WEBVTT 00:00:00.000 --> 00:00:00.500 00:00:00.500 --> 00:00:04.050 PROFESSOR: I would like to review just briefly what 00:00:04.050 --> 00:00:06.689 we discussed last time. 00:00:06.689 --> 00:00:15.530 We gave very important results, and that was Green's Theorem. 00:00:15.530 --> 00:00:19.110 And I would like to know if you remember 00:00:19.110 --> 00:00:25.280 when I said about the settling for this problem. 00:00:25.280 --> 00:00:30.660 So we'll assume we have a domain without a hole, 00:00:30.660 --> 00:00:34.790 D. D is a domain without a hole inside, 00:00:34.790 --> 00:00:39.760 without punctures or holes. 00:00:39.760 --> 00:00:43.410 00:00:43.410 --> 00:00:47.930 There is a scientific name in mathematics for such a domain. 00:00:47.930 --> 00:00:50.530 This is going to be simply connected. 00:00:50.530 --> 00:00:56.080 00:00:56.080 --> 00:00:59.860 And this is a difficult topological theorem, 00:00:59.860 --> 00:01:02.040 but this is what we expect, OK? 00:01:02.040 --> 00:01:03.600 And what does it mean? 00:01:03.600 --> 00:01:04.720 What does it mean? 00:01:04.720 --> 00:01:14.570 It means that in the C being a Jordan curve was what? 00:01:14.570 --> 00:01:15.470 How? 00:01:15.470 --> 00:01:20.790 This was continuous, no self intersections. 00:01:20.790 --> 00:01:27.230 00:01:27.230 --> 00:01:34.660 In such a case, we set up M and N to be C1 functions. 00:01:34.660 --> 00:01:39.440 00:01:39.440 --> 00:01:43.480 And then we proceed through the path integral of C. 00:01:43.480 --> 00:01:46.690 Do you like this kind of C, or you prefer a straight C? 00:01:46.690 --> 00:01:57.200 The path integral of C of M of xy dx class, N of xy, dy, 00:01:57.200 --> 00:01:59.010 everything is in plane. 00:01:59.010 --> 00:02:03.640 I'm sorry that I cannot repeat that, 00:02:03.640 --> 00:02:06.690 but we discussed that time, is in the plane of 2. 00:02:06.690 --> 00:02:10.139 And then what-- do you remember in terms 00:02:10.139 --> 00:02:16.120 of how this path integral, [INAUDIBLE] inside, 00:02:16.120 --> 00:02:21.660 is connected to a double integral over the whole domain. 00:02:21.660 --> 00:02:26.400 In particular, do you remember-- this is easy to memorize-- 00:02:26.400 --> 00:02:28.270 but do you remember what's inside? 00:02:28.270 --> 00:02:36.800 Because for the final, you are expected to know his result. 00:02:36.800 --> 00:02:38.640 STUDENT: [INAUDIBLE] 00:02:38.640 --> 00:02:43.170 PROFESSOR: N sub X. 00:02:43.170 --> 00:02:44.540 STUDENT: Minus M sub Y. 00:02:44.540 --> 00:02:47.120 PROFESSOR: Minus M sub Y. [INAUDIBLE] 00:02:47.120 --> 00:02:53.420 must M-- M and N-- M sub Y. Here is the Y. Of course this 00:02:53.420 --> 00:02:58.630 would be dA in plane, and in the-- if you 00:02:58.630 --> 00:03:02.490 want to represent this in the general format, 00:03:02.490 --> 00:03:05.630 the MdX minus the MdY. 00:03:05.630 --> 00:03:08.300 Feel free to do that. 00:03:08.300 --> 00:03:13.060 One was a correlary or a consequence. 00:03:13.060 --> 00:03:22.080 This theorem was that if I were to take this big M to be 00:03:22.080 --> 00:03:26.330 the minus Y as a function, then this function N will 00:03:26.330 --> 00:03:29.570 be plus X, what will I get? 00:03:29.570 --> 00:03:37.932 I would get that minus YdX plus NdY will be what? 00:03:37.932 --> 00:03:40.127 00:03:40.127 --> 00:03:40.960 STUDENT: [INAUDIBLE] 00:03:40.960 --> 00:03:42.293 PROFESSOR: Two times, excellent. 00:03:42.293 --> 00:03:43.590 You are very awake. 00:03:43.590 --> 00:03:45.000 So I wanted to catch you. 00:03:45.000 --> 00:03:46.210 I couldn't catch you. 00:03:46.210 --> 00:03:48.320 I thought you would say the A of the domain, 00:03:48.320 --> 00:03:49.500 but you said it right. 00:03:49.500 --> 00:03:52.780 You said Y is the area of the domain. 00:03:52.780 --> 00:03:55.700 You probably already in your mind 00:03:55.700 --> 00:04:01.820 did the math saying X sub X is one, minus Y sub 1 is 1. 00:04:01.820 --> 00:04:05.740 1 plus 1 is two, so the two part [INAUDIBLE]. 00:04:05.740 --> 00:04:08.970 OK, so what did we do with it? 00:04:08.970 --> 00:04:10.221 We just stared at it? 00:04:10.221 --> 00:04:10.720 No. 00:04:10.720 --> 00:04:11.845 We didn't just stare at it. 00:04:11.845 --> 00:04:14.470 We did something nice with it last time. 00:04:14.470 --> 00:04:21.260 We proved that, finally, that the area, this radius R 00:04:21.260 --> 00:04:23.740 will be pi R squared, and we also 00:04:23.740 --> 00:04:28.100 proved that the area [INAUDIBLE] is what? 00:04:28.100 --> 00:04:30.655 I'm testing you to see if you remember. 00:04:30.655 --> 00:04:31.280 STUDENT: AB pi. 00:04:31.280 --> 00:04:31.620 PROFESSOR: AB pi. 00:04:31.620 --> 00:04:32.340 Very good. 00:04:32.340 --> 00:04:33.500 Or pi AB. 00:04:33.500 --> 00:04:38.030 It's more, I like it the way you said it, AB pi, 00:04:38.030 --> 00:04:41.810 because pi is a transcendental number, and you go around 00:04:41.810 --> 00:04:44.590 and it's like partly variable to put at the end. 00:04:44.590 --> 00:04:47.490 And the real numbers that could be anything, 00:04:47.490 --> 00:04:54.630 so [INAUDIBLE] they are the semi axes of the ellipse. 00:04:54.630 --> 00:04:59.470 So we gain new knowledge and we are ready to move forward. 00:04:59.470 --> 00:05:02.220 And we're going to move forward to something 00:05:02.220 --> 00:05:10.590 called section 13.5, which is the surface integral. 00:05:10.590 --> 00:05:12.660 We will come back to Green's Theorem 00:05:12.660 --> 00:05:15.530 because there are generalizations 00:05:15.530 --> 00:05:18.680 of the Green's Theorem to more complicate the case. 00:05:18.680 --> 00:05:21.180 But in order to see those, we have 00:05:21.180 --> 00:05:23.650 to learn a little bit more. 00:05:23.650 --> 00:05:27.610 In mathematics, you need to know many things, many pieces 00:05:27.610 --> 00:05:31.840 of the puzzle, and then you put them together 00:05:31.840 --> 00:05:34.280 to get the whole picture. 00:05:34.280 --> 00:05:37.650 All right, so what is 13.5 about? 00:05:37.650 --> 00:05:39.220 This is just review. 00:05:39.220 --> 00:05:46.900 13.5, if should be looking like a friend, old friend, to you. 00:05:46.900 --> 00:05:48.624 And I'll show you in a minute why this 00:05:48.624 --> 00:05:49.915 is called the surface integral. 00:05:49.915 --> 00:05:54.440 00:05:54.440 --> 00:05:57.950 I saw that US natives don't pronounce integral, 00:05:57.950 --> 00:05:59.640 they pronounce in-negral. 00:05:59.640 --> 00:06:04.410 And everybody that I heard in romance language-speaking 00:06:04.410 --> 00:06:10.230 countries like Spanish, Italian, Portuguese, 00:06:10.230 --> 00:06:13.750 they put the T there out, very visibly. 00:06:13.750 --> 00:06:16.705 So it doesn't matter. 00:06:16.705 --> 00:06:19.510 Even some accent difference in different parts 00:06:19.510 --> 00:06:22.046 of the United States pronounce it differently. 00:06:22.046 --> 00:06:25.040 So what is the surface integral about? 00:06:25.040 --> 00:06:32.250 It's about integrating a smooth function, not a vector value, 00:06:32.250 --> 00:06:34.530 but a real value function. 00:06:34.530 --> 00:06:40.820 Let's say you have G or XY being a nice interglobal function 00:06:40.820 --> 00:06:46.350 over some surfaces. 00:06:46.350 --> 00:06:50.230 And you say, I'm going to take it, 00:06:50.230 --> 00:07:03.360 double integral, over S of GDS, where DS will be area level. 00:07:03.360 --> 00:07:08.600 00:07:08.600 --> 00:07:11.840 I had a student one time who looked at two different books 00:07:11.840 --> 00:07:16.210 and said, I have a problem with this, [INAUDIBLE]. 00:07:16.210 --> 00:07:23.330 In one book it shows a big, fat snake over S. 00:07:23.330 --> 00:07:25.965 And in another book, a double integral over it, 00:07:25.965 --> 00:07:28.820 and I don't know which one it is because I don't understand. 00:07:28.820 --> 00:07:32.370 No matter how you denote it, it's still a double integral. 00:07:32.370 --> 00:07:33.260 You know why? 00:07:33.260 --> 00:07:35.230 Because it's an integral over a surface. 00:07:35.230 --> 00:07:41.420 The same thing, integral over a surface or a domain plane, 00:07:41.420 --> 00:07:45.050 or anything two-dimensional will be a double integral. 00:07:45.050 --> 00:07:47.510 So it doesn't matter how you denote it. 00:07:47.510 --> 00:07:49.820 In the end, it's going to be a double integral. 00:07:49.820 --> 00:07:54.420 Now, what in the world do we mean by that? 00:07:54.420 --> 00:07:56.810 DS is an old friend of yours, and I don't know 00:07:56.810 --> 00:07:59.930 if you remember him at all. 00:07:59.930 --> 00:08:08.300 He was infinitesimal element on some curved or linear patch. 00:08:08.300 --> 00:08:11.650 Imagine your favorite surface. 00:08:11.650 --> 00:08:13.720 Let's assume it's a graph. 00:08:13.720 --> 00:08:16.450 It doesn't have to be a graph, but let's assume it's a graph. 00:08:16.450 --> 00:08:19.700 And that's your favorite surface S. 00:08:19.700 --> 00:08:23.170 And then you draw coordinate lines, 00:08:23.170 --> 00:08:25.716 and you are looking at a patch. 00:08:25.716 --> 00:08:28.580 00:08:28.580 --> 00:08:32.080 And this patch looks small, but it's not small enough. 00:08:32.080 --> 00:08:35.030 I want this to be infinitesimally small. 00:08:35.030 --> 00:08:41.370 Imagine that these curvature lines become closer and closer 00:08:41.370 --> 00:08:42.400 to one another. 00:08:42.400 --> 00:08:46.000 And then we look in the directions of DX and DY, 00:08:46.000 --> 00:08:48.980 and then you say, wait a minute, I'm not in plane. 00:08:48.980 --> 00:08:54.920 If I were in plane, DA will be DX, DY. 00:08:54.920 --> 00:08:59.240 If you work with [INAUDIBLE], I will be DX with DY. 00:08:59.240 --> 00:09:00.910 So we've matched the orientation. 00:09:00.910 --> 00:09:02.720 If you would change DY, [INAUDIBLE] 00:09:02.720 --> 00:09:04.760 put the minus in front. 00:09:04.760 --> 00:09:08.310 But this happens because-- thank God this 00:09:08.310 --> 00:09:16.080 will be a rectangular 1 patch in plane, in the plane of 2. 00:09:16.080 --> 00:09:18.300 But what if you were on the surface? 00:09:18.300 --> 00:09:21.930 On the surface, you don't have this animal. 00:09:21.930 --> 00:09:25.570 You will have-- which animal-- I'm testing your knowledge. 00:09:25.570 --> 00:09:27.430 I'm doing review with you. 00:09:27.430 --> 00:09:29.140 For sure, you will see something that 00:09:29.140 --> 00:09:31.250 involves the S in the final. 00:09:31.250 --> 00:09:33.840 Have you started browsing through those finals 00:09:33.840 --> 00:09:34.750 I sent you? 00:09:34.750 --> 00:09:37.320 Just out of curiosity. 00:09:37.320 --> 00:09:40.960 And do they look awful to you? 00:09:40.960 --> 00:09:42.090 They look awful to you. 00:09:42.090 --> 00:09:43.060 Come on. 00:09:43.060 --> 00:09:45.070 I'm going to work with you on some of those. 00:09:45.070 --> 00:09:47.500 I don't want you to have-- I don't want 00:09:47.500 --> 00:09:49.190 you to be afraid of this final. 00:09:49.190 --> 00:09:51.200 Because compared to other exams that you'll 00:09:51.200 --> 00:09:54.080 have in other courses, where a lot of memorization 00:09:54.080 --> 00:09:57.660 is emphasized, this should not be a problem. 00:09:57.660 --> 00:10:00.800 So you could go over the types of problems 00:10:00.800 --> 00:10:04.380 that are significant in this course, 00:10:04.380 --> 00:10:08.420 you will not have any-- you shouldn't have any problem. 