WEBVTT 00:00:00.030 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.770 Your support will help MIT OpenCourseWare 00:00:05.770 --> 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 --> 00:00:16.880 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.880 --> 00:00:19.390 at ocw.mit.edu. 00:00:19.390 --> 00:00:21.770 PROFESSOR STRANG: So. 00:00:21.770 --> 00:00:27.370 Today's lecture is partly cleaning up some pieces 00:00:27.370 --> 00:00:31.530 left from 1-D problems. 00:00:31.530 --> 00:00:35.270 3.1 was second order equations, 3.2 00:00:35.270 --> 00:00:38.280 was fourth order equations for beams. 00:00:38.280 --> 00:00:41.800 And just a few more comments to make. 00:00:41.800 --> 00:00:51.370 But then I do want to say something about splines. 00:00:51.370 --> 00:00:54.610 And I included a homework question that 00:00:54.610 --> 00:00:59.910 asked you to use the spline command and get a result. 00:00:59.910 --> 00:01:07.460 I'm not planning to make that a serious topic in the course. 00:01:07.460 --> 00:01:09.290 In other words, you're not like going 00:01:09.290 --> 00:01:14.040 to be responsible on an exam for a discussion of splines. 00:01:14.040 --> 00:01:22.640 But it's such a major event, and major requirement in scientific 00:01:22.640 --> 00:01:24.900 computing -- if you're given some values, 00:01:24.900 --> 00:01:28.491 put a curve through them -- that I have to say something about 00:01:28.491 --> 00:01:28.990 that. 00:01:28.990 --> 00:01:31.720 So that's interpolation. 00:01:31.720 --> 00:01:34.380 Because you'll have this all the time. 00:01:34.380 --> 00:01:39.490 If your experiment produces some values 00:01:39.490 --> 00:01:44.550 at specific times or a finite number of values, 00:01:44.550 --> 00:01:46.990 and you want to fit a curve through those points. 00:01:46.990 --> 00:01:48.320 How do you do it? 00:01:48.320 --> 00:01:51.970 So splines give a very, very good answer. 00:01:51.970 --> 00:02:02.540 And then, the next and big topic of the course 00:02:02.540 --> 00:02:04.700 we'll make a beginning on. 00:02:04.700 --> 00:02:09.170 Which is gradient, divergence, leading to Laplace's equation. 00:02:09.170 --> 00:02:14.430 Problems in 2-D. So partial differential equations 00:02:14.430 --> 00:02:15.600 are going to show up. 00:02:15.600 --> 00:02:19.000 OK. 00:02:19.000 --> 00:02:20.860 Some small points. 00:02:20.860 --> 00:02:27.490 First point, about the MATLAB homework for Monday. 00:02:27.490 --> 00:02:34.880 A question came by email. 00:02:34.880 --> 00:02:43.960 I put the spike and also the jump in c, in the coefficient, 00:02:43.960 --> 00:02:45.910 not at mesh point. 00:02:45.910 --> 00:02:47.570 If you've started on that problem, 00:02:47.570 --> 00:02:55.100 you probably noticed that, and maybe a little swearing 00:02:55.100 --> 00:03:01.350 went on. [LAUGHTER] Because it makes it not so easy. 00:03:01.350 --> 00:03:05.160 And then the question came, did I really 00:03:05.160 --> 00:03:10.020 expect you to compute, use the exact position 00:03:10.020 --> 00:03:15.720 and figure out-- When you have some integrals to do, 00:03:15.720 --> 00:03:20.120 they'll change at the place where c changes values. 00:03:20.120 --> 00:03:24.570 Where we have integrals of c, if c has one of those jumps, 00:03:24.570 --> 00:03:28.030 on the quiz the jump came right at a neat point, 00:03:28.030 --> 00:03:32.690 so it was clear. c was one on one side and two on the other. 00:03:32.690 --> 00:03:36.600 Now the jump is going to come at an in-between point. 00:03:36.600 --> 00:03:42.170 So the answer is yes. 00:03:42.170 --> 00:03:45.100 I had a nice email this morning from somebody who 00:03:45.100 --> 00:03:48.280 has done that MATLAB homework. 00:03:48.280 --> 00:03:55.130 And he said, quote, it's a good problem. 00:03:55.130 --> 00:03:59.070 I learned more from it than I did from the other problems 00:03:59.070 --> 00:04:00.390 in the pset. 00:04:00.390 --> 00:04:06.240 So I'll blame it on him. 00:04:06.240 --> 00:04:08.340 I'll stay with that problem, and ask 00:04:08.340 --> 00:04:14.700 you to do your best to deal with the calculation. 00:04:14.700 --> 00:04:17.370 It's second order and I used linear elements. 00:04:17.370 --> 00:04:20.610 I didn't want to push you up to these cubic elements, 00:04:20.610 --> 00:04:24.790 even though those would give much better accuracy. 00:04:24.790 --> 00:04:29.950 One reason I didn't put the spike and the jump right 00:04:29.950 --> 00:04:36.710 at the node is that, if I did, the finite element solution 00:04:36.710 --> 00:04:38.390 would be exactly right. 00:04:38.390 --> 00:04:43.180 Because the correct solution is going to be piecewise linear. 00:04:43.180 --> 00:04:49.890 And if I put the breaks between pieces right at node points, 00:04:49.890 --> 00:04:53.540 well, then I have a function that's in my finite element 00:04:53.540 --> 00:04:55.360 space, in my linear space. 00:04:55.360 --> 00:04:57.410 So it'll come out exactly. 00:04:57.410 --> 00:05:00.940 So I thought well, let's at least have some chance 00:05:00.940 --> 00:05:05.550 to see the error between the finite element 00:05:05.550 --> 00:05:07.120 solution and the exact one. 00:05:07.120 --> 00:05:08.730 You see what I'm saying? 00:05:08.730 --> 00:05:13.690 I'm just anticipating, that the exact solution 00:05:13.690 --> 00:05:14.850 will-- Let's see. 00:05:14.850 --> 00:05:18.820 Something happened at 1/3 and something happened at 2/3. 00:05:18.820 --> 00:05:21.460 I'm afraid I don't remember the boundary conditions. 00:05:21.460 --> 00:05:24.780 Does anybody remember what I took for boundary conditions? 00:05:24.780 --> 00:05:27.620 Probably free-fixed, or something. 00:05:27.620 --> 00:05:33.220 So maybe, if it was free-fixed, probably the solution 00:05:33.220 --> 00:05:35.710 would be maybe something like that. 00:05:35.710 --> 00:05:36.520 I don't know. 00:05:36.520 --> 00:05:38.010 Whatever. 00:05:38.010 --> 00:05:42.530 Piecewise linear with breaks where there's a jump in c, 00:05:42.530 --> 00:05:46.380 or where the point load is hitting. 00:05:46.380 --> 00:05:48.110 Yeah. 00:05:48.110 --> 00:05:50.800 I don't know if that picture is accurate. 00:05:50.800 --> 00:05:56.990 But my point was, if I chose those points to be also mesh 00:05:56.990 --> 00:06:00.380 points, then the finite element solution would be exact, 00:06:00.380 --> 00:06:03.610 and we wouldn't learn anything about accuracy. 00:06:03.610 --> 00:06:04.210 OK. 00:06:04.210 --> 00:06:06.670 So anyway, for that MATLAB problem, 00:06:06.670 --> 00:06:13.610 I'm hoping you can do the integrals, which 00:06:13.610 --> 00:06:20.300 will mean noticing which integral-- If that doesn't 00:06:20.300 --> 00:06:25.870 fall, so if the mesh points are like this, 00:06:25.870 --> 00:06:30.230 in the finite element integral over that mesh, 00:06:30.230 --> 00:06:32.530 over that interval, you're going to have 00:06:32.530 --> 00:06:34.900 to split it into two pieces. 