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PROFESSOR: Hi.
11
00:00:27,590 --> 00:00:30,760
In the reading assignment for
today, you'll notice in the
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00:00:30,760 --> 00:00:35,000
textbook that the topic is
double integrals in terms of
13
00:00:35,000 --> 00:00:36,570
polar coordinates.
14
00:00:36,570 --> 00:00:39,110
Now, the trouble with polar
coordinates is that aside from
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00:00:39,110 --> 00:00:42,450
straight lines, they're perhaps
the only coordinate
16
00:00:42,450 --> 00:00:46,110
system that we've studied in
terms of Euclidean geometry in
17
00:00:46,110 --> 00:00:47,650
a high school class.
18
00:00:47,650 --> 00:00:51,180
And consequently, if we were to
have a change of variables
19
00:00:51,180 --> 00:00:55,430
other than strict linear changes
of variables or polar
20
00:00:55,430 --> 00:00:59,360
coordinates, it might be
difficult to geometrically try
21
00:00:59,360 --> 00:01:02,860
to determine what the new double
integral looks like
22
00:01:02,860 --> 00:01:04,900
with respect to the
new variables.
23
00:01:04,900 --> 00:01:08,570
The text, you may recall, when
you-- or not you may recall,
24
00:01:08,570 --> 00:01:11,200
you haven't read it yet-- but
when you read the text, you'll
25
00:01:11,200 --> 00:01:13,500
notice at the end that Professor
Thomas says there is
26
00:01:13,500 --> 00:01:17,180
a technique called the Jacobian,
multiplying by the
27
00:01:17,180 --> 00:01:20,770
Jacobian determinant, that tells
you how to transfer a
28
00:01:20,770 --> 00:01:24,460
double integral from x- and
y-coordinates into another
29
00:01:24,460 --> 00:01:25,630
coordinate system.
30
00:01:25,630 --> 00:01:29,410
At any rate, with that prolog
as background, our aim in
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00:01:29,410 --> 00:01:33,290
today's lecture is to show
more generally how the
32
00:01:33,290 --> 00:01:37,820
Jacobian sneaks into the study
of multiple integrals.
33
00:01:37,820 --> 00:01:41,840
In particular, we call today's
lecture multiple integration
34
00:01:41,840 --> 00:01:43,280
and the Jacobian.
35
00:01:43,280 --> 00:01:46,620
And by way of review, let me
pick a problem that we've
36
00:01:46,620 --> 00:01:50,450
solved in the past in great
detail, but perhaps from a
37
00:01:50,450 --> 00:01:52,530
slightly different perspective,
that will lead
38
00:01:52,530 --> 00:01:55,980
into where the Jacobian matrix
and the Jacobian determinant
39
00:01:55,980 --> 00:01:57,260
comes from.
40
00:01:57,260 --> 00:02:00,460
Recall that when we want to
compute the definite integral
41
00:02:00,460 --> 00:02:04,900
1 to 3 2x squared root of x
squared plus 1 dx, we make the
42
00:02:04,900 --> 00:02:08,380
substitution u equals
x squared plus 1.
43
00:02:08,380 --> 00:02:12,290
Or inverting this, x equals
positive square
44
00:02:12,290 --> 00:02:13,450
root of u minus 1.
45
00:02:13,450 --> 00:02:17,280
And I emphasize the positive to
point out that in general
46
00:02:17,280 --> 00:02:21,280
the inverse of a squaring
function is not one to one.
47
00:02:21,280 --> 00:02:23,780
See a square root is usually
double valued.
48
00:02:23,780 --> 00:02:27,620
But notice that with the
restriction that x must be on
49
00:02:27,620 --> 00:02:31,430
the integral from 1 through
3, x cannot be negative.
50
00:02:31,430 --> 00:02:35,090
And therefore, we certainly
can assume that locally,
51
00:02:35,090 --> 00:02:38,640
meaning in the region in which
we're interested in, that x is
52
00:02:38,640 --> 00:02:41,040
the positive square
root of u minus 1.
53
00:02:41,040 --> 00:02:45,190
From this we saw that
du was 2xdx.
54
00:02:45,190 --> 00:02:47,910
We then went back to
this equation here.
55
00:02:47,910 --> 00:02:51,640
We replaced 2xdx by
its value du.
56
00:02:51,640 --> 00:02:54,910
We replaced the square root
of x squared plus 1 by the
57
00:02:54,910 --> 00:02:56,800
square root of u.
58
00:02:56,800 --> 00:03:01,270
And then noticing that when x
equaled 1, u equaled 2 and
59
00:03:01,270 --> 00:03:06,420
when x equaled 3, u equaled 10,
we wound up with the fact
60
00:03:06,420 --> 00:03:10,080
that the number named by this
definite integral was the same
61
00:03:10,080 --> 00:03:13,350
as the number named by this
definite integral.
62
00:03:13,350 --> 00:03:15,990
Now the only thing that I'd like
to say here as an aside
63
00:03:15,990 --> 00:03:20,680
is the following, there is
sometimes a tendency to think
64
00:03:20,680 --> 00:03:24,930
of dx as just being a symbol
over here, that we think of it
65
00:03:24,930 --> 00:03:29,240
as saying all we want is a
function whose derivative with
66
00:03:29,240 --> 00:03:31,770
respect to x is this.
67
00:03:31,770 --> 00:03:34,120
And that other than that, it
makes no difference what we
68
00:03:34,120 --> 00:03:34,990
put in here.
69
00:03:34,990 --> 00:03:37,600
What I would like you to
see at this time and--
70
00:03:37,600 --> 00:03:40,020
to review at this time because
we know it happens--
71
00:03:40,020 --> 00:03:43,570
is that if we made the
substitutions mentioned here,
72
00:03:43,570 --> 00:03:46,460
forgetting about the dx-- in
other words, if we replaced x
73
00:03:46,460 --> 00:03:49,800
by the positive square root of
u minus 1, if we replace x
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00:03:49,800 --> 00:03:55,250
squared plus 1 by u, and if we
replace the limits 1 to 3 by 2
75
00:03:55,250 --> 00:04:00,160
to 10 and then just tacked on
the du to indicate that we
76
00:04:00,160 --> 00:04:03,500
were doing this problem with
respect to u, the resulting
77
00:04:03,500 --> 00:04:07,420
definite integral would
not be equivalent to
78
00:04:07,420 --> 00:04:08,740
the original one.
79
00:04:08,740 --> 00:04:11,580
That is not to say that this
couldn't be computed.
80
00:04:11,580 --> 00:04:17,620
What I mean is this number is
incorrect if by this number
81
00:04:17,620 --> 00:04:21,240
you mean the value of this
definite integral here.
