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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

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00:00:32,000 --> 00:00:35,000
coordinates.
But, more generally,

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00:00:35,000 --> 00:00:40,000
there's a lot of different
changes of variables that you

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00:00:40,000 --> 00:00:44,000
might want to do.
OK, so today we're going to see

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00:00:44,000 --> 00:00:48,000
how to change variables,
if you want,

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00:00:48,000 --> 00:00:52,000
how to do substitutions in
double integrals.

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00:01:02,000 --> 00:01:10,000
OK, so let me start with a
simple example.

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00:01:10,000 --> 00:01:21,000
Let's say that we want to find
the area of an ellipse with

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00:01:21,000 --> 00:01:28,000
semi-axes a and b.
OK, so that means an ellipse is

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00:01:28,000 --> 00:01:38,000
just like a squished circle.
And so, there's a and there's b.

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00:01:38,000 --> 00:01:44,000
And, the equation of that
ellipse is x over a squared plus

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00:01:44,000 --> 00:01:49,000
y over b squared equals one.
That's the curve,

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00:01:49,000 --> 00:01:53,000
and the inside region is where
this is less than one.

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00:01:53,000 --> 00:01:58,000
OK, so it's just like a circle
that where you have rescaled x

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00:01:58,000 --> 00:02:01,000
and y differently.
So, let's say we want to find

23
00:02:01,000 --> 00:02:03,000
the area of it.
Maybe you know what the area is.

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00:02:03,000 --> 00:02:11,000
But let's do it as a double
integral.

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00:02:11,000 --> 00:02:14,000
So, you know,
if you find that the area is

26
00:02:14,000 --> 00:02:19,000
too easy, you can integrate any
function other than ellipse,

27
00:02:19,000 --> 00:02:23,000
if you prefer.
But, let's do it just with area.

28
00:02:23,000 --> 00:02:27,000
So, we know that we want to
integrate just the area element,

29
00:02:27,000 --> 00:02:30,000
let's say, dx dy over the
origin inside the ellipse.

30
00:02:30,000 --> 00:02:37,000
That's x over a2 plus y over b2
less than 1.

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00:02:37,000 --> 00:02:41,000
Now, we can try to set this up
in terms of x and y coordinates,

32
00:02:41,000 --> 00:02:46,000
you know, set up the bounds by
solving for first four x as a

33
00:02:46,000 --> 00:02:49,000
function of y if we do it this
order and,

34
00:02:49,000 --> 00:02:52,000
well, do the usual stuff.
That doesn't look very

35
00:02:52,000 --> 00:02:55,000
pleasant, and it's certainly not
the best way to do it.

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00:02:55,000 --> 00:02:57,000
OK, if this were a circle,
we would switch to polar

37
00:02:57,000 --> 00:02:59,000
coordinates.
Well, we can't quite do that

38
00:02:59,000 --> 00:03:00,000
yet.
But, you know,

39
00:03:00,000 --> 00:03:01,000
an ellipse is just a squished
circle.

40
00:03:01,000 --> 00:03:08,000
So, maybe we want to actually
first rescale x and y by a and

41
00:03:08,000 --> 00:03:11,000
b.
So, to do that,

42
00:03:11,000 --> 00:03:19,000
what we'd like to do is set x
over a to be u,

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00:03:19,000 --> 00:03:24,000
and y over b be v.
So, we'll have two new

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00:03:24,000 --> 00:03:28,000
variables, u and v,
and we'll try to redo our

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00:03:28,000 --> 00:03:32,000
integral in terms of u and v.
So, how do we do the

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00:03:32,000 --> 00:03:36,000
substitution?
So, in terms of u and v,

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00:03:36,000 --> 00:03:39,000
the condition,
the region that we are

48
00:03:39,000 --> 00:03:43,000
integrating on will become u^2
v^2 is less than 1,

49
00:03:43,000 --> 00:03:45,000
which is arguably nicer than
the ellipse.

50
00:03:45,000 --> 00:03:50,000
That's why we are doing it.
But, we need to know what to do

51
00:03:50,000 --> 00:03:53,000
with dx and dy.
Well, here, the answer is

52
00:03:53,000 --> 00:03:56,000
pretty easy because we just
change x and y separately.

53
00:03:56,000 --> 00:03:59,000
We do two independent
substitutions.

54
00:03:59,000 --> 00:04:10,000
OK, so if we set u equals x
over a, that means du is one

55
00:04:10,000 --> 00:04:18,000
over adx.
And here, dv is one over bdy.

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00:04:18,000 --> 00:04:26,000
So, it's very tempting to
write, and here we actually can

57
00:04:26,000 --> 00:04:34,000
write, in this particular case,
that dudv is (1/ab)dxdy,

58
00:04:34,000 --> 00:04:42,000
OK?
So, let me rewrite that.

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00:04:42,000 --> 00:04:54,000
OK, so I get dudv equals
(1/ab)dxdy, or equivalently dxdy

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00:04:54,000 --> 00:05:05,000
is ab times dudv.
OK, so in my double integral,

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00:05:05,000 --> 00:05:15,000
I'm going to write (ab)dudv.
OK, so now, my double integral

62
00:05:15,000 --> 00:05:18,000
becomes, well,
the double integral of a

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00:05:18,000 --> 00:05:23,000
constant in terms of u and v.
So, I can take the constant out.

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00:05:23,000 --> 00:05:31,000
I will get ab times double
integral over u^2 v^2<1 of du

65
00:05:31,000 --> 00:05:34,000
dv.
And, that is an integral that

66
00:05:34,000 --> 00:05:37,000
we know how to do.
Well, it's just the area of a

67
00:05:37,000 --> 00:05:40,000
unit circle.
So, we can just say,

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00:05:40,000 --> 00:05:50,000
this is ab times the area of
the unit disk,

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00:05:50,000 --> 00:05:54,000
which we know to be pi,
or if somehow you had some

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00:05:54,000 --> 00:05:57,000
function to integrate,
then you could have somehow

71
00:05:57,000 --> 00:05:59,000
switched to polar coordinates,
you know, setting u equals r

72
00:05:59,000 --> 00:06:02,000
times cos(theta),
v equals r times sin(theta),

73
00:06:02,000 --> 00:06:07,000
and then doing it in polar
coordinates.

74
00:06:07,000 --> 00:06:11,000
OK, so here the substitution
worked pretty easy.

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00:06:11,000 --> 00:06:14,000
The question is,
if we do a change of variables

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00:06:14,000 --> 00:06:18,000
where the relation between x and
y and u and v is more

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00:06:18,000 --> 00:06:20,000
complicated, what can we do?
Can we still do this,

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00:06:20,000 --> 00:06:22,000
or do we have to be more
careful?

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00:06:22,000 --> 00:06:23,000
And, actually,
we have to be more careful.

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00:06:23,000 --> 00:06:26,000
So, that's what we are going to
see next.

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00:06:26,000 --> 00:06:33,000
Any question about this, first?
No?

82
00:06:33,000 --> 00:06:38,000
OK.
OK, so, see the general problem

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00:06:38,000 --> 00:06:41,000
when we try to do this is to
figure out what is the scale

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00:06:41,000 --> 00:06:48,000
factor?
What's the relation between

85
00:06:48,000 --> 00:06:57,000
dxdy and dudv?
We need to find the scaling

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00:06:57,000 --> 00:07:07,000
factor.
So, we need to find dxdy versus

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00:07:07,000 --> 00:07:12,000
dudv.
So, let's do another example

88
00:07:12,000 --> 00:07:18,000
that's still pretty easy,
but a little bit less easy.

