WEBVTT 00:00:00.000 --> 00:00:02.916 PROFESSOR: Do you have any kind of questions? 00:00:02.916 --> 00:00:05.346 There were a few questions about the homework. 00:00:05.346 --> 00:00:11.081 Casey, you have that problem we need to the minus D? 00:00:11.081 --> 00:00:11.664 STUDENT: Yeah. 00:00:11.664 --> 00:00:12.622 PROFESSOR: The minus D? 00:00:12.622 --> 00:00:15.714 Let's do that in class, because there were several people who 00:00:15.714 --> 00:00:17.520 faced that problem. 00:00:17.520 --> 00:00:22.120 You said you faced it, and you got it and can I cheat? 00:00:22.120 --> 00:00:25.870 Can I take your work so I can present it at the board? 00:00:25.870 --> 00:00:27.214 I'm serious about it. 00:00:27.214 --> 00:00:27.880 STUDENT: OK, um. 00:00:27.880 --> 00:00:31.364 PROFESSOR: So I know we've done this together. 00:00:31.364 --> 00:00:33.690 I don't even remember the problem. 00:00:33.690 --> 00:00:35.410 How was it? 00:00:35.410 --> 00:00:37.060 Homework problem. 00:00:37.060 --> 00:00:39.530 STUDENT: She knows it. 00:00:39.530 --> 00:00:40.030 [INAUDIBLE] 00:00:40.030 --> 00:00:43.502 00:00:43.502 --> 00:00:44.990 STUDENT: But she knows it. 00:00:44.990 --> 00:00:47.800 PROFESSOR: They'll work with you to be minus M. Can you tell me, 00:00:47.800 --> 00:00:48.462 Casey? 00:00:48.462 --> 00:00:50.446 Can I tell the statement? 00:00:50.446 --> 00:00:52.926 STUDENT: Well, a black guy's [INAUDIBLE]. 00:00:52.926 --> 00:00:55.902 PROFESSOR: If you find it, give it to me and I'll give you $2. 00:00:55.902 --> 00:00:57.390 STUDENT: It's your problem. 00:00:57.390 --> 00:00:58.382 [INAUDIBLE] 00:00:58.382 --> 00:00:59.870 PROFESSOR: How is it's this one. 00:00:59.870 --> 00:01:00.862 STUDENT: No. 00:01:00.862 --> 00:01:01.612 PROFESSOR: Oh, no. 00:01:01.612 --> 00:01:02.505 It's not this one. 00:01:02.505 --> 00:01:03.838 STUDENT: Can I have it from you? 00:01:03.838 --> 00:01:05.822 I won't give you anything. 00:01:05.822 --> 00:01:07.310 STUDENT: Um, it's doable. 00:01:07.310 --> 00:01:09.790 [INAUDIBLE] 00:01:09.790 --> 00:01:13.338 PROFESSOR: So can somebody with me now, that's my handwriting. 00:01:13.338 --> 00:01:14.254 STUDENT: Yeah, I know. 00:01:14.254 --> 00:01:14.750 It is weird. 00:01:14.750 --> 00:01:14.998 PROFESSOR: OK. 00:01:14.998 --> 00:01:15.742 All right. 00:01:15.742 --> 00:01:17.726 So the problem says-- 00:01:17.726 --> 00:01:18.222 STUDENT: Do you have a problem with me? 00:01:18.222 --> 00:01:18.718 PROFESSOR: Any-- 00:01:18.718 --> 00:01:20.968 STUDENT: And then we're all here for her. [INAUDIBLE]. 00:01:20.968 --> 00:01:23.182 Doesn't it feel like [INAUDIBLE] kind of a bit? 00:01:23.182 --> 00:01:25.390 PROFESSOR: It's the one that has X of D equals 00:01:25.390 --> 00:01:28.582 into the minus the cosign D. 00:01:28.582 --> 00:01:30.046 STUDENT: Oh, [INAUDIBLE]. 00:01:30.046 --> 00:01:32.486 PROFESSOR: Y of T, and you go by your exclamation. 00:01:32.486 --> 00:01:37.370 I understand that you love this problem. 00:01:37.370 --> 00:01:43.645 And so you've had this type of pathing to grow compute. 00:01:43.645 --> 00:01:48.796 The pathing to grow with respect to the [INAUDIBLE] fellow man 00:01:48.796 --> 00:01:51.400 well meant that in life is slowly 00:01:51.400 --> 00:01:53.780 because nobody [INAUDIBLE] with you. 00:01:53.780 --> 00:01:58.190 00:01:58.190 --> 00:02:04.070 And to go over C of the integer will 00:02:04.070 --> 00:02:07.520 be a very nice friend of yours, [INAUDIBLE] explain it, 00:02:07.520 --> 00:02:11.210 but of course, they are both functions of T in general, 00:02:11.210 --> 00:02:13.910 and you will have the DS element, 00:02:13.910 --> 00:02:15.300 and what does this mean? 00:02:15.300 --> 00:02:17.462 S is [INAUDIBLE]. 00:02:17.462 --> 00:02:22.260 It means that you are archic element should 00:02:22.260 --> 00:02:25.512 be expressed in terms of what? 00:02:25.512 --> 00:02:27.630 Who in the world is the archeling infinite 00:02:27.630 --> 00:02:29.850 decimal element. 00:02:29.850 --> 00:02:32.040 It's the speed times the t. 00:02:32.040 --> 00:02:33.050 STUDENT: Say it again? 00:02:33.050 --> 00:02:34.730 PROFESSOR: It's the speed. 00:02:34.730 --> 00:02:36.400 STUDENT: And what was the speed? 00:02:36.400 --> 00:02:41.790 R in front of T. [INAUDIBLE]. 00:02:41.790 --> 00:02:42.760 Right? 00:02:42.760 --> 00:02:46.090 So you will have to transform this path integral 00:02:46.090 --> 00:02:49.590 into an integral, respected T, where T takes values 00:02:49.590 --> 00:02:52.610 from a T0 to a T1. 00:02:52.610 --> 00:02:59.212 And I don't want to give you your notebook back. 00:02:59.212 --> 00:02:59.920 STUDENT: It's OK. 00:02:59.920 --> 00:03:02.295 PROFESSOR: OK, now I'll do the same thing all over again, 00:03:02.295 --> 00:03:06.590 and you control me and then if I do something wrong, 00:03:06.590 --> 00:03:07.860 you've done me. 00:03:07.860 --> 00:03:12.740 And what were the-- what was the path? 00:03:12.740 --> 00:03:15.076 Specified as what? 00:03:15.076 --> 00:03:18.004 STUDENT: XYZ? [INAUDIBLE]. 00:03:18.004 --> 00:03:19.956 PROFESSOR: Yeah, the path was-- 00:03:19.956 --> 00:03:22.400 STUDENT: [INAUDIBLE]. 00:03:22.400 --> 00:03:24.910 PROFESSOR: T equals from zero to pi over 2. 00:03:24.910 --> 00:03:26.750 I have to write it down. 00:03:26.750 --> 00:03:29.690 00:03:29.690 --> 00:03:34.750 So let us write the-- [INAUDIBLE] are from the T. 00:03:34.750 --> 00:03:44.100 The speed square root of is from the T squared plus Y 00:03:44.100 --> 00:03:49.620 prime of T squared, because the sampling occurred. 00:03:49.620 --> 00:03:53.650 Before we do that, we have to go ahead and compute 00:03:53.650 --> 00:03:54.795 X prime and Y prime. 00:03:54.795 --> 00:03:58.662 00:03:58.662 --> 00:04:01.092 And of course that's product rule, 00:04:01.092 --> 00:04:03.036 and I need a better marker. 00:04:03.036 --> 00:04:04.008 STUDENT: [INAUDIBLE]. 00:04:04.008 --> 00:04:04.980 PROFESSOR: Yes, sir? 00:04:04.980 --> 00:04:06.771 STUDENT: Do you think the arc too is really 00:04:06.771 --> 00:04:08.880 taken as an arc [INAUDIBLE]? 00:04:08.880 --> 00:04:09.380 This 00:04:09.380 --> 00:04:10.380 PROFESSOR: This is the-- 00:04:10.380 --> 00:04:13.070 STUDENT: Because we take-- I would consider it as a path 00:04:13.070 --> 00:04:15.530 function that looks like an arc, or, like, thinking 00:04:15.530 --> 00:04:18.728 that it's missing one rule, and that's about it. 00:04:18.728 --> 00:04:19.720 That's fine. 00:04:19.720 --> 00:04:21.620 PROFESSOR: No, no, no no, no, no, no, no. 00:04:21.620 --> 00:04:22.120 no. 00:04:22.120 --> 00:04:23.090 OK, let me explain. 00:04:23.090 --> 00:04:28.090 So suppose you are [INAUDIBLE] arc in plane 00:04:28.090 --> 00:04:29.798 and this is your r of t. 00:04:29.798 --> 00:04:30.629 STUDENT: Oh, OK. 00:04:30.629 --> 00:04:32.670 PROFESSOR: And that's called the position vector, 00:04:32.670 --> 00:04:35.661 and that's x of t, y of t. 00:04:35.661 --> 00:04:36.160 OK. 00:04:36.160 --> 00:04:38.014 What is your velocity vector? 00:04:38.014 --> 00:04:41.730 Velocity vector would be in tangent to the curve. 00:04:41.730 --> 00:04:44.974 Suppose you go in this direction, counterclockwise, 00:04:44.974 --> 00:04:48.310 and then our prime of t will be this guy. 00:04:48.310 --> 00:04:50.705 And it's gonna be x prime, y prime. 00:04:50.705 --> 00:04:53.350 And we have to find its magnitude. 00:04:53.350 --> 00:04:55.700 And its magnitude will be this animal. 00:04:55.700 --> 00:04:59.012 So the only thing here is tricky because you 00:04:59.012 --> 00:05:03.480 will have to do this carefully, and there 00:05:03.480 --> 00:05:05.710 will be a simplification coming from the plus 00:05:05.710 --> 00:05:07.810 and minus of the binomial. 00:05:07.810 --> 00:05:11.250 So a few people missed it because of that reason. 00:05:11.250 --> 00:05:13.780 So let's see what we have-- minus 00:05:13.780 --> 00:05:17.410 e to the minus t, first prime, times second one 00:05:17.410 --> 00:05:24.844 prime plus the first one prime times the second prime. 00:05:24.844 --> 00:05:25.770 Good. 00:05:25.770 --> 00:05:28.066 We are done with this first guy. 00:05:28.066 --> 00:05:34.443 The second guy will be minus e to the minus t sin t. 00:05:34.443 --> 00:05:36.910 Why do I do this? 00:05:36.910 --> 00:05:40.730 Because I'm afraid that this being on the final. 00:05:40.730 --> 00:05:43.410 Well, it's good practice. 00:05:43.410 --> 00:05:47.228 You may expect something a little bit 00:05:47.228 --> 00:05:53.014 similar to that, so why don't we do this as part of our review, 00:05:53.014 --> 00:05:55.324 which will be a very good idea. 00:05:55.324 --> 00:05:58.290 We are gonna do lots of review this week and next 00:05:58.290 --> 00:06:01.710 week already, because the final is coming close 00:06:01.710 --> 00:06:09.832 and you have to go over everything that you've covered. 00:06:09.832 --> 00:06:16.428 Let's square them, and add them together. 00:06:16.428 --> 00:06:22.720 00:06:22.720 --> 00:06:23.740 OK. 00:06:23.740 --> 00:06:31.740 When we add them together, this guy, the speed [INAUDIBLE] 00:06:31.740 --> 00:06:33.880 is going to-- bless you. 00:06:33.880 --> 00:06:36.990 It is going-- it's not going to bless, it's going-- OK, 00:06:36.990 --> 00:06:41.050 you are being blessed, and now let's look at that. 00:06:41.050 --> 00:06:46.120 You have e to the minus 2t cosine squared 00:06:46.120 --> 00:06:48.170 and e to the minus 2t sine squared, 00:06:48.170 --> 00:06:50.672 and when you add those parts, the sine 00:06:50.672 --> 00:06:53.120 squared plus cosine squared stick together. 00:06:53.120 --> 00:06:56.808 They form a block called 1. 00:06:56.808 --> 00:06:58.580 Do you guys agree with me? 00:06:58.580 --> 00:07:02.960 So what we have as the first result of that 00:07:02.960 --> 00:07:04.882 would be this guy. 00:07:04.882 --> 00:07:09.550 But then, when you take twice the product 00:07:09.550 --> 00:07:12.810 of these guys in the binomial formula, 00:07:12.810 --> 00:07:15.560 and twice the product of these guys, what do you notice? 00:07:15.560 --> 00:07:19.820 We have exactly the same individuals inside, 00:07:19.820 --> 00:07:23.250 but when you do twice the product of these two red ones, 00:07:23.250 --> 00:07:26.800 you have minus, minus, plus. 00:07:26.800 --> 00:07:29.195 But when you do twice the product of these guys, 00:07:29.195 --> 00:07:31.870 you have minus, plus, minus. 00:07:31.870 --> 00:07:33.410 So they will cancel out. 00:07:33.410 --> 00:07:37.058 The part in the middle will cancel out. 00:07:37.058 --> 00:07:41.454 And finally, when I square this part and that part, 00:07:41.454 --> 00:07:42.620 what's going to happen them? 00:07:42.620 --> 00:07:47.460 And I'm gonna shut up because I want you to give me the answer. 00:07:47.460 --> 00:07:50.570 Square this animal, square this animal, add them together, 00:07:50.570 --> 00:07:51.950 what do you have? 00:07:51.950 --> 00:07:54.345 STUDENT: Squared, squared total-- [INTERPOSING VOICES] 00:07:54.345 --> 00:07:58.770 PROFESSOR: T to the minus 2t, so exactly the same as this guy. 00:07:58.770 --> 00:08:02.033 So all I know under the square root, 00:08:02.033 --> 00:08:12.230 I'm gonna get square root of 2 times e to the minus 2t. 00:08:12.230 --> 00:08:15.052 Which is e to the minus t square root of 2. 00:08:15.052 --> 00:08:17.966 Am I right, [INAUDIBLE] that's what we got last time? 00:08:17.966 --> 00:08:18.942 All right. 00:08:18.942 --> 00:08:25.680 So I know who this will be. 00:08:25.680 --> 00:08:30.450 I don't know who this will be, but I'm gonna need your help. 00:08:30.450 --> 00:08:34.710 Here I write it, x squared of t plus y squared of t 00:08:34.710 --> 00:08:38.740 in terms of t, squaring them and adding them together. 00:08:38.740 --> 00:08:42.700 It's gonna be again a piece of cake, because you've got it. 00:08:42.700 --> 00:08:45.670 How much is it? 00:08:45.670 --> 00:08:47.650 I'm waiting for you to tell me. 00:08:47.650 --> 00:08:50.620 This is this one. 00:08:50.620 --> 00:08:52.120 [INAUDIBLE] 00:08:52.120 --> 00:08:52.840 E to the? 00:08:52.840 --> 00:08:54.060 STUDENT: Minus t 00:08:54.060 --> 00:08:56.135 PROFESSOR: And anything else? 00:08:56.135 --> 00:08:59.100 STUDENT: Was it 2? 00:08:59.100 --> 00:08:59.600 [INAUDIBLE] 00:08:59.600 --> 00:09:04.060 00:09:04.060 --> 00:09:05.914 PROFESSOR: Why it times 2? 00:09:05.914 --> 00:09:07.375 STUDENT: Times 2 in the last one. 00:09:07.375 --> 00:09:09.958 Because we had an e to the minus 2t plus an e to the minus 2t. 00:09:09.958 --> 00:09:13.706 PROFESSOR: So I took this guy and squared it, 00:09:13.706 --> 00:09:15.505 and I took this guy and squared it-- 00:09:15.505 --> 00:09:17.130 STUDENT: No, we don't have [INAUDIBLE]. 00:09:17.130 --> 00:09:19.432 PROFESSOR: And I sum them up. 00:09:19.432 --> 00:09:21.520 And I close the issue. 00:09:21.520 --> 00:09:28.397 Unless I have sine squared plus cosine squared, which is 1, 00:09:28.397 --> 00:09:30.782 so we adjust it to the minus 2t. 00:09:30.782 --> 00:09:32.230 Agree with me? 00:09:32.230 --> 00:09:34.090 All right, now we have all the ingredients. 00:09:34.090 --> 00:09:35.715 Do we have all the ingredients we need? 00:09:35.715 --> 00:09:37.696 We have this, we have that, we have that. 00:09:37.696 --> 00:09:40.730 And we should just go ahead and solve the problem. 00:09:40.730 --> 00:09:48.106 So, integral from 0 to pi over 2, this friend of yours, e 00:09:48.106 --> 00:09:52.490 to the minus 2t plus [INAUDIBLE]. 00:09:52.490 --> 00:09:59.026 The speed was over there, e to the minus t times square root 00:09:59.026 --> 00:10:00.002 of 2. 00:10:00.002 --> 00:10:01.260 That was the speed. 00:10:01.260 --> 00:10:05.384 [INAUDIBLE] magnitude, dt. 00:10:05.384 --> 00:10:07.376 Is this what we got? 00:10:07.376 --> 00:10:08.372 All right. 00:10:08.372 --> 00:10:10.364 Now, we are almost done, in the sense 00:10:10.364 --> 00:10:14.348 that we should wrap things up. 00:10:14.348 --> 00:10:16.860 Square root 2 gets out. 00:10:16.860 --> 00:10:22.889 And then integral of it to the minus 3t from zero to pi over 2 00:10:22.889 --> 00:10:24.308 is our friend. 00:10:24.308 --> 00:10:26.673 We know how to deal with him. 00:10:26.673 --> 00:10:28.570 We have dt. 00:10:28.570 --> 00:10:35.240 So when you integrate that, what do you have? 00:10:35.240 --> 00:10:36.680 Let me erase-- 00:10:36.680 --> 00:10:40.302 STUDENT: Negative square root 2 over 3 [INAUDIBLE] 3t. 00:10:40.302 --> 00:10:41.010 PROFESSOR: Right. 00:10:41.010 --> 00:10:42.960 So let me erase this part. 00:10:42.960 --> 00:10:47.600 00:10:47.600 --> 00:10:50.610 So we have-- first we have to copy this guy. 00:10:50.610 --> 00:10:54.728 Then we have e to the minus 3t divided by minus 3 00:10:54.728 --> 00:10:58.150 because that is the antiderivative. 00:10:58.150 --> 00:11:04.950 And we take that into t equals zero and t equals pi over 2. 00:11:04.950 --> 00:11:10.650 Square root of 2 says I'm going out, and actually minus 3 00:11:10.650 --> 00:11:12.450 says also I'm going out. 00:11:12.450 --> 00:11:16.474 So he doesn't want to be involved in this discussion. 00:11:16.474 --> 00:11:17.788 [INAUDIBLE] 00:11:17.788 --> 00:11:23.650 Now, e to the minus 3 pi over 2 is the first thing we got. 00:11:23.650 --> 00:11:26.520 And then minus e to the 0. 00:11:26.520 --> 00:11:27.803 What's e to the 0? 00:11:27.803 --> 00:11:28.610 STUDENT: 1. 00:11:28.610 --> 00:11:30.310 PROFESSOR: 1. 00:11:30.310 --> 00:11:35.050 So in the end, you have to change the sign. 00:11:35.050 --> 00:11:40.540 You have root 2 over 3 times bracket notation when 00:11:40.540 --> 00:11:44.020 you type this in WeBWorK because based 00:11:44.020 --> 00:11:49.150 on your syntax, if your syntax is bad, you are-- for example, 00:11:49.150 --> 00:11:53.480 here we have to put ^ minus 3 pi over 2. 00:11:53.480 --> 00:11:54.906 Are you guys with me? 00:11:54.906 --> 00:11:57.030 Do you understand the words coming out of my mouth? 00:11:57.030 --> 00:12:01.840 So here you have to type the right syntax, 00:12:01.840 --> 00:12:03.743 and you did, and you got-- 00:12:03.743 --> 00:12:05.162 STUDENT: And I didn't [INAUDIBLE] 00:12:05.162 --> 00:12:08.000 but I need to type the decimal answer. 00:12:08.000 --> 00:12:10.332 In terms of decimal places. 00:12:10.332 --> 00:12:11.540 PROFESSOR: This is a problem. 