WEBVTT 00:00:00.000 --> 00:00:00.500 00:00:00.500 --> 00:00:01.900 PROFESSOR: Plus 1. 00:00:01.900 --> 00:00:04.730 And next would be between-- this is where 00:00:04.730 --> 00:00:06.900 most people have the problem. 00:00:06.900 --> 00:00:09.000 They thought x is any real number. 00:00:09.000 --> 00:00:10.830 No-- no, no, no, no, no. 00:00:10.830 --> 00:00:12.420 You wanted a segment. 00:00:12.420 --> 00:00:15.750 x has the values between this value, 00:00:15.750 --> 00:00:18.190 whatever value's on this axis and that value. 00:00:18.190 --> 00:00:23.730 So x equals 1, x equals 2 are the end points. 00:00:23.730 --> 00:00:29.810 How do you write a parameterized equation? 00:00:29.810 --> 00:00:32.479 And that should help you very much on the web work 00:00:32.479 --> 00:00:38.337 homework on that problem for such a function. 00:00:38.337 --> 00:00:39.545 Well, you say, wait a minute. 00:00:39.545 --> 00:00:41.675 Magdalena, this is a linear function. 00:00:41.675 --> 00:00:42.610 It's a piece of cake. 00:00:42.610 --> 00:00:45.230 I have just x plus 1. 00:00:45.230 --> 00:00:47.690 I know how to deal with that. 00:00:47.690 --> 00:00:49.750 Yes, but I'm asking you something else. 00:00:49.750 --> 00:00:53.770 Rather than writing the explicit equation 00:00:53.770 --> 00:00:58.870 in Cartesian coordinates x and y, tell me what time it is. 00:00:58.870 --> 00:01:01.110 And then I'm going to travel in time. 00:01:01.110 --> 00:01:06.590 I want to travel in time, in space-time, on the segment, 00:01:06.590 --> 00:01:08.190 right? 00:01:08.190 --> 00:01:13.700 So why if x equals x plus 1 has what 00:01:13.700 --> 00:01:15.910 is that-- what parameterization has infinitely 00:01:15.910 --> 00:01:18.080 many parameterization? 00:01:18.080 --> 00:01:22.620 Somebody will say, ha, you told us that it has infinitely many. 00:01:22.620 --> 00:01:24.200 Why do you insist on one? 00:01:24.200 --> 00:01:27.765 Which one is the most natural and the easiest to grasp? 00:01:27.765 --> 00:01:30.425 STUDENT: Zero to one. 00:01:30.425 --> 00:01:32.790 PROFESSOR: Zero to one is not a parameterization. 00:01:32.790 --> 00:01:34.560 STUDENT: Times zero one. 00:01:34.560 --> 00:01:39.280 PROFESSOR: So, so, so what is the parametric equation 00:01:39.280 --> 00:01:41.190 of a curve in general? 00:01:41.190 --> 00:01:46.530 If I have a curve, y equals-- oh, I'll start with x. 00:01:46.530 --> 00:01:49.850 X equals x of t and y equals y of t 00:01:49.850 --> 00:01:56.700 represent the two parametric questions that 00:01:56.700 --> 00:02:00.080 give that curve's equation in plane-- 00:02:00.080 --> 00:02:03.620 in plane where the i of t belongs 00:02:03.620 --> 00:02:05.270 to a certain interval i. 00:02:05.270 --> 00:02:06.720 That's the mysterious interval. 00:02:06.720 --> 00:02:10.100 I don't really care about that in general. 00:02:10.100 --> 00:02:15.280 In my case, which one is the most natural parametrization, 00:02:15.280 --> 00:02:17.940 guys? 00:02:17.940 --> 00:02:19.690 Take x to be time. 00:02:19.690 --> 00:02:20.790 Say again, Magdalena. 00:02:20.790 --> 00:02:23.040 Take x to be time. 00:02:23.040 --> 00:02:26.870 And that will make your life easier. 00:02:26.870 --> 00:02:29.550 I take x to be time. 00:02:29.550 --> 00:02:33.460 And then y would be time plus 1. 00:02:33.460 --> 00:02:34.570 And I'm happy. 