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

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00:00:28 --> 00:00:32
xy coordinates,
also how to switch to polar

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

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

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

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

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

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00:00:52 --> 00:01:02
 
 

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

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

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

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

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

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

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

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

23
00:01:58 --> 00:02:01
and y differently.
So, let's say we want to find

24
00:02:01 --> 00:02:03
the area of it.
Maybe you know what the area is.

25
00:02:03 --> 00:02:11
But let's do it as a double
integral.

26
00:02:11 --> 00:02:14
So, you know,
if you find that the area is

27
00:02:14 --> 00:02:19
too easy, you can integrate any
function other than ellipse,

28
00:02:19 --> 00:02:23
if you prefer.
But, let's do it just with area.

29
00:02:23 --> 00:02:27
So, we know that we want to
integrate just the area element,

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

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

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

33
00:02:41 --> 00:02:46
you know, set up the bounds by
solving for first four x as a

34
00:02:46 --> 00:02:49
function of y if we do it this
order and,

35
00:02:49 --> 00:02:52
well, do the usual stuff.
That doesn't look very

36
00:02:52 --> 00:02:55
pleasant, and it's certainly not
the best way to do it.

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

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

39
00:02:59 --> 00:03:00
yet.
But, you know,

40
00:03:00 --> 00:03:01
an ellipse is just a squished
circle.

41
00:03:01 --> 00:03:08
So, maybe we want to actually
first rescale x and y by a and

42
00:03:08 --> 00:03:11
b.
So, to do that,

43
00:03:11 --> 00:03:19
what we'd like to do is set x
over a to be u,

44
00:03:19 --> 00:03:24
and y over b be v.
So, we'll have two new

45
00:03:24 --> 00:03:28
variables, u and v,
and we'll try to redo our

46
00:03:28 --> 00:03:32
integral in terms of u and v.
So, how do we do the

47
00:03:32 --> 00:03:36
substitution?
So, in terms of u and v, 

48
00:03:36 --> 00:03:39
the condition, 
the region that we are

49
00:03:39 --> 00:03:43
integrating on will become u^2
v^2 is less than 1,

50
00:03:43 --> 00:03:45
which is arguably nicer than
the ellipse.

51
00:03:45 --> 00:03:50
That's why we are doing it.
But, we need to know what to do

52
00:03:50 --> 00:03:53
with dx and dy.
Well, here, the answer is

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

54
00:03:56 --> 00:03:59
We do two independent
substitutions.

55
00:03:59 --> 00:04:10
OK, so if we set u equals x
over a, that means du is one

56
00:04:10 --> 00:04:18
over adx.
And here, dv is one over bdy.

57
00:04:18 --> 00:04:26
So, it's very tempting to
write, and here we actually can

58
00:04:26 --> 00:04:34
write, in this particular case,
that dudv is (1/ab)dxdy,

59
00:04:34 --> 00:04:42
OK?
So, let me rewrite that.

60
00:04:42 --> 00:04:54
OK, so I get dudv equals
(1/ab)dxdy, or equivalently dxdy

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

62
00:05:05 --> 00:05:15
I'm going to write (ab)dudv.
OK, so now, my double integral

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

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

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

66
00:05:31 --> 00:05:34
dv.
And, that is an integral that

67
00:05:34 --> 00:05:37
we know how to do.
Well, it's just the area of a

68
00:05:37 --> 00:05:40
unit circle.
So, we can just say, 

69
00:05:40 --> 00:05:50
this is ab times the area of
the unit disk,

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

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

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

73
00:05:59 --> 00:06:02
times cos(theta),
v equals r times sin(theta), 

74
00:06:02 --> 00:06:07
and then doing it in polar
coordinates.

75
00:06:07 --> 00:06:11
OK, so here the substitution
worked pretty easy.

76
00:06:11 --> 00:06:14
The question is,
if we do a change of variables

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

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

79
00:06:20 --> 00:06:22
or do we have to be more
careful?

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

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

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

83
00:06:33 --> 00:06:38
OK.
OK, so, see the general problem

84
00:06:38 --> 00:06:41
when we try to do this is to
figure out what is the scale

85
00:06:41 --> 00:06:48
factor?
What's the relation between

86
00:06:48 --> 00:06:57
dxdy and dudv?
We need to find the scaling

87
00:06:57 --> 00:07:07
factor.
So, we need to find dxdy versus

88
00:07:07 --> 00:07:12
dudv.
So, let's do another example

89
00:07:12 --> 00:07:18
that's still pretty easy,
but a little bit less easy.

90
00:07:18 --> 00:07:24
OK, so let's say that for some
reason, we want to do the change

91
00:07:24 --> 00:07:27
of variables:
u equals 3x-2y,

92
00:07:27 --> 00:07:31
and v equals x y.
Why would we want to do that?

93
00:07:31 --> 00:07:34
Well, that might be to simplify
the integrand because we are

94
00:07:34 --> 00:07:38
integrating a function that
happens to be actually involving

95
00:07:38 --> 00:07:42
these guys rather than x and y.
Or, it might be to simplify the

96
00:07:42 --> 00:07:45
bounds because maybe we are
integrating over a region whose

97
00:07:45 --> 00:07:49
equation in xy coordinates is
very hard to write down.

98
00:07:49 --> 00:07:51
But, it becomes much easier in
terms of u and v.

99
00:07:51 --> 00:07:57
And then, the bounds would be
much easier to set up with u and

100
00:07:57 --> 00:08:02
v.
Anyway, so, whatever the reason

101
00:08:02 --> 00:08:12
might be, typically it would be
to simplify the integrant or the

102
00:08:12 --> 00:08:18
bounds.
Well, how do we convert dxdy to

103
00:08:18 --> 00:08:21
dudv?
So, we want to understand,

104
00:08:21 --> 00:08:27
what's the relation between,
let's call dA the area element

105
00:08:27 --> 00:08:31
in xy coordinates.
So, dA is dxdy,

106
00:08:31 --> 00:08:34
maybe dydx depending on the
order.

107
00:08:34 --> 00:08:39
And, the area element in uv
coordinates, let me call that dA

108
00:08:39 --> 00:08:42
prime just to make it look
different.

109
00:08:42 --> 00:08:50
So, that would just be dudv,
or dvdu depending on which

110
00:08:50 --> 00:08:55
order I will want to set it up
in.

