1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high-quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:16,765 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,765 --> 00:00:17,390 at ocw.mit.edu. 8 00:00:26,830 --> 00:00:27,590 PROFESSOR: Hi. 9 00:00:27,590 --> 00:00:29,960 In the reading assignment for today, 10 00:00:29,960 --> 00:00:32,700 you'll notice in the textbook that the topic 11 00:00:32,700 --> 00:00:36,556 is double integrals in terms of polar coordinates. 12 00:00:36,556 --> 00:00:38,180 Now, the trouble with polar coordinates 13 00:00:38,180 --> 00:00:39,970 is that, aside from straight lines, 14 00:00:39,970 --> 00:00:43,250 they're perhaps the only coordinate system that we've 15 00:00:43,250 --> 00:00:46,760 studied in terms of Euclidean geometry in a high school 16 00:00:46,760 --> 00:00:47,650 class. 17 00:00:47,650 --> 00:00:51,180 And consequently, if we were to have a change of variables 18 00:00:51,180 --> 00:00:54,840 other than strict linear changes of variables 19 00:00:54,840 --> 00:00:57,070 or polar coordinates, it might be 20 00:00:57,070 --> 00:01:00,150 difficult to geometrically try to determine 21 00:01:00,150 --> 00:01:03,480 what the new double integral looks like with respect 22 00:01:03,480 --> 00:01:04,900 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 --> 00:01:11,036 you haven't read it yet-- but when you read the text, 25 00:01:11,036 --> 00:01:13,160 you'll notice at the end that Professor Thomas says 26 00:01:13,160 --> 00:01:16,290 there is a technique called the Jacobian, 27 00:01:16,290 --> 00:01:18,600 multiplying by the Jacobian determinant, 28 00:01:18,600 --> 00:01:22,300 that tells you how to transfer a double integral 29 00:01:22,300 --> 00:01:25,630 from x- and y-coordinates into another coordinate system. 30 00:01:25,630 --> 00:01:28,760 At any rate, with that prolog as background, 31 00:01:28,760 --> 00:01:31,560 our aim in today's lecture is to show, 32 00:01:31,560 --> 00:01:34,910 more generally, how the Jacobian sneaks 33 00:01:34,910 --> 00:01:37,820 into the study of multiple integrals. 34 00:01:37,820 --> 00:01:41,840 In particular, we call today's lecture multiple integration 35 00:01:41,840 --> 00:01:43,280 and the Jacobian. 36 00:01:43,280 --> 00:01:46,290 And by way of review, let me pick a problem 37 00:01:46,290 --> 00:01:48,830 that we've solved in the past in great detail, 38 00:01:48,830 --> 00:01:52,120 but perhaps from a slightly different perspective, that 39 00:01:52,120 --> 00:01:54,490 will lead into where the Jacobian matrix 40 00:01:54,490 --> 00:01:57,260 and the Jacobian determinant comes from. 41 00:01:57,260 --> 00:02:00,670 Recall that when we wanted to compute the definite integral 1 42 00:02:00,670 --> 00:02:04,330 to 3 2x squared root of x squared plus 1 dx, 43 00:02:04,330 --> 00:02:08,380 we made the substitution u equals x squared plus 1. 44 00:02:08,380 --> 00:02:12,440 Or, inverting this, x equals positive square root 45 00:02:12,440 --> 00:02:13,450 of u minus 1. 46 00:02:13,450 --> 00:02:15,690 And I emphasize the positive to point out 47 00:02:15,690 --> 00:02:19,890 that in general, the inverse of a squaring function 48 00:02:19,890 --> 00:02:21,280 is not one-to-one. 49 00:02:21,280 --> 00:02:23,780 See, the square root is usually double valued. 50 00:02:23,780 --> 00:02:27,400 But notice that with the restriction that x must be 51 00:02:27,400 --> 00:02:31,430 on the interval from 1 through 3, x cannot be negative. 52 00:02:31,430 --> 00:02:35,090 And therefore, we certainly can assume that locally, 53 00:02:35,090 --> 00:02:36,710 meaning in the region in which we're 54 00:02:36,710 --> 00:02:39,930 interested in, that x is the positive square root of u 55 00:02:39,930 --> 00:02:41,040 minus 1. 56 00:02:41,040 --> 00:02:45,190 From this we saw that du was 2x*dx. 57 00:02:45,190 --> 00:02:47,910 We then went back to this equation here. 58 00:02:47,910 --> 00:02:51,640 We replaced 2x*dx by its value du. 59 00:02:51,640 --> 00:02:53,730 We replaced the square root of x squared 60 00:02:53,730 --> 00:02:56,800 plus 1 by the square root of u. 61 00:02:56,800 --> 00:03:01,100 And then noticing that when x equaled 1, u equaled 2 62 00:03:01,100 --> 00:03:06,420 and when x equaled 3, u equaled 10, we wound up with the fact 63 00:03:06,420 --> 00:03:09,250 that the number named by this definite integral 64 00:03:09,250 --> 00:03:13,350 was the same as the number named by this definite integral. 65 00:03:13,350 --> 00:03:15,990 Now the only thing that I'd like to say here, as an aside, 66 00:03:15,990 --> 00:03:20,090 is the following: there is sometimes a tendency 67 00:03:20,090 --> 00:03:24,030 to think of dx as just being a symbol over here, 68 00:03:24,030 --> 00:03:26,630 that we think of it as saying all we want 69 00:03:26,630 --> 00:03:31,770 is a function whose derivative with respect to x is this. 70 00:03:31,770 --> 00:03:33,840 And that other than that, it makes no difference 71 00:03:33,840 --> 00:03:34,990 what we put in here. 72 00:03:34,990 --> 00:03:37,220 What I would like you to see at this time 73 00:03:37,220 --> 00:03:40,020 and-- to review at this time because we know it happens-- 74 00:03:40,020 --> 00:03:43,570 is that if we made the substitutions mentioned here, 75 00:03:43,570 --> 00:03:45,470 forgetting about the dx-- in other words, 76 00:03:45,470 --> 00:03:47,810 if we replaced x by the positive square root of u 77 00:03:47,810 --> 00:03:51,710 minus 1, if we replace x squared plus 1 by u, 78 00:03:51,710 --> 00:03:56,350 and if we replace the limits 1 to 3 by 2 to 10, 79 00:03:56,350 --> 00:04:00,600 and then just tacked on the du to indicate that we were doing 80 00:04:00,600 --> 00:04:04,880 this problem with respect to u, the resulting definite integral 81 00:04:04,880 --> 00:04:08,740 would not be equivalent to the original one. 82 00:04:08,740 --> 00:04:11,580 That is not to say that this couldn't be computed. 