0:00:00.000,0:00:00.580 0:00:00.580,0:00:02.330 PROFESSOR: Any[br]questions about theory 0:00:02.330,0:00:05.300 that gave you headaches[br]regarding homework 0:00:05.300,0:00:07.740 you'd like to talk about? 0:00:07.740,0:00:10.950 Anything related[br]to what we covered 0:00:10.950,0:00:13.778 from chapter nine and today? 0:00:13.778,0:00:15.610 STUDENT: Can we[br]do some problems? 0:00:15.610,0:00:17.410 PROFESSOR: I can[br]fix from problems 0:00:17.410,0:00:20.103 like the ones in the[br]homework, but also I 0:00:20.103,0:00:24.547 can have you tell me what[br]bothers you in the homework. 0:00:24.547,0:00:25.880 STUDENT: Oh, I have [INAUDIBLE]. 0:00:25.880,0:00:29.010 PROFESSOR: What bothered[br]me about my own homework 0:00:29.010,0:00:35.330 was that I realized that I[br]did not remind you something 0:00:35.330,0:00:37.470 I assume you should[br]know, which is 0:00:37.470,0:00:42.800 the equation of a sphere of[br]given center and given radius. 0:00:42.800,0:00:47.500 And since I trust you so much,[br]I said, OK they know about it. 0:00:47.500,0:00:50.930 And then somebody asked[br]me by email what that was, 0:00:50.930,0:00:52.440 and I said, oh, yeah. 0:00:52.440,0:00:54.240 I did not review that in class. 0:00:54.240,0:01:14.750 So review the equation[br]in r3 form that's x, y, z 0:01:14.750,0:01:35.520 of the sphere of radius r and[br]center p of coordinates x0, y0, 0:01:35.520,0:01:37.580 z0. 0:01:37.580,0:01:43.460 One of you asked me by email,[br]does-- of course you do, 0:01:43.460,0:01:48.010 and then if you know it,[br]can you help me-- can you 0:01:48.010,0:01:50.475 help remind what that was? 0:01:50.475,0:01:54.840 0:01:54.840,0:01:56.830 STUDENT: x minus x0-- 0:01:56.830,0:02:06.990 PROFESSOR: x minus x0 squared[br]plus y minus y0 squared 0:02:06.990,0:02:15.060 plus z minus z0 squared[br]equals R squared. 0:02:15.060,0:02:16.282 OK? 0:02:16.282,0:02:18.230 When you ask, for[br]example, what is 0:02:18.230,0:02:22.922 the equation of a units sphere,[br]what do I mean by unit sphere? 0:02:22.922,0:02:23.630 STUDENT: Radius-- 0:02:23.630,0:02:28.330 PROFESSOR: Radius 1, and[br]center 0, standard unit sphere, 0:02:28.330,0:02:30.080 will be. 0:02:30.080,0:02:35.100 There is a notation for that[br]in mathematics called s2. 0:02:35.100,0:02:36.980 I'll tell you why its called s2. 0:02:36.980,0:02:41.310 x squared plus y squared[br]plus z squared equals 1. 0:02:41.310,0:02:44.340 0:02:44.340,0:02:48.030 s2 stands for the dimension. 0:02:48.030,0:02:53.000 That means the number[br]of the-- the number 0:02:53.000,0:02:54.545 of degrees of freedom. 0:02:54.545,0:03:00.170 0:03:00.170,0:03:04.320 you have on a certain manifold. 0:03:04.320,0:03:07.630 0:03:07.630,0:03:08.900 What is a manifold? 0:03:08.900,0:03:10.410 It's a geometric structure. 0:03:10.410,0:03:12.800 I'm not going to[br]go into details. 0:03:12.800,0:03:15.640 It's a geometric structure[br]with some special properties. 0:03:15.640,0:03:18.700 0:03:18.700,0:03:21.150 I'm not talking about[br]other fields of algebra, 0:03:21.150,0:03:22.410 anthropology. 0:03:22.410,0:03:26.505 I'm just talking about geometry[br]and calculus math 3, which 0:03:26.505,0:03:28.930 is multivariable calculus. 0:03:28.930,0:03:35.200 Now, how do I think[br]of degrees of freedom? 0:03:35.200,0:03:37.450 Look at the table. 0:03:37.450,0:03:42.910 What freedom do I have to move[br]along one of these sticks? 0:03:42.910,0:03:45.440 I have one degree of[br]freedom in the sense 0:03:45.440,0:03:50.000 that it's given by a[br]parameter like time. 0:03:50.000,0:03:50.670 Right? 0:03:50.670,0:03:53.020 It's a 1-parameter[br]manifold in the sense 0:03:53.020,0:03:55.340 that maybe I have[br]a line, maybe I 0:03:55.340,0:04:00.940 have the trajectory of the[br]parking space in terms of time. 0:04:00.940,0:04:05.110 The freedom that the bag has[br]is to move according to time, 0:04:05.110,0:04:09.680 and that's considered only[br]one degree of freedom. 0:04:09.680,0:04:13.620 Now if you were on a[br]plane or another surface, 0:04:13.620,0:04:18.010 why would you have more[br]than one degrees of freedom? 0:04:18.010,0:04:23.690 Well, I can move towards[br]you, or I can move this way. 0:04:23.690,0:04:26.720 I can draw a grid the way[br]the x and y coordinate. 0:04:26.720,0:04:29.980 And those are my[br]degrees of freedom. 0:04:29.980,0:04:32.960 Practically, the basis[br]IJ gives me that kind 0:04:32.960,0:04:35.840 of two degrees of freedom. 0:04:35.840,0:04:36.340 Right? 0:04:36.340,0:04:41.670 If I'm in three coordinates, I[br]have without other constraints, 0:04:41.670,0:04:43.380 because I could be[br]in three coordinates 0:04:43.380,0:04:46.740 and constrained to be on[br]a cylinder, in which case 0:04:46.740,0:04:48.860 I still have two[br]degrees of freedom. 0:04:48.860,0:04:53.440 But if I am a bug[br]who is free to fly, 0:04:53.440,0:04:59.820 I have the freedom to go with[br]three degrees of freedom, 0:04:59.820,0:05:00.830 right? 0:05:00.830,0:05:03.700 I have three degrees of[br]freedom, but if the bug 0:05:03.700,0:05:09.080 is moving-- not flying,[br]moving on a surface, 0:05:09.080,0:05:11.560 then he has two[br]degrees of freedom. 0:05:11.560,0:05:16.635 So to again review, lines[br]and curves in general 0:05:16.635,0:05:20.190 are one dimensional[br]things, because you 0:05:20.190,0:05:21.460 have one degree of freedom. 0:05:21.460,0:05:24.170 Two dimensional[br]things are surfaces, 0:05:24.170,0:05:28.013 three dimensional things are[br]spaces, like the Euclidean 0:05:28.013,0:05:31.800 space, and we are not[br]going to go beyond, 0:05:31.800,0:05:33.660 at least for the time[br]being, we are not 0:05:33.660,0:05:36.230 going to go beyond that. 0:05:36.230,0:05:41.350 However, where anybody is[br]interested in relativity, 0:05:41.350,0:05:47.300 say or let's say four[br]dimensional spaces, or things 0:05:47.300,0:05:52.200 of x, y, z spatial coordinates[br]and t as a fourth coordinate, 0:05:52.200,0:05:56.651 then we can go into higher[br]dimensions, as well. 0:05:56.651,0:05:57.151 OK. 0:05:57.151,0:06:01.079 0:06:01.079,0:06:02.825 I want to ask you a question. 0:06:02.825,0:06:07.550 If somebody gives you on[br]WeBWorK or outside of WeBWorK, 0:06:07.550,0:06:11.490 on the first quiz or[br]on the final exam, 0:06:11.490,0:06:18.520 let's say you have[br]this equation, 0:06:18.520,0:06:28.530 x squared plus y squared plus[br]z squared plus 2x plus 2y 0:06:28.530,0:06:32.470 equals 9. 0:06:32.470,0:06:37.190 What is this identified as? 0:06:37.190,0:06:38.350 It's a quadric. 0:06:38.350,0:06:40.930 Why would this be a quadric? 0:06:40.930,0:06:43.290 Well, there is no x, y, y, z. 0:06:43.290,0:06:45.780 Those terms are missing. 0:06:45.780,0:06:54.560 But I have something of the[br]type of quadric x squared 0:06:54.560,0:06:59.510 plus By squared plus[br]c squared plus dxy 0:06:59.510,0:07:07.810 plus exz plus fyz plus, those[br]are, oh my God, so many. 0:07:07.810,0:07:09.010 Degree two. 0:07:09.010,0:07:16.300 Degree one I would[br]have ax plus by plus cz 0:07:16.300,0:07:22.210 plus a little d constant, and[br]whew, that was a long one. 0:07:22.210,0:07:23.050 Right? 