00:06:34.900 --> 00:06:36.600 That's it. 00:06:36.600 --> 00:06:41.080 So see if you can do that split, do the split into two pieces. 00:06:41.080 --> 00:06:41.810 OK. 00:06:41.810 --> 00:06:46.040 So that's a comment on that homework. 00:06:46.040 --> 00:06:47.480 OK. 00:06:47.480 --> 00:06:49.740 What else do I need to comment on, when 00:06:49.740 --> 00:06:51.310 I'm sort of catching up here? 00:06:51.310 --> 00:06:55.050 Oh, about boundary conditions. 00:06:55.050 --> 00:06:57.130 I thought I would just repeat clearly 00:06:57.130 --> 00:07:04.870 on the board the rules about and the difference 00:07:04.870 --> 00:07:13.250 between fixed and free when we're doing this weak form 00:07:13.250 --> 00:07:17.580 approach to the problem. 00:07:17.580 --> 00:07:21.800 And then I gave the other names: a fixed condition, 00:07:21.800 --> 00:07:25.300 in this theory it's often called an essential boundary 00:07:25.300 --> 00:07:29.070 condition, but many people just name 00:07:29.070 --> 00:07:33.280 Dirichlet as the person's name that's associated 00:07:33.280 --> 00:07:34.700 with such conditions. 00:07:34.700 --> 00:07:37.770 Like u(0)=0. 00:07:37.770 --> 00:07:39.020 Right, fixed. 00:07:39.020 --> 00:07:43.360 And then free conditions are natural conditions 00:07:43.360 --> 00:07:45.850 in the finite element method, and that 00:07:45.850 --> 00:07:48.550 means that we don't have to impose them. 00:07:48.550 --> 00:07:49.600 Right. 00:07:49.600 --> 00:07:51.770 I discussed this in the earlier lecture. 00:07:51.770 --> 00:07:54.180 I just thought I'd bring it back here, 00:07:54.180 --> 00:07:59.920 because it's easy to describe in one line. 00:07:59.920 --> 00:08:09.160 Now somebody asked, on Wednesday, a good question. 00:08:09.160 --> 00:08:17.650 Suppose u' is given, but not given as zero. 00:08:17.650 --> 00:08:28.490 So suppose we have a condition like u'=A, let's say. 00:08:28.490 --> 00:08:30.080 What to do? 00:08:30.080 --> 00:08:32.320 So I'll just put question mark. 00:08:32.320 --> 00:08:36.360 How do I deal with-- How do I? 00:08:36.360 --> 00:08:40.890 How does the weak form approach and the Galerkin approach 00:08:40.890 --> 00:08:47.840 deal with a non-zero, natural condition? 00:08:47.840 --> 00:08:52.760 A non-zero Neumann condition, that'd be the right word. 00:08:52.760 --> 00:08:56.230 So this would be, in a second order problem, 00:08:56.230 --> 00:08:59.680 this would be a Neumann condition. 00:08:59.680 --> 00:09:02.780 And I'm thinking, what do you do if A isn't zero. 00:09:02.780 --> 00:09:06.810 Somehow that has to show up in your finite element equations, 00:09:06.810 --> 00:09:07.850 right? 00:09:07.850 --> 00:09:10.770 So I guess the right way is to go back 00:09:10.770 --> 00:09:12.320 to the way they came from. 00:09:12.320 --> 00:09:15.140 So you remember how the weak form came? 00:09:15.140 --> 00:09:18.060 I took the differential equation, 00:09:18.060 --> 00:09:20.320 I multiplied by any test function, 00:09:20.320 --> 00:09:23.070 and I integrated by parts. 00:09:23.070 --> 00:09:25.460 That's where the weak form started. 00:09:25.460 --> 00:09:28.680 So the integration by parts gave me 00:09:28.680 --> 00:09:38.250 this nice symmetric integral. 00:09:38.250 --> 00:09:41.010 It's symmetric in u and v. It only 00:09:41.010 --> 00:09:44.760 requires one derivative of u, so that I'm 00:09:44.760 --> 00:09:48.720 allowed to use hat functions. 00:09:48.720 --> 00:09:51.910 And I'm allowed to use hat functions for v, 00:09:51.910 --> 00:09:54.650 because their derivatives have just a jump, 00:09:54.650 --> 00:09:58.890 and a jump function I can integrate, no problem. 00:09:58.890 --> 00:10:04.110 Now what about this u'=A? 00:10:04.110 --> 00:10:08.640 And I guess we're seeing it there. 00:10:08.640 --> 00:10:13.380 If u' was zero, if we had a totally free end, 00:10:13.380 --> 00:10:16.540 u' was zero, nothing happening there, 00:10:16.540 --> 00:10:21.690 then that term would drop out of the integration by parts. 00:10:21.690 --> 00:10:24.460 And you see why I don't have to impose anything 00:10:24.460 --> 00:10:27.960 on v, because that term is already accounted 00:10:27.960 --> 00:10:31.120 for by the u' going away. 00:10:31.120 --> 00:10:38.330 Now, what about if u' is given, but not given zero? 00:10:38.330 --> 00:10:41.500 Suppose it's given the value A. All I want to say 00:10:41.500 --> 00:10:51.070 is, then this term has to stay. 00:10:51.070 --> 00:10:52.690 I have to pay attention to it. 00:10:52.690 --> 00:10:56.530 What happens when Galerkin takes over? 00:10:56.530 --> 00:11:02.600 Galerkin will put in one of his test functions. 00:11:02.600 --> 00:11:06.880 Maybe I just indicate that by changing the v to a cap V. 00:11:06.880 --> 00:11:08.120 It'll be one of them. 00:11:08.120 --> 00:11:11.780 There'll be n of them. 00:11:11.780 --> 00:11:16.060 Each V_i, each test guy, gives me an equation. 00:11:16.060 --> 00:11:22.140 And of course u is now going to-- The Galerkin idea is that 00:11:22.140 --> 00:11:25.870 instead of any u, I only have capital U's. 00:11:25.870 --> 00:11:32.130 Combinations of these phis. 00:11:32.130 --> 00:11:37.550 I'm not going to say anything very big here, or very clear. 00:11:37.550 --> 00:11:46.800 All I want to say is that if u prime is given at an end point, 00:11:46.800 --> 00:11:56.290 and given by A, then this whole stuff is equaling something 00:11:56.290 --> 00:11:57.980 with an f, right? 00:11:57.980 --> 00:12:00.460 f*V_i*dx. 00:12:00.460 --> 00:12:03.520 I should remember the other side of the equation. 00:12:03.520 --> 00:12:07.610 If u' is given, then c at that end point, 00:12:07.610 --> 00:12:14.120 times the given value of A, times the V will be -- 00:12:14.120 --> 00:12:22.610 whatever value of V that is -- will show up in equation i. 00:12:22.610 --> 00:12:25.180 That will be a term that goes on the right-hand side. 00:12:25.180 --> 00:12:29.730 That's all I'm saying, and all you would expect. 00:12:29.730 --> 00:12:35.740 That any time I'm given some data, 00:12:35.740 --> 00:12:39.030 I'm given some non-zero boundary conditions, 00:12:39.030 --> 00:12:42.530 or I'm given some non-zero f in the inside, 00:12:42.530 --> 00:12:49.080 that stuff is going to show up on the right-hand side. 00:12:49.080 --> 00:12:58.650 It'll show up as part of the F. The vector of these. 00:12:58.650 --> 00:13:03.430 So up to now, F has just come from little f(x), integrated 00:13:03.430 --> 00:13:07.820 against V. And all I'm saying is that if we 00:13:07.820 --> 00:13:13.690 had one of these guys, then that would contribute to big F, 00:13:13.690 --> 00:13:16.010 also. 00:13:16.010 --> 00:13:17.140 Yeah. 00:13:17.140 --> 00:13:22.580 You can see, it's not beautiful. 