83 00:04:11,580 --> 00:04:17,620 What I mean is this number is incorrect if by this number 84 00:04:17,620 --> 00:04:21,240 you mean the value of this definite integral here. 85 00:04:21,240 --> 00:04:24,180 And notice that from a pictorial point of view, 86 00:04:24,180 --> 00:04:29,450 all we're really saying is that the integral 1 to 3 2x squared 87 00:04:29,450 --> 00:04:34,410 root of x squared plus 1 dx is the area of the region R, where 88 00:04:34,410 --> 00:04:38,470 R is that region in the xy-plane bounded between the lines 89 00:04:38,470 --> 00:04:43,540 x equals 1, x equals 3, below by the x-axis, 90 00:04:43,540 --> 00:04:46,500 and above by the curve y equals 2x times 91 00:04:46,500 --> 00:04:48,180 the square root of x squared plus 1. 92 00:04:48,180 --> 00:04:50,870 And I've simply put the values of these endpoints 93 00:04:50,870 --> 00:04:54,980 in here-- namely when x is 1, y is 2 square roots of 2, 94 00:04:54,980 --> 00:04:57,600 when x is 3, y is 6 square roots of 10-- 95 00:04:57,600 --> 00:04:59,180 to give you sort of an orientation 96 00:04:59,180 --> 00:05:00,900 of this particular curve. 97 00:05:00,900 --> 00:05:04,460 On the other hand, that other integral that was incorrect-- 98 00:05:04,460 --> 00:05:07,290 integral from 2 to 10, et cetera-- 99 00:05:07,290 --> 00:05:10,510 is the area of the region S where 100 00:05:10,510 --> 00:05:14,760 S is the region that's obtained by taking that integral 101 00:05:14,760 --> 00:05:19,570 from 1 to 3 along the x-axis, mapping it by u 102 00:05:19,570 --> 00:05:21,560 equals x squared plus 1-- in other words, 103 00:05:21,560 --> 00:05:23,974 it goes from 2 to 10. 104 00:05:23,974 --> 00:05:25,890 And in fact, you don't even have to know that. 105 00:05:25,890 --> 00:05:28,390 All I'm saying is, if you just read this thing mechanically, 106 00:05:28,390 --> 00:05:32,840 in the yu-plane, this would be the area of the region S 107 00:05:32,840 --> 00:05:38,050 where S is bounded vertically by the lines u equal 2 108 00:05:38,050 --> 00:05:40,930 and u equal 10, below by the u-axis 109 00:05:40,930 --> 00:05:43,870 and above by the curve y equals twice square root 110 00:05:43,870 --> 00:05:46,040 u minus 1 square root of u. 111 00:05:46,040 --> 00:05:48,330 And it should be clear by inspection 112 00:05:48,330 --> 00:05:50,390 that there is no reason to expect 113 00:05:50,390 --> 00:05:53,160 that the area of the region R is the same 114 00:05:53,160 --> 00:05:58,270 as the area of the region S even though both R and S have areas. 115 00:05:58,270 --> 00:06:01,120 Now, what the whole geometrical impact 116 00:06:01,120 --> 00:06:05,430 is on this technique of integration, 117 00:06:05,430 --> 00:06:08,103 techniques of integration, what the whole geometric impact is 118 00:06:08,103 --> 00:06:09,790 is this. 119 00:06:09,790 --> 00:06:14,110 This is a difficult integral to evaluate to find the area. 120 00:06:14,110 --> 00:06:17,530 Hopefully, one would hope that we 121 00:06:17,530 --> 00:06:22,290 could find a way of scaling an element of area 122 00:06:22,290 --> 00:06:27,430 here to correspond to an element of area here 123 00:06:27,430 --> 00:06:30,070 which was easier to compute. 124 00:06:30,070 --> 00:06:32,560 And that since the mapping was one to one, 125 00:06:32,560 --> 00:06:36,410 by adding up the appropriately scaled pieces here, 126 00:06:36,410 --> 00:06:38,760 we equivalently add up the pieces 127 00:06:38,760 --> 00:06:40,780 to find the area of the region R. 128 00:06:40,780 --> 00:06:43,420 Now, because I know that sounds vague to you, 129 00:06:43,420 --> 00:06:46,170 I am going to do that in much more detail. 130 00:06:46,170 --> 00:06:48,840 For the time being, let me point out, though, 131 00:06:48,840 --> 00:06:51,470 that if I want to view this as a mapping, 132 00:06:51,470 --> 00:06:57,690 the interesting thing is that any point on the x-axis 133 00:06:57,690 --> 00:07:01,780 maps into the corresponding point on the u-axis 134 00:07:01,780 --> 00:07:03,410 by the mapping what? 135 00:07:03,410 --> 00:07:05,780 u equals x squared plus 1. 136 00:07:05,780 --> 00:07:09,740 But it's important to point out that the values of x and u 137 00:07:09,740 --> 00:07:11,540 were not independent. 138 00:07:11,540 --> 00:07:14,700 They were chosen to obey the identity u equals 139 00:07:14,700 --> 00:07:17,850 x squared plus 1, or x is the positive square root 140 00:07:17,850 --> 00:07:19,160 of u minus 1. 141 00:07:19,160 --> 00:07:21,430 So what that means geometrically is 142 00:07:21,430 --> 00:07:26,210 that whatever height was above a point in the region R 143 00:07:26,210 --> 00:07:28,920 along the x-axis-- whatever height was here-- 144 00:07:28,920 --> 00:07:31,600 that height is the same when that point is 145 00:07:31,600 --> 00:07:33,290 moved to the region S. 146 00:07:33,290 --> 00:07:36,280 Because that again sounds like a difficult mouthful, 147 00:07:36,280 --> 00:07:37,330 let me write that. 148 00:07:37,330 --> 00:07:41,380 All I'm saying is, notice that for the u corresponding 149 00:07:41,380 --> 00:07:46,720 to a given x, 2x times the square root of x squared plus 1 150 00:07:46,720 --> 00:07:50,680 is equal to 2 times the square root of u minus 1 151 00:07:50,680 --> 00:07:52,100 times the square root of u. 152 00:07:52,100 --> 00:07:53,080 How do I know that? 153 00:07:53,080 --> 00:07:55,990 Well, I know that because I picked x to be 154 00:07:55,990 --> 00:07:58,190 the square root of u minus 1. 155 00:07:58,190 --> 00:08:01,320 Or equivalently what? u equals x squared plus 1. 156 00:08:01,320 --> 00:08:04,250 This says that I can replace x by the square root of u 157 00:08:04,250 --> 00:08:05,260 minus 1. 158 00:08:05,260 --> 00:08:08,380 This says I can replace x squared plus 1 by u. 159 00:08:08,380 --> 00:08:12,320 Consequently, as long as the x matches with the image u, 160 00:08:12,320 --> 00:08:15,810 this number is the same as this number. 