82
00:04:21,240 --> 00:04:24,460
And notice that from a pictorial
point of view, all
83
00:04:24,460 --> 00:04:29,450
we're really saying is that the
integral 1 to 3 2x squared
84
00:04:29,450 --> 00:04:34,410
root of x squared plus 1 dx is
the area of the region R when
85
00:04:34,410 --> 00:04:38,100
R is that region in the xy-plane
bounded between the
86
00:04:38,100 --> 00:04:44,100
lines x equals 1, x equals 3,
below by the x-axis and above
87
00:04:44,100 --> 00:04:47,270
by the curve y equals 2x times
the square root of x
88
00:04:47,270 --> 00:04:48,180
squared plus 1.
89
00:04:48,180 --> 00:04:51,620
And I've simply put the values
of these endpoints in here--
90
00:04:51,620 --> 00:04:55,510
namely when x is 1, y is 2
square roots of 2, when x is
91
00:04:55,510 --> 00:04:57,600
3, y is 6 square roots of 10--
92
00:04:57,600 --> 00:04:59,260
to give you sort of
an orientation of
93
00:04:59,260 --> 00:05:00,900
this particular curve.
94
00:05:00,900 --> 00:05:04,460
On the other hand, that other
integral that was incorrect--
95
00:05:04,460 --> 00:05:07,290
integral from 2 to
10, et cetera--
96
00:05:07,290 --> 00:05:11,840
is the area of the region S
where S is the region that's
97
00:05:11,840 --> 00:05:16,090
obtained by taking that integral
from 1 to 3 along the
98
00:05:16,090 --> 00:05:21,240
x-axis, mapping it by u equals
x squared plus 1, in other
99
00:05:21,240 --> 00:05:24,420
words it goes from 2 to 10.
100
00:05:24,420 --> 00:05:25,890
And in fact, you don't even
have to know that.
101
00:05:25,890 --> 00:05:27,890
All I'm saying is if you just
read this thing mechanically
102
00:05:27,890 --> 00:05:33,110
in the yu-plane, this would be
the area of the region S where
103
00:05:33,110 --> 00:05:38,610
S is bounded vertically by the
lines u equal 2 and u equal
104
00:05:38,610 --> 00:05:42,980
10, below by the u-axis and
above by the curve y equals
105
00:05:42,980 --> 00:05:46,040
twice square root u minus
1 square root of u.
106
00:05:46,040 --> 00:05:48,980
And it should be clear by
inspection that there is no
107
00:05:48,980 --> 00:05:53,340
reason to expect that the area
of the region R is the same as
108
00:05:53,340 --> 00:05:56,260
the area of the region
S even though both
109
00:05:56,260 --> 00:05:58,270
R and S have areas.
110
00:05:58,270 --> 00:06:02,040
Now, what the whole geometrical
impact is on this
111
00:06:02,040 --> 00:06:06,610
technique of integration,
techniques of integration,
112
00:06:06,610 --> 00:06:09,790
what the whole geometric
impact is is this.
113
00:06:09,790 --> 00:06:14,110
This is a difficult integral to
evaluate to find the area.
114
00:06:14,110 --> 00:06:21,020
Hopefully, one would hope that
we could find a way of scaling
115
00:06:21,020 --> 00:06:26,400
an element of area here to
correspond to an element of
116
00:06:26,400 --> 00:06:30,070
area here which was
easier to compute.
117
00:06:30,070 --> 00:06:33,820
And that since the mapping was
one to one, by adding up the
118
00:06:33,820 --> 00:06:38,240
appropriately scaled pieces
here, we equivalently add up
119
00:06:38,240 --> 00:06:40,780
the pieces to find the
area of the region R.
120
00:06:40,780 --> 00:06:43,670
Now, because I know that sounds
vague to you, I am
121
00:06:43,670 --> 00:06:46,170
going to do that in
much more detail.
122
00:06:46,170 --> 00:06:49,220
For the time being let me point
out, though, that if I
123
00:06:49,220 --> 00:06:52,500
want to view this as a mapping,
the interesting thing
124
00:06:52,500 --> 00:06:58,660
is that any point on the
x-axis maps into the
125
00:06:58,660 --> 00:07:03,410
corresponding point on the
u-axis by the mapping what?
126
00:07:03,410 --> 00:07:05,780
u equals x squared plus 1.
127
00:07:05,780 --> 00:07:09,740
But it's important to point out
that the values of x and u
128
00:07:09,740 --> 00:07:11,540
were not independent.
129
00:07:11,540 --> 00:07:15,290
They were chosen to obey the
identity u equals x squared
130
00:07:15,290 --> 00:07:19,160
plus 1, or x is the positive
square root of u minus 1.
131
00:07:19,160 --> 00:07:22,070
So what that means geometrically
is that whatever
132
00:07:22,070 --> 00:07:26,620
height was above a point in
the region R along the
133
00:07:26,620 --> 00:07:28,920
x-axis-- whatever height
was here--
134
00:07:28,920 --> 00:07:32,120
that height is the same when
that point is moved to the
135
00:07:32,120 --> 00:07:33,290
region S.
136
00:07:33,290 --> 00:07:36,610
Because that again sounds like
a difficult mouthful, let me
137
00:07:36,610 --> 00:07:37,330
write that.
138
00:07:37,330 --> 00:07:41,700
All I'm saying is notice that
for the u corresponding to a
139
00:07:41,700 --> 00:07:47,020
given x, 2x times the square
root of x squared plus 1 is
140
00:07:47,020 --> 00:07:51,050
equal to 2 times the square root
of u minus 1 times the
141
00:07:51,050 --> 00:07:52,100
square root of u.
142
00:07:52,100 --> 00:07:53,080
How do I know that?
143
00:07:53,080 --> 00:07:56,650
Well, I know that because I
picked x to be the square root
144
00:07:56,650 --> 00:07:58,190
of u minus 1.
145
00:07:58,190 --> 00:08:01,320
Or equivalently what? u equals
x squared plus 1.
146
00:08:01,320 --> 00:08:03,920
This says that I can replace
x by the square
147
00:08:03,920 --> 00:08:05,260
root of u minus 1.
148
00:08:05,260 --> 00:08:08,380
This says I can replace
x squared plus 1 by u.
149
00:08:08,380 --> 00:08:12,830
Consequently, as long as the x
matches with the image u, this
150
00:08:12,830 --> 00:08:15,810
number is the same
as this number.
151
00:08:15,810 --> 00:08:19,140
In other words again, in terms
of a picture, if I start with
152
00:08:19,140 --> 00:08:25,520
the point on the x-axis x equals
2 and I'm looking at
153
00:08:25,520 --> 00:08:29,830
the point p being the point on
the region R directly above x
154
00:08:29,830 --> 00:08:34,380
equals 2, since 2 gets mapped
into 5 by the mapping u equals
155
00:08:34,380 --> 00:08:37,179
x squared plus 1-- see 2
squared plus 1 is 5--
156
00:08:37,179 --> 00:08:40,679
what was the height that went
with the point 2 over here?
