0:00:00.000,0:00:00.510 0:00:00.510,0:00:04.890 I love this model. 0:00:04.890,0:00:06.330 Again, thank you, Casey. 0:00:06.330,0:00:09.860 I'm not going to take[br]any credit for that. 0:00:09.860,0:00:11.630 So if you want to[br]imagine the stool 0:00:11.630,0:00:16.725 I was talking about as[br]a bamboo object, that 0:00:16.725,0:00:21.110 is about the same thing,[br]at the same scale, compared 0:00:21.110,0:00:28.130 to the diameter and the height,[br]scaled or dialated five times. 0:00:28.130,0:00:30.450 Uniform, no alterations. 0:00:30.450,0:00:35.040 And one can sit on it,[br][? and circle, ?] to sit on it. 0:00:35.040,0:00:39.710 Now, as you see this is[br]a doubly ruled surface. 0:00:39.710,0:00:41.350 And you say, oh wait a minute. 0:00:41.350,0:00:45.450 You said rule surface, why all[br]of a sudden, why doubly ruled 0:00:45.450,0:00:46.210 surface? 0:00:46.210,0:00:51.520 Because it is a surface[br]that is ruled and generated 0:00:51.520,0:00:58.150 by two different one[br]parameter families. 0:00:58.150,0:00:59.690 Each of them has a[br]certain parameter 0:00:59.690,0:01:02.600 and that gives them continuity. 0:01:02.600,0:01:04.319 So you have two[br]families of lines. 0:01:04.319,0:01:06.860 0:01:06.860,0:01:09.995 One family is in this direction. 0:01:09.995,0:01:11.330 Do you see it? 0:01:11.330,0:01:14.852 So these lines-- this[br]line is in motion. 0:01:14.852,0:01:16.935 It moves to the right, to[br]the right, to the right, 0:01:16.935,0:01:19.970 and it generated. 0:01:19.970,0:01:22.540 And the other family[br]of lines is this one 0:01:22.540,0:01:24.630 in the other direction. 0:01:24.630,0:01:28.230 You have a continuity[br]parameter for each of them. 0:01:28.230,0:01:33.990 So you have to imagine[br]some real parameter going 0:01:33.990,0:01:36.750 along the entire[br][? infinite real ?] axis. 0:01:36.750,0:01:40.460 Or along a circle which would[br]be about the same thing. 0:01:40.460,0:01:46.155 But in any case, you have[br]a one parameter family 0:01:46.155,0:01:49.130 and another one[br]parameter family. 0:01:49.130,0:01:52.300 Both of them are[br]together generating 0:01:52.300,0:01:55.980 this beautiful[br]one-sheeted hyperboloid. 0:01:55.980,0:02:00.890 It's incredible because you[br]see where these sort of round, 0:02:00.890,0:02:07.080 but if you go towards the[br]ends, it's topologically 0:02:07.080,0:02:08.449 a cylinder or a tube. 0:02:08.449,0:02:15.370 But if you look towards[br]the end, the two ends 0:02:15.370,0:02:18.810 will look more straight. 0:02:18.810,0:02:23.790 And you will see the[br]straight lines more clearly. 0:02:23.790,0:02:27.680 So imagine that you[br]have a continuation 0:02:27.680,0:02:31.210 to infinity in this direction,[br]and in the other direction. 0:02:31.210,0:02:36.112 And this actually should be an[br]infinite surface in your model. 0:02:36.112,0:02:38.550 You're just cutting it[br]between two z planes, 0:02:38.550,0:02:41.996 so you have a patch of a[br]one-sheeted hyperboloid. 0:02:41.996,0:02:43.370 Yeah, the one-sheeted[br]hyperboloid 0:02:43.370,0:02:47.220 that we wrote last time,[br]do you guys remember 0:02:47.220,0:02:49.990 x squared over a[br]squared plus y squared 0:02:49.990,0:02:53.370 over b squared minus z squared?[br]z should be this [INAUDIBLE]. 0:02:53.370,0:02:56.430 Minus z squared over[br]c squared minus 1 0:02:56.430,0:03:00.950 equals 0 is an[br]infinite surface area. 0:03:00.950,0:03:04.995 At both ends you keep going. 0:03:04.995,0:03:06.480 Very beautiful. 0:03:06.480,0:03:08.380 Thank you so much. 0:03:08.380,0:03:09.370 I appreciate. 0:03:09.370,0:03:12.240 And keep the brownies. 0:03:12.240,0:03:14.030 No, then I have to pay more. 0:03:14.030,0:03:16.700 Than I have to pay money. 0:03:16.700,0:03:18.944 STUDENT: It's made[br]out of [INAUDIBLE]. 0:03:18.944,0:03:20.318 PROFESSOR: When[br]is your birthday? 0:03:20.318,0:03:23.390 [LAUGHTER] 0:03:23.390,0:03:24.000 Really? 0:03:24.000,0:03:24.140 When is it? 0:03:24.140,0:03:25.014 STUDENT: February 29. 0:03:25.014,0:03:27.975 PROFESSOR: Oh, it's coming. 0:03:27.975,0:03:28.808 [INTERPOSING VOICES] 0:03:28.808,0:03:32.670 0:03:32.670,0:03:35.558 STUDENT: It's coming[br]in a year, too. 0:03:35.558,0:03:38.960 PROFESSOR: That was a smart one. 0:03:38.960,0:03:41.120 Anyway, I'll remember that. 0:03:41.120,0:03:42.870 I appreciate the gift very much. 0:03:42.870,0:03:45.470 And I will cherish[br]it and I'll use it 0:03:45.470,0:03:46.945 with both my[br]undergraduate students 0:03:46.945,0:03:50.382 and my graduate students who[br]are just learning about-- some 0:03:50.382,0:03:54.100 of them don't know the[br]one-sheeted hyperboloid model, 0:03:54.100,0:03:57.130 but they will learn about it. 0:03:57.130,0:04:00.080 Coming back to our lesson. 0:04:00.080,0:04:05.020 I announced Section 10.1. 0:04:05.020,0:04:06.785 Say goodbye to[br]quadrant for a while. 0:04:06.785,0:04:09.640 I know you love them,[br]but they will be there 0:04:09.640,0:04:11.450 for you in Chapter 11. 0:04:11.450,0:04:13.099 They will wait for you. 0:04:13.099,0:04:20.584 Now, let's go to Section[br]10.1 of Chapter 10. 0:04:20.584,0:04:23.079 Chapter 10 is a[br]beautiful chapter. 0:04:23.079,0:04:26.580 As you know very well,[br]I announced last time, 0:04:26.580,0:04:29.173 it is about[br]vector-valued functions. 0:04:29.173,0:04:40.800 0:04:40.800,0:04:43.222 And you say, oh[br]my god, I've never 0:04:43.222,0:04:45.677 heard about vector-valued[br]functions before. 0:04:45.677,0:04:48.623 You deal with them every day. 0:04:48.623,0:04:51.569 Every time you move,[br]you are dealing 0:04:51.569,0:04:54.790 with a vector-valued[br]function, which 0:04:54.790,0:05:01.640 is the displacement, which[br]takes values in a subset in R3. 0:05:01.640,0:05:06.820 So let's try and see what[br]you should understand 0:05:06.820,0:05:10.015 when you start Section 10.1. 0:05:10.015,0:05:14.380 Because the book is pretty[br]good, not that I'm a co-author. 0:05:14.380,0:05:18.880 But it was meant to be really[br]written for the students 0:05:18.880,0:05:22.510 and explain concepts[br]really well. 0:05:22.510,0:05:26.380 How many of you took physics? 0:05:26.380,0:05:29.020 OK, quite a lot of[br]you took physics. 0:05:29.020,0:05:33.840 Now, one of my students[br]in a previous honors class 0:05:33.840,0:05:38.480 told me he enjoyed my[br]class greatly in general. 0:05:38.480,0:05:41.475 The most [INAUDIBLE] thing[br]he had from my class, he 0:05:41.475,0:05:45.750 learned from my class was the[br]motion of the drunken bug. 0:05:45.750,0:05:47.880 And I said, did I say that? 0:05:47.880,0:05:49.690 Absolutely, you said that. 0:05:49.690,0:05:54.410 So apparently I had[br]started one of my lessons 0:05:54.410,0:05:59.830 with imagine you have a fly[br]who went into your coffee mug. 0:05:59.830,0:06:00.497 I think I did. 0:06:00.497,0:06:03.520 He reproduced the whole[br]thing the way I said it. 0:06:03.520,0:06:05.630 It was quite spontaneous. 0:06:05.630,0:06:10.430 So imagine your coffee mug had[br]some Baileys Irish Creme in it. 0:06:10.430,0:06:15.830 And the fly was really[br]happy after she got up. 0:06:15.830,0:06:17.890 She managed to get up. 0:06:17.890,0:06:21.260 And the trajectory of the[br]fly was something more 0:06:21.260,0:06:23.330 like a helix. 0:06:23.330,0:06:25.790 And this is how I actually[br]introduced the helix 0:06:25.790,0:06:27.280 in my classroom. 0:06:27.280,0:06:29.830 And I thought, OK,[br]is that unusual? 0:06:29.830,0:06:30.330 Very. 0:06:30.330,0:06:33.200 And I said, but that's[br]an honors class. 0:06:33.200,0:06:36.080 Everything is supposed[br]to be unusual, right? 0:06:36.080,0:06:50.190 So let's think about the[br]position vector or some sort 0:06:50.190,0:06:53.630 of vector-valued function that[br]you're familiar with already 0:06:53.630,0:06:55.412 from physics. 0:06:55.412,0:06:57.490 He is one of your best friends. 0:06:57.490,0:07:01.180 You have a function r of t. 0:07:01.180,0:07:10.120 And I will point out that r is[br]practically the position vector 0:07:10.120,0:07:16.000 measure that time t, or[br]observed at time t in R3. 0:07:16.000,0:07:19.150 So he takes values in R3. 0:07:19.150,0:07:20.070 How? 0:07:20.070,0:07:23.930 As the mathematician, because[br]I like to write mathematically 0:07:23.930,0:07:26.740 all the notion I[br]have, r is defined 0:07:26.740,0:07:34.470 on I was a sub-interval[br]of R with values in R3. 0:07:34.470,0:07:38.696 And he asked me, my student[br]said, what is this I? 0:07:38.696,0:07:40.850 Well, this I could[br]be any interval, 0:07:40.850,0:07:43.290 but let's assume for[br]the time being it's 0:07:43.290,0:07:46.410 just an open[br]interval of the type 0:07:46.410,0:07:52.180 a, b, where a and b are[br]real numbers, a less than b. 0:07:52.180,0:07:58.310 So this is practically the time[br]for my bug from the moment, 0:07:58.310,0:08:03.360 let's say a equals 0 when[br]she or he starts flying up, 0:08:03.360,0:08:06.915 until the moment she[br]completely freaks 0:08:06.915,0:08:11.740 out or drops from the[br]maximum point she reached. 0:08:11.740,0:08:13.400 And she eventually dies. 0:08:13.400,0:08:15.340 Or maybe she doesn't die. 0:08:15.340,0:08:19.280 Maybe she's just drunk and she[br]will wake up after a while. 0:08:19.280,0:08:24.400 OK, so what do I mean by[br]this displacement vector? 0:08:24.400,0:08:25.500 I mean, a function-- 0:08:25.500,0:08:26.480 STUDENT: Is that Tc? 0:08:26.480,0:08:27.480 Do you have [INAUDIBLE]? 0:08:27.480,0:08:28.771 PROFESSOR: This is r, little r. 0:08:28.771,0:08:30.510 STUDENT: I know, but the Tc. 0:08:30.510,0:08:32.010 PROFESSOR: Tc? 0:08:32.010,0:08:33.427 STUDENT: Or is that an I? 0:08:33.427,0:08:34.010 PROFESSOR: No. 0:08:34.010,0:08:37.460 This is I interval,[br]which is the same as a, 0:08:37.460,0:08:41.720 b open interval, like[br]from 2 to 7, included. 0:08:41.720,0:08:46.180 This is inclusion[br][INAUDIBLE] included in R. 0:08:46.180,0:08:51.520 So I mean R is the real number[br]set and a, b is my interval. 0:08:51.520,0:08:55.010 0:08:55.010,0:08:58.460 OK, so r of t is[br]going to be what? 0:08:58.460,0:09:01.680 x of t, y of t, z of t. 0:09:01.680,0:09:05.180 The book tells you, hey, guys--[br]it doesn't say hey, guys, 0:09:05.180,0:09:09.800 but it's quite informal-- if[br]you live in Rn, if your image is 0:09:09.800,0:09:12.720 in Rn, instead of x[br]of t, y of t, z of t, 0:09:12.720,0:09:19.030 you are going to get something[br]like x1 of t, y1 of t. 0:09:19.030,0:09:23.560 x1 of t, x2 of t,[br]x3 of t, et cetera. 