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00:07:18,000 --> 00:07:24,000
OK, so let's say that for some
reason, we want to do the change

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00:07:24,000 --> 00:07:27,000
of variables:
u equals 3x-2y,

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00:07:27,000 --> 00:07:31,000
and v equals x y.
Why would we want to do that?

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00:07:31,000 --> 00:07:34,000
Well, that might be to simplify
the integrand because we are

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00:07:34,000 --> 00:07:38,000
integrating a function that
happens to be actually involving

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00:07:38,000 --> 00:07:42,000
these guys rather than x and y.
Or, it might be to simplify the

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00:07:42,000 --> 00:07:45,000
bounds because maybe we are
integrating over a region whose

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00:07:45,000 --> 00:07:49,000
equation in xy coordinates is
very hard to write down.

97
00:07:49,000 --> 00:07:51,000
But, it becomes much easier in
terms of u and v.

98
00:07:51,000 --> 00:07:57,000
And then, the bounds would be
much easier to set up with u and

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00:07:57,000 --> 00:08:02,000
v.
Anyway, so, whatever the reason

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00:08:02,000 --> 00:08:12,000
might be, typically it would be
to simplify the integrant or the

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00:08:12,000 --> 00:08:18,000
bounds.
Well, how do we convert dxdy to

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00:08:18,000 --> 00:08:21,000
dudv?
So, we want to understand,

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00:08:21,000 --> 00:08:27,000
what's the relation between,
let's call dA the area element

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00:08:27,000 --> 00:08:31,000
in xy coordinates.
So, dA is dxdy,

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00:08:31,000 --> 00:08:34,000
maybe dydx depending on the
order.

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00:08:34,000 --> 00:08:39,000
And, the area element in uv
coordinates, let me call that dA

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00:08:39,000 --> 00:08:42,000
prime just to make it look
different.

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00:08:42,000 --> 00:08:50,000
So, that would just be dudv,
or dvdu depending on which

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00:08:50,000 --> 00:08:55,000
order I will want to set it up
in.

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00:08:55,000 --> 00:09:01,000
So, to find this relation,
it's probably best to draw a

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00:09:01,000 --> 00:09:09,000
picture to see what happens.
Let's consider a small piece of

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00:09:09,000 --> 00:09:18,000
the xy plane with area delta(A)
corresponding to just a box with

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00:09:18,000 --> 00:09:24,000
sides delta(y) and delta(x).
OK, and let's try to figure out

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00:09:24,000 --> 00:09:27,000
what it will look like in terms
of u and v.

115
00:09:27,000 --> 00:09:29,000
And then, we'll say,
well, when we integrate,

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00:09:29,000 --> 00:09:32,000
we're really summing the value
of the function of a lot of

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00:09:32,000 --> 00:09:36,000
small boxes times their area.
But, the problem is that the

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00:09:36,000 --> 00:09:40,000
area of the box in here is not
the same as the area of the box

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00:09:40,000 --> 00:09:47,000
in uv coordinates.
There, maybe it will look like,

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00:09:47,000 --> 00:09:49,000
actually,
if you see that these are

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linear changes of variables,
you know that the rectangle

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00:09:52,000 --> 00:09:55,000
will become a parallelogram
after the change of variables.

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00:09:55,000 --> 00:10:00,000
So, the area of a parallelogram
delta(A) prime,

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00:10:00,000 --> 00:10:05,000
well, we will have to figure
out how they are related so that

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00:10:05,000 --> 00:10:09,000
we can decide what conversion
factor,

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00:10:09,000 --> 00:10:13,000
what's the exchange rate
between these two currencies for

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00:10:13,000 --> 00:10:20,000
area?
OK, any questions at this point?

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00:10:20,000 --> 00:10:27,000
No? Still with me mostly?
I see a lot of tired faces.

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00:10:27,000 --> 00:10:34,000
Yes?
Why is delta(A) prime a

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00:10:34,000 --> 00:10:37,000
parallelogram?
That's a very good question.

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00:10:37,000 --> 00:10:41,000
Well, see, if I look at the
side of a rectangle,

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00:10:41,000 --> 00:10:45,000
say there's a vertical side,
it means I'm going to increase

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00:10:45,000 --> 00:10:49,000
y, keeping x the same.
If I look at the formulas for u

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00:10:49,000 --> 00:10:52,000
and v, they are linear formulas
in terms of x and y.

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00:10:52,000 --> 00:10:56,000
So, if I just increase y,
see that u is going to decrease

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00:10:56,000 --> 00:10:58,000
at a rate of two.
v is going to increase at a

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00:10:58,000 --> 00:11:02,000
rate of one at constant rates.
And, it doesn't matter whether

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00:11:02,000 --> 00:11:04,000
I was looking at this site or at
that site.

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00:11:04,000 --> 00:11:06,000
So, basically straight lines
become straight lines.

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00:11:06,000 --> 00:11:09,000
And if they are parallel,
they stay parallel.

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00:11:09,000 --> 00:11:11,000
So, if you just look at what
the transformation,

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00:11:11,000 --> 00:11:14,000
from xy to uv does,
it does this kind of thing.

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00:11:14,000 --> 00:11:17,000
Actually, this transformation
here you can express by a

144
00:11:17,000 --> 00:11:18,000
matrix.
And, remember,

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00:11:18,000 --> 00:11:20,000
we've seen what matrices do the
pictures.

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00:11:20,000 --> 00:11:24,000
We just take straight lines to
straight lines.

147
00:11:24,000 --> 00:11:29,000
They keep the notion of being
parallel, but of course they

148
00:11:29,000 --> 00:11:32,000
mess up lengths,
angles, and all that.

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00:11:32,000 --> 00:11:38,000
OK, so let's see.
So, let's try to figure out,

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00:11:38,000 --> 00:11:42,000
what is the area of this guy?
Well, in fact,

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00:11:42,000 --> 00:11:46,000
what I've been saying about
this transformation being

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00:11:46,000 --> 00:11:49,000
linear,
and transforming all of the

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00:11:49,000 --> 00:11:53,000
vertical lines in the same way,
all the horizontal lines in the

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00:11:53,000 --> 00:11:54,000
same way,
it tells me,

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00:11:54,000 --> 00:11:57,000
also, I should have a constant
scaling factor,

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00:11:57,000 --> 00:12:00,000
right, because how much I've
scaled my rectangle doesn't

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00:12:00,000 --> 00:12:03,000
depend on where my rectangle is.
If I move my rectangle to

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00:12:03,000 --> 00:12:05,000
somewhere else,
I have a rectangle of the same

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00:12:05,000 --> 00:12:08,000
size, same shape,
it will become a parallelogram

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00:12:08,000 --> 00:12:10,000
of the same size,
same shape somewhere else.

161
00:12:10,000 --> 00:12:13,000
So, in fact,
I can just take the simplest

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00:12:13,000 --> 00:12:16,000
rectangle I can think of and see
how its area changes.

