[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:00.50,Default,,0000,0000,0000,, Dialogue: 0,0:00:00.50,0:00:04.05,Default,,0000,0000,0000,,PROFESSOR: I would like to\Nreview just briefly what Dialogue: 0,0:00:04.05,0:00:06.69,Default,,0000,0000,0000,,we discussed last time. Dialogue: 0,0:00:06.69,0:00:15.53,Default,,0000,0000,0000,,We gave very important results,\Nand that was Green's Theorem. Dialogue: 0,0:00:15.53,0:00:19.11,Default,,0000,0000,0000,,And I would like to\Nknow if you remember Dialogue: 0,0:00:19.11,0:00:25.28,Default,,0000,0000,0000,,when I said about the\Nsettling for this problem. Dialogue: 0,0:00:25.28,0:00:30.66,Default,,0000,0000,0000,,So we'll assume we have\Na domain without a hole, Dialogue: 0,0:00:30.66,0:00:34.79,Default,,0000,0000,0000,,D. D is a domain\Nwithout a hole inside, Dialogue: 0,0:00:34.79,0:00:39.76,Default,,0000,0000,0000,,without punctures or holes. Dialogue: 0,0:00:39.76,0:00:43.41,Default,,0000,0000,0000,, Dialogue: 0,0:00:43.41,0:00:47.93,Default,,0000,0000,0000,,There is a scientific name in\Nmathematics for such a domain. Dialogue: 0,0:00:47.93,0:00:50.53,Default,,0000,0000,0000,,This is going to be\Nsimply connected. Dialogue: 0,0:00:50.53,0:00:56.08,Default,,0000,0000,0000,, Dialogue: 0,0:00:56.08,0:00:59.86,Default,,0000,0000,0000,,And this is a difficult\Ntopological theorem, Dialogue: 0,0:00:59.86,0:01:02.04,Default,,0000,0000,0000,,but this is what we expect, OK? Dialogue: 0,0:01:02.04,0:01:03.60,Default,,0000,0000,0000,,And what does it mean? Dialogue: 0,0:01:03.60,0:01:04.72,Default,,0000,0000,0000,,What does it mean? Dialogue: 0,0:01:04.72,0:01:14.57,Default,,0000,0000,0000,,It means that in the C being\Na Jordan curve was what? Dialogue: 0,0:01:14.57,0:01:15.47,Default,,0000,0000,0000,,How? Dialogue: 0,0:01:15.47,0:01:20.79,Default,,0000,0000,0000,,This was continuous,\Nno self intersections. Dialogue: 0,0:01:20.79,0:01:27.23,Default,,0000,0000,0000,, Dialogue: 0,0:01:27.23,0:01:34.66,Default,,0000,0000,0000,,In such a case, we set up\NM and N to be C1 functions. Dialogue: 0,0:01:34.66,0:01:39.44,Default,,0000,0000,0000,, Dialogue: 0,0:01:39.44,0:01:43.48,Default,,0000,0000,0000,,And then we proceed through\Nthe path integral of C. Dialogue: 0,0:01:43.48,0:01:46.69,Default,,0000,0000,0000,,Do you like this kind of C,\Nor you prefer a straight C? Dialogue: 0,0:01:46.69,0:01:57.20,Default,,0000,0000,0000,,The path integral of C of M\Nof xy dx class, N of xy, dy, Dialogue: 0,0:01:57.20,0:01:59.01,Default,,0000,0000,0000,,everything is in plane. Dialogue: 0,0:01:59.01,0:02:03.64,Default,,0000,0000,0000,,I'm sorry that I\Ncannot repeat that, Dialogue: 0,0:02:03.64,0:02:06.69,Default,,0000,0000,0000,,but we discussed that\Ntime, is in the plane of 2. Dialogue: 0,0:02:06.69,0:02:10.14,Default,,0000,0000,0000,,And then what-- do\Nyou remember in terms Dialogue: 0,0:02:10.14,0:02:16.12,Default,,0000,0000,0000,,of how this path integral,\N[INAUDIBLE] inside, Dialogue: 0,0:02:16.12,0:02:21.66,Default,,0000,0000,0000,,is connected to a double\Nintegral over the whole domain. Dialogue: 0,0:02:21.66,0:02:26.40,Default,,0000,0000,0000,,In particular, do you remember--\Nthis is easy to memorize-- Dialogue: 0,0:02:26.40,0:02:28.27,Default,,0000,0000,0000,,but do you remember\Nwhat's inside? Dialogue: 0,0:02:28.27,0:02:36.80,Default,,0000,0000,0000,,Because for the final, you are\Nexpected to know his result. Dialogue: 0,0:02:36.80,0:02:38.64,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:02:38.64,0:02:43.17,Default,,0000,0000,0000,,PROFESSOR: N sub X. Dialogue: 0,0:02:43.17,0:02:44.54,Default,,0000,0000,0000,,STUDENT: Minus M sub Y. Dialogue: 0,0:02:44.54,0:02:47.12,Default,,0000,0000,0000,,PROFESSOR: Minus M\Nsub Y. [INAUDIBLE] Dialogue: 0,0:02:47.12,0:02:53.42,Default,,0000,0000,0000,,must M-- M and N-- M sub Y.\NHere is the Y. Of course this Dialogue: 0,0:02:53.42,0:02:58.63,Default,,0000,0000,0000,,would be dA in plane,\Nand in the-- if you Dialogue: 0,0:02:58.63,0:03:02.49,Default,,0000,0000,0000,,want to represent this\Nin the general format, Dialogue: 0,0:03:02.49,0:03:05.63,Default,,0000,0000,0000,,the MdX minus the MdY. Dialogue: 0,0:03:05.63,0:03:08.30,Default,,0000,0000,0000,,Feel free to do that. Dialogue: 0,0:03:08.30,0:03:13.06,Default,,0000,0000,0000,,One was a correlary\Nor a consequence. Dialogue: 0,0:03:13.06,0:03:22.08,Default,,0000,0000,0000,,This theorem was that if I\Nwere to take this big M to be Dialogue: 0,0:03:22.08,0:03:26.33,Default,,0000,0000,0000,,the minus Y as a function,\Nthen this function N will Dialogue: 0,0:03:26.33,0:03:29.57,Default,,0000,0000,0000,,be plus X, what will I get? Dialogue: 0,0:03:29.57,0:03:37.93,Default,,0000,0000,0000,,I would get that minus\NYdX plus NdY will be what? Dialogue: 0,0:03:37.93,0:03:40.13,Default,,0000,0000,0000,, Dialogue: 0,0:03:40.13,0:03:40.96,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:03:40.96,0:03:42.29,Default,,0000,0000,0000,,PROFESSOR: Two times, excellent. Dialogue: 0,0:03:42.29,0:03:43.59,Default,,0000,0000,0000,,You are very awake. Dialogue: 0,0:03:43.59,0:03:45.00,Default,,0000,0000,0000,,So I wanted to catch you. Dialogue: 0,0:03:45.00,0:03:46.21,Default,,0000,0000,0000,,I couldn't catch you. Dialogue: 0,0:03:46.21,0:03:48.32,Default,,0000,0000,0000,,I thought you would say\Nthe A of the domain, Dialogue: 0,0:03:48.32,0:03:49.50,Default,,0000,0000,0000,,but you said it right. Dialogue: 0,0:03:49.50,0:03:52.78,Default,,0000,0000,0000,,You said Y is the\Narea of the domain. Dialogue: 0,0:03:52.78,0:03:55.70,Default,,0000,0000,0000,,You probably\Nalready in your mind Dialogue: 0,0:03:55.70,0:04:01.82,Default,,0000,0000,0000,,did the math saying X sub X\Nis one, minus Y sub 1 is 1. Dialogue: 0,0:04:01.82,0:04:05.74,Default,,0000,0000,0000,,1 plus 1 is two, so the\Ntwo part [INAUDIBLE]. Dialogue: 0,0:04:05.74,0:04:08.97,Default,,0000,0000,0000,,OK, so what did we do with it? Dialogue: 0,0:04:08.97,0:04:10.22,Default,,0000,0000,0000,,We just stared at it? Dialogue: 0,0:04:10.22,0:04:10.72,Default,,0000,0000,0000,,No. Dialogue: 0,0:04:10.72,0:04:11.84,Default,,0000,0000,0000,,We didn't just stare at it. Dialogue: 0,0:04:11.84,0:04:14.47,Default,,0000,0000,0000,,We did something nice\Nwith it last time. Dialogue: 0,0:04:14.47,0:04:21.26,Default,,0000,0000,0000,,We proved that, finally,\Nthat the area, this radius R Dialogue: 0,0:04:21.26,0:04:23.74,Default,,0000,0000,0000,,will be pi R\Nsquared, and we also Dialogue: 0,0:04:23.74,0:04:28.10,Default,,0000,0000,0000,,proved that the area\N[INAUDIBLE] is what? Dialogue: 0,0:04:28.10,0:04:30.66,Default,,0000,0000,0000,,I'm testing you to\Nsee if you remember. Dialogue: 0,0:04:30.66,0:04:31.28,Default,,0000,0000,0000,,STUDENT: AB pi. Dialogue: 0,0:04:31.28,0:04:31.62,Default,,0000,0000,0000,,PROFESSOR: AB pi. Dialogue: 0,0:04:31.62,0:04:32.34,Default,,0000,0000,0000,,Very good. Dialogue: 0,0:04:32.34,0:04:33.50,Default,,0000,0000,0000,,Or pi AB. Dialogue: 0,0:04:33.50,0:04:38.03,Default,,0000,0000,0000,,It's more, I like it the\Nway you said it, AB pi, Dialogue: 0,0:04:38.03,0:04:41.81,Default,,0000,0000,0000,,because pi is a transcendental\Nnumber, and you go around Dialogue: 0,0:04:41.81,0:04:44.59,Default,,0000,0000,0000,,and it's like partly\Nvariable to put at the end. Dialogue: 0,0:04:44.59,0:04:47.49,Default,,0000,0000,0000,,And the real numbers\Nthat could be anything, Dialogue: 0,0:04:47.49,0:04:54.63,Default,,0000,0000,0000,,so [INAUDIBLE] they are the\Nsemi axes of the ellipse. Dialogue: 0,0:04:54.63,0:04:59.47,Default,,0000,0000,0000,,So we gain new knowledge and\Nwe are ready to move forward. Dialogue: 0,0:04:59.47,0:05:02.22,Default,,0000,0000,0000,,And we're going to move\Nforward to something Dialogue: 0,0:05:02.22,0:05:10.59,Default,,0000,0000,0000,,called section 13.5, which\Nis the surface integral. Dialogue: 0,0:05:10.59,0:05:12.66,Default,,0000,0000,0000,,We will come back\Nto Green's Theorem Dialogue: 0,0:05:12.66,0:05:15.53,Default,,0000,0000,0000,,because there are\Ngeneralizations Dialogue: 0,0:05:15.53,0:05:18.68,Default,,0000,0000,0000,,of the Green's Theorem to\Nmore complicate the case. Dialogue: 0,0:05:18.68,0:05:21.18,Default,,0000,0000,0000,,But in order to\Nsee those, we have Dialogue: 0,0:05:21.18,0:05:23.65,Default,,0000,0000,0000,,to learn a little bit more. Dialogue: 0,0:05:23.65,0:05:27.61,Default,,0000,0000,0000,,In mathematics, you need to\Nknow many things, many pieces Dialogue: 0,0:05:27.61,0:05:31.84,Default,,0000,0000,0000,,of the puzzle, and then\Nyou put them together Dialogue: 0,0:05:31.84,0:05:34.28,Default,,0000,0000,0000,,to get the whole picture. Dialogue: 0,0:05:34.28,0:05:37.65,Default,,0000,0000,0000,,All right, so what\Nis 13.5 about? Dialogue: 0,0:05:37.65,0:05:39.22,Default,,0000,0000,0000,,This is just review. Dialogue: 0,0:05:39.22,0:05:46.90,Default,,0000,0000,0000,,13.5, if should be looking like\Na friend, old friend, to you. Dialogue: 0,0:05:46.90,0:05:48.62,Default,,0000,0000,0000,,And I'll show you\Nin a minute why this Dialogue: 0,0:05:48.62,0:05:49.92,Default,,0000,0000,0000,,is called the surface integral. Dialogue: 0,0:05:49.92,0:05:54.44,Default,,0000,0000,0000,, Dialogue: 0,0:05:54.44,0:05:57.95,Default,,0000,0000,0000,,I saw that US natives\Ndon't pronounce integral, Dialogue: 0,0:05:57.95,0:05:59.64,Default,,0000,0000,0000,,they pronounce in-negral. Dialogue: 0,0:05:59.64,0:06:04.41,Default,,0000,0000,0000,,And everybody that I heard\Nin romance language-speaking Dialogue: 0,0:06:04.41,0:06:10.23,Default,,0000,0000,0000,,countries like Spanish,\NItalian, Portuguese, Dialogue: 0,0:06:10.23,0:06:13.75,Default,,0000,0000,0000,,they put the T there\Nout, very visibly. Dialogue: 0,0:06:13.75,0:06:16.70,Default,,0000,0000,0000,,So it doesn't matter. Dialogue: 0,0:06:16.70,0:06:19.51,Default,,0000,0000,0000,,Even some accent difference\Nin different parts Dialogue: 0,0:06:19.51,0:06:22.05,Default,,0000,0000,0000,,of the United States\Npronounce it differently. Dialogue: 0,0:06:22.05,0:06:25.04,Default,,0000,0000,0000,,So what is the surface\Nintegral about? Dialogue: 0,0:06:25.04,0:06:32.25,Default,,0000,0000,0000,,It's about integrating a smooth\Nfunction, not a vector value, Dialogue: 0,0:06:32.25,0:06:34.53,Default,,0000,0000,0000,,but a real value function. Dialogue: 0,0:06:34.53,0:06:40.82,Default,,0000,0000,0000,,Let's say you have G or XY being\Na nice interglobal function Dialogue: 0,0:06:40.82,0:06:46.35,Default,,0000,0000,0000,,over some surfaces. Dialogue: 0,0:06:46.35,0:06:50.23,Default,,0000,0000,0000,,And you say, I'm\Ngoing to take it, Dialogue: 0,0:06:50.23,0:07:03.36,Default,,0000,0000,0000,,double integral, over S of GDS,\Nwhere DS will be area level. Dialogue: 0,0:07:03.36,0:07:08.60,Default,,0000,0000,0000,, Dialogue: 0,0:07:08.60,0:07:11.84,Default,,0000,0000,0000,,I had a student one time who\Nlooked at two different books Dialogue: 0,0:07:11.84,0:07:16.21,Default,,0000,0000,0000,,and said, I have a problem\Nwith this, [INAUDIBLE]. Dialogue: 0,0:07:16.21,0:07:23.33,Default,,0000,0000,0000,,In one book it shows a\Nbig, fat snake over S. Dialogue: 0,0:07:23.33,0:07:25.96,Default,,0000,0000,0000,,And in another book, a\Ndouble integral over it, Dialogue: 0,0:07:25.96,0:07:28.82,Default,,0000,0000,0000,,and I don't know which one it\Nis because I don't understand. Dialogue: 0,0:07:28.82,0:07:32.37,Default,,0000,0000,0000,,No matter how you denote it,\Nit's still a double integral. Dialogue: 0,0:07:32.37,0:07:33.26,Default,,0000,0000,0000,,You know why? Dialogue: 0,0:07:33.26,0:07:35.23,Default,,0000,0000,0000,,Because it's an\Nintegral over a surface. Dialogue: 0,0:07:35.23,0:07:41.42,Default,,0000,0000,0000,,The same thing, integral over\Na surface or a domain plane, Dialogue: 0,0:07:41.42,0:07:45.05,Default,,0000,0000,0000,,or anything two-dimensional\Nwill be a double integral. Dialogue: 0,0:07:45.05,0:07:47.51,Default,,0000,0000,0000,,So it doesn't matter\Nhow you denote it. Dialogue: 0,0:07:47.51,0:07:49.82,Default,,0000,0000,0000,,In the end, it's going\Nto be a double integral. Dialogue: 0,0:07:49.82,0:07:54.42,Default,,0000,0000,0000,,Now, what in the world\Ndo we mean by that? Dialogue: 0,0:07:54.42,0:07:56.81,Default,,0000,0000,0000,,DS is an old friend of\Nyours, and I don't know Dialogue: 0,0:07:56.81,0:07:59.93,Default,,0000,0000,0000,,if you remember him at all. Dialogue: 0,0:07:59.93,0:08:08.30,Default,,0000,0000,0000,,He was infinitesimal element\Non some curved or linear patch. Dialogue: 0,0:08:08.30,0:08:11.65,Default,,0000,0000,0000,,Imagine your favorite surface. Dialogue: 0,0:08:11.65,0:08:13.72,Default,,0000,0000,0000,,Let's assume it's a graph. Dialogue: 0,0:08:13.72,0:08:16.45,Default,,0000,0000,0000,,It doesn't have to be a graph,\Nbut let's assume it's a graph. Dialogue: 0,0:08:16.45,0:08:19.70,Default,,0000,0000,0000,,And that's your\Nfavorite surface S. Dialogue: 0,0:08:19.70,0:08:23.17,Default,,0000,0000,0000,,And then you draw\Ncoordinate lines, Dialogue: 0,0:08:23.17,0:08:25.72,Default,,0000,0000,0000,,and you are looking at a patch. Dialogue: 0,0:08:25.72,0:08:28.58,Default,,0000,0000,0000,, Dialogue: 0,0:08:28.58,0:08:32.08,Default,,0000,0000,0000,,And this patch looks small,\Nbut it's not small enough. Dialogue: 0,0:08:32.08,0:08:35.03,Default,,0000,0000,0000,,I want this to be\Ninfinitesimally small. Dialogue: 0,0:08:35.03,0:08:41.37,Default,,0000,0000,0000,,Imagine that these curvature\Nlines become closer and closer Dialogue: 0,0:08:41.37,0:08:42.40,Default,,0000,0000,0000,,to one another. Dialogue: 0,0:08:42.40,0:08:46.00,Default,,0000,0000,0000,,And then we look in the\Ndirections of DX and DY, Dialogue: 0,0:08:46.00,0:08:48.98,Default,,0000,0000,0000,,and then you say, wait a\Nminute, I'm not in plane. Dialogue: 0,0:08:48.98,0:08:54.92,Default,,0000,0000,0000,,If I were in plane,\NDA will be DX, DY. Dialogue: 0,0:08:54.92,0:08:59.24,Default,,0000,0000,0000,,If you work with [INAUDIBLE],\NI will be DX with DY. Dialogue: 0,0:08:59.24,0:09:00.91,Default,,0000,0000,0000,,So we've matched\Nthe orientation. Dialogue: 0,0:09:00.91,0:09:02.72,Default,,0000,0000,0000,,If you would change\NDY, [INAUDIBLE] Dialogue: 0,0:09:02.72,0:09:04.76,Default,,0000,0000,0000,,put the minus in front. Dialogue: 0,0:09:04.76,0:09:08.31,Default,,0000,0000,0000,,But this happens\Nbecause-- thank God this Dialogue: 0,0:09:08.31,0:09:16.08,Default,,0000,0000,0000,,will be a rectangular 1 patch\Nin plane, in the plane of 2. Dialogue: 0,0:09:16.08,0:09:18.30,Default,,0000,0000,0000,,But what if you\Nwere on the surface? Dialogue: 0,0:09:18.30,0:09:21.93,Default,,0000,0000,0000,,On the surface, you\Ndon't have this animal. Dialogue: 0,0:09:21.93,0:09:25.57,Default,,0000,0000,0000,,You will have-- which animal--\NI'm testing your knowledge. Dialogue: 0,0:09:25.57,0:09:27.43,Default,,0000,0000,0000,,I'm doing review with you. Dialogue: 0,0:09:27.43,0:09:29.14,Default,,0000,0000,0000,,For sure, you will\Nsee something that Dialogue: 0,0:09:29.14,0:09:31.25,Default,,0000,0000,0000,,involves the S in the final. Dialogue: 0,0:09:31.25,0:09:33.84,Default,,0000,0000,0000,,Have you started browsing\Nthrough those finals Dialogue: 0,0:09:33.84,0:09:34.75,Default,,0000,0000,0000,,I sent you? Dialogue: 0,0:09:34.75,0:09:37.32,Default,,0000,0000,0000,,Just out of curiosity. Dialogue: 0,0:09:37.32,0:09:40.96,Default,,0000,0000,0000,,And do they look awful to you? Dialogue: 0,0:09:40.96,0:09:42.09,Default,,0000,0000,0000,,They look awful to you. Dialogue: 0,0:09:42.09,0:09:43.06,Default,,0000,0000,0000,,Come on. Dialogue: 0,0:09:43.06,0:09:45.07,Default,,0000,0000,0000,,I'm going to work with\Nyou on some of those. Dialogue: 0,0:09:45.07,0:09:47.50,Default,,0000,0000,0000,,I don't want you to\Nhave-- I don't want Dialogue: 0,0:09:47.50,0:09:49.19,Default,,0000,0000,0000,,you to be afraid of this final. Dialogue: 0,0:09:49.19,0:09:51.20,Default,,0000,0000,0000,,Because compared to\Nother exams that you'll Dialogue: 0,0:09:51.20,0:09:54.08,Default,,0000,0000,0000,,have in other courses,\Nwhere a lot of memorization Dialogue: 0,0:09:54.08,0:09:57.66,Default,,0000,0000,0000,,is emphasized, this\Nshould not be a problem. Dialogue: 0,0:09:57.66,0:10:00.80,Default,,0000,0000,0000,,So you could go over\Nthe types of problems Dialogue: 0,0:10:00.80,0:10:04.38,Default,,0000,0000,0000,,that are significant\Nin this course, Dialogue: 0,0:10:04.38,0:10:08.42,Default,,0000,0000,0000,,you will not have any-- you\Nshouldn't have any problem. Dialogue: 0,0:10:08.42,0:10:10.81,Default,,0000,0000,0000,,And I