00:12:11.540 --> 00:12:13.700 It shouldn't be like that. 00:12:13.700 --> 00:12:18.040 Sometimes unfortunately-- well, fortunately it rarely 00:12:18.040 --> 00:12:27.420 happens that WeBWorK program does not take your answer 00:12:27.420 --> 00:12:28.480 in a certain format. 00:12:28.480 --> 00:12:30.620 Maybe the pi screws everything up. 00:12:30.620 --> 00:12:31.640 I don't know. 00:12:31.640 --> 00:12:34.630 But if you do this with your calculator, 00:12:34.630 --> 00:12:38.460 eventually, you can What was the approximate answer you got, 00:12:38.460 --> 00:12:39.740 [INAUDIBLE]? 00:12:39.740 --> 00:12:42.447 STUDENT: 0.467 00:12:42.447 --> 00:12:43.780 PROFESSOR: And blah, blah, blah. 00:12:43.780 --> 00:12:44.321 I don't know. 00:12:44.321 --> 00:12:48.180 I think WeBWorK only cares for the first two decimals 00:12:48.180 --> 00:12:49.375 to be correct. 00:12:49.375 --> 00:12:50.325 As I remember. 00:12:50.325 --> 00:12:51.275 I don't know. 00:12:51.275 --> 00:12:53.790 Now I have to ask the programmer. 00:12:53.790 --> 00:12:56.020 So this would be approximately-- you 00:12:56.020 --> 00:12:57.580 plug in the approximate answer. 00:12:57.580 --> 00:12:59.499 I solved the problem, so I should give myself 00:12:59.499 --> 00:13:02.520 the credit, plus a piece of candy, 00:13:02.520 --> 00:13:07.300 but I hope I was able to save you from some grief 00:13:07.300 --> 00:13:10.910 because you have so much review going on that you shouldn't 00:13:10.910 --> 00:13:14.290 spend time on problems that you headache 00:13:14.290 --> 00:13:16.260 for computational reasons. 00:13:16.260 --> 00:13:18.520 Actually, I have computational reasons, 00:13:18.520 --> 00:13:22.010 because we are not androids and we are not computers. 00:13:22.010 --> 00:13:27.190 What we can do is think of a problem 00:13:27.190 --> 00:13:31.230 and let the software solve the problem for us. 00:13:31.230 --> 00:13:36.020 So our strength does not consist in how fast we can compute, 00:13:36.020 --> 00:13:42.306 but on how well we can solve a problem so that the calculator 00:13:42.306 --> 00:13:45.880 or computer can carry on. 00:13:45.880 --> 00:13:46.880 All right. 00:13:46.880 --> 00:13:49.880 00:13:49.880 --> 00:13:55.610 I know I covered up to 13.6, and let 00:13:55.610 --> 00:13:57.900 me remind you what we covered. 00:13:57.900 --> 00:14:02.634 We covered some beautiful sections that were called 13.4. 00:14:02.634 --> 00:14:04.410 This was Green's theorem. 00:14:04.410 --> 00:14:07.875 And now, I'm really proud of you that all of you 00:14:07.875 --> 00:14:10.350 know Green's theorem very well. 00:14:10.350 --> 00:14:17.440 And the surface integral, which was 13.5. 00:14:17.440 --> 00:14:21.940 And then I promised you that today we'd 00:14:21.940 --> 00:14:25.874 move on to 13.6, which is Stokes' theorem, 00:14:25.874 --> 00:14:29.250 and I'm gonna do that. 00:14:29.250 --> 00:14:34.380 But before I do that, I want to attract your attention 00:14:34.380 --> 00:14:49.920 to a fact that this is a bigger result that incorporates 13.4. 00:14:49.920 --> 00:14:53.470 So Stokes' theorem is a more general 00:14:53.470 --> 00:14:58.410 result. So let me make a diagram, like a Venn diagram. 00:14:58.410 --> 00:15:01.600 This is all the cases of Stokes' theorem, 00:15:01.600 --> 00:15:03.501 and Green's is one of them. 00:15:03.501 --> 00:15:07.110 00:15:07.110 --> 00:15:11.490 And this is something you've learned, and you did very well, 00:15:11.490 --> 00:15:14.650 and we only considered this theorem 00:15:14.650 --> 00:15:17.560 on a domain that's interconnected. 00:15:17.560 --> 00:15:19.520 It has no holes in it. 00:15:19.520 --> 00:15:21.920 Green's theorem can be also taught 00:15:21.920 --> 00:15:25.080 on something like a doughnut, but that's not 00:15:25.080 --> 00:15:26.970 the purpose of this course. 00:15:26.970 --> 00:15:28.540 You have it in the book. 00:15:28.540 --> 00:15:30.110 It's very sophisticated. 00:15:30.110 --> 00:15:37.470 13.6 starts at-- oh, my God, I don't know the pages. 00:15:37.470 --> 00:15:39.770 And being a co-author of this book 00:15:39.770 --> 00:15:42.640 means that I should remember the pages. 00:15:42.640 --> 00:15:44.200 All right, there it is. 00:15:44.200 --> 00:15:50.760 13.6 is that page 1075. 00:15:50.760 --> 00:15:56.695 OK, and let's see what this theorem is about. 00:15:56.695 --> 00:16:01.227 00:16:01.227 --> 00:16:05.400 I'm gonna state it as first Stokes' theorem, 00:16:05.400 --> 00:16:08.850 and then I will see why Green's theorem is a particular case. 00:16:08.850 --> 00:16:11.110 We don't know yet why that is. 00:16:11.110 --> 00:16:13.620 Well, assume you have a force field, 00:16:13.620 --> 00:16:15.250 may the force be with you. 00:16:15.250 --> 00:16:30.260 This is a big vector-valued function over a domain in R 3 00:16:30.260 --> 00:16:39.150 that includes a surface s. 00:16:39.150 --> 00:16:41.975 We don't say much about the surface S 00:16:41.975 --> 00:16:45.650 because we try to avoid the terminology, 00:16:45.650 --> 00:16:48.080 but you guys should assume that this 00:16:48.080 --> 00:17:00.190 is a simply connected surface patch with a boundary 00:17:00.190 --> 00:17:10.760 c, such that c is a Jordan curve. 00:17:10.760 --> 00:17:13.960 00:17:13.960 --> 00:17:18.720 We use the word Jordan curve as a boundary of the surface, 00:17:18.720 --> 00:17:21.060 but we don't say simply connected. 00:17:21.060 --> 00:17:23.431 And I'm going to ask you, what in the world 00:17:23.431 --> 00:17:25.955 do we mean when we simply connected? 00:17:25.955 --> 00:17:27.920 I've used this before. 00:17:27.920 --> 00:17:30.880 I just want to test your memory and attention. 00:17:30.880 --> 00:17:33.170 Do you remember what that meant? 00:17:33.170 --> 00:17:39.170 I have some sort of little hill, or s could be a flat disc, 00:17:39.170 --> 00:17:41.330 or it could be a patch of the plane, 00:17:41.330 --> 00:17:46.170 or it could be just any kind of surface that 00:17:46.170 --> 00:17:49.142 is bounded by Jordan curve c. 00:17:49.142 --> 00:17:50.100 What is a Jordan curve? 00:17:50.100 --> 00:17:52.480 But what can we say about c? 00:17:52.480 --> 00:17:56.140 00:17:56.140 --> 00:18:00.500 So c would be nice piecewise continuous-- we 00:18:00.500 --> 00:18:02.380 assumed it continuous actually. 00:18:02.380 --> 00:18:03.320 Most cases-- 00:18:03.320 --> 00:18:05.361 STUDENT: It has to connect to itself, doesn't it? 00:18:05.361 --> 00:18:07.660 PROFESSOR: No self-intersections. 00:18:07.660 --> 00:18:10.570 So we knew that from before, but what does it mean, 00:18:10.570 --> 00:18:15.440 simply connected for us? 00:18:15.440 --> 00:18:16.395 I said it before. 00:18:16.395 --> 00:18:19.930 I don't know how attentive you were. 00:18:19.930 --> 00:18:22.600 Connectedness makes you think of something. 00:18:22.600 --> 00:18:23.850 No holes in it. 00:18:23.850 --> 00:18:26.290 So that means no holes. 00:18:26.290 --> 00:18:27.260 No punctures. 00:18:27.260 --> 00:18:31.200 No holes, no punctures. 00:18:31.200 --> 00:18:33.070 So why-- don't draw it. 00:18:33.070 --> 00:18:35.250 I will draw it so you can laugh. 00:18:35.250 --> 00:18:37.320 Assume that the dog came here and took 00:18:37.320 --> 00:18:39.420 a bite of this surface. 00:18:39.420 --> 00:18:41.720 And now you have a hole in it. 00:18:41.720 --> 00:18:43.710 Well, you're not supposed to have a hole in it, 00:18:43.710 --> 00:18:45.960 so tell the dog to go away. 00:18:45.960 --> 00:18:48.820 So you're not gonna have any problems, any puncture, 00:18:48.820 --> 00:18:52.600 any hole, any problem with this. 00:18:52.600 --> 00:18:57.780 Now the surface is assumed to be a regular surface, 00:18:57.780 --> 00:18:59.830 and we've seen that before. 00:18:59.830 --> 00:19:03.000 And since it's a regular surface, that means 00:19:03.000 --> 00:19:05.150 it's immersed in the ambient space, 00:19:05.150 --> 00:19:09.100 and you have an N orientation. 00:19:09.100 --> 00:19:16.860 Orientation which is the unit normal to the surface. 00:19:16.860 --> 00:19:19.630 00:19:19.630 --> 00:19:20.880 Can you draw it, Magdalena? 00:19:20.880 --> 00:19:24.180 Yes, in a minute, I will draw it. 00:19:24.180 --> 00:19:29.560 At every point you have an N unit normal. 00:19:29.560 --> 00:19:32.560 What was the unit normal for you when 00:19:32.560 --> 00:19:35.540 you parametrize the surface? 00:19:35.540 --> 00:19:40.210 That was the stick that has length 1 perpendicular 00:19:40.210 --> 00:19:41.860 to the tangent length, right? 00:19:41.860 --> 00:19:44.840 So if you wanted to do it for general R, 00:19:44.840 --> 00:19:48.860 you would take those R sub u or sub v the partial [INAUDIBLE]. 00:19:48.860 --> 00:19:53.220 And draw the cross product, and this 00:19:53.220 --> 00:19:55.060 is what I'm trying to do now. 00:19:55.060 --> 00:19:57.570 And just make the length be 1. 00:19:57.570 --> 00:20:02.170 So if the surface is regular, I can parametrize it 00:20:02.170 --> 00:20:06.650 as [INAUDIBLE] will exist in that orientation. 00:20:06.650 --> 00:20:09.000 I want something more. 00:20:09.000 --> 00:20:17.360 I want N orientation to be compatible to the direction 00:20:17.360 --> 00:20:26.450 of travel on c-- along c. 00:20:26.450 --> 00:20:29.540 Along, not on, but on is not bad either. 00:20:29.540 --> 00:20:32.460 So assume that this is a hill, and I'm 00:20:32.460 --> 00:20:34.170 running around the boundary. 00:20:34.170 --> 00:20:37.340 Look, I'm just running around the boundary, which is c. 00:20:37.340 --> 00:20:40.270 Am I running in a particular direction that 00:20:40.270 --> 00:20:42.870 tells you I'm a mathematician? 00:20:42.870 --> 00:20:44.660 It tells you that I'm a weirdo. 00:20:44.660 --> 00:20:45.520 Yes. 00:20:45.520 --> 00:20:49.240 So in what kind of direction am I running? 00:20:49.240 --> 00:20:50.610 Counterclockwise. 00:20:50.610 --> 00:20:51.110 Why? 00:20:51.110 --> 00:20:52.740 Because I'm a nerd. 00:20:52.740 --> 00:20:54.560 Like Sheldon or something. 00:20:54.560 --> 00:20:59.410 So let's go around, and so what does 00:20:59.410 --> 00:21:05.294 it mean I am compatible with the orientation? 00:21:05.294 --> 00:21:07.010 Think of the right hand rule. 00:21:07.010 --> 00:21:10.210 Or forget about right hand rule, I hate that word. 00:21:10.210 --> 00:21:12.940 Let's think faucet. 00:21:12.940 --> 00:21:17.340 So if your motion is along the c, 00:21:17.340 --> 00:21:20.950 so that it's like you are unscrewing the faucet, 00:21:20.950 --> 00:21:22.650 it's going up. 00:21:22.650 --> 00:21:25.050 That should mean that your orientation 00:21:25.050 --> 00:21:28.660 n should go up, or, not down, in the other direction. 00:21:28.660 --> 00:21:32.750 So if I take c to be my orientation around the curve, 00:21:32.750 --> 00:21:36.000 then the orientation of the surface should go up. 00:21:36.000 --> 00:21:41.150 Am I allowed to go around the opposite direction on the c. 00:21:41.150 --> 00:21:42.880 Yes I am. 00:21:42.880 --> 00:21:46.020 That's, how it this called, inverse trigonometric, 00:21:46.020 --> 00:21:48.035 or how do we call such a thing. 00:21:48.035 --> 00:21:49.398 STUDENT: Clockwise? 00:21:49.398 --> 00:21:52.146 PROFESSOR: Clockwise, you guessed it. 00:21:52.146 --> 00:21:53.520 OK, clockwise. 00:21:53.520 --> 00:21:57.310 If I would go clockwise in plane, 00:21:57.310 --> 00:22:00.980 then the N should be pointing down. 00:22:00.980 --> 00:22:04.090 So it should be oriented just the opposite way 00:22:04.090 --> 00:22:06.410 on the surface S. 00:22:06.410 --> 00:22:10.590 All right, that's sort of easy to understand now 00:22:10.590 --> 00:22:12.794 because most of you are engineers 00:22:12.794 --> 00:22:17.240 and you deal with this kind of stuff every day. 00:22:17.240 --> 00:22:18.960 What is Stokes' theorem? 00:22:18.960 --> 00:22:23.490 Stokes' theorem says well, in that case, the path integral 00:22:23.490 --> 00:22:30.211 over c of FdR, F dot dR. 00:22:30.211 --> 00:22:32.040 What the heck is this? 00:22:32.040 --> 00:22:35.250 I'm not gonna finish the sentence, because I'm mean. 00:22:35.250 --> 00:22:37.930 There is a sentence there, an equation, but I'm mean. 00:22:37.930 --> 00:22:43.400 So I'm asking you first, what in the world is this? 00:22:43.400 --> 00:22:45.590 F is the may the force be with you. 00:22:45.590 --> 00:22:47.880 R is the vector position. 00:22:47.880 --> 00:22:49.770 What is this animal? 00:22:49.770 --> 00:22:51.912 The book doesn't tell you. 00:22:51.912 --> 00:22:54.210 This is the work that you know so well. 00:22:54.210 --> 00:22:55.750 All right. 00:22:55.750 --> 00:22:59.910 So you may hear math majors saying they don't care. 00:22:59.910 --> 00:23:02.420 They don't care because they're not engineers or physicists, 00:23:02.420 --> 00:23:04.710 but work is very important. 00:23:04.710 --> 00:23:10.588 The work along the curve will be equal to-- now 00:23:10.588 --> 00:23:14.004 comes the beauty-- the beautiful part. 00:23:14.004 --> 00:23:17.270 This is a double integral over the surface [INAUDIBLE] 00:23:17.270 --> 00:23:22.240 with respect to the area element dS. 00:23:22.240 --> 00:23:26.580 Oh, guess what, you wouldn't know unless somebody taught you 00:23:26.580 --> 00:23:29.810 before coming to class, this is going 00:23:29.810 --> 00:23:34.220 to be curl F. What is curl? 00:23:34.220 --> 00:23:35.140 It's a vector. 00:23:35.140 --> 00:23:39.075 So I have to do dot product with another vector. 00:23:39.075 --> 00:23:40.780 And that vector is N. 00:23:40.780 --> 00:23:45.410 Some people read the book ahead of time, which is great. 00:23:45.410 --> 00:23:52.370 I would say 0.5% or less of the students read ahead 00:23:52.370 --> 00:23:53.170 in a textbook. 00:23:53.170 --> 00:23:56.650 I used to do that when I was young. 00:23:56.650 --> 00:24:00.296 I didn't always have the time to do it, 00:24:00.296 --> 00:24:04.828 but whenever I had the possibility I did it. 00:24:04.828 --> 00:24:11.950 Now a quiz for you. 00:24:11.950 --> 00:24:15.830 No, don't take any sheets out, but a quiz for you. 00:24:15.830 --> 00:24:18.930 Could you prove to me, just based 00:24:18.930 --> 00:24:24.710 on this thing that looks weird, that Green's theorem is 00:24:24.710 --> 00:24:28.190 a particular case of this? 00:24:28.190 --> 00:24:32.160 So prove-- where should I put it? 00:24:32.160 --> 00:24:35.155 That was Stokes' theorem. 00:24:35.155 --> 00:24:36.940 Stokes' theorem. 00:24:36.940 --> 00:24:39.830 And I'll say exercise number one, 00:24:39.830 --> 00:24:42.812 sometimes I put this in the final exam, 00:24:42.812 --> 00:24:46.040 so I consider this to be important. 00:24:46.040 --> 00:25:06.560 Prove that Green's theorem is nothing but a particular case 00:25:06.560 --> 00:25:08.036 of Stokes' theorem. 00:25:08.036 --> 00:25:12.464 00:25:12.464 --> 00:25:17.384 And I make a face in the sense that I'm trying to build trust. 00:25:17.384 --> 00:25:18.860 Maybe you don't trust me. 00:25:18.860 --> 00:25:21.500 But I-- let's do this together. 00:25:21.500 --> 00:25:23.820 Let's prove together that this is what it is. 00:25:23.820 --> 00:25:31.160 Now, the thing is, if I were to give you a test right now 00:25:31.160 --> 00:25:33.580 on Green's theorem, how many of you would 00:25:33.580 --> 00:25:35.630 know what Green's theorem said? 00:25:35.630 --> 00:25:39.900 So I'll put it here in an-- open an icon. 00:25:39.900 --> 00:25:44.170 Imagine this would be an icon-- or a window, 00:25:44.170 --> 00:25:48.090 a window on the computer screen. 00:25:48.090 --> 00:25:52.898 Like a tutorial reminding you what Green's theorem was. 00:25:52.898 --> 00:25:54.854 So Green's theorem said what? 00:25:54.854 --> 00:25:57.790 00:25:57.790 --> 00:26:00.910 We have to-- bless you. 00:26:00.910 --> 00:26:05.770 So Zander started the theorem by-- we 00:26:05.770 --> 00:26:12.660 have a domain D that was also simply connected. 00:26:12.660 --> 00:26:13.695 What does it mean? 00:26:13.695 --> 00:26:15.515 No punctures, no holes. 00:26:15.515 --> 00:26:16.425 No holes. 00:26:16.425 --> 00:26:20.500 Even if you have a puncture that's a point, 00:26:20.500 --> 00:26:22.090 that's still a hole. 00:26:22.090 --> 00:26:27.670 You may not see it, but if anybody punctured the portion 00:26:27.670 --> 00:26:30.230 of a plane, you are in trouble. 00:26:30.230 --> 00:26:32.190 So there are no such things. 00:26:32.190 --> 00:26:34.040 And c is a Jordan curve. 