161 00:08:15,810 --> 00:08:18,100 In other words again, in terms of a picture, 162 00:08:18,100 --> 00:08:24,390 if I start with the point on the x-axis x equals 2 163 00:08:24,390 --> 00:08:27,930 and I'm looking at the point P, being the point on the region 164 00:08:27,930 --> 00:08:31,530 R directly above x equals 2, since 2 165 00:08:31,530 --> 00:08:34,049 gets mapped into 5 by the mapping u 166 00:08:34,049 --> 00:08:36,020 equals x squared plus 1-- see 2 squared 167 00:08:36,020 --> 00:08:39,480 plus 1 is 5-- what was the height that went with the point 168 00:08:39,480 --> 00:08:40,679 2 over here? 169 00:08:40,679 --> 00:08:42,640 The height that went with the point 2 over here 170 00:08:42,640 --> 00:08:44,100 was simply what? 171 00:08:44,100 --> 00:08:47,100 4 square roots of 5. 172 00:08:47,100 --> 00:08:50,790 And I claim that that's the same as this, because when x was 2, 173 00:08:50,790 --> 00:08:52,070 u is 5. 174 00:08:52,070 --> 00:08:54,890 This is 2 square roots of 5. 175 00:08:54,890 --> 00:08:56,480 5 minus 1 is 4. 176 00:08:56,480 --> 00:08:58,030 Square root of 4 is 2. 177 00:08:58,030 --> 00:09:01,510 This is 4, therefore-- 2 times 2-- 4 square roots of 5. 178 00:09:01,510 --> 00:09:04,210 It makes no difference whether your using the x or the u, 179 00:09:04,210 --> 00:09:07,480 but that the point keeps the same height. 180 00:09:07,480 --> 00:09:11,880 It shifted laterally, but it does not distort the height. 181 00:09:11,880 --> 00:09:16,500 Which means, now, if we want to view this not as a mapping 182 00:09:16,500 --> 00:09:22,070 from xy-plane into the uy-plane but more traditionally in terms 183 00:09:22,070 --> 00:09:25,870 of the xy-plane into the uv-plane, that's what 184 00:09:25,870 --> 00:09:27,570 this v is in parentheses here. 185 00:09:27,570 --> 00:09:31,780 It means that I might want to name the y-axis the v-axis just 186 00:09:31,780 --> 00:09:34,680 so that I can use my identification established 187 00:09:34,680 --> 00:09:36,400 in the previous block in our course, 188 00:09:36,400 --> 00:09:40,820 when we talked about mapping the xy-plane into the uv-plane. 189 00:09:40,820 --> 00:09:43,370 That is not an accent mark over the v. That just 190 00:09:43,370 --> 00:09:45,840 happens to be an interruption of this arrow that 191 00:09:45,840 --> 00:09:47,450 connects 2 to 5. 192 00:09:47,450 --> 00:09:50,070 But at any rate, in terms of mappings, 193 00:09:50,070 --> 00:09:54,100 notice that the region R is mapped onto the region S 194 00:09:54,100 --> 00:09:59,450 by the mapping u equals x squared plus 1 and v equals y. 195 00:09:59,450 --> 00:10:01,980 In other words, u equals x squared plus 1. 196 00:10:01,980 --> 00:10:06,290 But the y-coordinate is the same as the v-coordinate. 197 00:10:06,290 --> 00:10:07,870 Just changing the name of the axis 198 00:10:07,870 --> 00:10:11,010 here to correspond to the uv-plane. 199 00:10:11,010 --> 00:10:14,890 And now, the idea is simply this. 200 00:10:14,890 --> 00:10:17,270 Since-- at least in the domain that we're 201 00:10:17,270 --> 00:10:20,880 interested in-- the mapping u equals x squared plus 1, 202 00:10:20,880 --> 00:10:26,190 v equals y maps R onto S in a one-to-one manner, 203 00:10:26,190 --> 00:10:29,250 each increment of area delta A sub S 204 00:10:29,250 --> 00:10:33,460 matches one and only one delta A sub R. 205 00:10:33,460 --> 00:10:36,670 Let me give you an example of what I mean by this. 206 00:10:36,670 --> 00:10:39,210 Let's suppose I start with the region S 207 00:10:39,210 --> 00:10:42,450 and let me arbitrarily divide it up into a number-- 208 00:10:42,450 --> 00:10:45,460 the interval from 2 to 10-- into a number of pieces here. 209 00:10:45,460 --> 00:10:48,140 Oh, I guess in the diagram that I've used here, 210 00:10:48,140 --> 00:10:51,510 I've divided this up into four pieces, all right. 211 00:10:51,510 --> 00:10:55,780 So I get these four little rectangles. 212 00:10:55,780 --> 00:10:58,860 An approximation for the area of the region S 213 00:10:58,860 --> 00:11:01,500 would be the sum of the areas of these four rectangles. 214 00:11:01,500 --> 00:11:04,890 What I'm saying is that under our mapping, 215 00:11:04,890 --> 00:11:11,190 these four regions induce four mutually exclusive regions that 216 00:11:11,190 --> 00:11:14,460 cover all of R. In fact, since v equals 217 00:11:14,460 --> 00:11:18,210 y, the way we do this mechanically is-- for example, 218 00:11:18,210 --> 00:11:21,920 let's focus on just one of these little elements over here. 219 00:11:21,920 --> 00:11:28,610 Let's suppose I want to find how to match this shaded rectangle 220 00:11:28,610 --> 00:11:33,710 with a suitable rectangle of R. What I said 221 00:11:33,710 --> 00:11:36,650 is whatever the v-value is over here, 222 00:11:36,650 --> 00:11:39,230 it must be the same as the y-value 223 00:11:39,230 --> 00:11:42,360 of the point that mapped into this point on the axis. 224 00:11:42,360 --> 00:11:45,440 In other words, if I call this point here u_0, 225 00:11:45,440 --> 00:11:50,280 that comes from some point here which I'll call x_0. 226 00:11:50,280 --> 00:11:51,990 See u_0 comes from x_0. 227 00:11:51,990 --> 00:11:54,560 What must the height above this point be? 228 00:11:54,560 --> 00:11:58,610 Since the transformation does not change the y-value at all, 229 00:11:58,610 --> 00:12:02,310 since v equals y, what this means is I can now draw a line 230 00:12:02,310 --> 00:12:06,040 parallel to the u-axis here, come over to here, 231 00:12:06,040 --> 00:12:09,720 and that locates where on the curve 232 00:12:09,720 --> 00:12:11,710 I'm going to locate the point x_0. 233 00:12:11,710 --> 00:12:13,930 See, in other words, I just come across 234 00:12:13,930 --> 00:12:15,560 like this, either of these two ways. 235 00:12:15,560 --> 00:12:18,960 This is how I locate the x_0 that matches the u_0. 236 00:12:18,960 --> 00:12:22,290 I take its height, take that same height over to this curve, 237 00:12:22,290 --> 00:12:23,870 and project down. 238 00:12:23,870 --> 00:12:27,270 Notice, of course, that the delta x 239 00:12:27,270 --> 00:12:30,030 that measures the difference between these two points 240 00:12:30,030 --> 00:12:33,640 on the x-axis is not the same as the delta u that 241 00:12:33,640 --> 00:12:35,870 measures the distance between these two points. 