00:13:22.580 --> 00:13:25.590 But we have to realize what we would do. 00:13:25.590 --> 00:13:28.520 OK, that's not my favorite topic. 00:13:28.520 --> 00:13:32.420 But I'll just say quit on that one. 00:13:32.420 --> 00:13:39.280 I'm not expecting you to do problems or anything that 00:13:39.280 --> 00:13:40.900 involved that possibility. 00:13:40.900 --> 00:13:42.960 Just to see that it could happen. 00:13:42.960 --> 00:13:44.250 OK. 00:13:44.250 --> 00:13:45.220 Quit. 00:13:45.220 --> 00:13:47.770 Now, interpolation. 00:13:47.770 --> 00:13:49.840 All right. 00:13:49.840 --> 00:13:54.030 So I guess if I had to list problems 00:13:54.030 --> 00:13:59.725 that people face all the time and need numerical guidance on, 00:13:59.725 --> 00:14:01.950 this is one of them. 00:14:01.950 --> 00:14:07.950 And so let's say we're in 1-D -- 1-D is certainly a lot easier 00:14:07.950 --> 00:14:11.370 -- so, suppose I have this. 00:14:11.370 --> 00:14:15.390 I'll call this x, just to have a name for the variable. 00:14:15.390 --> 00:14:21.360 Suppose I have some values. 00:14:21.360 --> 00:14:28.520 Say six values. 00:14:28.520 --> 00:14:30.720 And I've measured them, I've worked hard 00:14:30.720 --> 00:14:32.660 to find those six values. 00:14:32.660 --> 00:14:37.310 But now you may say, well I'm thinking 00:14:37.310 --> 00:14:43.320 I have some function F(x) here. 00:14:43.320 --> 00:14:45.770 But right now I don't have a function. 00:14:45.770 --> 00:14:48.390 Right now I just have six values of that function. 00:14:48.390 --> 00:14:55.030 You know, what is the stretching constant, what is the c(x). 00:14:57.650 --> 00:15:05.760 If you had a physical experiment with an actual spring, 00:15:05.760 --> 00:15:09.630 you might measure, you might put on different forces 00:15:09.630 --> 00:15:15.260 and measure the stretching, and have six values of that. 00:15:15.260 --> 00:15:18.980 How do I fit those with a curve? 00:15:18.980 --> 00:15:23.720 In other words, how do I know-- Interpolation means, inter- 00:15:23.720 --> 00:15:29.440 means asking, what about between the points? 00:15:29.440 --> 00:15:32.440 What value should it have there? 00:15:32.440 --> 00:15:37.100 OK, well there's one simple rule would be, 00:15:37.100 --> 00:15:43.050 just interpolate linearly between. 00:15:43.050 --> 00:15:46.970 OK. 00:15:46.970 --> 00:15:49.100 That's certainly pretty stable. 00:15:49.100 --> 00:15:51.880 It will not get out of control. 00:15:51.880 --> 00:16:00.590 But it's not very accurate. 00:16:00.590 --> 00:16:03.850 For the function that's probably lying behind this, 00:16:03.850 --> 00:16:07.230 this isn't that great of a representation. 00:16:07.230 --> 00:16:10.490 OK, you say, I want something smoother. 00:16:10.490 --> 00:16:14.990 Well another idea that naturally comes up 00:16:14.990 --> 00:16:24.910 is, the other extreme would be-- So that was completely local. 00:16:24.910 --> 00:16:28.700 Just, every two values determine the broken line. 00:16:28.700 --> 00:16:30.930 The second idea that's completely natural 00:16:30.930 --> 00:16:34.390 would be, fit a polynomial. 00:16:34.390 --> 00:16:36.270 Fit a polynomial through those points, 00:16:36.270 --> 00:16:39.950 and then you have a nice, totally simple function. 00:16:39.950 --> 00:16:41.370 Nice curve. 00:16:41.370 --> 00:16:46.410 And so what degree polynomial would we be looking for here? 00:16:46.410 --> 00:16:53.460 If I have six points, I could fit a polynomial of degree-- 00:16:53.460 --> 00:16:54.830 What do you think? 00:16:54.830 --> 00:16:58.120 I want to have six coefficients, so I 00:16:58.120 --> 00:17:02.100 can get six equations to make the polynomial go 00:17:02.100 --> 00:17:03.660 through those six points. 00:17:03.660 --> 00:17:08.080 So my polynomial P(x), which actually 00:17:08.080 --> 00:17:11.620 goes through those points, would be some polynomial of the form, 00:17:11.620 --> 00:17:15.830 it's got some constant term, and it's got some linear term. 00:17:15.830 --> 00:17:19.070 And how far am I going to go? 00:17:19.070 --> 00:17:19.920 Fifth, right. 00:17:19.920 --> 00:17:22.020 That'll give me six coefficients. 00:17:22.020 --> 00:17:22.970 OK. 00:17:22.970 --> 00:17:26.740 So I could put a fifth degree polynomial, that 00:17:26.740 --> 00:17:29.360 gives me a_0, a_1, up to a_5. 00:17:29.360 --> 00:17:33.620 Six coefficients through six points. 00:17:33.620 --> 00:17:36.700 Well, six isn't too bad. 00:17:36.700 --> 00:17:44.080 But if six became 60, don't do it. 00:17:44.080 --> 00:17:45.040 That's the message. 00:17:45.040 --> 00:17:47.970 Don't do it. 00:17:47.970 --> 00:17:55.960 If I'm going on up to, say I have an a_59 x to the 59th, 00:17:55.960 --> 00:17:58.070 And that fits my 60 points. 00:17:58.070 --> 00:18:01.030 And you say, well look, I've got 60 values, 00:18:01.030 --> 00:18:04.570 that should be way better than having only six. 00:18:04.570 --> 00:18:08.920 But the polynomial that would come out of that, 00:18:08.920 --> 00:18:13.180 even if these values were pretty smooth, 00:18:13.180 --> 00:18:17.460 the polynomial that fits -- well I'd have to put in a whole lot 00:18:17.460 --> 00:18:20.070 more to get 60, and I won't try -- 00:18:20.070 --> 00:18:26.170 the polynomial would go crazy. 00:18:26.170 --> 00:18:28.950 Can I just draw something crazy? 00:18:28.950 --> 00:18:31.840 I mean, whatever. 00:18:31.840 --> 00:18:34.350 It's unstable. 00:18:34.350 --> 00:18:37.430 And there are classical examples of that. 00:18:37.430 --> 00:18:41.290 In fact, the famous example is the function one over one 00:18:41.290 --> 00:18:44.560 plus x squared. 00:18:44.560 --> 00:18:48.530 And later, there's a figure in the book, I think it's 00:18:48.530 --> 00:18:56.780 in section 5.4, which shows the result. 00:18:56.780 --> 00:19:00.770 That's a terrific function. 00:19:00.770 --> 00:19:04.090 It's infinitely differentiable. 00:19:04.090 --> 00:19:06.530 I would even call it an analytic function. 00:19:06.530 --> 00:19:07.660 It's great. 00:19:07.660 --> 00:19:13.170 But if I tried to fit it with a high-degree polynomial 00:19:13.170 --> 00:19:18.360 at points, that polynomial gets out of hand. 00:19:18.360 --> 00:19:22.690 And a figure is in-- the figure there. 00:19:22.690 --> 00:19:34.220 So, MATLAB or any computing system 00:19:34.220 --> 00:19:39.590 would have a subroutine that does interpolation. 00:19:39.590 --> 00:19:43.450 And this is named after Lagrange. 00:19:43.450 --> 00:19:46.100 So this is called Lagrange interpolation, 00:19:46.100 --> 00:19:48.410 fitting a polynomial. 