00:26:34.040 --> 00:26:41.470 00:26:41.470 --> 00:26:44.460 And then you say, OK, how is it, how was F? 00:26:44.460 --> 00:26:46.420 F was a c 1. 00:26:46.420 --> 00:26:48.690 What does it mean that if is a c 1? 00:26:48.690 --> 00:26:52.550 F is a vector-valued function that's differentiable, 00:26:52.550 --> 00:26:56.020 and its derivatives are continuous, 00:26:56.020 --> 00:26:57.520 partial derivatives. 00:26:57.520 --> 00:27:07.930 And so you think F of xy will be M of xy I plus n of xyj 00:27:07.930 --> 00:27:14.850 is a vector field, so it's a multivariable, 00:27:14.850 --> 00:27:16.550 so I have two variables. 00:27:16.550 --> 00:27:18.400 OK? 00:27:18.400 --> 00:27:21.130 So you think, OK, I know what this is. 00:27:21.130 --> 00:27:23.100 Like, this would be a force. 00:27:23.100 --> 00:27:25.590 If this were a force, I would get 00:27:25.590 --> 00:27:33.740 the vector-- the work-- how can I write this again? 00:27:33.740 --> 00:27:35.210 We didn't write it like that. 00:27:35.210 --> 00:27:38.180 We wrote it as Mdx plus Ndy, which 00:27:38.180 --> 00:27:40.921 is the same thing as before. 00:27:40.921 --> 00:27:41.420 Why? 00:27:41.420 --> 00:27:45.780 Because Mr. F is MI plus NJ. 00:27:45.780 --> 00:27:47.195 Not k, Magdalena. 00:27:47.195 --> 00:27:50.420 You were too nice, but you didn't want to shout at me. 00:27:50.420 --> 00:27:56.610 And dR was what? dR was dxI plus dyJ, right? 00:27:56.610 --> 00:28:00.560 So when you do this, the product which is called work, 00:28:00.560 --> 00:28:04.420 the integral will read Mdx plus Ndy, 00:28:04.420 --> 00:28:08.315 and this is what it was in Green's theorem. 00:28:08.315 --> 00:28:10.580 And what did we claim it was? 00:28:10.580 --> 00:28:13.450 Now, you know it, because you've done a lot of homework. 00:28:13.450 --> 00:28:16.570 You're probably sick and tired of Green's theorem and you say, 00:28:16.570 --> 00:28:20.540 I understand that work-- a path integral can be expressed 00:28:20.540 --> 00:28:23.930 as a double integral some way. 00:28:23.930 --> 00:28:25.600 Do you know this by heart? 00:28:25.600 --> 00:28:28.920 You proved this to me last time you know it by heart. 00:28:28.920 --> 00:28:33.020 That was N sub x minus M sub y. 00:28:33.020 --> 00:28:40.080 And we memorized it-- dA over this is a planar domain. 00:28:40.080 --> 00:28:43.750 It's a domain in plane d, [INAUDIBLE]. 00:28:43.750 --> 00:28:45.760 I said it, but I didn't write it down. 00:28:45.760 --> 00:28:50.654 So double integral over d, N sub x minus M sub y dA. 00:28:50.654 --> 00:28:51.320 We've done that. 00:28:51.320 --> 00:28:55.330 That was section-- what section was that, guys? 00:28:55.330 --> 00:28:56.820 13.4. 00:28:56.820 --> 00:29:00.830 Yeah, for sure, you will have a problem on the final like that. 00:29:00.830 --> 00:29:03.800 Do not expect lots of problems. 00:29:03.800 --> 00:29:05.320 Do not expect 25 problems. 00:29:05.320 --> 00:29:06.700 You will not have the time. 00:29:06.700 --> 00:29:09.198 So you will have some 15, 16 problems. 00:29:09.198 --> 00:29:12.110 This will be one of them. 00:29:12.110 --> 00:29:14.560 You mastered this Green's theorem. 00:29:14.560 --> 00:29:18.640 When you sent me questions from WeBWorK 00:29:18.640 --> 00:29:23.070 I realized that you were able to solve the problems where 00:29:23.070 --> 00:29:26.025 these would be easy to manipulate, 00:29:26.025 --> 00:29:27.140 like constants and so on. 00:29:27.140 --> 00:29:29.410 That's a beautiful case. 00:29:29.410 --> 00:29:33.190 There was one that gave [INAUDIBLE] a headache, 00:29:33.190 --> 00:29:35.318 and then I decided-- number 22, right? 00:29:35.318 --> 00:29:40.410 Where this was more complicated as an integrant in y, and guys, 00:29:40.410 --> 00:29:43.130 your domain was like that. 00:29:43.130 --> 00:29:46.680 And then normally to integrate with respect to y and then x, 00:29:46.680 --> 00:29:50.770 you would have had to split this integral into two integrals-- 00:29:50.770 --> 00:29:53.002 one over a part of the triangle, the other one 00:29:53.002 --> 00:29:54.770 over part of the triangle. 00:29:54.770 --> 00:30:00.060 So the easier way was to do it how? 00:30:00.060 --> 00:30:04.320 To do it like that, with horizontal integrals. 00:30:04.320 --> 00:30:05.360 And we've done that. 00:30:05.360 --> 00:30:07.920 I told you-- I gave you too much, actually, 00:30:07.920 --> 00:30:12.780 I served it to you on a plate, the proof-- solution 00:30:12.780 --> 00:30:13.752 of that problem. 00:30:13.752 --> 00:30:15.210 But you have many others. 00:30:15.210 --> 00:30:20.745 Now, how do we prove that this individual equation that 00:30:20.745 --> 00:30:23.360 looks so sophisticated is nothing 00:30:23.360 --> 00:30:29.520 but that for the case when S is a planar patch? 00:30:29.520 --> 00:30:33.920 If S is like a hill, yeah, then we believe it. 00:30:33.920 --> 00:30:37.560 But what if S is the domain d in plane Well, 00:30:37.560 --> 00:30:43.080 then this S is exactly this d. 00:30:43.080 --> 00:30:44.850 So it reduces to d. 00:30:44.850 --> 00:30:46.630 So you say, wait a minute, doesn't it 00:30:46.630 --> 00:30:48.170 have to be curvilinear? 00:30:48.170 --> 00:30:49.710 Nope. 00:30:49.710 --> 00:30:56.290 Any surface that is bounded by c verifies Stokes' theorem. 00:30:56.290 --> 00:30:57.570 Say it again, Magdalena. 00:30:57.570 --> 00:31:01.770 Any surface S that is regular, so I'm 00:31:01.770 --> 00:31:04.560 within the conditions of the theorem, that 00:31:04.560 --> 00:31:07.300 is bounded by a Jordan curve, will satisfy the theorem. 00:31:07.300 --> 00:31:09.340 So let's see what I've become. 00:31:09.340 --> 00:31:11.520 That should became a friend of yours, 00:31:11.520 --> 00:31:14.540 and we already know who this guy is. 00:31:14.540 --> 00:31:24.660 So the integral FdR is your friend integral Mdx plus Ndy 00:31:24.660 --> 00:31:26.790 that's staring at you over c. 00:31:26.790 --> 00:31:31.410 It's an integral over one form, and it says that's work. 00:31:31.410 --> 00:31:35.120 And the right-hand side, it's a little bit more complicated. 00:31:35.120 --> 00:31:36.950 So we have to think. 00:31:36.950 --> 00:31:38.300 We have to think. 00:31:38.300 --> 00:31:41.740 It's not about computation, it's about how good we 00:31:41.740 --> 00:31:45.790 are at identifying everybody. 00:31:45.790 --> 00:31:50.130 If I go, for this particular case, S is d, right? 00:31:50.130 --> 00:31:51.560 Right. 00:31:51.560 --> 00:31:56.965 So I have a double integral over D. Sometimes you ask me, 00:31:56.965 --> 00:32:02.000 but I saw that over a domain that's a two-dimensional domain 00:32:02.000 --> 00:32:05.420 people wrote only one snake, and it looks fat, 00:32:05.420 --> 00:32:08.030 like somebody fed the snake too much. 00:32:08.030 --> 00:32:10.730 Mathematicians are lazy people. 00:32:10.730 --> 00:32:14.070 They don't want to write always double snake, triple snake. 00:32:14.070 --> 00:32:16.570 So sometimes they say, I have an integral 00:32:16.570 --> 00:32:18.030 over an n-dimensional domain. 00:32:18.030 --> 00:32:20.136 I'll make it a fat snake. 00:32:20.136 --> 00:32:22.860 And that should be enough. 00:32:22.860 --> 00:32:25.680 Curl F N-- we have to do this together. 00:32:25.680 --> 00:32:26.460 Is it hard? 00:32:26.460 --> 00:32:27.390 I don't know. 00:32:27.390 --> 00:32:30.880 You have to help me. 00:32:30.880 --> 00:32:33.221 So what in the world was that? 00:32:33.221 --> 00:32:34.470 I pretend I forgot everything. 00:32:34.470 --> 00:32:35.207 I have amnesia. 00:32:35.207 --> 00:32:36.082 STUDENT: [INAUDIBLE]. 00:32:36.082 --> 00:32:39.040 00:32:39.040 --> 00:32:41.190 PROFESSOR: Yeah, so actually some of you 00:32:41.190 --> 00:32:45.790 told me by email that you prefer that. 00:32:45.790 --> 00:32:49.240 I really like it that you-- maybe I 00:32:49.240 --> 00:32:52.106 should have started a Facebook group or something. 00:32:52.106 --> 00:32:55.640 Because instead of the personal email interaction 00:32:55.640 --> 00:32:59.660 between me and you, everybody could see this. 00:32:59.660 --> 00:33:03.810 So some of you tell me, I like better this notation, 00:33:03.810 --> 00:33:05.950 because I use it in my engineering 00:33:05.950 --> 00:33:12.160 course, curl F. OK, good, it's up to you what you want to use. 00:33:12.160 --> 00:33:15.520 d/dx, d/dy-- I mean it. 00:33:15.520 --> 00:33:18.820 In principle, in r3, but I'm really lucky. 00:33:18.820 --> 00:33:23.134 Because in this case, F is in r2, value in r2. 00:33:23.134 --> 00:33:24.115 STUDENT: You mean d/dz? 00:33:24.115 --> 00:33:24.740 PROFESSOR: Huh? 00:33:24.740 --> 00:33:26.140 STUDENT: d/dx, d/dy, d/dz. 00:33:26.140 --> 00:33:27.050 PROFESSOR: I'm sorry. 00:33:27.050 --> 00:33:29.782 You are so on the ball. 00:33:29.782 --> 00:33:30.859 Thank you, Alexander. 00:33:30.859 --> 00:33:33.150 STUDENT: No, I thought I had completely misunderstood-- 00:33:33.150 --> 00:33:36.280 PROFESSOR: No, no, no, no, I wrote it twice. 00:33:36.280 --> 00:33:41.860 So M and N and 0, M is a function of x and y only. 00:33:41.860 --> 00:33:44.420 N of course-- do I have to write that? 00:33:44.420 --> 00:33:46.550 No, I'm just being silly. 00:33:46.550 --> 00:33:49.789 And what do I get in this case? 00:33:49.789 --> 00:33:50.664 STUDENT: [INAUDIBLE]. 00:33:50.664 --> 00:33:56.080 00:33:56.080 --> 00:33:59.146 PROFESSOR: I times this guy-- how much is this guy? 00:33:59.146 --> 00:33:59.869 STUDENT: 0. 00:33:59.869 --> 00:34:00.410 PROFESSOR: 0. 00:34:00.410 --> 00:34:01.700 Why is that 0? 00:34:01.700 --> 00:34:05.150 Because this contains no z, and I prime with respect to z. 00:34:05.150 --> 00:34:11.500 So that is nonsense, 0i minus 0j. 00:34:11.500 --> 00:34:12.190 Why is that? 00:34:12.190 --> 00:34:16.550 Because 0 minus something that doesn't depend on z. 00:34:16.550 --> 00:34:22.520 So plus, finally-- the only guy that matters there 00:34:22.520 --> 00:34:28.409 is [INAUDIBLE], which is this, which is that. 00:34:28.409 --> 00:34:31.859 So because I have derivative of N 00:34:31.859 --> 00:34:34.520 with respect to h minus derivative of M 00:34:34.520 --> 00:34:35.563 with respect to y. 00:34:35.563 --> 00:34:39.260 00:34:39.260 --> 00:34:42.320 And now I stare at it, and I say, times k. 00:34:42.320 --> 00:34:45.945 That's the only guy that's not 0, the only component. 00:34:45.945 --> 00:34:49.270 Now I'm going to go ahead and multiply this 00:34:49.270 --> 00:34:51.090 in the top product with him. 00:34:51.090 --> 00:34:54.614 But we have to be smart and think, N is what? 00:34:54.614 --> 00:34:56.117 STUDENT: [INAUDIBLE]. 00:34:56.117 --> 00:34:57.700 PROFESSOR: It's normal to the surface. 00:34:57.700 --> 00:35:01.930 But the surface is a patch of a plane. 00:35:01.930 --> 00:35:03.460 The normal would be trivial. 00:35:03.460 --> 00:35:05.376 What will the normal be? 00:35:05.376 --> 00:35:10.890 The vector field of all pencils that are k-- k. 00:35:10.890 --> 00:35:12.820 It's all k, k everywhere. 00:35:12.820 --> 00:35:14.550 All over the domain is k. 00:35:14.550 --> 00:35:16.270 So N becomes k. 00:35:16.270 --> 00:35:17.623 Where is it? 00:35:17.623 --> 00:35:21.080 There, N becomes k. 00:35:21.080 --> 00:35:24.200 So when you multiply in the dot product 00:35:24.200 --> 00:35:26.624 this guy with this guy, what do you have? 00:35:26.624 --> 00:35:27.620 STUDENT: [INAUDIBLE]. 00:35:27.620 --> 00:35:35.010 PROFESSOR: N sub x minus M sub y dA. 00:35:35.010 --> 00:35:37.890 00:35:37.890 --> 00:35:41.680 QED-- what does it mean, QED? 00:35:41.680 --> 00:35:44.100 $1, which I don't have, for the person 00:35:44.100 --> 00:35:46.740 who will tell me what that is. 00:35:46.740 --> 00:35:50.961 00:35:50.961 --> 00:36:05.310 Latin-- quod erat demonstrandum, which was to be proved, yes? 00:36:05.310 --> 00:36:06.390 So I'm done. 00:36:06.390 --> 00:36:10.155 When people put QED, that means they are done with the proof. 00:36:10.155 --> 00:36:13.810 But now since mathematicians are a little bit illiterate, 00:36:13.810 --> 00:36:17.340 they don't know much about philosophy or linguistics. 00:36:17.340 --> 00:36:19.440 Now many of them, instead of QED, 00:36:19.440 --> 00:36:22.540 they put a little square box. 00:36:22.540 --> 00:36:24.610 And we do the same in our books. 00:36:24.610 --> 00:36:28.230 So that means I'm done with the proof. 00:36:28.230 --> 00:36:30.520 Let's go home, but not that. 00:36:30.520 --> 00:36:39.590 So we proved that for the particular case of the planar 00:36:39.590 --> 00:36:43.550 domains, Stokes' theorem becomes Green's theorem. 00:36:43.550 --> 00:36:46.480 And actually this is the curl. 00:36:46.480 --> 00:36:49.580 And this-- well, not the curl. 00:36:49.580 --> 00:36:55.140 But you have the curl of F multiplied with dot product 00:36:55.140 --> 00:36:58.180 with k and this green fellow is exactly 00:36:58.180 --> 00:37:02.215 the same as N sub x minus N sub y 00:37:02.215 --> 00:37:06.750 smooth function, real value function. 00:37:06.750 --> 00:37:10.140 All right, am I done? 00:37:10.140 --> 00:37:13.450 Yes, with this Exercise 1, which is a proof, I'm done. 00:37:13.450 --> 00:37:17.700 You haven't seen many proofs in calculus. 00:37:17.700 --> 00:37:21.060 You've seen some from me that we never cover. 00:37:21.060 --> 00:37:24.850 We don't do epsilon delta in regular classes of calculus, 00:37:24.850 --> 00:37:25.980 only in honors. 00:37:25.980 --> 00:37:28.510 And not in all the honors you've seen some proofs 00:37:28.510 --> 00:37:29.980 with epsilon delta. 00:37:29.980 --> 00:37:35.650 You've seen one or two proofs from me occasionally. 00:37:35.650 --> 00:37:40.461 And this was one simple proof that I wanted to work with you. 00:37:40.461 --> 00:37:42.996 Now, do you know if you're ever going 00:37:42.996 --> 00:37:47.260 to see proofs in math classes, out of curiosity? 00:37:47.260 --> 00:37:51.170 US It depends how much math you want to take. 00:37:51.170 --> 00:37:55.950 If you're a math major, you take a course called 3310. 00:37:55.950 --> 00:37:58.720 That's called Introduction to Proofs. 00:37:58.720 --> 00:38:02.090 If you are not a math major, but assume 00:38:02.090 --> 00:38:06.200 you are in this dual program-- we 00:38:06.200 --> 00:38:11.290 have a beautiful and tough dual major, mathematics and computer 00:38:11.290 --> 00:38:14.860 science, 162 hours. 00:38:14.860 --> 00:38:17.710 Then you see everything you would normally 00:38:17.710 --> 00:38:19.760 see for an engineering major. 00:38:19.760 --> 00:38:22.400 But in addition, you see a few more courses 00:38:22.400 --> 00:38:24.560 that have excellent proofs. 00:38:24.560 --> 00:38:28.740 And one of them is linear algebra, Linear Algebra 2360. 00:38:28.740 --> 00:38:32.690 We do a few proofs-- depends who teaches that. 00:38:32.690 --> 00:38:35.020 And in 3310 also you see some proofs 00:38:35.020 --> 00:38:37.720 like, by way of contradiction, let's prove this and that. 00:38:37.720 --> 00:38:41.220 OK, so it's sort of fun. 00:38:41.220 --> 00:38:45.930 But we don't attempt long and nasty, complicated proofs 00:38:45.930 --> 00:38:49.035 until you are in graduate school, normally. 00:38:49.035 --> 00:38:52.590 Some of you will do graduate studies. 00:38:52.590 --> 00:38:54.320 Some of you-- I know four of you-- 00:38:54.320 --> 00:38:57.510 want to go to medical school. 00:38:57.510 --> 00:39:02.424 And then many of you hopefully will get a graduate program 00:39:02.424 --> 00:39:06.400 in engineering. 00:39:06.400 --> 00:39:11.210 OK, let's see another example for this section. 00:39:11.210 --> 00:39:15.260 I don't particularly like all the examples 00:39:15.260 --> 00:39:17.680 we have in the book. 00:39:17.680 --> 00:39:21.672 But I have my favorites. 00:39:21.672 --> 00:39:31.096 And I'm going to go ahead and choose one. 00:39:31.096 --> 00:39:34.600 00:39:34.600 --> 00:39:38.500 There is one that's a little bit complicated. 00:39:38.500 --> 00:39:40.650 And you asked me about it. 00:39:40.650 --> 00:39:45.740 And I wanted to talk about this one. 00:39:45.740 --> 00:39:51.005 Because it gave several of you a headache. 00:39:51.005 --> 00:40:00.060 There is Example 1, which says-- what does it say? 00:40:00.060 --> 00:40:07.460 Evaluate fat integral over C of 1 over 2 i 00:40:07.460 --> 00:40:23.340 squared dx plus zdy plus xdz where C is the intersection 00:40:23.340 --> 00:40:47.210 curve between the plane x plus z equals 1 and the ellipsoid x 00:40:47.210 --> 00:40:50.410 squared plus 2y squared plus z squared 00:40:50.410 --> 00:41:01.306 equals 1 that's oriented counterclockwise as viewed 00:41:01.306 --> 00:41:03.348 from the above picture. 