242 00:12:35,870 --> 00:12:39,960 But what I want you to see is that these four rectangles here 243 00:12:39,960 --> 00:12:43,100 induce four rectangles here. 244 00:12:43,100 --> 00:12:45,410 And what I would like to be able to do 245 00:12:45,410 --> 00:12:50,040 is somehow be able, hopefully, to find out 246 00:12:50,040 --> 00:12:53,900 how to express a typical rectangle here 247 00:12:53,900 --> 00:12:57,240 as a scaled version of one of these rectangles, 248 00:12:57,240 --> 00:12:59,740 hoping that when I then take the sum 249 00:12:59,740 --> 00:13:03,030 the resulting summation leads to an integral 250 00:13:03,030 --> 00:13:04,990 which is easy to evaluate. 251 00:13:04,990 --> 00:13:07,120 And before I get into that, to show you 252 00:13:07,120 --> 00:13:08,690 what does happen here-- I think I'm 253 00:13:08,690 --> 00:13:11,210 making this longer than it may really seem-- but let me just 254 00:13:11,210 --> 00:13:12,850 get onto the next step. 255 00:13:12,850 --> 00:13:17,410 What I want to do next is to blow up these two shaded areas, 256 00:13:17,410 --> 00:13:19,750 so I can look at them in more detail. 257 00:13:19,750 --> 00:13:22,550 What I have is a region which I'll call delta 258 00:13:22,550 --> 00:13:25,550 A sub S in the uv-plane and a delta 259 00:13:25,550 --> 00:13:30,280 A sub R in the xy-plane, where the mapping is, again, what? 260 00:13:30,280 --> 00:13:33,750 u equals x squared plus 1, v equals y. 261 00:13:33,750 --> 00:13:36,690 This piece matches with this piece. 262 00:13:36,690 --> 00:13:37,400 All right. 263 00:13:37,400 --> 00:13:41,010 Now, for small delta u, notice that by definition 264 00:13:41,010 --> 00:13:44,070 of derivative, delta x divided by delta u 265 00:13:44,070 --> 00:13:46,170 is approximately dx/du. 266 00:13:46,170 --> 00:13:50,330 Consequently, I can say that delta x is approximately dx/du 267 00:13:50,330 --> 00:13:51,890 times du. 268 00:13:51,890 --> 00:13:53,400 Now, the thing that I really want 269 00:13:53,400 --> 00:13:56,330 is not the area of a piece of S, I 270 00:13:56,330 --> 00:14:00,390 want a portion of the area of R. I want delta A sub R. 271 00:14:00,390 --> 00:14:06,020 Notice that delta A sub R has as its height y sub 0 272 00:14:06,020 --> 00:14:08,770 and as its width delta x. 273 00:14:08,770 --> 00:14:13,710 Notice also that since v sub 0 equals y sub 0-- 274 00:14:13,710 --> 00:14:18,370 see y equals v in this transformation-- notice 275 00:14:18,370 --> 00:14:26,201 that the area delta A_R is v_0 times delta x. 276 00:14:26,201 --> 00:14:26,700 OK. 277 00:14:26,700 --> 00:14:30,660 We also know that delta x is approximately dx/du times delta 278 00:14:30,660 --> 00:14:31,350 u. 279 00:14:31,350 --> 00:14:33,620 So making this substitution, I see 280 00:14:33,620 --> 00:14:37,260 that my little increment of area in the region R 281 00:14:37,260 --> 00:14:38,630 is precisely what? 282 00:14:38,630 --> 00:14:44,420 It's v_0 times the replacement for delta x, dx/du times 283 00:14:44,420 --> 00:14:45,720 delta u. 284 00:14:45,720 --> 00:14:50,840 Let me also notice that the region delta A_S over here 285 00:14:50,840 --> 00:14:55,410 has as its height v_0, as its base delta u. 286 00:14:55,410 --> 00:15:00,210 So the area of this rectangle is v_0 delta u. 287 00:15:00,210 --> 00:15:03,550 Let me, therefore, group these two factors together, 288 00:15:03,550 --> 00:15:06,650 rewrite this term in this fashion, 289 00:15:06,650 --> 00:15:11,310 noticing that v_0 delta u is delta A sub S. And I now 290 00:15:11,310 --> 00:15:16,240 have the very interesting result that delta A sub R is not 291 00:15:16,240 --> 00:15:19,500 delta A sub S. But the correction factor is what? 292 00:15:19,500 --> 00:15:23,970 It's dx/du multiplying delta A sub S where, 293 00:15:23,970 --> 00:15:26,340 for the sake of argument over a small enough strip here, 294 00:15:26,340 --> 00:15:31,060 let me assume that I've chosen dx/du to be evaluated at u 295 00:15:31,060 --> 00:15:33,400 equals u_0, say. 296 00:15:33,400 --> 00:15:35,540 At any rate, what we are saying is, 297 00:15:35,540 --> 00:15:37,750 to find all of the area of region R, 298 00:15:37,750 --> 00:15:40,170 we want to add up all of these delta A sub 299 00:15:40,170 --> 00:15:44,120 R's as the maximum delta x sub k goes to 0. 300 00:15:44,120 --> 00:15:48,850 But from what we've just seen, a typical delta A sub R 301 00:15:48,850 --> 00:15:52,700 is a dx/du, evaluated at u equals u_0, 302 00:15:52,700 --> 00:15:55,300 times delta A sub S. And therefore, 303 00:15:55,300 --> 00:15:59,360 to find delta A sub R, this limit is precisely the same 304 00:15:59,360 --> 00:16:00,750 as this limit. 305 00:16:00,750 --> 00:16:04,420 Now the point is that delta A sub S is certainly 306 00:16:04,420 --> 00:16:07,280 just as messy as delta A sub R, in general. 307 00:16:07,280 --> 00:16:10,550 It may also happened that when I scale delta A sub 308 00:16:10,550 --> 00:16:15,800 S by multiplying it by dx/du, the result is even more messy 309 00:16:15,800 --> 00:16:17,990 than the original expression. 310 00:16:17,990 --> 00:16:21,350 But it's also possible that dx/du 311 00:16:21,350 --> 00:16:23,950 happens to be the factor that wipes 312 00:16:23,950 --> 00:16:27,440 out the nasty part of delta A sub S. You see, what I'm saying 313 00:16:27,440 --> 00:16:30,820 is, if there is a one-to-one correspondence-- which there 314 00:16:30,820 --> 00:16:34,690 is-- between the delta A_S's and the delta A_R's, 315 00:16:34,690 --> 00:16:37,910 if this expression here happens to be convenient, 316 00:16:37,910 --> 00:16:43,760 I can find this sum simply by computing this sum. 317 00:16:43,760 --> 00:16:45,850 And that's why, in techniques of integration, 318 00:16:45,850 --> 00:16:47,710 that's precisely what we look for. 319 00:16:47,710 --> 00:16:50,940 We look for the change of variable, the substitution, 320 00:16:50,940 --> 00:16:52,820 that makes this thing simplify. 