1 00:00:00,000 --> 00:00:00,580 2 00:00:00,580 --> 00:00:02,330 PROFESSOR: Any questions about theory 3 00:00:02,330 --> 00:00:05,300 that gave you headaches regarding homework 4 00:00:05,300 --> 00:00:07,740 you'd like to talk about? 5 00:00:07,740 --> 00:00:10,950 Anything related to what we covered 6 00:00:10,950 --> 00:00:13,778 from chapter nine and today? 7 00:00:13,778 --> 00:00:15,610 STUDENT: Can we do some problems? 8 00:00:15,610 --> 00:00:17,410 PROFESSOR: I can fix from problems 9 00:00:17,410 --> 00:00:20,103 like the ones in the homework, but also I 10 00:00:20,103 --> 00:00:24,547 can have you tell me what bothers you in the homework. 11 00:00:24,547 --> 00:00:25,880 STUDENT: Oh, I have [INAUDIBLE]. 12 00:00:25,880 --> 00:00:29,010 PROFESSOR: What bothered me about my own homework 13 00:00:29,010 --> 00:00:35,330 was that I realized that I did not remind you something 14 00:00:35,330 --> 00:00:37,470 I assume you should know, which is 15 00:00:37,470 --> 00:00:42,800 the equation of a sphere of given center and given radius. 16 00:00:42,800 --> 00:00:47,500 And since I trust you so much, I said, OK they know about it. 17 00:00:47,500 --> 00:00:50,930 And then somebody asked me by email what that was, 18 00:00:50,930 --> 00:00:52,440 and I said, oh, yeah. 19 00:00:52,440 --> 00:00:54,240 I did not review that in class. 20 00:00:54,240 --> 00:01:14,750 So review the equation in r3 form that's x, y, z 21 00:01:14,750 --> 00:01:35,520 of the sphere of radius r and center p of coordinates x0, y0, 22 00:01:35,520 --> 00:01:37,580 z0. 23 00:01:37,580 --> 00:01:43,460 One of you asked me by email, does-- of course you do, 24 00:01:43,460 --> 00:01:48,010 and then if you know it, can you help me-- can you 25 00:01:48,010 --> 00:01:50,475 help remind what that was? 26 00:01:50,475 --> 00:01:54,840 27 00:01:54,840 --> 00:01:56,830 STUDENT: x minus x0-- 28 00:01:56,830 --> 00:02:06,990 PROFESSOR: x minus x0 squared plus y minus y0 squared 29 00:02:06,990 --> 00:02:15,060 plus z minus z0 squared equals R squared. 30 00:02:15,060 --> 00:02:16,282 OK? 31 00:02:16,282 --> 00:02:18,230 When you ask, for example, what is 32 00:02:18,230 --> 00:02:22,922 the equation of a units sphere, what do I mean by unit sphere? 33 00:02:22,922 --> 00:02:23,630 STUDENT: Radius-- 34 00:02:23,630 --> 00:02:28,330 PROFESSOR: Radius 1, and center 0, standard unit sphere, 35 00:02:28,330 --> 00:02:30,080 will be. 36 00:02:30,080 --> 00:02:35,100 There is a notation for that in mathematics called s2. 37 00:02:35,100 --> 00:02:36,980 I'll tell you why its called s2. 38 00:02:36,980 --> 00:02:41,310 x squared plus y squared plus z squared equals 1. 39 00:02:41,310 --> 00:02:44,340 40 00:02:44,340 --> 00:02:48,030 s2 stands for the dimension. 41 00:02:48,030 --> 00:02:53,000 That means the number of the-- the number 42 00:02:53,000 --> 00:02:54,545 of degrees of freedom. 43 00:02:54,545 --> 00:03:00,170 44 00:03:00,170 --> 00:03:04,320 you have on a certain manifold. 45 00:03:04,320 --> 00:03:07,630 46 00:03:07,630 --> 00:03:08,900 What is a manifold? 47 00:03:08,900 --> 00:03:10,410 It's a geometric structure. 48 00:03:10,410 --> 00:03:12,800 I'm not going to go into details. 49 00:03:12,800 --> 00:03:15,640 It's a geometric structure with some special properties. 50 00:03:15,640 --> 00:03:18,700 51 00:03:18,700 --> 00:03:21,150 I'm not talking about other fields of algebra, 52 00:03:21,150 --> 00:03:22,410 anthropology. 53 00:03:22,410 --> 00:03:26,505 I'm just talking about geometry and calculus math 3, which 54 00:03:26,505 --> 00:03:28,930 is multivariable calculus. 55 00:03:28,930 --> 00:03:35,200 Now, how do I think of degrees of freedom? 56 00:03:35,200 --> 00:03:37,450 Look at the table. 57 00:03:37,450 --> 00:03:42,910 What freedom do I have to move along one of these sticks? 58 00:03:42,910 --> 00:03:45,440 I have one degree of freedom in the sense 59 00:03:45,440 --> 00:03:50,000 that it's given by a parameter like time. 60 00:03:50,000 --> 00:03:50,670 Right? 61 00:03:50,670 --> 00:03:53,020 It's a 1-parameter manifold in the sense 62 00:03:53,020 --> 00:03:55,340 that maybe I have a line, maybe I 63 00:03:55,340 --> 00:04:00,940 have the trajectory of the parking space in terms of time. 64 00:04:00,940 --> 00:04:05,110 The freedom that the bag has is to move according to time, 65 00:04:05,110 --> 00:04:09,680 and that's considered only one degree of freedom. 66 00:04:09,680 --> 00:04:13,620 Now if you were on a plane or another surface, 67 00:04:13,620 --> 00:04:18,010 why would you have more than one degrees of freedom? 68 00:04:18,010 --> 00:04:23,690 Well, I can move towards you, or I can move this way. 69 00:04:23,690 --> 00:04:26,720 I can draw a grid the way the x and y coordinate. 70 00:04:26,720 --> 00:04:29,980 And those are my degrees of freedom. 71 00:04:29,980 --> 00:04:32,960 Practically, the basis IJ gives me that kind 72 00:04:32,960 --> 00:04:35,840 of two degrees of freedom. 73 00:04:35,840 --> 00:04:36,340 Right? 74 00:04:36,340 --> 00:04:41,670 If I'm in three coordinates, I have without other constraints, 75 00:04:41,670 --> 00:04:43,380 because I could be in three coordinates 76 00:04:43,380 --> 00:04:46,740 and constrained to be on a cylinder, in which case 77 00:04:46,740 --> 00:04:48,860 I still have two degrees of freedom. 78 00:04:48,860 --> 00:04:53,440 But if I am a bug who is free to fly, 79 00:04:53,440 --> 00:04:59,820 I have the freedom to go with three degrees of freedom, 80 00:04:59,820 --> 00:05:00,830 right? 81 00:05:00,830 --> 00:05:03,700 I have three degrees of freedom, but if the bug 82 00:05:03,700 --> 00:05:09,080 is moving-- not flying, moving on a surface, 83 00:05:09,080 --> 00:05:11,560 then he has two degrees of freedom. 84 00:05:11,560 --> 00:05:16,635 So to again review, lines and curves in general 85 00:05:16,635 --> 00:05:20,190 are one dimensional things, because you 86 00:05:20,190 --> 00:05:21,460 have one degree of freedom. 87 00:05:21,460 --> 00:05:24,170 Two dimensional things are surfaces, 88 00:05:24,170 --> 00:05:28,013 three dimensional things are spaces, like the Euclidean 89 00:05:28,013 --> 00:05:31,800 space, and we are not going to go beyond, 90 00:05:31,800 --> 00:05:33,660 at least for the time being, we are not 91 00:05:33,660 --> 00:05:36,230 going to go beyond that. 92 00:05:36,230 --> 00:05:41,350 However, where anybody is interested in relativity, 93 00:05:41,350 --> 00:05:47,300 say or let's say four dimensional spaces, or things 94 00:05:47,300 --> 00:05:52,200 of x, y, z spatial coordinates and t as a fourth coordinate, 95 00:05:52,200 --> 00:05:56,651 then we can go into higher dimensions, as well. 96 00:05:56,651 --> 00:05:57,151 OK. 97 00:05:57,151 --> 00:06:01,079 98 00:06:01,079 --> 00:06:02,825 I want to ask you a question. 99 00:06:02,825 --> 00:06:07,550 If somebody gives you on WeBWorK or outside of WeBWorK, 100 00:06:07,550 --> 00:06:11,490 on the first quiz or on the final exam, 101 00:06:11,490 --> 00:06:18,520 let's say you have this equation, 102 00:06:18,520 --> 00:06:28,530 x squared plus y squared plus z squared plus 2x plus 2y 103 00:06:28,530 --> 00:06:32,470 equals 9. 104 00:06:32,470 --> 00:06:37,190 What is this identified as? 105 00:06:37,190 --> 00:06:38,350 It's a quadric. 106 00:06:38,350 --> 00:06:40,930 Why would this be a quadric? 107 00:06:40,930 --> 00:06:43,290 Well, there is no x, y, y, z. 108 00:06:43,290 --> 00:06:45,780 Those terms are missing. 109 00:06:45,780 --> 00:06:54,560 But I have something of the type of quadric x squared 110 00:06:54,560 --> 00:06:59,510 plus By squared plus c squared plus dxy 111 00:06:59,510 --> 00:07:07,810 plus exz plus fyz plus, those are, oh my God, so many. 112 00:07:07,810 --> 00:07:09,010 Degree two. 113 00:07:09,010 --> 00:07:16,300 Degree one I would have ax plus by plus cz 114 00:07:16,300 --> 00:07:22,210 plus a little d constant, and whew, that was a long one. 115 00:07:22,210 --> 00:07:23,050 Right? 