WEBVTT 00:00:00.030 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.742 Commons license. 00:00:03.742 --> 00:00:05.450 Your support will help MIT OpenCourseWare 00:00:05.450 --> 00:00:10.050 continue to offer high-quality educational resources for free. 00:00:10.050 --> 00:00:12.530 To make a donation, or to view additional materials 00:00:12.530 --> 00:00:15.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.210 --> 00:00:21.370 at ocw.mit.edu. 00:00:21.370 --> 00:00:24.550 PROFESSOR STRANG: OK, it's Laplace again today. 00:00:24.550 --> 00:00:26.020 Laplace's equation. 00:00:26.020 --> 00:00:31.710 And trying to describe-- That's a big area that's 00:00:31.710 --> 00:00:35.370 a lot of people have worked on for centuries. 00:00:35.370 --> 00:00:38.900 And for the early centuries, there 00:00:38.900 --> 00:00:42.070 were always analysis methods. 00:00:42.070 --> 00:00:45.460 And that's what we got started on last time. 00:00:45.460 --> 00:00:47.490 And we'll do a bit more. 00:00:47.490 --> 00:00:49.470 There's no way we could do everything 00:00:49.470 --> 00:00:53.030 that people have worked on years and years, 00:00:53.030 --> 00:00:55.830 trying to find ideas about solving. 00:00:55.830 --> 00:00:58.970 But we can get the idea. 00:00:58.970 --> 00:01:01.390 And this part, then, is in the section 00:01:01.390 --> 00:01:03.210 called Laplace's equation. 00:01:03.210 --> 00:01:08.050 And the exam Wednesday would include 00:01:08.050 --> 00:01:12.070 some of these constructions. 00:01:12.070 --> 00:01:14.070 So this is what we did last time, we 00:01:14.070 --> 00:01:19.440 identified a whole family of solutions to Laplace's equation 00:01:19.440 --> 00:01:23.590 as polynomials in x and y. 00:01:23.590 --> 00:01:28.120 Of increasing degree n, and then when we wrote them in polar 00:01:28.120 --> 00:01:31.880 form they were fantastic. r^n*cos(n*theta) 00:01:31.880 --> 00:01:34.360 and r^n*sin(n*theta). 00:01:34.360 --> 00:01:39.350 So my idea is just, we've got them, now let's use them. 00:01:39.350 --> 00:01:44.900 So how to use these solutions? 00:01:44.900 --> 00:01:48.380 Because we can take combinations of them, 00:01:48.380 --> 00:01:51.590 we can create series of sines and cosines. 00:01:51.590 --> 00:01:53.750 So I'll do that first. 00:01:53.750 --> 00:01:54.340 Series. 00:01:54.340 --> 00:01:58.660 And then Green's function, that's the name you remember, 00:01:58.660 --> 00:02:00.240 we've seen it before. 00:02:00.240 --> 00:02:04.180 That's the solution when the right side is a delta function. 00:02:04.180 --> 00:02:08.600 When we have Poisson's equation with a delta there. 00:02:08.600 --> 00:02:10.930 So that an important one. 00:02:10.930 --> 00:02:17.790 And then well, a big part of two-dimensional and 00:02:17.790 --> 00:02:20.930 three-dimensional problems is that the region itself, 00:02:20.930 --> 00:02:23.600 not just the equation but the region itself, 00:02:23.600 --> 00:02:26.160 can be all over the place. 00:02:26.160 --> 00:02:29.660 We'll solve when the region is nice, 00:02:29.660 --> 00:02:33.470 like a circle or a square, and then there 00:02:33.470 --> 00:02:39.360 is a way to, in principle, to get other regions. 00:02:39.360 --> 00:02:43.630 To change from a crazy region to a circle or a square, 00:02:43.630 --> 00:02:45.470 and then solve it there. 00:02:45.470 --> 00:02:48.240 So that's called conformal mapping. 00:02:48.240 --> 00:02:52.410 And I can't let the whole course go without saying 00:02:52.410 --> 00:02:53.920 a word or two about that. 00:02:53.920 --> 00:03:00.680 But somehow among numerical methods, 00:03:00.680 --> 00:03:05.880 it's conformal mapping-- There are packages 00:03:05.880 --> 00:03:07.400 that do conformal mapping. 00:03:07.400 --> 00:03:12.020 But they're not the central way to solve 00:03:12.020 --> 00:03:13.720 these equations numerically. 00:03:13.720 --> 00:03:16.630 Finite differences, finite elements are. 00:03:16.630 --> 00:03:19.430 And that's what's coming next week. 00:03:19.430 --> 00:03:27.010 So this is the future; this is the present, right there. 00:03:27.010 --> 00:03:31.270 So, can I just start with an example or two? 00:03:31.270 --> 00:03:36.680 Like, how would you solve Laplace's equation in a circle? 00:03:36.680 --> 00:03:39.640 So, in a circle. 00:03:39.640 --> 00:03:42.370 This is the idea here. 00:03:42.370 --> 00:03:44.390 I have Laplace's equation. 00:03:44.390 --> 00:03:47.620 OK, so I've got a whole lot of solutions. 00:03:47.620 --> 00:03:51.300 And I've even got chalk to write them down. 00:03:51.300 --> 00:03:56.050 OK, so here's my circle, might as well make it 00:03:56.050 --> 00:03:59.550 the unit circle. 00:03:59.550 --> 00:04:01.390 Radius one. 00:04:01.390 --> 00:04:04.530 And inside here is Laplace. 00:04:07.340 --> 00:04:09.630 u_xx+u_yy=0. 00:04:09.630 --> 00:04:12.140 No sources inside. 00:04:12.140 --> 00:04:14.190 So we have to have sources from somewhere, 00:04:14.190 --> 00:04:16.640 and they will come from the boundary. 00:04:16.640 --> 00:04:20.590 So on the boundary, we keep-- Let 00:04:20.590 --> 00:04:23.490 me think of u as temperature. 00:04:23.490 --> 00:04:26.820 So I set the temperature on the boundary. u 00:04:26.820 --> 00:04:30.620 equals some u_0, some known function. 00:04:30.620 --> 00:04:31.980 This is given. 00:04:31.980 --> 00:04:33.680 This is the boundary condition. 00:04:33.680 --> 00:04:36.780 This is the given boundary condition. 00:04:36.780 --> 00:04:41.950 And it's a function of-- I'm going to use polar coordinates. 00:04:41.950 --> 00:04:44.690 Polar coordinates are natural for a circle. 00:04:44.690 --> 00:04:52.830 So this is at r=1, so maybe I should say on the boundary, 00:04:52.830 --> 00:04:59.490 which is r=1, and going around the angle theta, is given. 00:04:59.490 --> 00:05:02.270 This is u_0(theta). 00:05:02.270 --> 00:05:08.090 That's my given boundary condition. 00:05:08.090 --> 00:05:10.850 This problem is named after Dirichlet, 00:05:10.850 --> 00:05:15.800 because it's like giving fixed conditions and not 00:05:15.800 --> 00:05:17.720 Neumann conditions. 00:05:17.720 --> 00:05:21.800 OK, so I'm just looking for a combination. 00:05:21.800 --> 00:05:24.440 For a function that solves Laplace's equation 00:05:24.440 --> 00:05:29.050 inside the circle, and takes on some values 00:05:29.050 --> 00:05:30.790 around the boundary. 00:05:30.790 --> 00:05:34.710 And of course the boundary values 00:05:34.710 --> 00:05:39.030 might be plus one on the top and minus one on the bottom. 00:05:39.030 --> 00:05:43.180 Or the boundary condition might vary around, 00:05:43.180 --> 00:05:46.390 it might, variable then come around. 00:05:46.390 --> 00:05:50.970 But notice that this is a periodic function. 00:05:50.970 --> 00:05:57.170 This is 2pi-periodic, because the problem's the same. 00:05:57.170 --> 00:06:01.020 If I increase theta by 2pi I've come back to the same point. 