sent you three samples. Dialogue: 0,0:10:10.81,0:10:13.74,Default,,0000,0000,0000,,Didn't I send you three\Nsamples with solutions? Dialogue: 0,0:10:13.74,0:10:17.31,Default,,0000,0000,0000,,Those are going to help\Nyou once you read the exam Dialogue: 0,0:10:17.31,0:10:19.90,Default,,0000,0000,0000,,and you can go ahead\Nand try the exam Dialogue: 0,0:10:19.90,0:10:22.04,Default,,0000,0000,0000,,or go ahead, read the solutions. Dialogue: 0,0:10:22.04,0:10:26.78,Default,,0000,0000,0000,,If I give you more of that, then\Nyou should be doctors in those, Dialogue: 0,0:10:26.78,0:10:30.32,Default,,0000,0000,0000,,and you would be able to\Nsolve them yourselves. Dialogue: 0,0:10:30.32,0:10:33.24,Default,,0000,0000,0000,,What about this one? Dialogue: 0,0:10:33.24,0:10:34.57,Default,,0000,0000,0000,,This is not DA. Dialogue: 0,0:10:34.57,0:10:37.96,Default,,0000,0000,0000,,It's a DA times something. Dialogue: 0,0:10:37.96,0:10:44.07,Default,,0000,0000,0000,,There is some factor in front\Nof that, and why is that? Dialogue: 0,0:10:44.07,0:10:46.60,Default,,0000,0000,0000,,In case of Z equals\NF of X and Y, Dialogue: 0,0:10:46.60,0:10:49.43,Default,,0000,0000,0000,,you should know that by heart,\Nand I know that some of you Dialogue: 0,0:10:49.43,0:10:50.40,Default,,0000,0000,0000,,know it. Dialogue: 0,0:10:50.40,0:10:52.50,Default,,0000,0000,0000,,You just have to ring\Nthe bell, and I'll Dialogue: 0,0:10:52.50,0:10:53.85,Default,,0000,0000,0000,,start ringing the bell. Dialogue: 0,0:10:53.85,0:10:55.87,Default,,0000,0000,0000,,Look at my first step. Dialogue: 0,0:10:55.87,0:10:57.78,Default,,0000,0000,0000,,And now you know, right? Dialogue: 0,0:10:57.78,0:10:59.47,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] 1-- Dialogue: 0,0:10:59.47,0:11:00.65,Default,,0000,0000,0000,,PROFESSOR: I start with 1. Dialogue: 0,0:11:00.65,0:11:01.70,Default,,0000,0000,0000,,You said it right. Dialogue: 0,0:11:01.70,0:11:02.83,Default,,0000,0000,0000,,1 plus-- Dialogue: 0,0:11:02.83,0:11:03.59,Default,,0000,0000,0000,,STUDENT: F of X. Dialogue: 0,0:11:03.59,0:11:04.82,Default,,0000,0000,0000,,PROFESSOR: F of X-- Dialogue: 0,0:11:04.82,0:11:06.18,Default,,0000,0000,0000,,STUDENT: F squared. Dialogue: 0,0:11:06.18,0:11:07.64,Default,,0000,0000,0000,,PROFESSOR: Squared plus-- Dialogue: 0,0:11:07.64,0:11:08.47,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:11:08.47,0:11:09.62,Default,,0000,0000,0000,,PROFESSOR: --SY squared. Dialogue: 0,0:11:09.62,0:11:11.27,Default,,0000,0000,0000,,So this what you're doing. Dialogue: 0,0:11:11.27,0:11:13.02,Default,,0000,0000,0000,,What are you going to do? Dialogue: 0,0:11:13.02,0:11:14.69,Default,,0000,0000,0000,,You're going to\Ndo wait a minute. Dialogue: 0,0:11:14.69,0:11:18.13,Default,,0000,0000,0000,,This animal of mine,\Nthat looks so scary, Dialogue: 0,0:11:18.13,0:11:21.43,Default,,0000,0000,0000,,this is nothing but what? Dialogue: 0,0:11:21.43,0:11:27.30,Default,,0000,0000,0000,,It's the same thing as, not\Nthe picture, my picture. Dialogue: 0,0:11:27.30,0:11:32.15,Default,,0000,0000,0000,,It's going to be double integral\Nover a plane or domain D. Dialogue: 0,0:11:32.15,0:11:34.26,Default,,0000,0000,0000,,Well, I just said\Ngoodbye to the picture, Dialogue: 0,0:11:34.26,0:11:37.25,Default,,0000,0000,0000,,but I find you are really smart. Dialogue: 0,0:11:37.25,0:11:41.15,Default,,0000,0000,0000,,I would have drawn the\N[INAUDIBLE] of a picture here. Dialogue: 0,0:11:41.15,0:11:43.76,Default,,0000,0000,0000,,This is S and this\Nis D. What is D? Dialogue: 0,0:11:43.76,0:11:46.81,Default,,0000,0000,0000,,It's the projection,\Nprojects the shadow. Dialogue: 0,0:11:46.81,0:11:51.10,Default,,0000,0000,0000,,The projection of S\Non the plane XY when Dialogue: 0,0:11:51.10,0:11:53.92,Default,,0000,0000,0000,,I have to deal with a graph. Dialogue: 0,0:11:53.92,0:11:57.38,Default,,0000,0000,0000,,So when I have to deal with a\Ngraph, my life is really easy. Dialogue: 0,0:11:57.38,0:12:06.02,Default,,0000,0000,0000,,And I said I'd get double\Nintegral over D of G of God Dialogue: 0,0:12:06.02,0:12:11.68,Default,,0000,0000,0000,,knows what in the end will\Nbe a function of X and Y. OK? Dialogue: 0,0:12:11.68,0:12:15.90,Default,,0000,0000,0000,,And here I'm going to have\Nsquare root of this animal. Dialogue: 0,0:12:15.90,0:12:18.33,Default,,0000,0000,0000,,Let me change it,\NF sub X squared Dialogue: 0,0:12:18.33,0:12:20.93,Default,,0000,0000,0000,,like-- because in this\None it is like that. Dialogue: 0,0:12:20.93,0:12:21.50,Default,,0000,0000,0000,,Plus 1. Dialogue: 0,0:12:21.50,0:12:24.28,Default,,0000,0000,0000,,It doesn't matter\Nwhere I put the 1. Dialogue: 0,0:12:24.28,0:12:25.12,Default,,0000,0000,0000,,DXDY. Dialogue: 0,0:12:25.12,0:12:31.06,Default,,0000,0000,0000,,DXDY will be like the area of an\Ninfinitesimally small rectangle Dialogue: 0,0:12:31.06,0:12:34.25,Default,,0000,0000,0000,,based on displacement\NDX and displacement DY Dialogue: 0,0:12:34.25,0:12:35.37,Default,,0000,0000,0000,,and disintegration. Dialogue: 0,0:12:35.37,0:12:37.35,Default,,0000,0000,0000,,So this is DA. Dialogue: 0,0:12:37.35,0:12:42.82,Default,,0000,0000,0000,,Make the distinction\Nbetween the DA and the DX. Dialogue: 0,0:12:42.82,0:12:46.95,Default,,0000,0000,0000,,Can I draw the two animals? Dialogue: 0,0:12:46.95,0:12:50.75,Default,,0000,0000,0000,,Let me try again. Dialogue: 0,0:12:50.75,0:12:55.49,Default,,0000,0000,0000,,So you have the\Ndirection of X and Y. Dialogue: 0,0:12:55.49,0:12:59.30,Default,,0000,0000,0000,,You have to be imaginative and\Nsee that some coordinate lines Dialogue: 0,0:12:59.30,0:13:02.99,Default,,0000,0000,0000,,are [INAUDIBLE] for fixing Y. Dialogue: 0,0:13:02.99,0:13:06.88,Default,,0000,0000,0000,,When I fix Y, I sliced a\Nlot like that very nicely. Dialogue: 0,0:13:06.88,0:13:11.71,Default,,0000,0000,0000,,That's the same piece of\Ncheese that I've been dreaming Dialogue: 0,0:13:11.71,0:13:13.15,Default,,0000,0000,0000,,because I didn't have lunch. Dialogue: 0,0:13:13.15,0:13:15.46,Default,,0000,0000,0000,,I was too busy not\Nto have any lunch. Dialogue: 0,0:13:15.46,0:13:17.40,Default,,0000,0000,0000,,So you slice it\Nup like that where Dialogue: 0,0:13:17.40,0:13:20.59,Default,,0000,0000,0000,,Y equals constant to slice\Nit up like that for X Dialogue: 0,0:13:20.59,0:13:21.70,Default,,0000,0000,0000,,equals constant. Dialogue: 0,0:13:21.70,0:13:24.95,Default,,0000,0000,0000,,What you get are so-called\Ncoordinate lines. Dialogue: 0,0:13:24.95,0:13:27.44,Default,,0000,0000,0000,,So the coordinate\Nlines are [INAUDIBLE]. Dialogue: 0,0:13:27.44,0:13:32.08,Default,,0000,0000,0000,,Y equals my zeros, and\NX equals the zeros. Dialogue: 0,0:13:32.08,0:13:35.81,Default,,0000,0000,0000,,And when they get to be\Nmany dense and refined, Dialogue: 0,0:13:35.81,0:13:42.31,Default,,0000,0000,0000,,your curvilinear element is\Nthis-- between two curves Dialogue: 0,0:13:42.31,0:13:46.01,Default,,0000,0000,0000,,like this two curves like that. Dialogue: 0,0:13:46.01,0:13:48.19,Default,,0000,0000,0000,,Shrunk in the limit. Dialogue: 0,0:13:48.19,0:13:50.64,Default,,0000,0000,0000,,It's an infinitesimal element. Dialogue: 0,0:13:50.64,0:13:54.47,Default,,0000,0000,0000,,This shadow is going\Nto be a rectangle. Dialogue: 0,0:13:54.47,0:13:55.91,Default,,0000,0000,0000,,Say that again, Magdalena. Dialogue: 0,0:13:55.91,0:13:59.41,Default,,0000,0000,0000,,This is not just\Ndelta X and delta Y. Dialogue: 0,0:13:59.41,0:14:03.65,Default,,0000,0000,0000,,This is DX and DY\Nbecause I shrink them Dialogue: 0,0:14:03.65,0:14:06.66,Default,,0000,0000,0000,,until it become\Ninfinitesimally small. Dialogue: 0,0:14:06.66,0:14:11.16,Default,,0000,0000,0000,,So you can imagine,\Nwhich one is bigger? Dialogue: 0,0:14:11.16,0:14:13.98,Default,,0000,0000,0000,,DS is bigger, or DA is bigger? Dialogue: 0,0:14:13.98,0:14:14.90,Default,,0000,0000,0000,,STUDENT: DS is bigger. Dialogue: 0,0:14:14.90,0:14:16.08,Default,,0000,0000,0000,,PROFESSOR: DS is bigger. Dialogue: 0,0:14:16.08,0:14:16.92,Default,,0000,0000,0000,,DS is bigger. Dialogue: 0,0:14:16.92,0:14:18.64,Default,,0000,0000,0000,,And can I see it's true? Dialogue: 0,0:14:18.64,0:14:19.14,Default,,0000,0000,0000,,Yes. Dialogue: 0,0:14:19.14,0:14:22.88,Default,,0000,0000,0000,,Because for God's sake, this\Nis greater than 1, right? Dialogue: 0,0:14:22.88,0:14:28.11,Default,,0000,0000,0000,,And if I multiply the\Nlittle orange area, by that, Dialogue: 0,0:14:28.11,0:14:32.74,Default,,0000,0000,0000,,I'm going to get this,\Nwhich is greater than 1. Dialogue: 0,0:14:32.74,0:14:38.14,Default,,0000,0000,0000,,They could be equal when\Nboth would be plainer, right? Dialogue: 0,0:14:38.14,0:14:41.32,Default,,0000,0000,0000,,If you have a plane or surface\Non top of a plane or surface, Dialogue: 0,0:14:41.32,0:14:43.16,Default,,0000,0000,0000,,then you have two\Ntiny rectangles Dialogue: 0,0:14:43.16,0:14:48.11,Default,,0000,0000,0000,,and you have like a prism\Nbetween them, goes down. Dialogue: 0,0:14:48.11,0:14:50.03,Default,,0000,0000,0000,,But in general, the\Ncurve in your [INAUDIBLE] Dialogue: 0,0:14:50.03,0:14:52.95,Default,,0000,0000,0000,,here-- let me make\Nhim more curvilinear. Dialogue: 0,0:14:52.95,0:14:55.31,Default,,0000,0000,0000,,He looks so-- so square. Dialogue: 0,0:14:55.31,0:15:01.46,Default,,0000,0000,0000,,But he's between two lines,\Nbut he's a curvilinear. Dialogue: 0,0:15:01.46,0:15:07.01,Default,,0000,0000,0000,,Dinah says that he belongs to a\Ncurved surface, not a flat one. Dialogue: 0,0:15:07.01,0:15:08.02,Default,,0000,0000,0000,,All right. Dialogue: 0,0:15:08.02,0:15:11.24,Default,,0000,0000,0000,,When he could be flat,\Nthese guys go away. Dialogue: 0,0:15:11.24,0:15:12.64,Default,,0000,0000,0000,,Zero and zero. Dialogue: 0,0:15:12.64,0:15:15.38,Default,,0000,0000,0000,,And that would be it. Dialogue: 0,0:15:15.38,0:15:23.75,Default,,0000,0000,0000,,If somebody else, they--\Nwell, this is hard to imagine, Dialogue: 0,0:15:23.75,0:15:29.97,Default,,0000,0000,0000,,but what if it could\Nbe a tiny-- this Dialogue: 0,0:15:29.97,0:15:32.03,Default,,0000,0000,0000,,would not be curvilinear, right? Dialogue: 0,0:15:32.03,0:15:37.29,Default,,0000,0000,0000,,But it would be something like\Na rectangular patch of a plane. Dialogue: 0,0:15:37.29,0:15:40.17,Default,,0000,0000,0000,,You have a grid in that plane. Dialogue: 0,0:15:40.17,0:15:46.40,Default,,0000,0000,0000,,And then it's just-- DS\Nwould be itself a rectangle. Dialogue: 0,0:15:46.40,0:15:48.86,Default,,0000,0000,0000,,When you project\Nthat rectangle here, Dialogue: 0,0:15:48.86,0:15:51.84,Default,,0000,0000,0000,,it will still be a rectangle. Dialogue: 0,0:15:51.84,0:15:54.90,Default,,0000,0000,0000,,When we were little-- I mean,\Nlittle, we were in K-12, Dialogue: 0,0:15:54.90,0:15:59.23,Default,,0000,0000,0000,,we're smart in math better\Nthan other people in class-- Dialogue: 0,0:15:59.23,0:16:03.26,Default,,0000,0000,0000,,did you ever have to do\Nanything with the two areas? Dialogue: 0,0:16:03.26,0:16:04.68,Default,,0000,0000,0000,,I did. Dialogue: 0,0:16:04.68,0:16:05.67,Default,,0000,0000,0000,,This was the shadow. Dialogue: 0,0:16:05.67,0:16:08.55,Default,,0000,0000,0000,,The projection in this\Nwas that [INAUDIBLE]. Dialogue: 0,0:16:08.55,0:16:11.51,Default,,0000,0000,0000,,And do you know what\Nthe relationship Dialogue: 0,0:16:11.51,0:16:14.19,Default,,0000,0000,0000,,would be if I have a plane. Dialogue: 0,0:16:14.19,0:16:17.52,Default,,0000,0000,0000,,I'm doing that for-- actually,\NI'm doing that for Casey Dialogue: 0,0:16:17.52,0:16:20.03,Default,,0000,0000,0000,,because she has something\Nsimilar to that. Dialogue: 0,0:16:20.03,0:16:22.91,Default,,0000,0000,0000,,So imagine that\Nyou have to project Dialogue: 0,0:16:22.91,0:16:27.65,Default,,0000,0000,0000,,a rectangle that's in plane to\Na rectangle that is the shadow. Dialogue: 0,0:16:27.65,0:16:30.34,Default,,0000,0000,0000,,The rectangle is on the ground. Dialogue: 0,0:16:30.34,0:16:33.56,Default,,0000,0000,0000,,The flat ground. Dialogue: 0,0:16:33.56,0:16:35.97,Default,,0000,0000,0000,,What's the relationship\Nbetween the two ends? Dialogue: 0,0:16:35.97,0:16:40.47,Default,,0000,0000,0000,, Dialogue: 0,0:16:40.47,0:16:42.88,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:16:42.88,0:16:44.67,Default,,0000,0000,0000,,PROFESSOR: No matter\Nwhat it is, but assume Dialogue: 0,0:16:44.67,0:16:48.12,Default,,0000,0000,0000,,it's like a rectangle up\Nhere and the shadow is also Dialogue: 0,0:16:48.12,0:16:50.20,Default,,0000,0000,0000,,a rectangle down here. Dialogue: 0,0:16:50.20,0:16:52.29,Default,,0000,0000,0000,,Obviously, the rectangle\Ndown here, the shadow Dialogue: 0,0:16:52.29,0:16:55.81,Default,,0000,0000,0000,,will be much smaller than\Nthis because this is oblique. Dialogue: 0,0:16:55.81,0:16:56.93,Default,,0000,0000,0000,,It's an oblique. Dialogue: 0,0:16:56.93,0:16:59.54,Default,,0000,0000,0000,,And assume that I\Nhave this plane making Dialogue: 0,0:16:59.54,0:17:03.55,Default,,0000,0000,0000,,an angle, a fixed angle with\Nthis laying on the table. Dialogue: 0,0:17:03.55,0:17:05.32,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:17:05.32,0:17:06.20,Default,,0000,0000,0000,,PROFESSOR: Excellent. Dialogue: 0,0:17:06.20,0:17:06.99,Default,,0000,0000,0000,,STUDENT: --cosine-- Dialogue: 0,0:17:06.99,0:17:08.72,Default,,0000,0000,0000,,PROFESSOR: Which one\Nis cosine of what? Dialogue: 0,0:17:08.72,0:17:12.07,Default,,0000,0000,0000,,So the S would be the\Nthe equal sign of theta, Dialogue: 0,0:17:12.07,0:17:18.62,Default,,0000,0000,0000,,or the A will be the\NS cosine of theta? Dialogue: 0,0:17:18.62,0:17:19.94,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] DA. Dialogue: 0,0:17:19.94,0:17:22.56,Default,,0000,0000,0000,,PROFESSOR: DA is the S\Ncosine of theta, a very smart Dialogue: 0,0:17:22.56,0:17:23.06,Default,,0000,0000,0000,,[INAUDIBLE]. Dialogue: 0,0:17:23.06,0:17:24.68,Default,,0000,0000,0000,,How does she know [INAUDIBLE]? Dialogue: 0,0:17:24.68,0:17:25.87,Default,,0000,0000,0000,,STUDENT: Because it's\Ngot to be less than one. Dialogue: 0,0:17:25.87,0:17:27.41,Default,,0000,0000,0000,,PROFESSOR: It's less\Nthan one, right? Dialogue: 0,0:17:27.41,0:17:29.48,Default,,0000,0000,0000,,Cosine theta is\Nbetween zero and one, Dialogue: 0,0:17:29.48,0:17:31.58,Default,,0000,0000,0000,,so you think which one is less. Dialogue: 0,0:17:31.58,0:17:33.63,Default,,0000,0000,0000,,All right, very good. Dialogue: 0,0:17:33.63,0:17:36.70,Default,,0000,0000,0000,,So when you have a\Nsimple example like that, Dialogue: 0,0:17:36.70,0:17:40.31,Default,,0000,0000,0000,,you were back to\Nyour K-12, and you Dialogue: 0,0:17:40.31,0:17:42.13,Default,,0000,0000,0000,,were happy-- I\Njust meant we were Dialogue: 0,0:17:42.13,0:17:44.58,Default,,0000,0000,0000,,avoiding three years of exams. Dialogue: 0,0:17:44.58,0:17:47.58,Default,,0000,0000,0000,,We only have [INAUDIBLE]. Dialogue: 0,0:17:47.58,0:17:49.99,Default,,0000,0000,0000,,But now exams became\Nserious, and look. Dialogue: 0,0:17:49.99,0:17:52.97,Default,,0000,0000,0000,,This is curvilinear\Nelemental variant. Dialogue: 0,0:17:52.97,0:17:58.92,Default,,0000,0000,0000,,So let me write it how\Npeople call the S's then. Dialogue: 0,0:17:58.92,0:18:01.17,Default,,0000,0000,0000,,Some people call it\Ncurvilinear elemental variant. Dialogue: 0,0:18:01.17,0:18:01.75,Default,,0000,0000,0000,,Yeah? Dialogue: 0,0:18:01.75,0:18:06.28,Default,,0000,0000,0000,,Many engineers I\Ntalk to do that. Dialogue: 0,0:18:06.28,0:18:09.30,Default,,0000,0000,0000,, Dialogue: 0,0:18:09.30,0:18:11.67,Default,,0000,0000,0000,,Now, I think we should just\Ncall it surface area element. Dialogue: 0,0:18:11.67,0:18:14.70,Default,,0000,0000,0000,, Dialogue: 0,0:18:14.70,0:18:18.21,Default,,0000,0000,0000,,[? I'm ?] a physicist, so you\Nalso say surface area element. Dialogue: 0,0:18:18.21,0:18:22.55,Default,,0000,0000,0000,,So I think we should just\Nlearn each other's language. Dialogue: 0,0:18:22.55,0:18:24.57,Default,,0000,0000,0000,,We are doing the same things. Dialogue: 0,0:18:24.57,0:18:27.93,Default,,0000,0000,0000,,We just-- we have a language\Nbarrier between-- it's Dialogue: 0,0:18:27.93,0:18:31.10,Default,,0000,0000,0000,,not writing interdisciplinary,\Nso if we could establish Dialogue: 0,0:18:31.10,0:18:35.30,Default,,0000,0000,0000,,a little bit more work in\Ncommon, because there are so Dialogue: 0,0:18:35.30,0:18:38.38,Default,,0000,0000,0000,,many applications to\Nengineering of this