1 00:00:00,000 --> 00:00:00,510 2 00:00:00,510 --> 00:00:04,890 I love this model. 3 00:00:04,890 --> 00:00:06,330 Again, thank you, Casey. 4 00:00:06,330 --> 00:00:09,860 I'm not going to take any credit for that. 5 00:00:09,860 --> 00:00:11,630 So if you want to imagine the stool 6 00:00:11,630 --> 00:00:16,725 I was talking about as a bamboo object, that 7 00:00:16,725 --> 00:00:21,110 is about the same thing, at the same scale, compared 8 00:00:21,110 --> 00:00:28,130 to the diameter and the height, scaled or dialated five times. 9 00:00:28,130 --> 00:00:30,450 Uniform, no alterations. 10 00:00:30,450 --> 00:00:35,040 And one can sit on it, [? and circle, ?] to sit on it. 11 00:00:35,040 --> 00:00:39,710 Now, as you see this is a doubly ruled surface. 12 00:00:39,710 --> 00:00:41,350 And you say, oh wait a minute. 13 00:00:41,350 --> 00:00:45,450 You said rule surface, why all of a sudden, why doubly ruled 14 00:00:45,450 --> 00:00:46,210 surface? 15 00:00:46,210 --> 00:00:51,520 Because it is a surface that is ruled and generated 16 00:00:51,520 --> 00:00:58,150 by two different one parameter families. 17 00:00:58,150 --> 00:00:59,690 Each of them has a certain parameter 18 00:00:59,690 --> 00:01:02,600 and that gives them continuity. 19 00:01:02,600 --> 00:01:04,319 So you have two families of lines. 20 00:01:04,319 --> 00:01:06,860 21 00:01:06,860 --> 00:01:09,995 One family is in this direction. 22 00:01:09,995 --> 00:01:11,330 Do you see it? 23 00:01:11,330 --> 00:01:14,852 So these lines-- this line is in motion. 24 00:01:14,852 --> 00:01:16,935 It moves to the right, to the right, to the right, 25 00:01:16,935 --> 00:01:19,970 and it generated. 26 00:01:19,970 --> 00:01:22,540 And the other family of lines is this one 27 00:01:22,540 --> 00:01:24,630 in the other direction. 28 00:01:24,630 --> 00:01:28,230 You have a continuity parameter for each of them. 29 00:01:28,230 --> 00:01:33,990 So you have to imagine some real parameter going 30 00:01:33,990 --> 00:01:36,750 along the entire [? infinite real ?] axis. 31 00:01:36,750 --> 00:01:40,460 Or along a circle which would be about the same thing. 32 00:01:40,460 --> 00:01:46,155 But in any case, you have a one parameter family 33 00:01:46,155 --> 00:01:49,130 and another one parameter family. 34 00:01:49,130 --> 00:01:52,300 Both of them are together generating 35 00:01:52,300 --> 00:01:55,980 this beautiful one-sheeted hyperboloid. 36 00:01:55,980 --> 00:02:00,890 It's incredible because you see where these sort of round, 37 00:02:00,890 --> 00:02:07,080 but if you go towards the ends, it's topologically 38 00:02:07,080 --> 00:02:08,449 a cylinder or a tube. 39 00:02:08,449 --> 00:02:15,370 But if you look towards the end, the two ends 40 00:02:15,370 --> 00:02:18,810 will look more straight. 41 00:02:18,810 --> 00:02:23,790 And you will see the straight lines more clearly. 42 00:02:23,790 --> 00:02:27,680 So imagine that you have a continuation 43 00:02:27,680 --> 00:02:31,210 to infinity in this direction, and in the other direction. 44 00:02:31,210 --> 00:02:36,112 And this actually should be an infinite surface in your model. 45 00:02:36,112 --> 00:02:38,550 You're just cutting it between two z planes, 46 00:02:38,550 --> 00:02:41,996 so you have a patch of a one-sheeted hyperboloid. 47 00:02:41,996 --> 00:02:43,370 Yeah, the one-sheeted hyperboloid 48 00:02:43,370 --> 00:02:47,220 that we wrote last time, do you guys remember 49 00:02:47,220 --> 00:02:49,990 x squared over a squared plus y squared 50 00:02:49,990 --> 00:02:53,370 over b squared minus z squared? z should be this [INAUDIBLE]. 51 00:02:53,370 --> 00:02:56,430 Minus z squared over c squared minus 1 52 00:02:56,430 --> 00:03:00,950 equals 0 is an infinite surface area. 53 00:03:00,950 --> 00:03:04,995 At both ends you keep going. 54 00:03:04,995 --> 00:03:06,480 Very beautiful. 55 00:03:06,480 --> 00:03:08,380 Thank you so much. 56 00:03:08,380 --> 00:03:09,370 I appreciate. 57 00:03:09,370 --> 00:03:12,240 And keep the brownies. 58 00:03:12,240 --> 00:03:14,030 No, then I have to pay more. 59 00:03:14,030 --> 00:03:16,700 Than I have to pay money. 60 00:03:16,700 --> 00:03:18,944 STUDENT: It's made out of [INAUDIBLE]. 61 00:03:18,944 --> 00:03:20,318 PROFESSOR: When is your birthday? 62 00:03:20,318 --> 00:03:23,390 [LAUGHTER] 63 00:03:23,390 --> 00:03:24,000 Really? 64 00:03:24,000 --> 00:03:24,140 When is it? 65 00:03:24,140 --> 00:03:25,014 STUDENT: February 29. 66 00:03:25,014 --> 00:03:27,975 PROFESSOR: Oh, it's coming. 67 00:03:27,975 --> 00:03:28,808 [INTERPOSING VOICES] 68 00:03:28,808 --> 00:03:32,670 69 00:03:32,670 --> 00:03:35,558 STUDENT: It's coming in a year, too. 70 00:03:35,558 --> 00:03:38,960 PROFESSOR: That was a smart one. 71 00:03:38,960 --> 00:03:41,120 Anyway, I'll remember that. 72 00:03:41,120 --> 00:03:42,870 I appreciate the gift very much. 73 00:03:42,870 --> 00:03:45,470 And I will cherish it and I'll use it 74 00:03:45,470 --> 00:03:46,945 with both my undergraduate students 75 00:03:46,945 --> 00:03:50,382 and my graduate students who are just learning about-- some 76 00:03:50,382 --> 00:03:54,100 of them don't know the one-sheeted hyperboloid model, 77 00:03:54,100 --> 00:03:57,130 but they will learn about it. 78 00:03:57,130 --> 00:04:00,080 Coming back to our lesson. 79 00:04:00,080 --> 00:04:05,020 I announced Section 10.1. 80 00:04:05,020 --> 00:04:06,785 Say goodbye to quadrant for a while. 81 00:04:06,785 --> 00:04:09,640 I know you love them, but they will be there 82 00:04:09,640 --> 00:04:11,450 for you in Chapter 11. 83 00:04:11,450 --> 00:04:13,099 They will wait for you. 84 00:04:13,099 --> 00:04:20,584 Now, let's go to Section 10.1 of Chapter 10. 85 00:04:20,584 --> 00:04:23,079 Chapter 10 is a beautiful chapter. 86 00:04:23,079 --> 00:04:26,580 As you know very well, I announced last time, 87 00:04:26,580 --> 00:04:29,173 it is about vector-valued functions. 88 00:04:29,173 --> 00:04:40,800 89 00:04:40,800 --> 00:04:43,222 And you say, oh my god, I've never 90 00:04:43,222 --> 00:04:45,677 heard about vector-valued functions before. 91 00:04:45,677 --> 00:04:48,623 You deal with them every day. 92 00:04:48,623 --> 00:04:51,569 Every time you move, you are dealing 93 00:04:51,569 --> 00:04:54,790 with a vector-valued function, which 94 00:04:54,790 --> 00:05:01,640 is the displacement, which takes values in a subset in R3. 95 00:05:01,640 --> 00:05:06,820 So let's try and see what you should understand 96 00:05:06,820 --> 00:05:10,015 when you start Section 10.1. 97 00:05:10,015 --> 00:05:14,380 Because the book is pretty good, not that I'm a co-author. 98 00:05:14,380 --> 00:05:18,880 But it was meant to be really written for the students 99 00:05:18,880 --> 00:05:22,510 and explain concepts really well. 100 00:05:22,510 --> 00:05:26,380 How many of you took physics? 101 00:05:26,380 --> 00:05:29,020 OK, quite a lot of you took physics. 102 00:05:29,020 --> 00:05:33,840 Now, one of my students in a previous honors class 103 00:05:33,840 --> 00:05:38,480 told me he enjoyed my class greatly in general. 104 00:05:38,480 --> 00:05:41,475 The most [INAUDIBLE] thing he had from my class, he 105 00:05:41,475 --> 00:05:45,750 learned from my class was the motion of the drunken bug. 106 00:05:45,750 --> 00:05:47,880 And I said, did I say that? 107 00:05:47,880 --> 00:05:49,690 Absolutely, you said that. 108 00:05:49,690 --> 00:05:54,410 So apparently I had started one of my lessons 109 00:05:54,410 --> 00:05:59,830 with imagine you have a fly who went into your coffee mug. 110 00:05:59,830 --> 00:06:00,497 I think I did. 111 00:06:00,497 --> 00:06:03,520 He reproduced the whole thing the way I said it. 112 00:06:03,520 --> 00:06:05,630 It was quite spontaneous. 113 00:06:05,630 --> 00:06:10,430 So imagine your coffee mug had some Baileys Irish Creme in it. 114 00:06:10,430 --> 00:06:15,830 And the fly was really happy after she got up. 115 00:06:15,830 --> 00:06:17,890 She managed to get up. 116 00:06:17,890 --> 00:06:21,260 And the trajectory of the fly was something more 117 00:06:21,260 --> 00:06:23,330 like a helix. 118 00:06:23,330 --> 00:06:25,790 And this is how I actually introduced the helix 119 00:06:25,790 --> 00:06:27,280 in my classroom. 120 00:06:27,280 --> 00:06:29,830 And I thought, OK, is that unusual? 121 00:06:29,830 --> 00:06:30,330 Very. 122 00:06:30,330 --> 00:06:33,200 And I said, but that's an honors class. 123 00:06:33,200 --> 00:06:36,080 Everything is supposed to be unusual, right? 124 00:06:36,080 --> 00:06:50,190 So let's think about the position vector or some sort 125 00:06:50,190 --> 00:06:53,630 of vector-valued function that you're familiar with already 126 00:06:53,630 --> 00:06:55,412 from physics. 127 00:06:55,412 --> 00:06:57,490 He is one of your best friends. 128 00:06:57,490 --> 00:07:01,180 You have a function r of t. 129 00:07:01,180 --> 00:07:10,120 And I will point out that r is practically the position vector 130 00:07:10,120 --> 00:07:16,000 measure that time t, or observed at time t in R3. 131 00:07:16,000 --> 00:07:19,150 So he takes values in R3. 132 00:07:19,150 --> 00:07:20,070 How? 133 00:07:20,070 --> 00:07:23,930 As the mathematician, because I like to write mathematically 134 00:07:23,930 --> 00:07:26,740 all the notion I have, r is defined 135 00:07:26,740 --> 00:07:34,470 on I was a sub-interval of R with values in R3. 136 00:07:34,470 --> 00:07:38,696 And he asked me, my student said, what is this I? 137 00:07:38,696 --> 00:07:40,850 Well, this I could be any interval, 138 00:07:40,850 --> 00:07:43,290 but let's assume for the time being it's 139 00:07:43,290 --> 00:07:46,410 just an open interval of the type 140 00:07:46,410 --> 00:07:52,180 a, b, where a and b are real numbers, a less than b. 141 00:07:52,180 --> 00:07:58,310 So this is practically the time for my bug from the moment, 142 00:07:58,310 --> 00:08:03,360 let's say a equals 0 when she or he starts flying up, 143 00:08:03,360 --> 00:08:06,915 until the moment she completely freaks 144 00:08:06,915 --> 00:08:11,740 out or drops from the maximum point she reached. 145 00:08:11,740 --> 00:08:13,400 And she eventually dies. 146 00:08:13,400 --> 00:08:15,340 Or maybe she doesn't die. 147 00:08:15,340 --> 00:08:19,280 Maybe she's just drunk and she will wake up after a while. 148 00:08:19,280 --> 00:08:24,400 OK, so what do I mean by this displacement vector? 149 00:08:24,400 --> 00:08:25,500 I mean, a function-- 150 00:08:25,500 --> 00:08:26,480 STUDENT: Is that Tc? 151 00:08:26,480 --> 00:08:27,480 Do you have [INAUDIBLE]? 152 00:08:27,480 --> 00:08:28,771 PROFESSOR: This is r, little r. 153 00:08:28,771 --> 00:08:30,510 STUDENT: I know, but the Tc. 154 00:08:30,510 --> 00:08:32,010 PROFESSOR: Tc? 155 00:08:32,010 --> 00:08:33,427 STUDENT: Or is that an I? 156 00:08:33,427 --> 00:08:34,010 PROFESSOR: No. 157 00:08:34,010 --> 00:08:37,460 This is I interval, which is the same as a, 158 00:08:37,460 --> 00:08:41,720 b open interval, like from 2 to 7, included. 159 00:08:41,720 --> 00:08:46,180 This is inclusion [INAUDIBLE] included in R. 160 00:08:46,180 --> 00:08:51,520 So I mean R is the real number set and a, b is my interval. 161 00:08:51,520 --> 00:08:55,010 162 00:08:55,010 --> 00:08:58,460 OK, so r of t is going to be what? 163 00:08:58,460 --> 00:09:01,680 x of t, y of t, z of t. 164 00:09:01,680 --> 00:09:05,180 The book tells you, hey, guys-- it doesn't say hey, guys, 165 00:09:05,180 --> 00:09:09,800 but it's quite informal-- if you live in Rn, if your image is 166 00:09:09,800 --> 00:09:12,720 in Rn, instead of x of t, y of t, z of t, 167 00:09:12,720 --> 00:09:19,030 you are going to get something like x1 of t, y1 of t. 168 00:09:19,030 --> 00:09:23,560 x1 of t, x2 of t, x3 of t, et cetera. 169 00:09:23,560 --> 00:09:26,480 What do we assume about R? 170 00:09:26,480 --> 00:09:28,589 We have to assume something about it, right? 171 00:09:28,589 --> 00:09:30,130 STUDENT: It's a function [INAUDIBLE]. 