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