321 00:16:52,820 --> 00:16:54,760 And that's precisely what happened 322 00:16:54,760 --> 00:16:55,940 in this particular example. 323 00:16:55,940 --> 00:16:58,910 Keep in mind that what I've written down over here 324 00:16:58,910 --> 00:17:03,160 is true for any substitution where x is some function of u, 325 00:17:03,160 --> 00:17:05,680 not just for x equals u squared plus 1. 326 00:17:05,680 --> 00:17:07,119 I could've done this any time. 327 00:17:07,119 --> 00:17:09,369 But what I claim is that if x weren't 328 00:17:09,369 --> 00:17:12,819 equal to-- what was it-- the square root of u minus 1, 329 00:17:12,819 --> 00:17:16,625 this wouldn't have turned out to be a very nice expression. 330 00:17:16,625 --> 00:17:18,250 And you see, this is going to be called 331 00:17:18,250 --> 00:17:20,150 the one-dimensional Jacobian later on. 332 00:17:20,150 --> 00:17:22,990 This is the correction factor, the scaling factor, you see. 333 00:17:22,990 --> 00:17:24,770 Let's see how that worked out. 334 00:17:24,770 --> 00:17:30,020 Notice that delta A sub S was v_0 times delta u. 335 00:17:30,020 --> 00:17:33,180 And notice that by definition of what the curve looked 336 00:17:33,180 --> 00:17:38,060 like in the uv-plane, v_0 is twice the square root of u_0 337 00:17:38,060 --> 00:17:41,200 minus 1 times the square root of u_0. 338 00:17:41,200 --> 00:17:43,850 On the other hand, what is dx/du? 339 00:17:43,850 --> 00:17:46,550 Since u is equal to x squared plus 1, 340 00:17:46,550 --> 00:17:50,620 it's easy to show that dx/du is 1 over twice 341 00:17:50,620 --> 00:17:52,710 the square root of u minus 1. 342 00:17:52,710 --> 00:17:55,190 So in particular, when u equals u_0, 343 00:17:55,190 --> 00:18:00,320 dx/du is 1 over 2 square root of u_0 minus 1. 344 00:18:00,320 --> 00:18:04,340 Notice now, even though delta A sub S is quite messy, 345 00:18:04,340 --> 00:18:07,660 when I multiply it by this particular scaling factor, 346 00:18:07,660 --> 00:18:09,920 look at what that scaling factor wipes out. 347 00:18:09,920 --> 00:18:13,740 The 2 square root of u_0 minus 1 wipes this out. 348 00:18:13,740 --> 00:18:17,660 All I have left is the square root of u_0 times delta u. 349 00:18:17,660 --> 00:18:21,200 When I form the definite integral summing this thing up, 350 00:18:21,200 --> 00:18:24,230 it's trivial, you see, to see that that simply 351 00:18:24,230 --> 00:18:25,630 comes out to be what? 352 00:18:25,630 --> 00:18:30,120 The definite integral from 2 to 10 square root of u du. 353 00:18:30,120 --> 00:18:34,370 And since this particular sum was equal to A sub R, 354 00:18:34,370 --> 00:18:37,830 that is the mapping interpretation 355 00:18:37,830 --> 00:18:41,230 of why the area of the region R can be evaluated 356 00:18:41,230 --> 00:18:42,900 by this particular integral. 357 00:18:42,900 --> 00:18:45,790 Now, in a sense, all of this has been review, 358 00:18:45,790 --> 00:18:48,640 even though the pitch has been slightly different. 359 00:18:48,640 --> 00:18:53,010 Let me now generalize this to a bona fide mapping 360 00:18:53,010 --> 00:18:55,800 of two-dimensional space into two-dimensional space. 361 00:18:55,800 --> 00:18:57,990 And the reason I use the word bona fide 362 00:18:57,990 --> 00:19:01,450 is that when you say let u equal x squared plus 1 363 00:19:01,450 --> 00:19:04,070 and let v equal y, you really haven't 364 00:19:04,070 --> 00:19:06,860 got a general mapping of two-dimensional space 365 00:19:06,860 --> 00:19:08,230 into two-dimensional space. 366 00:19:08,230 --> 00:19:11,480 You've essentially let the y-axis remain fixed. 367 00:19:11,480 --> 00:19:13,880 So let me talk about something more general. 368 00:19:13,880 --> 00:19:15,660 Let's suppose I have an arbitrary region 369 00:19:15,660 --> 00:19:20,330 R and an arbitrary function f bar which maps R 370 00:19:20,330 --> 00:19:24,320 onto S in a one-to-one manner. 371 00:19:24,320 --> 00:19:26,530 By the way, notice, the whole idea is this. 372 00:19:26,530 --> 00:19:29,070 When I want the area of the region R, 373 00:19:29,070 --> 00:19:33,260 it's going to involve a dx*dy inside the double integral. 374 00:19:33,260 --> 00:19:35,660 When I want the area of the region S, 375 00:19:35,660 --> 00:19:39,790 that's what's going to involve a delta u times delta v. Now 376 00:19:39,790 --> 00:19:42,300 the reason that delta u times delta v 377 00:19:42,300 --> 00:19:45,240 is indeed a bona fide element of area 378 00:19:45,240 --> 00:19:48,730 when we're breaking up S lies in the fact that, 379 00:19:48,730 --> 00:19:51,660 in the uv-plane, delta u times delta 380 00:19:51,660 --> 00:19:59,060 v is the actual area of an increment of area in S, 381 00:19:59,060 --> 00:20:03,370 when we break up S by lines parallel to the u and v axis. 382 00:20:03,370 --> 00:20:05,310 On the other hand, notice that if we 383 00:20:05,310 --> 00:20:08,940 see what the line u equals a constant comes from, back 384 00:20:08,940 --> 00:20:12,280 in the xy-plane, u is a function of x and y. 385 00:20:12,280 --> 00:20:14,457 And to say that u of x, y equals a constant 386 00:20:14,457 --> 00:20:16,290 does not mean that you have a straight line. 387 00:20:16,290 --> 00:20:19,010 You could have some pretty squiggly lines over here. 388 00:20:19,010 --> 00:20:22,560 In other words, it might be a very funny-looking line 389 00:20:22,560 --> 00:20:25,490 that maps into a straight line, a straight vertical line, 390 00:20:25,490 --> 00:20:28,820 in the uv-plane with respect to the mapping f bar. 391 00:20:28,820 --> 00:20:31,030 And in a similar way, the lines v 392 00:20:31,030 --> 00:20:34,690 equal a constant-- the lines v equal 393 00:20:34,690 --> 00:20:38,350 a constant-- in the xy-plane, are read what? v of x, y 394 00:20:38,350 --> 00:20:39,450 equals a constant. 395 00:20:39,450 --> 00:20:41,610 That's a general curve in the xy-plane. 