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OK, so far we've learned how to
do double integrals in terms of

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xy coordinates,
also how to switch to polar

00:00:32.000 --> 00:00:35.000
coordinates.
But, more generally,

00:00:35.000 --> 00:00:40.000
there's a lot of different
changes of variables that you

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might want to do.
OK, so today we're going to see

00:00:44.000 --> 00:00:48.000
how to change variables,
if you want,

00:00:48.000 --> 00:00:52.000
how to do substitutions in
double integrals.

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OK, so let me start with a
simple example.

00:01:10.000 --> 00:01:21.000
Let's say that we want to find
the area of an ellipse with

00:01:21.000 --> 00:01:28.000
semi-axes a and b.
OK, so that means an ellipse is

00:01:28.000 --> 00:01:38.000
just like a squished circle.
And so, there's a and there's b.

00:01:38.000 --> 00:01:44.000
And, the equation of that
ellipse is x over a squared plus

00:01:44.000 --> 00:01:49.000
y over b squared equals one.
That's the curve,

00:01:49.000 --> 00:01:53.000
and the inside region is where
this is less than one.

00:01:53.000 --> 00:01:58.000
OK, so it's just like a circle
that where you have rescaled x

00:01:58.000 --> 00:02:01.000
and y differently.
So, let's say we want to find

00:02:01.000 --> 00:02:03.000
the area of it.
Maybe you know what the area is.

00:02:03.000 --> 00:02:11.000
But let's do it as a double
integral.

00:02:11.000 --> 00:02:14.000
So, you know,
if you find that the area is

00:02:14.000 --> 00:02:19.000
too easy, you can integrate any
function other than ellipse,

00:02:19.000 --> 00:02:23.000
if you prefer.
But, let's do it just with area.

00:02:23.000 --> 00:02:27.000
So, we know that we want to
integrate just the area element,

00:02:27.000 --> 00:02:30.000
let's say, dx dy over the
origin inside the ellipse.

00:02:30.000 --> 00:02:37.000
That's x over a2 plus y over b2
less than 1.

00:02:37.000 --> 00:02:41.000
Now, we can try to set this up
in terms of x and y coordinates,

00:02:41.000 --> 00:02:46.000
you know, set up the bounds by
solving for first four x as a

00:02:46.000 --> 00:02:49.000
function of y if we do it this
order and,

00:02:49.000 --> 00:02:52.000
well, do the usual stuff.
That doesn't look very

00:02:52.000 --> 00:02:55.000
pleasant, and it's certainly not
the best way to do it.

00:02:55.000 --> 00:02:57.000
OK, if this were a circle,
we would switch to polar

00:02:57.000 --> 00:02:59.000
coordinates.
Well, we can't quite do that

00:02:59.000 --> 00:03:00.000
yet.
But, you know,

00:03:00.000 --> 00:03:01.000
an ellipse is just a squished
circle.

00:03:01.000 --> 00:03:08.000
So, maybe we want to actually
first rescale x and y by a and

00:03:08.000 --> 00:03:11.000
b.
So, to do that,

00:03:11.000 --> 00:03:19.000
what we'd like to do is set x
over a to be u,

00:03:19.000 --> 00:03:24.000
and y over b be v.
So, we'll have two new

00:03:24.000 --> 00:03:28.000
variables, u and v,
and we'll try to redo our

00:03:28.000 --> 00:03:32.000
integral in terms of u and v.
So, how do we do the

00:03:32.000 --> 00:03:36.000
substitution?
So, in terms of u and v,

00:03:36.000 --> 00:03:39.000
the condition,
the region that we are

00:03:39.000 --> 00:03:43.000
integrating on will become u^2
v^2 is less than 1,

00:03:43.000 --> 00:03:45.000
which is arguably nicer than
the ellipse.

00:03:45.000 --> 00:03:50.000
That's why we are doing it.
But, we need to know what to do

00:03:50.000 --> 00:03:53.000
with dx and dy.
Well, here, the answer is

00:03:53.000 --> 00:03:56.000
pretty easy because we just
change x and y separately.

00:03:56.000 --> 00:03:59.000
We do two independent
substitutions.

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OK, so if we set u equals x
over a, that means du is one

00:04:10.000 --> 00:04:18.000
over adx.
And here, dv is one over bdy.

00:04:18.000 --> 00:04:26.000
So, it's very tempting to
write, and here we actually can

00:04:26.000 --> 00:04:34.000
write, in this particular case,
that dudv is (1/ab)dxdy,

00:04:34.000 --> 00:04:42.000
OK?
So, let me rewrite that.

00:04:42.000 --> 00:04:54.000
OK, so I get dudv equals
(1/ab)dxdy, or equivalently dxdy

00:04:54.000 --> 00:05:05.000
is ab times dudv.
OK, so in my double integral,

00:05:05.000 --> 00:05:15.000
I'm going to write (ab)dudv.
OK, so now, my double integral

00:05:15.000 --> 00:05:18.000
becomes, well,
the double integral of a

00:05:18.000 --> 00:05:23.000
constant in terms of u and v.
So, I can take the constant out.

00:05:23.000 --> 00:05:31.000
I will get ab times double
integral over u^2 v^2<1 of du

00:05:31.000 --> 00:05:34.000
dv.
And, that is an integral that

00:05:34.000 --> 00:05:37.000
we know how to do.
Well, it's just the area of a

00:05:37.000 --> 00:05:40.000
unit circle.
So, we can just say,

00:05:40.000 --> 00:05:50.000
this is ab times the area of
the unit disk,

00:05:50.000 --> 00:05:54.000
which we know to be pi,
or if somehow you had some

00:05:54.000 --> 00:05:57.000
function to integrate,
then you could have somehow

00:05:57.000 --> 00:05:59.000
switched to polar coordinates,
you know, setting u equals r

00:05:59.000 --> 00:06:02.000
times cos(theta),
v equals r times sin(theta),

00:06:02.000 --> 00:06:07.000
and then doing it in polar
coordinates.

00:06:07.000 --> 00:06:11.000
OK, so here the substitution
worked pretty easy.

00:06:11.000 --> 00:06:14.000
The question is,
if we do a change of variables

00:06:14.000 --> 00:06:18.000
where the relation between x and
y and u and v is more

00:06:18.000 --> 00:06:20.000
complicated, what can we do?
Can we still do this,

00:06:20.000 --> 00:06:22.000
or do we have to be more
careful?

00:06:22.000 --> 00:06:23.000
And, actually,
we have to be more careful.

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So, that's what we are going to
see next.

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Any question about this, first?
No?

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OK.
OK, so, see the general problem

00:06:38.000 --> 00:06:41.000
when we try to do this is to
figure out what is the scale

00:06:41.000 --> 00:06:48.000
factor?
What's the relation between

00:06:48.000 --> 00:06:57.000
dxdy and dudv?
We need to find the scaling

00:06:57.000 --> 00:07:07.000
factor.
So, we need to find dxdy versus

00:07:07.000 --> 00:07:12.000
dudv.
So, let's do another example

00:07:12.000 --> 00:07:18.000
that's still pretty easy,
but a little bit less easy.