0:07:23.050,0:07:27.470 Now, is this of the[br]type of a project? 0:07:27.470,0:07:28.200 Yes, it is. 0:07:28.200,0:07:31.840 Of course there are some terms[br]that are missing, good for us. 0:07:31.840,0:07:35.580 How are you going to try to[br]identify the type of quadric 0:07:35.580,0:07:37.570 by looking at this? 0:07:37.570,0:07:41.410 As you said very well,[br]I think it's-- you say, 0:07:41.410,0:07:45.740 I think of a sphere, maybe I can[br]complete the squares, you said. 0:07:45.740,0:07:49.240 How do we complete the squares? 0:07:49.240,0:07:52.900 x squared plus 2x plus[br]some missing number, 0:07:52.900,0:07:57.010 a magic number-- yes sir? 0:07:57.010,0:08:02.050 STUDENT: So, basically I'll[br]have to take x plus 2 times 4 0:08:02.050,0:08:04.450 will go outside. 0:08:04.450,0:08:07.840 It's like x min-- x plus 2-- 0:08:07.840,0:08:09.233 PROFESSOR: Why x plus 2? 0:08:09.233,0:08:10.316 STUDENT: Because it's 2x-- 0:08:10.316,0:08:12.140 STUDENT: It's 2x. 0:08:12.140,0:08:14.750 PROFESSOR: But if[br]I take x plus 2, 0:08:14.750,0:08:18.320 then that's going to give[br]me x squared plus 4x plus 4, 0:08:18.320,0:08:19.550 so it's not a good idea. 0:08:19.550,0:08:20.880 STUDENT: On the x plus 1 0:08:20.880,0:08:21.890 PROFESSOR: x plus 1. 0:08:21.890,0:08:25.700 So I'm going to complete[br]x plus 1 squared. 0:08:25.700,0:08:29.424 What did I invent[br]that wasn't there? 0:08:29.424,0:08:30.195 STUDENT: 1. 0:08:30.195,0:08:31.820 PROFESSOR: I invented[br]the 1, and I have 0:08:31.820,0:08:34.179 to compensate for my invention. 0:08:34.179,0:08:38.080 I added the 1, created[br]the 1 out of nothing, 0:08:38.080,0:08:41.150 so I have to compensate[br]by subtracting it. 0:08:41.150,0:08:42.400 How much is from here to here? 0:08:42.400,0:08:45.560 Is it exactly the[br]thing that I underlined 0:08:45.560,0:08:49.380 with a wiggly line, a[br]light wiggly line thing, 0:08:49.380,0:08:52.980 plus what is the[br]blue wiggly line, 0:08:52.980,0:09:01.730 the blue wiggly line[br]that doesn't show-- 0:09:01.730,0:09:06.370 I have y plus 1 squared, and[br]again, I have to compensate 0:09:06.370,0:09:08.470 for what I invented. 0:09:08.470,0:09:13.350 I created a 1 out of nothing,[br]so this is y squared plus 2y. 0:09:13.350,0:09:17.590 0:09:17.590,0:09:21.858 The z squared is all by himself,[br]and he's crying, I'm so lonely, 0:09:21.858,0:09:26.170 I don't know, there is[br]nobody like me over there. 0:09:26.170,0:09:29.620 So in the end, I can rewrite[br]the whole thing as x plus 1 0:09:29.620,0:09:33.640 squared plus y plus 1[br]squared plus z squared, if I 0:09:33.640,0:09:39.141 want to work them out in[br]this format, equals what? 0:09:39.141,0:09:39.640 STUDENT: 10. 0:09:39.640,0:09:40.990 0:09:40.990,0:09:41.890 PROFESSOR: 11. 0:09:41.890,0:09:47.380 11 is the square[br]root of 11 squared. 0:09:47.380,0:09:49.310 Like my son said the other day. 0:09:49.310,0:09:52.560 So that the radius[br]would be square foot 0:09:52.560,0:09:57.310 11 of a sphere of what circle? 0:09:57.310,0:10:00.130 What is the-- or the[br]sphere of what center? 0:10:00.130,0:10:00.880 STUDENT: Minus 1-- 0:10:00.880,0:10:03.960 PROFESSOR: Minus[br]1, minus 1, and 0. 0:10:03.960,0:10:06.079 So I don't want to insult you. 0:10:06.079,0:10:07.870 Of course you know how[br]to complete squares. 0:10:07.870,0:10:10.630 0:10:10.630,0:10:15.690 However, I have discovered in an[br]upper level class at some point 0:10:15.690,0:10:19.460 that my students didn't know[br]how to complete squares, which 0:10:19.460,0:10:22.480 was very, very heartbreaking. 0:10:22.480,0:10:24.748 All right, now. 0:10:24.748,0:10:28.440 0:10:28.440,0:10:39.430 Any questions regarding--[br]while I have a few of yours, 0:10:39.430,0:10:41.350 I'm going to wait[br]a little bit longer 0:10:41.350,0:10:43.650 until I give[br]everybody the chance 0:10:43.650,0:10:47.940 to complete the extra credit. 0:10:47.940,0:10:52.770 I have the question[br]by email saying, 0:10:52.770,0:10:56.900 you mentioned that[br]genius guy in your class. 0:10:56.900,0:11:00.430 This is a 1-sheeted hyperboloid. 0:11:00.430,0:11:04.250 x squared plus y squared minus[br]z squared minus 1 equals 0. 0:11:04.250,0:11:08.190 The question was, by[br]email, how in the world, 0:11:08.190,0:11:19.590 did he figure out what the two[br]families of generatrices are? 0:11:19.590,0:11:22.040 So you have one family[br]and another family, 0:11:22.040,0:11:27.509 and both together generate[br]the 1-sheeted hyperboloid. 0:11:27.509,0:11:30.200 0:11:30.200,0:11:33.070 Let me give you a little[br]bit more of a hint, 0:11:33.070,0:11:35.820 but I'm still going to stop. 0:11:35.820,0:11:41.870 So last time I said, he[br]noticed you can root together 0:11:41.870,0:11:46.970 the y squared minus 1 and the[br]x squared minus z squared, 0:11:46.970,0:11:48.352 and you can separate them. 0:11:48.352,0:11:52.850 So you're going to have x[br]squared minus z squared equals 0:11:52.850,0:11:53.800 1 minus y squared. 0:11:53.800,0:11:56.420 0:11:56.420,0:12:00.350 You can't hide the[br]difference of two squares 0:12:00.350,0:12:03.990 as product of sum[br]and difference. 0:12:03.990,0:12:13.390 x plus z times x minus z equals[br]1 plus y times 1 minus y. 0:12:13.390,0:12:19.520 So how can you[br]eventually arrange stuff 0:12:19.520,0:12:25.350 to be giving due[br]to the lines that 0:12:25.350,0:12:28.810 are sitting on the surface? 0:12:28.810,0:12:30.910 The lines that are[br]sitting on the surface 0:12:30.910,0:12:33.710 are infinitely many,[br]and I would like 0:12:33.710,0:12:39.320 at least a 1-parameter[br]family of such lines. 0:12:39.320,0:12:40.830 You can have choices. 0:12:40.830,0:12:43.230 One of the choices[br]would be-- this 0:12:43.230,0:12:46.180 is a product, of[br]two numbers, right? 0:12:46.180,0:12:51.320 So you can write it as an[br]equality of two fractions. 0:12:51.320,0:12:56.090 So you would have something[br]like x plus z on top, x minus 0:12:56.090,0:12:57.420 z below. 0:12:57.420,0:13:01.570 Observe that you are[br]creating singularities here. 0:13:01.570,0:13:11.690 So you have to take x minus[br]z case equals 0 separately, 0:13:11.690,0:13:16.540 and then you have, let's[br]say you have 1 minus y here, 0:13:16.540,0:13:21.120 and 1 plus y here. 0:13:21.120,0:13:26.870 What else do you have to impose[br]when you impose x minus z 0:13:26.870,0:13:27.940 equals 0. 0:13:27.940,0:13:29.750 You cannot have 7 over 0. 0:13:29.750,0:13:31.500 That is undefined. 0:13:31.500,0:13:34.630 but if you have 0 over[br]0, that's still possible. 0:13:34.630,0:13:38.430 So whenever you take x[br]minus z equals 0 separately, 0:13:38.430,0:13:42.600 that will imply that the[br]numerator corresponding to it 0:13:42.600,0:13:44.840 will also have to be 0. 0:13:44.840,0:13:47.430 And together these[br]guys are friends. 