00:10:08.420 --> 00:10:10.810 And I sent you three samples. 00:10:10.810 --> 00:10:13.740 Didn't I send you three samples with solutions? 00:10:13.740 --> 00:10:17.312 Those are going to help you once you read the exam 00:10:17.312 --> 00:10:19.900 and you can go ahead and try the exam 00:10:19.900 --> 00:10:22.040 or go ahead, read the solutions. 00:10:22.040 --> 00:10:26.780 If I give you more of that, then you should be doctors in those, 00:10:26.780 --> 00:10:30.320 and you would be able to solve them yourselves. 00:10:30.320 --> 00:10:33.240 What about this one? 00:10:33.240 --> 00:10:34.570 This is not DA. 00:10:34.570 --> 00:10:37.960 It's a DA times something. 00:10:37.960 --> 00:10:44.070 There is some factor in front of that, and why is that? 00:10:44.070 --> 00:10:46.600 In case of Z equals F of X and Y, 00:10:46.600 --> 00:10:49.430 you should know that by heart, and I know that some of you 00:10:49.430 --> 00:10:50.400 know it. 00:10:50.400 --> 00:10:52.495 You just have to ring the bell, and I'll 00:10:52.495 --> 00:10:53.850 start ringing the bell. 00:10:53.850 --> 00:10:55.870 Look at my first step. 00:10:55.870 --> 00:10:57.783 And now you know, right? 00:10:57.783 --> 00:10:59.470 STUDENT: [INAUDIBLE] 1-- 00:10:59.470 --> 00:11:00.650 PROFESSOR: I start with 1. 00:11:00.650 --> 00:11:01.700 You said it right. 00:11:01.700 --> 00:11:02.830 1 plus-- 00:11:02.830 --> 00:11:03.590 STUDENT: F of X. 00:11:03.590 --> 00:11:04.815 PROFESSOR: F of X-- 00:11:04.815 --> 00:11:06.180 STUDENT: F squared. 00:11:06.180 --> 00:11:07.637 PROFESSOR: Squared plus-- 00:11:07.637 --> 00:11:08.470 STUDENT: [INAUDIBLE] 00:11:08.470 --> 00:11:09.620 PROFESSOR: --SY squared. 00:11:09.620 --> 00:11:11.270 So this what you're doing. 00:11:11.270 --> 00:11:13.020 What are you going to do? 00:11:13.020 --> 00:11:14.690 You're going to do wait a minute. 00:11:14.690 --> 00:11:18.130 This animal of mine, that looks so scary, 00:11:18.130 --> 00:11:21.431 this is nothing but what? 00:11:21.431 --> 00:11:27.300 It's the same thing as, not the picture, my picture. 00:11:27.300 --> 00:11:32.150 It's going to be double integral over a plane or domain D. 00:11:32.150 --> 00:11:34.260 Well, I just said goodbye to the picture, 00:11:34.260 --> 00:11:37.250 but I find you are really smart. 00:11:37.250 --> 00:11:41.150 I would have drawn the [INAUDIBLE] of a picture here. 00:11:41.150 --> 00:11:43.760 This is S and this is D. What is D? 00:11:43.760 --> 00:11:46.810 It's the projection, projects the shadow. 00:11:46.810 --> 00:11:51.100 The projection of S on the plane XY when 00:11:51.100 --> 00:11:53.920 I have to deal with a graph. 00:11:53.920 --> 00:11:57.380 So when I have to deal with a graph, my life is really easy. 00:11:57.380 --> 00:12:06.020 And I said I'd get double integral over D of G of God 00:12:06.020 --> 00:12:11.680 knows what in the end will be a function of X and Y. OK? 00:12:11.680 --> 00:12:15.900 And here I'm going to have square root of this animal. 00:12:15.900 --> 00:12:18.330 Let me change it, F sub X squared 00:12:18.330 --> 00:12:20.930 like-- because in this one it is like that. 00:12:20.930 --> 00:12:21.500 Plus 1. 00:12:21.500 --> 00:12:24.280 It doesn't matter where I put the 1. 00:12:24.280 --> 00:12:25.120 DXDY. 00:12:25.120 --> 00:12:31.060 DXDY will be like the area of an infinitesimally small rectangle 00:12:31.060 --> 00:12:34.250 based on displacement DX and displacement DY 00:12:34.250 --> 00:12:35.370 and disintegration. 00:12:35.370 --> 00:12:37.350 So this is DA. 00:12:37.350 --> 00:12:42.820 Make the distinction between the DA and the DX. 00:12:42.820 --> 00:12:46.950 Can I draw the two animals? 00:12:46.950 --> 00:12:50.750 Let me try again. 00:12:50.750 --> 00:12:55.490 So you have the direction of X and Y. 00:12:55.490 --> 00:12:59.300 You have to be imaginative and see that some coordinate lines 00:12:59.300 --> 00:13:02.990 are [INAUDIBLE] for fixing Y. 00:13:02.990 --> 00:13:06.880 When I fix Y, I sliced a lot like that very nicely. 00:13:06.880 --> 00:13:11.710 That's the same piece of cheese that I've been dreaming 00:13:11.710 --> 00:13:13.150 because I didn't have lunch. 00:13:13.150 --> 00:13:15.460 I was too busy not to have any lunch. 00:13:15.460 --> 00:13:17.400 So you slice it up like that where 00:13:17.400 --> 00:13:20.590 Y equals constant to slice it up like that for X 00:13:20.590 --> 00:13:21.700 equals constant. 00:13:21.700 --> 00:13:24.950 What you get are so-called coordinate lines. 00:13:24.950 --> 00:13:27.440 So the coordinate lines are [INAUDIBLE]. 00:13:27.440 --> 00:13:32.085 Y equals my zeros, and X equals the zeros. 00:13:32.085 --> 00:13:35.810 And when they get to be many dense and refined, 00:13:35.810 --> 00:13:42.310 your curvilinear element is this-- between two curves 00:13:42.310 --> 00:13:46.010 like this two curves like that. 00:13:46.010 --> 00:13:48.190 Shrunk in the limit. 00:13:48.190 --> 00:13:50.640 It's an infinitesimal element. 00:13:50.640 --> 00:13:54.470 This shadow is going to be a rectangle. 00:13:54.470 --> 00:13:55.910 Say that again, Magdalena. 00:13:55.910 --> 00:13:59.410 This is not just delta X and delta Y. 00:13:59.410 --> 00:14:03.650 This is DX and DY because I shrink them 00:14:03.650 --> 00:14:06.660 until it become infinitesimally small. 00:14:06.660 --> 00:14:11.160 So you can imagine, which one is bigger? 00:14:11.160 --> 00:14:13.984 DS is bigger, or DA is bigger? 00:14:13.984 --> 00:14:14.900 STUDENT: DS is bigger. 00:14:14.900 --> 00:14:16.080 PROFESSOR: DS is bigger. 00:14:16.080 --> 00:14:16.920 DS is bigger. 00:14:16.920 --> 00:14:18.640 And can I see it's true? 00:14:18.640 --> 00:14:19.140 Yes. 00:14:19.140 --> 00:14:22.875 Because for God's sake, this is greater than 1, right? 00:14:22.875 --> 00:14:28.110 And if I multiply the little orange area, by that, 00:14:28.110 --> 00:14:32.740 I'm going to get this, which is greater than 1. 00:14:32.740 --> 00:14:38.140 They could be equal when both would be plainer, right? 00:14:38.140 --> 00:14:41.320 If you have a plane or surface on top of a plane or surface, 00:14:41.320 --> 00:14:43.160 then you have two tiny rectangles 00:14:43.160 --> 00:14:48.110 and you have like a prism between them, goes down. 00:14:48.110 --> 00:14:50.030 But in general, the curve in your [INAUDIBLE] 00:14:50.030 --> 00:14:52.950 here-- let me make him more curvilinear. 00:14:52.950 --> 00:14:55.310 He looks so-- so square. 00:14:55.310 --> 00:15:01.460 But he's between two lines, but he's a curvilinear. 00:15:01.460 --> 00:15:07.010 Dinah says that he belongs to a curved surface, not a flat one. 00:15:07.010 --> 00:15:08.020 All right. 00:15:08.020 --> 00:15:11.240 When he could be flat, these guys go away. 00:15:11.240 --> 00:15:12.640 Zero and zero. 00:15:12.640 --> 00:15:15.380 And that would be it. 00:15:15.380 --> 00:15:23.750 If somebody else, they-- well, this is hard to imagine, 00:15:23.750 --> 00:15:29.970 but what if it could be a tiny-- this 00:15:29.970 --> 00:15:32.030 would not be curvilinear, right? 00:15:32.030 --> 00:15:37.290 But it would be something like a rectangular patch of a plane. 00:15:37.290 --> 00:15:40.170 You have a grid in that plane. 00:15:40.170 --> 00:15:46.400 And then it's just-- DS would be itself a rectangle. 00:15:46.400 --> 00:15:48.860 When you project that rectangle here, 00:15:48.860 --> 00:15:51.840 it will still be a rectangle. 00:15:51.840 --> 00:15:54.900 When we were little-- I mean, little, we were in K-12, 00:15:54.900 --> 00:15:59.230 we're smart in math better than other people in class-- 00:15:59.230 --> 00:16:03.260 did you ever have to do anything with the two areas? 00:16:03.260 --> 00:16:04.680 I did. 00:16:04.680 --> 00:16:05.670 This was the shadow. 00:16:05.670 --> 00:16:08.550 The projection in this was that [INAUDIBLE]. 00:16:08.550 --> 00:16:11.510 And do you know what the relationship 00:16:11.510 --> 00:16:14.190 would be if I have a plane. 00:16:14.190 --> 00:16:17.520 I'm doing that for-- actually, I'm doing that for Casey 00:16:17.520 --> 00:16:20.030 because she has something similar to that. 00:16:20.030 --> 00:16:22.910 So imagine that you have to project 00:16:22.910 --> 00:16:27.650 a rectangle that's in plane to a rectangle that is the shadow. 00:16:27.650 --> 00:16:30.340 The rectangle is on the ground. 00:16:30.340 --> 00:16:33.560 The flat ground. 00:16:33.560 --> 00:16:35.970 What's the relationship between the two ends? 00:16:35.970 --> 00:16:40.470 00:16:40.470 --> 00:16:42.879 STUDENT: [INAUDIBLE] 00:16:42.879 --> 00:16:44.670 PROFESSOR: No matter what it is, but assume 00:16:44.670 --> 00:16:48.120 it's like a rectangle up here and the shadow is also 00:16:48.120 --> 00:16:50.205 a rectangle down here. 00:16:50.205 --> 00:16:52.290 Obviously, the rectangle down here, the shadow 00:16:52.290 --> 00:16:55.810 will be much smaller than this because this is oblique. 00:16:55.810 --> 00:16:56.930 It's an oblique. 00:16:56.930 --> 00:16:59.540 And assume that I have this plane making 00:16:59.540 --> 00:17:03.550 an angle, a fixed angle with this laying on the table. 00:17:03.550 --> 00:17:05.325 STUDENT: [INAUDIBLE] 00:17:05.325 --> 00:17:06.200 PROFESSOR: Excellent. 00:17:06.200 --> 00:17:06.991 STUDENT: --cosine-- 00:17:06.991 --> 00:17:08.720 PROFESSOR: Which one is cosine of what? 00:17:08.720 --> 00:17:12.069 So the S would be the the equal sign of theta, 00:17:12.069 --> 00:17:18.619 or the A will be the S cosine of theta? 00:17:18.619 --> 00:17:19.940 STUDENT: [INAUDIBLE] DA. 00:17:19.940 --> 00:17:22.560 PROFESSOR: DA is the S cosine of theta, a very smart 00:17:22.560 --> 00:17:23.060 [INAUDIBLE]. 00:17:23.060 --> 00:17:24.680 How does she know [INAUDIBLE]? 00:17:24.680 --> 00:17:25.868 STUDENT: Because it's got to be less than one. 00:17:25.868 --> 00:17:27.410 PROFESSOR: It's less than one, right? 00:17:27.410 --> 00:17:29.480 Cosine theta is between zero and one, 00:17:29.480 --> 00:17:31.585 so you think which one is less. 00:17:31.585 --> 00:17:33.630 All right, very good. 00:17:33.630 --> 00:17:36.700 So when you have a simple example like that, 00:17:36.700 --> 00:17:40.310 you were back to your K-12, and you 00:17:40.310 --> 00:17:42.130 were happy-- I just meant we were 00:17:42.130 --> 00:17:44.580 avoiding three years of exams. 00:17:44.580 --> 00:17:47.580 We only have [INAUDIBLE]. 00:17:47.580 --> 00:17:49.990 But now exams became serious, and look. 00:17:49.990 --> 00:17:52.970 This is curvilinear elemental variant. 00:17:52.970 --> 00:17:58.920 So let me write it how people call the S's then. 00:17:58.920 --> 00:18:01.170 Some people call it curvilinear elemental variant. 00:18:01.170 --> 00:18:01.750 Yeah? 00:18:01.750 --> 00:18:06.280 Many engineers I talk to do that. 00:18:06.280 --> 00:18:09.295 00:18:09.295 --> 00:18:11.670 Now, I think we should just call it surface area element. 00:18:11.670 --> 00:18:14.700 00:18:14.700 --> 00:18:18.213 [? I'm ?] a physicist, so you also say surface area element. 00:18:18.213 --> 00:18:22.550 So I think we should just learn each other's language. 00:18:22.550 --> 00:18:24.570 We are doing the same things. 