WEBVTT 00:00:00.030 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.710 Commons license. 00:00:03.710 --> 00:00:05.450 Your support will help MIT OpenCourseWare 00:00:05.450 --> 00:00:09.395 continue to offer high quality educational resources for free. 00:00:09.395 --> 00:00:11.520 To make a donation, or to view additional materials 00:00:11.520 --> 00:00:15.246 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.246 --> 00:00:15.870 at ocw.mit.edu. 00:00:20.880 --> 00:00:25.210 PROFESSOR STRANG: So I'm ready for anything, hope I am. 00:00:25.210 --> 00:00:27.030 Questions about any topic. 00:00:27.030 --> 00:00:27.530 Yes. 00:00:27.530 --> 00:00:28.816 AUDIENCE: [INAUDIBLE] 00:00:28.816 --> 00:00:31.190 PROFESSOR STRANG: I feel this is like a White House press 00:00:31.190 --> 00:00:31.730 conference. 00:00:31.730 --> 00:00:35.760 I think there's always somebody in the front row who 00:00:35.760 --> 00:00:38.610 gets to ask the first question, and then 00:00:38.610 --> 00:00:41.350 gets to say thank you Mr. President at the end, 00:00:41.350 --> 00:00:47.450 and then I'm off. 00:00:47.450 --> 00:00:50.800 Yes. 00:00:50.800 --> 00:00:53.100 I'm tempted by the way to ask you all, 00:00:53.100 --> 00:00:55.060 are you going to vote next Tuesday 00:00:55.060 --> 00:00:57.780 and of course I'd like to know who you vote for, 00:00:57.780 --> 00:01:00.130 and I'd like to give you my advice. 00:01:00.130 --> 00:01:04.050 But I don't know that that's proper. 00:01:04.050 --> 00:01:06.510 If anybody wants advice, they can email. 00:01:06.510 --> 00:01:08.460 But please vote. 00:01:08.460 --> 00:01:09.320 Please vote. 00:01:09.320 --> 00:01:10.380 Yeah. 00:01:10.380 --> 00:01:13.900 Alright, question here and then we'll have, well, yeah. 00:01:13.900 --> 00:01:19.360 AUDIENCE: [INAUDIBLE] 00:01:19.360 --> 00:01:21.660 PROFESSOR STRANG: Oh, OK, so those were just posted. 00:01:21.660 --> 00:01:24.930 Like, I see. 00:01:24.930 --> 00:01:29.400 3.3 number two. 00:01:29.400 --> 00:01:33.970 AUDIENCE: [INAUDIBLE] 00:01:33.970 --> 00:01:40.130 PROFESSOR STRANG: OK, so this was a case -- yeah, right. 00:01:40.130 --> 00:01:42.640 Oh, OK. 00:01:42.640 --> 00:01:50.690 We could be wrong so this is 3.3 number two, 00:01:50.690 --> 00:01:55.970 asks you about the flow field. 00:01:55.970 --> 00:02:00.540 Which has no flow in the x direction, 00:02:00.540 --> 00:02:03.530 the velocity in the x direction is zero. 00:02:03.530 --> 00:02:06.660 The velocity in the y direction is x. 00:02:06.660 --> 00:02:10.210 OK, so suppose we just take that as a full field 00:02:10.210 --> 00:02:13.820 and try to understand, is it a gradient? 00:02:13.820 --> 00:02:16.260 I mean, so what are the questions I would ask? 00:02:16.260 --> 00:02:17.940 Is it a gradient of anything? 00:02:17.940 --> 00:02:23.290 Because we're now thinking v and w are pretty much the same guy. 00:02:23.290 --> 00:02:26.060 So is it a gradient, yes or no? 00:02:26.060 --> 00:02:29.110 If so, what's the potential? 00:02:29.110 --> 00:02:32.160 Is it divergence-free, yes or no? 00:02:32.160 --> 00:02:34.810 If it is, what's the stream function? 00:02:34.810 --> 00:02:40.510 And of course if the answer to both tests was yes, then 00:02:40.510 --> 00:02:44.110 we would be talking about Laplace's equation. 00:02:44.110 --> 00:02:49.600 I suspect for this example the answer, 00:02:49.600 --> 00:02:52.480 at least to one of the two questions, will be no. 00:02:52.480 --> 00:02:58.160 So we won't have the two pieces coming together into Laplace. 00:02:58.160 --> 00:02:59.820 OK, so first of all. 00:02:59.820 --> 00:03:01.860 Is it a gradient? 00:03:01.860 --> 00:03:08.300 What's the test for, so my two questions are, 00:03:08.300 --> 00:03:13.160 is v the gradient of some u? 00:03:13.160 --> 00:03:17.560 And what's the test for that? 00:03:17.560 --> 00:03:19.930 You remember if it is a gradient -- 00:03:19.930 --> 00:03:21.860 and see if I can remember myself. 00:03:21.860 --> 00:03:28.130 If it is a gradient, then this is du/dx, and this is du/dy, 00:03:28.130 --> 00:03:33.050 and the condition that v_1 and v_2 would have to satisfy 00:03:33.050 --> 00:03:36.090 is that the y derivative of that would 00:03:36.090 --> 00:03:38.820 have to equal the x derivative of that, 00:03:38.820 --> 00:03:42.840 because on the right-hand side they are the same. 00:03:42.840 --> 00:03:45.550 u_xy is the same as u_yx. 00:03:45.550 --> 00:03:53.300 So I would look at -- so let me write that again. 00:03:53.300 --> 00:04:04.620 I need dv_1/dy to equal dv_2/dx, and is that true in this 00:04:04.620 --> 00:04:06.690 example? 00:04:06.690 --> 00:04:09.780 What's dv_1/dy? 00:04:09.780 --> 00:04:10.546 Zero. 00:04:10.546 --> 00:04:11.170 What's dv_2/dx? 00:04:14.610 --> 00:04:15.600 One. 00:04:15.600 --> 00:04:17.960 So the answer's no. 00:04:17.960 --> 00:04:19.360 OK. 00:04:19.360 --> 00:04:22.870 So, test failed. 00:04:22.870 --> 00:04:29.480 Alright, the second question is does it possibly sit over 00:04:29.480 --> 00:04:31.680 in the divergence-free world? 00:04:31.680 --> 00:04:36.310 Is the divergence of, now I'll call it w. equal zero? 00:04:36.310 --> 00:04:39.420 So the answer was no to that question 00:04:39.420 --> 00:04:43.720 but now I think the answer to this question will be yes. 00:04:43.720 --> 00:04:47.550 Because what's the divergence of this thing? 00:04:47.550 --> 00:04:51.020 It's the x derivative of that, which is certainly 00:04:51.020 --> 00:04:55.780 zero, plus the y derivative of that, which is certainly zero. 00:04:55.780 --> 00:04:58.260 So the answer is yes. 00:04:58.260 --> 00:05:04.290 So there's no potential but there is a stream function, 00:05:04.290 --> 00:05:04.980 right? 00:05:04.980 --> 00:05:08.200 Because the stream function comes in with this test. 00:05:08.200 --> 00:05:12.390 So let's remember what, just from today's lecture, what 00:05:12.390 --> 00:05:13.980 was the stream function? 00:05:13.980 --> 00:05:26.550 From dw_1/dx+dw_2/dy=0, that'll be satisfied if w_1 is the y 00:05:26.550 --> 00:05:28.160 derivative of the stream function, 00:05:28.160 --> 00:05:32.220 and w_2 is minus the x derivative. 00:05:32.220 --> 00:05:38.730 Because then this matches the x derivative of this 00:05:38.730 --> 00:05:41.980 plus the y derivative of this, which is the divergence; 00:05:41.980 --> 00:05:44.980 on the right-hand side I would get zero. 00:05:44.980 --> 00:05:47.040 So there's got to be an s, and what is it? 00:05:47.040 --> 00:05:49.150 Probably not hard to find. 00:05:49.150 --> 00:05:50.930 Let's see. 00:05:50.930 --> 00:05:53.540 Here w_1 is zero, so that tells me 00:05:53.540 --> 00:05:57.600 s doesn't depend on y at all. w_2 00:05:57.600 --> 00:06:02.860 is x, so x is supposed to be minus the x derivative 00:06:02.860 --> 00:06:05.930 of the stream function, so what is the stream function now? 00:06:05.930 --> 00:06:10.330 Have I got room to put it here? 00:06:10.330 --> 00:06:11.820 Just about. 