0:09:23.560,0:09:26.480 What do we assume about R? 0:09:26.480,0:09:28.589 We have to assume[br]something about it, right? 0:09:28.589,0:09:30.130 STUDENT: It's a[br]function [INAUDIBLE]. 0:09:30.130,0:09:33.340 PROFESSOR: It's a function[br]that is differentiable 0:09:33.340,0:09:36.630 most of the times, right? 0:09:36.630,0:09:37.900 What does it mean smooth? 0:09:37.900,0:09:42.910 I saw that your books[br]before college level 0:09:42.910,0:09:44.300 never mention smooth. 0:09:44.300,0:09:48.550 A smooth function is a[br]function that is differentiable 0:09:48.550,0:09:51.400 and whose first[br]derivative is continuous. 0:09:51.400,0:09:55.710 Some mathematicians even assume[br]that you have c infinity, which 0:09:55.710,0:10:00.320 means you have a function that's[br]infinitely many differentiable. 0:10:00.320,0:10:02.570 So you have first derivative,[br]second derivative, third 0:10:02.570,0:10:03.778 derivative, fifth derivative. 0:10:03.778,0:10:05.060 Somebody stop me. 0:10:05.060,0:10:09.240 All the derivatives exist[br]and they are all continuous. 0:10:09.240,0:10:13.270 By smooth, I will[br]assume c1 in this case. 0:10:13.270,0:10:16.000 I know it's not accurate,[br]but let's assume c1. 0:10:16.000,0:10:18.745 What does it mean? 0:10:18.745,0:10:23.660 Differentiable function whose[br]derivative is continuous. 0:10:23.660,0:10:30.944 0:10:30.944,0:10:35.220 And I will assume[br]one more thing. 0:10:35.220,0:10:37.360 That is not enough for me. 0:10:37.360,0:10:41.900 I will also assume that[br]r prime of t in this case 0:10:41.900,0:10:49.704 is different from 0 for[br]every t in the interval I. 0:10:49.704,0:10:53.780 Could somebody tell me in[br]everyday words what that means? 0:10:53.780,0:10:55.736 We call that regular function. 0:10:55.736,0:10:56.235 [INAUDIBLE] 0:10:56.235,0:11:00.238 0:11:00.238,0:11:01.770 You have a brownie [INAUDIBLE]. 0:11:01.770,0:11:03.390 I have no brownies with me. 0:11:03.390,0:11:05.550 But if you answer, so what-- 0:11:05.550,0:11:08.220 STUDENT: So that means you've[br]got no relative mins or maxes, 0:11:08.220,0:11:11.402 and you never-- the[br]object never stops moving. 0:11:11.402,0:11:15.250 PROFESSOR: Well, actually,[br]you can have relative mins 0:11:15.250,0:11:18.470 and maxes in some way. 0:11:18.470,0:11:22.630 I'm talking about something[br]like that, r prime. 0:11:22.630,0:11:28.330 0:11:28.330,0:11:29.850 This is r of t. 0:11:29.850,0:11:33.380 And r prime of t[br]is the derivative. 0:11:33.380,0:11:34.980 It's never going to stop. 0:11:34.980,0:11:36.000 The velocity. 0:11:36.000,0:11:38.030 I'm talking about this[br]piece of information. 0:11:38.030,0:11:42.490 Velocity [INAUDIBLE] 0 means[br]that drunken bug between time 0:11:42.490,0:11:45.830 a and time b never stops. 0:11:45.830,0:11:50.620 He stops at the end, but the end[br]is b, is outside [INAUDIBLE]. 0:11:50.620,0:11:54.740 So he stops at b and he falls. 0:11:54.740,0:11:56.110 So I don't stop. 0:11:56.110,0:11:59.080 I move on from time a to time b. 0:11:59.080,0:12:01.386 I don't stop at all. 0:12:01.386,0:12:02.832 Yes, sir. 0:12:02.832,0:12:06.628 STUDENT: Wouldn't the derivative[br]of that line at some point 0:12:06.628,0:12:08.030 equal 0 where it flattens out? 0:12:08.030,0:12:10.890 PROFESSOR: Let me[br]draw very well. 0:12:10.890,0:12:14.600 So at time r of t, this[br]is the position vector. 0:12:14.600,0:12:16.552 What is the derivative? 0:12:16.552,0:12:19.580 The derivative represents[br]the velocity vector. 0:12:19.580,0:12:24.250 A beautiful thing about the[br]velocity vector r prime of t 0:12:24.250,0:12:26.910 is that it has a[br]beautiful property. 0:12:26.910,0:12:30.130 It's always tangent[br]to the trajectory. 0:12:30.130,0:12:32.610 So at every point[br]you're going to have 0:12:32.610,0:12:36.468 a velocity vector that is[br]tangent to the trajectory. 0:12:36.468,0:12:37.950 [INAUDIBLE] in physics. 0:12:37.950,0:12:41.902 This r prime of t[br]should never become 0. 0:12:41.902,0:12:46.590 So you will never have a[br]point instead of a segment 0:12:46.590,0:12:51.070 when it comes to r prime. 0:12:51.070,0:12:52.000 So you don't stop. 0:12:52.000,0:12:58.560 0:12:58.560,0:13:00.060 You are going to[br]say, wait a minute? 0:13:00.060,0:13:04.450 But are you always going to[br]consider curves, regular curves 0:13:04.450,0:13:06.220 in space? 0:13:06.220,0:13:10.490 Regular curves in space. 0:13:10.490,0:13:15.720 And by space, I know you guys[br]mean the Euclidean three space. 0:13:15.720,0:13:20.460 Actually, many times I will[br]consider curves in plane. 0:13:20.460,0:13:22.840 And the plane is[br]part of the space. 0:13:22.840,0:13:25.680 And you say, give us an example. 0:13:25.680,0:13:28.210 I will give you an[br]example right now. 0:13:28.210,0:13:30.385 You're going to laugh[br]how simple that is. 0:13:30.385,0:13:33.180 0:13:33.180,0:13:37.240 Now, I have another bug[br]who is really happy, 0:13:37.240,0:13:39.880 but it's not drunk at all. 0:13:39.880,0:13:46.580 And this bug knows how to[br]circle around a certain point 0:13:46.580,0:13:49.310 at the same speed. 0:13:49.310,0:13:51.800 So very organized bug. 0:13:51.800,0:13:52.495 Yes, sir. 0:13:52.495,0:13:55.014 STUDENT: Where did[br]you get c prime? 0:13:55.014,0:13:55.680 PROFESSOR: What? 0:13:55.680,0:14:01.390 STUDENT: You have c prime is[br]differentiable, is [INAUDIBLE]. 0:14:01.390,0:14:02.070 PROFESSOR: c1. 0:14:02.070,0:14:03.001 STUDENT: c1. 0:14:03.001,0:14:03.750 PROFESSOR: OK. c1. 0:14:03.750,0:14:09.246 This is the notation for any[br]function that is differentiable 0:14:09.246,0:14:11.860 and whose derivative[br]is continuous. 0:14:11.860,0:14:16.850 So again, give an[br]example of a c1 function. 0:14:16.850,0:14:18.330 STUDENT: x squared. 0:14:18.330,0:14:19.220 PROFESSOR: Yeah. 0:14:19.220,0:14:20.670 On some real interval. 0:14:20.670,0:14:27.300 How about absolute value[br]of x over the real line? 0:14:27.300,0:14:29.506 What's the problem with that? 0:14:29.506,0:14:30.940 [INTERPOSING VOICES] 0:14:30.940,0:14:33.280 PROFESSOR: It's not[br]differentiable at 0. 0:14:33.280,0:14:36.640 OK, so we'll talk a little[br]bit later about smoothness. 0:14:36.640,0:14:39.848 It's a little bit[br]delicate as a notion. 0:14:39.848,0:14:42.690 It's really beautiful[br]on the other side. 0:14:42.690,0:14:49.830 Let's find the nice picture[br]trajectory for the bug. 0:14:49.830,0:14:51.460 This is a ladybug. 0:14:51.460,0:14:54.170 I cannot draw her, anyway. 0:14:54.170,0:14:56.870 She is moving along this circle. 0:14:56.870,0:15:00.560 And I'll give you[br]the law of motion. 0:15:00.560,0:15:06.960 And that reminds me of a[br]student who told me, what 0:15:06.960,0:15:08.630 do I care about law of motion? 0:15:08.630,0:15:10.790 He never had me as a[br]teacher, obviously. 0:15:10.790,0:15:14.080 But he was telling me,[br]well, after I graduated, 0:15:14.080,0:15:18.460 I always thought, what do I[br]care about the law of motion? 0:15:18.460,0:15:20.650 I mean, I took calculus. 0:15:20.650,0:15:24.240 Everything was about[br]the law of motion. 0:15:24.240,0:15:27.340 I'm sorry, you should care[br]about the law of motion. 0:15:27.340,0:15:30.280 Once you're not there anymore,[br]absolutely you don't care. 0:15:30.280,0:15:33.076 But why do you want to[br][INAUDIBLE] doing calculus? 0:15:33.076,0:15:34.700 When you bring[br][INAUDIBLE] to calculus, 0:15:34.700,0:15:37.460 when you walk into[br]calculus, it's law of motion 0:15:37.460,0:15:39.990 everywhere whether[br]you like it or not. 0:15:39.990,0:15:49.060 So let's try cosine t[br]sine t and z to b 1. 0:15:49.060,0:15:52.240 Let's make it 1 to[br]make your life easier. 0:15:52.240,0:15:54.470 What kind of curve[br]is this and why am I 0:15:54.470,0:15:58.290 claiming that the ladybug[br]following this curve 0:15:58.290,0:16:00.620 is moving at a constant speed? 0:16:00.620,0:16:01.490 Oh my god. 0:16:01.490,0:16:02.619 Go ahead, Alexander. 0:16:02.619,0:16:03.660 STUDENT: That's a circle. 0:16:03.660,0:16:05.120 PROFESSOR: That's the circle. 0:16:05.120,0:16:06.520 It's more than a circle. 0:16:06.520,0:16:07.920 It's a parametrized circle. 0:16:07.920,0:16:10.490 It's a vector-valued function. 0:16:10.490,0:16:15.460 Now, like every mathematician[br]I should specify the domain. 0:16:15.460,0:16:18.140 I am just winding[br]around one time, 0:16:18.140,0:16:20.680 and I stop where I started. 0:16:20.680,0:16:24.430 So I better be smart and[br]realize time is not infinity. 0:16:24.430,0:16:25.540 It could be. 0:16:25.540,0:16:28.130 I'm wrapping around the[br]circle infinitely many times. 0:16:28.130,0:16:30.320 They do that in[br]topology actually when 0:16:30.320,0:16:34.310 you're going to be--[br]seniors takes topology. 0:16:34.310,0:16:38.420 But I'm not going around[br]in circles only one time. 0:16:38.420,0:16:41.060 So my time will[br]start at 0 when I 0:16:41.060,0:16:44.820 start my motion and[br]end at 2 pi seconds 0:16:44.820,0:16:47.930 if the time is in seconds 0:16:47.930,0:16:52.100 So I say r is defined[br]on the interval I which 0:16:52.100,0:16:53.670 is-- say it again, Magdalena. 0:16:53.670,0:16:55.210 You just said it. 0:16:55.210,0:16:56.110 STUDENT: 0. 0:16:56.110,0:16:58.070 PROFESSOR: 0 to pi. 0:16:58.070,0:17:01.050 If you want to take[br]0 together, fine. 0:17:01.050,0:17:05.920 But for consistency, let's[br]take it like before, 0 to 2 pi. 0:17:05.920,0:17:07.775 I'm actually[br]excluding the origin. 0:17:07.775,0:17:10.550 0:17:10.550,0:17:12.200 And with values in R3. 0:17:12.200,0:17:17.618 Although, this is a [? plane ?][br]curve, z will be constant. 0:17:17.618,0:17:19.868 Do I care about that very much? 0:17:19.868,0:17:21.970 You will see the beauty of it. 0:17:21.970,0:17:25.770 I have the velocity vector[br]being really pretty. 0:17:25.770,0:17:28.078 What is the velocity vector? 0:17:28.078,0:17:30.180 STUDENT: [INAUDIBLE]. 0:17:30.180,0:17:31.610 PROFESSOR: Negative sign t. 0:17:31.610,0:17:32.475 Thank you. 0:17:32.475,0:17:33.350 STUDENT: [INAUDIBLE]. 0:17:33.350,0:17:35.860 PROFESSOR: Cosine t. 0:17:35.860,0:17:37.200 And 0, finally. 0:17:37.200,0:17:40.930 Because as you saw[br]very well in the book, 0:17:40.930,0:17:43.940 the way we compute[br]the velocity vector 0:17:43.940,0:17:47.311 is by taking x of[br]t, y of t, z of t 0:17:47.311,0:17:50.130 and differentiating[br]them in terms of time. 0:17:50.130,0:17:54.480 0:17:54.480,0:17:55.000 Good. 0:17:55.000,0:17:58.260 Is this a regular function? 0:17:58.260,0:18:02.520 As the bug moves between[br]time 0 and time equals 2 pi, 0:18:02.520,0:18:06.588 is the bug ever going to[br]stop between these times? 