116 00:07:23,050 --> 00:07:27,470 Now, is this of the type of a project? 117 00:07:27,470 --> 00:07:28,200 Yes, it is. 118 00:07:28,200 --> 00:07:31,840 Of course there are some terms that are missing, good for us. 119 00:07:31,840 --> 00:07:35,580 How are you going to try to identify the type of quadric 120 00:07:35,580 --> 00:07:37,570 by looking at this? 121 00:07:37,570 --> 00:07:41,410 As you said very well, I think it's-- you say, 122 00:07:41,410 --> 00:07:45,740 I think of a sphere, maybe I can complete the squares, you said. 123 00:07:45,740 --> 00:07:49,240 How do we complete the squares? 124 00:07:49,240 --> 00:07:52,900 x squared plus 2x plus some missing number, 125 00:07:52,900 --> 00:07:57,010 a magic number-- yes sir? 126 00:07:57,010 --> 00:08:02,050 STUDENT: So, basically I'll have to take x plus 2 times 4 127 00:08:02,050 --> 00:08:04,450 will go outside. 128 00:08:04,450 --> 00:08:07,840 It's like x min-- x plus 2-- 129 00:08:07,840 --> 00:08:09,233 PROFESSOR: Why x plus 2? 130 00:08:09,233 --> 00:08:10,316 STUDENT: Because it's 2x-- 131 00:08:10,316 --> 00:08:12,140 STUDENT: It's 2x. 132 00:08:12,140 --> 00:08:14,750 PROFESSOR: But if I take x plus 2, 133 00:08:14,750 --> 00:08:18,320 then that's going to give me x squared plus 4x plus 4, 134 00:08:18,320 --> 00:08:19,550 so it's not a good idea. 135 00:08:19,550 --> 00:08:20,880 STUDENT: On the x plus 1 136 00:08:20,880 --> 00:08:21,890 PROFESSOR: x plus 1. 137 00:08:21,890 --> 00:08:25,700 So I'm going to complete x plus 1 squared. 138 00:08:25,700 --> 00:08:29,424 What did I invent that wasn't there? 139 00:08:29,424 --> 00:08:30,195 STUDENT: 1. 140 00:08:30,195 --> 00:08:31,820 PROFESSOR: I invented the 1, and I have 141 00:08:31,820 --> 00:08:34,179 to compensate for my invention. 142 00:08:34,179 --> 00:08:38,080 I added the 1, created the 1 out of nothing, 143 00:08:38,080 --> 00:08:41,150 so I have to compensate by subtracting it. 144 00:08:41,150 --> 00:08:42,400 How much is from here to here? 145 00:08:42,400 --> 00:08:45,560 Is it exactly the thing that I underlined 146 00:08:45,560 --> 00:08:49,380 with a wiggly line, a light wiggly line thing, 147 00:08:49,380 --> 00:08:52,980 plus what is the blue wiggly line, 148 00:08:52,980 --> 00:09:01,730 the blue wiggly line that doesn't show-- 149 00:09:01,730 --> 00:09:06,370 I have y plus 1 squared, and again, I have to compensate 150 00:09:06,370 --> 00:09:08,470 for what I invented. 151 00:09:08,470 --> 00:09:13,350 I created a 1 out of nothing, so this is y squared plus 2y. 152 00:09:13,350 --> 00:09:17,590 153 00:09:17,590 --> 00:09:21,858 The z squared is all by himself, and he's crying, I'm so lonely, 154 00:09:21,858 --> 00:09:26,170 I don't know, there is nobody like me over there. 155 00:09:26,170 --> 00:09:29,620 So in the end, I can rewrite the whole thing as x plus 1 156 00:09:29,620 --> 00:09:33,640 squared plus y plus 1 squared plus z squared, if I 157 00:09:33,640 --> 00:09:39,141 want to work them out in this format, equals what? 158 00:09:39,141 --> 00:09:39,640 STUDENT: 10. 159 00:09:39,640 --> 00:09:40,990 160 00:09:40,990 --> 00:09:41,890 PROFESSOR: 11. 161 00:09:41,890 --> 00:09:47,380 11 is the square root of 11 squared. 162 00:09:47,380 --> 00:09:49,310 Like my son said the other day. 163 00:09:49,310 --> 00:09:52,560 So that the radius would be square foot 164 00:09:52,560 --> 00:09:57,310 11 of a sphere of what circle? 165 00:09:57,310 --> 00:10:00,130 What is the-- or the sphere of what center? 166 00:10:00,130 --> 00:10:00,880 STUDENT: Minus 1-- 167 00:10:00,880 --> 00:10:03,960 PROFESSOR: Minus 1, minus 1, and 0. 168 00:10:03,960 --> 00:10:06,079 So I don't want to insult you. 169 00:10:06,079 --> 00:10:07,870 Of course you know how to complete squares. 170 00:10:07,870 --> 00:10:10,630 171 00:10:10,630 --> 00:10:15,690 However, I have discovered in an upper level class at some point 172 00:10:15,690 --> 00:10:19,460 that my students didn't know how to complete squares, which 173 00:10:19,460 --> 00:10:22,480 was very, very heartbreaking. 174 00:10:22,480 --> 00:10:24,748 All right, now. 175 00:10:24,748 --> 00:10:28,440 176 00:10:28,440 --> 00:10:39,430 Any questions regarding-- while I have a few of yours, 177 00:10:39,430 --> 00:10:41,350 I'm going to wait a little bit longer 178 00:10:41,350 --> 00:10:43,650 until I give everybody the chance 179 00:10:43,650 --> 00:10:47,940 to complete the extra credit. 180 00:10:47,940 --> 00:10:52,770 I have the question by email saying, 181 00:10:52,770 --> 00:10:56,900 you mentioned that genius guy in your class. 182 00:10:56,900 --> 00:11:00,430 This is a 1-sheeted hyperboloid. 183 00:11:00,430 --> 00:11:04,250 x squared plus y squared minus z squared minus 1 equals 0. 184 00:11:04,250 --> 00:11:08,190 The question was, by email, how in the world, 185 00:11:08,190 --> 00:11:19,590 did he figure out what the two families of generatrices are? 186 00:11:19,590 --> 00:11:22,040 So you have one family and another family, 187 00:11:22,040 --> 00:11:27,509 and both together generate the 1-sheeted hyperboloid. 188 00:11:27,509 --> 00:11:30,200 189 00:11:30,200 --> 00:11:33,070 Let me give you a little bit more of a hint, 190 00:11:33,070 --> 00:11:35,820 but I'm still going to stop. 191 00:11:35,820 --> 00:11:41,870 So last time I said, he noticed you can root together 192 00:11:41,870 --> 00:11:46,970 the y squared minus 1 and the x squared minus z squared, 193 00:11:46,970 --> 00:11:48,352 and you can separate them. 194 00:11:48,352 --> 00:11:52,850 So you're going to have x squared minus z squared equals 195 00:11:52,850 --> 00:11:53,800 1 minus y squared. 196 00:11:53,800 --> 00:11:56,420 197 00:11:56,420 --> 00:12:00,350 You can't hide the difference of two squares 198 00:12:00,350 --> 00:12:03,990 as product of sum and difference. 199 00:12:03,990 --> 00:12:13,390 x plus z times x minus z equals 1 plus y times 1 minus y. 200 00:12:13,390 --> 00:12:19,520 So how can you eventually arrange stuff 201 00:12:19,520 --> 00:12:25,350 to be giving due to the lines that 202 00:12:25,350 --> 00:12:28,810 are sitting on the surface? 203 00:12:28,810 --> 00:12:30,910 The lines that are sitting on the surface 204 00:12:30,910 --> 00:12:33,710 are infinitely many, and I would like 205 00:12:33,710 --> 00:12:39,320 at least a 1-parameter family of such lines. 206 00:12:39,320 --> 00:12:40,830 You can have choices. 207 00:12:40,830 --> 00:12:43,230 One of the choices would be-- this 208 00:12:43,230 --> 00:12:46,180 is a product, of two numbers, right? 209 00:12:46,180 --> 00:12:51,320 So you can write it as an equality of two fractions. 210 00:12:51,320 --> 00:12:56,090 So you would have something like x plus z on top, x minus 211 00:12:56,090 --> 00:12:57,420 z below. 212 00:12:57,420 --> 00:13:01,570 Observe that you are creating singularities here. 213 00:13:01,570 --> 00:13:11,690 So you have to take x minus z case equals 0 separately, 214 00:13:11,690 --> 00:13:16,540 and then you have, let's say you have 1 minus y here, 215 00:13:16,540 --> 00:13:21,120 and 1 plus y here. 216 00:13:21,120 --> 00:13:26,870 What else do you have to impose when you impose x minus z 217 00:13:26,870 --> 00:13:27,940 equals 0. 218 00:13:27,940 --> 00:13:29,750 You cannot have 7 over 0. 219 00:13:29,750 --> 00:13:31,500 That is undefined. 220 00:13:31,500 --> 00:13:34,630 but if you have 0 over 0, that's still possible. 221 00:13:34,630 --> 00:13:38,430 So whenever you take x minus z equals 0 separately, 222 00:13:38,430 --> 00:13:42,600 that will imply that the numerator corresponding to it 223 00:13:42,600 --> 00:13:44,840 will also have to be 0. 224 00:13:44,840 --> 00:13:47,430 And together these guys are friends. 225 00:13:47,430 --> 00:13:49,420 What are they? 226 00:13:49,420 --> 00:13:49,920 2-- 227 00:13:49,920 --> 00:13:51,211 STUDENT: A system of equations. 