00:07:18.000 --> 00:07:24.000
OK, so let's say that for some
reason, we want to do the change

00:07:24.000 --> 00:07:27.000
of variables:
u equals 3x-2y,

00:07:27.000 --> 00:07:31.000
and v equals x y.
Why would we want to do that?

00:07:31.000 --> 00:07:34.000
Well, that might be to simplify
the integrand because we are

00:07:34.000 --> 00:07:38.000
integrating a function that
happens to be actually involving

00:07:38.000 --> 00:07:42.000
these guys rather than x and y.
Or, it might be to simplify the

00:07:42.000 --> 00:07:45.000
bounds because maybe we are
integrating over a region whose

00:07:45.000 --> 00:07:49.000
equation in xy coordinates is
very hard to write down.

00:07:49.000 --> 00:07:51.000
But, it becomes much easier in
terms of u and v.

00:07:51.000 --> 00:07:57.000
And then, the bounds would be
much easier to set up with u and

00:07:57.000 --> 00:08:02.000
v.
Anyway, so, whatever the reason

00:08:02.000 --> 00:08:12.000
might be, typically it would be
to simplify the integrant or the

00:08:12.000 --> 00:08:18.000
bounds.
Well, how do we convert dxdy to

00:08:18.000 --> 00:08:21.000
dudv?
So, we want to understand,

00:08:21.000 --> 00:08:27.000
what's the relation between,
let's call dA the area element

00:08:27.000 --> 00:08:31.000
in xy coordinates.
So, dA is dxdy,

00:08:31.000 --> 00:08:34.000
maybe dydx depending on the
order.

00:08:34.000 --> 00:08:39.000
And, the area element in uv
coordinates, let me call that dA

00:08:39.000 --> 00:08:42.000
prime just to make it look
different.

00:08:42.000 --> 00:08:50.000
So, that would just be dudv,
or dvdu depending on which

00:08:50.000 --> 00:08:55.000
order I will want to set it up
in.

00:08:55.000 --> 00:09:01.000
So, to find this relation,
it's probably best to draw a

00:09:01.000 --> 00:09:09.000
picture to see what happens.
Let's consider a small piece of

00:09:09.000 --> 00:09:18.000
the xy plane with area delta(A)
corresponding to just a box with

00:09:18.000 --> 00:09:24.000
sides delta(y) and delta(x).
OK, and let's try to figure out

00:09:24.000 --> 00:09:27.000
what it will look like in terms
of u and v.

00:09:27.000 --> 00:09:29.000
And then, we'll say,
well, when we integrate,

00:09:29.000 --> 00:09:32.000
we're really summing the value
of the function of a lot of

00:09:32.000 --> 00:09:36.000
small boxes times their area.
But, the problem is that the

00:09:36.000 --> 00:09:40.000
area of the box in here is not
the same as the area of the box

00:09:40.000 --> 00:09:47.000
in uv coordinates.
There, maybe it will look like,

00:09:47.000 --> 00:09:49.000
actually,
if you see that these are

00:09:49.000 --> 00:09:52.000
linear changes of variables,
you know that the rectangle

00:09:52.000 --> 00:09:55.000
will become a parallelogram
after the change of variables.

00:09:55.000 --> 00:10:00.000
So, the area of a parallelogram
delta(A) prime,

00:10:00.000 --> 00:10:05.000
well, we will have to figure
out how they are related so that

00:10:05.000 --> 00:10:09.000
we can decide what conversion
factor,

00:10:09.000 --> 00:10:13.000
what's the exchange rate
between these two currencies for

00:10:13.000 --> 00:10:20.000
area?
OK, any questions at this point?

00:10:20.000 --> 00:10:27.000
No? Still with me mostly?
I see a lot of tired faces.

00:10:27.000 --> 00:10:34.000
Yes?
Why is delta(A) prime a

00:10:34.000 --> 00:10:37.000
parallelogram?
That's a very good question.

00:10:37.000 --> 00:10:41.000
Well, see, if I look at the
side of a rectangle,

00:10:41.000 --> 00:10:45.000
say there's a vertical side,
it means I'm going to increase

00:10:45.000 --> 00:10:49.000
y, keeping x the same.
If I look at the formulas for u

00:10:49.000 --> 00:10:52.000
and v, they are linear formulas
in terms of x and y.

00:10:52.000 --> 00:10:56.000
So, if I just increase y,
see that u is going to decrease

00:10:56.000 --> 00:10:58.000
at a rate of two.
v is going to increase at a

00:10:58.000 --> 00:11:02.000
rate of one at constant rates.
And, it doesn't matter whether

00:11:02.000 --> 00:11:04.000
I was looking at this site or at
that site.

00:11:04.000 --> 00:11:06.000
So, basically straight lines
become straight lines.

00:11:06.000 --> 00:11:09.000
And if they are parallel,
they stay parallel.

00:11:09.000 --> 00:11:11.000
So, if you just look at what
the transformation,

00:11:11.000 --> 00:11:14.000
from xy to uv does,
it does this kind of thing.

00:11:14.000 --> 00:11:17.000
Actually, this transformation
here you can express by a

00:11:17.000 --> 00:11:18.000
matrix.
And, remember,

00:11:18.000 --> 00:11:20.000
we've seen what matrices do the
pictures.

00:11:20.000 --> 00:11:24.000
We just take straight lines to
straight lines.

00:11:24.000 --> 00:11:29.000
They keep the notion of being
parallel, but of course they

00:11:29.000 --> 00:11:32.000
mess up lengths,
angles, and all that.

00:11:32.000 --> 00:11:38.000
OK, so let's see.
So, let's try to figure out,

00:11:38.000 --> 00:11:42.000
what is the area of this guy?
Well, in fact,

00:11:42.000 --> 00:11:46.000
what I've been saying about
this transformation being

00:11:46.000 --> 00:11:49.000
linear,
and transforming all of the

00:11:49.000 --> 00:11:53.000
vertical lines in the same way,
all the horizontal lines in the

00:11:53.000 --> 00:11:54.000
same way,
it tells me,

00:11:54.000 --> 00:11:57.000
also, I should have a constant
scaling factor,

00:11:57.000 --> 00:12:00.000
right, because how much I've
scaled my rectangle doesn't

00:12:00.000 --> 00:12:03.000
depend on where my rectangle is.
If I move my rectangle to

00:12:03.000 --> 00:12:05.000
somewhere else,
I have a rectangle of the same

00:12:05.000 --> 00:12:08.000
size, same shape,
it will become a parallelogram

00:12:08.000 --> 00:12:10.000
of the same size,
same shape somewhere else.

00:12:10.000 --> 00:12:13.000
So, in fact,
I can just take the simplest

00:12:13.000 --> 00:12:16.000
rectangle I can think of and see
how its area changes.