0:18:06.588,0:18:07.427 STUDENT: No. 0:18:07.427,0:18:08.010 PROFESSOR: No. 0:18:08.010,0:18:08.676 How do you know? 0:18:08.676,0:18:10.510 You guys are faster[br]than me, right? 0:18:10.510,0:18:11.270 What did you do? 0:18:11.270,0:18:12.810 You did the speed. 0:18:12.810,0:18:14.210 What's the relationship? 0:18:14.210,0:18:16.539 What's the difference[br]between velocity and speed? 0:18:16.539,0:18:18.580 STUDENT: Speed is the[br]absolute value [INAUDIBLE]. 0:18:18.580,0:18:19.280 PROFESSOR: Wonderful. 0:18:19.280,0:18:20.076 This is very good. 0:18:20.076,0:18:21.950 You should tell everybody[br]that because people 0:18:21.950,0:18:23.745 confuse that left and right. 0:18:23.745,0:18:26.792 So the velocity is[br]a vector, like you 0:18:26.792,0:18:28.180 learned in engineering. 0:18:28.180,0:18:29.960 You learned in physics. 0:18:29.960,0:18:31.090 Velocity is a vector. 0:18:31.090,0:18:32.080 It changes direction. 0:18:32.080,0:18:33.960 I'm going to Amarillo this way. 0:18:33.960,0:18:34.900 I'm driving. 0:18:34.900,0:18:37.340 The velocity will be a[br]vector pointing this way. 0:18:37.340,0:18:40.790 As I come back, will[br]point the opposite way. 0:18:40.790,0:18:43.815 The speed will be a[br]scalar, not a vector. 0:18:43.815,0:18:46.460 It's a magnitude of[br]a velocity vector. 0:18:46.460,0:18:48.060 So say it again, Magdalena. 0:18:48.060,0:18:49.240 What is the speed? 0:18:49.240,0:18:55.540 The speed is the magnitude[br]of the velocity vector. 0:18:55.540,0:18:58.585 It's a scalar. 0:18:58.585,0:19:00.780 Speed. 0:19:00.780,0:19:02.820 Speed. 0:19:02.820,0:19:06.040 I heard that before in[br]cars, in the movie Cars. 0:19:06.040,0:19:10.700 Anyway, r prime of t magnitude. 0:19:10.700,0:19:12.440 In magnitude. 0:19:12.440,0:19:17.360 Remember, there is a big[br]difference between the velocity 0:19:17.360,0:19:19.095 as the notion. 0:19:19.095,0:19:22.770 Velocity is a vector. 0:19:22.770,0:19:25.570 The speed is a[br]magnitude, is a scalar. 0:19:25.570,0:19:27.800 I'm going to go[br]ahead and erase that 0:19:27.800,0:19:33.190 and I'm going to ask[br]you what the speed is 0:19:33.190,0:19:36.200 for my fellow over here. 0:19:36.200,0:19:40.612 What is the speed[br]of a trajectory 0:19:40.612,0:19:46.350 of the bug who is sober and[br]moves at the constant speed? 0:19:46.350,0:19:46.850 OK. 0:19:46.850,0:19:49.210 As I already told[br]you, it's constant. 0:19:49.210,0:19:50.390 What is that constant? 0:19:50.390,0:19:53.800 0:19:53.800,0:19:56.590 What's the constant speed[br]I was talking about? 0:19:56.590,0:19:58.790 STUDENT: [INAUDIBLE]. 0:19:58.790,0:20:01.840 PROFESSOR: I say the[br]magnitude of that. 0:20:01.840,0:20:04.360 I'm too lazy to write it down. 0:20:04.360,0:20:06.225 It's a Tuesday, almost morning. 0:20:06.225,0:20:10.240 So I go square root[br]of minus I squared 0:20:10.240,0:20:12.050 plus cosine squared plus 0. 0:20:12.050,0:20:13.780 I don't need to write that down. 0:20:13.780,0:20:15.270 You write it down. 0:20:15.270,0:20:16.580 And how much is that? 0:20:16.580,0:20:17.455 STUDENT: [INAUDIBLE]. 0:20:17.455,0:20:18.170 PROFESSOR: 1. 0:20:18.170,0:20:24.600 So I love this curve because[br]in mathematician slang, 0:20:24.600,0:20:28.800 especially in [? a geometer's ?][br]slang-- and my area 0:20:28.800,0:20:30.240 is differential geometry. 0:20:30.240,0:20:35.250 So in a way, I do calculus in[br]R3 every day on a daily basis. 0:20:35.250,0:20:37.030 So I have what? 0:20:37.030,0:20:42.572 This is a special kind of curve. 0:20:42.572,0:20:46.380 It's a curve parameterized[br]in arc length. 0:20:46.380,0:20:57.840 So definition, we say[br]that a curve in R3, 0:20:57.840,0:21:10.130 or Rn, well anyway, is[br]parameterized in arc length. 0:21:10.130,0:21:12.940 0:21:12.940,0:21:13.440 When? 0:21:13.440,0:21:14.450 Say it again, Magdalena. 0:21:14.450,0:21:32.270 Whenever, if and only if,[br]its speed is constantly 1. 0:21:32.270,0:21:36.150 0:21:36.150,0:21:40.630 So this is an example[br]where the speed is 1. 0:21:40.630,0:21:45.548 In such cases, we avoid[br]the notation with t. 0:21:45.548,0:21:46.470 You say, oh my god. 0:21:46.470,0:21:47.460 Why? 0:21:47.460,0:21:50.100 When the curve is[br]parameterized in arc length, 0:21:50.100,0:21:54.970 from now on the we[br]will actually try 0:21:54.970,0:21:58.920 to use s whatever we[br]know it's an arc length. 0:21:58.920,0:22:00.850 We use s instead of t. 0:22:00.850,0:22:05.210 So I'm sorry for the people[br]who cannot change that, 0:22:05.210,0:22:08.700 but you should all be[br]able t change that. 0:22:08.700,0:22:12.550 So everything will be[br]in s because we just 0:22:12.550,0:22:15.450 discovered[br][? Discovery Channel, ?] we 0:22:15.450,0:22:19.400 just discovered that speed is 1. 0:22:19.400,0:22:24.300 So there is something[br]special about this s. 0:22:24.300,0:22:29.380 0:22:29.380,0:22:32.720 In this example-- oh, you[br]can rewrite the whole example 0:22:32.720,0:22:37.370 if you want in s so you don't[br]have to smudge the paper. 0:22:37.370,0:22:38.830 OK, it's beautiful. 0:22:38.830,0:22:41.420 So I am already arc length. 0:22:41.420,0:22:43.950 And in that case, I'm going[br]to call my time parameter 0:22:43.950,0:22:46.190 little s. s comes from special. 0:22:46.190,0:22:48.460 No, s comes from[br]speed [INAUDIBLE]. 0:22:48.460,0:22:51.590 STUDENT: So you use s[br]when it's [INAUDIBLE]? 0:22:51.590,0:22:57.140 PROFESSOR: We use s whenever the[br]speed of that curve will be 1. 0:22:57.140,0:22:58.140 STUDENT: So [INAUDIBLE]. 0:22:58.140,0:23:00.473 PROFESSOR: And we call that[br]arc length parameterization. 0:23:00.473,0:23:02.970 0:23:02.970,0:23:06.240 I'm moving into the duration[br]of your final thoughts. 0:23:06.240,0:23:07.961 Yes, sir. 0:23:07.961,0:23:09.502 STUDENT: When we[br]get the question, so 0:23:09.502,0:23:10.627 before solving [INAUDIBLE]. 0:23:10.627,0:23:13.140 0:23:13.140,0:23:14.460 PROFESSOR: We don't know. 0:23:14.460,0:23:17.920 That's why it was our[br]discovery that, hey, at the end 0:23:17.920,0:23:22.430 it is an arc length, so I better[br]change [INAUDIBLE] t into s 0:23:22.430,0:23:26.920 because that will help me in[br]the future remember to do that. 0:23:26.920,0:23:30.380 Every time I have arc length,[br]that it means speed 1. 0:23:30.380,0:23:33.390 I will call it s instead of y. 0:23:33.390,0:23:34.920 There is a reason for that. 0:23:34.920,0:23:36.550 I'm going to erase[br]the definition 0:23:36.550,0:23:42.865 and I'm going to give[br]you the-- more or less, 0:23:42.865,0:23:45.580 the explanation that my[br]physics professor gave me. 0:23:45.580,0:23:50.170 Because as a freshman,[br]my mathematics professor 0:23:50.170,0:23:54.450 in that area, in geometry,[br]was not very, very active. 0:23:54.450,0:23:57.330 But practically, what my physics[br]professor told me is that, 0:23:57.330,0:24:05.900 hey, I would like to have[br]some sort of a uniform tangent 0:24:05.900,0:24:09.830 vector, something that is[br]standardized to be in speed 1. 0:24:09.830,0:24:15.860 So I would like that tangent[br]vector to be important to us. 0:24:15.860,0:24:19.845 And if r is an[br]arc length, then r 0:24:19.845,0:24:24.110 prime would be that unit[br]vector that I'm talking about. 0:24:24.110,0:24:30.790 So he introduced for any r of[br]t, which is x of t, y of t, 0:24:30.790,0:24:32.120 z of t. 0:24:32.120,0:24:36.580 My physics professor introduced[br]the following terminology. 0:24:36.580,0:24:42.870 The tangent unit vector[br]for a regular curve-- 0:24:42.870,0:24:46.630 he was very well-organized[br]I might add about him-- 0:24:46.630,0:24:52.670 is by definition r[br]prime of t as a vector 0:24:52.670,0:24:54.380 divided by the[br]speed of the vector. 0:24:54.380,0:24:56.250 So what is he doing? 0:24:56.250,0:24:58.535 He is unitarizing the velocity. 0:24:58.535,0:25:00.130 Say it again, Magdalena. 0:25:00.130,0:25:03.210 He has unitarized[br]the velocity in order 0:25:03.210,0:25:08.500 to make research more consistent[br]from the viewpoint of Frenet 0:25:08.500,0:25:10.040 frame. 0:25:10.040,0:25:12.520 So in Frenet frame, you[br]will see-- you probably 0:25:12.520,0:25:14.310 learned about the[br]Frenet frame if you 0:25:14.310,0:25:18.260 are a mechanics major, or some[br]solid mechanics or physics 0:25:18.260,0:25:19.130 major. 0:25:19.130,0:25:22.600 The Frenet frame is[br]an orthogonal frame 0:25:22.600,0:25:28.750 moving along a line in time[br]where the three components are 0:25:28.750,0:25:33.200 t, and the principal normal[br]vector, and b the [INAUDIBLE]. 0:25:33.200,0:25:35.510 We only know of the[br]first of them, which 0:25:35.510,0:25:38.650 is T, which is a unit vector. 0:25:38.650,0:25:40.320 Say it again who it was. 0:25:40.320,0:25:44.980 It was the velocity vector[br]divided by its magnitude. 0:25:44.980,0:25:47.330 So the velocity vector could[br]be any wild, crazy vector 0:25:47.330,0:25:54.560 that's tangent to the trajectory[br]at the point where you are. 0:25:54.560,0:25:58.120 His magnitude varies from[br]one point to the other. 0:25:58.120,0:25:59.910 He's absolutely crazy. 0:25:59.910,0:26:01.430 He or she, the velocity vector. 0:26:01.430,0:26:02.290 Yes, sir. 0:26:02.290,0:26:03.165 STUDENT: [INAUDIBLE]. 0:26:03.165,0:26:06.849 0:26:06.849,0:26:07.515 PROFESSOR: Here? 0:26:07.515,0:26:08.332 Here? 0:26:08.332,0:26:09.415 STUDENT: Yeah, down there. 0:26:09.415,0:26:10.623 PROFESSOR: D-E-F, definition. 0:26:10.623,0:26:12.815 That's how a mathematician[br]defines things. 0:26:12.815,0:26:18.340 So to define you write def[br]on top of an equality sign 0:26:18.340,0:26:20.960 or double dot equal. 0:26:20.960,0:26:23.630 That's a formal way a[br]mathematician introduces 0:26:23.630,0:26:24.780 a definition. 0:26:24.780,0:26:27.600 Well, he was a physicist,[br]but he does math. 0:26:27.600,0:26:29.330 So what do we do? 0:26:29.330,0:26:32.500 We say all the blue[br]guys that are not equal, 0:26:32.500,0:26:34.580 divide yourselves[br]by your magnitude. 0:26:34.580,0:26:39.720 And I'm going to have[br]the T here is next one, 0:26:39.720,0:26:42.750 the T here is next one,[br]the T here is next one. 0:26:42.750,0:26:43.580 They are all equal. 0:26:43.580,0:26:51.200 So that T changes direction, but[br]its magnitude will always be 1. 0:26:51.200,0:26:51.700 Right? 0:26:51.700,0:26:55.000 Know that the magnitude--[br]that's what unit vector means, 0:26:55.000,0:26:58.200 the magnitude is 1. 