00:18:24.570 --> 00:18:27.930 We just-- we have a language barrier between-- it's 00:18:27.930 --> 00:18:31.100 not writing interdisciplinary, so if we could establish 00:18:31.100 --> 00:18:35.300 a little bit more work in common, because there are so 00:18:35.300 --> 00:18:38.380 many applications to engineering of this thing, 00:18:38.380 --> 00:18:41.150 you have no idea yet. 00:18:41.150 --> 00:18:47.150 OK, let's pick a problem like the ones we wrote in the book, 00:18:47.150 --> 00:18:50.680 and see how hard it gets. 00:18:50.680 --> 00:18:53.640 It shouldn't get very hard. 00:18:53.640 --> 00:18:57.030 I'll start with one, the only one, that is naturally 00:18:57.030 --> 00:19:00.320 coming to your mind right now, which would 00:19:00.320 --> 00:19:03.690 be the one where G would be 1. 00:19:03.690 --> 00:19:06.458 Somebody has to tell me what that would be. 00:19:06.458 --> 00:19:10.020 00:19:10.020 --> 00:19:13.800 So guys, what if G would be 1? 00:19:13.800 --> 00:19:14.915 STUDENT: [INAUDIBLE] 00:19:14.915 --> 00:19:15.790 PROFESSOR: Very good. 00:19:15.790 --> 00:19:17.520 It would be the A of the surface. 00:19:17.520 --> 00:19:21.740 I'm going to look for some simple application. 00:19:21.740 --> 00:19:22.550 Nothing is simple. 00:19:22.550 --> 00:19:27.600 Why did we make this problem, this book, so complicated? 00:19:27.600 --> 00:19:28.480 OK, it' s good. 00:19:28.480 --> 00:19:35.360 We can pick-- I can make up a problem like this one. 00:19:35.360 --> 00:19:39.440 00:19:39.440 --> 00:19:41.420 But I can do a better job. 00:19:41.420 --> 00:19:43.490 I can give you an better example. 00:19:43.490 --> 00:19:46.625 I'm looking at the example 1 in section 13.5. 00:19:46.625 --> 00:19:49.035 I'll give you something like that 00:19:49.035 --> 00:19:51.180 if I were to write an exam 1. 00:19:51.180 --> 00:19:56.270 I put on it something like Z equals 00:19:56.270 --> 00:19:58.450 X squared plus 1 squared. 00:19:58.450 --> 00:20:02.130 You know is my favorite eggshell which is a [INAUDIBLE]. 00:20:02.130 --> 00:20:16.490 00:20:16.490 --> 00:20:19.820 And somebody says, I'm not interested 00:20:19.820 --> 00:20:25.430 in the whole surface, which is infinitely large. 00:20:25.430 --> 00:20:29.880 I'm only interested in a piece of a surface that 00:20:29.880 --> 00:20:39.820 is above the disk D of center O and radius 1. 00:20:39.820 --> 00:20:41.210 So say, what, Magdalena? 00:20:41.210 --> 00:20:48.080 Say that I want just that part of the surface 00:20:48.080 --> 00:20:54.030 that he's sitting above the disk of center O and radius 1. 00:20:54.030 --> 00:20:58.420 And I want to know how to set up the surface integral. 00:20:58.420 --> 00:21:02.194 Set up main surface area integral. 00:21:02.194 --> 00:21:07.420 00:21:07.420 --> 00:21:10.000 And of course, when you first see that you freak out 00:21:10.000 --> 00:21:13.280 for a second, and then you say, no, no, that's not a problem. 00:21:13.280 --> 00:21:15.770 I know how to do that. 00:21:15.770 --> 00:21:18.810 So example 1 out of this section would 00:21:18.810 --> 00:21:27.010 be a double integral over your S. You have to call it names. 00:21:27.010 --> 00:21:30.400 S. 1 instead of G and DS. 00:21:30.400 --> 00:21:33.530 00:21:33.530 --> 00:21:35.590 But then you say wait a minute. 00:21:35.590 --> 00:21:40.380 I know that is true, but I have to change it accordingly. 00:21:40.380 --> 00:21:42.140 The same thing is here. 00:21:42.140 --> 00:21:47.930 So I'm going to have it over D. And D is the shadow, 00:21:47.930 --> 00:21:52.770 DS is the plane of what? 00:21:52.770 --> 00:21:55.440 Of 1 times. 00:21:55.440 --> 00:21:59.790 I know I'm silly saying 1 times, but that's what it is. 00:21:59.790 --> 00:22:07.400 Square root of-- S of X squared plus S of Y squared plus 1. 00:22:07.400 --> 00:22:13.930 DS, DY or DA as Rachel said, somebody said. 00:22:13.930 --> 00:22:15.457 Aaron said. 00:22:15.457 --> 00:22:17.415 I don't know, you just whispered, I should say. 00:22:17.415 --> 00:22:20.330 00:22:20.330 --> 00:22:21.550 All right. 00:22:21.550 --> 00:22:28.050 So first of all, this looks a little bit bad. 00:22:28.050 --> 00:22:31.000 It makes me a little bit nervous. 00:22:31.000 --> 00:22:34.890 But in the end, with your help, I'm going to do it. 00:22:34.890 --> 00:22:37.941 And I'm going to do it by using what kind of coordinates? 00:22:37.941 --> 00:22:38.440 I'm-- 00:22:38.440 --> 00:22:39.560 STUDENT: [INAUDIBLE] 00:22:39.560 --> 00:22:42.000 PROFESSOR: Former coordinates of the Y and Z. 00:22:42.000 --> 00:22:43.260 It would be a killer. 00:22:43.260 --> 00:22:48.800 Double, double, square root of 1 plus-- who's telling me 00:22:48.800 --> 00:22:49.933 what's coming next? 00:22:49.933 --> 00:22:50.880 STUDENT: 4X squared. 00:22:50.880 --> 00:22:52.610 4X squared, excellent. 00:22:52.610 --> 00:22:55.830 4R squared, you say. 00:22:55.830 --> 00:22:56.947 STUDENT: [INAUDIBLE] 00:22:56.947 --> 00:22:57.530 PROFESSOR: OK. 00:22:57.530 --> 00:23:01.010 Let me write it with X and Y, and then 00:23:01.010 --> 00:23:02.670 realize that this is our square. 00:23:02.670 --> 00:23:04.670 How about that? 00:23:04.670 --> 00:23:08.140 And then I have DX, DY over the domain D, 00:23:08.140 --> 00:23:12.340 and now I finally become smart and say I just 00:23:12.340 --> 00:23:13.730 fooled around here. 00:23:13.730 --> 00:23:17.450 I want to do it in four coordinates finally. 00:23:17.450 --> 00:23:22.210 And that means I'll say zero to 2 pi for theta. 00:23:22.210 --> 00:23:26.130 So that theta will be the last of the [INAUDIBLE]. 00:23:26.130 --> 00:23:28.520 R will be from zero to 1. 00:23:28.520 --> 00:23:33.110 00:23:33.110 --> 00:23:35.030 And So what? 00:23:35.030 --> 00:23:40.630 This is an ugly, fairly ugly, I just [INAUDIBLE]. 00:23:40.630 --> 00:23:42.580 I don't know what I'm going to do yet. 00:23:42.580 --> 00:23:44.590 I reduced our confusion, right? 00:23:44.590 --> 00:23:46.480 But I'm not done. 00:23:46.480 --> 00:23:47.412 STUDENT: R. 00:23:47.412 --> 00:23:51.890 PROFESSOR: R. Never forget it. 00:23:51.890 --> 00:23:56.940 So if I didn't have this R, I would be horrible. 00:23:56.940 --> 00:23:58.580 Why would it be horrible? 00:23:58.580 --> 00:24:01.250 Imagine you couldn't have the R. 00:24:01.250 --> 00:24:02.550 STUDENT: [INAUDIBLE] 00:24:02.550 --> 00:24:05.860 PROFESSOR: We have to look that this thing in integral table 00:24:05.860 --> 00:24:10.190 or some-- use the calculator, which we are not allowed 00:24:10.190 --> 00:24:12.110 to do in this kind of course. 00:24:12.110 --> 00:24:14.350 So what do we do? 00:24:14.350 --> 00:24:16.890 We say it's a new substitution. 00:24:16.890 --> 00:24:19.070 I have an R. That's a blessing. 00:24:19.070 --> 00:24:23.200 So U equals 4 squared plus 1. 00:24:23.200 --> 00:24:29.890 DU equals 8R, DR. I think R, DR is a block. 00:24:29.890 --> 00:24:34.890 And I know that's what I'm going to do is a U substitution. 00:24:34.890 --> 00:24:36.260 And I'm almost there. 00:24:36.260 --> 00:24:39.280 00:24:39.280 --> 00:24:41.380 It's a pretty good example, but the one 00:24:41.380 --> 00:24:48.160 you have as a first example in this section, 13.5, 00:24:48.160 --> 00:24:50.070 it's a little bit too computational. 00:24:50.070 --> 00:24:52.760 It's not smart at all. 00:24:52.760 --> 00:24:57.810 It has a similar function over a rectangle, something like that. 00:24:57.810 --> 00:24:59.560 But it's a little bit too confrontational. 00:24:59.560 --> 00:25:01.290 We are looking for something that 00:25:01.290 --> 00:25:04.920 is not going-- examples that are going to be easy to do and not 00:25:04.920 --> 00:25:09.380 involve too much heavy competition by him, because you 00:25:09.380 --> 00:25:10.640 do everything by him. 00:25:10.640 --> 00:25:14.730 Not-- like you don't have a calculator, et cetera. 00:25:14.730 --> 00:25:19.590 And the exam is very limited in time, DU over 8. 00:25:19.590 --> 00:25:22.580 So you say OK, I'm know what that is. 00:25:22.580 --> 00:25:28.230 That's going to be the A of S. And that is going to be 2 pi. 00:25:28.230 --> 00:25:32.380 Why can't I be so confident and pull 2 pi out? 00:25:32.380 --> 00:25:33.340 STUDENT: [INAUDIBLE] 00:25:33.340 --> 00:25:36.310 PROFESSOR: Because there is no dependence on theta. 00:25:36.310 --> 00:25:38.490 All right? 00:25:38.490 --> 00:25:41.570 So I have that one. 00:25:41.570 --> 00:25:46.340 And then you go all right, integral, square of you 00:25:46.340 --> 00:25:52.290 times the U over 8-- 1 over 8DU. 00:25:52.290 --> 00:25:56.430 And I have to be careful because when R is zero-- 00:25:56.430 --> 00:26:00.230 if I put zero and 1 here like some of my students, 00:26:00.230 --> 00:26:04.780 I'm dead meat, because I'm going to lose a lot of credit, right? 00:26:04.780 --> 00:26:06.570 So I have to pay attention. 00:26:06.570 --> 00:26:09.100 R is 0, and U equals? 00:26:09.100 --> 00:26:09.890 STUDENT: 1. 00:26:09.890 --> 00:26:10.730 PROFESSOR: 1. 00:26:10.730 --> 00:26:12.970 R equals 1. 00:26:12.970 --> 00:26:16.760 U equals 5. 00:26:16.760 --> 00:26:23.120 And I worked this out and I should be done. 00:26:23.120 --> 00:26:27.560 And that's-- you should expect something like that. 00:26:27.560 --> 00:26:33.630 Nice, not computational, you kind of looking. 00:26:33.630 --> 00:26:36.800 What is integral of square of U? 00:26:36.800 --> 00:26:38.460 STUDENT: [INAUDIBLE] 00:26:38.460 --> 00:26:41.690 PROFESSOR: So you have-- you do the three halves, 00:26:41.690 --> 00:26:43.940 and you pull out the 2/3, right? 00:26:43.940 --> 00:26:45.160 That's what you do. 00:26:45.160 --> 00:26:49.110 And then you go between U equals 1 down, and U equals 5 up. 00:26:49.110 --> 00:26:51.895 And it's like one of those examples we worked before. 00:26:51.895 --> 00:26:53.270 Remember, and more important, you 00:26:53.270 --> 00:26:58.460 had something like that for surface area? 00:26:58.460 --> 00:27:00.230 Oh, my god. 00:27:00.230 --> 00:27:01.240 4 over 8. 00:27:01.240 --> 00:27:03.108 How much is 4 over 8? 00:27:03.108 --> 00:27:04.020 STUDENT: [INAUDIBLE] 00:27:04.020 --> 00:27:04.853 PROFESSOR: One half. 00:27:04.853 --> 00:27:08.760 00:27:08.760 --> 00:27:09.260 Right? 00:27:09.260 --> 00:27:15.330 So we will have 1 over 6, and write pi times 5 00:27:15.330 --> 00:27:20.280 to the three halves minus 1. 00:27:20.280 --> 00:27:21.600 So do I like it? 00:27:21.600 --> 00:27:22.970 I would leave it like that. 00:27:22.970 --> 00:27:23.470 I'm fine. 00:27:23.470 --> 00:27:24.580 I'll forget about it. 00:27:24.580 --> 00:27:26.640 I have people who care. 00:27:26.640 --> 00:27:32.042 I don't care how some people write it-- 5 with 5 minus 1 00:27:32.042 --> 00:27:33.500 because they think it looks better. 00:27:33.500 --> 00:27:34.000 It doesn't. 00:27:34.000 --> 00:27:37.150 That's the scientific equation, and I'm fine with it. 00:27:37.150 --> 00:27:38.416 Right? 00:27:38.416 --> 00:27:39.260 OK. 00:27:39.260 --> 00:27:43.320 So expect something like-- maybe I'm talking too much, 00:27:43.320 --> 00:27:46.750 but maybe it's a good thing to tell you what to expect 00:27:46.750 --> 00:27:48.500 because we have to [INAUDIBLE]. 00:27:48.500 --> 00:27:50.500 At the same time, we're teaching new things 00:27:50.500 --> 00:27:54.970 as staff instructors doing review of what's important. 