157
00:08:40,679 --> 00:08:42,220
The height that went with
the point 2 over
158
00:08:42,220 --> 00:08:44,100
here was simply what?
159
00:08:44,100 --> 00:08:47,100
4 square roots of 5.
160
00:08:47,100 --> 00:08:49,720
And I claim that that's the same
as this, because when x
161
00:08:49,720 --> 00:08:52,070
was 2, u is 5.
162
00:08:52,070 --> 00:08:54,890
This is 2 square roots of 5.
163
00:08:54,890 --> 00:08:56,480
5 minus 1 is 4.
164
00:08:56,480 --> 00:08:58,030
Square root of 4 is 2.
165
00:08:58,030 --> 00:08:59,370
This is 4, therefore--
166
00:08:59,370 --> 00:09:00,010
2 times 2--
167
00:09:00,010 --> 00:09:01,510
4 square roots of 5.
168
00:09:01,510 --> 00:09:04,480
It makes no difference whether
your using the x or the u, but
169
00:09:04,480 --> 00:09:07,480
that the point keeps
the same height.
170
00:09:07,480 --> 00:09:11,880
It shifted laterally, but it
does not distort the height.
171
00:09:11,880 --> 00:09:16,500
Which means now if we want to
view this not as a mapping
172
00:09:16,500 --> 00:09:20,540
from xy-plane into the
uy-plane but more
173
00:09:20,540 --> 00:09:25,600
traditionally in terms of the
xy-plane into the uv-plane,
174
00:09:25,600 --> 00:09:27,570
that's what this v is
in parentheses here.
175
00:09:27,570 --> 00:09:31,520
It means that I might want to
name the y-axis the v-axis
176
00:09:31,520 --> 00:09:34,810
just so that I can use my
identification established in
177
00:09:34,810 --> 00:09:37,650
the previous block in our course
when we talked about
178
00:09:37,650 --> 00:09:40,820
mapping the xy-plane
into the uv-plane.
179
00:09:40,820 --> 00:09:43,700
That is not an accent mark over
the v. That just happens
180
00:09:43,700 --> 00:09:47,450
to be an interruption of this
arrow that connects 2 to 5.
181
00:09:47,450 --> 00:09:50,830
But at any rate, in terms of
mappings, notice that the
182
00:09:50,830 --> 00:09:55,720
region R is mapped onto the
region S by the mapping u
183
00:09:55,720 --> 00:09:59,450
equals x squared plus
1 and v equals y.
184
00:09:59,450 --> 00:10:01,980
In other words, u equals
x squared plus 1.
185
00:10:01,980 --> 00:10:06,290
But the y-coordinate is the
same as the v-coordinate.
186
00:10:06,290 --> 00:10:08,470
Just changing the name
of the axis here to
187
00:10:08,470 --> 00:10:11,010
correspond to the uv-plane.
188
00:10:11,010 --> 00:10:14,890
And now, the idea
is simply this.
189
00:10:14,890 --> 00:10:15,810
Since--
190
00:10:15,810 --> 00:10:18,570
at least in the domain that
we're interested in-- the
191
00:10:18,570 --> 00:10:23,400
mapping u equals x squared plus
1 v equals y maps R onto
192
00:10:23,400 --> 00:10:28,460
S in a one to one manner, each
increment of area delta A sub
193
00:10:28,460 --> 00:10:33,460
S matches one and only
one delta A sub R.
194
00:10:33,460 --> 00:10:36,670
Let me give you an example
of what I mean by this.
195
00:10:36,670 --> 00:10:39,920
Let's suppose I start with
the region S and let me
196
00:10:39,920 --> 00:10:42,450
arbitrarily divide it
up into a number--
197
00:10:42,450 --> 00:10:43,760
the interval from 2 to 10--
198
00:10:43,760 --> 00:10:45,460
into a number of pieces here.
199
00:10:45,460 --> 00:10:48,360
Oh, I guess in the diagram
that I've used here, I've
200
00:10:48,360 --> 00:10:51,510
divided this up into four
pieces, all right.
201
00:10:51,510 --> 00:10:55,780
So I get these four
little rectangles.
202
00:10:55,780 --> 00:10:59,190
An approximation for the area
of the region S would be the
203
00:10:59,190 --> 00:11:01,500
sum of the areas of these
four rectangles.
204
00:11:01,500 --> 00:11:06,370
What I'm saying is that under
our mapping, these four
205
00:11:06,370 --> 00:11:11,760
regions induce four mutually
exclusive regions that cover
206
00:11:11,760 --> 00:11:15,990
all of R. In fact, since v
equals y, the way we do this
207
00:11:15,990 --> 00:11:17,750
mechanically is--
208
00:11:17,750 --> 00:11:20,580
for example, let's focus on
just one of these little
209
00:11:20,580 --> 00:11:21,920
elements over here.
210
00:11:21,920 --> 00:11:27,680
Let's suppose I want to find
how to match this shaded
211
00:11:27,680 --> 00:11:34,240
rectangle with a suitable
rectangle of R. What I said is
212
00:11:34,240 --> 00:11:37,950
whatever the v-value is over
here, it must be the same as
213
00:11:37,950 --> 00:11:41,380
the y-value of the point
that mapped into this
214
00:11:41,380 --> 00:11:42,360
point on the axis.
215
00:11:42,360 --> 00:11:46,440
In other words, if I call this
point here u0, that comes from
216
00:11:46,440 --> 00:11:50,280
some point here which
I'll call x0.
217
00:11:50,280 --> 00:11:51,990
See u0 comes from x0.
218
00:11:51,990 --> 00:11:54,560
What must the height above
this point be?
219
00:11:54,560 --> 00:11:58,610
Since the transformation does
not change the y-value at all
220
00:11:58,610 --> 00:12:02,310
since v equals y, what this
means is I can now draw a line
221
00:12:02,310 --> 00:12:06,420
parallel to the u-axis here,
come over to here, and that
222
00:12:06,420 --> 00:12:11,710
locates where on the curve I'm
going to locate the point x0.
223
00:12:11,710 --> 00:12:14,600
See, in other words, I just come
across like this, either
224
00:12:14,600 --> 00:12:15,560
of these two ways.
225
00:12:15,560 --> 00:12:18,960
This is how I locate the
x0 that matches the u0.
226
00:12:18,960 --> 00:12:21,690
I take its height, take that
same height over to this
227
00:12:21,690 --> 00:12:23,870
curve, and project down.
228
00:12:23,870 --> 00:12:27,960
Notice, of course, that the
delta x that measures the
229
00:12:27,960 --> 00:12:31,610
difference between these two
points on the x-axis is not
230
00:12:31,610 --> 00:12:35,060
the same as the delta u that
measures the distance between
231
00:12:35,060 --> 00:12:35,870
these two points.
232
00:12:35,870 --> 00:12:39,290
But what I want you to see is
that these four rectangles
233
00:12:39,290 --> 00:12:43,100
here induce four rectangles
here.