228 00:13:51,211 --> 00:13:53,130 PROFESSOR: It's a system of equations. 229 00:13:53,130 --> 00:13:57,840 They both represent planes, and the intersection of two planes 230 00:13:57,840 --> 00:14:00,530 is a line. 231 00:14:00,530 --> 00:14:07,790 It's a particular line, which is part of the family-- which 232 00:14:07,790 --> 00:14:10,340 is part of a family. 233 00:14:10,340 --> 00:14:13,320 234 00:14:13,320 --> 00:14:14,100 OK. 235 00:14:14,100 --> 00:14:21,100 Now, on the other hand, in case you have 1 plus y equals 0-- 236 00:14:21,100 --> 00:14:25,400 so if it happens that you have this extreme case 237 00:14:25,400 --> 00:14:28,590 that the denominator will be 0, you absolutely 238 00:14:28,590 --> 00:14:34,830 have to impose x plus z to be 0, and then you have another life. 239 00:14:34,830 --> 00:14:36,605 It's not easy for me to draw those, 240 00:14:36,605 --> 00:14:39,390 but I could if you asked me privately 241 00:14:39,390 --> 00:14:43,050 to draw those and show you what the lines look like. 242 00:14:43,050 --> 00:14:43,660 OK? 243 00:14:43,660 --> 00:14:46,180 All right. 244 00:14:46,180 --> 00:14:51,650 So you have two special lines that are part of that picture. 245 00:14:51,650 --> 00:14:55,530 They are embedded in the surface. 246 00:14:55,530 --> 00:15:00,420 How do you find a family of planes? 247 00:15:00,420 --> 00:15:03,160 Oh my god, I only had one choice, 248 00:15:03,160 --> 00:15:06,076 but I could have yet another choice 249 00:15:06,076 --> 00:15:08,320 of how to pick the parameters. 250 00:15:08,320 --> 00:15:11,540 Let's take lambda to be a real number parameter. 251 00:15:11,540 --> 00:15:15,743 252 00:15:15,743 --> 00:15:20,220 And lambda could be anything-- if lambda is 0, 253 00:15:20,220 --> 00:15:22,010 what have I got to have, guys? 254 00:15:22,010 --> 00:15:22,800 STUDENT: The top. 255 00:15:22,800 --> 00:15:24,530 PROFESSOR: The top guys will be 0, 256 00:15:24,530 --> 00:15:29,140 and I still have 1 minus y equals 0, a plane, intersected 257 00:15:29,140 --> 00:15:33,920 with x plus z equals 0, another plane, so still a line. 258 00:15:33,920 --> 00:15:37,300 So lambda equals 0 will give me yet another line, which 259 00:15:37,300 --> 00:15:39,080 is not written big. 260 00:15:39,080 --> 00:15:41,104 Are you guys with me? 261 00:15:41,104 --> 00:15:42,490 Could lambda ever go to infinity? 262 00:15:42,490 --> 00:15:45,610 263 00:15:45,610 --> 00:15:49,440 Lambda wants to go to infinity, and when does lambda 264 00:15:49,440 --> 00:15:50,410 go to infinity? 265 00:15:50,410 --> 00:15:52,118 STUDENT: When the bottoms would equal 0-- 266 00:15:52,118 --> 00:15:55,472 PROFESSOR: When both the bottoms would be 0. 267 00:15:55,472 --> 00:16:01,740 268 00:16:01,740 --> 00:16:06,440 So this is-- I can call it L infinity, the line of infinity. 269 00:16:06,440 --> 00:16:07,080 You see? 270 00:16:07,080 --> 00:16:09,570 But still those would be two planes. 271 00:16:09,570 --> 00:16:11,800 There's an intersection, it's a line. 272 00:16:11,800 --> 00:16:12,480 OK. 273 00:16:12,480 --> 00:16:19,950 Can we write this family-- just one family of lines? 274 00:16:19,950 --> 00:16:24,080 A line is always an intersection of two planes, right? 275 00:16:24,080 --> 00:16:28,150 So which are the planes that I'm talking about? 276 00:16:28,150 --> 00:16:32,600 x plus z equals lambda times 1 plus y. 277 00:16:32,600 --> 00:16:36,540 This is not in the book, because, oh my God, this is 278 00:16:36,540 --> 00:16:38,650 too hard for the book, right? 279 00:16:38,650 --> 00:16:43,340 But it's a nice example to look at in an honors class. 280 00:16:43,340 --> 00:16:47,472 1 minus y equals lambda times x minus z. 281 00:16:47,472 --> 00:16:48,680 It's not in the book. 282 00:16:48,680 --> 00:16:55,300 It's not in any book that I know of at the level of calculus. 283 00:16:55,300 --> 00:16:57,080 All right, OK. 284 00:16:57,080 --> 00:16:58,590 What are these animals? 285 00:16:58,590 --> 00:17:00,130 The first animal is a plane. 286 00:17:00,130 --> 00:17:02,530 The second animal is a plane. 287 00:17:02,530 --> 00:17:05,040 How many planes are in the picture? 288 00:17:05,040 --> 00:17:11,630 For each lambda, you have a-- for each lambda value in R, 289 00:17:11,630 --> 00:17:15,670 you have a couple of planes that intersect along your line. 290 00:17:15,670 --> 00:17:18,454 This is the line L lambda. 291 00:17:18,454 --> 00:17:21,560 And shut up, Magdalena, you told people too much. 292 00:17:21,560 --> 00:17:25,220 If you still want them to do this for 2 extra credit points, 293 00:17:25,220 --> 00:17:28,210 give them the chance to finish the exercise. 294 00:17:28,210 --> 00:17:32,010 So I zip my lips, but only after I ask you, 295 00:17:32,010 --> 00:17:33,660 how do you think you are going to get 296 00:17:33,660 --> 00:17:37,980 the other family of rulers? 297 00:17:37,980 --> 00:17:42,210 The ruling guys are two families, you see? 298 00:17:42,210 --> 00:17:47,235 So this family is going in one direction. 299 00:17:47,235 --> 00:17:48,693 How am I going to get two families? 300 00:17:48,693 --> 00:17:51,556 301 00:17:51,556 --> 00:17:55,030 I have another choice that-- how did I take this? 302 00:17:55,030 --> 00:17:57,880 More or less, I made my choice. 303 00:17:57,880 --> 00:18:02,270 Just like having two people that would 304 00:18:02,270 --> 00:18:04,920 be prospective job candidates. 305 00:18:04,920 --> 00:18:07,627 You pick one of them. 306 00:18:07,627 --> 00:18:10,235 STUDENT: Now, we can put 1 minus y in the denominator. 307 00:18:10,235 --> 00:18:12,520 The denominator in place of 1 plus y. 308 00:18:12,520 --> 00:18:16,680 PROFESSOR: So I could have done-- I could have taken this, 309 00:18:16,680 --> 00:18:19,580 and put 1 plus y here, and 1 minus y here. 310 00:18:19,580 --> 00:18:21,460 I'm going to let you do the rest, 311 00:18:21,460 --> 00:18:24,890 and get the second family of generators 312 00:18:24,890 --> 00:18:28,460 for the whole surface. 313 00:18:28,460 --> 00:18:29,170 That's enough. 314 00:18:29,170 --> 00:18:31,180 You're not missing your credit. 315 00:18:31,180 --> 00:18:35,760 Just, you wanted help, and I helped you. 316 00:18:35,760 --> 00:18:39,930 And I'm not mad whatsoever when you ask me things. 317 00:18:39,930 --> 00:18:44,050 The email I got sounded like-- says, this is not in the book, 318 00:18:44,050 --> 00:18:47,170 or in any book, or on the internet. 319 00:18:47,170 --> 00:18:48,790 How shall I approach this? 320 00:18:48,790 --> 00:18:51,170 How shall I start thinking about this problem? 321 00:18:51,170 --> 00:18:54,330 This is a completely legitimate question. 322 00:18:54,330 --> 00:18:58,280 How do I start on this problem? 323 00:18:58,280 --> 00:18:59,204 OK. 324 00:18:59,204 --> 00:19:02,240 On the homework-- maybe it's too easy-- you have two or three 325 00:19:02,240 --> 00:19:04,830 examples involving spheres. 326 00:19:04,830 --> 00:19:06,870 Those will be too easy for you. 327 00:19:06,870 --> 00:19:10,865 I only gave you a very thin among of homework this time. 328 00:19:10,865 --> 00:19:15,176 You Have plenty of time until Monday at 1:30 or something PM. 329 00:19:15,176 --> 00:19:18,790 330 00:19:18,790 --> 00:19:21,280 I would like to draw a little bit more, 331 00:19:21,280 --> 00:19:25,020 because in this homework and the next homework, 332 00:19:25,020 --> 00:19:33,160 I'm building something special called the Frenet Trihedron. 333 00:19:33,160 --> 00:19:36,580 And I told you a little bit about this Frenet Trihedron, 334 00:19:36,580 --> 00:19:39,700 but I didn't tell you much. 335 00:19:39,700 --> 00:19:42,600 336 00:19:42,600 --> 00:19:45,910 Many textbooks in multivariable calculus 337 00:19:45,910 --> 00:19:48,810 don't say much about it, which I think is a shame. 338 00:19:48,810 --> 00:19:52,670 339 00:19:52,670 --> 00:19:57,000 You have a position vector that gives you 340 00:19:57,000 --> 00:19:59,251 the equation of a regular curve. 