00:12:16.000 --> 00:12:18.000
And, if you don't believe me,
then try with any other

00:12:18.000 --> 00:12:21.000
rectangle.
You will see it works exactly

00:12:21.000 --> 00:12:28.000
the same way.
OK, so I claim that the area

00:12:28.000 --> 00:12:41.000
scaling factor -- -- here in
this case doesn't depend on the

00:12:41.000 --> 00:12:53.000
choice of the rectangle.
And I should say that because

00:12:53.000 --> 00:13:05.000
we are actually doing a linear
change of variables -- So,

00:13:05.000 --> 00:13:08.000
you know, somehow,
the exchange rate between uv

00:13:08.000 --> 00:13:10.000
and xy is going to be the same
everywhere.

00:13:10.000 --> 00:13:14.000
So, let's try to see what
happens to the simplest

00:13:14.000 --> 00:13:19.000
rectangle I can think of,
namely, just the unit square.

00:13:19.000 --> 00:13:21.000
And, you know,
if you don't trust me,

00:13:21.000 --> 00:13:24.000
then while I'm doing this one,
do it with a different

00:13:24.000 --> 00:13:26.000
rectangle.
Do the same calculation,

00:13:26.000 --> 00:13:30.000
and see that you will get the
same conversion ratio.

00:13:30.000 --> 00:13:37.000
So, let's say that I take a
unit square -- -- so,

00:13:37.000 --> 00:13:45.000
something that goes from zero
to one both in x and y

00:13:45.000 --> 00:13:49.000
directions.
OK, and let's try to figure out

00:13:49.000 --> 00:13:51.000
what it looks like on the other
side.

00:13:51.000 --> 00:13:58.000
So, here the area is one.
Let's try to draw it in terms

00:13:58.000 --> 00:14:00.000
of u and v coordinates,
OK?

00:14:00.000 --> 00:14:05.000
So, here we have x equals 0,
y equals 0.

00:14:05.000 --> 00:14:13.000
Well, that tells us u and v are
going to be 0.

00:14:13.000 --> 00:14:17.000
Next, let's look at this corner.
Well, in xy coordinates,

00:14:17.000 --> 00:14:20.000
this is one zero.
If you plug x equals 1,

00:14:20.000 --> 00:14:24.000
y equals 0, you get u equals 3;
v equals 1.

00:14:24.000 --> 00:14:38.000
So, that goes somewhere here.
And so, this edge of the square

00:14:38.000 --> 00:14:44.000
will become this line here,
OK?

00:14:44.000 --> 00:14:49.000
Next, let's look at that point.
So that point here was (0,1).

00:14:49.000 --> 00:15:01.000
If I plug x equals zero y
equals one I will get (-2,1).

00:15:01.000 --> 00:15:11.000
So, this edge goes here.
Then, if you put x equals one,

00:15:11.000 --> 00:15:14.000
y equals one,
you will get u equals 1,

00:15:14.000 --> 00:15:22.000
v equals 2.
So, I want (1,2).

00:15:22.000 --> 00:15:28.000
And, these edges will go to
these edges here.

00:15:28.000 --> 00:15:31.000
And, you see,
it does look like a

00:15:31.000 --> 00:15:38.000
parallelogram.
OK, so now what the area of

00:15:38.000 --> 00:15:44.000
this parallelogram?
Well, we can get that by taking

00:15:44.000 --> 00:15:47.000
the determinant of these two
vectors.

00:15:47.000 --> 00:15:53.000
So, one of them is ,
and the other one is

00:15:53.000 --> 00:15:57.000
.
That will be 3 2.

00:15:57.000 --> 00:16:01.000
That's 5.
OK, this parallelogram is

00:16:01.000 --> 00:16:04.000
apparently five times the size
of this square.

00:16:04.000 --> 00:16:07.000
Here, it looks like it's less
because I somehow changed my

00:16:07.000 --> 00:16:10.000
scale.
I mean, my unit length is

00:16:10.000 --> 00:16:15.000
smaller here than here.
But, it should be a lot bigger

00:16:15.000 --> 00:16:16.000
than that.
OK,

00:16:16.000 --> 00:16:19.000
and if you do the same
calculations not with zero and

00:16:19.000 --> 00:16:21.000
one,
but with x and x plus delta x,

00:16:21.000 --> 00:16:24.000
and so on,
you will still find that the

00:16:24.000 --> 00:16:27.000
area has been multiplied by
five.

00:16:27.000 --> 00:16:36.000
So, that tells us,
actually for any other

00:16:36.000 --> 00:16:47.000
rectangle, area is also
multiplied by five.

00:16:47.000 --> 00:16:52.000
So, that tells us that dA
prime, the area element in uv

00:16:52.000 --> 00:16:57.000
coordinate is worth five times
more than the area element in

00:16:57.000 --> 00:16:59.000
the xy coordinate.

00:17:11.000 --> 00:17:17.000
So, that means du dv is worth
five times dx dy.

00:17:17.000 --> 00:17:30.000
What's so funny?
What?

00:17:30.000 --> 00:17:40.000
Oh.
[LAUGHTER] OK, rectangle.

00:17:40.000 --> 00:17:45.000
OK, is that OK now?
Did I misspell other words?

00:17:45.000 --> 00:17:48.000
No?
OK, it's really hard to see

00:17:48.000 --> 00:17:54.000
when you are up close.
It's much easier from a

00:17:54.000 --> 00:17:58.000
distance.
OK, so yeah,

00:17:58.000 --> 00:18:05.000
so we've said our
transformation multiplies areas

00:18:05.000 --> 00:18:09.000
by five.
And so, dudv is five times dxdy.

00:18:09.000 --> 00:18:14.000
So, if I'm integrating some
function, dx dy,

00:18:14.000 --> 00:18:20.000
then when I switch to uv
coordinates, I will have to

00:18:20.000 --> 00:18:26.000
replace that by one fifth dudv.
OK, and of course I would also,

00:18:26.000 --> 00:18:29.000
here my function would probably
involve x and y.

00:18:29.000 --> 00:18:33.000
I will replace them by u's and
v's.

00:18:33.000 --> 00:18:35.000
And, the bounds,
well, the shape of my origin in

00:18:35.000 --> 00:18:39.000
the xy coordinates I will have
to switch to some shape in the

00:18:39.000 --> 00:18:42.000
uv coordinates.
And, that's also something that

00:18:42.000 --> 00:18:46.000
might be easy or might be tricky
depending on what origin we are

00:18:46.000 --> 00:18:50.000
looking at.
So, usually we will do changes

00:18:50.000 --> 00:18:54.000
of variables to actually
simplify the region so it

00:18:54.000 --> 00:18:58.000
becomes easier to set up the
bounds.

00:18:58.000 --> 00:19:05.000
So, anyway, so this is kind of
an illustration of a general

00:19:05.000 --> 00:19:07.000
case.
And, why is that?