00:06:11.820 --> 00:06:16.440 What will work? 00:06:16.440 --> 00:06:21.060 So again, here's w_1. 00:06:21.060 --> 00:06:25.420 The y derivative of s is zero. w_2 tells me 00:06:25.420 --> 00:06:28.710 that the x derivative of s is minus x. 00:06:28.710 --> 00:06:31.660 So all I'm looking for is a function that only depends 00:06:31.660 --> 00:06:36.520 on x, has no dependence on y, and its derivative 00:06:36.520 --> 00:06:38.530 should be minus x. 00:06:38.530 --> 00:06:41.260 So what's the function? 00:06:41.260 --> 00:06:45.070 Minus a half of x squared. 00:06:45.070 --> 00:06:54.430 Yeah, so this gives me s equal minus a half of x squared. 00:06:54.430 --> 00:06:58.830 Alright, so you're saying that, so there 00:06:58.830 --> 00:07:02.560 is a stream function, right. 00:07:02.560 --> 00:07:04.780 And what does that travel along? 00:07:04.780 --> 00:07:09.110 That travels along steam, that means that the flow buzzes 00:07:09.110 --> 00:07:10.800 along streamlines. 00:07:10.800 --> 00:07:12.650 And what are the streamlines? 00:07:12.650 --> 00:07:16.220 They're the lines where s is constant. 00:07:16.220 --> 00:07:19.100 Equipotentials were the lines where u is constant, 00:07:19.100 --> 00:07:22.050 but here we don't have a u in this problem. 00:07:22.050 --> 00:07:25.300 Streamlines are lines where the s is constant, 00:07:25.300 --> 00:07:28.030 so minus 1/2 x squared is constant, 00:07:28.030 --> 00:07:31.270 what's the picture look like? 00:07:31.270 --> 00:07:35.910 Picture then, for that, well, and you 00:07:35.910 --> 00:07:42.600 know what the flow is doing at a typical point here. 00:07:42.600 --> 00:07:46.390 Say x=3, y=1. 00:07:46.390 --> 00:07:49.630 Let me draw the little arrow. 00:07:49.630 --> 00:07:55.150 With a big chalk Which way is the flow going? 00:07:55.150 --> 00:08:01.940 Well, the x component is zero, the y component is x. 00:08:01.940 --> 00:08:05.730 So the flow is going up there, right? 00:08:05.730 --> 00:08:10.270 Here the y component's x. 00:08:10.270 --> 00:08:16.330 This whole line is all traveling up with the same velocity. 00:08:16.330 --> 00:08:19.770 If I drop a leaf there, it buzzes up that straight line. 00:08:19.770 --> 00:08:21.570 So that's the streamline. 00:08:21.570 --> 00:08:27.940 And its velocity is, that's x equal -- if, say, 00:08:27.940 --> 00:08:33.590 the velocity is three then this speed is three. 00:08:33.590 --> 00:08:35.440 It's going up that line. 00:08:35.440 --> 00:08:38.160 So that's a streamline. 00:08:38.160 --> 00:08:41.780 And sure enough, on that line minus 1/2 x squared 00:08:41.780 --> 00:08:42.950 is a constant. 00:08:42.950 --> 00:08:45.850 So you see we're not talking parabolas 00:08:45.850 --> 00:08:51.200 here because our curve is not y equals minus 1/2 x squared, 00:08:51.200 --> 00:08:54.140 it's minus 1/2 x squared equal constant. 00:08:54.140 --> 00:08:55.330 Yeah, that's what we want. 00:08:55.330 --> 00:09:02.630 So, but now having got so far, let me take x=1, say. 00:09:02.630 --> 00:09:04.650 What's the flow like on that? 00:09:04.650 --> 00:09:11.340 So there's a streamline with s equal constant. 00:09:11.340 --> 00:09:14.070 And the velocity on that is zero, 00:09:14.070 --> 00:09:17.750 so nothing is going in that -- horizontally. 00:09:17.750 --> 00:09:25.120 And now it's one, so the flow is slower up this line. 00:09:25.120 --> 00:09:26.640 OK, slower flow. 00:09:26.640 --> 00:09:30.140 This was faster flow. 00:09:30.140 --> 00:09:34.610 And then the question that's in that homework problem is, 00:09:34.610 --> 00:09:38.570 is there any rotation in this flow? 00:09:38.570 --> 00:09:45.300 We think about rotation, we have an image of rotational flow. 00:09:45.300 --> 00:09:49.370 And that could be the next example, we could figure out. 00:09:49.370 --> 00:09:54.660 A flow that goes around in circles, right? 00:09:54.660 --> 00:09:56.440 Those would be the streamlines. 00:09:56.440 --> 00:10:03.830 So this would be like pure rotation, shall I call it. 00:10:03.830 --> 00:10:05.330 But I don't have that there. 00:10:05.330 --> 00:10:08.580 I just want to draw the other picture, in which 00:10:08.580 --> 00:10:16.410 the streamlines are circles. 00:10:16.410 --> 00:10:19.790 To have another nice, clean, beautiful example. 00:10:19.790 --> 00:10:21.910 OK, but here we don't have. 00:10:21.910 --> 00:10:24.070 Our streamlines are straight lines, 00:10:24.070 --> 00:10:26.910 and yet we have rotation. 00:10:26.910 --> 00:10:30.490 That's the point here. 00:10:30.490 --> 00:10:32.180 Why do I say we have rotation? 00:10:32.180 --> 00:10:36.530 Because the test for rotation was that original test 00:10:36.530 --> 00:10:43.710 of looking at, which I just wrote the answer to be no here. 00:10:43.710 --> 00:10:47.120 So if it's not a gradient, the reason 00:10:47.120 --> 00:10:48.530 is there's some rotation. 00:10:48.530 --> 00:10:51.230 Gradient fields don't have any rotation. 00:10:51.230 --> 00:10:59.680 The rotation is this thing that comes out, 00:10:59.680 --> 00:11:04.000 yeah it's this difference. dv_2/dx, 00:11:04.000 --> 00:11:06.720 it's the difference between those that tells us 00:11:06.720 --> 00:11:08.190 the rotation. 00:11:08.190 --> 00:11:12.260 And that was not zero, right? 00:11:12.260 --> 00:11:18.600 For this example dv_1/dy was zero, because v_1 is zero. 00:11:18.600 --> 00:11:23.670 dv_2/dx was one, because v_2 is x. 00:11:23.670 --> 00:11:25.590 So there's some rotation here. 00:11:25.590 --> 00:11:30.100 And in other words the test for being a gradient 00:11:30.100 --> 00:11:31.800 is no rotation. 00:11:31.800 --> 00:11:33.720 This fails that test. 00:11:33.720 --> 00:11:35.260 But how is it rotating? 00:11:35.260 --> 00:11:41.000 How can it be rotating when the all the flow is just 00:11:41.000 --> 00:11:43.530 traveling vertically? 00:11:43.530 --> 00:11:48.150 I guess I give you this example because it meant something 00:11:48.150 --> 00:11:49.820 to me. 00:11:49.820 --> 00:11:54.350 My image of rotation was this simpleminded type of flow. 00:11:54.350 --> 00:11:56.940 You know, like a phonograph record or something. 00:11:56.940 --> 00:11:59.160 This would be called a sheer flow. 00:11:59.160 --> 00:12:03.950 A very important type of flow. 00:12:03.950 --> 00:12:08.540 And actually, you'll realize that if x is negative 00:12:08.540 --> 00:12:13.480 then the flow in the second component, the velocity, 00:12:13.480 --> 00:12:14.470 is now negative. 00:12:14.470 --> 00:12:17.300 So it would be the streamline, the flow 00:12:17.300 --> 00:12:19.820 would be going down this way. 00:12:19.820 --> 00:12:22.420 And this point wouldn't move at all. 00:12:22.420 --> 00:12:24.670 This would be, well I don't know if it's a streamline, 00:12:24.670 --> 00:12:30.200 it's a stagnant streamline, right? x=0. 00:12:30.200 --> 00:12:33.510 On that line, there's no velocity. [0, 0]. 