0:26:58.200,0:27:00.706 Why am I so happy about that? 0:27:00.706,0:27:03.940 Well let me tell[br]you that we can have 0:27:03.940,0:27:07.480 another parametrization[br]and another parametrization 0:27:07.480,0:27:11.120 and another parametrization[br]of the same curve. 0:27:11.120,0:27:12.260 Say what? 0:27:12.260,0:27:14.820 The parametrization of[br]a curve is not unique? 0:27:14.820,0:27:15.717 No. 0:27:15.717,0:27:18.810 There are infinitely[br]many parametrizations 0:27:18.810,0:27:21.930 for a physical curve. 0:27:21.930,0:27:34.158 There are infinitely[br]many parametrizations 0:27:34.158,0:27:39.730 for an even physical curve. 0:27:39.730,0:27:43.058 0:27:43.058,0:27:45.030 Like [INAUDIBLE][br]the regular one? 0:27:45.030,0:27:47.495 Well let me give you[br]another example that 0:27:47.495,0:27:51.475 says that this is[br]currently R of T 0:27:51.475,0:27:58.290 equals cosine 5T sine 5T and 1. 0:27:58.290,0:27:59.060 Why 1? 0:27:59.060,0:28:02.560 I still want to have[br]the same physical curve. 0:28:02.560,0:28:03.690 What's different, guys? 0:28:03.690,0:28:06.740 Look at that and then[br]say oh OK, is this 0:28:06.740,0:28:11.890 the same curve as[br]a physical curve? 0:28:11.890,0:28:13.410 What's different in this case? 0:28:13.410,0:28:14.800 I'm still here. 0:28:14.800,0:28:16.620 It's still the[br][? red ?] physical curve 0:28:16.620,0:28:18.221 I'm moving along. 0:28:18.221,0:28:18.970 What is different? 0:28:18.970,0:28:19.886 STUDENT: The velocity. 0:28:19.886,0:28:20.960 PROFESSOR: The velocity. 0:28:20.960,0:28:23.880 The velocity and[br]actually the speed. 0:28:23.880,0:28:29.380 I'm moving faster or slower, I[br]don't know, we have to decide. 0:28:29.380,0:28:34.230 Now how do I realize[br]how many times 0:28:34.230,0:28:36.380 I'm moving along this curve? 0:28:36.380,0:28:39.740 I can be smart and say[br]hey, I'm not stupid. 0:28:39.740,0:28:43.420 I know how to move only one[br]time and stop where I started. 0:28:43.420,0:28:47.040 So if I start with[br]my T in the interval 0:28:47.040,0:28:52.630 zero-- I start at[br]zero, where do I stop? 0:28:52.630,0:28:54.456 I can hear your brain buzzing. 0:28:54.456,0:28:55.330 STUDENT: [INAUDIBLE]. 0:28:55.330,0:28:57.901 PROFESSOR: 2pi over 5. 0:28:57.901,0:28:58.840 Why is that? 0:28:58.840,0:29:00.260 Excellent answer. 0:29:00.260,0:29:03.354 STUDENT: Because when you[br]plug it in, it's [INAUDIBLE]. 0:29:03.354,0:29:05.150 PROFESSOR: 5 times 2pi over 5. 0:29:05.150,0:29:06.130 That's where I stop. 0:29:06.130,0:29:08.364 So this is not the same[br]interval as before. 0:29:08.364,0:29:09.678 Are you guys with me? 0:29:09.678,0:29:16.650 This is a new guy, which[br]is called J. Oh, all right. 0:29:16.650,0:29:19.460 So there is a[br]relationship between the T 0:29:19.460,0:29:23.480 and the S. That's why I[br]use different notations. 0:29:23.480,0:29:26.740 And I wish my teachers[br]started it just 0:29:26.740,0:29:29.794 like that when I took math[br]analysis as a freshman, 0:29:29.794,0:29:30.630 or calculus. 0:29:30.630,0:29:32.330 That's calculus. 0:29:32.330,0:29:35.700 Because what they started[br]with was a diagram. 0:29:35.700,0:29:37.300 What kind of diagram? 0:29:37.300,0:29:41.840 Say OK, the[br]parametrizations are both 0:29:41.840,0:29:45.060 starting from[br]different intervals. 0:29:45.060,0:29:47.550 And first I have[br]the parametrization 0:29:47.550,0:29:50.290 from I going to our 3. 0:29:50.290,0:29:53.350 And that's called-- how[br]did we baptize that? 0:29:53.350,0:29:57.640 R. And the other[br]one, from J to R3, 0:29:57.640,0:30:01.790 we call that big R.[br]They're both vectors. 0:30:01.790,0:30:05.170 And hey guys, we[br]should have some sort 0:30:05.170,0:30:08.720 of correspondence[br]functions between I 0:30:08.720,0:30:14.145 and J that are both 1 to 1, and[br]they are 1 being [INAUDIBLE] 0:30:14.145,0:30:16.420 the other. 0:30:16.420,0:30:18.310 I swear to God,[br]when they started 0:30:18.310,0:30:21.000 with this theoretical[br]model, I didn't understand 0:30:21.000,0:30:23.190 the motivation at all. 0:30:23.190,0:30:25.220 At all. 0:30:25.220,0:30:27.620 Now with an example,[br]I can get you 0:30:27.620,0:30:30.890 closer to the motivation[br]of such a diagram. 0:30:30.890,0:30:34.700 So where does our[br]primary S live? 0:30:34.700,0:30:39.130 S lives in I, and[br]T lives in J. So I 0:30:39.130,0:30:42.790 have to have a correspondence[br]that takes S to T or T to S. 0:30:42.790,0:30:46.352 STUDENT: Wait I[br]thought since R of T 0:30:46.352,0:30:48.477 is also pretty much[br][INAUDIBLE] that we should also 0:30:48.477,0:30:49.590 use S [INAUDIBLE]. 0:30:49.590,0:30:53.040 PROFESSOR: It's very--[br]actually it's very easy. 0:30:53.040,0:30:55.880 This is 5T. 0:30:55.880,0:31:02.250 And we cannot use S[br]instead of this T, 0:31:02.250,0:31:05.080 because if we use S[br]instead of this T, 0:31:05.080,0:31:07.690 and we compute the[br]speed, we get 5. 0:31:07.690,0:31:10.940 So it cannot be called S.[br]This is very important. 0:31:10.940,0:31:15.135 So T is not an arc[br]length parameter. 0:31:15.135,0:31:18.170 I wonder what the speed[br]will be for this guy. 0:31:18.170,0:31:20.470 So who wants to[br]compute R prime of T? 0:31:20.470,0:31:22.580 Nobody, but I'll force you to. 0:31:22.580,0:31:26.520 And the magnitude of that[br]will be god knows what. 0:31:26.520,0:31:27.740 I claim it's 5. 0:31:27.740,0:31:30.110 Maybe I'm wrong. 0:31:30.110,0:31:31.270 I did this in my head. 0:31:31.270,0:31:33.150 I have to do it on paper, right. 0:31:33.150,0:31:35.310 So I have what? 0:31:35.310,0:31:38.730 I have to differentiate[br]component-wise. 0:31:38.730,0:31:42.250 And I have [INAUDIBLE] that,[br]because I'm running out of gas. 0:31:42.250,0:31:43.030 STUDENT: Minus 5-- 0:31:43.030,0:31:45.590 PROFESSOR: Minus 5, very good. 0:31:45.590,0:31:47.720 Sine of 5T. 0:31:47.720,0:31:49.320 What have we applied? 0:31:49.320,0:31:51.510 In case you don't[br]know that, out. 0:31:51.510,0:31:52.770 That was Calc 1. 0:31:52.770,0:31:53.590 Chain rule. 0:31:53.590,0:31:55.420 Right? 0:31:55.420,0:32:00.080 So 5 times cosine 5T. 0:32:00.080,0:32:03.720 And finally, 1[br]prime, which is 0. 0:32:03.720,0:32:09.850 Now let's be brave and[br]write the whole thing down. 0:32:09.850,0:32:13.150 I know I'm lazy today, but I'm[br]going to have to do something. 0:32:13.150,0:32:13.650 Right? 0:32:13.650,0:32:18.390 So I'll say minus 5[br]sine 5T is all squared. 0:32:18.390,0:32:20.880 Let me take it and square it. 0:32:20.880,0:32:23.886 Because I see one[br]face is confused. 0:32:23.886,0:32:26.840 And since one face[br]is confused, it 0:32:26.840,0:32:29.970 doesn't matter that the[br]others are not confused. 0:32:29.970,0:32:31.060 OK? 0:32:31.060,0:32:36.280 So I have square root of this[br]plus square of [INAUDIBLE] plus 0:32:36.280,0:32:38.540 [INAUDIBLE] computing[br]the magnitude. 0:32:38.540,0:32:39.750 What do I get out of here? 0:32:39.750,0:32:40.482 STUDENT: Five. 0:32:40.482,0:32:41.170 PROFESSOR: Five. 0:32:41.170,0:32:41.780 Excellent. 0:32:41.780,0:32:45.430 This is 5 sine squared[br]plus 5 cosine squared. 0:32:45.430,0:32:49.670 Now yes, then I have 5 times 1. 0:32:49.670,0:32:54.550 So I have square root[br]of 25 here will be 5. 0:32:54.550,0:32:55.700 What is 5? 0:32:55.700,0:33:03.480 5 is the speed of the [? bug ?][br]along the same physical curve 0:33:03.480,0:33:04.640 the other way around. 0:33:04.640,0:33:06.990 The second time around. 0:33:06.990,0:33:10.210 Now can you tell me the[br]relationship between T and S? 0:33:10.210,0:33:12.578 They are related. 0:33:12.578,0:33:19.220 They are like if you're my[br]uncle, then I'm your niece. 0:33:19.220,0:33:21.290 It's the same way. 0:33:21.290,0:33:23.015 It depends where you look at. 0:33:23.015,0:33:26.040 T is a function of S,[br]and S is a function of T. 0:33:26.040,0:33:32.220 So it has to be a 1 to 1[br]correspondence between the two. 0:33:32.220,0:33:38.235 Now any ideas of how I what[br]to compute the-- how do I 0:33:38.235,0:33:43.180 want to write the[br]relationship between them. 0:33:43.180,0:33:46.400 Well, S is a[br]function of T, right? 0:33:46.400,0:33:50.534 I just don't know what[br]function of T that is. 0:33:50.534,0:33:52.450 And I wish my professor[br]had started like that, 0:33:52.450,0:33:54.710 but he started[br]with this diagram. 0:33:54.710,0:33:58.890 So simply here you[br]have S equals S of T, 0:33:58.890,0:34:01.380 and here you have[br]T equals T of S, 0:34:01.380,0:34:03.170 the inverse of that function. 0:34:03.170,0:34:05.820 And when you-- when[br]somebody starts that 0:34:05.820,0:34:09.560 without an example as a[br]general diagram philosophy, 0:34:09.560,0:34:12.050 then it's really, really tough. 0:34:12.050,0:34:13.050 All right? 0:34:13.050,0:34:16.050 So I'd like to know[br]who S of T-- how 0:34:16.050,0:34:19.530 in the world do I want[br]to define that S of T. 0:34:19.530,0:34:25.570 He spoonfed us S of T. I don't[br]want to spoonfeed you anything. 0:34:25.570,0:34:27.728 Because this is[br]honors class, and you 0:34:27.728,0:34:30.929 should be able to figure[br]this out yourselves. 0:34:30.929,0:34:35.840 So who is big R of T? 0:34:35.840,0:34:42.199 Big R of T should[br]be, what, should 0:34:42.199,0:34:44.820 be the same thing in[br]the end as R of S. 0:34:44.820,0:34:56.690 But I should say maybe it's[br]R of function T of S, right? 0:34:56.690,0:34:59.655 Which is the same[br]thing as R of S. So 0:34:59.655,0:35:05.820 what should be the[br]relationship between T and S? 0:35:05.820,0:35:11.280 We have to call them-- one of[br]them should be T equals T of S. 0:35:11.280,0:35:12.520 How about this function? 0:35:12.520,0:35:15.590 Give it a Greek name,[br]what do you want. 0:35:15.590,0:35:16.120 Alpha? 0:35:16.120,0:35:16.670 Beta? 0:35:16.670,0:35:16.800 What? 0:35:16.800,0:35:17.675 STUDENT: [INAUDIBLE]. 0:35:17.675,0:35:19.250 PROFESSOR: Alpha? 0:35:19.250,0:35:19.790 Beta? 0:35:19.790,0:35:20.289 Alpha? 0:35:20.289,0:35:21.700 I don't know. 0:35:21.700,0:35:25.520 So S going to T, alpha. 0:35:25.520,0:35:27.270 And this is going[br]to be alpha inverse. 0:35:27.270,0:35:30.644 0:35:30.644,0:35:32.090 Right? 0:35:32.090,0:35:37.305 So T equals alpha of S.[br]It's more elegant to call it 0:35:37.305,0:35:44.980 like that than T of S. T[br]equals alpha of S. Alpha of S. 0:35:44.980,0:35:49.080 So from this thing,[br]I realize that I 0:35:49.080,0:35:54.296 get that R composed with[br]alpha equals R. Say what? 0:35:54.296,0:35:54.796 Magdalena? 0:35:54.796,0:35:57.170 Yeah, yeah, that[br]was pre-calculus. 