530
00:41:58,000 --> 00:42:03,000
We actually have seen how to
prove it.

531
00:42:03,000 --> 00:42:10,000
And now the analog of that in
space is Stokes' theorem.

532
00:42:10,000 --> 00:42:15,000
Stokes says if I have a closed
curve in space,

533
00:42:15,000 --> 00:42:20,000
now I have to decide what kind
of thing it bounds.

534
00:42:20,000 --> 00:42:23,000
And the answer is it will have
to bound some surface,

535
00:42:23,000 --> 00:42:28,000
but I have a choice of surface.
I choose my favorite surface

536
00:42:28,000 --> 00:42:31,000
bounded by C.
I guess I will just draw it

537
00:42:31,000 --> 00:42:33,000
like that.
And I have to choose a

538
00:42:33,000 --> 00:42:36,000
compatible orientation.
Remember, we have seen this

539
00:42:36,000 --> 00:42:39,000
right hand rule for choosing how
to orient the surface.

540
00:42:39,000 --> 00:42:43,000
I believe, in this case,
if I take C like that then the

541
00:42:43,000 --> 00:42:49,000
normal vector has to go up.
And then it tells me how to

542
00:42:49,000 --> 00:42:53,000
compute the work done by F along
C.

543
00:42:53,000 --> 00:43:01,000
Namely, that becomes the double
integral over that surface S of

544
00:43:01,000 --> 00:43:08,000
curl F, which I will write as
dell cross F dot n dS.

545
00:43:08,000 --> 00:43:10,000
This line integral is a usual
line integral,

546
00:43:10,000 --> 00:43:14,000
but if for some reason we don't
want to compute it directly we

547
00:43:14,000 --> 00:43:18,000
can actually replace it by a
surface integral over any

548
00:43:18,000 --> 00:43:22,000
surface bounded by the curve.
It might be that a problem will

549
00:43:22,000 --> 00:43:23,000
tell you which surface you have
to consider.

550
00:43:23,000 --> 00:43:28,000
It might be that you will be
left to choose the simplest

551
00:43:28,000 --> 00:43:33,000
possible surface you can think
of that is somehow having this

552
00:43:33,000 --> 00:43:36,000
curve as its boundary.
And so now, remember,

553
00:43:36,000 --> 00:43:40,000
curl of a vector field in space
is going to be another vector

554
00:43:40,000 --> 00:43:42,000
expression.
It has three components.

555
00:43:42,000 --> 00:43:45,000
And the way you compute it is
not by remembering the actual

556
00:43:45,000 --> 00:43:48,000
formula,
which is really complicated by,

557
00:43:48,000 --> 00:43:54,000
but by instead computing the
cross-product between dell and

558
00:43:54,000 --> 00:44:01,000
F.
You set up the cross-product.

559
00:44:01,000 --> 00:44:03,000
And, of course,
it is a highly symbolic

560
00:44:03,000 --> 00:44:07,000
cross-product.
I mean it is not an

561
00:44:07,000 --> 00:44:16,000
cross-product of actual vectors
but it works the same way.

562
00:44:16,000 --> 00:44:20,000
Both of these formulas
basically relate work on a curve

563
00:44:20,000 --> 00:44:24,000
with what happens to the curl on
the surface that is enclosed by

564
00:44:24,000 --> 00:44:27,000
this curve,
that is bounded by this curve.

565
00:44:27,000 --> 00:44:30,000
And in this one you have less
freedom of choice because you

566
00:44:30,000 --> 00:44:34,000
don't have somehow a z direction
in which you could move your

567
00:44:34,000 --> 00:44:36,000
surface.
There is only possible choice

568
00:44:36,000 --> 00:44:39,000
of surface.
There is only one thing that is

569
00:44:39,000 --> 00:44:42,000
enclosed by this curve in the
plane.

570
00:44:42,000 --> 00:44:45,000
In both cases,
these things tell you that you

571
00:44:45,000 --> 00:44:48,000
can think of curl as measuring
how much the field fails to be

572
00:44:48,000 --> 00:44:51,000
conservative.
See, if your field was

573
00:44:51,000 --> 00:44:55,000
conservative -- If a curl was
zero then the right-hand side

574
00:44:55,000 --> 00:44:57,000
would just be zero.
And that would be fortunate

575
00:44:57,000 --> 00:45:01,000
because if a curl is zero then
your field is less conservative.

576
00:45:01,000 --> 00:45:02,000
That means it comes from a
potential.

577
00:45:02,000 --> 00:45:05,000
That means when you go along a
closed curve,

578
00:45:05,000 --> 00:45:09,000
well, the change of value of a
potential should be zero.