00:12:33.510 --> 00:12:37.000 So this is all just staying there. 00:12:37.000 --> 00:12:40.000 These lines are moving, this line moving faster, 00:12:40.000 --> 00:12:41.950 this line would be moving even faster. 00:12:41.950 --> 00:12:44.920 This line's going the other way. 00:12:44.920 --> 00:12:46.650 Faster and faster the other way. 00:12:46.650 --> 00:12:49.190 It's a important flow. 00:12:49.190 --> 00:12:52.650 You know, in earthquakes and things like that. 00:12:52.650 --> 00:12:57.920 This happens, when one plate shears with respect to another. 00:12:57.920 --> 00:12:59.210 So that's shearing. 00:12:59.210 --> 00:13:05.940 The word shearing means that a line that was -- 00:13:05.940 --> 00:13:09.810 that line after a while is tilted. 00:13:09.810 --> 00:13:12.050 This is going faster than this. 00:13:12.050 --> 00:13:12.710 Yes. 00:13:12.710 --> 00:13:16.440 AUDIENCE: [INAUDIBLE] 00:13:16.440 --> 00:13:20.530 PROFESSOR STRANG: Right, being a constant. 00:13:20.530 --> 00:13:24.690 AUDIENCE: [INAUDIBLE] That's true. 00:13:24.690 --> 00:13:28.370 Ah, well, OK. 00:13:28.370 --> 00:13:32.290 Let's see. 00:13:32.290 --> 00:13:33.590 Well, how do I fix that? 00:13:33.590 --> 00:13:39.770 AUDIENCE: [INAUDIBLE] 00:13:39.770 --> 00:13:41.810 PROFESSOR STRANG: Yes. 00:13:41.810 --> 00:13:42.870 That's a good question. 00:13:42.870 --> 00:13:48.680 Should I have allowed in my stream function, 00:13:48.680 --> 00:13:51.320 I mean that's a stream function. 00:13:51.320 --> 00:13:54.730 Because it satisfies the equations that stream functions 00:13:54.730 --> 00:13:57.580 are -- I could have thrown in a constant, yeah. 00:13:57.580 --> 00:13:59.910 So your pointing out a difficulty 00:13:59.910 --> 00:14:02.880 makes me think I should have thrown in a constant. 00:14:02.880 --> 00:14:07.510 So if I throw in constants then I could get other lines, yeah. 00:14:07.510 --> 00:14:09.630 Thanks, that's a good point. 00:14:09.630 --> 00:14:13.600 I just want to see, do you see rotation in this flow, 00:14:13.600 --> 00:14:15.380 in this shear flow? 00:14:15.380 --> 00:14:16.600 And I think you do. 00:14:16.600 --> 00:14:17.840 If you think about it. 00:14:17.840 --> 00:14:22.680 Suppose you put a little leaf, or a little penny or something 00:14:22.680 --> 00:14:23.920 right there. 00:14:23.920 --> 00:14:27.150 OK, is it going to turn? 00:14:27.150 --> 00:14:31.880 It'll flow along, but as it flows, is it going to turn? 00:14:31.880 --> 00:14:40.410 In other words, is there some difference in the speed on one 00:14:40.410 --> 00:14:41.550 side compared to the other? 00:14:41.550 --> 00:14:45.280 I mean, it's what makes a curveball curve, right? 00:14:45.280 --> 00:14:50.140 When the pitcher throws the ball, he imparts a spin to it, 00:14:50.140 --> 00:14:52.090 and that gives a different pressure 00:14:52.090 --> 00:14:56.630 on the two sides of the ball, and the ball moves. 00:14:56.630 --> 00:14:59.330 I think that's going to happen here. 00:14:59.330 --> 00:15:02.540 Maybe you see it, and I'm just talking. 00:15:02.540 --> 00:15:05.220 I mean, this side is going faster than this side. 00:15:05.220 --> 00:15:11.180 So the net result is that even though the thing is traveling 00:15:11.180 --> 00:15:15.440 up and down, it's turning. 00:15:15.440 --> 00:15:18.550 It's turning because the right-hand side is going faster 00:15:18.550 --> 00:15:19.910 than the left-hand side. 00:15:19.910 --> 00:15:22.750 So it does have a rotation. 00:15:22.750 --> 00:15:28.290 This quantity, this difference between dv_1/dy and dv_2/dx, 00:15:28.290 --> 00:15:31.510 which is the component of the curl, 00:15:31.510 --> 00:15:33.700 maybe the sign should be the opposite, 00:15:33.700 --> 00:15:37.650 maybe I think it should be minus this plus this or something. 00:15:37.650 --> 00:15:40.990 Point is that it's not zero. 00:15:40.990 --> 00:15:43.800 So there is curl, there is rotation. 00:15:43.800 --> 00:15:44.830 OK. 00:15:44.830 --> 00:15:48.600 I was going to ask about this picture, too. 00:15:48.600 --> 00:15:50.870 And then I'll open to more examples. 00:15:50.870 --> 00:15:53.600 I just feel examples are good. 00:15:53.600 --> 00:15:57.430 Simple velocity fields, like 0x. 00:16:00.040 --> 00:16:04.440 Just to think through, OK, what does that mean? 00:16:04.440 --> 00:16:06.310 Is it curl free? 00:16:06.310 --> 00:16:09.650 Another way of saying is it a gradient field would be to say 00:16:09.650 --> 00:16:11.900 is it curl free? 00:16:11.900 --> 00:16:15.260 Irrotational is the right word here. 00:16:15.260 --> 00:16:18.960 Test one, is it irrotational, answer no. 00:16:18.960 --> 00:16:22.320 Is it divergence-free, is it source-free, 00:16:22.320 --> 00:16:25.690 the answer was yes, for this example. 00:16:25.690 --> 00:16:28.830 If we pick another example I could reverse those, 00:16:28.830 --> 00:16:31.810 or another example -- I can probably come up with 00:16:31.810 --> 00:16:33.590 an example here. 00:16:33.590 --> 00:16:39.160 Let's see, what if I wanted the streamlines to be circles, 00:16:39.160 --> 00:16:46.110 what would be a good velocity field that goes in circles? 00:16:46.110 --> 00:16:47.020 Let's see. 00:16:47.020 --> 00:16:52.120 At a typical point, if I want the velocity to be going that 00:16:52.120 --> 00:16:58.030 way, here's the vector, the position vector, 00:16:58.030 --> 00:17:01.260 the radial vector that goes -- so what are the components 00:17:01.260 --> 00:17:02.880 of this vector? 00:17:02.880 --> 00:17:04.890 Just [x, y]. 00:17:04.890 --> 00:17:12.300 So now if I want the velocity field to go other way, 00:17:12.300 --> 00:17:18.560 what would be a good thing with rotation? 00:17:18.560 --> 00:17:20.640 [-y, x] would sound good. 00:17:20.640 --> 00:17:21.460 v=[-y, x]. 00:17:24.440 --> 00:17:29.390 So are we expecting this to be a gradient of anything? 00:17:29.390 --> 00:17:32.730 I'm not. 00:17:32.730 --> 00:17:35.380 We've built in rotation here. 00:17:35.380 --> 00:17:38.940 I'm expecting the curl of this thing, this quantity, 00:17:38.940 --> 00:17:44.720 I think I take the x derivative -- 00:17:44.720 --> 00:17:48.805 I look at the y derivative of this and compare it with the x 00:17:48.805 --> 00:17:49.910 derivative of that. 00:17:49.910 --> 00:17:52.450 And they're not the same; in fact one is minus one 00:17:52.450 --> 00:17:54.070 and the other's plus one. 00:17:54.070 --> 00:17:56.420 So I've got rotation here. 00:17:56.420 --> 00:18:03.340 I've got sort of two, is the component of the curl. 00:18:03.340 --> 00:18:09.180 So let's just write it down. dv_2/dx, 00:18:09.180 --> 00:18:16.210 this vorticity that measures the turning speed is one from 00:18:16.210 --> 00:18:20.620 dv_2/dx, minus one is two. 00:18:20.620 --> 00:18:23.940 So it's not a gradient of anything. 