0:35:57.170,0:36:01.110 R composed with alpha[br]equals little r. 0:36:01.110,0:36:09.141 So how do I get a little r[br]by composing R with alpha? 0:36:09.141,0:36:12.087 How do we say that? 0:36:12.087,0:36:17.488 Alpha followed by R.[br]R composed with alpha. 0:36:17.488,0:36:22.030 R of alpha of S equals[br]R of S. Say it again. 0:36:22.030,0:36:30.770 R of alpha of S, which is T--[br]this T is alpha of S-- equals 0:36:30.770,0:36:31.420 R. 0:36:31.420,0:36:39.190 This is the composition[br]that we learned in pre-calc. 0:36:39.190,0:36:40.925 Who can find me the[br]definition of S? 0:36:40.925,0:36:44.372 Because this may be[br]a little bit hard. 0:36:44.372,0:36:46.580 This may be a little bit hard. 0:36:46.580,0:36:48.901 STUDENT: S [INAUDIBLE]. 0:36:48.901,0:36:52.430 PROFESSOR: Eh, yeah,[br]let me write it down. 0:36:52.430,0:36:56.868 I want to find out[br]what S of T is. 0:36:56.868,0:36:59.940 0:36:59.940,0:37:11.302 Equals what in terms of the[br]function R of T. The one 0:37:11.302,0:37:13.787 that's given here. 0:37:13.787,0:37:14.781 Why is that? 0:37:14.781,0:37:22.760 0:37:22.760,0:37:26.060 Let's try some sort[br]of chain rule, right? 0:37:26.060,0:37:28.698 So what do I know I have? 0:37:28.698,0:37:29.820 I have that. 0:37:29.820,0:37:32.740 Look at that. 0:37:32.740,0:37:39.070 R prime of S, which[br]is the velocity of-- I 0:37:39.070,0:37:43.840 erased it-- the velocity of R[br]with respect to the arc length 0:37:43.840,0:37:46.560 parameter is going to be what? 0:37:46.560,0:37:52.090 R of alpha of S prime[br]with respect to S, right? 0:37:52.090,0:37:53.835 So I should put DDS. 0:37:53.835,0:37:55.430 Well I'm a little bit lazy. 0:37:55.430,0:37:58.190 Let's do it again. 0:37:58.190,0:38:06.070 DDS, R of alpha of S. 0:38:06.070,0:38:07.930 OK. 0:38:07.930,0:38:11.060 And what do I have in this case? 0:38:11.060,0:38:18.562 Well, I have R prime of-- who is[br]alpha of S. T, [INAUDIBLE] of T 0:38:18.562,0:38:27.035 and alpha of S times[br]R prime of alpha 0:38:27.035,0:38:30.400 of S times the prime outside. 0:38:30.400,0:38:32.300 How do we prime[br]in the chain rule? 0:38:32.300,0:38:35.220 From the outside to the[br]inside, one at a time. 0:38:35.220,0:38:38.760 So I differentiated the[br]outer shell, R prime, 0:38:38.760,0:38:39.910 and then times what? 0:38:39.910,0:38:41.390 Chain rule, guys. 0:38:41.390,0:38:44.890 Alpha prime of S. Very good. 0:38:44.890,0:38:50.490 Alpha prime of S. 0:38:50.490,0:38:51.100 All right. 0:38:51.100,0:38:54.750 So I would like[br]to understand how 0:38:54.750,0:39:02.640 I want to compute-- how I want[br]to define S of T. If I take 0:39:02.640,0:39:06.590 this in absolute value, R[br]prime of S in absolute value 0:39:06.590,0:39:11.990 equals R prime of T in absolute[br]value times alpha prime of S 0:39:11.990,0:39:14.562 in absolute value. 0:39:14.562,0:39:15.145 What do I get? 0:39:15.145,0:39:20.510 0:39:20.510,0:39:22.406 Who is R prime of S? 0:39:22.406,0:39:26.160 This is my original[br]function in arc length, 0:39:26.160,0:39:28.660 and that's the[br]speed in arc length. 0:39:28.660,0:39:30.980 What was the speed[br]in arc length? 0:39:30.980,0:39:31.820 STUDENT: One. 0:39:31.820,0:39:33.900 PROFESSOR: One. 0:39:33.900,0:39:37.082 And what is the speed[br]in not in arc length? 0:39:37.082,0:39:38.474 STUDENT: Five. 0:39:38.474,0:39:41.810 PROFESSOR: In that case,[br]this is going to be five. 0:39:41.810,0:39:46.325 And so what is this[br]alpha prime of S guy? 0:39:46.325,0:39:47.200 STUDENT: [INAUDIBLE]. 0:39:47.200,0:39:51.015 PROFESSOR: It's going to be 1/5. 0:39:51.015,0:39:52.440 OK. 0:39:52.440,0:39:52.960 All right. 0:39:52.960,0:39:56.125 Actually alpha of S,[br]who is that going to be? 0:39:56.125,0:40:03.900 Alpha of S. 0:40:03.900,0:40:06.610 Do you notice the[br]correspondence? 0:40:06.610,0:40:12.070 We simply have to re-define[br]this as S. That's how it goes. 0:40:12.070,0:40:14.627 That five times[br]is nothing but S. 0:40:14.627,0:40:17.012 STUDENT: How did you[br]get the [INAUDIBLE]? 0:40:17.012,0:40:21.450 PROFESSOR: Because 1[br]equals 5 times what? 0:40:21.450,0:40:26.205 1, which is arc length[br]speed, equals 5 times what? 0:40:26.205,0:40:26.705 1/5. 0:40:26.705,0:40:27.600 STUDENT: Yeah, but then[br]where'd you get the 1? 0:40:27.600,0:40:29.058 PROFESSOR: That's[br]one way to do it. 0:40:29.058,0:40:32.290 Oh, this is by definition,[br]because little r means 0:40:32.290,0:40:35.600 curve in arc length, and little[br]s is the arc length parameter. 0:40:35.600,0:40:39.170 By definition, that[br]means you get speed 1. 0:40:39.170,0:40:40.830 This was our assumption. 0:40:40.830,0:40:44.140 So we could've gotten[br]that much faster saying 0:40:44.140,0:40:46.220 oh, well, forget[br]about this diagram 0:40:46.220,0:40:48.750 that you introduced-- and[br]it's also in the book. 0:40:48.750,0:40:52.960 Simply take 5T to BS, 5T to BS. 0:40:52.960,0:40:56.320 Then I get my old[br]friend, the curve. 0:40:56.320,0:40:59.200 The arc length[br]parameter is the curve. 0:40:59.200,0:41:04.520 So this is the same as cosine[br]of S, sine of S, and 1. 0:41:04.520,0:41:07.650 So what is the correspondence[br]between S and T? 0:41:07.650,0:41:10.590 0:41:10.590,0:41:14.930 Since S is 5T in[br]this example, I'll 0:41:14.930,0:41:16.397 put it-- where shall I put it. 0:41:16.397,0:41:19.810 I'll put it here. 0:41:19.810,0:41:22.640 S is 5T. 0:41:22.640,0:41:24.784 I'll say S of T is 5T. 0:41:24.784,0:41:28.088 0:41:28.088,0:41:32.240 and T of S, what[br]is T in terms of S? 0:41:32.240,0:41:37.050 T in terms of S is S over 5. 0:41:37.050,0:41:39.905 So instead of T of[br]S, we call this alpha 0:41:39.905,0:41:47.798 of S. So the correspondence[br]between S and T, what is T? 0:41:47.798,0:41:51.970 T is exactly S over[br]5 in this example. 0:41:51.970,0:41:52.640 Say it again. 0:41:52.640,0:41:55.190 T is exactly S over 5. 0:41:55.190,0:41:57.644 So alpha of S would be S over 5. 0:41:57.644,0:42:01.770 In this case, alpha prime of[br]S would simply be 1 over 5. 0:42:01.770,0:42:04.410 Oh, so that's how I got it. 0:42:04.410,0:42:06.360 That's another way to get it. 0:42:06.360,0:42:07.500 Much faster. 0:42:07.500,0:42:09.290 Much simpler. 0:42:09.290,0:42:13.640 So just think of replacing[br]5T by the S knowing 0:42:13.640,0:42:19.020 that you put S here, the whole[br]thing will have speed of 1. 0:42:19.020,0:42:19.610 All right. 0:42:19.610,0:42:21.560 So what do I do? 0:42:21.560,0:42:24.640 I say OK, alpha prime[br]of S is 1 over 5. 0:42:24.640,0:42:28.265 The whole chain rule also[br]spit out alpha prime of S 0:42:28.265,0:42:29.500 to B1 over 5. 0:42:29.500,0:42:32.540 Now I understand the[br]relationship between S and T. 0:42:32.540,0:42:33.685 It's very simple. 0:42:33.685,0:42:39.800 S is 5T in this example,[br]or T equals S over 5. 0:42:39.800,0:42:40.300 OK? 0:42:40.300,0:42:46.430 So if somebody gives you a curve[br]that looks like cosine 5T, sine 0:42:46.430,0:42:52.400 5T, 1, and that is in speed[br]5, as we were able to find, 0:42:52.400,0:42:56.800 how do you re-parametrize[br]that in arc length? 0:42:56.800,0:43:01.490 You just change[br]something inside so 0:43:01.490,0:43:08.190 that you make this curve be[br]representative-- representable 0:43:08.190,0:43:12.328 as little r of S.[br]This is in arc length. 0:43:12.328,0:43:13.795 In arc length. 0:43:13.795,0:43:17.700 0:43:17.700,0:43:18.200 OK. 0:43:18.200,0:43:20.330 Finally, this is[br]just an example. 0:43:20.330,0:43:23.680 Can you tell me how that[br]arc length parameter 0:43:23.680,0:43:25.870 is introduced in general? 0:43:25.870,0:43:29.712 What is S of T by definition? 0:43:29.712,0:43:34.200 What if I have[br]something really wild? 0:43:34.200,0:43:36.410 How do I get to that[br]S of T by definition? 0:43:36.410,0:43:38.948 0:43:38.948,0:43:41.358 What is S of T in terms[br]of the function R? 0:43:41.358,0:43:45.230 STUDENT: [INAUDIBLE] velocity[br][? of the ?] [INAUDIBLE]? 0:43:45.230,0:43:47.840 PROFESSOR: S prime of T will[br]be one of the [INAUDIBLE]. 0:43:47.840,0:43:48.670 STUDENT: Yes. 0:43:48.670,0:43:49.340 PROFESSOR: OK. 0:43:49.340,0:43:58.770 So let's see what we[br]have if we define S of T 0:43:58.770,0:44:12.460 as being integral from 0 to[br]T of the speed R prime of T. 0:44:12.460,0:44:14.330 And instead of T, we put tau. 0:44:14.330,0:44:14.830 Right? 0:44:14.830,0:44:15.830 P tau. 0:44:15.830,0:44:18.330 STUDENT: What is that? 0:44:18.330,0:44:20.450 PROFESSOR: We cannot[br]put T, T, and T. 0:44:20.450,0:44:21.264 STUDENT: Oh. 0:44:21.264,0:44:22.080 PROFESSOR: OK? 0:44:22.080,0:44:25.700 So tau is the Greek T[br]that runs between zero 0:44:25.700,0:44:29.490 and T. This is the[br]definition of S 0:44:29.490,0:44:44.196 of T. General definition[br]of the arc length parameter 0:44:44.196,0:44:49.652 that is according to the chain[br]rule, given by the chain rule. 0:44:49.652,0:44:57.110 0:44:57.110,0:45:00.040 Can we verify really[br]quickly in our case, 0:45:00.040,0:45:02.500 is it easy to see that[br]in our case it's correct? 0:45:02.500,0:45:03.260 STUDENT: Yeah. 0:45:03.260,0:45:05.530 PROFESSOR: Oh yeah,[br]S of T will be, 0:45:05.530,0:45:08.440 in our case,[br]integral from 0 to T. 0:45:08.440,0:45:14.340 We are lucky our prime of tau[br]is a constant, which is 5. 0:45:14.340,0:45:16.360 So I'm going to[br]have integral from 0 0:45:16.360,0:45:20.725 to T absolute value of[br]5 [INAUDIBLE] d tau. 0:45:20.725,0:45:23.100 And what in the world[br]is absolute value of 5? 0:45:23.100,0:45:27.808 It's 5 integral from 0[br]to T [? of the ?] tau. 0:45:27.808,0:45:30.990 What is integral from[br]0 to T of the tau? 0:45:30.990,0:45:33.660 T. 5T. 0:45:33.660,0:45:36.940 So S is 5T. 0:45:36.940,0:45:39.534 And that's what I[br]said before, right? 0:45:39.534,0:45:41.840 S is 5T. 0:45:41.840,0:45:46.720 S equals 5T, and[br]T equals S over 5. 0:45:46.720,0:45:51.295 So this thing, in general,[br]is told to us by who? 0:45:51.295,0:45:53.160 It has to match the chain rule. 0:45:53.160,0:45:55.152 It matches the chain rule. 0:45:55.152,0:46:19.580 0:46:19.580,0:46:20.100 OK. 0:46:20.100,0:46:24.720 So again, why does that[br]match the chain rule? 0:46:24.720,0:46:31.290 We have that-- we[br]have R-- or how 0:46:31.290,0:46:34.550 should I start, the little f,[br]the little r, little r of S, 0:46:34.550,0:46:35.528 right? 