00:41:03.348 --> 00:41:06.040 And I need to draw the picture. 00:41:06.040 --> 00:41:09.630 The picture looks really ugly. 00:41:09.630 --> 00:41:11.590 You have this ellipsoid. 00:41:11.590 --> 00:41:17.470 00:41:17.470 --> 00:41:23.920 And when you draw this intersection 00:41:23.920 --> 00:41:29.780 between this plane and the ellipsoid, it looks horrible. 00:41:29.780 --> 00:41:32.740 And the hint of this problem-- well, 00:41:32.740 --> 00:41:38.460 if you were to be given such a thing on an exam, 00:41:38.460 --> 00:41:41.990 the hint would be that a projection-- 00:41:41.990 --> 00:41:42.990 look at the picture. 00:41:42.990 --> 00:41:50.815 The projection of the curve of intersection on the ground-- 00:41:50.815 --> 00:41:53.610 ground means the plane on the equator. 00:41:53.610 --> 00:41:54.660 How shall I say that? 00:41:54.660 --> 00:42:01.010 The x, y plane is this. 00:42:01.010 --> 00:42:02.730 It looks horrible. 00:42:02.730 --> 00:42:10.400 00:42:10.400 --> 00:42:12.110 And it looks like an egg. 00:42:12.110 --> 00:42:13.568 It's not supposed to be an egg, OK? 00:42:13.568 --> 00:42:14.670 It's a circle. 00:42:14.670 --> 00:42:16.780 I'm sorry if it looks like an egg. 00:42:16.780 --> 00:42:20.540 00:42:20.540 --> 00:42:24.892 OK, and that would be the only hint you would get. 00:42:24.892 --> 00:42:29.430 You would be asked to figure out this circle 00:42:29.430 --> 00:42:31.745 in polar coordinates. 00:42:31.745 --> 00:42:37.395 And I'm not sure if all of you would know how to do that. 00:42:37.395 --> 00:42:41.070 And this is what worried me. 00:42:41.070 --> 00:42:44.570 So before we do everything, before everything, 00:42:44.570 --> 00:42:49.930 can we express this in polar coordinates? 00:42:49.930 --> 00:42:54.280 How are you going to set up something in r theta 00:42:54.280 --> 00:42:59.159 for the same domain inside this disc? 00:42:59.159 --> 00:43:00.034 STUDENT: [INAUDIBLE]. 00:43:00.034 --> 00:43:27.807 00:43:27.807 --> 00:43:29.390 PROFESSOR: So if we were, for example, 00:43:29.390 --> 00:43:32.820 to say x is r cosine theta, can we do that? 00:43:32.820 --> 00:43:35.600 And i to be r sine theta, what would 00:43:35.600 --> 00:43:37.655 we get instead of this equation? 00:43:37.655 --> 00:43:41.390 Because it looks horrible. 00:43:41.390 --> 00:43:45.190 We would get-- this equation, let's brush it up a little bit 00:43:45.190 --> 00:43:45.810 first. 00:43:45.810 --> 00:43:48.430 It's x squared plus y squared. 00:43:48.430 --> 00:43:49.450 And that's nice. 00:43:49.450 --> 00:43:57.320 But then it's minus twice-- it's just x plus 1/4 00:43:57.320 --> 00:44:00.840 equals 1/4, the heck with it. 00:44:00.840 --> 00:44:02.440 My son says, don't say "heck." 00:44:02.440 --> 00:44:03.270 That's a bad word. 00:44:03.270 --> 00:44:04.700 I didn't know that. 00:44:04.700 --> 00:44:07.530 But he says that he's being told in school it's a bad word. 00:44:07.530 --> 00:44:09.870 So he must know what he's talking about. 00:44:09.870 --> 00:44:13.390 So this is r squared. 00:44:13.390 --> 00:44:15.265 And x is r cosine theta. 00:44:15.265 --> 00:44:20.970 Aha, so there we almost did it in the sense 00:44:20.970 --> 00:44:24.170 that r squared equals r cosine theta is 00:44:24.170 --> 00:44:27.460 the polar equation, equation of the circle 00:44:27.460 --> 00:44:28.620 in polar coordinates. 00:44:28.620 --> 00:44:31.690 But we hate r. 00:44:31.690 --> 00:44:33.280 Let's simplify by an r. 00:44:33.280 --> 00:44:36.950 Because r is positive-- cannot be 0, right? 00:44:36.950 --> 00:44:38.190 It would be a point. 00:44:38.190 --> 00:44:44.200 So divide by r and get r equals cosine theta. 00:44:44.200 --> 00:44:45.770 So what is r equals cosine theta? 00:44:45.770 --> 00:44:49.590 r equals cosine theta is your worst nightmare. 00:44:49.590 --> 00:44:52.020 So I'm going to make a face. 00:44:52.020 --> 00:44:55.263 That was your worst nightmare in Calculus II. 00:44:55.263 --> 00:44:58.522 And I was just talking to a few colleagues in Calculus II 00:44:58.522 --> 00:45:01.280 telling me that the students don't know that, 00:45:01.280 --> 00:45:06.270 and they have a big hard time with that. 00:45:06.270 --> 00:45:12.270 So the equation of this circle is r equals cosine theta. 00:45:12.270 --> 00:45:16.400 So if I were to express this domain, 00:45:16.400 --> 00:45:19.466 which in Cartesian coordinates would be written-- 00:45:19.466 --> 00:45:22.378 I don't know if you want to-- as double integral, We'd? 00:45:22.378 --> 00:45:24.710 Have to do the vertical strip thingy. 00:45:24.710 --> 00:45:28.620 But if I want to do it in polar coordinates, 00:45:28.620 --> 00:45:33.562 I'm going to say, I start-- well, 00:45:33.562 --> 00:45:36.420 you have to tell me what you think. 00:45:36.420 --> 00:45:42.330 00:45:42.330 --> 00:45:44.930 We have an r that starts with the origin. 00:45:44.930 --> 00:45:50.500 And that's dr. How far does r go? 00:45:50.500 --> 00:45:55.942 For the domain inside, r goes between 0 and cosine theta. 00:45:55.942 --> 00:45:57.650 STUDENT: Why were you able to divide by r 00:45:57.650 --> 00:45:59.060 if it could have equaled 0? 00:45:59.060 --> 00:46:00.143 PROFESSOR: We already did. 00:46:00.143 --> 00:46:02.960 STUDENT: Yes, but then you just said 00:46:02.960 --> 00:46:05.200 you could only do that because it never equaled 0. 00:46:05.200 --> 00:46:08.840 PROFESSOR: Right, and for 0 we pull out one point 00:46:08.840 --> 00:46:12.550 where we take the angle that we want. 00:46:12.550 --> 00:46:15.760 We will still get the same thing. 00:46:15.760 --> 00:46:16.635 STUDENT: [INAUDIBLE]. 00:46:16.635 --> 00:46:19.882 00:46:19.882 --> 00:46:22.140 PROFESSOR: No, r will be any-- 00:46:22.140 --> 00:46:24.739 00:46:24.739 --> 00:46:25.530 STUDENT: Oh, I see. 00:46:25.530 --> 00:46:27.730 PROFESSOR: Yeah, so little r, what 00:46:27.730 --> 00:46:30.396 is the r of any little point inside? 00:46:30.396 --> 00:46:33.702 The r of any little point inside is 00:46:33.702 --> 00:46:37.880 between 0 and N cosine theta. 00:46:37.880 --> 00:46:42.430 Cosine theta would be the r corresponding to the boundary. 00:46:42.430 --> 00:46:45.300 Say it again-- so every point on the boundary 00:46:45.300 --> 00:46:49.010 will have that r equals cosine theta. 00:46:49.010 --> 00:46:55.920 The points inside the domain-- and this is on the circle, 00:46:55.920 --> 00:46:59.170 on C. This is the circle. 00:46:59.170 --> 00:47:03.125 Let's call it C ground. 00:47:03.125 --> 00:47:08.060 That is the C. 00:47:08.060 --> 00:47:13.710 So the r, the points inside have one property, 00:47:13.710 --> 00:47:16.320 that their r is between 0 and cosine theta. 00:47:16.320 --> 00:47:18.940 If I take r theta with this property, 00:47:18.940 --> 00:47:21.480 I should be able to get all the domain. 00:47:21.480 --> 00:47:27.360 But theta, you have to be a little bit careful about theta. 00:47:27.360 --> 00:47:30.299 STUDENT: It goes from pi over 2 to negative pi over 2. 00:47:30.299 --> 00:47:31.340 PROFESSOR: Actually, yes. 00:47:31.340 --> 00:47:38.140 So you have theta will be between minus pi over 2 00:47:38.140 --> 00:47:39.230 and pi over 2. 00:47:39.230 --> 00:47:42.100 And you have to think a little bit about how 00:47:42.100 --> 00:47:43.740 you set up the double integral. 00:47:43.740 --> 00:47:44.840 But you're not there yet. 00:47:44.840 --> 00:47:48.170 So when we'll be there at the double integral 00:47:48.170 --> 00:47:50.070 we will have to think about it. 00:47:50.070 --> 00:47:54.020 00:47:54.020 --> 00:47:56.012 What else did I want? 00:47:56.012 --> 00:48:06.000 00:48:06.000 --> 00:48:12.230 All right, so did I give you the right form of F? 00:48:12.230 --> 00:48:13.688 Yes. 00:48:13.688 --> 00:48:18.065 I'd like you to compute curl F and N all by yourselves. 00:48:18.065 --> 00:48:18.880 So compute. 00:48:18.880 --> 00:48:24.270 00:48:24.270 --> 00:48:26.740 This is going to be F1. 00:48:26.740 --> 00:48:28.788 This is going to be F2. 00:48:28.788 --> 00:48:31.178 This is going to be F3. 00:48:31.178 --> 00:48:34.660 And I'd like you to realize that this 00:48:34.660 --> 00:48:42.091 is nothing but integral over C F dR. So who is this animal? 00:48:42.091 --> 00:48:43.444 This is the work, guys. 00:48:43.444 --> 00:48:46.150 00:48:46.150 --> 00:48:49.930 All right, so I should be able to set up 00:48:49.930 --> 00:48:53.530 some integral, double integral, over a surface 00:48:53.530 --> 00:49:02.320 where I have curl F times N dS. 00:49:02.320 --> 00:49:06.135 So what I want you to do is simply-- maybe 00:49:06.135 --> 00:49:08.060 I'm a little bit too lazy. 00:49:08.060 --> 00:49:10.490 Take the curl of F and tell me what it is. 00:49:10.490 --> 00:49:14.720 Take the unit normal vector field and tell me what it is. 00:49:14.720 --> 00:49:22.630 00:49:22.630 --> 00:49:24.462 And then we will figure out the rest. 00:49:24.462 --> 00:49:27.840 00:49:27.840 --> 00:49:29.855 So you say, wait a minute, Magdalena, now 00:49:29.855 --> 00:49:33.860 you want me to look at this Stokes' theorem over what 00:49:33.860 --> 00:49:34.500 surface? 00:49:34.500 --> 00:49:37.260 Because C is the red boundary. 00:49:37.260 --> 00:49:44.080 So you want me to look at this surface, right, the cap? 00:49:44.080 --> 00:49:46.360 So the surface could be the cap. 00:49:46.360 --> 00:49:49.310 But what did I tell you before? 00:49:49.310 --> 00:49:56.700 I told you that Stokes' theorem works for any kind of domain 00:49:56.700 --> 00:50:02.490 that is bounded by the curve C. So is this the way 00:50:02.490 --> 00:50:05.890 you're going to do it-- take the cap, put the normals, 00:50:05.890 --> 00:50:09.500 find the normals, and do all the horrible computation? 00:50:09.500 --> 00:50:12.450 Or you will simplify your life and understand 00:50:12.450 --> 00:50:17.700 that this is exactly the same as the integral evaluated 00:50:17.700 --> 00:50:22.050 over any surface bounded by C. 00:50:22.050 --> 00:50:25.210 Well, this horrible thing is going to kill us. 00:50:25.210 --> 00:50:27.466 So what's the simplest way to do this? 00:50:27.466 --> 00:50:32.360 00:50:32.360 --> 00:50:35.150 It would be to do it over another surface. 00:50:35.150 --> 00:50:37.060 It doesn't matter what surface you have. 00:50:37.060 --> 00:50:41.180 This is the C. You can take any surface that's bounded by C. 00:50:41.180 --> 00:50:43.050 You can take this balloon. 00:50:43.050 --> 00:50:44.540 You can take this one. 00:50:44.540 --> 00:50:47.182 You can take the disc bounded by C. 00:50:47.182 --> 00:50:49.612 You can take any surface that's bounded by C. 00:50:49.612 --> 00:50:53.986 So in particular, what if you take 00:50:53.986 --> 00:51:00.030 the surface inside this red disc, 00:51:00.030 --> 00:51:03.730 the planar surface inside that red disc? 00:51:03.730 --> 00:51:07.660 OK, do you see it? 00:51:07.660 --> 00:51:10.460 OK, that's going to be part of a plane. 00:51:10.460 --> 00:51:12.250 What is that plane? 00:51:12.250 --> 00:51:15.180 x plus z equals 1. 00:51:15.180 --> 00:51:18.064 So you guys have to tell me who N will be 00:51:18.064 --> 00:51:19.992 and who the curl will be. 00:51:19.992 --> 00:51:25.300 00:51:25.300 --> 00:51:30.230 And let me show you again with my hands what you have. 00:51:30.230 --> 00:51:34.430 You have a surface that's curvilinear and round 00:51:34.430 --> 00:51:35.410 and has boundary C. 00:51:35.410 --> 00:51:39.540 The boundary is C. You have another surface that's 00:51:39.540 --> 00:51:45.320 an ellipse that has C as a boundary. 00:51:45.320 --> 00:51:47.140 And this is sitting in a plane. 00:51:47.140 --> 00:51:50.800 And I want-- it's very hard to model with my hands. 00:51:50.800 --> 00:51:51.460 But this is it. 00:51:51.460 --> 00:51:52.272 You see it? 00:51:52.272 --> 00:51:53.084 You see it? 00:51:53.084 --> 00:51:56.780 OK, when you project this on the ground, 00:51:56.780 --> 00:52:00.690 this is going to become that circle that I just erased, 00:52:00.690 --> 00:52:02.650 so this and that. 00:52:02.650 --> 00:52:04.740 We have a surface integral. 00:52:04.740 --> 00:52:08.650 Remember, you have dS here up, and you have dA here down-- 00:52:08.650 --> 00:52:10.490 dS here up, dA here down. 00:52:10.490 --> 00:52:13.370 So that shouldn't be hard to do at all. 00:52:13.370 --> 00:52:17.740 Now what is N? 00:52:17.740 --> 00:52:22.080 N, for such an individual, will be really nice and sassy. 00:52:22.080 --> 00:52:27.010 x plus z equals 1. 00:52:27.010 --> 00:52:33.820 So what is the normal to the plane x plus z? 00:52:33.820 --> 00:52:34.320 [INAUDIBLE] 00:52:34.320 --> 00:52:38.820 00:52:38.820 --> 00:52:46.940 So who is this normal for D curl F times [INAUDIBLE] 00:52:46.940 --> 00:52:54.570 but N d-- I don't know, another S, S tilde. 00:52:54.570 --> 00:53:01.295 So for this kind of surface, I have another dS, dS tilde. 00:53:01.295 --> 00:53:04.654 So who's going to tell me who N is? 00:53:04.654 --> 00:53:07.380 00:53:07.380 --> 00:53:13.730 Well, it should be x plus z equals 1. 00:53:13.730 --> 00:53:14.660 What do we keep? 00:53:14.660 --> 00:53:16.136 What do we throw away? 00:53:16.136 --> 00:53:18.420 The plane is x plus z equals 1. 00:53:18.420 --> 00:53:19.296 What's the normal? 00:53:19.296 --> 00:53:22.870 00:53:22.870 --> 00:53:27.820 So the plane is x plus 0y plus 1z equals 1. 00:53:27.820 --> 00:53:29.808 What's the normal to the plane? 00:53:29.808 --> 00:53:31.724 STUDENT: Is it i plus k over square root of 2? 00:53:31.724 --> 00:53:33.720 PROFESSOR: i plus k, very good, but why 00:53:33.720 --> 00:53:37.670 does Alexander say the over square root of 2? 00:53:37.670 --> 00:53:40.090 Because it says, remember guys, that that 00:53:40.090 --> 00:53:42.740 has to be a unit normal. 00:53:42.740 --> 00:53:47.900 We cannot take i plus k based on being perpendicular to the x 00:53:47.900 --> 00:53:48.750 plus z. 00:53:48.750 --> 00:53:51.200 Because you need to normalize it. 00:53:51.200 --> 00:53:52.020 So he did. 00:53:52.020 --> 00:53:57.632 So he got i plus k over square root of 2. 00:53:57.632 --> 00:54:00.464 How much is curl F? 00:54:00.464 --> 00:54:02.352 You have to do this by yourselves. 00:54:02.352 --> 00:54:04.030 I'll just give it to you. 00:54:04.030 --> 00:54:06.180 I'll give you three minutes, and then I'll 00:54:06.180 --> 00:54:10.392 check your work based on the answers that we have. 00:54:10.392 --> 00:54:16.200 00:54:16.200 --> 00:54:19.070 And in the end, I'll have to do the dot product and keep going. 00:54:19.070 --> 00:55:32.650 00:55:32.650 --> 00:55:33.840 Is it hard? 00:55:33.840 --> 00:55:36.822 I should do it along with you guys. 00:55:36.822 --> 00:55:41.792 I have i jk d/dx, d/dy, d/dz. 00:55:41.792 --> 00:55:45.790 00:55:45.790 --> 00:55:50.710 Who were the guys? y squared over 2 was F1. 00:55:50.710 --> 00:55:53.364 z was F2. 00:55:53.364 --> 00:55:54.663 x was F3. 00:55:54.663 --> 00:55:58.882 00:55:58.882 --> 00:56:00.866 And let's see what you got. 00:56:00.866 --> 00:56:05.330 00:56:05.330 --> 00:56:09.320 I'm checking to see if you get the same thing. 00:56:09.320 --> 00:56:11.350 Minus psi is the first guy. 00:56:11.350 --> 00:56:13.996 [INAUDIBLE] the next one? 00:56:13.996 --> 00:56:14.950 STUDENT: Minus j. 00:56:14.950 --> 00:56:16.704 PROFESSOR: Minus j. 00:56:16.704 --> 00:56:19.020 STUDENT: Minus yk. 00:56:19.020 --> 00:56:20.540 PROFESSOR: yk. 00:56:20.540 --> 00:56:23.700 And I think that's what it is, yes. 00:56:23.700 --> 00:56:28.120 So when you do the integral, what are you going to get? 00:56:28.120 --> 00:56:29.958 I'm going to erase this here. 00:56:29.958 --> 00:56:35.326 00:56:35.326 --> 00:56:41.450 You have your N. And your N is nice. 00:56:41.450 --> 00:56:43.430 What was it again, Alexander? 00:56:43.430 --> 00:56:47.170 i plus k over square root of 2, right? 00:56:47.170 --> 00:56:49.950 So let's write down the integral W 00:56:49.950 --> 00:56:55.540 will be-- double integral over the domain. 00:56:55.540 --> 00:57:02.830 Now, in our case, the domain is this domain, this one here. 00:57:02.830 --> 00:57:07.120 Let's call it-- do you want to call it D or D star or D tilde? 