00:19:48.410 --> 00:19:53.260 And if the degree is small, and maybe degree five 00:19:53.260 --> 00:19:58.620 would be okay, then that's an important thing 00:19:58.620 --> 00:20:02.010 to be able to do. 00:20:02.010 --> 00:20:05.270 In other words, you're finding these six coefficients 00:20:05.270 --> 00:20:06.520 from these six heights. 00:20:06.520 --> 00:20:10.210 You've got some six by six matrix that connects the six 00:20:10.210 --> 00:20:13.700 heights to the six a's. 00:20:13.700 --> 00:20:18.760 But when you get up to 60, that matrix stops behaving well. 00:20:18.760 --> 00:20:23.550 It becomes very ill conditioned, and your coefficients 00:20:23.550 --> 00:20:25.100 go all over the place. 00:20:25.100 --> 00:20:29.440 OK, so my question is, what do you do? 00:20:29.440 --> 00:20:34.850 And one answer, one good answer is fit those points 00:20:34.850 --> 00:20:39.490 not with straight lines, that's too crude; not 00:20:39.490 --> 00:20:43.640 with a single polynomial, that's too unstable; 00:20:43.640 --> 00:20:45.790 but fit it with splines. 00:20:45.790 --> 00:20:50.030 So interpolation by splines. 00:20:50.030 --> 00:20:51.040 OK. 00:20:51.040 --> 00:20:57.440 So it's just sort of appearing in this section 00:20:57.440 --> 00:21:07.470 3.2-- I mean by that, and people often do mean, 00:21:07.470 --> 00:21:13.510 when they say spline, they mean cubic spline. 00:21:13.510 --> 00:21:18.190 So you could have splines of other degrees. 00:21:18.190 --> 00:21:22.290 But often, sort of the natural choice is the cubic. 00:21:22.290 --> 00:21:27.800 So can I just briefly describe what that does. 00:21:27.800 --> 00:21:32.020 What the cubic spline would be like. 00:21:32.020 --> 00:21:35.490 So let me draw in again these six points 00:21:35.490 --> 00:21:38.120 that I'm going to fit. 00:21:38.120 --> 00:21:46.000 And just say, for these few minutes, what's a cubic spline. 00:21:46.000 --> 00:21:48.680 I touched on that Wednesday, and now 00:21:48.680 --> 00:21:51.060 I just want to say a little bit more. 00:21:51.060 --> 00:21:56.940 And then the homework asked you to actually do it 00:21:56.940 --> 00:22:02.600 with values of a particular function. 00:22:02.600 --> 00:22:05.540 And see how close does the cubic spline come 00:22:05.540 --> 00:22:09.440 to the given function. 00:22:09.440 --> 00:22:12.260 OK, so what's a cubic spline? 00:22:12.260 --> 00:22:14.090 What does that word, spline, what 00:22:14.090 --> 00:22:16.110 should that mean in your mind? 00:22:16.110 --> 00:22:22.020 So a cubic spline is a cubic in each piece. 00:22:22.020 --> 00:22:24.800 A cubic in each piece. 00:22:24.800 --> 00:22:29.700 Now we've met cubics in each piece as finite elements. 00:22:29.700 --> 00:22:32.950 And part of this short discussion 00:22:32.950 --> 00:22:39.860 is to keep those two slightly different ideas separate. 00:22:39.860 --> 00:22:47.030 The finite element idea was also piecewise cubic. 00:22:47.030 --> 00:22:50.930 But it used values and slopes. 00:22:50.930 --> 00:22:52.560 So it was completely local. 00:22:52.560 --> 00:22:56.500 If I had a value, as I have here, six values, 00:22:56.500 --> 00:23:02.540 and if I also had six slopes, if I knew the slopes, then 00:23:02.540 --> 00:23:06.430 I would use those cubic elements. 00:23:06.430 --> 00:23:12.980 And I would fit a-- If I knew the height and slope there, 00:23:12.980 --> 00:23:14.770 and I knew the height and slope there, 00:23:14.770 --> 00:23:16.560 there would be exactly one cubic. 00:23:16.560 --> 00:23:19.340 Because I would have four conditions, 00:23:19.340 --> 00:23:21.660 two conditions there, two conditions there, 00:23:21.660 --> 00:23:22.470 would be four. 00:23:22.470 --> 00:23:25.990 I could fit a cubic between, another cubic there, 00:23:25.990 --> 00:23:27.550 another cubic there. 00:23:27.550 --> 00:23:32.380 And because I'm using the same slope from the left 00:23:32.380 --> 00:23:35.930 and from the right, the slope would be good. 00:23:35.930 --> 00:23:37.830 It would be continuous. 00:23:37.830 --> 00:23:42.030 The second derivative, the curvature, 00:23:42.030 --> 00:23:46.180 would not be continuous with those cubic elements. 00:23:46.180 --> 00:23:47.570 And that's the difference. 00:23:47.570 --> 00:23:53.230 So the difference is, splines have continuous, 00:23:53.230 --> 00:23:56.160 no jump in-- Let me just put it this way. 00:23:56.160 --> 00:24:03.160 No jump in the function. 00:24:03.160 --> 00:24:05.930 I'll use S, maybe, for spline. 00:24:05.930 --> 00:24:09.070 No jump in its slope. 00:24:09.070 --> 00:24:12.490 And no jump in the second derivative. 00:24:12.490 --> 00:24:14.460 So that's the difference. 00:24:14.460 --> 00:24:20.560 That the spline functions, the only 00:24:20.560 --> 00:24:26.880 jumps you see for those are jumps in the third derivative. 00:24:26.880 --> 00:24:29.820 So that makes this extremely smooth. 00:24:29.820 --> 00:24:33.240 So if I just try to draw now what a spline would do, 00:24:33.240 --> 00:24:37.330 it'll go through those points, coming out of the spline fit 00:24:37.330 --> 00:24:39.000 MATLAB command. 00:24:39.000 --> 00:24:45.570 And it'll be as smooth as I drew it. 00:24:45.570 --> 00:24:53.140 There is a change in the third derivative at these points. 00:24:53.140 --> 00:24:58.230 Actually, have you ever seen this word, "spline," before? 00:24:58.230 --> 00:25:05.730 It came out of naval engineering. 00:25:05.730 --> 00:25:15.460 When people were designing the shape of the ship. 00:25:15.460 --> 00:25:22.930 A naval architect is fitting the proposed shape of a ship-- 00:25:22.930 --> 00:25:30.340 this was before MATLAB, yeah, before life started. 00:25:30.340 --> 00:25:37.610 [LAUGHTER] They had little physical, slightly 00:25:37.610 --> 00:25:43.310 bendable-- I'll call them splines. 00:25:43.310 --> 00:25:44.677 I guess that's what they were. 00:25:44.677 --> 00:25:46.010 That's where the word came from. 00:25:46.010 --> 00:25:50.070 Some physical thing which was like a curve that you use-- I 00:25:50.070 --> 00:25:54.950 don't know if you guys ever did mechanical drawing. 00:25:54.950 --> 00:25:58.070 That was a freshman subject when I came to MIT. 00:25:58.070 --> 00:25:59.120 Mechanical drawing. 00:25:59.120 --> 00:26:04.960 I was terrible, terrible, at mechanical drawing. 00:26:04.960 --> 00:26:06.570 I don't know. 00:26:06.570 --> 00:26:13.040 I had a friend who helped. l probably helped him 00:26:13.040 --> 00:26:14.480 in some other course. 00:26:14.480 --> 00:26:16.730 Anyway, whatever. 00:26:16.730 --> 00:26:20.010 So there were physical things, curves you used to use. 00:26:20.010 --> 00:26:22.450 And these spline curves were used, 00:26:22.450 --> 00:26:27.700 and maybe still are used, by naval architects 00:26:27.700 --> 00:26:29.740 in creating a drawing. 