00:19:07.000 --> 00:19:10.000
Well, here it looks very easy.
We are just using linear

00:19:10.000 --> 00:19:14.000
formulas, and somehow the
relation between dx dy and du dv

00:19:14.000 --> 00:19:17.000
is the same everywhere.
If you take actually more

00:19:17.000 --> 00:19:21.000
complicated changes of variables
that's not true because usually

00:19:21.000 --> 00:19:25.000
you will expect that there are
some places where the rescaling

00:19:25.000 --> 00:19:28.000
is enlarging things,
and some of other places where

00:19:28.000 --> 00:19:31.000
things are shrunk,
so, certainly the exchange rate

00:19:31.000 --> 00:19:35.000
between dudv and dxdy will
fluctuate from point to point.

00:19:35.000 --> 00:19:37.000
It's the same as if you're
trying to change dollars to

00:19:37.000 --> 00:19:39.000
euros.
It depends on where you do it.

00:19:39.000 --> 00:19:43.000
You will get a better rate or a
worse one.

00:19:43.000 --> 00:19:47.000
So, of course,
we'll get a formula where

00:19:47.000 --> 00:19:52.000
actually this scaling factor
depends on x and y or on u and

00:19:52.000 --> 00:19:54.000
v.
But, if you fix a point,

00:19:54.000 --> 00:19:57.000
then we have linear
approximation.

00:19:57.000 --> 00:20:00.000
And, linear approximation tells
us, oh, we can do as if our

00:20:00.000 --> 00:20:02.000
function is just a linear
function of x and y.

00:20:02.000 --> 00:20:06.000
So then, we can do it the same
way we did here.

00:20:06.000 --> 00:20:18.000
OK, so let's try to think about
that.

00:20:18.000 --> 00:20:22.000
So, in the general case,
well, that means we will

00:20:22.000 --> 00:20:26.000
replace x and y by new
coordinates, u and v.

00:20:26.000 --> 00:20:30.000
And, u and v will be some
functions of x and y.

00:20:30.000 --> 00:20:34.000
So, well,
we'll have an approximation

00:20:34.000 --> 00:20:37.000
formula which tells us that the
change in u,

00:20:37.000 --> 00:20:40.000
if I change x or y a little
bit,

00:20:40.000 --> 00:20:45.000
will be roughly (u sub x times
change in x) (u sub y times

00:20:45.000 --> 00:20:50.000
change in y).
And, the change in v will be

00:20:50.000 --> 00:20:57.000
roughly (v sub x delta x) (v sub
y delta y).

00:20:57.000 --> 00:21:03.000
Or, the other way to say it,
if you want in matrix form is

00:21:03.000 --> 00:21:08.000
delta u delta v is,
sorry, approximately equal to

00:21:08.000 --> 00:21:12.000
matrix |u sub x,
u sub y, v sub x,

00:21:12.000 --> 00:21:20.000
v sub y| times matrix |delta x,
delta y|,

00:21:20.000 --> 00:21:26.000
OK?
So, if we look at that,

00:21:26.000 --> 00:21:32.000
what it tells us, in fact,
is that if we take a small

00:21:32.000 --> 00:21:40.000
rectangle in xy coordinates,
so that means we have a certain

00:21:40.000 --> 00:21:44.000
point, x, y,
and then we have a certain

00:21:44.000 --> 00:21:51.000
width.
This is going to be too small.

00:21:51.000 --> 00:21:56.000
Well, so, I have my width,
delta x.

00:21:56.000 --> 00:22:06.000
I have my height, delta y.
This is going to correspond to

00:22:06.000 --> 00:22:14.000
a small uv parallelogram.
And, what the shape and the

00:22:14.000 --> 00:22:20.000
size of the parallelogram are
depends on the partial

00:22:20.000 --> 00:22:24.000
derivatives of u and v.
So, in particular,

00:22:24.000 --> 00:22:26.000
it depends on at which point we
are.

00:22:26.000 --> 00:22:30.000
But still, at a given point,
it's a bit like that.

00:22:30.000 --> 00:22:35.000
And, so if we do the same
argument as before,

00:22:35.000 --> 00:22:41.000
what we will see is that the
scaling factor is actually the

00:22:41.000 --> 00:22:45.000
determinant of this
transformation.

00:22:45.000 --> 00:22:50.000
So, that's one thing that maybe
we didn't emphasize enough when

00:22:50.000 --> 00:22:53.000
we did matrices at the beginning
of a semester.

00:22:53.000 --> 00:22:57.000
But, when you have a linear
transformation between

00:22:57.000 --> 00:23:01.000
variables, the determinant of
that transformation represents

00:23:01.000 --> 00:23:05.000
how it scales areas.
OK, so one way to think about

00:23:05.000 --> 00:23:09.000
it is just to try it and see
what happens.

00:23:09.000 --> 00:23:12.000
Take this side.
This side in x,

00:23:12.000 --> 00:23:16.000
y coordinates corresponds to
delta x and zero.

00:23:16.000 --> 00:23:20.000
And, now, if you take the image
of that, if you see what happens

00:23:20.000 --> 00:23:24.000
to delta u and delta v,
that will be basically u sub x

00:23:24.000 --> 00:23:28.000
delta x and v sub x delta x.
There's no delta y.

00:23:28.000 --> 00:23:33.000
For the other side,
OK, so maybe I should do it

00:23:33.000 --> 00:23:36.000
actually.
So, you know,

00:23:36.000 --> 00:23:40.000
if we move in the x,
y coordinates by delta x and

00:23:40.000 --> 00:23:45.000
zero,
then delta u and delta v will

00:23:45.000 --> 00:23:50.000
be approximately u sub x delta
x,

00:23:50.000 --> 00:24:02.000
and v sub x delta x.
And, on the other hand,

00:24:02.000 --> 00:24:04.000
if you move in the other
direction along the other side

00:24:04.000 --> 00:24:08.000
of your rectangle,
zero and delta y,

00:24:08.000 --> 00:24:13.000
then the change in u and the
change in v will correspond to,

00:24:13.000 --> 00:24:16.000
well, how does u change?
That's u sub y delta y,

00:24:16.000 --> 00:24:20.000
and v changes by v sub y delta
y.

00:24:20.000 --> 00:24:22.000
And so, now,
if you take the determinant of

00:24:22.000 --> 00:24:25.000
these two vectors,
OK, so these are the sides of

00:24:25.000 --> 00:24:29.000
your parallelogram up here.
And, if you take these sides to

00:24:29.000 --> 00:24:31.000
get the area of the
parallelogram,

00:24:31.000 --> 00:24:33.000
you'll need to take the
determinant.

00:24:33.000 --> 00:24:41.000
And, the determinant will be
the determinant of this matrix

00:24:41.000 --> 00:24:48.000
times delta x times delta y.
So, the area in uv coordinates

00:24:48.000 --> 00:24:53.000
will be the determinant of a
matrix times delta x,

00:24:53.000 --> 00:24:57.000
delta y.
And so,

00:24:57.000 --> 00:25:02.000
what I'm trying to say is that
when you have a general change

00:25:02.000 --> 00:25:06.000
of variables,
du dv versus dx dy is given by

00:25:06.000 --> 00:25:11.000
the determinant of this matrix
of partial derivatives.