00:57:07.120 --> 00:57:08.060 I don't know what. 00:57:08.060 --> 00:57:10.655 Because we use to call the domain on the ground 00:57:10.655 --> 00:57:15.329 D. Let's put here D star. 00:57:15.329 --> 00:57:21.100 So over D star, and the cap doesn't exist in your life 00:57:21.100 --> 00:57:21.710 anymore. 00:57:21.710 --> 00:57:23.540 You said, bye-bye bubble. 00:57:23.540 --> 00:57:27.210 I can do the whole computation on D star. 00:57:27.210 --> 00:57:29.000 I get the same answer. 00:57:29.000 --> 00:57:32.400 So you help me right? 00:57:32.400 --> 00:57:37.700 I get minus 1 times 1 over root 2. 00:57:37.700 --> 00:57:39.630 Am I right? 00:57:39.630 --> 00:57:48.832 A 0 for the middle term, and a minus y times 1 over root 2, 00:57:48.832 --> 00:57:53.540 good-- this is the whole thing over here. 00:57:53.540 --> 00:57:57.450 My worry is about dS star. 00:57:57.450 --> 00:58:00.170 What was dS star? 00:58:00.170 --> 00:58:06.880 dS star is the area limit for the plane-- are 00:58:06.880 --> 00:58:10.760 limit for how can I call this? 00:58:10.760 --> 00:58:15.361 For disc, for D star, not for D. 00:58:15.361 --> 00:58:16.970 It's a little bit complicated. 00:58:16.970 --> 00:58:18.966 D is a projection. 00:58:18.966 --> 00:58:22.700 So who reminds me how we did it? 00:58:22.700 --> 00:58:28.760 dS star was what times dA? 00:58:28.760 --> 00:58:30.710 This is the surface area. 00:58:30.710 --> 00:58:35.930 And if you have a surface that's nice-- your surface is nice. 00:58:35.930 --> 00:58:39.150 STUDENT: It's area, so r? 00:58:39.150 --> 00:58:41.616 PROFESSOR: What was this equation of this surface 00:58:41.616 --> 00:58:42.550 up here? 00:58:42.550 --> 00:58:48.304 This is the ellipse that goes projected on the surface. 00:58:48.304 --> 00:58:50.240 STUDENT: Cosine of theta. 00:58:50.240 --> 00:58:52.180 PROFESSOR: The equation of the plane, see? 00:58:52.180 --> 00:58:53.580 The equation of the plane. 00:58:53.580 --> 00:58:54.630 So I erased it. 00:58:54.630 --> 00:58:57.410 So was it x plus z equals 1? 00:58:57.410 --> 00:58:58.230 STUDENT: Yes. 00:58:58.230 --> 00:59:01.660 PROFESSOR: So z must be 1 minus x. 00:59:01.660 --> 00:59:09.750 So this is going to be the square root of 1 plus-- minus 1 00:59:09.750 --> 00:59:10.970 is the first partial. 00:59:10.970 --> 00:59:12.722 Are you guys with me? 00:59:12.722 --> 00:59:19.440 Partial with respect to x of this guy is minus 1 squared 00:59:19.440 --> 00:59:22.785 plus the partial of this with respect to y 00:59:22.785 --> 00:59:25.710 is missing 0 squared. 00:59:25.710 --> 00:59:28.240 And then comes dA, and who is dA? 00:59:28.240 --> 00:59:32.800 dA is dxdy in the floor plane. 00:59:32.800 --> 00:59:40.130 This is the [INAUDIBLE] that projects onto the floor. 00:59:40.130 --> 00:59:44.340 Good, ds star is going to be then square root 2dA. 00:59:44.340 --> 00:59:46.250 Again, the old trick that I taught 00:59:46.250 --> 00:59:49.600 you guys is that this will always 00:59:49.600 --> 00:59:53.810 have to simplify with [INAUDIBLE] on the bottom 00:59:53.810 --> 00:59:56.420 of the N. Say what? 00:59:56.420 --> 00:59:57.740 Magdalena, say it again. 00:59:57.740 --> 01:00:02.770 Square root of 2DA, this is that magic square root 01:00:02.770 --> 01:00:06.460 of 1 plus [INAUDIBLE]. 01:00:06.460 --> 01:00:10.430 This guy, no matter what exercise you are doing, 01:00:10.430 --> 01:00:20.210 will always simplify with the bottom of N [INAUDIBLE], 01:00:20.210 --> 01:00:22.326 so you can do this simplification 01:00:22.326 --> 01:00:23.730 from the beginning. 01:00:23.730 --> 01:00:26.100 And so in the end, what are you going to have? 01:00:26.100 --> 01:00:35.800 You're going to have W is minus y minus 1 01:00:35.800 --> 01:00:40.175 over the domain D in the plane that this will claim. 01:00:40.175 --> 01:00:43.430 01:00:43.430 --> 01:00:45.620 The square root of [INAUDIBLE], and then you'll 01:00:45.620 --> 01:00:50.800 have dA, which is dxdy 01:00:50.800 --> 01:00:56.020 OK, at this point suppose that you are taking the 5. 01:00:56.020 --> 01:00:58.510 And this is why I got to this point 01:00:58.510 --> 01:01:04.650 because I wanted to emphasize this. 01:01:04.650 --> 01:01:09.170 Whether you stop here or you do one more step, 01:01:09.170 --> 01:01:11.360 I would be happy. 01:01:11.360 --> 01:01:12.780 Let's see what I mean. 01:01:12.780 --> 01:01:22.140 So you would have minus who is y r cosine theta minus 1. 01:01:22.140 --> 01:01:24.565 dA will become instead of dxdy, you have-- 01:01:24.565 --> 01:01:25.440 STUDENT: [INAUDIBLE]. 01:01:25.440 --> 01:01:28.430 PROFESSOR: r, very good. r dr is theta. 01:01:28.430 --> 01:01:31.200 01:01:31.200 --> 01:01:33.766 So you're thinking-- 01:01:33.766 --> 01:01:35.170 STUDENT: [INAUDIBLE]. 01:01:35.170 --> 01:01:40.125 PROFESSOR: --well, so you're thinking-- 01:01:40.125 --> 01:01:47.610 I'm looking here what we have-- r was from 0 to cosine theta, 01:01:47.610 --> 01:01:51.050 and theta is from minus [INAUDIBLE]. 01:01:51.050 --> 01:01:54.200 Please stop here, all right? 01:01:54.200 --> 01:01:59.330 So in the exam, we will not expect-- on some integrals who 01:01:59.330 --> 01:02:02.460 are not expected to go on and do them, 01:02:02.460 --> 01:02:04.531 which they set up the integral and leave it. 01:02:04.531 --> 01:02:05.030 Yes, sir? 01:02:05.030 --> 01:02:08.260 STUDENT: Why did you throw r cosine theta for y? 01:02:08.260 --> 01:02:10.600 PROFESSOR: OK, because let me remind you, 01:02:10.600 --> 01:02:15.290 when you project the image of this ellipse on the plane, 01:02:15.290 --> 01:02:20.240 we got this fellow, which is drawn in the book 01:02:20.240 --> 01:02:22.780 as being this. 01:02:22.780 --> 01:02:27.135 So we said, I want to see how I set this up 01:02:27.135 --> 01:02:28.800 in [INAUDIBLE] coordinates. 01:02:28.800 --> 01:02:31.700 The equation of the plane of the circle 01:02:31.700 --> 01:02:34.280 was r equals cosine theta, and this was calculus too. 01:02:34.280 --> 01:02:37.780 That's why we actually [INAUDIBLE]. 01:02:37.780 --> 01:02:40.120 So if somebody would ask you guys, 01:02:40.120 --> 01:02:45.880 compute me instead of an area over the domain, what 01:02:45.880 --> 01:02:48.812 if you compute for me the linear area of the domain? 01:02:48.812 --> 01:02:50.108 How would you do that? 01:02:50.108 --> 01:02:57.530 Well, double integral of 1 or whatever-that-is integral 01:02:57.530 --> 01:03:04.630 of r drd theta, instead of 1, you 01:03:04.630 --> 01:03:07.980 can have some other ugly integral looking at you. 01:03:07.980 --> 01:03:10.880 I put the stop here. 01:03:10.880 --> 01:03:15.810 Theta is between minus pie over 2n pi over 2 01:03:15.810 --> 01:03:23.170 because I'm moving from here to here, from here to here, OK? 01:03:23.170 --> 01:03:27.655 Nr is between 0 and the margin. 01:03:27.655 --> 01:03:29.380 Who is on the margin? 01:03:29.380 --> 01:03:29.940 I started 0. 01:03:29.940 --> 01:03:31.370 I ended cosine theta. 01:03:31.370 --> 01:03:33.720 I started 0, ended cosine theta. 01:03:33.720 --> 01:03:38.570 Cosine theta happens online for the boundary, 01:03:38.570 --> 01:03:39.777 so that's what you do. 01:03:39.777 --> 01:03:40.860 Do we want you to do that? 01:03:40.860 --> 01:03:43.130 No, we want you to leave it. 01:03:43.130 --> 01:03:43.630 Yes? 01:03:43.630 --> 01:03:49.500 STUDENT: He was asking why you had a negative y minus 1 r sine 01:03:49.500 --> 01:03:52.055 theta, not r cosine theta. 01:03:52.055 --> 01:03:53.950 PROFESSOR: You are so right. 01:03:53.950 --> 01:04:00.680 I forgot that x was r cosine theta, and y was r sine theta. 01:04:00.680 --> 01:04:02.180 You are correct. 01:04:02.180 --> 01:04:05.350 And you have the group good observation. 01:04:05.350 --> 01:04:09.900 So r was [INAUDIBLE] cosine theta. 01:04:09.900 --> 01:04:13.930 And x was r cosine theta. 01:04:13.930 --> 01:04:16.920 y was r sine theta. 01:04:16.920 --> 01:04:19.440 Very good. 01:04:19.440 --> 01:04:23.410 OK, so if you get something like that, we will now 01:04:23.410 --> 01:04:28.935 want you to go on, we will want you to stop. 01:04:28.935 --> 01:04:34.410 Let me show you one where we wanted to go on, 01:04:34.410 --> 01:04:37.510 and we indicate it like this, example 3. 01:04:37.510 --> 01:04:44.280 01:04:44.280 --> 01:04:48.550 So here, we just dont' want you to show some work, 01:04:48.550 --> 01:04:52.702 we wanted to actually get the exact answer. 01:04:52.702 --> 01:04:56.993 And I'll draw the picture, and don't be afraid of it. 01:04:56.993 --> 01:04:58.534 It's going to look a little bit ugly. 01:04:58.534 --> 01:05:01.936 01:05:01.936 --> 01:05:07.960 You have the surface Z equals 1 minus x 01:05:07.960 --> 01:05:11.960 squared minus 2y squared. 01:05:11.960 --> 01:05:19.630 And you have to evaluate over double the integral 01:05:19.630 --> 01:05:23.511 of the surface S. This is the surface. 01:05:23.511 --> 01:05:24.510 Let me draw the surface. 01:05:24.510 --> 01:05:30.308 We will have to understand what kind of surface that is. 01:05:30.308 --> 01:05:34.619 01:05:34.619 --> 01:05:42.250 Double integral of curl F [INAUDIBLE] dS evaluated 01:05:42.250 --> 01:05:55.600 where F equals xI plus y squared J plus-- this looks like a Z e 01:05:55.600 --> 01:05:57.592 to the xy. 01:05:57.592 --> 01:05:58.415 It's very tiny. 01:05:58.415 --> 01:06:00.430 I bet you won't see it. 01:06:00.430 --> 01:06:07.640 [INAUDIBLE] xy k and S. Is that part of the surface? 01:06:07.640 --> 01:06:13.590 01:06:13.590 --> 01:06:16.504 Let me change the marker so the video can see better. 01:06:16.504 --> 01:06:19.910 01:06:19.910 --> 01:06:22.188 Z-- this is a bad marker. 01:06:22.188 --> 01:06:26.530 01:06:26.530 --> 01:06:33.800 Z equals-- what was it, guys? 01:06:33.800 --> 01:06:40.984 1 minus x squared minus 2y squared with Z positive or 0. 01:06:40.984 --> 01:06:44.470 01:06:44.470 --> 01:06:48.210 And the [? thing ?] is I think we may give you 01:06:48.210 --> 01:06:50.130 this hint on the exam. 01:06:50.130 --> 01:06:56.060 Think of the Stokes theorem and the typical-- think 01:06:56.060 --> 01:07:06.456 of the Stokes theorem and the typical tools. 01:07:06.456 --> 01:07:09.935 You have learned them. 01:07:09.935 --> 01:07:11.426 OK, what does it mean? 01:07:11.426 --> 01:07:13.800 We have like an eggshell, which is coming 01:07:13.800 --> 01:07:15.870 from the parabola [INAUDIBLE]. 01:07:15.870 --> 01:07:20.270 This parabola [INAUDIBLE] is S minus x squared 01:07:20.270 --> 01:07:23.960 minus 2y squared, and we call that S, 01:07:23.960 --> 01:07:27.320 but you see, we have two surfaces that are 01:07:27.320 --> 01:07:29.830 in this picture bounded by c. 01:07:29.830 --> 01:07:33.580 The other one is the domain D, and it's a simple problem 01:07:33.580 --> 01:07:37.590 because your domain D is sitting on the xy plane. 01:07:37.590 --> 01:07:43.326 So it's a blessing that you already know what D will be. 01:07:43.326 --> 01:07:48.260 D will be those pairs xy with what property? 01:07:48.260 --> 01:07:52.710 Can you guys tell me what D will be? 01:07:52.710 --> 01:07:54.900 Z should be 0, right? 01:07:54.900 --> 01:07:58.010 If you impose it to be 0, then this 01:07:58.010 --> 01:08:02.050 has to satisfy x squared plus 2y squared less than 01:08:02.050 --> 01:08:03.480 or equal to 1. 01:08:03.480 --> 01:08:05.510 Who is the C? 01:08:05.510 --> 01:08:08.090 C are the points on the boundary, 01:08:08.090 --> 01:08:13.250 which means exactly x squared plus 2y squared is equal to 1. 01:08:13.250 --> 01:08:15.250 What in the world is this curve? 01:08:15.250 --> 01:08:16.580 STUDENT: [INAUDIBLE]. 01:08:16.580 --> 01:08:18.050 PROFESSOR: It's an ellipse. 01:08:18.050 --> 01:08:19.260 Is it an ugly ellipse? 01:08:19.260 --> 01:08:21.500 Uh, not really. 01:08:21.500 --> 01:08:22.348 It's a nice ellipse. 01:08:22.348 --> 01:08:26.020 01:08:26.020 --> 01:08:28.850 OK, what do they give us? 01:08:28.850 --> 01:08:32.444 They give us xy squared and Z times 01:08:32.444 --> 01:08:37.591 e to the xy, so this is F1, F2, and F3. 01:08:37.591 --> 01:08:42.569 01:08:42.569 --> 01:08:46.590 So the surface itself is just the part 01:08:46.590 --> 01:08:51.694 that corresponds to Z positive, not all the surface 01:08:51.694 --> 01:08:55.069 because the whole surface will be infinitely large. 01:08:55.069 --> 01:08:59.649 It's a paraboloid that keeps going down to minus infinite, 01:08:59.649 --> 01:09:01.942 so you only take this part. 01:09:01.942 --> 01:09:07.899 It's a finite patch that I stop. 01:09:07.899 --> 01:09:12.310 So this is a problem that's amazingly simple 01:09:12.310 --> 01:09:13.990 once you solve it one time. 01:09:13.990 --> 01:09:20.970 You don't even have to show your work much in the actual exam, 01:09:20.970 --> 01:09:22.710 and I'll show you why. 01:09:22.710 --> 01:09:26.830 So Stokes theorem tells you what in this case? 01:09:26.830 --> 01:09:29.305 Let's review what Stokes theorem says. 01:09:29.305 --> 01:09:34.250 Stokes theorem says, OK, you have the work performed 01:09:34.250 --> 01:09:37.550 by the four steps that's given to you as a vector value 01:09:37.550 --> 01:09:42.029 function along the path C, which is given to you 01:09:42.029 --> 01:09:44.380 as this wonderful ellipse. 01:09:44.380 --> 01:09:48.505 Let me put C like I did it before, C. This is not L, 01:09:48.505 --> 01:09:53.210 it's C, which what is that? 01:09:53.210 --> 01:09:57.540 It's the same as double integral over S, 01:09:57.540 --> 01:10:05.440 the round paraboloid [INAUDIBLE] like church roof, S curl F 01:10:05.440 --> 01:10:09.261 times N dS. 01:10:09.261 --> 01:10:11.310 But what does it say, this happens 01:10:11.310 --> 01:10:15.066 for any-- for every, for any, do you know the sign? 01:10:15.066 --> 01:10:15.940 STUDENT: [INAUDIBLE]. 01:10:15.940 --> 01:10:22.950 PROFESSOR: Surface is bounded by C. 01:10:22.950 --> 01:10:28.134 And here is that winking emoticon from-- how 01:10:28.134 --> 01:10:30.370 is that in Facebook? 01:10:30.370 --> 01:10:31.700 Something like that? 01:10:31.700 --> 01:10:35.430 A wink would be a good hint on the final. 01:10:35.430 --> 01:10:38.180 What are you going to do when you see that wink? 01:10:38.180 --> 01:10:40.520 If it's not on the final, I will wink at you 01:10:40.520 --> 01:10:43.480 until you understand what I'm trying to say. 01:10:43.480 --> 01:10:48.560 It means that you can change the surface to any other surface 01:10:48.560 --> 01:10:51.920 that has the boundary C. What's the simplest 01:10:51.920 --> 01:10:53.216 surface you may think of? 01:10:53.216 --> 01:10:54.090 STUDENT: [INAUDIBLE]. 01:10:54.090 --> 01:10:57.180 PROFESSOR: The D. So I'm going to say, 01:10:57.180 --> 01:11:02.960 double integral over D. Curl left, God knows what that is. 01:11:02.960 --> 01:11:06.120 We still have to do some work here. 01:11:06.120 --> 01:11:08.880 I'm making a sad face because I really 01:11:08.880 --> 01:11:12.021 wanted no work whatsoever. 01:11:12.021 --> 01:11:17.444 N becomes-- we've done this argument three times today. 01:11:17.444 --> 01:11:18.110 STUDENT: It's k. 01:11:18.110 --> 01:11:19.140 PROFESSOR: It's a k. 01:11:19.140 --> 01:11:21.380 That is your blessing. 01:11:21.380 --> 01:11:24.310 That's what you have to indicate on the exam 01:11:24.310 --> 01:11:27.780 that N is k when I look at the plane or domain. 01:11:27.780 --> 01:11:28.800 STUDENT: And dS is DA. 01:11:28.800 --> 01:11:31.880 PROFESSOR: And dS is dA. 01:11:31.880 --> 01:11:34.539 It's much simpler than before because you 01:11:34.539 --> 01:11:35.717 don't have to project. 01:11:35.717 --> 01:11:37.130 You are already projecting. 01:11:37.130 --> 01:11:38.543 You are all to the floor. 01:11:38.543 --> 01:11:40.430 You are on the ground. 01:11:40.430 --> 01:11:42.430 What else do you have to do? 01:11:42.430 --> 01:11:47.440 Not much, you just have to be patient and compute with me 01:11:47.440 --> 01:11:49.290 something I don't like to. 01:11:49.290 --> 01:11:51.960 Last time I asked you to do it by yourself, 01:11:51.960 --> 01:11:55.890 but now I shouldn't be lazy. 01:11:55.890 --> 01:11:58.480 I have to help you. 01:11:58.480 --> 01:11:59.780 You have to help me. 01:11:59.780 --> 01:12:08.954 i j k z dx z dx z dz of what? 01:12:08.954 --> 01:12:13.930 x y squared and this horrible guy. 01:12:13.930 --> 01:12:18.845 01:12:18.845 --> 01:12:20.825 What do we get? 01:12:20.825 --> 01:12:24.290 01:12:24.290 --> 01:12:28.290 Well, it's not so obvious anymore. 