00:02:34.570 --> 00:02:39.380 So the way they asked you to enter your answer in web work 00:02:39.380 --> 00:02:45.050 was as r of t equals-- and it's blinking, blinking, 00:02:45.050 --> 00:02:46.885 interactive field for you. 00:02:46.885 --> 00:02:48.820 You say, OK, t? 00:02:48.820 --> 00:02:51.490 T what? 00:02:51.490 --> 00:02:53.660 And I'm not going to solve your problem. 00:02:53.660 --> 00:02:55.530 But your problem is similar. 00:02:55.530 --> 00:02:56.030 Why? 00:02:56.030 --> 00:03:04.020 Because r of t, which is the vector equation of your y 00:03:04.020 --> 00:03:09.670 or curve would give you the position vector, which is what? 00:03:09.670 --> 00:03:10.265 Wait a second. 00:03:10.265 --> 00:03:14.440 Let me finish. x of t times i plus y of t times 00:03:14.440 --> 00:03:17.320 j is the definition I gave last time. 00:03:17.320 --> 00:03:18.280 Go ahead. 00:03:18.280 --> 00:03:20.900 STUDENT: Where'd you get r of t and what is it? 00:03:20.900 --> 00:03:23.550 PROFESSOR: I already discussed it last time. 00:03:23.550 --> 00:03:27.300 So since I'm reviewing today, just 00:03:27.300 --> 00:03:30.020 reviewing today chapter 10, I really 00:03:30.020 --> 00:03:32.340 don't mind going over with you. 00:03:32.340 --> 00:03:34.780 But please keep in mind this is the first 00:03:34.780 --> 00:03:37.629 and the last time I'm going to review things 00:03:37.629 --> 00:03:38.420 with you last time. 00:03:38.420 --> 00:03:44.150 So what did you say a position vector is for a curve? 00:03:44.150 --> 00:03:46.953 When we talked about the drunken bug, 00:03:46.953 --> 00:03:50.720 we say the drunken bug is following a trajectory. 00:03:50.720 --> 00:03:53.870 He or she is struggling in time. 00:03:53.870 --> 00:04:00.620 I have a given frame xyz system of coordinates-- system 00:04:00.620 --> 00:04:03.650 of axes of coordinates with a certain origin. 00:04:03.650 --> 00:04:07.660 Thank God for this origin because you cannot refer 00:04:07.660 --> 00:04:11.070 to a position vector unless you have a frame-- 00:04:11.070 --> 00:04:14.190 an original frame, a position frame, initial frame. 00:04:14.190 --> 00:04:21.856 So r of t represents the vector that originates at the origin o 00:04:21.856 --> 00:04:28.480 and ends exactly at the position of your particle at time t. 00:04:28.480 --> 00:04:30.925 If you want, if you hate bugs, this 00:04:30.925 --> 00:04:35.370 is just the particle from physics that travels in time t. 00:04:35.370 --> 00:04:35.990 So-- 00:04:35.990 --> 00:04:39.731 STUDENT: OK, so the r of t is represented in the parent 00:04:39.731 --> 00:04:40.230 equation 00:04:40.230 --> 00:04:41.500 PROFESSOR: Yes, sir. 00:04:41.500 --> 00:04:42.530 Exactly. 00:04:42.530 --> 00:04:45.820 In a plane where z is 0-- so you imagine 00:04:45.820 --> 00:04:48.520 the z-axis coming at z0. 00:04:48.520 --> 00:04:50.520 This is the xy plane. 00:04:50.520 --> 00:04:52.960 And I'm very happy I have on the floor. 00:04:52.960 --> 00:04:54.570 This bug is on the floor. 00:04:54.570 --> 00:04:56.469 He doesn't want to know what's the dimension. 00:04:56.469 --> 00:04:57.510 So what's he going to do? 00:04:57.510 --> 00:05:02.480 He's going to say plus 0 times k that I don't care about 00:05:02.480 --> 00:05:05.919 because the position vector will be given by-- 00:05:05.919 --> 00:05:06.460 STUDENT: So-- 00:05:06.460 --> 00:05:07.876 PROFESSOR: --or for a plane curve. 