00:26:29.740 --> 00:26:33.120 But I'm assuming that those things are now 00:26:33.120 --> 00:26:37.510 all computerized, and the spline command is used. 00:26:37.510 --> 00:26:39.320 Anyway, the result is that. 00:26:39.320 --> 00:26:42.860 Now, I have to draw one picture of a spline. 00:26:42.860 --> 00:26:44.470 Of the most important spline. 00:26:44.470 --> 00:26:48.960 And then I'm done with splines. 00:26:48.960 --> 00:26:56.320 So, what I'm going to draw is now a B-spline. 00:26:56.320 --> 00:27:02.850 And that's a basic spline. 00:27:02.850 --> 00:27:04.810 It could be one of our functions, 00:27:04.810 --> 00:27:09.710 it could be one of our functions phi(x). 00:27:09.710 --> 00:27:12.430 One of our trial functions. 00:27:12.430 --> 00:27:15.170 It could be, and let me comment on that. 00:27:15.170 --> 00:27:22.600 So let me remind myself, good or bad, with a question. 00:27:22.600 --> 00:27:24.170 And let me try to answer that. 00:27:24.170 --> 00:27:27.140 But let me first draw the B-spline. 00:27:27.140 --> 00:27:32.070 OK, so it's like a hat function. 00:27:32.070 --> 00:27:38.330 Actually a hat function is a linear spline. 00:27:38.330 --> 00:27:42.450 A hat function is that low level spline 00:27:42.450 --> 00:27:44.870 that's linear between pieces. 00:27:44.870 --> 00:27:53.940 Now the B-spline is going to have the value one there. 00:27:53.940 --> 00:27:57.340 It's going to have no jump in the function. 00:27:57.340 --> 00:27:59.730 So the function will go through there. 00:27:59.730 --> 00:28:01.420 No jump in the slope. 00:28:01.420 --> 00:28:04.370 No jump in the second derivative. 00:28:04.370 --> 00:28:07.180 And I want to get down to zero. 00:28:07.180 --> 00:28:13.660 I want to get down to zero. 00:28:13.660 --> 00:28:16.840 I want it to be as local as I can make it. 00:28:16.840 --> 00:28:22.750 So I want it to get to zero. 00:28:22.750 --> 00:28:27.280 Here's the point. 00:28:27.280 --> 00:28:31.070 With those cubic finite elements, 00:28:31.070 --> 00:28:33.110 I got them down to zero. 00:28:33.110 --> 00:28:35.830 They came in here with zero slope, 00:28:35.830 --> 00:28:42.340 and then they continued as zero, no problem. 00:28:42.340 --> 00:28:50.390 If I'm wanting the second derivative also 00:28:50.390 --> 00:28:53.560 to be zero, to be continuous, I won't be 00:28:53.560 --> 00:28:55.360 able to do it in one interval. 00:28:55.360 --> 00:28:59.310 It's going to take me a total of four intervals. 00:28:59.310 --> 00:29:01.870 This one can come down to some point, 00:29:01.870 --> 00:29:07.770 here, where another cubic starts. 00:29:07.770 --> 00:29:10.130 Maybe I should do the one here. 00:29:10.130 --> 00:29:14.450 Going up, you remember the allowed cubic spline 00:29:14.450 --> 00:29:21.680 would be an x cubed over six, some multiple of x cubed. 00:29:21.680 --> 00:29:24.980 I don't know what multiple it'll take, maybe x cubed over six 00:29:24.980 --> 00:29:25.570 is right. 00:29:25.570 --> 00:29:28.310 It'll come up to some point here. 00:29:28.310 --> 00:29:36.990 Now I've got to get it beginning to curve downwards. 00:29:36.990 --> 00:29:40.780 So I'm going to have to change the third derivative. 00:29:40.780 --> 00:29:45.730 Change to another cubic there, change to another cubic 00:29:45.730 --> 00:29:49.420 there, and a final cubic here. 00:29:49.420 --> 00:29:52.250 So there's a picture of a B-spline. 00:29:52.250 --> 00:29:59.500 And we could figure out, by requiring all these continuity 00:29:59.500 --> 00:30:03.470 conditions, we could figure out the formula for the cubic 00:30:03.470 --> 00:30:05.130 in these four pieces. 00:30:05.130 --> 00:30:07.350 And it would be symmetric, of course, 00:30:07.350 --> 00:30:09.200 across that center point. 00:30:09.200 --> 00:30:13.980 But I think I won't try to do it. 00:30:13.980 --> 00:30:18.580 I'll just leave that idea, then. 00:30:18.580 --> 00:30:20.080 There's a figure in the book showing 00:30:20.080 --> 00:30:24.910 a picture of the B-spline. 00:30:24.910 --> 00:30:28.880 So those are functions, extremely valuable 00:30:28.880 --> 00:30:32.010 in this interpolation problem. 00:30:32.010 --> 00:30:35.620 Because all splines are combinations of B-splines. 00:30:35.620 --> 00:30:38.570 Yeah, now let me pull this topic together. 00:30:38.570 --> 00:30:44.260 Every spline, every spline function, 00:30:44.260 --> 00:30:52.700 is combination of these B-splines. 00:30:52.700 --> 00:30:58.810 Let B_i(x) be the one centered on node i. 00:30:58.810 --> 00:31:03.450 And then there's a neighbor centered on node i+1. 00:31:03.450 --> 00:31:07.200 And a neighbor to the left centered on node i-1. 00:31:07.200 --> 00:31:09.220 What's the point? 00:31:09.220 --> 00:31:19.280 The point is that, at a typical node, i+1, 00:31:19.280 --> 00:31:23.030 three of these functions will be non-zero. 00:31:23.030 --> 00:31:28.110 With these very local hat functions at that node, 00:31:28.110 --> 00:31:31.240 the only one of the phi functions that wasn't zero 00:31:31.240 --> 00:31:32.950 is the one I've drawn. 00:31:32.950 --> 00:31:35.770 All the others were zero there, right? 00:31:35.770 --> 00:31:38.650 All the other hats were zero. 00:31:38.650 --> 00:31:40.200 There was just the one. 00:31:40.200 --> 00:31:44.100 But now there'll be a B-spline starting from here, 00:31:44.100 --> 00:31:47.780 going up to here, coming down from here. 00:31:47.780 --> 00:31:51.390 There'll be the B-spline starting from here, going up 00:31:51.390 --> 00:31:54.540 to here, up, down, so on. 00:31:54.540 --> 00:31:57.880 So at a typical node, I'm getting 00:31:57.880 --> 00:32:01.700 the one centered at that node, and also 00:32:01.700 --> 00:32:05.440 the one to the left, which is on its way down, 00:32:05.440 --> 00:32:08.500 and also the one to the right, which is on its way up. 00:32:08.500 --> 00:32:12.460 In other words, it's not as local 00:32:12.460 --> 00:32:17.880 as the other construction. 00:32:17.880 --> 00:32:26.090 So I would say good, because it's smooth, but bad 00:32:26.090 --> 00:32:30.020 because it's not fully local. 00:32:30.020 --> 00:32:32.420 Not completely local. 00:32:32.420 --> 00:32:35.230 It's reasonably local, in that there are only 00:32:35.230 --> 00:32:39.400 three functions that are affecting these points. 00:32:39.400 --> 00:32:44.090 If I use those polynomials, they weren't local at all. 00:32:44.090 --> 00:32:50.630 All the points, it was-- A typical x to the fifth 00:32:50.630 --> 00:32:52.970 is not zero at any of the points. 00:32:52.970 --> 00:32:56.220 These B-splines are zero at a lot of points, 00:32:56.220 --> 00:32:59.940 but three of them, one, two, and three, 00:32:59.940 --> 00:33:03.170 will be non-zero at a typical mesh point. 00:33:03.170 --> 00:33:13.550 OK, so they're not popular, as a result, for finite elements. 00:33:13.550 --> 00:33:16.750 I'm just wanting to be sure you make the distinction. 