01:12:28.290 --> 01:12:29.276 STUDENT: [INAUDIBLE]. 01:12:29.276 --> 01:12:34.720 PROFESSOR: It's Z prime this guy with respect to y Zx, 01:12:34.720 --> 01:12:35.660 very good. 01:12:35.660 --> 01:12:43.500 The x into the xy times i, and I don't care about the rest. 01:12:43.500 --> 01:12:45.000 Why don't I care about the rest? 01:12:45.000 --> 01:12:49.100 Because when I prime y squared with respect to Z goes away. 01:12:49.100 --> 01:12:51.940 So I'm done with the first term. 01:12:51.940 --> 01:12:56.640 I'm going very slow as you can see, but I don't care. 01:12:56.640 --> 01:12:57.945 So I'm going to erase more. 01:12:57.945 --> 01:13:00.520 01:13:00.520 --> 01:13:07.400 Next guy, minus and then we'll make an observation. 01:13:07.400 --> 01:13:11.090 The same thing here, I go [INAUDIBLE]. 01:13:11.090 --> 01:13:17.715 So I have x to the Z Zy e to the xy. 01:13:17.715 --> 01:13:22.070 Are you guys with me, or am I talking nonsense? 01:13:22.070 --> 01:13:25.970 So what am I saying? 01:13:25.970 --> 01:13:31.560 I'm saying that I expand with respect to the j element here. 01:13:31.560 --> 01:13:33.560 I have a minus because of that, and then I 01:13:33.560 --> 01:13:36.400 have the derivative of this animal with respect 01:13:36.400 --> 01:13:42.806 to x, which is Zy into the x y j, correct? 01:13:42.806 --> 01:13:43.620 STUDENT: Yes. 01:13:43.620 --> 01:13:48.220 PROFESSOR: Finally, last but not least, and actually that's 01:13:48.220 --> 01:13:50.950 the most important guy, and I'll tell you in a second why. 01:13:50.950 --> 01:13:52.190 What is the last guy? 01:13:52.190 --> 01:13:53.065 STUDENT: [INAUDIBLE]. 01:13:53.065 --> 01:13:54.060 PROFESSOR: 0. 01:13:54.060 --> 01:13:59.870 So one of you will hopefully realize what 01:13:59.870 --> 01:14:01.880 I'm going to ask you right now. 01:14:01.880 --> 01:14:08.820 No matter what I got here, this was-- what is that called? 01:14:08.820 --> 01:14:14.530 Work that is not necessary, it's some stupid word. 01:14:14.530 --> 01:14:16.830 So why is that not necessary? 01:14:16.830 --> 01:14:24.055 Why could I have said star i plus start j-- God knows 01:14:24.055 --> 01:14:27.404 what that is-- plus 0k. 01:14:27.404 --> 01:14:32.870 Because in the end, I have to multiply that product with k, 01:14:32.870 --> 01:14:36.736 so no matter what we do here, and we sweat a lot. 01:14:36.736 --> 01:14:39.001 And so no matter what we put here 01:14:39.001 --> 01:14:41.940 it would not have made a difference because I have 01:14:41.940 --> 01:14:46.640 to take this whole curl and multiply as a dot with k, 01:14:46.640 --> 01:14:52.630 and what matters is only what's left over. 01:14:52.630 --> 01:14:57.493 So my observation is this whole thing is how much? 01:14:57.493 --> 01:14:58.200 STUDENT: 0. 01:14:58.200 --> 01:14:59.260 PROFESSOR: 0, thank God. 01:14:59.260 --> 01:15:01.350 So the answer is 0. 01:15:01.350 --> 01:15:03.100 And we've given this problem where 01:15:03.100 --> 01:15:09.176 the answer is 0 about four times on four different finals. 01:15:09.176 --> 01:15:12.210 The thing is that many students won't study, 01:15:12.210 --> 01:15:14.730 and they didn't know the trick. 01:15:14.730 --> 01:15:18.630 When you have a surface like that, that bounds the curve 01:15:18.630 --> 01:15:21.440 C. Instead of doing Stokes over the surface, 01:15:21.440 --> 01:15:25.660 you do Stokes over the domain and plane, and you'll get zero. 01:15:25.660 --> 01:15:28.800 So poor kids, they went ahead and tried 01:15:28.800 --> 01:15:31.740 to compute this from scratch for the surface, 01:15:31.740 --> 01:15:33.750 and they got nowhere. 01:15:33.750 --> 01:15:36.955 And then I started the fights with, of course, [INAUDIBLE], 01:15:36.955 --> 01:15:38.990 but they don't want to give them any credit. 01:15:38.990 --> 01:15:41.600 And I wanted to give them at least some credit 01:15:41.600 --> 01:15:43.790 for knowing the theorem, the statement, 01:15:43.790 --> 01:15:51.040 and trying to do something for the nasty surface, the roof 01:15:51.040 --> 01:15:53.396 that is a paraboloid. 01:15:53.396 --> 01:15:55.680 They've done something, so in the end, 01:15:55.680 --> 01:15:57.710 I said I want to do whatever I want, 01:15:57.710 --> 01:15:59.700 and I gave partial credit. 01:15:59.700 --> 01:16:03.130 But normally, I was told not to give partial credit 01:16:03.130 --> 01:16:08.310 for this kind of a thing because the whole key of the problem 01:16:08.310 --> 01:16:11.990 is to be smart, understand the idea, 01:16:11.990 --> 01:16:16.890 and get 0 without doing any work, and that was nice. 01:16:16.890 --> 01:16:17.666 Yes, sir? 01:16:17.666 --> 01:16:19.665 STUDENT: Does that mean that all we would really 01:16:19.665 --> 01:16:22.490 needed to do compute the curl is the k part? 01:16:22.490 --> 01:16:24.790 Because if k would have been something, 01:16:24.790 --> 01:16:26.440 then there would have been a dot on it. 01:16:26.440 --> 01:16:28.270 PROFESSOR: Exactly, but only if-- guys, 01:16:28.270 --> 01:16:35.340 no matter what, if we give you, if your surface has a planer 01:16:35.340 --> 01:16:37.216 boundary-- say it again? 01:16:37.216 --> 01:16:38.797 If your surface, no matter what it 01:16:38.797 --> 01:16:44.890 is-- it could look like geography-- if your surface has 01:16:44.890 --> 01:16:49.590 a boundary in the plane xy like it is in geography, 01:16:49.590 --> 01:16:52.520 imagine you have a hill or something, 01:16:52.520 --> 01:16:54.270 and that's the sea level. 01:16:54.270 --> 01:16:58.454 And around the hill you have the rim of the [INAUDIBLE]. 01:16:58.454 --> 01:17:00.340 OK, that's your planar curve. 01:17:00.340 --> 01:17:03.520 Then you can reduce to the plane, 01:17:03.520 --> 01:17:07.130 and all the arguments will be like that. 01:17:07.130 --> 01:17:15.170 So the thing is you get 0 when the curl has 0 here, 01:17:15.170 --> 01:17:17.170 and there is [INAUDIBLE]. 01:17:17.170 --> 01:17:18.080 Say it again? 01:17:18.080 --> 01:17:22.720 When the F is given to you so that the last component 01:17:22.720 --> 01:17:26.970 of the curl is zero, you will get 0 for the work. 01:17:26.970 --> 01:17:32.300 Otherwise, you can get something else, but not bad at all. 01:17:32.300 --> 01:17:35.790 You can get something that-- let's do 01:17:35.790 --> 01:17:40.083 another example like that where you have a simplification. 01:17:40.083 --> 01:17:43.210 I'm going to go ahead and erase the whole-- 01:17:43.210 --> 01:17:49.725 STUDENT: So, let's say if I knew the [INAUDIBLE] equal to 0, 01:17:49.725 --> 01:17:50.687 so I-- 01:17:50.687 --> 01:17:55.016 PROFESSOR: Eh, you cannot know unless you look at the F first. 01:17:55.016 --> 01:17:55.980 You see-- 01:17:55.980 --> 01:17:58.100 STUDENT: Let's say that I put the F on stop, 01:17:58.100 --> 01:18:00.940 and I put the equation, which is F d r, 01:18:00.940 --> 01:18:05.042 and I put the curl F [INAUDIBLE], so 01:18:05.042 --> 01:18:06.775 and then I said-- I looked at it. 01:18:06.775 --> 01:18:08.755 I said, oh, it's a 0. 01:18:08.755 --> 01:18:10.735 PROFESSOR: If you see that's a big 0, 01:18:10.735 --> 01:18:12.710 you can go at them to 0 at the end. 01:18:12.710 --> 01:18:13.210 STUDENT: OK. 01:18:13.210 --> 01:18:15.685 PROFESSOR: Because the dot product between k, 01:18:15.685 --> 01:18:17.170 that's what matters. 01:18:17.170 --> 01:18:21.130 The dot product between k and the last component of the curl. 01:18:21.130 --> 01:18:23.605 And in the end, integral of 0 is 0. 01:18:23.605 --> 01:18:26.080 And that is the lesson. 01:18:26.080 --> 01:18:28.555 STUDENT: We should also have N equal to k 01:18:28.555 --> 01:18:30.570 if we don't have that. 01:18:30.570 --> 01:18:35.470 PROFESSOR: Yeah, so I'm saying if-- um, that's a problem. 01:18:35.470 --> 01:18:39.170 This is not going to happen, but assume that somebody gives you 01:18:39.170 --> 01:18:45.040 a hill that looks like that, and this is not a planar curve. 01:18:45.040 --> 01:18:47.990 This would be a really nasty curve in space. 01:18:47.990 --> 01:18:50.000 You cannot do that anymore. 01:18:50.000 --> 01:18:54.120 You have to apply [INAUDIBLE] for the general surface. 01:18:54.120 --> 01:19:01.150 But if your boundary sees a planar boundary [INAUDIBLE], 01:19:01.150 --> 01:19:05.608 then you can do that, and simplify your life. 01:19:05.608 --> 01:19:07.440 So let me give you another example. 01:19:07.440 --> 01:19:11.126 01:19:11.126 --> 01:19:15.690 This time it's not going to be-- OK, you will see the surprise. 01:19:15.690 --> 01:19:49.806 01:19:49.806 --> 01:19:53.520 And you have a sphere, and you have a spherical cap, 01:19:53.520 --> 01:20:00.490 the sphere of radius R, and this is going to be, let's say, 01:20:00.490 --> 01:20:01.810 R to be 5. 01:20:01.810 --> 01:20:06.690 And this is z equals 3. 01:20:06.690 --> 01:20:08.642 You have the surface. 01:20:08.642 --> 01:20:15.000 Somebody gives you the surface S. 01:20:15.000 --> 01:20:22.570 That is the spherical cap of the sphere x squared 01:20:22.570 --> 01:20:30.522 plus y squared plus z squared equals 25 above the plane z 01:20:30.522 --> 01:20:31.494 equals 3. 01:20:31.494 --> 01:20:37.820 01:20:37.820 --> 01:20:52.750 Compute double integral of F times-- 01:20:52.750 --> 01:20:55.558 how did we phrase this if we phrase it as a-- 01:20:55.558 --> 01:20:58.360 STUDENT: Curl FN? 01:20:58.360 --> 01:21:00.340 PROFESSOR: No, he said, curl FN. 01:21:00.340 --> 01:21:10.060 I'm sorry, if we rephrase it as work curl FN 01:21:10.060 --> 01:21:15.160 over S, whereas this is the spherical cap. 01:21:15.160 --> 01:21:20.140 This is S. 01:21:20.140 --> 01:21:23.090 So you're going to have this on the final. 01:21:23.090 --> 01:21:24.600 First thing is, stay calm. 01:21:24.600 --> 01:21:26.182 Don't freak out. 01:21:26.182 --> 01:21:29.830 This is a typical-- you have to say, OK. 01:21:29.830 --> 01:21:30.830 She prepared me well. 01:21:30.830 --> 01:21:32.650 I did review, [INAUDIBLE]. 01:21:32.650 --> 01:21:35.228 For God's sake, I'm going to do fine. 01:21:35.228 --> 01:21:37.520 Just keep in mind that no matter what we do, 01:21:37.520 --> 01:21:40.260 it's not going to involve a heavy computation like we 01:21:40.260 --> 01:21:42.470 saw in that horrible first example 01:21:42.470 --> 01:21:44.670 I gave you-- second example I gave you. 01:21:44.670 --> 01:21:48.530 So the whole idea is to make your life easier rather than 01:21:48.530 --> 01:21:49.030 harder. 01:21:49.030 --> 01:21:50.850 So what's the first thing you do? 01:21:50.850 --> 01:21:56.160 You take curl F, and you want to see what that will be. 01:21:56.160 --> 01:22:05.130 i j k is going to be d dx, d dy, d dz. 01:22:05.130 --> 01:22:07.170 And you say, all right, then I'll 01:22:07.170 --> 01:22:16.444 have x squared yz xy squared z and xy z squared. 01:22:16.444 --> 01:22:19.811 And then you say, well, this look ugly, right? 01:22:19.811 --> 01:22:21.740 That's what you're going to say. 01:22:21.740 --> 01:22:31.070 So what times i minus what times j plus what times k remains up 01:22:31.070 --> 01:22:35.120 to you to clue the computation, and you say, wait a minute. 01:22:35.120 --> 01:22:38.980 The first minor is it math? 01:22:38.980 --> 01:22:42.640 No, the first minor-- minor is the name of such a determinant 01:22:42.640 --> 01:22:44.900 is just a silly path. 01:22:44.900 --> 01:22:47.470 So you do x yz squared with respect to y, 01:22:47.470 --> 01:23:01.860 it's xz squared minus prime with respect to z dz xy squared. 01:23:01.860 --> 01:23:05.630 Next guy, what do we have? 01:23:05.630 --> 01:23:06.890 Who tells me? 01:23:06.890 --> 01:23:08.640 He's sort of significant but not really-- 01:23:08.640 --> 01:23:09.473 STUDENT: yz squared? 01:23:09.473 --> 01:23:11.126 PROFESSOR:yz squared, good. 01:23:11.126 --> 01:23:13.102 It's symmetric in a way. 01:23:13.102 --> 01:23:16.560 01:23:16.560 --> 01:23:19.350 x squared y, right guys? 01:23:19.350 --> 01:23:20.274 Are you with me? 01:23:20.274 --> 01:23:20.940 STUDENT: Mh-hmm. 01:23:20.940 --> 01:23:22.606 PROFESSOR: And for the k, you will have? 01:23:22.606 --> 01:23:24.004 STUDENT: y squared z. 01:23:24.004 --> 01:23:25.952 PROFESSOR: y squared z. 01:23:25.952 --> 01:23:30.050 STUDENT: And x squared z. 01:23:30.050 --> 01:23:36.470 PROFESSOR: z squared z because if you look at this guy-- 01:23:36.470 --> 01:23:38.410 so we [INAUDIBLE] again. 01:23:38.410 --> 01:23:43.340 And you say, well, I have derivative with respect 01:23:43.340 --> 01:23:45.490 to x is y squared z. 01:23:45.490 --> 01:23:49.110 The derivative with respect to y is 01:23:49.110 --> 01:23:56.314 x squared y squared z and then minus x squared z. 01:23:56.314 --> 01:24:00.554 Then I have-- [INAUDIBLE]. 01:24:00.554 --> 01:24:05.090 01:24:05.090 --> 01:24:05.790 I did, right? 01:24:05.790 --> 01:24:08.020 So this is squared. 01:24:08.020 --> 01:24:12.000 What matters is that I check what I'm going to do, 01:24:12.000 --> 01:24:17.510 so now I say, my c is a boundaries. 01:24:17.510 --> 01:24:21.650 That's a circle, so the meaning of this integral given 01:24:21.650 --> 01:24:24.550 by Stokes is actually a path integral 01:24:24.550 --> 01:24:27.600 along the c at the level z equals 3. 01:24:27.600 --> 01:24:30.810 I'm at the third floor looking at the world from up there. 01:24:30.810 --> 01:24:34.770 I have the circle on the third floor z equals 3. 01:24:34.770 --> 01:24:40.230 And then I say, that's going to be F dot dR God knows what. 01:24:40.230 --> 01:24:42.210 That was originally the work. 01:24:42.210 --> 01:24:44.620 And Stokes theorem says, no matter 01:24:44.620 --> 01:24:46.770 what surfaces you're going to take 01:24:46.770 --> 01:24:50.250 to have a regular surface without controversy, 01:24:50.250 --> 01:24:54.690 without holes that bounded family by the circles c, 01:24:54.690 --> 01:24:57.100 you're going to be in business. 01:24:57.100 --> 01:25:03.040 So I say, the heck with the S. I want the D, 01:25:03.040 --> 01:25:07.820 and I want that D to be colorful because life is great enough. 01:25:07.820 --> 01:25:10.670 Let's make it D. 01:25:10.670 --> 01:25:13.620 That D has what meaning? 01:25:13.620 --> 01:25:14.850 z equals 3. 01:25:14.850 --> 01:25:18.370 I'm at the level three, but also x 01:25:18.370 --> 01:25:23.012 squared plus y squared must be less than or equal to sum. 01:25:23.012 --> 01:25:25.442 Could anybody tell me what that is? 01:25:25.442 --> 01:25:26.900 STUDENT: 16. 01:25:26.900 --> 01:25:28.550 PROFESSOR: So how do I know? 01:25:28.550 --> 01:25:31.960 I will just plug in a 3 here. 01:25:31.960 --> 01:25:34.240 3 squared is 9. 01:25:34.240 --> 01:25:40.660 25 minus 9, so I get x squared plus y squared equals 16. 01:25:40.660 --> 01:25:45.646 So from here to here, how much do I have? 01:25:45.646 --> 01:25:46.145 STUDENT: 4. 01:25:46.145 --> 01:25:47.960 PROFESSOR: 4, right? 01:25:47.960 --> 01:25:52.950 So that little radius of that yellow domain, [INAUDIBLE]. 01:25:52.950 --> 01:25:56.850 01:25:56.850 --> 01:26:00.095 OK, so let's write down the thing. 01:26:00.095 --> 01:26:08.810 Let's go with D. This domain is going to be called D. 01:26:08.810 --> 01:26:12.230 And then I have this curl F. 01:26:12.230 --> 01:26:15.420 And who is N? 01:26:15.420 --> 01:26:16.250 N is k. 01:26:16.250 --> 01:26:16.750 Why? 01:26:16.750 --> 01:26:20.660 Because I'm in a plane that's upstairs, 01:26:20.660 --> 01:26:24.040 and I have dA because whether the plane is 01:26:24.040 --> 01:26:27.210 upstairs or downstairs on the first floor, 01:26:27.210 --> 01:26:29.162 dA will still be dxdy. 01:26:29.162 --> 01:26:34.042 01:26:34.042 --> 01:26:40.890 OK, so now let's compute what we have backwards. 01:26:40.890 --> 01:26:44.080 So this times k will give me double integral 01:26:44.080 --> 01:26:52.400 over D of y squared times. 01:26:52.400 --> 01:26:55.780 Who is z? 01:26:55.780 --> 01:26:58.311 I'm in a domain, d, where z is fixed. 01:26:58.311 --> 01:26:58.810 STUDENT: 3. 01:26:58.810 --> 01:27:01.360 PROFESSOR: z is 3. 01:27:01.360 --> 01:27:05.520 Minus x squared times 3. 01:27:05.520 --> 01:27:07.050 And-- 01:27:07.050 --> 01:27:07.837 STUDENT: dx2. 01:27:07.837 --> 01:27:09.670 PROFESSOR: I just came up with this problem. 01:27:09.670 --> 01:27:11.390 If I were to write it for the final, 01:27:11.390 --> 01:27:13.340 I would write it even simpler. 