0:46:35.528,0:46:41.396 Little r of S is[br]little r of S of T. 0:46:41.396,0:46:45.370 How do I differentiate[br]that with respect to T? 0:46:45.370,0:46:53.240 Well DDT of R will be R[br]primed with respect to S. 0:46:53.240,0:47:01.720 So I'll say DRDS of[br]S of T times DSDT. 0:47:01.720,0:47:04.510 0:47:04.510,0:47:06.190 Now what is DSDT? 0:47:06.190,0:47:09.220 DSDT was the derivative of that. 0:47:09.220,0:47:15.870 It's exactly the speed[br]absolute value of R prime of T. 0:47:15.870,0:47:18.190 So when you prime[br]here, S prime of T 0:47:18.190,0:47:22.950 will be exactly that,[br]with T replacing tau. 0:47:22.950,0:47:24.450 We learned that in Calc 1. 0:47:24.450,0:47:26.530 I know it's been a long time. 0:47:26.530,0:47:28.704 I can feel you're[br]a little bit rusty. 0:47:28.704,0:47:29.620 But it doesn't matter. 0:47:29.620,0:47:32.820 So S prime of T,[br]DSDT will simply 0:47:32.820,0:47:36.220 be absolute value[br]of R prime of T. 0:47:36.220,0:47:40.671 That's the speed of[br]the original curve. 0:47:40.671,0:47:43.580 This one. 0:47:43.580,0:47:46.180 OK? 0:47:46.180,0:47:46.910 All right. 0:47:46.910,0:47:58.965 So here, when I look at[br]DRDS, this is going to be 1. 0:47:58.965,0:48:02.250 0:48:02.250,0:48:06.340 And if you think of[br]this as a function of T, 0:48:06.340,0:48:11.737 you have DR of S of[br]T. Who is R of S of T? 0:48:11.737,0:48:15.230 This is R-- big[br]R-- of T. So this 0:48:15.230,0:48:21.717 is the DRDT Which is exactly[br]the same as R prime of T 0:48:21.717,0:48:24.720 when you put the absolute[br]values [INAUDIBLE]. 0:48:24.720,0:48:26.470 It has to fit. 0:48:26.470,0:48:33.080 So indeed, you have R prime[br]of T, R prime of T, and 1. 0:48:33.080,0:48:35.131 It's an identity. 0:48:35.131,0:48:38.910 If I didn't put DSDT to[br][? P, ?] our prime of T 0:48:38.910,0:48:42.216 in absolute value,[br]it wouldn't work out. 0:48:42.216,0:48:48.350 DSDT has to be R prime[br]of T in absolute value. 0:48:48.350,0:48:51.475 And this is how we[br]got, again-- are 0:48:51.475,0:48:54.420 you going to remember[br]this without having 0:48:54.420,0:48:56.220 to re-do the whole thing? 0:48:56.220,0:49:11.310 Integral from 0 to T of R[br]prime of T or tau d tau. 0:49:11.310,0:49:13.735 When you prime this[br]guy with respect to T 0:49:13.735,0:49:17.570 as soon as it's positive--[br]when it is positive-- assume-- 0:49:17.570,0:49:20.140 why is this positive, S of T? 0:49:20.140,0:49:23.690 Because you integrate from[br]time 0 to another time 0:49:23.690,0:49:24.670 a positive number. 0:49:24.670,0:49:29.110 So it has to be[br]positive derivative. 0:49:29.110,0:49:30.490 It's an increasing function. 0:49:30.490,0:49:34.210 This function is increasing. 0:49:34.210,0:49:37.360 So DSDT again will be the speed. 0:49:37.360,0:49:38.570 Say it again, Magdalena? 0:49:38.570,0:49:44.190 DSDT will be the speed[br]of the original line. 0:49:44.190,0:49:47.110 DSDT in our case was 5. 0:49:47.110,0:49:48.050 Right? 0:49:48.050,0:49:50.200 DSDT was 5. 0:49:50.200,0:49:54.590 S was 5 times T.[br]S was 5 times T. 0:49:54.590,0:49:55.090 All right. 0:49:55.090,0:49:58.030 That was a simple[br]example, sort of, kind of. 0:49:58.030,0:49:59.990 What do we want to remember? 0:49:59.990,0:50:03.620 We remember the formula[br]of the arc length. 0:50:03.620,0:50:05.530 Formula of arc length. 0:50:05.530,0:50:08.720 0:50:08.720,0:50:11.310 So the formula of[br]arc length exists 0:50:11.310,0:50:15.440 in this form because of[br]the chain rule [INAUDIBLE] 0:50:15.440,0:50:18.595 from this diagram. 0:50:18.595,0:50:24.840 So always remember, we have[br]a composition of functions. 0:50:24.840,0:50:27.512 We use that composition of[br]function for the chain rule 0:50:27.512,0:50:28.964 to re-parametrize it. 0:50:28.964,0:50:30.900 And finally, the drunken bug. 0:50:30.900,0:50:34.059 0:50:34.059,0:50:35.350 what did I take [INAUDIBLE] 14? 0:50:35.350,0:50:37.110 R of t. 0:50:37.110,0:50:44.481 Let's say this is 2[br]cosine t, 2 sine t. 0:50:44.481,0:50:46.460 Let me make it more beautiful. 0:50:46.460,0:50:53.500 Let me put 4-- 4, 4, and 3t. 0:50:53.500,0:50:56.740 Can anybody tell[br]me why I did that? 0:50:56.740,0:50:59.990 Maybe you can guess my mind. 0:50:59.990,0:51:04.000 Find the following things. 0:51:04.000,0:51:11.312 The unit vector T, by[br]definition R prime over R prime 0:51:11.312,0:51:16.450 of t in absolute value. 0:51:16.450,0:51:22.210 Find the speed of[br]this motion R of t. 0:51:22.210,0:51:24.710 This is a law of motion. 0:51:24.710,0:51:32.426 And reparametrize in arclength--[br]this curve in arclength. 0:51:32.426,0:51:36.650 0:51:36.650,0:51:39.910 And you go, oh my God, I[br]have a problem with a, b,c. 0:51:39.910,0:51:43.260 The is a typical problem for[br]the final exam, by the way. 0:51:43.260,0:51:46.290 This problem popped up on[br]many, many final exams. 0:51:46.290,0:51:47.262 Is it hard? 0:51:47.262,0:51:49.210 Is it easy? 0:51:49.210,0:51:53.380 First of all, how did I[br]know what it looked like? 0:51:53.380,0:51:57.090 I should give at[br]least an explanation. 0:51:57.090,0:52:00.890 If instead of 3t I[br]would have 3, then I 0:52:00.890,0:52:04.720 would have the plane[br]z equals 3 constant. 0:52:04.720,0:52:07.610 And then I'll say, I'm moving[br]in circles, in circles, 0:52:07.610,0:52:11.160 in circles, in circles,[br]with t as a real parameter, 0:52:11.160,0:52:13.560 and I'm not evolving. 0:52:13.560,0:52:16.955 But this is like, what, this[br]like in in the avatar OK? 0:52:16.955,0:52:22.060 So I'm performing the circular[br]motion, but at the same time 0:52:22.060,0:52:25.070 going on a different level. 0:52:25.070,0:52:26.720 Assume another life. 0:52:26.720,0:52:31.117 I'm starting another life[br]on the next spiritual level. 0:52:31.117,0:52:34.020 OK, I have no religious[br]beliefs in that area, 0:52:34.020,0:52:36.300 but it's a good physical[br]example to give. 0:52:36.300,0:52:38.290 So I go circular. 0:52:38.290,0:52:41.530 Instead of going again[br]circular and again circular, 0:52:41.530,0:52:45.470 I go, oh, I go up and[br]up and up, and this 3t 0:52:45.470,0:52:49.210 tells me I should also[br]evolve on the vertical. 0:52:49.210,0:52:50.330 Ah-hah. 0:52:50.330,0:52:55.370 So instead of circular motion[br]I get a helicoidal motion. 0:52:55.370,0:52:56.140 This is a helix. 0:52:56.140,0:52:58.650 0:52:58.650,0:53:01.920 Could somebody tell me how I'm[br]going to draw such a helix? 0:53:01.920,0:53:02.555 Is it hard? 0:53:02.555,0:53:04.280 Is it easy? 0:53:04.280,0:53:05.390 This helix-- yes, sir. 0:53:05.390,0:53:08.380 0:53:08.380,0:53:09.190 Yes. 0:53:09.190,0:53:10.910 STUDENT: [INAUDIBLE] 0:53:10.910,0:53:12.350 PROFESSOR: It's like a tornado. 0:53:12.350,0:53:14.410 It's like a tornado,[br]hurricane, but how 0:53:14.410,0:53:18.435 do I draw the cylinder on[br]which this helix exists? 0:53:18.435,0:53:22.500 I have to be a smart girl and[br]remember what I learned before. 0:53:22.500,0:53:25.410 What is x squared[br]plus y squared? 0:53:25.410,0:53:28.750 Suppose that z is not[br]playing in the picture. 0:53:28.750,0:53:32.560 If I take Mr. x and Mr. y[br]and I square them and I add 0:53:32.560,0:53:34.705 them together, what do I get? 0:53:34.705,0:53:35.746 STUDENT: It's the radius. 0:53:35.746,0:53:38.260 PROFESSOR: What is[br]the radius squared? 0:53:38.260,0:53:38.910 4 squared. 0:53:38.910,0:53:41.295 I'm gonna write 4[br]squared because it's 0:53:41.295,0:53:43.130 easier than writing 16. 0:53:43.130,0:53:44.350 Thank you for your help. 0:53:44.350,0:53:51.370 So I simply have to go ahead and[br]draw the frame first, x, y, z, 0:53:51.370,0:53:54.900 and then I'll say, OK, smart. 0:53:54.900,0:53:57.790 R is 4. 0:53:57.790,0:53:59.610 The radius should be 4. 0:53:59.610,0:54:02.240 This is the cylinder[br]where I'm at. 0:54:02.240,0:54:06.570 Where do I start[br]my physical motion? 0:54:06.570,0:54:10.134 This bug is drunk,[br]but sort of not. 0:54:10.134,0:54:11.517 I don't know. 0:54:11.517,0:54:16.020 It's a bug that can keep[br]the same radius, which 0:54:16.020,0:54:16.987 is quite something. 0:54:16.987,0:54:17.820 STUDENT: It's tipsy. 0:54:17.820,0:54:19.600 PROFESSOR: Yeah,[br]exactly, tipsy one. 0:54:19.600,0:54:22.540 So how about t equals 0. 0:54:22.540,0:54:24.910 Where do I start my motion? 0:54:24.910,0:54:26.900 At 4, 0, 0. 0:54:26.900,0:54:28.505 Where is 4, 0, 0? 0:54:28.505,0:54:29.290 Over here. 0:54:29.290,0:54:31.590 So that's my first[br]point where the bug 0:54:31.590,0:54:33.079 will start at t equals 0. 0:54:33.079,0:54:34.370 STUDENT: How'd you get 4, 0, 0? 0:54:34.370,0:54:36.300 PROFESSOR: Because I'm--[br]very good question. 0:54:36.300,0:54:38.640 I'm on x, y, z axes. 0:54:38.640,0:54:42.050 4, y is 0, z is 0. 0:54:42.050,0:54:46.650 I plug in t, would be 0,[br]and I get 4 times 1, 4 times 0:54:46.650,0:54:50.620 0, 3 times 0, so I[br]know I'm starting here. 0:54:50.620,0:54:55.880 And when I move, I move[br]along the cylinder like that. 0:54:55.880,0:55:00.240 Can somebody tell me at[br]what time I'm gonna be here? 0:55:00.240,0:55:03.910 Not at 1:50, but what time am[br]I going to be at this point? 0:55:03.910,0:55:08.090 And then I continue, and I go[br]up, and I continue and I go up. 0:55:08.090,0:55:09.710 STUDENT: [INAUDIBLE] 0:55:09.710,0:55:11.030 PROFESSOR: Pi over 2. 0:55:11.030,0:55:12.670 Excellent. 0:55:12.670,0:55:14.480 And can you-- can[br]you tell me what 0:55:14.480,0:55:16.970 point it is in space in R 3? 0:55:16.970,0:55:18.170 Plug in pi over 2. 0:55:18.170,0:55:19.620 You can do it faster than me. 0:55:19.620,0:55:20.330 STUDENT: 0. 0:55:20.330,0:55:23.940 PROFESSOR: 0, 4 and 3 pi over 2. 0:55:23.940,0:55:25.580 And I keep going. 0:55:25.580,0:55:28.850 So this is the helicoidal[br]motion I'm talking about. 0:55:28.850,0:55:31.690 The unit vector-- is it easy[br]to write it on the final? 0:55:31.690,0:55:33.150 Can do that in no time. 0:55:33.150,0:55:39.140 So we get like, let's say, 30%,[br]30%, 30%, and 10% for drawing. 0:55:39.140,0:55:40.510 How about that? 0:55:40.510,0:55:44.210 That would be a typical[br]grid for the problem. 0:55:44.210,0:55:49.900 So t will be minus 4 sine t. 0:55:49.900,0:55:53.871 If I make a mistake, are[br]you gonna shout, please? 0:55:53.871,0:55:58.660 4 cosine t and 3[br]divided by what? 0:55:58.660,0:56:00.950 What is the tangent unit vector? 0:56:00.950,0:56:03.840 At every point in[br]space, I'm gonna 0:56:03.840,0:56:05.580 have this tangent unit vector. 0:56:05.580,0:56:08.140 It has to have[br]length 1, and it has 0:56:08.140,0:56:11.100 to be tangent to my trajectory. 0:56:11.100,0:56:12.200 I'll draw him. 0:56:12.200,0:56:15.670 So he gives me a[br]field, a vector field-- 0:56:15.670,0:56:19.080 this is beautiful-- T[br]of t is a vector field. 