00:18:23.940 --> 00:18:27.180 If the x derivative of u is minus y, 00:18:27.180 --> 00:18:30.890 then the y derivative can't be plus x, no good. 00:18:30.890 --> 00:18:35.580 OK, what about, is it divergence-free? 00:18:38.330 --> 00:18:41.350 Do I need a source to keep this flow going? 00:18:41.350 --> 00:18:42.700 Well, what's the test? 00:18:42.700 --> 00:18:47.650 In other words, is there a stream function for this guy? 00:18:47.650 --> 00:18:50.320 I think probably there is. 00:18:50.320 --> 00:18:54.900 What's the test to know if there is a stream function? 00:18:54.900 --> 00:18:58.100 I take the divergence, I'm over on the right-hand side 00:18:58.100 --> 00:18:59.260 of my picture now. 00:18:59.260 --> 00:19:01.170 I take the divergence. 00:19:01.170 --> 00:19:07.130 Divergence of this v is the x derivative of that 00:19:07.130 --> 00:19:11.280 plus the y derivative of that, good, zero. 00:19:11.280 --> 00:19:13.440 So there is a stream function. 00:19:13.440 --> 00:19:15.620 And what is it? 00:19:15.620 --> 00:19:20.740 Well, I'm pretty sure that these streamlines are circles, 00:19:20.740 --> 00:19:22.525 I think the stream function is going 00:19:22.525 --> 00:19:27.670 to be x squared plus y squared. 00:19:27.670 --> 00:19:28.920 Yep. 00:19:28.920 --> 00:19:38.420 Then, am I right that the y derivative of that will be 2y. 00:19:38.420 --> 00:19:41.990 That's not looking too good. 00:19:41.990 --> 00:19:45.590 What do I want here? 00:19:45.590 --> 00:19:49.520 Here's my v, which is the same as w. 00:19:49.520 --> 00:19:54.630 And what I'm looking for is to get these guys correct. 00:19:54.630 --> 00:19:57.330 So -- and I should be able to do it. 00:19:57.330 --> 00:19:59.230 And what would s be? 00:19:59.230 --> 00:20:02.410 I haven't got s quite right. 00:20:02.410 --> 00:20:04.760 I think if I multiply by negative 1/2, 00:20:04.760 --> 00:20:06.490 that might have done it. 00:20:06.490 --> 00:20:13.860 Yeah, because now the y derivative is now minus y. 00:20:13.860 --> 00:20:15.050 Great. 00:20:15.050 --> 00:20:20.410 And the x derivative of s is minus x, 00:20:20.410 --> 00:20:22.800 and then I should take a minus that, so I 00:20:22.800 --> 00:20:25.890 should want a plus x, which is what I've got. 00:20:25.890 --> 00:20:27.950 So those are the streamlines. 00:20:27.950 --> 00:20:30.380 Circles. 00:20:30.380 --> 00:20:35.530 So I have circle, the flow is going around in a circle. 00:20:35.530 --> 00:20:40.550 I don't have to -- I don't need any source to keep it going. 00:20:40.550 --> 00:20:48.400 But it's not a gradient. 00:20:48.400 --> 00:20:57.760 So this is like a sample test, to take a simple flow field, 00:20:57.760 --> 00:21:00.340 apply the two tests, and I guess we 00:21:00.340 --> 00:21:05.310 should complete with an example that passes both tests. 00:21:05.310 --> 00:21:07.500 Right? 00:21:07.500 --> 00:21:09.740 Let me open to any other question, 00:21:09.740 --> 00:21:13.460 and then we could cook up an example that passes 00:21:13.460 --> 00:21:16.380 both tests before we stop. 00:21:16.380 --> 00:21:17.850 I'll stop talking first, though. 00:21:17.850 --> 00:21:23.360 Just listen for a question on any topic. 00:21:23.360 --> 00:21:26.100 Or is it useful just to take fields like this 00:21:26.100 --> 00:21:27.620 and go through those steps? 00:21:27.620 --> 00:21:28.610 It probably is. 00:21:28.610 --> 00:21:32.020 It's certainly good for me. 00:21:32.020 --> 00:21:36.290 OK, what's a field that will satisfy everybody, 00:21:36.290 --> 00:21:40.710 that will be a gradient field and also 00:21:40.710 --> 00:21:45.030 divergence-free, so that we'll have solutions 00:21:45.030 --> 00:21:48.460 to Laplace's equation. 00:21:48.460 --> 00:21:49.610 Let's see. 00:21:49.610 --> 00:21:59.040 Well we had some solutions to Laplace's equation there. 00:21:59.040 --> 00:22:03.880 You know if I make it linear it's real easy. 00:22:03.880 --> 00:22:11.450 If I make it quadratic -- huh. 00:22:11.450 --> 00:22:15.740 Can I anticipate a little what's coming Friday? 00:22:15.740 --> 00:22:20.720 I so recommend to come to Friday's lecture, but -- 00:22:20.720 --> 00:22:21.750 so what's coming? 00:22:21.750 --> 00:22:23.310 What did we do today? 00:22:23.310 --> 00:22:27.760 We discovered that we got solutions to Laplace's equation 00:22:27.760 --> 00:22:31.440 from all, by real and imaginary parts of all these guys. 00:22:31.440 --> 00:22:33.510 Those were terrific. 00:22:33.510 --> 00:22:38.180 And then we could take combinations of those. 00:22:38.180 --> 00:22:40.610 So here's what's coming Friday. 00:22:40.610 --> 00:22:46.800 When I take combinations of these guys I get some function 00:22:46.800 --> 00:22:54.030 of this magic complex -- of this magic combination x+iy. 00:22:54.030 --> 00:22:56.940 Some function, any function. 00:22:56.940 --> 00:22:59.850 Any reasonable function, and we'll say what reasonable 00:22:59.850 --> 00:23:04.820 means, of x+iy, its real part and its imaginary part are 00:23:04.820 --> 00:23:06.330 going to be great. 00:23:06.330 --> 00:23:10.460 This is like the center of a big, big part of mathematics. 00:23:10.460 --> 00:23:11.660 Functions of x+iy. 00:23:14.560 --> 00:23:20.030 And by nice I mean that these series converge. 00:23:20.030 --> 00:23:22.240 So that we have really a nice function. 00:23:22.240 --> 00:23:25.780 Let me take the first function that comes to mind. e^(x+iy). 00:23:28.510 --> 00:23:31.850 So let me take this to be e^(x+iy). 00:23:34.690 --> 00:23:37.571 OK. 00:23:37.571 --> 00:23:38.070 Right. 00:23:38.070 --> 00:23:40.590 So you remember, I'm aiming to get solutions 00:23:40.590 --> 00:23:43.930 to Laplace's equation, because that will give me automatically 00:23:43.930 --> 00:23:46.220 the two pieces both working. 00:23:46.220 --> 00:23:50.870 So I claim that the real part of that, and the imaginary part, 00:23:50.870 --> 00:23:54.860 those are my twins, u and s, both solve -- 00:23:54.860 --> 00:23:59.200 so u is going to be the real part of this function. 00:23:59.200 --> 00:24:02.740 And s is going to be the imaginary part of it. 00:24:02.740 --> 00:24:06.360 And I claim that those will both solve Laplace's equation. 00:24:06.360 --> 00:24:09.150 We can plug it in and see that it does. 00:24:09.150 --> 00:24:13.320 And that they will have, they're twinned 00:24:13.320 --> 00:24:15.820 by the Cauchy-Riemann equations. 00:24:15.820 --> 00:24:18.950 So how am I going to simplify that, 00:24:18.950 --> 00:24:22.710 so that I can identify what's the real part of that thing 00:24:22.710 --> 00:24:25.170 and what's the imaginary part? 00:24:25.170 --> 00:24:30.950 This is actually, that's a good question. 00:24:30.950 --> 00:24:36.960 I don't know how much you've run into i, in the past. 00:24:36.960 --> 00:24:40.350 Are you happy with something like that? 00:24:40.350 --> 00:24:42.360 How could you find the real part of it, 00:24:42.360 --> 00:24:44.540 how could you simplify it? 00:24:44.540 --> 00:24:51.820 How else could I write e to the something? 