341 00:19:59,251 --> 00:20:05,530 342 00:20:05,530 --> 00:20:07,930 x of t, y of t, z of t. 343 00:20:07,930 --> 00:20:09,830 Again, what was a regular curve? 344 00:20:09,830 --> 00:20:13,870 I'm just doing review of what we did last time. 345 00:20:13,870 --> 00:20:17,240 A very nice curve that is differentiable 346 00:20:17,240 --> 00:20:21,840 and whose derivative is continuous everywhere 347 00:20:21,840 --> 00:20:23,030 on the interval. 348 00:20:23,030 --> 00:20:30,920 But moreover, the r prime of t never becomes 0. 349 00:20:30,920 --> 00:20:35,520 So continuously differentiable, and r prime of t 350 00:20:35,520 --> 00:20:40,980 never becomes 0 for any-- do you know this name, 351 00:20:40,980 --> 00:20:43,201 any for every or for any? 352 00:20:43,201 --> 00:20:43,700 OK. 353 00:20:43,700 --> 00:20:46,970 This is the symbolistics of mathematics. 354 00:20:46,970 --> 00:20:50,090 You know because you are as nerdy as me. 355 00:20:50,090 --> 00:20:51,970 But everybody else doesn't. 356 00:20:51,970 --> 00:20:53,650 You guys will learn. 357 00:20:53,650 --> 00:20:55,710 This is what mathematicians like. 358 00:20:55,710 --> 00:21:00,260 You see, mathematicians hate writing lots of words down. 359 00:21:00,260 --> 00:21:05,010 If we liked writing essays and lots of blah, blah, blah, 360 00:21:05,010 --> 00:21:06,850 we would do something else. 361 00:21:06,850 --> 00:21:08,760 We wouldn't do mathematics. 362 00:21:08,760 --> 00:21:11,490 We would do debates, we would do politics, 363 00:21:11,490 --> 00:21:14,400 we would do other things. 364 00:21:14,400 --> 00:21:16,880 Mathematicians like ideas, but when 365 00:21:16,880 --> 00:21:19,420 it comes to writing them down, they 366 00:21:19,420 --> 00:21:23,450 want to right them down in the most compact way possible. 367 00:21:23,450 --> 00:21:26,485 That's why they created sort of their own language, 368 00:21:26,485 --> 00:21:30,730 and they have all sorts of logical quantifiers. 369 00:21:30,730 --> 00:21:34,060 And it's like your secret language 370 00:21:34,060 --> 00:21:37,310 when it comes to your less nerdy friends. 371 00:21:37,310 --> 00:21:45,044 So you go for every-- for any or for every-- 372 00:21:45,044 --> 00:21:45,960 do you know this sign? 373 00:21:45,960 --> 00:21:48,860 374 00:21:48,860 --> 00:21:49,535 There exists. 375 00:21:49,535 --> 00:21:54,690 376 00:21:54,690 --> 00:21:57,080 And do you know this thing? 377 00:21:57,080 --> 00:22:01,025 Because one of the-- huh? 378 00:22:01,025 --> 00:22:02,660 STUDENT: Is that factorial? 379 00:22:02,660 --> 00:22:05,450 PROFESSOR: Factorial, but in logic, 380 00:22:05,450 --> 00:22:09,020 that means there exists a unique-- a unique. 381 00:22:09,020 --> 00:22:11,320 So there exists a unique. 382 00:22:11,320 --> 00:22:14,720 There exists a unique number. 383 00:22:14,720 --> 00:22:18,780 There is a unique number. 384 00:22:18,780 --> 00:22:20,870 So we have our own language. 385 00:22:20,870 --> 00:22:23,560 Of course, empty set, everybody knows that. 386 00:22:23,560 --> 00:22:28,160 And it's used in mathematical logic a lot. 387 00:22:28,160 --> 00:22:33,340 You know most of the symbols from unit intersection, 388 00:22:33,340 --> 00:22:35,480 or, and. 389 00:22:35,480 --> 00:22:38,570 I'm going to use some of those as well. 390 00:22:38,570 --> 00:22:40,480 Coming back to the Frenet Trihedron, 391 00:22:40,480 --> 00:22:44,065 we have that velocity vector at every point. 392 00:22:44,065 --> 00:22:44,940 We are happy with it. 393 00:22:44,940 --> 00:22:49,060 We have our prime of t that is referred from 0. 394 00:22:49,060 --> 00:22:51,080 I said I want to make it uniform, 395 00:22:51,080 --> 00:22:53,810 and then I divided by the magnitude, 396 00:22:53,810 --> 00:22:57,860 and I have this wonderful t vector we just talked about. 397 00:22:57,860 --> 00:23:03,580 Mr. t is r prime over the magnitude of r prime, which 398 00:23:03,580 --> 00:23:06,720 is called it's peak right? 399 00:23:06,720 --> 00:23:09,860 We divide by its peak. 400 00:23:09,860 --> 00:23:12,612 What's the name of t, again? 401 00:23:12,612 --> 00:23:13,570 STUDENT: Tangent unit-- 402 00:23:13,570 --> 00:23:16,310 PROFESSOR: Tangent unit vector, very good. 403 00:23:16,310 --> 00:23:19,690 How did you remember that so quickly? 404 00:23:19,690 --> 00:23:22,300 Tangent unit vector. 405 00:23:22,300 --> 00:23:28,070 There is also another guy who is famous. 406 00:23:28,070 --> 00:23:32,755 I wanted to make him green, but let's see 407 00:23:32,755 --> 00:23:35,170 if I can make him blue. 408 00:23:35,170 --> 00:23:42,380 t is defined-- should I write the f on top of here? 409 00:23:42,380 --> 00:23:43,502 Do you know what that is? 410 00:23:43,502 --> 00:23:45,630 STUDENT: I thought n was the normal vector. 411 00:23:45,630 --> 00:23:48,137 PROFESSOR: t prime divided by the length of-- 412 00:23:48,137 --> 00:23:48,720 STUDENT: Wait. 413 00:23:48,720 --> 00:23:53,340 I thought the vector n was the normal. 414 00:23:53,340 --> 00:23:56,450 PROFESSOR: n-- there are many normals. 415 00:23:56,450 --> 00:24:01,440 It's a very good thing, because we don't say that in the book. 416 00:24:01,440 --> 00:24:04,970 OK, this is the t along my r. 417 00:24:04,970 --> 00:24:09,110 Now when I go through a point, this is the normal plane, 418 00:24:09,110 --> 00:24:09,980 right? 419 00:24:09,980 --> 00:24:14,540 There are many normals to the surface-- to the curve. 420 00:24:14,540 --> 00:24:15,760 Which one am I taking? 421 00:24:15,760 --> 00:24:19,670 All of them are perpendicular to the direction, right? 422 00:24:19,670 --> 00:24:20,269 STUDENT: tf. 423 00:24:20,269 --> 00:24:22,060 PROFESSOR: So I take this one, or this one, 424 00:24:22,060 --> 00:24:25,460 or this one, or this one, or this one, or this one, there. 425 00:24:25,460 --> 00:24:26,830 I have to make up my mind. 426 00:24:26,830 --> 00:24:31,390 And that's how people came up with the so-called principal 427 00:24:31,390 --> 00:24:33,450 unit normal. 428 00:24:33,450 --> 00:24:36,080 And this is the one I'm talking about. 429 00:24:36,080 --> 00:24:39,020 And you are right, it is normal. 430 00:24:39,020 --> 00:24:42,334 Principal unit normal. 431 00:24:42,334 --> 00:24:45,290 Remember this very well for your exam, 432 00:24:45,290 --> 00:24:48,270 because it's a very important notion. 433 00:24:48,270 --> 00:24:50,100 How do I get to that? 434 00:24:50,100 --> 00:24:54,040 I take t, I differentiate it, and I divide 435 00:24:54,040 --> 00:24:59,020 by the lengths of t prime. 436 00:24:59,020 --> 00:25:07,220 Now, can you prove to me that indeed this fellow 437 00:25:07,220 --> 00:25:09,710 is perpendicular to t? 438 00:25:09,710 --> 00:25:12,380 Can you do that? 439 00:25:12,380 --> 00:25:14,452 STUDENT: That n is perpendicular to t? 440 00:25:14,452 --> 00:25:16,470 PROFESSOR: Mm-hmm. 441 00:25:16,470 --> 00:25:17,694 So a little exercise. 442 00:25:17,694 --> 00:25:22,634 443 00:25:22,634 --> 00:25:30,590 Prove that-- Prove that I don't have a good marker anymore. 444 00:25:30,590 --> 00:25:37,510 Prove that n, the unit principal vector field, 445 00:25:37,510 --> 00:25:44,840 is perpendicular-- you see, I'm a mathematician. 446 00:25:44,840 --> 00:25:48,690 I swear, I hate to write down the whole word perpendicular. 447 00:25:48,690 --> 00:25:51,610 I would love to say, perpendicular. 448 00:25:51,610 --> 00:25:57,840 That's how I write perpendicular really fast-- to t fore 449 00:25:57,840 --> 00:26:01,470 every value of t. 450 00:26:01,470 --> 00:26:03,010 For every value of t. 451 00:26:03,010 --> 00:26:03,910 OK. 452 00:26:03,910 --> 00:26:06,340 How in the world can I do that? 453 00:26:06,340 --> 00:26:08,560 I have to think about it. 454 00:26:08,560 --> 00:26:11,820 This is hard. 455 00:26:11,820 --> 00:26:12,980 Wish me luck. 456 00:26:12,980 --> 00:26:15,800 So do I know anything about Mr. t? 457 00:26:15,800 --> 00:26:18,150 What do I know about Mr. t? 458 00:26:18,150 --> 00:26:20,390 I'll take it and I'll differentiate it later. 