0:13:47.430,0:13:49.420 What are they? 0:13:49.420,0:13:49.920 2-- 0:13:49.920,0:13:51.211 STUDENT: A system of equations. 0:13:51.211,0:13:53.130 PROFESSOR: It's a[br]system of equations. 0:13:53.130,0:13:57.840 They both represent planes, and[br]the intersection of two planes 0:13:57.840,0:14:00.530 is a line. 0:14:00.530,0:14:07.790 It's a particular line, which[br]is part of the family-- which 0:14:07.790,0:14:10.340 is part of a family. 0:14:10.340,0:14:13.320 0:14:13.320,0:14:14.100 OK. 0:14:14.100,0:14:21.100 Now, on the other hand, in case[br]you have 1 plus y equals 0-- 0:14:21.100,0:14:25.400 so if it happens that you[br]have this extreme case 0:14:25.400,0:14:28.590 that the denominator[br]will be 0, you absolutely 0:14:28.590,0:14:34.830 have to impose x plus z to be 0,[br]and then you have another life. 0:14:34.830,0:14:36.605 It's not easy for[br]me to draw those, 0:14:36.605,0:14:39.390 but I could if you[br]asked me privately 0:14:39.390,0:14:43.050 to draw those and show you[br]what the lines look like. 0:14:43.050,0:14:43.660 OK? 0:14:43.660,0:14:46.180 All right. 0:14:46.180,0:14:51.650 So you have two special lines[br]that are part of that picture. 0:14:51.650,0:14:55.530 They are embedded[br]in the surface. 0:14:55.530,0:15:00.420 How do you find a[br]family of planes? 0:15:00.420,0:15:03.160 Oh my god, I only[br]had one choice, 0:15:03.160,0:15:06.076 but I could have[br]yet another choice 0:15:06.076,0:15:08.320 of how to pick the parameters. 0:15:08.320,0:15:11.540 Let's take lambda to be[br]a real number parameter. 0:15:11.540,0:15:15.743 0:15:15.743,0:15:20.220 And lambda could be[br]anything-- if lambda is 0, 0:15:20.220,0:15:22.010 what have I got to have, guys? 0:15:22.010,0:15:22.800 STUDENT: The top. 0:15:22.800,0:15:24.530 PROFESSOR: The top[br]guys will be 0, 0:15:24.530,0:15:29.140 and I still have 1 minus y[br]equals 0, a plane, intersected 0:15:29.140,0:15:33.920 with x plus z equals 0,[br]another plane, so still a line. 0:15:33.920,0:15:37.300 So lambda equals 0 will give[br]me yet another line, which 0:15:37.300,0:15:39.080 is not written big. 0:15:39.080,0:15:41.104 Are you guys with me? 0:15:41.104,0:15:42.490 Could lambda ever[br]go to infinity? 0:15:42.490,0:15:45.610 0:15:45.610,0:15:49.440 Lambda wants to go to[br]infinity, and when does lambda 0:15:49.440,0:15:50.410 go to infinity? 0:15:50.410,0:15:52.118 STUDENT: When the[br]bottoms would equal 0-- 0:15:52.118,0:15:55.472 PROFESSOR: When both[br]the bottoms would be 0. 0:15:55.472,0:16:01.740 0:16:01.740,0:16:06.440 So this is-- I can call it L[br]infinity, the line of infinity. 0:16:06.440,0:16:07.080 You see? 0:16:07.080,0:16:09.570 But still those[br]would be two planes. 0:16:09.570,0:16:11.800 There's an intersection,[br]it's a line. 0:16:11.800,0:16:12.480 OK. 0:16:12.480,0:16:19.950 Can we write this family--[br]just one family of lines? 0:16:19.950,0:16:24.080 A line is always an intersection[br]of two planes, right? 0:16:24.080,0:16:28.150 So which are the planes[br]that I'm talking about? 0:16:28.150,0:16:32.600 x plus z equals[br]lambda times 1 plus y. 0:16:32.600,0:16:36.540 This is not in the book,[br]because, oh my God, this is 0:16:36.540,0:16:38.650 too hard for the book, right? 0:16:38.650,0:16:43.340 But it's a nice example to[br]look at in an honors class. 0:16:43.340,0:16:47.472 1 minus y equals[br]lambda times x minus z. 0:16:47.472,0:16:48.680 It's not in the book. 0:16:48.680,0:16:55.300 It's not in any book that I know[br]of at the level of calculus. 0:16:55.300,0:16:57.080 All right, OK. 0:16:57.080,0:16:58.590 What are these animals? 0:16:58.590,0:17:00.130 The first animal is a plane. 0:17:00.130,0:17:02.530 The second animal is a plane. 0:17:02.530,0:17:05.040 How many planes[br]are in the picture? 0:17:05.040,0:17:11.630 For each lambda, you have a--[br]for each lambda value in R, 0:17:11.630,0:17:15.670 you have a couple of planes[br]that intersect along your line. 0:17:15.670,0:17:18.454 This is the line L lambda. 0:17:18.454,0:17:21.560 And shut up, Magdalena,[br]you told people too much. 0:17:21.560,0:17:25.220 If you still want them to do[br]this for 2 extra credit points, 0:17:25.220,0:17:28.210 give them the chance[br]to finish the exercise. 0:17:28.210,0:17:32.010 So I zip my lips, but[br]only after I ask you, 0:17:32.010,0:17:33.660 how do you think[br]you are going to get 0:17:33.660,0:17:37.980 the other family of rulers? 0:17:37.980,0:17:42.210 The ruling guys are[br]two families, you see? 0:17:42.210,0:17:47.235 So this family is[br]going in one direction. 0:17:47.235,0:17:48.693 How am I going to[br]get two families? 0:17:48.693,0:17:51.556 0:17:51.556,0:17:55.030 I have another choice[br]that-- how did I take this? 0:17:55.030,0:17:57.880 More or less, I made my choice. 0:17:57.880,0:18:02.270 Just like having two[br]people that would 0:18:02.270,0:18:04.920 be prospective job candidates. 0:18:04.920,0:18:07.627 You pick one of them. 0:18:07.627,0:18:10.235 STUDENT: Now, we can put 1[br]minus y in the denominator. 0:18:10.235,0:18:12.520 The denominator in[br]place of 1 plus y. 0:18:12.520,0:18:16.680 PROFESSOR: So I could have[br]done-- I could have taken this, 0:18:16.680,0:18:19.580 and put 1 plus y here,[br]and 1 minus y here. 0:18:19.580,0:18:21.460 I'm going to let[br]you do the rest, 0:18:21.460,0:18:24.890 and get the second[br]family of generators 0:18:24.890,0:18:28.460 for the whole surface. 0:18:28.460,0:18:29.170 That's enough. 0:18:29.170,0:18:31.180 You're not missing your credit. 0:18:31.180,0:18:35.760 Just, you wanted help,[br]and I helped you. 0:18:35.760,0:18:39.930 And I'm not mad whatsoever[br]when you ask me things. 0:18:39.930,0:18:44.050 The email I got sounded like--[br]says, this is not in the book, 0:18:44.050,0:18:47.170 or in any book, or[br]on the internet. 0:18:47.170,0:18:48.790 How shall I approach this? 0:18:48.790,0:18:51.170 How shall I start thinking[br]about this problem? 0:18:51.170,0:18:54.330 This is a completely[br]legitimate question. 0:18:54.330,0:18:58.280 How do I start on this problem? 0:18:58.280,0:18:59.204 OK. 0:18:59.204,0:19:02.240 On the homework-- maybe it's[br]too easy-- you have two or three 0:19:02.240,0:19:04.830 examples involving spheres. 0:19:04.830,0:19:06.870 Those will be too easy for you. 0:19:06.870,0:19:10.865 I only gave you a very thin[br]among of homework this time. 0:19:10.865,0:19:15.176 You Have plenty of time until[br]Monday at 1:30 or something PM. 0:19:15.176,0:19:18.790 0:19:18.790,0:19:21.280 I would like to draw[br]a little bit more, 0:19:21.280,0:19:25.020 because in this homework[br]and the next homework, 0:19:25.020,0:19:33.160 I'm building something special[br]called the Frenet Trihedron. 0:19:33.160,0:19:36.580 And I told you a little bit[br]about this Frenet Trihedron, 0:19:36.580,0:19:39.700 but I didn't tell you much. 0:19:39.700,0:19:42.600 0:19:42.600,0:19:45.910 Many textbooks in[br]multivariable calculus 0:19:45.910,0:19:48.810 don't say much about it,[br]which I think is a shame. 