00:06:01.020 --> 00:06:04.030 So it's got to have that value. 00:06:04.030 --> 00:06:12.180 OK, so well let me give you a couple of examples first. 00:06:12.180 --> 00:06:18.120 Suppose u_0, suppose this function-- Example 1, easy. 00:06:18.120 --> 00:06:21.580 Suppose u_0 is sin(3theta). 00:06:25.890 --> 00:06:30.030 So that means I've got a region here, 00:06:30.030 --> 00:06:33.300 I'm prescribing its temperature on the boundary. 00:06:33.300 --> 00:06:36.370 And I want to say what does it look like inside? 00:06:36.370 --> 00:06:40.090 And I'm prescribing right now the sin(3theta), 00:06:40.090 --> 00:06:45.310 so there theta is zero, so it's zero, the boundary condition's 00:06:45.310 --> 00:06:50.610 zero there, climbs to one, back to zero, down to minus one. 00:06:50.610 --> 00:06:51.940 back to zero. 00:06:51.940 --> 00:06:57.170 Three times, and again comes back to zero again there. 00:06:57.170 --> 00:07:02.550 So, I'm looking for a solution to Laplace's equation -- 00:07:02.550 --> 00:07:05.110 and I've got a pretty good list -- 00:07:05.110 --> 00:07:08.990 that will match u_0 when r is one. 00:07:08.990 --> 00:07:11.440 So that's the boundary, r is one. 00:07:11.440 --> 00:07:13.690 So you can tell me what it is. 00:07:13.690 --> 00:07:15.450 So you can solve this problem, right away. 00:07:15.450 --> 00:07:26.550 The answer is u of r and theta is, what function will--? 00:07:26.550 --> 00:07:29.750 Remember I've got my eye on that list. 00:07:29.750 --> 00:07:31.610 You too, right? 00:07:31.610 --> 00:07:34.680 I'm just trying to get one that when r is one 00:07:34.680 --> 00:07:37.190 it will match sin(3theta). 00:07:37.190 --> 00:07:40.570 What's the good guy? 00:07:40.570 --> 00:07:43.760 It'll be on that list. 00:07:43.760 --> 00:07:46.120 Which of those, by itself, here I 00:07:46.120 --> 00:07:53.480 don't need a series because I've got such a neat u_0 function. 00:07:53.480 --> 00:07:57.250 I'll get it right with one answer, 00:07:57.250 --> 00:08:01.090 and what is that answer? 00:08:01.090 --> 00:08:02.200 I look there. 00:08:02.200 --> 00:08:09.870 I say what do I do, so that at r=1, I'll match sin(3theta). 00:08:09.870 --> 00:08:12.740 I'll use r cubed sin(3theta). 00:08:12.740 --> 00:08:15.210 So the good winner will be r cubed sin(3theta). 00:08:17.750 --> 00:08:19.670 That solves Laplace's equation. 00:08:19.670 --> 00:08:24.850 We checked it out, it's the imaginary part of x+iy cubed. 00:08:24.850 --> 00:08:28.160 We could write it in x and y coordinates if we 00:08:28.160 --> 00:08:31.180 wanted but we don't want to. 00:08:31.180 --> 00:08:36.750 And it matches when r is one, it gives us sin(3theta). 00:08:36.750 --> 00:08:42.560 That's it. 00:08:42.560 --> 00:08:46.210 And of course I could take any one. 00:08:46.210 --> 00:08:48.740 Now suppose I'm trying to match something that's 00:08:48.740 --> 00:08:50.510 not as simple as sin(3theta). 00:08:53.920 --> 00:08:57.740 In that case, I may have to use all of them. 00:08:57.740 --> 00:09:01.730 I mean, it's very, very fluky that one term 00:09:01.730 --> 00:09:02.920 is going to do it. 00:09:02.920 --> 00:09:06.430 Usually, so my main examples would be 00:09:06.430 --> 00:09:08.090 I'll have to match all of them. 00:09:08.090 --> 00:09:09.770 So what do I do? 00:09:09.770 --> 00:09:19.360 At r=1, so my general solution is a combination of these guys 00:09:19.360 --> 00:09:21.270 I worked so hard to get. 00:09:21.270 --> 00:09:23.970 The solution is of this form. 00:09:23.970 --> 00:09:27.310 It's some a_0, the constant. 00:09:27.310 --> 00:09:33.080 And then a_1*r*cos(theta), and b_1*r*sin(theta). 00:09:36.520 --> 00:09:42.540 And a_2 r squared cos(2theta). 00:09:42.540 --> 00:09:45.160 And so on. 00:09:45.160 --> 00:09:48.120 I'm just taking any combination of, 00:09:48.120 --> 00:09:51.970 I'm using the a's as the coefficients 00:09:51.970 --> 00:09:54.670 for the cosine guys. 00:09:54.670 --> 00:09:59.510 And the b's, b_1, b_2, b_3 would be the coefficients 00:09:59.510 --> 00:10:01.510 for the sine one. 00:10:01.510 --> 00:10:04.020 OK, that's my general solution. 00:10:04.020 --> 00:10:06.130 That solves Laplace's equation. 00:10:06.130 --> 00:10:09.500 Every term did, so every combination will. 00:10:09.500 --> 00:10:12.520 Now, set r=1. 00:10:12.520 --> 00:10:18.050 To match that r=1-- and match the boundary. 00:10:18.050 --> 00:10:24.000 And match u_0(theta), the required temperature around 00:10:24.000 --> 00:10:26.850 the-- on the boundary. 00:10:26.850 --> 00:10:30.230 The boundary being where r is one. 00:10:30.230 --> 00:10:32.010 So this is set r=1. 00:10:32.010 --> 00:10:40.570 So then u_0(theta), this given thing, has to match this, 00:10:40.570 --> 00:10:41.930 when r is one. 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, 00:10:51.200 --> 00:10:52.210 so it's just cos(theta). 00:10:54.790 --> 00:11:00.540 Now, b_1, r is one, so I just have sin(theta). 00:11:00.540 --> 00:11:06.690 And I have an a_2*cos(2theta), and a b_2*sin(2theta), 00:11:06.690 --> 00:11:12.890 and so on. 00:11:12.890 --> 00:11:16.330 Here, just let me put it together now. 00:11:16.330 --> 00:11:21.630 I'm given any temperature distribution 00:11:21.630 --> 00:11:24.600 around the boundary. 00:11:24.600 --> 00:11:27.900 It's in equilibrium, the temperature, 00:11:27.900 --> 00:11:34.050 where if the temperature's high near that point 00:11:34.050 --> 00:11:37.870 and low over here the temperature 00:11:37.870 --> 00:11:41.120 inside will gradually go from that high point, 00:11:41.120 --> 00:11:45.120 dot dot dot dot, to the lower one. 00:11:45.120 --> 00:11:47.170 By matching on the boundary. 00:11:47.170 --> 00:11:49.750 And this is the match on the boundary. 00:11:49.750 --> 00:12:02.410 Now, this is really a lead in to the last part of this course. 00:12:02.410 --> 00:12:07.120 So whose name is associated with a series like that? 00:12:07.120 --> 00:12:08.450 Fourier. 00:12:08.450 --> 00:12:12.760 You recognize that as what's called a Fourier series. 00:12:12.760 --> 00:12:17.240 So the idea is, I'm given these boundary values. 00:12:17.240 --> 00:12:22.070 I find their expansion in sines and cosines, 00:12:22.070 --> 00:12:25.570 and that's what we'll do in November. 00:12:25.570 --> 00:12:29.140 And then I've got it. 00:12:29.140 --> 00:12:32.150 Then I know the a's and the b's. 00:12:32.150 --> 00:12:37.340 And then basically I just put in the r's. r and r squareds and r 00:12:37.340 --> 00:12:38.690 cubeds and so on. 00:12:38.690 --> 00:12:43.950 So then I've got the answer inside. 00:12:43.950 --> 00:12:48.510 In principle it's so easy. 00:12:48.510 --> 00:12:51.410 So, why is it easy, though? 00:12:51.410 --> 00:12:54.600 First, it's easy because it's a circle we're working in. 00:12:54.600 --> 00:12:59.750 If I was in an ellipse or a strange shape, forget it. 00:12:59.750 --> 00:13:03.270 I mean, so this is quite special. 