111
00:08:55 --> 00:09:01
So, to find this relation,
it's probably best to draw a

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

113
00:09:09 --> 00:09:18
the xy plane with area delta(A)
corresponding to just a box with

114
00:09:18 --> 00:09:24
sides delta(y) and delta(x).
OK, and let's try to figure out

115
00:09:24 --> 00:09:27
what it will look like in terms
of u and v.

116
00:09:27 --> 00:09:29
And then, we'll say,
well, when we integrate,

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

118
00:09:32 --> 00:09:36
small boxes times their area.
But, the problem is that the

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

120
00:09:40 --> 00:09:47
in uv coordinates.
There, maybe it will look like, 

121
00:09:47 --> 00:09:49
actually, 
if you see that these are

122
00:09:49 --> 00:09:52
linear changes of variables,
you know that the rectangle

123
00:09:52 --> 00:09:55
will become a parallelogram
after the change of variables.

124
00:09:55 --> 00:10:00
So, the area of a parallelogram
delta(A) prime,

125
00:10:00 --> 00:10:05
well, we will have to figure
out how they are related so that

126
00:10:05 --> 00:10:09
we can decide what conversion
factor,

127
00:10:09 --> 00:10:13
what's the exchange rate
between these two currencies for

128
00:10:13 --> 00:10:20
area?
OK, any questions at this point?

129
00:10:20 --> 00:10:27
No? Still with me mostly?
I see a lot of tired faces.

130
00:10:27 --> 00:10:34
Yes?
Why is delta(A) prime a

131
00:10:34 --> 00:10:37
parallelogram?
That's a very good question.

132
00:10:37 --> 00:10:41
Well, see, if I look at the
side of a rectangle,

133
00:10:41 --> 00:10:45
say there's a vertical side,
it means I'm going to increase

134
00:10:45 --> 00:10:49
y, keeping x the same.
If I look at the formulas for u

135
00:10:49 --> 00:10:52
and v, they are linear formulas
in terms of x and y.

136
00:10:52 --> 00:10:56
So, if I just increase y,
see that u is going to decrease

137
00:10:56 --> 00:10:58
at a rate of two.
v is going to increase at a

138
00:10:58 --> 00:11:02
rate of one at constant rates.
And, it doesn't matter whether

139
00:11:02 --> 00:11:04
I was looking at this site or at
that site.

140
00:11:04 --> 00:11:06
So, basically straight lines
become straight lines.

141
00:11:06 --> 00:11:09
And if they are parallel,
they stay parallel.

142
00:11:09 --> 00:11:11
So, if you just look at what
the transformation,

143
00:11:11 --> 00:11:14
from xy to uv does,
it does this kind of thing.

144
00:11:14 --> 00:11:17
Actually, this transformation
here you can express by a

145
00:11:17 --> 00:11:18
matrix.
And, remember,

146
00:11:18 --> 00:11:20
we've seen what matrices do the
pictures.

147
00:11:20 --> 00:11:24
We just take straight lines to
straight lines.

148
00:11:24 --> 00:11:29
They keep the notion of being
parallel, but of course they

149
00:11:29 --> 00:11:32
mess up lengths,
angles, and all that.

150
00:11:32 --> 00:11:38
OK, so let's see.
So, let's try to figure out,

151
00:11:38 --> 00:11:42
what is the area of this guy?
Well, in fact,

152
00:11:42 --> 00:11:46
what I've been saying about
this transformation being

153
00:11:46 --> 00:11:49
linear,
and transforming all of the

154
00:11:49 --> 00:11:53
vertical lines in the same way,
all the horizontal lines in the

155
00:11:53 --> 00:11:54
same way,
it tells me,

156
00:11:54 --> 00:11:57
also, I should have a constant
scaling factor,

157
00:11:57 --> 00:12:00
right, because how much I've
scaled my rectangle doesn't

158
00:12:00 --> 00:12:03
depend on where my rectangle is.
If I move my rectangle to

159
00:12:03 --> 00:12:05
somewhere else,
I have a rectangle of the same

160
00:12:05 --> 00:12:08
size, same shape,
it will become a parallelogram

161
00:12:08 --> 00:12:10
of the same size,
same shape somewhere else.

162
00:12:10 --> 00:12:13
So, in fact,
I can just take the simplest

163
00:12:13 --> 00:12:16
rectangle I can think of and see
how its area changes.

164
00:12:16 --> 00:12:18
And, if you don't believe me,
then try with any other

165
00:12:18 --> 00:12:21
rectangle.
You will see it works exactly

166
00:12:21 --> 00:12:28
the same way.
OK, so I claim that the area

167
00:12:28 --> 00:12:41
scaling factor -- -- here in
this case doesn't depend on the

168
00:12:41 --> 00:12:53
choice of the rectangle.
And I should say that because

169
00:12:53 --> 00:13:05
we are actually doing a linear
change of variables -- So,

170
00:13:05 --> 00:13:08
you know, somehow,
the exchange rate between uv

171
00:13:08 --> 00:13:10
and xy is going to be the same
everywhere.

172
00:13:10 --> 00:13:14
So, let's try to see what
happens to the simplest

173
00:13:14 --> 00:13:19
rectangle I can think of,
namely, just the unit square.

174
00:13:19 --> 00:13:21
And, you know,
if you don't trust me,

175
00:13:21 --> 00:13:24
then while I'm doing this one,
do it with a different

176
00:13:24 --> 00:13:26
rectangle.
Do the same calculation,

177
00:13:26 --> 00:13:30
and see that you will get the
same conversion ratio.

178
00:13:30 --> 00:13:37
So, let's say that I take a
unit square -- -- so,

179
00:13:37 --> 00:13:45
something that goes from zero
to one both in x and y

180
00:13:45 --> 00:13:49
directions.
OK, and let's try to figure out

181
00:13:49 --> 00:13:51
what it looks like on the other
side.

182
00:13:51 --> 00:13:58
So, here the area is one.
Let's try to draw it in terms

183
00:13:58 --> 00:14:00
of u and v coordinates,
OK?

184
00:14:00 --> 00:14:05
So, here we have x equals 0,
y equals 0.

185
00:14:05 --> 00:14:13
Well, that tells us u and v are
going to be 0.

186
00:14:13 --> 00:14:17
Next, let's look at this corner.
Well, in xy coordinates,

187
00:14:17 --> 00:14:20
this is one zero.
If you plug x equals 1,

188
00:14:20 --> 00:14:24
y equals 0, you get u equals 3;
v equals 1.