459 00:26:20,390 --> 00:26:25,050 It Mr. t is magic in the sense that he's a unit vector. 460 00:26:25,050 --> 00:26:27,860 I'm going to write that down. 461 00:26:27,860 --> 00:26:31,910 t in absolute value equals 1. 462 00:26:31,910 --> 00:26:33,120 It's beautiful. 463 00:26:33,120 --> 00:26:36,920 If I squared that-- and you're going to say, 464 00:26:36,920 --> 00:26:38,600 why would you want to square that? 465 00:26:38,600 --> 00:26:40,500 You're going to see in a minute. 466 00:26:40,500 --> 00:26:43,140 If I squared that, then I'm going 467 00:26:43,140 --> 00:26:50,580 to have the dot product between t and itself equals 1. 468 00:26:50,580 --> 00:26:53,130 469 00:26:53,130 --> 00:26:56,740 Can somebody tell me why the dot product between t and itself 470 00:26:56,740 --> 00:27:00,850 is the square of a length of t? 471 00:27:00,850 --> 00:27:04,930 What's the definition of the dot product? 472 00:27:04,930 --> 00:27:08,290 Magnitude of the first vector, times the magnitude 473 00:27:08,290 --> 00:27:11,340 of the second vector-- there i am already-- 474 00:27:11,340 --> 00:27:15,410 times the cosine of the angle between the two vectors 475 00:27:15,410 --> 00:27:17,480 Duh, that's 0. 476 00:27:17,480 --> 00:27:20,496 So cosine of 0 is 1, I'm done. 477 00:27:20,496 --> 00:27:21,390 Right? 478 00:27:21,390 --> 00:27:26,580 Now, I have a vector function times a vector function-- 479 00:27:26,580 --> 00:27:31,270 this is crazy, right-- equals 1. 480 00:27:31,270 --> 00:27:34,250 I'm going to go ahead and differentiate. 481 00:27:34,250 --> 00:27:37,690 Keep in mind that this is a product. 482 00:27:37,690 --> 00:27:39,770 What's the product? 483 00:27:39,770 --> 00:27:42,330 One of my professors, colleagues, 484 00:27:42,330 --> 00:27:44,980 was telling me, now, let's be serious. 485 00:27:44,980 --> 00:27:49,370 In five years, how many of your engineering majors 486 00:27:49,370 --> 00:27:51,250 will remember the product? 487 00:27:51,250 --> 00:27:53,180 I really was thinking about this. 488 00:27:53,180 --> 00:27:56,680 I hope everybody, if they were my students, 489 00:27:56,680 --> 00:27:59,180 because we are going to have enough practice. 490 00:27:59,180 --> 00:28:01,780 So the prime rule in Calc 1 said that if you 491 00:28:01,780 --> 00:28:05,125 have f of t times g of t, you have a product. 492 00:28:05,125 --> 00:28:08,320 You prime that product, and never write 493 00:28:08,320 --> 00:28:12,530 f prime times g prime unless you want me to call you around 2 AM 494 00:28:12,530 --> 00:28:15,255 to say you should never do that. 495 00:28:15,255 --> 00:28:20,100 496 00:28:20,100 --> 00:28:23,760 So how does the product rule work? 497 00:28:23,760 --> 00:28:27,500 The first one prime times the second unprime 498 00:28:27,500 --> 00:28:32,310 plus the first one unprime times the second prime. 499 00:28:32,310 --> 00:28:34,505 My students know the product rule. 500 00:28:34,505 --> 00:28:37,480 I don't care if the rest of the world doesn't. 501 00:28:37,480 --> 00:28:40,070 I don't care about any community college who 502 00:28:40,070 --> 00:28:42,590 would say, I don't want the product rule to be known, 503 00:28:42,590 --> 00:28:44,760 you can differentiate with a calculator. 504 00:28:44,760 --> 00:28:46,000 That's a no, no, no. 505 00:28:46,000 --> 00:28:50,108 You don't know calculus if you don't know the product rule. 506 00:28:50,108 --> 00:28:53,480 So the product rule is a blessing from God. 507 00:28:53,480 --> 00:28:58,020 It helps everywhere in physics, in mechanics, in engineering. 508 00:28:58,020 --> 00:29:00,990 It really helps in differential geometry 509 00:29:00,990 --> 00:29:03,755 with the directional derivative, the Lie derivative. 510 00:29:03,755 --> 00:29:07,910 It helps you understand all the upper level mathematics. 511 00:29:07,910 --> 00:29:11,700 Now here you have t prime, the first prime times 512 00:29:11,700 --> 00:29:16,300 the second unprime, plus the first unprime times the second 513 00:29:16,300 --> 00:29:17,390 prime. 514 00:29:17,390 --> 00:29:21,040 It's the same as for regular scalar functions. 515 00:29:21,040 --> 00:29:23,750 What's the derivative of 1? 516 00:29:23,750 --> 00:29:24,250 STUDENT: 0. 517 00:29:24,250 --> 00:29:25,610 PROFESSOR: 0. 518 00:29:25,610 --> 00:29:26,860 Look at this guy! 519 00:29:26,860 --> 00:29:29,270 Doesn't he look funny? 520 00:29:29,270 --> 00:29:32,630 It is the dot product community. 521 00:29:32,630 --> 00:29:34,255 Yes it is, by definition. 522 00:29:34,255 --> 00:29:40,120 So you have twice T times T prime equals 0. 523 00:29:40,120 --> 00:29:44,050 This 2 is-- stinking guy, let's divide by 2. 524 00:29:44,050 --> 00:29:45,076 Forget about that. 525 00:29:45,076 --> 00:29:46,590 What does this say? 526 00:29:46,590 --> 00:29:53,840 The dot product of T times-- I mean by T prime is 0. 527 00:29:53,840 --> 00:29:57,220 When are two vectors giving you dot product 0? 528 00:29:57,220 --> 00:29:58,720 STUDENT: When they're perpendicular. 529 00:29:58,720 --> 00:29:59,550 530 00:29:59,550 --> 00:30:01,900 PROFESSOR: So if both of them are non-zero, 531 00:30:01,900 --> 00:30:03,370 they have to be like that. 532 00:30:03,370 --> 00:30:06,330 They have to be like this, perpendicular, right? 533 00:30:06,330 --> 00:30:11,762 So it follows that t has to be perpendicular to T prime. 534 00:30:11,762 --> 00:30:15,550 And now, that's why n is perpendicular to t. 535 00:30:15,550 --> 00:30:19,370 But, because n is collinear to t prime. 536 00:30:19,370 --> 00:30:20,100 Hello. 537 00:30:20,100 --> 00:30:22,500 n is collinear to t prime. 538 00:30:22,500 --> 00:30:25,660 So this is t prime. 539 00:30:25,660 --> 00:30:27,840 Is t prime unitary? 540 00:30:27,840 --> 00:30:29,420 I'm going to measure it. 541 00:30:29,420 --> 00:30:30,720 No it's not. 542 00:30:30,720 --> 00:30:31,620 t prime. 543 00:30:31,620 --> 00:30:33,650 So if I want to make it unitary, I'm 544 00:30:33,650 --> 00:30:36,430 going to chop my-- no, I'm not going to chop. 545 00:30:36,430 --> 00:30:40,160 I just take it, t prime, and divide by its magnitude. 546 00:30:40,160 --> 00:30:43,450 Then I'm going to get that vector n, which is unitary. 547 00:30:43,450 --> 00:30:47,790 So from here it follows that t and n are indeed perpendicular, 548 00:30:47,790 --> 00:30:52,670 and your colleague over there said, hey, it has to be normal. 549 00:30:52,670 --> 00:30:55,140 That's perpendicular to t, but which one? 550 00:30:55,140 --> 00:30:58,180 A special one, because I have many normals. 551 00:30:58,180 --> 00:31:02,310 Now, this special one is easy to find like that. 552 00:31:02,310 --> 00:31:05,726 Where shall I put here-- I'll draw him very nicely. 553 00:31:05,726 --> 00:31:08,880 554 00:31:08,880 --> 00:31:10,000 I'll draw him. 555 00:31:10,000 --> 00:31:13,410 Now you guys have to imagine-- am I drawing 556 00:31:13,410 --> 00:31:14,580 well enough for you? 557 00:31:14,580 --> 00:31:15,900 I don't even know. 558 00:31:15,900 --> 00:31:17,780 t and n should be perpendicular. 559 00:31:17,780 --> 00:31:21,960 Can you imagine them having that 90 degree angle between them? 560 00:31:21,960 --> 00:31:22,460 OK. 561 00:31:22,460 --> 00:31:26,710 Now there is a magic one that you don't even have to define. 562 00:31:26,710 --> 00:31:28,670 And yes sir? 563 00:31:28,670 --> 00:31:31,180 STUDENT: In this thing, can [INAUDIBLE] 564 00:31:31,180 --> 00:31:34,410 this T vector [INAUDIBLE] written by the definition 565 00:31:34,410 --> 00:31:36,170 thing? 566 00:31:36,170 --> 00:31:37,829 PROFESSOR: No. 567 00:31:37,829 --> 00:31:39,370 STUDENT: N vector times the magnitude 568 00:31:39,370 --> 00:31:42,060 of t vector derivative? 