0:19:48.810,0:19:52.670 0:19:52.670,0:19:57.000 You have a position[br]vector that gives you 0:19:57.000,0:19:59.251 the equation of a regular curve. 0:19:59.251,0:20:05.530 0:20:05.530,0:20:07.930 x of t, y of t, z of t. 0:20:07.930,0:20:09.830 Again, what was a regular curve? 0:20:09.830,0:20:13.870 I'm just doing review of[br]what we did last time. 0:20:13.870,0:20:17.240 A very nice curve[br]that is differentiable 0:20:17.240,0:20:21.840 and whose derivative is[br]continuous everywhere 0:20:21.840,0:20:23.030 on the interval. 0:20:23.030,0:20:30.920 But moreover, the r prime[br]of t never becomes 0. 0:20:30.920,0:20:35.520 So continuously differentiable,[br]and r prime of t 0:20:35.520,0:20:40.980 never becomes 0 for any--[br]do you know this name, 0:20:40.980,0:20:43.201 any for every or for any? 0:20:43.201,0:20:43.700 OK. 0:20:43.700,0:20:46.970 This is the symbolistics[br]of mathematics. 0:20:46.970,0:20:50.090 You know because you[br]are as nerdy as me. 0:20:50.090,0:20:51.970 But everybody else doesn't. 0:20:51.970,0:20:53.650 You guys will learn. 0:20:53.650,0:20:55.710 This is what[br]mathematicians like. 0:20:55.710,0:21:00.260 You see, mathematicians hate[br]writing lots of words down. 0:21:00.260,0:21:05.010 If we liked writing essays[br]and lots of blah, blah, blah, 0:21:05.010,0:21:06.850 we would do something else. 0:21:06.850,0:21:08.760 We wouldn't do mathematics. 0:21:08.760,0:21:11.490 We would do debates,[br]we would do politics, 0:21:11.490,0:21:14.400 we would do other things. 0:21:14.400,0:21:16.880 Mathematicians like[br]ideas, but when 0:21:16.880,0:21:19.420 it comes to writing[br]them down, they 0:21:19.420,0:21:23.450 want to right them down in[br]the most compact way possible. 0:21:23.450,0:21:26.485 That's why they created[br]sort of their own language, 0:21:26.485,0:21:30.730 and they have all sorts[br]of logical quantifiers. 0:21:30.730,0:21:34.060 And it's like your[br]secret language 0:21:34.060,0:21:37.310 when it comes to your[br]less nerdy friends. 0:21:37.310,0:21:45.044 So you go for every--[br]for any or for every-- 0:21:45.044,0:21:45.960 do you know this sign? 0:21:45.960,0:21:48.860 0:21:48.860,0:21:49.535 There exists. 0:21:49.535,0:21:54.690 0:21:54.690,0:21:57.080 And do you know this thing? 0:21:57.080,0:22:01.025 Because one of the-- huh? 0:22:01.025,0:22:02.660 STUDENT: Is that factorial? 0:22:02.660,0:22:05.450 PROFESSOR: Factorial,[br]but in logic, 0:22:05.450,0:22:09.020 that means there exists[br]a unique-- a unique. 0:22:09.020,0:22:11.320 So there exists a unique. 0:22:11.320,0:22:14.720 There exists a unique number. 0:22:14.720,0:22:18.780 There is a unique number. 0:22:18.780,0:22:20.870 So we have our own language. 0:22:20.870,0:22:23.560 Of course, empty set,[br]everybody knows that. 0:22:23.560,0:22:28.160 And it's used in[br]mathematical logic a lot. 0:22:28.160,0:22:33.340 You know most of the symbols[br]from unit intersection, 0:22:33.340,0:22:35.480 or, and. 0:22:35.480,0:22:38.570 I'm going to use some[br]of those as well. 0:22:38.570,0:22:40.480 Coming back to the[br]Frenet Trihedron, 0:22:40.480,0:22:44.065 we have that velocity[br]vector at every point. 0:22:44.065,0:22:44.940 We are happy with it. 0:22:44.940,0:22:49.060 We have our prime of t[br]that is referred from 0. 0:22:49.060,0:22:51.080 I said I want to[br]make it uniform, 0:22:51.080,0:22:53.810 and then I divided[br]by the magnitude, 0:22:53.810,0:22:57.860 and I have this wonderful t[br]vector we just talked about. 0:22:57.860,0:23:03.580 Mr. t is r prime over the[br]magnitude of r prime, which 0:23:03.580,0:23:06.720 is called it's peak right? 0:23:06.720,0:23:09.860 We divide by its peak. 0:23:09.860,0:23:12.612 What's the name of t, again? 0:23:12.612,0:23:13.570 STUDENT: Tangent unit-- 0:23:13.570,0:23:16.310 PROFESSOR: Tangent[br]unit vector, very good. 0:23:16.310,0:23:19.690 How did you remember[br]that so quickly? 0:23:19.690,0:23:22.300 Tangent unit vector. 0:23:22.300,0:23:28.070 There is also another[br]guy who is famous. 0:23:28.070,0:23:32.755 I wanted to make him[br]green, but let's see 0:23:32.755,0:23:35.170 if I can make him blue. 0:23:35.170,0:23:42.380 t is defined-- should I[br]write the f on top of here? 0:23:42.380,0:23:43.502 Do you know what that is? 0:23:43.502,0:23:45.630 STUDENT: I thought n[br]was the normal vector. 0:23:45.630,0:23:48.137 PROFESSOR: t prime[br]divided by the length of-- 0:23:48.137,0:23:48.720 STUDENT: Wait. 0:23:48.720,0:23:53.340 I thought the vector[br]n was the normal. 0:23:53.340,0:23:56.450 PROFESSOR: n-- there[br]are many normals. 0:23:56.450,0:24:01.440 It's a very good thing, because[br]we don't say that in the book. 0:24:01.440,0:24:04.970 OK, this is the t along my r. 0:24:04.970,0:24:09.110 Now when I go through a point,[br]this is the normal plane, 0:24:09.110,0:24:09.980 right? 0:24:09.980,0:24:14.540 There are many normals to[br]the surface-- to the curve. 0:24:14.540,0:24:15.760 Which one am I taking? 0:24:15.760,0:24:19.670 All of them are perpendicular[br]to the direction, right? 0:24:19.670,0:24:20.269 STUDENT: tf. 0:24:20.269,0:24:22.060 PROFESSOR: So I take[br]this one, or this one, 0:24:22.060,0:24:25.460 or this one, or this one, or[br]this one, or this one, there. 0:24:25.460,0:24:26.830 I have to make up my mind. 0:24:26.830,0:24:31.390 And that's how people came up[br]with the so-called principal 0:24:31.390,0:24:33.450 unit normal. 0:24:33.450,0:24:36.080 And this is the one[br]I'm talking about. 0:24:36.080,0:24:39.020 And you are right, it is normal. 0:24:39.020,0:24:42.334 Principal unit normal. 0:24:42.334,0:24:45.290 Remember this very[br]well for your exam, 0:24:45.290,0:24:48.270 because it's a very[br]important notion. 0:24:48.270,0:24:50.100 How do I get to that? 0:24:50.100,0:24:54.040 I take t, I differentiate[br]it, and I divide 0:24:54.040,0:24:59.020 by the lengths of t prime. 0:24:59.020,0:25:07.220 Now, can you prove to me[br]that indeed this fellow 0:25:07.220,0:25:09.710 is perpendicular to t? 0:25:09.710,0:25:12.380 Can you do that? 0:25:12.380,0:25:14.452 STUDENT: That n is[br]perpendicular to t? 0:25:14.452,0:25:16.470 PROFESSOR: Mm-hmm. 0:25:16.470,0:25:17.694 So a little exercise. 0:25:17.694,0:25:22.634 0:25:22.634,0:25:30.590 Prove that-- Prove that I don't[br]have a good marker anymore. 0:25:30.590,0:25:37.510 Prove that n, the unit[br]principal vector field, 0:25:37.510,0:25:44.840 is perpendicular-- you[br]see, I'm a mathematician. 0:25:44.840,0:25:48.690 I swear, I hate to write down[br]the whole word perpendicular. 0:25:48.690,0:25:51.610 I would love to[br]say, perpendicular. 0:25:51.610,0:25:57.840 That's how I write perpendicular[br]really fast-- to t fore 0:25:57.840,0:26:01.470 every value of t. 0:26:01.470,0:26:03.010 For every value of t. 0:26:03.010,0:26:03.910 OK. 0:26:03.910,0:26:06.340 How in the world can I do that? 0:26:06.340,0:26:08.560 I have to think about it. 0:26:08.560,0:26:11.820 This is hard. 0:26:11.820,0:26:12.980 Wish me luck. 0:26:12.980,0:26:15.800 So do I know[br]anything about Mr. t? 0:26:15.800,0:26:18.150 What do I know about Mr. t? 0:26:18.150,0:26:20.390 I'll take it and I'll[br]differentiate it later. 