189
00:14:24 --> 00:14:38
So, that goes somewhere here.
And so, this edge of the square

190
00:14:38 --> 00:14:44
will become this line here,
OK?

191
00:14:44 --> 00:14:49
Next, let's look at that point.
So that point here was (0,1).

192
00:14:49 --> 00:15:01
If I plug x equals zero y
equals one I will get (-2,1).

193
00:15:01 --> 00:15:11
So, this edge goes here.
Then, if you put x equals one,

194
00:15:11 --> 00:15:14
y equals one,
you will get u equals 1,

195
00:15:14 --> 00:15:22
v equals 2.
So, I want (1,2).

196
00:15:22 --> 00:15:28
And, these edges will go to
these edges here.

197
00:15:28 --> 00:15:31
And, you see,
it does look like a

198
00:15:31 --> 00:15:38
parallelogram.
OK, so now what the area of

199
00:15:38 --> 00:15:44
this parallelogram?
Well, we can get that by taking

200
00:15:44 --> 00:15:47
the determinant of these two
vectors.

201
00:15:47 --> 00:15:53
So, one of them is ,
and the other one is

202
00:15:53 --> 00:15:57
.
That will be 3 2.

203
00:15:57 --> 00:16:01
That's 5.
OK, this parallelogram is

204
00:16:01 --> 00:16:04
apparently five times the size
of this square.

205
00:16:04 --> 00:16:07
Here, it looks like it's less
because I somehow changed my

206
00:16:07 --> 00:16:10
scale.
I mean, my unit length is

207
00:16:10 --> 00:16:15
smaller here than here.
But, it should be a lot bigger

208
00:16:15 --> 00:16:16
than that.
OK, 

209
00:16:16 --> 00:16:19
and if you do the same
calculations not with zero and

210
00:16:19 --> 00:16:21
one,
but with x and x plus delta x,

211
00:16:21 --> 00:16:24
and so on,
you will still find that the

212
00:16:24 --> 00:16:27
area has been multiplied by
five.

213
00:16:27 --> 00:16:36
So, that tells us,
actually for any other

214
00:16:36 --> 00:16:47
rectangle, area is also
multiplied by five.

215
00:16:47 --> 00:16:52
So, that tells us that dA
prime, the area element in uv

216
00:16:52 --> 00:16:57
coordinate is worth five times
more than the area element in

217
00:16:57 --> 00:16:59
the xy coordinate.

218
00:16:59 --> 00:17:11

219
00:17:11 --> 00:17:17
So, that means du dv is worth
five times dx dy.

220
00:17:17 --> 00:17:30
What's so funny?
What?

221
00:17:30 --> 00:17:40
Oh.
[LAUGHTER] OK, rectangle.

222
00:17:40 --> 00:17:45
OK, is that OK now?
Did I misspell other words?

223
00:17:45 --> 00:17:48
No?
OK, it's really hard to see

224
00:17:48 --> 00:17:54
when you are up close.
It's much easier from a

225
00:17:54 --> 00:17:58
distance.
OK, so yeah,

226
00:17:58 --> 00:18:05
so we've said our
transformation multiplies areas

227
00:18:05 --> 00:18:09
by five.
And so, dudv is five times dxdy.

228
00:18:09 --> 00:18:14
So, if I'm integrating some
function, dx dy,

229
00:18:14 --> 00:18:20
then when I switch to uv
coordinates, I will have to

230
00:18:20 --> 00:18:26
replace that by one fifth dudv.
OK, and of course I would also,

231
00:18:26 --> 00:18:29
here my function would probably
involve x and y.

232
00:18:29 --> 00:18:33
I will replace them by u's and
v's.

233
00:18:33 --> 00:18:35
And, the bounds,
well, the shape of my origin in

234
00:18:35 --> 00:18:39
the xy coordinates I will have
to switch to some shape in the

235
00:18:39 --> 00:18:42
uv coordinates.
And, that's also something that

236
00:18:42 --> 00:18:46
might be easy or might be tricky
depending on what origin we are

237
00:18:46 --> 00:18:50
looking at.
So, usually we will do changes

238
00:18:50 --> 00:18:54
of variables to actually
simplify the region so it

239
00:18:54 --> 00:18:58
becomes easier to set up the
bounds.

240
00:18:58 --> 00:19:05
So, anyway, so this is kind of
an illustration of a general

241
00:19:05 --> 00:19:07
case.
And, why is that?

242
00:19:07 --> 00:19:10
Well, here it looks very easy.
We are just using linear

243
00:19:10 --> 00:19:14
formulas, and somehow the
relation between dx dy and du dv

244
00:19:14 --> 00:19:17
is the same everywhere.
If you take actually more

245
00:19:17 --> 00:19:21
complicated changes of variables
that's not true because usually

246
00:19:21 --> 00:19:25
you will expect that there are
some places where the rescaling

247
00:19:25 --> 00:19:28
is enlarging things,
and some of other places where

248
00:19:28 --> 00:19:31
things are shrunk,
so, certainly the exchange rate

249
00:19:31 --> 00:19:35
between dudv and dxdy will
fluctuate from point to point.

250
00:19:35 --> 00:19:37
It's the same as if you're
trying to change dollars to

251
00:19:37 --> 00:19:39
euros.
It depends on where you do it.

252
00:19:39 --> 00:19:43
You will get a better rate or a
worse one.

253
00:19:43 --> 00:19:47
So, of course,
we'll get a formula where

254
00:19:47 --> 00:19:52
actually this scaling factor
depends on x and y or on u and

255
00:19:52 --> 00:19:54
v.
But, if you fix a point,

256
00:19:54 --> 00:19:57
then we have linear
approximation.

257
00:19:57 --> 00:20:00
And, linear approximation tells
us, oh, we can do as if our

258
00:20:00 --> 00:20:02
function is just a linear
function of x and y.

259
00:20:02 --> 00:20:06
So then, we can do it the same
way we did here.

260
00:20:06 --> 00:20:18
OK, so let's try to think about
that.

261
00:20:18 --> 00:20:22
So, in the general case,
well, that means we will

262
00:20:22 --> 00:20:26
replace x and y by new
coordinates, u and v.

263
00:20:26 --> 00:20:30
And, u and v will be some
functions of x and y.