00:25:11.000 --> 00:25:13.000
It doesn't matter in which
order you write it.

00:25:13.000 --> 00:25:16.000
I mean, you can put in rows or
columns.

00:25:16.000 --> 00:25:18.000
If you transpose a matrix,
that doesn't change the

00:25:18.000 --> 00:25:21.000
determinant.
It's just any sensible matrix

00:25:21.000 --> 00:25:24.000
that you can write will have the
correct determinant.

00:26:02.000 --> 00:26:07.000
OK, so what we need to know is
the following thing.

00:26:07.000 --> 00:26:11.000
So,
we define something called the

00:26:11.000 --> 00:26:16.000
Jacobian of a change of
variables and used the letter J,

00:26:16.000 --> 00:26:21.000
or maybe a more useful notation
is partial of u,

00:26:21.000 --> 00:26:24.000
v over partial of x,
y.

00:26:24.000 --> 00:26:27.000
That's a very strange notation.
I mean, that doesn't mean that

00:26:27.000 --> 00:26:30.000
we are actually taking the
partial derivatives of anything.

00:26:30.000 --> 00:26:34.000
OK, it's just a notation to
remind us that this has to do

00:26:34.000 --> 00:26:37.000
with the ratio between dudv and
dxdy.

00:26:37.000 --> 00:26:42.000
And, it's obtained using the
partial derivatives of u and v

00:26:42.000 --> 00:26:50.000
with respect to x and y.
So, it's the determinant of the

00:26:50.000 --> 00:26:55.000
matrix |u sub x,
u sub y, v sub x,

00:26:55.000 --> 00:27:02.000
v sub y|, the matrix that I had
up there.

00:27:02.000 --> 00:27:10.000
OK, and what we need to know is
that du dv is equal to the

00:27:10.000 --> 00:27:17.000
absolute value of J dx dy.
Or, if you prefer to see it in

00:27:17.000 --> 00:27:23.000
the easier to remember version,
it's (absolute value of d of

00:27:23.000 --> 00:27:27.000
(u, v) over partial xy) times dx
dy.

00:27:27.000 --> 00:27:32.000
OK, so this is just what you
need to remember,

00:27:32.000 --> 00:27:38.000
and it says that the area in uv
coordinates is worth,

00:27:38.000 --> 00:27:42.000
well, the ratio to the xy
coordinates is given by this

00:27:42.000 --> 00:27:46.000
Jacobian determinant except for
one small thing.

00:27:46.000 --> 00:27:48.000
It's given by,
actually, the absolute value of

00:27:48.000 --> 00:27:52.000
this guy.
OK, so what's going on here?

00:27:52.000 --> 00:27:56.000
What's going on here is when we
are saying the determinant of

00:27:56.000 --> 00:27:59.000
the transformation tells us how
the area is multiplied,

00:27:59.000 --> 00:28:02.000
there's a small catch.
Remember, the determinants are

00:28:02.000 --> 00:28:06.000
equal to areas up to sine.
Sometimes, the determinant is

00:28:06.000 --> 00:28:10.000
negative because of reversing
the orientation of things.

00:28:10.000 --> 00:28:13.000
But, the area is still the same.
Area is always positive.

00:28:13.000 --> 00:28:17.000
So, the area elements are
actually related by the absolute

00:28:17.000 --> 00:28:23.000
value of this guy.
OK, so if you find -10 as your

00:28:23.000 --> 00:28:29.000
answer, then du dv is still ten
times dx dy.

00:28:29.000 --> 00:28:33.000
OK, so I didn't put it all
together because then you would

00:28:33.000 --> 00:28:36.000
have two sets of vertical bars.
See, this is a vertical bar for

00:28:36.000 --> 00:28:38.000
absolute value.
This is vertical bar for

00:28:38.000 --> 00:28:42.000
determinant.
They're not the same.

00:28:42.000 --> 00:28:46.000
That's the one thing to
remember.

00:28:46.000 --> 00:28:54.000
OK, any questions about this?
No?

00:28:54.000 --> 00:29:06.000
OK.
So, actually let's do our first

00:29:06.000 --> 00:29:12.000
example of that.
Let's check what we had for

00:29:12.000 --> 00:29:16.000
polar coordinates.
Last time I told you if we have

00:29:16.000 --> 00:29:19.000
dx dy we could switch it to r dr
d theta.

00:29:19.000 --> 00:29:25.000
And, we had some argument for
that by looking at the area of a

00:29:25.000 --> 00:29:31.000
small circular sector.
But, let's check again using

00:29:31.000 --> 00:29:37.000
this new method.
So, in polar coordinates I'm

00:29:37.000 --> 00:29:44.000
setting x equals r cosine theta,
y equals r sine theta.

00:29:44.000 --> 00:29:48.000
So, the Jacobian for this
change of variables,

00:29:48.000 --> 00:29:54.000
so let's say I'm trying to find
the partial derivatives of x,

00:29:54.000 --> 00:29:58.000
y with respect to r,
theta.

00:29:58.000 --> 00:30:04.000
Well, what is,
OK, let me actually write them

00:30:04.000 --> 00:30:10.000
here again for you.
And, so what does that become?

00:30:10.000 --> 00:30:17.000
Partial x over partial r is
just cosine theta.

00:30:17.000 --> 00:30:25.000
Partial x over partial theta is
negative r sine theta.

00:30:25.000 --> 00:30:27.000
Sorry, I guess I'm going to run
out of space here.

00:30:27.000 --> 00:30:33.000
So, let me do it underneath.
So, we said x sub r is cosine

00:30:33.000 --> 00:30:36.000
theta;
x sub theta is negative r sine

00:30:36.000 --> 00:30:41.000
theta.
y sub r is sine;

00:30:41.000 --> 00:30:49.000
y sub theta is r cosine.
And now, if we compute this

00:30:49.000 --> 00:30:58.000
determinant, we'll get (r cosine
squared theta) (r sine squared

00:30:58.000 --> 00:31:02.000
theta).
And, that simplifies to r.

00:31:02.000 --> 00:31:08.000
So, dx dy is,
well, absolute value of r dr d

00:31:08.000 --> 00:31:12.000
theta.
But, remember that r is always

00:31:12.000 --> 00:31:18.000
positive.
So, it's r dr d theta.

00:31:18.000 --> 00:31:26.000
OK, so that's another way to
justify how we did double

00:31:26.000 --> 00:31:34.000
integrals in polar coordinates.
OK, any questions on that?

00:31:34.000 --> 00:31:47.000
Where?
Yeah, OK.

00:31:47.000 --> 00:31:52.000
Yeah, so this one seems to be
switching.

00:31:52.000 --> 00:31:56.000
Well, it depends what you do.
So, OK, actually here's an

00:31:56.000 --> 00:32:00.000
important thing that I didn't
quite say.

00:32:00.000 --> 00:32:04.000
So, I said, you know,
we are going to switch from xy

00:32:04.000 --> 00:32:07.000
to uv.
We can also switch from uv to

00:32:07.000 --> 00:32:09.000
xy.
And, this conversion ratio,

00:32:09.000 --> 00:32:12.000
the Jacobian,
works both ways.