172 00:09:30,130 --> 00:09:33,340 PROFESSOR: It's a function that is differentiable 173 00:09:33,340 --> 00:09:36,630 most of the times, right? 174 00:09:36,630 --> 00:09:37,900 What does it mean smooth? 175 00:09:37,900 --> 00:09:42,910 I saw that your books before college level 176 00:09:42,910 --> 00:09:44,300 never mention smooth. 177 00:09:44,300 --> 00:09:48,550 A smooth function is a function that is differentiable 178 00:09:48,550 --> 00:09:51,400 and whose first derivative is continuous. 179 00:09:51,400 --> 00:09:55,710 Some mathematicians even assume that you have c infinity, which 180 00:09:55,710 --> 00:10:00,320 means you have a function that's infinitely many differentiable. 181 00:10:00,320 --> 00:10:02,570 So you have first derivative, second derivative, third 182 00:10:02,570 --> 00:10:03,778 derivative, fifth derivative. 183 00:10:03,778 --> 00:10:05,060 Somebody stop me. 184 00:10:05,060 --> 00:10:09,240 All the derivatives exist and they are all continuous. 185 00:10:09,240 --> 00:10:13,270 By smooth, I will assume c1 in this case. 186 00:10:13,270 --> 00:10:16,000 I know it's not accurate, but let's assume c1. 187 00:10:16,000 --> 00:10:18,745 What does it mean? 188 00:10:18,745 --> 00:10:23,660 Differentiable function whose derivative is continuous. 189 00:10:23,660 --> 00:10:30,944 190 00:10:30,944 --> 00:10:35,220 And I will assume one more thing. 191 00:10:35,220 --> 00:10:37,360 That is not enough for me. 192 00:10:37,360 --> 00:10:41,900 I will also assume that r prime of t in this case 193 00:10:41,900 --> 00:10:49,704 is different from 0 for every t in the interval I. 194 00:10:49,704 --> 00:10:53,780 Could somebody tell me in everyday words what that means? 195 00:10:53,780 --> 00:10:55,736 We call that regular function. 196 00:10:55,736 --> 00:10:56,235 [INAUDIBLE] 197 00:10:56,235 --> 00:11:00,238 198 00:11:00,238 --> 00:11:01,770 You have a brownie [INAUDIBLE]. 199 00:11:01,770 --> 00:11:03,390 I have no brownies with me. 200 00:11:03,390 --> 00:11:05,550 But if you answer, so what-- 201 00:11:05,550 --> 00:11:08,220 STUDENT: So that means you've got no relative mins or maxes, 202 00:11:08,220 --> 00:11:11,402 and you never-- the object never stops moving. 203 00:11:11,402 --> 00:11:15,250 PROFESSOR: Well, actually, you can have relative mins 204 00:11:15,250 --> 00:11:18,470 and maxes in some way. 205 00:11:18,470 --> 00:11:22,630 I'm talking about something like that, r prime. 206 00:11:22,630 --> 00:11:28,330 207 00:11:28,330 --> 00:11:29,850 This is r of t. 208 00:11:29,850 --> 00:11:33,380 And r prime of t is the derivative. 209 00:11:33,380 --> 00:11:34,980 It's never going to stop. 210 00:11:34,980 --> 00:11:36,000 The velocity. 211 00:11:36,000 --> 00:11:38,030 I'm talking about this piece of information. 212 00:11:38,030 --> 00:11:42,490 Velocity [INAUDIBLE] 0 means that drunken bug between time 213 00:11:42,490 --> 00:11:45,830 a and time b never stops. 214 00:11:45,830 --> 00:11:50,620 He stops at the end, but the end is b, is outside [INAUDIBLE]. 215 00:11:50,620 --> 00:11:54,740 So he stops at b and he falls. 216 00:11:54,740 --> 00:11:56,110 So I don't stop. 217 00:11:56,110 --> 00:11:59,080 I move on from time a to time b. 218 00:11:59,080 --> 00:12:01,386 I don't stop at all. 219 00:12:01,386 --> 00:12:02,832 Yes, sir. 220 00:12:02,832 --> 00:12:06,628 STUDENT: Wouldn't the derivative of that line at some point 221 00:12:06,628 --> 00:12:08,030 equal 0 where it flattens out? 222 00:12:08,030 --> 00:12:10,890 PROFESSOR: Let me draw very well. 223 00:12:10,890 --> 00:12:14,600 So at time r of t, this is the position vector. 224 00:12:14,600 --> 00:12:16,552 What is the derivative? 225 00:12:16,552 --> 00:12:19,580 The derivative represents the velocity vector. 226 00:12:19,580 --> 00:12:24,250 A beautiful thing about the velocity vector r prime of t 227 00:12:24,250 --> 00:12:26,910 is that it has a beautiful property. 228 00:12:26,910 --> 00:12:30,130 It's always tangent to the trajectory. 229 00:12:30,130 --> 00:12:32,610 So at every point you're going to have 230 00:12:32,610 --> 00:12:36,468 a velocity vector that is tangent to the trajectory. 231 00:12:36,468 --> 00:12:37,950 [INAUDIBLE] in physics. 232 00:12:37,950 --> 00:12:41,902 This r prime of t should never become 0. 233 00:12:41,902 --> 00:12:46,590 So you will never have a point instead of a segment 234 00:12:46,590 --> 00:12:51,070 when it comes to r prime. 235 00:12:51,070 --> 00:12:52,000 So you don't stop. 236 00:12:52,000 --> 00:12:58,560 237 00:12:58,560 --> 00:13:00,060 You are going to say, wait a minute? 238 00:13:00,060 --> 00:13:04,450 But are you always going to consider curves, regular curves 239 00:13:04,450 --> 00:13:06,220 in space? 240 00:13:06,220 --> 00:13:10,490 Regular curves in space. 241 00:13:10,490 --> 00:13:15,720 And by space, I know you guys mean the Euclidean three space. 242 00:13:15,720 --> 00:13:20,460 Actually, many times I will consider curves in plane. 243 00:13:20,460 --> 00:13:22,840 And the plane is part of the space. 244 00:13:22,840 --> 00:13:25,680 And you say, give us an example. 245 00:13:25,680 --> 00:13:28,210 I will give you an example right now. 246 00:13:28,210 --> 00:13:30,385 You're going to laugh how simple that is. 247 00:13:30,385 --> 00:13:33,180 248 00:13:33,180 --> 00:13:37,240 Now, I have another bug who is really happy, 249 00:13:37,240 --> 00:13:39,880 but it's not drunk at all. 250 00:13:39,880 --> 00:13:46,580 And this bug knows how to circle around a certain point 251 00:13:46,580 --> 00:13:49,310 at the same speed. 252 00:13:49,310 --> 00:13:51,800 So very organized bug. 253 00:13:51,800 --> 00:13:52,495 Yes, sir. 254 00:13:52,495 --> 00:13:55,014 STUDENT: Where did you get c prime? 255 00:13:55,014 --> 00:13:55,680 PROFESSOR: What? 256 00:13:55,680 --> 00:14:01,390 STUDENT: You have c prime is differentiable, is [INAUDIBLE]. 257 00:14:01,390 --> 00:14:02,070 PROFESSOR: c1. 258 00:14:02,070 --> 00:14:03,001 STUDENT: c1. 259 00:14:03,001 --> 00:14:03,750 PROFESSOR: OK. c1. 260 00:14:03,750 --> 00:14:09,246 This is the notation for any function that is differentiable 261 00:14:09,246 --> 00:14:11,860 and whose derivative is continuous. 262 00:14:11,860 --> 00:14:16,850 So again, give an example of a c1 function. 263 00:14:16,850 --> 00:14:18,330 STUDENT: x squared. 264 00:14:18,330 --> 00:14:19,220 PROFESSOR: Yeah. 265 00:14:19,220 --> 00:14:20,670 On some real interval. 266 00:14:20,670 --> 00:14:27,300 How about absolute value of x over the real line? 267 00:14:27,300 --> 00:14:29,506 What's the problem with that? 268 00:14:29,506 --> 00:14:30,940 [INTERPOSING VOICES] 269 00:14:30,940 --> 00:14:33,280 PROFESSOR: It's not differentiable at 0. 270 00:14:33,280 --> 00:14:36,640 OK, so we'll talk a little bit later about smoothness. 271 00:14:36,640 --> 00:14:39,848 It's a little bit delicate as a notion. 272 00:14:39,848 --> 00:14:42,690 It's really beautiful on the other side. 273 00:14:42,690 --> 00:14:49,830 Let's find the nice picture trajectory for the bug. 274 00:14:49,830 --> 00:14:51,460 This is a ladybug. 275 00:14:51,460 --> 00:14:54,170 I cannot draw her, anyway. 276 00:14:54,170 --> 00:14:56,870 She is moving along this circle. 277 00:14:56,870 --> 00:15:00,560 And I'll give you the law of motion. 278 00:15:00,560 --> 00:15:06,960 And that reminds me of a student who told me, what 279 00:15:06,960 --> 00:15:08,630 do I care about law of motion? 280 00:15:08,630 --> 00:15:10,790 He never had me as a teacher, obviously. 281 00:15:10,790 --> 00:15:14,080 But he was telling me, well, after I graduated, 282 00:15:14,080 --> 00:15:18,460 I always thought, what do I care about the law of motion? 283 00:15:18,460 --> 00:15:20,650 I mean, I took calculus. 284 00:15:20,650 --> 00:15:24,240 Everything was about the law of motion. 285 00:15:24,240 --> 00:15:27,340 I'm sorry, you should care about the law of motion. 286 00:15:27,340 --> 00:15:30,280 Once you're not there anymore, absolutely you don't care. 287 00:15:30,280 --> 00:15:33,076 But why do you want to [INAUDIBLE] doing calculus? 288 00:15:33,076 --> 00:15:34,700 When you bring [INAUDIBLE] to calculus, 289 00:15:34,700 --> 00:15:37,460 when you walk into calculus, it's law of motion 290 00:15:37,460 --> 00:15:39,990 everywhere whether you like it or not. 291 00:15:39,990 --> 00:15:49,060 So let's try cosine t sine t and z to b 1. 292 00:15:49,060 --> 00:15:52,240 Let's make it 1 to make your life easier. 293 00:15:52,240 --> 00:15:54,470 What kind of curve is this and why am I 294 00:15:54,470 --> 00:15:58,290 claiming that the ladybug following this curve 295 00:15:58,290 --> 00:16:00,620 is moving at a constant speed? 296 00:16:00,620 --> 00:16:01,490 Oh my god. 297 00:16:01,490 --> 00:16:02,619 Go ahead, Alexander. 298 00:16:02,619 --> 00:16:03,660 STUDENT: That's a circle. 299 00:16:03,660 --> 00:16:05,120 PROFESSOR: That's the circle. 300 00:16:05,120 --> 00:16:06,520 It's more than a circle. 301 00:16:06,520 --> 00:16:07,920 It's a parametrized circle. 302 00:16:07,920 --> 00:16:10,490 It's a vector-valued function. 303 00:16:10,490 --> 00:16:15,460 Now, like every mathematician I should specify the domain. 304 00:16:15,460 --> 00:16:18,140 I am just winding around one time, 305 00:16:18,140 --> 00:16:20,680 and I stop where I started. 306 00:16:20,680 --> 00:16:24,430 So I better be smart and realize time is not infinity. 307 00:16:24,430 --> 00:16:25,540 It could be. 308 00:16:25,540 --> 00:16:28,130 I'm wrapping around the circle infinitely many times. 309 00:16:28,130 --> 00:16:30,320 They do that in topology actually when 310 00:16:30,320 --> 00:16:34,310 you're going to be-- seniors takes topology. 311 00:16:34,310 --> 00:16:38,420 But I'm not going around in circles only one time. 312 00:16:38,420 --> 00:16:41,060 So my time will start at 0 when I 313 00:16:41,060 --> 00:16:44,820 start my motion and end at 2 pi seconds 314 00:16:44,820 --> 00:16:47,930 if the time is in seconds 315 00:16:47,930 --> 00:16:52,100 So I say r is defined on the interval I which 316 00:16:52,100 --> 00:16:53,670 is-- say it again, Magdalena. 317 00:16:53,670 --> 00:16:55,210 You just said it. 318 00:16:55,210 --> 00:16:56,110 STUDENT: 0. 319 00:16:56,110 --> 00:16:58,070 PROFESSOR: 0 to pi. 320 00:16:58,070 --> 00:17:01,050 If you want to take 0 together, fine. 321 00:17:01,050 --> 00:17:05,920 But for consistency, let's take it like before, 0 to 2 pi. 322 00:17:05,920 --> 00:17:07,775 I'm actually excluding the origin. 323 00:17:07,775 --> 00:17:10,550 324 00:17:10,550 --> 00:17:12,200 And with values in R3. 325 00:17:12,200 --> 00:17:17,618 Although, this is a [? plane ?] curve, z will be constant. 326 00:17:17,618 --> 00:17:19,868 Do I care about that very much? 327 00:17:19,868 --> 00:17:21,970 You will see the beauty of it. 328 00:17:21,970 --> 00:17:25,770 I have the velocity vector being really pretty. 329 00:17:25,770 --> 00:17:28,078 What is the velocity vector? 330 00:17:28,078 --> 00:17:30,180 STUDENT: [INAUDIBLE]. 331 00:17:30,180 --> 00:17:31,610 PROFESSOR: Negative sign t. 332 00:17:31,610 --> 00:17:32,475 Thank you. 333 00:17:32,475 --> 00:17:33,350 STUDENT: [INAUDIBLE]. 334 00:17:33,350 --> 00:17:35,860 PROFESSOR: Cosine t. 335 00:17:35,860 --> 00:17:37,200 And 0, finally. 336 00:17:37,200 --> 00:17:40,930 Because as you saw very well in the book, 337 00:17:40,930 --> 00:17:43,940 the way we compute the velocity vector 338 00:17:43,940 --> 00:17:47,311 is by taking x of t, y of t, z of t 339 00:17:47,311 --> 00:17:50,130 and differentiating them in terms of time. 340 00:17:50,130 --> 00:17:54,480 341 00:17:54,480 --> 00:17:55,000 Good. 342 00:17:55,000 --> 00:17:58,260 Is this a regular function? 343 00:17:58,260 --> 00:18:02,520 As the bug moves between time 0 and time equals 2 pi, 344 00:18:02,520 --> 00:18:06,588 is the bug ever going to stop between these times? 345 00:18:06,588 --> 00:18:07,427 STUDENT: No. 346 00:18:07,427 --> 00:18:08,010 PROFESSOR: No. 347 00:18:08,010 --> 00:18:08,676 How do you know? 348 00:18:08,676 --> 00:18:10,510 You guys are faster than me, right? 