thing, Dialogue: 0,0:18:38.38,0:18:41.15,Default,,0000,0000,0000,,you have no idea yet. Dialogue: 0,0:18:41.15,0:18:47.15,Default,,0000,0000,0000,,OK, let's pick a problem like\Nthe ones we wrote in the book, Dialogue: 0,0:18:47.15,0:18:50.68,Default,,0000,0000,0000,,and see how hard it gets. Dialogue: 0,0:18:50.68,0:18:53.64,Default,,0000,0000,0000,,It shouldn't get very hard. Dialogue: 0,0:18:53.64,0:18:57.03,Default,,0000,0000,0000,,I'll start with one, the\Nonly one, that is naturally Dialogue: 0,0:18:57.03,0:19:00.32,Default,,0000,0000,0000,,coming to your mind\Nright now, which would Dialogue: 0,0:19:00.32,0:19:03.69,Default,,0000,0000,0000,,be the one where G would be 1. Dialogue: 0,0:19:03.69,0:19:06.46,Default,,0000,0000,0000,,Somebody has to tell\Nme what that would be. Dialogue: 0,0:19:06.46,0:19:10.02,Default,,0000,0000,0000,, Dialogue: 0,0:19:10.02,0:19:13.80,Default,,0000,0000,0000,,So guys, what if G would be 1? Dialogue: 0,0:19:13.80,0:19:14.92,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:19:14.92,0:19:15.79,Default,,0000,0000,0000,,PROFESSOR: Very good. Dialogue: 0,0:19:15.79,0:19:17.52,Default,,0000,0000,0000,,It would be the\NA of the surface. Dialogue: 0,0:19:17.52,0:19:21.74,Default,,0000,0000,0000,,I'm going to look for\Nsome simple application. Dialogue: 0,0:19:21.74,0:19:22.55,Default,,0000,0000,0000,,Nothing is simple. Dialogue: 0,0:19:22.55,0:19:27.60,Default,,0000,0000,0000,,Why did we make this problem,\Nthis book, so complicated? Dialogue: 0,0:19:27.60,0:19:28.48,Default,,0000,0000,0000,,OK, it' s good. Dialogue: 0,0:19:28.48,0:19:35.36,Default,,0000,0000,0000,,We can pick-- I can make\Nup a problem like this one. Dialogue: 0,0:19:35.36,0:19:39.44,Default,,0000,0000,0000,, Dialogue: 0,0:19:39.44,0:19:41.42,Default,,0000,0000,0000,,But I can do a better job. Dialogue: 0,0:19:41.42,0:19:43.49,Default,,0000,0000,0000,,I can give you an\Nbetter example. Dialogue: 0,0:19:43.49,0:19:46.62,Default,,0000,0000,0000,,I'm looking at the\Nexample 1 in section 13.5. Dialogue: 0,0:19:46.62,0:19:49.04,Default,,0000,0000,0000,,I'll give you\Nsomething like that Dialogue: 0,0:19:49.04,0:19:51.18,Default,,0000,0000,0000,,if I were to write an exam 1. Dialogue: 0,0:19:51.18,0:19:56.27,Default,,0000,0000,0000,,I put on it something\Nlike Z equals Dialogue: 0,0:19:56.27,0:19:58.45,Default,,0000,0000,0000,,X squared plus 1 squared. Dialogue: 0,0:19:58.45,0:20:02.13,Default,,0000,0000,0000,,You know is my favorite\Neggshell which is a [INAUDIBLE]. Dialogue: 0,0:20:02.13,0:20:16.49,Default,,0000,0000,0000,, Dialogue: 0,0:20:16.49,0:20:19.82,Default,,0000,0000,0000,,And somebody says,\NI'm not interested Dialogue: 0,0:20:19.82,0:20:25.43,Default,,0000,0000,0000,,in the whole surface,\Nwhich is infinitely large. Dialogue: 0,0:20:25.43,0:20:29.88,Default,,0000,0000,0000,,I'm only interested in a\Npiece of a surface that Dialogue: 0,0:20:29.88,0:20:39.82,Default,,0000,0000,0000,,is above the disk D of\Ncenter O and radius 1. Dialogue: 0,0:20:39.82,0:20:41.21,Default,,0000,0000,0000,,So say, what, Magdalena? Dialogue: 0,0:20:41.21,0:20:48.08,Default,,0000,0000,0000,,Say that I want just\Nthat part of the surface Dialogue: 0,0:20:48.08,0:20:54.03,Default,,0000,0000,0000,,that he's sitting above the\Ndisk of center O and radius 1. Dialogue: 0,0:20:54.03,0:20:58.42,Default,,0000,0000,0000,,And I want to know how to\Nset up the surface integral. Dialogue: 0,0:20:58.42,0:21:02.19,Default,,0000,0000,0000,,Set up main surface\Narea integral. Dialogue: 0,0:21:02.19,0:21:07.42,Default,,0000,0000,0000,, Dialogue: 0,0:21:07.42,0:21:10.00,Default,,0000,0000,0000,,And of course, when you\Nfirst see that you freak out Dialogue: 0,0:21:10.00,0:21:13.28,Default,,0000,0000,0000,,for a second, and then you say,\Nno, no, that's not a problem. Dialogue: 0,0:21:13.28,0:21:15.77,Default,,0000,0000,0000,,I know how to do that. Dialogue: 0,0:21:15.77,0:21:18.81,Default,,0000,0000,0000,,So example 1 out of\Nthis section would Dialogue: 0,0:21:18.81,0:21:27.01,Default,,0000,0000,0000,,be a double integral over your\NS. You have to call it names. Dialogue: 0,0:21:27.01,0:21:30.40,Default,,0000,0000,0000,,S. 1 instead of G and DS. Dialogue: 0,0:21:30.40,0:21:33.53,Default,,0000,0000,0000,, Dialogue: 0,0:21:33.53,0:21:35.59,Default,,0000,0000,0000,,But then you say wait a minute. Dialogue: 0,0:21:35.59,0:21:40.38,Default,,0000,0000,0000,,I know that is true, but I\Nhave to change it accordingly. Dialogue: 0,0:21:40.38,0:21:42.14,Default,,0000,0000,0000,,The same thing is here. Dialogue: 0,0:21:42.14,0:21:47.93,Default,,0000,0000,0000,,So I'm going to have it\Nover D. And D is the shadow, Dialogue: 0,0:21:47.93,0:21:52.77,Default,,0000,0000,0000,,DS is the plane of what? Dialogue: 0,0:21:52.77,0:21:55.44,Default,,0000,0000,0000,,Of 1 times. Dialogue: 0,0:21:55.44,0:21:59.79,Default,,0000,0000,0000,,I know I'm silly saying 1\Ntimes, but that's what it is. Dialogue: 0,0:21:59.79,0:22:07.40,Default,,0000,0000,0000,,Square root of-- S of X squared\Nplus S of Y squared plus 1. Dialogue: 0,0:22:07.40,0:22:13.93,Default,,0000,0000,0000,,DS, DY or DA as Rachel\Nsaid, somebody said. Dialogue: 0,0:22:13.93,0:22:15.46,Default,,0000,0000,0000,,Aaron said. Dialogue: 0,0:22:15.46,0:22:17.42,Default,,0000,0000,0000,,I don't know, you just\Nwhispered, I should say. Dialogue: 0,0:22:17.42,0:22:20.33,Default,,0000,0000,0000,, Dialogue: 0,0:22:20.33,0:22:21.55,Default,,0000,0000,0000,,All right. Dialogue: 0,0:22:21.55,0:22:28.05,Default,,0000,0000,0000,,So first of all, this\Nlooks a little bit bad. Dialogue: 0,0:22:28.05,0:22:31.00,Default,,0000,0000,0000,,It makes me a\Nlittle bit nervous. Dialogue: 0,0:22:31.00,0:22:34.89,Default,,0000,0000,0000,,But in the end, with your\Nhelp, I'm going to do it. Dialogue: 0,0:22:34.89,0:22:37.94,Default,,0000,0000,0000,,And I'm going to do it by\Nusing what kind of coordinates? Dialogue: 0,0:22:37.94,0:22:38.44,Default,,0000,0000,0000,,I'm-- Dialogue: 0,0:22:38.44,0:22:39.56,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:22:39.56,0:22:42.00,Default,,0000,0000,0000,,PROFESSOR: Former\Ncoordinates of the Y and Z. Dialogue: 0,0:22:42.00,0:22:43.26,Default,,0000,0000,0000,,It would be a killer. Dialogue: 0,0:22:43.26,0:22:48.80,Default,,0000,0000,0000,,Double, double, square root\Nof 1 plus-- who's telling me Dialogue: 0,0:22:48.80,0:22:49.93,Default,,0000,0000,0000,,what's coming next? Dialogue: 0,0:22:49.93,0:22:50.88,Default,,0000,0000,0000,,STUDENT: 4X squared. Dialogue: 0,0:22:50.88,0:22:52.61,Default,,0000,0000,0000,,4X squared, excellent. Dialogue: 0,0:22:52.61,0:22:55.83,Default,,0000,0000,0000,,4R squared, you say. Dialogue: 0,0:22:55.83,0:22:56.95,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:22:56.95,0:22:57.53,Default,,0000,0000,0000,,PROFESSOR: OK. Dialogue: 0,0:22:57.53,0:23:01.01,Default,,0000,0000,0000,,Let me write it with\NX and Y, and then Dialogue: 0,0:23:01.01,0:23:02.67,Default,,0000,0000,0000,,realize that this is our square. Dialogue: 0,0:23:02.67,0:23:04.67,Default,,0000,0000,0000,,How about that? Dialogue: 0,0:23:04.67,0:23:08.14,Default,,0000,0000,0000,,And then I have DX,\NDY over the domain D, Dialogue: 0,0:23:08.14,0:23:12.34,Default,,0000,0000,0000,,and now I finally become\Nsmart and say I just Dialogue: 0,0:23:12.34,0:23:13.73,Default,,0000,0000,0000,,fooled around here. Dialogue: 0,0:23:13.73,0:23:17.45,Default,,0000,0000,0000,,I want to do it in four\Ncoordinates finally. Dialogue: 0,0:23:17.45,0:23:22.21,Default,,0000,0000,0000,,And that means I'll say\Nzero to 2 pi for theta. Dialogue: 0,0:23:22.21,0:23:26.13,Default,,0000,0000,0000,,So that theta will be the\Nlast of the [INAUDIBLE]. Dialogue: 0,0:23:26.13,0:23:28.52,Default,,0000,0000,0000,,R will be from zero to 1. Dialogue: 0,0:23:28.52,0:23:33.11,Default,,0000,0000,0000,, Dialogue: 0,0:23:33.11,0:23:35.03,Default,,0000,0000,0000,,And So what? Dialogue: 0,0:23:35.03,0:23:40.63,Default,,0000,0000,0000,,This is an ugly, fairly\Nugly, I just [INAUDIBLE]. Dialogue: 0,0:23:40.63,0:23:42.58,Default,,0000,0000,0000,,I don't know what\NI'm going to do yet. Dialogue: 0,0:23:42.58,0:23:44.59,Default,,0000,0000,0000,,I reduced our confusion, right? Dialogue: 0,0:23:44.59,0:23:46.48,Default,,0000,0000,0000,,But I'm not done. Dialogue: 0,0:23:46.48,0:23:47.41,Default,,0000,0000,0000,,STUDENT: R. Dialogue: 0,0:23:47.41,0:23:51.89,Default,,0000,0000,0000,,PROFESSOR: R. Never forget it. Dialogue: 0,0:23:51.89,0:23:56.94,Default,,0000,0000,0000,,So if I didn't have this\NR, I would be horrible. Dialogue: 0,0:23:56.94,0:23:58.58,Default,,0000,0000,0000,,Why would it be horrible? Dialogue: 0,0:23:58.58,0:24:01.25,Default,,0000,0000,0000,,Imagine you couldn't have the R. Dialogue: 0,0:24:01.25,0:24:02.55,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:24:02.55,0:24:05.86,Default,,0000,0000,0000,,PROFESSOR: We have to look that\Nthis thing in integral table Dialogue: 0,0:24:05.86,0:24:10.19,Default,,0000,0000,0000,,or some-- use the calculator,\Nwhich we are not allowed Dialogue: 0,0:24:10.19,0:24:12.11,Default,,0000,0000,0000,,to do in this kind of course. Dialogue: 0,0:24:12.11,0:24:14.35,Default,,0000,0000,0000,,So what do we do? Dialogue: 0,0:24:14.35,0:24:16.89,Default,,0000,0000,0000,,We say it's a new substitution. Dialogue: 0,0:24:16.89,0:24:19.07,Default,,0000,0000,0000,,I have an R. That's a blessing. Dialogue: 0,0:24:19.07,0:24:23.20,Default,,0000,0000,0000,,So U equals 4 squared plus 1. Dialogue: 0,0:24:23.20,0:24:29.89,Default,,0000,0000,0000,,DU equals 8R, DR. I\Nthink R, DR is a block. Dialogue: 0,0:24:29.89,0:24:34.89,Default,,0000,0000,0000,,And I know that's what I'm\Ngoing to do is a U substitution. Dialogue: 0,0:24:34.89,0:24:36.26,Default,,0000,0000,0000,,And I'm almost there. Dialogue: 0,0:24:36.26,0:24:39.28,Default,,0000,0000,0000,, Dialogue: 0,0:24:39.28,0:24:41.38,Default,,0000,0000,0000,,It's a pretty good\Nexample, but the one Dialogue: 0,0:24:41.38,0:24:48.16,Default,,0000,0000,0000,,you have as a first example\Nin this section, 13.5, Dialogue: 0,0:24:48.16,0:24:50.07,Default,,0000,0000,0000,,it's a little bit\Ntoo computational. Dialogue: 0,0:24:50.07,0:24:52.76,Default,,0000,0000,0000,,It's not smart at all. Dialogue: 0,0:24:52.76,0:24:57.81,Default,,0000,0000,0000,,It has a similar function over a\Nrectangle, something like that. Dialogue: 0,0:24:57.81,0:24:59.56,Default,,0000,0000,0000,,But it's a little bit\Ntoo confrontational. Dialogue: 0,0:24:59.56,0:25:01.29,Default,,0000,0000,0000,,We are looking\Nfor something that Dialogue: 0,0:25:01.29,0:25:04.92,Default,,0000,0000,0000,,is not going-- examples that are\Ngoing to be easy to do and not Dialogue: 0,0:25:04.92,0:25:09.38,Default,,0000,0000,0000,,involve too much heavy\Ncompetition by him, because you Dialogue: 0,0:25:09.38,0:25:10.64,Default,,0000,0000,0000,,do everything by him. Dialogue: 0,0:25:10.64,0:25:14.73,Default,,0000,0000,0000,,Not-- like you don't have\Na calculator, et cetera. Dialogue: 0,0:25:14.73,0:25:19.59,Default,,0000,0000,0000,,And the exam is very\Nlimited in time, DU over 8. Dialogue: 0,0:25:19.59,0:25:22.58,Default,,0000,0000,0000,,So you say OK, I'm\Nknow what that is. Dialogue: 0,0:25:22.58,0:25:28.23,Default,,0000,0000,0000,,That's going to be the A of S.\NAnd that is going to be 2 pi. Dialogue: 0,0:25:28.23,0:25:32.38,Default,,0000,0000,0000,,Why can't I be so confident\Nand pull 2 pi out? Dialogue: 0,0:25:32.38,0:25:33.34,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:25:33.34,0:25:36.31,Default,,0000,0000,0000,,PROFESSOR: Because there\Nis no dependence on theta. Dialogue: 0,0:25:36.31,0:25:38.49,Default,,0000,0000,0000,,All right? Dialogue: 0,0:25:38.49,0:25:41.57,Default,,0000,0000,0000,,So I have that one. Dialogue: 0,0:25:41.57,0:25:46.34,Default,,0000,0000,0000,,And then you go all right,\Nintegral, square of you Dialogue: 0,0:25:46.34,0:25:52.29,Default,,0000,0000,0000,,times the U over 8-- 1 over 8DU. Dialogue: 0,0:25:52.29,0:25:56.43,Default,,0000,0000,0000,,And I have to be careful\Nbecause when R is zero-- Dialogue: 0,0:25:56.43,0:26:00.23,Default,,0000,0000,0000,,if I put zero and 1 here\Nlike some of my students, Dialogue: 0,0:26:00.23,0:26:04.78,Default,,0000,0000,0000,,I'm dead meat, because I'm going\Nto lose a lot of credit, right? Dialogue: 0,0:26:04.78,0:26:06.57,Default,,0000,0000,0000,,So I have to pay attention. Dialogue: 0,0:26:06.57,0:26:09.10,Default,,0000,0000,0000,,R is 0, and U equals? Dialogue: 0,0:26:09.10,0:26:09.89,Default,,0000,0000,0000,,STUDENT: 1. Dialogue: 0,0:26:09.89,0:26:10.73,Default,,0000,0000,0000,,PROFESSOR: 1. Dialogue: 0,0:26:10.73,0:26:12.97,Default,,0000,0000,0000,,R equals 1. Dialogue: 0,0:26:12.97,0:26:16.76,Default,,0000,0000,0000,,U equals 5. Dialogue: 0,0:26:16.76,0:26:23.12,Default,,0000,0000,0000,,And I worked this out\Nand I should be done. Dialogue: 0,0:26:23.12,0:26:27.56,Default,,0000,0000,0000,,And that's-- you should\Nexpect something like that. Dialogue: 0,0:26:27.56,0:26:33.63,Default,,0000,0000,0000,,Nice, not computational,\Nyou kind of looking. Dialogue: 0,0:26:33.63,0:26:36.80,Default,,0000,0000,0000,,What is integral of square of U? Dialogue: 0,0:26:36.80,0:26:38.46,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:26:38.46,0:26:41.69,Default,,0000,0000,0000,,PROFESSOR: So you have--\Nyou do the three halves, Dialogue: 0,0:26:41.69,0:26:43.94,Default,,0000,0000,0000,,and you pull out the 2/3, right? Dialogue: 0,0:26:43.94,0:26:45.16,Default,,0000,0000,0000,,That's what you do. Dialogue: 0,0:26:45.16,0:26:49.11,Default,,0000,0000,0000,,And then you go between U equals\N1 down, and U equals 5 up. Dialogue: 0,0:26:49.11,0:26:51.90,Default,,0000,0000,0000,,And it's like one of those\Nexamples we worked before. Dialogue: 0,0:26:51.90,0:26:53.27,Default,,0000,0000,0000,,Remember, and more\Nimportant, you Dialogue: 0,0:26:53.27,0:26:58.46,Default,,0000,0000,0000,,had something like\Nthat for surface area? Dialogue: 0,0:26:58.46,0:27:00.23,Default,,0000,0000,0000,,Oh, my god. Dialogue: 0,0:27:00.23,0:27:01.24,Default,,0000,0000,0000,,4 over 8. Dialogue: 0,0:27:01.24,0:27:03.11,Default,,0000,0000,0000,,How much is 4 over 8? Dialogue: 0,0:27:03.11,0:27:04.02,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:27:04.02,0:27:04.85,Default,,0000,0000,0000,,PROFESSOR: One half. Dialogue: 0,0:27:04.85,0:27:08.76,Default,,0000,0000,0000,, Dialogue: 0,0:27:08.76,0:27:09.26,Default,,0000,0000,0000,,Right? Dialogue: 0,0:27:09.26,0:27:15.33,Default,,0000,0000,0000,,So we will have 1 over\N6, and write pi times 5 Dialogue: 0,0:27:15.33,0:27:20.28,Default,,0000,0000,0000,,to the three halves minus 1. Dialogue: 0,0:27:20.28,0:27:21.60,Default,,0000,0000,0000,,So do I like it? Dialogue: 0,0:27:21.60,0:27:22.97,Default,,0000,0000,0000,,I would leave it like that. Dialogue: 0,0:27:22.97,0:27:23.47,Default,,0000,0000,0000,,I'm fine. Dialogue: 0,0:27:23.47,0:27:24.58,Default,,0000,0000,0000,,I'll forget about it. Dialogue: 0,0:27:24.58,0:27:26.64,Default,,0000,0000,0000,,I have people who care. Dialogue: 0,0:27:26.64,0:27:32.04,Default,,0000,0000,0000,,I don't care how some people\Nwrite it-- 5 with 5 minus 1 Dialogue: 0,0:27:32.04,0:27:33.50,Default,,0000,0000,0000,,because they think\Nit looks better. Dialogue: 0,0:27:33.50,0:27:34.00,Default,,0000,0000,0000,,It doesn't. Dialogue: 0,0:27:34.00,0:27:37.15,Default,,0000,0000,0000,,That's the scientific\Nequation, and I'm fine with it. Dialogue: 0,0:27:37.15,0:27:38.42,Default,,0000,0000,0000,,Right? Dialogue: 0,0:27:38.42,0:27:39.26,Default,,0000,0000,0000,,OK. Dialogue: 0,0:27:39.26,0:27:43.32,Default,,0000,0000,0000,,So expect something like--\Nmaybe I'm talking too much, Dialogue: 0,0:27:43.32,0:27:46.75,Default,,0000,0000,0000,,but maybe it's a good thing\Nto tell you what to expect Dialogue: 0,0:27:46.75,0:27:48.50,Default,,0000,0000,0000,,because we have to [INAUDIBLE]. Dialogue: 0,0:27:48.50,0:27:50.50,Default,,0000,0000,0000,,At the same time, we're\Nteaching new things Dialogue: 0,0:27:50.50,0:27:54.97,Default,,0000,0000,0000,,as staff instructors doing\Nreview of what's important. Dialogue: 0,0:27:54.97,0:28:03.05,Default,,0000,0000,0000,,I'm thinking if I'm doing things\Nright and at the same pace, Dialogue: 0,0:28:03.05,0:28:10.11,Default,,0000,0000,0000,,I should be finished\Nwith chapter 13 Dialogue: 0,0:28:10.11,0:28:12.23,Default,,0000,0000,0000,,at the end of next week. Dialogue: 0,0:28:12.23,0:28:15.46,Default,,0000,0000,0000,,Because after 13.5,\Nwe have 13.6 which Dialogue: 0,0:28:15.46,0:28:18.12,Default,,0000,0000,0000,,is a