396 00:20:41,610 --> 00:20:44,780 What we're saying is that, since this mapping is one to one, 397 00:20:44,780 --> 00:20:49,410 when I break up this region into elementary elements 398 00:20:49,410 --> 00:20:52,800 of rectangles, that will induce a breaking up 399 00:20:52,800 --> 00:20:55,980 of this region into little elements here. 400 00:20:55,980 --> 00:20:58,100 But notice that the resulting elements-- say 401 00:20:58,100 --> 00:21:00,830 we take a piece like this, and we see 402 00:21:00,830 --> 00:21:02,180 where that piece comes from. 403 00:21:02,180 --> 00:21:03,840 Let's say that that particular piece 404 00:21:03,840 --> 00:21:06,340 happened to come from here. 405 00:21:06,340 --> 00:21:09,770 Say that that was the one-to-one correspondence. 406 00:21:09,770 --> 00:21:12,620 You can take delta u times delta v here. 407 00:21:12,620 --> 00:21:15,490 But notice that delta u times delta v 408 00:21:15,490 --> 00:21:19,930 simply means multiplying two edges which 409 00:21:19,930 --> 00:21:22,150 aren't straight lines, which aren't necessarily 410 00:21:22,150 --> 00:21:23,920 perpendicular, and hence in no way 411 00:21:23,920 --> 00:21:27,530 should represent what the area of this little element here is. 412 00:21:27,530 --> 00:21:31,120 The key point is we do not want the area of the region S. 413 00:21:31,120 --> 00:21:33,750 We want the area of the region R. 414 00:21:33,750 --> 00:21:35,670 And what we're hoping that we can 415 00:21:35,670 --> 00:21:38,760 do is that by making the change of variables that 416 00:21:38,760 --> 00:21:41,310 maps the region R in the xy-plane 417 00:21:41,310 --> 00:21:44,910 into the region S in the uv-plane, 418 00:21:44,910 --> 00:21:48,790 that we somehow find a convenient way of scaling 419 00:21:48,790 --> 00:21:51,790 an individual element of area here 420 00:21:51,790 --> 00:21:56,180 with respect to one here, and define the area of this region 421 00:21:56,180 --> 00:21:59,130 by adding up the appropriate pieces here. 422 00:21:59,130 --> 00:22:02,550 And, again, let me show you what that means in terms 423 00:22:02,550 --> 00:22:04,960 of just an enlargement again. 424 00:22:04,960 --> 00:22:06,830 You see, in the same as I did before, 425 00:22:06,830 --> 00:22:09,580 let me take this little piece over here and really blow it 426 00:22:09,580 --> 00:22:10,300 up. 427 00:22:10,300 --> 00:22:12,990 Let me take this little piece that 428 00:22:12,990 --> 00:22:14,700 is the back-map of this-- in other words, 429 00:22:14,700 --> 00:22:18,220 the piece that maps into this-- let me blow that up. 430 00:22:18,220 --> 00:22:20,830 And the idea is this. 431 00:22:20,830 --> 00:22:24,790 If I pick delta u and delta v sufficiently small, 432 00:22:24,790 --> 00:22:29,490 notice that the area of the back-map of delta A sub S 433 00:22:29,490 --> 00:22:32,520 is approximately the area of a parallelogram. 434 00:22:32,520 --> 00:22:35,440 You see, come back to this statement 435 00:22:35,440 --> 00:22:37,250 after I've explained the picture. 436 00:22:37,250 --> 00:22:40,770 What I'm saying is I start with an element of area delta A sub 437 00:22:40,770 --> 00:22:43,200 S in the uv-plane. 438 00:22:43,200 --> 00:22:44,820 You see, I pick its vertices. 439 00:22:44,820 --> 00:22:49,090 I'll call them A bar, B bar, D bar, C bar. 440 00:22:49,090 --> 00:22:51,490 That's a little rectangle over here. 441 00:22:51,490 --> 00:22:53,530 Because the mapping is one-to-one, 442 00:22:53,530 --> 00:22:57,240 I know that there is one and only one point in the xy-plane 443 00:22:57,240 --> 00:22:59,480 that maps into A bar. 444 00:22:59,480 --> 00:22:59,980 See that? 445 00:22:59,980 --> 00:23:01,820 Let's call that point A. There is 446 00:23:01,820 --> 00:23:05,330 one and only one point in the xy-plane that maps into B bar. 447 00:23:05,330 --> 00:23:07,820 Let's call that B. One and only one 448 00:23:07,820 --> 00:23:10,870 point in the xy-plane that maps into C bar. 449 00:23:10,870 --> 00:23:14,080 Let's call that point C. And let me leave the point D out 450 00:23:14,080 --> 00:23:15,230 for a moment. 451 00:23:15,230 --> 00:23:19,050 Now the idea is if we call the coordinates of A bar u_0 comma 452 00:23:19,050 --> 00:23:23,760 v_0, then because this is a line parallel to the axis-- 453 00:23:23,760 --> 00:23:29,230 call this dimension delta u-- B bar is u_0 plus delta u comma 454 00:23:29,230 --> 00:23:30,400 v_0. 455 00:23:30,400 --> 00:23:32,490 C bar-- call this dimension delta 456 00:23:32,490 --> 00:23:37,480 v-- is u_0 comma v_0 plus delta v. Now the point 457 00:23:37,480 --> 00:23:41,470 is, there is no reason why the image of B 458 00:23:41,470 --> 00:23:48,310 bar-- namely the point B-- has to be on the line that 459 00:23:48,310 --> 00:23:51,120 joins A parallel to the x-axis. 460 00:23:51,120 --> 00:23:53,470 In other words, B is up here someplace, 461 00:23:53,470 --> 00:23:55,290 C is up here some place. 462 00:23:55,290 --> 00:23:58,440 In other words, again, there's no reason why these back-maps 463 00:23:58,440 --> 00:24:01,190 give me a rectangle over here. 464 00:24:01,190 --> 00:24:03,320 The point is that B has some coordinates. 465 00:24:03,320 --> 00:24:06,060 It's x_0 plus some increment involving 466 00:24:06,060 --> 00:24:09,550 x-- let me call that delta x_1-- and its y-coordinate 467 00:24:09,550 --> 00:24:12,550 is y_0 plus delta y_1. 468 00:24:12,550 --> 00:24:17,190 C is the point x_0 plus some increment delta x_2 comma 469 00:24:17,190 --> 00:24:19,740 y_0 plus delta y_2. 470 00:24:19,740 --> 00:24:22,960 And what I'm saying is now, if I were to just look 471 00:24:22,960 --> 00:24:30,010 at the parallelogram which had AB and AC as adjacent sides, 472 00:24:30,010 --> 00:24:33,810 it's very easy for me to find the area of that parallelogram. 