349 00:18:10,510 --> 00:18:11,270 What did you do? 350 00:18:11,270 --> 00:18:12,810 You did the speed. 351 00:18:12,810 --> 00:18:14,210 What's the relationship? 352 00:18:14,210 --> 00:18:16,539 What's the difference between velocity and speed? 353 00:18:16,539 --> 00:18:18,580 STUDENT: Speed is the absolute value [INAUDIBLE]. 354 00:18:18,580 --> 00:18:19,280 PROFESSOR: Wonderful. 355 00:18:19,280 --> 00:18:20,076 This is very good. 356 00:18:20,076 --> 00:18:21,950 You should tell everybody that because people 357 00:18:21,950 --> 00:18:23,745 confuse that left and right. 358 00:18:23,745 --> 00:18:26,792 So the velocity is a vector, like you 359 00:18:26,792 --> 00:18:28,180 learned in engineering. 360 00:18:28,180 --> 00:18:29,960 You learned in physics. 361 00:18:29,960 --> 00:18:31,090 Velocity is a vector. 362 00:18:31,090 --> 00:18:32,080 It changes direction. 363 00:18:32,080 --> 00:18:33,960 I'm going to Amarillo this way. 364 00:18:33,960 --> 00:18:34,900 I'm driving. 365 00:18:34,900 --> 00:18:37,340 The velocity will be a vector pointing this way. 366 00:18:37,340 --> 00:18:40,790 As I come back, will point the opposite way. 367 00:18:40,790 --> 00:18:43,815 The speed will be a scalar, not a vector. 368 00:18:43,815 --> 00:18:46,460 It's a magnitude of a velocity vector. 369 00:18:46,460 --> 00:18:48,060 So say it again, Magdalena. 370 00:18:48,060 --> 00:18:49,240 What is the speed? 371 00:18:49,240 --> 00:18:55,540 The speed is the magnitude of the velocity vector. 372 00:18:55,540 --> 00:18:58,585 It's a scalar. 373 00:18:58,585 --> 00:19:00,780 Speed. 374 00:19:00,780 --> 00:19:02,820 Speed. 375 00:19:02,820 --> 00:19:06,040 I heard that before in cars, in the movie Cars. 376 00:19:06,040 --> 00:19:10,700 Anyway, r prime of t magnitude. 377 00:19:10,700 --> 00:19:12,440 In magnitude. 378 00:19:12,440 --> 00:19:17,360 Remember, there is a big difference between the velocity 379 00:19:17,360 --> 00:19:19,095 as the notion. 380 00:19:19,095 --> 00:19:22,770 Velocity is a vector. 381 00:19:22,770 --> 00:19:25,570 The speed is a magnitude, is a scalar. 382 00:19:25,570 --> 00:19:27,800 I'm going to go ahead and erase that 383 00:19:27,800 --> 00:19:33,190 and I'm going to ask you what the speed is 384 00:19:33,190 --> 00:19:36,200 for my fellow over here. 385 00:19:36,200 --> 00:19:40,612 What is the speed of a trajectory 386 00:19:40,612 --> 00:19:46,350 of the bug who is sober and moves at the constant speed? 387 00:19:46,350 --> 00:19:46,850 OK. 388 00:19:46,850 --> 00:19:49,210 As I already told you, it's constant. 389 00:19:49,210 --> 00:19:50,390 What is that constant? 390 00:19:50,390 --> 00:19:53,800 391 00:19:53,800 --> 00:19:56,590 What's the constant speed I was talking about? 392 00:19:56,590 --> 00:19:58,790 STUDENT: [INAUDIBLE]. 393 00:19:58,790 --> 00:20:01,840 PROFESSOR: I say the magnitude of that. 394 00:20:01,840 --> 00:20:04,360 I'm too lazy to write it down. 395 00:20:04,360 --> 00:20:06,225 It's a Tuesday, almost morning. 396 00:20:06,225 --> 00:20:10,240 So I go square root of minus I squared 397 00:20:10,240 --> 00:20:12,050 plus cosine squared plus 0. 398 00:20:12,050 --> 00:20:13,780 I don't need to write that down. 399 00:20:13,780 --> 00:20:15,270 You write it down. 400 00:20:15,270 --> 00:20:16,580 And how much is that? 401 00:20:16,580 --> 00:20:17,455 STUDENT: [INAUDIBLE]. 402 00:20:17,455 --> 00:20:18,170 PROFESSOR: 1. 403 00:20:18,170 --> 00:20:24,600 So I love this curve because in mathematician slang, 404 00:20:24,600 --> 00:20:28,800 especially in [? a geometer's ?] slang-- and my area 405 00:20:28,800 --> 00:20:30,240 is differential geometry. 406 00:20:30,240 --> 00:20:35,250 So in a way, I do calculus in R3 every day on a daily basis. 407 00:20:35,250 --> 00:20:37,030 So I have what? 408 00:20:37,030 --> 00:20:42,572 This is a special kind of curve. 409 00:20:42,572 --> 00:20:46,380 It's a curve parameterized in arc length. 410 00:20:46,380 --> 00:20:57,840 So definition, we say that a curve in R3, 411 00:20:57,840 --> 00:21:10,130 or Rn, well anyway, is parameterized in arc length. 412 00:21:10,130 --> 00:21:12,940 413 00:21:12,940 --> 00:21:13,440 When? 414 00:21:13,440 --> 00:21:14,450 Say it again, Magdalena. 415 00:21:14,450 --> 00:21:32,270 Whenever, if and only if, its speed is constantly 1. 416 00:21:32,270 --> 00:21:36,150 417 00:21:36,150 --> 00:21:40,630 So this is an example where the speed is 1. 418 00:21:40,630 --> 00:21:45,548 In such cases, we avoid the notation with t. 419 00:21:45,548 --> 00:21:46,470 You say, oh my god. 420 00:21:46,470 --> 00:21:47,460 Why? 421 00:21:47,460 --> 00:21:50,100 When the curve is parameterized in arc length, 422 00:21:50,100 --> 00:21:54,970 from now on the we will actually try 423 00:21:54,970 --> 00:21:58,920 to use s whatever we know it's an arc length. 424 00:21:58,920 --> 00:22:00,850 We use s instead of t. 425 00:22:00,850 --> 00:22:05,210 So I'm sorry for the people who cannot change that, 426 00:22:05,210 --> 00:22:08,700 but you should all be able t change that. 427 00:22:08,700 --> 00:22:12,550 So everything will be in s because we just 428 00:22:12,550 --> 00:22:15,450 discovered [? Discovery Channel, ?] we 429 00:22:15,450 --> 00:22:19,400 just discovered that speed is 1. 430 00:22:19,400 --> 00:22:24,300 So there is something special about this s. 431 00:22:24,300 --> 00:22:29,380 432 00:22:29,380 --> 00:22:32,720 In this example-- oh, you can rewrite the whole example 433 00:22:32,720 --> 00:22:37,370 if you want in s so you don't have to smudge the paper. 434 00:22:37,370 --> 00:22:38,830 OK, it's beautiful. 435 00:22:38,830 --> 00:22:41,420 So I am already arc length. 436 00:22:41,420 --> 00:22:43,950 And in that case, I'm going to call my time parameter 437 00:22:43,950 --> 00:22:46,190 little s. s comes from special. 438 00:22:46,190 --> 00:22:48,460 No, s comes from speed [INAUDIBLE]. 439 00:22:48,460 --> 00:22:51,590 STUDENT: So you use s when it's [INAUDIBLE]? 440 00:22:51,590 --> 00:22:57,140 PROFESSOR: We use s whenever the speed of that curve will be 1. 441 00:22:57,140 --> 00:22:58,140 STUDENT: So [INAUDIBLE]. 442 00:22:58,140 --> 00:23:00,473 PROFESSOR: And we call that arc length parameterization. 443 00:23:00,473 --> 00:23:02,970 444 00:23:02,970 --> 00:23:06,240 I'm moving into the duration of your final thoughts. 445 00:23:06,240 --> 00:23:07,961 Yes, sir. 446 00:23:07,961 --> 00:23:09,502 STUDENT: When we get the question, so 447 00:23:09,502 --> 00:23:10,627 before solving [INAUDIBLE]. 448 00:23:10,627 --> 00:23:13,140 449 00:23:13,140 --> 00:23:14,460 PROFESSOR: We don't know. 450 00:23:14,460 --> 00:23:17,920 That's why it was our discovery that, hey, at the end 451 00:23:17,920 --> 00:23:22,430 it is an arc length, so I better change [INAUDIBLE] t into s 452 00:23:22,430 --> 00:23:26,920 because that will help me in the future remember to do that. 453 00:23:26,920 --> 00:23:30,380 Every time I have arc length, that it means speed 1. 454 00:23:30,380 --> 00:23:33,390 I will call it s instead of y. 455 00:23:33,390 --> 00:23:34,920 There is a reason for that. 456 00:23:34,920 --> 00:23:36,550 I'm going to erase the definition 457 00:23:36,550 --> 00:23:42,865 and I'm going to give you the-- more or less, 458 00:23:42,865 --> 00:23:45,580 the explanation that my physics professor gave me. 459 00:23:45,580 --> 00:23:50,170 Because as a freshman, my mathematics professor 460 00:23:50,170 --> 00:23:54,450 in that area, in geometry, was not very, very active. 461 00:23:54,450 --> 00:23:57,330 But practically, what my physics professor told me is that, 462 00:23:57,330 --> 00:24:05,900 hey, I would like to have some sort of a uniform tangent 463 00:24:05,900 --> 00:24:09,830 vector, something that is standardized to be in speed 1. 464 00:24:09,830 --> 00:24:15,860 So I would like that tangent vector to be important to us. 465 00:24:15,860 --> 00:24:19,845 And if r is an arc length, then r 466 00:24:19,845 --> 00:24:24,110 prime would be that unit vector that I'm talking about. 467 00:24:24,110 --> 00:24:30,790 So he introduced for any r of t, which is x of t, y of t, 468 00:24:30,790 --> 00:24:32,120 z of t. 469 00:24:32,120 --> 00:24:36,580 My physics professor introduced the following terminology. 470 00:24:36,580 --> 00:24:42,870 The tangent unit vector for a regular curve-- 471 00:24:42,870 --> 00:24:46,630 he was very well-organized I might add about him-- 472 00:24:46,630 --> 00:24:52,670 is by definition r prime of t as a vector 473 00:24:52,670 --> 00:24:54,380 divided by the speed of the vector. 474 00:24:54,380 --> 00:24:56,250 So what is he doing? 475 00:24:56,250 --> 00:24:58,535 He is unitarizing the velocity. 476 00:24:58,535 --> 00:25:00,130 Say it again, Magdalena. 477 00:25:00,130 --> 00:25:03,210 He has unitarized the velocity in order 478 00:25:03,210 --> 00:25:08,500 to make research more consistent from the viewpoint of Frenet 479 00:25:08,500 --> 00:25:10,040 frame. 480 00:25:10,040 --> 00:25:12,520 So in Frenet frame, you will see-- you probably 481 00:25:12,520 --> 00:25:14,310 learned about the Frenet frame if you 482 00:25:14,310 --> 00:25:18,260 are a mechanics major, or some solid mechanics or physics 483 00:25:18,260 --> 00:25:19,130 major. 484 00:25:19,130 --> 00:25:22,600 The Frenet frame is an orthogonal frame 485 00:25:22,600 --> 00:25:28,750 moving along a line in time where the three components are 486 00:25:28,750 --> 00:25:33,200 t, and the principal normal vector, and b the [INAUDIBLE]. 487 00:25:33,200 --> 00:25:35,510 We only know of the first of them, which 488 00:25:35,510 --> 00:25:38,650 is T, which is a unit vector. 489 00:25:38,650 --> 00:25:40,320 Say it again who it was. 490 00:25:40,320 --> 00:25:44,980 It was the velocity vector divided by its magnitude. 491 00:25:44,980 --> 00:25:47,330 So the velocity vector could be any wild, crazy vector 492 00:25:47,330 --> 00:25:54,560 that's tangent to the trajectory at the point where you are. 493 00:25:54,560 --> 00:25:58,120 His magnitude varies from one point to the other. 494 00:25:58,120 --> 00:25:59,910 He's absolutely crazy. 495 00:25:59,910 --> 00:26:01,430 He or she, the velocity vector. 496 00:26:01,430 --> 00:26:02,290 Yes, sir. 497 00:26:02,290 --> 00:26:03,165 STUDENT: [INAUDIBLE]. 498 00:26:03,165 --> 00:26:06,849 499 00:26:06,849 --> 00:26:07,515 PROFESSOR: Here? 500 00:26:07,515 --> 00:26:08,332 Here? 501 00:26:08,332 --> 00:26:09,415 STUDENT: Yeah, down there. 502 00:26:09,415 --> 00:26:10,623 PROFESSOR: D-E-F, definition. 503 00:26:10,623 --> 00:26:12,815 That's how a mathematician defines things. 504 00:26:12,815 --> 00:26:18,340 So to define you write def on top of an equality sign 505 00:26:18,340 --> 00:26:20,960 or double dot equal. 506 00:26:20,960 --> 00:26:23,630 That's a formal way a mathematician introduces 507 00:26:23,630 --> 00:26:24,780 a definition. 508 00:26:24,780 --> 00:26:27,600 Well, he was a physicist, but he does math. 509 00:26:27,600 --> 00:26:29,330 So what do we do? 510 00:26:29,330 --> 00:26:32,500 We say all the blue guys that are not equal, 511 00:26:32,500 --> 00:26:34,580 divide yourselves by your magnitude. 512 00:26:34,580 --> 00:26:39,720 And I'm going to have the T here is next one, 513 00:26:39,720 --> 00:26:42,750 the T here is next one, the T here is next one. 514 00:26:42,750 --> 00:26:43,580 They are all equal. 515 00:26:43,580 --> 00:26:51,200 So that T changes direction, but its magnitude will always be 1. 516 00:26:51,200 --> 00:26:51,700 Right? 517 00:26:51,700 --> 00:26:55,000 Know that the magnitude-- that's what unit vector means, 518 00:26:55,000 --> 00:26:58,200 the magnitude is 1. 519 00:26:58,200 --> 00:27:00,706 Why am I so happy about that? 520 00:27:00,706 --> 00:27:03,940 Well let me tell you that we can have 521 00:27:03,940 --> 00:27:07,480 another parametrization and another parametrization 522 00:27:07,480 --> 00:27:11,120 and another parametrization of the same curve. 