00:13:03.270 --> 00:13:09.750 And secondly, it's easy because these functions are so nice. 00:13:09.750 --> 00:13:14.080 Fourier works with the best functions ever. 00:13:14.080 --> 00:13:16.000 These sines and cosines. 00:13:16.000 --> 00:13:20.310 So I'll find a way to find those coefficients, 00:13:20.310 --> 00:13:24.460 the a's and the b's. 00:13:24.460 --> 00:13:26.710 Even though there are lots of them, 00:13:26.710 --> 00:13:29.580 I'll be able to pick them off one at the time, 00:13:29.580 --> 00:13:31.540 the a's and b's. 00:13:31.540 --> 00:13:35.610 Once I know the a's and b's, I know the answer. 00:13:35.610 --> 00:13:38.860 So do you see this is in principle a great way 00:13:38.860 --> 00:13:39.880 to solve it? 00:13:39.880 --> 00:13:42.660 In fact, it's the way we used over here, 00:13:42.660 --> 00:13:47.440 when my u_0 was sin(3theta), then the only term in its 00:13:47.440 --> 00:13:50.710 Fourier series was 1*sin(3theta). 00:13:50.710 --> 00:13:55.910 And then the solution was 1 r cubed sin(3theta). 00:13:55.910 --> 00:13:59.370 So you can learn things from this. 00:13:59.370 --> 00:14:03.280 For example, oh, what can you learn? 00:14:03.280 --> 00:14:08.830 One thing I noticed, an important feature 00:14:08.830 --> 00:14:15.800 of Laplace's equation is that this solution inside the circle 00:14:15.800 --> 00:14:22.110 gets very smooth. 00:14:22.110 --> 00:14:25.700 The boundary conditions could be like a delta function. 00:14:25.700 --> 00:14:28.530 I could say that on the boundary, 00:14:28.530 --> 00:14:31.440 the temperature is zero everywhere except at that 00:14:31.440 --> 00:14:35.110 point it spikes. 00:14:35.110 --> 00:14:36.630 So I could take u_0-- 00:14:36.630 --> 00:14:41.050 So example 2, and I won't do it in full, 00:14:41.050 --> 00:14:46.670 would be u_0 on the boundary equal a delta function. 00:14:46.670 --> 00:14:49.510 A spike at that one point. 00:14:49.510 --> 00:14:55.150 So all the heat is coming from the source at that one point. 00:14:55.150 --> 00:14:59.330 Like I've got a fire going there. 00:14:59.330 --> 00:15:03.750 Keeping the rest of the boundary frozen, 00:15:03.750 --> 00:15:07.630 the heat's kind of going to come inside. 00:15:07.630 --> 00:15:09.750 So then how would I proceed? 00:15:09.750 --> 00:15:16.660 Well, if I have this boundary value as a delta function, 00:15:16.660 --> 00:15:20.780 I look for its Fourier series, and it's 00:15:20.780 --> 00:15:22.820 a very important, beautiful, Fourier 00:15:22.820 --> 00:15:25.350 series for a delta function. 00:15:25.350 --> 00:15:27.940 Would you want to know it? 00:15:27.940 --> 00:15:30.690 I mean, we'll know it well in November. 00:15:30.690 --> 00:15:32.420 Would you want to know it in October? 00:15:32.420 --> 00:15:37.980 This is Halloween, I guess, so delta-- I'll 00:15:37.980 --> 00:15:39.330 tell you what it is. 00:15:39.330 --> 00:15:43.060 Since you insist. 00:15:43.060 --> 00:15:50.170 delta(theta) I think will-- I think there's a 1/(2pi) 00:15:50.170 --> 00:15:53.980 or something. 00:15:53.980 --> 00:15:55.010 Ah, shoot. 00:15:55.010 --> 00:15:57.470 We'll get it exactly right. 00:15:57.470 --> 00:16:06.090 It's something like 1 and 2 cos(theta) and 2cos(2theta), 00:16:06.090 --> 00:16:08.270 I'm not sure about the 2pi. 00:16:08.270 --> 00:16:14.740 2cos(2theta), and 2cos(3theta), and so on. 00:16:14.740 --> 00:16:18.340 We'll know it well when we get there. 00:16:18.340 --> 00:16:22.584 What I notice about this delta function-- Of course you're 00:16:22.584 --> 00:16:24.250 going to expect the delta function being 00:16:24.250 --> 00:16:26.840 somehow a little bit strange. 00:16:26.840 --> 00:16:32.190 At theta=0, what does that series add up to? 00:16:32.190 --> 00:16:35.160 Just so you begin to get a hang of Fourier series. 00:16:35.160 --> 00:16:40.860 At theta=0, what does that series look like? 00:16:40.860 --> 00:16:44.510 Well, all these cosine thetas are? 00:16:44.510 --> 00:16:45.350 One. 00:16:45.350 --> 00:16:50.460 So this series at theta=0 is 1+2+2+2+2... 00:16:50.460 --> 00:16:51.900 It's infinite. 00:16:51.900 --> 00:16:54.860 And that's what we want. 00:16:54.860 --> 00:16:59.080 The delta function is infinite at theta=0. 00:16:59.080 --> 00:17:02.330 And it's periodic, of course, so that if I go around 00:17:02.330 --> 00:17:07.060 to theta=2pi I'll come back to zero again. 00:17:07.060 --> 00:17:12.180 At theta=pi, you could sort of see, well, yeah, 00:17:12.180 --> 00:17:14.770 theta=pi is a sort of interesting point. 00:17:14.770 --> 00:17:18.000 At theta is pi, what's the cosine? 00:17:18.000 --> 00:17:20.040 Is negative one, right? 00:17:20.040 --> 00:17:23.530 But then the cos(2pi) will be plus one. 00:17:23.530 --> 00:17:28.540 So at theta=pi, I think I'm getting a one minus a two plus 00:17:28.540 --> 00:17:31.160 a two, minus a two, plus a two. 00:17:31.160 --> 00:17:35.650 You see, it's doing its best to cancel itself out and give me 00:17:35.650 --> 00:17:41.260 the zero that I want, the theta=pi over on the left side 00:17:41.260 --> 00:17:42.450 of the circle. 00:17:42.450 --> 00:17:49.150 Anyway, so that's an extreme example. 00:17:49.150 --> 00:17:52.150 But now, what's the temperature inside? 00:17:52.150 --> 00:17:54.540 Can you just follow the same rule? 00:17:54.540 --> 00:17:56.380 What will be the temperature inside? 00:17:56.380 --> 00:18:00.610 If that's the delta function, if that's 00:18:00.610 --> 00:18:04.510 the right series, whatever, it might be a 4pi, 00:18:04.510 --> 00:18:10.210 I'm not sure, for that. 00:18:10.210 --> 00:18:13.420 Now, you can tell me what's the solution, what's 00:18:13.420 --> 00:18:16.720 the temperature distribution inside a circle when 00:18:16.720 --> 00:18:26.040 one point on the boundary has a heat source, a delta function. 00:18:26.040 --> 00:18:27.310 What do I do? 00:18:27.310 --> 00:18:30.420 How do I match this with this guy? 00:18:30.420 --> 00:18:33.060 I just put in the r's, right? 00:18:33.060 --> 00:18:36.860 If this is what it's supposed to match when r is one, 00:18:36.860 --> 00:18:44.840 then when r is-- So maybe I'll put it under here. 00:18:44.840 --> 00:18:53.710 So the u(r,theta), from the delta guy, 00:18:53.710 --> 00:18:57.800 is just put in the r's. 00:18:57.800 --> 00:19:06.960 1+2r*cos(theta), and 2 r squared cos(2theta), and so on. 00:19:06.960 --> 00:19:17.230 OK, and eventually 2 r to the 100th cos(100theta), and more. 00:19:17.230 --> 00:19:24.970 OK, I write this out, you could say why did he write this down? 00:19:24.970 --> 00:19:29.510 I wanted to make this point that the important feature 00:19:29.510 --> 00:19:32.430 of the solution to Laplace's equation 00:19:32.430 --> 00:19:37.290 is how smooth it gets when you go inside the region. 00:19:37.290 --> 00:19:38.670 And why is that? 00:19:38.670 --> 00:19:45.970 Because at r=1/2, this term is practically gone, right? 