473 00:24:33,810 --> 00:24:36,790 Namely, to find the area of a parallelogram in vector form, 474 00:24:36,790 --> 00:24:39,660 I just take the magnitude of the cross-product of the two 475 00:24:39,660 --> 00:24:42,330 vectors AB and AC. 476 00:24:42,330 --> 00:24:44,400 You see, what's wrong with this is 477 00:24:44,400 --> 00:24:47,280 that that particular parallelogram is not 478 00:24:47,280 --> 00:24:51,300 the exact image of the back-map of delta A_S. 479 00:24:51,300 --> 00:24:55,480 Sure, A bar maps exactly into A. C bar maps 480 00:24:55,480 --> 00:25:00,860 exactly into C. B bar maps exactly into B. 481 00:25:00,860 --> 00:25:05,690 But the point is that these points along A bar C bar 482 00:25:05,690 --> 00:25:11,500 do not map into the straight line from A to C, in general. 483 00:25:11,500 --> 00:25:14,680 In other words, what characterizes this? 484 00:25:14,680 --> 00:25:17,570 This is characterized by delta u equals 0. 485 00:25:17,570 --> 00:25:21,280 And when u is written in terms of x and y, 486 00:25:21,280 --> 00:25:24,900 that doesn't say that delta x or delta y is 0. 487 00:25:24,900 --> 00:25:28,280 So, the true image of this might be 488 00:25:28,280 --> 00:25:32,380 what I've represented with this dotted array here. 489 00:25:32,380 --> 00:25:34,920 In other words, what the true back-map of delta A is 490 00:25:34,920 --> 00:25:38,860 is this dotted thing, where D is now this vertex here. 491 00:25:38,860 --> 00:25:42,620 You see, there's no guarantee that the back-map of D bar 492 00:25:42,620 --> 00:25:46,090 is going to be the fourth vertex of this parallelogram. 493 00:25:46,090 --> 00:25:47,780 But what the key point is-- and this 494 00:25:47,780 --> 00:25:50,140 is where that linearity is so important-- 495 00:25:50,140 --> 00:25:53,320 is that if the transformation is smooth enough, continuously 496 00:25:53,320 --> 00:25:56,370 differentiable, then what it does mean 497 00:25:56,370 --> 00:26:00,460 is that as long as delta u and delta v are sufficiently small, 498 00:26:00,460 --> 00:26:04,200 the true area of the region delta 499 00:26:04,200 --> 00:26:07,000 A_R that we're looking for is approximately 500 00:26:07,000 --> 00:26:09,790 equal to the area of this parallelogram. 501 00:26:09,790 --> 00:26:11,940 And by approximately equal, I mean what? 502 00:26:11,940 --> 00:26:14,730 That the error goes to 0 as we take the limit 503 00:26:14,730 --> 00:26:16,305 in forming the double sum. 504 00:26:16,305 --> 00:26:19,990 In other words, again, the key point is this. 505 00:26:19,990 --> 00:26:25,610 The back-map of delta A sub S yields delta A_R, 506 00:26:25,610 --> 00:26:28,880 but that delta A_R is approximately 507 00:26:28,880 --> 00:26:30,590 this parallelogram. 508 00:26:30,590 --> 00:26:32,930 And the area of this parallelogram 509 00:26:32,930 --> 00:26:36,840 is exactly the magnitude of AB-- the vector 510 00:26:36,840 --> 00:26:39,915 AB-- crossed with AC. 511 00:26:39,915 --> 00:26:41,440 In other words, the approximation 512 00:26:41,440 --> 00:26:46,350 comes in because this is exactly the area of the parallelogram. 513 00:26:46,350 --> 00:26:50,490 But delta A sub R is only approximately equal to the area 514 00:26:50,490 --> 00:26:51,700 of the parallelogram. 515 00:26:51,700 --> 00:26:54,770 At any rate, notice in terms of i and j components, 516 00:26:54,770 --> 00:26:58,420 how easy it is to compute AB and AC. 517 00:26:58,420 --> 00:27:01,180 You see, what are the components of the vector from A to B? 518 00:27:01,180 --> 00:27:03,370 The i component is delta x_1. 519 00:27:03,370 --> 00:27:04,840 See, this minus this. 520 00:27:04,840 --> 00:27:10,020 The j component is this minus this, right. 521 00:27:10,020 --> 00:27:12,700 In other words, AC has what as its components? 522 00:27:12,700 --> 00:27:15,700 It's this minus this, namely delta x_2. 523 00:27:15,700 --> 00:27:17,990 This minus this is the y-component. 524 00:27:17,990 --> 00:27:19,249 That's delta y_2. 525 00:27:19,249 --> 00:27:20,790 In other words-- to write this out so 526 00:27:20,790 --> 00:27:23,750 you don't have to listen to how fast I'm talking-- 527 00:27:23,750 --> 00:27:26,020 AB is this particular vector. 528 00:27:26,020 --> 00:27:28,500 AC is this particular vector. 529 00:27:28,500 --> 00:27:31,040 Remembering that when we take a cross-product, 530 00:27:31,040 --> 00:27:37,060 i crossed i is 0, j cross j is 0, i cross j is k, 531 00:27:37,060 --> 00:27:39,142 and j cross i is minus k. 532 00:27:39,142 --> 00:27:40,600 Remember, for the cross-product, we 533 00:27:40,600 --> 00:27:42,571 can't change the order of the terms 534 00:27:42,571 --> 00:27:44,070 from the order in which they appear. 535 00:27:44,070 --> 00:27:45,940 We then determined what? 536 00:27:45,940 --> 00:27:50,100 That AB cross AC is delta x_1 delta y_2 537 00:27:50,100 --> 00:27:54,740 minus delta x_2 delta y_1 times the vector k, the unit vector 538 00:27:54,740 --> 00:27:56,440 in the z direction. 539 00:27:56,440 --> 00:28:00,722 Now, delta x_1 is exactly-- see, remember, 540 00:28:00,722 --> 00:28:02,430 we're looking at the parallelogram, which 541 00:28:02,430 --> 00:28:03,260 is a straight line. 542 00:28:03,260 --> 00:28:05,590 If you want to think of it in terms of the region R, 543 00:28:05,590 --> 00:28:08,740 by differentials delta x_1 would be what? 544 00:28:08,740 --> 00:28:11,700 It's approximately the partial of x with respect to u 545 00:28:11,700 --> 00:28:14,950 times delta u plus the partial of x with respect to v times 546 00:28:14,950 --> 00:28:19,700 delta v. Similarly, delta y_1 is y sub u times delta 547 00:28:19,700 --> 00:28:22,690 u plus y sub v times delta v. 548 00:28:22,690 --> 00:28:27,940 But now keep in mind where delta x_1 and delta y_1 come from. 549 00:28:27,940 --> 00:28:32,070 Delta x_1 and delta y_1 come from the back mapping 550 00:28:32,070 --> 00:28:35,540 from A bar B bar back to AB. 551 00:28:35,540 --> 00:28:41,240 And along A bar B bar, notice that v is always equal to v_0. 552 00:28:41,240 --> 00:28:43,700 So that means that delta v is 0. 