00:24:51.820 --> 00:24:52.830 Exactly. 00:24:52.830 --> 00:24:54.880 Think of it as the product of two, 00:24:54.880 --> 00:24:57.790 so the key fact about exponentials is that 00:24:57.790 --> 00:25:00.650 that's the same as e^x times e^(iy). 00:25:03.680 --> 00:25:07.260 The exponents add, so that's the first thing always 00:25:07.260 --> 00:25:09.280 to think about as a possibility. 00:25:09.280 --> 00:25:10.950 Now, what am I going to do? 00:25:10.950 --> 00:25:13.770 I still want to get a real part. 00:25:13.770 --> 00:25:18.330 This is clearly all real, right? e^x is real. 00:25:18.330 --> 00:25:22.960 So it's this part that's going to give me the two pieces. 00:25:22.960 --> 00:25:25.310 So this is going to be e^x times -- now, 00:25:25.310 --> 00:25:28.850 what do I put for e^(iy)? 00:25:28.850 --> 00:25:32.820 cos(y), good. 00:25:32.820 --> 00:25:35.820 Plus i*sin(y), good. 00:25:35.820 --> 00:25:38.090 And now I can read off, no problem. 00:25:38.090 --> 00:25:43.990 What is this real part that I was looking for? e^x*cos(y). 00:25:48.380 --> 00:25:52.787 And the imaginary part is just what's multiplying the i, 00:25:52.787 --> 00:25:53.620 it's the e^x*sin(y). 00:25:57.840 --> 00:26:02.080 OK, so what's my claim? 00:26:02.080 --> 00:26:05.990 I claim that that function solves Laplace's equation. 00:26:05.990 --> 00:26:08.330 And this one too. 00:26:08.330 --> 00:26:10.310 And that they're twinned. 00:26:10.310 --> 00:26:13.890 And that they give streamlines and equipotentials 00:26:13.890 --> 00:26:22.670 that meet at right angles, it's another pair. 00:26:22.670 --> 00:26:25.290 Plug that into Laplace's equation. 00:26:25.290 --> 00:26:33.835 So let me do u_xx+u_yy, just to satisfy that it is going 00:26:33.835 --> 00:26:35.710 to come out zero. 00:26:35.710 --> 00:26:39.380 So what's the xx derivative, the second x derivative 00:26:39.380 --> 00:26:41.880 of that function? 00:26:41.880 --> 00:26:44.620 Take its derivative with respect to x, and then do it again, 00:26:44.620 --> 00:26:46.480 and what do you have? 00:26:46.480 --> 00:26:47.010 Same. 00:26:47.010 --> 00:26:55.330 Didn't change. e^x is just -- and now what about the second y 00:26:55.330 --> 00:26:57.060 derivative? 00:26:57.060 --> 00:26:59.350 So now e^x is just a constant. 00:26:59.350 --> 00:27:01.330 What's the second derivative of cos(y)? 00:27:03.880 --> 00:27:05.080 Negative cos(y). 00:27:05.080 --> 00:27:05.580 Right. 00:27:05.580 --> 00:27:07.870 Because the first derivative is negative sine, 00:27:07.870 --> 00:27:10.160 the second derivative is negative cosine. 00:27:10.160 --> 00:27:14.300 So the second derivative is e^x, it didn't change, 00:27:14.300 --> 00:27:17.440 times cos(y) with a minus sine. 00:27:17.440 --> 00:27:23.390 And you see what -- did I write sine? 00:27:23.390 --> 00:27:25.710 I meant to write cosine. 00:27:25.710 --> 00:27:27.430 Cancel that from the tape. 00:27:27.430 --> 00:27:29.171 OK, right. 00:27:29.171 --> 00:27:29.670 Yeah. 00:27:29.670 --> 00:27:33.065 So the second x derivative was just e^x, e^x, 00:27:33.065 --> 00:27:34.900 cos(y) didn't move. 00:27:34.900 --> 00:27:35.770 Sorry. 00:27:35.770 --> 00:27:37.100 That was frightening. 00:27:37.100 --> 00:27:41.850 OK, and then now here's the second y derivative. 00:27:41.850 --> 00:27:44.630 In other words, it gives zero. 00:27:44.630 --> 00:27:48.760 Gives zero, and this one would too. 00:27:48.760 --> 00:28:01.380 Now, I don't really have an idea of what the picture is like. 00:28:01.380 --> 00:28:03.890 But it's important. 00:28:03.890 --> 00:28:08.780 We've got a flow field here, and it's from e^z, e^(x+iy). 00:28:08.780 --> 00:28:12.010 Exponential has gotta be an important function. 00:28:12.010 --> 00:28:15.040 So it's got to be somehow interesting. 00:28:15.040 --> 00:28:20.140 What do you think -- so what would the -- 00:28:20.140 --> 00:28:23.610 what would the equipotential lines looks like? 00:28:23.610 --> 00:28:25.760 Oh, boy. 00:28:25.760 --> 00:28:29.210 e^x*cos(y) equal a constant. 00:28:29.210 --> 00:28:31.740 My gosh. 00:28:31.740 --> 00:28:33.350 e^x*cos(y). 00:28:33.350 --> 00:28:37.240 So let's see. 00:28:37.240 --> 00:28:42.610 I don't know how to draw this picture, but one thing I know 00:28:42.610 --> 00:28:48.840 is that if I changed y by 2pi, I would get another copy 00:28:48.840 --> 00:28:50.650 of this curve, right? 00:28:50.650 --> 00:28:54.060 If I changed y by -- every time you see cosine or sine, 00:28:54.060 --> 00:28:55.990 you think hey, that's periodic. 00:28:55.990 --> 00:28:57.730 If I change it by 2pi. 00:28:57.730 --> 00:29:10.510 So I'm thinking that y between zero and 2 pi, so here's y=0. 00:29:10.510 --> 00:29:18.390 And y=2pi, I'm thinking that my flow probably somehow stays 00:29:18.390 --> 00:29:20.850 in a strip. 00:29:20.850 --> 00:29:21.970 Like that. 00:29:21.970 --> 00:29:25.780 And then the whole thing just repeats, and repeats, 00:29:25.780 --> 00:29:26.430 and repeats. 00:29:26.430 --> 00:29:31.260 So I'm thinking, really this is flow in an infinite strip. 00:29:31.260 --> 00:29:33.510 Infinite pipe or something like that. 00:29:33.510 --> 00:29:38.160 You can imagine that there could be applications. 00:29:38.160 --> 00:29:40.040 But I still haven't drawn the curve. 00:29:40.040 --> 00:29:46.080 I just think, let's see, what would it look like when y is 00:29:46.080 --> 00:29:50.300 a little -- suppose I'm trying to draw the picture 00:29:50.300 --> 00:29:55.380 of e^x*cos(y)=1, whatever. 00:29:55.380 --> 00:30:00.970 OK, I'll just attempt to draw that curve. 00:30:00.970 --> 00:30:09.180 Just, so if y was a little bit off of zero, 00:30:09.180 --> 00:30:14.460 the cosine would be, yeah, how's it going to go? 00:30:14.460 --> 00:30:24.980 If y is just a little off zero, tell me 00:30:24.980 --> 00:30:30.080 any points on this curve? 00:30:30.080 --> 00:30:36.560 I can see that e^x is going to be a big number. 00:30:36.560 --> 00:30:39.400 Is (0,0) on the curve? 00:30:39.400 --> 00:30:40.140 Good. 00:30:40.140 --> 00:30:44.310 Got one point. 00:30:44.310 --> 00:30:47.320 Alright. 00:30:47.320 --> 00:30:52.120 Now, suppose y is a little bit more than zero. 00:30:52.120 --> 00:30:55.190 So suppose y goes up a little bit. 00:30:55.190 --> 00:30:59.920 Then what? (1,0) or something? 00:30:59.920 --> 00:31:00.420 Yeah. 00:31:00.420 --> 00:31:04.530 I suppose (1,0)? 00:31:04.530 --> 00:31:08.410 No, no. 00:31:08.410 --> 00:31:16.790 So if y goes up a little, then x would go out a little bit. 00:31:16.790 --> 00:31:18.080 So what's happening? 00:31:18.080 --> 00:31:26.800 So cos(y), so the cos(y) is going to drop from one to zero, 00:31:26.800 --> 00:31:28.680 right? 