569 00:31:42,060 --> 00:31:47,050 PROFESSOR: So technically you have 570 00:31:47,050 --> 00:31:50,764 t prime would be the magnitude of t prime times n. 571 00:31:50,764 --> 00:31:51,640 STUDENT: Yes. 572 00:31:51,640 --> 00:31:54,440 PROFESSOR: But keep in mind that sometimes is tricky, 573 00:31:54,440 --> 00:31:57,310 because this is, in general, not a constant. 574 00:31:57,310 --> 00:31:59,470 Always keep it in mind, it's not a constant. 575 00:31:59,470 --> 00:32:02,000 We'll have some examples later. 576 00:32:02,000 --> 00:32:04,690 There is a magic guy called binormal. 577 00:32:04,690 --> 00:32:09,710 That binormal is the normal to both t and n. 578 00:32:09,710 --> 00:32:12,430 And he's defined as t plus n because it's 579 00:32:12,430 --> 00:32:14,445 normal to both of them. 580 00:32:14,445 --> 00:32:18,370 So I'm going to write this b vector is t cross n. 581 00:32:18,370 --> 00:32:22,002 Now I'm asking you to draw it. 582 00:32:22,002 --> 00:32:23,710 Can anybody come to the board and draw it 583 00:32:23,710 --> 00:32:26,960 for 0.01 extra credit? 584 00:32:26,960 --> 00:32:29,642 Yes, sir? 585 00:32:29,642 --> 00:32:30,660 STUDENT: [INAUDIBLE] 586 00:32:30,660 --> 00:32:35,000 PROFESSOR: Draw that on the picture like t and n, t and n, 587 00:32:35,000 --> 00:32:38,470 t is the-- who the heck is t? t is the red one, 588 00:32:38,470 --> 00:32:40,860 and blue is the n. 589 00:32:40,860 --> 00:32:42,930 So does it go down or up? 590 00:32:42,930 --> 00:32:46,460 We should be perpendicular to both of them. 591 00:32:46,460 --> 00:32:49,340 Is b unitary or not? 592 00:32:49,340 --> 00:32:51,820 If you have two unit vectors, will the cross product 593 00:32:51,820 --> 00:32:53,060 be a unit vector? 594 00:32:53,060 --> 00:32:56,370 595 00:32:56,370 --> 00:33:00,200 Only if the two vectors are perpendicular, 596 00:33:00,200 --> 00:33:04,680 it is going to be, right? 597 00:33:04,680 --> 00:33:12,420 So you have-- well, I think it goes that-- 598 00:33:12,420 --> 00:33:13,760 in which direction does it go? 599 00:33:13,760 --> 00:33:14,620 Because 600 00:33:14,620 --> 00:33:16,360 STUDENT: It should not be how we have it. 601 00:33:16,360 --> 00:33:17,276 PROFESSOR: No, no, no. 602 00:33:17,276 --> 00:33:18,370 Because this is-- 603 00:33:18,370 --> 00:33:19,270 STUDENT: Yeah. 604 00:33:19,270 --> 00:33:19,820 I'm using-- 605 00:33:19,820 --> 00:33:22,935 PROFESSOR: So t goes over n, so I'm 606 00:33:22,935 --> 00:33:27,470 going to try-- it is like that, sort of. 607 00:33:27,470 --> 00:33:28,710 STUDENT: Into the chord? 608 00:33:28,710 --> 00:33:31,452 PROFESSOR: So again, it's not very clear because 609 00:33:31,452 --> 00:33:33,860 of my stinking art, here. 610 00:33:33,860 --> 00:33:36,000 It's really not nice art. 611 00:33:36,000 --> 00:33:39,830 t, and this is n. 612 00:33:39,830 --> 00:33:43,540 And if I go t going over n. 613 00:33:43,540 --> 00:33:47,627 T going over n goes up or down? 614 00:33:47,627 --> 00:33:48,210 STUDENT: Down. 615 00:33:48,210 --> 00:33:49,084 PROFESSOR: Goes down. 616 00:33:49,084 --> 00:33:52,310 So it's going to look more like this, feet. 617 00:33:52,310 --> 00:33:54,670 Now guys, when we-- thank you so much. 618 00:33:54,670 --> 00:33:57,970 So you've like a 0.01 extra credit. 619 00:33:57,970 --> 00:34:00,610 OK. 620 00:34:00,610 --> 00:34:03,400 Tangent, normal, and binormal form a corner. 621 00:34:03,400 --> 00:34:04,320 Yes, sir? 622 00:34:04,320 --> 00:34:07,460 STUDENT: Is rt-- rt is the function at the-- 623 00:34:07,460 --> 00:34:09,159 for the flag that's flying? 624 00:34:09,159 --> 00:34:12,120 PROFESSOR: The r of t is the position vector 625 00:34:12,120 --> 00:34:15,030 of the flag that was flying that he was drunk. 626 00:34:15,030 --> 00:34:21,350 STUDENT: Why wasn't the derivative of it perpendicular? 627 00:34:21,350 --> 00:34:24,420 Why isn't t perpendicular to rt? 628 00:34:24,420 --> 00:34:26,199 PROFESSOR: If-- well, good question. 629 00:34:26,199 --> 00:34:28,770 630 00:34:28,770 --> 00:34:30,600 We'll talk about it. 631 00:34:30,600 --> 00:34:34,690 If the length of r would be a constant, 632 00:34:34,690 --> 00:34:38,690 can we prove that r and r prime are perpendicular? 633 00:34:38,690 --> 00:34:40,985 Let's do that as another exercise. 634 00:34:40,985 --> 00:34:42,880 All right? 635 00:34:42,880 --> 00:34:46,190 So tnb looks like a corner. 636 00:34:46,190 --> 00:34:51,810 Look at the corner that the video cannot see over there. 637 00:34:51,810 --> 00:34:53,659 TN and B are mutually octagonal. 638 00:34:53,659 --> 00:34:56,300 639 00:34:56,300 --> 00:34:57,720 I'm going to draw them. 640 00:34:57,720 --> 00:35:00,810 This is an arbitrary point on a curve, 641 00:35:00,810 --> 00:35:04,410 and this is t, which is always tangent to the curve, 642 00:35:04,410 --> 00:35:05,610 and this is n. 643 00:35:05,610 --> 00:35:08,490 Let's say that's the unit principle normal. 644 00:35:08,490 --> 00:35:11,109 And t cross n will go, again, down. 645 00:35:11,109 --> 00:35:11,650 I don't know. 646 00:35:11,650 --> 00:35:14,560 I have an obsession with me going down. 647 00:35:14,560 --> 00:35:16,480 This is called the Frenet Trihedron. 648 00:35:16,480 --> 00:35:20,220 649 00:35:20,220 --> 00:35:23,000 And I have a proposal for a problem 650 00:35:23,000 --> 00:35:34,950 that maybe I should give my students in the future. 651 00:35:34,950 --> 00:35:46,450 Show that for a circle, playing in space, I don't know. 652 00:35:46,450 --> 00:36:07,210 The position vector and the velocity vector are always how? 653 00:36:07,210 --> 00:36:07,710 Friends. 654 00:36:07,710 --> 00:36:09,140 Let's say friends. 655 00:36:09,140 --> 00:36:12,470 No, come on, I'm kidding. 656 00:36:12,470 --> 00:36:13,270 How are they? 657 00:36:13,270 --> 00:36:15,391 STUDENT: Perpendicular. 658 00:36:15,391 --> 00:36:16,640 PROFESSOR: How do you do that? 659 00:36:16,640 --> 00:36:18,360 Is it hard? 660 00:36:18,360 --> 00:36:20,880 We should be smart enough to do that, right? 661 00:36:20,880 --> 00:36:21,980 I have a circle. 662 00:36:21,980 --> 00:36:26,380 That circle has what-- what is the property of a circle? 663 00:36:26,380 --> 00:36:29,880 Euclid defined that-- this is one of the axioms of Euclid. 664 00:36:29,880 --> 00:36:32,430 Does anybody know which axiom? 665 00:36:32,430 --> 00:36:35,790 That there exists such a set of points 666 00:36:35,790 --> 00:36:38,660 that are all at the same distance from a given point 667 00:36:38,660 --> 00:36:40,630 called center. 668 00:36:40,630 --> 00:36:42,850 So that is a circle, right? 669 00:36:42,850 --> 00:36:44,270 That's what Mr. Euclid said. 670 00:36:44,270 --> 00:36:45,400 He was a genius. 671 00:36:45,400 --> 00:36:51,980 So no matter where I put that circle, I can take r of t 672 00:36:51,980 --> 00:36:55,290 in magnitude measured from the origin 673 00:36:55,290 --> 00:36:57,190 from the center of the circle. 674 00:36:57,190 --> 00:37:01,200 Keep in mind, always the center of the circle. 675 00:37:01,200 --> 00:37:06,000 I put it at the origin of the space-- origin of the universe. 676 00:37:06,000 --> 00:37:08,420 No, origin of the space, actually. 677 00:37:08,420 --> 00:37:12,840 R of T magnitude would be a constant. 678 00:37:12,840 --> 00:37:13,995 Give me a constant, guys. 679 00:37:13,995 --> 00:37:14,495 OK? 680 00:37:14,495 --> 00:37:16,480 It doesn't matter. 681 00:37:16,480 --> 00:37:18,460 Let me draw. 682 00:37:18,460 --> 00:37:20,270 I want to draw in plane, OK? 683 00:37:20,270 --> 00:37:25,980 Because I'm getting tired. x y, and this is r of t, 684 00:37:25,980 --> 00:37:30,210 and the magnitude of this r of t is the radius of the circle. 685 00:37:30,210 --> 00:37:32,540 Right? 686 00:37:32,540 --> 00:37:36,677 So let's say, this is the radius of the circle. 687 00:37:36,677 --> 00:37:40,880 688 00:37:40,880 --> 00:37:43,930 How in the world do I prove the same idea? 689 00:37:43,930 --> 00:37:47,985 Who helps me prove that r is always 690 00:37:47,985 --> 00:37:50,610 perpendicular to r prime? 