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00:12:16,000 --> 00:12:18,000
And, if you don't believe me,
then try with any other

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00:12:18,000 --> 00:12:21,000
rectangle.
You will see it works exactly

165
00:12:21,000 --> 00:12:28,000
the same way.
OK, so I claim that the area

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00:12:28,000 --> 00:12:41,000
scaling factor -- -- here in
this case doesn't depend on the

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00:12:41,000 --> 00:12:53,000
choice of the rectangle.
And I should say that because

168
00:12:53,000 --> 00:13:05,000
we are actually doing a linear
change of variables -- So,

169
00:13:05,000 --> 00:13:08,000
you know, somehow,
the exchange rate between uv

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00:13:08,000 --> 00:13:10,000
and xy is going to be the same
everywhere.

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00:13:10,000 --> 00:13:14,000
So, let's try to see what
happens to the simplest

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00:13:14,000 --> 00:13:19,000
rectangle I can think of,
namely, just the unit square.

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00:13:19,000 --> 00:13:21,000
And, you know,
if you don't trust me,

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00:13:21,000 --> 00:13:24,000
then while I'm doing this one,
do it with a different

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00:13:24,000 --> 00:13:26,000
rectangle.
Do the same calculation,

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00:13:26,000 --> 00:13:30,000
and see that you will get the
same conversion ratio.

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00:13:30,000 --> 00:13:37,000
So, let's say that I take a
unit square -- -- so,

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00:13:37,000 --> 00:13:45,000
something that goes from zero
to one both in x and y

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00:13:45,000 --> 00:13:49,000
directions.
OK, and let's try to figure out

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00:13:49,000 --> 00:13:51,000
what it looks like on the other
side.

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00:13:51,000 --> 00:13:58,000
So, here the area is one.
Let's try to draw it in terms

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00:13:58,000 --> 00:14:00,000
of u and v coordinates,
OK?

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00:14:00,000 --> 00:14:05,000
So, here we have x equals 0,
y equals 0.

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00:14:05,000 --> 00:14:13,000
Well, that tells us u and v are
going to be 0.

185
00:14:13,000 --> 00:14:17,000
Next, let's look at this corner.
Well, in xy coordinates,

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00:14:17,000 --> 00:14:20,000
this is one zero.
If you plug x equals 1,

187
00:14:20,000 --> 00:14:24,000
y equals 0, you get u equals 3;
v equals 1.

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00:14:24,000 --> 00:14:38,000
So, that goes somewhere here.
And so, this edge of the square

189
00:14:38,000 --> 00:14:44,000
will become this line here,
OK?

190
00:14:44,000 --> 00:14:49,000
Next, let's look at that point.
So that point here was (0,1).

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00:14:49,000 --> 00:15:01,000
If I plug x equals zero y
equals one I will get (-2,1).

192
00:15:01,000 --> 00:15:11,000
So, this edge goes here.
Then, if you put x equals one,

193
00:15:11,000 --> 00:15:14,000
y equals one,
you will get u equals 1,

194
00:15:14,000 --> 00:15:22,000
v equals 2.
So, I want (1,2).

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00:15:22,000 --> 00:15:28,000
And, these edges will go to
these edges here.

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00:15:28,000 --> 00:15:31,000
And, you see,
it does look like a

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00:15:31,000 --> 00:15:38,000
parallelogram.
OK, so now what the area of

198
00:15:38,000 --> 00:15:44,000
this parallelogram?
Well, we can get that by taking

199
00:15:44,000 --> 00:15:47,000
the determinant of these two
vectors.

200
00:15:47,000 --> 00:15:53,000
So, one of them is ,
and the other one is

201
00:15:53,000 --> 00:15:57,000
.
That will be 3 2.

202
00:15:57,000 --> 00:16:01,000
That's 5.
OK, this parallelogram is

203
00:16:01,000 --> 00:16:04,000
apparently five times the size
of this square.

204
00:16:04,000 --> 00:16:07,000
Here, it looks like it's less
because I somehow changed my

205
00:16:07,000 --> 00:16:10,000
scale.
I mean, my unit length is

206
00:16:10,000 --> 00:16:15,000
smaller here than here.
But, it should be a lot bigger

207
00:16:15,000 --> 00:16:16,000
than that.
OK,

208
00:16:16,000 --> 00:16:19,000
and if you do the same
calculations not with zero and

209
00:16:19,000 --> 00:16:21,000
one,
but with x and x plus delta x,

210
00:16:21,000 --> 00:16:24,000
and so on,
you will still find that the

211
00:16:24,000 --> 00:16:27,000
area has been multiplied by
five.

212
00:16:27,000 --> 00:16:36,000
So, that tells us,
actually for any other

213
00:16:36,000 --> 00:16:47,000
rectangle, area is also
multiplied by five.

214
00:16:47,000 --> 00:16:52,000
So, that tells us that dA
prime, the area element in uv

215
00:16:52,000 --> 00:16:57,000
coordinate is worth five times
more than the area element in

216
00:16:57,000 --> 00:16:59,000
the xy coordinate.

217
00:17:11,000 --> 00:17:17,000
So, that means du dv is worth
five times dx dy.

218
00:17:17,000 --> 00:17:30,000
What's so funny?
What?

219
00:17:30,000 --> 00:17:40,000
Oh.
[LAUGHTER] OK, rectangle.

220
00:17:40,000 --> 00:17:45,000
OK, is that OK now?
Did I misspell other words?

221
00:17:45,000 --> 00:17:48,000
No?
OK, it's really hard to see

222
00:17:48,000 --> 00:17:54,000
when you are up close.
It's much easier from a

223
00:17:54,000 --> 00:17:58,000
distance.
OK, so yeah,

224
00:17:58,000 --> 00:18:05,000
so we've said our
transformation multiplies areas

225
00:18:05,000 --> 00:18:09,000
by five.
And so, dudv is five times dxdy.

226
00:18:09,000 --> 00:18:14,000
So, if I'm integrating some
function, dx dy,

227
00:18:14,000 --> 00:18:20,000
then when I switch to uv
coordinates, I will have to

228
00:18:20,000 --> 00:18:26,000
replace that by one fifth dudv.
OK, and of course I would also,

229
00:18:26,000 --> 00:18:29,000
here my function would probably
involve x and y.

230
00:18:29,000 --> 00:18:33,000
I will replace them by u's and
v's.

231
00:18:33,000 --> 00:18:35,000
And, the bounds,
well, the shape of my origin in

232
00:18:35,000 --> 00:18:39,000
the xy coordinates I will have
to switch to some shape in the

233
00:18:39,000 --> 00:18:42,000
uv coordinates.
And, that's also something that

234
00:18:42,000 --> 00:18:46,000
might be easy or might be tricky
depending on what origin we are

235
00:18:46,000 --> 00:18:50,000
looking at.
So, usually we will do changes

236
00:18:50,000 --> 00:18:54,000
of variables to actually
simplify the region so it

237
00:18:54,000 --> 00:18:58,000
becomes easier to set up the
bounds.

238
00:18:58,000 --> 00:19:05,000
So, anyway, so this is kind of
an illustration of a general

239
00:19:05,000 --> 00:19:07,000
case.
And, why is that?

240
00:19:07,000 --> 00:19:10,000
Well, here it looks very easy.
We are just using linear

241
00:19:10,000 --> 00:19:14,000
formulas, and somehow the
relation between dx dy and du dv

242
00:19:14,000 --> 00:19:17,000
is the same everywhere.
If you take actually more

243
00:19:17,000 --> 00:19:21,000
complicated changes of variables
that's not true because usually

244
00:19:21,000 --> 00:19:25,000
you will expect that there are
some places where the rescaling

245
00:19:25,000 --> 00:19:28,000
is enlarging things,
and some of other places where

246
00:19:28,000 --> 00:19:31,000
things are shrunk,
so, certainly the exchange rate

247
00:19:31,000 --> 00:19:35,000
between dudv and dxdy will
fluctuate from point to point.