generalization\Nof Green's Theorem. Dialogue: 0,0:28:18.12,0:28:20.81,Default,,0000,0000,0000,,13.6 as you recall is\Ncalled Stokes' Theorem. Dialogue: 0,0:28:20.81,0:28:25.09,Default,,0000,0000,0000,,13.7 is also a generalization\Nof Green's Theorem. Dialogue: 0,0:28:25.09,0:28:27.19,Default,,0000,0000,0000,,And they are all related. Dialogue: 0,0:28:27.19,0:28:31.50,Default,,0000,0000,0000,,It's like the trinity\Non [INAUDIBLE]. Dialogue: 0,0:28:31.50,0:28:33.17,Default,,0000,0000,0000,,That's the Divergence Theorem. Dialogue: 0,0:28:33.17,0:28:37.97,Default,,0000,0000,0000,,That is the last section,\N13.7, Divergence Theorem. Dialogue: 0,0:28:37.97,0:28:41.58,Default,,0000,0000,0000,,So if I am going\Nat the right pace, Dialogue: 0,0:28:41.58,0:28:44.09,Default,,0000,0000,0000,,by-- what is next\Nwee on Thursday? Dialogue: 0,0:28:44.09,0:28:47.100,Default,,0000,0000,0000,,The-- 23rd? Dialogue: 0,0:28:47.100,0:28:50.84,Default,,0000,0000,0000,,I should be more or less\Ndone with the chapter. Dialogue: 0,0:28:50.84,0:28:54.28,Default,,0000,0000,0000,,And I'm thinking I have\Nall the time in the world Dialogue: 0,0:28:54.28,0:28:57.41,Default,,0000,0000,0000,,to review with you\Nfrom that moment on. Dialogue: 0,0:28:57.41,0:28:59.49,Default,,0000,0000,0000,,In which sense are\Nwe going to review? Dialogue: 0,0:28:59.49,0:29:04.74,Default,,0000,0000,0000,,We are going to review\Nby solving past finals. Dialogue: 0,0:29:04.74,0:29:05.61,Default,,0000,0000,0000,,Right? Dialogue: 0,0:29:05.61,0:29:08.88,Default,,0000,0000,0000,,That's what we are-- that's\Nwhat I'm planning to do. Dialogue: 0,0:29:08.88,0:29:12.38,Default,,0000,0000,0000,,I'm going to erase this\Nand move on to something Dialogue: 0,0:29:12.38,0:29:15.34,Default,,0000,0000,0000,,more spectacular. Dialogue: 0,0:29:15.34,0:29:16.25,Default,,0000,0000,0000,,Many-- OK. Dialogue: 0,0:29:16.25,0:29:18.31,Default,,0000,0000,0000,,This second part that\NI want to teach you Dialogue: 0,0:29:18.31,0:29:23.03,Default,,0000,0000,0000,,now about, many instructors\Nin regular courses Dialogue: 0,0:29:23.03,0:29:28.59,Default,,0000,0000,0000,,just skip it because they do\Nnot want to teach you-- not you, Dialogue: 0,0:29:28.59,0:29:29.61,Default,,0000,0000,0000,,you are honor students. Dialogue: 0,0:29:29.61,0:29:31.32,Default,,0000,0000,0000,,But they don't want\Nto teach the students Dialogue: 0,0:29:31.32,0:29:36.07,Default,,0000,0000,0000,,about some more general\Nways to look at a surface. Dialogue: 0,0:29:36.07,0:29:40.24,Default,,0000,0000,0000,,Remember, guys, a surface\Nthat is written like that Dialogue: 0,0:29:40.24,0:29:43.11,Default,,0000,0000,0000,,is called a graph. Dialogue: 0,0:29:43.11,0:29:47.56,Default,,0000,0000,0000,,But not all the\Nsurfaces were graphs. Dialogue: 0,0:29:47.56,0:29:56.82,Default,,0000,0000,0000,,And actually for a surface\NS, what the most general way Dialogue: 0,0:29:56.82,0:30:00.06,Default,,0000,0000,0000,,to represent the presentation\Nwould be a parameterization. Dialogue: 0,0:30:00.06,0:30:05.56,Default,,0000,0000,0000,, Dialogue: 0,0:30:05.56,0:30:10.45,Default,,0000,0000,0000,,And I'll do a little bit\Nof a review for those. Dialogue: 0,0:30:10.45,0:30:17.64,Default,,0000,0000,0000,,R-- little R or big R--\Nbig R, because that's Dialogue: 0,0:30:17.64,0:30:20.54,Default,,0000,0000,0000,,the position vector the\Nway I serve it to you Dialogue: 0,0:30:20.54,0:30:24.35,Default,,0000,0000,0000,,on a plate, whether,\Nfor curves in space. Dialogue: 0,0:30:24.35,0:30:29.11,Default,,0000,0000,0000,,I say that's R of P. And when\Nwe moved on curves to surfaces, Dialogue: 0,0:30:29.11,0:30:34.43,Default,,0000,0000,0000,,I said you move your path\Ntwo directions of motion. Dialogue: 0,0:30:34.43,0:30:37.01,Default,,0000,0000,0000,,You have two-- what are\Nthose called in mechanics? Dialogue: 0,0:30:37.01,0:30:38.44,Default,,0000,0000,0000,,Degrees of freedom. Dialogue: 0,0:30:38.44,0:30:40.57,Default,,0000,0000,0000,,So you have two\Ndegrees of freedom Dialogue: 0,0:30:40.57,0:30:42.40,Default,,0000,0000,0000,,like latitude and longitude. Dialogue: 0,0:30:42.40,0:30:46.96,Default,,0000,0000,0000,,Then R belongs--\Nthe position vector Dialogue: 0,0:30:46.96,0:30:52.71,Default,,0000,0000,0000,,is a function of two variables,\Nand it belongs to R3, Dialogue: 0,0:30:52.71,0:30:54.32,Default,,0000,0000,0000,,because it's a vector in R3. Dialogue: 0,0:30:54.32,0:30:58.22,Default,,0000,0000,0000,,And want to have-- imagine\Nthat my hand is a surface. Dialogue: 0,0:30:58.22,0:30:59.55,Default,,0000,0000,0000,,Well, OK. Dialogue: 0,0:30:59.55,0:31:02.40,Default,,0000,0000,0000,,This is the position vector, I'm\Njust kind of sweeping my hand, Dialogue: 0,0:31:02.40,0:31:04.93,Default,,0000,0000,0000,,going this way, one\Ndegree of freedom. Dialogue: 0,0:31:04.93,0:31:07.33,Default,,0000,0000,0000,,Or going that way, the\Nother degree of freedom. Dialogue: 0,0:31:07.33,0:31:10.60,Default,,0000,0000,0000,,This is what\Nparameterization is. Dialogue: 0,0:31:10.60,0:31:16.64,Default,,0000,0000,0000,,So for a sphere, if you want to\Nparameterize the whole sphere-- Dialogue: 0,0:31:16.64,0:31:20.02,Default,,0000,0000,0000,,I'll be done in a second. Dialogue: 0,0:31:20.02,0:31:23.42,Default,,0000,0000,0000,,I need you to see\Nif you remember Dialogue: 0,0:31:23.42,0:31:25.40,Default,,0000,0000,0000,,how to parameterize a sphere. Dialogue: 0,0:31:25.40,0:31:26.45,Default,,0000,0000,0000,,I'm testing you. Dialogue: 0,0:31:26.45,0:31:28.40,Default,,0000,0000,0000,,I'm mean today. Dialogue: 0,0:31:28.40,0:31:29.75,Default,,0000,0000,0000,,So examples. Dialogue: 0,0:31:29.75,0:31:32.38,Default,,0000,0000,0000,,Example 1 is\Nparameterize a sphere. Dialogue: 0,0:31:32.38,0:31:36.66,Default,,0000,0000,0000,, Dialogue: 0,0:31:36.66,0:31:38.31,Default,,0000,0000,0000,,Was it hard? Dialogue: 0,0:31:38.31,0:31:40.55,Default,,0000,0000,0000,,That was a long\Ntime ago, my god. Dialogue: 0,0:31:40.55,0:31:45.57,Default,,0000,0000,0000,,X, Y, and Z are what? Dialogue: 0,0:31:45.57,0:31:47.88,Default,,0000,0000,0000,,Latitude from Santa Clause. Dialogue: 0,0:31:47.88,0:31:52.00,Default,,0000,0000,0000,,Always latitude from\Nthe North Pole is 5. Dialogue: 0,0:31:52.00,0:31:54.39,Default,,0000,0000,0000,,Longitude is from zero to 5. Dialogue: 0,0:31:54.39,0:31:56.56,Default,,0000,0000,0000,,The meridian is zero to 5. Dialogue: 0,0:31:56.56,0:31:59.85,Default,,0000,0000,0000,,That was theta, the\Nparameter of theta. Dialogue: 0,0:31:59.85,0:32:04.94,Default,,0000,0000,0000,,R was the distance\Nfrom this to a point. Dialogue: 0,0:32:04.94,0:32:08.26,Default,,0000,0000,0000,,But R was allowed to be\Nfrom-- take many values. Dialogue: 0,0:32:08.26,0:32:11.87,Default,,0000,0000,0000,,Now if I'm moving on\Na sphere of radius Dialogue: 0,0:32:11.87,0:32:15.63,Default,,0000,0000,0000,,A-- let me make\Nthat radius a just Dialogue: 0,0:32:15.63,0:32:16.80,Default,,0000,0000,0000,,to make your life miserable. Dialogue: 0,0:32:16.80,0:32:19.84,Default,,0000,0000,0000,,Assume that A would\Nbe a sample, A. Dialogue: 0,0:32:19.84,0:32:22.31,Default,,0000,0000,0000,,How am I going to write\Nthat parameterization? Dialogue: 0,0:32:22.31,0:32:25.08,Default,,0000,0000,0000,,STUDENT: X equals\NA plus [INAUDIBLE]? Dialogue: 0,0:32:25.08,0:32:28.96,Default,,0000,0000,0000,,PROFESSOR: A something,\NA something, A something. Dialogue: 0,0:32:28.96,0:32:30.15,Default,,0000,0000,0000,,STUDENT: A [INAUDIBLE] Dialogue: 0,0:32:30.15,0:32:31.11,Default,,0000,0000,0000,,PROFESSOR: He is right. Dialogue: 0,0:32:31.11,0:32:32.07,Default,,0000,0000,0000,,I have to move on. Dialogue: 0,0:32:32.07,0:32:32.90,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:32:32.90,0:32:34.51,Default,,0000,0000,0000,,PROFESSOR: Go slow. Dialogue: 0,0:32:34.51,0:32:37.37,Default,,0000,0000,0000,,So I have-- the last\None-- you were right, Dialogue: 0,0:32:37.37,0:32:40.25,Default,,0000,0000,0000,,Buddy, you have the\Nmemory of a medical doctor Dialogue: 0,0:32:40.25,0:32:42.34,Default,,0000,0000,0000,,and some day you will\Nbe a medical doctor. Dialogue: 0,0:32:42.34,0:32:44.51,Default,,0000,0000,0000,,Not everybody has a good memory. Dialogue: 0,0:32:44.51,0:32:49.98,Default,,0000,0000,0000,,So the way you can do that\Nis, wait a minute, this is pi, Dialogue: 0,0:32:49.98,0:32:50.67,Default,,0000,0000,0000,,right? Dialogue: 0,0:32:50.67,0:32:52.25,Default,,0000,0000,0000,,This [INAUDIBLE]. Dialogue: 0,0:32:52.25,0:32:55.33,Default,,0000,0000,0000,,If you want the Z, you\Nstart with that first. Dialogue: 0,0:32:55.33,0:33:00.59,Default,,0000,0000,0000,,And since Z is adjacent, you\Ngo R, cosine, sine, phi equals Dialogue: 0,0:33:00.59,0:33:01.80,Default,,0000,0000,0000,,sine phi. Dialogue: 0,0:33:01.80,0:33:05.08,Default,,0000,0000,0000,,Now we started with X\Nbecause he's worked on this Dialogue: 0,0:33:05.08,0:33:06.12,Default,,0000,0000,0000,,and remembers everything. Dialogue: 0,0:33:06.12,0:33:07.59,Default,,0000,0000,0000,,He has it memorized. Dialogue: 0,0:33:07.59,0:33:10.43,Default,,0000,0000,0000,,Sine phi for both. Dialogue: 0,0:33:10.43,0:33:13.52,Default,,0000,0000,0000,,And times what in both cases? Dialogue: 0,0:33:13.52,0:33:15.31,Default,,0000,0000,0000,,He's just the guy\Nwho's not here. Dialogue: 0,0:33:15.31,0:33:17.02,Default,,0000,0000,0000,,So sine phi. Dialogue: 0,0:33:17.02,0:33:20.82,Default,,0000,0000,0000,,It helps to memorize N\Ncosine theta, and sine theta. Dialogue: 0,0:33:20.82,0:33:22.60,Default,,0000,0000,0000,,Is that really easy to memorize? Dialogue: 0,0:33:22.60,0:33:26.27,Default,,0000,0000,0000,,So where phi was the\Nlatitude from the North Dialogue: 0,0:33:26.27,0:33:30.08,Default,,0000,0000,0000,,Pole between zero\Nand phi, it theta Dialogue: 0,0:33:30.08,0:33:35.54,Default,,0000,0000,0000,,was the longitude-- excuse\Nme, guys-- longitude from zero Dialogue: 0,0:33:35.54,0:33:41.41,Default,,0000,0000,0000,,to 2 pi, all around one more. Dialogue: 0,0:33:41.41,0:33:43.92,Default,,0000,0000,0000,,So you say wait a\Nminute, Magdalena, Dialogue: 0,0:33:43.92,0:33:45.19,Default,,0000,0000,0000,,these are Euler's angle. Dialogue: 0,0:33:45.19,0:33:47.28,Default,,0000,0000,0000,,What do they call in mechanics? Dialogue: 0,0:33:47.28,0:33:49.67,Default,,0000,0000,0000,,I think they call\Nthem Euler angles. Dialogue: 0,0:33:49.67,0:33:53.41,Default,,0000,0000,0000,,But anyway, for\Nphi theta, we call Dialogue: 0,0:33:53.41,0:33:55.62,Default,,0000,0000,0000,,them latitude and longitude. Dialogue: 0,0:33:55.62,0:33:59.99,Default,,0000,0000,0000,,I'll replace them, because look,\NI want R to be in terms of U,V. Dialogue: 0,0:33:59.99,0:34:02.39,Default,,0000,0000,0000,,So in mathematics, it's\Nnot about location. Dialogue: 0,0:34:02.39,0:34:05.47,Default,,0000,0000,0000,,We can call them\Nwhatever we want. Dialogue: 0,0:34:05.47,0:34:10.41,Default,,0000,0000,0000,,Mathematics is about the freedom\Nto call people names-- no-- Dialogue: 0,0:34:10.41,0:34:13.76,Default,,0000,0000,0000,,to call things names\Nand people names-- Dialogue: 0,0:34:13.76,0:34:15.91,Default,,0000,0000,0000,,STUDENT: Could U not equal zero? Dialogue: 0,0:34:15.91,0:34:16.54,Default,,0000,0000,0000,,PROFESSOR: Who? Dialogue: 0,0:34:16.54,0:34:17.84,Default,,0000,0000,0000,,STUDENT: U. Dialogue: 0,0:34:17.84,0:34:18.53,Default,,0000,0000,0000,,PROFESSOR: Yes. Dialogue: 0,0:34:18.53,0:34:19.08,Default,,0000,0000,0000,,So U can-- Dialogue: 0,0:34:19.08,0:34:19.91,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:34:19.91,0:34:24.52,Default,,0000,0000,0000,,PROFESSOR: --yeah, but\Nwhy didn't I write zero? Dialogue: 0,0:34:24.52,0:34:25.47,Default,,0000,0000,0000,,Well-- Dialogue: 0,0:34:25.47,0:34:26.93,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nmakes sense. Dialogue: 0,0:34:26.93,0:34:30.33,Default,,0000,0000,0000,,PROFESSOR: --because,\Nyeah, you can take both. Dialogue: 0,0:34:30.33,0:34:33.04,Default,,0000,0000,0000,,If I want to study\Ndifferentiability, Dialogue: 0,0:34:33.04,0:34:36.50,Default,,0000,0000,0000,,I usually have to take it less\Nthan and less than and less Dialogue: 0,0:34:36.50,0:34:39.05,Default,,0000,0000,0000,,than and less than because we\Nstudied differentiability on Dialogue: 0,0:34:39.05,0:34:40.19,Default,,0000,0000,0000,,[INAUDIBLE]. Dialogue: 0,0:34:40.19,0:34:43.31,Default,,0000,0000,0000,,But right now, I can take them\Nfrom the North Pole itself Dialogue: 0,0:34:43.31,0:34:47.09,Default,,0000,0000,0000,,to the South Pole itself-- so. Dialogue: 0,0:34:47.09,0:34:50.61,Default,,0000,0000,0000,,I'm not deleting any meridian. Dialogue: 0,0:34:50.61,0:34:53.74,Default,,0000,0000,0000,,If I were-- suppose\NI were to delete it. Dialogue: 0,0:34:53.74,0:34:55.96,Default,,0000,0000,0000,,By the way, what does this mean? Dialogue: 0,0:34:55.96,0:34:57.03,Default,,0000,0000,0000,,I'm just kidding. Dialogue: 0,0:34:57.03,0:34:58.03,Default,,0000,0000,0000,,I'll put it back. Dialogue: 0,0:34:58.03,0:35:00.91,Default,,0000,0000,0000,,But Alex had a smart\Nquestion over there, Dialogue: 0,0:35:00.91,0:35:04.03,Default,,0000,0000,0000,,and he made me thinking. Dialogue: 0,0:35:04.03,0:35:06.60,Default,,0000,0000,0000,,It's a dangerous thing\Nwhen people make you think. Dialogue: 0,0:35:06.60,0:35:10.26,Default,,0000,0000,0000,,So it goes from zero to 2 pi. Dialogue: 0,0:35:10.26,0:35:12.05,Default,,0000,0000,0000,,Why would that be? Dialogue: 0,0:35:12.05,0:35:15.10,Default,,0000,0000,0000,,Imagine you have all the\Nmeridians in the world Dialogue: 0,0:35:15.10,0:35:17.68,Default,,0000,0000,0000,,except for one. Dialogue: 0,0:35:17.68,0:35:22.47,Default,,0000,0000,0000,,From the sphere, you cut it and\Nremove the Greenwich meridian, Dialogue: 0,0:35:22.47,0:35:25.56,Default,,0000,0000,0000,,the one that passes\Nthrough Greenwich Village. Dialogue: 0,0:35:25.56,0:35:29.96,Default,,0000,0000,0000,,The one-- not the one in New\NYork, the one next to London, Dialogue: 0,0:35:29.96,0:35:31.40,Default,,0000,0000,0000,,right? Dialogue: 0,0:35:31.40,0:35:32.50,Default,,0000,0000,0000,,So put it back. Dialogue: 0,0:35:32.50,0:35:34.80,Default,,0000,0000,0000,,Put that meridian back. Dialogue: 0,0:35:34.80,0:35:38.66,Default,,0000,0000,0000,,It's like you take an\Norange, and you make a slice. Dialogue: 0,0:35:38.66,0:35:40.24,Default,,0000,0000,0000,,I am-- OK. Dialogue: 0,0:35:40.24,0:35:43.05,Default,,0000,0000,0000,,Stop with the fruit\Nbecause I'm hungry. Dialogue: 0,0:35:43.05,0:35:47.18,Default,,0000,0000,0000,,Now, example two. Dialogue: 0,0:35:47.18,0:35:52.73,Default,,0000,0000,0000,,Now, imagine another surface\Narea you're used to, the what? Dialogue: 0,0:35:52.73,0:35:58.49,Default,,0000,0000,0000,,The paraboloid is one of our\Nfavorite guys this semester. Dialogue: 0,0:35:58.49,0:36:00.17,Default,,0000,0000,0000,,X squared plus Y squared. Dialogue: 0,0:36:00.17,0:36:02.13,Default,,0000,0000,0000,,What is the\Nparameterization of that? Dialogue: 0,0:36:02.13,0:36:06.95,Default,,0000,0000,0000,, Dialogue: 0,0:36:06.95,0:36:09.68,Default,,0000,0000,0000,,Well, if I write it\Nlike that, it's a graph. Dialogue: 0,0:36:09.68,0:36:11.60,Default,,0000,0000,0000,,But if I don't want to\Nwrite it as a graph, Dialogue: 0,0:36:11.60,0:36:14.30,Default,,0000,0000,0000,,I have to write\Nit as a parameter. Dialogue: 0,0:36:14.30,0:36:16.07,Default,,0000,0000,0000,,What am I going to do? Dialogue: 0,0:36:16.07,0:36:19.38,Default,,0000,0000,0000,,I really know X to be U, right? Dialogue: 0,0:36:19.38,0:36:21.38,Default,,0000,0000,0000,,That's the simplest\Nchoice possible. Dialogue: 0,0:36:21.38,0:36:25.30,Default,,0000,0000,0000,,Y could be V. And then Z will\Nbe U squared plus V squared. Dialogue: 0,0:36:25.30,0:36:26.36,Default,,0000,0000,0000,,And there I am. Dialogue: 0,0:36:26.36,0:36:26.86,Default,,0000,0000,0000,,[SNEEZE] Dialogue: 0,0:36:26.86,0:36:29.11,Default,,0000,0000,0000,,So I'm going to write-- bless\Nyour heart, [INAUDIBLE]. Dialogue: 0,0:36:29.11,0:36:31.53,Default,,0000,0000,0000,, Dialogue: 0,0:36:31.53,0:36:38.32,Default,,0000,0000,0000,,V plus J plus U squared\Nplus V squared, K. So this Dialogue: 0,0:36:38.32,0:36:44.41,Default,,0000,0000,0000,,is the parameterization\Nof a paraboloid. Dialogue: 0,0:36:44.41,0:36:46.73,Default,,0000,0000,0000,,That one of them--\Nthere are infinitely Dialogue: 0,0:36:46.73,0:36:48.65,Default,,0000,0000,0000,,many-- the one that comes\Nto mind because it's Dialogue: 0,0:36:48.65,0:36:52.60,Default,,0000,0000,0000,,the easiest one to think about. Dialogue: 