01:27:13.340 --> 01:27:17.280 But let's see, 3 and 3, and then nothing. 01:27:17.280 --> 01:27:20.400 And then da, dx dy, right? 01:27:20.400 --> 01:27:23.300 Over the d, which is x squared plus y 01:27:23.300 --> 01:27:26.964 squared this is [INAUDIBLE] over 16. 01:27:26.964 --> 01:27:28.940 How do I solve such a integral? 01:27:28.940 --> 01:27:30.916 I'm going to make it nicer. 01:27:30.916 --> 01:27:32.970 OK. 01:27:32.970 --> 01:27:34.910 How would I solve such an integral? 01:27:34.910 --> 01:27:39.638 Is it a painful thing? 01:27:39.638 --> 01:27:41.509 STUDENT: [INAUDIBLE]. 01:27:41.509 --> 01:27:43.050 PROFESSOR: Well, they're coordinates. 01:27:43.050 --> 01:27:45.160 And somebody's going to help me. 01:27:45.160 --> 01:27:47.090 And as soon as we are done, we are done. 01:27:47.090 --> 01:27:49.980 3 gets out. 01:27:49.980 --> 01:27:53.170 And instead of x squared plus y squared is then 16, 01:27:53.170 --> 01:28:00.270 I have r between 0 and 4, theta between 0 and 2pi. 01:28:00.270 --> 01:28:03.120 I have to take advantage of everything 01:28:03.120 --> 01:28:06.710 I've learned all the semester. 01:28:06.710 --> 01:28:07.880 Knowledge is power. 01:28:07.880 --> 01:28:08.780 What's missing? 01:28:08.780 --> 01:28:10.300 r. 01:28:10.300 --> 01:28:12.005 A 3 gets out. 01:28:12.005 --> 01:28:14.720 And here I have to be just smart and pay attention 01:28:14.720 --> 01:28:15.573 to what you told me. 01:28:15.573 --> 01:28:18.970 Because you told me, Magdalena, why is our sine theta not 01:28:18.970 --> 01:28:20.240 our cosine theta? 01:28:20.240 --> 01:28:21.170 [INAUDIBLE] 01:28:21.170 --> 01:28:27.190 This is r squared, sine squared theta minus r 01:28:27.190 --> 01:28:30.940 squared cosine squared theta. 01:28:30.940 --> 01:28:34.820 So what have I taught you about integrals 01:28:34.820 --> 01:28:37.750 that can be expressed as products of a function of theta 01:28:37.750 --> 01:28:39.830 and function of r? 01:28:39.830 --> 01:28:41.480 That they have a blessing from God. 01:28:41.480 --> 01:28:49.640 So you have 3 integral from 0 to 2pi minus 1. 01:28:49.640 --> 01:28:52.160 I have my plan when it comes to this guy. 01:28:52.160 --> 01:28:55.480 Because it goes on my nerves. 01:28:55.480 --> 01:28:56.100 All right. 01:28:56.100 --> 01:28:57.080 Do you see this? 01:28:57.080 --> 01:28:57.830 [INAUDIBLE] theta. 01:28:57.830 --> 01:28:58.614 STUDENT: That's the [INAUDIBLE]. 01:28:58.614 --> 01:28:59.447 [INTERPOSING VOICES] 01:28:59.447 --> 01:29:01.735 PROFESSOR: Do you know what I'm coming up with? 01:29:01.735 --> 01:29:02.780 [INTERPOSING VOICES] 01:29:02.780 --> 01:29:05.260 PROFESSOR: Cosine of a double angle. 01:29:05.260 --> 01:29:06.280 Very good. 01:29:06.280 --> 01:29:07.950 I'm proud of you guys. 01:29:07.950 --> 01:29:10.520 If I were to test-- oh, there was a test. 01:29:10.520 --> 01:29:14.020 But [INAUDIBLE] next for the whole nation. 01:29:14.020 --> 01:29:18.520 Only about 10% of the students remembered that 01:29:18.520 --> 01:29:22.180 by the end of the calculus series. 01:29:22.180 --> 01:29:25.800 But I think that's not-- that doesn't show weakness 01:29:25.800 --> 01:29:27.200 of the [INAUDIBLE] programs. 01:29:27.200 --> 01:29:32.580 It shows a weakness in the trigonometry classes 01:29:32.580 --> 01:29:36.560 that are either missing from high school or whatever. 01:29:36.560 --> 01:29:38.785 So you know that you want in power. 01:29:38.785 --> 01:29:45.364 Now, times what integral from 0 to 4? 01:29:45.364 --> 01:29:46.252 STUDENT: r squared. 01:29:46.252 --> 01:29:47.140 Or r cubed. 01:29:47.140 --> 01:29:51.309 PROFESSOR: r cubed, which again is wonderful that we have. 01:29:51.309 --> 01:29:55.221 And we should be able to compute the whole thing easily. 01:29:55.221 --> 01:29:57.769 Now if I'm smart, 01:29:57.769 --> 01:29:58.644 STUDENT: [INAUDIBLE]. 01:29:58.644 --> 01:29:59.727 PROFESSOR: How can we see? 01:29:59.727 --> 01:30:03.534 STUDENT: Because the cosine to the integral is sine to theta. 01:30:03.534 --> 01:30:04.512 And [INAUDIBLE]. 01:30:04.512 --> 01:30:05.490 PROFESSOR: Right. 01:30:05.490 --> 01:30:09.600 So the sine to theta, whether I put it here or here, 01:30:09.600 --> 01:30:10.630 is still going to be 0. 01:30:10.630 --> 01:30:12.580 The whole thing will be 0. 01:30:12.580 --> 01:30:13.810 So I play the game. 01:30:13.810 --> 01:30:18.120 Maybe I should've given such a problem when we 01:30:18.120 --> 01:30:20.570 wrote this edition of the book. 01:30:20.570 --> 01:30:24.145 I think it's nicer than the computational one 01:30:24.145 --> 01:30:25.680 you saw before. 01:30:25.680 --> 01:30:30.720 But I told you this trick so you remember it for the final. 01:30:30.720 --> 01:30:33.160 And you are to promise that you'll remember it. 01:30:33.160 --> 01:30:40.090 And that was the whole essence of understanding 01:30:40.090 --> 01:30:44.610 that the Stokes' theorem can become Green's theorem very 01:30:44.610 --> 01:30:47.160 easily when you work with a surface that's 01:30:47.160 --> 01:30:51.120 a domain in plane, a planar domain. [INAUDIBLE]. 01:30:51.120 --> 01:30:53.441 Are you done with this? 01:30:53.441 --> 01:30:53.940 OK. 01:30:53.940 --> 01:31:02.060 01:31:02.060 --> 01:31:05.340 So you say, OK, so what else? 01:31:05.340 --> 01:31:07.580 This was something that's sort of fun. 01:31:07.580 --> 01:31:09.010 I understand it. 01:31:09.010 --> 01:31:12.180 Is there anything left in this whole chapter? 01:31:12.180 --> 01:31:16.160 Fortunately or unfortunately, there is only one section left. 01:31:16.160 --> 01:31:18.559 And I'm going to go over it today. 01:31:18.559 --> 01:31:21.100 STUDENT: Can I ask you a quick question about [INAUDIBLE] 6-- 01:31:21.100 --> 01:31:21.350 PROFESSOR: Yes, sire. 01:31:21.350 --> 01:31:22.659 STUDENT: --before you move on? 01:31:22.659 --> 01:31:23.450 PROFESSOR: Move on? 01:31:23.450 --> 01:31:24.525 STUDENT: I was an idiot. 01:31:24.525 --> 01:31:25.650 PROFESSOR: No, you are not. 01:31:25.650 --> 01:31:27.441 STUDENT: And when I was writing these down, 01:31:27.441 --> 01:31:29.070 I missed the variable. 01:31:29.070 --> 01:31:32.645 So I have the integral of fdr over c 01:31:32.645 --> 01:31:36.627 equals double integral over f, curl f dot n. 01:31:36.627 --> 01:31:37.210 PROFESSOR: ds. 01:31:37.210 --> 01:31:39.966 STUDENT: I didn't write down what c was. 01:31:39.966 --> 01:31:41.465 I didn't write down what this c was. 01:31:41.465 --> 01:31:45.530 PROFESSOR: The c was the whatever boundary 01:31:45.530 --> 01:31:49.090 you had there of the surface s. 01:31:49.090 --> 01:31:51.115 And that was in the beginning when 01:31:51.115 --> 01:31:55.650 we defined the sphere, when we gave the general statement 01:31:55.650 --> 01:31:59.480 for the function. 01:31:59.480 --> 01:32:02.245 So I'm going to try and draw a potato. 01:32:02.245 --> 01:32:05.140 We don't do a very good job in the book 01:32:05.140 --> 01:32:07.520 drawing the solid body. 01:32:07.520 --> 01:32:11.626 But I'll try and draw a very nice solid body. 01:32:11.626 --> 01:32:12.126 Let's see. 01:32:12.126 --> 01:32:17.940 01:32:17.940 --> 01:32:21.570 You have a solid body. 01:32:21.570 --> 01:32:26.490 Imagine it as a potato, topologically a sphere. 01:32:26.490 --> 01:32:28.240 It's a balloon that you blow. 01:32:28.240 --> 01:32:29.290 It's a closed surface. 01:32:29.290 --> 01:32:31.560 It closes in itself. 01:32:31.560 --> 01:32:36.050 And we call that r in the book. 01:32:36.050 --> 01:32:43.360 It's a solid region enclosed by the closed surfaces. 01:32:43.360 --> 01:32:46.650 01:32:46.650 --> 01:32:48.862 Sometimes we call such a surface compact 01:32:48.862 --> 01:32:51.640 for some topological reasons. 01:32:51.640 --> 01:32:53.850 Let's put s. 01:32:53.850 --> 01:32:55.896 s is the boundary of r. 01:32:55.896 --> 01:33:00.660 01:33:00.660 --> 01:33:05.053 We as you know our old friend to be a vector value function. 01:33:05.053 --> 01:33:09.750 01:33:09.750 --> 01:33:14.230 And again, if you like a force field, 01:33:14.230 --> 01:33:15.770 think of it as a force field. 01:33:15.770 --> 01:33:17.520 Now, I'm not going to tell you what it is. 01:33:17.520 --> 01:33:21.480 It's [INAUDIBLE] function differential [INAUDIBLE] 01:33:21.480 --> 01:33:24.930 the partial here is continuous. 01:33:24.930 --> 01:33:28.660 The magic thing is that this surface must be orientable. 01:33:28.660 --> 01:33:34.185 And if we are going to immerse it, it's a regular surface. 01:33:34.185 --> 01:33:35.646 Then of course, n exists. 01:33:35.646 --> 01:33:40.770 And your [INAUDIBLE], guys, doesn't have to be outwards. 01:33:40.770 --> 01:33:44.552 It could be inwards [INAUDIBLE]. 01:33:44.552 --> 01:33:51.890 Let's make the convention that n will be outwards by convention. 01:33:51.890 --> 01:33:54.990 So we have to have an agreement like they do in politics, 01:33:54.990 --> 01:33:57.840 between Fidel Castro and Obama. 01:33:57.840 --> 01:34:00.910 By convention, whether we like it or not, 01:34:00.910 --> 01:34:07.172 let's assume the normal will be pointing out. 01:34:07.172 --> 01:34:09.580 Then something magic happens. 01:34:09.580 --> 01:34:13.040 And that magic thing, I'm not going to tell you what it is. 01:34:13.040 --> 01:34:16.400 But you should tell me if you remember 01:34:16.400 --> 01:34:20.760 what the double integral was in this case, intolerance 01:34:20.760 --> 01:34:22.660 of physics. 01:34:22.660 --> 01:34:23.740 Shut up, Magdalena. 01:34:23.740 --> 01:34:26.710 Don't tell them everything. 01:34:26.710 --> 01:34:29.230 Let people remember what this was. 01:34:29.230 --> 01:34:31.530 So what is the second term? 01:34:31.530 --> 01:34:34.670 This is the so-called famous divergence theorem. 01:34:34.670 --> 01:34:37.700 01:34:37.700 --> 01:34:39.850 So this is the divergence. 01:34:39.850 --> 01:34:42.680 If you don't remember that, we will review it. 01:34:42.680 --> 01:34:44.550 dV is the volume integral. 01:34:44.550 --> 01:34:48.350 I have a [INAUDIBLE] integral over the solid potato, 01:34:48.350 --> 01:34:49.023 of course. 01:34:49.023 --> 01:34:53.024 What is this animal [INAUDIBLE]? 01:34:53.024 --> 01:34:53.790 OK. 01:34:53.790 --> 01:34:59.590 Take some milk and strain it and make cheese. 01:34:59.590 --> 01:35:02.950 And you have that kind of piece of cloth. 01:35:02.950 --> 01:35:04.900 And you hang it. 01:35:04.900 --> 01:35:09.330 And the water goes through that piece of cloth. 01:35:09.330 --> 01:35:12.570 [INAUDIBLE] have this kind of suggestive image 01:35:12.570 --> 01:35:14.430 should make you think of something we 01:35:14.430 --> 01:35:16.030 talked about before. 01:35:16.030 --> 01:35:20.920 Whether that was fluid dynamics or electromagnetism, 01:35:20.920 --> 01:35:24.890 [INAUDIBLE], this has the same name. 01:35:24.890 --> 01:35:32.220 f is some sort of field, vector [INAUDIBLE] field. 01:35:32.220 --> 01:35:34.816 N is the outer normal in this case. 01:35:34.816 --> 01:35:40.170 What is the meaning of that, for a dollar, which I don't have? 01:35:40.170 --> 01:35:41.460 It's a four-letter word. 01:35:41.460 --> 01:35:42.640 It's an F word. 01:35:42.640 --> 01:35:43.526 STUDENT: Flux. 01:35:43.526 --> 01:35:44.400 PROFESSOR: Very good. 01:35:44.400 --> 01:35:45.390 I'm proud of you. 01:35:45.390 --> 01:35:47.780 Who said it first? 01:35:47.780 --> 01:35:49.339 Aaron said it first? 01:35:49.339 --> 01:35:50.130 I owe you a dollar. 01:35:50.130 --> 01:35:51.427 You can stop by my office. 01:35:51.427 --> 01:35:52.385 I'll give you a dollar. 01:35:52.385 --> 01:35:53.744 STUDENT: He said it five minutes ago. 01:35:53.744 --> 01:35:55.166 PROFESSOR: So the flux-- He did? 01:35:55.166 --> 01:35:56.114 STUDENT: Yeah, he did. 01:35:56.114 --> 01:35:57.536 Silently. 01:35:57.536 --> 01:36:00.380 PROFESSOR: Aaron is a mindreader. 01:36:00.380 --> 01:36:01.328 OK. 01:36:01.328 --> 01:36:03.059 So the flux in the left-hand side. 01:36:03.059 --> 01:36:04.600 This thing you don't know what it is. 01:36:04.600 --> 01:36:06.508 But it's some sort of potato. 01:36:06.508 --> 01:36:08.416 What is the divergence of something? 01:36:08.416 --> 01:36:11.290 01:36:11.290 --> 01:36:20.125 So if somebody gives you the vector field F1, [INAUDIBLE], 01:36:20.125 --> 01:36:26.830 where these are functions of xyz, [INAUDIBLE]. 01:36:26.830 --> 01:36:29.810 What is the divergence of F by definition? 01:36:29.810 --> 01:36:34.810 Remember section 13.1? 01:36:34.810 --> 01:36:36.924 Keep it in mind for the final. 01:36:36.924 --> 01:36:40.660 01:36:40.660 --> 01:36:41.950 So what do we do? 01:36:41.950 --> 01:36:44.840 Differentiate the first component respect 01:36:44.840 --> 01:36:50.100 to x plus differentiate the second component 01:36:50.100 --> 01:36:52.815 respect to y plus differentiate the third component 01:36:52.815 --> 01:36:54.780 respect to c, sum them up. 01:36:54.780 --> 01:36:56.165 And that's your divergence. 01:36:56.165 --> 01:36:58.940 01:36:58.940 --> 01:37:01.540 OK? 01:37:01.540 --> 01:37:05.428 How do engineers write divergence? 01:37:05.428 --> 01:37:08.460 Not like a mathematician or like a geometer. 01:37:08.460 --> 01:37:09.950 I'm doing differential geometry. 01:37:09.950 --> 01:37:11.280 How do they write? 01:37:11.280 --> 01:37:12.530 STUDENT: Kinds of [INAUDIBLE]. 01:37:12.530 --> 01:37:16.850 PROFESSOR: [INAUDIBLE] dot if. 01:37:16.850 --> 01:37:19.409 This is how engineers write divergence. 01:37:19.409 --> 01:37:22.283 And when they write curl, how do they write it? 01:37:22.283 --> 01:37:24.678 They write [INAUDIBLE] cross product. 01:37:24.678 --> 01:37:25.915 Because it has a meaning. 01:37:25.915 --> 01:37:29.982 If you think about operator, you have ddx 01:37:29.982 --> 01:37:34.730 applied to F1, ddy applied to F2, ddz applied to F3. 01:37:34.730 --> 01:37:39.165 So it's like having the dot product between ddx, ddy, 01:37:39.165 --> 01:37:43.640 ddz operators, which would be the [INAUDIBLE] operator 01:37:43.640 --> 01:37:46.510 acting on F1, F2, F3. 01:37:46.510 --> 01:37:49.850 So you go first first, plus second second, 01:37:49.850 --> 01:37:52.340 plus third third, right? 01:37:52.340 --> 01:37:58.426 It's exactly the same idea that you inherited from dot product. 01:37:58.426 --> 01:38:02.300 Now let's see the last two problems of this semester. 01:38:02.300 --> 01:38:04.470 except for step the review. 01:38:04.470 --> 01:38:07.090 But the review's another story. 01:38:07.090 --> 01:38:12.940 So I'm going to pick one of your favorite problems. 01:38:12.940 --> 01:38:21.780 01:38:21.780 --> 01:38:22.280 OK. 01:38:22.280 --> 01:38:26.478 Example one, remember your favorite tetrahedron. 01:38:26.478 --> 01:38:28.370 I'm going to erase it. 01:38:28.370 --> 01:38:31.681 01:38:31.681 --> 01:38:35.730 Instead of the potato, you can have something like a pyramid. 01:38:35.730 --> 01:38:37.575 And you have example one. 01:38:37.575 --> 01:38:40.275 01:38:40.275 --> 01:38:44.170 Let's say, [INAUDIBLE] we have that. 01:38:44.170 --> 01:38:46.070 Somebody gives you the F. 01:38:46.070 --> 01:38:48.730 I'm going to make it nice and sassy. 01:38:48.730 --> 01:38:52.995 Because the final is coming and I want simple examples. 01:38:52.995 --> 01:38:56.180 And don't expect anything [INAUDIBLE] really nice 01:38:56.180 --> 01:38:59.030 examples also on the final. 01:38:59.030 --> 01:39:12.740 Apply divergence theorem in order 01:39:12.740 --> 01:39:22.460 to compute double integral of F dot n ds over s, 01:39:22.460 --> 01:39:37.992 where s is the surface of the tetrahedron in the picture. 01:39:37.992 --> 01:39:40.880 And that's your favorite tetrahedron. 01:39:40.880 --> 01:39:43.950 We've done that like a million times. 01:39:43.950 --> 01:39:51.130 Somebody gave you a-- shall I put 1 or a? 01:39:51.130 --> 01:39:55.920 1, because [INAUDIBLE] is [INAUDIBLE] is [INAUDIBLE]. 01:39:55.920 --> 01:39:59.315 So you have the plane x plus y plus z. 01:39:59.315 --> 01:40:02.780 Plus 1 you intersect with the axis'. 01:40:02.780 --> 01:40:05.500 The coordinates, you take the place of coordinates 01:40:05.500 --> 01:40:08.000 and you form a tetrahedron. 01:40:08.000 --> 01:40:10.500 Next tetrahedron is a little bit beautiful 01:40:10.500 --> 01:40:14.645 that it has 90 degree angles at the vertex. 01:40:14.645 --> 01:40:18.300 And it has a name, OABC. 01:40:18.300 --> 01:40:20.940 OABC is the tetrahedron. 