264
00:20:30 --> 00:20:34
So, well, 
we'll have an approximation

265
00:20:34 --> 00:20:37
formula which tells us that the
change in u,

266
00:20:37 --> 00:20:40
if I change x or y a little
bit,

267
00:20:40 --> 00:20:45
will be roughly (u sub x times
change in x) (u sub y times

268
00:20:45 --> 00:20:50
change in y).
And, the change in v will be

269
00:20:50 --> 00:20:57
roughly (v sub x delta x) (v sub
y delta y).

270
00:20:57 --> 00:21:03
Or, the other way to say it, 
if you want in matrix form is

271
00:21:03 --> 00:21:08
delta u delta v is,
sorry, approximately equal to

272
00:21:08 --> 00:21:12
matrix |u sub x,
u sub y, v sub x,

273
00:21:12 --> 00:21:20
v sub y| times matrix |delta x,
delta y|,

274
00:21:20 --> 00:21:26
OK?
So, if we look at that, 

275
00:21:26 --> 00:21:32
what it tells us, in fact, 
is that if we take a small

276
00:21:32 --> 00:21:40
rectangle in xy coordinates,
so that means we have a certain

277
00:21:40 --> 00:21:44
point, x, y,
and then we have a certain

278
00:21:44 --> 00:21:51
width.
This is going to be too small.

279
00:21:51 --> 00:21:56
Well, so, I have my width,
delta x.

280
00:21:56 --> 00:22:06
I have my height, delta y.
This is going to correspond to

281
00:22:06 --> 00:22:14
a small uv parallelogram.
And, what the shape and the

282
00:22:14 --> 00:22:20
size of the parallelogram are
depends on the partial

283
00:22:20 --> 00:22:24
derivatives of u and v.
So, in particular,

284
00:22:24 --> 00:22:26
it depends on at which point we
are.

285
00:22:26 --> 00:22:30
But still, at a given point,
it's a bit like that.

286
00:22:30 --> 00:22:35
And, so if we do the same
argument as before,

287
00:22:35 --> 00:22:41
what we will see is that the
scaling factor is actually the

288
00:22:41 --> 00:22:45
determinant of this
transformation.

289
00:22:45 --> 00:22:50
So, that's one thing that maybe
we didn't emphasize enough when

290
00:22:50 --> 00:22:53
we did matrices at the beginning
of a semester.

291
00:22:53 --> 00:22:57
But, when you have a linear
transformation between

292
00:22:57 --> 00:23:01
variables, the determinant of
that transformation represents

293
00:23:01 --> 00:23:05
how it scales areas.
OK, so one way to think about

294
00:23:05 --> 00:23:09
it is just to try it and see
what happens.

295
00:23:09 --> 00:23:12
Take this side.
This side in x,

296
00:23:12 --> 00:23:16
y coordinates corresponds to
delta x and zero.

297
00:23:16 --> 00:23:20
And, now, if you take the image
of that, if you see what happens

298
00:23:20 --> 00:23:24
to delta u and delta v,
that will be basically u sub x

299
00:23:24 --> 00:23:28
delta x and v sub x delta x.
There's no delta y.

300
00:23:28 --> 00:23:33
For the other side,
OK, so maybe I should do it

301
00:23:33 --> 00:23:36
actually.
So, you know,

302
00:23:36 --> 00:23:40
if we move in the x,
y coordinates by delta x and

303
00:23:40 --> 00:23:45
zero,
then delta u and delta v will

304
00:23:45 --> 00:23:50
be approximately u sub x delta
x,

305
00:23:50 --> 00:24:02
and v sub x delta x.
And, on the other hand, 

306
00:24:02 --> 00:24:04
if you move in the other
direction along the other side

307
00:24:04 --> 00:24:08
of your rectangle,
zero and delta y, 

308
00:24:08 --> 00:24:13
then the change in u and the
change in v will correspond to,

309
00:24:13 --> 00:24:16
well, how does u change?
That's u sub y delta y,

310
00:24:16 --> 00:24:20
and v changes by v sub y delta
y.

311
00:24:20 --> 00:24:22
And so, now,
if you take the determinant of

312
00:24:22 --> 00:24:25
these two vectors,
OK, so these are the sides of

313
00:24:25 --> 00:24:29
your parallelogram up here.
And, if you take these sides to

314
00:24:29 --> 00:24:31
get the area of the
parallelogram,

315
00:24:31 --> 00:24:33
you'll need to take the
determinant.

316
00:24:33 --> 00:24:41
And, the determinant will be
the determinant of this matrix

317
00:24:41 --> 00:24:48
times delta x times delta y.
So, the area in uv coordinates

318
00:24:48 --> 00:24:53
will be the determinant of a
matrix times delta x,

319
00:24:53 --> 00:24:57
delta y.
And so, 

320
00:24:57 --> 00:25:02
what I'm trying to say is that
when you have a general change

321
00:25:02 --> 00:25:06
of variables,
du dv versus dx dy is given by

322
00:25:06 --> 00:25:11
the determinant of this matrix
of partial derivatives.

323
00:25:11 --> 00:25:13
It doesn't matter in which
order you write it.

324
00:25:13 --> 00:25:16
I mean, you can put in rows or
columns.

325
00:25:16 --> 00:25:18
If you transpose a matrix,
that doesn't change the

326
00:25:18 --> 00:25:21
determinant.
It's just any sensible matrix

327
00:25:21 --> 00:25:24
that you can write will have the
correct determinant.

328
00:25:24 --> 00:26:02
 
 

329
00:26:02 --> 00:26:07
OK, so what we need to know is
the following thing.

330
00:26:07 --> 00:26:11
So, 
we define something called the

331
00:26:11 --> 00:26:16
Jacobian of a change of
variables and used the letter J,

332
00:26:16 --> 00:26:21
or maybe a more useful notation
is partial of u,

333
00:26:21 --> 00:26:24
v over partial of x,
y.

334
00:26:24 --> 00:26:27
That's a very strange notation.
I mean, that doesn't mean that

335
00:26:27 --> 00:26:30
we are actually taking the
partial derivatives of anything.

336
00:26:30 --> 00:26:34
OK, it's just a notation to
remind us that this has to do

337
00:26:34 --> 00:26:37
with the ratio between dudv and
dxdy.

