1 00:00:00,000 --> 00:00:00,030 2 00:00:00,030 --> 00:00:02,330 The following content is provided under a Creative 3 00:00:02,330 --> 00:00:03,742 Commons license. 4 00:00:03,742 --> 00:00:05,450 Your support will help MIT OpenCourseWare 5 00:00:05,450 --> 00:00:10,050 continue to offer high-quality educational resources for free. 6 00:00:10,050 --> 00:00:12,530 To make a donation, or to view additional materials 7 00:00:12,530 --> 00:00:15,210 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,210 --> 00:00:21,370 at ocw.mit.edu. 9 00:00:21,370 --> 00:00:24,550 PROFESSOR STRANG: OK, it's Laplace again today. 10 00:00:24,550 --> 00:00:26,020 Laplace's equation. 11 00:00:26,020 --> 00:00:31,710 And trying to describe-- That's a big area that's 12 00:00:31,710 --> 00:00:35,370 a lot of people have worked on for centuries. 13 00:00:35,370 --> 00:00:38,900 And for the early centuries, there 14 00:00:38,900 --> 00:00:42,070 were always analysis methods. 15 00:00:42,070 --> 00:00:45,460 And that's what we got started on last time. 16 00:00:45,460 --> 00:00:47,490 And we'll do a bit more. 17 00:00:47,490 --> 00:00:49,470 There's no way we could do everything 18 00:00:49,470 --> 00:00:53,030 that people have worked on years and years, 19 00:00:53,030 --> 00:00:55,830 trying to find ideas about solving. 20 00:00:55,830 --> 00:00:58,970 But we can get the idea. 21 00:00:58,970 --> 00:01:01,390 And this part, then, is in the section 22 00:01:01,390 --> 00:01:03,210 called Laplace's equation. 23 00:01:03,210 --> 00:01:08,050 And the exam Wednesday would include 24 00:01:08,050 --> 00:01:12,070 some of these constructions. 25 00:01:12,070 --> 00:01:14,070 So this is what we did last time, we 26 00:01:14,070 --> 00:01:19,440 identified a whole family of solutions to Laplace's equation 27 00:01:19,440 --> 00:01:23,590 as polynomials in x and y. 28 00:01:23,590 --> 00:01:28,120 Of increasing degree n, and then when we wrote them in polar 29 00:01:28,120 --> 00:01:31,880 form they were fantastic. r^n*cos(n*theta) 30 00:01:31,880 --> 00:01:34,360 and r^n*sin(n*theta). 31 00:01:34,360 --> 00:01:39,350 So my idea is just, we've got them, now let's use them. 32 00:01:39,350 --> 00:01:44,900 So how to use these solutions? 33 00:01:44,900 --> 00:01:48,380 Because we can take combinations of them, 34 00:01:48,380 --> 00:01:51,590 we can create series of sines and cosines. 35 00:01:51,590 --> 00:01:53,750 So I'll do that first. 36 00:01:53,750 --> 00:01:54,340 Series. 37 00:01:54,340 --> 00:01:58,660 And then Green's function, that's the name you remember, 38 00:01:58,660 --> 00:02:00,240 we've seen it before. 39 00:02:00,240 --> 00:02:04,180 That's the solution when the right side is a delta function. 40 00:02:04,180 --> 00:02:08,600 When we have Poisson's equation with a delta there. 41 00:02:08,600 --> 00:02:10,930 So that an important one. 42 00:02:10,930 --> 00:02:17,790 And then well, a big part of two-dimensional and 43 00:02:17,790 --> 00:02:20,930 three-dimensional problems is that the region itself, 44 00:02:20,930 --> 00:02:23,600 not just the equation but the region itself, 45 00:02:23,600 --> 00:02:26,160 can be all over the place. 46 00:02:26,160 --> 00:02:29,660 We'll solve when the region is nice, 47 00:02:29,660 --> 00:02:33,470 like a circle or a square, and then there 48 00:02:33,470 --> 00:02:39,360 is a way to, in principle, to get other regions. 49 00:02:39,360 --> 00:02:43,630 To change from a crazy region to a circle or a square, 50 00:02:43,630 --> 00:02:45,470 and then solve it there. 51 00:02:45,470 --> 00:02:48,240 So that's called conformal mapping. 52 00:02:48,240 --> 00:02:52,410 And I can't let the whole course go without saying 53 00:02:52,410 --> 00:02:53,920 a word or two about that. 54 00:02:53,920 --> 00:03:00,680 But somehow among numerical methods, 55 00:03:00,680 --> 00:03:05,880 it's conformal mapping-- There are packages 56 00:03:05,880 --> 00:03:07,400 that do conformal mapping. 57 00:03:07,400 --> 00:03:12,020 But they're not the central way to solve 58 00:03:12,020 --> 00:03:13,720 these equations numerically. 59 00:03:13,720 --> 00:03:16,630 Finite differences, finite elements are. 60 00:03:16,630 --> 00:03:19,430 And that's what's coming next week. 61 00:03:19,430 --> 00:03:27,010 So this is the future; this is the present, right there. 62 00:03:27,010 --> 00:03:31,270 So, can I just start with an example or two? 63 00:03:31,270 --> 00:03:36,680 Like, how would you solve Laplace's equation in a circle? 64 00:03:36,680 --> 00:03:39,640 So, in a circle. 65 00:03:39,640 --> 00:03:42,370 This is the idea here. 66 00:03:42,370 --> 00:03:44,390 I have Laplace's equation. 67 00:03:44,390 --> 00:03:47,620 OK, so I've got a whole lot of solutions. 68 00:03:47,620 --> 00:03:51,300 And I've even got chalk to write them down. 69 00:03:51,300 --> 00:03:56,050 OK, so here's my circle, might as well make it 70 00:03:56,050 --> 00:03:59,550 the unit circle. 71 00:03:59,550 --> 00:04:01,390 Radius one. 72 00:04:01,390 --> 00:04:04,530 And inside here is Laplace. 73 00:04:04,530 --> 00:04:07,340 74 00:04:07,340 --> 00:04:09,630 u_xx+u_yy=0. 75 00:04:09,630 --> 00:04:12,140 No sources inside. 76 00:04:12,140 --> 00:04:14,190 So we have to have sources from somewhere, 77 00:04:14,190 --> 00:04:16,640 and they will come from the boundary. 78 00:04:16,640 --> 00:04:20,590 So on the boundary, we keep-- Let 79 00:04:20,590 --> 00:04:23,490 me think of u as temperature. 80 00:04:23,490 --> 00:04:26,820 So I set the temperature on the boundary. u 81 00:04:26,820 --> 00:04:30,620 equals some u_0, some known function. 82 00:04:30,620 --> 00:04:31,980 This is given. 83 00:04:31,980 --> 00:04:33,680 This is the boundary condition. 84 00:04:33,680 --> 00:04:36,780 This is the given boundary condition. 85 00:04:36,780 --> 00:04:41,950 And it's a function of-- I'm going to use polar coordinates. 86 00:04:41,950 --> 00:04:44,690 Polar coordinates are natural for a circle. 87 00:04:44,690 --> 00:04:52,830 So this is at r=1, so maybe I should say on the boundary, 88 00:04:52,830 --> 00:04:59,490 which is r=1, and going around the angle theta, is given. 89 00:04:59,490 --> 00:05:02,270 This is u_0(theta). 90 00:05:02,270 --> 00:05:08,090 That's my given boundary condition. 91 00:05:08,090 --> 00:05:10,850 This problem is named after Dirichlet, 92 00:05:10,850 --> 00:05:15,800 because it's like giving fixed conditions and not 93 00:05:15,800 --> 00:05:17,720 Neumann conditions. 94 00:05:17,720 --> 00:05:21,800 OK, so I'm just looking for a combination. 95 00:05:21,800 --> 00:05:24,440 For a function that solves Laplace's equation 96 00:05:24,440 --> 00:05:29,050 inside the circle, and takes on some values 97 00:05:29,050 --> 00:05:30,790 around the boundary. 98 00:05:30,790 --> 00:05:34,710 And of course the boundary values 99 00:05:34,710 --> 00:05:39,030 might be plus one on the top and minus one on the bottom. 100 00:05:39,030 --> 00:05:43,180 Or the boundary condition might vary around, 101 00:05:43,180 --> 00:05:46,390 it might, variable then come around. 102 00:05:46,390 --> 00:05:50,970 But notice that this is a periodic function. 103 00:05:50,970 --> 00:05:57,170 This is 2pi-periodic, because the problem's the same. 104 00:05:57,170 --> 00:06:01,020 If I increase theta by 2pi I've come back to the same point. 105 00:06:01,020 --> 00:06:04,030 So it's got to have that value. 