234
00:12:43,100 --> 00:12:47,570
And what I would like to be able
to do is somehow be able,
235
00:12:47,570 --> 00:12:52,300
hopefully, to find out how to
express a typical rectangle
236
00:12:52,300 --> 00:12:57,240
here as a scaled version of
one of these rectangles,
237
00:12:57,240 --> 00:13:00,830
hoping that when I then take
the sum the resulting
238
00:13:00,830 --> 00:13:04,990
summation leads to an integral
which is easy to evaluate.
239
00:13:04,990 --> 00:13:07,120
And before I get into
that to show you
240
00:13:07,120 --> 00:13:08,330
what does happen here--
241
00:13:08,330 --> 00:13:10,690
I think I'm making this longer
than it may really seem--
242
00:13:10,690 --> 00:13:12,850
but let me just get onto
the next step.
243
00:13:12,850 --> 00:13:17,410
What I want to do next is to
blow up these two shaded area
244
00:13:17,410 --> 00:13:19,750
so I can look at them
in more detail.
245
00:13:19,750 --> 00:13:23,680
What I have is a region which
I'll call delta A sub S in the
246
00:13:23,680 --> 00:13:28,820
uv-plane and a delta A sub R
in the xy-plane where the
247
00:13:28,820 --> 00:13:31,650
mapping is, again, what?
u equals x squared
248
00:13:31,650 --> 00:13:33,750
plus 1 v equals y.
249
00:13:33,750 --> 00:13:36,690
This piece matches
with this piece.
250
00:13:36,690 --> 00:13:37,400
All right.
251
00:13:37,400 --> 00:13:41,100
Now, for small delta u notice
that by definition of
252
00:13:41,100 --> 00:13:46,170
derivative, delta x divided by
delta u is approximately dxdu.
253
00:13:46,170 --> 00:13:48,210
Consequently, I can say
that delta x is
254
00:13:48,210 --> 00:13:51,890
approximately dxdu times du.
255
00:13:51,890 --> 00:13:55,150
Now the thing that I really
want is not the area of a
256
00:13:55,150 --> 00:13:59,150
piece of S, I want a portion of
the area of R. I want delta
257
00:13:59,150 --> 00:14:04,950
A sub R. Notice that delta A sub
R has as its height y sub
258
00:14:04,950 --> 00:14:08,770
0 and as it's width delta x.
259
00:14:08,770 --> 00:14:14,520
Notice also that since v sub 0
equals y sub 0-- see y equals
260
00:14:14,520 --> 00:14:17,890
v in this transformation--
261
00:14:17,890 --> 00:14:26,300
notice that the area delta
AR is v0 times delta x.
262
00:14:26,300 --> 00:14:26,700
OK.
263
00:14:26,700 --> 00:14:29,370
We also know that delta
x is approximately
264
00:14:29,370 --> 00:14:31,350
dxdu times delta u.
265
00:14:31,350 --> 00:14:34,610
So making this substitution, I
see that my little increment
266
00:14:34,610 --> 00:14:38,630
of area in the region
R is precisely what?
267
00:14:38,630 --> 00:14:43,370
It's v0 times the replacement
for delta x,
268
00:14:43,370 --> 00:14:45,720
dxdu times delta u.
269
00:14:45,720 --> 00:14:51,210
Let me also notice that the
region delta AS over here has
270
00:14:51,210 --> 00:14:55,410
as its height v0, as
its base delta u.
271
00:14:55,410 --> 00:15:00,210
So the area of this
rectangle is v0du.
272
00:15:00,210 --> 00:15:04,080
Let me, therefore, group these
two factors together, rewrite
273
00:15:04,080 --> 00:15:09,270
this term in this fashion,
noticing that v0 delta u is
274
00:15:09,270 --> 00:15:13,850
delta A sub S. And I now have
the very interesting result
275
00:15:13,850 --> 00:15:17,990
that delta A sub R is not
delta A sub S. But the
276
00:15:17,990 --> 00:15:19,500
correction factor is what?
277
00:15:19,500 --> 00:15:24,530
It's dxdu multiplying delta A
sub S where, for the sake of
278
00:15:24,530 --> 00:15:27,630
argument over a small enough
strip here, let we assume that
279
00:15:27,630 --> 00:15:33,400
I've chosen dxdu to be evaluated
at u equals u0, say.
280
00:15:33,400 --> 00:15:36,610
At any rate, what we are saying
is to find all of the
281
00:15:36,610 --> 00:15:39,890
area of region R, we want to
add up all of these delta A
282
00:15:39,890 --> 00:15:44,120
sub Rs as the maximum delta
x sub k goes to 0.
283
00:15:44,120 --> 00:15:49,330
But from what we've just seen,
a typical delta A sub R is a
284
00:15:49,330 --> 00:15:54,890
dxdu evaluated at u equals u0
times delta A sub S. And
285
00:15:54,890 --> 00:15:59,000
therefore, to find delta A sub
R, this limit is precisely the
286
00:15:59,000 --> 00:16:00,750
same as this limit.
287
00:16:00,750 --> 00:16:04,830
Now the point is that delta A
sub S is certainly just as
288
00:16:04,830 --> 00:16:07,280
messy as delta A sub
R, in general.
289
00:16:07,280 --> 00:16:10,990
It may also happened that when
I scale delta A sub S by
290
00:16:10,990 --> 00:16:16,000
multiplying it by dxdu, the
result is even more messy than
291
00:16:16,000 --> 00:16:17,990
the original expression.
292
00:16:17,990 --> 00:16:23,240
But it's also possible that dxdu
happens to be the factor
293
00:16:23,240 --> 00:16:26,820
that wipes out the nasty part
of delta A sub S. You see,
294
00:16:26,820 --> 00:16:30,430
what I'm saying is if there is
a one to one correspondence--
295
00:16:30,430 --> 00:16:31,370
which there is--
296
00:16:31,370 --> 00:16:35,830
between the delta ASes and the
delta ARs, if this expression
297
00:16:35,830 --> 00:16:42,210
here happens to be convenient,
I can find this sum simply by
298
00:16:42,210 --> 00:16:43,760
computing this sum.
299
00:16:43,760 --> 00:16:46,090
And that's why in techniques
of integration, that's
300
00:16:46,090 --> 00:16:47,710
precisely what we look for.
301
00:16:47,710 --> 00:16:50,940
We look for the change of
variable, the substitution,
302
00:16:50,940 --> 00:16:52,820
that makes this thing
simplify.
303
00:16:52,820 --> 00:16:54,820
And that's precisely
what happened in
304
00:16:54,820 --> 00:16:55,940
this particular example.
305
00:16:55,940 --> 00:16:59,980
Keep in mind that what I've
written down over here is true
306
00:16:59,980 --> 00:17:03,565
for any substitution when x is
some function of u, not just
307
00:17:03,565 --> 00:17:05,680
for x equals u squared plus 1.