523 00:27:11,120 --> 00:27:12,260 Say what? 524 00:27:12,260 --> 00:27:14,820 The parametrization of a curve is not unique? 525 00:27:14,820 --> 00:27:15,717 No. 526 00:27:15,717 --> 00:27:18,810 There are infinitely many parametrizations 527 00:27:18,810 --> 00:27:21,930 for a physical curve. 528 00:27:21,930 --> 00:27:34,158 There are infinitely many parametrizations 529 00:27:34,158 --> 00:27:39,730 for an even physical curve. 530 00:27:39,730 --> 00:27:43,058 531 00:27:43,058 --> 00:27:45,030 Like [INAUDIBLE] the regular one? 532 00:27:45,030 --> 00:27:47,495 Well let me give you another example that 533 00:27:47,495 --> 00:27:51,475 says that this is currently R of T 534 00:27:51,475 --> 00:27:58,290 equals cosine 5T sine 5T and 1. 535 00:27:58,290 --> 00:27:59,060 Why 1? 536 00:27:59,060 --> 00:28:02,560 I still want to have the same physical curve. 537 00:28:02,560 --> 00:28:03,690 What's different, guys? 538 00:28:03,690 --> 00:28:06,740 Look at that and then say oh OK, is this 539 00:28:06,740 --> 00:28:11,890 the same curve as a physical curve? 540 00:28:11,890 --> 00:28:13,410 What's different in this case? 541 00:28:13,410 --> 00:28:14,800 I'm still here. 542 00:28:14,800 --> 00:28:16,620 It's still the [? red ?] physical curve 543 00:28:16,620 --> 00:28:18,221 I'm moving along. 544 00:28:18,221 --> 00:28:18,970 What is different? 545 00:28:18,970 --> 00:28:19,886 STUDENT: The velocity. 546 00:28:19,886 --> 00:28:20,960 PROFESSOR: The velocity. 547 00:28:20,960 --> 00:28:23,880 The velocity and actually the speed. 548 00:28:23,880 --> 00:28:29,380 I'm moving faster or slower, I don't know, we have to decide. 549 00:28:29,380 --> 00:28:34,230 Now how do I realize how many times 550 00:28:34,230 --> 00:28:36,380 I'm moving along this curve? 551 00:28:36,380 --> 00:28:39,740 I can be smart and say hey, I'm not stupid. 552 00:28:39,740 --> 00:28:43,420 I know how to move only one time and stop where I started. 553 00:28:43,420 --> 00:28:47,040 So if I start with my T in the interval 554 00:28:47,040 --> 00:28:52,630 zero-- I start at zero, where do I stop? 555 00:28:52,630 --> 00:28:54,456 I can hear your brain buzzing. 556 00:28:54,456 --> 00:28:55,330 STUDENT: [INAUDIBLE]. 557 00:28:55,330 --> 00:28:57,901 PROFESSOR: 2pi over 5. 558 00:28:57,901 --> 00:28:58,840 Why is that? 559 00:28:58,840 --> 00:29:00,260 Excellent answer. 560 00:29:00,260 --> 00:29:03,354 STUDENT: Because when you plug it in, it's [INAUDIBLE]. 561 00:29:03,354 --> 00:29:05,150 PROFESSOR: 5 times 2pi over 5. 562 00:29:05,150 --> 00:29:06,130 That's where I stop. 563 00:29:06,130 --> 00:29:08,364 So this is not the same interval as before. 564 00:29:08,364 --> 00:29:09,678 Are you guys with me? 565 00:29:09,678 --> 00:29:16,650 This is a new guy, which is called J. Oh, all right. 566 00:29:16,650 --> 00:29:19,460 So there is a relationship between the T 567 00:29:19,460 --> 00:29:23,480 and the S. That's why I use different notations. 568 00:29:23,480 --> 00:29:26,740 And I wish my teachers started it just 569 00:29:26,740 --> 00:29:29,794 like that when I took math analysis as a freshman, 570 00:29:29,794 --> 00:29:30,630 or calculus. 571 00:29:30,630 --> 00:29:32,330 That's calculus. 572 00:29:32,330 --> 00:29:35,700 Because what they started with was a diagram. 573 00:29:35,700 --> 00:29:37,300 What kind of diagram? 574 00:29:37,300 --> 00:29:41,840 Say OK, the parametrizations are both 575 00:29:41,840 --> 00:29:45,060 starting from different intervals. 576 00:29:45,060 --> 00:29:47,550 And first I have the parametrization 577 00:29:47,550 --> 00:29:50,290 from I going to our 3. 578 00:29:50,290 --> 00:29:53,350 And that's called-- how did we baptize that? 579 00:29:53,350 --> 00:29:57,640 R. And the other one, from J to R3, 580 00:29:57,640 --> 00:30:01,790 we call that big R. They're both vectors. 581 00:30:01,790 --> 00:30:05,170 And hey guys, we should have some sort 582 00:30:05,170 --> 00:30:08,720 of correspondence functions between I 583 00:30:08,720 --> 00:30:14,145 and J that are both 1 to 1, and they are 1 being [INAUDIBLE] 584 00:30:14,145 --> 00:30:16,420 the other. 585 00:30:16,420 --> 00:30:18,310 I swear to God, when they started 586 00:30:18,310 --> 00:30:21,000 with this theoretical model, I didn't understand 587 00:30:21,000 --> 00:30:23,190 the motivation at all. 588 00:30:23,190 --> 00:30:25,220 At all. 589 00:30:25,220 --> 00:30:27,620 Now with an example, I can get you 590 00:30:27,620 --> 00:30:30,890 closer to the motivation of such a diagram. 591 00:30:30,890 --> 00:30:34,700 So where does our primary S live? 592 00:30:34,700 --> 00:30:39,130 S lives in I, and T lives in J. So I 593 00:30:39,130 --> 00:30:42,790 have to have a correspondence that takes S to T or T to S. 594 00:30:42,790 --> 00:30:46,352 STUDENT: Wait I thought since R of T 595 00:30:46,352 --> 00:30:48,477 is also pretty much [INAUDIBLE] that we should also 596 00:30:48,477 --> 00:30:49,590 use S [INAUDIBLE]. 597 00:30:49,590 --> 00:30:53,040 PROFESSOR: It's very-- actually it's very easy. 598 00:30:53,040 --> 00:30:55,880 This is 5T. 599 00:30:55,880 --> 00:31:02,250 And we cannot use S instead of this T, 600 00:31:02,250 --> 00:31:05,080 because if we use S instead of this T, 601 00:31:05,080 --> 00:31:07,690 and we compute the speed, we get 5. 602 00:31:07,690 --> 00:31:10,940 So it cannot be called S. This is very important. 603 00:31:10,940 --> 00:31:15,135 So T is not an arc length parameter. 604 00:31:15,135 --> 00:31:18,170 I wonder what the speed will be for this guy. 605 00:31:18,170 --> 00:31:20,470 So who wants to compute R prime of T? 606 00:31:20,470 --> 00:31:22,580 Nobody, but I'll force you to. 607 00:31:22,580 --> 00:31:26,520 And the magnitude of that will be god knows what. 608 00:31:26,520 --> 00:31:27,740 I claim it's 5. 609 00:31:27,740 --> 00:31:30,110 Maybe I'm wrong. 610 00:31:30,110 --> 00:31:31,270 I did this in my head. 611 00:31:31,270 --> 00:31:33,150 I have to do it on paper, right. 612 00:31:33,150 --> 00:31:35,310 So I have what? 613 00:31:35,310 --> 00:31:38,730 I have to differentiate component-wise. 614 00:31:38,730 --> 00:31:42,250 And I have [INAUDIBLE] that, because I'm running out of gas. 615 00:31:42,250 --> 00:31:43,030 STUDENT: Minus 5-- 616 00:31:43,030 --> 00:31:45,590 PROFESSOR: Minus 5, very good. 617 00:31:45,590 --> 00:31:47,720 Sine of 5T. 618 00:31:47,720 --> 00:31:49,320 What have we applied? 619 00:31:49,320 --> 00:31:51,510 In case you don't know that, out. 620 00:31:51,510 --> 00:31:52,770 That was Calc 1. 621 00:31:52,770 --> 00:31:53,590 Chain rule. 622 00:31:53,590 --> 00:31:55,420 Right? 623 00:31:55,420 --> 00:32:00,080 So 5 times cosine 5T. 624 00:32:00,080 --> 00:32:03,720 And finally, 1 prime, which is 0. 625 00:32:03,720 --> 00:32:09,850 Now let's be brave and write the whole thing down. 626 00:32:09,850 --> 00:32:13,150 I know I'm lazy today, but I'm going to have to do something. 627 00:32:13,150 --> 00:32:13,650 Right? 628 00:32:13,650 --> 00:32:18,390 So I'll say minus 5 sine 5T is all squared. 629 00:32:18,390 --> 00:32:20,880 Let me take it and square it. 630 00:32:20,880 --> 00:32:23,886 Because I see one face is confused. 631 00:32:23,886 --> 00:32:26,840 And since one face is confused, it 632 00:32:26,840 --> 00:32:29,970 doesn't matter that the others are not confused. 633 00:32:29,970 --> 00:32:31,060 OK? 634 00:32:31,060 --> 00:32:36,280 So I have square root of this plus square of [INAUDIBLE] plus 635 00:32:36,280 --> 00:32:38,540 [INAUDIBLE] computing the magnitude. 636 00:32:38,540 --> 00:32:39,750 What do I get out of here? 637 00:32:39,750 --> 00:32:40,482 STUDENT: Five. 638 00:32:40,482 --> 00:32:41,170 PROFESSOR: Five. 639 00:32:41,170 --> 00:32:41,780 Excellent. 640 00:32:41,780 --> 00:32:45,430 This is 5 sine squared plus 5 cosine squared. 641 00:32:45,430 --> 00:32:49,670 Now yes, then I have 5 times 1. 642 00:32:49,670 --> 00:32:54,550 So I have square root of 25 here will be 5. 643 00:32:54,550 --> 00:32:55,700 What is 5? 644 00:32:55,700 --> 00:33:03,480 5 is the speed of the [? bug ?] along the same physical curve 645 00:33:03,480 --> 00:33:04,640 the other way around. 646 00:33:04,640 --> 00:33:06,990 The second time around. 647 00:33:06,990 --> 00:33:10,210 Now can you tell me the relationship between T and S? 648 00:33:10,210 --> 00:33:12,578 They are related. 649 00:33:12,578 --> 00:33:19,220 They are like if you're my uncle, then I'm your niece. 650 00:33:19,220 --> 00:33:21,290 It's the same way. 651 00:33:21,290 --> 00:33:23,015 It depends where you look at. 652 00:33:23,015 --> 00:33:26,040 T is a function of S, and S is a function of T. 653 00:33:26,040 --> 00:33:32,220 So it has to be a 1 to 1 correspondence between the two. 654 00:33:32,220 --> 00:33:38,235 Now any ideas of how I what to compute the-- how do I 655 00:33:38,235 --> 00:33:43,180 want to write the relationship between them. 656 00:33:43,180 --> 00:33:46,400 Well, S is a function of T, right? 657 00:33:46,400 --> 00:33:50,534 I just don't know what function of T that is. 658 00:33:50,534 --> 00:33:52,450 And I wish my professor had started like that, 659 00:33:52,450 --> 00:33:54,710 but he started with this diagram. 660 00:33:54,710 --> 00:33:58,890 So simply here you have S equals S of T, 661 00:33:58,890 --> 00:34:01,380 and here you have T equals T of S, 662 00:34:01,380 --> 00:34:03,170 the inverse of that function. 663 00:34:03,170 --> 00:34:05,820 And when you-- when somebody starts that 664 00:34:05,820 --> 00:34:09,560 without an example as a general diagram philosophy, 665 00:34:09,560 --> 00:34:12,050 then it's really, really tough. 666 00:34:12,050 --> 00:34:13,050 All right? 667 00:34:13,050 --> 00:34:16,050 So I'd like to know who S of T-- how 668 00:34:16,050 --> 00:34:19,530 in the world do I want to define that S of T. 669 00:34:19,530 --> 00:34:25,570 He spoonfed us S of T. I don't want to spoonfeed you anything. 670 00:34:25,570 --> 00:34:27,728 Because this is honors class, and you 671 00:34:27,728 --> 00:34:30,929 should be able to figure this out yourselves. 672 00:34:30,929 --> 00:34:35,840 So who is big R of T? 673 00:34:35,840 --> 00:34:42,199 Big R of T should be, what, should 674 00:34:42,199 --> 00:34:44,820 be the same thing in the end as R of S. 675 00:34:44,820 --> 00:34:56,690 But I should say maybe it's R of function T of S, right? 676 00:34:56,690 --> 00:34:59,655 Which is the same thing as R of S. So 677 00:34:59,655 --> 00:35:05,820 what should be the relationship between T and S? 678 00:35:05,820 --> 00:35:11,280 We have to call them-- one of them should be T equals T of S. 679 00:35:11,280 --> 00:35:12,520 How about this function? 680 00:35:12,520 --> 00:35:15,590 Give it a Greek name, what do you want. 681 00:35:15,590 --> 00:35:16,120 Alpha? 682 00:35:16,120 --> 00:35:16,670 Beta? 683 00:35:16,670 --> 00:35:16,800 What? 684 00:35:16,800 --> 00:35:17,675 STUDENT: [INAUDIBLE]. 685 00:35:17,675 --> 00:35:19,250 PROFESSOR: Alpha? 686 00:35:19,250 --> 00:35:19,790 Beta? 687 00:35:19,790 --> 00:35:20,289 Alpha? 688 00:35:20,289 --> 00:35:21,700 I don't know. 689 00:35:21,700 --> 00:35:25,520 So S going to T, alpha. 690 00:35:25,520 --> 00:35:27,270 And this is going to be alpha inverse. 691 00:35:27,270 --> 00:35:30,644 692 00:35:30,644 --> 00:35:32,090 Right? 693 00:35:32,090 --> 00:35:37,305 So T equals alpha of S. It's more elegant to call it 694 00:35:37,305 --> 00:35:44,980 like that than T of S. T equals alpha of S. Alpha of S. 695 00:35:44,980 --> 00:35:49,080 So from this thing, I realize that I 696 00:35:49,080 --> 00:35:54,296 get that R composed with alpha equals R. Say what? 697 00:35:54,296 --> 00:35:54,796 Magdalena? 698 00:35:54,796 --> 00:35:57,170 Yeah, yeah, that was pre-calculus. 699 00:35:57,170 --> 00:36:01,110 R composed with alpha equals little r. 700 00:36:01,110 --> 00:36:09,141 So how do I get a little r by composing R with alpha? 701 00:36:09,141 --> 00:36:12,087 How do we say that? 702 00:36:12,087 --> 00:36:17,488 Alpha followed by R. R composed with alpha. 