248
00:19:35,000 --> 00:19:37,000
It's the same as if you're
trying to change dollars to

249
00:19:37,000 --> 00:19:39,000
euros.
It depends on where you do it.

250
00:19:39,000 --> 00:19:43,000
You will get a better rate or a
worse one.

251
00:19:43,000 --> 00:19:47,000
So, of course,
we'll get a formula where

252
00:19:47,000 --> 00:19:52,000
actually this scaling factor
depends on x and y or on u and

253
00:19:52,000 --> 00:19:54,000
v.
But, if you fix a point,

254
00:19:54,000 --> 00:19:57,000
then we have linear
approximation.

255
00:19:57,000 --> 00:20:00,000
And, linear approximation tells
us, oh, we can do as if our

256
00:20:00,000 --> 00:20:02,000
function is just a linear
function of x and y.

257
00:20:02,000 --> 00:20:06,000
So then, we can do it the same
way we did here.

258
00:20:06,000 --> 00:20:18,000
OK, so let's try to think about
that.

259
00:20:18,000 --> 00:20:22,000
So, in the general case,
well, that means we will

260
00:20:22,000 --> 00:20:26,000
replace x and y by new
coordinates, u and v.

261
00:20:26,000 --> 00:20:30,000
And, u and v will be some
functions of x and y.

262
00:20:30,000 --> 00:20:34,000
So, well,
we'll have an approximation

263
00:20:34,000 --> 00:20:37,000
formula which tells us that the
change in u,

264
00:20:37,000 --> 00:20:40,000
if I change x or y a little
bit,

265
00:20:40,000 --> 00:20:45,000
will be roughly (u sub x times
change in x) (u sub y times

266
00:20:45,000 --> 00:20:50,000
change in y).
And, the change in v will be

267
00:20:50,000 --> 00:20:57,000
roughly (v sub x delta x) (v sub
y delta y).

268
00:20:57,000 --> 00:21:03,000
Or, the other way to say it,
if you want in matrix form is

269
00:21:03,000 --> 00:21:08,000
delta u delta v is,
sorry, approximately equal to

270
00:21:08,000 --> 00:21:12,000
matrix |u sub x,
u sub y, v sub x,

271
00:21:12,000 --> 00:21:20,000
v sub y| times matrix |delta x,
delta y|,

272
00:21:20,000 --> 00:21:26,000
OK?
So, if we look at that,

273
00:21:26,000 --> 00:21:32,000
what it tells us, in fact,
is that if we take a small

274
00:21:32,000 --> 00:21:40,000
rectangle in xy coordinates,
so that means we have a certain

275
00:21:40,000 --> 00:21:44,000
point, x, y,
and then we have a certain

276
00:21:44,000 --> 00:21:51,000
width.
This is going to be too small.

277
00:21:51,000 --> 00:21:56,000
Well, so, I have my width,
delta x.

278
00:21:56,000 --> 00:22:06,000
I have my height, delta y.
This is going to correspond to

279
00:22:06,000 --> 00:22:14,000
a small uv parallelogram.
And, what the shape and the

280
00:22:14,000 --> 00:22:20,000
size of the parallelogram are
depends on the partial

281
00:22:20,000 --> 00:22:24,000
derivatives of u and v.
So, in particular,

282
00:22:24,000 --> 00:22:26,000
it depends on at which point we
are.

283
00:22:26,000 --> 00:22:30,000
But still, at a given point,
it's a bit like that.

284
00:22:30,000 --> 00:22:35,000
And, so if we do the same
argument as before,

285
00:22:35,000 --> 00:22:41,000
what we will see is that the
scaling factor is actually the

286
00:22:41,000 --> 00:22:45,000
determinant of this
transformation.

287
00:22:45,000 --> 00:22:50,000
So, that's one thing that maybe
we didn't emphasize enough when

288
00:22:50,000 --> 00:22:53,000
we did matrices at the beginning
of a semester.

289
00:22:53,000 --> 00:22:57,000
But, when you have a linear
transformation between

290
00:22:57,000 --> 00:23:01,000
variables, the determinant of
that transformation represents

291
00:23:01,000 --> 00:23:05,000
how it scales areas.
OK, so one way to think about

292
00:23:05,000 --> 00:23:09,000
it is just to try it and see
what happens.

293
00:23:09,000 --> 00:23:12,000
Take this side.
This side in x,

294
00:23:12,000 --> 00:23:16,000
y coordinates corresponds to
delta x and zero.

295
00:23:16,000 --> 00:23:20,000
And, now, if you take the image
of that, if you see what happens

296
00:23:20,000 --> 00:23:24,000
to delta u and delta v,
that will be basically u sub x

297
00:23:24,000 --> 00:23:28,000
delta x and v sub x delta x.
There's no delta y.

298
00:23:28,000 --> 00:23:33,000
For the other side,
OK, so maybe I should do it

299
00:23:33,000 --> 00:23:36,000
actually.
So, you know,

300
00:23:36,000 --> 00:23:40,000
if we move in the x,
y coordinates by delta x and

301
00:23:40,000 --> 00:23:45,000
zero,
then delta u and delta v will

302
00:23:45,000 --> 00:23:50,000
be approximately u sub x delta
x,

303
00:23:50,000 --> 00:24:02,000
and v sub x delta x.
And, on the other hand,

304
00:24:02,000 --> 00:24:04,000
if you move in the other
direction along the other side

305
00:24:04,000 --> 00:24:08,000
of your rectangle,
zero and delta y,

306
00:24:08,000 --> 00:24:13,000
then the change in u and the
change in v will correspond to,

307
00:24:13,000 --> 00:24:16,000
well, how does u change?
That's u sub y delta y,

308
00:24:16,000 --> 00:24:20,000
and v changes by v sub y delta
y.

309
00:24:20,000 --> 00:24:22,000
And so, now,
if you take the determinant of

310
00:24:22,000 --> 00:24:25,000
these two vectors,
OK, so these are the sides of

311
00:24:25,000 --> 00:24:29,000
your parallelogram up here.
And, if you take these sides to

312
00:24:29,000 --> 00:24:31,000
get the area of the
parallelogram,

313
00:24:31,000 --> 00:24:33,000
you'll need to take the
determinant.

314
00:24:33,000 --> 00:24:41,000
And, the determinant will be
the determinant of this matrix

315
00:24:41,000 --> 00:24:48,000
times delta x times delta y.
So, the area in uv coordinates

316
00:24:48,000 --> 00:24:53,000
will be the determinant of a
matrix times delta x,

317
00:24:53,000 --> 00:24:57,000
delta y.
And so,

318
00:24:57,000 --> 00:25:02,000
what I'm trying to say is that
when you have a general change

319
00:25:02,000 --> 00:25:06,000
of variables,
du dv versus dx dy is given by

320
00:25:06,000 --> 00:25:11,000
the determinant of this matrix
of partial derivatives.

321
00:25:11,000 --> 00:25:13,000
It doesn't matter in which
order you write it.