00:27:54.970 --> 00:28:03.050 I'm thinking if I'm doing things right and at the same pace, 00:28:03.050 --> 00:28:10.110 I should be finished with chapter 13 00:28:10.110 --> 00:28:12.230 at the end of next week. 00:28:12.230 --> 00:28:15.460 Because after 13.5, we have 13.6 which 00:28:15.460 --> 00:28:18.125 is a generalization of Green's Theorem. 00:28:18.125 --> 00:28:20.810 13.6 as you recall is called Stokes' Theorem. 00:28:20.810 --> 00:28:25.090 13.7 is also a generalization of Green's Theorem. 00:28:25.090 --> 00:28:27.190 And they are all related. 00:28:27.190 --> 00:28:31.500 It's like the trinity on [INAUDIBLE]. 00:28:31.500 --> 00:28:33.170 That's the Divergence Theorem. 00:28:33.170 --> 00:28:37.970 That is the last section, 13.7, Divergence Theorem. 00:28:37.970 --> 00:28:41.580 So if I am going at the right pace, 00:28:41.580 --> 00:28:44.090 by-- what is next wee on Thursday? 00:28:44.090 --> 00:28:47.995 The-- 23rd? 00:28:47.995 --> 00:28:50.840 I should be more or less done with the chapter. 00:28:50.840 --> 00:28:54.280 And I'm thinking I have all the time in the world 00:28:54.280 --> 00:28:57.410 to review with you from that moment on. 00:28:57.410 --> 00:28:59.490 In which sense are we going to review? 00:28:59.490 --> 00:29:04.740 We are going to review by solving past finals. 00:29:04.740 --> 00:29:05.610 Right? 00:29:05.610 --> 00:29:08.881 That's what we are-- that's what I'm planning to do. 00:29:08.881 --> 00:29:12.380 I'm going to erase this and move on to something 00:29:12.380 --> 00:29:15.340 more spectacular. 00:29:15.340 --> 00:29:16.250 Many-- OK. 00:29:16.250 --> 00:29:18.310 This second part that I want to teach you 00:29:18.310 --> 00:29:23.030 now about, many instructors in regular courses 00:29:23.030 --> 00:29:28.590 just skip it because they do not want to teach you-- not you, 00:29:28.590 --> 00:29:29.610 you are honor students. 00:29:29.610 --> 00:29:31.318 But they don't want to teach the students 00:29:31.318 --> 00:29:36.070 about some more general ways to look at a surface. 00:29:36.070 --> 00:29:40.235 Remember, guys, a surface that is written like that 00:29:40.235 --> 00:29:43.110 is called a graph. 00:29:43.110 --> 00:29:47.560 But not all the surfaces were graphs. 00:29:47.560 --> 00:29:56.820 And actually for a surface S, what the most general way 00:29:56.820 --> 00:30:00.058 to represent the presentation would be a parameterization. 00:30:00.058 --> 00:30:05.560 00:30:05.560 --> 00:30:10.450 And I'll do a little bit of a review for those. 00:30:10.450 --> 00:30:17.640 R-- little R or big R-- big R, because that's 00:30:17.640 --> 00:30:20.540 the position vector the way I serve it to you 00:30:20.540 --> 00:30:24.350 on a plate, whether, for curves in space. 00:30:24.350 --> 00:30:29.113 I say that's R of P. And when we moved on curves to surfaces, 00:30:29.113 --> 00:30:34.430 I said you move your path two directions of motion. 00:30:34.430 --> 00:30:37.010 You have two-- what are those called in mechanics? 00:30:37.010 --> 00:30:38.440 Degrees of freedom. 00:30:38.440 --> 00:30:40.570 So you have two degrees of freedom 00:30:40.570 --> 00:30:42.400 like latitude and longitude. 00:30:42.400 --> 00:30:46.960 Then R belongs-- the position vector 00:30:46.960 --> 00:30:52.710 is a function of two variables, and it belongs to R3, 00:30:52.710 --> 00:30:54.320 because it's a vector in R3. 00:30:54.320 --> 00:30:58.220 And want to have-- imagine that my hand is a surface. 00:30:58.220 --> 00:30:59.550 Well, OK. 00:30:59.550 --> 00:31:02.400 This is the position vector, I'm just kind of sweeping my hand, 00:31:02.400 --> 00:31:04.930 going this way, one degree of freedom. 00:31:04.930 --> 00:31:07.330 Or going that way, the other degree of freedom. 00:31:07.330 --> 00:31:10.600 This is what parameterization is. 00:31:10.600 --> 00:31:16.645 So for a sphere, if you want to parameterize the whole sphere-- 00:31:16.645 --> 00:31:20.020 I'll be done in a second. 00:31:20.020 --> 00:31:23.420 I need you to see if you remember 00:31:23.420 --> 00:31:25.400 how to parameterize a sphere. 00:31:25.400 --> 00:31:26.450 I'm testing you. 00:31:26.450 --> 00:31:28.400 I'm mean today. 00:31:28.400 --> 00:31:29.750 So examples. 00:31:29.750 --> 00:31:32.376 Example 1 is parameterize a sphere. 00:31:32.376 --> 00:31:36.660 00:31:36.660 --> 00:31:38.310 Was it hard? 00:31:38.310 --> 00:31:40.550 That was a long time ago, my god. 00:31:40.550 --> 00:31:45.570 X, Y, and Z are what? 00:31:45.570 --> 00:31:47.880 Latitude from Santa Clause. 00:31:47.880 --> 00:31:52.000 Always latitude from the North Pole is 5. 00:31:52.000 --> 00:31:54.390 Longitude is from zero to 5. 00:31:54.390 --> 00:31:56.560 The meridian is zero to 5. 00:31:56.560 --> 00:31:59.850 That was theta, the parameter of theta. 00:31:59.850 --> 00:32:04.940 R was the distance from this to a point. 00:32:04.940 --> 00:32:08.260 But R was allowed to be from-- take many values. 00:32:08.260 --> 00:32:11.870 Now if I'm moving on a sphere of radius 00:32:11.870 --> 00:32:15.634 A-- let me make that radius a just 00:32:15.634 --> 00:32:16.800 to make your life miserable. 00:32:16.800 --> 00:32:19.840 Assume that A would be a sample, A. 00:32:19.840 --> 00:32:22.311 How am I going to write that parameterization? 00:32:22.311 --> 00:32:25.080 STUDENT: X equals A plus [INAUDIBLE]? 00:32:25.080 --> 00:32:28.960 PROFESSOR: A something, A something, A something. 00:32:28.960 --> 00:32:30.150 STUDENT: A [INAUDIBLE] 00:32:30.150 --> 00:32:31.110 PROFESSOR: He is right. 00:32:31.110 --> 00:32:32.067 I have to move on. 00:32:32.067 --> 00:32:32.900 STUDENT: [INAUDIBLE] 00:32:32.900 --> 00:32:34.510 PROFESSOR: Go slow. 00:32:34.510 --> 00:32:37.370 So I have-- the last one-- you were right, 00:32:37.370 --> 00:32:40.250 Buddy, you have the memory of a medical doctor 00:32:40.250 --> 00:32:42.340 and some day you will be a medical doctor. 00:32:42.340 --> 00:32:44.510 Not everybody has a good memory. 00:32:44.510 --> 00:32:49.980 So the way you can do that is, wait a minute, this is pi, 00:32:49.980 --> 00:32:50.670 right? 00:32:50.670 --> 00:32:52.250 This [INAUDIBLE]. 00:32:52.250 --> 00:32:55.330 If you want the Z, you start with that first. 00:32:55.330 --> 00:33:00.588 And since Z is adjacent, you go R, cosine, sine, phi equals 00:33:00.588 --> 00:33:01.800 sine phi. 00:33:01.800 --> 00:33:05.079 Now we started with X because he's worked on this 00:33:05.079 --> 00:33:06.120 and remembers everything. 00:33:06.120 --> 00:33:07.590 He has it memorized. 00:33:07.590 --> 00:33:10.430 Sine phi for both. 00:33:10.430 --> 00:33:13.520 And times what in both cases? 00:33:13.520 --> 00:33:15.310 He's just the guy who's not here. 00:33:15.310 --> 00:33:17.020 So sine phi. 00:33:17.020 --> 00:33:20.820 It helps to memorize N cosine theta, and sine theta. 00:33:20.820 --> 00:33:22.600 Is that really easy to memorize? 00:33:22.600 --> 00:33:26.270 So where phi was the latitude from the North 00:33:26.270 --> 00:33:30.080 Pole between zero and phi, it theta 00:33:30.080 --> 00:33:35.540 was the longitude-- excuse me, guys-- longitude from zero 00:33:35.540 --> 00:33:41.410 to 2 pi, all around one more. 00:33:41.410 --> 00:33:43.920 So you say wait a minute, Magdalena, 00:33:43.920 --> 00:33:45.190 these are Euler's angle. 00:33:45.190 --> 00:33:47.280 What do they call in mechanics? 00:33:47.280 --> 00:33:49.670 I think they call them Euler angles. 00:33:49.670 --> 00:33:53.410 But anyway, for phi theta, we call 00:33:53.410 --> 00:33:55.620 them latitude and longitude. 00:33:55.620 --> 00:33:59.990 I'll replace them, because look, I want R to be in terms of U,V. 00:33:59.990 --> 00:34:02.390 So in mathematics, it's not about location. 00:34:02.390 --> 00:34:05.470 We can call them whatever we want. 00:34:05.470 --> 00:34:10.409 Mathematics is about the freedom to call people names-- no-- 00:34:10.409 --> 00:34:13.760 to call things names and people names-- 00:34:13.760 --> 00:34:15.909 STUDENT: Could U not equal zero? 00:34:15.909 --> 00:34:16.540 PROFESSOR: Who? 00:34:16.540 --> 00:34:17.840 STUDENT: U. 00:34:17.840 --> 00:34:18.530 PROFESSOR: Yes. 00:34:18.530 --> 00:34:19.076 So U can-- 00:34:19.076 --> 00:34:19.909 STUDENT: [INAUDIBLE] 00:34:19.909 --> 00:34:24.520 PROFESSOR: --yeah, but why didn't I write zero? 00:34:24.520 --> 00:34:25.469 Well-- 00:34:25.469 --> 00:34:26.929 STUDENT: [INAUDIBLE] makes sense. 00:34:26.929 --> 00:34:30.330 PROFESSOR: --because, yeah, you can take both. 00:34:30.330 --> 00:34:33.040 If I want to study differentiability, 00:34:33.040 --> 00:34:36.500 I usually have to take it less than and less than and less 00:34:36.500 --> 00:34:39.050 than and less than because we studied differentiability on 00:34:39.050 --> 00:34:40.190 [INAUDIBLE]. 00:34:40.190 --> 00:34:43.310 But right now, I can take them from the North Pole itself 00:34:43.310 --> 00:34:47.090 to the South Pole itself-- so. 00:34:47.090 --> 00:34:50.610 I'm not deleting any meridian. 00:34:50.610 --> 00:34:53.739 If I were-- suppose I were to delete it. 00:34:53.739 --> 00:34:55.960 By the way, what does this mean? 00:34:55.960 --> 00:34:57.030 I'm just kidding. 00:34:57.030 --> 00:34:58.030 I'll put it back. 00:34:58.030 --> 00:35:00.910 But Alex had a smart question over there, 00:35:00.910 --> 00:35:04.030 and he made me thinking. 00:35:04.030 --> 00:35:06.600 It's a dangerous thing when people make you think. 00:35:06.600 --> 00:35:10.260 So it goes from zero to 2 pi. 00:35:10.260 --> 00:35:12.050 Why would that be? 00:35:12.050 --> 00:35:15.100 Imagine you have all the meridians in the world 00:35:15.100 --> 00:35:17.680 except for one. 00:35:17.680 --> 00:35:22.470 From the sphere, you cut it and remove the Greenwich meridian, 00:35:22.470 --> 00:35:25.560 the one that passes through Greenwich Village. 00:35:25.560 --> 00:35:29.960 The one-- not the one in New York, the one next to London, 00:35:29.960 --> 00:35:31.400 right? 00:35:31.400 --> 00:35:32.500 So put it back. 00:35:32.500 --> 00:35:34.800 Put that meridian back. 00:35:34.800 --> 00:35:38.660 It's like you take an orange, and you make a slice. 00:35:38.660 --> 00:35:40.240 I am-- OK. 00:35:40.240 --> 00:35:43.050 Stop with the fruit because I'm hungry. 00:35:43.050 --> 00:35:47.180 Now, example two. 00:35:47.180 --> 00:35:52.730 Now, imagine another surface area you're used to, the what? 00:35:52.730 --> 00:35:58.490 The paraboloid is one of our favorite guys this semester. 00:35:58.490 --> 00:36:00.170 X squared plus Y squared. 00:36:00.170 --> 00:36:02.127 What is the parameterization of that? 00:36:02.127 --> 00:36:06.950 00:36:06.950 --> 00:36:09.680 Well, if I write it like that, it's a graph. 00:36:09.680 --> 00:36:11.600 But if I don't want to write it as a graph, 00:36:11.600 --> 00:36:14.300 I have to write it as a parameter. 00:36:14.300 --> 00:36:16.070 What am I going to do? 00:36:16.070 --> 00:36:19.380 I really know X to be U, right? 00:36:19.380 --> 00:36:21.380 That's the simplest choice possible. 00:36:21.380 --> 00:36:25.300 Y could be V. And then Z will be U squared plus V squared. 00:36:25.300 --> 00:36:26.360 And there I am. 00:36:26.360 --> 00:36:26.860 [SNEEZE] 00:36:26.860 --> 00:36:29.110 So I'm going to write-- bless your heart, [INAUDIBLE]. 00:36:29.110 --> 00:36:31.530 00:36:31.530 --> 00:36:38.320 V plus J plus U squared plus V squared, K. So this 00:36:38.320 --> 00:36:44.410 is the parameterization of a paraboloid. 