0:26:20.390,0:26:25.050 It Mr. t is magic in the[br]sense that he's a unit vector. 0:26:25.050,0:26:27.860 I'm going to write that down. 0:26:27.860,0:26:31.910 t in absolute value equals 1. 0:26:31.910,0:26:33.120 It's beautiful. 0:26:33.120,0:26:36.920 If I squared that-- and[br]you're going to say, 0:26:36.920,0:26:38.600 why would you want[br]to square that? 0:26:38.600,0:26:40.500 You're going to see in a minute. 0:26:40.500,0:26:43.140 If I squared that,[br]then I'm going 0:26:43.140,0:26:50.580 to have the dot product[br]between t and itself equals 1. 0:26:50.580,0:26:53.130 0:26:53.130,0:26:56.740 Can somebody tell me why the[br]dot product between t and itself 0:26:56.740,0:27:00.850 is the square of a length of t? 0:27:00.850,0:27:04.930 What's the definition[br]of the dot product? 0:27:04.930,0:27:08.290 Magnitude of the first[br]vector, times the magnitude 0:27:08.290,0:27:11.340 of the second vector--[br]there i am already-- 0:27:11.340,0:27:15.410 times the cosine of the[br]angle between the two vectors 0:27:15.410,0:27:17.480 Duh, that's 0. 0:27:17.480,0:27:20.496 So cosine of 0 is 1, I'm done. 0:27:20.496,0:27:21.390 Right? 0:27:21.390,0:27:26.580 Now, I have a vector function[br]times a vector function-- 0:27:26.580,0:27:31.270 this is crazy, right-- equals 1. 0:27:31.270,0:27:34.250 I'm going to go ahead[br]and differentiate. 0:27:34.250,0:27:37.690 Keep in mind that[br]this is a product. 0:27:37.690,0:27:39.770 What's the product? 0:27:39.770,0:27:42.330 One of my professors,[br]colleagues, 0:27:42.330,0:27:44.980 was telling me, now,[br]let's be serious. 0:27:44.980,0:27:49.370 In five years, how many[br]of your engineering majors 0:27:49.370,0:27:51.250 will remember the product? 0:27:51.250,0:27:53.180 I really was[br]thinking about this. 0:27:53.180,0:27:56.680 I hope everybody, if[br]they were my students, 0:27:56.680,0:27:59.180 because we are going to[br]have enough practice. 0:27:59.180,0:28:01.780 So the prime rule in[br]Calc 1 said that if you 0:28:01.780,0:28:05.125 have f of t times g of[br]t, you have a product. 0:28:05.125,0:28:08.320 You prime that product,[br]and never write 0:28:08.320,0:28:12.530 f prime times g prime unless you[br]want me to call you around 2 AM 0:28:12.530,0:28:15.255 to say you should never do that. 0:28:15.255,0:28:20.100 0:28:20.100,0:28:23.760 So how does the[br]product rule work? 0:28:23.760,0:28:27.500 The first one prime[br]times the second unprime 0:28:27.500,0:28:32.310 plus the first one unprime[br]times the second prime. 0:28:32.310,0:28:34.505 My students know[br]the product rule. 0:28:34.505,0:28:37.480 I don't care if the rest[br]of the world doesn't. 0:28:37.480,0:28:40.070 I don't care about any[br]community college who 0:28:40.070,0:28:42.590 would say, I don't want the[br]product rule to be known, 0:28:42.590,0:28:44.760 you can differentiate[br]with a calculator. 0:28:44.760,0:28:46.000 That's a no, no, no. 0:28:46.000,0:28:50.108 You don't know calculus if you[br]don't know the product rule. 0:28:50.108,0:28:53.480 So the product rule is[br]a blessing from God. 0:28:53.480,0:28:58.020 It helps everywhere in physics,[br]in mechanics, in engineering. 0:28:58.020,0:29:00.990 It really helps in[br]differential geometry 0:29:00.990,0:29:03.755 with the directional[br]derivative, the Lie derivative. 0:29:03.755,0:29:07.910 It helps you understand all[br]the upper level mathematics. 0:29:07.910,0:29:11.700 Now here you have t prime,[br]the first prime times 0:29:11.700,0:29:16.300 the second unprime, plus the[br]first unprime times the second 0:29:16.300,0:29:17.390 prime. 0:29:17.390,0:29:21.040 It's the same as for[br]regular scalar functions. 0:29:21.040,0:29:23.750 What's the derivative of 1? 0:29:23.750,0:29:24.250 STUDENT: 0. 0:29:24.250,0:29:25.610 PROFESSOR: 0. 0:29:25.610,0:29:26.860 Look at this guy! 0:29:26.860,0:29:29.270 Doesn't he look funny? 0:29:29.270,0:29:32.630 It is the dot product community. 0:29:32.630,0:29:34.255 Yes it is, by definition. 0:29:34.255,0:29:40.120 So you have twice T[br]times T prime equals 0. 0:29:40.120,0:29:44.050 This 2 is-- stinking[br]guy, let's divide by 2. 0:29:44.050,0:29:45.076 Forget about that. 0:29:45.076,0:29:46.590 What does this say? 0:29:46.590,0:29:53.840 The dot product of T times--[br]I mean by T prime is 0. 0:29:53.840,0:29:57.220 When are two vectors[br]giving you dot product 0? 0:29:57.220,0:29:58.720 STUDENT: When they're[br]perpendicular. 0:29:58.720,0:29:59.550 0:29:59.550,0:30:01.900 PROFESSOR: So if both[br]of them are non-zero, 0:30:01.900,0:30:03.370 they have to be like that. 0:30:03.370,0:30:06.330 They have to be like this,[br]perpendicular, right? 0:30:06.330,0:30:11.762 So it follows that t has to[br]be perpendicular to T prime. 0:30:11.762,0:30:15.550 And now, that's why n[br]is perpendicular to t. 0:30:15.550,0:30:19.370 But, because n is[br]collinear to t prime. 0:30:19.370,0:30:20.100 Hello. 0:30:20.100,0:30:22.500 n is collinear to t prime. 0:30:22.500,0:30:25.660 So this is t prime. 0:30:25.660,0:30:27.840 Is t prime unitary? 0:30:27.840,0:30:29.420 I'm going to measure it. 0:30:29.420,0:30:30.720 No it's not. 0:30:30.720,0:30:31.620 t prime. 0:30:31.620,0:30:33.650 So if I want to[br]make it unitary, I'm 0:30:33.650,0:30:36.430 going to chop my-- no,[br]I'm not going to chop. 0:30:36.430,0:30:40.160 I just take it, t prime,[br]and divide by its magnitude. 0:30:40.160,0:30:43.450 Then I'm going to get that[br]vector n, which is unitary. 0:30:43.450,0:30:47.790 So from here it follows that t[br]and n are indeed perpendicular, 0:30:47.790,0:30:52.670 and your colleague over there[br]said, hey, it has to be normal. 0:30:52.670,0:30:55.140 That's perpendicular[br]to t, but which one? 0:30:55.140,0:30:58.180 A special one, because[br]I have many normals. 0:30:58.180,0:31:02.310 Now, this special one is[br]easy to find like that. 0:31:02.310,0:31:05.726 Where shall I put here--[br]I'll draw him very nicely. 0:31:05.726,0:31:08.880 0:31:08.880,0:31:10.000 I'll draw him. 0:31:10.000,0:31:13.410 Now you guys have to[br]imagine-- am I drawing 0:31:13.410,0:31:14.580 well enough for you? 0:31:14.580,0:31:15.900 I don't even know. 0:31:15.900,0:31:17.780 t and n should be perpendicular. 0:31:17.780,0:31:21.960 Can you imagine them having that[br]90 degree angle between them? 0:31:21.960,0:31:22.460 OK. 0:31:22.460,0:31:26.710 Now there is a magic one that[br]you don't even have to define. 0:31:26.710,0:31:28.670 And yes sir? 0:31:28.670,0:31:31.180 STUDENT: In this[br]thing, can [INAUDIBLE] 0:31:31.180,0:31:34.410 this T vector [INAUDIBLE][br]written by the definition 0:31:34.410,0:31:36.170 thing? 