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OK, so far we've learned how to
do double integrals in terms of
8
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
11
00:00:40 --> 00:00:44
might want to do.
OK, so today we're going to see
12
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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15
00:01:02 --> 00:01:10
OK, so let me start with a
simple example.
16
00:01:10 --> 00:01:21
Let's say that we want to find
the area of an ellipse with
17
00:01:21 --> 00:01:28
semi-axes a and b.
OK, so that means an ellipse is
18
00:01:28 --> 00:01:38
just like a squished circle.
And so, there's a and there's b.
19
00:01:38 --> 00:01:44
And, the equation of that
ellipse is x over a squared plus
20
00:01:44 --> 00:01:49
y over b squared equals one.
That's the curve,
21
00:01:49 --> 00:01:53
and the inside region is where
this is less than one.
22
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.
31
00:02:30 --> 00:02:37
That's x over a2 plus y over b2
less than 1.
32
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.
37
00:02:55 --> 00:02:57
OK, if this were a circle,
we would switch to polar
38
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
61
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
63
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,
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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
71
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,
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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.
81
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
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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
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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,
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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