00:36:44.410 --> 00:36:46.734 That one of them-- there are infinitely 00:36:46.734 --> 00:36:48.650 many-- the one that comes to mind because it's 00:36:48.650 --> 00:36:52.600 the easiest one to think about. 00:36:52.600 --> 00:36:53.560 STUDENT: [INAUDIBLE]. 00:36:53.560 --> 00:36:54.520 PROFESSOR: Good. 00:36:54.520 --> 00:36:59.060 For a minute, guys, you didn't need me. 00:36:59.060 --> 00:37:02.390 You didn't need me at all to come up with those. 00:37:02.390 --> 00:37:05.870 But maybe you would need me to remember, or maybe not-- 00:37:05.870 --> 00:37:08.480 to remind you of the helicoid. 00:37:08.480 --> 00:37:09.810 Helicoid. 00:37:09.810 --> 00:37:13.250 Did you go to the, as I told you to go 00:37:13.250 --> 00:37:15.710 to the [INAUDIBLE] spectrum-- what was that called? 00:37:15.710 --> 00:37:15.860 The-- 00:37:15.860 --> 00:37:16.530 STUDENT: Science spectrum. 00:37:16.530 --> 00:37:17.760 PROFESSOR: Science spectrum. 00:37:17.760 --> 00:37:24.620 And dip into soap solution the thingy was-- a metal 00:37:24.620 --> 00:37:27.950 rod with a-- with a what? 00:37:27.950 --> 00:37:34.210 With; a helix made of metal so the soap 00:37:34.210 --> 00:37:37.040 film would take which shape? 00:37:37.040 --> 00:37:42.230 The shape of this spiral that's going to go inside here, right? 00:37:42.230 --> 00:37:45.630 That's called a helicoid. 00:37:45.630 --> 00:37:46.150 OK. 00:37:46.150 --> 00:37:46.720 All right. 00:37:46.720 --> 00:37:47.595 You're not mad at me. 00:37:47.595 --> 00:37:48.094 STUDENT: No. 00:37:48.094 --> 00:37:49.010 PROFESSOR: OK, good. 00:37:49.010 --> 00:37:54.410 So in this case, R of UV will be what? 00:37:54.410 --> 00:37:57.690 It was a long time ago, once upon a time I gave it to you. 00:37:57.690 --> 00:38:00.290 It's extremely hard to memorize if you don't work 00:38:00.290 --> 00:38:03.560 with it on a regular basis. 00:38:03.560 --> 00:38:06.940 If it were a helix, what would it be? 00:38:06.940 --> 00:38:09.970 If it were a helix, it would be R of T right? 00:38:09.970 --> 00:38:14.320 It would be like equal sign T, A sine T, BT. 00:38:14.320 --> 00:38:17.190 Say it again, Magdalena, that was a long time ago, 00:38:17.190 --> 00:38:17.970 chapter 10. 00:38:17.970 --> 00:38:19.690 Chapter 10. 00:38:19.690 --> 00:38:23.960 Equal sign, T, A sine T, MBT, standard helix. 00:38:23.960 --> 00:38:25.350 This is not going to be that. 00:38:25.350 --> 00:38:33.310 It's going to be-- U cosine B. U sine B. Look at the picture. 00:38:33.310 --> 00:38:37.691 And imagine that these guys are extended to infinity. 00:38:37.691 --> 00:38:39.190 It's not just the stairs themselves, 00:38:39.190 --> 00:38:41.460 or whatever they are. 00:38:41.460 --> 00:38:46.350 There are infinite lines, straight lines, and busy. 00:38:46.350 --> 00:38:47.760 This is done. 00:38:47.760 --> 00:38:49.140 NB is a positive constant. 00:38:49.140 --> 00:38:51.760 00:38:51.760 --> 00:38:56.520 But your parameters are U and V. Any other guy 00:38:56.520 --> 00:39:00.400 that comes to mind, I'm out of imagination right now. 00:39:00.400 --> 00:39:04.520 You can do a torus on the fold that looks like a donut. 00:39:04.520 --> 00:39:05.740 You will have two parameters. 00:39:05.740 --> 00:39:08.450 Imagine a donut. 00:39:08.450 --> 00:39:10.970 How do you-- I'm not going to write that. 00:39:10.970 --> 00:39:13.480 Eventually I could give you that as an extra credit thing. 00:39:13.480 --> 00:39:18.660 What are the two degrees of freedom of moving on the donut, 00:39:18.660 --> 00:39:21.422 assuming that you would like to move in circles? 00:39:21.422 --> 00:39:25.417 00:39:25.417 --> 00:39:26.250 STUDENT: [INAUDIBLE] 00:39:26.250 --> 00:39:29.750 00:39:29.750 --> 00:39:32.010 PROFESSOR: Let me draw a donut, because I'm hungry, 00:39:32.010 --> 00:39:34.720 and I really-- I cannot help it. 00:39:34.720 --> 00:39:38.450 I just have to-- this is called a torus in mathematics. 00:39:38.450 --> 00:39:42.744 And you'll have-- one degree of freedom will be like this, 00:39:42.744 --> 00:39:44.660 the other degree of freedom will be like that. 00:39:44.660 --> 00:39:46.880 This is U and B. Instead of U and B, 00:39:46.880 --> 00:39:50.110 mathematicians, apologists, geometers, 00:39:50.110 --> 00:39:53.370 they call those angles phi and theta because they really 00:39:53.370 --> 00:39:56.100 are between zero and 2 pi. 00:39:56.100 --> 00:39:59.800 It has a rotation like that along the donut. 00:39:59.800 --> 00:40:02.180 You can cut, slice the donut, or if they 00:40:02.180 --> 00:40:04.450 don't put cheese filling in it. 00:40:04.450 --> 00:40:08.790 That was a bad idea not having anything to eat. 00:40:08.790 --> 00:40:15.300 And the other angle will be your 2 pi along this little circle. 00:40:15.300 --> 00:40:18.380 So you still have two degrees of freedom on a donut. 00:40:18.380 --> 00:40:19.230 It's a surface. 00:40:19.230 --> 00:40:20.210 You can write the parameterization. 00:40:20.210 --> 00:40:20.860 Yes? 00:40:20.860 --> 00:40:22.443 STUDENT: Why is a pie this way around. 00:40:22.443 --> 00:40:25.870 Why is it like [INAUDIBLE]. 00:40:25.870 --> 00:40:27.710 PROFESSOR: It doesn't have to be. 00:40:27.710 --> 00:40:29.710 STUDENT: Or is it just kind of like [INAUDIBLE]? 00:40:29.710 --> 00:40:31.420 PROFESSOR: That's what they call it. 00:40:31.420 --> 00:40:32.480 Yeah. 00:40:32.480 --> 00:40:35.230 So they are between 2 and 2 pi. 00:40:35.230 --> 00:40:41.780 While I erase-- or should I-- enough expectation 00:40:41.780 --> 00:40:44.380 in terms of parameterization, I have to night 00:40:44.380 --> 00:40:47.280 teach you something about that. 00:40:47.280 --> 00:40:53.480 If somebody would say I'm giving you a patch of a surface, 00:40:53.480 --> 00:40:57.440 but that patch of a surface is in a frame-- 00:40:57.440 --> 00:41:01.700 it's a nice parameterization. 00:41:01.700 --> 00:41:04.110 This is the P on the surface. 00:41:04.110 --> 00:41:07.520 00:41:07.520 --> 00:41:10.510 And you say, well, the parameterization 00:41:10.510 --> 00:41:14.190 is going to be R of U and V equals 00:41:14.190 --> 00:41:22.110 X of UVI plus Y of UVJ plus Z of UVK. 00:41:22.110 --> 00:41:25.090 00:41:25.090 --> 00:41:28.006 And suppose that somebody says this is you favorite test. 00:41:28.006 --> 00:41:30.880 00:41:30.880 --> 00:41:35.990 Find V. Well, that would be absurd. 00:41:35.990 --> 00:41:37.420 My god, how do we do that? 00:41:37.420 --> 00:41:48.670 Find the flux corresponding to-- do 00:41:48.670 --> 00:41:51.840 we say restart-- just a second-- just 00:41:51.840 --> 00:41:54.130 to restart with applications. 00:41:54.130 --> 00:41:54.630 [INAUDIBLE] 00:41:54.630 --> 00:41:57.270 00:41:57.270 --> 00:42:01.490 We don't say what kind of vector field that it is, 00:42:01.490 --> 00:42:06.820 but we will say plus corresponding to the vector 00:42:06.820 --> 00:42:08.417 field. 00:42:08.417 --> 00:42:09.000 F [INAUDIBLE]. 00:42:09.000 --> 00:42:14.380 00:42:14.380 --> 00:42:16.300 And this vector field, I'll tell you 00:42:16.300 --> 00:42:22.450 in a second what's expected from this to be a vector field. 00:42:22.450 --> 00:42:35.130 Through, on the surface, we find on the surface-- yes. 00:42:35.130 --> 00:42:39.150 Mathematicians say define normal surface S. 00:42:39.150 --> 00:42:44.460 But a physicist will say flux through, 00:42:44.460 --> 00:42:55.350 the flux corresponding to F through the surface. 00:42:55.350 --> 00:42:58.702 00:42:58.702 --> 00:43:01.170 Yes. 00:43:01.170 --> 00:43:03.940 So you'll say why would that be, and what is the flux? 00:43:03.940 --> 00:43:11.955 By definition, how should we denote it? 00:43:11.955 --> 00:43:16.370 Let's make a beautiful script F. That's crazy, right? 00:43:16.370 --> 00:43:21.590 And then it goes doubling over the surface F test. 00:43:21.590 --> 00:43:24.910 Is anybody mechanical engineering here? 00:43:24.910 --> 00:43:28.380 Do you know the flux formula? 00:43:28.380 --> 00:43:33.980 It's going to be [INAUDIBLE] over S of F, this magic thing. 00:43:33.980 --> 00:43:36.890 Not DN, DS. 00:43:36.890 --> 00:43:38.960 Do you know what N means? 00:43:38.960 --> 00:43:41.250 What it is N for mechanical engineering, 00:43:41.250 --> 00:43:42.940 [INAUDIBLE] engineers? 00:43:42.940 --> 00:44:01.070 N to would be the unit normal vector field to the surface S. 00:44:01.070 --> 00:44:03.050 How would you want to imagine that? 00:44:03.050 --> 00:44:07.110 You would have a surface, and you have this velocity vectors 00:44:07.110 --> 00:44:13.350 here at the bottom that goes to S. And this field goes up. 00:44:13.350 --> 00:44:17.460 You'll have a force and acceleration, velocity, 00:44:17.460 --> 00:44:19.850 you have everything going this way. 00:44:19.850 --> 00:44:22.800 And you want to find out what happens. 00:44:22.800 --> 00:44:26.720 You introduce this notion of flux through the surface. 00:44:26.720 --> 00:44:29.400 Another way to have a flux through the surface 00:44:29.400 --> 00:44:31.460 maybe through the same surface but associated 00:44:31.460 --> 00:44:34.040 through another kind of concept-- 00:44:34.040 --> 00:44:36.180 if there could be something else. 00:44:36.180 --> 00:44:39.960 In electromagnetism, F would be something else, some other type 00:44:39.960 --> 00:44:41.075 of vector field. 00:44:41.075 --> 00:44:42.145 Yes, sir. 00:44:42.145 --> 00:44:43.020 STUDENT: [INAUDIBLE]. 00:44:43.020 --> 00:44:45.550 PROFESSOR: So find out, by the way until next time, 00:44:45.550 --> 00:44:49.390 if you were an electrical engineering major, what 00:44:49.390 --> 00:44:51.930 would flux be for you guys? 00:44:51.930 --> 00:44:55.910 Two surfaces, one would be the meaning of the vector field 00:44:55.910 --> 00:44:58.280 F for you, and why would you care 00:44:58.280 --> 00:45:01.240 about the electromagnetic flux or something like that. 00:45:01.240 --> 00:45:02.920 I don't want to talk too much about it. 00:45:02.920 --> 00:45:07.150 It's for you to do the search and find out. 00:45:07.150 --> 00:45:09.880 So suppose that somebody gives you 00:45:09.880 --> 00:45:13.300 this notion that says you have a parameteric surface. 00:45:13.300 --> 00:45:19.050 Give an application of that and find out 00:45:19.050 --> 00:45:24.166 how you're going be deal with it. 00:45:24.166 --> 00:45:27.840 I'll give you an example that shouldn't be too hard. 00:45:27.840 --> 00:45:32.700 00:45:32.700 --> 00:45:34.440 I'll make up my own example. 00:45:34.440 --> 00:45:38.469 And looks like example 6, but it's going to be different. 00:45:38.469 --> 00:45:47.391 00:45:47.391 --> 00:45:47.891 Example. 00:45:47.891 --> 00:45:51.330 00:45:51.330 --> 00:45:58.600 Find the flux F if F will be a simple function. 00:45:58.600 --> 00:46:05.235 Let's say something equals X, I plus Y,J Z, K at every point X, 00:46:05.235 --> 00:46:13.030 Y-- at every point of the space XYZ. 00:46:13.030 --> 00:46:16.380 That means you could have this vector field defined everywhere 00:46:16.380 --> 00:46:18.360 in space in [INAUDIBLE]. 00:46:18.360 --> 00:46:22.660 But you only care about this acting on the surface. 00:46:22.660 --> 00:46:25.064 So it's acting on the surface. 00:46:25.064 --> 00:46:28.000 00:46:28.000 --> 00:46:30.330 And then what will the flux be? 00:46:30.330 --> 00:46:34.570 On the surface, which surface? 00:46:34.570 --> 00:46:41.012 My favorite one, Z equals X squared plus Y squared. 