338
00:26:37 --> 00:26:42
And, it's obtained using the
partial derivatives of u and v

339
00:26:42 --> 00:26:50
with respect to x and y.
So, it's the determinant of the

340
00:26:50 --> 00:26:55
matrix |u sub x,
u sub y, v sub x,

341
00:26:55 --> 00:27:02
v sub y|, the matrix that I had
up there.

342
00:27:02 --> 00:27:10
OK, and what we need to know is
that du dv is equal to the

343
00:27:10 --> 00:27:17
absolute value of J dx dy.
Or, if you prefer to see it in

344
00:27:17 --> 00:27:23
the easier to remember version,
it's (absolute value of d of

345
00:27:23 --> 00:27:27
(u, v) over partial xy) times dx
dy.

346
00:27:27 --> 00:27:32
OK, so this is just what you
need to remember,

347
00:27:32 --> 00:27:38
and it says that the area in uv
coordinates is worth,

348
00:27:38 --> 00:27:42
well, the ratio to the xy
coordinates is given by this

349
00:27:42 --> 00:27:46
Jacobian determinant except for
one small thing.

350
00:27:46 --> 00:27:48
It's given by,
actually, the absolute value of

351
00:27:48 --> 00:27:52
this guy.
OK, so what's going on here?

352
00:27:52 --> 00:27:56
What's going on here is when we
are saying the determinant of

353
00:27:56 --> 00:27:59
the transformation tells us how
the area is multiplied,

354
00:27:59 --> 00:28:02
there's a small catch.
Remember, the determinants are

355
00:28:02 --> 00:28:06
equal to areas up to sine.
Sometimes, the determinant is

356
00:28:06 --> 00:28:10
negative because of reversing
the orientation of things.

357
00:28:10 --> 00:28:13
But, the area is still the same.
Area is always positive.

358
00:28:13 --> 00:28:17
So, the area elements are
actually related by the absolute

359
00:28:17 --> 00:28:23
value of this guy.
OK, so if you find -10 as your

360
00:28:23 --> 00:28:29
answer, then du dv is still ten
times dx dy.

361
00:28:29 --> 00:28:33
OK, so I didn't put it all
together because then you would

362
00:28:33 --> 00:28:36
have two sets of vertical bars.
See, this is a vertical bar for

363
00:28:36 --> 00:28:38
absolute value.
This is vertical bar for

364
00:28:38 --> 00:28:42
determinant.
They're not the same.

365
00:28:42 --> 00:28:46
That's the one thing to
remember.

366
00:28:46 --> 00:28:54
OK, any questions about this?
No?

367
00:28:54 --> 00:29:06
OK.
So, actually let's do our first

368
00:29:06 --> 00:29:12
example of that.
Let's check what we had for

369
00:29:12 --> 00:29:16
polar coordinates.
Last time I told you if we have

370
00:29:16 --> 00:29:19
dx dy we could switch it to r dr
d theta.

371
00:29:19 --> 00:29:25
And, we had some argument for
that by looking at the area of a

372
00:29:25 --> 00:29:31
small circular sector.
But, let's check again using

373
00:29:31 --> 00:29:37
this new method.
So, in polar coordinates I'm

374
00:29:37 --> 00:29:44
setting x equals r cosine theta,
y equals r sine theta.

375
00:29:44 --> 00:29:48
So, the Jacobian for this
change of variables,

376
00:29:48 --> 00:29:54
so let's say I'm trying to find
the partial derivatives of x,

377
00:29:54 --> 00:29:58
y with respect to r,
theta.

378
00:29:58 --> 00:30:04
Well, what is,
OK, let me actually write them

379
00:30:04 --> 00:30:10
here again for you.
And, so what does that become?

380
00:30:10 --> 00:30:17
Partial x over partial r is
just cosine theta.

381
00:30:17 --> 00:30:25
Partial x over partial theta is
negative r sine theta.

382
00:30:25 --> 00:30:27
Sorry, I guess I'm going to run
out of space here.

383
00:30:27 --> 00:30:33
So, let me do it underneath.
So, we said x sub r is cosine

384
00:30:33 --> 00:30:36
theta;
x sub theta is negative r sine

385
00:30:36 --> 00:30:41
theta.
y sub r is sine;

386
00:30:41 --> 00:30:49
y sub theta is r cosine.
And now, if we compute this

387
00:30:49 --> 00:30:58
determinant, we'll get (r cosine
squared theta) (r sine squared

388
00:30:58 --> 00:31:02
theta).
And, that simplifies to r.

389
00:31:02 --> 00:31:08
So, dx dy is,
well, absolute value of r dr d

390
00:31:08 --> 00:31:12
theta.
But, remember that r is always

391
00:31:12 --> 00:31:18
positive.
So, it's r dr d theta.

392
00:31:18 --> 00:31:26
OK, so that's another way to
justify how we did double

393
00:31:26 --> 00:31:34
integrals in polar coordinates.
OK, any questions on that?

394
00:31:34 --> 00:31:47
Where?
Yeah, OK.

395
00:31:47 --> 00:31:52
Yeah, so this one seems to be
switching.

396
00:31:52 --> 00:31:56
Well, it depends what you do.
So, OK, actually here's an

397
00:31:56 --> 00:32:00
important thing that I didn't
quite say.

398
00:32:00 --> 00:32:04
So, I said, you know,
we are going to switch from xy

399
00:32:04 --> 00:32:07
to uv.
We can also switch from uv to

400
00:32:07 --> 00:32:09
xy.
And, this conversion ratio,

401
00:32:09 --> 00:32:12
the Jacobian,
works both ways.

402
00:32:12 --> 00:32:16
Once you have found the ratio
between du dv and dx dy,

403
00:32:16 --> 00:32:20
then it works one way or it
works the other way.

404
00:32:20 --> 00:32:22
I mean, here,
of course, we get the answer in

405
00:32:22 --> 00:32:26
terms of r.
So, this would let us switch

406
00:32:26 --> 00:32:30
from xy to r theta.
But, we can also switch from r

407
00:32:30 --> 00:32:33
theta to xy.
Just, we'd write dr d theta

408
00:32:33 --> 00:32:37
equals (1 over r) times dx dy.
And then we'd have,

409
00:32:37 --> 00:32:41
of course, to replace r by its
formula in xy coordinates.