0,0:36:52.60,0:36:53.56,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:36:53.56,0:36:54.52,Default,,0000,0000,0000,,PROFESSOR: Good. Dialogue: 0,0:36:54.52,0:36:59.06,Default,,0000,0000,0000,,For a minute, guys,\Nyou didn't need me. Dialogue: 0,0:36:59.06,0:37:02.39,Default,,0000,0000,0000,,You didn't need me at all\Nto come up with those. Dialogue: 0,0:37:02.39,0:37:05.87,Default,,0000,0000,0000,,But maybe you would need me\Nto remember, or maybe not-- Dialogue: 0,0:37:05.87,0:37:08.48,Default,,0000,0000,0000,,to remind you of the helicoid. Dialogue: 0,0:37:08.48,0:37:09.81,Default,,0000,0000,0000,,Helicoid. Dialogue: 0,0:37:09.81,0:37:13.25,Default,,0000,0000,0000,,Did you go to the,\Nas I told you to go Dialogue: 0,0:37:13.25,0:37:15.71,Default,,0000,0000,0000,,to the [INAUDIBLE] spectrum--\Nwhat was that called? Dialogue: 0,0:37:15.71,0:37:15.86,Default,,0000,0000,0000,,The-- Dialogue: 0,0:37:15.86,0:37:16.53,Default,,0000,0000,0000,,STUDENT: Science spectrum. Dialogue: 0,0:37:16.53,0:37:17.76,Default,,0000,0000,0000,,PROFESSOR: Science spectrum. Dialogue: 0,0:37:17.76,0:37:24.62,Default,,0000,0000,0000,,And dip into soap solution\Nthe thingy was-- a metal Dialogue: 0,0:37:24.62,0:37:27.95,Default,,0000,0000,0000,,rod with a-- with a what? Dialogue: 0,0:37:27.95,0:37:34.21,Default,,0000,0000,0000,,With; a helix made\Nof metal so the soap Dialogue: 0,0:37:34.21,0:37:37.04,Default,,0000,0000,0000,,film would take which shape? Dialogue: 0,0:37:37.04,0:37:42.23,Default,,0000,0000,0000,,The shape of this spiral that's\Ngoing to go inside here, right? Dialogue: 0,0:37:42.23,0:37:45.63,Default,,0000,0000,0000,,That's called a helicoid. Dialogue: 0,0:37:45.63,0:37:46.15,Default,,0000,0000,0000,,OK. Dialogue: 0,0:37:46.15,0:37:46.72,Default,,0000,0000,0000,,All right. Dialogue: 0,0:37:46.72,0:37:47.60,Default,,0000,0000,0000,,You're not mad at me. Dialogue: 0,0:37:47.60,0:37:48.09,Default,,0000,0000,0000,,STUDENT: No. Dialogue: 0,0:37:48.09,0:37:49.01,Default,,0000,0000,0000,,PROFESSOR: OK, good. Dialogue: 0,0:37:49.01,0:37:54.41,Default,,0000,0000,0000,,So in this case, R\Nof UV will be what? Dialogue: 0,0:37:54.41,0:37:57.69,Default,,0000,0000,0000,,It was a long time ago, once\Nupon a time I gave it to you. Dialogue: 0,0:37:57.69,0:38:00.29,Default,,0000,0000,0000,,It's extremely hard to\Nmemorize if you don't work Dialogue: 0,0:38:00.29,0:38:03.56,Default,,0000,0000,0000,,with it on a regular basis. Dialogue: 0,0:38:03.56,0:38:06.94,Default,,0000,0000,0000,,If it were a helix,\Nwhat would it be? Dialogue: 0,0:38:06.94,0:38:09.97,Default,,0000,0000,0000,,If it were a helix, it\Nwould be R of T right? Dialogue: 0,0:38:09.97,0:38:14.32,Default,,0000,0000,0000,,It would be like equal\Nsign T, A sine T, BT. Dialogue: 0,0:38:14.32,0:38:17.19,Default,,0000,0000,0000,,Say it again, Magdalena,\Nthat was a long time ago, Dialogue: 0,0:38:17.19,0:38:17.97,Default,,0000,0000,0000,,chapter 10. Dialogue: 0,0:38:17.97,0:38:19.69,Default,,0000,0000,0000,,Chapter 10. Dialogue: 0,0:38:19.69,0:38:23.96,Default,,0000,0000,0000,,Equal sign, T, A sine\NT, MBT, standard helix. Dialogue: 0,0:38:23.96,0:38:25.35,Default,,0000,0000,0000,,This is not going to be that. Dialogue: 0,0:38:25.35,0:38:33.31,Default,,0000,0000,0000,,It's going to be-- U cosine B.\NU sine B. Look at the picture. Dialogue: 0,0:38:33.31,0:38:37.69,Default,,0000,0000,0000,,And imagine that these guys\Nare extended to infinity. Dialogue: 0,0:38:37.69,0:38:39.19,Default,,0000,0000,0000,,It's not just the\Nstairs themselves, Dialogue: 0,0:38:39.19,0:38:41.46,Default,,0000,0000,0000,,or whatever they are. Dialogue: 0,0:38:41.46,0:38:46.35,Default,,0000,0000,0000,,There are infinite lines,\Nstraight lines, and busy. Dialogue: 0,0:38:46.35,0:38:47.76,Default,,0000,0000,0000,,This is done. Dialogue: 0,0:38:47.76,0:38:49.14,Default,,0000,0000,0000,,NB is a positive constant. Dialogue: 0,0:38:49.14,0:38:51.76,Default,,0000,0000,0000,, Dialogue: 0,0:38:51.76,0:38:56.52,Default,,0000,0000,0000,,But your parameters are\NU and V. Any other guy Dialogue: 0,0:38:56.52,0:39:00.40,Default,,0000,0000,0000,,that comes to mind, I'm out\Nof imagination right now. Dialogue: 0,0:39:00.40,0:39:04.52,Default,,0000,0000,0000,,You can do a torus on the\Nfold that looks like a donut. Dialogue: 0,0:39:04.52,0:39:05.74,Default,,0000,0000,0000,,You will have two parameters. Dialogue: 0,0:39:05.74,0:39:08.45,Default,,0000,0000,0000,,Imagine a donut. Dialogue: 0,0:39:08.45,0:39:10.97,Default,,0000,0000,0000,,How do you-- I'm not\Ngoing to write that. Dialogue: 0,0:39:10.97,0:39:13.48,Default,,0000,0000,0000,,Eventually I could give you\Nthat as an extra credit thing. Dialogue: 0,0:39:13.48,0:39:18.66,Default,,0000,0000,0000,,What are the two degrees of\Nfreedom of moving on the donut, Dialogue: 0,0:39:18.66,0:39:21.42,Default,,0000,0000,0000,,assuming that you would\Nlike to move in circles? Dialogue: 0,0:39:21.42,0:39:25.42,Default,,0000,0000,0000,, Dialogue: 0,0:39:25.42,0:39:26.25,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:39:26.25,0:39:29.75,Default,,0000,0000,0000,, Dialogue: 0,0:39:29.75,0:39:32.01,Default,,0000,0000,0000,,PROFESSOR: Let me draw a\Ndonut, because I'm hungry, Dialogue: 0,0:39:32.01,0:39:34.72,Default,,0000,0000,0000,,and I really-- I cannot help it. Dialogue: 0,0:39:34.72,0:39:38.45,Default,,0000,0000,0000,,I just have to-- this is\Ncalled a torus in mathematics. Dialogue: 0,0:39:38.45,0:39:42.74,Default,,0000,0000,0000,,And you'll have-- one degree\Nof freedom will be like this, Dialogue: 0,0:39:42.74,0:39:44.66,Default,,0000,0000,0000,,the other degree of\Nfreedom will be like that. Dialogue: 0,0:39:44.66,0:39:46.88,Default,,0000,0000,0000,,This is U and B.\NInstead of U and B, Dialogue: 0,0:39:46.88,0:39:50.11,Default,,0000,0000,0000,,mathematicians,\Napologists, geometers, Dialogue: 0,0:39:50.11,0:39:53.37,Default,,0000,0000,0000,,they call those angles phi\Nand theta because they really Dialogue: 0,0:39:53.37,0:39:56.10,Default,,0000,0000,0000,,are between zero and 2 pi. Dialogue: 0,0:39:56.10,0:39:59.80,Default,,0000,0000,0000,,It has a rotation like\Nthat along the donut. Dialogue: 0,0:39:59.80,0:40:02.18,Default,,0000,0000,0000,,You can cut, slice\Nthe donut, or if they Dialogue: 0,0:40:02.18,0:40:04.45,Default,,0000,0000,0000,,don't put cheese filling in it. Dialogue: 0,0:40:04.45,0:40:08.79,Default,,0000,0000,0000,,That was a bad idea not\Nhaving anything to eat. Dialogue: 0,0:40:08.79,0:40:15.30,Default,,0000,0000,0000,,And the other angle will be your\N2 pi along this little circle. Dialogue: 0,0:40:15.30,0:40:18.38,Default,,0000,0000,0000,,So you still have two degrees\Nof freedom on a donut. Dialogue: 0,0:40:18.38,0:40:19.23,Default,,0000,0000,0000,,It's a surface. Dialogue: 0,0:40:19.23,0:40:20.21,Default,,0000,0000,0000,,You can write the\Nparameterization. Dialogue: 0,0:40:20.21,0:40:20.86,Default,,0000,0000,0000,,Yes? Dialogue: 0,0:40:20.86,0:40:22.44,Default,,0000,0000,0000,,STUDENT: Why is a\Npie this way around. Dialogue: 0,0:40:22.44,0:40:25.87,Default,,0000,0000,0000,,Why is it like [INAUDIBLE]. Dialogue: 0,0:40:25.87,0:40:27.71,Default,,0000,0000,0000,,PROFESSOR: It\Ndoesn't have to be. Dialogue: 0,0:40:27.71,0:40:29.71,Default,,0000,0000,0000,,STUDENT: Or is it just\Nkind of like [INAUDIBLE]? Dialogue: 0,0:40:29.71,0:40:31.42,Default,,0000,0000,0000,,PROFESSOR: That's\Nwhat they call it. Dialogue: 0,0:40:31.42,0:40:32.48,Default,,0000,0000,0000,,Yeah. Dialogue: 0,0:40:32.48,0:40:35.23,Default,,0000,0000,0000,,So they are between 2 and 2 pi. Dialogue: 0,0:40:35.23,0:40:41.78,Default,,0000,0000,0000,,While I erase-- or should\NI-- enough expectation Dialogue: 0,0:40:41.78,0:40:44.38,Default,,0000,0000,0000,,in terms of parameterization,\NI have to night Dialogue: 0,0:40:44.38,0:40:47.28,Default,,0000,0000,0000,,teach you something about that. Dialogue: 0,0:40:47.28,0:40:53.48,Default,,0000,0000,0000,,If somebody would say I'm\Ngiving you a patch of a surface, Dialogue: 0,0:40:53.48,0:40:57.44,Default,,0000,0000,0000,,but that patch of a\Nsurface is in a frame-- Dialogue: 0,0:40:57.44,0:41:01.70,Default,,0000,0000,0000,,it's a nice parameterization. Dialogue: 0,0:41:01.70,0:41:04.11,Default,,0000,0000,0000,,This is the P on the surface. Dialogue: 0,0:41:04.11,0:41:07.52,Default,,0000,0000,0000,, Dialogue: 0,0:41:07.52,0:41:10.51,Default,,0000,0000,0000,,And you say, well,\Nthe parameterization Dialogue: 0,0:41:10.51,0:41:14.19,Default,,0000,0000,0000,,is going to be R\Nof U and V equals Dialogue: 0,0:41:14.19,0:41:22.11,Default,,0000,0000,0000,,X of UVI plus Y of\NUVJ plus Z of UVK. Dialogue: 0,0:41:22.11,0:41:25.09,Default,,0000,0000,0000,, Dialogue: 0,0:41:25.09,0:41:28.01,Default,,0000,0000,0000,,And suppose that somebody says\Nthis is you favorite test. Dialogue: 0,0:41:28.01,0:41:30.88,Default,,0000,0000,0000,, Dialogue: 0,0:41:30.88,0:41:35.99,Default,,0000,0000,0000,,Find V. Well, that\Nwould be absurd. Dialogue: 0,0:41:35.99,0:41:37.42,Default,,0000,0000,0000,,My god, how do we do that? Dialogue: 0,0:41:37.42,0:41:48.67,Default,,0000,0000,0000,,Find the flux\Ncorresponding to-- do Dialogue: 0,0:41:48.67,0:41:51.84,Default,,0000,0000,0000,,we say restart--\Njust a second-- just Dialogue: 0,0:41:51.84,0:41:54.13,Default,,0000,0000,0000,,to restart with applications. Dialogue: 0,0:41:54.13,0:41:54.63,Default,,0000,0000,0000,,[INAUDIBLE] Dialogue: 0,0:41:54.63,0:41:57.27,Default,,0000,0000,0000,, Dialogue: 0,0:41:57.27,0:42:01.49,Default,,0000,0000,0000,,We don't say what kind of\Nvector field that it is, Dialogue: 0,0:42:01.49,0:42:06.82,Default,,0000,0000,0000,,but we will say plus\Ncorresponding to the vector Dialogue: 0,0:42:06.82,0:42:08.42,Default,,0000,0000,0000,,field. Dialogue: 0,0:42:08.42,0:42:09.00,Default,,0000,0000,0000,,F [INAUDIBLE]. Dialogue: 0,0:42:09.00,0:42:14.38,Default,,0000,0000,0000,, Dialogue: 0,0:42:14.38,0:42:16.30,Default,,0000,0000,0000,,And this vector\Nfield, I'll tell you Dialogue: 0,0:42:16.30,0:42:22.45,Default,,0000,0000,0000,,in a second what's expected\Nfrom this to be a vector field. Dialogue: 0,0:42:22.45,0:42:35.13,Default,,0000,0000,0000,,Through, on the surface, we\Nfind on the surface-- yes. Dialogue: 0,0:42:35.13,0:42:39.15,Default,,0000,0000,0000,,Mathematicians say\Ndefine normal surface S. Dialogue: 0,0:42:39.15,0:42:44.46,Default,,0000,0000,0000,,But a physicist will\Nsay flux through, Dialogue: 0,0:42:44.46,0:42:55.35,Default,,0000,0000,0000,,the flux corresponding\Nto F through the surface. Dialogue: 0,0:42:55.35,0:42:58.70,Default,,0000,0000,0000,, Dialogue: 0,0:42:58.70,0:43:01.17,Default,,0000,0000,0000,,Yes. Dialogue: 0,0:43:01.17,0:43:03.94,Default,,0000,0000,0000,,So you'll say why would that\Nbe, and what is the flux? Dialogue: 0,0:43:03.94,0:43:11.96,Default,,0000,0000,0000,,By definition, how\Nshould we denote it? Dialogue: 0,0:43:11.96,0:43:16.37,Default,,0000,0000,0000,,Let's make a beautiful script\NF. That's crazy, right? Dialogue: 0,0:43:16.37,0:43:21.59,Default,,0000,0000,0000,,And then it goes doubling\Nover the surface F test. Dialogue: 0,0:43:21.59,0:43:24.91,Default,,0000,0000,0000,,Is anybody mechanical\Nengineering here? Dialogue: 0,0:43:24.91,0:43:28.38,Default,,0000,0000,0000,,Do you know the flux formula? Dialogue: 0,0:43:28.38,0:43:33.98,Default,,0000,0000,0000,,It's going to be [INAUDIBLE]\Nover S of F, this magic thing. Dialogue: 0,0:43:33.98,0:43:36.89,Default,,0000,0000,0000,,Not DN, DS. Dialogue: 0,0:43:36.89,0:43:38.96,Default,,0000,0000,0000,,Do you know what N means? Dialogue: 0,0:43:38.96,0:43:41.25,Default,,0000,0000,0000,,What it is N for\Nmechanical engineering, Dialogue: 0,0:43:41.25,0:43:42.94,Default,,0000,0000,0000,,[INAUDIBLE] engineers? Dialogue: 0,0:43:42.94,0:44:01.07,Default,,0000,0000,0000,,N to would be the unit normal\Nvector field to the surface S. Dialogue: 0,0:44:01.07,0:44:03.05,Default,,0000,0000,0000,,How would you want\Nto imagine that? Dialogue: 0,0:44:03.05,0:44:07.11,Default,,0000,0000,0000,,You would have a surface, and\Nyou have this velocity vectors Dialogue: 0,0:44:07.11,0:44:13.35,Default,,0000,0000,0000,,here at the bottom that goes\Nto S. And this field goes up. Dialogue: 0,0:44:13.35,0:44:17.46,Default,,0000,0000,0000,,You'll have a force and\Nacceleration, velocity, Dialogue: 0,0:44:17.46,0:44:19.85,Default,,0000,0000,0000,,you have everything\Ngoing this way. Dialogue: 0,0:44:19.85,0:44:22.80,Default,,0000,0000,0000,,And you want to find\Nout what happens. Dialogue: 0,0:44:22.80,0:44:26.72,Default,,0000,0000,0000,,You introduce this notion\Nof flux through the surface. Dialogue: 0,0:44:26.72,0:44:29.40,Default,,0000,0000,0000,,Another way to have a\Nflux through the surface Dialogue: 0,0:44:29.40,0:44:31.46,Default,,0000,0000,0000,,maybe through the same\Nsurface but associated Dialogue: 0,0:44:31.46,0:44:34.04,Default,,0000,0000,0000,,through another\Nkind of concept-- Dialogue: 0,0:44:34.04,0:44:36.18,Default,,0000,0000,0000,,if there could be\Nsomething else. Dialogue: 0,0:44:36.18,0:44:39.96,Default,,0000,0000,0000,,In electromagnetism, F would be\Nsomething else, some other type Dialogue: 0,0:44:39.96,0:44:41.08,Default,,0000,0000,0000,,of vector field. Dialogue: 0,0:44:41.08,0:44:42.14,Default,,0000,0000,0000,,Yes, sir. Dialogue: 0,0:44:42.14,0:44:43.02,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:44:43.02,0:44:45.55,Default,,0000,0000,0000,,PROFESSOR: So find out, by\Nthe way until next time, Dialogue: 0,0:44:45.55,0:44:49.39,Default,,0000,0000,0000,,if you were an electrical\Nengineering major, what Dialogue: 0,0:44:49.39,0:44:51.93,Default,,0000,0000,0000,,would flux be for you guys? Dialogue: 0,0:44:51.93,0:44:55.91,Default,,0000,0000,0000,,Two surfaces, one would be the\Nmeaning of the vector field Dialogue: 0,0:44:55.91,0:44:58.28,Default,,0000,0000,0000,,F for you, and\Nwhy would you care Dialogue: 0,0:44:58.28,0:45:01.24,Default,,0000,0000,0000,,about the electromagnetic\Nflux or something like that. Dialogue: 0,0:45:01.24,0:45:02.92,Default,,0000,0000,0000,,I don't want to talk\Ntoo much about it. Dialogue: 0,0:45:02.92,0:45:07.15,Default,,0000,0000,0000,,It's for you to do the\Nsearch and find out. Dialogue: 0,0:45:07.15,0:45:09.88,Default,,0000,0000,0000,,So suppose that\Nsomebody gives you Dialogue: 0,0:45:09.88,0:45:13.30,Default,,0000,0000,0000,,this notion that says you\Nhave a parameteric surface. Dialogue: 0,0:45:13.30,0:45:19.05,Default,,0000,0000,0000,,Give an application\Nof that and find out Dialogue: 0,0:45:19.05,0:45:24.17,Default,,0000,0000,0000,,how you're going\Nbe deal with it. Dialogue: 0,0:45:24.17,0:45:27.84,Default,,0000,0000,0000,,I'll give you an example\Nthat shouldn't be too hard. Dialogue: 0,0:45:27.84,0:45:32.70,Default,,0000,0000,0000,, Dialogue: 0,0:45:32.70,0:45:34.44,Default,,0000,0000,0000,,I'll make up my own example. Dialogue: 0,0:45:34.44,0:45:38.47,Default,,0000,0000,0000,,And looks like example 6, but\Nit's going to be different. Dialogue: 0,0:45:38.47,0:45:47.39,Default,,0000,0000,0000,, Dialogue: 0,0:45:47.39,0:45:47.89,Default,,0000,0000,0000,,Example. Dialogue: 0,0:45:47.89,0:45:51.33,Default,,0000,0000,0000,, Dialogue: 0,0:45:51.33,0:45:58.60,Default,,0000,0000,0000,,Find the flux F if F will\Nbe a simple function. Dialogue: 0,0:45:58.60,0:46:05.24,Default,,0000,0000,0000,,Let's say something equals X, I\Nplus Y,J Z, K at every point X, Dialogue: 0,0:46:05.24,0:46:13.03,Default,,0000,0000,0000,,Y-- at every point\Nof the space XYZ. Dialogue: 0,0:46:13.03,0:46:16.38,Default,,0000,0000,0000,,That means you could have this\Nvector field defined everywhere Dialogue: 0,0:46:16.38,0:46:18.36,Default,,0000,0000,0000,,in space in [INAUDIBLE]. Dialogue: 0,0:46:18.36,0:46:22.66,Default,,0000,0000,0000,,But you only care about\Nthis acting on the surface. Dialogue: 0,0:46:22.66,0:46:25.06,Default,,0000,0000,0000,,So it's acting on the surface. Dialogue: 0,0:46:25.06,0:46:28.00,Default,,0000,0000,0000,, Dialogue: 0,0:46:28.00,0:46:30.33,Default,,0000,0000,0000,,And then what will the flux be? Dialogue: 0,0:46:30.33,0:46:34.57,Default,,0000,0000,0000,,On the surface, which surface? Dialogue: 0,0:46:34.57,0:46:41.01,Default,,0000,0000,0000,,My favorite one, Z equals\NX squared plus Y squared. Dialogue: 0,0:46:41.01,0:46:47.29,Default,,0000,0000,0000,, Dialogue: 0,0:46:47.29,0:46:49.26,Default,,0000,0000,0000,,First of all, you say\Nwait, wait, Magdalena, Dialogue: 0,0:46:49.26,0:46:50.99,Default,,0000,0000,0000,,do you want to do it like that? Dialogue: 0,0:46:50.99,0:46:54.33,Default,,0000,0000,0000,,Do you want to say F\Nover XY to be a graph? Dialogue: 0,0:46:54.33,0:46:58.80,Default,,0000,0000,0000,,Or do you want to consider it\Nas a parameterized surface? Dialogue: 0,0:46:58.80,0:47:01.92,Default,,0000,0000,0000,,And that means it's the same\Nthing, equivalent to or if Dialogue: 0,0:47:01.92,0:47:09.00,Default,,0000,0000,0000,,and only if, who tells me again\Nwhat R was for such a surface? Dialogue: 0,0:47:09.00,0:47:09.50,Default,,0000,0000,0000,,STUDENT: XI. Dialogue: 0,0:47:09.50,0:47:11.38,Default,,0000,0000,0000,,PROFESSOR: X is\NU. Y is V, so U-- Dialogue: 0,0:47:11.38,0:47:12.21,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:47:12.21,0:47:15.75,Default,,0000,0000,0000,,PROFESSOR: --I, that\Nwould be J, then good. Dialogue: 0,0:47:15.75,0:47:19.72,Default,,0000,0000,0000,,U squared plus U squared UK. Dialogue: 0,0:47:19.72,0:47:23.30,Default,,0000,0000,0000,,Well, when you say that,\Nwe have-- first of all, Dialogue: 0,0:47:23.30,0:47:26.92,Default,,0000,0000,0000,,we have no idea what\Nthe heck we need to do, Dialogue: 0,0:47:26.92,0:47:32.45,Default,,0000,0000,0000,,because do we want to do it\Nin this form like a graph? Dialogue: 0,0:47:32.45,0:47:34.06,Default,,0000,0000,0000,,Or do we want to do\Nit parameterized? Dialogue: 0,0:47:34.06,0:47:37.34,Default,,0000,0000,0000,,We have to set up\Nformulas for the flats. Dialogue: 0,0:47:37.34,0:47:38.65,Default,,0000,0000,0000,,It's not so easy. Dialogue: 0,0:47:38.65,0:47:43.25,Default,,0000,0000,0000,,So assume that we are brave\Nenough and we start everything. Dialogue: 0,0:47:43.25,0:47:48.50,Default,,0000,0000,0000,,I want to understand what\Nflux really is as an integral. Dialogue: 0,0:47:48.50,0:47:55.51,Default,,0000,0000,0000,,And let me set it up for the\Nfirst case, the case of Z Dialogue: 0,0:47:55.51,0:47:58.39,Default,,0000,0000,0000,,equals F of X and Y.