691 00:37:50,610 --> 00:37:53,857 Which way do you want to move, counterclockwise or clockwise? 692 00:37:53,857 --> 00:37:54,940 STUDENT: Counterclockwise. 693 00:37:54,940 --> 00:37:56,106 PROFESSOR: Counterclockwise. 694 00:37:56,106 --> 00:37:58,160 Because if you are a real scientist, 695 00:37:58,160 --> 00:38:00,110 I'm proud of you guys. 696 00:38:00,110 --> 00:38:01,950 It's clear from the picture that r prime 697 00:38:01,950 --> 00:38:04,790 would be perpendicular to r. 698 00:38:04,790 --> 00:38:05,825 Why is that? 699 00:38:05,825 --> 00:38:07,593 How am I going to do that? 700 00:38:07,593 --> 00:38:11,510 Now, mimic everything I-- don't look at your notes, 701 00:38:11,510 --> 00:38:17,260 and try to tell me how I show that quickly. 702 00:38:17,260 --> 00:38:18,290 What am I going to do? 703 00:38:18,290 --> 00:38:23,380 So all I know, all that gave me was r of t 704 00:38:23,380 --> 00:38:27,680 equals k in magnitude constant. 705 00:38:27,680 --> 00:38:30,530 For every t, this same constant. 706 00:38:30,530 --> 00:38:31,510 What's next? 707 00:38:31,510 --> 00:38:34,922 What do I want to do next? 708 00:38:34,922 --> 00:38:36,310 STUDENT: Square it? 709 00:38:36,310 --> 00:38:38,390 PROFESSOR: Square it, differentiate it. 710 00:38:38,390 --> 00:38:40,454 I can also go ahead and differentiate it 711 00:38:40,454 --> 00:38:41,995 without squaring it, but that's going 712 00:38:41,995 --> 00:38:47,350 to be a little bit of more pain. 713 00:38:47,350 --> 00:38:52,170 So square it, differentiate it. 714 00:38:52,170 --> 00:38:53,390 I'm too lazy. 715 00:38:53,390 --> 00:38:56,140 When I differentiate, what am I going to get? 716 00:38:56,140 --> 00:39:05,320 From the product rule, twice r dot r primed of t equals 0. 717 00:39:05,320 --> 00:39:06,870 Well, I'm done. 718 00:39:06,870 --> 00:39:12,000 Because it means that for every t that radius-- not 719 00:39:12,000 --> 00:39:13,270 the radius, guys, I'm sorry. 720 00:39:13,270 --> 00:39:16,870 The position vector will be perpendicular to the velocity 721 00:39:16,870 --> 00:39:17,560 vector. 722 00:39:17,560 --> 00:39:22,180 Now, if I draw the trajectory of my drunken flag 723 00:39:22,180 --> 00:39:24,660 this [INAUDIBLE] is not true, right? 724 00:39:24,660 --> 00:39:27,430 This is crazy. 725 00:39:27,430 --> 00:39:29,924 Of course this is r, and this is r prime, 726 00:39:29,924 --> 00:39:35,590 and there is an arbitrary angle between r and r prime. 727 00:39:35,590 --> 00:39:38,260 The good thing is that the arbitrary angle always 728 00:39:38,260 --> 00:39:41,320 exists, and is continuous as a function. 729 00:39:41,320 --> 00:39:43,450 I never have that angle disappear. 730 00:39:43,450 --> 00:39:46,900 That's way I want that prime never to become 0. 731 00:39:46,900 --> 00:39:49,330 Because if the bag was stopping its motion, 732 00:39:49,330 --> 00:39:54,020 goodbye angle, goodbye analysis, right? 733 00:39:54,020 --> 00:39:54,730 OK. 734 00:39:54,730 --> 00:39:55,540 Very nice. 735 00:39:55,540 --> 00:39:57,470 So don't give me more ideas. 736 00:39:57,470 --> 00:39:59,940 You smart people, if you give me more ideas, 737 00:39:59,940 --> 00:40:02,550 I'm going to come up with all sorts of problems. 738 00:40:02,550 --> 00:40:05,000 And this is actually one of the first problems 739 00:40:05,000 --> 00:40:08,980 you learn in a graduate level geometry class. 740 00:40:08,980 --> 00:40:13,930 741 00:40:13,930 --> 00:40:16,670 Let me give you another piece of information 742 00:40:16,670 --> 00:40:20,240 that you're going to love, which could 743 00:40:20,240 --> 00:40:22,390 be one of those types of combined 744 00:40:22,390 --> 00:40:25,300 problems on a final exam or midterm, 745 00:40:25,300 --> 00:40:29,900 A, B, C, D, E. The curvature of a curve 746 00:40:29,900 --> 00:40:33,930 is a measure of how the curve will bend. 747 00:40:33,930 --> 00:40:35,720 Say what? 748 00:40:35,720 --> 00:40:45,970 The curvature of a curve is a measure 749 00:40:45,970 --> 00:40:48,542 of the bending of that curve. 750 00:40:48,542 --> 00:40:58,800 751 00:40:58,800 --> 00:41:04,380 By definition, you have to take it like that. 752 00:41:04,380 --> 00:41:20,730 If the curve is parameterized in arc length-- somebody 753 00:41:20,730 --> 00:41:23,070 remind me what that is. 754 00:41:23,070 --> 00:41:24,960 What does it mean? 755 00:41:24,960 --> 00:41:32,440 That is r of s such that-- what does it mean, 756 00:41:32,440 --> 00:41:33,644 parameterizing arc length-- 757 00:41:33,644 --> 00:41:34,560 STUDENT: r prime of s. 758 00:41:34,560 --> 00:41:37,320 PROFESSOR: r primed of s in magnitude is 1. 759 00:41:37,320 --> 00:41:37,910 The speed 1. 760 00:41:37,910 --> 00:41:39,200 It's a speed 1 curve. 761 00:41:39,200 --> 00:41:42,510 762 00:41:42,510 --> 00:42:00,690 Then, the curvature of this curve is defined as k of s 763 00:42:00,690 --> 00:42:06,320 equals the magnitude of the acceleration vector 764 00:42:06,320 --> 00:42:09,460 will respect the S. Say what, Magdalena? 765 00:42:09,460 --> 00:42:12,890 I can also write it magnitude of d-- 766 00:42:12,890 --> 00:42:17,070 oh my gosh, second derivative with respect s of r. 767 00:42:17,070 --> 00:42:20,340 I'll do it right now. d2r ds2. 768 00:42:20,340 --> 00:42:21,910 And I know you get a headache when 769 00:42:21,910 --> 00:42:25,940 I solve, when I write that, because you are not used to it. 770 00:42:25,940 --> 00:42:33,450 A quick and beautiful example that can be on the homework, 771 00:42:33,450 --> 00:42:39,380 and would also be on the exam, maybe on all the exams, 772 00:42:39,380 --> 00:42:41,864 I don't know. 773 00:42:41,864 --> 00:42:48,140 Compute the curvature of a circle of radius a Say what? 774 00:42:48,140 --> 00:43:02,670 Compute the curvature of a circle of radius a And you say, 775 00:43:02,670 --> 00:43:03,620 wait a minute. 776 00:43:03,620 --> 00:43:07,040 For a circle of radius a in plane-- 777 00:43:07,040 --> 00:43:08,990 why can I assume it's in plane? 778 00:43:08,990 --> 00:43:12,590 Because if the circle is a planar curve, 779 00:43:12,590 --> 00:43:15,860 I can always assume it to be in plane. 780 00:43:15,860 --> 00:43:19,030 And it has radius a I can find infinitely many 781 00:43:19,030 --> 00:43:20,160 parameterizations. 782 00:43:20,160 --> 00:43:23,100 So what, am I crazy? 783 00:43:23,100 --> 00:43:25,444 Well, yes, I am, but that's another story. 784 00:43:25,444 --> 00:43:28,330 Now, if I want to parameterize, I 785 00:43:28,330 --> 00:43:31,170 have to parameterize in arc length. 786 00:43:31,170 --> 00:43:34,020 If I do anything else, that means I'm stupid. 787 00:43:34,020 --> 00:43:38,960 So, r of s will be what? 788 00:43:38,960 --> 00:43:41,390 Can somebody tell me how I parameterize 789 00:43:41,390 --> 00:43:46,280 a curve in arc length for a-- what is this guy? 790 00:43:46,280 --> 00:43:49,210 A circle of radius a. 791 00:43:49,210 --> 00:43:50,520 Yeah, I cannot do it. 792 00:43:50,520 --> 00:43:52,470 I'm not smart enough. 793 00:43:52,470 --> 00:43:59,390 So I'll say R of T will be a cosine t, a sine t and 0. 794 00:43:59,390 --> 00:44:03,150 And here I stop, because I had a headache. 795 00:44:03,150 --> 00:44:08,510 t is from 0 to 2 pi, and I think this a is making 796 00:44:08,510 --> 00:44:14,980 my life miserable, because it's telling me, 797 00:44:14,980 --> 00:44:17,410 you don't have speed 1, Magdalena. 798 00:44:17,410 --> 00:44:19,200 Drive to Amarillo and back, you're 799 00:44:19,200 --> 00:44:22,050 not going to get speed 1. 800 00:44:22,050 --> 00:44:23,320 Why don't I have speed 1? 801 00:44:23,320 --> 00:44:23,950 Think about it. 802 00:44:23,950 --> 00:44:24,930 Bear with me. 803 00:44:24,930 --> 00:44:28,546 Minus a sine t equals sine t, 0. 804 00:44:28,546 --> 00:44:29,420 Bad. 805 00:44:29,420 --> 00:44:31,150 What is the speed? 