00:33:16.750 --> 00:33:20.900 They're very popular for the interpolation job. 00:33:20.900 --> 00:33:24.750 They're very popular for the interpolation job. 00:33:24.750 --> 00:33:30.680 I've got some combination of these at particular mesh 00:33:30.680 --> 00:33:31.210 points. 00:33:31.210 --> 00:33:36.470 It's supposed to agree with F at those mesh points. 00:33:36.470 --> 00:33:38.450 This is my system to solve. 00:33:38.450 --> 00:33:41.880 I create these functions, these B-splines. 00:33:41.880 --> 00:33:45.290 I'm given F at typical points. 00:33:45.290 --> 00:33:51.640 And I choose a combination which matches F at those points. 00:33:51.640 --> 00:33:54.550 Yeah, OK. 00:33:54.550 --> 00:34:00.990 Is there a question or discussion on this? 00:34:00.990 --> 00:34:05.550 I don't like to not say anything about such an important problem 00:34:05.550 --> 00:34:09.380 as putting curves through points. 00:34:09.380 --> 00:34:16.121 But I don't want to make it a course on splines. 00:34:16.121 --> 00:34:16.620 Yes? 00:34:16.620 --> 00:34:18.620 AUDIENCE: [UNINTELLIGIBLE] 00:34:18.620 --> 00:34:24.720 PROFESSOR STRANG: I don't know that I'll-- Oh, OK, yes. 00:34:24.720 --> 00:34:25.730 All right. 00:34:25.730 --> 00:34:30.010 Two or three comments, and that suggests a good question, 00:34:30.010 --> 00:34:32.700 a very good question. 00:34:32.700 --> 00:34:36.240 Can I make a separate comment that if I 00:34:36.240 --> 00:34:40.910 go into two dimensions, this gets much tougher. 00:34:40.910 --> 00:34:44.160 What happens if I had a function of x and y? 00:34:44.160 --> 00:34:46.940 So that I've got points on a surface, 00:34:46.940 --> 00:34:49.930 and I'm trying to fit a surface to it. 00:34:49.930 --> 00:34:55.630 Just one message first: that's not as easy. 00:34:55.630 --> 00:35:00.970 It's pretty easy if those points are on a rectangular grid. 00:35:00.970 --> 00:35:04.650 So this is like typical. 00:35:04.650 --> 00:35:06.830 And then I'll come to your Nyquist question, 00:35:06.830 --> 00:35:09.400 which is a very good one. 00:35:09.400 --> 00:35:14.350 Can I just-- because we're coming in to 2-D now. 00:35:14.350 --> 00:35:17.310 Suppose I-- There's two dimensions. 00:35:17.310 --> 00:35:18.830 I have a grid. 00:35:18.830 --> 00:35:20.890 Let's suppose it's a nice grid. 00:35:20.890 --> 00:35:27.080 And at every point, at every grid point, I have a height. 00:35:27.080 --> 00:35:28.620 And I'm fitting a surface. 00:35:28.620 --> 00:35:30.310 Right? 00:35:30.310 --> 00:35:38.480 My function is now a function of x and y. x is here, y is here, 00:35:38.480 --> 00:35:44.400 F is the surface coming out of board, in the third dimension. 00:35:44.400 --> 00:35:48.300 And fitting those points, if they're regularly spaced 00:35:48.300 --> 00:35:50.660 like that, my life would be OK. 00:35:50.660 --> 00:35:56.080 I could use sort of products of splines in 1-D. 00:35:56.080 --> 00:36:04.880 I could use products of basis-- of splines 00:36:04.880 --> 00:36:09.270 in the x-direction times splines in the y-direction. 00:36:09.270 --> 00:36:13.260 And it would be pretty successful. 00:36:13.260 --> 00:36:18.680 It would be, not quite as nice, but almost OK. 00:36:18.680 --> 00:36:28.650 But if I had irregularly spaced points from a general grid, 00:36:28.650 --> 00:36:30.220 it's not as easy. 00:36:30.220 --> 00:36:33.880 And I won't-- People have obviously had to figure out how 00:36:33.880 --> 00:36:38.480 to do it, and to repeat again, that's what the CAD/CAM world 00:36:38.480 --> 00:36:42.010 is having to do all the time, is fit a curve, 00:36:42.010 --> 00:36:44.190 fit a surface through points. 00:36:44.190 --> 00:36:45.950 It's significant. 00:36:45.950 --> 00:36:49.250 Now, you asked about Nyquist. 00:36:49.250 --> 00:36:51.420 So that's a good question. 00:36:51.420 --> 00:36:59.150 Can I just say, I have to say what-- So 00:36:59.150 --> 00:37:00.830 what's a function called? 00:37:00.830 --> 00:37:04.210 Band-limited. 00:37:04.210 --> 00:37:07.330 How many have heard Nyquist's name? 00:37:07.330 --> 00:37:10.900 Okay, some of you may know a lot more than I about it. 00:37:10.900 --> 00:37:18.140 But let me just get some context for band-limited functions. 00:37:18.140 --> 00:37:21.010 Okay. 00:37:21.010 --> 00:37:23.180 This is actually a topic that will 00:37:23.180 --> 00:37:27.180 belong in the third part of our course, in the Fourier part. 00:37:27.180 --> 00:37:35.240 So this is a Fourier idea. 00:37:35.240 --> 00:37:38.930 So I'll come back to it, actually. 00:37:38.930 --> 00:37:47.740 So right here I'm just going to say, very briefly, 00:37:47.740 --> 00:37:51.890 how it might connect us to what I've said today. 00:37:51.890 --> 00:37:57.340 But then let's make a plan to, when 00:37:57.340 --> 00:38:04.560 we've got the idea of Fourier coefficients. 00:38:04.560 --> 00:38:11.770 What is a band-limited function? 00:38:11.770 --> 00:38:16.920 You know the Fourier idea is to take F(x), 00:38:16.920 --> 00:38:23.970 and write it as some combination of pure frequencies. e^(i*K*x), 00:38:23.970 --> 00:38:27.420 let me say. e^(i*K*x). 00:38:27.420 --> 00:38:30.570 So this is Fourier that's coming. 00:38:30.570 --> 00:38:33.300 I take the function F(x), and I write it-- 00:38:33.300 --> 00:38:36.160 I could think of it as a combination 00:38:36.160 --> 00:38:41.390 of pure exponentials, pure frequencies. 00:38:41.390 --> 00:38:44.210 That would be a Fourier series. 00:38:44.210 --> 00:38:47.660 So that's a Fourier series because I'm 00:38:47.660 --> 00:38:50.590 using-- K has integer values. 00:38:50.590 --> 00:38:55.110 The frequencies are zero, one, two, three, whatever. 00:38:55.110 --> 00:39:01.820 Now that we'll use, so that Fourier series 00:39:01.820 --> 00:39:05.030 comes for functions that are periodic. 00:39:05.030 --> 00:39:06.640 They repeat every 2pi. 00:39:06.640 --> 00:39:10.410 Because those functions, if I increase x by 2pi, 00:39:10.410 --> 00:39:11.650 don't change. 00:39:11.650 --> 00:39:13.970 So I'm repeating every 2pi. 00:39:13.970 --> 00:39:21.920 Now I have to say a word about the other possibility, which 00:39:21.920 --> 00:39:26.150 would be to have all frequencies. 00:39:26.150 --> 00:39:27.950 I'll integrate now, dK. 00:39:27.950 --> 00:39:32.150 Instead of summing on K, I'll integrate on K. 00:39:32.150 --> 00:39:35.010 What does band-limited mean? 00:39:35.010 --> 00:39:41.770 Band-limited means that only frequencies in a certain range, 00:39:41.770 --> 00:39:44.970 say a range around zero, are included. 00:39:44.970 --> 00:39:49.560 So a band-limited function would be a function whose frequencies 00:39:49.560 --> 00:39:55.490 go from some value, say minus omega to omega, 00:39:55.490 --> 00:39:58.380 instead of going from minus infinity to infinity. 