106 00:06:04,030 --> 00:06:12,180 OK, so well let me give you a couple of examples first. 107 00:06:12,180 --> 00:06:18,120 Suppose u_0, suppose this function-- Example 1, easy. 108 00:06:18,120 --> 00:06:21,580 Suppose u_0 is sin(3theta). 109 00:06:21,580 --> 00:06:25,890 110 00:06:25,890 --> 00:06:30,030 So that means I've got a region here, 111 00:06:30,030 --> 00:06:33,300 I'm prescribing its temperature on the boundary. 112 00:06:33,300 --> 00:06:36,370 And I want to say what does it look like inside? 113 00:06:36,370 --> 00:06:40,090 And I'm prescribing right now the sin(3theta), 114 00:06:40,090 --> 00:06:45,310 so there theta is zero, so it's zero, the boundary condition's 115 00:06:45,310 --> 00:06:50,610 zero there, climbs to one, back to zero, down to minus one. 116 00:06:50,610 --> 00:06:51,940 back to zero. 117 00:06:51,940 --> 00:06:57,170 Three times, and again comes back to zero again there. 118 00:06:57,170 --> 00:07:02,550 So, I'm looking for a solution to Laplace's equation -- 119 00:07:02,550 --> 00:07:05,110 and I've got a pretty good list -- 120 00:07:05,110 --> 00:07:08,990 that will match u_0 when r is one. 121 00:07:08,990 --> 00:07:11,440 So that's the boundary, r is one. 122 00:07:11,440 --> 00:07:13,690 So you can tell me what it is. 123 00:07:13,690 --> 00:07:15,450 So you can solve this problem, right away. 124 00:07:15,450 --> 00:07:26,550 The answer is u of r and theta is, what function will--? 125 00:07:26,550 --> 00:07:29,750 Remember I've got my eye on that list. 126 00:07:29,750 --> 00:07:31,610 You too, right? 127 00:07:31,610 --> 00:07:34,680 I'm just trying to get one that when r is one 128 00:07:34,680 --> 00:07:37,190 it will match sin(3theta). 129 00:07:37,190 --> 00:07:40,570 What's the good guy? 130 00:07:40,570 --> 00:07:43,760 It'll be on that list. 131 00:07:43,760 --> 00:07:46,120 Which of those, by itself, here I 132 00:07:46,120 --> 00:07:53,480 don't need a series because I've got such a neat u_0 function. 133 00:07:53,480 --> 00:07:57,250 I'll get it right with one answer, 134 00:07:57,250 --> 00:08:01,090 and what is that answer? 135 00:08:01,090 --> 00:08:02,200 I look there. 136 00:08:02,200 --> 00:08:09,870 I say what do I do, so that at r=1, I'll match sin(3theta). 137 00:08:09,870 --> 00:08:12,740 I'll use r cubed sin(3theta). 138 00:08:12,740 --> 00:08:15,210 So the good winner will be r cubed sin(3theta). 139 00:08:15,210 --> 00:08:17,750 140 00:08:17,750 --> 00:08:19,670 That solves Laplace's equation. 141 00:08:19,670 --> 00:08:24,850 We checked it out, it's the imaginary part of x+iy cubed. 142 00:08:24,850 --> 00:08:28,160 We could write it in x and y coordinates if we 143 00:08:28,160 --> 00:08:31,180 wanted but we don't want to. 144 00:08:31,180 --> 00:08:36,750 And it matches when r is one, it gives us sin(3theta). 145 00:08:36,750 --> 00:08:42,560 That's it. 146 00:08:42,560 --> 00:08:46,210 And of course I could take any one. 147 00:08:46,210 --> 00:08:48,740 Now suppose I'm trying to match something that's 148 00:08:48,740 --> 00:08:50,510 not as simple as sin(3theta). 149 00:08:50,510 --> 00:08:53,920 150 00:08:53,920 --> 00:08:57,740 In that case, I may have to use all of them. 151 00:08:57,740 --> 00:09:01,730 I mean, it's very, very fluky that one term 152 00:09:01,730 --> 00:09:02,920 is going to do it. 153 00:09:02,920 --> 00:09:06,430 Usually, so my main examples would be 154 00:09:06,430 --> 00:09:08,090 I'll have to match all of them. 155 00:09:08,090 --> 00:09:09,770 So what do I do? 156 00:09:09,770 --> 00:09:19,360 At r=1, so my general solution is a combination of these guys 157 00:09:19,360 --> 00:09:21,270 I worked so hard to get. 158 00:09:21,270 --> 00:09:23,970 The solution is of this form. 159 00:09:23,970 --> 00:09:27,310 It's some a_0, the constant. 160 00:09:27,310 --> 00:09:33,080 And then a_1*r*cos(theta), and b_1*r*sin(theta). 161 00:09:33,080 --> 00:09:36,520 162 00:09:36,520 --> 00:09:42,540 And a_2 r squared cos(2theta). 163 00:09:42,540 --> 00:09:45,160 And so on. 164 00:09:45,160 --> 00:09:48,120 I'm just taking any combination of, 165 00:09:48,120 --> 00:09:51,970 I'm using the a's as the coefficients 166 00:09:51,970 --> 00:09:54,670 for the cosine guys. 167 00:09:54,670 --> 00:09:59,510 And the b's, b_1, b_2, b_3 would be the coefficients 168 00:09:59,510 --> 00:10:01,510 for the sine one. 169 00:10:01,510 --> 00:10:04,020 OK, that's my general solution. 170 00:10:04,020 --> 00:10:06,130 That solves Laplace's equation. 171 00:10:06,130 --> 00:10:09,500 Every term did, so every combination will. 172 00:10:09,500 --> 00:10:12,520 Now, set r=1. 173 00:10:12,520 --> 00:10:18,050 To match that r=1-- and match the boundary. 174 00:10:18,050 --> 00:10:24,000 And match u_0(theta), the required temperature around 175 00:10:24,000 --> 00:10:26,850 the-- on the boundary. 176 00:10:26,850 --> 00:10:30,230 The boundary being where r is one. 177 00:10:30,230 --> 00:10:32,010 So this is set r=1. 178 00:10:32,010 --> 00:10:40,570 So then u_0(theta), this given thing, has to match this, 179 00:10:40,570 --> 00:10:41,930 when r is one. 180 00:10:41,930 --> 00:10:51,200 So it's a_0 plus a_1, now what do I write here? r is one, 181 00:10:51,200 --> 00:10:52,210 so it's just cos(theta). 182 00:10:52,210 --> 00:10:54,790 183 00:10:54,790 --> 00:11:00,540 Now, b_1, r is one, so I just have sin(theta). 184 00:11:00,540 --> 00:11:06,690 And I have an a_2*cos(2theta), and a b_2*sin(2theta), 185 00:11:06,690 --> 00:11:12,890 and so on. 186 00:11:12,890 --> 00:11:16,330 Here, just let me put it together now. 187 00:11:16,330 --> 00:11:21,630 I'm given any temperature distribution 188 00:11:21,630 --> 00:11:24,600 around the boundary. 189 00:11:24,600 --> 00:11:27,900 It's in equilibrium, the temperature, 190 00:11:27,900 --> 00:11:34,050 where if the temperature's high near that point 191 00:11:34,050 --> 00:11:37,870 and low over here the temperature 192 00:11:37,870 --> 00:11:41,120 inside will gradually go from that high point, 193 00:11:41,120 --> 00:11:45,120 dot dot dot dot, to the lower one. 194 00:11:45,120 --> 00:11:47,170 By matching on the boundary. 195 00:11:47,170 --> 00:11:49,750 And this is the match on the boundary. 196 00:11:49,750 --> 00:12:02,410 Now, this is really a lead in to the last part of this course. 197 00:12:02,410 --> 00:12:07,120 So whose name is associated with a series like that? 198 00:12:07,120 --> 00:12:08,450 Fourier. 199 00:12:08,450 --> 00:12:12,760 You recognize that as what's called a Fourier series. 200 00:12:12,760 --> 00:12:17,240 So the idea is, I'm given these boundary values. 201 00:12:17,240 --> 00:12:22,070 I find their expansion in sines and cosines, 202 00:12:22,070 --> 00:12:25,570 and that's what we'll do in November. 203 00:12:25,570 --> 00:12:29,140 And then I've got it. 204 00:12:29,140 --> 00:12:32,150 Then I know the a's and the b's. 205 00:12:32,150 --> 00:12:37,340 And then basically I just put in the r's. r and r squareds and r 206 00:12:37,340 --> 00:12:38,690 cubeds and so on. 207 00:12:38,690 --> 00:12:43,950 So then I've got the answer inside. 208 00:12:43,950 --> 00:12:48,510 In principle it's so easy. 209 00:12:48,510 --> 00:12:51,410 So, why is it easy, though? 