00:19:45.970 --> 00:19:50.930 If I go halfway into the circle, this term is practically gone. 00:19:50.930 --> 00:19:53.040 1/2 to the hundredth power. 00:19:53.040 --> 00:19:56.090 And if I go to the center of the circle, it's completely gone. 00:19:56.090 --> 00:19:59.110 In fact, what's the value at the center of the circle? 00:19:59.110 --> 00:20:07.580 What's the temperature at the center? 00:20:07.580 --> 00:20:08.600 1/2pi. 00:20:08.600 --> 00:20:11.470 This is the only term that's remaining. 00:20:11.470 --> 00:20:19.940 And it's the average, around the circle. 00:20:19.940 --> 00:20:23.830 That makes physical sense, I guess. 00:20:23.830 --> 00:20:27.030 Since the whole thing's completely isotropic, 00:20:27.030 --> 00:20:35.330 we've got a perfect circle, the value 00:20:35.330 --> 00:20:37.690 at the center of the circle is always 00:20:37.690 --> 00:20:39.750 the average going around. 00:20:39.750 --> 00:20:43.880 The constant term in the Fourier series, this guy. 00:20:43.880 --> 00:20:46.790 We'll get to know that one very well. 00:20:46.790 --> 00:20:50.480 That's the average. 00:20:50.480 --> 00:20:55.100 You're just seeing a little bit of Fourier series early, here. 00:20:55.100 --> 00:21:02.230 But my point is that you could have high oscillation 00:21:02.230 --> 00:21:04.770 around the boundary, that damps out 00:21:04.770 --> 00:21:07.480 because of these powers of r. 00:21:07.480 --> 00:21:13.300 And inside the circle it's only the low order terms 00:21:13.300 --> 00:21:22.090 that begin to take over. 00:21:22.090 --> 00:21:24.360 This is the kind of trick you have, or not 00:21:24.360 --> 00:21:27.660 trick but the kind of method that you 00:21:27.660 --> 00:21:34.750 can use for solving Laplace's equation by an infinite series. 00:21:34.750 --> 00:21:38.220 Of course, a person who wants a number 00:21:38.220 --> 00:21:41.440 can complain that, wait a minute, 00:21:41.440 --> 00:21:43.640 how do I use that infinite series? 00:21:43.640 --> 00:21:47.860 Well, of course, if you wanted to know the temperature 00:21:47.860 --> 00:21:50.120 at a particular point you'd have to plug in 00:21:50.120 --> 00:21:52.700 that value of r, that value of theta, 00:21:52.700 --> 00:21:57.770 add up the terms until you hope that they become so 00:21:57.770 --> 00:22:00.450 small that you can ignore them. 00:22:00.450 --> 00:22:03.810 So infinite series is one form of a solution. 00:22:03.810 --> 00:22:12.510 And somehow these are examples-- I should use the words 00:22:12.510 --> 00:22:14.910 separation of variables. 00:22:14.910 --> 00:22:20.830 Separation of variables is the golden idea 00:22:20.830 --> 00:22:22.800 in this analysis stuff. 00:22:22.800 --> 00:22:26.910 Separation of variables means I got the r part separated 00:22:26.910 --> 00:22:29.760 from the theta part. 00:22:29.760 --> 00:22:33.800 And that worked great, worked well for a circle. 00:22:33.800 --> 00:22:39.230 Let's see, maybe for a square I could try to separate x from y. 00:22:39.230 --> 00:22:46.800 Maybe there's a homework problem, a solution that 00:22:46.800 --> 00:22:52.130 separates x from y, I think is something like-- So this 00:22:52.130 --> 00:22:53.910 would be another family. 00:22:53.910 --> 00:23:00.810 Good for squares, something like sin(kx) sinh times, 00:23:00.810 --> 00:23:08.070 so this separation is something in x times something in y. 00:23:08.070 --> 00:23:10.670 Again I'm just mentioning things. 00:23:10.670 --> 00:23:14.390 I think that that solves Laplace's equation 00:23:14.390 --> 00:23:17.710 because if I take two x derivatives, 00:23:17.710 --> 00:23:22.560 that'll bring down k squared, but it'll flip the sign, right? 00:23:22.560 --> 00:23:25.820 These two derivatives of the sine will be a minus. 00:23:25.820 --> 00:23:30.380 And if I take-- I need a ky there. 00:23:30.380 --> 00:23:35.240 And if I took two derivatives of this hyperbolic sine, 00:23:35.240 --> 00:23:40.570 you remember that's the e^(ky) and the e^(-ky). 00:23:40.570 --> 00:23:42.650 The two derivatives of that will bring out 00:23:42.650 --> 00:23:45.800 a k squared with a plus sign. 00:23:45.800 --> 00:23:49.390 So two x derivatives bring out the minus k squared, 00:23:49.390 --> 00:23:52.260 two y derivatives bring out a plus k squared 00:23:52.260 --> 00:23:56.440 and together that solves Laplace's equation. 00:23:56.440 --> 00:23:58.680 We'll check that in our homework problem. 00:23:58.680 --> 00:24:02.410 So there would be an example, good for a square. 00:24:02.410 --> 00:24:13.950 So, there's hope to do an exact solution in a special region. 00:24:13.950 --> 00:24:19.700 Now, what's this Green's function idea? 00:24:19.700 --> 00:24:26.480 OK, that's now this is another thing. 00:24:26.480 --> 00:24:35.770 So last time we appreciated that this combination x+iy was 00:24:35.770 --> 00:24:37.650 magic. 00:24:37.650 --> 00:24:42.880 The idea was that we could take any function of x+iy, 00:24:42.880 --> 00:24:47.390 and it solves Laplace's equation. 00:24:47.390 --> 00:24:52.420 Can we just see, sort of very crudely why that is? 00:24:52.420 --> 00:25:00.310 We saw the pattern, we saw x+iy to the nth. 00:25:00.310 --> 00:25:04.520 Sort of, we went as far as n=3, checked it all out. 00:25:04.520 --> 00:25:08.120 But now, really if I want to be able to-- why does that 00:25:08.120 --> 00:25:13.820 solve Laplace's equation for any n? 00:25:13.820 --> 00:25:16.970 Should I just plug that into Laplace's equation? 00:25:16.970 --> 00:25:24.740 What happens if I take the two x derivatives of this thing? 00:25:24.740 --> 00:25:28.070 So this going to be a typical function of x+iy, 00:25:28.070 --> 00:25:29.710 typically nice one. 00:25:29.710 --> 00:25:32.080 If I take two x derivatives, I want 00:25:32.080 --> 00:25:34.520 to plug it in and see that it really 00:25:34.520 --> 00:25:36.920 does solve Laplace's equation. 00:25:36.920 --> 00:25:40.730 So two x derivatives of that will give me what? 00:25:40.730 --> 00:25:44.050 The first x derivative will bring down an n 00:25:44.050 --> 00:25:47.420 times this thing to the n-1. 00:25:47.420 --> 00:25:51.890 And then the next x derivative will bring down an n-1 times 00:25:51.890 --> 00:25:55.490 this thing to the n-2. 00:25:55.490 --> 00:25:59.550 So that'll be the u_xx. 00:25:59.550 --> 00:26:05.240 And what about u_yy? 00:26:05.240 --> 00:26:08.580 This is my u. 00:26:08.580 --> 00:26:14.260 I'm sort of just checking that yes, this-- See again, 00:26:14.260 --> 00:26:17.440 see if it still works Friday what worked Wednesday. 00:26:17.440 --> 00:26:23.260 That this x+iy is magic and functions of it like powers, 00:26:23.260 --> 00:26:26.230 exponentials, logarithms, whatever, 00:26:26.230 --> 00:26:28.530 all solve Laplace's equation. 00:26:28.530 --> 00:26:33.270 OK, so we did u_xx, and we got-- easy. 00:26:33.270 --> 00:26:35.820 Now, what happens with u_yy? 00:26:35.820 --> 00:26:36.860 Do you see the point? 