00:39:58.380 --> 00:40:05.760 This would be band-limited frequencies 00:40:05.760 --> 00:40:12.280 between minus omega and omega. 00:40:12.280 --> 00:40:18.730 So that's another kind of smooth function. 00:40:18.730 --> 00:40:22.910 That's another way-- Functions that 00:40:22.910 --> 00:40:27.790 have only low frequencies are associated with smoothness. 00:40:27.790 --> 00:40:31.660 High frequencies are associated with fast oscillations. 00:40:31.660 --> 00:40:36.160 So the class of band-limited functions 00:40:36.160 --> 00:40:41.750 gives me another, a Fourier way, to talk about smoothness. 00:40:41.750 --> 00:40:45.180 So for us, smoothness was something about how many 00:40:45.180 --> 00:40:47.760 derivatives were continuous. 00:40:47.760 --> 00:40:50.900 That's the sort of smoothness in the x domain. 00:40:50.900 --> 00:40:53.040 How many derivatives. 00:40:53.040 --> 00:40:55.320 Smoothness in the frequency domain 00:40:55.320 --> 00:40:59.260 is, how fast do the frequencies drop off. 00:40:59.260 --> 00:41:03.200 And here, band-limited means they drop like a shot. 00:41:03.200 --> 00:41:09.270 Band-limited means that the frequencies in the function, 00:41:09.270 --> 00:41:13.660 that these e^(i*K*x)'s are not there for high frequencies. 00:41:13.660 --> 00:41:18.400 High frequencies are out for a band-limited function. 00:41:18.400 --> 00:41:26.070 And then Shannon has a formula for the natural way 00:41:26.070 --> 00:41:32.300 to fit-- So our same interpolation problem. 00:41:32.300 --> 00:41:36.210 So now, completing the answer to your question. 00:41:36.210 --> 00:41:38.820 So I have these points. 00:41:38.820 --> 00:41:45.400 So Shannon could fit a function, and it would look smooth, 00:41:45.400 --> 00:41:47.590 through those points. 00:41:47.590 --> 00:41:53.010 And his function would be band-limited. 00:41:53.010 --> 00:41:55.340 So it would be smooth again. 00:41:55.340 --> 00:42:02.430 It would be-- yeah, it would be smooth. 00:42:02.430 --> 00:42:06.250 In some way, this Shannon band-limited stuff 00:42:06.250 --> 00:42:12.180 is the limit of splines as the spline degree goes way up. 00:42:12.180 --> 00:42:16.480 So we did hat functions, degree one splines. 00:42:16.480 --> 00:42:23.740 Cubic splines I recommended as a pretty reliable construction. 00:42:23.740 --> 00:42:26.820 But you could do fifth degrees splines, seventh degree 00:42:26.820 --> 00:42:29.320 splines, you could keep going. 00:42:29.320 --> 00:42:32.120 And in the limit, you would get this. 00:42:32.120 --> 00:42:36.030 So maybe that's some partial answer 00:42:36.030 --> 00:42:42.350 to the connection between splines and this Fourier world. 00:42:42.350 --> 00:42:43.690 OK. 00:42:43.690 --> 00:42:45.090 Thanks. 00:42:45.090 --> 00:42:48.640 So these are topics now-- Wow, today's lecture 00:42:48.640 --> 00:42:52.850 is kind of-- Can I do one really important thing now 00:42:52.850 --> 00:42:54.330 in today's lecture? 00:42:54.330 --> 00:42:56.290 For you to remember? 00:42:56.290 --> 00:42:58.050 Gradient and divergence? 00:42:58.050 --> 00:42:59.700 I don't want you to spend the weekend 00:42:59.700 --> 00:43:10.660 without thinking about gradient and divergence. [LAUGHTER] OK. 00:43:10.660 --> 00:43:14.410 Here's the idea. 00:43:14.410 --> 00:43:19.500 For lots and lots of applications, for a region, 00:43:19.500 --> 00:43:29.370 let's say, in the plane, I have the same-- What 00:43:29.370 --> 00:43:31.960 physical example shall I pick now? 00:43:31.960 --> 00:43:35.530 Maybe I'll let u be the temperature. 00:43:35.530 --> 00:43:37.420 So instead of being displacement, 00:43:37.420 --> 00:43:41.400 let me make it temperature. u. 00:43:41.400 --> 00:43:42.830 OK. 00:43:42.830 --> 00:43:46.310 Then I have a temperature gradient. 00:43:46.310 --> 00:43:47.290 A slope. 00:43:47.290 --> 00:43:51.480 But now, the whole point is, that's a function of x and y. 00:43:51.480 --> 00:43:55.600 Then I have a temperature gradient. 00:43:55.600 --> 00:43:59.010 And if I'm consistent with the notation, that'll be e(x,y). 00:44:01.860 --> 00:44:08.440 And then you'll expect that there's some c(x,y). 00:44:08.440 --> 00:44:13.640 Some operator c that tells me how much heat flows. 00:44:13.640 --> 00:44:17.100 This will tell me something about the the, 00:44:17.100 --> 00:44:18.960 the thermal conductivity. 00:44:18.960 --> 00:44:20.180 Right? 00:44:20.180 --> 00:44:27.300 Really, as I speak about this framework, 00:44:27.300 --> 00:44:31.470 I'm just uttering the correct words. 00:44:31.470 --> 00:44:34.260 Having started with temperature, this thing 00:44:34.260 --> 00:44:36.180 should be a temperature gradient. 00:44:36.180 --> 00:44:39.360 Then I should have some physical thermal conductivity, 00:44:39.360 --> 00:44:43.110 different for different metals or different materials. 00:44:43.110 --> 00:44:48.560 And I'll have a heat flow, w(x,y). 00:44:48.560 --> 00:44:53.150 And everybody knows that w will be c times e. 00:44:53.150 --> 00:44:57.610 And then there will be some A transpose. 00:44:57.610 --> 00:45:01.860 Of course, there will be some A transpose here, 00:45:01.860 --> 00:45:03.920 and some A here. 00:45:03.920 --> 00:45:07.740 And up here I'll have a balance equation. 00:45:07.740 --> 00:45:13.730 OK. 00:45:13.730 --> 00:45:20.650 I just want to think, what's the operator A? 00:45:20.650 --> 00:45:24.190 If we can focus on that question, 00:45:24.190 --> 00:45:28.340 then that's what's going to occupy us for, certainly 00:45:28.340 --> 00:45:31.010 the whole of next week. 00:45:31.010 --> 00:45:35.960 So I've actually used the word gradient. 00:45:35.960 --> 00:45:39.280 We have functions of two variables. 00:45:39.280 --> 00:45:42.730 We're looking for the change, the rate of change, 00:45:42.730 --> 00:45:44.890 the steepness of those functions. 00:45:44.890 --> 00:45:52.640 So this A, Au, is going to give me two derivatives. 00:45:52.640 --> 00:45:56.650 I've got two variables, there are two first derivatives. 00:45:56.650 --> 00:46:01.160 Both of them are important. 00:46:01.160 --> 00:46:03.500 That's what the A is. 00:46:03.500 --> 00:46:09.000 For the next big example in the course. 00:46:09.000 --> 00:46:11.300 The final major example of the course 00:46:11.300 --> 00:46:15.800 is, when A acts on a function of two variables, 00:46:15.800 --> 00:46:19.420 because I'm in a region in the plane, 00:46:19.420 --> 00:46:23.450 to find its rate of change. 00:46:23.450 --> 00:46:26.730 And this is called the gradient. 00:46:26.730 --> 00:46:34.030 The shorthand is the gradient of u. 00:46:34.030 --> 00:46:37.720 So we have to understand that. 00:46:37.720 --> 00:46:41.870 We have to understand what the gradient is. 00:46:41.870 --> 00:46:45.460 And, of course, we want to know its transpose. 