\NAnd I'm happy with it Dialogue: 0,0:47:58.39,0:48:01.30,Default,,0000,0000,0000,,because that's\Nthe simplest case. Dialogue: 0,0:48:01.30,0:48:03.93,Default,,0000,0000,0000,,Who's going to teach\Nme what I have to do? Dialogue: 0,0:48:03.93,0:48:05.24,Default,,0000,0000,0000,,You are confusing. Dialogue: 0,0:48:05.24,0:48:09.92,Default,,0000,0000,0000,,I have double integral over S\Nminus theory of F in general. Dialogue: 0,0:48:09.92,0:48:12.81,Default,,0000,0000,0000,,This is a general\Nvector value field. Dialogue: 0,0:48:12.81,0:48:15.81,Default,,0000,0000,0000,, Dialogue: 0,0:48:15.81,0:48:16.94,Default,,0000,0000,0000,,It could be anything. Dialogue: 0,0:48:16.94,0:48:18.22,Default,,0000,0000,0000,,Could be anything. Dialogue: 0,0:48:18.22,0:48:23.21,Default,,0000,0000,0000,,But then I have to [INAUDIBLE],\Nbecause N corresponds Dialogue: 0,0:48:23.21,0:48:27.18,Default,,0000,0000,0000,,to the normal to the surface. Dialogue: 0,0:48:27.18,0:48:29.27,Default,,0000,0000,0000,,So I-- it's not so easy, right? Dialogue: 0,0:48:29.27,0:48:30.94,Default,,0000,0000,0000,,I have to be a little bit smart. Dialogue: 0,0:48:30.94,0:48:31.99,Default,,0000,0000,0000,,If I'm not smart-- Dialogue: 0,0:48:31.99,0:48:32.82,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:48:32.82,0:48:35.12,Default,,0000,0000,0000,, Dialogue: 0,0:48:35.12,0:48:36.79,Default,,0000,0000,0000,,PROFESSOR: That-- you\Nare getting close. Dialogue: 0,0:48:36.79,0:48:40.87,Default,,0000,0000,0000,,So guys, the normal\Ntwo-way surface-- somebody Dialogue: 0,0:48:40.87,0:48:42.55,Default,,0000,0000,0000,,gave you a surface, OK? Dialogue: 0,0:48:42.55,0:48:46.10,Default,,0000,0000,0000,,And normal to a surface\Nis normal to the plane-- Dialogue: 0,0:48:46.10,0:48:49.59,Default,,0000,0000,0000,,the tangent plane\Nof the surface. Dialogue: 0,0:48:49.59,0:48:51.60,Default,,0000,0000,0000,,So how did we get that? Dialogue: 0,0:48:51.60,0:48:53.72,Default,,0000,0000,0000,,There were many ways to do it. Dialogue: 0,0:48:53.72,0:48:56.19,Default,,0000,0000,0000,,Either you write\Nthe tangent plane Dialogue: 0,0:48:56.19,0:49:02.70,Default,,0000,0000,0000,,and you know it by heart--\Nthat was Z minus Z zero Dialogue: 0,0:49:02.70,0:49:07.46,Default,,0000,0000,0000,,equals-- what the heck was\Nthat-- S of X times X minus X Dialogue: 0,0:49:07.46,0:49:10.71,Default,,0000,0000,0000,,equals-- plus X of Y\Ntimes Y minus Y zero. Dialogue: 0,0:49:10.71,0:49:14.11,Default,,0000,0000,0000,,And from here you collect--\Nwhat do you collect? Dialogue: 0,0:49:14.11,0:49:16.14,Default,,0000,0000,0000,,You move everybody--\Nit's a moving sale. Dialogue: 0,0:49:16.14,0:49:19.55,Default,,0000,0000,0000,,You move everybody to the\Nleft hand side and that's it. Dialogue: 0,0:49:19.55,0:49:21.20,Default,,0000,0000,0000,,[INAUDIBLE] moving sale. Dialogue: 0,0:49:21.20,0:49:22.85,Default,,0000,0000,0000,,OK? Dialogue: 0,0:49:22.85,0:49:26.90,Default,,0000,0000,0000,,And everybody will be\Ngiving you some components. Dialogue: 0,0:49:26.90,0:49:31.03,Default,,0000,0000,0000,,You're going to have minus S\Nof X-- S minus X zero-- minus S Dialogue: 0,0:49:31.03,0:49:36.62,Default,,0000,0000,0000,,of Y, Y minus Y zero, plus\N1-- this is really funny. Dialogue: 0,0:49:36.62,0:49:39.70,Default,,0000,0000,0000,,1 times Z minus Z, Z. Dialogue: 0,0:49:39.70,0:49:43.37,Default,,0000,0000,0000,,Your normal will\Nbe given by what? Dialogue: 0,0:49:43.37,0:49:46.30,Default,,0000,0000,0000,,The normal-- how do\Nyou collect the normal? Dialogue: 0,0:49:46.30,0:49:47.13,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:49:47.13,0:49:52.30,Default,,0000,0000,0000,,PROFESSOR: Pi is A, B, C. A,\NB, and C will be the normal. Dialogue: 0,0:49:52.30,0:49:54.30,Default,,0000,0000,0000,,Except it's not unitary. Dialogue: 0,0:49:54.30,0:49:57.82,Default,,0000,0000,0000,,And the mechanical engineer\Ntells you, yeah, you're Dialogue: 0,0:49:57.82,0:50:00.99,Default,,0000,0000,0000,,stupid-- well, they\Nnever say that. Dialogue: 0,0:50:00.99,0:50:06.08,Default,,0000,0000,0000,,They will stay look, you have\Nto be a little more careful. Dialogue: 0,0:50:06.08,0:50:07.68,Default,,0000,0000,0000,,Not say they are equal. Dialogue: 0,0:50:07.68,0:50:08.94,Default,,0000,0000,0000,,What do they mean? Dialogue: 0,0:50:08.94,0:50:10.54,Default,,0000,0000,0000,,They say for us,\Nin fluid mechanics, Dialogue: 0,0:50:10.54,0:50:14.98,Default,,0000,0000,0000,,solid mechanics, when we write\NN, we mean you mean vector. Dialogue: 0,0:50:14.98,0:50:16.15,Default,,0000,0000,0000,,You are almost there. Dialogue: 0,0:50:16.15,0:50:16.91,Default,,0000,0000,0000,,What's missing? Dialogue: 0,0:50:16.91,0:50:18.24,Default,,0000,0000,0000,,STUDENT: Magnitude. [INAUDIBLE]. Dialogue: 0,0:50:18.24,0:50:19.74,Default,,0000,0000,0000,,PROFESSOR: Very\Ngood, the magnitude. Dialogue: 0,0:50:19.74,0:50:24.46,Default,,0000,0000,0000,,So they will say, go ahead and\Nyou [INAUDIBLE] the magnitude. Dialogue: 0,0:50:24.46,0:50:28.99,Default,,0000,0000,0000,,And you are lucky now that\Nyou know what N will be. Dialogue: 0,0:50:28.99,0:50:29.98,Default,,0000,0000,0000,,On the other hand-- Dialogue: 0,0:50:29.98,0:50:31.13,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:50:31.13,0:50:32.35,Default,,0000,0000,0000,,PROFESSOR: This is excellent. Dialogue: 0,0:50:32.35,0:50:35.60,Default,,0000,0000,0000,,The one on the bottom-- Alex\Nis thinking like in chess, two Dialogue: 0,0:50:35.60,0:50:37.90,Default,,0000,0000,0000,,or three moves ahead. Dialogue: 0,0:50:37.90,0:50:40.69,Default,,0000,0000,0000,,You should get two extra\Ncredit points with that. Dialogue: 0,0:50:40.69,0:50:41.48,Default,,0000,0000,0000,,STUDENT: All right. Dialogue: 0,0:50:41.48,0:50:42.73,Default,,0000,0000,0000,,PROFESSOR: You already got it. Dialogue: 0,0:50:42.73,0:50:46.36,Default,,0000,0000,0000,,DS is 1 plus S of X squared\Nplus F of X squared. Dialogue: 0,0:50:46.36,0:50:51.75,Default,,0000,0000,0000,,The 1 on the bottom and the\N1 on the top will simplify. Dialogue: 0,0:50:51.75,0:50:53.00,Default,,0000,0000,0000,,So say it again, Magdalena. Dialogue: 0,0:50:53.00,0:50:54.98,Default,,0000,0000,0000,,Let me write it down here. Dialogue: 0,0:50:54.98,0:51:01.44,Default,,0000,0000,0000,,1 S of X, minus S of Y\N1 over all this animal, Dialogue: 0,0:51:01.44,0:51:04.94,Default,,0000,0000,0000,,S of X squared plus S\Nof Y squared plus 1. Dialogue: 0,0:51:04.94,0:51:08.85,Default,,0000,0000,0000,,This is the thinking\Nlike the early element Dialogue: 0,0:51:08.85,0:51:13.47,Default,,0000,0000,0000,,times the early element\Nwill be the same thing. Dialogue: 0,0:51:13.47,0:51:17.59,Default,,0000,0000,0000,,I'll write it twice even if you\Nlaugh at me because we are just Dialogue: 0,0:51:17.59,0:51:19.43,Default,,0000,0000,0000,,learning together,\Nand now you finally Dialogue: 0,0:51:19.43,0:51:21.35,Default,,0000,0000,0000,,see-- everybody can\Nsee that desimplifies. Dialogue: 0,0:51:21.35,0:51:23.97,Default,,0000,0000,0000,, Dialogue: 0,0:51:23.97,0:51:27.85,Default,,0000,0000,0000,,So it's going to be easy to\Nsolve this integral in the end, Dialogue: 0,0:51:27.85,0:51:29.13,Default,,0000,0000,0000,,right? Dialogue: 0,0:51:29.13,0:51:32.12,Default,,0000,0000,0000,,So let's do the\Nproblem, finally. Dialogue: 0,0:51:32.12,0:51:35.59,Default,,0000,0000,0000,,I'm going to erase it. Dialogue: 0,0:51:35.59,0:51:40.05,Default,,0000,0000,0000,,Let's do this problem\Njust for us, at any point. Dialogue: 0,0:51:40.05,0:51:42.71,Default,,0000,0000,0000,,I didn't say where. Dialogue: 0,0:51:42.71,0:51:46.78,Default,,0000,0000,0000,,Over the same thing. Dialogue: 0,0:51:46.78,0:51:49.94,Default,,0000,0000,0000,,The DS was over V01. Dialogue: 0,0:51:49.94,0:51:53.02,Default,,0000,0000,0000,,So the picture is\Nthe same as before. Dialogue: 0,0:51:53.02,0:51:57.56,Default,,0000,0000,0000,,The S will be the\Nnutshell, the eggshell-- Dialogue: 0,0:51:57.56,0:52:02.78,Default,,0000,0000,0000,,I don't know what it was--\Nover the domain D plane. Dialogue: 0,0:52:02.78,0:52:07.62,Default,,0000,0000,0000,,The domain D plane\Nwas D of zero 1. Dialogue: 0,0:52:07.62,0:52:12.16,Default,,0000,0000,0000,,And I say that I need\Nto use another color. Dialogue: 0,0:52:12.16,0:52:17.43,Default,,0000,0000,0000,,This it's going to be my\Nshell, my surface S. Z Dialogue: 0,0:52:17.43,0:52:20.47,Default,,0000,0000,0000,,equals X squared\Nplus [INAUDIBLE]. Dialogue: 0,0:52:20.47,0:52:23.14,Default,,0000,0000,0000,,How do you compute the flux? Dialogue: 0,0:52:23.14,0:52:25.51,Default,,0000,0000,0000,,Well, this is that. Dialogue: 0,0:52:25.51,0:52:28.27,Default,,0000,0000,0000,,So if we have to be a\Nlittle bit careful and smart Dialogue: 0,0:52:28.27,0:52:33.16,Default,,0000,0000,0000,,and say double integral over\NS, and now without rushing, Dialogue: 0,0:52:33.16,0:52:35.73,Default,,0000,0000,0000,,we have to do a good job. Dialogue: 0,0:52:35.73,0:52:39.21,Default,,0000,0000,0000,,First of all, how do\Nyou do the dot product? Dialogue: 0,0:52:39.21,0:52:40.86,Default,,0000,0000,0000,,The dot product-- Dialogue: 0,0:52:40.86,0:52:43.51,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:52:43.51,0:52:44.22,Default,,0000,0000,0000,,PROFESSOR: Right. Dialogue: 0,0:52:44.22,0:52:46.53,Default,,0000,0000,0000,,So first component\Ntimes first component, Dialogue: 0,0:52:46.53,0:52:50.29,Default,,0000,0000,0000,,a second component,\Nsecond component times Dialogue: 0,0:52:50.29,0:52:52.53,Default,,0000,0000,0000,,second component plus\Nthat component times Dialogue: 0,0:52:52.53,0:52:53.71,Default,,0000,0000,0000,,third component. Dialogue: 0,0:52:53.71,0:53:01.27,Default,,0000,0000,0000,,So if 1 is X, F2 is 1. Dialogue: 0,0:53:01.27,0:53:02.17,Default,,0000,0000,0000,,Good. Dialogue: 0,0:53:02.17,0:53:04.29,Default,,0000,0000,0000,,Z, though, he's not free. Dialogue: 0,0:53:04.29,0:53:05.27,Default,,0000,0000,0000,,He's married. Dialogue: 0,0:53:05.27,0:53:07.04,Default,,0000,0000,0000,,Why is he married? Dialogue: 0,0:53:07.04,0:53:07.92,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:53:07.92,0:53:10.58,Default,,0000,0000,0000,,PROFESSOR: Because he\Ndepends on X and Y. Dialogue: 0,0:53:10.58,0:53:14.23,Default,,0000,0000,0000,,So Z was even here,\Nbecause I'm on the surface. Dialogue: 0,0:53:14.23,0:53:16.69,Default,,0000,0000,0000,,I don't care what F does\Naway from the surface, Dialogue: 0,0:53:16.69,0:53:20.05,Default,,0000,0000,0000,,but when he sticks\Nto the surface, when Dialogue: 0,0:53:20.05,0:53:24.03,Default,,0000,0000,0000,,he's origin is on\Nthe surface, then he Dialogue: 0,0:53:24.03,0:53:27.86,Default,,0000,0000,0000,,has to listen to the surface. Dialogue: 0,0:53:27.86,0:53:29.88,Default,,0000,0000,0000,,And that Z is not independent. Dialogue: 0,0:53:29.88,0:53:33.11,Default,,0000,0000,0000,,The Z is X squared\Nby Y squared here. Dialogue: 0,0:53:33.11,0:53:36.74,Default,,0000,0000,0000,,In a bracket, we are\Nover the surface. Dialogue: 0,0:53:36.74,0:53:41.11,Default,,0000,0000,0000,,That product minus S\Nof X, minus S of Y. I Dialogue: 0,0:53:41.11,0:53:42.96,Default,,0000,0000,0000,,know you're going to\Nlaugh at me because I Dialogue: 0,0:53:42.96,0:53:44.67,Default,,0000,0000,0000,,haven't written where they are. Dialogue: 0,0:53:44.67,0:53:46.37,Default,,0000,0000,0000,,But that's what I\Nneed your help for. Dialogue: 0,0:53:46.37,0:53:46.87,Default,,0000,0000,0000,,DA. Dialogue: 0,0:53:46.87,0:53:49.39,Default,,0000,0000,0000,, Dialogue: 0,0:53:49.39,0:53:51.81,Default,,0000,0000,0000,,Who are they? Dialogue: 0,0:53:51.81,0:53:53.76,Default,,0000,0000,0000,,Who is this guy? Dialogue: 0,0:53:53.76,0:53:54.80,Default,,0000,0000,0000,,STUDENT: The [INAUDIBLE]. Dialogue: 0,0:53:54.80,0:53:55.47,Default,,0000,0000,0000,,PROFESSOR: What? Dialogue: 0,0:53:55.47,0:53:56.58,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:53:56.58,0:53:58.00,Default,,0000,0000,0000,,PROFESSOR: Negative 2X. Dialogue: 0,0:53:58.00,0:53:58.86,Default,,0000,0000,0000,,Is it? Dialogue: 0,0:53:58.86,0:53:59.85,Default,,0000,0000,0000,,STUDENT: No. Dialogue: 0,0:53:59.85,0:54:01.35,Default,,0000,0000,0000,,PROFESSOR: How about this guy? Dialogue: 0,0:54:01.35,0:54:02.43,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:54:02.43,0:54:03.39,Default,,0000,0000,0000,,PROFESSOR: Negative 2Y. Dialogue: 0,0:54:03.39,0:54:05.75,Default,,0000,0000,0000,,How about this guy? Dialogue: 0,0:54:05.75,0:54:06.71,Default,,0000,0000,0000,,I'm just kidding. Dialogue: 0,0:54:06.71,0:54:07.68,Default,,0000,0000,0000,,OK. Dialogue: 0,0:54:07.68,0:54:14.23,Default,,0000,0000,0000,,So finally we should be able\Nto compute this integral. Dialogue: 0,0:54:14.23,0:54:15.65,Default,,0000,0000,0000,,That looks awful. Dialogue: 0,0:54:15.65,0:54:17.56,Default,,0000,0000,0000,,Over D. Dialogue: 0,0:54:17.56,0:54:20.58,Default,,0000,0000,0000,,So instead of S, we\Nhave the D, which Dialogue: 0,0:54:20.58,0:54:25.17,Default,,0000,0000,0000,,is the disk of\Nradius one in plane. Dialogue: 0,0:54:25.17,0:54:29.74,Default,,0000,0000,0000,,And we say, OK, I have,\Noh my god, it's OK. Dialogue: 0,0:54:29.74,0:54:31.72,Default,,0000,0000,0000,,This times that is how much? Dialogue: 0,0:54:31.72,0:54:32.59,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:54:32.59,0:54:33.89,Default,,0000,0000,0000,,PROFESSOR: Minus 2X squared. Dialogue: 0,0:54:33.89,0:54:35.19,Default,,0000,0000,0000,,Right? Dialogue: 0,0:54:35.19,0:54:35.88,Default,,0000,0000,0000,,There. Dialogue: 0,0:54:35.88,0:54:37.12,Default,,0000,0000,0000,,Take the green. Dialogue: 0,0:54:37.12,0:54:40.87,Default,,0000,0000,0000,,This times that is how much? Dialogue: 0,0:54:40.87,0:54:44.25,Default,,0000,0000,0000,,Minus the Y squared. Dialogue: 0,0:54:44.25,0:54:50.78,Default,,0000,0000,0000,,And this times that is finally\Njust X squared plus Y squared. Dialogue: 0,0:54:50.78,0:54:53.44,Default,,0000,0000,0000,,Very nice think. Dialogue: 0,0:54:53.44,0:54:55.56,Default,,0000,0000,0000,,I think that at\Nfirst, but now I see Dialogue: 0,0:54:55.56,0:54:59.11,Default,,0000,0000,0000,,that life is beautiful\Nagain-- DX, DY-- Dialogue: 0,0:54:59.11,0:55:02.77,Default,,0000,0000,0000,,that I can go ahead and do it. Dialogue: 0,0:55:02.77,0:55:05.05,Default,,0000,0000,0000,,I can get a hold of this. Dialogue: 0,0:55:05.05,0:55:11.13,Default,,0000,0000,0000,,And inside that, what do I--\Nwhat am I left with in the end? Dialogue: 0,0:55:11.13,0:55:12.05,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:55:12.05,0:55:14.65,Default,,0000,0000,0000,,PROFESSOR: Minus 2\Ntimes this animal, Dialogue: 0,0:55:14.65,0:55:18.55,Default,,0000,0000,0000,,called X squared plus Y squared,\Nwhich is