00:26:36.860 --> 00:26:39.060 AUDIENCE: [INAUDIBLE] 00:26:39.060 --> 00:26:41.140 PROFESSOR STRANG: Sorry opposite sign. 00:26:41.140 --> 00:26:44.710 And why does the sign come out opposite? 00:26:44.710 --> 00:26:46.260 Because of that guy. 00:26:46.260 --> 00:26:48.300 Yeah, it's the chain rule, right? 00:26:48.300 --> 00:26:50.900 The derivative of this with respect to y 00:26:50.900 --> 00:26:55.250 will give me an n times this thing to one lower power. 00:26:55.250 --> 00:26:58.430 Times the derivative of what's inside. 00:26:58.430 --> 00:27:01.470 And the derivative of what's inside is an i. 00:27:01.470 --> 00:27:03.730 And then the second derivative will bring down 00:27:03.730 --> 00:27:10.720 an n-1, this guy will be down to n-2, another i will come out 00:27:10.720 --> 00:27:14.140 and-- Just what you want, right? 00:27:14.140 --> 00:27:17.170 Because the i squared is minus one, those cancel. 00:27:17.170 --> 00:27:21.070 When those are equal opposite signs. 00:27:21.070 --> 00:27:29.490 And we get u_xx+u_yy equaling 0. 00:27:29.490 --> 00:27:31.140 So that works. 00:27:31.140 --> 00:27:34.490 And, actually, the same idea would work for any function 00:27:34.490 --> 00:27:36.000 of x+iy. 00:27:36.000 --> 00:27:42.150 The two x derivatives just give f''. 00:27:42.150 --> 00:27:46.520 Two y derivatives will give f'' but the chain rule will bring 00:27:46.520 --> 00:27:49.560 out i both times and we've got it. 00:27:49.560 --> 00:28:00.640 OK, I think we just need another couple of examples. 00:28:00.640 --> 00:28:05.000 And this of course could be in polar coordinates, 00:28:05.000 --> 00:28:08.650 f of re^(i*theta). 00:28:08.650 --> 00:28:12.220 That's just, everybody recognizes re^(i*theta) is 00:28:12.220 --> 00:28:13.760 the same as x+iy? 00:28:13.760 --> 00:28:17.610 Better just be sure we've got that. x is 00:28:17.610 --> 00:28:23.160 some point here in the complex plane. iy takes us up to here. 00:28:23.160 --> 00:28:26.080 So there's x+iy. 00:28:26.080 --> 00:28:30.090 That's x+iy there, but it's also-- 00:28:30.090 --> 00:28:32.370 So let me put those in better. 00:28:32.370 --> 00:28:37.120 So there's x and there's y. 00:28:37.120 --> 00:28:39.230 Everybody knows this picture, right? 00:28:39.230 --> 00:28:43.260 This x and this y, now if I want to go to polar coordinates, 00:28:43.260 --> 00:28:48.770 that angle is theta, this x is r*cos(theta), 00:28:48.770 --> 00:28:55.320 this y is r*sin(theta), and this guy is re^(i*theta). 00:28:55.320 --> 00:28:55.910 re^(i*theta). 00:29:00.380 --> 00:29:05.640 r*cos(theta) plus i*r*sin(theta) is the same as re^i*theta. 00:29:05.640 --> 00:29:07.830 That's utterly fundamental. 00:29:07.830 --> 00:29:12.750 Everybody's responsible for that picture 00:29:12.750 --> 00:29:18.140 of putting the complex numbers into their beautiful polar 00:29:18.140 --> 00:29:18.640 form. 00:29:18.640 --> 00:29:25.950 That's what made our r to the nth cos(n*theta) all so simple. 00:29:25.950 --> 00:29:30.230 Now, what was I aiming to do? 00:29:30.230 --> 00:29:33.160 Give a particular f. 00:29:33.160 --> 00:29:38.750 Now I want to give a particular function f, or maybe 00:29:38.750 --> 00:29:40.620 a couple of choices. 00:29:40.620 --> 00:29:44.100 A couple of functions f, and see that their real parts 00:29:44.100 --> 00:29:50.090 and their imaginary parts solve Laplace's equation. 00:29:50.090 --> 00:30:00.460 Let me take first a one that works completely. 00:30:00.460 --> 00:30:05.520 Take the real part and the imaginary part-- 00:30:05.520 --> 00:30:07.380 Let me take e^(x+iy). 00:30:10.980 --> 00:30:15.240 It's a function of x+iy, extremely nice function 00:30:15.240 --> 00:30:20.210 of x+iy, and we can figure out its real and imaginary parts, 00:30:20.210 --> 00:30:25.130 and we get two solutions to Laplace's equation. 00:30:25.130 --> 00:30:31.320 The good way is to write this thing as e^x times e^(iy). 00:30:31.320 --> 00:30:39.140 And again we'll write it as e^x times cos(y)+i*sin(y). 00:30:39.140 --> 00:30:42.750 So now I can see that the real part-- 00:30:42.750 --> 00:30:45.100 I can see what the real part is, and I can 00:30:45.100 --> 00:30:46.830 see what the imaginary part is. 00:30:46.830 --> 00:30:49.060 The real part will be, that's real. 00:30:49.060 --> 00:30:54.930 And that's real. so this will so give me e^x*cos(y). 00:30:54.930 --> 00:30:59.120 And the imaginary part will be e^x*sin(y). 00:30:59.120 --> 00:31:03.080 You see it. 00:31:03.080 --> 00:31:08.080 And those will solve Laplace's equation. 00:31:08.080 --> 00:31:13.970 Can I give a name to this whole field of analysis? 00:31:13.970 --> 00:31:20.430 This e^z is an analytic -- I should just use that word -- 00:31:20.430 --> 00:31:25.110 an analytic function. 00:31:25.110 --> 00:31:27.830 And these guys, the real and imaginary parts, 00:31:27.830 --> 00:31:34.700 are two harmonic functions. 00:31:34.700 --> 00:31:37.550 Maybe it's not so important to know the word harmonic 00:31:37.550 --> 00:31:38.360 function. 00:31:38.360 --> 00:31:40.670 But analytic function, yeah, I would 00:31:40.670 --> 00:31:44.860 say that's an important word. 00:31:44.860 --> 00:31:48.070 Actually, what does it mean? 00:31:48.070 --> 00:31:52.480 It's a function of z. 00:31:52.480 --> 00:31:57.900 So we're in the complex plane here now. 00:31:57.900 --> 00:32:04.570 It's a function of z, e^z, and it can be written as a power 00:32:04.570 --> 00:32:09.930 series, of course, one plus z plus 1 over 2 factorial z 00:32:09.930 --> 00:32:13.210 squared and all those guys. 00:32:13.210 --> 00:32:15.080 So it has a power series. 00:32:15.080 --> 00:32:19.450 That makes it a combination of our special ones. 00:32:19.450 --> 00:32:25.700 The great thing about that series is it converges. 00:32:25.700 --> 00:32:29.470 So an analytic function, an analytic function 00:32:29.470 --> 00:32:34.290 is the sum of a power series that converges. 00:32:34.290 --> 00:32:35.540 And this one does. 00:32:35.540 --> 00:32:37.340 So there's an example. 00:32:37.340 --> 00:32:40.530 Yeah, so the whole theory of analytic functions 00:32:40.530 --> 00:32:44.710 is actually, that's Chapter 5 of the textbook. 00:32:44.710 --> 00:32:53.490 And we won't get beyond this point, I think, in one semester 00:32:53.490 --> 00:32:57.280 with analytic functions. 00:32:57.280 --> 00:32:59.000 So what am I saying, though? 00:32:59.000 --> 00:33:02.020 I'm saying that the theory of analytic functions 00:33:02.020 --> 00:33:05.300 is closely tied to Laplace's equation. 00:33:05.300 --> 00:33:08.090 Because the real and the imaginary parts 00:33:08.090 --> 00:33:13.110 give me this pair u and s that satisfy, they each satisfy 00:33:13.110 --> 00:33:14.870 Laplace's equation. 00:33:14.870 --> 00:33:19.510 And they're connected by the Cauchy-Riemann equations. 