553 00:28:43,700 --> 00:28:47,350 That means, therefore, that because delta v is 0, 554 00:28:47,350 --> 00:28:51,440 delta x_1 and delta y_1 are simply this. 555 00:28:51,440 --> 00:28:52,750 See this drops out. 556 00:28:52,750 --> 00:28:55,420 Similarly, delta x_2 and delta y_2 557 00:28:55,420 --> 00:28:58,370 come from the back-map of A bar A bar. 558 00:28:58,370 --> 00:29:01,530 Along A bar C bar, u is equal to u_0. 559 00:29:01,530 --> 00:29:03,130 So delta u is zero. 560 00:29:03,130 --> 00:29:05,730 So writing this out, these drop out. 561 00:29:05,730 --> 00:29:09,680 And now, with these terms being 0, with these terms being 0, 562 00:29:09,680 --> 00:29:13,500 I could very simply compute the product delta x_1 delta 563 00:29:13,500 --> 00:29:16,360 y_2 minus delta x_2 delta y_1. 564 00:29:16,360 --> 00:29:21,280 And if I do that, you see, right away what I obtain is what? 565 00:29:21,280 --> 00:29:23,910 It's this times this. 566 00:29:23,910 --> 00:29:29,290 I now, OK, collect the terms here. 567 00:29:29,290 --> 00:29:32,530 In other words, I want the delta u and the delta v together. 568 00:29:32,530 --> 00:29:34,840 The multiplier out front here is x 569 00:29:34,840 --> 00:29:39,560 sub u times y sub v. In a similar way, delta x_2 times 570 00:29:39,560 --> 00:29:42,930 delta y_1 is x sub v y sub u times delta u 571 00:29:42,930 --> 00:29:48,370 delta v. Therefore, putting that into here, the magnitude of AB 572 00:29:48,370 --> 00:29:52,390 cross AC is simply this expression here, 573 00:29:52,390 --> 00:29:56,630 noticing that the k vector drops out because its magnitude is 1. 574 00:29:56,630 --> 00:30:01,810 Notice that delta u times delta v is precisely delta A sub S 575 00:30:01,810 --> 00:30:05,110 and that x sub u times y sub v minus x 576 00:30:05,110 --> 00:30:09,270 sub v times y sub u is precisely the determinant 577 00:30:09,270 --> 00:30:12,010 of our old friend the Jacobian matrix, 578 00:30:12,010 --> 00:30:14,850 the Jacobian matrix of x and y with respect to u 579 00:30:14,850 --> 00:30:19,040 and v. In other words, delta A sub R is approximately 580 00:30:19,040 --> 00:30:22,350 equal to the determinant of the Jacobian matrix of x 581 00:30:22,350 --> 00:30:25,820 and y with respect to u and v times delta A_S. 582 00:30:25,820 --> 00:30:30,230 And if I now perform this double sum, you see this becomes what? 583 00:30:30,230 --> 00:30:33,390 The area of the region R-- just write that in, that's really 584 00:30:33,390 --> 00:30:38,140 the area of the region R-- is a double integral over R dx*dy. 585 00:30:38,140 --> 00:30:39,610 And that's the same as what? 586 00:30:39,610 --> 00:30:45,980 The double integral over S times du*dv multiplied by the scaling 587 00:30:45,980 --> 00:30:48,710 factor of the Jacobian determinant. 588 00:30:48,710 --> 00:30:53,490 By the way, notice I dropped the determinant symbol over here. 589 00:30:53,490 --> 00:30:57,830 The reason for that is that many textbooks, including our own, 590 00:30:57,830 --> 00:31:02,770 use this notation not to name the Jacobian matrix but to name 591 00:31:02,770 --> 00:31:04,250 the Jacobian determinant. 592 00:31:04,250 --> 00:31:07,940 I have been using this to name the Jacobian matrix. 593 00:31:07,940 --> 00:31:11,480 From this point on, I will now switch to become uniform 594 00:31:11,480 --> 00:31:12,390 with the text. 595 00:31:12,390 --> 00:31:14,510 And unless otherwise specified, I 596 00:31:14,510 --> 00:31:17,890 will write this rather than put the determinant symbol in. 597 00:31:17,890 --> 00:31:20,690 From now on in our course, when I write this 598 00:31:20,690 --> 00:31:26,330 I am referring to the Jacobian determinant, OK. 599 00:31:26,330 --> 00:31:27,730 But the thing is this. 600 00:31:27,730 --> 00:31:30,580 Notice that the given mapping might straighten out 601 00:31:30,580 --> 00:31:33,180 the region R into a nicer-looking region 602 00:31:33,180 --> 00:31:36,840 S. But to offset this, it may also 603 00:31:36,840 --> 00:31:40,290 turn out that the resulting integrand here 604 00:31:40,290 --> 00:31:42,280 is much worse than the integrand here. 605 00:31:42,280 --> 00:31:47,670 Here the multiplier of dx*dy was the simple number 1, wasn't it? 606 00:31:47,670 --> 00:31:51,010 Here, what's multiplying du*dv, no matter how nice S is, 607 00:31:51,010 --> 00:31:54,620 is this expression here, which may be quite messy. 608 00:31:54,620 --> 00:31:57,440 And therefore, in most practical applications 609 00:31:57,440 --> 00:32:00,380 where one solves multiple integrals 610 00:32:00,380 --> 00:32:02,750 by a change of variable, one not only 611 00:32:02,750 --> 00:32:05,590 wants a change of variables that straightens out 612 00:32:05,590 --> 00:32:07,750 the region into a nicer-looking one, 613 00:32:07,750 --> 00:32:09,680 he wants a combination of two things. 614 00:32:09,680 --> 00:32:12,210 He would like a nicer-looking region. 615 00:32:12,210 --> 00:32:14,610 And more importantly, even if he can't 616 00:32:14,610 --> 00:32:16,830 get a nicer-looking region, at least 617 00:32:16,830 --> 00:32:20,890 if he gets a correction factor, a Jacobian determinant, that 618 00:32:20,890 --> 00:32:23,580 gives him something that's easy to integrate, 619 00:32:23,580 --> 00:32:24,960 he'll settle for that. 620 00:32:24,960 --> 00:32:27,580 And what that means, hopefully, will become clearer 621 00:32:27,580 --> 00:32:30,890 as we go through the exercises and the reading material. 622 00:32:30,890 --> 00:32:32,540 At any rate, I think that's all I 623 00:32:32,540 --> 00:32:35,690 want to say about a supplement to Professor Thomas's 624 00:32:35,690 --> 00:32:37,910 treatment of polar coordinates at this time. 625 00:32:37,910 --> 00:32:39,960 And until next time, then, good bye. 626 00:32:46,160 --> 00:32:48,520 Funding for the publication of this video 627 00:32:48,520 --> 00:32:53,410 was provided by the Gabrielle and Paul Rosenbaum Foundation. 628 00:32:53,410 --> 00:32:57,580 Help OCW continue to provide free and open access to MIT 629 00:32:57,580 --> 00:33:01,998 courses by making a donation at ocw.mit.edu/donate.