703 00:36:17,488 --> 00:36:22,030 R of alpha of S equals R of S. Say it again. 704 00:36:22,030 --> 00:36:30,770 R of alpha of S, which is T-- this T is alpha of S-- equals 705 00:36:30,770 --> 00:36:31,420 R. 706 00:36:31,420 --> 00:36:39,190 This is the composition that we learned in pre-calc. 707 00:36:39,190 --> 00:36:40,925 Who can find me the definition of S? 708 00:36:40,925 --> 00:36:44,372 Because this may be a little bit hard. 709 00:36:44,372 --> 00:36:46,580 This may be a little bit hard. 710 00:36:46,580 --> 00:36:48,901 STUDENT: S [INAUDIBLE]. 711 00:36:48,901 --> 00:36:52,430 PROFESSOR: Eh, yeah, let me write it down. 712 00:36:52,430 --> 00:36:56,868 I want to find out what S of T is. 713 00:36:56,868 --> 00:36:59,940 714 00:36:59,940 --> 00:37:11,302 Equals what in terms of the function R of T. The one 715 00:37:11,302 --> 00:37:13,787 that's given here. 716 00:37:13,787 --> 00:37:14,781 Why is that? 717 00:37:14,781 --> 00:37:22,760 718 00:37:22,760 --> 00:37:26,060 Let's try some sort of chain rule, right? 719 00:37:26,060 --> 00:37:28,698 So what do I know I have? 720 00:37:28,698 --> 00:37:29,820 I have that. 721 00:37:29,820 --> 00:37:32,740 Look at that. 722 00:37:32,740 --> 00:37:39,070 R prime of S, which is the velocity of-- I 723 00:37:39,070 --> 00:37:43,840 erased it-- the velocity of R with respect to the arc length 724 00:37:43,840 --> 00:37:46,560 parameter is going to be what? 725 00:37:46,560 --> 00:37:52,090 R of alpha of S prime with respect to S, right? 726 00:37:52,090 --> 00:37:53,835 So I should put DDS. 727 00:37:53,835 --> 00:37:55,430 Well I'm a little bit lazy. 728 00:37:55,430 --> 00:37:58,190 Let's do it again. 729 00:37:58,190 --> 00:38:06,070 DDS, R of alpha of S. 730 00:38:06,070 --> 00:38:07,930 OK. 731 00:38:07,930 --> 00:38:11,060 And what do I have in this case? 732 00:38:11,060 --> 00:38:18,562 Well, I have R prime of-- who is alpha of S. T, [INAUDIBLE] of T 733 00:38:18,562 --> 00:38:27,035 and alpha of S times R prime of alpha 734 00:38:27,035 --> 00:38:30,400 of S times the prime outside. 735 00:38:30,400 --> 00:38:32,300 How do we prime in the chain rule? 736 00:38:32,300 --> 00:38:35,220 From the outside to the inside, one at a time. 737 00:38:35,220 --> 00:38:38,760 So I differentiated the outer shell, R prime, 738 00:38:38,760 --> 00:38:39,910 and then times what? 739 00:38:39,910 --> 00:38:41,390 Chain rule, guys. 740 00:38:41,390 --> 00:38:44,890 Alpha prime of S. Very good. 741 00:38:44,890 --> 00:38:50,490 Alpha prime of S. 742 00:38:50,490 --> 00:38:51,100 All right. 743 00:38:51,100 --> 00:38:54,750 So I would like to understand how 744 00:38:54,750 --> 00:39:02,640 I want to compute-- how I want to define S of T. If I take 745 00:39:02,640 --> 00:39:06,590 this in absolute value, R prime of S in absolute value 746 00:39:06,590 --> 00:39:11,990 equals R prime of T in absolute value times alpha prime of S 747 00:39:11,990 --> 00:39:14,562 in absolute value. 748 00:39:14,562 --> 00:39:15,145 What do I get? 749 00:39:15,145 --> 00:39:20,510 750 00:39:20,510 --> 00:39:22,406 Who is R prime of S? 751 00:39:22,406 --> 00:39:26,160 This is my original function in arc length, 752 00:39:26,160 --> 00:39:28,660 and that's the speed in arc length. 753 00:39:28,660 --> 00:39:30,980 What was the speed in arc length? 754 00:39:30,980 --> 00:39:31,820 STUDENT: One. 755 00:39:31,820 --> 00:39:33,900 PROFESSOR: One. 756 00:39:33,900 --> 00:39:37,082 And what is the speed in not in arc length? 757 00:39:37,082 --> 00:39:38,474 STUDENT: Five. 758 00:39:38,474 --> 00:39:41,810 PROFESSOR: In that case, this is going to be five. 759 00:39:41,810 --> 00:39:46,325 And so what is this alpha prime of S guy? 760 00:39:46,325 --> 00:39:47,200 STUDENT: [INAUDIBLE]. 761 00:39:47,200 --> 00:39:51,015 PROFESSOR: It's going to be 1/5. 762 00:39:51,015 --> 00:39:52,440 OK. 763 00:39:52,440 --> 00:39:52,960 All right. 764 00:39:52,960 --> 00:39:56,125 Actually alpha of S, who is that going to be? 765 00:39:56,125 --> 00:40:03,900 Alpha of S. 766 00:40:03,900 --> 00:40:06,610 Do you notice the correspondence? 767 00:40:06,610 --> 00:40:12,070 We simply have to re-define this as S. That's how it goes. 768 00:40:12,070 --> 00:40:14,627 That five times is nothing but S. 769 00:40:14,627 --> 00:40:17,012 STUDENT: How did you get the [INAUDIBLE]? 770 00:40:17,012 --> 00:40:21,450 PROFESSOR: Because 1 equals 5 times what? 771 00:40:21,450 --> 00:40:26,205 1, which is arc length speed, equals 5 times what? 772 00:40:26,205 --> 00:40:26,705 1/5. 773 00:40:26,705 --> 00:40:27,600 STUDENT: Yeah, but then where'd you get the 1? 774 00:40:27,600 --> 00:40:29,058 PROFESSOR: That's one way to do it. 775 00:40:29,058 --> 00:40:32,290 Oh, this is by definition, because little r means 776 00:40:32,290 --> 00:40:35,600 curve in arc length, and little s is the arc length parameter. 777 00:40:35,600 --> 00:40:39,170 By definition, that means you get speed 1. 778 00:40:39,170 --> 00:40:40,830 This was our assumption. 779 00:40:40,830 --> 00:40:44,140 So we could've gotten that much faster saying 780 00:40:44,140 --> 00:40:46,220 oh, well, forget about this diagram 781 00:40:46,220 --> 00:40:48,750 that you introduced-- and it's also in the book. 782 00:40:48,750 --> 00:40:52,960 Simply take 5T to BS, 5T to BS. 783 00:40:52,960 --> 00:40:56,320 Then I get my old friend, the curve. 784 00:40:56,320 --> 00:40:59,200 The arc length parameter is the curve. 785 00:40:59,200 --> 00:41:04,520 So this is the same as cosine of S, sine of S, and 1. 786 00:41:04,520 --> 00:41:07,650 So what is the correspondence between S and T? 787 00:41:07,650 --> 00:41:10,590 788 00:41:10,590 --> 00:41:14,930 Since S is 5T in this example, I'll 789 00:41:14,930 --> 00:41:16,397 put it-- where shall I put it. 790 00:41:16,397 --> 00:41:19,810 I'll put it here. 791 00:41:19,810 --> 00:41:22,640 S is 5T. 792 00:41:22,640 --> 00:41:24,784 I'll say S of T is 5T. 793 00:41:24,784 --> 00:41:28,088 794 00:41:28,088 --> 00:41:32,240 and T of S, what is T in terms of S? 795 00:41:32,240 --> 00:41:37,050 T in terms of S is S over 5. 796 00:41:37,050 --> 00:41:39,905 So instead of T of S, we call this alpha 797 00:41:39,905 --> 00:41:47,798 of S. So the correspondence between S and T, what is T? 798 00:41:47,798 --> 00:41:51,970 T is exactly S over 5 in this example. 799 00:41:51,970 --> 00:41:52,640 Say it again. 800 00:41:52,640 --> 00:41:55,190 T is exactly S over 5. 801 00:41:55,190 --> 00:41:57,644 So alpha of S would be S over 5. 802 00:41:57,644 --> 00:42:01,770 In this case, alpha prime of S would simply be 1 over 5. 803 00:42:01,770 --> 00:42:04,410 Oh, so that's how I got it. 804 00:42:04,410 --> 00:42:06,360 That's another way to get it. 805 00:42:06,360 --> 00:42:07,500 Much faster. 806 00:42:07,500 --> 00:42:09,290 Much simpler. 807 00:42:09,290 --> 00:42:13,640 So just think of replacing 5T by the S knowing 808 00:42:13,640 --> 00:42:19,020 that you put S here, the whole thing will have speed of 1. 809 00:42:19,020 --> 00:42:19,610 All right. 810 00:42:19,610 --> 00:42:21,560 So what do I do? 811 00:42:21,560 --> 00:42:24,640 I say OK, alpha prime of S is 1 over 5. 812 00:42:24,640 --> 00:42:28,265 The whole chain rule also spit out alpha prime of S 813 00:42:28,265 --> 00:42:29,500 to B1 over 5. 814 00:42:29,500 --> 00:42:32,540 Now I understand the relationship between S and T. 815 00:42:32,540 --> 00:42:33,685 It's very simple. 816 00:42:33,685 --> 00:42:39,800 S is 5T in this example, or T equals S over 5. 817 00:42:39,800 --> 00:42:40,300 OK? 818 00:42:40,300 --> 00:42:46,430 So if somebody gives you a curve that looks like cosine 5T, sine 819 00:42:46,430 --> 00:42:52,400 5T, 1, and that is in speed 5, as we were able to find, 820 00:42:52,400 --> 00:42:56,800 how do you re-parametrize that in arc length? 821 00:42:56,800 --> 00:43:01,490 You just change something inside so 822 00:43:01,490 --> 00:43:08,190 that you make this curve be representative-- representable 823 00:43:08,190 --> 00:43:12,328 as little r of S. This is in arc length. 824 00:43:12,328 --> 00:43:13,795 In arc length. 825 00:43:13,795 --> 00:43:17,700 826 00:43:17,700 --> 00:43:18,200 OK. 827 00:43:18,200 --> 00:43:20,330 Finally, this is just an example. 828 00:43:20,330 --> 00:43:23,680 Can you tell me how that arc length parameter 829 00:43:23,680 --> 00:43:25,870 is introduced in general? 830 00:43:25,870 --> 00:43:29,712 What is S of T by definition? 831 00:43:29,712 --> 00:43:34,200 What if I have something really wild? 832 00:43:34,200 --> 00:43:36,410 How do I get to that S of T by definition? 833 00:43:36,410 --> 00:43:38,948 834 00:43:38,948 --> 00:43:41,358 What is S of T in terms of the function R? 835 00:43:41,358 --> 00:43:45,230 STUDENT: [INAUDIBLE] velocity [? of the ?] [INAUDIBLE]? 836 00:43:45,230 --> 00:43:47,840 PROFESSOR: S prime of T will be one of the [INAUDIBLE]. 837 00:43:47,840 --> 00:43:48,670 STUDENT: Yes. 838 00:43:48,670 --> 00:43:49,340 PROFESSOR: OK. 839 00:43:49,340 --> 00:43:58,770 So let's see what we have if we define S of T 840 00:43:58,770 --> 00:44:12,460 as being integral from 0 to T of the speed R prime of T. 841 00:44:12,460 --> 00:44:14,330 And instead of T, we put tau. 842 00:44:14,330 --> 00:44:14,830 Right? 843 00:44:14,830 --> 00:44:15,830 P tau. 844 00:44:15,830 --> 00:44:18,330 STUDENT: What is that? 845 00:44:18,330 --> 00:44:20,450 PROFESSOR: We cannot put T, T, and T. 846 00:44:20,450 --> 00:44:21,264 STUDENT: Oh. 847 00:44:21,264 --> 00:44:22,080 PROFESSOR: OK? 848 00:44:22,080 --> 00:44:25,700 So tau is the Greek T that runs between zero 849 00:44:25,700 --> 00:44:29,490 and T. This is the definition of S 850 00:44:29,490 --> 00:44:44,196 of T. General definition of the arc length parameter 851 00:44:44,196 --> 00:44:49,652 that is according to the chain rule, given by the chain rule. 852 00:44:49,652 --> 00:44:57,110 853 00:44:57,110 --> 00:45:00,040 Can we verify really quickly in our case, 854 00:45:00,040 --> 00:45:02,500 is it easy to see that in our case it's correct? 855 00:45:02,500 --> 00:45:03,260 STUDENT: Yeah. 856 00:45:03,260 --> 00:45:05,530 PROFESSOR: Oh yeah, S of T will be, 857 00:45:05,530 --> 00:45:08,440 in our case, integral from 0 to T. 858 00:45:08,440 --> 00:45:14,340 We are lucky our prime of tau is a constant, which is 5. 859 00:45:14,340 --> 00:45:16,360 So I'm going to have integral from 0 860 00:45:16,360 --> 00:45:20,725 to T absolute value of 5 [INAUDIBLE] d tau. 861 00:45:20,725 --> 00:45:23,100 And what in the world is absolute value of 5? 862 00:45:23,100 --> 00:45:27,808 It's 5 integral from 0 to T [? of the ?] tau. 863 00:45:27,808 --> 00:45:30,990 What is integral from 0 to T of the tau? 864 00:45:30,990 --> 00:45:33,660 T. 5T. 865 00:45:33,660 --> 00:45:36,940 So S is 5T. 866 00:45:36,940 --> 00:45:39,534 And that's what I said before, right? 867 00:45:39,534 --> 00:45:41,840 S is 5T. 868 00:45:41,840 --> 00:45:46,720 S equals 5T, and T equals S over 5. 869 00:45:46,720 --> 00:45:51,295 So this thing, in general, is told to us by who? 870 00:45:51,295 --> 00:45:53,160 It has to match the chain rule. 871 00:45:53,160 --> 00:45:55,152 It matches the chain rule. 872 00:45:55,152 --> 00:46:19,580 873 00:46:19,580 --> 00:46:20,100 OK. 874 00:46:20,100 --> 00:46:24,720 So again, why does that match the chain rule? 875 00:46:24,720 --> 00:46:31,290 We have that-- we have R-- or how 876 00:46:31,290 --> 00:46:34,550 should I start, the little f, the little r, little r of S, 877 00:46:34,550 --> 00:46:35,528 right? 878 00:46:35,528 --> 00:46:41,396 Little r of S is little r of S of T. 879 00:46:41,396 --> 00:46:45,370 How do I differentiate that with respect to T? 880 00:46:45,370 --> 00:46:53,240 Well DDT of R will be R primed with respect to S. 881 00:46:53,240 --> 00:47:01,720 So I'll say DRDS of S of T times DSDT. 882 00:47:01,720 --> 00:47:04,510 883 00:47:04,510 --> 00:47:06,190 Now what is DSDT? 884 00:47:06,190 --> 00:47:09,220 DSDT was the derivative of that. 885 00:47:09,220 --> 00:47:15,870 It's exactly the speed absolute value of R prime of T. 886 00:47:15,870 --> 00:47:18,190 So when you prime here, S prime of T 887 00:47:18,190 --> 00:47:22,950 will be exactly that, with T replacing tau. 888 00:47:22,950 --> 00:47:24,450 We learned that in Calc 1. 