00:46:45.460 --> 00:46:48.900 So can I just think, what should be 00:46:48.900 --> 00:46:54.410 the transpose of the gradient? 00:46:54.410 --> 00:46:55.000 OK. 00:46:55.000 --> 00:46:57.510 I'll take that picture out. 00:46:57.510 --> 00:46:58.470 OK. 00:46:58.470 --> 00:47:03.740 Thinking now about the transpose of the gradient. 00:47:03.740 --> 00:47:06.070 OK. 00:47:06.070 --> 00:47:14.560 So A itself is, you could say, is d/dx and d/dy. 00:47:14.560 --> 00:47:20.000 You notice how I'm separating out A from Au. 00:47:20.000 --> 00:47:22.030 When this acts on a function, this 00:47:22.030 --> 00:47:26.040 is the gradient operator that acts on a function, u, 00:47:26.040 --> 00:47:29.700 to produce du/dx and du/dy. 00:47:29.700 --> 00:47:30.510 OK. 00:47:30.510 --> 00:47:33.670 Now what's the transpose of this? 00:47:33.670 --> 00:47:35.480 OK. 00:47:35.480 --> 00:47:38.000 You can guess what it should be, and then we'll 00:47:38.000 --> 00:47:40.960 see that yes, that guess is correct. 00:47:40.960 --> 00:47:44.860 So what should A transpose look like? 00:47:44.860 --> 00:47:54.540 If there's any justice, A transpose should be-- OK, 00:47:54.540 --> 00:47:58.560 this is two by one. 00:47:58.560 --> 00:48:06.310 The transpose you would expect to be a row, a row vector. 00:48:06.310 --> 00:48:08.930 I should have the transpose of that there. 00:48:08.930 --> 00:48:11.160 And what is the transpose of that? 00:48:11.160 --> 00:48:13.660 Just tell me. 00:48:13.660 --> 00:48:15.370 Because we have an idea. 00:48:15.370 --> 00:48:19.790 What should be the transpose of d/dx? 00:48:19.790 --> 00:48:20.650 Negative d/dx. 00:48:24.040 --> 00:48:26.710 And what should be the transpose of d/dy? 00:48:26.710 --> 00:48:28.690 It should be negative d/dy. 00:48:28.690 --> 00:48:30.550 Those are two different pieces. 00:48:30.550 --> 00:48:32.460 This is not run together. 00:48:32.460 --> 00:48:35.020 There's a big space in there. 00:48:35.020 --> 00:48:37.750 That's two pieces. 00:48:37.750 --> 00:48:44.810 So what is A transpose applied to a-- So, heat flow w. 00:48:44.810 --> 00:48:50.840 I want to say, what is A transpose applied to w? 00:48:50.840 --> 00:48:54.900 You're going to see this again, but we'll just 00:48:54.900 --> 00:48:58.210 take these minutes to show it for the first time. 00:48:58.210 --> 00:49:03.030 So A transpose-- Wait a minute. 00:49:03.030 --> 00:49:08.090 Is w a function, or is it a vector? 00:49:08.090 --> 00:49:11.540 Yeah we've got to get that straight before we start. 00:49:11.540 --> 00:49:14.220 Here's an ordinary function, a scalar function. 00:49:14.220 --> 00:49:18.330 Just whatever, x squared plus y squared. 00:49:18.330 --> 00:49:21.520 What is e? 00:49:21.520 --> 00:49:25.660 Suppose this is x squared plus y squared. 00:49:25.660 --> 00:49:27.750 Let's have a specific example. 00:49:27.750 --> 00:49:31.590 What would e then be? 00:49:31.590 --> 00:49:33.630 It's got two components, right? 00:49:33.630 --> 00:49:38.910 It's got an x derivative and a y derivative. e is a vector, 00:49:38.910 --> 00:49:41.870 [2x, 2y]. 00:49:41.870 --> 00:49:44.270 Then I multiply by c. 00:49:44.270 --> 00:49:46.730 So this w has two components. 00:49:46.730 --> 00:49:50.870 This has got a w_1(x,y), w_2(x,y). 00:49:53.910 --> 00:49:56.670 So just keep things straight. 00:49:56.670 --> 00:50:00.340 So it's that that I want to apply A transpose to. 00:50:00.340 --> 00:50:08.550 So A transpose w is minus d/dx, minus d/dy. 00:50:08.550 --> 00:50:10.210 And everything's coming out right 00:50:10.210 --> 00:50:14.450 because it's applied to a function w_1 and w_2. 00:50:14.450 --> 00:50:17.740 w is a vector field. 00:50:17.740 --> 00:50:20.350 It's not a scalar field, it's a vector field. 00:50:20.350 --> 00:50:24.050 And the result is just what it should be: 00:50:24.050 --> 00:50:31.290 minus dw_1/dx minus dw_2/dy. 00:50:31.290 --> 00:50:37.580 OK, good. 00:50:37.580 --> 00:50:40.620 This has got to come out of integration by parts, right? 00:50:40.620 --> 00:50:43.290 So we'll have to think: what does integration by parts 00:50:43.290 --> 00:50:45.910 mean in two variables? 00:50:45.910 --> 00:50:50.570 And it's a famous formula named after Gauss and Green. 00:50:50.570 --> 00:50:52.330 The Green's formula, often. 00:50:52.330 --> 00:50:56.520 But do you recognize what I'm looking at here? 00:50:56.520 --> 00:51:01.180 This is so important it has a name. 00:51:01.180 --> 00:51:02.650 And what's the name? 00:51:02.650 --> 00:51:04.260 And then we're ready to go. 00:51:04.260 --> 00:51:09.470 What's the name of, if I take a vector field, like [2x, 2y]. 00:51:09.470 --> 00:51:11.840 Let me take [2x, 2y]. 00:51:11.840 --> 00:51:17.570 as a specific example. 00:51:17.570 --> 00:51:21.480 Suppose w is [2x,2y]. 00:51:21.480 --> 00:51:26.150 That was multiplied by, let me take c to be one. 00:51:26.150 --> 00:51:33.140 Then what is A transpose w? 00:51:33.140 --> 00:51:34.800 Specifically. 00:51:34.800 --> 00:51:39.010 What am I getting out of it? 00:51:39.010 --> 00:51:43.860 What do I get from here? 00:51:43.860 --> 00:51:48.060 That's minus the x derivative of the first guy, 00:51:48.060 --> 00:51:50.060 and the x derivative of that guy is two, 00:51:50.060 --> 00:51:52.830 so I'm getting a minus two. 00:51:52.830 --> 00:51:58.650 And this is minus the y derivative of the second guy, 00:51:58.650 --> 00:52:00.970 so that's another minus two. 00:52:00.970 --> 00:52:02.250 So I'm getting a number. 00:52:02.250 --> 00:52:03.990 It happens to be a number here. 00:52:03.990 --> 00:52:09.340 I chose such a simple function it came out to be a number. 00:52:09.340 --> 00:52:11.100 And what's the name? 00:52:11.100 --> 00:52:15.550 So this is minus the what of w? 00:52:15.550 --> 00:52:20.470 Just tell me, what's the name everybody uses 00:52:20.470 --> 00:52:22.890 for that operation? 00:52:22.890 --> 00:52:24.770 The divergence. 00:52:24.770 --> 00:52:28.040 Minus the divergence of w. 00:52:28.040 --> 00:52:37.950 So I what I'm saying here is that this-- I'm 00:52:37.950 --> 00:52:41.240 saying it because somehow I remember 00:52:41.240 --> 00:52:43.470 studying vector calculus. 00:52:43.470 --> 00:52:48.070 And in that process, I learned about the gradient, 00:52:48.070 --> 00:52:51.360 and I learned about the divergence. 00:52:51.360 --> 00:52:56.120 But I never learned that one was the transpose of the other. 00:52:56.120 --> 00:53:03.420 I think, looking back, that was criminal. 00:53:03.420 --> 00:53:07.500 To describe those-- With a minus sign, of course. 00:53:07.500 --> 00:53:16.230 I learned Green's formula, but now we'll see what it means. 00:53:16.230 --> 00:53:17.880 OK, that's next week's job. 00:53:17.880 --> 00:53:20.501 Have a great weekend and see you Monday.