308
00:17:05,680 --> 00:17:07,119
I could've done this any time.
309
00:17:07,119 --> 00:17:10,750
But what I claim is that
if x weren't equal to--
310
00:17:10,750 --> 00:17:13,310
what was it-- the square root
of u minus 1, this wouldn't
311
00:17:13,310 --> 00:17:16,810
have turned out to be a
very nice expression.
312
00:17:16,810 --> 00:17:19,069
And you see this is going to be
called the one-dimensional
313
00:17:19,069 --> 00:17:20,150
Jacobian later on.
314
00:17:20,150 --> 00:17:22,990
This is the correction factor,
the scaling factor, you see.
315
00:17:22,990 --> 00:17:24,770
Let's see how that worked out.
316
00:17:24,770 --> 00:17:30,020
Notice that delta A sub S
was v0 times delta u.
317
00:17:30,020 --> 00:17:33,390
And notice that by definition of
what the curve looked like
318
00:17:33,390 --> 00:17:39,080
in the uv-plane, v0 is twice the
square root of u0 minus 1
319
00:17:39,080 --> 00:17:41,200
times the square root of u0.
320
00:17:41,200 --> 00:17:43,850
On the other hand,
what is dxdu?
321
00:17:43,850 --> 00:17:48,240
Since u is equal to x squared
plus 1, it's easy to show that
322
00:17:48,240 --> 00:17:52,710
dxdu is 1 over twice the square
root of u minus 1.
323
00:17:52,710 --> 00:17:58,080
So in particular when u equals
u0, dxdu is 1 over 2 square
324
00:17:58,080 --> 00:18:00,320
root of u0 minus 1.
325
00:18:00,320 --> 00:18:04,760
Notice now, even though delta A
sub S is quite messy, when I
326
00:18:04,760 --> 00:18:08,080
multiply it by this particular
scaling factor, look at what
327
00:18:08,080 --> 00:18:09,920
that scaling factor wipes out.
328
00:18:09,920 --> 00:18:13,740
The 2 square root of u0 minus
1 wipes this out.
329
00:18:13,740 --> 00:18:17,660
All I have left is the square
root of u0 times delta u.
330
00:18:17,660 --> 00:18:20,610
When I form the definite
integral summing this thing
331
00:18:20,610 --> 00:18:24,710
up, it's trivial, you see, to
see that that simply comes out
332
00:18:24,710 --> 00:18:25,630
to be what?
333
00:18:25,630 --> 00:18:30,120
The definite integral from 2
to 10 square root of u du.
334
00:18:30,120 --> 00:18:34,900
And since this particular sum
was equal to A sub R, that is
335
00:18:34,900 --> 00:18:39,920
the mapping interpretation of
why the area of the region R
336
00:18:39,920 --> 00:18:42,900
can be evaluated by this
particular integral.
337
00:18:42,900 --> 00:18:46,310
Now in a sense, all of this has
been review even though
338
00:18:46,310 --> 00:18:48,640
the pitch has been slightly
different.
339
00:18:48,640 --> 00:18:53,220
Let me now generalized this
to a bona fide mapping of
340
00:18:53,220 --> 00:18:55,800
two-dimensional space into
two-dimensional space.
341
00:18:55,800 --> 00:18:59,320
And the reason I use the word
bona fide is that when you say
342
00:18:59,320 --> 00:19:03,680
let u equal x squared plus 1 and
let v equal y, you really
343
00:19:03,680 --> 00:19:07,040
haven't got a general mapping of
two-dimensional space into
344
00:19:07,040 --> 00:19:08,230
two-dimensional space.
345
00:19:08,230 --> 00:19:11,480
You've essentially let the
y-axis remain fixed.
346
00:19:11,480 --> 00:19:13,880
So let me talk about something
more general.
347
00:19:13,880 --> 00:19:16,920
Let's suppose I have an
arbitrary region R and an
348
00:19:16,920 --> 00:19:21,910
arbitrary function f bar
which maps R onto S in
349
00:19:21,910 --> 00:19:24,320
a one to one manner.
350
00:19:24,320 --> 00:19:26,530
By the way, notice the
whole idea is this.
351
00:19:26,530 --> 00:19:29,520
When I want the area of the
region R, it's going to
352
00:19:29,520 --> 00:19:33,260
involve a dxdy inside
the double integral.
353
00:19:33,260 --> 00:19:36,410
When I want the area of the
region S, that's what's going
354
00:19:36,410 --> 00:19:40,490
to involve a delta u times delta
v. Now the reason that
355
00:19:40,490 --> 00:19:44,710
delta u times delta v is indeed
a bona fide element of
356
00:19:44,710 --> 00:19:49,010
area when we're breaking up S
lies in the fact that in the
357
00:19:49,010 --> 00:19:54,260
us-plane delta u times delta
v is the actual area of an
358
00:19:54,260 --> 00:20:01,890
increment of area in S when we
break up S by lines parallel
359
00:20:01,890 --> 00:20:03,370
to the u and v axis.
360
00:20:03,370 --> 00:20:06,840
On the other hand, notice that
if we see what the line u
361
00:20:06,840 --> 00:20:10,780
equals a constant comes from,
back in the xy-plane u is a
362
00:20:10,780 --> 00:20:12,280
function of x and y.
363
00:20:12,280 --> 00:20:15,320
And to say that u of xy equals
a constant does not mean that
364
00:20:15,320 --> 00:20:16,290
you have a straight line.
365
00:20:16,290 --> 00:20:19,010
You could have some pretty
squiggly lines over here.
366
00:20:19,010 --> 00:20:22,770
In other words, it might be a
very funny looking line that
367
00:20:22,770 --> 00:20:25,760
maps into a straight line, a
straight vertical line, in the
368
00:20:25,760 --> 00:20:28,820
uv-plane with respect to
the mapping f bar.
369
00:20:28,820 --> 00:20:34,010
And in a similar way, the lines
v equal a constant--
370
00:20:34,010 --> 00:20:35,470
the lines v equal a constant--
371
00:20:35,470 --> 00:20:39,450
in the xy-plane I read what?
v of xy equals a constant.
372
00:20:39,450 --> 00:20:41,610
That's a general curve
in the xy-plane.
373
00:20:41,610 --> 00:20:44,170
What we're saying is that since
this mapping is one to
374
00:20:44,170 --> 00:20:48,940
one, when I break up this
region into elementary
375
00:20:48,940 --> 00:20:52,910
elements of rectangles, that
will induce a breaking up of
376
00:20:52,910 --> 00:20:55,980
this region into little
elements here.
377
00:20:55,980 --> 00:20:58,670
But notice that the resulting
elements-- say we take a piece
378
00:20:58,670 --> 00:21:02,180
like this, and we see where
that piece comes from.
379
00:21:02,180 --> 00:21:04,140
Let's say that particular
piece happened
380
00:21:04,140 --> 00:21:06,340
to come from here.