going to be R squared. Dialogue: 0,0:55:18.55,0:55:23.23,Default,,0000,0000,0000,,So the flux-- the flux for\Nthis problem in the end Dialogue: 0,0:55:23.23,0:55:25.64,Default,,0000,0000,0000,,is going to be very\Nnice and sassy. Dialogue: 0,0:55:25.64,0:55:26.31,Default,,0000,0000,0000,,Look at that. Dialogue: 0,0:55:26.31,0:55:28.14,Default,,0000,0000,0000,,F would be-- Dialogue: 0,0:55:28.14,0:55:29.52,Default,,0000,0000,0000,,STUDENT: There\Nwould not be any-- Dialogue: 0,0:55:29.52,0:55:30.35,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,0:55:30.35,0:55:36.58,Default,,0000,0000,0000,, Dialogue: 0,0:55:36.58,0:55:37.39,Default,,0000,0000,0000,,PROFESSOR: What? Dialogue: 0,0:55:37.39,0:55:40.99,Default,,0000,0000,0000,,STUDENT: You've got\Nminus 2 and the plus 1. Dialogue: 0,0:55:40.99,0:55:42.03,Default,,0000,0000,0000,,PROFESSOR: Oh, thank God. Dialogue: 0,0:55:42.03,0:55:43.73,Default,,0000,0000,0000,,Thank God you exist. Dialogue: 0,0:55:43.73,0:55:47.25,Default,,0000,0000,0000,,So I thought about\Nit before, but then I Dialogue: 0,0:55:47.25,0:55:49.04,Default,,0000,0000,0000,,said-- I don't know why. Dialogue: 0,0:55:49.04,0:55:50.12,Default,,0000,0000,0000,,I messed up. Dialogue: 0,0:55:50.12,0:55:52.54,Default,,0000,0000,0000,,So we have minus R squared. Dialogue: 0,0:55:52.54,0:55:53.53,Default,,0000,0000,0000,,Very good. Dialogue: 0,0:55:53.53,0:55:54.81,Default,,0000,0000,0000,,It's easy. Dialogue: 0,0:55:54.81,0:56:01.95,Default,,0000,0000,0000,,Times an R from the\NJacobian, DR is theta. Dialogue: 0,0:56:01.95,0:56:04.55,Default,,0000,0000,0000,,And theta is between 0 and 2 pi. Dialogue: 0,0:56:04.55,0:56:08.02,Default,,0000,0000,0000,,And R between 0 and 1. Dialogue: 0,0:56:08.02,0:56:11.05,Default,,0000,0000,0000,,And now I will need\Na plumber to tell me Dialogue: 0,0:56:11.05,0:56:12.79,Default,,0000,0000,0000,,what I do the limits\Nof the integrals, Dialogue: 0,0:56:12.79,0:56:17.75,Default,,0000,0000,0000,,because I think I'm getting\Na negative answer, so. Dialogue: 0,0:56:17.75,0:56:20.53,Default,,0000,0000,0000,, Dialogue: 0,0:56:20.53,0:56:21.69,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:56:21.69,0:56:23.15,Default,,0000,0000,0000,,PROFESSOR: I'll do\Nit, and then you Dialogue: 0,0:56:23.15,0:56:26.70,Default,,0000,0000,0000,,tell me why I got what I got. Dialogue: 0,0:56:26.70,0:56:29.71,Default,,0000,0000,0000,,I have a minus\Npulled out by nature. Dialogue: 0,0:56:29.71,0:56:32.85,Default,,0000,0000,0000,,And then I have integral-- Dialogue: 0,0:56:32.85,0:56:34.00,Default,,0000,0000,0000,,STUDENT: R [INAUDIBLE]. Dialogue: 0,0:56:34.00,0:56:35.82,Default,,0000,0000,0000,,PROFESSOR: R to the\Nfourth of a fourth. Dialogue: 0,0:56:35.82,0:56:36.59,Default,,0000,0000,0000,,Very good. Dialogue: 0,0:56:36.59,0:56:40.45,Default,,0000,0000,0000,,But you have your [INAUDIBLE] so\Nwhen I do between zero and 1-- Dialogue: 0,0:56:40.45,0:56:41.53,Default,,0000,0000,0000,,STUDENT: It's [INAUDIBLE]. Dialogue: 0,0:56:41.53,0:56:43.83,Default,,0000,0000,0000,,PROFESSOR: 1 over\N4-- you are too Dialogue: 0,0:56:43.83,0:56:49.40,Default,,0000,0000,0000,,fast-- as 2 pi-- that's a\Ngood thing-- minus pi over 2, Dialogue: 0,0:56:49.40,0:56:50.89,Default,,0000,0000,0000,,you said, Gus. Dialogue: 0,0:56:50.89,0:56:53.16,Default,,0000,0000,0000,,And I could see it\Ncoming straight at me Dialogue: 0,0:56:53.16,0:56:55.32,Default,,0000,0000,0000,,and hit me between the eyes. Dialogue: 0,0:56:55.32,0:56:57.72,Default,,0000,0000,0000,,What is the problem. Dialogue: 0,0:56:57.72,0:56:59.75,Default,,0000,0000,0000,,Is there a problem? Dialogue: 0,0:56:59.75,0:57:02.82,Default,,0000,0000,0000,,Without an area as a flux, would\Nthat say, what is the negative? Dialogue: 0,0:57:02.82,0:57:03.81,Default,,0000,0000,0000,,Yes. Dialogue: 0,0:57:03.81,0:57:05.63,Default,,0000,0000,0000,,How can I make it positive? Dialogue: 0,0:57:05.63,0:57:07.48,Default,,0000,0000,0000,,This is my question. Dialogue: 0,0:57:07.48,0:57:08.73,Default,,0000,0000,0000,,STUDENT: Change the direction. Dialogue: 0,0:57:08.73,0:57:10.62,Default,,0000,0000,0000,,PROFESSOR: Change\Nthe direction of who? Dialogue: 0,0:57:10.62,0:57:12.22,Default,,0000,0000,0000,,STUDENT: The flux. Dialogue: 0,0:57:12.22,0:57:13.36,Default,,0000,0000,0000,,PROFESSOR: The flux. Dialogue: 0,0:57:13.36,0:57:14.85,Default,,0000,0000,0000,,I could change the direction. Dialogue: 0,0:57:14.85,0:57:17.64,Default,,0000,0000,0000,,So what is it that\Ndoesn't match? Dialogue: 0,0:57:17.64,0:57:19.98,Default,,0000,0000,0000,,[INAUDIBLE] Dialogue: 0,0:57:19.98,0:57:22.64,Default,,0000,0000,0000,,If I want to keep-- the\Nflux will be the same. Dialogue: 0,0:57:22.64,0:57:25.27,Default,,0000,0000,0000,,When I can change the\Norientation of the service. Dialogue: 0,0:57:25.27,0:57:27.82,Default,,0000,0000,0000,,And instead I get a minus then. Dialogue: 0,0:57:27.82,0:57:36.41,Default,,0000,0000,0000,,My N was it sticking\Nin-- oh, my god. Dialogue: 0,0:57:36.41,0:57:40.34,Default,,0000,0000,0000,,So is it sticking\Nin or sticking out? Dialogue: 0,0:57:40.34,0:57:40.99,Default,,0000,0000,0000,,Look at it. Dialogue: 0,0:57:40.99,0:57:41.90,Default,,0000,0000,0000,,Think about it. Dialogue: 0,0:57:41.90,0:57:47.66,Default,,0000,0000,0000,,I have minus the positive guy\Nminus another positive guy, Dialogue: 0,0:57:47.66,0:57:48.98,Default,,0000,0000,0000,,and 1 sticking out. Dialogue: 0,0:57:48.98,0:57:51.01,Default,,0000,0000,0000,,But it goes with\Nthe holes inside. Dialogue: 0,0:57:51.01,0:57:54.02,Default,,0000,0000,0000,,This is the paraboloid\N[INAUDIBLE]. Dialogue: 0,0:57:54.02,0:57:57.47,Default,,0000,0000,0000,,If I have something I minus I\Nminus J, does it go out or in? Dialogue: 0,0:57:57.47,0:57:57.97,Default,,0000,0000,0000,,STUDENT: In. Dialogue: 0,0:57:57.97,0:57:59.54,Default,,0000,0000,0000,,PROFESSOR: It goes in. Dialogue: 0,0:57:59.54,0:58:00.80,Default,,0000,0000,0000,,It goes in, and it'll be up. Dialogue: 0,0:58:00.80,0:58:02.72,Default,,0000,0000,0000,,So it's going to be like\Nall these normals are Dialogue: 0,0:58:02.72,0:58:07.17,Default,,0000,0000,0000,,going to be like a vector\Nfield like that, like amoebas. Dialogue: 0,0:58:07.17,0:58:09.74,Default,,0000,0000,0000,,But they are pointing\Ntowards inside. Dialogue: 0,0:58:09.74,0:58:11.09,Default,,0000,0000,0000,,Do I like that? Dialogue: 0,0:58:11.09,0:58:13.16,Default,,0000,0000,0000,,Yes, because I'm a\Ncrazy mathematician. Dialogue: 0,0:58:13.16,0:58:16.74,Default,,0000,0000,0000,,Does the engineer like that? Dialogue: 0,0:58:16.74,0:58:17.71,Default,,0000,0000,0000,,No. Dialogue: 0,0:58:17.71,0:58:18.66,Default,,0000,0000,0000,,Why? Dialogue: 0,0:58:18.66,0:58:21.79,Default,,0000,0000,0000,,The flux is pointing in or out? Dialogue: 0,0:58:21.79,0:58:23.15,Default,,0000,0000,0000,,The flux. Dialogue: 0,0:58:23.15,0:58:23.65,Default,,0000,0000,0000,,The flux. Dialogue: 0,0:58:23.65,0:58:25.90,Default,,0000,0000,0000,,The flux, the flux\Nis pointing out. Dialogue: 0,0:58:25.90,0:58:27.59,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,0:58:27.59,0:58:31.37,Default,,0000,0000,0000,,X plus Y-- X plus I plus J.\NIt's like this pointing out. Dialogue: 0,0:58:31.37,0:58:33.54,Default,,0000,0000,0000,,So the flux get\Nout of the surface. Dialogue: 0,0:58:33.54,0:58:36.79,Default,,0000,0000,0000,,It's like to pour water\Ninside, and the water's Dialogue: 0,0:58:36.79,0:58:41.58,Default,,0000,0000,0000,,just a net-- not a net, but\Nlike something that holds it in. Dialogue: 0,0:58:41.58,0:58:42.95,Default,,0000,0000,0000,,And like a-- Dialogue: 0,0:58:42.95,0:58:44.12,Default,,0000,0000,0000,,STUDENT: Like a [INAUDIBLE]? Dialogue: 0,0:58:44.12,0:58:45.76,Default,,0000,0000,0000,,PROFESSOR: --pasta strainer. Dialogue: 0,0:58:45.76,0:58:47.64,Default,,0000,0000,0000,,And the water goes up\N[SPRAYING NOISE], well, Dialogue: 0,0:58:47.64,0:58:48.67,Default,,0000,0000,0000,,like a jet. Dialogue: 0,0:58:48.67,0:58:49.32,Default,,0000,0000,0000,,Like that. Dialogue: 0,0:58:49.32,0:58:52.58,Default,,0000,0000,0000,,So that is your flux\Ngoing through the surface. Dialogue: 0,0:58:52.58,0:58:56.19,Default,,0000,0000,0000,,Are you happy that I took\Nthe normal pointing inside? Dialogue: 0,0:58:56.19,0:58:56.79,Default,,0000,0000,0000,,No. Dialogue: 0,0:58:56.79,0:58:57.78,Default,,0000,0000,0000,,That was crazy. Dialogue: 0,0:58:57.78,0:59:02.82,Default,,0000,0000,0000,,So here comes you, the\Nmechanical engineer majoring Dialogue: 0,0:59:02.82,0:59:06.10,Default,,0000,0000,0000,,in solid or [INAUDIBLE]\Nand say Magdalena, Dialogue: 0,0:59:06.10,0:59:09.25,Default,,0000,0000,0000,,you should have taken\Nthe outer normal, Dialogue: 0,0:59:09.25,0:59:12.29,Default,,0000,0000,0000,,because look at the\Nflux pointing out. Dialogue: 0,0:59:12.29,0:59:14.21,Default,,0000,0000,0000,,Take the outer of normal,\Nand things are going Dialogue: 0,0:59:14.21,0:59:16.59,Default,,0000,0000,0000,,to looks right and nice again. Dialogue: 0,0:59:16.59,0:59:18.61,Default,,0000,0000,0000,,So if I were to\Nchange the normal, Dialogue: 0,0:59:18.61,0:59:21.40,Default,,0000,0000,0000,,I would put the\Nplus, plus, minus. Dialogue: 0,0:59:21.40,0:59:23.50,Default,,0000,0000,0000,,I'll take the outer normal. Dialogue: 0,0:59:23.50,0:59:26.01,Default,,0000,0000,0000,,And in the end I\Nget plus 5 over 2. Dialogue: 0,0:59:26.01,0:59:29.10,Default,,0000,0000,0000,,So no remark. Dialogue: 0,0:59:29.10,0:59:36.28,Default,,0000,0000,0000,,If I change N to minus N, this\Nwould become the outer normal. Dialogue: 0,0:59:36.28,0:59:40.68,Default,,0000,0000,0000,,Then the flux would\Nbecome pi over 2. solar Dialogue: 0,0:59:40.68,0:59:42.73,Default,,0000,0000,0000,,flux depends on the what? Dialogue: 0,0:59:42.73,0:59:44.98,Default,,0000,0000,0000,,The match between\Nthe flux, the angles, Dialogue: 0,0:59:44.98,0:59:48.99,Default,,0000,0000,0000,,sort of between the flux if\Nfunction, vector [INAUDIBLE] Dialogue: 0,0:59:48.99,0:59:53.34,Default,,0000,0000,0000,,function, and the normal\Nthat I take to the surface. Dialogue: 0,0:59:53.34,0:59:54.00,Default,,0000,0000,0000,,Right? Dialogue: 0,0:59:54.00,0:59:59.20,Default,,0000,0000,0000,,I can change the normal and\NI get the opposite answer. Dialogue: 0,0:59:59.20,1:00:01.50,Default,,0000,0000,0000,,In absolute values,\Nthe same flux. Dialogue: 0,1:00:01.50,1:00:05.77,Default,,0000,0000,0000,,So flux should be equal\N[INAUDIBLE] the absolute value. Dialogue: 0,1:00:05.77,1:00:09.06,Default,,0000,0000,0000,,Unlike the area that should\Nbe always a positive number. Dialogue: 0,1:00:09.06,1:00:11.43,Default,,0000,0000,0000,,Volume, that should always\Nbe a positive number. Dialogue: 0,1:00:11.43,1:00:14.94,Default,,0000,0000,0000,,So if I get a limited area,\Nthat means I messed up. Dialogue: 0,1:00:14.94,1:00:17.04,Default,,0000,0000,0000,,If I get a negative\Non all of them, Dialogue: 0,1:00:17.04,1:00:20.92,Default,,0000,0000,0000,,it means messed up in my\Ncomputation somewhere. Dialogue: 0,1:00:20.92,1:00:23.07,Default,,0000,0000,0000,,But that doesn't mean\NI messed up here. Dialogue: 0,1:00:23.07,1:00:24.81,Default,,0000,0000,0000,,I just chose the other normal. Dialogue: 0,1:00:24.81,1:00:26.20,Default,,0000,0000,0000,,It's possible. Dialogue: 0,1:00:26.20,1:00:30.72,Default,,0000,0000,0000,,So the flux can be taken as\Nis and put in absolute value. Dialogue: 0,1:00:30.72,1:00:31.44,Default,,0000,0000,0000,,All right. Dialogue: 0,1:00:31.44,1:00:33.70,Default,,0000,0000,0000,,OK. Dialogue: 0,1:00:33.70,1:00:36.98,Default,,0000,0000,0000,,We have to think of it like\Nthe surface, and stuff that Dialogue: 0,1:00:36.98,1:00:40.32,Default,,0000,0000,0000,,goes through surface\Nin electric circuits. Dialogue: 0,1:00:40.32,1:00:42.70,Default,,0000,0000,0000,,Can you do some research\Nfor you about flux Dialogue: 0,1:00:42.70,1:00:45.12,Default,,0000,0000,0000,,and electrical engineering? Dialogue: 0,1:00:45.12,1:00:50.28,Default,,0000,0000,0000,,And next time somebody\Ntells me a story about it. Dialogue: 0,1:00:50.28,1:00:52.71,Default,,0000,0000,0000,,Who is-- again-- who is\Nelectrical engineering major Dialogue: 0,1:00:52.71,1:00:54.28,Default,,0000,0000,0000,,here? Dialogue: 0,1:00:54.28,1:00:56.16,Default,,0000,0000,0000,,Oh, so five people. Dialogue: 0,1:00:56.16,1:00:58.22,Default,,0000,0000,0000,,You're going to get four\Nextra credit points. Dialogue: 0,1:00:58.22,1:00:59.92,Default,,0000,0000,0000,,You guys are jealous. Dialogue: 0,1:00:59.92,1:01:03.51,Default,,0000,0000,0000,,I'm going to give you four extra\Ncredit points if in 10 minutes Dialogue: 0,1:01:03.51,1:01:08.49,Default,,0000,0000,0000,,you can tell us a little bit\Nabout where flux can be seen. Dialogue: 0,1:01:08.49,1:01:10.47,Default,,0000,0000,0000,,Well, you don't have\Nto come to the board. Dialogue: 0,1:01:10.47,1:01:13.07,Default,,0000,0000,0000,,You can just talk to us\Nfrom outside if you want, Dialogue: 0,1:01:13.07,1:01:14.80,Default,,0000,0000,0000,,or down inside the classroom. Dialogue: 0,1:01:14.80,1:01:16.95,Default,,0000,0000,0000,,Tell us where the\Nnotion of flux appears Dialogue: 0,1:01:16.95,1:01:19.91,Default,,0000,0000,0000,,in the electric\Ncircuits and why it Dialogue: 0,1:01:19.91,1:01:24.29,Default,,0000,0000,0000,,would be important for\NCalculus 3 as well. Dialogue: 0,1:01:24.29,1:01:25.19,Default,,0000,0000,0000,,OK. Dialogue: 0,1:01:25.19,1:01:28.71,Default,,0000,0000,0000,,Now a big question\Nbefore I let you go. Dialogue: 0,1:01:28.71,1:01:34.55,Default,,0000,0000,0000,,Can I have a flux\Nthat corresponds Dialogue: 0,1:01:34.55,1:01:36.12,Default,,0000,0000,0000,,to a parameterization? Dialogue: 0,1:01:36.12,1:01:41.56,Default,,0000,0000,0000,,That is my big worry, that\NI have to do that as well. Dialogue: 0,1:01:41.56,1:01:45.14,Default,,0000,0000,0000,,Eventually, could I\Nhave solved this problem Dialogue: 0,1:01:45.14,1:01:47.95,Default,,0000,0000,0000,,if the surface that\Nis parameterized Dialogue: 0,1:01:47.95,1:01:54.08,Default,,0000,0000,0000,,was my friend--\Nwho was my friend? Dialogue: 0,1:01:54.08,1:01:54.98,Default,,0000,0000,0000,,I don't remember. Dialogue: 0,1:01:54.98,1:01:59.35,Default,,0000,0000,0000,,UI plus VJ plus U\Nsquared plus-- you Dialogue: 0,1:01:59.35,1:02:03.49,Default,,0000,0000,0000,,gave it to me-- OK, that\Nwas the previous example, Dialogue: 0,1:02:03.49,1:02:05.45,Default,,0000,0000,0000,,and that's the last\Nexample on the board. Dialogue: 0,1:02:05.45,1:02:12.61,Default,,0000,0000,0000,,So you have double integral\Nof force field times NDS. Dialogue: 0,1:02:12.61,1:02:16.93,Default,,0000,0000,0000,,Now, what if I say I don't\Nwant to do it like this-- Z Dialogue: 0,1:02:16.93,1:02:19.46,Default,,0000,0000,0000,,equals F of XY. Dialogue: 0,1:02:19.46,1:02:21.41,Default,,0000,0000,0000,,So I don't want to\Ndo it like that. Dialogue: 0,1:02:21.41,1:02:25.07,Default,,0000,0000,0000,,I want to do it in\Na different way. Dialogue: 0,1:02:25.07,1:02:33.72,Default,,0000,0000,0000,,That means you pulling out of\Nyour brain some old memories. Dialogue: 0,1:02:33.72,1:02:35.26,Default,,0000,0000,0000,,F was F, right? Dialogue: 0,1:02:35.26,1:02:37.48,Default,,0000,0000,0000,,You need to leave F alone,\Npoor fellow, because he Dialogue: 0,1:02:37.48,1:02:39.90,Default,,0000,0000,0000,,has no better way to do it. Dialogue: 0,1:02:39.90,1:02:43.90,Default,,0000,0000,0000,,This is becoming\Ncomplicated, the [INAUDIBLE] Dialogue: 0,1:02:43.90,1:02:44.86,Default,,0000,0000,0000,,mechanical engineering. Dialogue: 0,1:02:44.86,1:02:47.86,Default,,0000,0000,0000,, Dialogue: 0,1:02:47.86,1:02:52.53,Default,,0000,0000,0000,,And what's given to you\Nbefore, but you don't remember? Dialogue: 0,1:02:52.53,1:02:55.81,Default,,0000,0000,0000,,R was given to you\Nas position vector. Dialogue: 0,1:02:55.81,1:02:59.71,Default,,0000,0000,0000,,R sub U and R sub V,\Nyou may not remember-- Dialogue: 0,1:02:59.71,1:03:03.48,Default,,0000,0000,0000,,that was a long time ago-- we\Nproved that R sub U and R sub Dialogue: 0,1:03:03.48,1:03:05.45,Default,,0000,0000,0000,,V were on the surface. Dialogue: 0,1:03:05.45,1:03:07.19,Default,,0000,0000,0000,,They are both tensions\Nof the surface. Dialogue: 0,1:03:07.19,1:03:09.36,Default,,0000,0000,0000,,It was a long time ago. Dialogue: 0,1:03:09.36,1:03:11.28,Default,,0000,0000,0000,,So the normal is\N[INAUDIBLE], and that's Dialogue: 0,1:03:11.28,1:03:14.05,Default,,0000,0000,0000,,exactly what I wanted to\Nsay the normal will be. Dialogue: 0,1:03:14.05,1:03:17.82,Default,,0000,0000,0000,,Not quite pressed product,\Nbut just like before, Dialogue: 0,1:03:17.82,1:03:21.24,Default,,0000,0000,0000,,pressed product\Ndivided by the norm, Dialogue: 0,1:03:21.24,1:03:28.47,Default,,0000,0000,0000,,because then the unit normal\Nvector has to be length 1. Dialogue: 0,1:03:28.47,1:03:30.66,Default,,0000,0000,0000,,So I have to divide\Nby the number. Dialogue: 0,1:03:30.66,1:03:31.16,Default,,0000,0000,0000,,[SNEEZE] Dialogue: 0,1:03:31.16,1:03:32.02,Default,,0000,0000,0000,,The DS-- Dialogue: 0,1:03:32.02,1:03:32.81,Default,,0000,0000,0000,,STUDENT: Thank you. Dialogue: 0,1:03:32.81,1:03:36.02,Default,,0000,0000,0000,,PROFESSOR: --is going to--\NOK, now it's up to you guys. Dialogue: 0,1:03:36.02,1:03:36.78,Default,,0000,0000,0000,,You're smart. Dialogue: 0,1:03:36.78,1:03:39.36,Default,,0000,0000,0000,,You know what I want to say. Dialogue: 0,1:03:39.36,1:03:43.77,Default,,0000,0000,0000,,So I'll pretend that you\Nknow what DS is in terms Dialogue: 0,1:03:43.77,1:03:45.03,Default,,0000,0000,0000,,of the parameterization. Dialogue: 0,1:03:45.03,1:03:46.52,Default,,0000,0000,0000,,What's coming? Dialogue: 0,1:03:46.52,1:03:47.79,Default,,0000,0000,0000,,We said that. Dialogue: 0,1:03:47.79,1:03:49.34,Default,,0000,0000,0000,,It was a long time ago. Dialogue: 0,1:03:49.34,1:03:51.50,Default,,0000,0000,0000,,You can guess it by\Njust being smart-- Dialogue: 0,1:03:51.50,1:03:51.