579
00:45:09,000 --> 00:45:13,000
Another way to say it is path
independence tells you no work.

580
00:45:13,000 --> 00:45:16,000
And, of course,
if you have a vector field that

581
00:45:16,000 --> 00:45:20,000
is not a gradient field then the
curl is not necessarily zero and

582
00:45:20,000 --> 00:45:23,000
then you get a more interesting
answer.

583
00:45:23,000 --> 00:45:27,000
Finally, let's move onto the
theorems about flux.

584
00:45:27,000 --> 00:45:43,000
That is Green for flux and that
is the divergence theorem.

585
00:45:43,000 --> 00:45:54,000
Flux theorems.
Here I say that will be

586
00:45:54,000 --> 00:46:01,000
divergence.
And here it will be Green again.

587
00:46:01,000 --> 00:46:07,000
Green's theorem for flux says I
have a closed curve that goes

588
00:46:07,000 --> 00:46:10,000
counterclockwise around some
region.

589
00:46:10,000 --> 00:46:13,000
In particular,
counterclockwise means that the

590
00:46:13,000 --> 00:46:16,000
normal vector will be going out
of the region.

591
00:46:16,000 --> 00:46:20,000
And then it tells us that the
flux out of the region,

592
00:46:20,000 --> 00:46:24,000
through the curve C,
so that will be the line

593
00:46:24,000 --> 00:46:30,000
integral of F dot n ds is equal
to the integral of a region

594
00:46:30,000 --> 00:46:37,000
inside of div F dA.
And remember the divergence of

595
00:46:37,000 --> 00:46:42,000
M, N is just Mx plus Ny.
This one here,

596
00:46:42,000 --> 00:46:46,000
the divergence theorem,
tells you something similar but

597
00:46:46,000 --> 00:46:49,000
now for a region of space
bounded by a closed surface.

598
00:46:49,000 --> 00:46:55,000
So if you have some region of
space and you call its boundary

599
00:46:55,000 --> 00:47:00,000
surface S and you let n be the
normal vector that goes out of

600
00:47:00,000 --> 00:47:07,000
the region R.
You orient S outwards.

601
00:47:07,000 --> 00:47:15,000
Then the flux out of the region
through S is going to be the

602
00:47:15,000 --> 00:47:23,000
same as the triple integral over
the region of divergence F dV.

603
00:47:23,000 --> 00:47:31,000
Remember, the divergence of a
vector field with components P,

604
00:47:31,000 --> 00:47:36,000
Q, R is Px plus Qy plus Rz.
What do these two theorems say?

605
00:47:36,000 --> 00:47:39,000
Well, they say essentially the
same thing.

606
00:47:39,000 --> 00:47:43,000
They say the total flux out of
a region is equal to the

607
00:47:43,000 --> 00:47:46,000
integral of divergence over
whatever is inside.

608
00:47:46,000 --> 00:47:49,000
And the reason for that is,
again, we have seen for a

609
00:47:49,000 --> 00:47:52,000
velocity field that divergence
measures how much things are

610
00:47:52,000 --> 00:47:55,000
expanding or how much stuff is
being created.

611
00:47:55,000 --> 00:47:59,000
It tells you the amount of
sources per unit portion of the

612
00:47:59,000 --> 00:48:01,000
region.
When you sum that over

613
00:48:01,000 --> 00:48:03,000
everything,
you get the amount of fluid

614
00:48:03,000 --> 00:48:05,000
that is being,
you know, the total amount of

615
00:48:05,000 --> 00:48:08,000
sources inside here,
and that tells us how much

616
00:48:08,000 --> 00:48:10,000
stuff has to go out per unit
time.

617
00:48:10,000 --> 00:48:12,000
That is basically the
interpretation.

618
00:48:12,000 --> 00:48:16,000
In a way, I would be tempted to
say that this table of four

619
00:48:16,000 --> 00:48:20,000
theorems is somehow the crucial
point of 18.02.

620
00:48:20,000 --> 00:48:23,000
And you would do well to
remember them.

621
00:48:23,000 --> 00:48:26,000
However, I would like also to
point out that these theorems

622
00:48:26,000 --> 00:48:28,000
are completely useless if you
don't know how to compute any of

623
00:48:28,000 --> 00:48:31,000
the integrals that are in there.
So all the stuff that was

624
00:48:31,000 --> 00:48:35,000
around there before is actually
somehow more fundamental.

625
00:48:35,000 --> 00:48:38,000
And if you don't know how to
compute the double or triple

626
00:48:38,000 --> 00:48:41,000
integrals then this is of little
use to you.

627
00:48:41,000 --> 00:48:48,000
That is the end.
I guess I have to wish you

628
00:48:48,000 --> 00:48:52,000
happy holidays.