410
00:32:41 --> 00:32:43
Usually, we don't do that.
Usually, we actually start with

411
00:32:43 --> 00:32:47
xy and switch to polar.
But, 

412
00:32:47 --> 00:32:50
so in general, 
when you have this formula

413
00:32:50 --> 00:32:54
relating du dv with dx dy,
you can use it both ways, 

414
00:32:54 --> 00:33:00
either to switch from du dv to
dx dy or the other way around.

415
00:33:00 --> 00:33:04
And, the thing that I'm not
telling you that now I should

416
00:33:04 --> 00:33:08
probably tell you is I could
define two Jacobians because if

417
00:33:08 --> 00:33:12
I solve for xy in terms of uv
instead of uv in terms of xy,

418
00:33:12 --> 00:33:15
then I can compute two
different Jacobians.

419
00:33:15 --> 00:33:19
I can compute partial uv over
partial xy, or I can compute

420
00:33:19 --> 00:33:24
partial xy over partial uv if I
have the formulas both ways.

421
00:33:24 --> 00:33:27
Well, the good news is these
guys are the inverse of each

422
00:33:27 --> 00:33:29
other.
So, the two formulas that you

423
00:33:29 --> 00:33:31
might get are consistent.

424
00:33:31 --> 00:33:59

425
00:33:59 --> 00:34:16
OK, so useful remark -- So,
say that you can compute both

426
00:34:16 --> 00:34:23
-- -- these guys.
Well, then actually,

427
00:34:23 --> 00:34:26
the product will just be 1.
So, they are the inverse of

428
00:34:26 --> 00:34:28
each other.
So, it doesn't matter which one

429
00:34:28 --> 00:34:34
you compute.
You can compute whichever one

430
00:34:34 --> 00:34:45
is the easiest to compute no
matter which one of the two you

431
00:34:45 --> 00:34:48
need.
And, one way to see that is

432
00:34:48 --> 00:34:50
that, in fact,
we're looking at the

433
00:34:50 --> 00:34:53
determinant of these matrices
that tell us the relation in

434
00:34:53 --> 00:34:56
variables.
So, if one of them tells you

435
00:34:56 --> 00:34:58
how delta u delta v relate to
delta x delta y,

436
00:34:58 --> 00:35:00
the other one does the opposite
thing.

437
00:35:00 --> 00:35:03
It means they are the inverse
matrices.

438
00:35:03 --> 00:35:06
And, the determinant of the
inverse matrix is the inverse of

439
00:35:06 --> 00:35:10
the determinant.
So, they are really

440
00:35:10 --> 00:35:14
interchangeable.
I mean, you can just compute

441
00:35:14 --> 00:35:17
whichever one is easiest.
So here, if you wanted, 

442
00:35:17 --> 00:35:22
dr d theta in terms of dx dy, 
it's easier to do this and then

443
00:35:22 --> 00:35:27
move the r over there than to
first solve for r and theta as

444
00:35:27 --> 00:35:31
functions of x and y and then do
the entire thing again.

445
00:35:31 --> 00:35:42
But, you can do it if you want.
I mean, it works.

446
00:35:42 --> 00:35:45
Oh yeah, the other useful
remark, so, I mentioned it,

447
00:35:45 --> 00:35:49
but let me emphasize again.
So, now, the ratio between du

448
00:35:49 --> 00:35:51
dv and dx dy,
it's not a constant anymore,

449
00:35:51 --> 00:35:54
although there it used to be
five.

450
00:35:54 --> 00:35:56
But now, it's become r,
or anything.

451
00:35:56 --> 00:35:58
In general, it will be a
function that depends on the

452
00:35:58 --> 00:36:01
variables.
So, it's not true that you can

453
00:36:01 --> 00:36:04
just say, oh,
I'll put a constant times du

454
00:36:04 --> 00:36:14
dv.
Yes?

455
00:36:14 --> 00:36:17
It would still work the same.
You could imagine drawing a

456
00:36:17 --> 00:36:20
picture where r and theta are
the Cartesian coordinates,

457
00:36:20 --> 00:36:22
and your picture would be
completely messed up.

458
00:36:22 --> 00:36:26
It would be a very strange
thing to do to try to draw,

459
00:36:26 --> 00:36:30
you know, I'm going to do it,
but don't take notes on that.

460
00:36:30 --> 00:36:32
You could try to draw picture
like that, and then a circle

461
00:36:32 --> 00:36:34
would start looking like,
you know, a disk would look

462
00:36:34 --> 00:36:35
like that.
It would be very

463
00:36:35 --> 00:36:37
counterintuitive.
But, you could do it.

464
00:36:37 --> 00:36:41
And that would be equivalent to
what we did with a previous

465
00:36:41 --> 00:36:43
change of variables.
So, in this case,

466
00:36:43 --> 00:36:47
certainly you would never draw
a picture like that.

467
00:36:47 --> 00:36:59
But, you could do it.
OK, so now let's do a complete

468
00:36:59 --> 00:37:07
example to see how things fit
together, how we do everything.

469
00:37:07 --> 00:37:10
So, let's say that we want to
compute, so I have to warn you,

470
00:37:10 --> 00:37:12
it's going to be a very silly
example.

471
00:37:12 --> 00:37:16
It's an example where it's much
easier to compute things without

472
00:37:16 --> 00:37:19
the change of variables.
But, you know,

473
00:37:19 --> 00:37:24
it's good practice in the sense
that we're going to make it so

474
00:37:24 --> 00:37:29
complicated that if we can do
this one, then we can do that

475
00:37:29 --> 00:37:31
one.
So, let's say that we want to

476
00:37:31 --> 00:37:33
compute this.
And, of course,

477
00:37:33 --> 00:37:35
it's very easy to compute it
directly.

478
00:37:35 --> 00:37:42
But let's say that for some
evil reason we want to do that

479
00:37:42 --> 00:37:49
by changing variables to u
equals x and v equals xy.

480
00:37:49 --> 00:37:55
OK, that's a very strange idea,
but let's do it anyway.

481
00:37:55 --> 00:37:58
I mean, normally,
you would only do this kind of

482
00:37:58 --> 00:38:01
substitution if either it
simplifies a lot the function

483
00:38:01 --> 00:38:03
you are integrating,
or it simplifies a lot the

484
00:38:03 --> 00:38:06
region on which you are
integrating.

485
00:38:06 --> 00:38:12
And here, neither happens.
But anyway, so the first thing

486
00:38:12 --> 00:38:16
we have to do here is figure out
what we are going to be

487
00:38:16 --> 00:38:18
integrating.
OK, so to do that,

488
00:38:18 --> 00:38:23
we should figure out what dx dy
will become in terms of u and v.