322
00:25:13,000 --> 00:25:16,000
I mean, you can put in rows or
columns.

323
00:25:16,000 --> 00:25:18,000
If you transpose a matrix,
that doesn't change the

324
00:25:18,000 --> 00:25:21,000
determinant.
It's just any sensible matrix

325
00:25:21,000 --> 00:25:24,000
that you can write will have the
correct determinant.

326
00:26:02,000 --> 00:26:07,000
OK, so what we need to know is
the following thing.

327
00:26:07,000 --> 00:26:11,000
So,
we define something called the

328
00:26:11,000 --> 00:26:16,000
Jacobian of a change of
variables and used the letter J,

329
00:26:16,000 --> 00:26:21,000
or maybe a more useful notation
is partial of u,

330
00:26:21,000 --> 00:26:24,000
v over partial of x,
y.

331
00:26:24,000 --> 00:26:27,000
That's a very strange notation.
I mean, that doesn't mean that

332
00:26:27,000 --> 00:26:30,000
we are actually taking the
partial derivatives of anything.

333
00:26:30,000 --> 00:26:34,000
OK, it's just a notation to
remind us that this has to do

334
00:26:34,000 --> 00:26:37,000
with the ratio between dudv and
dxdy.

335
00:26:37,000 --> 00:26:42,000
And, it's obtained using the
partial derivatives of u and v

336
00:26:42,000 --> 00:26:50,000
with respect to x and y.
So, it's the determinant of the

337
00:26:50,000 --> 00:26:55,000
matrix |u sub x,
u sub y, v sub x,

338
00:26:55,000 --> 00:27:02,000
v sub y|, the matrix that I had
up there.

339
00:27:02,000 --> 00:27:10,000
OK, and what we need to know is
that du dv is equal to the

340
00:27:10,000 --> 00:27:17,000
absolute value of J dx dy.
Or, if you prefer to see it in

341
00:27:17,000 --> 00:27:23,000
the easier to remember version,
it's (absolute value of d of

342
00:27:23,000 --> 00:27:27,000
(u, v) over partial xy) times dx
dy.

343
00:27:27,000 --> 00:27:32,000
OK, so this is just what you
need to remember,

344
00:27:32,000 --> 00:27:38,000
and it says that the area in uv
coordinates is worth,

345
00:27:38,000 --> 00:27:42,000
well, the ratio to the xy
coordinates is given by this

346
00:27:42,000 --> 00:27:46,000
Jacobian determinant except for
one small thing.

347
00:27:46,000 --> 00:27:48,000
It's given by,
actually, the absolute value of

348
00:27:48,000 --> 00:27:52,000
this guy.
OK, so what's going on here?

349
00:27:52,000 --> 00:27:56,000
What's going on here is when we
are saying the determinant of

350
00:27:56,000 --> 00:27:59,000
the transformation tells us how
the area is multiplied,

351
00:27:59,000 --> 00:28:02,000
there's a small catch.
Remember, the determinants are

352
00:28:02,000 --> 00:28:06,000
equal to areas up to sine.
Sometimes, the determinant is

353
00:28:06,000 --> 00:28:10,000
negative because of reversing
the orientation of things.

354
00:28:10,000 --> 00:28:13,000
But, the area is still the same.
Area is always positive.

355
00:28:13,000 --> 00:28:17,000
So, the area elements are
actually related by the absolute

356
00:28:17,000 --> 00:28:23,000
value of this guy.
OK, so if you find -10 as your

357
00:28:23,000 --> 00:28:29,000
answer, then du dv is still ten
times dx dy.

358
00:28:29,000 --> 00:28:33,000
OK, so I didn't put it all
together because then you would

359
00:28:33,000 --> 00:28:36,000
have two sets of vertical bars.
See, this is a vertical bar for

360
00:28:36,000 --> 00:28:38,000
absolute value.
This is vertical bar for

361
00:28:38,000 --> 00:28:42,000
determinant.
They're not the same.

362
00:28:42,000 --> 00:28:46,000
That's the one thing to
remember.

363
00:28:46,000 --> 00:28:54,000
OK, any questions about this?
No?

364
00:28:54,000 --> 00:29:06,000
OK.
So, actually let's do our first

365
00:29:06,000 --> 00:29:12,000
example of that.
Let's check what we had for

366
00:29:12,000 --> 00:29:16,000
polar coordinates.
Last time I told you if we have

367
00:29:16,000 --> 00:29:19,000
dx dy we could switch it to r dr
d theta.

368
00:29:19,000 --> 00:29:25,000
And, we had some argument for
that by looking at the area of a

369
00:29:25,000 --> 00:29:31,000
small circular sector.
But, let's check again using

370
00:29:31,000 --> 00:29:37,000
this new method.
So, in polar coordinates I'm

371
00:29:37,000 --> 00:29:44,000
setting x equals r cosine theta,
y equals r sine theta.

372
00:29:44,000 --> 00:29:48,000
So, the Jacobian for this
change of variables,

373
00:29:48,000 --> 00:29:54,000
so let's say I'm trying to find
the partial derivatives of x,

374
00:29:54,000 --> 00:29:58,000
y with respect to r,
theta.

375
00:29:58,000 --> 00:30:04,000
Well, what is,
OK, let me actually write them

376
00:30:04,000 --> 00:30:10,000
here again for you.
And, so what does that become?

377
00:30:10,000 --> 00:30:17,000
Partial x over partial r is
just cosine theta.

378
00:30:17,000 --> 00:30:25,000
Partial x over partial theta is
negative r sine theta.

379
00:30:25,000 --> 00:30:27,000
Sorry, I guess I'm going to run
out of space here.

380
00:30:27,000 --> 00:30:33,000
So, let me do it underneath.
So, we said x sub r is cosine

381
00:30:33,000 --> 00:30:36,000
theta;
x sub theta is negative r sine

382
00:30:36,000 --> 00:30:41,000
theta.
y sub r is sine;

383
00:30:41,000 --> 00:30:49,000
y sub theta is r cosine.
And now, if we compute this

384
00:30:49,000 --> 00:30:58,000
determinant, we'll get (r cosine
squared theta) (r sine squared

385
00:30:58,000 --> 00:31:02,000
theta).
And, that simplifies to r.

386
00:31:02,000 --> 00:31:08,000
So, dx dy is,
well, absolute value of r dr d

387
00:31:08,000 --> 00:31:12,000
theta.
But, remember that r is always

388
00:31:12,000 --> 00:31:18,000
positive.
So, it's r dr d theta.

389
00:31:18,000 --> 00:31:26,000
OK, so that's another way to
justify how we did double

390
00:31:26,000 --> 00:31:34,000
integrals in polar coordinates.
OK, any questions on that?

391
00:31:34,000 --> 00:31:47,000
Where?
Yeah, OK.

392
00:31:47,000 --> 00:31:52,000
Yeah, so this one seems to be
switching.

393
00:31:52,000 --> 00:31:56,000
Well, it depends what you do.
So, OK, actually here's an

394
00:31:56,000 --> 00:32:00,000
important thing that I didn't
quite say.

395
00:32:00,000 --> 00:32:04,000
So, I said, you know,
we are going to switch from xy

396
00:32:04,000 --> 00:32:07,000
to uv.
We can also switch from uv to

397
00:32:07,000 --> 00:32:09,000
xy.
And, this conversion ratio,

398
00:32:09,000 --> 00:32:12,000
the Jacobian,
works both ways.