210 00:12:51,410 --> 00:12:54,600 First, it's easy because it's a circle we're working in. 211 00:12:54,600 --> 00:12:59,750 If I was in an ellipse or a strange shape, forget it. 212 00:12:59,750 --> 00:13:03,270 I mean, so this is quite special. 213 00:13:03,270 --> 00:13:09,750 And secondly, it's easy because these functions are so nice. 214 00:13:09,750 --> 00:13:14,080 Fourier works with the best functions ever. 215 00:13:14,080 --> 00:13:16,000 These sines and cosines. 216 00:13:16,000 --> 00:13:20,310 So I'll find a way to find those coefficients, 217 00:13:20,310 --> 00:13:24,460 the a's and the b's. 218 00:13:24,460 --> 00:13:26,710 Even though there are lots of them, 219 00:13:26,710 --> 00:13:29,580 I'll be able to pick them off one at the time, 220 00:13:29,580 --> 00:13:31,540 the a's and b's. 221 00:13:31,540 --> 00:13:35,610 Once I know the a's and b's, I know the answer. 222 00:13:35,610 --> 00:13:38,860 So do you see this is in principle a great way 223 00:13:38,860 --> 00:13:39,880 to solve it? 224 00:13:39,880 --> 00:13:42,660 In fact, it's the way we used over here, 225 00:13:42,660 --> 00:13:47,440 when my u_0 was sin(3theta), then the only term in its 226 00:13:47,440 --> 00:13:50,710 Fourier series was 1*sin(3theta). 227 00:13:50,710 --> 00:13:55,910 And then the solution was 1 r cubed sin(3theta). 228 00:13:55,910 --> 00:13:59,370 So you can learn things from this. 229 00:13:59,370 --> 00:14:03,280 For example, oh, what can you learn? 230 00:14:03,280 --> 00:14:08,830 One thing I noticed, an important feature 231 00:14:08,830 --> 00:14:15,800 of Laplace's equation is that this solution inside the circle 232 00:14:15,800 --> 00:14:22,110 gets very smooth. 233 00:14:22,110 --> 00:14:25,700 The boundary conditions could be like a delta function. 234 00:14:25,700 --> 00:14:28,530 I could say that on the boundary, 235 00:14:28,530 --> 00:14:31,440 the temperature is zero everywhere except at that 236 00:14:31,440 --> 00:14:35,110 point it spikes. 237 00:14:35,110 --> 00:14:36,630 So I could take u_0-- 238 00:14:36,630 --> 00:14:41,050 So example 2, and I won't do it in full, 239 00:14:41,050 --> 00:14:46,670 would be u_0 on the boundary equal a delta function. 240 00:14:46,670 --> 00:14:49,510 A spike at that one point. 241 00:14:49,510 --> 00:14:55,150 So all the heat is coming from the source at that one point. 242 00:14:55,150 --> 00:14:59,330 Like I've got a fire going there. 243 00:14:59,330 --> 00:15:03,750 Keeping the rest of the boundary frozen, 244 00:15:03,750 --> 00:15:07,630 the heat's kind of going to come inside. 245 00:15:07,630 --> 00:15:09,750 So then how would I proceed? 246 00:15:09,750 --> 00:15:16,660 Well, if I have this boundary value as a delta function, 247 00:15:16,660 --> 00:15:20,780 I look for its Fourier series, and it's 248 00:15:20,780 --> 00:15:22,820 a very important, beautiful, Fourier 249 00:15:22,820 --> 00:15:25,350 series for a delta function. 250 00:15:25,350 --> 00:15:27,940 Would you want to know it? 251 00:15:27,940 --> 00:15:30,690 I mean, we'll know it well in November. 252 00:15:30,690 --> 00:15:32,420 Would you want to know it in October? 253 00:15:32,420 --> 00:15:37,980 This is Halloween, I guess, so delta-- I'll 254 00:15:37,980 --> 00:15:39,330 tell you what it is. 255 00:15:39,330 --> 00:15:43,060 Since you insist. 256 00:15:43,060 --> 00:15:50,170 delta(theta) I think will-- I think there's a 1/(2pi) 257 00:15:50,170 --> 00:15:53,980 or something. 258 00:15:53,980 --> 00:15:55,010 Ah, shoot. 259 00:15:55,010 --> 00:15:57,470 We'll get it exactly right. 260 00:15:57,470 --> 00:16:06,090 It's something like 1 and 2 cos(theta) and 2cos(2theta), 261 00:16:06,090 --> 00:16:08,270 I'm not sure about the 2pi. 262 00:16:08,270 --> 00:16:14,740 2cos(2theta), and 2cos(3theta), and so on. 263 00:16:14,740 --> 00:16:18,340 We'll know it well when we get there. 264 00:16:18,340 --> 00:16:22,584 What I notice about this delta function-- Of course you're 265 00:16:22,584 --> 00:16:24,250 going to expect the delta function being 266 00:16:24,250 --> 00:16:26,840 somehow a little bit strange. 267 00:16:26,840 --> 00:16:32,190 At theta=0, what does that series add up to? 268 00:16:32,190 --> 00:16:35,160 Just so you begin to get a hang of Fourier series. 269 00:16:35,160 --> 00:16:40,860 At theta=0, what does that series look like? 270 00:16:40,860 --> 00:16:44,510 Well, all these cosine thetas are? 271 00:16:44,510 --> 00:16:45,350 One. 272 00:16:45,350 --> 00:16:50,460 So this series at theta=0 is 1+2+2+2+2... 273 00:16:50,460 --> 00:16:51,900 It's infinite. 274 00:16:51,900 --> 00:16:54,860 And that's what we want. 275 00:16:54,860 --> 00:16:59,080 The delta function is infinite at theta=0. 276 00:16:59,080 --> 00:17:02,330 And it's periodic, of course, so that if I go around 277 00:17:02,330 --> 00:17:07,060 to theta=2pi I'll come back to zero again. 278 00:17:07,060 --> 00:17:12,180 At theta=pi, you could sort of see, well, yeah, 279 00:17:12,180 --> 00:17:14,770 theta=pi is a sort of interesting point. 280 00:17:14,770 --> 00:17:18,000 At theta is pi, what's the cosine? 281 00:17:18,000 --> 00:17:20,040 Is negative one, right? 282 00:17:20,040 --> 00:17:23,530 But then the cos(2pi) will be plus one. 283 00:17:23,530 --> 00:17:28,540 So at theta=pi, I think I'm getting a one minus a two plus 284 00:17:28,540 --> 00:17:31,160 a two, minus a two, plus a two. 285 00:17:31,160 --> 00:17:35,650 You see, it's doing its best to cancel itself out and give me 286 00:17:35,650 --> 00:17:41,260 the zero that I want, the theta=pi over on the left side 287 00:17:41,260 --> 00:17:42,450 of the circle. 288 00:17:42,450 --> 00:17:49,150 Anyway, so that's an extreme example. 289 00:17:49,150 --> 00:17:52,150 But now, what's the temperature inside? 290 00:17:52,150 --> 00:17:54,540 Can you just follow the same rule? 291 00:17:54,540 --> 00:17:56,380 What will be the temperature inside? 292 00:17:56,380 --> 00:18:00,610 If that's the delta function, if that's 293 00:18:00,610 --> 00:18:04,510 the right series, whatever, it might be a 4pi, 294 00:18:04,510 --> 00:18:10,210 I'm not sure, for that. 295 00:18:10,210 --> 00:18:13,420 Now, you can tell me what's the solution, what's 296 00:18:13,420 --> 00:18:16,720 the temperature distribution inside a circle when 297 00:18:16,720 --> 00:18:26,040 one point on the boundary has a heat source, a delta function. 298 00:18:26,040 --> 00:18:27,310 What do I do? 299 00:18:27,310 --> 00:18:30,420 How do I match this with this guy? 300 00:18:30,420 --> 00:18:33,060 I just put in the r's, right? 301 00:18:33,060 --> 00:18:36,860 If this is what it's supposed to match when r is one, 302 00:18:36,860 --> 00:18:44,840 then when r is-- So maybe I'll put it under here. 303 00:18:44,840 --> 00:18:53,710 So the u(r,theta), from the delta guy, 304 00:18:53,710 --> 00:18:57,800 is just put in the r's. 305 00:18:57,800 --> 00:19:06,960 1+2r*cos(theta), and 2 r squared cos(2theta), and so on. 306 00:19:06,960 --> 00:19:17,230 OK, and eventually 2 r to the 100th cos(100theta), and more. 307 00:19:17,230 --> 00:19:24,970 OK, I write this out, you could say why did he write this down? 308 00:19:24,970 --> 00:19:29,510 I wanted to make this point that the important feature 309 00:19:29,510 --> 00:19:32,430 of the solution to Laplace's equation 310 00:19:32,430 --> 00:19:37,290 is how smooth it gets when you go inside the region. 