806 00:44:31,150 --> 00:44:32,750 a. 807 00:44:32,750 --> 00:44:35,570 If you do the math, the speed will be a. 808 00:44:35,570 --> 00:44:39,650 So length of our prime of t will be a. 809 00:44:39,650 --> 00:44:40,760 Somebody help me. 810 00:44:40,760 --> 00:44:42,300 Get me out of trouble. 811 00:44:42,300 --> 00:44:43,220 Who is this? 812 00:44:43,220 --> 00:44:45,400 I want to do it in arc length. 813 00:44:45,400 --> 00:44:48,250 Otherwise, how can I do the curvature? 814 00:44:48,250 --> 00:44:50,790 So somebody tell me how to get to s. 815 00:44:50,790 --> 00:44:52,270 What the heck is that? 816 00:44:52,270 --> 00:44:58,610 s of t is integral from 0 to t of-- who tells me? 817 00:44:58,610 --> 00:44:59,840 The speed, right? 818 00:44:59,840 --> 00:45:04,390 Was it not the displacement, the arc length traveled along, 819 00:45:04,390 --> 00:45:08,036 and the curve is integral in time of the speed. 820 00:45:08,036 --> 00:45:11,950 821 00:45:11,950 --> 00:45:13,030 OK? 822 00:45:13,030 --> 00:45:16,420 So I have-- what is that? 823 00:45:16,420 --> 00:45:17,560 Speed is? 824 00:45:17,560 --> 00:45:18,360 STUDENT: Um-- 825 00:45:18,360 --> 00:45:18,980 PROFESSOR: a. 826 00:45:18,980 --> 00:45:22,850 So a time t, am I right, guys? s is a times t. 827 00:45:22,850 --> 00:45:24,650 So what do I have to do? 828 00:45:24,650 --> 00:45:30,950 Take Mr. t, shake his hand, and replace him with s over a. 829 00:45:30,950 --> 00:45:31,810 OK. 830 00:45:31,810 --> 00:45:39,740 So instead of r of t, I'll say-- what other letters do I have? 831 00:45:39,740 --> 00:45:40,480 Not r. 832 00:45:40,480 --> 00:45:41,443 Rho of s. 833 00:45:41,443 --> 00:45:42,400 I love rho. 834 00:45:42,400 --> 00:45:44,420 Rho is the Greek [INAUDIBLE]. 835 00:45:44,420 --> 00:45:46,350 Is this finally an arc length? 836 00:45:46,350 --> 00:45:51,000 Cosine of-- what is t, guys, again? 837 00:45:51,000 --> 00:45:53,140 s over a. 838 00:45:53,140 --> 00:45:58,220 s over a, a sine s over a, and 0. 839 00:45:58,220 --> 00:46:01,710 This is the parameterization in arc length. 840 00:46:01,710 --> 00:46:08,470 This is an arc length parameterization of the circle. 841 00:46:08,470 --> 00:46:11,470 And then what is this definition of curvature? 842 00:46:11,470 --> 00:46:13,760 It's here. 843 00:46:13,760 --> 00:46:16,930 Do that rho once, twice. 844 00:46:16,930 --> 00:46:19,510 Prime it twice, and do the length. 845 00:46:19,510 --> 00:46:20,760 So rho prime. 846 00:46:20,760 --> 00:46:25,060 Oh my God is it hard. 847 00:46:25,060 --> 00:46:29,420 a times minus sine of s over a. 848 00:46:29,420 --> 00:46:30,270 Am I done, though? 849 00:46:30,270 --> 00:46:30,959 Chain rule. 850 00:46:30,959 --> 00:46:32,000 Pay attention, Magdalena. 851 00:46:32,000 --> 00:46:34,070 Don't screwed up with this one. 852 00:46:34,070 --> 00:46:36,370 1 over a. 853 00:46:36,370 --> 00:46:37,730 Good. 854 00:46:37,730 --> 00:46:38,580 Next. 855 00:46:38,580 --> 00:46:41,716 a cosine of s over a. 856 00:46:41,716 --> 00:46:42,670 Chain rule. 857 00:46:42,670 --> 00:46:44,300 Don't forget, multiply by 1 over a. 858 00:46:44,300 --> 00:46:46,860 OK, that makes my life easier. 859 00:46:46,860 --> 00:46:47,820 We simplify. 860 00:46:47,820 --> 00:46:51,590 Thank God a simplifies here, a simplifies there, 861 00:46:51,590 --> 00:46:54,180 so that is that derivative. 862 00:46:54,180 --> 00:46:55,870 What's the second derivative? 863 00:46:55,870 --> 00:47:01,217 Rho double prime of s will be-- somebody help me, OK? 864 00:47:01,217 --> 00:47:02,716 Because this is a lot of derivation. 865 00:47:02,716 --> 00:47:03,470 STUDENT: --cosine-- 866 00:47:03,470 --> 00:47:04,553 PROFESSOR: Thank you, sir. 867 00:47:04,553 --> 00:47:06,810 Minus cosine of s over a. 868 00:47:06,810 --> 00:47:07,810 STUDENT: Times 1 over a. 869 00:47:07,810 --> 00:47:13,330 PROFESSOR: Times 1 over a, comma, minus sine of s over a. 870 00:47:13,330 --> 00:47:15,600 That's all I have left in my life, right? 871 00:47:15,600 --> 00:47:19,860 Minus sine of s over a times 1 over a from the chain rule. 872 00:47:19,860 --> 00:47:22,690 I have to pay attention and see. 873 00:47:22,690 --> 00:47:24,350 What's the magnitude of this? 874 00:47:24,350 --> 00:47:28,644 The magnitude of this of this animal will be the curvature. 875 00:47:28,644 --> 00:47:30,060 Oh, my God. 876 00:47:30,060 --> 00:47:32,320 So what is k? 877 00:47:32,320 --> 00:47:35,120 k of s will be-- could somebody tell me 878 00:47:35,120 --> 00:47:39,610 what magnitude I get after I square all these individuals, 879 00:47:39,610 --> 00:47:42,914 sum them up, and take the square root of them? 880 00:47:42,914 --> 00:47:44,360 STUDENT: [INAUDIBLE] 881 00:47:44,360 --> 00:47:49,550 PROFESSOR: Square root of 1 over 1 squared. 882 00:47:49,550 --> 00:47:50,740 And I get 1 over a. 883 00:47:50,740 --> 00:47:53,236 You are too fast for me, you teach me that. 884 00:47:53,236 --> 00:47:54,110 No, I'm just kidding. 885 00:47:54,110 --> 00:47:56,290 I knew it was 1 over a. 886 00:47:56,290 --> 00:47:59,320 Now, how did engineers know that? 887 00:47:59,320 --> 00:48:02,170 Actually, for hundreds of years, mathematicians, engineers, 888 00:48:02,170 --> 00:48:04,460 and physicists knew that. 889 00:48:04,460 --> 00:48:07,470 And that's the last thing I want to teach you today. 890 00:48:07,470 --> 00:48:09,910 We have two circles. 891 00:48:09,910 --> 00:48:17,390 This is of, let's say, radius 1/2, and this is radius 2. 892 00:48:17,390 --> 00:48:21,030 The engineer, mathematician, physicist, whoever they are, 893 00:48:21,030 --> 00:48:26,465 they knew that the curvature is inverse proportional 894 00:48:26,465 --> 00:48:28,250 to the radius. 895 00:48:28,250 --> 00:48:30,440 That radius is 1/2. 896 00:48:30,440 --> 00:48:33,516 The curvature will be 2 in this case. 897 00:48:33,516 --> 00:48:38,072 The radius is 2, the curvature will be 1/2. 898 00:48:38,072 --> 00:48:41,890 Does that make sense, this inverse proportionality? 899 00:48:41,890 --> 00:48:45,660 The bigger the radius, the lesser the curvature, 900 00:48:45,660 --> 00:48:47,670 that less bent you are. 901 00:48:47,670 --> 00:48:49,600 The more fat-- well, OK. 902 00:48:49,600 --> 00:48:53,080 I'm not going to say anything politically incorrect. 903 00:48:53,080 --> 00:48:58,540 So this is really curved because the radius is really small. 904 00:48:58,540 --> 00:49:02,620 This less curved, almost-- at infinity, 905 00:49:02,620 --> 00:49:05,550 this curvature becomes 0, because 906 00:49:05,550 --> 00:49:09,075 at infinity, that radius explodes to plus infinity bag 907 00:49:09,075 --> 00:49:10,090 theory. 908 00:49:10,090 --> 00:49:13,600 Then you have 1 over infinity will be 0, 909 00:49:13,600 --> 00:49:19,064 and that will be the curvature of a circle of infinite radius. 910 00:49:19,064 --> 00:49:20,420 Right? 911 00:49:20,420 --> 00:49:22,530 So we learned something today. 912 00:49:22,530 --> 00:49:25,489 We learned about the curvature of a circle, which 913 00:49:25,489 --> 00:49:26,030 is something. 914 00:49:26,030 --> 00:49:30,930 But this is the same way for any curve. 915 00:49:30,930 --> 00:49:31,836 You reparameterize. 916 00:49:31,836 --> 00:49:34,460 Now you understand why you need to reparameterize in arc length 917 00:49:34,460 --> 00:49:35,880 s. 918 00:49:35,880 --> 00:49:37,854 You take the acceleration in arc length. 919 00:49:37,854 --> 00:49:38,770 You get the magnitude. 920 00:49:38,770 --> 00:49:41,520 That measures how bent the curve is. 921 00:49:41,520 --> 00:49:47,210 Next time, you're going to do how bent the helix is. 922 00:49:47,210 --> 00:49:47,750 OK? 923 00:49:47,750 --> 00:49:49,442 At every point. 924 00:49:49,442 --> 00:49:51,240 Enjoy your WeBWorK homework. 925 00:49:51,240 --> 00:49:54,935 Ask me anytime, and ask me also Thursday. 926 00:49:54,935 --> 00:49:58,710 Do not have a block about your homework questions. 927 00:49:58,710 --> 00:50:05,120 You can ask me anytime by email, or in person. 928 00:50:05,120 --> 00:50:10,489