00:46:41.012 --> 00:46:47.290 00:46:47.290 --> 00:46:49.260 First of all, you say wait, wait, Magdalena, 00:46:49.260 --> 00:46:50.990 do you want to do it like that? 00:46:50.990 --> 00:46:54.330 Do you want to say F over XY to be a graph? 00:46:54.330 --> 00:46:58.800 Or do you want to consider it as a parameterized surface? 00:46:58.800 --> 00:47:01.915 And that means it's the same thing, equivalent to or if 00:47:01.915 --> 00:47:09.000 and only if, who tells me again what R was for such a surface? 00:47:09.000 --> 00:47:09.500 STUDENT: XI. 00:47:09.500 --> 00:47:11.380 PROFESSOR: X is U. Y is V, so U-- 00:47:11.380 --> 00:47:12.213 STUDENT: [INAUDIBLE] 00:47:12.213 --> 00:47:15.750 PROFESSOR: --I, that would be J, then good. 00:47:15.750 --> 00:47:19.720 U squared plus U squared UK. 00:47:19.720 --> 00:47:23.300 Well, when you say that, we have-- first of all, 00:47:23.300 --> 00:47:26.920 we have no idea what the heck we need to do, 00:47:26.920 --> 00:47:32.450 because do we want to do it in this form like a graph? 00:47:32.450 --> 00:47:34.060 Or do we want to do it parameterized? 00:47:34.060 --> 00:47:37.340 We have to set up formulas for the flats. 00:47:37.340 --> 00:47:38.650 It's not so easy. 00:47:38.650 --> 00:47:43.250 So assume that we are brave enough and we start everything. 00:47:43.250 --> 00:47:48.500 I want to understand what flux really is as an integral. 00:47:48.500 --> 00:47:55.510 And let me set it up for the first case, the case of Z 00:47:55.510 --> 00:47:58.390 equals F of X and Y. And I'm happy with it 00:47:58.390 --> 00:48:01.300 because that's the simplest case. 00:48:01.300 --> 00:48:03.930 Who's going to teach me what I have to do? 00:48:03.930 --> 00:48:05.240 You are confusing. 00:48:05.240 --> 00:48:09.920 I have double integral over S minus theory of F in general. 00:48:09.920 --> 00:48:12.810 This is a general vector value field. 00:48:12.810 --> 00:48:15.810 00:48:15.810 --> 00:48:16.940 It could be anything. 00:48:16.940 --> 00:48:18.220 Could be anything. 00:48:18.220 --> 00:48:23.210 But then I have to [INAUDIBLE], because N corresponds 00:48:23.210 --> 00:48:27.180 to the normal to the surface. 00:48:27.180 --> 00:48:29.270 So I-- it's not so easy, right? 00:48:29.270 --> 00:48:30.935 I have to be a little bit smart. 00:48:30.935 --> 00:48:31.987 If I'm not smart-- 00:48:31.987 --> 00:48:32.820 STUDENT: [INAUDIBLE] 00:48:32.820 --> 00:48:35.124 00:48:35.124 --> 00:48:36.790 PROFESSOR: That-- you are getting close. 00:48:36.790 --> 00:48:40.870 So guys, the normal two-way surface-- somebody 00:48:40.870 --> 00:48:42.550 gave you a surface, OK? 00:48:42.550 --> 00:48:46.100 And normal to a surface is normal to the plane-- 00:48:46.100 --> 00:48:49.590 the tangent plane of the surface. 00:48:49.590 --> 00:48:51.600 So how did we get that? 00:48:51.600 --> 00:48:53.720 There were many ways to do it. 00:48:53.720 --> 00:48:56.190 Either you write the tangent plane 00:48:56.190 --> 00:49:02.700 and you know it by heart-- that was Z minus Z zero 00:49:02.700 --> 00:49:07.460 equals-- what the heck was that-- S of X times X minus X 00:49:07.460 --> 00:49:10.710 equals-- plus X of Y times Y minus Y zero. 00:49:10.710 --> 00:49:14.110 And from here you collect-- what do you collect? 00:49:14.110 --> 00:49:16.140 You move everybody-- it's a moving sale. 00:49:16.140 --> 00:49:19.550 You move everybody to the left hand side and that's it. 00:49:19.550 --> 00:49:21.200 [INAUDIBLE] moving sale. 00:49:21.200 --> 00:49:22.850 OK? 00:49:22.850 --> 00:49:26.900 And everybody will be giving you some components. 00:49:26.900 --> 00:49:31.030 You're going to have minus S of X-- S minus X zero-- minus S 00:49:31.030 --> 00:49:36.620 of Y, Y minus Y zero, plus 1-- this is really funny. 00:49:36.620 --> 00:49:39.700 1 times Z minus Z, Z. 00:49:39.700 --> 00:49:43.370 Your normal will be given by what? 00:49:43.370 --> 00:49:46.297 The normal-- how do you collect the normal? 00:49:46.297 --> 00:49:47.130 STUDENT: [INAUDIBLE] 00:49:47.130 --> 00:49:52.300 PROFESSOR: Pi is A, B, C. A, B, and C will be the normal. 00:49:52.300 --> 00:49:54.300 Except it's not unitary. 00:49:54.300 --> 00:49:57.820 And the mechanical engineer tells you, yeah, you're 00:49:57.820 --> 00:50:00.990 stupid-- well, they never say that. 00:50:00.990 --> 00:50:06.080 They will stay look, you have to be a little more careful. 00:50:06.080 --> 00:50:07.680 Not say they are equal. 00:50:07.680 --> 00:50:08.940 What do they mean? 00:50:08.940 --> 00:50:10.540 They say for us, in fluid mechanics, 00:50:10.540 --> 00:50:14.980 solid mechanics, when we write N, we mean you mean vector. 00:50:14.980 --> 00:50:16.150 You are almost there. 00:50:16.150 --> 00:50:16.907 What's missing? 00:50:16.907 --> 00:50:18.240 STUDENT: Magnitude. [INAUDIBLE]. 00:50:18.240 --> 00:50:19.739 PROFESSOR: Very good, the magnitude. 00:50:19.739 --> 00:50:24.460 So they will say, go ahead and you [INAUDIBLE] the magnitude. 00:50:24.460 --> 00:50:28.990 And you are lucky now that you know what N will be. 00:50:28.990 --> 00:50:29.980 On the other hand-- 00:50:29.980 --> 00:50:31.130 STUDENT: [INAUDIBLE]. 00:50:31.130 --> 00:50:32.350 PROFESSOR: This is excellent. 00:50:32.350 --> 00:50:35.600 The one on the bottom-- Alex is thinking like in chess, two 00:50:35.600 --> 00:50:37.900 or three moves ahead. 00:50:37.900 --> 00:50:40.689 You should get two extra credit points with that. 00:50:40.689 --> 00:50:41.480 STUDENT: All right. 00:50:41.480 --> 00:50:42.730 PROFESSOR: You already got it. 00:50:42.730 --> 00:50:46.360 DS is 1 plus S of X squared plus F of X squared. 00:50:46.360 --> 00:50:51.750 The 1 on the bottom and the 1 on the top will simplify. 00:50:51.750 --> 00:50:53.000 So say it again, Magdalena. 00:50:53.000 --> 00:50:54.980 Let me write it down here. 00:50:54.980 --> 00:51:01.440 1 S of X, minus S of Y 1 over all this animal, 00:51:01.440 --> 00:51:04.940 S of X squared plus S of Y squared plus 1. 00:51:04.940 --> 00:51:08.850 This is the thinking like the early element 00:51:08.850 --> 00:51:13.470 times the early element will be the same thing. 00:51:13.470 --> 00:51:17.590 I'll write it twice even if you laugh at me because we are just 00:51:17.590 --> 00:51:19.430 learning together, and now you finally 00:51:19.430 --> 00:51:21.350 see-- everybody can see that desimplifies. 00:51:21.350 --> 00:51:23.970 00:51:23.970 --> 00:51:27.850 So it's going to be easy to solve this integral in the end, 00:51:27.850 --> 00:51:29.130 right? 00:51:29.130 --> 00:51:32.120 So let's do the problem, finally. 00:51:32.120 --> 00:51:35.590 I'm going to erase it. 00:51:35.590 --> 00:51:40.050 Let's do this problem just for us, at any point. 00:51:40.050 --> 00:51:42.710 I didn't say where. 00:51:42.710 --> 00:51:46.780 Over the same thing. 00:51:46.780 --> 00:51:49.940 The DS was over V01. 00:51:49.940 --> 00:51:53.025 So the picture is the same as before. 00:51:53.025 --> 00:51:57.560 The S will be the nutshell, the eggshell-- 00:51:57.560 --> 00:52:02.780 I don't know what it was-- over the domain D plane. 00:52:02.780 --> 00:52:07.620 The domain D plane was D of zero 1. 00:52:07.620 --> 00:52:12.160 And I say that I need to use another color. 00:52:12.160 --> 00:52:17.430 This it's going to be my shell, my surface S. Z 00:52:17.430 --> 00:52:20.470 equals X squared plus [INAUDIBLE]. 00:52:20.470 --> 00:52:23.140 How do you compute the flux? 00:52:23.140 --> 00:52:25.510 Well, this is that. 00:52:25.510 --> 00:52:28.270 So if we have to be a little bit careful and smart 00:52:28.270 --> 00:52:33.160 and say double integral over S, and now without rushing, 00:52:33.160 --> 00:52:35.730 we have to do a good job. 00:52:35.730 --> 00:52:39.210 First of all, how do you do the dot product? 00:52:39.210 --> 00:52:40.856 The dot product-- 00:52:40.856 --> 00:52:43.512 STUDENT: [INAUDIBLE] 00:52:43.512 --> 00:52:44.220 PROFESSOR: Right. 00:52:44.220 --> 00:52:46.530 So first component times first component, 00:52:46.530 --> 00:52:50.290 a second component, second component times 00:52:50.290 --> 00:52:52.530 second component plus that component times 00:52:52.530 --> 00:52:53.710 third component. 00:52:53.710 --> 00:53:01.270 So if 1 is X, F2 is 1. 00:53:01.270 --> 00:53:02.170 Good. 00:53:02.170 --> 00:53:04.290 Z, though, he's not free. 00:53:04.290 --> 00:53:05.270 He's married. 00:53:05.270 --> 00:53:07.045 Why is he married? 00:53:07.045 --> 00:53:07.920 STUDENT: [INAUDIBLE]. 00:53:07.920 --> 00:53:10.580 PROFESSOR: Because he depends on X and Y. 00:53:10.580 --> 00:53:14.230 So Z was even here, because I'm on the surface. 00:53:14.230 --> 00:53:16.690 I don't care what F does away from the surface, 00:53:16.690 --> 00:53:20.050 but when he sticks to the surface, when 00:53:20.050 --> 00:53:24.030 he's origin is on the surface, then he 00:53:24.030 --> 00:53:27.860 has to listen to the surface. 00:53:27.860 --> 00:53:29.880 And that Z is not independent. 00:53:29.880 --> 00:53:33.110 The Z is X squared by Y squared here. 00:53:33.110 --> 00:53:36.740 In a bracket, we are over the surface. 00:53:36.740 --> 00:53:41.110 That product minus S of X, minus S of Y. I 00:53:41.110 --> 00:53:42.960 know you're going to laugh at me because I 00:53:42.960 --> 00:53:44.670 haven't written where they are. 00:53:44.670 --> 00:53:46.370 But that's what I need your help for. 00:53:46.370 --> 00:53:46.870 DA. 00:53:46.870 --> 00:53:49.390 00:53:49.390 --> 00:53:51.810 Who are they? 00:53:51.810 --> 00:53:53.759 Who is this guy? 00:53:53.759 --> 00:53:54.800 STUDENT: The [INAUDIBLE]. 00:53:54.800 --> 00:53:55.466 PROFESSOR: What? 00:53:55.466 --> 00:53:56.580 STUDENT: [INAUDIBLE]. 00:53:56.580 --> 00:53:58.000 PROFESSOR: Negative 2X. 00:53:58.000 --> 00:53:58.856 Is it? 00:53:58.856 --> 00:53:59.850 STUDENT: No. 00:53:59.850 --> 00:54:01.350 PROFESSOR: How about this guy? 00:54:01.350 --> 00:54:02.430 STUDENT: [INAUDIBLE]. 00:54:02.430 --> 00:54:03.388 PROFESSOR: Negative 2Y. 00:54:03.388 --> 00:54:05.750 How about this guy? 00:54:05.750 --> 00:54:06.710 I'm just kidding. 00:54:06.710 --> 00:54:07.680 OK. 00:54:07.680 --> 00:54:14.230 So finally we should be able to compute this integral. 00:54:14.230 --> 00:54:15.650 That looks awful. 00:54:15.650 --> 00:54:17.560 Over D. 00:54:17.560 --> 00:54:20.575 So instead of S, we have the D, which 00:54:20.575 --> 00:54:25.170 is the disk of radius one in plane. 00:54:25.170 --> 00:54:29.740 And we say, OK, I have, oh my god, it's OK. 00:54:29.740 --> 00:54:31.716 This times that is how much? 00:54:31.716 --> 00:54:32.590 STUDENT: [INAUDIBLE]. 00:54:32.590 --> 00:54:33.890 PROFESSOR: Minus 2X squared. 00:54:33.890 --> 00:54:35.190 Right? 00:54:35.190 --> 00:54:35.880 There. 00:54:35.880 --> 00:54:37.120 Take the green. 00:54:37.120 --> 00:54:40.870 This times that is how much? 00:54:40.870 --> 00:54:44.250 Minus the Y squared. 00:54:44.250 --> 00:54:50.780 And this times that is finally just X squared plus Y squared. 00:54:50.780 --> 00:54:53.440 Very nice think. 00:54:53.440 --> 00:54:55.560 I think that at first, but now I see 00:54:55.560 --> 00:54:59.110 that life is beautiful again-- DX, DY-- 00:54:59.110 --> 00:55:02.770 that I can go ahead and do it. 00:55:02.770 --> 00:55:05.053 I can get a hold of this. 00:55:05.053 --> 00:55:11.134 And inside that, what do I-- what am I left with in the end? 00:55:11.134 --> 00:55:12.050 STUDENT: [INAUDIBLE]. 