311 00:19:37,290 --> 00:19:38,670 And why is that? 312 00:19:38,670 --> 00:19:45,970 Because at r=1/2, this term is practically gone, right? 313 00:19:45,970 --> 00:19:50,930 If I go halfway into the circle, this term is practically gone. 314 00:19:50,930 --> 00:19:53,040 1/2 to the hundredth power. 315 00:19:53,040 --> 00:19:56,090 And if I go to the center of the circle, it's completely gone. 316 00:19:56,090 --> 00:19:59,110 In fact, what's the value at the center of the circle? 317 00:19:59,110 --> 00:20:07,580 What's the temperature at the center? 318 00:20:07,580 --> 00:20:08,600 1/2pi. 319 00:20:08,600 --> 00:20:11,470 This is the only term that's remaining. 320 00:20:11,470 --> 00:20:19,940 And it's the average, around the circle. 321 00:20:19,940 --> 00:20:23,830 That makes physical sense, I guess. 322 00:20:23,830 --> 00:20:27,030 Since the whole thing's completely isotropic, 323 00:20:27,030 --> 00:20:35,330 we've got a perfect circle, the value 324 00:20:35,330 --> 00:20:37,690 at the center of the circle is always 325 00:20:37,690 --> 00:20:39,750 the average going around. 326 00:20:39,750 --> 00:20:43,880 The constant term in the Fourier series, this guy. 327 00:20:43,880 --> 00:20:46,790 We'll get to know that one very well. 328 00:20:46,790 --> 00:20:50,480 That's the average. 329 00:20:50,480 --> 00:20:55,100 You're just seeing a little bit of Fourier series early, here. 330 00:20:55,100 --> 00:21:02,230 But my point is that you could have high oscillation 331 00:21:02,230 --> 00:21:04,770 around the boundary, that damps out 332 00:21:04,770 --> 00:21:07,480 because of these powers of r. 333 00:21:07,480 --> 00:21:13,300 And inside the circle it's only the low order terms 334 00:21:13,300 --> 00:21:22,090 that begin to take over. 335 00:21:22,090 --> 00:21:24,360 This is the kind of trick you have, or not 336 00:21:24,360 --> 00:21:27,660 trick but the kind of method that you 337 00:21:27,660 --> 00:21:34,750 can use for solving Laplace's equation by an infinite series. 338 00:21:34,750 --> 00:21:38,220 Of course, a person who wants a number 339 00:21:38,220 --> 00:21:41,440 can complain that, wait a minute, 340 00:21:41,440 --> 00:21:43,640 how do I use that infinite series? 341 00:21:43,640 --> 00:21:47,860 Well, of course, if you wanted to know the temperature 342 00:21:47,860 --> 00:21:50,120 at a particular point you'd have to plug in 343 00:21:50,120 --> 00:21:52,700 that value of r, that value of theta, 344 00:21:52,700 --> 00:21:57,770 add up the terms until you hope that they become so 345 00:21:57,770 --> 00:22:00,450 small that you can ignore them. 346 00:22:00,450 --> 00:22:03,810 So infinite series is one form of a solution. 347 00:22:03,810 --> 00:22:12,510 And somehow these are examples-- I should use the words 348 00:22:12,510 --> 00:22:14,910 separation of variables. 349 00:22:14,910 --> 00:22:20,830 Separation of variables is the golden idea 350 00:22:20,830 --> 00:22:22,800 in this analysis stuff. 351 00:22:22,800 --> 00:22:26,910 Separation of variables means I got the r part separated 352 00:22:26,910 --> 00:22:29,760 from the theta part. 353 00:22:29,760 --> 00:22:33,800 And that worked great, worked well for a circle. 354 00:22:33,800 --> 00:22:39,230 Let's see, maybe for a square I could try to separate x from y. 355 00:22:39,230 --> 00:22:46,800 Maybe there's a homework problem, a solution that 356 00:22:46,800 --> 00:22:52,130 separates x from y, I think is something like-- So this 357 00:22:52,130 --> 00:22:53,910 would be another family. 358 00:22:53,910 --> 00:23:00,810 Good for squares, something like sin(kx) sinh times, 359 00:23:00,810 --> 00:23:08,070 so this separation is something in x times something in y. 360 00:23:08,070 --> 00:23:10,670 Again I'm just mentioning things. 361 00:23:10,670 --> 00:23:14,390 I think that that solves Laplace's equation 362 00:23:14,390 --> 00:23:17,710 because if I take two x derivatives, 363 00:23:17,710 --> 00:23:22,560 that'll bring down k squared, but it'll flip the sign, right? 364 00:23:22,560 --> 00:23:25,820 These two derivatives of the sine will be a minus. 365 00:23:25,820 --> 00:23:30,380 And if I take-- I need a ky there. 366 00:23:30,380 --> 00:23:35,240 And if I took two derivatives of this hyperbolic sine, 367 00:23:35,240 --> 00:23:40,570 you remember that's the e^(ky) and the e^(-ky). 368 00:23:40,570 --> 00:23:42,650 The two derivatives of that will bring out 369 00:23:42,650 --> 00:23:45,800 a k squared with a plus sign. 370 00:23:45,800 --> 00:23:49,390 So two x derivatives bring out the minus k squared, 371 00:23:49,390 --> 00:23:52,260 two y derivatives bring out a plus k squared 372 00:23:52,260 --> 00:23:56,440 and together that solves Laplace's equation. 373 00:23:56,440 --> 00:23:58,680 We'll check that in our homework problem. 374 00:23:58,680 --> 00:24:02,410 So there would be an example, good for a square. 375 00:24:02,410 --> 00:24:13,950 So, there's hope to do an exact solution in a special region. 376 00:24:13,950 --> 00:24:19,700 Now, what's this Green's function idea? 377 00:24:19,700 --> 00:24:26,480 OK, that's now this is another thing. 378 00:24:26,480 --> 00:24:35,770 So last time we appreciated that this combination x+iy was 379 00:24:35,770 --> 00:24:37,650 magic. 380 00:24:37,650 --> 00:24:42,880 The idea was that we could take any function of x+iy, 381 00:24:42,880 --> 00:24:47,390 and it solves Laplace's equation. 382 00:24:47,390 --> 00:24:52,420 Can we just see, sort of very crudely why that is? 383 00:24:52,420 --> 00:25:00,310 We saw the pattern, we saw x+iy to the nth. 384 00:25:00,310 --> 00:25:04,520 Sort of, we went as far as n=3, checked it all out. 385 00:25:04,520 --> 00:25:08,120 But now, really if I want to be able to-- why does that 386 00:25:08,120 --> 00:25:13,820 solve Laplace's equation for any n? 387 00:25:13,820 --> 00:25:16,970 Should I just plug that into Laplace's equation? 388 00:25:16,970 --> 00:25:24,740 What happens if I take the two x derivatives of this thing? 389 00:25:24,740 --> 00:25:28,070 So this going to be a typical function of x+iy, 390 00:25:28,070 --> 00:25:29,710 typically nice one. 391 00:25:29,710 --> 00:25:32,080 If I take two x derivatives, I want 392 00:25:32,080 --> 00:25:34,520 to plug it in and see that it really 393 00:25:34,520 --> 00:25:36,920 does solve Laplace's equation. 394 00:25:36,920 --> 00:25:40,730 So two x derivatives of that will give me what? 395 00:25:40,730 --> 00:25:44,050 The first x derivative will bring down an n 396 00:25:44,050 --> 00:25:47,420 times this thing to the n-1. 397 00:25:47,420 --> 00:25:51,890 And then the next x derivative will bring down an n-1 times 398 00:25:51,890 --> 00:25:55,490 this thing to the n-2. 399 00:25:55,490 --> 00:25:59,550 So that'll be the u_xx. 400 00:25:59,550 --> 00:26:05,240 And what about u_yy? 401 00:26:05,240 --> 00:26:08,580 This is my u. 402 00:26:08,580 --> 00:26:14,260 I'm sort of just checking that yes, this-- See again, 403 00:26:14,260 --> 00:26:17,440 see if it still works Friday what worked Wednesday. 404 00:26:17,440 --> 00:26:23,260 That this x+iy is magic and functions of it like powers, 405 00:26:23,260 --> 00:26:26,230 exponentials, logarithms, whatever, 406 00:26:26,230 --> 00:26:28,530 all solve Laplace's equation. 407 00:26:28,530 --> 00:26:33,270 OK, so we did u_xx, and we got-- easy. 408 00:26:33,270 --> 00:26:35,820 Now, what happens with u_yy? 409 00:26:35,820 --> 00:26:36,860 Do you see the point? 