00:31:28.680 --> 00:31:30.020 To start with. 00:31:30.020 --> 00:31:34.360 Then, if this cos(y) is dropping from one to zero then this e^x 00:31:34.360 --> 00:31:39.650 has got to climb up, to to keep the product one. 00:31:39.650 --> 00:31:40.530 So I'll move out. 00:31:40.530 --> 00:31:42.390 So somehow it'll move out. 00:31:42.390 --> 00:31:59.230 I think maybe when y reaches pi/2, 00:31:59.230 --> 00:32:06.100 then the cosine has got down to zero. 00:32:06.100 --> 00:32:09.270 We could work on this for a while. 00:32:09.270 --> 00:32:11.440 Or we could let MATLAB draw it. 00:32:11.440 --> 00:32:17.590 But I think that we would see these -- and I could do better. 00:32:17.590 --> 00:32:22.900 I'm feeling pretty humiliated to not have a better picture here. 00:32:22.900 --> 00:32:25.190 Suppose y is a little less than zero, 00:32:25.190 --> 00:32:26.990 do we get anything interesting there? 00:32:26.990 --> 00:32:30.610 Oh well, the cosine is an even function. 00:32:30.610 --> 00:32:33.740 So I think the thing might, is it just 00:32:33.740 --> 00:32:38.600 going to turn around like that? 00:32:38.600 --> 00:32:43.450 So that y and minus y -- for a certain x, 00:32:43.450 --> 00:32:47.299 the y value and the minus y will both be on the curve 00:32:47.299 --> 00:32:49.590 because the cosine doesn't know whether it's the cosine 00:32:49.590 --> 00:32:52.170 of of a plus or a minus. 00:32:52.170 --> 00:32:58.250 Yeah, I think we would get curves of that sort. 00:32:58.250 --> 00:33:04.210 And then the other curves, s equal constant, 00:33:04.210 --> 00:33:08.530 the streamlines will somehow go vertically. 00:33:08.530 --> 00:33:14.090 Maybe I'll just not use the whole time 00:33:14.090 --> 00:33:18.300 to work on that particular curve. 00:33:18.300 --> 00:33:20.220 We'd have to prepare it. 00:33:20.220 --> 00:33:25.690 The point is, you see how incredibly easily we produce 00:33:25.690 --> 00:33:28.450 solutions to Laplace's equation that you 00:33:28.450 --> 00:33:31.190 wouldn't have thought of, and I wouldn't have thought of. 00:33:31.190 --> 00:33:34.340 So that would be one way to produce solutions. 00:33:34.340 --> 00:33:39.410 I might even repeat this one in class Friday, or I might not. 00:33:39.410 --> 00:33:43.090 Let me suggest another couple of possibilities 00:33:43.090 --> 00:33:45.690 that I will do in class. 00:33:45.690 --> 00:33:49.930 Can I just give you a couple of other functions f. 00:33:49.930 --> 00:33:54.220 In fact, I'll just erase that one and put in some other ones. 00:33:54.220 --> 00:33:58.980 Suppose I took the function 1/(x+iy). 00:34:07.370 --> 00:34:11.080 So that's a function of this magic combination, x+iy. 00:34:11.080 --> 00:34:15.460 What's its real part and what's its imaginary part? 00:34:15.460 --> 00:34:19.470 Do you know how to split that guy into real and imaginary? 00:34:19.470 --> 00:34:21.380 There's a little trick, if you remember 00:34:21.380 --> 00:34:24.660 from learning complex numbers. 00:34:24.660 --> 00:34:26.500 Do you remember the trick? 00:34:26.500 --> 00:34:32.130 The problem is that this thing is down in the denominator, 00:34:32.130 --> 00:34:33.070 right? 00:34:33.070 --> 00:34:34.610 We don't want it there. 00:34:34.610 --> 00:34:38.570 Because we can't split the real and imaginary parts down there. 00:34:38.570 --> 00:34:41.560 So I would like to rewrite it in a way that 00:34:41.560 --> 00:34:45.690 gets something real down in the denominator, 00:34:45.690 --> 00:34:48.340 moves all the i stuff up in the numerator 00:34:48.340 --> 00:34:50.290 where I can separate it. 00:34:50.290 --> 00:34:52.830 How do I do it? 00:34:52.830 --> 00:34:59.540 Multiply both sides by, both top and bottom, by x-iy. 00:34:59.540 --> 00:35:03.310 Good. 00:35:03.310 --> 00:35:05.320 So what does that put down here now? 00:35:05.320 --> 00:35:07.140 That's a number times its conjugate 00:35:07.140 --> 00:35:11.550 and that's going to produce x squared. 00:35:11.550 --> 00:35:13.620 Minus or plus? 00:35:13.620 --> 00:35:16.450 Plus y squared, right. 00:35:16.450 --> 00:35:19.860 The number times its conjugate is the length squared. 00:35:19.860 --> 00:35:27.090 And now we just have x-iy, and now it's obvious what the u is. 00:35:27.090 --> 00:35:30.490 This is real now, so the u is just 00:35:30.490 --> 00:35:35.550 x over x squared plus y squared, and the s 00:35:35.550 --> 00:35:41.930 is the minus y over x squared plus y squared. 00:35:41.930 --> 00:35:44.760 That's a very interesting flow. 00:35:44.760 --> 00:35:47.070 That's an interesting flow, and we 00:35:47.070 --> 00:35:48.800 could do its picture and so on. 00:35:48.800 --> 00:35:50.980 And in fact it would be a nicer picture 00:35:50.980 --> 00:35:57.400 than the one we stopped on. 00:35:57.400 --> 00:36:03.170 What should I notice about this flow? 00:36:03.170 --> 00:36:08.760 Of course, the flow is automatically irrotational; 00:36:08.760 --> 00:36:13.860 the curl is zero because there is a potential. 00:36:13.860 --> 00:36:17.560 A gradient of a potential, the gradient of a potential 00:36:17.560 --> 00:36:22.710 is going to be free of rotation. 00:36:22.710 --> 00:36:30.550 And there will be streamlines, all those good things. 00:36:30.550 --> 00:36:34.090 There's one bad point about the flow, though. 00:36:34.090 --> 00:36:36.180 Which is where? 00:36:36.180 --> 00:36:37.450 At (0,0). 00:36:37.450 --> 00:36:42.690 The whole thing falls apart, at the origin this falls apart. 00:36:42.690 --> 00:36:46.240 So this is a great flow except at the origin, 00:36:46.240 --> 00:36:49.420 it's very problematic. 00:36:49.420 --> 00:36:51.080 It's singular at the origin. 00:36:51.080 --> 00:36:56.370 So if we drew the pictures we would see something strange. 00:36:56.370 --> 00:36:59.530 This is going to zero at the origin. 00:36:59.530 --> 00:37:03.800 So, yeah, we have trouble at the origin but an important flow 00:37:03.800 --> 00:37:05.450 otherwise, yep. 00:37:05.450 --> 00:37:07.170 And I'll just mention the third but I 00:37:07.170 --> 00:37:08.590 won't do anything with it. 00:37:08.590 --> 00:37:12.720 Because it's such a neat one that I 00:37:12.720 --> 00:37:15.540 have to save it for Friday. 00:37:15.540 --> 00:37:19.500 The other natural function to think of is the logarithm. 00:37:19.500 --> 00:37:22.890 The logarithm of x+iy. 00:37:22.890 --> 00:37:28.870 Split that into u and s. 00:37:28.870 --> 00:37:31.320 What kind of a thing do we have here? 00:37:31.320 --> 00:37:32.760 What kind of singularity? 00:37:32.760 --> 00:37:38.080 Yeah, let me just do two moments on this example, 00:37:38.080 --> 00:37:41.560 and then leave it for Friday because the whole class has 00:37:41.560 --> 00:37:44.090 to see it. 00:37:44.090 --> 00:37:47.530 Is there a singularity for this guy? 00:37:47.530 --> 00:37:56.300 Is there a point (x,y) where the logarithm is not great? 00:37:56.300 --> 00:37:59.460 At the origin, again. 00:37:59.460 --> 00:38:01.660 We'll again have a singularity at the origin. 00:38:01.660 --> 00:38:05.260 Something strange is happening at the origin. 00:38:05.260 --> 00:38:08.730 And what we'll find is there's a delta function there. 