889 00:47:24,450 --> 00:47:26,530 I know it's been a long time. 890 00:47:26,530 --> 00:47:28,704 I can feel you're a little bit rusty. 891 00:47:28,704 --> 00:47:29,620 But it doesn't matter. 892 00:47:29,620 --> 00:47:32,820 So S prime of T, DSDT will simply 893 00:47:32,820 --> 00:47:36,220 be absolute value of R prime of T. 894 00:47:36,220 --> 00:47:40,671 That's the speed of the original curve. 895 00:47:40,671 --> 00:47:43,580 This one. 896 00:47:43,580 --> 00:47:46,180 OK? 897 00:47:46,180 --> 00:47:46,910 All right. 898 00:47:46,910 --> 00:47:58,965 So here, when I look at DRDS, this is going to be 1. 899 00:47:58,965 --> 00:48:02,250 900 00:48:02,250 --> 00:48:06,340 And if you think of this as a function of T, 901 00:48:06,340 --> 00:48:11,737 you have DR of S of T. Who is R of S of T? 902 00:48:11,737 --> 00:48:15,230 This is R-- big R-- of T. So this 903 00:48:15,230 --> 00:48:21,717 is the DRDT Which is exactly the same as R prime of T 904 00:48:21,717 --> 00:48:24,720 when you put the absolute values [INAUDIBLE]. 905 00:48:24,720 --> 00:48:26,470 It has to fit. 906 00:48:26,470 --> 00:48:33,080 So indeed, you have R prime of T, R prime of T, and 1. 907 00:48:33,080 --> 00:48:35,131 It's an identity. 908 00:48:35,131 --> 00:48:38,910 If I didn't put DSDT to [? P, ?] our prime of T 909 00:48:38,910 --> 00:48:42,216 in absolute value, it wouldn't work out. 910 00:48:42,216 --> 00:48:48,350 DSDT has to be R prime of T in absolute value. 911 00:48:48,350 --> 00:48:51,475 And this is how we got, again-- are 912 00:48:51,475 --> 00:48:54,420 you going to remember this without having 913 00:48:54,420 --> 00:48:56,220 to re-do the whole thing? 914 00:48:56,220 --> 00:49:11,310 Integral from 0 to T of R prime of T or tau d tau. 915 00:49:11,310 --> 00:49:13,735 When you prime this guy with respect to T 916 00:49:13,735 --> 00:49:17,570 as soon as it's positive-- when it is positive-- assume-- 917 00:49:17,570 --> 00:49:20,140 why is this positive, S of T? 918 00:49:20,140 --> 00:49:23,690 Because you integrate from time 0 to another time 919 00:49:23,690 --> 00:49:24,670 a positive number. 920 00:49:24,670 --> 00:49:29,110 So it has to be positive derivative. 921 00:49:29,110 --> 00:49:30,490 It's an increasing function. 922 00:49:30,490 --> 00:49:34,210 This function is increasing. 923 00:49:34,210 --> 00:49:37,360 So DSDT again will be the speed. 924 00:49:37,360 --> 00:49:38,570 Say it again, Magdalena? 925 00:49:38,570 --> 00:49:44,190 DSDT will be the speed of the original line. 926 00:49:44,190 --> 00:49:47,110 DSDT in our case was 5. 927 00:49:47,110 --> 00:49:48,050 Right? 928 00:49:48,050 --> 00:49:50,200 DSDT was 5. 929 00:49:50,200 --> 00:49:54,590 S was 5 times T. S was 5 times T. 930 00:49:54,590 --> 00:49:55,090 All right. 931 00:49:55,090 --> 00:49:58,030 That was a simple example, sort of, kind of. 932 00:49:58,030 --> 00:49:59,990 What do we want to remember? 933 00:49:59,990 --> 00:50:03,620 We remember the formula of the arc length. 934 00:50:03,620 --> 00:50:05,530 Formula of arc length. 935 00:50:05,530 --> 00:50:08,720 936 00:50:08,720 --> 00:50:11,310 So the formula of arc length exists 937 00:50:11,310 --> 00:50:15,440 in this form because of the chain rule [INAUDIBLE] 938 00:50:15,440 --> 00:50:18,595 from this diagram. 939 00:50:18,595 --> 00:50:24,840 So always remember, we have a composition of functions. 940 00:50:24,840 --> 00:50:27,512 We use that composition of function for the chain rule 941 00:50:27,512 --> 00:50:28,964 to re-parametrize it. 942 00:50:28,964 --> 00:50:30,900 And finally, the drunken bug. 943 00:50:30,900 --> 00:50:34,059 944 00:50:34,059 --> 00:50:35,350 what did I take [INAUDIBLE] 14? 945 00:50:35,350 --> 00:50:37,110 R of t. 946 00:50:37,110 --> 00:50:44,481 Let's say this is 2 cosine t, 2 sine t. 947 00:50:44,481 --> 00:50:46,460 Let me make it more beautiful. 948 00:50:46,460 --> 00:50:53,500 Let me put 4-- 4, 4, and 3t. 949 00:50:53,500 --> 00:50:56,740 Can anybody tell me why I did that? 950 00:50:56,740 --> 00:50:59,990 Maybe you can guess my mind. 951 00:50:59,990 --> 00:51:04,000 Find the following things. 952 00:51:04,000 --> 00:51:11,312 The unit vector T, by definition R prime over R prime 953 00:51:11,312 --> 00:51:16,450 of t in absolute value. 954 00:51:16,450 --> 00:51:22,210 Find the speed of this motion R of t. 955 00:51:22,210 --> 00:51:24,710 This is a law of motion. 956 00:51:24,710 --> 00:51:32,426 And reparametrize in arclength-- this curve in arclength. 957 00:51:32,426 --> 00:51:36,650 958 00:51:36,650 --> 00:51:39,910 And you go, oh my God, I have a problem with a, b,c. 959 00:51:39,910 --> 00:51:43,260 The is a typical problem for the final exam, by the way. 960 00:51:43,260 --> 00:51:46,290 This problem popped up on many, many final exams. 961 00:51:46,290 --> 00:51:47,262 Is it hard? 962 00:51:47,262 --> 00:51:49,210 Is it easy? 963 00:51:49,210 --> 00:51:53,380 First of all, how did I know what it looked like? 964 00:51:53,380 --> 00:51:57,090 I should give at least an explanation. 965 00:51:57,090 --> 00:52:00,890 If instead of 3t I would have 3, then I 966 00:52:00,890 --> 00:52:04,720 would have the plane z equals 3 constant. 967 00:52:04,720 --> 00:52:07,610 And then I'll say, I'm moving in circles, in circles, 968 00:52:07,610 --> 00:52:11,160 in circles, in circles, with t as a real parameter, 969 00:52:11,160 --> 00:52:13,560 and I'm not evolving. 970 00:52:13,560 --> 00:52:16,955 But this is like, what, this like in in the avatar OK? 971 00:52:16,955 --> 00:52:22,060 So I'm performing the circular motion, but at the same time 972 00:52:22,060 --> 00:52:25,070 going on a different level. 973 00:52:25,070 --> 00:52:26,720 Assume another life. 974 00:52:26,720 --> 00:52:31,117 I'm starting another life on the next spiritual level. 975 00:52:31,117 --> 00:52:34,020 OK, I have no religious beliefs in that area, 976 00:52:34,020 --> 00:52:36,300 but it's a good physical example to give. 977 00:52:36,300 --> 00:52:38,290 So I go circular. 978 00:52:38,290 --> 00:52:41,530 Instead of going again circular and again circular, 979 00:52:41,530 --> 00:52:45,470 I go, oh, I go up and up and up, and this 3t 980 00:52:45,470 --> 00:52:49,210 tells me I should also evolve on the vertical. 981 00:52:49,210 --> 00:52:50,330 Ah-hah. 982 00:52:50,330 --> 00:52:55,370 So instead of circular motion I get a helicoidal motion. 983 00:52:55,370 --> 00:52:56,140 This is a helix. 984 00:52:56,140 --> 00:52:58,650 985 00:52:58,650 --> 00:53:01,920 Could somebody tell me how I'm going to draw such a helix? 986 00:53:01,920 --> 00:53:02,555 Is it hard? 987 00:53:02,555 --> 00:53:04,280 Is it easy? 988 00:53:04,280 --> 00:53:05,390 This helix-- yes, sir. 989 00:53:05,390 --> 00:53:08,380 990 00:53:08,380 --> 00:53:09,190 Yes. 991 00:53:09,190 --> 00:53:10,910 STUDENT: [INAUDIBLE] 992 00:53:10,910 --> 00:53:12,350 PROFESSOR: It's like a tornado. 993 00:53:12,350 --> 00:53:14,410 It's like a tornado, hurricane, but how 994 00:53:14,410 --> 00:53:18,435 do I draw the cylinder on which this helix exists? 995 00:53:18,435 --> 00:53:22,500 I have to be a smart girl and remember what I learned before. 996 00:53:22,500 --> 00:53:25,410 What is x squared plus y squared? 997 00:53:25,410 --> 00:53:28,750 Suppose that z is not playing in the picture. 998 00:53:28,750 --> 00:53:32,560 If I take Mr. x and Mr. y and I square them and I add 999 00:53:32,560 --> 00:53:34,705 them together, what do I get? 1000 00:53:34,705 --> 00:53:35,746 STUDENT: It's the radius. 1001 00:53:35,746 --> 00:53:38,260 PROFESSOR: What is the radius squared? 1002 00:53:38,260 --> 00:53:38,910 4 squared. 1003 00:53:38,910 --> 00:53:41,295 I'm gonna write 4 squared because it's 1004 00:53:41,295 --> 00:53:43,130 easier than writing 16. 1005 00:53:43,130 --> 00:53:44,350 Thank you for your help. 1006 00:53:44,350 --> 00:53:51,370 So I simply have to go ahead and draw the frame first, x, y, z, 1007 00:53:51,370 --> 00:53:54,900 and then I'll say, OK, smart. 1008 00:53:54,900 --> 00:53:57,790 R is 4. 1009 00:53:57,790 --> 00:53:59,610 The radius should be 4. 1010 00:53:59,610 --> 00:54:02,240 This is the cylinder where I'm at. 1011 00:54:02,240 --> 00:54:06,570 Where do I start my physical motion? 1012 00:54:06,570 --> 00:54:10,134 This bug is drunk, but sort of not. 1013 00:54:10,134 --> 00:54:11,517 I don't know. 1014 00:54:11,517 --> 00:54:16,020 It's a bug that can keep the same radius, which 1015 00:54:16,020 --> 00:54:16,987 is quite something. 1016 00:54:16,987 --> 00:54:17,820 STUDENT: It's tipsy. 1017 00:54:17,820 --> 00:54:19,600 PROFESSOR: Yeah, exactly, tipsy one. 1018 00:54:19,600 --> 00:54:22,540 So how about t equals 0. 1019 00:54:22,540 --> 00:54:24,910 Where do I start my motion? 1020 00:54:24,910 --> 00:54:26,900 At 4, 0, 0. 1021 00:54:26,900 --> 00:54:28,505 Where is 4, 0, 0? 1022 00:54:28,505 --> 00:54:29,290 Over here. 1023 00:54:29,290 --> 00:54:31,590 So that's my first point where the bug 1024 00:54:31,590 --> 00:54:33,079 will start at t equals 0. 1025 00:54:33,079 --> 00:54:34,370 STUDENT: How'd you get 4, 0, 0? 1026 00:54:34,370 --> 00:54:36,300 PROFESSOR: Because I'm-- very good question. 1027 00:54:36,300 --> 00:54:38,640 I'm on x, y, z axes. 1028 00:54:38,640 --> 00:54:42,050 4, y is 0, z is 0. 1029 00:54:42,050 --> 00:54:46,650 I plug in t, would be 0, and I get 4 times 1, 4 times 1030 00:54:46,650 --> 00:54:50,620 0, 3 times 0, so I know I'm starting here. 1031 00:54:50,620 --> 00:54:55,880 And when I move, I move along the cylinder like that. 1032 00:54:55,880 --> 00:55:00,240 Can somebody tell me at what time I'm gonna be here? 1033 00:55:00,240 --> 00:55:03,910 Not at 1:50, but what time am I going to be at this point? 1034 00:55:03,910 --> 00:55:08,090 And then I continue, and I go up, and I continue and I go up. 1035 00:55:08,090 --> 00:55:09,710 STUDENT: [INAUDIBLE] 1036 00:55:09,710 --> 00:55:11,030 PROFESSOR: Pi over 2. 1037 00:55:11,030 --> 00:55:12,670 Excellent. 1038 00:55:12,670 --> 00:55:14,480 And can you-- can you tell me what 1039 00:55:14,480 --> 00:55:16,970 point it is in space in R 3? 1040 00:55:16,970 --> 00:55:18,170 Plug in pi over 2. 1041 00:55:18,170 --> 00:55:19,620 You can do it faster than me. 1042 00:55:19,620 --> 00:55:20,330 STUDENT: 0. 1043 00:55:20,330 --> 00:55:23,940 PROFESSOR: 0, 4 and 3 pi over 2. 1044 00:55:23,940 --> 00:55:25,580 And I keep going. 1045 00:55:25,580 --> 00:55:28,850 So this is the helicoidal motion I'm talking about. 1046 00:55:28,850 --> 00:55:31,690 The unit vector-- is it easy to write it on the final? 1047 00:55:31,690 --> 00:55:33,150 Can do that in no time. 1048 00:55:33,150 --> 00:55:39,140 So we get like, let's say, 30%, 30%, 30%, and 10% for drawing. 1049 00:55:39,140 --> 00:55:40,510 How about that? 1050 00:55:40,510 --> 00:55:44,210 That would be a typical grid for the problem. 1051 00:55:44,210 --> 00:55:49,900 So t will be minus 4 sine t. 1052 00:55:49,900 --> 00:55:53,871 If I make a mistake, are you gonna shout, please? 1053 00:55:53,871 --> 00:55:58,660 4 cosine t and 3 divided by what? 1054 00:55:58,660 --> 00:56:00,950 What is the tangent unit vector? 1055 00:56:00,950 --> 00:56:03,840 At every point in space, I'm gonna 1056 00:56:03,840 --> 00:56:05,580 have this tangent unit vector. 1057 00:56:05,580 --> 00:56:08,140 It has to have length 1, and it has 1058 00:56:08,140 --> 00:56:11,100 to be tangent to my trajectory. 1059 00:56:11,100 --> 00:56:12,200 I'll draw him. 1060 00:56:12,200 --> 00:56:15,670 So he gives me a field, a vector field-- 1061 00:56:15,670 --> 00:56:19,080 this is beautiful-- T of t is a vector field. 1062 00:56:19,080 --> 00:56:20,850 At every point of the trajectory, 1063 00:56:20,850 --> 00:56:23,460 I have only one such vector. 