00:33:19.510 --> 00:33:25.280 Boy, it's a lot of mathematics coming real fast here. 00:33:25.280 --> 00:33:29.130 Now I'd like to take one more example. 00:33:29.130 --> 00:33:33.600 Instead of the exponential, can we take the logarithm. 00:33:33.600 --> 00:33:38.820 I want to take the log of x+iy, and I want you to split it 00:33:38.820 --> 00:33:41.120 into its real and imaginary parts, 00:33:41.120 --> 00:33:44.180 and get the u and the s that go with that. 00:33:44.180 --> 00:33:48.810 So this was like the nicest possible. 00:33:48.810 --> 00:33:52.780 We got a series of, e^z is good for every z, 00:33:52.780 --> 00:33:55.540 the series converges, fantastic. 00:33:55.540 --> 00:33:58.990 It's an analytic function everywhere. 00:33:58.990 --> 00:34:01.440 Best possible. 00:34:01.440 --> 00:34:06.380 Now we go to one that's not best possible but nevertheless 00:34:06.380 --> 00:34:08.340 highly valuable. 00:34:08.340 --> 00:34:11.620 OK, so e^z, I've done. 00:34:11.620 --> 00:34:17.590 Let me erase e^z, take log z. 00:34:17.590 --> 00:34:23.760 OK, so now I'm not doing e^z any more. 00:34:23.760 --> 00:34:28.425 And I want to find the logarithm, OK. 00:34:28.425 --> 00:34:30.050 So, what's the deal with the logarithm? 00:34:30.050 --> 00:34:32.550 Real and imaginary parts. 00:34:32.550 --> 00:34:42.450 Now I'm going to take the log of x+iy. 00:34:42.450 --> 00:34:50.570 That is a function of x+iy, except at one point it has 00:34:50.570 --> 00:34:52.910 a problem, right? 00:34:52.910 --> 00:34:57.970 There's a point where this is not going to be analytic, 00:34:57.970 --> 00:35:06.130 and there's going to be a special point in the flow which 00:35:06.130 --> 00:35:07.840 is singular somehow. 00:35:07.840 --> 00:35:13.090 But away from that point, we have a nice-looking function, 00:35:13.090 --> 00:35:17.480 the logarithm of x+iy, and now I'd like to get its real 00:35:17.480 --> 00:35:19.040 and imaginary parts. 00:35:19.040 --> 00:35:22.050 I'd like to know the u and the s. 00:35:22.050 --> 00:35:23.900 But nobody in their right mind wants 00:35:23.900 --> 00:35:26.690 to take the logarithm of a sum, right? 00:35:26.690 --> 00:35:32.350 That's a very foolish thing to try to do, the log of a sum. 00:35:32.350 --> 00:35:35.590 What's the good way to get somewhere with this? 00:35:35.590 --> 00:35:39.140 Real and imaginary part. 00:35:39.140 --> 00:35:44.560 I can take the log of a product. 00:35:44.560 --> 00:35:48.000 So the polar is way better again. 00:35:48.000 --> 00:35:53.230 I want to write this as a log of r e to the-- I 00:35:53.230 --> 00:35:55.180 want to write it that way. 00:35:55.180 --> 00:36:00.870 And now what's the log of a product? 00:36:00.870 --> 00:36:03.310 The sum of the two pieces. 00:36:03.310 --> 00:36:13.060 So I have log r, and the log of e^(i*theta), which is? 00:36:13.060 --> 00:36:14.230 Which is i*theta. 00:36:14.230 --> 00:36:17.690 Boy, look, this is fantastic. 00:36:17.690 --> 00:36:21.330 Fantastic except at zero. 00:36:21.330 --> 00:36:27.160 I mean, it's fantastic but it's got a big problem at zero. 00:36:27.160 --> 00:36:30.890 But it's an extremely important example. 00:36:30.890 --> 00:36:33.840 So what's the real part? 00:36:33.840 --> 00:36:36.530 It's sitting there. 00:36:36.530 --> 00:36:38.140 This is my u. 00:36:38.140 --> 00:36:42.220 This is my u(r,theta), my u(x,y), whatever you want, 00:36:42.220 --> 00:36:45.820 is the log of r. 00:36:45.820 --> 00:36:51.570 The log of the square root of x squared plus y squared. 00:36:51.570 --> 00:36:56.970 I claim that again by this magic combination, this log, 00:36:56.970 --> 00:37:00.680 this-- r is the square root of x squared plus y squared. 00:37:00.680 --> 00:37:04.520 I claim if you substitute that into Laplace's equation 00:37:04.520 --> 00:37:05.300 you get zero. 00:37:05.300 --> 00:37:07.940 It works. 00:37:07.940 --> 00:37:12.510 And what's the imaginary part, the s? 00:37:12.510 --> 00:37:18.340 The twin is the imaginary part, which is theta. 00:37:18.340 --> 00:37:23.170 Oh, what is theta in x, if I wanted it in x and y? 00:37:23.170 --> 00:37:25.860 What would theta be? 00:37:25.860 --> 00:37:28.930 It's the arctan, it's the angle whose 00:37:28.930 --> 00:37:36.070 tangent is something. y/x, so if I really want it 00:37:36.070 --> 00:37:42.100 in rectangular xy stuff, it's the angle whose tangent is y/x. 00:37:42.100 --> 00:37:44.520 And again, if you remember in calculus 00:37:44.520 --> 00:37:46.860 how to take derivatives of this thing 00:37:46.860 --> 00:37:50.140 and you plug it into Laplace's equation you get zero. 00:37:50.140 --> 00:37:53.190 It works. 00:37:53.190 --> 00:37:57.220 So that's a great solution except where? 00:37:57.220 --> 00:37:58.610 At zero. 00:37:58.610 --> 00:38:00.520 Except at zero. 00:38:00.520 --> 00:38:11.310 And this doesn't tell us what's happening at zero. 00:38:11.310 --> 00:38:13.280 It's an excellent solution. 00:38:13.280 --> 00:38:18.230 What's the picture? 00:38:18.230 --> 00:38:24.610 So by Wednesday's exam I'm not expecting 00:38:24.610 --> 00:38:30.070 you to be an expert on the theory of analytic functions. 00:38:30.070 --> 00:38:35.370 I don't expect you to know any conformal mappings. 00:38:35.370 --> 00:38:43.040 By Wednesday, God, that's-- But, I do expect you to have these 00:38:43.040 --> 00:38:44.680 pictures in mind. 00:38:44.680 --> 00:38:48.465 So when I draw those axes, what picture is it 00:38:48.465 --> 00:38:50.950 that I'm planning on? 00:38:50.950 --> 00:38:52.920 I'm planning on the equipotentials u 00:38:52.920 --> 00:39:02.460 equal constant, and the, who are the other guys? 00:39:02.460 --> 00:39:04.900 The streamlines. 00:39:04.900 --> 00:39:08.050 The places where the stream functions-- So here 00:39:08.050 --> 00:39:10.460 is the potential function. 00:39:10.460 --> 00:39:14.980 So what are the equipotential curves? 00:39:14.980 --> 00:39:16.950 For that guy? 00:39:16.950 --> 00:39:18.570 Circles. 00:39:18.570 --> 00:39:22.200 This is a constant when r is a constant, 00:39:22.200 --> 00:39:28.980 so the equipotential functions would be circles. 00:39:28.980 --> 00:39:31.730 I don't want to draw that circle with radius zero, though. 00:39:31.730 --> 00:39:34.950 I'm nervous about that one. 00:39:34.950 --> 00:39:37.000 But all the others are great. 00:39:37.000 --> 00:39:40.430 And what are the streamlines, now? 00:39:40.430 --> 00:39:47.140 The streamlines are, well, what will the streamlines be? 00:39:47.140 --> 00:39:51.620 If I've drawn one family, you can tell me the other family. 00:39:51.620 --> 00:39:54.140 The streamlines will be? 00:39:54.140 --> 00:39:57.170 Radial lines. 00:39:57.170 --> 00:39:59.860 Because they're going to be perpendicular to this. 00:39:59.860 --> 00:40:06.680 And so what do I get, this is the stream function, theta. 00:40:06.680 --> 00:40:08.290 So what's a streamline? 00:40:08.290 --> 00:40:10.740 The stream function should be a constant. 