00:38:08.730 --> 00:38:14.220 We're feeding in, we have a source right at the origin 00:38:14.220 --> 00:38:20.410 and then it's flowing out on, I think on radial lines. 00:38:20.410 --> 00:38:23.600 I think the streamlines go out from the origin 00:38:23.600 --> 00:38:26.180 and the equipotentials go around the origin. 00:38:26.180 --> 00:38:28.220 Yeah, it's a great example. 00:38:28.220 --> 00:38:31.200 So that's another one to come. 00:38:31.200 --> 00:38:34.290 OK, so examples like these are -- 00:38:34.290 --> 00:38:41.820 I mean generations of thinking went into solutions 00:38:41.820 --> 00:38:44.380 of Laplace's equation. 00:38:44.380 --> 00:38:50.550 And 2-D particularly where we have this special combination. 00:38:50.550 --> 00:38:55.210 I wish we had such a combination in 3-D but we simply don't. 00:38:55.210 --> 00:38:58.780 We can discuss Laplace's equation in 3-D of course, 00:38:58.780 --> 00:38:59.590 very important. 00:38:59.590 --> 00:39:03.980 But I mean, wave equation, this fact that we're talking to each 00:39:03.980 --> 00:39:16.520 other, is got the Laplacian in 3-D, but there's no x+iy magic. 00:39:16.520 --> 00:39:21.170 OK, that's some u's and s's and v's and w's. 00:39:21.170 --> 00:39:25.880 What else is on your mind? 00:39:25.880 --> 00:39:30.420 Questions? 00:39:30.420 --> 00:39:33.200 I could ask this question, oh, here's something 00:39:33.200 --> 00:39:37.910 I did not do in class. 00:39:37.910 --> 00:39:40.070 I think I wrote down the divergence theorem. 00:39:40.070 --> 00:39:42.380 So can we start by doing that? 00:39:42.380 --> 00:39:47.350 Let me write down the divergence theorem, with your help. 00:39:47.350 --> 00:39:55.270 And then use it. 00:39:55.270 --> 00:39:58.280 So what does the divergence theorem -- we're in 2-D. 00:39:58.280 --> 00:40:03.230 So this is 2-D, just the similar theorem. 00:40:03.230 --> 00:40:08.950 So what does the theorem say, that if I take the divergence 00:40:08.950 --> 00:40:19.710 of some w, some vector field, then if I integrate that over 00:40:19.710 --> 00:40:24.370 some region -- so I have some region here and at every point 00:40:24.370 --> 00:40:26.950 there's a flow w. 00:40:26.950 --> 00:40:33.700 And I look at the divergence of w and I integrate that, dx dy, 00:40:33.700 --> 00:40:37.500 so that's a double integral over a region, 00:40:37.500 --> 00:40:44.950 I will get, what's the right-hand side? 00:40:44.950 --> 00:40:48.040 What does the divergence measure? 00:40:48.040 --> 00:40:53.380 So I'm really asking like just memory, what is the divergence, 00:40:53.380 --> 00:40:54.740 it's an identity. 00:40:54.740 --> 00:41:00.990 It's integration by parts in some way, as we'll see. 00:41:00.990 --> 00:41:06.470 But what do you remember for the divergence theorem? 00:41:06.470 --> 00:41:09.070 You get what? 00:41:09.070 --> 00:41:14.020 It measures how much flux out, right? 00:41:14.020 --> 00:41:16.750 So when we measure the flux out by integrating 00:41:16.750 --> 00:41:21.700 around the boundary, how much is getting through the boundary? 00:41:21.700 --> 00:41:24.870 And what's the flow through the boundary? 00:41:24.870 --> 00:41:28.430 I take w, but that's a vector. 00:41:28.430 --> 00:41:32.340 And I'm looking for what component of w? 00:41:32.340 --> 00:41:36.230 The normal component, the component of w, w dot n, 00:41:36.230 --> 00:41:42.040 the component of w that's headed out. n is defined to be, 00:41:42.040 --> 00:41:46.100 whatever the boundary is -- here I've made it look like a circle 00:41:46.100 --> 00:41:47.650 but I shouldn't have. 00:41:47.650 --> 00:41:51.730 Let me make it a little wobblier or something. 00:41:51.730 --> 00:41:57.990 So the normal component at any, there, look at that point. 00:41:57.990 --> 00:42:01.630 The normal direction through the boundary, 00:42:01.630 --> 00:42:05.390 down in that crazy point, is this way. 00:42:05.390 --> 00:42:08.090 So the normal is going this way here. 00:42:08.090 --> 00:42:10.100 Here, it's going over this way. 00:42:10.100 --> 00:42:13.680 It's perpendicular to the boundary, OK? 00:42:13.680 --> 00:42:17.160 And then we integrate around the boundary. 00:42:17.160 --> 00:42:20.880 Alright. 00:42:20.880 --> 00:42:27.440 So that's the identity of that, that's the divergence theorem. 00:42:27.440 --> 00:42:32.280 Now, let's see. 00:42:32.280 --> 00:42:33.510 Could you, yeah. 00:42:33.510 --> 00:42:38.260 So we have a minute. 00:42:38.260 --> 00:42:41.990 You want to take a particular w and see 00:42:41.990 --> 00:42:45.770 if this would be correct? 00:42:45.770 --> 00:42:50.710 How about w=w=[0, x], our first example? 00:42:50.710 --> 00:42:52.830 Suppose I tried w=[0, x]. 00:42:52.830 --> 00:42:55.640 I just want to see if the divergence -- 00:42:55.640 --> 00:42:58.980 what the flux is through the boundary. 00:42:58.980 --> 00:43:03.130 What what region shall I take for the -- 00:43:03.130 --> 00:43:05.470 so the divergence theorem has two inputs. 00:43:05.470 --> 00:43:07.320 It has a flow field. 00:43:07.320 --> 00:43:11.580 And let me take w to be [0, x], just so it's a shear. 00:43:11.580 --> 00:43:14.260 And a region. 00:43:14.260 --> 00:43:18.650 And of course the integral might not be that much fun to do, 00:43:18.650 --> 00:43:21.010 unless we make the region nice. 00:43:21.010 --> 00:43:29.900 What do you take as a nice region for -- actually, 00:43:29.900 --> 00:43:32.320 it doesn't matter what the region is. 00:43:32.320 --> 00:43:33.510 Take any old region. 00:43:33.510 --> 00:43:36.580 For the moment. 00:43:36.580 --> 00:43:38.570 What's the answer? 00:43:38.570 --> 00:43:41.040 For this particular flow, w=w=[0, x]? 00:43:45.870 --> 00:43:48.550 Zero. 00:43:48.550 --> 00:43:49.570 That's the cool part. 00:43:49.570 --> 00:43:54.500 If the answer's zero then work is suspended. 00:43:54.500 --> 00:43:56.840 And why is it zero? 00:43:56.840 --> 00:44:00.620 Because the divergence of this particular w, 00:44:00.620 --> 00:44:03.550 the x derivative of that plus the y derivative of that, 00:44:03.550 --> 00:44:04.050 is zero. 00:44:04.050 --> 00:44:07.800 This has divergence everywhere zero. 00:44:07.800 --> 00:44:10.740 So integrating is no problem at all, 00:44:10.740 --> 00:44:15.000 so that would be zero, for this flow field. 00:44:15.000 --> 00:44:17.290 For this divergence-free field. 00:44:17.290 --> 00:44:24.610 Zero for that because div w is zero. 00:44:24.610 --> 00:44:26.290 But is that correct? 00:44:26.290 --> 00:44:29.880 What does that tell me, these flows are -- 00:44:29.880 --> 00:44:33.870 we saw what the flow is like. 00:44:33.870 --> 00:44:35.990 Say there's the origin. 00:44:35.990 --> 00:44:40.230 It doesn't have to be a circle, it looks like a circle. 00:44:40.230 --> 00:44:42.700 Do you see why the flux is zero? 00:44:42.700 --> 00:44:46.540 There is flow through the boundary, right? 