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:03:51.74,1:03:52.78,Default,,0000,0000,0000,,PROFESSOR: --or you can-- Dialogue: 0,1:03:52.78,1:03:53.83,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:03:53.83,1:03:55.29,Default,,0000,0000,0000,,PROFESSOR: Yes, exactly. Dialogue: 0,1:03:55.29,1:03:57.80,Default,,0000,0000,0000,,And you got another\None extra credit point. Dialogue: 0,1:03:57.80,1:04:01.99,Default,,0000,0000,0000,, Dialogue: 0,1:04:01.99,1:04:04.58,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:04:04.58,1:04:06.62,Default,,0000,0000,0000,,PROFESSOR: So since before,\Nthey were simplified, Dialogue: 0,1:04:06.62,1:04:07.63,Default,,0000,0000,0000,,for god's sake. Dialogue: 0,1:04:07.63,1:04:10.64,Default,,0000,0000,0000,,Now we have the new kind\Nof writing area element DS. Dialogue: 0,1:04:10.64,1:04:13.25,Default,,0000,0000,0000,,They also have to simplify. Dialogue: 0,1:04:13.25,1:04:15.51,Default,,0000,0000,0000,,It wasn't hard to see. Dialogue: 0,1:04:15.51,1:04:18.33,Default,,0000,0000,0000,,So you could have\Ndone it like that. Dialogue: 0,1:04:18.33,1:04:22.60,Default,,0000,0000,0000,,You could have done\Nit like that, how? Dialogue: 0,1:04:22.60,1:04:25.51,Default,,0000,0000,0000,,Somebody need to help me,\Nbecause I have no idea what Dialogue: 0,1:04:25.51,1:04:28.05,Default,,0000,0000,0000,,I'm going to do here. Dialogue: 0,1:04:28.05,1:04:29.38,Default,,0000,0000,0000,,Do we get the same thing or not? Dialogue: 0,1:04:29.38,1:04:30.26,Default,,0000,0000,0000,,This is the question. Dialogue: 0,1:04:30.26,1:04:31.80,Default,,0000,0000,0000,,And I'm going to\Nfinish with that, Dialogue: 0,1:04:31.80,1:04:33.85,Default,,0000,0000,0000,,but I don't want to\Ngo home-- I'm not Dialogue: 0,1:04:33.85,1:04:37.89,Default,,0000,0000,0000,,going to let you go home\Nuntil you finish this. Dialogue: 0,1:04:37.89,1:04:42.27,Default,,0000,0000,0000,,F was a simple,\Nbeautiful vector field. Dialogue: 0,1:04:42.27,1:04:45.49,Default,,0000,0000,0000,,Given-- like that. Dialogue: 0,1:04:45.49,1:04:46.76,Default,,0000,0000,0000,,This is a force. Dialogue: 0,1:04:46.76,1:04:48.72,Default,,0000,0000,0000,,May the force be\Nwith you like that. Dialogue: 0,1:04:48.72,1:04:54.01,Default,,0000,0000,0000,,But we changed it in U,V because\Nwe are acting on the surface S, Dialogue: 0,1:04:54.01,1:04:56.05,Default,,0000,0000,0000,,what is the pressure\Nin V, right? Dialogue: 0,1:04:56.05,1:05:00.70,Default,,0000,0000,0000,,So you have UI plus VJ\Nplus-- you gave it to me-- Dialogue: 0,1:05:00.70,1:05:03.68,Default,,0000,0000,0000,,U squared plus V squared. Dialogue: 0,1:05:03.68,1:05:05.49,Default,,0000,0000,0000,,Am I right, or am\NI talking nonsense? Dialogue: 0,1:05:05.49,1:05:08.01,Default,,0000,0000,0000,, Dialogue: 0,1:05:08.01,1:05:08.65,Default,,0000,0000,0000,,All right. Dialogue: 0,1:05:08.65,1:05:12.85,Default,,0000,0000,0000,,So now again I have\Nto be seeing them. Dialogue: 0,1:05:12.85,1:05:14.14,Default,,0000,0000,0000,,Am I getting the same thing? Dialogue: 0,1:05:14.14,1:05:16.28,Default,,0000,0000,0000,,If I'm not getting\Nthe same thing, Dialogue: 0,1:05:16.28,1:05:19.28,Default,,0000,0000,0000,,I can just go home and\Nget drunk and be sad. Dialogue: 0,1:05:19.28,1:05:22.71,Default,,0000,0000,0000,,But I have to get\Nthe same thing. Dialogue: 0,1:05:22.71,1:05:27.02,Default,,0000,0000,0000,,Otherwise, there is something\Nwrong with my setup. Dialogue: 0,1:05:27.02,1:05:32.62,Default,,0000,0000,0000,,So I have to have U, V.\NU squared plus V squared. Dialogue: 0,1:05:32.62,1:05:34.50,Default,,0000,0000,0000,,Close. Dialogue: 0,1:05:34.50,1:05:37.00,Default,,0000,0000,0000,,Dot product. Dialogue: 0,1:05:37.00,1:05:41.84,Default,,0000,0000,0000,,This guy over on top-- say what? Dialogue: 0,1:05:41.84,1:05:48.12,Default,,0000,0000,0000,,Magdalena, this guy over on top\Nhas to be-- has to be a what? Dialogue: 0,1:05:48.12,1:05:50.04,Default,,0000,0000,0000,,Well, I didn't say what it was. Dialogue: 0,1:05:50.04,1:05:52.72,Default,,0000,0000,0000,,I should do it now. Dialogue: 0,1:05:52.72,1:05:53.46,Default,,0000,0000,0000,,Right? Dialogue: 0,1:05:53.46,1:05:57.16,Default,,0000,0000,0000,,So how will we do that? Dialogue: 0,1:05:57.16,1:06:07.96,Default,,0000,0000,0000,,We were saying R of\NUV will be UI plus VJ Dialogue: 0,1:06:07.96,1:06:10.50,Default,,0000,0000,0000,,plus U squared plus V squared. Dialogue: 0,1:06:10.50,1:06:11.10,Default,,0000,0000,0000,,OK. Dialogue: 0,1:06:11.10,1:06:14.92,Default,,0000,0000,0000,,So R sub U will be--\Nyou teach me quickly, Dialogue: 0,1:06:14.92,1:06:18.15,Default,,0000,0000,0000,,and R sub [INAUDIBLE]\Nis-- voila. Dialogue: 0,1:06:18.15,1:06:19.77,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:06:19.77,1:06:21.56,Default,,0000,0000,0000,,PROFESSOR: 1-- Dialogue: 0,1:06:21.56,1:06:23.53,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:06:23.53,1:06:28.12,Default,,0000,0000,0000,,PROFESSOR: Plus zero--\Nthank you-- plus 2U, OK. Dialogue: 0,1:06:28.12,1:06:34.83,Default,,0000,0000,0000,,0 plus 1J plus 2VK Am I done? Dialogue: 0,1:06:34.83,1:06:35.33,Default,,0000,0000,0000,,I'm done. Dialogue: 0,1:06:35.33,1:06:35.90,Default,,0000,0000,0000,,No, I'm not done. Dialogue: 0,1:06:35.90,1:06:36.82,Default,,0000,0000,0000,,What do I have to do? Dialogue: 0,1:06:36.82,1:06:38.59,Default,,0000,0000,0000,,Cross them. Dialogue: 0,1:06:38.59,1:06:41.47,Default,,0000,0000,0000,, Dialogue: 0,1:06:41.47,1:06:44.50,Default,,0000,0000,0000,,Cross multiply IJK. Dialogue: 0,1:06:44.50,1:06:47.36,Default,,0000,0000,0000,,This looks nice. Dialogue: 0,1:06:47.36,1:06:49.24,Default,,0000,0000,0000,,Look, it's not so ugly. Dialogue: 0,1:06:49.24,1:06:51.48,Default,,0000,0000,0000,,I thought it would\Nbe uglier, right? Dialogue: 0,1:06:51.48,1:06:51.98,Default,,0000,0000,0000,,OK. Dialogue: 0,1:06:51.98,1:06:54.69,Default,,0000,0000,0000,,What it is? Dialogue: 0,1:06:54.69,1:06:57.52,Default,,0000,0000,0000,,What it this thing? Dialogue: 0,1:06:57.52,1:06:58.39,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:06:58.39,1:07:06.18,Default,,0000,0000,0000,,PROFESSOR: Minus the\NU, I. Minus-- plus. Dialogue: 0,1:07:06.18,1:07:09.13,Default,,0000,0000,0000,,Minus, plus 1. Dialogue: 0,1:07:09.13,1:07:11.46,Default,,0000,0000,0000,,2V minus because it's-- Dialogue: 0,1:07:11.46,1:07:12.08,Default,,0000,0000,0000,,STUDENT: Minus. Dialogue: 0,1:07:12.08,1:07:13.45,Default,,0000,0000,0000,,PROFESSOR: --minus in front. Dialogue: 0,1:07:13.45,1:07:15.36,Default,,0000,0000,0000,,Right. Dialogue: 0,1:07:15.36,1:07:17.51,Default,,0000,0000,0000,,So I'm alternating. Dialogue: 0,1:07:17.51,1:07:19.98,Default,,0000,0000,0000,,And 1K. Dialogue: 0,1:07:19.98,1:07:24.04,Default,,0000,0000,0000,,So again, I get minus\NX of S minus XY and 1, Dialogue: 0,1:07:24.04,1:07:26.42,Default,,0000,0000,0000,,and again, I'm pointing\Nin, and that's bad. Dialogue: 0,1:07:26.42,1:07:29.52,Default,,0000,0000,0000,,So my normal will point\Ninside the surface Dialogue: 0,1:07:29.52,1:07:33.67,Default,,0000,0000,0000,,like needles that are\Nperpendicular to the surface Dialogue: 0,1:07:33.67,1:07:34.84,Default,,0000,0000,0000,,pointing inside. Dialogue: 0,1:07:34.84,1:07:36.00,Default,,0000,0000,0000,,But that's OK. Dialogue: 0,1:07:36.00,1:07:39.05,Default,,0000,0000,0000,,In the end, I take\Neverything in absolute value. Dialogue: 0,1:07:39.05,1:07:39.55,Default,,0000,0000,0000,,Right? Dialogue: 0,1:07:39.55,1:07:47.64,Default,,0000,0000,0000,, Dialogue: 0,1:07:47.64,1:07:51.10,Default,,0000,0000,0000,,So again, I do the same math. Dialogue: 0,1:07:51.10,1:07:54.76,Default,,0000,0000,0000,,So I get minus-- I don't\Nwant to do it anymore. Dialogue: 0,1:07:54.76,1:07:59.41,Default,,0000,0000,0000,,Minus 2A squared, minus 2B\Nsquared, plus your squared, Dialogue: 0,1:07:59.41,1:08:01.58,Default,,0000,0000,0000,,plus this squared,\Nthen you save me Dialogue: 0,1:08:01.58,1:08:04.87,Default,,0000,0000,0000,,and you said minus 2\Nsquared [INAUDIBLE] squared. Dialogue: 0,1:08:04.87,1:08:06.26,Default,,0000,0000,0000,,DUDV. Dialogue: 0,1:08:06.26,1:08:15.02,Default,,0000,0000,0000,,But DUDV means that UV is a\Npair, a point in this, guys. Dialogue: 0,1:08:15.02,1:08:16.12,Default,,0000,0000,0000,,UV. Dialogue: 0,1:08:16.12,1:08:18.96,Default,,0000,0000,0000,,It's a pair in the\Ndisk of radius one. Dialogue: 0,1:08:18.96,1:08:21.99,Default,,0000,0000,0000,,So I'm getting exactly,\Nwhat exactly the same thing Dialogue: 0,1:08:21.99,1:08:22.94,Default,,0000,0000,0000,,as before. Dialogue: 0,1:08:22.94,1:08:25.64,Default,,0000,0000,0000,,Because this is\Nminus R squared, so I Dialogue: 0,1:08:25.64,1:08:32.91,Default,,0000,0000,0000,,get integral, integral, minus\NR squared times R. DR, D theta. Dialogue: 0,1:08:32.91,1:08:35.66,Default,,0000,0000,0000,,From zero to 1,\Nfrom zero to 2 pi, Dialogue: 0,1:08:35.66,1:08:38.16,Default,,0000,0000,0000,,and I get the same\Nanswer, which was? Dialogue: 0,1:08:38.16,1:08:39.49,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:08:39.49,1:08:39.80,Default,,0000,0000,0000,,PROFESSOR: Minus what? Dialogue: 0,1:08:39.80,1:08:40.30,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:08:40.30,1:08:40.59,Default,,0000,0000,0000,,PROFESSOR: Pi over-- Dialogue: 0,1:08:40.59,1:08:41.68,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:08:41.68,1:08:42.47,Default,,0000,0000,0000,,PROFESSOR: You see? Dialogue: 0,1:08:42.47,1:08:46.02,Default,,0000,0000,0000,,I already forgot. Dialogue: 0,1:08:46.02,1:08:48.36,Default,,0000,0000,0000,,STUDENT: 2. Dialogue: 0,1:08:48.36,1:08:49.91,Default,,0000,0000,0000,,PROFESSOR: So what\Nmatters is that we Dialogue: 0,1:08:49.91,1:08:52.29,Default,,0000,0000,0000,,take the flux in\Nabsolute value because it Dialogue: 0,1:08:52.29,1:08:54.42,Default,,0000,0000,0000,,depends on the\Norientation of the normal. Dialogue: 0,1:08:54.42,1:08:57.19,Default,,0000,0000,0000,,If we take the\Nnormal [INAUDIBLE]. Dialogue: 0,1:08:57.19,1:09:02.95,Default,,0000,0000,0000,,Please, one thing I want you\Nto do when you go home now, Dialogue: 0,1:09:02.95,1:09:06.03,Default,,0000,0000,0000,,open the book which\Nmaybe you rarely do, Dialogue: 0,1:09:06.03,1:09:08.53,Default,,0000,0000,0000,,but now it's\Nreally-- the material Dialogue: 0,1:09:08.53,1:09:10.13,Default,,0000,0000,0000,,became complicated enough. Dialogue: 0,1:09:10.13,1:09:14.17,Default,,0000,0000,0000,,We are not just doing math,\Ncalculus, we are doing physics, Dialogue: 0,1:09:14.17,1:09:17.64,Default,,0000,0000,0000,,we are doing mechanics, we are\Ndealing with surface integrals Dialogue: 0,1:09:17.64,1:09:19.07,Default,,0000,0000,0000,,and flux. Dialogue: 0,1:09:19.07,1:09:27.12,Default,,0000,0000,0000,,I want you to open the book\Nat page-- I don't know. Dialogue: 0,1:09:27.12,1:09:32.79,Default,,0000,0000,0000,,At surface integrals\Nstarts at page 1,063. Dialogue: 0,1:09:32.79,1:09:34.96,Default,,0000,0000,0000,,Section 13.5. Dialogue: 0,1:09:34.96,1:09:38.25,Default,,0000,0000,0000,,And it keeps going like that,\Npretty pictures of surfaces Dialogue: 0,1:09:38.25,1:09:40.68,Default,,0000,0000,0000,,and fluxes and so on. Dialogue: 0,1:09:40.68,1:09:41.84,Default,,0000,0000,0000,,Vector fields. Dialogue: 0,1:09:41.84,1:09:43.66,Default,,0000,0000,0000,,And it keeps going like that. Dialogue: 0,1:09:43.66,1:09:48.89,Default,,0000,0000,0000,,But it doesn't cover anything\Nnew except what I said today. Dialogue: 0,1:09:48.89,1:09:51.31,Default,,0000,0000,0000,,It's just that it shows\Nyou examples that are not Dialogue: 0,1:09:51.31,1:09:54.81,Default,,0000,0000,0000,,as beautiful as the ones I\Ngave, but they are essentially Dialogue: 0,1:09:54.81,1:09:58.19,Default,,0000,0000,0000,,the same, only a little\Nbit nastier to complete. Dialogue: 0,1:09:58.19,1:10:02.48,Default,,0000,0000,0000,,So up to 1,072. Dialogue: 0,1:10:02.48,1:10:05.74,Default,,0000,0000,0000,,So that is what you're\Ngoing to do this weekend, Dialogue: 0,1:10:05.74,1:10:06.76,Default,,0000,0000,0000,,plus the homework. Dialogue: 0,1:10:06.76,1:10:07.80,Default,,0000,0000,0000,,Keep on the homework. Dialogue: 0,1:10:07.80,1:10:11.07,Default,,0000,0000,0000,,Now, if you get stuck\NSaturday, Sunday, Dialogue: 0,1:10:11.07,1:10:13.98,Default,,0000,0000,0000,,whenever you try your homework\Nyou get stuck, what do you do? Dialogue: 0,1:10:13.98,1:10:14.81,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:10:14.81,1:10:15.81,Default,,0000,0000,0000,,PROFESSOR: You email me. Dialogue: 0,1:10:15.81,1:10:18.46,Default,,0000,0000,0000,,So you say what in the world\Nis going on with this problem Dialogue: 0,1:10:18.46,1:10:25.04,Default,,0000,0000,0000,,because I tried it seven\Ntimes and-- 88 times. Dialogue: 0,1:10:25.04,1:10:27.18,Default,,0000,0000,0000,,And then you got\Nthe brownie points. Dialogue: 0,1:10:27.18,1:10:28.01,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:10:28.01,1:10:30.20,Default,,0000,0000,0000,,PROFESSOR: [INAUDIBLE] problem. Dialogue: 0,1:10:30.20,1:10:31.47,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] by 32. Dialogue: 0,1:10:31.47,1:10:33.24,Default,,0000,0000,0000,,PROFESSOR: There\Nwas a problem, guys. Dialogue: 0,1:10:33.24,1:10:35.11,Default,,0000,0000,0000,,There are not so many problems. Dialogue: 0,1:10:35.11,1:10:38.93,Default,,0000,0000,0000,,But the only part, serious\Npart that we would catch, Dialogue: 0,1:10:38.93,1:10:41.67,Default,,0000,0000,0000,,he found it first, and\Nhe tried it 88 times. Dialogue: 0,1:10:41.67,1:10:44.40,Default,,0000,0000,0000,, Dialogue: 0,1:10:44.40,1:10:47.16,Default,,0000,0000,0000,,I'll never forget you, though,\Nbecause you are unique, Dialogue: 0,1:10:47.16,1:10:49.97,Default,,0000,0000,0000,,and that-- I appreciated\Nthat very much. Dialogue: 0,1:10:49.97,1:10:55.38,Default,,0000,0000,0000,,So doing this weekend, do\Nnot hesitate to pester. Dialogue: 0,1:10:55.38,1:10:58.40,Default,,0000,0000,0000,,I will answer all the web\Nwork problems you have. Dialogue: 0,1:10:58.40,1:11:00.10,Default,,0000,0000,0000,,I want you to do well. Dialogue: 0,1:11:00.10,1:11:01.98,Default,,0000,0000,0000,,Next week is the last\Nweek on new theory, Dialogue: 0,1:11:01.98,1:11:04.38,Default,,0000,0000,0000,,and then we start\Nworking for the final, Dialogue: 0,1:11:04.38,1:11:09.06,Default,,0000,0000,0000,,so by the time of the final,\Nyou'll be [INAUDIBLE]. Dialogue: 0,1:11:09.06,1:11:10.32,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]? Dialogue: 0,1:11:10.32,1:11:11.15,Default,,0000,0000,0000,,PROFESSOR: Yes, sir. Dialogue: 0,1:11:11.15,1:11:13.38,Default,,0000,0000,0000,,Oh, I appreciated\Nthat you did that. Dialogue: 0,1:11:13.38,1:11:14.21,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] Dialogue: 0,1:11:14.21,1:11:18.20,Default,,0000,0000,0000,, Dialogue: 0,1:11:18.20,1:11:20.59,Default,,0000,0000,0000,,PROFESSOR: Again,\NI forgot these. Dialogue: 0,1:11:20.59,1:11:24.92,Default,,0000,0000,0000,, Dialogue: 0,1:11:24.92,1:11:28.46,Default,,0000,0000,0000,,With the extra points you\Ngot, you shouldn't care. Dialogue: 0,1:11:28.46,1:11:28.96,Default,,0000,0000,0000,,