489
00:38:23 --> 00:38:26
So, that's what we've just seen
using the Jacobian.

490
00:38:26 --> 00:38:32
OK, so the first thing to do is
find the area element.

491
00:38:32 --> 00:38:33
And, for that,
we use the Jacobian.

492
00:38:33 --> 00:38:36
So, well, let's see,
the one that we can do easily

493
00:38:36 --> 00:38:40
is partials of u and v with
respect to x and y.

494
00:38:40 --> 00:38:42
I mean, the other one is not
very hard because here you can

495
00:38:42 --> 00:38:45
solve easily.
But, the one that's given to

496
00:38:45 --> 00:38:49
you is partial of u and v with
respect to x and y,

497
00:38:49 --> 00:38:55
so partial u partial x is one.
Partial u partial y is zero.

498
00:38:55 --> 00:39:03
Partial v partial x is y.
And partial v partial y is x.

499
00:39:03 --> 00:39:17
So that's just x.
So, that means that du dv is x

500
00:39:17 --> 00:39:20
dx dy.
Well, it would be absolute

501
00:39:20 --> 00:39:23
value of x, but x is positive in
our origin.

502
00:39:23 --> 00:39:35
So, at least we don't have to
worry about that.

503
00:39:35 --> 00:39:45
OK, so now that we have that,
we can try to look at the

504
00:39:45 --> 00:39:55
integrand in terms of u and v.
OK, so we were integrating x

505
00:39:55 --> 00:40:00
squared y dx dy.
So, let's switch it.

506
00:40:00 --> 00:40:09
Well, let's first switch the dx
dy that becomes one over x du

507
00:40:09 --> 00:40:15
dv.
So, that's actually xy du dv.

508
00:40:15 --> 00:40:18
And, what is xy in terms of u
and v?

509
00:40:18 --> 00:40:20
Well, here at least we had a
little bit of luck.

510
00:40:20 --> 00:40:26
xy is just v.
So, that's v du dv.

511
00:40:26 --> 00:40:32
So, in fact,
what we'll be computing is a

512
00:40:32 --> 00:40:40
double integral over some
mysterious region of v du dv.

513
00:40:40 --> 00:40:44
Now, last but not least,
we'll have to find what are the

514
00:40:44 --> 00:40:49
bounds for u and v in the new
integral so that we know how to

515
00:40:49 --> 00:40:50
evaluate this.

516
00:40:50 --> 00:41:14

517
00:41:14 --> 00:41:17
In fact, well,
we could do it du dv or dv du.

518
00:41:17 --> 00:41:23
We don't know yet.
Oh, amazing.

519
00:41:23 --> 00:41:31
It went all the way down this
time.

520
00:41:31 --> 00:41:43
OK, so it could be dv du if
that's easier.

521
00:41:43 --> 00:41:46
So, let's try to find the
bounds.

522
00:41:46 --> 00:41:52
In this case,
that's the hardest part.

523
00:41:52 --> 00:42:00
OK, so let me draw a picture in
xy coordinates and try to

524
00:42:00 --> 00:42:06
understand things using that.
OK, so x and y go from zero to

525
00:42:06 --> 00:42:08
one.
The region that we want to

526
00:42:08 --> 00:42:11
integrate over was just this
square.

527
00:42:11 --> 00:42:16
Let's try to figure out how u
and v vary there.

528
00:42:16 --> 00:42:23
So, let's say that we're going
to do it du dv.

529
00:42:23 --> 00:42:32
OK, so What we want to
understand is how u and v vary

530
00:42:32 --> 00:42:36
in here.
What's going to happen?

531
00:42:36 --> 00:42:40
So, the way we can think about
it is we try to figure out how

532
00:42:40 --> 00:42:43
we are slicing our origin.
OK, so here,

533
00:42:43 --> 00:42:46
we are integrating first over
u.

534
00:42:46 --> 00:42:51
That means we start by keeping
u constant, no,

535
00:42:51 --> 00:42:55
by keeping v constant as u
changes.

536
00:42:55 --> 00:43:03
OK, so u changes as v is
constant.

537
00:43:03 --> 00:43:06
What does it mean that I'm
keeping v constant.

538
00:43:06 --> 00:43:09
Well, what is v?
v is xy.

539
00:43:09 --> 00:43:13
So, that means I keep xy equals
constant.

540
00:43:13 --> 00:43:16
What does the curve xy equals
constant look like?

541
00:43:16 --> 00:43:22
Well, it's just a hyperbola.
y equals constant over x.

542
00:43:22 --> 00:43:28
So, if I look at the various
values of v that I can take,

543
00:43:28 --> 00:43:33
for each value of v,
if I fix a value of v,

544
00:43:33 --> 00:43:38
I will be moving on one of
these red curves.

545
00:43:38 --> 00:43:42
OK, and u, well,
u is the same thing as x.

546
00:43:42 --> 00:43:47
So, that means u will increase.
Here, maybe it will be 0.1 and

547
00:43:47 --> 00:43:51
it will increase all the way to
one here.

548
00:43:51 --> 00:43:59
OK, so we are just traveling on
each of these slices.

549
00:43:59 --> 00:44:03
Now, so the question we must
answer here is for a given value

550
00:44:03 --> 00:44:08
of v, what are the bounds for u?
So, I'm traveling on my curve,

551
00:44:08 --> 00:44:11
v equals constant,
and trying to figure out,

552
00:44:11 --> 00:44:14
when do I enter my origin?
When do I leave it?

553
00:44:14 --> 00:44:18
Well, I enter it when I go
through this side.

554
00:44:18 --> 00:44:24
So, the question is,
what's the value of u here?

555
00:44:24 --> 00:44:29
Well, we don't know that very
easily until we look at these

556
00:44:29 --> 00:44:32
formulas.
So, u equals x,

557
00:44:32 --> 00:44:36
OK, but we don't know what x is
at that point.

558
00:44:36 --> 00:44:42
v equals x and v equals xy.
What do we go here?

559
00:44:42 --> 00:44:44
Well, we don't know x,
but we know y certainly.

560
00:44:44 --> 00:44:49
OK, so let's forget about
trying to find u.

561
00:44:49 --> 00:44:53
And, let's say,
for now, we know y equals one.