1064 00:56:23,460 --> 00:56:27,067 That's what we mean by vector field. 1065 00:56:27,067 --> 00:56:29,989 What's the magnitude? 1066 00:56:29,989 --> 00:56:31,450 It's buzzing. 1067 00:56:31,450 --> 00:56:33,400 It's buzzing. 1068 00:56:33,400 --> 00:56:35,220 How did you do it? 1069 00:56:35,220 --> 00:56:40,140 4, 16 times sine squared plus cosine squared. 1070 00:56:40,140 --> 00:56:42,400 16 plus 9 is 25. 1071 00:56:42,400 --> 00:56:46,120 Square root of 25 is 5. 1072 00:56:46,120 --> 00:56:47,870 Are you guys with me? 1073 00:56:47,870 --> 00:56:49,860 Do I have to write this down? 1074 00:56:49,860 --> 00:56:51,775 Are you guys sure? 1075 00:56:51,775 --> 00:56:53,108 STUDENT: You plugged in 0 for t? 1076 00:56:53,108 --> 00:56:56,390 Is that what you did when you [INAUDIBLE] 1077 00:56:56,390 --> 00:56:58,980 PROFESSOR: No, I plugged 0 for t when I started. 1078 00:56:58,980 --> 00:57:01,680 But when I'm computing, I don't plug anything, 1079 00:57:01,680 --> 00:57:03,940 I just do it in general. 1080 00:57:03,940 --> 00:57:07,710 I said 16 sine squared plus 16 cosine squared 1081 00:57:07,710 --> 00:57:10,400 is 16 times 1 plus 9. 1082 00:57:10,400 --> 00:57:13,410 My son would know this one and he's 10, right? 1083 00:57:13,410 --> 00:57:16,030 16 plus 9 square root of 25. 1084 00:57:16,030 --> 00:57:17,810 And I taught him about square roots. 1085 00:57:17,810 --> 00:57:20,590 So square root of 25, he knows that's 5. 1086 00:57:20,590 --> 00:57:22,252 And if he knows that's 5, then you 1087 00:57:22,252 --> 00:57:24,470 should do that in a minute-- in a second. 1088 00:57:24,470 --> 00:57:25,330 All right. 1089 00:57:25,330 --> 00:57:32,430 So t will simply be-- if you don't simplify 1/5 minus 4 sine 1090 00:57:32,430 --> 00:57:37,465 t 4 cosine t 3 in the final, it wouldn't be a big deal, 1091 00:57:37,465 --> 00:57:39,315 I would give you still partial credit, 1092 00:57:39,315 --> 00:57:42,260 but what if we raise this as a multiple choice? 1093 00:57:42,260 --> 00:57:46,520 Then you have to be able to find where the 5 is. 1094 00:57:46,520 --> 00:57:47,270 What is the speed? 1095 00:57:47,270 --> 00:57:49,410 Was that hard for you to find? 1096 00:57:49,410 --> 00:57:50,990 Where is the speed hiding? 1097 00:57:50,990 --> 00:57:53,800 It's exactly the denominator of R. 1098 00:57:53,800 --> 00:57:57,070 This is the speed of the curve in t. 1099 00:57:57,070 --> 00:57:58,570 And that was 5. 1100 00:57:58,570 --> 00:58:01,190 You told me the speed was 5, and I'm very happy. 1101 00:58:01,190 --> 00:58:07,800 So you got 30%, 30%, 10% from the picture-- no, this picture. 1102 00:58:07,800 --> 00:58:09,297 This picture's no good. 1103 00:58:09,297 --> 00:58:13,193 STUDENT: What does the first word of c say? 1104 00:58:13,193 --> 00:58:15,287 Question c, what does the first word say? 1105 00:58:15,287 --> 00:58:16,370 PROFESSOR: The first what? 1106 00:58:16,370 --> 00:58:17,730 STUDENT: The word. 1107 00:58:17,730 --> 00:58:19,380 PROFESSOR: Reparametrize. 1108 00:58:19,380 --> 00:58:23,070 Reparametrize this curve in arclength. 1109 00:58:23,070 --> 00:58:26,430 Oh my God, so according to that chain rule, 1110 00:58:26,430 --> 00:58:31,430 could you guys remember-- if you remember, what is the s of t? 1111 00:58:31,430 --> 00:58:39,082 If I want to reparametrize in arclength integral from 0 1112 00:58:39,082 --> 00:58:45,580 to t of the speed, how is the speed defined? 1113 00:58:45,580 --> 00:58:49,040 Absolute value of r prime of t. 1114 00:58:49,040 --> 00:58:54,370 dt, but I don't like t, I write-- I write tau. 1115 00:58:54,370 --> 00:58:56,610 Like Dr. [? Solinger, ?] you know him, 1116 00:58:56,610 --> 00:58:59,370 he's one of my colleagues, calls that-- that's 1117 00:58:59,370 --> 00:59:00,885 the dummy dummy variable. 1118 00:59:00,885 --> 00:59:03,770 In many books, tau is the dummy variable. 1119 00:59:03,770 --> 00:59:08,485 Or you can-- some people even put t by inclusive notation. 1120 00:59:08,485 --> 00:59:09,900 All right? 1121 00:59:09,900 --> 00:59:12,580 So in my case, what is s of t? 1122 00:59:12,580 --> 00:59:14,070 It should be easy. 1123 00:59:14,070 --> 00:59:18,670 Because although this not a circular motion, 1124 00:59:18,670 --> 00:59:20,610 I still have constant speed. 1125 00:59:20,610 --> 00:59:23,590 So who is that special speed? 1126 00:59:23,590 --> 00:59:24,350 5. 1127 00:59:24,350 --> 00:59:31,400 Integral from 0 to t5 d tau, and that is 5t, am I right? 1128 00:59:31,400 --> 00:59:32,050 5t. 1129 00:59:32,050 --> 00:59:36,980 So-- so if I want to reparametrize this helix, 1130 00:59:36,980 --> 00:59:41,596 keeping in mind that s is simply 5t, 1131 00:59:41,596 --> 00:59:47,392 what do I have to do to get 100% on this problem? 1132 00:59:47,392 --> 00:59:57,590 All I have to do is say little r of s, which represents actually 1133 00:59:57,590 --> 01:00:00,690 big R of t of s. 1134 01:00:00,690 --> 01:00:02,380 Are you guys with me? 1135 01:00:02,380 --> 01:00:04,451 Do you have to write all this story down? 1136 01:00:04,451 --> 01:00:04,950 No. 1137 01:00:04,950 --> 01:00:07,650 But that will remind you of the diagram. 1138 01:00:07,650 --> 01:00:12,000 So I have R of t of s. 1139 01:00:12,000 --> 01:00:13,268 Or alpha of s. 1140 01:00:13,268 --> 01:00:15,360 And this is t of s. 1141 01:00:15,360 --> 01:00:16,390 t of s. 1142 01:00:16,390 --> 01:00:19,642 R of t of s is R of s, right? 1143 01:00:19,642 --> 01:00:21,100 Do you have to remind me? 1144 01:00:21,100 --> 01:00:21,600 No. 1145 01:00:21,600 --> 01:00:23,240 The heck with the diagram. 1146 01:00:23,240 --> 01:00:26,900 As long as you understood it was about a composition 1147 01:00:26,900 --> 01:00:28,250 of functions. 1148 01:00:28,250 --> 01:00:30,645 And then R of s will simply be what? 1149 01:00:30,645 --> 01:00:33,060 How do we do that fast? 1150 01:00:33,060 --> 01:00:37,430 We replaced t by s over 5. 1151 01:00:37,430 --> 01:00:38,790 Where from? 1152 01:00:38,790 --> 01:00:42,280 Little s equals 5t, we just computed it. 1153 01:00:42,280 --> 01:00:43,680 Little s equals 5t. 1154 01:00:43,680 --> 01:00:44,780 That's all you need to do. 1155 01:00:44,780 --> 01:00:49,110 To pull out t, replace the third sub s. 1156 01:00:49,110 --> 01:00:52,866 So what is the function t in terms of s? 1157 01:00:52,866 --> 01:00:55,096 It's s over 5. 1158 01:00:55,096 --> 01:00:59,600 What is the function t, what's the parameter t, in terms of s? 1159 01:00:59,600 --> 01:01:01,480 s over 5. 1160 01:01:01,480 --> 01:01:07,140 And finally, at the end, 3 times what is the stinking t? 1161 01:01:07,140 --> 01:01:09,140 s over 5. 1162 01:01:09,140 --> 01:01:11,002 I'm done. 1163 01:01:11,002 --> 01:01:16,245 I got 100% I don't want to say how much time it's 1164 01:01:16,245 --> 01:01:18,180 gonna take me to do it, but I think 1165 01:01:18,180 --> 01:01:20,480 I can do it in like, 2 or 3 minutes, 5 minutes. 1166 01:01:20,480 --> 01:01:24,290 If I know the problem I'll do it in a few minutes. 1167 01:01:24,290 --> 01:01:26,640 If I waste too much time thinking, 1168 01:01:26,640 --> 01:01:28,710 I'm not gonna do it at all. 1169 01:01:28,710 --> 01:01:30,470 So what do you have to remember? 1170 01:01:30,470 --> 01:01:35,010 You have to remember the formula that says s of t, 1171 01:01:35,010 --> 01:01:40,780 the arclength parameter-- the arclength parameter 1172 01:01:40,780 --> 01:01:46,740 equals integral from 0 to t is 0 to t of the speed. 1173 01:01:46,740 --> 01:01:53,140 Does this element of information remind you of something? 1174 01:01:53,140 --> 01:01:56,460 Of course, s will be the arclength, practically. 1175 01:01:56,460 --> 01:01:58,450 What kind of parameter is that? 1176 01:01:58,450 --> 01:02:03,510 Is you're measuring how big-- how much you travel. 1177 01:02:03,510 --> 01:02:06,960 s of t is the time you travel-- the distance 1178 01:02:06,960 --> 01:02:10,679 you travel in time t. 1179 01:02:10,679 --> 01:02:15,570 1180 01:02:15,570 --> 01:02:20,010 So it's a space-time continuum. 1181 01:02:20,010 --> 01:02:23,600 It's a space-time relationship. 1182 01:02:23,600 --> 01:02:27,400 So it's the space you travel in times t. 1183 01:02:27,400 --> 01:02:30,440 Now, if I drive to Amarillo at 60 miles an hour, 1184 01:02:30,440 --> 01:02:35,005 I'm happy and sassy, and I say OK, it's gonna be s of t. 1185 01:02:35,005 --> 01:02:37,670 My displacement to Amarillo is given 1186 01:02:37,670 --> 01:02:41,540 by this linear law, 60 times t. 1187 01:02:41,540 --> 01:02:42,910 Suppose I'm on cruise control. 1188 01:02:42,910 --> 01:02:44,370 But I've never on cruise control. 1189 01:02:44,370 --> 01:02:47,260 1190 01:02:47,260 --> 01:02:50,750 So this is going to be very variable. 1191 01:02:50,750 --> 01:02:54,674 And the only way you can compute this displacement or distance 1192 01:02:54,674 --> 01:02:57,140 traveled, it'll be as an integral. 1193 01:02:57,140 --> 01:03:01,330 From time 0, when I start driving, to time t of my speed, 1194 01:03:01,330 --> 01:03:02,220 and that's it. 1195 01:03:02,220 --> 01:03:04,360 That's all you have to remember. 1196 01:03:04,360 --> 01:03:08,200 It's actually-- mathematics should not be memorized. 1197 01:03:08,200 --> 01:03:11,520 It should be sort of understood, just like physics. 1198 01:03:11,520 --> 01:03:15,170 What if you take your first test, quiz, 1199 01:03:15,170 --> 01:03:18,810 whatever, on WeBWorK or in person, and you freak out. 1200 01:03:18,810 --> 01:03:22,925 You get such a problem, and you blank. 1201 01:03:22,925 --> 01:03:24,950 You just blank. 1202 01:03:24,950 --> 01:03:27,670 What do you do? 1203 01:03:27,670 --> 01:03:31,622 You sort of know this, but you have a blank. 1204 01:03:31,622 --> 01:03:34,100 Always tell me, right? 1205 01:03:34,100 --> 01:03:36,026 Always email, say I'm freaking out here. 1206 01:03:36,026 --> 01:03:38,690 I don't know what's the matter with me. 1207 01:03:38,690 --> 01:03:46,400 Don't cut our correspondence, either by speaking or by email. 1208 01:03:46,400 --> 01:03:48,810 Very few of you email me. 1209 01:03:48,810 --> 01:03:51,570 I'd like you to be more like my friends, 1210 01:03:51,570 --> 01:03:53,250 and I would be more like your tutor, 1211 01:03:53,250 --> 01:03:55,400 and when you encounter an obstacle, 1212 01:03:55,400 --> 01:03:58,340 you email me and I email you back. 1213 01:03:58,340 --> 01:04:00,660 This is what I want. 1214 01:04:00,660 --> 01:04:03,645 The WeBWorK, this is what I want our model of interaction 1215 01:04:03,645 --> 01:04:05,750 to become. 1216 01:04:05,750 --> 01:04:06,880 Don't be shy. 1217 01:04:06,880 --> 01:04:10,600 Many of you are shy even to ask questions in the classroom. 1218 01:04:10,600 --> 01:04:12,500 And I'm not going to let you be shy. 1219 01:04:12,500 --> 01:04:16,640 At 2 o'clock I'm going to let you ask all the questions you 1220 01:04:16,640 --> 01:04:19,690 have about homework, and we will do 1221 01:04:19,690 --> 01:04:21,250 more homework-like questions. 1222 01:04:21,250 --> 01:04:24,060 I want to imitate some WeBWorK questions. 1223 01:04:24,060 --> 01:04:27,810 And we will work them out. 1224 01:04:27,810 --> 01:04:32,310 So any questions right now? 1225 01:04:32,310 --> 01:04:32,840 Yes, sir. 1226 01:04:32,840 --> 01:04:35,684 STUDENT: You emailed-- did you email us this weekend 1227 01:04:35,684 --> 01:04:37,580 the numbers for WeBWorK? 1228 01:04:37,580 --> 01:04:41,040 PROFESSOR: I emailed you the WeBWorK assignment completely. 1229 01:04:41,040 --> 01:04:44,913 I mean, the link-- you get in and you of see it. 1230 01:04:44,913 --> 01:04:48,340 STUDENT: Which email did you send that to? 1231 01:04:48,340 --> 01:04:49,660 PROFESSOR: To your TTU. 1232 01:04:49,660 --> 01:04:51,860 All the emails go to your TTU. 1233 01:04:51,860 --> 01:04:56,400 You have one week starting yesterday until, 1234 01:04:56,400 --> 01:04:58,140 was it the 2nd? 1235 01:04:58,140 --> 01:05:00,010 I gave you a little bit more time. 1236 01:05:00,010 --> 01:05:03,008 So it's due on the 2nd of February at, 1237 01:05:03,008 --> 01:05:03,982 I forgot what time. 1238 01:05:03,982 --> 01:05:05,443 1 o'clock or something. 