00:55:12.050 --> 00:55:14.650 PROFESSOR: Minus 2 times this animal, 00:55:14.650 --> 00:55:18.550 called X squared plus Y squared, which is going to be R squared. 00:55:18.550 --> 00:55:23.230 So the flux-- the flux for this problem in the end 00:55:23.230 --> 00:55:25.640 is going to be very nice and sassy. 00:55:25.640 --> 00:55:26.310 Look at that. 00:55:26.310 --> 00:55:28.145 F would be-- 00:55:28.145 --> 00:55:29.520 STUDENT: There would not be any-- 00:55:29.520 --> 00:55:30.353 STUDENT: [INAUDIBLE] 00:55:30.353 --> 00:55:36.580 00:55:36.580 --> 00:55:37.390 PROFESSOR: What? 00:55:37.390 --> 00:55:40.989 STUDENT: You've got minus 2 and the plus 1. 00:55:40.989 --> 00:55:42.030 PROFESSOR: Oh, thank God. 00:55:42.030 --> 00:55:43.730 Thank God you exist. 00:55:43.730 --> 00:55:47.250 So I thought about it before, but then I 00:55:47.250 --> 00:55:49.040 said-- I don't know why. 00:55:49.040 --> 00:55:50.120 I messed up. 00:55:50.120 --> 00:55:52.540 So we have minus R squared. 00:55:52.540 --> 00:55:53.530 Very good. 00:55:53.530 --> 00:55:54.810 It's easy. 00:55:54.810 --> 00:56:01.950 Times an R from the Jacobian, DR is theta. 00:56:01.950 --> 00:56:04.550 And theta is between 0 and 2 pi. 00:56:04.550 --> 00:56:08.020 And R between 0 and 1. 00:56:08.020 --> 00:56:11.050 And now I will need a plumber to tell me 00:56:11.050 --> 00:56:12.790 what I do the limits of the integrals, 00:56:12.790 --> 00:56:17.752 because I think I'm getting a negative answer, so. 00:56:17.752 --> 00:56:20.527 00:56:20.527 --> 00:56:21.692 STUDENT: [INAUDIBLE]. 00:56:21.692 --> 00:56:23.150 PROFESSOR: I'll do it, and then you 00:56:23.150 --> 00:56:26.700 tell me why I got what I got. 00:56:26.700 --> 00:56:29.710 I have a minus pulled out by nature. 00:56:29.710 --> 00:56:32.850 And then I have integral-- 00:56:32.850 --> 00:56:34.000 STUDENT: R [INAUDIBLE]. 00:56:34.000 --> 00:56:35.820 PROFESSOR: R to the fourth of a fourth. 00:56:35.820 --> 00:56:36.590 Very good. 00:56:36.590 --> 00:56:40.447 But you have your [INAUDIBLE] so when I do between zero and 1-- 00:56:40.447 --> 00:56:41.530 STUDENT: It's [INAUDIBLE]. 00:56:41.530 --> 00:56:43.830 PROFESSOR: 1 over 4-- you are too 00:56:43.830 --> 00:56:49.400 fast-- as 2 pi-- that's a good thing-- minus pi over 2, 00:56:49.400 --> 00:56:50.890 you said, Gus. 00:56:50.890 --> 00:56:53.160 And I could see it coming straight at me 00:56:53.160 --> 00:56:55.320 and hit me between the eyes. 00:56:55.320 --> 00:56:57.720 What is the problem. 00:56:57.720 --> 00:56:59.750 Is there a problem? 00:56:59.750 --> 00:57:02.820 Without an area as a flux, would that say, what is the negative? 00:57:02.820 --> 00:57:03.810 Yes. 00:57:03.810 --> 00:57:05.630 How can I make it positive? 00:57:05.630 --> 00:57:07.480 This is my question. 00:57:07.480 --> 00:57:08.730 STUDENT: Change the direction. 00:57:08.730 --> 00:57:10.624 PROFESSOR: Change the direction of who? 00:57:10.624 --> 00:57:12.220 STUDENT: The flux. 00:57:12.220 --> 00:57:13.360 PROFESSOR: The flux. 00:57:13.360 --> 00:57:14.850 I could change the direction. 00:57:14.850 --> 00:57:17.640 So what is it that doesn't match? 00:57:17.640 --> 00:57:19.980 [INAUDIBLE] 00:57:19.980 --> 00:57:22.645 If I want to keep-- the flux will be the same. 00:57:22.645 --> 00:57:25.270 When I can change the orientation of the service. 00:57:25.270 --> 00:57:27.820 And instead I get a minus then. 00:57:27.820 --> 00:57:36.410 My N was it sticking in-- oh, my god. 00:57:36.410 --> 00:57:40.340 So is it sticking in or sticking out? 00:57:40.340 --> 00:57:40.990 Look at it. 00:57:40.990 --> 00:57:41.900 Think about it. 00:57:41.900 --> 00:57:47.660 I have minus the positive guy minus another positive guy, 00:57:47.660 --> 00:57:48.980 and 1 sticking out. 00:57:48.980 --> 00:57:51.010 But it goes with the holes inside. 00:57:51.010 --> 00:57:54.020 This is the paraboloid [INAUDIBLE]. 00:57:54.020 --> 00:57:57.470 If I have something I minus I minus J, does it go out or in? 00:57:57.470 --> 00:57:57.970 STUDENT: In. 00:57:57.970 --> 00:57:59.540 PROFESSOR: It goes in. 00:57:59.540 --> 00:58:00.804 It goes in, and it'll be up. 00:58:00.804 --> 00:58:02.720 So it's going to be like all these normals are 00:58:02.720 --> 00:58:07.170 going to be like a vector field like that, like amoebas. 00:58:07.170 --> 00:58:09.740 But they are pointing towards inside. 00:58:09.740 --> 00:58:11.090 Do I like that? 00:58:11.090 --> 00:58:13.160 Yes, because I'm a crazy mathematician. 00:58:13.160 --> 00:58:16.740 Does the engineer like that? 00:58:16.740 --> 00:58:17.710 No. 00:58:17.710 --> 00:58:18.660 Why? 00:58:18.660 --> 00:58:21.790 The flux is pointing in or out? 00:58:21.790 --> 00:58:23.150 The flux. 00:58:23.150 --> 00:58:23.650 The flux. 00:58:23.650 --> 00:58:25.900 The flux, the flux is pointing out. 00:58:25.900 --> 00:58:27.590 Are you guys with me? 00:58:27.590 --> 00:58:31.370 X plus Y-- X plus I plus J. It's like this pointing out. 00:58:31.370 --> 00:58:33.540 So the flux get out of the surface. 00:58:33.540 --> 00:58:36.790 It's like to pour water inside, and the water's 00:58:36.790 --> 00:58:41.580 just a net-- not a net, but like something that holds it in. 00:58:41.580 --> 00:58:42.954 And like a-- 00:58:42.954 --> 00:58:44.120 STUDENT: Like a [INAUDIBLE]? 00:58:44.120 --> 00:58:45.765 PROFESSOR: --pasta strainer. 00:58:45.765 --> 00:58:47.640 And the water goes up [SPRAYING NOISE], well, 00:58:47.640 --> 00:58:48.670 like a jet. 00:58:48.670 --> 00:58:49.320 Like that. 00:58:49.320 --> 00:58:52.580 So that is your flux going through the surface. 00:58:52.580 --> 00:58:56.190 Are you happy that I took the normal pointing inside? 00:58:56.190 --> 00:58:56.790 No. 00:58:56.790 --> 00:58:57.780 That was crazy. 00:58:57.780 --> 00:59:02.820 So here comes you, the mechanical engineer majoring 00:59:02.820 --> 00:59:06.100 in solid or [INAUDIBLE] and say Magdalena, 00:59:06.100 --> 00:59:09.250 you should have taken the outer normal, 00:59:09.250 --> 00:59:12.294 because look at the flux pointing out. 00:59:12.294 --> 00:59:14.210 Take the outer of normal, and things are going 00:59:14.210 --> 00:59:16.590 to looks right and nice again. 00:59:16.590 --> 00:59:18.610 So if I were to change the normal, 00:59:18.610 --> 00:59:21.400 I would put the plus, plus, minus. 00:59:21.400 --> 00:59:23.500 I'll take the outer normal. 00:59:23.500 --> 00:59:26.010 And in the end I get plus 5 over 2. 00:59:26.010 --> 00:59:29.100 So no remark. 00:59:29.100 --> 00:59:36.280 If I change N to minus N, this would become the outer normal. 00:59:36.280 --> 00:59:40.680 Then the flux would become pi over 2. solar 00:59:40.680 --> 00:59:42.730 flux depends on the what? 00:59:42.730 --> 00:59:44.980 The match between the flux, the angles, 00:59:44.980 --> 00:59:48.990 sort of between the flux if function, vector [INAUDIBLE] 00:59:48.990 --> 00:59:53.340 function, and the normal that I take to the surface. 00:59:53.340 --> 00:59:54.000 Right? 00:59:54.000 --> 00:59:59.200 I can change the normal and I get the opposite answer. 00:59:59.200 --> 01:00:01.505 In absolute values, the same flux. 01:00:01.505 --> 01:00:05.770 So flux should be equal [INAUDIBLE] the absolute value. 01:00:05.770 --> 01:00:09.060 Unlike the area that should be always a positive number. 01:00:09.060 --> 01:00:11.430 Volume, that should always be a positive number. 01:00:11.430 --> 01:00:14.940 So if I get a limited area, that means I messed up. 01:00:14.940 --> 01:00:17.040 If I get a negative on all of them, 01:00:17.040 --> 01:00:20.920 it means messed up in my computation somewhere. 01:00:20.920 --> 01:00:23.070 But that doesn't mean I messed up here. 01:00:23.070 --> 01:00:24.810 I just chose the other normal. 01:00:24.810 --> 01:00:26.200 It's possible. 01:00:26.200 --> 01:00:30.720 So the flux can be taken as is and put in absolute value. 01:00:30.720 --> 01:00:31.440 All right. 01:00:31.440 --> 01:00:33.700 OK. 01:00:33.700 --> 01:00:36.980 We have to think of it like the surface, and stuff that 01:00:36.980 --> 01:00:40.320 goes through surface in electric circuits. 01:00:40.320 --> 01:00:42.700 Can you do some research for you about flux 01:00:42.700 --> 01:00:45.120 and electrical engineering? 01:00:45.120 --> 01:00:50.280 And next time somebody tells me a story about it. 01:00:50.280 --> 01:00:52.710 Who is-- again-- who is electrical engineering major 01:00:52.710 --> 01:00:54.280 here? 01:00:54.280 --> 01:00:56.160 Oh, so five people. 01:00:56.160 --> 01:00:58.220 You're going to get four extra credit points. 01:00:58.220 --> 01:00:59.920 You guys are jealous. 01:00:59.920 --> 01:01:03.510 I'm going to give you four extra credit points if in 10 minutes 01:01:03.510 --> 01:01:08.490 you can tell us a little bit about where flux can be seen. 01:01:08.490 --> 01:01:10.470 Well, you don't have to come to the board. 01:01:10.470 --> 01:01:13.070 You can just talk to us from outside if you want, 01:01:13.070 --> 01:01:14.800 or down inside the classroom. 01:01:14.800 --> 01:01:16.950 Tell us where the notion of flux appears 01:01:16.950 --> 01:01:19.910 in the electric circuits and why it 01:01:19.910 --> 01:01:24.290 would be important for Calculus 3 as well. 01:01:24.290 --> 01:01:25.190 OK. 01:01:25.190 --> 01:01:28.710 Now a big question before I let you go. 01:01:28.710 --> 01:01:34.550 Can I have a flux that corresponds 01:01:34.550 --> 01:01:36.120 to a parameterization? 01:01:36.120 --> 01:01:41.560 That is my big worry, that I have to do that as well. 01:01:41.560 --> 01:01:45.140 Eventually, could I have solved this problem 01:01:45.140 --> 01:01:47.950 if the surface that is parameterized 01:01:47.950 --> 01:01:54.080 was my friend-- who was my friend? 01:01:54.080 --> 01:01:54.980 I don't remember. 01:01:54.980 --> 01:01:59.350 UI plus VJ plus U squared plus-- you 01:01:59.350 --> 01:02:03.490 gave it to me-- OK, that was the previous example, 01:02:03.490 --> 01:02:05.450 and that's the last example on the board. 01:02:05.450 --> 01:02:12.610 So you have double integral of force field times NDS. 01:02:12.610 --> 01:02:16.930 Now, what if I say I don't want to do it like this-- Z 01:02:16.930 --> 01:02:19.460 equals F of XY. 01:02:19.460 --> 01:02:21.406 So I don't want to do it like that. 01:02:21.406 --> 01:02:25.070 I want to do it in a different way. 01:02:25.070 --> 01:02:33.720 That means you pulling out of your brain some old memories. 01:02:33.720 --> 01:02:35.260 F was F, right? 01:02:35.260 --> 01:02:37.480 You need to leave F alone, poor fellow, because he 01:02:37.480 --> 01:02:39.900 has no better way to do it. 01:02:39.900 --> 01:02:43.902 This is becoming complicated, the [INAUDIBLE] 01:02:43.902 --> 01:02:44.860 mechanical engineering. 01:02:44.860 --> 01:02:47.862 01:02:47.862 --> 01:02:52.530 And what's given to you before, but you don't remember? 01:02:52.530 --> 01:02:55.810 R was given to you as position vector. 01:02:55.810 --> 01:02:59.710 R sub U and R sub V, you may not remember-- 01:02:59.710 --> 01:03:03.475 that was a long time ago-- we proved that R sub U and R sub 01:03:03.475 --> 01:03:05.450 V were on the surface. 01:03:05.450 --> 01:03:07.190 They are both tensions of the surface. 