00:05:07.876 --> 00:05:09.470 STUDENT: So if this was in 3D space 00:05:09.470 --> 00:05:14.180 and we had three equations so it was like z equals-- 00:05:14.180 --> 00:05:19.392 is equal to 2y plus x plus 1, then it would be-- then how 00:05:19.392 --> 00:05:20.605 would we do that? 00:05:20.605 --> 00:05:23.400 PROFESSOR: Let me remind us in general the way I pointed it 00:05:23.400 --> 00:05:24.780 out last. 00:05:24.780 --> 00:05:27.800 R of t in general as a position vector, 00:05:27.800 --> 00:05:29.540 we said many things about it. 00:05:29.540 --> 00:05:33.590 We said it is a smooth function. 00:05:33.590 --> 00:05:36.170 What does it mean differential role 00:05:36.170 --> 00:05:38.810 with derivative continuous? 00:05:38.810 --> 00:05:41.072 What did-- actually, that's c1. 00:05:41.072 --> 00:05:42.030 What else did they say? 00:05:42.030 --> 00:05:43.520 He said it's a regular. 00:05:43.520 --> 00:05:45.540 It's a regular vector function. 00:05:45.540 --> 00:05:46.700 What does it mean? 00:05:46.700 --> 00:05:49.040 It never stops, not even for a second. 00:05:49.040 --> 00:05:51.950 Well, the velocity of that is zero. 00:05:51.950 --> 00:05:53.850 When we introduced it-- all right, 00:05:53.850 --> 00:05:56.015 I cannot teach the whole thing all over again, 00:05:56.015 --> 00:05:59.990 but I'll be happy to do review just today. 00:05:59.990 --> 00:06:05.420 It's going to be x of ti plus y of tj plus z over k. 00:06:05.420 --> 00:06:07.220 That is a way to write it like that. 00:06:07.220 --> 00:06:13.220 Or the simpler way to write it as x of t, y of t, z of t. 00:06:13.220 --> 00:06:15.830 Now, if it involves using different notation, 00:06:15.830 --> 00:06:17.720 I want to warn you about that. 00:06:17.720 --> 00:06:21.700 Some people like to put braces like angular brackets. 00:06:21.700 --> 00:06:25.460 Or some people like because it's a vector. 00:06:25.460 --> 00:06:29.486 And that's the way they define vector Some people like just 00:06:29.486 --> 00:06:30.236 round parentheses. 00:06:30.236 --> 00:06:31.710 This is more practically. 00:06:31.710 --> 00:06:34.450 These are the coordinates of a position vector 00:06:34.450 --> 00:06:37.240 with respect to the ijk frame. 00:06:37.240 --> 00:06:40.420 So since we talked about this already, 00:06:40.420 --> 00:06:43.020 some simple examples have been given. 00:06:43.020 --> 00:06:45.160 One of them was a circling plane, 00:06:45.160 --> 00:06:48.070 another circling plane of a different speed, 00:06:48.070 --> 00:06:49.500 a segment of a line. 00:06:49.500 --> 00:06:50.970 This is the segment of a line. 00:06:50.970 --> 00:06:52.260 What else have we discussed? 00:06:52.260 --> 00:06:54.360 We discuss about something wilder, 00:06:54.360 --> 00:06:57.690 which was the helix at different speeds? 00:06:57.690 --> 00:07:01.190 All right, so very good question for him was-- so 00:07:01.190 --> 00:07:02.830 is this x of tt? 00:07:02.830 --> 00:07:03.380 Yes. 00:07:03.380 --> 00:07:05.420 Is this y of tt plus 1? 00:07:05.420 --> 00:07:05.920 Yes. 00:07:05.920 --> 00:07:08.760 Is this z of t 0 in my case? 00:07:08.760 --> 00:07:09.722 Precisely 00:07:09.722 --> 00:07:13.166 STUDENT: So if you gave value to z, 00:07:13.166 --> 00:07:16.610 what would you chose to make t parameterized? 00:07:16.610 --> 00:07:20.314 PROFESSOR: OK, t in general, if you are moving, 00:07:20.314 --> 00:07:22.480 you have an infinite motion that comes from nowhere, 00:07:22.480 --> 00:07:24.220 goes nowhere, right? 00:07:24.220 --> 00:07:28.770 OK, then you can say t is between minus 00:07:28.770 --> 00:07:29.920 infinity plus infinity. 