0:56:19.080,0:56:20.850 At every point of[br]the trajectory, 0:56:20.850,0:56:23.460 I have only one such vector. 0:56:23.460,0:56:27.067 That's what we mean[br]by vector field. 0:56:27.067,0:56:29.989 What's the magnitude? 0:56:29.989,0:56:31.450 It's buzzing. 0:56:31.450,0:56:33.400 It's buzzing. 0:56:33.400,0:56:35.220 How did you do it? 0:56:35.220,0:56:40.140 4, 16 times sine squared[br]plus cosine squared. 0:56:40.140,0:56:42.400 16 plus 9 is 25. 0:56:42.400,0:56:46.120 Square root of 25 is 5. 0:56:46.120,0:56:47.870 Are you guys with me? 0:56:47.870,0:56:49.860 Do I have to write this down? 0:56:49.860,0:56:51.775 Are you guys sure? 0:56:51.775,0:56:53.108 STUDENT: You plugged in 0 for t? 0:56:53.108,0:56:56.390 Is that what you did[br]when you [INAUDIBLE] 0:56:56.390,0:56:58.980 PROFESSOR: No, I plugged[br]0 for t when I started. 0:56:58.980,0:57:01.680 But when I'm computing,[br]I don't plug anything, 0:57:01.680,0:57:03.940 I just do it in general. 0:57:03.940,0:57:07.710 I said 16 sine squared[br]plus 16 cosine squared 0:57:07.710,0:57:10.400 is 16 times 1 plus 9. 0:57:10.400,0:57:13.410 My son would know this[br]one and he's 10, right? 0:57:13.410,0:57:16.030 16 plus 9 square root of 25. 0:57:16.030,0:57:17.810 And I taught him[br]about square roots. 0:57:17.810,0:57:20.590 So square root of 25,[br]he knows that's 5. 0:57:20.590,0:57:22.252 And if he knows[br]that's 5, then you 0:57:22.252,0:57:24.470 should do that in a[br]minute-- in a second. 0:57:24.470,0:57:25.330 All right. 0:57:25.330,0:57:32.430 So t will simply be-- if you[br]don't simplify 1/5 minus 4 sine 0:57:32.430,0:57:37.465 t 4 cosine t 3 in the final,[br]it wouldn't be a big deal, 0:57:37.465,0:57:39.315 I would give you[br]still partial credit, 0:57:39.315,0:57:42.260 but what if we raise this[br]as a multiple choice? 0:57:42.260,0:57:46.520 Then you have to be able[br]to find where the 5 is. 0:57:46.520,0:57:47.270 What is the speed? 0:57:47.270,0:57:49.410 Was that hard for you to find? 0:57:49.410,0:57:50.990 Where is the speed hiding? 0:57:50.990,0:57:53.800 It's exactly the[br]denominator of R. 0:57:53.800,0:57:57.070 This is the speed[br]of the curve in t. 0:57:57.070,0:57:58.570 And that was 5. 0:57:58.570,0:58:01.190 You told me the speed was[br]5, and I'm very happy. 0:58:01.190,0:58:07.800 So you got 30%, 30%, 10% from[br]the picture-- no, this picture. 0:58:07.800,0:58:09.297 This picture's no good. 0:58:09.297,0:58:13.193 STUDENT: What does the[br]first word of c say? 0:58:13.193,0:58:15.287 Question c, what does[br]the first word say? 0:58:15.287,0:58:16.370 PROFESSOR: The first what? 0:58:16.370,0:58:17.730 STUDENT: The word. 0:58:17.730,0:58:19.380 PROFESSOR: Reparametrize. 0:58:19.380,0:58:23.070 Reparametrize this[br]curve in arclength. 0:58:23.070,0:58:26.430 Oh my God, so according[br]to that chain rule, 0:58:26.430,0:58:31.430 could you guys remember-- if you[br]remember, what is the s of t? 0:58:31.430,0:58:39.082 If I want to reparametrize[br]in arclength integral from 0 0:58:39.082,0:58:45.580 to t of the speed, how[br]is the speed defined? 0:58:45.580,0:58:49.040 Absolute value of r prime of t. 0:58:49.040,0:58:54.370 dt, but I don't like t,[br]I write-- I write tau. 0:58:54.370,0:58:56.610 Like Dr. [? Solinger, ?][br]you know him, 0:58:56.610,0:58:59.370 he's one of my colleagues,[br]calls that-- that's 0:58:59.370,0:59:00.885 the dummy dummy variable. 0:59:00.885,0:59:03.770 In many books, tau is[br]the dummy variable. 0:59:03.770,0:59:08.485 Or you can-- some people even[br]put t by inclusive notation. 0:59:08.485,0:59:09.900 All right? 0:59:09.900,0:59:12.580 So in my case, what is s of t? 0:59:12.580,0:59:14.070 It should be easy. 0:59:14.070,0:59:18.670 Because although this[br]not a circular motion, 0:59:18.670,0:59:20.610 I still have constant speed. 0:59:20.610,0:59:23.590 So who is that special speed? 0:59:23.590,0:59:24.350 5. 0:59:24.350,0:59:31.400 Integral from 0 to t5 d tau,[br]and that is 5t, am I right? 0:59:31.400,0:59:32.050 5t. 0:59:32.050,0:59:36.980 So-- so if I want to[br]reparametrize this helix, 0:59:36.980,0:59:41.596 keeping in mind[br]that s is simply 5t, 0:59:41.596,0:59:47.392 what do I have to do to[br]get 100% on this problem? 0:59:47.392,0:59:57.590 All I have to do is say little r[br]of s, which represents actually 0:59:57.590,1:00:00.690 big R of t of s. 1:00:00.690,1:00:02.380 Are you guys with me? 1:00:02.380,1:00:04.451 Do you have to write[br]all this story down? 1:00:04.451,1:00:04.950 No. 1:00:04.950,1:00:07.650 But that will remind[br]you of the diagram. 1:00:07.650,1:00:12.000 So I have R of t of s. 1:00:12.000,1:00:13.268 Or alpha of s. 1:00:13.268,1:00:15.360 And this is t of s. 1:00:15.360,1:00:16.390 t of s. 1:00:16.390,1:00:19.642 R of t of s is R of s, right? 1:00:19.642,1:00:21.100 Do you have to remind me? 1:00:21.100,1:00:21.600 No. 1:00:21.600,1:00:23.240 The heck with the diagram. 1:00:23.240,1:00:26.900 As long as you understood[br]it was about a composition 1:00:26.900,1:00:28.250 of functions. 1:00:28.250,1:00:30.645 And then R of s[br]will simply be what? 1:00:30.645,1:00:33.060 How do we do that fast? 1:00:33.060,1:00:37.430 We replaced t by s over 5. 1:00:37.430,1:00:38.790 Where from? 1:00:38.790,1:00:42.280 Little s equals 5t,[br]we just computed it. 1:00:42.280,1:00:43.680 Little s equals 5t. 1:00:43.680,1:00:44.780 That's all you need to do. 1:00:44.780,1:00:49.110 To pull out t, replace[br]the third sub s. 1:00:49.110,1:00:52.866 So what is the function[br]t in terms of s? 1:00:52.866,1:00:55.096 It's s over 5. 1:00:55.096,1:00:59.600 What is the function t, what's[br]the parameter t, in terms of s? 1:00:59.600,1:01:01.480 s over 5. 1:01:01.480,1:01:07.140 And finally, at the end, 3[br]times what is the stinking t? 1:01:07.140,1:01:09.140 s over 5. 1:01:09.140,1:01:11.002 I'm done. 1:01:11.002,1:01:16.245 I got 100% I don't want[br]to say how much time it's 1:01:16.245,1:01:18.180 gonna take me to[br]do it, but I think 1:01:18.180,1:01:20.480 I can do it in like, 2[br]or 3 minutes, 5 minutes. 1:01:20.480,1:01:24.290 If I know the problem I'll[br]do it in a few minutes. 1:01:24.290,1:01:26.640 If I waste too[br]much time thinking, 1:01:26.640,1:01:28.710 I'm not gonna do it at all. 1:01:28.710,1:01:30.470 So what do you have to remember? 1:01:30.470,1:01:35.010 You have to remember the[br]formula that says s of t, 1:01:35.010,1:01:40.780 the arclength parameter--[br]the arclength parameter 1:01:40.780,1:01:46.740 equals integral from 0 to[br]t is 0 to t of the speed. 1:01:46.740,1:01:53.140 Does this element of information[br]remind you of something? 1:01:53.140,1:01:56.460 Of course, s will be the[br]arclength, practically. 1:01:56.460,1:01:58.450 What kind of parameter is that? 1:01:58.450,1:02:03.510 Is you're measuring how[br]big-- how much you travel. 1:02:03.510,1:02:06.960 s of t is the time you[br]travel-- the distance 1:02:06.960,1:02:10.679 you travel in time t. 1:02:10.679,1:02:15.570 1:02:15.570,1:02:20.010 So it's a space-time continuum. 1:02:20.010,1:02:23.600 It's a space-time relationship. 1:02:23.600,1:02:27.400 So it's the space you[br]travel in times t. 1:02:27.400,1:02:30.440 Now, if I drive to Amarillo[br]at 60 miles an hour, 1:02:30.440,1:02:35.005 I'm happy and sassy, and I[br]say OK, it's gonna be s of t. 1:02:35.005,1:02:37.670 My displacement to[br]Amarillo is given 1:02:37.670,1:02:41.540 by this linear law, 60 times t. 1:02:41.540,1:02:42.910 Suppose I'm on cruise control. 1:02:42.910,1:02:44.370 But I've never on[br]cruise control. 1:02:44.370,1:02:47.260 1:02:47.260,1:02:50.750 So this is going to[br]be very variable. 1:02:50.750,1:02:54.674 And the only way you can compute[br]this displacement or distance 1:02:54.674,1:02:57.140 traveled, it'll[br]be as an integral. 1:02:57.140,1:03:01.330 From time 0, when I start[br]driving, to time t of my speed, 1:03:01.330,1:03:02.220 and that's it. 1:03:02.220,1:03:04.360 That's all you have to remember. 1:03:04.360,1:03:08.200 It's actually-- mathematics[br]should not be memorized. 1:03:08.200,1:03:11.520 It should be sort of[br]understood, just like physics. 1:03:11.520,1:03:15.170 What if you take your[br]first test, quiz, 1:03:15.170,1:03:18.810 whatever, on WeBWorK or in[br]person, and you freak out. 1:03:18.810,1:03:22.925 You get such a[br]problem, and you blank. 1:03:22.925,1:03:24.950 You just blank. 1:03:24.950,1:03:27.670 What do you do? 1:03:27.670,1:03:31.622 You sort of know this,[br]but you have a blank. 1:03:31.622,1:03:34.100 Always tell me, right? 1:03:34.100,1:03:36.026 Always email, say I'm[br]freaking out here. 1:03:36.026,1:03:38.690 I don't know what's[br]the matter with me. 1:03:38.690,1:03:46.400 Don't cut our correspondence,[br]either by speaking or by email. 1:03:46.400,1:03:48.810 Very few of you email me. 1:03:48.810,1:03:51.570 I'd like you to be[br]more like my friends, 1:03:51.570,1:03:53.250 and I would be more[br]like your tutor, 1:03:53.250,1:03:55.400 and when you[br]encounter an obstacle, 1:03:55.400,1:03:58.340 you email me and[br]I email you back. 1:03:58.340,1:04:00.660 This is what I want. 1:04:00.660,1:04:03.645 The WeBWorK, this is what I[br]want our model of interaction 1:04:03.645,1:04:05.750 to become. 1:04:05.750,1:04:06.880 Don't be shy. 1:04:06.880,1:04:10.600 Many of you are shy even to[br]ask questions in the classroom. 1:04:10.600,1:04:12.500 And I'm not going[br]to let you be shy. 1:04:12.500,1:04:16.640 At 2 o'clock I'm going to let[br]you ask all the questions you 1:04:16.640,1:04:19.690 have about homework,[br]and we will do 1:04:19.690,1:04:21.250 more homework-like questions. 1:04:21.250,1:04:24.060 I want to imitate some[br]WeBWorK questions. 1:04:24.060,1:04:27.810 And we will work them out. 1:04:27.810,1:04:32.310 So any questions right now? 1:04:32.310,1:04:32.840 Yes, sir. 1:04:32.840,1:04:35.684 STUDENT: You emailed-- did[br]you email us this weekend 1:04:35.684,1:04:37.580 the numbers for WeBWorK? 1:04:37.580,1:04:41.040 PROFESSOR: I emailed you the[br]WeBWorK assignment completely. 1:04:41.040,1:04:44.913 I mean, the link-- you[br]get in and you of see it. 1:04:44.913,1:04:48.340 STUDENT: Which email[br]did you send that to? 1:04:48.340,1:04:49.660 PROFESSOR: To your TTU. 1:04:49.660,1:04:51.860 All the emails go to your TTU. 1:04:51.860,1:04:56.400 You have one week[br]starting yesterday until, 1:04:56.400,1:04:58.140 was it the 2nd? 1:04:58.140,1:05:00.010 I gave you a little[br]bit more time. 1:05:00.010,1:05:03.008 So it's due on the[br]2nd of February at, 1:05:03.008,1:05:03.982 I forgot what time. 1:05:03.982,1:05:05.443 1 o'clock or something. 