410 00:26:36,860 --> 00:26:39,060 AUDIENCE: [INAUDIBLE] 411 00:26:39,060 --> 00:26:41,140 PROFESSOR STRANG: Sorry opposite sign. 412 00:26:41,140 --> 00:26:44,710 And why does the sign come out opposite? 413 00:26:44,710 --> 00:26:46,260 Because of that guy. 414 00:26:46,260 --> 00:26:48,300 Yeah, it's the chain rule, right? 415 00:26:48,300 --> 00:26:50,900 The derivative of this with respect to y 416 00:26:50,900 --> 00:26:55,250 will give me an n times this thing to one lower power. 417 00:26:55,250 --> 00:26:58,430 Times the derivative of what's inside. 418 00:26:58,430 --> 00:27:01,470 And the derivative of what's inside is an i. 419 00:27:01,470 --> 00:27:03,730 And then the second derivative will bring down 420 00:27:03,730 --> 00:27:10,720 an n-1, this guy will be down to n-2, another i will come out 421 00:27:10,720 --> 00:27:14,140 and-- Just what you want, right? 422 00:27:14,140 --> 00:27:17,170 Because the i squared is minus one, those cancel. 423 00:27:17,170 --> 00:27:21,070 When those are equal opposite signs. 424 00:27:21,070 --> 00:27:29,490 And we get u_xx+u_yy equaling 0. 425 00:27:29,490 --> 00:27:31,140 So that works. 426 00:27:31,140 --> 00:27:34,490 And, actually, the same idea would work for any function 427 00:27:34,490 --> 00:27:36,000 of x+iy. 428 00:27:36,000 --> 00:27:42,150 The two x derivatives just give f''. 429 00:27:42,150 --> 00:27:46,520 Two y derivatives will give f'' but the chain rule will bring 430 00:27:46,520 --> 00:27:49,560 out i both times and we've got it. 431 00:27:49,560 --> 00:28:00,640 OK, I think we just need another couple of examples. 432 00:28:00,640 --> 00:28:05,000 And this of course could be in polar coordinates, 433 00:28:05,000 --> 00:28:08,650 f of re^(i*theta). 434 00:28:08,650 --> 00:28:12,220 That's just, everybody recognizes re^(i*theta) is 435 00:28:12,220 --> 00:28:13,760 the same as x+iy? 436 00:28:13,760 --> 00:28:17,610 Better just be sure we've got that. x is 437 00:28:17,610 --> 00:28:23,160 some point here in the complex plane. iy takes us up to here. 438 00:28:23,160 --> 00:28:26,080 So there's x+iy. 439 00:28:26,080 --> 00:28:30,090 That's x+iy there, but it's also-- 440 00:28:30,090 --> 00:28:32,370 So let me put those in better. 441 00:28:32,370 --> 00:28:37,120 So there's x and there's y. 442 00:28:37,120 --> 00:28:39,230 Everybody knows this picture, right? 443 00:28:39,230 --> 00:28:43,260 This x and this y, now if I want to go to polar coordinates, 444 00:28:43,260 --> 00:28:48,770 that angle is theta, this x is r*cos(theta), 445 00:28:48,770 --> 00:28:55,320 this y is r*sin(theta), and this guy is re^(i*theta). 446 00:28:55,320 --> 00:28:55,910 re^(i*theta). 447 00:28:55,910 --> 00:29:00,380 448 00:29:00,380 --> 00:29:05,640 r*cos(theta) plus i*r*sin(theta) is the same as re^i*theta. 449 00:29:05,640 --> 00:29:07,830 That's utterly fundamental. 450 00:29:07,830 --> 00:29:12,750 Everybody's responsible for that picture 451 00:29:12,750 --> 00:29:18,140 of putting the complex numbers into their beautiful polar 452 00:29:18,140 --> 00:29:18,640 form. 453 00:29:18,640 --> 00:29:25,950 That's what made our r to the nth cos(n*theta) all so simple. 454 00:29:25,950 --> 00:29:30,230 Now, what was I aiming to do? 455 00:29:30,230 --> 00:29:33,160 Give a particular f. 456 00:29:33,160 --> 00:29:38,750 Now I want to give a particular function f, or maybe 457 00:29:38,750 --> 00:29:40,620 a couple of choices. 458 00:29:40,620 --> 00:29:44,100 A couple of functions f, and see that their real parts 459 00:29:44,100 --> 00:29:50,090 and their imaginary parts solve Laplace's equation. 460 00:29:50,090 --> 00:30:00,460 Let me take first a one that works completely. 461 00:30:00,460 --> 00:30:05,520 Take the real part and the imaginary part-- 462 00:30:05,520 --> 00:30:07,380 Let me take e^(x+iy). 463 00:30:07,380 --> 00:30:10,980 464 00:30:10,980 --> 00:30:15,240 It's a function of x+iy, extremely nice function 465 00:30:15,240 --> 00:30:20,210 of x+iy, and we can figure out its real and imaginary parts, 466 00:30:20,210 --> 00:30:25,130 and we get two solutions to Laplace's equation. 467 00:30:25,130 --> 00:30:31,320 The good way is to write this thing as e^x times e^(iy). 468 00:30:31,320 --> 00:30:39,140 And again we'll write it as e^x times cos(y)+i*sin(y). 469 00:30:39,140 --> 00:30:42,750 So now I can see that the real part-- 470 00:30:42,750 --> 00:30:45,100 I can see what the real part is, and I can 471 00:30:45,100 --> 00:30:46,830 see what the imaginary part is. 472 00:30:46,830 --> 00:30:49,060 The real part will be, that's real. 473 00:30:49,060 --> 00:30:54,930 And that's real. so this will so give me e^x*cos(y). 474 00:30:54,930 --> 00:30:59,120 And the imaginary part will be e^x*sin(y). 475 00:30:59,120 --> 00:31:03,080 You see it. 476 00:31:03,080 --> 00:31:08,080 And those will solve Laplace's equation. 477 00:31:08,080 --> 00:31:13,970 Can I give a name to this whole field of analysis? 478 00:31:13,970 --> 00:31:20,430 This e^z is an analytic -- I should just use that word -- 479 00:31:20,430 --> 00:31:25,110 an analytic function. 480 00:31:25,110 --> 00:31:27,830 And these guys, the real and imaginary parts, 481 00:31:27,830 --> 00:31:34,700 are two harmonic functions. 482 00:31:34,700 --> 00:31:37,550 Maybe it's not so important to know the word harmonic 483 00:31:37,550 --> 00:31:38,360 function. 484 00:31:38,360 --> 00:31:40,670 But analytic function, yeah, I would 485 00:31:40,670 --> 00:31:44,860 say that's an important word. 486 00:31:44,860 --> 00:31:48,070 Actually, what does it mean? 487 00:31:48,070 --> 00:31:52,480 It's a function of z. 488 00:31:52,480 --> 00:31:57,900 So we're in the complex plane here now. 489 00:31:57,900 --> 00:32:04,570 It's a function of z, e^z, and it can be written as a power 490 00:32:04,570 --> 00:32:09,930 series, of course, one plus z plus 1 over 2 factorial z 491 00:32:09,930 --> 00:32:13,210 squared and all those guys. 492 00:32:13,210 --> 00:32:15,080 So it has a power series. 493 00:32:15,080 --> 00:32:19,450 That makes it a combination of our special ones. 494 00:32:19,450 --> 00:32:25,700 The great thing about that series is it converges. 495 00:32:25,700 --> 00:32:29,470 So an analytic function, an analytic function 496 00:32:29,470 --> 00:32:34,290 is the sum of a power series that converges. 497 00:32:34,290 --> 00:32:35,540 And this one does. 498 00:32:35,540 --> 00:32:37,340 So there's an example. 499 00:32:37,340 --> 00:32:40,530 Yeah, so the whole theory of analytic functions 500 00:32:40,530 --> 00:32:44,710 is actually, that's Chapter 5 of the textbook. 501 00:32:44,710 --> 00:32:53,490 And we won't get beyond this point, I think, in one semester 502 00:32:53,490 --> 00:32:57,280 with analytic functions. 503 00:32:57,280 --> 00:32:59,000 So what am I saying, though? 504 00:32:59,000 --> 00:33:02,020 I'm saying that the theory of analytic functions 505 00:33:02,020 --> 00:33:05,300 is closely tied to Laplace's equation. 506 00:33:05,300 --> 00:33:08,090 Because the real and the imaginary parts 507 00:33:08,090 --> 00:33:13,110 give me this pair u and s that satisfy, they each satisfy 508 00:33:13,110 --> 00:33:14,870 Laplace's equation. 509 00:33:14,870 --> 00:33:19,510 And they're connected by the Cauchy-Riemann equations. 