0:31:36.170,0:31:37.829 PROFESSOR: No. 0:31:37.829,0:31:39.370 STUDENT: N vector[br]times the magnitude 0:31:39.370,0:31:42.060 of t vector derivative? 0:31:42.060,0:31:47.050 PROFESSOR: So[br]technically you have 0:31:47.050,0:31:50.764 t prime would be the[br]magnitude of t prime times n. 0:31:50.764,0:31:51.640 STUDENT: Yes. 0:31:51.640,0:31:54.440 PROFESSOR: But keep in mind[br]that sometimes is tricky, 0:31:54.440,0:31:57.310 because this is, in[br]general, not a constant. 0:31:57.310,0:31:59.470 Always keep it in mind,[br]it's not a constant. 0:31:59.470,0:32:02.000 We'll have some examples later. 0:32:02.000,0:32:04.690 There is a magic[br]guy called binormal. 0:32:04.690,0:32:09.710 That binormal is the[br]normal to both t and n. 0:32:09.710,0:32:12.430 And he's defined as[br]t plus n because it's 0:32:12.430,0:32:14.445 normal to both of them. 0:32:14.445,0:32:18.370 So I'm going to write this[br]b vector is t cross n. 0:32:18.370,0:32:22.002 Now I'm asking you to draw it. 0:32:22.002,0:32:23.710 Can anybody come to[br]the board and draw it 0:32:23.710,0:32:26.960 for 0.01 extra credit? 0:32:26.960,0:32:29.642 Yes, sir? 0:32:29.642,0:32:30.660 STUDENT: [INAUDIBLE] 0:32:30.660,0:32:35.000 PROFESSOR: Draw that on the[br]picture like t and n, t and n, 0:32:35.000,0:32:38.470 t is the-- who the heck[br]is t? t is the red one, 0:32:38.470,0:32:40.860 and blue is the n. 0:32:40.860,0:32:42.930 So does it go down or up? 0:32:42.930,0:32:46.460 We should be perpendicular[br]to both of them. 0:32:46.460,0:32:49.340 Is b unitary or not? 0:32:49.340,0:32:51.820 If you have two unit vectors,[br]will the cross product 0:32:51.820,0:32:53.060 be a unit vector? 0:32:53.060,0:32:56.370 0:32:56.370,0:33:00.200 Only if the two vectors[br]are perpendicular, 0:33:00.200,0:33:04.680 it is going to be, right? 0:33:04.680,0:33:12.420 So you have-- well, I[br]think it goes that-- 0:33:12.420,0:33:13.760 in which direction does it go? 0:33:13.760,0:33:14.620 Because 0:33:14.620,0:33:16.360 STUDENT: It should[br]not be how we have it. 0:33:16.360,0:33:17.276 PROFESSOR: No, no, no. 0:33:17.276,0:33:18.370 Because this is-- 0:33:18.370,0:33:19.270 STUDENT: Yeah. 0:33:19.270,0:33:19.820 I'm using-- 0:33:19.820,0:33:22.935 PROFESSOR: So t[br]goes over n, so I'm 0:33:22.935,0:33:27.470 going to try-- it is[br]like that, sort of. 0:33:27.470,0:33:28.710 STUDENT: Into the chord? 0:33:28.710,0:33:31.452 PROFESSOR: So again, it's[br]not very clear because 0:33:31.452,0:33:33.860 of my stinking art, here. 0:33:33.860,0:33:36.000 It's really not nice art. 0:33:36.000,0:33:39.830 t, and this is n. 0:33:39.830,0:33:43.540 And if I go t going over n. 0:33:43.540,0:33:47.627 T going over n goes up or down? 0:33:47.627,0:33:48.210 STUDENT: Down. 0:33:48.210,0:33:49.084 PROFESSOR: Goes down. 0:33:49.084,0:33:52.310 So it's going to look[br]more like this, feet. 0:33:52.310,0:33:54.670 Now guys, when we--[br]thank you so much. 0:33:54.670,0:33:57.970 So you've like a[br]0.01 extra credit. 0:33:57.970,0:34:00.610 OK. 0:34:00.610,0:34:03.400 Tangent, normal, and[br]binormal form a corner. 0:34:03.400,0:34:04.320 Yes, sir? 0:34:04.320,0:34:07.460 STUDENT: Is rt-- rt is[br]the function at the-- 0:34:07.460,0:34:09.159 for the flag that's flying? 0:34:09.159,0:34:12.120 PROFESSOR: The r of t[br]is the position vector 0:34:12.120,0:34:15.030 of the flag that was[br]flying that he was drunk. 0:34:15.030,0:34:21.350 STUDENT: Why wasn't the[br]derivative of it perpendicular? 0:34:21.350,0:34:24.420 Why isn't t perpendicular to rt? 0:34:24.420,0:34:26.199 PROFESSOR: If--[br]well, good question. 0:34:26.199,0:34:28.770 0:34:28.770,0:34:30.600 We'll talk about it. 0:34:30.600,0:34:34.690 If the length of r[br]would be a constant, 0:34:34.690,0:34:38.690 can we prove that r and r[br]prime are perpendicular? 0:34:38.690,0:34:40.985 Let's do that as[br]another exercise. 0:34:40.985,0:34:42.880 All right? 0:34:42.880,0:34:46.190 So tnb looks like a corner. 0:34:46.190,0:34:51.810 Look at the corner that the[br]video cannot see over there. 0:34:51.810,0:34:53.659 TN and B are mutually octagonal. 0:34:53.659,0:34:56.300 0:34:56.300,0:34:57.720 I'm going to draw them. 0:34:57.720,0:35:00.810 This is an arbitrary[br]point on a curve, 0:35:00.810,0:35:04.410 and this is t, which is[br]always tangent to the curve, 0:35:04.410,0:35:05.610 and this is n. 0:35:05.610,0:35:08.490 Let's say that's the[br]unit principle normal. 0:35:08.490,0:35:11.109 And t cross n will[br]go, again, down. 0:35:11.109,0:35:11.650 I don't know. 0:35:11.650,0:35:14.560 I have an obsession[br]with me going down. 0:35:14.560,0:35:16.480 This is called the[br]Frenet Trihedron. 0:35:16.480,0:35:20.220 0:35:20.220,0:35:23.000 And I have a proposal[br]for a problem 0:35:23.000,0:35:34.950 that maybe I should give[br]my students in the future. 0:35:34.950,0:35:46.450 Show that for a circle,[br]playing in space, I don't know. 0:35:46.450,0:36:07.210 The position vector and the[br]velocity vector are always how? 0:36:07.210,0:36:07.710 Friends. 0:36:07.710,0:36:09.140 Let's say friends. 0:36:09.140,0:36:12.470 No, come on, I'm kidding. 0:36:12.470,0:36:13.270 How are they? 0:36:13.270,0:36:15.391 STUDENT: Perpendicular. 0:36:15.391,0:36:16.640 PROFESSOR: How do you do that? 0:36:16.640,0:36:18.360 Is it hard? 0:36:18.360,0:36:20.880 We should be smart[br]enough to do that, right? 0:36:20.880,0:36:21.980 I have a circle. 0:36:21.980,0:36:26.380 That circle has what-- what[br]is the property of a circle? 0:36:26.380,0:36:29.880 Euclid defined that-- this is[br]one of the axioms of Euclid. 0:36:29.880,0:36:32.430 Does anybody know which axiom? 0:36:32.430,0:36:35.790 That there exists[br]such a set of points 0:36:35.790,0:36:38.660 that are all at the same[br]distance from a given point 0:36:38.660,0:36:40.630 called center. 0:36:40.630,0:36:42.850 So that is a circle, right? 0:36:42.850,0:36:44.270 That's what Mr. Euclid said. 0:36:44.270,0:36:45.400 He was a genius. 0:36:45.400,0:36:51.980 So no matter where I put that[br]circle, I can take r of t 0:36:51.980,0:36:55.290 in magnitude measured[br]from the origin 0:36:55.290,0:36:57.190 from the center of the circle. 0:36:57.190,0:37:01.200 Keep in mind, always the[br]center of the circle. 0:37:01.200,0:37:06.000 I put it at the origin of the[br]space-- origin of the universe. 0:37:06.000,0:37:08.420 No, origin of the[br]space, actually. 0:37:08.420,0:37:12.840 R of T magnitude[br]would be a constant. 0:37:12.840,0:37:13.995 Give me a constant, guys. 0:37:13.995,0:37:14.495 OK? 0:37:14.495,0:37:16.480 It doesn't matter. 0:37:16.480,0:37:18.460 Let me draw. 0:37:18.460,0:37:20.270 I want to draw in plane, OK? 0:37:20.270,0:37:25.980 Because I'm getting tired.[br]x y, and this is r of t, 0:37:25.980,0:37:30.210 and the magnitude of this r of[br]t is the radius of the circle. 0:37:30.210,0:37:32.540 Right? 0:37:32.540,0:37:36.677 So let's say, this is[br]the radius of the circle. 0:37:36.677,0:37:40.880 0:37:40.880,0:37:43.930 How in the world do I[br]prove the same idea? 0:37:43.930,0:37:47.985 Who helps me prove[br]that r is always 0:37:47.985,0:37:50.610 perpendicular to r prime? 