00:32:12.000 --> 00:32:16.000
Once you have found the ratio
between du dv and dx dy,

00:32:16.000 --> 00:32:20.000
then it works one way or it
works the other way.

00:32:20.000 --> 00:32:22.000
I mean, here,
of course, we get the answer in

00:32:22.000 --> 00:32:26.000
terms of r.
So, this would let us switch

00:32:26.000 --> 00:32:30.000
from xy to r theta.
But, we can also switch from r

00:32:30.000 --> 00:32:33.000
theta to xy.
Just, we'd write dr d theta

00:32:33.000 --> 00:32:37.000
equals (1 over r) times dx dy.
And then we'd have,

00:32:37.000 --> 00:32:41.000
of course, to replace r by its
formula in xy coordinates.

00:32:41.000 --> 00:32:43.000
Usually, we don't do that.
Usually, we actually start with

00:32:43.000 --> 00:32:47.000
xy and switch to polar.
But,

00:32:47.000 --> 00:32:50.000
so in general,
when you have this formula

00:32:50.000 --> 00:32:54.000
relating du dv with dx dy,
you can use it both ways,

00:32:54.000 --> 00:33:00.000
either to switch from du dv to
dx dy or the other way around.

00:33:00.000 --> 00:33:04.000
And, the thing that I'm not
telling you that now I should

00:33:04.000 --> 00:33:08.000
probably tell you is I could
define two Jacobians because if

00:33:08.000 --> 00:33:12.000
I solve for xy in terms of uv
instead of uv in terms of xy,

00:33:12.000 --> 00:33:15.000
then I can compute two
different Jacobians.

00:33:15.000 --> 00:33:19.000
I can compute partial uv over
partial xy, or I can compute

00:33:19.000 --> 00:33:24.000
partial xy over partial uv if I
have the formulas both ways.

00:33:24.000 --> 00:33:27.000
Well, the good news is these
guys are the inverse of each

00:33:27.000 --> 00:33:29.000
other.
So, the two formulas that you

00:33:29.000 --> 00:33:31.000
might get are consistent.

00:33:59.000 --> 00:34:16.000
OK, so useful remark -- So,
say that you can compute both

00:34:16.000 --> 00:34:23.000
-- -- these guys.
Well, then actually,

00:34:23.000 --> 00:34:26.000
the product will just be 1.
So, they are the inverse of

00:34:26.000 --> 00:34:28.000
each other.
So, it doesn't matter which one

00:34:28.000 --> 00:34:34.000
you compute.
You can compute whichever one

00:34:34.000 --> 00:34:45.000
is the easiest to compute no
matter which one of the two you

00:34:45.000 --> 00:34:48.000
need.
And, one way to see that is

00:34:48.000 --> 00:34:50.000
that, in fact,
we're looking at the

00:34:50.000 --> 00:34:53.000
determinant of these matrices
that tell us the relation in

00:34:53.000 --> 00:34:56.000
variables.
So, if one of them tells you

00:34:56.000 --> 00:34:58.000
how delta u delta v relate to
delta x delta y,

00:34:58.000 --> 00:35:00.000
the other one does the opposite
thing.

00:35:00.000 --> 00:35:03.000
It means they are the inverse
matrices.

00:35:03.000 --> 00:35:06.000
And, the determinant of the
inverse matrix is the inverse of

00:35:06.000 --> 00:35:10.000
the determinant.
So, they are really

00:35:10.000 --> 00:35:14.000
interchangeable.
I mean, you can just compute

00:35:14.000 --> 00:35:17.000
whichever one is easiest.
So here, if you wanted,

00:35:17.000 --> 00:35:22.000
dr d theta in terms of dx dy,
it's easier to do this and then

00:35:22.000 --> 00:35:27.000
move the r over there than to
first solve for r and theta as

00:35:27.000 --> 00:35:31.000
functions of x and y and then do
the entire thing again.

00:35:31.000 --> 00:35:42.000
But, you can do it if you want.
I mean, it works.

00:35:42.000 --> 00:35:45.000
Oh yeah, the other useful
remark, so, I mentioned it,

00:35:45.000 --> 00:35:49.000
but let me emphasize again.
So, now, the ratio between du

00:35:49.000 --> 00:35:51.000
dv and dx dy,
it's not a constant anymore,

00:35:51.000 --> 00:35:54.000
although there it used to be
five.

00:35:54.000 --> 00:35:56.000
But now, it's become r,
or anything.

00:35:56.000 --> 00:35:58.000
In general, it will be a
function that depends on the

00:35:58.000 --> 00:36:01.000
variables.
So, it's not true that you can

00:36:01.000 --> 00:36:04.000
just say, oh,
I'll put a constant times du

00:36:04.000 --> 00:36:14.000
dv.
Yes?

00:36:14.000 --> 00:36:17.000
It would still work the same.
You could imagine drawing a

00:36:17.000 --> 00:36:20.000
picture where r and theta are
the Cartesian coordinates,

00:36:20.000 --> 00:36:22.000
and your picture would be
completely messed up.

00:36:22.000 --> 00:36:26.000
It would be a very strange
thing to do to try to draw,

00:36:26.000 --> 00:36:30.000
you know, I'm going to do it,
but don't take notes on that.

00:36:30.000 --> 00:36:32.000
You could try to draw picture
like that, and then a circle

00:36:32.000 --> 00:36:34.000
would start looking like,
you know, a disk would look

00:36:34.000 --> 00:36:35.000
like that.
It would be very

00:36:35.000 --> 00:36:37.000
counterintuitive.
But, you could do it.

00:36:37.000 --> 00:36:41.000
And that would be equivalent to
what we did with a previous

00:36:41.000 --> 00:36:43.000
change of variables.
So, in this case,

00:36:43.000 --> 00:36:47.000
certainly you would never draw
a picture like that.

00:36:47.000 --> 00:36:59.000
But, you could do it.
OK, so now let's do a complete

00:36:59.000 --> 00:37:07.000
example to see how things fit
together, how we do everything.

00:37:07.000 --> 00:37:10.000
So, let's say that we want to
compute, so I have to warn you,

00:37:10.000 --> 00:37:12.000
it's going to be a very silly
example.

00:37:12.000 --> 00:37:16.000
It's an example where it's much
easier to compute things without

00:37:16.000 --> 00:37:19.000
the change of variables.
But, you know,

00:37:19.000 --> 00:37:24.000
it's good practice in the sense
that we're going to make it so

00:37:24.000 --> 00:37:29.000
complicated that if we can do
this one, then we can do that

00:37:29.000 --> 00:37:31.000
one.
So, let's say that we want to

00:37:31.000 --> 00:37:33.000
compute this.
And, of course,

00:37:33.000 --> 00:37:35.000
it's very easy to compute it
directly.

00:37:35.000 --> 00:37:42.000
But let's say that for some
evil reason we want to do that

00:37:42.000 --> 00:37:49.000
by changing variables to u
equals x and v equals xy.