399
00:32:12,000 --> 00:32:16,000
Once you have found the ratio
between du dv and dx dy,

400
00:32:16,000 --> 00:32:20,000
then it works one way or it
works the other way.

401
00:32:20,000 --> 00:32:22,000
I mean, here,
of course, we get the answer in

402
00:32:22,000 --> 00:32:26,000
terms of r.
So, this would let us switch

403
00:32:26,000 --> 00:32:30,000
from xy to r theta.
But, we can also switch from r

404
00:32:30,000 --> 00:32:33,000
theta to xy.
Just, we'd write dr d theta

405
00:32:33,000 --> 00:32:37,000
equals (1 over r) times dx dy.
And then we'd have,

406
00:32:37,000 --> 00:32:41,000
of course, to replace r by its
formula in xy coordinates.

407
00:32:41,000 --> 00:32:43,000
Usually, we don't do that.
Usually, we actually start with

408
00:32:43,000 --> 00:32:47,000
xy and switch to polar.
But,

409
00:32:47,000 --> 00:32:50,000
so in general,
when you have this formula

410
00:32:50,000 --> 00:32:54,000
relating du dv with dx dy,
you can use it both ways,

411
00:32:54,000 --> 00:33:00,000
either to switch from du dv to
dx dy or the other way around.

412
00:33:00,000 --> 00:33:04,000
And, the thing that I'm not
telling you that now I should

413
00:33:04,000 --> 00:33:08,000
probably tell you is I could
define two Jacobians because if

414
00:33:08,000 --> 00:33:12,000
I solve for xy in terms of uv
instead of uv in terms of xy,

415
00:33:12,000 --> 00:33:15,000
then I can compute two
different Jacobians.

416
00:33:15,000 --> 00:33:19,000
I can compute partial uv over
partial xy, or I can compute

417
00:33:19,000 --> 00:33:24,000
partial xy over partial uv if I
have the formulas both ways.

418
00:33:24,000 --> 00:33:27,000
Well, the good news is these
guys are the inverse of each

419
00:33:27,000 --> 00:33:29,000
other.
So, the two formulas that you

420
00:33:29,000 --> 00:33:31,000
might get are consistent.

421
00:33:59,000 --> 00:34:16,000
OK, so useful remark -- So,
say that you can compute both

422
00:34:16,000 --> 00:34:23,000
-- -- these guys.
Well, then actually,

423
00:34:23,000 --> 00:34:26,000
the product will just be 1.
So, they are the inverse of

424
00:34:26,000 --> 00:34:28,000
each other.
So, it doesn't matter which one

425
00:34:28,000 --> 00:34:34,000
you compute.
You can compute whichever one

426
00:34:34,000 --> 00:34:45,000
is the easiest to compute no
matter which one of the two you

427
00:34:45,000 --> 00:34:48,000
need.
And, one way to see that is

428
00:34:48,000 --> 00:34:50,000
that, in fact,
we're looking at the

429
00:34:50,000 --> 00:34:53,000
determinant of these matrices
that tell us the relation in

430
00:34:53,000 --> 00:34:56,000
variables.
So, if one of them tells you

431
00:34:56,000 --> 00:34:58,000
how delta u delta v relate to
delta x delta y,

432
00:34:58,000 --> 00:35:00,000
the other one does the opposite
thing.

433
00:35:00,000 --> 00:35:03,000
It means they are the inverse
matrices.

434
00:35:03,000 --> 00:35:06,000
And, the determinant of the
inverse matrix is the inverse of

435
00:35:06,000 --> 00:35:10,000
the determinant.
So, they are really

436
00:35:10,000 --> 00:35:14,000
interchangeable.
I mean, you can just compute

437
00:35:14,000 --> 00:35:17,000
whichever one is easiest.
So here, if you wanted,

438
00:35:17,000 --> 00:35:22,000
dr d theta in terms of dx dy,
it's easier to do this and then

439
00:35:22,000 --> 00:35:27,000
move the r over there than to
first solve for r and theta as

440
00:35:27,000 --> 00:35:31,000
functions of x and y and then do
the entire thing again.

441
00:35:31,000 --> 00:35:42,000
But, you can do it if you want.
I mean, it works.

442
00:35:42,000 --> 00:35:45,000
Oh yeah, the other useful
remark, so, I mentioned it,

443
00:35:45,000 --> 00:35:49,000
but let me emphasize again.
So, now, the ratio between du

444
00:35:49,000 --> 00:35:51,000
dv and dx dy,
it's not a constant anymore,

445
00:35:51,000 --> 00:35:54,000
although there it used to be
five.

446
00:35:54,000 --> 00:35:56,000
But now, it's become r,
or anything.

447
00:35:56,000 --> 00:35:58,000
In general, it will be a
function that depends on the

448
00:35:58,000 --> 00:36:01,000
variables.
So, it's not true that you can

449
00:36:01,000 --> 00:36:04,000
just say, oh,
I'll put a constant times du

450
00:36:04,000 --> 00:36:14,000
dv.
Yes?

451
00:36:14,000 --> 00:36:17,000
It would still work the same.
You could imagine drawing a

452
00:36:17,000 --> 00:36:20,000
picture where r and theta are
the Cartesian coordinates,

453
00:36:20,000 --> 00:36:22,000
and your picture would be
completely messed up.

454
00:36:22,000 --> 00:36:26,000
It would be a very strange
thing to do to try to draw,

455
00:36:26,000 --> 00:36:30,000
you know, I'm going to do it,
but don't take notes on that.

456
00:36:30,000 --> 00:36:32,000
You could try to draw picture
like that, and then a circle

457
00:36:32,000 --> 00:36:34,000
would start looking like,
you know, a disk would look

458
00:36:34,000 --> 00:36:35,000
like that.
It would be very

459
00:36:35,000 --> 00:36:37,000
counterintuitive.
But, you could do it.

460
00:36:37,000 --> 00:36:41,000
And that would be equivalent to
what we did with a previous

461
00:36:41,000 --> 00:36:43,000
change of variables.
So, in this case,

462
00:36:43,000 --> 00:36:47,000
certainly you would never draw
a picture like that.

463
00:36:47,000 --> 00:36:59,000
But, you could do it.
OK, so now let's do a complete

464
00:36:59,000 --> 00:37:07,000
example to see how things fit
together, how we do everything.

465
00:37:07,000 --> 00:37:10,000
So, let's say that we want to
compute, so I have to warn you,

466
00:37:10,000 --> 00:37:12,000
it's going to be a very silly
example.

467
00:37:12,000 --> 00:37:16,000
It's an example where it's much
easier to compute things without

468
00:37:16,000 --> 00:37:19,000
the change of variables.
But, you know,

469
00:37:19,000 --> 00:37:24,000
it's good practice in the sense
that we're going to make it so

470
00:37:24,000 --> 00:37:29,000
complicated that if we can do
this one, then we can do that

471
00:37:29,000 --> 00:37:31,000
one.
So, let's say that we want to

472
00:37:31,000 --> 00:37:33,000
compute this.
And, of course,

473
00:37:33,000 --> 00:37:35,000
it's very easy to compute it
directly.

474
00:37:35,000 --> 00:37:42,000
But let's say that for some
evil reason we want to do that

475
00:37:42,000 --> 00:37:49,000
by changing variables to u
equals x and v equals xy.