01:40:20.940 --> 01:40:24.800 And the surface of the tetrahedron is s. 01:40:24.800 --> 01:40:28.640 How are you going to do this problem? 01:40:28.640 --> 01:40:32.068 You're going to say, oh my god, I don't know. 01:40:32.068 --> 01:40:32.748 It's not hard. 01:40:32.748 --> 01:40:34.498 STUDENT: It looks like you're going to use 01:40:34.498 --> 01:40:35.956 the formula you just gave us. 01:40:35.956 --> 01:40:37.414 PROFESSOR: The divergence theorem. 01:40:37.414 --> 01:40:39.580 STUDENT: And the divergence for that is really easy. 01:40:39.580 --> 01:40:40.816 It's just a constant. 01:40:40.816 --> 01:40:41.788 PROFESSOR: Right. 01:40:41.788 --> 01:40:45.820 And we have to give a name to the tetrahedron, [INAUDIBLE] 01:40:45.820 --> 01:40:47.480 T, with the solid tetrahedron. 01:40:47.480 --> 01:40:53.010 01:40:53.010 --> 01:40:59.350 And its area, its surface is this. 01:40:59.350 --> 01:41:02.640 Instead of a potato, you have the solid tetrahedron. 01:41:02.640 --> 01:41:03.920 So what do you write? 01:41:03.920 --> 01:41:10.095 Exactly what [INAUDIBLE] told you, triple integral over T. 01:41:10.095 --> 01:41:11.009 Of what? 01:41:11.009 --> 01:41:16.270 The divergence of F, because that's the divergence theorem, 01:41:16.270 --> 01:41:16.770 dv. 01:41:16.770 --> 01:41:19.950 01:41:19.950 --> 01:41:20.950 Well, it should be easy. 01:41:20.950 --> 01:41:24.816 Because just as you said, divergence of F 01:41:24.816 --> 01:41:25.808 would be a constant. 01:41:25.808 --> 01:41:27.300 How come? 01:41:27.300 --> 01:41:30.730 Differentiate this with respect to x, 2. 01:41:30.730 --> 01:41:32.690 This with respect to y, 3. 01:41:32.690 --> 01:41:36.512 This with respect to z, 5. 01:41:36.512 --> 01:41:40.816 Last time I checked this was 10 when I was [INAUDIBLE]. 01:41:40.816 --> 01:41:43.800 01:41:43.800 --> 01:41:48.485 So 10 says I'm going for a walk. 01:41:48.485 --> 01:41:54.038 And then triple integral of the volume of 1dv over T, 01:41:54.038 --> 01:41:55.034 what is this? 01:41:55.034 --> 01:41:59.018 01:41:59.018 --> 01:42:02.010 [INTERPOSING VOICES] 01:42:02.010 --> 01:42:06.630 PROFESSOR: Well, because I taught you how to cheat, yes. 01:42:06.630 --> 01:42:14.435 But what if I were to ask you to express this as-- 01:42:14.435 --> 01:42:15.310 STUDENT: [INAUDIBLE]? 01:42:15.310 --> 01:42:15.976 PROFESSOR: Yeah. 01:42:15.976 --> 01:42:17.040 Integrate one at a time. 01:42:17.040 --> 01:42:22.380 So you have 1dz, dy, dx-- I'm doing review with you-- 01:42:22.380 --> 01:42:27.644 from 0 to 1 minus x minus y from 0 to-- 01:42:27.644 --> 01:42:29.100 STUDENT: [INTERPOSING VOICES] 01:42:29.100 --> 01:42:33.430 PROFESSOR: --1 minus x from zero to 1. 01:42:33.430 --> 01:42:36.472 And how did I teach you how to cheat? 01:42:36.472 --> 01:42:39.700 I taught you that in this case you shouldn't 01:42:39.700 --> 01:42:42.180 bother to compute that. 01:42:42.180 --> 01:42:44.790 Remember that you were in school and we 01:42:44.790 --> 01:42:47.310 learned the volume of a tetrahedron 01:42:47.310 --> 01:42:54.340 was the area of the base times the height divided by 3, 01:42:54.340 --> 01:43:01.700 which was one half times 1 divided by 3. 01:43:01.700 --> 01:43:03.380 So you guys right. 01:43:03.380 --> 01:43:06.350 The answer is 10 times 1 over 6. 01:43:06.350 --> 01:43:07.520 Do I leave it like that? 01:43:07.520 --> 01:43:08.019 No. 01:43:08.019 --> 01:43:09.710 Because it's not nice. 01:43:09.710 --> 01:43:11.130 So the answer is 5/3. 01:43:11.130 --> 01:43:13.740 01:43:13.740 --> 01:43:19.280 Expect something like that on the final, 01:43:19.280 --> 01:43:21.100 something very similar. 01:43:21.100 --> 01:43:24.970 So you'll have to apply the divergence theorem 01:43:24.970 --> 01:43:26.130 and do a good job. 01:43:26.130 --> 01:43:29.102 And of course, you have to be careful. 01:43:29.102 --> 01:43:30.880 But it shouldn't be hard. 01:43:30.880 --> 01:43:34.940 It's something that should be easy to do. 01:43:34.940 --> 01:43:38.700 Now, the last problem of the semester that I want to do 01:43:38.700 --> 01:43:41.410 is an application of the divergence theorem 01:43:41.410 --> 01:43:43.030 is over a cube. 01:43:43.030 --> 01:43:49.160 So I'm going to erase it, the whole thing. 01:43:49.160 --> 01:43:53.040 And I'm going to draw a cube, which is 01:43:53.040 --> 01:43:56.160 an open-topped box upside down. 01:43:56.160 --> 01:44:01.130 Say it again, an open-topped box upside down, which 01:44:01.130 --> 01:44:04.460 means somebody gives you a cubic box 01:44:04.460 --> 01:44:06.530 and tells you to turn it upside down. 01:44:06.530 --> 01:44:11.540 01:44:11.540 --> 01:44:13.900 And you have from here to here, 1. 01:44:13.900 --> 01:44:16.970 All the dimensions of the cube are 1. 01:44:16.970 --> 01:44:23.490 The top is missing, so there's faces missing. 01:44:23.490 --> 01:44:25.350 The bottom face is missing. 01:44:25.350 --> 01:44:27.930 Bottom face is missing. 01:44:27.930 --> 01:44:31.084 Let's call it-- you know, what shall we call it? 01:44:31.084 --> 01:44:38.180 01:44:38.180 --> 01:44:39.980 F1. 01:44:39.980 --> 01:44:44.140 Because it was the top, but now it's the bottom. 01:44:44.140 --> 01:44:45.080 OK? 01:44:45.080 --> 01:44:55.797 And the rest are F2, F3, F4, F5, and F6, which is the top. 01:44:55.797 --> 01:44:57.681 And I'm going to erase. 01:44:57.681 --> 01:45:00.520 01:45:00.520 --> 01:45:07.440 And the last thing before this section is to do the following. 01:45:07.440 --> 01:45:08.980 What do I want? 01:45:08.980 --> 01:45:17.634 Evaluate the flux double integral over s F dot n ds. 01:45:17.634 --> 01:45:20.940 You have to evaluate that. 01:45:20.940 --> 01:45:25.490 For the case when F-- I usually don't take the exact data 01:45:25.490 --> 01:45:26.100 from the book. 01:45:26.100 --> 01:45:27.500 But in this case, I want to. 01:45:27.500 --> 01:45:29.600 Because I know you'll read it. 01:45:29.600 --> 01:45:33.550 And I don't want you to have any difficulty with this problem. 01:45:33.550 --> 01:45:35.470 I hate the data myself. 01:45:35.470 --> 01:45:36.700 I didn't like it very much. 01:45:36.700 --> 01:45:41.530 01:45:41.530 --> 01:45:44.640 It's unit cube, OK. 01:45:44.640 --> 01:45:47.710 So x must be between 0 and 1. 01:45:47.710 --> 01:45:51.080 y must be between 0 and 1 including them. 01:45:51.080 --> 01:45:57.810 But z-- attention guys-- must be between 0 [INAUDIBLE], 01:45:57.810 --> 01:45:59.530 without 0. 01:45:59.530 --> 01:46:02.580 Because you remove the face on the ground. 01:46:02.580 --> 01:46:08.700 z is greater than 0 and less than or equal to 1. 01:46:08.700 --> 01:46:12.590 And do we want anything else? 01:46:12.590 --> 01:46:13.150 No. 01:46:13.150 --> 01:46:15.280 That is all. 01:46:15.280 --> 01:46:20.750 So let's compute the whole thing. 01:46:20.750 --> 01:46:29.429 Now, assume the box would be complete. 01:46:29.429 --> 01:46:35.700 01:46:35.700 --> 01:46:38.540 If the box were complete, then I would 01:46:38.540 --> 01:46:43.640 have the following, double integral over all the 01:46:43.640 --> 01:46:50.120 faces F2 union with F3 union with F4 union 01:46:50.120 --> 01:46:58.490 with F5 union with-- oh my God-- F6 of F 01:46:58.490 --> 01:47:05.236 dot 10 ds plus double integral over what's missing guys, F1? 01:47:05.236 --> 01:47:09.610 01:47:09.610 --> 01:47:14.956 Of n dot n ds-- F, Magdalena, that's the flux. 01:47:14.956 --> 01:47:17.900 F dot n ds. 01:47:17.900 --> 01:47:20.820 If it were complete, that would mean 01:47:20.820 --> 01:47:25.060 I have the double integral over all the six faces. 01:47:25.060 --> 01:47:29.820 In that case, this sum would be-- I can apply finally 01:47:29.820 --> 01:47:31.740 the divergence theorem. 01:47:31.740 --> 01:47:35.206 That would be triple integral of-- God 01:47:35.206 --> 01:47:42.570 knows what that is-- divergence of F dv over the cube. 01:47:42.570 --> 01:47:46.130 What do you want us to call the cube? 01:47:46.130 --> 01:47:47.490 STUDENT: C. 01:47:47.490 --> 01:47:51.250 PROFESSOR: C is usually what we denote for the curve. 01:47:51.250 --> 01:47:53.108 STUDENT: How about q? 01:47:53.108 --> 01:47:54.000 PROFESSOR: Beautiful. 01:47:54.000 --> 01:47:54.740 Sounds like. 01:47:54.740 --> 01:47:56.030 Oh, I like that. q. 01:47:56.030 --> 01:48:00.830 q is the cube inside the whole thing. 01:48:00.830 --> 01:48:03.415 Unfortunately, this is not very nicely 01:48:03.415 --> 01:48:09.900 picked just to make your life miserable. 01:48:09.900 --> 01:48:14.420 So you have dv x over y. 01:48:14.420 --> 01:48:16.790 There is no j, at least that. 01:48:16.790 --> 01:48:20.495 ddz minus this way. 01:48:20.495 --> 01:48:22.387 As soon as we are done with this, 01:48:22.387 --> 01:48:24.790 since I gave you no break, I'm going to let you go. 01:48:24.790 --> 01:48:26.050 So what do we have? 01:48:26.050 --> 01:48:29.280 y minus 2z. 01:48:29.280 --> 01:48:30.470 Does it look good? 01:48:30.470 --> 01:48:31.530 No. 01:48:31.530 --> 01:48:32.750 Does it look bad? 01:48:32.750 --> 01:48:36.920 No, not really bad either. 01:48:36.920 --> 01:48:39.360 If I were to solve the problem, I 01:48:39.360 --> 01:48:46.250 would have to say triple, triple, triple y minus 2z. 01:48:46.250 --> 01:48:53.140 And now-- oh my God-- dz, dy, dx. 01:48:53.140 --> 01:48:59.125 I sort of hate when a little bit of computation 0 to 1, 0 to 1, 01:48:59.125 --> 01:49:00.180 0 to 1. 01:49:00.180 --> 01:49:02.924 But this is for [INAUDIBLE] theorem. 01:49:02.924 --> 01:49:05.600 Is there anybody missing from the picture 01:49:05.600 --> 01:49:09.516 so I can reduce it to a double integral? 01:49:09.516 --> 01:49:10.230 STUDENT: x. 01:49:10.230 --> 01:49:11.229 PROFESSOR: x is missing. 01:49:11.229 --> 01:49:14.620 So I say there is no x inside. 01:49:14.620 --> 01:49:18.801 I go what is integral from 0 to 1 of 1dx? 01:49:18.801 --> 01:49:19.300 STUDENT: 1 01:49:19.300 --> 01:49:20.060 PROFESSOR: 1. 01:49:20.060 --> 01:49:24.457 So I will rewrite it as integral from 0 to1, 01:49:24.457 --> 01:49:30.260 integral from 0 to1, y minus 2z, dz, dy. 01:49:30.260 --> 01:49:32.022 Is this hard? 01:49:32.022 --> 01:49:33.187 Eh, no. 01:49:33.187 --> 01:49:34.520 But it's a little bit obnoxious. 01:49:34.520 --> 01:49:37.638 01:49:37.638 --> 01:49:41.510 When I integrate with respect to z, what do I get? 01:49:41.510 --> 01:49:44.420 01:49:44.420 --> 01:49:49.640 Yz minus z squared. 01:49:49.640 --> 01:49:58.774 No, not that-- between z equals 0 1 down, z equals 1 up to z 01:49:58.774 --> 01:50:03.080 equals 0 down dy. 01:50:03.080 --> 01:50:06.550 So z goes from 0 to 1 [INAUDIBLE]. 01:50:06.550 --> 01:50:09.530 When z is 0 down, I have nothing. 01:50:09.530 --> 01:50:11.990 STUDENT: Yeah, 1 minus-- y minus 1. 01:50:11.990 --> 01:50:15.780 PROFESSOR: 1. y minus 1, not so bad. 01:50:15.780 --> 01:50:18.210 Not so bad, dy. 01:50:18.210 --> 01:50:24.970 So I get y squared over 2 minus y. 01:50:24.970 --> 01:50:30.504 Between 0 and 1, what do I get? 01:50:30.504 --> 01:50:31.420 STUDENT: Negative 1/2. 01:50:31.420 --> 01:50:32.419 PROFESSOR: Negative 1/2. 01:50:32.419 --> 01:50:36.680 01:50:36.680 --> 01:50:37.260 All right. 01:50:37.260 --> 01:50:38.550 Let's see what we've got here. 01:50:38.550 --> 01:50:39.050 Yeah. 01:50:39.050 --> 01:50:40.590 They got [INAUDIBLE]. 01:50:40.590 --> 01:50:43.290 And now I'm asking you what's going to happen. 01:50:43.290 --> 01:50:46.590 Our contour is the open-topped box upside down. 01:50:46.590 --> 01:50:48.900 This is what we need. 01:50:48.900 --> 01:50:54.020 This is what we-- 01:50:54.020 --> 01:50:57.490 STUDENT: Couldn't you just the double integral? 01:50:57.490 --> 01:51:01.360 PROFESSOR: We just have to compute this fellow. 01:51:01.360 --> 01:51:03.860 We need to compute that fellow. 01:51:03.860 --> 01:51:06.935 So how do we do that? 01:51:06.935 --> 01:51:09.905 How do we do that? 01:51:09.905 --> 01:51:12.890 STUDENT: What is the problem asking for again? 01:51:12.890 --> 01:51:16.700 PROFESSOR: So the problem is asking over this flux, 01:51:16.700 --> 01:51:21.910 but only over the box' walls and the top. 01:51:21.910 --> 01:51:25.085 The top, one, two, three, four, without the bottom, 01:51:25.085 --> 01:51:26.830 which is missing. 01:51:26.830 --> 01:51:29.150 In order to apply divergence theorem, 01:51:29.150 --> 01:51:36.190 I have to put the bottom back and have a closed surface that 01:51:36.190 --> 01:51:38.330 is enclosing the whole cube. 01:51:38.330 --> 01:51:41.140 So this is what I want. 01:51:41.140 --> 01:51:42.740 This is what I know. 01:51:42.740 --> 01:51:43.680 How much is it guys? 01:51:43.680 --> 01:51:44.710 Minus 1/2. 01:51:44.710 --> 01:51:47.660 And this is, again, what I need. 01:51:47.660 --> 01:51:48.160 Right? 01:51:48.160 --> 01:51:53.417 That's the last thing I'm going to do today. 01:51:53.417 --> 01:51:54.250 [INTERPOSING VOICES] 01:51:54.250 --> 01:51:57.930 01:51:57.930 --> 01:52:00.293 STUDENT: F times k da. 01:52:00.293 --> 01:52:02.610 PROFESSOR: Let's compute it. 01:52:02.610 --> 01:52:06.000 k is a blessing, as you said, [INAUDIBLE]. 01:52:06.000 --> 01:52:07.490 It's actually minus k. 01:52:07.490 --> 01:52:09.050 Why is it minus k? 01:52:09.050 --> 01:52:10.820 Because it's upside down. 01:52:10.820 --> 01:52:12.130 And it's an altered normal. 01:52:12.130 --> 01:52:13.274 STUDENT: Oh, it is the [INAUDIBLE] normal. 01:52:13.274 --> 01:52:13.716 OK. 01:52:13.716 --> 01:52:14.158 Yeah. 01:52:14.158 --> 01:52:14.600 That's right. 01:52:14.600 --> 01:52:15.042 PROFESSOR: [INAUDIBLE]. 01:52:15.042 --> 01:52:15.541 So minus k. 01:52:15.541 --> 01:52:17.640 But it doesn't [INAUDIBLE]. 01:52:17.640 --> 01:52:18.610 The sign matters. 01:52:18.610 --> 01:52:20.560 So I have to be careful. 01:52:20.560 --> 01:52:25.620 F is-- z is 0, thank God. 01:52:25.620 --> 01:52:26.600 So that does away. 01:52:26.600 --> 01:52:36.000 So I have x y i dot product with minus k. 01:52:36.000 --> 01:52:37.520 What's the beauty of this? 01:52:37.520 --> 01:52:38.170 0. 01:52:38.170 --> 01:52:39.580 STUDENT: 0. 01:52:39.580 --> 01:52:40.610 PROFESSOR: Yay. 01:52:40.610 --> 01:52:42.060 0. 01:52:42.060 --> 01:52:46.030 So the answer to this problem is minus 1/2. 01:52:46.030 --> 01:52:50.880 So the answer is minus 1. 01:52:50.880 --> 01:52:52.685 And we are done with the last section 01:52:52.685 --> 01:52:57.480 of the book, which is 13.7. 01:52:57.480 --> 01:52:58.630 It was a long way. 01:52:58.630 --> 01:53:01.430 We came a long way to what I'm going 01:53:01.430 --> 01:53:05.521 to do next time and the times to come. 01:53:05.521 --> 01:53:08.370 First of all, ask me from now on you want a break or not. 01:53:08.370 --> 01:53:10.750 Because I didn't give you a break today. 01:53:10.750 --> 01:53:12.880 We are not in a hurry. 01:53:12.880 --> 01:53:15.180 I will pick up exams. 01:53:15.180 --> 01:53:18.150 And I will go over them together with you. 01:53:18.150 --> 01:53:21.620 And by the time we finish this review, 01:53:21.620 --> 01:53:26.210 we will have solved two or three finals completely. 01:53:26.210 --> 01:53:29.810 We will be [INAUDIBLE]. 01:53:29.810 --> 01:53:38.810 And so the final is on the 11th, May 11 at 10:30 in the morning. 01:53:38.810 --> 01:53:39.435 I think. 01:53:39.435 --> 01:53:40.310 STUDENT: It's the 11. 01:53:40.310 --> 01:53:41.226 The 12 is [INAUDIBLE]. 01:53:41.226 --> 01:53:43.610 The 12th is the other class. 01:53:43.610 --> 01:53:44.210 STUDENT: Yeah. 01:53:44.210 --> 01:53:44.810 I'm positive. 01:53:44.810 --> 01:53:46.310 PROFESSOR: We are switching the two classes. 01:53:46.310 --> 01:53:47.210 STUDENT: [INAUDIBLE]. 01:53:47.210 --> 01:53:51.410 PROFESSOR: And it's May 11 at 10:30 in the morning. 01:53:51.410 --> 01:53:55.610 On May 12, there are other math courses that have a final. 01:53:55.610 --> 01:53:59.510 But fortunately for them, they start at 4:30. 01:53:59.510 --> 01:54:02.210 I'm really blessed that I don't have that [INAUDIBLE]. 01:54:02.210 --> 01:54:05.510 They start at 4:30, and they end at 7:00. 01:54:05.510 --> 01:54:07.610 Can you imagine how frisky you feel when you 01:54:07.610 --> 01:54:09.110 take that final in the night? 01:54:09.110 --> 01:54:11.810 01:54:11.810 --> 01:54:13.010 Good luck with the homework. 01:54:13.010 --> 01:54:16.660 Ask me questions about the homework if you have them. 01:54:16.660 --> 01:54:19.777