00:37:49.000 --> 00:37:55.000
OK, that's a very strange idea,
but let's do it anyway.

00:37:55.000 --> 00:37:58.000
I mean, normally,
you would only do this kind of

00:37:58.000 --> 00:38:01.000
substitution if either it
simplifies a lot the function

00:38:01.000 --> 00:38:03.000
you are integrating,
or it simplifies a lot the

00:38:03.000 --> 00:38:06.000
region on which you are
integrating.

00:38:06.000 --> 00:38:12.000
And here, neither happens.
But anyway, so the first thing

00:38:12.000 --> 00:38:16.000
we have to do here is figure out
what we are going to be

00:38:16.000 --> 00:38:18.000
integrating.
OK, so to do that,

00:38:18.000 --> 00:38:23.000
we should figure out what dx dy
will become in terms of u and v.

00:38:23.000 --> 00:38:26.000
So, that's what we've just seen
using the Jacobian.

00:38:26.000 --> 00:38:32.000
OK, so the first thing to do is
find the area element.

00:38:32.000 --> 00:38:33.000
And, for that,
we use the Jacobian.

00:38:33.000 --> 00:38:36.000
So, well, let's see,
the one that we can do easily

00:38:36.000 --> 00:38:40.000
is partials of u and v with
respect to x and y.

00:38:40.000 --> 00:38:42.000
I mean, the other one is not
very hard because here you can

00:38:42.000 --> 00:38:45.000
solve easily.
But, the one that's given to

00:38:45.000 --> 00:38:49.000
you is partial of u and v with
respect to x and y,

00:38:49.000 --> 00:38:55.000
so partial u partial x is one.
Partial u partial y is zero.

00:38:55.000 --> 00:39:03.000
Partial v partial x is y.
And partial v partial y is x.

00:39:03.000 --> 00:39:17.000
So that's just x.
So, that means that du dv is x

00:39:17.000 --> 00:39:20.000
dx dy.
Well, it would be absolute

00:39:20.000 --> 00:39:23.000
value of x, but x is positive in
our origin.

00:39:23.000 --> 00:39:35.000
So, at least we don't have to
worry about that.

00:39:35.000 --> 00:39:45.000
OK, so now that we have that,
we can try to look at the

00:39:45.000 --> 00:39:55.000
integrand in terms of u and v.
OK, so we were integrating x

00:39:55.000 --> 00:40:00.000
squared y dx dy.
So, let's switch it.

00:40:00.000 --> 00:40:09.000
Well, let's first switch the dx
dy that becomes one over x du

00:40:09.000 --> 00:40:15.000
dv.
So, that's actually xy du dv.

00:40:15.000 --> 00:40:18.000
And, what is xy in terms of u
and v?

00:40:18.000 --> 00:40:20.000
Well, here at least we had a
little bit of luck.

00:40:20.000 --> 00:40:26.000
xy is just v.
So, that's v du dv.

00:40:26.000 --> 00:40:32.000
So, in fact,
what we'll be computing is a

00:40:32.000 --> 00:40:40.000
double integral over some
mysterious region of v du dv.

00:40:40.000 --> 00:40:44.000
Now, last but not least,
we'll have to find what are the

00:40:44.000 --> 00:40:49.000
bounds for u and v in the new
integral so that we know how to

00:40:49.000 --> 00:40:50.000
evaluate this.

00:41:14.000 --> 00:41:17.000
In fact, well,
we could do it du dv or dv du.

00:41:17.000 --> 00:41:23.000
We don't know yet.
Oh, amazing.

00:41:23.000 --> 00:41:31.000
It went all the way down this
time.

00:41:31.000 --> 00:41:43.000
OK, so it could be dv du if
that's easier.

00:41:43.000 --> 00:41:46.000
So, let's try to find the
bounds.

00:41:46.000 --> 00:41:52.000
In this case,
that's the hardest part.

00:41:52.000 --> 00:42:00.000
OK, so let me draw a picture in
xy coordinates and try to

00:42:00.000 --> 00:42:06.000
understand things using that.
OK, so x and y go from zero to

00:42:06.000 --> 00:42:08.000
one.
The region that we want to

00:42:08.000 --> 00:42:11.000
integrate over was just this
square.

00:42:11.000 --> 00:42:16.000
Let's try to figure out how u
and v vary there.

00:42:16.000 --> 00:42:23.000
So, let's say that we're going
to do it du dv.

00:42:23.000 --> 00:42:32.000
OK, so What we want to
understand is how u and v vary

00:42:32.000 --> 00:42:36.000
in here.
What's going to happen?

00:42:36.000 --> 00:42:40.000
So, the way we can think about
it is we try to figure out how

00:42:40.000 --> 00:42:43.000
we are slicing our origin.
OK, so here,

00:42:43.000 --> 00:42:46.000
we are integrating first over
u.

00:42:46.000 --> 00:42:51.000
That means we start by keeping
u constant, no,

00:42:51.000 --> 00:42:55.000
by keeping v constant as u
changes.

00:42:55.000 --> 00:43:03.000
OK, so u changes as v is
constant.

00:43:03.000 --> 00:43:06.000
What does it mean that I'm
keeping v constant.

00:43:06.000 --> 00:43:09.000
Well, what is v?
v is xy.

00:43:09.000 --> 00:43:13.000
So, that means I keep xy equals
constant.

00:43:13.000 --> 00:43:16.000
What does the curve xy equals
constant look like?

00:43:16.000 --> 00:43:22.000
Well, it's just a hyperbola.
y equals constant over x.

00:43:22.000 --> 00:43:28.000
So, if I look at the various
values of v that I can take,

00:43:28.000 --> 00:43:33.000
for each value of v,
if I fix a value of v,

00:43:33.000 --> 00:43:38.000
I will be moving on one of
these red curves.

00:43:38.000 --> 00:43:42.000
OK, and u, well,
u is the same thing as x.

00:43:42.000 --> 00:43:47.000
So, that means u will increase.
Here, maybe it will be 0.1 and

00:43:47.000 --> 00:43:51.000
it will increase all the way to
one here.

00:43:51.000 --> 00:43:59.000
OK, so we are just traveling on
each of these slices.

00:43:59.000 --> 00:44:03.000
Now, so the question we must
answer here is for a given value

00:44:03.000 --> 00:44:08.000
of v, what are the bounds for u?
So, I'm traveling on my curve,

00:44:08.000 --> 00:44:11.000
v equals constant,
and trying to figure out,

00:44:11.000 --> 00:44:14.000
when do I enter my origin?
When do I leave it?

00:44:14.000 --> 00:44:18.000
Well, I enter it when I go
through this side.

00:44:18.000 --> 00:44:24.000
So, the question is,
what's the value of u here?

00:44:24.000 --> 00:44:29.000
Well, we don't know that very
easily until we look at these

00:44:29.000 --> 00:44:32.000
formulas.
So, u equals x,

00:44:32.000 --> 00:44:36.000
OK, but we don't know what x is
at that point.