476
00:37:49,000 --> 00:37:55,000
OK, that's a very strange idea,
but let's do it anyway.

477
00:37:55,000 --> 00:37:58,000
I mean, normally,
you would only do this kind of

478
00:37:58,000 --> 00:38:01,000
substitution if either it
simplifies a lot the function

479
00:38:01,000 --> 00:38:03,000
you are integrating,
or it simplifies a lot the

480
00:38:03,000 --> 00:38:06,000
region on which you are
integrating.

481
00:38:06,000 --> 00:38:12,000
And here, neither happens.
But anyway, so the first thing

482
00:38:12,000 --> 00:38:16,000
we have to do here is figure out
what we are going to be

483
00:38:16,000 --> 00:38:18,000
integrating.
OK, so to do that,

484
00:38:18,000 --> 00:38:23,000
we should figure out what dx dy
will become in terms of u and v.

485
00:38:23,000 --> 00:38:26,000
So, that's what we've just seen
using the Jacobian.

486
00:38:26,000 --> 00:38:32,000
OK, so the first thing to do is
find the area element.

487
00:38:32,000 --> 00:38:33,000
And, for that,
we use the Jacobian.

488
00:38:33,000 --> 00:38:36,000
So, well, let's see,
the one that we can do easily

489
00:38:36,000 --> 00:38:40,000
is partials of u and v with
respect to x and y.

490
00:38:40,000 --> 00:38:42,000
I mean, the other one is not
very hard because here you can

491
00:38:42,000 --> 00:38:45,000
solve easily.
But, the one that's given to

492
00:38:45,000 --> 00:38:49,000
you is partial of u and v with
respect to x and y,

493
00:38:49,000 --> 00:38:55,000
so partial u partial x is one.
Partial u partial y is zero.

494
00:38:55,000 --> 00:39:03,000
Partial v partial x is y.
And partial v partial y is x.

495
00:39:03,000 --> 00:39:17,000
So that's just x.
So, that means that du dv is x

496
00:39:17,000 --> 00:39:20,000
dx dy.
Well, it would be absolute

497
00:39:20,000 --> 00:39:23,000
value of x, but x is positive in
our origin.

498
00:39:23,000 --> 00:39:35,000
So, at least we don't have to
worry about that.

499
00:39:35,000 --> 00:39:45,000
OK, so now that we have that,
we can try to look at the

500
00:39:45,000 --> 00:39:55,000
integrand in terms of u and v.
OK, so we were integrating x

501
00:39:55,000 --> 00:40:00,000
squared y dx dy.
So, let's switch it.

502
00:40:00,000 --> 00:40:09,000
Well, let's first switch the dx
dy that becomes one over x du

503
00:40:09,000 --> 00:40:15,000
dv.
So, that's actually xy du dv.

504
00:40:15,000 --> 00:40:18,000
And, what is xy in terms of u
and v?

505
00:40:18,000 --> 00:40:20,000
Well, here at least we had a
little bit of luck.

506
00:40:20,000 --> 00:40:26,000
xy is just v.
So, that's v du dv.

507
00:40:26,000 --> 00:40:32,000
So, in fact,
what we'll be computing is a

508
00:40:32,000 --> 00:40:40,000
double integral over some
mysterious region of v du dv.

509
00:40:40,000 --> 00:40:44,000
Now, last but not least,
we'll have to find what are the

510
00:40:44,000 --> 00:40:49,000
bounds for u and v in the new
integral so that we know how to

511
00:40:49,000 --> 00:40:50,000
evaluate this.

512
00:41:14,000 --> 00:41:17,000
In fact, well,
we could do it du dv or dv du.

513
00:41:17,000 --> 00:41:23,000
We don't know yet.
Oh, amazing.

514
00:41:23,000 --> 00:41:31,000
It went all the way down this
time.

515
00:41:31,000 --> 00:41:43,000
OK, so it could be dv du if
that's easier.

516
00:41:43,000 --> 00:41:46,000
So, let's try to find the
bounds.

517
00:41:46,000 --> 00:41:52,000
In this case,
that's the hardest part.

518
00:41:52,000 --> 00:42:00,000
OK, so let me draw a picture in
xy coordinates and try to

519
00:42:00,000 --> 00:42:06,000
understand things using that.
OK, so x and y go from zero to

520
00:42:06,000 --> 00:42:08,000
one.
The region that we want to

521
00:42:08,000 --> 00:42:11,000
integrate over was just this
square.

522
00:42:11,000 --> 00:42:16,000
Let's try to figure out how u
and v vary there.

523
00:42:16,000 --> 00:42:23,000
So, let's say that we're going
to do it du dv.

524
00:42:23,000 --> 00:42:32,000
OK, so What we want to
understand is how u and v vary

525
00:42:32,000 --> 00:42:36,000
in here.
What's going to happen?

526
00:42:36,000 --> 00:42:40,000
So, the way we can think about
it is we try to figure out how

527
00:42:40,000 --> 00:42:43,000
we are slicing our origin.
OK, so here,

528
00:42:43,000 --> 00:42:46,000
we are integrating first over
u.

529
00:42:46,000 --> 00:42:51,000
That means we start by keeping
u constant, no,

530
00:42:51,000 --> 00:42:55,000
by keeping v constant as u
changes.

531
00:42:55,000 --> 00:43:03,000
OK, so u changes as v is
constant.

532
00:43:03,000 --> 00:43:06,000
What does it mean that I'm
keeping v constant.

533
00:43:06,000 --> 00:43:09,000
Well, what is v?
v is xy.

534
00:43:09,000 --> 00:43:13,000
So, that means I keep xy equals
constant.

535
00:43:13,000 --> 00:43:16,000
What does the curve xy equals
constant look like?

536
00:43:16,000 --> 00:43:22,000
Well, it's just a hyperbola.
y equals constant over x.

537
00:43:22,000 --> 00:43:28,000
So, if I look at the various
values of v that I can take,

538
00:43:28,000 --> 00:43:33,000
for each value of v,
if I fix a value of v,

539
00:43:33,000 --> 00:43:38,000
I will be moving on one of
these red curves.

540
00:43:38,000 --> 00:43:42,000
OK, and u, well,
u is the same thing as x.

541
00:43:42,000 --> 00:43:47,000
So, that means u will increase.
Here, maybe it will be 0.1 and

542
00:43:47,000 --> 00:43:51,000
it will increase all the way to
one here.

543
00:43:51,000 --> 00:43:59,000
OK, so we are just traveling on
each of these slices.

544
00:43:59,000 --> 00:44:03,000
Now, so the question we must
answer here is for a given value

545
00:44:03,000 --> 00:44:08,000
of v, what are the bounds for u?
So, I'm traveling on my curve,

546
00:44:08,000 --> 00:44:11,000
v equals constant,
and trying to figure out,

547
00:44:11,000 --> 00:44:14,000
when do I enter my origin?
When do I leave it?

548
00:44:14,000 --> 00:44:18,000
Well, I enter it when I go
through this side.

549
00:44:18,000 --> 00:44:24,000
So, the question is,
what's the value of u here?

550
00:44:24,000 --> 00:44:29,000
Well, we don't know that very
easily until we look at these

551
00:44:29,000 --> 00:44:32,000
formulas.
So, u equals x,

552
00:44:32,000 --> 00:44:36,000
OK, but we don't know what x is
at that point.