0:37:50.610,0:37:53.857 Which way do you want to move,[br]counterclockwise or clockwise? 0:37:53.857,0:37:54.940 STUDENT: Counterclockwise. 0:37:54.940,0:37:56.106 PROFESSOR: Counterclockwise. 0:37:56.106,0:37:58.160 Because if you are[br]a real scientist, 0:37:58.160,0:38:00.110 I'm proud of you guys. 0:38:00.110,0:38:01.950 It's clear from the[br]picture that r prime 0:38:01.950,0:38:04.790 would be perpendicular to r. 0:38:04.790,0:38:05.825 Why is that? 0:38:05.825,0:38:07.593 How am I going to do that? 0:38:07.593,0:38:11.510 Now, mimic everything I--[br]don't look at your notes, 0:38:11.510,0:38:17.260 and try to tell me how[br]I show that quickly. 0:38:17.260,0:38:18.290 What am I going to do? 0:38:18.290,0:38:23.380 So all I know, all[br]that gave me was r of t 0:38:23.380,0:38:27.680 equals k in magnitude constant. 0:38:27.680,0:38:30.530 For every t, this same constant. 0:38:30.530,0:38:31.510 What's next? 0:38:31.510,0:38:34.922 What do I want to do next? 0:38:34.922,0:38:36.310 STUDENT: Square it? 0:38:36.310,0:38:38.390 PROFESSOR: Square[br]it, differentiate it. 0:38:38.390,0:38:40.454 I can also go ahead[br]and differentiate it 0:38:40.454,0:38:41.995 without squaring[br]it, but that's going 0:38:41.995,0:38:47.350 to be a little bit of more pain. 0:38:47.350,0:38:52.170 So square it, differentiate it. 0:38:52.170,0:38:53.390 I'm too lazy. 0:38:53.390,0:38:56.140 When I differentiate,[br]what am I going to get? 0:38:56.140,0:39:05.320 From the product rule, twice[br]r dot r primed of t equals 0. 0:39:05.320,0:39:06.870 Well, I'm done. 0:39:06.870,0:39:12.000 Because it means that for[br]every t that radius-- not 0:39:12.000,0:39:13.270 the radius, guys, I'm sorry. 0:39:13.270,0:39:16.870 The position vector will be[br]perpendicular to the velocity 0:39:16.870,0:39:17.560 vector. 0:39:17.560,0:39:22.180 Now, if I draw the[br]trajectory of my drunken flag 0:39:22.180,0:39:24.660 this [INAUDIBLE][br]is not true, right? 0:39:24.660,0:39:27.430 This is crazy. 0:39:27.430,0:39:29.924 Of course this is r,[br]and this is r prime, 0:39:29.924,0:39:35.590 and there is an arbitrary[br]angle between r and r prime. 0:39:35.590,0:39:38.260 The good thing is that[br]the arbitrary angle always 0:39:38.260,0:39:41.320 exists, and is[br]continuous as a function. 0:39:41.320,0:39:43.450 I never have that[br]angle disappear. 0:39:43.450,0:39:46.900 That's way I want that[br]prime never to become 0. 0:39:46.900,0:39:49.330 Because if the bag was[br]stopping its motion, 0:39:49.330,0:39:54.020 goodbye angle, goodbye[br]analysis, right? 0:39:54.020,0:39:54.730 OK. 0:39:54.730,0:39:55.540 Very nice. 0:39:55.540,0:39:57.470 So don't give me more ideas. 0:39:57.470,0:39:59.940 You smart people, if[br]you give me more ideas, 0:39:59.940,0:40:02.550 I'm going to come up with[br]all sorts of problems. 0:40:02.550,0:40:05.000 And this is actually one[br]of the first problems 0:40:05.000,0:40:08.980 you learn in a graduate[br]level geometry class. 0:40:08.980,0:40:13.930 0:40:13.930,0:40:16.670 Let me give you another[br]piece of information 0:40:16.670,0:40:20.240 that you're going[br]to love, which could 0:40:20.240,0:40:22.390 be one of those[br]types of combined 0:40:22.390,0:40:25.300 problems on a final[br]exam or midterm, 0:40:25.300,0:40:29.900 A, B, C, D, E. The[br]curvature of a curve 0:40:29.900,0:40:33.930 is a measure of how[br]the curve will bend. 0:40:33.930,0:40:35.720 Say what? 0:40:35.720,0:40:45.970 The curvature of a[br]curve is a measure 0:40:45.970,0:40:48.542 of the bending of that curve. 0:40:48.542,0:40:58.800 0:40:58.800,0:41:04.380 By definition, you have[br]to take it like that. 0:41:04.380,0:41:20.730 If the curve is parameterized[br]in arc length-- somebody 0:41:20.730,0:41:23.070 remind me what that is. 0:41:23.070,0:41:24.960 What does it mean? 0:41:24.960,0:41:32.440 That is r of s such[br]that-- what does it mean, 0:41:32.440,0:41:33.644 parameterizing arc length-- 0:41:33.644,0:41:34.560 STUDENT: r prime of s. 0:41:34.560,0:41:37.320 PROFESSOR: r primed of[br]s in magnitude is 1. 0:41:37.320,0:41:37.910 The speed 1. 0:41:37.910,0:41:39.200 It's a speed 1 curve. 0:41:39.200,0:41:42.510 0:41:42.510,0:42:00.690 Then, the curvature of this[br]curve is defined as k of s 0:42:00.690,0:42:06.320 equals the magnitude of[br]the acceleration vector 0:42:06.320,0:42:09.460 will respect the S.[br]Say what, Magdalena? 0:42:09.460,0:42:12.890 I can also write[br]it magnitude of d-- 0:42:12.890,0:42:17.070 oh my gosh, second derivative[br]with respect s of r. 0:42:17.070,0:42:20.340 I'll do it right now. d2r ds2. 0:42:20.340,0:42:21.910 And I know you get[br]a headache when 0:42:21.910,0:42:25.940 I solve, when I write that,[br]because you are not used to it. 0:42:25.940,0:42:33.450 A quick and beautiful example[br]that can be on the homework, 0:42:33.450,0:42:39.380 and would also be on the[br]exam, maybe on all the exams, 0:42:39.380,0:42:41.864 I don't know. 0:42:41.864,0:42:48.140 Compute the curvature of a[br]circle of radius a Say what? 0:42:48.140,0:43:02.670 Compute the curvature of a[br]circle of radius a And you say, 0:43:02.670,0:43:03.620 wait a minute. 0:43:03.620,0:43:07.040 For a circle of[br]radius a in plane-- 0:43:07.040,0:43:08.990 why can I assume it's in plane? 0:43:08.990,0:43:12.590 Because if the circle[br]is a planar curve, 0:43:12.590,0:43:15.860 I can always assume[br]it to be in plane. 0:43:15.860,0:43:19.030 And it has radius a I[br]can find infinitely many 0:43:19.030,0:43:20.160 parameterizations. 0:43:20.160,0:43:23.100 So what, am I crazy? 0:43:23.100,0:43:25.444 Well, yes, I am, but[br]that's another story. 0:43:25.444,0:43:28.330 Now, if I want to[br]parameterize, I 0:43:28.330,0:43:31.170 have to parameterize[br]in arc length. 0:43:31.170,0:43:34.020 If I do anything else,[br]that means I'm stupid. 0:43:34.020,0:43:38.960 So, r of s will be what? 0:43:38.960,0:43:41.390 Can somebody tell me[br]how I parameterize 0:43:41.390,0:43:46.280 a curve in arc length[br]for a-- what is this guy? 0:43:46.280,0:43:49.210 A circle of radius a. 0:43:49.210,0:43:50.520 Yeah, I cannot do it. 0:43:50.520,0:43:52.470 I'm not smart enough. 0:43:52.470,0:43:59.390 So I'll say R of T will be[br]a cosine t, a sine t and 0. 0:43:59.390,0:44:03.150 And here I stop, because[br]I had a headache. 0:44:03.150,0:44:08.510 t is from 0 to 2 pi, and[br]I think this a is making 0:44:08.510,0:44:14.980 my life miserable,[br]because it's telling me, 0:44:14.980,0:44:17.410 you don't have[br]speed 1, Magdalena. 0:44:17.410,0:44:19.200 Drive to Amarillo[br]and back, you're 0:44:19.200,0:44:22.050 not going to get speed 1. 0:44:22.050,0:44:23.320 Why don't I have speed 1? 0:44:23.320,0:44:23.950 Think about it. 0:44:23.950,0:44:24.930 Bear with me. 0:44:24.930,0:44:28.546 Minus a sine t equals sine t, 0. 0:44:28.546,0:44:29.420 Bad. 0:44:29.420,0:44:31.150 What is the speed? 