1:05:05.443,1:05:06.417 Yes, sir. 1:05:06.417,1:05:07.878 STUDENT: [INAUDIBLE][br]I was confused 1:05:07.878,1:05:10.313 at the beginning where you got[br]x squared plus y squared equals 1:05:10.313,1:05:10.813 4 squared. 1:05:10.813,1:05:13.597 Where did you get that? 1:05:13.597,1:05:14.180 PROFESSOR: Oh. 1:05:14.180,1:05:15.030 OK. 1:05:15.030,1:05:19.160 I eliminated the t between[br]the first two guys. 1:05:19.160,1:05:24.940 This is called eliminating a[br]parameter, which was the time 1:05:24.940,1:05:27.970 parameter between x and y. 1:05:27.970,1:05:32.090 When I do that, I get a[br]beautiful equation which 1:05:32.090,1:05:36.670 is x squared plus y squared[br]equals 16, which tells me, hey, 1:05:36.670,1:05:39.830 your curve sits on[br]the surface x squared 1:05:39.830,1:05:42.230 plus y squared equals 16. 1:05:42.230,1:05:44.320 It's not the same[br]with the surface, 1:05:44.320,1:05:47.520 because you have additional[br]constraints on the z. 1:05:47.520,1:05:52.370 So the z is constrained[br]to follow this thing. 1:05:52.370,1:05:59.910 Now, could anybody tell me how[br]I'm gonna write eventually-- 1:05:59.910,1:06:02.400 this is a harder[br]task, OK, but I'm 1:06:02.400,1:06:09.132 glad you asked because I[br]wanted to discuss that. 1:06:09.132,1:06:13.020 How do I express t[br]in terms of x and y? 1:06:13.020,1:06:16.790 I mean, I'm going to have an[br]intersection of two surfaces. 1:06:16.790,1:06:18.456 How? 1:06:18.456,1:06:21.010 This is just practically[br]differential geometry 1:06:21.010,1:06:24.250 or advanced calculus[br]at the same time. 1:06:24.250,1:06:27.970 x squared plus y squared[br]equals our first surface 1:06:27.970,1:06:31.950 that I'm thinking about, which[br]I'm sitting with my curve. 1:06:31.950,1:06:35.370 But I also have my curve[br]to be at the intersection 1:06:35.370,1:06:39.490 between the cylinder[br]and something else. 1:06:39.490,1:06:45.210 And it's hard to figure out how[br]I'm going to do the other one. 1:06:45.210,1:06:49.266 Can anybody figure[br]out how another 1:06:49.266,1:06:51.380 surface-- what is the surface? 1:06:51.380,1:06:56.380 A surface will have an implicit[br]equation of the type f of x, y, 1:06:56.380,1:06:58.000 z equals a constant. 1:06:58.000,1:07:01.140 So you have to sort of[br]eliminate your parameter t. 1:07:01.140,1:07:02.470 The heck with the time. 1:07:02.470,1:07:05.310 We don't care about time,[br]we only care about space. 1:07:05.310,1:07:07.370 So is there any other[br]way to eliminate 1:07:07.370,1:07:09.960 t between the equations? 1:07:09.960,1:07:13.860 I have to use the information[br]that I haven't used yet. 1:07:13.860,1:07:15.340 All right. 1:07:15.340,1:07:19.580 Now my question is[br]that, how can I do that? 1:07:19.580,1:07:22.620 z is beautiful. 1:07:22.620,1:07:23.770 3 is beautiful. 1:07:23.770,1:07:25.700 t drives me nuts. 1:07:25.700,1:07:30.280 How do I get the t out of[br]the first two equations? 1:07:30.280,1:07:32.780 [INTERPOSING VOICES] 1:07:32.780,1:07:35.820 Yeah, I divide them[br]one to the other one. 1:07:35.820,1:07:39.730 So if I-- for example,[br]I go y over x. 1:07:39.730,1:07:42.750 What is y over x? 1:07:42.750,1:07:45.423 It's tangent of t. 1:07:45.423,1:07:48.804 How do I pull Mr. t out? 1:07:48.804,1:07:51.650 Say t, get out. 1:07:51.650,1:07:54.720 Well, I have to think about[br]if I'm not losing anything. 1:07:54.720,1:07:58.450 But in principle, t would[br]be arctangent of y over x. 1:07:58.450,1:08:01.541 1:08:01.541,1:08:02.040 OK? 1:08:02.040,1:08:06.080 So, I'm having two[br]equations of this type. 1:08:06.080,1:08:08.497 I'm eliminating t[br]between the two. 1:08:08.497,1:08:10.457 I don't care about[br]the other one. 1:08:10.457,1:08:13.770 I only cared for you[br]to draw the cylinder. 1:08:13.770,1:08:17.250 So we can draw point[br]by point the helix. 1:08:17.250,1:08:18.859 I don't draw many points. 1:08:18.859,1:08:22.580 I draw only t equals 0,[br]where I'm starting over here, 1:08:22.580,1:08:25.160 t equals pi over 2, which[br][INAUDIBLE] gave me, 1:08:25.160,1:08:27.050 then what was it? 1:08:27.050,1:08:29.850 At pi I'm here, and so on. 1:08:29.850,1:08:33.652 So I move-- when[br]I move one time, 1:08:33.652,1:08:39.407 so let's say from 0 to[br]2 pi, I should be smart. 1:08:39.407,1:08:48.979 Pi over 2, pi, 3 pi over 2,[br]2 pi just on top of that. 1:08:48.979,1:08:52.435 It has to be on the same line. 1:08:52.435,1:08:54.622 On top of that--[br]on the cylinder. 1:08:54.622,1:08:55.830 They are all on the cylinder. 1:08:55.830,1:08:59.439 I'm not good enough to draw[br]them as being on the cylinder. 1:08:59.439,1:09:03.189 So I'm coming where I started[br]from, but on the higher 1:09:03.189,1:09:08.050 level of intelligence-- no, on[br]a higher level of experience. 1:09:08.050,1:09:08.840 Right? 1:09:08.840,1:09:13.995 That's kind of the idea[br]of evolving on the helix? 1:09:13.995,1:09:16.652 Any other questions? 1:09:16.652,1:09:17.584 Yes, sir. 1:09:17.584,1:09:19.760 STUDENT: So that[br]capital R of t is 1:09:19.760,1:09:23.886 you position vector, but what's[br]little r of t? [INAUDIBLE] 1:09:23.886,1:09:25.524 PROFESSOR: It's also[br]a position vector. 1:09:25.524,1:09:32.000 So practically it depends on[br]the type of parametrization 1:09:32.000,1:09:33.484 you are using. 1:09:33.484,1:09:36.249 1:09:36.249,1:09:39.679 The dependence of[br]time is crucial. 1:09:39.679,1:09:43.340 The dependence of the[br]time parameter is crucial. 1:09:43.340,1:09:50.578 So when you draw[br]this diagram, r of s 1:09:50.578,1:09:59.120 will practically be the same[br]as R of s of t-- R of t of s, 1:09:59.120,1:10:00.000 I'm sorry. 1:10:00.000,1:10:02.340 R of t of s. 1:10:02.340,1:10:05.620 So practically it's telling[br]me it's a combination. 1:10:05.620,1:10:11.800 Physically, it's the same[br]thing, but at a different time. 1:10:11.800,1:10:20.120 So you look at one vector[br]at time-- time is t here, 1:10:20.120,1:10:22.520 but s was 5t. 1:10:22.520,1:10:26.050 So I'm gonna be-- let[br]me give you an example. 1:10:26.050,1:10:29.890 So we had s was 5t, right? 1:10:29.890,1:10:32.560 I don't remember how it went. 1:10:32.560,1:10:36.260 So when I have[br]little r of s, that 1:10:36.260,1:10:42.770 means the same as[br]little r of 5t, 1:10:42.770,1:10:47.470 which means this kind of guy. 1:10:47.470,1:10:57.550 Now assume that I have something[br]like cosine 5t, sine 5t, and 0. 1:10:57.550,1:11:00.810 And what does this mean? 1:11:00.810,1:11:10.330 It means that R of 2 pi over[br]5 is the same as little r of 2 1:11:10.330,1:11:16.260 pi where R of t is cosine[br]of 5t, and little r of s 1:11:16.260,1:11:20.790 is cosine of s, sine s, 0. 1:11:20.790,1:11:23.860 So Mr. t says, I'm[br]running, I'm time. 1:11:23.860,1:11:29.290 I'm running from 0 to 2 pi[br]over 5, and that's when I stop. 1:11:29.290,1:11:31.455 And little s says,[br]I'm running too. 1:11:31.455,1:11:34.760 I'm also time, but I'm[br]a special kind of time, 1:11:34.760,1:11:38.395 and I'm running from 0 to[br]2 pi, and I stop at 2 pi 1:11:38.395,1:11:40.920 where the circle will stop. 1:11:40.920,1:11:44.390 Then physically,[br]the two vectors, 1:11:44.390,1:11:48.750 at two different moments[br]in time, are the same. 1:11:48.750,1:11:51.310 Where-- why-- why is that? 1:11:51.310,1:11:53.300 So I start here. 1:11:53.300,1:11:55.770 And I end here. 1:11:55.770,1:12:01.296 So physically, these two guys[br]have the same, the red vector, 1:12:01.296,1:12:05.453 but they are there at[br]different moments in time. 1:12:05.453,1:12:06.750 All right? 1:12:06.750,1:12:12.840 So imagine that you have sister. 1:12:12.840,1:12:17.765 And she is five times faster[br]than you in a competition. 1:12:17.765,1:12:20.980 It's a math competition,[br]athletic, it doesn't matter. 1:12:20.980,1:12:25.570 You both get there, but you[br]get there in different times, 1:12:25.570,1:12:27.480 in different amounts of time. 1:12:27.480,1:12:31.030 And unfortunately, this is--[br]I will do philosophy still 1:12:31.030,1:12:35.790 in mathematics-- this is the[br]situation with many of us 1:12:35.790,1:12:39.246 when it comes to[br]understanding a material, 1:12:39.246,1:12:42.470 like calculus or advanced[br]calculus or geometry. 1:12:42.470,1:12:47.420 We get to the understanding[br]in different times. 1:12:47.420,1:12:51.350 In my class-- I was[br]talking to my old-- 1:12:51.350,1:12:55.590 they are all old now,[br]all in their 40s-- 1:12:55.590,1:12:59.010 when did you[br]understand this helix 1:12:59.010,1:13:01.710 thing being on a cylinder? 1:13:01.710,1:13:04.000 Because I think I[br]understood it when 1:13:04.000,1:13:07.560 I was in third-- like a[br]junior level, sophomore level, 1:13:07.560,1:13:09.820 and I understood nothing[br]of this kind of stuff 1:13:09.820,1:13:14.830 in my freshman [INAUDIBLE][br]And one of my colleagues 1:13:14.830,1:13:17.998 who was really smart,[br]had a big background, 1:13:17.998,1:13:20.870 was in a Math[br]Olympiad, said, I think 1:13:20.870,1:13:22.985 I understood it as a freshman. 1:13:22.985,1:13:25.110 So then the other two that[br]I was talking-- actually 1:13:25.110,1:13:27.620 I never understood it. 1:13:27.620,1:13:32.000 So we all eventually get to[br]that point, that position, 1:13:32.000,1:13:34.880 but at a different[br]moment in time. 1:13:34.880,1:13:39.080 And it's also unfortunate it[br]happens about relationships. 1:13:39.080,1:13:42.290 You are in a relationship[br]with somebody, 1:13:42.290,1:13:44.700 and one is faster[br]than the other one. 1:13:44.700,1:13:46.760 One grows faster[br]than the other one. 1:13:46.760,1:13:50.430 Eventually both get to the[br]same level of understanding, 1:13:50.430,1:13:53.480 but since it's at[br]different moments in time, 1:13:53.480,1:13:55.860 the relationship could[br]break by the time 1:13:55.860,1:13:58.840 both reach that level[br]of understanding. 1:13:58.840,1:14:02.620 So physical phenomena,[br]really tricky. 1:14:02.620,1:14:05.490 It's-- physically you[br]see where everything is, 1:14:05.490,1:14:08.490 but you have to think[br]dynamically, in time. 1:14:08.490,1:14:11.300 Everything evolves in time. 1:14:11.300,1:14:15.166 Any other questions? 1:14:15.166,1:14:17.880 I'm gonna do problems[br]with you next time, 1:14:17.880,1:14:22.690 but you need a break because[br]your brain is overheated. 1:14:22.690,1:14:27.840 And so, we will take a[br]break of 10-12 minutes. 1:14:27.840,1:14:30.653