381
00:21:06,340 --> 00:21:09,770
Say that that was the one
to one correspondence.
382
00:21:09,770 --> 00:21:12,620
You can take delta u
times delta v here.
383
00:21:12,620 --> 00:21:16,470
But notice that delta u times
delta v simply means
384
00:21:16,470 --> 00:21:21,270
multiplying two edges which
aren't straight lines, which
385
00:21:21,270 --> 00:21:23,710
aren't necessarily
perpendicular, and hence in no
386
00:21:23,710 --> 00:21:25,970
way should represent what
the area of this little
387
00:21:25,970 --> 00:21:27,530
element here is.
388
00:21:27,530 --> 00:21:31,120
The key point is we do not want
the area of the region S.
389
00:21:31,120 --> 00:21:35,350
We want the area of the region
R. And what we're hoping that
390
00:21:35,350 --> 00:21:38,760
we can do is that by making the
change of variables that
391
00:21:38,760 --> 00:21:42,970
maps the region R in the
xy-plane into the region S in
392
00:21:42,970 --> 00:21:47,820
the uv-plane that we somehow
find a convenient way of
393
00:21:47,820 --> 00:21:52,810
scaling an individual element
of area here with respect to
394
00:21:52,810 --> 00:21:56,960
one here, and define the area
of this region by adding up
395
00:21:56,960 --> 00:21:59,130
the appropriate pieces here.
396
00:21:59,130 --> 00:22:03,070
And, again, let me show you what
that means in terms of
397
00:22:03,070 --> 00:22:04,960
just an enlargement again.
398
00:22:04,960 --> 00:22:07,530
You see, in the same as I did
before, let me take this
399
00:22:07,530 --> 00:22:10,300
little piece over here and
really blow it up.
400
00:22:10,300 --> 00:22:14,170
Let me take this little piece
that is the backmap of this--
401
00:22:14,170 --> 00:22:16,250
in other words, the piece that
maps into this-- let
402
00:22:16,250 --> 00:22:18,220
me blow that up.
403
00:22:18,220 --> 00:22:20,830
And the idea is this.
404
00:22:20,830 --> 00:22:25,260
If I pick delta u and delta v
sufficiently small, notice
405
00:22:25,260 --> 00:22:29,740
that the area of the backmap
off delta A sub S is
406
00:22:29,740 --> 00:22:32,520
approximately the area
of a parallelogram.
407
00:22:32,520 --> 00:22:36,520
You see, come back to this
statement after I've explained
408
00:22:36,520 --> 00:22:37,250
the picture.
409
00:22:37,250 --> 00:22:40,390
What I'm saying is I start with
an element of area delta
410
00:22:40,390 --> 00:22:43,200
A sub S in the uv-plane.
411
00:22:43,200 --> 00:22:44,820
You see I pick its vertices.
412
00:22:44,820 --> 00:22:49,090
I'll call them A bar,
B bar, D bar, C bar.
413
00:22:49,090 --> 00:22:51,490
That's a little rectangle
over here.
414
00:22:51,490 --> 00:22:54,780
Because the mapping is one to
one, I know that there is one
415
00:22:54,780 --> 00:22:59,500
and only one point in the
xy-plane that maps into A bar.
416
00:22:59,500 --> 00:22:59,980
See that?
417
00:22:59,980 --> 00:23:03,290
Let's call that point A. There
is one and only one point in
418
00:23:03,290 --> 00:23:05,330
the xy-plane that
maps into B bar.
419
00:23:05,330 --> 00:23:09,160
Let's call that B. One and only
one point in the xy-plane
420
00:23:09,160 --> 00:23:10,870
that maps into C bar.
421
00:23:10,870 --> 00:23:14,080
Let's call that point C. And let
me leave the point D out
422
00:23:14,080 --> 00:23:15,230
for a moment.
423
00:23:15,230 --> 00:23:19,050
Now the idea is if we call the
coordinates of A bar u0 comma
424
00:23:19,050 --> 00:23:23,760
v0, then because this is a line
parallel to the axis--
425
00:23:23,760 --> 00:23:26,670
call this dimension delta u--
426
00:23:26,670 --> 00:23:30,400
b bar is u0 plus delta
u comma v0.
427
00:23:30,400 --> 00:23:31,280
C bar--
428
00:23:31,280 --> 00:23:33,150
call this dimension delta v--
429
00:23:33,150 --> 00:23:39,130
is u0 comma v0 plus delta v. Now
the point is there is no
430
00:23:39,130 --> 00:23:42,675
reason why the image of B bar--
name of the point B--
431
00:23:42,675 --> 00:23:46,950
432
00:23:46,950 --> 00:23:49,190
has to be on the line
that joins A
433
00:23:49,190 --> 00:23:51,120
parallel to the x-axis.
434
00:23:51,120 --> 00:23:53,860
In other words, B is up
here someplace, C is
435
00:23:53,860 --> 00:23:55,290
up here some place.
436
00:23:55,290 --> 00:23:58,440
In other words, again, there's
no reason why these backmaps
437
00:23:58,440 --> 00:24:01,190
give me a rectangle over here.
438
00:24:01,190 --> 00:24:03,320
The point is that B has
some coordinates.
439
00:24:03,320 --> 00:24:06,490
It's x0 plus some increment
involving x-- let me
440
00:24:06,490 --> 00:24:08,130
call that delta x1--
441
00:24:08,130 --> 00:24:12,550
and its y-coordinate is
y0 plus delta y1.
442
00:24:12,550 --> 00:24:17,820
C is the point x0 plus some
increment delta x2 comma y0
443
00:24:17,820 --> 00:24:19,740
plus delta y2.
444
00:24:19,740 --> 00:24:23,190
And what I'm saying is now, if
I were to just look at the
445
00:24:23,190 --> 00:24:30,270
parallelogram which had AB and
AC as adjacent sides, it's
446
00:24:30,270 --> 00:24:33,810
very easy for me to find the
area of that parallelogram.
447
00:24:33,810 --> 00:24:36,790
Namely to find the area of a
parallelogram in vector form,
448
00:24:36,790 --> 00:24:39,660
I just take the magnitude of the
cross-product of the two
449
00:24:39,660 --> 00:24:42,330
vectors AB and AC.
450
00:24:42,330 --> 00:24:45,430
You see what's wrong with this
is that that particular
451
00:24:45,430 --> 00:24:49,060
parallelogram is not the
exact image of the
452
00:24:49,060 --> 00:24:51,300
backmap of delta AS.
453
00:24:51,300 --> 00:24:57,155
Sure, A bar maps exactly into A.
C bar maps exactly into C.
454
00:24:57,155 --> 00:25:03,010
B bar maps exactly into B. But
the point is that these points
455
00:25:03,010 --> 00:25:10,360
along A bar C bar do not map
into the straight line from A
456
00:25:10,360 --> 00:25:11,500
to C, in general.