510 00:33:19,510 --> 00:33:25,280 Boy, it's a lot of mathematics coming real fast here. 511 00:33:25,280 --> 00:33:29,130 Now I'd like to take one more example. 512 00:33:29,130 --> 00:33:33,600 Instead of the exponential, can we take the logarithm. 513 00:33:33,600 --> 00:33:38,820 I want to take the log of x+iy, and I want you to split it 514 00:33:38,820 --> 00:33:41,120 into its real and imaginary parts, 515 00:33:41,120 --> 00:33:44,180 and get the u and the s that go with that. 516 00:33:44,180 --> 00:33:48,810 So this was like the nicest possible. 517 00:33:48,810 --> 00:33:52,780 We got a series of, e^z is good for every z, 518 00:33:52,780 --> 00:33:55,540 the series converges, fantastic. 519 00:33:55,540 --> 00:33:58,990 It's an analytic function everywhere. 520 00:33:58,990 --> 00:34:01,440 Best possible. 521 00:34:01,440 --> 00:34:06,380 Now we go to one that's not best possible but nevertheless 522 00:34:06,380 --> 00:34:08,340 highly valuable. 523 00:34:08,340 --> 00:34:11,620 OK, so e^z, I've done. 524 00:34:11,620 --> 00:34:17,590 Let me erase e^z, take log z. 525 00:34:17,590 --> 00:34:23,760 OK, so now I'm not doing e^z any more. 526 00:34:23,760 --> 00:34:28,425 And I want to find the logarithm, OK. 527 00:34:28,425 --> 00:34:30,050 So, what's the deal with the logarithm? 528 00:34:30,050 --> 00:34:32,550 Real and imaginary parts. 529 00:34:32,550 --> 00:34:42,450 Now I'm going to take the log of x+iy. 530 00:34:42,450 --> 00:34:50,570 That is a function of x+iy, except at one point it has 531 00:34:50,570 --> 00:34:52,910 a problem, right? 532 00:34:52,910 --> 00:34:57,970 There's a point where this is not going to be analytic, 533 00:34:57,970 --> 00:35:06,130 and there's going to be a special point in the flow which 534 00:35:06,130 --> 00:35:07,840 is singular somehow. 535 00:35:07,840 --> 00:35:13,090 But away from that point, we have a nice-looking function, 536 00:35:13,090 --> 00:35:17,480 the logarithm of x+iy, and now I'd like to get its real 537 00:35:17,480 --> 00:35:19,040 and imaginary parts. 538 00:35:19,040 --> 00:35:22,050 I'd like to know the u and the s. 539 00:35:22,050 --> 00:35:23,900 But nobody in their right mind wants 540 00:35:23,900 --> 00:35:26,690 to take the logarithm of a sum, right? 541 00:35:26,690 --> 00:35:32,350 That's a very foolish thing to try to do, the log of a sum. 542 00:35:32,350 --> 00:35:35,590 What's the good way to get somewhere with this? 543 00:35:35,590 --> 00:35:39,140 Real and imaginary part. 544 00:35:39,140 --> 00:35:44,560 I can take the log of a product. 545 00:35:44,560 --> 00:35:48,000 So the polar is way better again. 546 00:35:48,000 --> 00:35:53,230 I want to write this as a log of r e to the-- I 547 00:35:53,230 --> 00:35:55,180 want to write it that way. 548 00:35:55,180 --> 00:36:00,870 And now what's the log of a product? 549 00:36:00,870 --> 00:36:03,310 The sum of the two pieces. 550 00:36:03,310 --> 00:36:13,060 So I have log r, and the log of e^(i*theta), which is? 551 00:36:13,060 --> 00:36:14,230 Which is i*theta. 552 00:36:14,230 --> 00:36:17,690 Boy, look, this is fantastic. 553 00:36:17,690 --> 00:36:21,330 Fantastic except at zero. 554 00:36:21,330 --> 00:36:27,160 I mean, it's fantastic but it's got a big problem at zero. 555 00:36:27,160 --> 00:36:30,890 But it's an extremely important example. 556 00:36:30,890 --> 00:36:33,840 So what's the real part? 557 00:36:33,840 --> 00:36:36,530 It's sitting there. 558 00:36:36,530 --> 00:36:38,140 This is my u. 559 00:36:38,140 --> 00:36:42,220 This is my u(r,theta), my u(x,y), whatever you want, 560 00:36:42,220 --> 00:36:45,820 is the log of r. 561 00:36:45,820 --> 00:36:51,570 The log of the square root of x squared plus y squared. 562 00:36:51,570 --> 00:36:56,970 I claim that again by this magic combination, this log, 563 00:36:56,970 --> 00:37:00,680 this-- r is the square root of x squared plus y squared. 564 00:37:00,680 --> 00:37:04,520 I claim if you substitute that into Laplace's equation 565 00:37:04,520 --> 00:37:05,300 you get zero. 566 00:37:05,300 --> 00:37:07,940 It works. 567 00:37:07,940 --> 00:37:12,510 And what's the imaginary part, the s? 568 00:37:12,510 --> 00:37:18,340 The twin is the imaginary part, which is theta. 569 00:37:18,340 --> 00:37:23,170 Oh, what is theta in x, if I wanted it in x and y? 570 00:37:23,170 --> 00:37:25,860 What would theta be? 571 00:37:25,860 --> 00:37:28,930 It's the arctan, it's the angle whose 572 00:37:28,930 --> 00:37:36,070 tangent is something. y/x, so if I really want it 573 00:37:36,070 --> 00:37:42,100 in rectangular xy stuff, it's the angle whose tangent is y/x. 574 00:37:42,100 --> 00:37:44,520 And again, if you remember in calculus 575 00:37:44,520 --> 00:37:46,860 how to take derivatives of this thing 576 00:37:46,860 --> 00:37:50,140 and you plug it into Laplace's equation you get zero. 577 00:37:50,140 --> 00:37:53,190 It works. 578 00:37:53,190 --> 00:37:57,220 So that's a great solution except where? 579 00:37:57,220 --> 00:37:58,610 At zero. 580 00:37:58,610 --> 00:38:00,520 Except at zero. 581 00:38:00,520 --> 00:38:11,310 And this doesn't tell us what's happening at zero. 582 00:38:11,310 --> 00:38:13,280 It's an excellent solution. 583 00:38:13,280 --> 00:38:18,230 What's the picture? 584 00:38:18,230 --> 00:38:24,610 So by Wednesday's exam I'm not expecting 585 00:38:24,610 --> 00:38:30,070 you to be an expert on the theory of analytic functions. 586 00:38:30,070 --> 00:38:35,370 I don't expect you to know any conformal mappings. 587 00:38:35,370 --> 00:38:43,040 By Wednesday, God, that's-- But, I do expect you to have these 588 00:38:43,040 --> 00:38:44,680 pictures in mind. 589 00:38:44,680 --> 00:38:48,465 So when I draw those axes, what picture is it 590 00:38:48,465 --> 00:38:50,950 that I'm planning on? 591 00:38:50,950 --> 00:38:52,920 I'm planning on the equipotentials u 592 00:38:52,920 --> 00:39:02,460 equal constant, and the, who are the other guys? 593 00:39:02,460 --> 00:39:04,900 The streamlines. 594 00:39:04,900 --> 00:39:08,050 The places where the stream functions-- So here 595 00:39:08,050 --> 00:39:10,460 is the potential function. 596 00:39:10,460 --> 00:39:14,980 So what are the equipotential curves? 597 00:39:14,980 --> 00:39:16,950 For that guy? 598 00:39:16,950 --> 00:39:18,570 Circles. 599 00:39:18,570 --> 00:39:22,200 This is a constant when r is a constant, 600 00:39:22,200 --> 00:39:28,980 so the equipotential functions would be circles. 601 00:39:28,980 --> 00:39:31,730 I don't want to draw that circle with radius zero, though. 602 00:39:31,730 --> 00:39:34,950 I'm nervous about that one. 603 00:39:34,950 --> 00:39:37,000 But all the others are great. 604 00:39:37,000 --> 00:39:40,430 And what are the streamlines, now? 605 00:39:40,430 --> 00:39:47,140 The streamlines are, well, what will the streamlines be? 606 00:39:47,140 --> 00:39:51,620 If I've drawn one family, you can tell me the other family. 607 00:39:51,620 --> 00:39:54,140 The streamlines will be? 608 00:39:54,140 --> 00:39:57,170 Radial lines. 609 00:39:57,170 --> 00:39:59,860 Because they're going to be perpendicular to this. 610 00:39:59,860 --> 00:40:06,680 And so what do I get, this is the stream function, theta. 611 00:40:06,680 --> 00:40:08,290 So what's a streamline? 612 00:40:08,290 --> 00:40:10,740 The stream function should be a constant. 