00:40:10.740 --> 00:40:12.140 Theta's a constant. 00:40:12.140 --> 00:40:15.980 That means I'm going out on rays. 00:40:15.980 --> 00:40:20.930 Those are all streamlines. 00:40:20.930 --> 00:40:23.570 Again, everything fantastic. 00:40:23.570 --> 00:40:26.720 If you look in a little region here 00:40:26.720 --> 00:40:31.690 you see just a beautiful picture of equipotentials 00:40:31.690 --> 00:40:35.560 and streamlines crossing them at right angles. 00:40:35.560 --> 00:40:37.300 Everything great. 00:40:37.300 --> 00:40:43.270 Just that point is obviously a problem. 00:40:43.270 --> 00:40:51.660 Now, and I'm suspecting that there's a source here. 00:40:51.660 --> 00:41:00.000 I think this flow, which is given by these guys, 00:41:00.000 --> 00:41:06.400 comes from some kind of a delta function right there. 00:41:06.400 --> 00:41:12.260 And the flow goes outwards. 00:41:12.260 --> 00:41:16.510 So I know u, I know v is the gradient of u, right? 00:41:16.510 --> 00:41:19.560 I could take the x and y derivatives, 00:41:19.560 --> 00:41:22.700 I'd know the velocity. 00:41:22.700 --> 00:41:26.270 I know the stream function, the divergence would be zero. 00:41:26.270 --> 00:41:32.200 Everything great, except at the origin. 00:41:32.200 --> 00:41:35.430 I think we've got some action at the origin. 00:41:35.430 --> 00:41:43.460 Because, here's the way to test it. 00:41:43.460 --> 00:41:49.100 I want to see what's happening at the origin. 00:41:49.100 --> 00:41:52.150 And I'm going to use the divergence theorem. 00:41:52.150 --> 00:41:52.680 Yeah. 00:41:52.680 --> 00:41:53.179 Yeah. 00:41:53.179 --> 00:41:55.160 I'm going to use the divergence theorem. 00:41:55.160 --> 00:42:00.780 So the divergence theorem says-- What is the divergence theorem? 00:42:00.780 --> 00:42:09.610 So this is the key thing that connects double integrals. 00:42:09.610 --> 00:42:14.230 Let me take a circle of radius R. So 00:42:14.230 --> 00:42:19.350 that's the circle of radius R. R could be big, or little. 00:42:19.350 --> 00:42:22.330 So I integrate over the circle of 00:42:22.330 --> 00:42:30.890 radius R. So what's the deal? v is the same as w. 00:42:30.890 --> 00:42:32.850 What does the divergence theorem tell me? 00:42:32.850 --> 00:42:36.860 It tells me that if I integrate, what do I integrate, 00:42:36.860 --> 00:42:43.210 the divergence of w? dx/dy, or r*dr*d theta. 00:42:45.960 --> 00:42:52.760 Then I get the flux. 00:42:52.760 --> 00:42:56.450 So this is a key identity. 00:42:56.450 --> 00:42:59.600 Fundamentally, more than just the key identity, 00:42:59.600 --> 00:43:01.320 it's central here. 00:43:01.320 --> 00:43:07.650 The total flow out of the region must make it 00:43:07.650 --> 00:43:09.000 through the boundary. 00:43:09.000 --> 00:43:11.460 So I integrate this boundary, and this boundary 00:43:11.460 --> 00:43:14.180 is a circle of radius R, and what do I 00:43:14.180 --> 00:43:19.390 integrate along that circle? 00:43:19.390 --> 00:43:23.670 What's the other side of the divergence theorem? 00:43:23.670 --> 00:43:30.280 w dot n. w dot n, around the boundary. 00:43:30.280 --> 00:43:37.730 And remember, I have this nice-- my curve here 00:43:37.730 --> 00:43:42.450 is this nice circle. 00:43:42.450 --> 00:43:44.600 So I'm going to integrate around that circle. 00:43:44.600 --> 00:43:52.130 First of all, what is n? 00:43:52.130 --> 00:43:56.990 By definition, n is the normal that points outward, 00:43:56.990 --> 00:43:58.250 straight out. 00:43:58.250 --> 00:44:02.520 So it's actually going out that way. 00:44:02.520 --> 00:44:05.540 At every point it's pointing straight out. 00:44:05.540 --> 00:44:09.130 And ds-- Yeah, I think we can figure out exactly 00:44:09.130 --> 00:44:18.420 what that right-hand side is. 00:44:18.420 --> 00:44:23.090 How do I get that right-hand side? 00:44:23.090 --> 00:44:29.670 I'm looking for w, and then I have to integrate. 00:44:29.670 --> 00:44:35.580 OK, here is my u. 00:44:35.580 --> 00:44:42.790 My u is log r. 00:44:42.790 --> 00:44:45.520 So what's the gradient of log r? 00:44:45.520 --> 00:44:47.690 It points outwards. 00:44:47.690 --> 00:44:49.430 And how large is the derivative? 00:44:49.430 --> 00:44:55.600 So the derivative of this log r is 1/r. 00:44:55.600 --> 00:45:03.400 I think that this comes down to, this is the integral. 00:45:03.400 --> 00:45:05.010 Around the circle. 00:45:05.010 --> 00:45:14.930 I think that this thing is 1/R. I went pretty quickly there, 00:45:14.930 --> 00:45:17.960 so I'll ask you to look in the book 00:45:17.960 --> 00:45:21.360 because this is such an important example it's 00:45:21.360 --> 00:45:26.150 done there in more detail. 00:45:26.150 --> 00:45:30.080 So I'm claiming that the derivative is 1/R, 00:45:30.080 --> 00:45:32.780 and that it points directly out. 00:45:32.780 --> 00:45:35.930 So the gradient points out. 00:45:35.930 --> 00:45:40.700 The normal points out, so that I just get exactly 1/R. Now, 00:45:40.700 --> 00:45:41.470 what is ds? 00:45:45.300 --> 00:45:47.840 For integrating around the circle 00:45:47.840 --> 00:45:54.410 what's a little tiny piece of arc on a circle? 00:45:54.410 --> 00:45:56.260 Of radius R? 00:45:56.260 --> 00:45:57.050 R d theta. 00:45:57.050 --> 00:45:57.830 Good man. 00:45:57.830 --> 00:46:03.120 R d theta. 00:46:03.120 --> 00:46:09.150 Now that's an integral I can do, right? 00:46:09.150 --> 00:46:11.200 And what do I get? 00:46:11.200 --> 00:46:13.630 2 pi. 00:46:13.630 --> 00:46:19.730 R cancels R, I'm integrating d theta around from zero to 2pi. 00:46:19.730 --> 00:46:20.620 The answer is 2pi. 00:46:24.160 --> 00:46:27.530 So what do I learn from that? 00:46:27.530 --> 00:46:35.700 I learn that somehow this source in the inside has strength 2pi. 00:46:35.700 --> 00:46:42.120 What's sitting in there is 2pi times a delta function. 00:46:42.120 --> 00:46:47.910 This is the solution to Laplace's equation 00:46:47.910 --> 00:46:50.770 except at that source term, so I really 00:46:50.770 --> 00:46:53.710 should say Poisson's equation. 00:46:53.710 --> 00:46:56.250 This has turned out to be the solution 00:46:56.250 --> 00:47:03.580 to Poisson with a delta, or with 2pi times a delta. 00:47:03.580 --> 00:47:08.990 We have just solved this important equation. 00:47:08.990 --> 00:47:12.690 Poisson's equation with a point source. 00:47:12.690 --> 00:47:16.160 And, of course, that's important because when 00:47:16.160 --> 00:47:17.900 you can solve with a point source, 00:47:17.900 --> 00:47:22.650 you can put together all sorts of sources. 00:47:22.650 --> 00:47:24.790 And this is called the Green's function. 00:47:24.790 --> 00:47:26.800 The Green's function is the solution 00:47:26.800 --> 00:47:29.540 when the source is a delta. 00:47:29.540 --> 00:47:33.740 So if I divide by 2pi, now I've got it. 00:47:33.740 --> 00:47:37.920 I divide this by 2pi and there is the Green's function. 