553
00:44:36,000 --> 00:44:42,000
v equals x and v equals xy.
What do we go here?

554
00:44:42,000 --> 00:44:44,000
Well, we don't know x,
but we know y certainly.

555
00:44:44,000 --> 00:44:49,000
OK, so let's forget about
trying to find u.

556
00:44:49,000 --> 00:44:53,000
And, let's say,
for now, we know y equals one.

557
00:44:53,000 --> 00:44:58,000
Well, if we set y equals one,
that tells us that u and v are

558
00:44:58,000 --> 00:45:03,000
both equal to x.
So, in terms of u and v,

559
00:45:03,000 --> 00:45:11,000
the equation of this uv
coordinate is u equals v.

560
00:45:11,000 --> 00:45:14,000
OK, I mean, the other way to do
it is, say that you know you

561
00:45:14,000 --> 00:45:17,000
want y equals one.
You want to know what is y in

562
00:45:17,000 --> 00:45:18,000
terms of u and v.
Well, it's easy.

563
00:45:18,000 --> 00:45:26,000
y is v over u.
So, let me actually add an

564
00:45:26,000 --> 00:45:31,000
extra step in case that's,
so, we know that y is v over u

565
00:45:31,000 --> 00:45:35,000
equals one.
So, that means u=v is my

566
00:45:35,000 --> 00:45:39,000
equation.
OK, so when I'm here,

567
00:45:39,000 --> 00:45:47,000
when I'm entering my region,
the value of u at this point is

568
00:45:47,000 --> 00:45:53,000
just v, u equals v.
That's the hard part.

569
00:45:53,000 --> 00:45:56,000
Now, we need to figure out,
so, we started u equals v.

570
00:45:56,000 --> 00:45:59,000
u increases,
increases, increases.

571
00:45:59,000 --> 00:46:01,000
Where does it exit?
It exits one when we are here.

572
00:46:01,000 --> 00:46:05,000
What's the value of u here?
One. That one is easier, right?

573
00:46:05,000 --> 00:46:10,000
This side here,
so, this side here is x equals

574
00:46:10,000 --> 00:46:13,000
one.
That means u equals one.

575
00:46:13,000 --> 00:46:20,000
So, we start at u equals one.
Now, we've done the inner

576
00:46:20,000 --> 00:46:24,000
integral.
What about the outer?

577
00:46:24,000 --> 00:46:28,000
So, we have to figure out,
what is the first and what is

578
00:46:28,000 --> 00:46:32,000
the last value of v that we'll
want to consider?

579
00:46:32,000 --> 00:46:36,000
Well, if you look at all these
hyperbola's, xy equals constant.

580
00:46:36,000 --> 00:46:39,000
What's the smallest value of xy
that we'll ever want to look at

581
00:46:39,000 --> 00:46:41,000
in here?
Zero, OK.

582
00:46:41,000 --> 00:46:49,000
Let me actually,
where's my yellow chalk?

583
00:46:49,000 --> 00:46:55,000
Is it, no, ah.
So, this one here,

584
00:46:55,000 --> 00:47:00,000
that's actually v=0.
So, we'll start at v equals

585
00:47:00,000 --> 00:47:02,000
zero.
And, what's the last hyperbola

586
00:47:02,000 --> 00:47:05,000
we want to look at?
Well, it's the one that's right

587
00:47:05,000 --> 00:47:07,000
there in the corner.
It's this one here.

588
00:47:07,000 --> 00:47:15,000
And, that's v equals one.
So, v goes from zero to one.

589
00:47:15,000 --> 00:47:17,000
OK, and now,
we can compute this.

590
00:47:17,000 --> 00:47:22,000
I mean, it's not particularly
easier than that one,

591
00:47:22,000 --> 00:47:26,000
but it's not harder either.
How else could we have gotten

592
00:47:26,000 --> 00:47:28,000
these bounds,
because that was quite evil.

593
00:47:28,000 --> 00:47:32,000
So, I would like to recommend
that you try this way in case it

594
00:47:32,000 --> 00:47:34,000
works well.
Just try to picture,

595
00:47:34,000 --> 00:47:38,000
what are the slices in terms of
u and v, and how you travel on

596
00:47:38,000 --> 00:47:40,000
them, where you enter,
where you leave,

597
00:47:40,000 --> 00:47:47,000
staying in the xy picture.
If that somehow doesn't work

598
00:47:47,000 --> 00:47:58,000
well, another way is to draw the
picture in the uv coordinates.

599
00:47:58,000 --> 00:48:04,000
So, switch to a uv picture.
So, what do I mean by that?

600
00:48:04,000 --> 00:48:09,000
Well, we had here a picture in
xy coordinates where we had our

601
00:48:09,000 --> 00:48:12,000
sides.
And, we are going to try to

602
00:48:12,000 --> 00:48:15,000
draw what it looks like in terms
of u and v.

603
00:48:15,000 --> 00:48:18,000
So, here we said this is x
equals one.

604
00:48:18,000 --> 00:48:24,000
That becomes u equals one.
So, we'll draw u equals one.

605
00:48:24,000 --> 00:48:30,000
This side we said is y equals
one becomes u equals v.

606
00:48:30,000 --> 00:48:33,000
That's what we've done over
there.

607
00:48:33,000 --> 00:48:39,000
OK, so u equals v.
Now, we have the two other

608
00:48:39,000 --> 00:48:41,000
sides to deal with.
Well, let's look at this one

609
00:48:41,000 --> 00:48:44,000
first.
So, that was x equals zero.

610
00:48:44,000 --> 00:48:48,000
What happens when x equals zero?
Well, both u and v are zero.

611
00:48:48,000 --> 00:48:51,000
So, this side actually gets
squished in the change of

612
00:48:51,000 --> 00:48:53,000
variables.
It's a bit strange,

613
00:48:53,000 --> 00:48:57,000
but it's a bit the same thing
as when you switch to polar

614
00:48:57,000 --> 00:49:00,000
coordinates at the origin,
r is zero but theta can be

615
00:49:00,000 --> 00:49:03,000
anything.
It's not always one point is

616
00:49:03,000 --> 00:49:07,000
one point.
So anyway, this is the origin,

617
00:49:07,000 --> 00:49:11,000
and then the last side,
y equals zero,

618
00:49:11,000 --> 00:49:15,000
and x varies just becomes v
equals zero.

619
00:49:15,000 --> 00:49:18,000
So, somehow,
in the change of variables,

620
00:49:18,000 --> 00:49:21,000
this square becomes this
triangle.

621
00:49:21,000 --> 00:49:24,000
And now, if we want to
integrate du dv,

622
00:49:24,000 --> 00:49:30,000
it means we are going to slice
by v equals constant.

623
00:49:30,000 --> 00:49:33,000
So, we are going to integrate
over slices like this,

624
00:49:33,000 --> 00:49:36,000
and you see for each value of
v, we go from u equals v to u

625
00:49:36,000 --> 00:49:41,000
equals one.
And, v goes from zero to one.

626
00:49:41,000 --> 00:49:44,000
OK, so you get the same bounds
just by drawing a different

627
00:49:44,000 --> 00:49:47,000
picture.
So, it's up to you to decide

628
00:49:47,000 --> 00:49:51,000
whether you prefer to think on
this picture or draw that one

629
00:49:51,000 --> 00:49:53,000
instead.
It depends on which problems

630
00:49:53,000 --> 00:49:55,000
you're doing.