613 00:40:10,740 --> 00:40:12,140 Theta's a constant. 614 00:40:12,140 --> 00:40:15,980 That means I'm going out on rays. 615 00:40:15,980 --> 00:40:20,930 Those are all streamlines. 616 00:40:20,930 --> 00:40:23,570 Again, everything fantastic. 617 00:40:23,570 --> 00:40:26,720 If you look in a little region here 618 00:40:26,720 --> 00:40:31,690 you see just a beautiful picture of equipotentials 619 00:40:31,690 --> 00:40:35,560 and streamlines crossing them at right angles. 620 00:40:35,560 --> 00:40:37,300 Everything great. 621 00:40:37,300 --> 00:40:43,270 Just that point is obviously a problem. 622 00:40:43,270 --> 00:40:51,660 Now, and I'm suspecting that there's a source here. 623 00:40:51,660 --> 00:41:00,000 I think this flow, which is given by these guys, 624 00:41:00,000 --> 00:41:06,400 comes from some kind of a delta function right there. 625 00:41:06,400 --> 00:41:12,260 And the flow goes outwards. 626 00:41:12,260 --> 00:41:16,510 So I know u, I know v is the gradient of u, right? 627 00:41:16,510 --> 00:41:19,560 I could take the x and y derivatives, 628 00:41:19,560 --> 00:41:22,700 I'd know the velocity. 629 00:41:22,700 --> 00:41:26,270 I know the stream function, the divergence would be zero. 630 00:41:26,270 --> 00:41:32,200 Everything great, except at the origin. 631 00:41:32,200 --> 00:41:35,430 I think we've got some action at the origin. 632 00:41:35,430 --> 00:41:43,460 Because, here's the way to test it. 633 00:41:43,460 --> 00:41:49,100 I want to see what's happening at the origin. 634 00:41:49,100 --> 00:41:52,150 And I'm going to use the divergence theorem. 635 00:41:52,150 --> 00:41:52,680 Yeah. 636 00:41:52,680 --> 00:41:53,179 Yeah. 637 00:41:53,179 --> 00:41:55,160 I'm going to use the divergence theorem. 638 00:41:55,160 --> 00:42:00,780 So the divergence theorem says-- What is the divergence theorem? 639 00:42:00,780 --> 00:42:09,610 So this is the key thing that connects double integrals. 640 00:42:09,610 --> 00:42:14,230 Let me take a circle of radius R. So 641 00:42:14,230 --> 00:42:19,350 that's the circle of radius R. R could be big, or little. 642 00:42:19,350 --> 00:42:22,330 So I integrate over the circle of 643 00:42:22,330 --> 00:42:30,890 radius R. So what's the deal? v is the same as w. 644 00:42:30,890 --> 00:42:32,850 What does the divergence theorem tell me? 645 00:42:32,850 --> 00:42:36,860 It tells me that if I integrate, what do I integrate, 646 00:42:36,860 --> 00:42:43,210 the divergence of w? dx/dy, or r*dr*d theta. 647 00:42:43,210 --> 00:42:45,960 648 00:42:45,960 --> 00:42:52,760 Then I get the flux. 649 00:42:52,760 --> 00:42:56,450 So this is a key identity. 650 00:42:56,450 --> 00:42:59,600 Fundamentally, more than just the key identity, 651 00:42:59,600 --> 00:43:01,320 it's central here. 652 00:43:01,320 --> 00:43:07,650 The total flow out of the region must make it 653 00:43:07,650 --> 00:43:09,000 through the boundary. 654 00:43:09,000 --> 00:43:11,460 So I integrate this boundary, and this boundary 655 00:43:11,460 --> 00:43:14,180 is a circle of radius R, and what do I 656 00:43:14,180 --> 00:43:19,390 integrate along that circle? 657 00:43:19,390 --> 00:43:23,670 What's the other side of the divergence theorem? 658 00:43:23,670 --> 00:43:30,280 w dot n. w dot n, around the boundary. 659 00:43:30,280 --> 00:43:37,730 And remember, I have this nice-- my curve here 660 00:43:37,730 --> 00:43:42,450 is this nice circle. 661 00:43:42,450 --> 00:43:44,600 So I'm going to integrate around that circle. 662 00:43:44,600 --> 00:43:52,130 First of all, what is n? 663 00:43:52,130 --> 00:43:56,990 By definition, n is the normal that points outward, 664 00:43:56,990 --> 00:43:58,250 straight out. 665 00:43:58,250 --> 00:44:02,520 So it's actually going out that way. 666 00:44:02,520 --> 00:44:05,540 At every point it's pointing straight out. 667 00:44:05,540 --> 00:44:09,130 And ds-- Yeah, I think we can figure out exactly 668 00:44:09,130 --> 00:44:18,420 what that right-hand side is. 669 00:44:18,420 --> 00:44:23,090 How do I get that right-hand side? 670 00:44:23,090 --> 00:44:29,670 I'm looking for w, and then I have to integrate. 671 00:44:29,670 --> 00:44:35,580 OK, here is my u. 672 00:44:35,580 --> 00:44:42,790 My u is log r. 673 00:44:42,790 --> 00:44:45,520 So what's the gradient of log r? 674 00:44:45,520 --> 00:44:47,690 It points outwards. 675 00:44:47,690 --> 00:44:49,430 And how large is the derivative? 676 00:44:49,430 --> 00:44:55,600 So the derivative of this log r is 1/r. 677 00:44:55,600 --> 00:45:03,400 I think that this comes down to, this is the integral. 678 00:45:03,400 --> 00:45:05,010 Around the circle. 679 00:45:05,010 --> 00:45:14,930 I think that this thing is 1/R. I went pretty quickly there, 680 00:45:14,930 --> 00:45:17,960 so I'll ask you to look in the book 681 00:45:17,960 --> 00:45:21,360 because this is such an important example it's 682 00:45:21,360 --> 00:45:26,150 done there in more detail. 683 00:45:26,150 --> 00:45:30,080 So I'm claiming that the derivative is 1/R, 684 00:45:30,080 --> 00:45:32,780 and that it points directly out. 685 00:45:32,780 --> 00:45:35,930 So the gradient points out. 686 00:45:35,930 --> 00:45:40,700 The normal points out, so that I just get exactly 1/R. Now, 687 00:45:40,700 --> 00:45:41,470 what is ds? 688 00:45:41,470 --> 00:45:45,300 689 00:45:45,300 --> 00:45:47,840 For integrating around the circle 690 00:45:47,840 --> 00:45:54,410 what's a little tiny piece of arc on a circle? 691 00:45:54,410 --> 00:45:56,260 Of radius R? 692 00:45:56,260 --> 00:45:57,050 R d theta. 693 00:45:57,050 --> 00:45:57,830 Good man. 694 00:45:57,830 --> 00:46:03,120 R d theta. 695 00:46:03,120 --> 00:46:09,150 Now that's an integral I can do, right? 696 00:46:09,150 --> 00:46:11,200 And what do I get? 697 00:46:11,200 --> 00:46:13,630 2 pi. 698 00:46:13,630 --> 00:46:19,730 R cancels R, I'm integrating d theta around from zero to 2pi. 699 00:46:19,730 --> 00:46:20,620 The answer is 2pi. 700 00:46:20,620 --> 00:46:24,160 701 00:46:24,160 --> 00:46:27,530 So what do I learn from that? 702 00:46:27,530 --> 00:46:35,700 I learn that somehow this source in the inside has strength 2pi. 703 00:46:35,700 --> 00:46:42,120 What's sitting in there is 2pi times a delta function. 704 00:46:42,120 --> 00:46:47,910 This is the solution to Laplace's equation 705 00:46:47,910 --> 00:46:50,770 except at that source term, so I really 706 00:46:50,770 --> 00:46:53,710 should say Poisson's equation. 707 00:46:53,710 --> 00:46:56,250 This has turned out to be the solution 708 00:46:56,250 --> 00:47:03,580 to Poisson with a delta, or with 2pi times a delta. 709 00:47:03,580 --> 00:47:08,990 We have just solved this important equation. 710 00:47:08,990 --> 00:47:12,690 Poisson's equation with a point source. 711 00:47:12,690 --> 00:47:16,160 And, of course, that's important because when 712 00:47:16,160 --> 00:47:17,900 you can solve with a point source, 713 00:47:17,900 --> 00:47:22,650 you can put together all sorts of sources. 714 00:47:22,650 --> 00:47:24,790 And this is called the Green's function. 715 00:47:24,790 --> 00:47:26,800 The Green's function is the solution 716 00:47:26,800 --> 00:47:29,540 when the source is a delta. 717 00:47:29,540 --> 00:47:33,740 So if I divide by 2pi, now I've got it. 718 00:47:33,740 --> 00:47:37,920 I divide this by 2pi and there is the Green's function. 