457
00:25:11,500 --> 00:25:14,680
In other words, what
characterizes this?
458
00:25:14,680 --> 00:25:17,570
This is characterized
by delta u equals 0.
459
00:25:17,570 --> 00:25:22,450
And when u is written in terms
of x and y that doesn't say
460
00:25:22,450 --> 00:25:24,900
that delta x or delta y is 0.
461
00:25:24,900 --> 00:25:29,460
So, the true image of this might
be what I've represented
462
00:25:29,460 --> 00:25:32,380
with this dotted array here.
463
00:25:32,380 --> 00:25:35,250
In other words, what the true
backmap of delta A is this
464
00:25:35,250 --> 00:25:38,860
dotted thing where D is
now this vertex here.
465
00:25:38,860 --> 00:25:42,840
You see, there's no guarantee
that the backmap of D bar is
466
00:25:42,840 --> 00:25:46,090
going to be the fourth vertex
of this parallelogram.
467
00:25:46,090 --> 00:25:48,160
But what the key point is--
and this is where that
468
00:25:48,160 --> 00:25:50,140
familiarity is so important--
469
00:25:50,140 --> 00:25:53,320
is that if the transformation is
smooth enough, continuously
470
00:25:53,320 --> 00:25:57,280
differentiable, then what it
does mean is that as long as
471
00:25:57,280 --> 00:26:01,320
delta u and delta v are
sufficiently small, the true
472
00:26:01,320 --> 00:26:05,960
area of the region delta AR
that we're looking for is
473
00:26:05,960 --> 00:26:09,790
approximately equal to the area
of this parallelogram.
474
00:26:09,790 --> 00:26:11,940
And by approximately
equal I mean what?
475
00:26:11,940 --> 00:26:15,420
That the arrow goes to 0 as we
take the limit in forming the
476
00:26:15,420 --> 00:26:16,305
double sum.
477
00:26:16,305 --> 00:26:19,990
In other words, again, the
key point is this.
478
00:26:19,990 --> 00:26:26,400
The backmap of delta A sub S
yields delta AR, but that
479
00:26:26,400 --> 00:26:30,590
delta AR is approximately
this parallelogram.
480
00:26:30,590 --> 00:26:34,390
And the area of this
parallelogram is exactly the
481
00:26:34,390 --> 00:26:36,380
magnitude of AB--
482
00:26:36,380 --> 00:26:37,960
the vector AB--
483
00:26:37,960 --> 00:26:39,915
crossed with AC.
484
00:26:39,915 --> 00:26:42,820
In other words, the
approximation comes in because
485
00:26:42,820 --> 00:26:46,350
this is exactly the area
of the parallelogram.
486
00:26:46,350 --> 00:26:50,090
But delta A sub R is only
approximately equal to the
487
00:26:50,090 --> 00:26:51,700
area of the parallelogram.
488
00:26:51,700 --> 00:26:55,360
At any rate, notice in terms of
i and j components how easy
489
00:26:55,360 --> 00:26:58,420
it is to compute AB and AC.
490
00:26:58,420 --> 00:26:59,790
You see, where are the
components of the
491
00:26:59,790 --> 00:27:01,180
vector from A to B?
492
00:27:01,180 --> 00:27:03,370
The i component is delta x1.
493
00:27:03,370 --> 00:27:04,840
See, this minus this.
494
00:27:04,840 --> 00:27:10,020
The j component is this
minus this, right.
495
00:27:10,020 --> 00:27:12,700
In other words, AC has what
as its components?
496
00:27:12,700 --> 00:27:15,700
It's this minus this,
namely delta x2.
497
00:27:15,700 --> 00:27:17,990
This minus this is
the y-component.
498
00:27:17,990 --> 00:27:19,400
That's delta y2.
499
00:27:19,400 --> 00:27:21,810
In other words-- to write this
out so you don't have to
500
00:27:21,810 --> 00:27:23,750
listen to how fast
I'm talking--
501
00:27:23,750 --> 00:27:26,020
AB is this particular vector.
502
00:27:26,020 --> 00:27:28,500
AC is this particular vector.
503
00:27:28,500 --> 00:27:31,950
Remembering that when we take
a cross-product, i crossed j
504
00:27:31,950 --> 00:27:37,600
is 0, j cross j is 0, i
cross j is k, and j
505
00:27:37,600 --> 00:27:39,170
cross i is minus k.
506
00:27:39,170 --> 00:27:41,210
Remember, for the cross-product
we can't change
507
00:27:41,210 --> 00:27:44,070
the order of the terms in the
order in which they appear.
508
00:27:44,070 --> 00:27:45,940
We then determined what?
509
00:27:45,940 --> 00:27:51,710
That AB cross AC is delta x1
delta y2 minus delta x2 delta
510
00:27:51,710 --> 00:27:54,380
y1 times the vector k, the unit
511
00:27:54,380 --> 00:27:56,440
vector in the z direction.
512
00:27:56,440 --> 00:28:00,330
Now, delta x1 is exactly--
513
00:28:00,330 --> 00:28:02,510
see, remember, we're looking at
the parallelogram, which is
514
00:28:02,510 --> 00:28:03,260
a straight line.
515
00:28:03,260 --> 00:28:05,760
If you want to think of it in
terms of the region R, by
516
00:28:05,760 --> 00:28:08,740
differentials delta
x1 would be what?
517
00:28:08,740 --> 00:28:12,030
It's approximately the partial
of x with respect to u times
518
00:28:12,030 --> 00:28:15,220
delta u plus the partial of x
with respect to v times delta
519
00:28:15,220 --> 00:28:21,040
v. Similarly, delta y1 is y sub
u times delta u plus y sub
520
00:28:21,040 --> 00:28:22,690
v times delta v.
521
00:28:22,690 --> 00:28:25,200
But now keep in mind
where delta x1 and
522
00:28:25,200 --> 00:28:27,940
delta y1 come from.
523
00:28:27,940 --> 00:28:32,700
Delta x1 and delta y1 come from
the back mapping from A
524
00:28:32,700 --> 00:28:35,540
bar B bar back to AB.
525
00:28:35,540 --> 00:28:41,240
And along A bar B bar, noticed
that v is always equal to v0.
526
00:28:41,240 --> 00:28:43,700
So that means that
delta v is 0.
527
00:28:43,700 --> 00:28:48,040
That means, therefore, that
because delta v is 0, delta x1
528
00:28:48,040 --> 00:28:51,440
and delta y1 are simply this.
529
00:28:51,440 --> 00:28:52,750
See this drops out.
530
00:28:52,750 --> 00:28:55,890
Similarly, delta x2 and
delta y2 come from the
531
00:28:55,890 --> 00:28:58,370
backmap of A bar A bar.
532
00:28:58,370 --> 00:29:01,530
Along A bar C bar,
u is equal to u0.