00:47:37.920 --> 00:47:43.070 I have to put that in bold letters. 00:47:43.070 --> 00:47:48.220 Green's function. 00:47:48.220 --> 00:47:52.130 It's the solution to the equation when 00:47:52.130 --> 00:47:54.530 the source is a delta and the answer 00:47:54.530 --> 00:47:59.870 is u is the log of r over 2pi. 00:47:59.870 --> 00:48:06.460 So that's the Green's function in 2-D. Physicists, 00:48:06.460 --> 00:48:10.320 you know, they live and die with these Green's function. 00:48:10.320 --> 00:48:12.510 Live, let's say, with Green's function. 00:48:12.510 --> 00:48:18.700 And they would want to know the Green's function in 3-D. 00:48:18.700 --> 00:48:21.590 So the Green's function in three dimensions 00:48:21.590 --> 00:48:23.400 also turns out beautifully. 00:48:23.400 --> 00:48:28.110 This is in, they would say, in free space. 00:48:28.110 --> 00:48:32.170 This is the Green's function when there's no other charges. 00:48:32.170 --> 00:48:36.940 Nothing is happening, except for the charge right at the center. 00:48:36.940 --> 00:48:44.510 And if I'm in two dimensions the Green's function is this log r. 00:48:44.510 --> 00:48:49.360 So it grows more slowly. 00:48:49.360 --> 00:48:51.550 It behaves like log r. 00:48:51.550 --> 00:48:56.670 And in 3-D I think the answer is 1/(4pi*r). 00:48:56.670 --> 00:49:02.260 It's just amazing that those Green's functions, 00:49:02.260 --> 00:49:09.730 when the right side is a delta, have such nice formulas. 00:49:09.730 --> 00:49:16.280 OK, let me take one moment here. 00:49:16.280 --> 00:49:22.080 I'll tell you what conformal mapping is about. 00:49:22.080 --> 00:49:26.300 But what's your take-home from this lecture? 00:49:26.300 --> 00:49:32.530 Your take-home is two methods that we can really 00:49:32.530 --> 00:49:35.530 use to get a formula for the answer. 00:49:35.530 --> 00:49:43.400 One method was for Laplace's equation in a circle. 00:49:43.400 --> 00:49:47.840 Get the boundary conditions in a series of sines and cosines, 00:49:47.840 --> 00:49:52.420 and then just put in the r's that we need. 00:49:52.420 --> 00:49:56.000 That's a simple, simple method. 00:49:56.000 --> 00:50:00.520 Provided we can get started with the Fourier series. 00:50:00.520 --> 00:50:05.480 The second method is, look at functions of x+iy, 00:50:05.480 --> 00:50:09.270 and try to pick one that matches your problem. 00:50:09.270 --> 00:50:14.830 And if your problem has a point source, at the origin, 00:50:14.830 --> 00:50:17.300 we found the one. 00:50:17.300 --> 00:50:20.350 So the literature for hundreds of years 00:50:20.350 --> 00:50:24.730 is aimed at solving other problems. 00:50:24.730 --> 00:50:27.350 If the point source is somewhere else, what happens? 00:50:27.350 --> 00:50:28.980 That's not hard. 00:50:28.980 --> 00:50:32.440 If it's not a point source but some other kind of source, 00:50:32.440 --> 00:50:37.770 or if the region is not a circle. 00:50:37.770 --> 00:50:42.110 Can I say in one final sentence just what to do, 00:50:42.110 --> 00:50:46.820 this conformal mapping idea, when the region is not 00:50:46.820 --> 00:50:51.010 a circle. 00:50:51.010 --> 00:50:54.380 Well, I can say it in one word, make it a circle. 00:50:54.380 --> 00:50:57.860 I mean, that's what Riemann said, you could do it. 00:50:57.860 --> 00:51:02.410 You could think of a function, so Riemann said that 00:51:02.410 --> 00:51:05.820 there's always some function of x+iy, 00:51:05.820 --> 00:51:10.210 let me call this Riemann's function capital F of x, y. 00:51:10.210 --> 00:51:13.770 So this is now the idea of conformal mapping. 00:51:13.770 --> 00:51:16.700 Change variables. 00:51:16.700 --> 00:51:19.640 Conformal mapping is a change of variables. 00:51:19.640 --> 00:51:24.530 He picked some function and let its real part be X and let 00:51:24.530 --> 00:51:29.670 its imaginary part be Y. Capital Y. OK, 00:51:29.670 --> 00:51:33.110 this is totally ridiculous to put conformal mapping 00:51:33.110 --> 00:51:34.970 in 30 seconds. 00:51:34.970 --> 00:51:42.070 But, never mind, let's just do it. 00:51:42.070 --> 00:51:44.490 The book describes conformal mappings 00:51:44.490 --> 00:51:49.360 and classical applied math courses do much more 00:51:49.360 --> 00:51:51.290 with conformal mapping. 00:51:51.290 --> 00:51:54.180 But the truth is, computationally 00:51:54.180 --> 00:51:59.130 they're not anything like as much used as these. 00:51:59.130 --> 00:52:00.460 So what's the idea? 00:52:00.460 --> 00:52:05.400 The idea is to find a neat function of x+iy, 00:52:05.400 --> 00:52:11.620 so that your crazy boundary becomes a circle. 00:52:11.620 --> 00:52:15.110 In the capital X, capital Y variables. 00:52:15.110 --> 00:52:19.030 So you're mapping the region, ellipse, 00:52:19.030 --> 00:52:23.310 whatever it looks like, by changing from little x, 00:52:23.310 --> 00:52:27.540 little y, where it was an ellipse, to capital X, capital 00:52:27.540 --> 00:52:29.350 Y, where it's a circle. 00:52:29.350 --> 00:52:30.960 And the point is Laplace's equation 00:52:30.960 --> 00:52:33.740 stays Laplace's equation. 00:52:33.740 --> 00:52:37.490 That change of variables does not mess up Laplace's equation. 00:52:37.490 --> 00:52:40.710 So that then you've got it in a circle. 00:52:40.710 --> 00:52:44.010 You solve it in a circle, for these guys. 00:52:44.010 --> 00:52:46.060 And then you go back. 00:52:46.060 --> 00:52:51.210 In a word, you're able to solve Laplace's equation in this 00:52:51.210 --> 00:52:57.880 crazy region because you never leave the magic x+iy. 00:52:57.880 --> 00:53:02.360 You find a combination with that magic x+iy that makes 00:53:02.360 --> 00:53:04.310 your region into a circle. 00:53:04.310 --> 00:53:10.710 In the circle we now know how to use capital X plus i capital 00:53:10.710 --> 00:53:15.110 Y. You're staying with that magic combination 00:53:15.110 --> 00:53:18.090 and getting the region to be what you like. 00:53:18.090 --> 00:53:21.540 So people know a lot of these conformal mappings. 00:53:21.540 --> 00:53:24.810 A famous one is the Joukowski one, 00:53:24.810 --> 00:53:31.160 that takes something that looks very like an airfoil, 00:53:31.160 --> 00:53:33.910 and you can get a circle out of it. 00:53:33.910 --> 00:53:37.890 So I'll put down Joukowski's name. 00:53:37.890 --> 00:53:48.950 So that's one that I trust Course 16 still finds valuable. 00:53:48.950 --> 00:53:55.780 It's a transformation that takes certain shapes 00:53:55.780 --> 00:53:58.390 and they include shapes that look like airfoils, 00:53:58.390 --> 00:54:00.240 and produce circles. 00:54:00.240 --> 00:54:07.740 OK, so sorry about such a quick presentation of such a 00:54:07.740 --> 00:54:10.080 basic subject. 00:54:10.080 --> 00:54:15.120 Conformal mapping, not on any exam, that'd be impossible. 00:54:15.120 --> 00:54:19.250 It's really this stuff that you're number one responsible 00:54:19.250 --> 00:54:20.293 for.