00:44:36.000 --> 00:44:42.000
v equals x and v equals xy.
What do we go here?

00:44:42.000 --> 00:44:44.000
Well, we don't know x,
but we know y certainly.

00:44:44.000 --> 00:44:49.000
OK, so let's forget about
trying to find u.

00:44:49.000 --> 00:44:53.000
And, let's say,
for now, we know y equals one.

00:44:53.000 --> 00:44:58.000
Well, if we set y equals one,
that tells us that u and v are

00:44:58.000 --> 00:45:03.000
both equal to x.
So, in terms of u and v,

00:45:03.000 --> 00:45:11.000
the equation of this uv
coordinate is u equals v.

00:45:11.000 --> 00:45:14.000
OK, I mean, the other way to do
it is, say that you know you

00:45:14.000 --> 00:45:17.000
want y equals one.
You want to know what is y in

00:45:17.000 --> 00:45:18.000
terms of u and v.
Well, it's easy.

00:45:18.000 --> 00:45:26.000
y is v over u.
So, let me actually add an

00:45:26.000 --> 00:45:31.000
extra step in case that's,
so, we know that y is v over u

00:45:31.000 --> 00:45:35.000
equals one.
So, that means u=v is my

00:45:35.000 --> 00:45:39.000
equation.
OK, so when I'm here,

00:45:39.000 --> 00:45:47.000
when I'm entering my region,
the value of u at this point is

00:45:47.000 --> 00:45:53.000
just v, u equals v.
That's the hard part.

00:45:53.000 --> 00:45:56.000
Now, we need to figure out,
so, we started u equals v.

00:45:56.000 --> 00:45:59.000
u increases,
increases, increases.

00:45:59.000 --> 00:46:01.000
Where does it exit?
It exits one when we are here.

00:46:01.000 --> 00:46:05.000
What's the value of u here?
One. That one is easier, right?

00:46:05.000 --> 00:46:10.000
This side here,
so, this side here is x equals

00:46:10.000 --> 00:46:13.000
one.
That means u equals one.

00:46:13.000 --> 00:46:20.000
So, we start at u equals one.
Now, we've done the inner

00:46:20.000 --> 00:46:24.000
integral.
What about the outer?

00:46:24.000 --> 00:46:28.000
So, we have to figure out,
what is the first and what is

00:46:28.000 --> 00:46:32.000
the last value of v that we'll
want to consider?

00:46:32.000 --> 00:46:36.000
Well, if you look at all these
hyperbola's, xy equals constant.

00:46:36.000 --> 00:46:39.000
What's the smallest value of xy
that we'll ever want to look at

00:46:39.000 --> 00:46:41.000
in here?
Zero, OK.

00:46:41.000 --> 00:46:49.000
Let me actually,
where's my yellow chalk?

00:46:49.000 --> 00:46:55.000
Is it, no, ah.
So, this one here,

00:46:55.000 --> 00:47:00.000
that's actually v=0.
So, we'll start at v equals

00:47:00.000 --> 00:47:02.000
zero.
And, what's the last hyperbola

00:47:02.000 --> 00:47:05.000
we want to look at?
Well, it's the one that's right

00:47:05.000 --> 00:47:07.000
there in the corner.
It's this one here.

00:47:07.000 --> 00:47:15.000
And, that's v equals one.
So, v goes from zero to one.

00:47:15.000 --> 00:47:17.000
OK, and now,
we can compute this.

00:47:17.000 --> 00:47:22.000
I mean, it's not particularly
easier than that one,

00:47:22.000 --> 00:47:26.000
but it's not harder either.
How else could we have gotten

00:47:26.000 --> 00:47:28.000
these bounds,
because that was quite evil.

00:47:28.000 --> 00:47:32.000
So, I would like to recommend
that you try this way in case it

00:47:32.000 --> 00:47:34.000
works well.
Just try to picture,

00:47:34.000 --> 00:47:38.000
what are the slices in terms of
u and v, and how you travel on

00:47:38.000 --> 00:47:40.000
them, where you enter,
where you leave,

00:47:40.000 --> 00:47:47.000
staying in the xy picture.
If that somehow doesn't work

00:47:47.000 --> 00:47:58.000
well, another way is to draw the
picture in the uv coordinates.

00:47:58.000 --> 00:48:04.000
So, switch to a uv picture.
So, what do I mean by that?

00:48:04.000 --> 00:48:09.000
Well, we had here a picture in
xy coordinates where we had our

00:48:09.000 --> 00:48:12.000
sides.
And, we are going to try to

00:48:12.000 --> 00:48:15.000
draw what it looks like in terms
of u and v.

00:48:15.000 --> 00:48:18.000
So, here we said this is x
equals one.

00:48:18.000 --> 00:48:24.000
That becomes u equals one.
So, we'll draw u equals one.

00:48:24.000 --> 00:48:30.000
This side we said is y equals
one becomes u equals v.

00:48:30.000 --> 00:48:33.000
That's what we've done over
there.

00:48:33.000 --> 00:48:39.000
OK, so u equals v.
Now, we have the two other

00:48:39.000 --> 00:48:41.000
sides to deal with.
Well, let's look at this one

00:48:41.000 --> 00:48:44.000
first.
So, that was x equals zero.

00:48:44.000 --> 00:48:48.000
What happens when x equals zero?
Well, both u and v are zero.

00:48:48.000 --> 00:48:51.000
So, this side actually gets
squished in the change of

00:48:51.000 --> 00:48:53.000
variables.
It's a bit strange,

00:48:53.000 --> 00:48:57.000
but it's a bit the same thing
as when you switch to polar

00:48:57.000 --> 00:49:00.000
coordinates at the origin,
r is zero but theta can be

00:49:00.000 --> 00:49:03.000
anything.
It's not always one point is

00:49:03.000 --> 00:49:07.000
one point.
So anyway, this is the origin,

00:49:07.000 --> 00:49:11.000
and then the last side,
y equals zero,

00:49:11.000 --> 00:49:15.000
and x varies just becomes v
equals zero.

00:49:15.000 --> 00:49:18.000
So, somehow,
in the change of variables,

00:49:18.000 --> 00:49:21.000
this square becomes this
triangle.

00:49:21.000 --> 00:49:24.000
And now, if we want to
integrate du dv,

00:49:24.000 --> 00:49:30.000
it means we are going to slice
by v equals constant.

00:49:30.000 --> 00:49:33.000
So, we are going to integrate
over slices like this,

00:49:33.000 --> 00:49:36.000
and you see for each value of
v, we go from u equals v to u

00:49:36.000 --> 00:49:41.000
equals one.
And, v goes from zero to one.

00:49:41.000 --> 00:49:44.000
OK, so you get the same bounds
just by drawing a different

00:49:44.000 --> 00:49:47.000
picture.
So, it's up to you to decide

00:49:47.000 --> 00:49:51.000
whether you prefer to think on
this picture or draw that one

00:49:51.000 --> 00:49:53.000
instead.
It depends on which problems

00:49:53.000 --> 00:49:55.000
you're doing.