533
00:29:01,530 --> 00:29:03,130
So delta u is zero.
534
00:29:03,130 --> 00:29:05,730
So writing this out,
these drop out.
535
00:29:05,730 --> 00:29:09,820
And now with these terms being
0, with these terms being 0, I
536
00:29:09,820 --> 00:29:14,140
could very simply compute the
product delta x1 delta y2
537
00:29:14,140 --> 00:29:16,360
minus delta x2 delta y1.
538
00:29:16,360 --> 00:29:18,740
And if I do that, you
see, right away
539
00:29:18,740 --> 00:29:21,280
what I obtain is what?
540
00:29:21,280 --> 00:29:23,910
It's this times this.
541
00:29:23,910 --> 00:29:29,290
I now, OK, collect
the terms here.
542
00:29:29,290 --> 00:29:31,040
In other words, I
want the delta u
543
00:29:31,040 --> 00:29:32,530
and the delta v together.
544
00:29:32,530 --> 00:29:37,260
The multiplier out front here is
x sub u times y sub v. In a
545
00:29:37,260 --> 00:29:42,180
similar way, delta x2 times
delta y1 is x sub v y sub u
546
00:29:42,180 --> 00:29:46,130
times delta u delta v.
Therefore, putting that into
547
00:29:46,130 --> 00:29:51,120
here, the magnitude of AB
cross AC is simply this
548
00:29:51,120 --> 00:29:54,740
expression here, noticing that
the k vector drops out because
549
00:29:54,740 --> 00:29:56,630
its magnitude is 1.
550
00:29:56,630 --> 00:30:00,490
Notice that delta u times delta
v is precisely delta A
551
00:30:00,490 --> 00:30:06,160
sub S and that x sub u times y
sub v minus x sub v times y
552
00:30:06,160 --> 00:30:10,500
sub u is precisely the
determinant of our old friend
553
00:30:10,500 --> 00:30:13,850
the Jacobian matrix, the
Jacobian matrix of x and y
554
00:30:13,850 --> 00:30:18,120
with respect to u and v. In
other words, delta A sub R is
555
00:30:18,120 --> 00:30:21,580
approximately equal to the
determinant of the Jacobian
556
00:30:21,580 --> 00:30:25,820
matrix of x and y with respect
to u and v times delta AS.
557
00:30:25,820 --> 00:30:29,430
And if I now perform this
double sum, you see this
558
00:30:29,430 --> 00:30:30,230
becomes what?
559
00:30:30,230 --> 00:30:32,490
The area of the region R--
560
00:30:32,490 --> 00:30:33,840
just write that in, that's
really the area
561
00:30:33,840 --> 00:30:34,900
of the region R--
562
00:30:34,900 --> 00:30:38,140
is a double integral
over R dxdy.
563
00:30:38,140 --> 00:30:39,610
And that's the same as what?
564
00:30:39,610 --> 00:30:45,460
The double integral over S times
dudv multiplied by the
565
00:30:45,460 --> 00:30:48,710
scaling factor of the Jacobian
determinant.
566
00:30:48,710 --> 00:30:51,640
By the way, notice
I dropped the
567
00:30:51,640 --> 00:30:53,490
determinant symbol over here.
568
00:30:53,490 --> 00:30:56,970
The reason for that is that many
textbooks, including our
569
00:30:56,970 --> 00:31:02,360
own, use this notation not to
name the Jacobian matrix but
570
00:31:02,360 --> 00:31:04,250
to name the Jacobian
determinant.
571
00:31:04,250 --> 00:31:07,940
I have been using this to name
the Jacobian matrix.
572
00:31:07,940 --> 00:31:10,920
From this point on, I will
now switch to become
573
00:31:10,920 --> 00:31:12,390
uniform with the text.
574
00:31:12,390 --> 00:31:15,550
And unless otherwise specified,
I will write this
575
00:31:15,550 --> 00:31:17,890
rather than put the determinant
symbol in.
576
00:31:17,890 --> 00:31:21,280
From now on in our course,
when I write this I am
577
00:31:21,280 --> 00:31:26,330
referring to the Jacobian
determinant, OK.
578
00:31:26,330 --> 00:31:27,730
But the thing is this.
579
00:31:27,730 --> 00:31:30,690
Notice that the given mapping
might straighten out the
580
00:31:30,690 --> 00:31:36,030
region R into a nicer looking
region S. But to offset this,
581
00:31:36,030 --> 00:31:40,500
it may also turn out the
resulting integrand here is
582
00:31:40,500 --> 00:31:42,280
much worse than the
integrand here.
583
00:31:42,280 --> 00:31:45,440
Here the multiplier of
dxdy was the simple
584
00:31:45,440 --> 00:31:47,670
number 1, wasn't it?
585
00:31:47,670 --> 00:31:51,230
Here, what's multiplying dudv,
no matter how nice S is, is
586
00:31:51,230 --> 00:31:54,620
this expression here, which
may be quite messy.
587
00:31:54,620 --> 00:31:57,780
And therefore, in most practical
applications where
588
00:31:57,780 --> 00:32:02,280
one solves multiple integrals
by a change of variable, one
589
00:32:02,280 --> 00:32:05,590
not only wants a change of
variables that straightens out
590
00:32:05,590 --> 00:32:09,160
the region into a nice looking
one, he wants a combination of
591
00:32:09,160 --> 00:32:09,680
two things.
592
00:32:09,680 --> 00:32:12,210
He would like a nice
looking region.
593
00:32:12,210 --> 00:32:15,550
And more importantly, even if
he can't get a nicer looking
594
00:32:15,550 --> 00:32:19,180
region, at least if he gets a
correction factor, a Jacobian
595
00:32:19,180 --> 00:32:22,400
determinant, that gives him
something that's easy to
596
00:32:22,400 --> 00:32:24,960
integrate, he'll settle
for that.
597
00:32:24,960 --> 00:32:27,580
And what that means, hopefully,
will become clearer
598
00:32:27,580 --> 00:32:30,890
as we go through the exercises
and the reading material.
599
00:32:30,890 --> 00:32:34,140
At any rate, I think that's
all I want to say about a
600
00:32:34,140 --> 00:32:36,400
supplement to Professor Thomas'
treatment of polar
601
00:32:36,400 --> 00:32:37,910
coordinates at this time.
602
00:32:37,910 --> 00:32:39,960
And until next time,
then, good bye.
603
00:32:39,960 --> 00:32:46,160
604
00:32:46,160 --> 00:32:49,350
Funding for the publication of
this video was provided by the
605
00:32:49,350 --> 00:32:53,410
Gabrielle and Paul Rosenbaum
Foundation.
606
00:32:53,410 --> 00:32:57,580
Help OCW continue to provide
free and open access to MIT
607
00:32:57,580 --> 00:33:01,998
courses by making a donation
at ocw.mit.edu/donate.
608
00:33:01,998 --> 00:33:06,560