562
00:44:53 --> 00:44:58
Well, if we set y equals one,
that tells us that u and v are

563
00:44:58 --> 00:45:03
both equal to x.
So, in terms of u and v,

564
00:45:03 --> 00:45:11
the equation of this uv
coordinate is u equals v.

565
00:45:11 --> 00:45:14
OK, I mean, the other way to do
it is, say that you know you

566
00:45:14 --> 00:45:17
want y equals one.
You want to know what is y in

567
00:45:17 --> 00:45:18
terms of u and v.
Well, it's easy.

568
00:45:18 --> 00:45:26
y is v over u.
So, let me actually add an

569
00:45:26 --> 00:45:31
extra step in case that's,
so, we know that y is v over u

570
00:45:31 --> 00:45:35
equals one.
So, that means u=v is my

571
00:45:35 --> 00:45:39
equation.
OK, so when I'm here,

572
00:45:39 --> 00:45:47
when I'm entering my region,
the value of u at this point is

573
00:45:47 --> 00:45:53
just v, u equals v.
That's the hard part.

574
00:45:53 --> 00:45:56
Now, we need to figure out,
so, we started u equals v.

575
00:45:56 --> 00:45:59
u increases,
increases, increases.

576
00:45:59 --> 00:46:01
Where does it exit?
It exits one when we are here.

577
00:46:01 --> 00:46:05
What's the value of u here?
One. That one is easier, right?

578
00:46:05 --> 00:46:10
This side here,
so, this side here is x equals

579
00:46:10 --> 00:46:13
one.
That means u equals one.

580
00:46:13 --> 00:46:20
So, we start at u equals one.
Now, we've done the inner

581
00:46:20 --> 00:46:24
integral.
What about the outer?

582
00:46:24 --> 00:46:28
So, we have to figure out,
what is the first and what is

583
00:46:28 --> 00:46:32
the last value of v that we'll
want to consider?

584
00:46:32 --> 00:46:36
Well, if you look at all these
hyperbola's, xy equals constant.

585
00:46:36 --> 00:46:39
What's the smallest value of xy
that we'll ever want to look at

586
00:46:39 --> 00:46:41
in here?
Zero, OK.

587
00:46:41 --> 00:46:49
Let me actually,
where's my yellow chalk?

588
00:46:49 --> 00:46:55
Is it, no, ah.
So, this one here,

589
00:46:55 --> 00:47:00
that's actually v=0.
So, we'll start at v equals

590
00:47:00 --> 00:47:02
zero.
And, what's the last hyperbola

591
00:47:02 --> 00:47:05
we want to look at?
Well, it's the one that's right

592
00:47:05 --> 00:47:07
there in the corner.
It's this one here.

593
00:47:07 --> 00:47:15
And, that's v equals one.
So, v goes from zero to one.

594
00:47:15 --> 00:47:17
OK, and now,
we can compute this.

595
00:47:17 --> 00:47:22
I mean, it's not particularly
easier than that one,

596
00:47:22 --> 00:47:26
but it's not harder either.
How else could we have gotten

597
00:47:26 --> 00:47:28
these bounds,
because that was quite evil.

598
00:47:28 --> 00:47:32
So, I would like to recommend
that you try this way in case it

599
00:47:32 --> 00:47:34
works well.
Just try to picture,

600
00:47:34 --> 00:47:38
what are the slices in terms of
u and v, and how you travel on

601
00:47:38 --> 00:47:40
them, where you enter,
where you leave,

602
00:47:40 --> 00:47:47
staying in the xy picture.
If that somehow doesn't work

603
00:47:47 --> 00:47:58
well, another way is to draw the
picture in the uv coordinates.

604
00:47:58 --> 00:48:04
So, switch to a uv picture.
So, what do I mean by that?

605
00:48:04 --> 00:48:09
Well, we had here a picture in
xy coordinates where we had our

606
00:48:09 --> 00:48:12
sides.
And, we are going to try to

607
00:48:12 --> 00:48:15
draw what it looks like in terms
of u and v.

608
00:48:15 --> 00:48:18
So, here we said this is x
equals one.

609
00:48:18 --> 00:48:24
That becomes u equals one.
So, we'll draw u equals one.

610
00:48:24 --> 00:48:30
This side we said is y equals
one becomes u equals v.

611
00:48:30 --> 00:48:33
That's what we've done over
there.

612
00:48:33 --> 00:48:39
OK, so u equals v.
Now, we have the two other

613
00:48:39 --> 00:48:41
sides to deal with.
Well, let's look at this one

614
00:48:41 --> 00:48:44
first.
So, that was x equals zero.

615
00:48:44 --> 00:48:48
What happens when x equals zero?
Well, both u and v are zero.

616
00:48:48 --> 00:48:51
So, this side actually gets
squished in the change of

617
00:48:51 --> 00:48:53
variables.
It's a bit strange,

618
00:48:53 --> 00:48:57
but it's a bit the same thing
as when you switch to polar

619
00:48:57 --> 00:49:00
coordinates at the origin,
r is zero but theta can be

620
00:49:00 --> 00:49:03
anything.
It's not always one point is

621
00:49:03 --> 00:49:07
one point.
So anyway, this is the origin,

622
00:49:07 --> 00:49:11
and then the last side,
y equals zero,

623
00:49:11 --> 00:49:15
and x varies just becomes v
equals zero.

624
00:49:15 --> 00:49:18
So, somehow,
in the change of variables,

625
00:49:18 --> 00:49:21
this square becomes this
triangle.

626
00:49:21 --> 00:49:24
And now, if we want to
integrate du dv,

627
00:49:24 --> 00:49:30
it means we are going to slice
by v equals constant.

628
00:49:30 --> 00:49:33
So, we are going to integrate
over slices like this,

629
00:49:33 --> 00:49:36
and you see for each value of
v, we go from u equals v to u

630
00:49:36 --> 00:49:41
equals one.
And, v goes from zero to one.

631
00:49:41 --> 00:49:44
OK, so you get the same bounds
just by drawing a different

632
00:49:44 --> 00:49:47
picture.
So, it's up to you to decide

633
00:49:47 --> 00:49:51
whether you prefer to think on
this picture or draw that one

634
00:49:51 --> 00:49:53
instead.
It depends on which problems

635
00:49:53 --> 00:49:55
you're doing.
 

636
00:49:55 --> 00:50:60