0:44:31.150,0:44:32.750 a. 0:44:32.750,0:44:35.570 If you do the math,[br]the speed will be a. 0:44:35.570,0:44:39.650 So length of our[br]prime of t will be a. 0:44:39.650,0:44:40.760 Somebody help me. 0:44:40.760,0:44:42.300 Get me out of trouble. 0:44:42.300,0:44:43.220 Who is this? 0:44:43.220,0:44:45.400 I want to do it in arc length. 0:44:45.400,0:44:48.250 Otherwise, how can[br]I do the curvature? 0:44:48.250,0:44:50.790 So somebody tell[br]me how to get to s. 0:44:50.790,0:44:52.270 What the heck is that? 0:44:52.270,0:44:58.610 s of t is integral from[br]0 to t of-- who tells me? 0:44:58.610,0:44:59.840 The speed, right? 0:44:59.840,0:45:04.390 Was it not the displacement,[br]the arc length traveled along, 0:45:04.390,0:45:08.036 and the curve is integral[br]in time of the speed. 0:45:08.036,0:45:11.950 0:45:11.950,0:45:13.030 OK? 0:45:13.030,0:45:16.420 So I have-- what is that? 0:45:16.420,0:45:17.560 Speed is? 0:45:17.560,0:45:18.360 STUDENT: Um-- 0:45:18.360,0:45:18.980 PROFESSOR: a. 0:45:18.980,0:45:22.850 So a time t, am I right,[br]guys? s is a times t. 0:45:22.850,0:45:24.650 So what do I have to do? 0:45:24.650,0:45:30.950 Take Mr. t, shake his hand,[br]and replace him with s over a. 0:45:30.950,0:45:31.810 OK. 0:45:31.810,0:45:39.740 So instead of r of t, I'll say--[br]what other letters do I have? 0:45:39.740,0:45:40.480 Not r. 0:45:40.480,0:45:41.443 Rho of s. 0:45:41.443,0:45:42.400 I love rho. 0:45:42.400,0:45:44.420 Rho is the Greek [INAUDIBLE]. 0:45:44.420,0:45:46.350 Is this finally an arc length? 0:45:46.350,0:45:51.000 Cosine of-- what[br]is t, guys, again? 0:45:51.000,0:45:53.140 s over a. 0:45:53.140,0:45:58.220 s over a, a sine[br]s over a, and 0. 0:45:58.220,0:46:01.710 This is the parameterization[br]in arc length. 0:46:01.710,0:46:08.470 This is an arc length[br]parameterization of the circle. 0:46:08.470,0:46:11.470 And then what is this[br]definition of curvature? 0:46:11.470,0:46:13.760 It's here. 0:46:13.760,0:46:16.930 Do that rho once, twice. 0:46:16.930,0:46:19.510 Prime it twice,[br]and do the length. 0:46:19.510,0:46:20.760 So rho prime. 0:46:20.760,0:46:25.060 Oh my God is it hard. 0:46:25.060,0:46:29.420 a times minus sine of s over a. 0:46:29.420,0:46:30.270 Am I done, though? 0:46:30.270,0:46:30.959 Chain rule. 0:46:30.959,0:46:32.000 Pay attention, Magdalena. 0:46:32.000,0:46:34.070 Don't screwed up with this one. 0:46:34.070,0:46:36.370 1 over a. 0:46:36.370,0:46:37.730 Good. 0:46:37.730,0:46:38.580 Next. 0:46:38.580,0:46:41.716 a cosine of s over a. 0:46:41.716,0:46:42.670 Chain rule. 0:46:42.670,0:46:44.300 Don't forget,[br]multiply by 1 over a. 0:46:44.300,0:46:46.860 OK, that makes my life easier. 0:46:46.860,0:46:47.820 We simplify. 0:46:47.820,0:46:51.590 Thank God a simplifies[br]here, a simplifies there, 0:46:51.590,0:46:54.180 so that is that derivative. 0:46:54.180,0:46:55.870 What's the second derivative? 0:46:55.870,0:47:01.217 Rho double prime of s will[br]be-- somebody help me, OK? 0:47:01.217,0:47:02.716 Because this is a[br]lot of derivation. 0:47:02.716,0:47:03.470 STUDENT: --cosine-- 0:47:03.470,0:47:04.553 PROFESSOR: Thank you, sir. 0:47:04.553,0:47:06.810 Minus cosine of s over a. 0:47:06.810,0:47:07.810 STUDENT: Times 1 over a. 0:47:07.810,0:47:13.330 PROFESSOR: Times 1 over a,[br]comma, minus sine of s over a. 0:47:13.330,0:47:15.600 That's all I have left[br]in my life, right? 0:47:15.600,0:47:19.860 Minus sine of s over a times[br]1 over a from the chain rule. 0:47:19.860,0:47:22.690 I have to pay attention and see. 0:47:22.690,0:47:24.350 What's the magnitude of this? 0:47:24.350,0:47:28.644 The magnitude of this of this[br]animal will be the curvature. 0:47:28.644,0:47:30.060 Oh, my God. 0:47:30.060,0:47:32.320 So what is k? 0:47:32.320,0:47:35.120 k of s will be--[br]could somebody tell me 0:47:35.120,0:47:39.610 what magnitude I get after I[br]square all these individuals, 0:47:39.610,0:47:42.914 sum them up, and take[br]the square root of them? 0:47:42.914,0:47:44.360 STUDENT: [INAUDIBLE] 0:47:44.360,0:47:49.550 PROFESSOR: Square root[br]of 1 over 1 squared. 0:47:49.550,0:47:50.740 And I get 1 over a. 0:47:50.740,0:47:53.236 You are too fast for[br]me, you teach me that. 0:47:53.236,0:47:54.110 No, I'm just kidding. 0:47:54.110,0:47:56.290 I knew it was 1 over a. 0:47:56.290,0:47:59.320 Now, how did[br]engineers know that? 0:47:59.320,0:48:02.170 Actually, for hundreds of years,[br]mathematicians, engineers, 0:48:02.170,0:48:04.460 and physicists knew that. 0:48:04.460,0:48:07.470 And that's the last thing[br]I want to teach you today. 0:48:07.470,0:48:09.910 We have two circles. 0:48:09.910,0:48:17.390 This is of, let's say, radius[br]1/2, and this is radius 2. 0:48:17.390,0:48:21.030 The engineer, mathematician,[br]physicist, whoever they are, 0:48:21.030,0:48:26.465 they knew that the curvature[br]is inverse proportional 0:48:26.465,0:48:28.250 to the radius. 0:48:28.250,0:48:30.440 That radius is 1/2. 0:48:30.440,0:48:33.516 The curvature will[br]be 2 in this case. 0:48:33.516,0:48:38.072 The radius is 2, the[br]curvature will be 1/2. 0:48:38.072,0:48:41.890 Does that make sense, this[br]inverse proportionality? 0:48:41.890,0:48:45.660 The bigger the radius,[br]the lesser the curvature, 0:48:45.660,0:48:47.670 that less bent you are. 0:48:47.670,0:48:49.600 The more fat-- well, OK. 0:48:49.600,0:48:53.080 I'm not going to say anything[br]politically incorrect. 0:48:53.080,0:48:58.540 So this is really curved because[br]the radius is really small. 0:48:58.540,0:49:02.620 This less curved,[br]almost-- at infinity, 0:49:02.620,0:49:05.550 this curvature[br]becomes 0, because 0:49:05.550,0:49:09.075 at infinity, that radius[br]explodes to plus infinity bag 0:49:09.075,0:49:10.090 theory. 0:49:10.090,0:49:13.600 Then you have 1 over[br]infinity will be 0, 0:49:13.600,0:49:19.064 and that will be the curvature[br]of a circle of infinite radius. 0:49:19.064,0:49:20.420 Right? 0:49:20.420,0:49:22.530 So we learned something today. 0:49:22.530,0:49:25.489 We learned about the[br]curvature of a circle, which 0:49:25.489,0:49:26.030 is something. 0:49:26.030,0:49:30.930 But this is the same[br]way for any curve. 0:49:30.930,0:49:31.836 You reparameterize. 0:49:31.836,0:49:34.460 Now you understand why you need[br]to reparameterize in arc length 0:49:34.460,0:49:35.880 s. 0:49:35.880,0:49:37.854 You take the acceleration[br]in arc length. 0:49:37.854,0:49:38.770 You get the magnitude. 0:49:38.770,0:49:41.520 That measures how[br]bent the curve is. 0:49:41.520,0:49:47.210 Next time, you're going to[br]do how bent the helix is. 0:49:47.210,0:49:47.750 OK? 0:49:47.750,0:49:49.442 At every point. 0:49:49.442,0:49:51.240 Enjoy your WeBWorK homework. 0:49:51.240,0:49:54.935 Ask me anytime, and[br]ask me also Thursday. 0:49:54.935,0:49:58.710 Do not have a block about[br]your homework questions. 0:49:58.710,0:50:05.120 You can ask me anytime[br]by email, or in person. 0:50:05.120,0:50:10.489