00:44:46.540 --> 00:44:52.130 Flow is going buzz, buzz, buzz up this line and out. 00:44:52.130 --> 00:44:54.920 And it's coming in here. 00:44:54.920 --> 00:44:57.740 So there that's what we discovered. 00:44:57.740 --> 00:45:04.080 This [0, x] is vertical flow. 00:45:04.080 --> 00:45:07.750 It hasn't gotten any horizontal component. 00:45:07.750 --> 00:45:10.320 It's got a vertical component, it's going out. 00:45:10.320 --> 00:45:13.940 And would we want to do this right-hand side? 00:45:13.940 --> 00:45:16.670 I don't think so, right. 00:45:16.670 --> 00:45:18.370 This right-hand side is asking me 00:45:18.370 --> 00:45:21.960 what, I have to find the normal direction on this, whatever 00:45:21.960 --> 00:45:23.650 curve that is. 00:45:23.650 --> 00:45:28.020 I have to take its dot product with the flow [0, x], 00:45:28.020 --> 00:45:33.180 so this is some quantity. 00:45:33.180 --> 00:45:36.400 And then I have to do this ds which I haven't even mentioned, 00:45:36.400 --> 00:45:38.350 ds is arc length around. 00:45:38.350 --> 00:45:40.680 I'm integrating around these pieces. 00:45:40.680 --> 00:45:48.690 But yet somehow we have some idea from that picture 00:45:48.690 --> 00:45:52.670 that the total flux is zero. 00:45:52.670 --> 00:45:57.210 How would you say it in words, if I say here's the flow field. 00:45:57.210 --> 00:46:01.690 There's a region, funny shape. 00:46:01.690 --> 00:46:03.880 The flux is zero through that boundary. 00:46:03.880 --> 00:46:07.700 And if I asked you why, what would you say? 00:46:07.700 --> 00:46:11.690 I mean, a math answer would be use the divergence theorem. 00:46:11.690 --> 00:46:19.300 But why from this picture does it look like we have zero flux? 00:46:19.300 --> 00:46:21.200 What comes in goes out, yeah. 00:46:21.200 --> 00:46:25.840 What's coming in the bottom here is going out the top. 00:46:25.840 --> 00:46:29.620 That's basically it. 00:46:29.620 --> 00:46:34.620 So we would get zero for that one. 00:46:34.620 --> 00:46:38.260 So I think the homework, the suggested homework 00:46:38.260 --> 00:46:42.430 maybe includes an example where the divergence isn't zero. 00:46:42.430 --> 00:46:44.910 And then you actually have to do these integrals. 00:46:44.910 --> 00:46:50.450 Just as practice for what do those integrals mean. 00:46:50.450 --> 00:46:55.690 Maybe I won't go through one now, but that's good practice. 00:46:55.690 --> 00:47:03.860 Take some simple w, but one with a non-zero divergence 00:47:03.860 --> 00:47:06.770 and then see if you can do either or both 00:47:06.770 --> 00:47:11.500 of the integrals that are supposed to come out equal. 00:47:11.500 --> 00:47:15.970 That's a good one Now, there's one thing I could -- 00:47:15.970 --> 00:47:18.990 any questions, or discussion? 00:47:18.990 --> 00:47:25.090 You guys are seeing these examples come up; 00:47:25.090 --> 00:47:28.080 it's the only way I would know to learn this subject is take 00:47:28.080 --> 00:47:30.210 simple v's and w's. 00:47:30.210 --> 00:47:36.140 And see what you can do with them. 00:47:36.140 --> 00:47:37.800 We've got the general principles, 00:47:37.800 --> 00:47:42.300 but then apply them to specific flows. 00:47:42.300 --> 00:47:44.920 AUDIENCE: [INAUDIBLE] PROFESSOR STRANG: Yes, thanks 00:47:44.920 --> 00:47:51.729 AUDIENCE: [INAUDIBLE] 00:47:51.729 --> 00:47:53.020 PROFESSOR STRANG: This theorem? 00:47:53.020 --> 00:47:58.356 AUDIENCE: [INAUDIBLE] 00:47:58.356 --> 00:47:59.730 PROFESSOR STRANG: Yeah if it was, 00:47:59.730 --> 00:48:02.350 well let's draw a funny shape. 00:48:02.350 --> 00:48:03.680 See what we think. 00:48:03.680 --> 00:48:11.470 I mean, with this flow, right? 00:48:11.470 --> 00:48:16.510 Let me just say, if the flow has some difficult divergence 00:48:16.510 --> 00:48:19.550 and the region is some mess, nobody's 00:48:19.550 --> 00:48:20.810 going to be able to do it. 00:48:20.810 --> 00:48:21.770 I mean, yeah. 00:48:21.770 --> 00:48:24.460 So don't think that these can all be done. 00:48:24.460 --> 00:48:29.150 The equality, it's like integrals in calculus. 00:48:29.150 --> 00:48:31.020 No problem to think of integrations that 00:48:31.020 --> 00:48:33.620 are just beyond human capacity. 00:48:33.620 --> 00:48:36.680 But the formulas still hold. 00:48:36.680 --> 00:48:43.510 Suppose my region was even like this? 00:48:43.510 --> 00:48:48.340 Would that still be, do we still see flow in equals flow out 00:48:48.340 --> 00:48:50.320 for this particular flow? 00:48:50.320 --> 00:48:53.780 I think, yeah, the flow's going this way. 00:48:53.780 --> 00:48:56.430 So it's coming in here, it's going out again. 00:48:56.430 --> 00:48:57.790 That contributes. 00:48:57.790 --> 00:49:00.660 Back in again here, and out again here. 00:49:00.660 --> 00:49:09.290 Yeah, I think our instinct would be correct there, yeah. 00:49:09.290 --> 00:49:11.420 All sorts of examples. 00:49:11.420 --> 00:49:16.890 I was going to, well, I'll maybe do it in class. 00:49:16.890 --> 00:49:20.750 This divergence theorem is the fundamental theorem 00:49:20.750 --> 00:49:22.920 of 2-D calculus, you could say. 00:49:22.920 --> 00:49:27.930 Or one of them. 00:49:27.930 --> 00:49:42.050 And to write these things and see what they lead to, yeah. 00:49:42.050 --> 00:49:44.500 I'll tell you what I was going to do. 00:49:44.500 --> 00:49:52.220 I was going to apply this to the vector field u times w. 00:49:52.220 --> 00:49:58.180 So u is a scalar, w is a vector, and therefore uw is a vector. 00:49:58.180 --> 00:50:06.230 It's got two components, uw_1, are you willing to do that one? 00:50:06.230 --> 00:50:16.820 So I apply this not to w itself, but to u times w. 00:50:16.820 --> 00:50:22.720 Which has two components, uw_1 and uw_2. 00:50:22.720 --> 00:50:25.460 OK, so I should take the divergence of uw. 00:50:25.460 --> 00:50:31.240 I mean, that's a vector field, uw, this guy. 00:50:31.240 --> 00:50:35.420 And I'll get the uw dot n. 00:50:35.420 --> 00:50:41.700 And it just turns out that this is the right way to do it. 00:50:41.700 --> 00:50:44.750 To see the fact that gradient and divergence 00:50:44.750 --> 00:50:47.070 are transposes of each other. 00:50:47.070 --> 00:50:48.100 Yeah, yeah. 00:50:48.100 --> 00:50:51.060 I maybe I won't do that calculation now, 00:50:51.060 --> 00:50:55.790 I'll just say that if you take the divergence theorem and you 00:50:55.790 --> 00:51:01.290 apply it to this guy, [uw 1, uw 2], 00:51:01.290 --> 00:51:10.190 and write out what it means, you get a very interesting formula. 00:51:10.190 --> 00:51:12.030 I'll just leave that there. 00:51:12.030 --> 00:51:14.570 So I'm ready for a final question if there is one, 00:51:14.570 --> 00:51:21.940 or otherwise keep going Friday with these Laplace equation 00:51:21.940 --> 00:51:23.210 solutions. 00:51:23.210 --> 00:51:25.780 Play with some vector fields. 00:51:25.780 --> 00:51:29.740 That's my best advice. 00:51:29.740 --> 00:51:34.090 And I'll see you Friday.