1239 01:05:05,443 --> 01:05:06,417 Yes, sir. 1240 01:05:06,417 --> 01:05:07,878 STUDENT: [INAUDIBLE] I was confused 1241 01:05:07,878 --> 01:05:10,313 at the beginning where you got x squared plus y squared equals 1242 01:05:10,313 --> 01:05:10,813 4 squared. 1243 01:05:10,813 --> 01:05:13,597 Where did you get that? 1244 01:05:13,597 --> 01:05:14,180 PROFESSOR: Oh. 1245 01:05:14,180 --> 01:05:15,030 OK. 1246 01:05:15,030 --> 01:05:19,160 I eliminated the t between the first two guys. 1247 01:05:19,160 --> 01:05:24,940 This is called eliminating a parameter, which was the time 1248 01:05:24,940 --> 01:05:27,970 parameter between x and y. 1249 01:05:27,970 --> 01:05:32,090 When I do that, I get a beautiful equation which 1250 01:05:32,090 --> 01:05:36,670 is x squared plus y squared equals 16, which tells me, hey, 1251 01:05:36,670 --> 01:05:39,830 your curve sits on the surface x squared 1252 01:05:39,830 --> 01:05:42,230 plus y squared equals 16. 1253 01:05:42,230 --> 01:05:44,320 It's not the same with the surface, 1254 01:05:44,320 --> 01:05:47,520 because you have additional constraints on the z. 1255 01:05:47,520 --> 01:05:52,370 So the z is constrained to follow this thing. 1256 01:05:52,370 --> 01:05:59,910 Now, could anybody tell me how I'm gonna write eventually-- 1257 01:05:59,910 --> 01:06:02,400 this is a harder task, OK, but I'm 1258 01:06:02,400 --> 01:06:09,132 glad you asked because I wanted to discuss that. 1259 01:06:09,132 --> 01:06:13,020 How do I express t in terms of x and y? 1260 01:06:13,020 --> 01:06:16,790 I mean, I'm going to have an intersection of two surfaces. 1261 01:06:16,790 --> 01:06:18,456 How? 1262 01:06:18,456 --> 01:06:21,010 This is just practically differential geometry 1263 01:06:21,010 --> 01:06:24,250 or advanced calculus at the same time. 1264 01:06:24,250 --> 01:06:27,970 x squared plus y squared equals our first surface 1265 01:06:27,970 --> 01:06:31,950 that I'm thinking about, which I'm sitting with my curve. 1266 01:06:31,950 --> 01:06:35,370 But I also have my curve to be at the intersection 1267 01:06:35,370 --> 01:06:39,490 between the cylinder and something else. 1268 01:06:39,490 --> 01:06:45,210 And it's hard to figure out how I'm going to do the other one. 1269 01:06:45,210 --> 01:06:49,266 Can anybody figure out how another 1270 01:06:49,266 --> 01:06:51,380 surface-- what is the surface? 1271 01:06:51,380 --> 01:06:56,380 A surface will have an implicit equation of the type f of x, y, 1272 01:06:56,380 --> 01:06:58,000 z equals a constant. 1273 01:06:58,000 --> 01:07:01,140 So you have to sort of eliminate your parameter t. 1274 01:07:01,140 --> 01:07:02,470 The heck with the time. 1275 01:07:02,470 --> 01:07:05,310 We don't care about time, we only care about space. 1276 01:07:05,310 --> 01:07:07,370 So is there any other way to eliminate 1277 01:07:07,370 --> 01:07:09,960 t between the equations? 1278 01:07:09,960 --> 01:07:13,860 I have to use the information that I haven't used yet. 1279 01:07:13,860 --> 01:07:15,340 All right. 1280 01:07:15,340 --> 01:07:19,580 Now my question is that, how can I do that? 1281 01:07:19,580 --> 01:07:22,620 z is beautiful. 1282 01:07:22,620 --> 01:07:23,770 3 is beautiful. 1283 01:07:23,770 --> 01:07:25,700 t drives me nuts. 1284 01:07:25,700 --> 01:07:30,280 How do I get the t out of the first two equations? 1285 01:07:30,280 --> 01:07:32,780 [INTERPOSING VOICES] 1286 01:07:32,780 --> 01:07:35,820 Yeah, I divide them one to the other one. 1287 01:07:35,820 --> 01:07:39,730 So if I-- for example, I go y over x. 1288 01:07:39,730 --> 01:07:42,750 What is y over x? 1289 01:07:42,750 --> 01:07:45,423 It's tangent of t. 1290 01:07:45,423 --> 01:07:48,804 How do I pull Mr. t out? 1291 01:07:48,804 --> 01:07:51,650 Say t, get out. 1292 01:07:51,650 --> 01:07:54,720 Well, I have to think about if I'm not losing anything. 1293 01:07:54,720 --> 01:07:58,450 But in principle, t would be arctangent of y over x. 1294 01:07:58,450 --> 01:08:01,541 1295 01:08:01,541 --> 01:08:02,040 OK? 1296 01:08:02,040 --> 01:08:06,080 So, I'm having two equations of this type. 1297 01:08:06,080 --> 01:08:08,497 I'm eliminating t between the two. 1298 01:08:08,497 --> 01:08:10,457 I don't care about the other one. 1299 01:08:10,457 --> 01:08:13,770 I only cared for you to draw the cylinder. 1300 01:08:13,770 --> 01:08:17,250 So we can draw point by point the helix. 1301 01:08:17,250 --> 01:08:18,859 I don't draw many points. 1302 01:08:18,859 --> 01:08:22,580 I draw only t equals 0, where I'm starting over here, 1303 01:08:22,580 --> 01:08:25,160 t equals pi over 2, which [INAUDIBLE] gave me, 1304 01:08:25,160 --> 01:08:27,050 then what was it? 1305 01:08:27,050 --> 01:08:29,850 At pi I'm here, and so on. 1306 01:08:29,850 --> 01:08:33,652 So I move-- when I move one time, 1307 01:08:33,652 --> 01:08:39,407 so let's say from 0 to 2 pi, I should be smart. 1308 01:08:39,407 --> 01:08:48,979 Pi over 2, pi, 3 pi over 2, 2 pi just on top of that. 1309 01:08:48,979 --> 01:08:52,435 It has to be on the same line. 1310 01:08:52,435 --> 01:08:54,622 On top of that-- on the cylinder. 1311 01:08:54,622 --> 01:08:55,830 They are all on the cylinder. 1312 01:08:55,830 --> 01:08:59,439 I'm not good enough to draw them as being on the cylinder. 1313 01:08:59,439 --> 01:09:03,189 So I'm coming where I started from, but on the higher 1314 01:09:03,189 --> 01:09:08,050 level of intelligence-- no, on a higher level of experience. 1315 01:09:08,050 --> 01:09:08,840 Right? 1316 01:09:08,840 --> 01:09:13,995 That's kind of the idea of evolving on the helix? 1317 01:09:13,995 --> 01:09:16,652 Any other questions? 1318 01:09:16,652 --> 01:09:17,584 Yes, sir. 1319 01:09:17,584 --> 01:09:19,760 STUDENT: So that capital R of t is 1320 01:09:19,760 --> 01:09:23,886 you position vector, but what's little r of t? [INAUDIBLE] 1321 01:09:23,886 --> 01:09:25,524 PROFESSOR: It's also a position vector. 1322 01:09:25,524 --> 01:09:32,000 So practically it depends on the type of parametrization 1323 01:09:32,000 --> 01:09:33,484 you are using. 1324 01:09:33,484 --> 01:09:36,249 1325 01:09:36,249 --> 01:09:39,679 The dependence of time is crucial. 1326 01:09:39,679 --> 01:09:43,340 The dependence of the time parameter is crucial. 1327 01:09:43,340 --> 01:09:50,578 So when you draw this diagram, r of s 1328 01:09:50,578 --> 01:09:59,120 will practically be the same as R of s of t-- R of t of s, 1329 01:09:59,120 --> 01:10:00,000 I'm sorry. 1330 01:10:00,000 --> 01:10:02,340 R of t of s. 1331 01:10:02,340 --> 01:10:05,620 So practically it's telling me it's a combination. 1332 01:10:05,620 --> 01:10:11,800 Physically, it's the same thing, but at a different time. 1333 01:10:11,800 --> 01:10:20,120 So you look at one vector at time-- time is t here, 1334 01:10:20,120 --> 01:10:22,520 but s was 5t. 1335 01:10:22,520 --> 01:10:26,050 So I'm gonna be-- let me give you an example. 1336 01:10:26,050 --> 01:10:29,890 So we had s was 5t, right? 1337 01:10:29,890 --> 01:10:32,560 I don't remember how it went. 1338 01:10:32,560 --> 01:10:36,260 So when I have little r of s, that 1339 01:10:36,260 --> 01:10:42,770 means the same as little r of 5t, 1340 01:10:42,770 --> 01:10:47,470 which means this kind of guy. 1341 01:10:47,470 --> 01:10:57,550 Now assume that I have something like cosine 5t, sine 5t, and 0. 1342 01:10:57,550 --> 01:11:00,810 And what does this mean? 1343 01:11:00,810 --> 01:11:10,330 It means that R of 2 pi over 5 is the same as little r of 2 1344 01:11:10,330 --> 01:11:16,260 pi where R of t is cosine of 5t, and little r of s 1345 01:11:16,260 --> 01:11:20,790 is cosine of s, sine s, 0. 1346 01:11:20,790 --> 01:11:23,860 So Mr. t says, I'm running, I'm time. 1347 01:11:23,860 --> 01:11:29,290 I'm running from 0 to 2 pi over 5, and that's when I stop. 1348 01:11:29,290 --> 01:11:31,455 And little s says, I'm running too. 1349 01:11:31,455 --> 01:11:34,760 I'm also time, but I'm a special kind of time, 1350 01:11:34,760 --> 01:11:38,395 and I'm running from 0 to 2 pi, and I stop at 2 pi 1351 01:11:38,395 --> 01:11:40,920 where the circle will stop. 1352 01:11:40,920 --> 01:11:44,390 Then physically, the two vectors, 1353 01:11:44,390 --> 01:11:48,750 at two different moments in time, are the same. 1354 01:11:48,750 --> 01:11:51,310 Where-- why-- why is that? 1355 01:11:51,310 --> 01:11:53,300 So I start here. 1356 01:11:53,300 --> 01:11:55,770 And I end here. 1357 01:11:55,770 --> 01:12:01,296 So physically, these two guys have the same, the red vector, 1358 01:12:01,296 --> 01:12:05,453 but they are there at different moments in time. 1359 01:12:05,453 --> 01:12:06,750 All right? 1360 01:12:06,750 --> 01:12:12,840 So imagine that you have sister. 1361 01:12:12,840 --> 01:12:17,765 And she is five times faster than you in a competition. 1362 01:12:17,765 --> 01:12:20,980 It's a math competition, athletic, it doesn't matter. 1363 01:12:20,980 --> 01:12:25,570 You both get there, but you get there in different times, 1364 01:12:25,570 --> 01:12:27,480 in different amounts of time. 1365 01:12:27,480 --> 01:12:31,030 And unfortunately, this is-- I will do philosophy still 1366 01:12:31,030 --> 01:12:35,790 in mathematics-- this is the situation with many of us 1367 01:12:35,790 --> 01:12:39,246 when it comes to understanding a material, 1368 01:12:39,246 --> 01:12:42,470 like calculus or advanced calculus or geometry. 1369 01:12:42,470 --> 01:12:47,420 We get to the understanding in different times. 1370 01:12:47,420 --> 01:12:51,350 In my class-- I was talking to my old-- 1371 01:12:51,350 --> 01:12:55,590 they are all old now, all in their 40s-- 1372 01:12:55,590 --> 01:12:59,010 when did you understand this helix 1373 01:12:59,010 --> 01:13:01,710 thing being on a cylinder? 1374 01:13:01,710 --> 01:13:04,000 Because I think I understood it when 1375 01:13:04,000 --> 01:13:07,560 I was in third-- like a junior level, sophomore level, 1376 01:13:07,560 --> 01:13:09,820 and I understood nothing of this kind of stuff 1377 01:13:09,820 --> 01:13:14,830 in my freshman [INAUDIBLE] And one of my colleagues 1378 01:13:14,830 --> 01:13:17,998 who was really smart, had a big background, 1379 01:13:17,998 --> 01:13:20,870 was in a Math Olympiad, said, I think 1380 01:13:20,870 --> 01:13:22,985 I understood it as a freshman. 1381 01:13:22,985 --> 01:13:25,110 So then the other two that I was talking-- actually 1382 01:13:25,110 --> 01:13:27,620 I never understood it. 1383 01:13:27,620 --> 01:13:32,000 So we all eventually get to that point, that position, 1384 01:13:32,000 --> 01:13:34,880 but at a different moment in time. 1385 01:13:34,880 --> 01:13:39,080 And it's also unfortunate it happens about relationships. 1386 01:13:39,080 --> 01:13:42,290 You are in a relationship with somebody, 1387 01:13:42,290 --> 01:13:44,700 and one is faster than the other one. 1388 01:13:44,700 --> 01:13:46,760 One grows faster than the other one. 1389 01:13:46,760 --> 01:13:50,430 Eventually both get to the same level of understanding, 1390 01:13:50,430 --> 01:13:53,480 but since it's at different moments in time, 1391 01:13:53,480 --> 01:13:55,860 the relationship could break by the time 1392 01:13:55,860 --> 01:13:58,840 both reach that level of understanding. 1393 01:13:58,840 --> 01:14:02,620 So physical phenomena, really tricky. 1394 01:14:02,620 --> 01:14:05,490 It's-- physically you see where everything is, 1395 01:14:05,490 --> 01:14:08,490 but you have to think dynamically, in time. 1396 01:14:08,490 --> 01:14:11,300 Everything evolves in time. 1397 01:14:11,300 --> 01:14:15,166 Any other questions? 1398 01:14:15,166 --> 01:14:17,880 I'm gonna do problems with you next time, 1399 01:14:17,880 --> 01:14:22,690 but you need a break because your brain is overheated. 1400 01:14:22,690 --> 01:14:27,840 And so, we will take a break of 10-12 minutes. 1401 01:14:27,840 --> 01:14:30,653