719 00:47:37,920 --> 00:47:43,070 I have to put that in bold letters. 720 00:47:43,070 --> 00:47:48,220 Green's function. 721 00:47:48,220 --> 00:47:52,130 It's the solution to the equation when 722 00:47:52,130 --> 00:47:54,530 the source is a delta and the answer 723 00:47:54,530 --> 00:47:59,870 is u is the log of r over 2pi. 724 00:47:59,870 --> 00:48:06,460 So that's the Green's function in 2-D. Physicists, 725 00:48:06,460 --> 00:48:10,320 you know, they live and die with these Green's function. 726 00:48:10,320 --> 00:48:12,510 Live, let's say, with Green's function. 727 00:48:12,510 --> 00:48:18,700 And they would want to know the Green's function in 3-D. 728 00:48:18,700 --> 00:48:21,590 So the Green's function in three dimensions 729 00:48:21,590 --> 00:48:23,400 also turns out beautifully. 730 00:48:23,400 --> 00:48:28,110 This is in, they would say, in free space. 731 00:48:28,110 --> 00:48:32,170 This is the Green's function when there's no other charges. 732 00:48:32,170 --> 00:48:36,940 Nothing is happening, except for the charge right at the center. 733 00:48:36,940 --> 00:48:44,510 And if I'm in two dimensions the Green's function is this log r. 734 00:48:44,510 --> 00:48:49,360 So it grows more slowly. 735 00:48:49,360 --> 00:48:51,550 It behaves like log r. 736 00:48:51,550 --> 00:48:56,670 And in 3-D I think the answer is 1/(4pi*r). 737 00:48:56,670 --> 00:49:02,260 It's just amazing that those Green's functions, 738 00:49:02,260 --> 00:49:09,730 when the right side is a delta, have such nice formulas. 739 00:49:09,730 --> 00:49:16,280 OK, let me take one moment here. 740 00:49:16,280 --> 00:49:22,080 I'll tell you what conformal mapping is about. 741 00:49:22,080 --> 00:49:26,300 But what's your take-home from this lecture? 742 00:49:26,300 --> 00:49:32,530 Your take-home is two methods that we can really 743 00:49:32,530 --> 00:49:35,530 use to get a formula for the answer. 744 00:49:35,530 --> 00:49:43,400 One method was for Laplace's equation in a circle. 745 00:49:43,400 --> 00:49:47,840 Get the boundary conditions in a series of sines and cosines, 746 00:49:47,840 --> 00:49:52,420 and then just put in the r's that we need. 747 00:49:52,420 --> 00:49:56,000 That's a simple, simple method. 748 00:49:56,000 --> 00:50:00,520 Provided we can get started with the Fourier series. 749 00:50:00,520 --> 00:50:05,480 The second method is, look at functions of x+iy, 750 00:50:05,480 --> 00:50:09,270 and try to pick one that matches your problem. 751 00:50:09,270 --> 00:50:14,830 And if your problem has a point source, at the origin, 752 00:50:14,830 --> 00:50:17,300 we found the one. 753 00:50:17,300 --> 00:50:20,350 So the literature for hundreds of years 754 00:50:20,350 --> 00:50:24,730 is aimed at solving other problems. 755 00:50:24,730 --> 00:50:27,350 If the point source is somewhere else, what happens? 756 00:50:27,350 --> 00:50:28,980 That's not hard. 757 00:50:28,980 --> 00:50:32,440 If it's not a point source but some other kind of source, 758 00:50:32,440 --> 00:50:37,770 or if the region is not a circle. 759 00:50:37,770 --> 00:50:42,110 Can I say in one final sentence just what to do, 760 00:50:42,110 --> 00:50:46,820 this conformal mapping idea, when the region is not 761 00:50:46,820 --> 00:50:51,010 a circle. 762 00:50:51,010 --> 00:50:54,380 Well, I can say it in one word, make it a circle. 763 00:50:54,380 --> 00:50:57,860 I mean, that's what Riemann said, you could do it. 764 00:50:57,860 --> 00:51:02,410 You could think of a function, so Riemann said that 765 00:51:02,410 --> 00:51:05,820 there's always some function of x+iy, 766 00:51:05,820 --> 00:51:10,210 let me call this Riemann's function capital F of x, y. 767 00:51:10,210 --> 00:51:13,770 So this is now the idea of conformal mapping. 768 00:51:13,770 --> 00:51:16,700 Change variables. 769 00:51:16,700 --> 00:51:19,640 Conformal mapping is a change of variables. 770 00:51:19,640 --> 00:51:24,530 He picked some function and let its real part be X and let 771 00:51:24,530 --> 00:51:29,670 its imaginary part be Y. Capital Y. OK, 772 00:51:29,670 --> 00:51:33,110 this is totally ridiculous to put conformal mapping 773 00:51:33,110 --> 00:51:34,970 in 30 seconds. 774 00:51:34,970 --> 00:51:42,070 But, never mind, let's just do it. 775 00:51:42,070 --> 00:51:44,490 The book describes conformal mappings 776 00:51:44,490 --> 00:51:49,360 and classical applied math courses do much more 777 00:51:49,360 --> 00:51:51,290 with conformal mapping. 778 00:51:51,290 --> 00:51:54,180 But the truth is, computationally 779 00:51:54,180 --> 00:51:59,130 they're not anything like as much used as these. 780 00:51:59,130 --> 00:52:00,460 So what's the idea? 781 00:52:00,460 --> 00:52:05,400 The idea is to find a neat function of x+iy, 782 00:52:05,400 --> 00:52:11,620 so that your crazy boundary becomes a circle. 783 00:52:11,620 --> 00:52:15,110 In the capital X, capital Y variables. 784 00:52:15,110 --> 00:52:19,030 So you're mapping the region, ellipse, 785 00:52:19,030 --> 00:52:23,310 whatever it looks like, by changing from little x, 786 00:52:23,310 --> 00:52:27,540 little y, where it was an ellipse, to capital X, capital 787 00:52:27,540 --> 00:52:29,350 Y, where it's a circle. 788 00:52:29,350 --> 00:52:30,960 And the point is Laplace's equation 789 00:52:30,960 --> 00:52:33,740 stays Laplace's equation. 790 00:52:33,740 --> 00:52:37,490 That change of variables does not mess up Laplace's equation. 791 00:52:37,490 --> 00:52:40,710 So that then you've got it in a circle. 792 00:52:40,710 --> 00:52:44,010 You solve it in a circle, for these guys. 793 00:52:44,010 --> 00:52:46,060 And then you go back. 794 00:52:46,060 --> 00:52:51,210 In a word, you're able to solve Laplace's equation in this 795 00:52:51,210 --> 00:52:57,880 crazy region because you never leave the magic x+iy. 796 00:52:57,880 --> 00:53:02,360 You find a combination with that magic x+iy that makes 797 00:53:02,360 --> 00:53:04,310 your region into a circle. 798 00:53:04,310 --> 00:53:10,710 In the circle we now know how to use capital X plus i capital 799 00:53:10,710 --> 00:53:15,110 Y. You're staying with that magic combination 800 00:53:15,110 --> 00:53:18,090 and getting the region to be what you like. 801 00:53:18,090 --> 00:53:21,540 So people know a lot of these conformal mappings. 802 00:53:21,540 --> 00:53:24,810 A famous one is the Joukowski one, 803 00:53:24,810 --> 00:53:31,160 that takes something that looks very like an airfoil, 804 00:53:31,160 --> 00:53:33,910 and you can get a circle out of it. 805 00:53:33,910 --> 00:53:37,890 So I'll put down Joukowski's name. 806 00:53:37,890 --> 00:53:48,950 So that's one that I trust Course 16 still finds valuable. 807 00:53:48,950 --> 00:53:55,780 It's a transformation that takes certain shapes 808 00:53:55,780 --> 00:53:58,390 and they include shapes that look like airfoils, 809 00:53:58,390 --> 00:54:00,240 and produce circles. 810 00:54:00,240 --> 00:54:07,740 OK, so sorry about such a quick presentation of such a 811 00:54:07,740 --> 00:54:10,080 basic subject. 812 00:54:10,080 --> 00:54:15,120 Conformal mapping, not on any exam, that'd be impossible. 813 00:54:15,120 --> 00:54:19,250 It's really this stuff that you're number one responsible 814 00:54:19,250 --> 00:54:20,293 for. 815 00:54:20,293 --> 00:54:20,793