01:03:07.190 --> 01:03:09.360 It was a long time ago. 01:03:09.360 --> 01:03:11.280 So the normal is [INAUDIBLE], and that's 01:03:11.280 --> 01:03:14.050 exactly what I wanted to say the normal will be. 01:03:14.050 --> 01:03:17.820 Not quite pressed product, but just like before, 01:03:17.820 --> 01:03:21.240 pressed product divided by the norm, 01:03:21.240 --> 01:03:28.470 because then the unit normal vector has to be length 1. 01:03:28.470 --> 01:03:30.660 So I have to divide by the number. 01:03:30.660 --> 01:03:31.160 [SNEEZE] 01:03:31.160 --> 01:03:32.019 The DS-- 01:03:32.019 --> 01:03:32.810 STUDENT: Thank you. 01:03:32.810 --> 01:03:36.020 PROFESSOR: --is going to-- OK, now it's up to you guys. 01:03:36.020 --> 01:03:36.780 You're smart. 01:03:36.780 --> 01:03:39.360 You know what I want to say. 01:03:39.360 --> 01:03:43.770 So I'll pretend that you know what DS is in terms 01:03:43.770 --> 01:03:45.030 of the parameterization. 01:03:45.030 --> 01:03:46.515 What's coming? 01:03:46.515 --> 01:03:47.790 We said that. 01:03:47.790 --> 01:03:49.340 It was a long time ago. 01:03:49.340 --> 01:03:51.500 You can guess it by just being smart-- 01:03:51.500 --> 01:03:51.740 STUDENT: [INAUDIBLE]. 01:03:51.740 --> 01:03:52.781 PROFESSOR: --or you can-- 01:03:52.781 --> 01:03:53.830 STUDENT: [INAUDIBLE]. 01:03:53.830 --> 01:03:55.290 PROFESSOR: Yes, exactly. 01:03:55.290 --> 01:03:57.800 And you got another one extra credit point. 01:03:57.800 --> 01:04:01.990 01:04:01.990 --> 01:04:04.579 STUDENT: [INAUDIBLE] 01:04:04.579 --> 01:04:06.620 PROFESSOR: So since before, they were simplified, 01:04:06.620 --> 01:04:07.630 for god's sake. 01:04:07.630 --> 01:04:10.640 Now we have the new kind of writing area element DS. 01:04:10.640 --> 01:04:13.250 They also have to simplify. 01:04:13.250 --> 01:04:15.510 It wasn't hard to see. 01:04:15.510 --> 01:04:18.330 So you could have done it like that. 01:04:18.330 --> 01:04:22.600 You could have done it like that, how? 01:04:22.600 --> 01:04:25.510 Somebody need to help me, because I have no idea what 01:04:25.510 --> 01:04:28.050 I'm going to do here. 01:04:28.050 --> 01:04:29.384 Do we get the same thing or not? 01:04:29.384 --> 01:04:30.258 This is the question. 01:04:30.258 --> 01:04:31.800 And I'm going to finish with that, 01:04:31.800 --> 01:04:33.850 but I don't want to go home-- I'm not 01:04:33.850 --> 01:04:37.890 going to let you go home until you finish this. 01:04:37.890 --> 01:04:42.270 F was a simple, beautiful vector field. 01:04:42.270 --> 01:04:45.490 Given-- like that. 01:04:45.490 --> 01:04:46.760 This is a force. 01:04:46.760 --> 01:04:48.720 May the force be with you like that. 01:04:48.720 --> 01:04:54.010 But we changed it in U,V because we are acting on the surface S, 01:04:54.010 --> 01:04:56.050 what is the pressure in V, right? 01:04:56.050 --> 01:05:00.700 So you have UI plus VJ plus-- you gave it to me-- 01:05:00.700 --> 01:05:03.680 U squared plus V squared. 01:05:03.680 --> 01:05:05.490 Am I right, or am I talking nonsense? 01:05:05.490 --> 01:05:08.010 01:05:08.010 --> 01:05:08.650 All right. 01:05:08.650 --> 01:05:12.850 So now again I have to be seeing them. 01:05:12.850 --> 01:05:14.140 Am I getting the same thing? 01:05:14.140 --> 01:05:16.280 If I'm not getting the same thing, 01:05:16.280 --> 01:05:19.280 I can just go home and get drunk and be sad. 01:05:19.280 --> 01:05:22.710 But I have to get the same thing. 01:05:22.710 --> 01:05:27.020 Otherwise, there is something wrong with my setup. 01:05:27.020 --> 01:05:32.620 So I have to have U, V. U squared plus V squared. 01:05:32.620 --> 01:05:34.500 Close. 01:05:34.500 --> 01:05:37.000 Dot product. 01:05:37.000 --> 01:05:41.840 This guy over on top-- say what? 01:05:41.840 --> 01:05:48.120 Magdalena, this guy over on top has to be-- has to be a what? 01:05:48.120 --> 01:05:50.040 Well, I didn't say what it was. 01:05:50.040 --> 01:05:52.720 I should do it now. 01:05:52.720 --> 01:05:53.460 Right? 01:05:53.460 --> 01:05:57.160 So how will we do that? 01:05:57.160 --> 01:06:07.960 We were saying R of UV will be UI plus VJ 01:06:07.960 --> 01:06:10.500 plus U squared plus V squared. 01:06:10.500 --> 01:06:11.100 OK. 01:06:11.100 --> 01:06:14.920 So R sub U will be-- you teach me quickly, 01:06:14.920 --> 01:06:18.146 and R sub [INAUDIBLE] is-- voila. 01:06:18.146 --> 01:06:19.770 STUDENT: [INAUDIBLE] 01:06:19.770 --> 01:06:21.564 PROFESSOR: 1-- 01:06:21.564 --> 01:06:23.530 STUDENT: [INAUDIBLE] 01:06:23.530 --> 01:06:28.120 PROFESSOR: Plus zero-- thank you-- plus 2U, OK. 01:06:28.120 --> 01:06:34.830 0 plus 1J plus 2VK Am I done? 01:06:34.830 --> 01:06:35.330 I'm done. 01:06:35.330 --> 01:06:35.900 No, I'm not done. 01:06:35.900 --> 01:06:36.820 What do I have to do? 01:06:36.820 --> 01:06:38.590 Cross them. 01:06:38.590 --> 01:06:41.470 01:06:41.470 --> 01:06:44.500 Cross multiply IJK. 01:06:44.500 --> 01:06:47.355 This looks nice. 01:06:47.355 --> 01:06:49.240 Look, it's not so ugly. 01:06:49.240 --> 01:06:51.481 I thought it would be uglier, right? 01:06:51.481 --> 01:06:51.980 OK. 01:06:51.980 --> 01:06:54.690 What it is? 01:06:54.690 --> 01:06:57.515 What it this thing? 01:06:57.515 --> 01:06:58.390 STUDENT: [INAUDIBLE]. 01:06:58.390 --> 01:07:06.180 PROFESSOR: Minus the U, I. Minus-- plus. 01:07:06.180 --> 01:07:09.130 Minus, plus 1. 01:07:09.130 --> 01:07:11.455 2V minus because it's-- 01:07:11.455 --> 01:07:12.080 STUDENT: Minus. 01:07:12.080 --> 01:07:13.450 PROFESSOR: --minus in front. 01:07:13.450 --> 01:07:15.360 Right. 01:07:15.360 --> 01:07:17.510 So I'm alternating. 01:07:17.510 --> 01:07:19.980 And 1K. 01:07:19.980 --> 01:07:24.040 So again, I get minus X of S minus XY and 1, 01:07:24.040 --> 01:07:26.420 and again, I'm pointing in, and that's bad. 01:07:26.420 --> 01:07:29.520 So my normal will point inside the surface 01:07:29.520 --> 01:07:33.670 like needles that are perpendicular to the surface 01:07:33.670 --> 01:07:34.840 pointing inside. 01:07:34.840 --> 01:07:36.000 But that's OK. 01:07:36.000 --> 01:07:39.050 In the end, I take everything in absolute value. 01:07:39.050 --> 01:07:39.550 Right? 01:07:39.550 --> 01:07:47.640 01:07:47.640 --> 01:07:51.100 So again, I do the same math. 01:07:51.100 --> 01:07:54.760 So I get minus-- I don't want to do it anymore. 01:07:54.760 --> 01:07:59.410 Minus 2A squared, minus 2B squared, plus your squared, 01:07:59.410 --> 01:08:01.580 plus this squared, then you save me 01:08:01.580 --> 01:08:04.870 and you said minus 2 squared [INAUDIBLE] squared. 01:08:04.870 --> 01:08:06.260 DUDV. 01:08:06.260 --> 01:08:15.020 But DUDV means that UV is a pair, a point in this, guys. 01:08:15.020 --> 01:08:16.120 UV. 01:08:16.120 --> 01:08:18.960 It's a pair in the disk of radius one. 01:08:18.960 --> 01:08:21.990 So I'm getting exactly, what exactly the same thing 01:08:21.990 --> 01:08:22.939 as before. 01:08:22.939 --> 01:08:25.640 Because this is minus R squared, so I 01:08:25.640 --> 01:08:32.910 get integral, integral, minus R squared times R. DR, D theta. 01:08:32.910 --> 01:08:35.660 From zero to 1, from zero to 2 pi, 01:08:35.660 --> 01:08:38.158 and I get the same answer, which was? 01:08:38.158 --> 01:08:39.490 STUDENT: [INAUDIBLE]. 01:08:39.490 --> 01:08:39.800 PROFESSOR: Minus what? 01:08:39.800 --> 01:08:40.300 STUDENT: [INAUDIBLE] 01:08:40.300 --> 01:08:40.590 PROFESSOR: Pi over-- 01:08:40.590 --> 01:08:41.679 STUDENT: [INAUDIBLE]. 01:08:41.679 --> 01:08:42.470 PROFESSOR: You see? 01:08:42.470 --> 01:08:46.024 I already forgot. 01:08:46.024 --> 01:08:48.359 STUDENT: 2. 01:08:48.359 --> 01:08:49.910 PROFESSOR: So what matters is that we 01:08:49.910 --> 01:08:52.290 take the flux in absolute value because it 01:08:52.290 --> 01:08:54.420 depends on the orientation of the normal. 01:08:54.420 --> 01:08:57.189 If we take the normal [INAUDIBLE]. 01:08:57.189 --> 01:09:02.950 Please, one thing I want you to do when you go home now, 01:09:02.950 --> 01:09:06.029 open the book which maybe you rarely do, 01:09:06.029 --> 01:09:08.529 but now it's really-- the material 01:09:08.529 --> 01:09:10.130 became complicated enough. 01:09:10.130 --> 01:09:14.170 We are not just doing math, calculus, we are doing physics, 01:09:14.170 --> 01:09:17.640 we are doing mechanics, we are dealing with surface integrals 01:09:17.640 --> 01:09:19.069 and flux. 01:09:19.069 --> 01:09:27.120 I want you to open the book at page-- I don't know. 01:09:27.120 --> 01:09:32.790 At surface integrals starts at page 1,063. 01:09:32.790 --> 01:09:34.960 Section 13.5. 01:09:34.960 --> 01:09:38.250 And it keeps going like that, pretty pictures of surfaces 01:09:38.250 --> 01:09:40.680 and fluxes and so on. 01:09:40.680 --> 01:09:41.840 Vector fields. 01:09:41.840 --> 01:09:43.660 And it keeps going like that. 01:09:43.660 --> 01:09:48.890 But it doesn't cover anything new except what I said today. 01:09:48.890 --> 01:09:51.310 It's just that it shows you examples that are not 01:09:51.310 --> 01:09:54.810 as beautiful as the ones I gave, but they are essentially 01:09:54.810 --> 01:09:58.190 the same, only a little bit nastier to complete. 01:09:58.190 --> 01:10:02.480 So up to 1,072. 01:10:02.480 --> 01:10:05.740 So that is what you're going to do this weekend, 01:10:05.740 --> 01:10:06.760 plus the homework. 01:10:06.760 --> 01:10:07.800 Keep on the homework. 01:10:07.800 --> 01:10:11.070 Now, if you get stuck Saturday, Sunday, 01:10:11.070 --> 01:10:13.977 whenever you try your homework you get stuck, what do you do? 01:10:13.977 --> 01:10:14.810 STUDENT: [INAUDIBLE] 01:10:14.810 --> 01:10:15.810 PROFESSOR: You email me. 01:10:15.810 --> 01:10:18.460 So you say what in the world is going on with this problem 01:10:18.460 --> 01:10:25.045 because I tried it seven times and-- 88 times. 01:10:25.045 --> 01:10:27.177 And then you got the brownie points. 01:10:27.177 --> 01:10:28.010 STUDENT: [INAUDIBLE] 01:10:28.010 --> 01:10:30.200 PROFESSOR: [INAUDIBLE] problem. 01:10:30.200 --> 01:10:31.470 STUDENT: [INAUDIBLE] by 32. 01:10:31.470 --> 01:10:33.240 PROFESSOR: There was a problem, guys. 01:10:33.240 --> 01:10:35.110 There are not so many problems. 01:10:35.110 --> 01:10:38.930 But the only part, serious part that we would catch, 01:10:38.930 --> 01:10:41.670 he found it first, and he tried it 88 times. 01:10:41.670 --> 01:10:44.400 01:10:44.400 --> 01:10:47.155 I'll never forget you, though, because you are unique, 01:10:47.155 --> 01:10:49.970 and that-- I appreciated that very much. 01:10:49.970 --> 01:10:55.380 So doing this weekend, do not hesitate to pester. 01:10:55.380 --> 01:10:58.400 I will answer all the web work problems you have. 01:10:58.400 --> 01:11:00.100 I want you to do well. 01:11:00.100 --> 01:11:01.980 Next week is the last week on new theory, 01:11:01.980 --> 01:11:04.380 and then we start working for the final, 01:11:04.380 --> 01:11:09.060 so by the time of the final, you'll be [INAUDIBLE]. 01:11:09.060 --> 01:11:10.317 STUDENT: [INAUDIBLE]? 01:11:10.317 --> 01:11:11.150 PROFESSOR: Yes, sir. 01:11:11.150 --> 01:11:13.377 Oh, I appreciated that you did that. 01:11:13.377 --> 01:11:14.210 STUDENT: [INAUDIBLE] 01:11:14.210 --> 01:11:18.200 01:11:18.200 --> 01:11:20.592 PROFESSOR: Again, I forgot these. 01:11:20.592 --> 01:11:24.920 01:11:24.920 --> 01:11:28.456 With the extra points you got, you shouldn't care. 01:11:28.456 --> 01:11:28.956