00:07:29.920 --> 00:07:31.050 And that's your i-- 00:07:31.050 --> 00:07:32.300 STUDENT: But what I'm saying-- 00:07:32.300 --> 00:07:36.510 PROFESSOR: But-- but in your case-- in your case, 00:07:36.510 --> 00:07:40.370 you think oh, I know where I'm starting. 00:07:40.370 --> 00:07:44.230 So to that equals to 1, t must be 1. 00:07:44.230 --> 00:07:47.060 So I start my movement at 1 second 00:07:47.060 --> 00:07:52.690 and I end my movement at 2 seconds where x will be 2, 00:07:52.690 --> 00:07:54.580 and y will be 3. 00:07:54.580 --> 00:07:57.431 STUDENT: Well, I mean-- so you said x equals t. 00:07:57.431 --> 00:07:59.816 You took that from the y equals x plus 1. 00:07:59.816 --> 00:08:02.439 If you had the third variable t, what would you-- 00:08:02.439 --> 00:08:03.980 PROFESSOR: It's not a third variable. 00:08:03.980 --> 00:08:05.860 It's the time parameter. 00:08:05.860 --> 00:08:08.770 So I work in three variables-- xyz in space. 00:08:08.770 --> 00:08:10.810 Those are my space coordinates. 00:08:10.810 --> 00:08:14.090 The space coordinates are function of time. 00:08:14.090 --> 00:08:17.130 So it's all about physics. 00:08:17.130 --> 00:08:19.690 So mathematics sometimes becomes physics. 00:08:19.690 --> 00:08:22.945 Thank God we are sisters, even step-sisters. 00:08:22.945 --> 00:08:24.760 X is a function of t. 00:08:24.760 --> 00:08:26.102 Y is a function of t. 00:08:26.102 --> 00:08:28.430 Z is a function of t. 00:08:28.430 --> 00:08:29.030 Right? 00:08:29.030 --> 00:08:30.740 Am I answering your question or maybe 00:08:30.740 --> 00:08:33.010 I didn't quite understand the-- 00:08:33.010 --> 00:08:35.474 STUDENT: Well, I understand how to parameterize 00:08:35.474 --> 00:08:36.826 the idea of a plane. 00:08:36.826 --> 00:08:39.179 How do you do it in space though? 00:08:39.179 --> 00:08:42.220 PROFESSOR: In space-- in space, you're already here. 00:08:42.220 --> 00:08:46.370 So if you want to ride this not in plane but in space, 00:08:46.370 --> 00:08:51.380 your parametric equation is ti plus t plus 1j plus 0k, 00:08:51.380 --> 00:08:54.430 for this example, anywhere in r3. 00:08:54.430 --> 00:08:56.570 We live in r3. 00:08:56.570 --> 00:08:58.380 All righty? 00:08:58.380 --> 00:09:00.850 We live in r3. 00:09:00.850 --> 00:09:03.430 OK, let me give you more examples. 00:09:03.430 --> 00:09:05.920 Because I think I'm running out of time. 00:09:05.920 --> 00:09:09.170 But I still have to cover the material, 00:09:09.170 --> 00:09:11.120 eventually get somewhere. 00:09:11.120 --> 00:09:15.740 However, I want you to see more examples that will help 00:09:15.740 --> 00:09:18.610 you grasp this notion better. 00:09:18.610 --> 00:09:25.190 So guys, imagine that we have space r3-- that 00:09:25.190 --> 00:09:28.634 could be rn-- in which I have an origin 00:09:28.634 --> 00:09:31.586 and I have a [INAUDIBLE]. 00:09:31.586 --> 00:09:35.030 And somebody gives me a position vector 00:09:35.030 --> 00:09:38.500 for a motion that's a regular curve. 00:09:38.500 --> 00:09:44.760 And that's x of tri plus y is tj plus z of tk. 00:09:44.760 --> 00:09:49.180 And since his question is a very valid one, 00:09:49.180 --> 00:09:52.760 let's see what happens in a later case. 00:09:52.760 --> 00:09:56.410 So I'm going to deviate a little from my lesson plan. 00:09:56.410 --> 00:09:59.690 And I say let us be flexible and compare 00:09:59.690 --> 00:10:02.029 that with the inner curve. 00:10:02.029 --> 00:10:04.025 Because in the process of comparison, 00:10:04.025 --> 00:10:06.520 you learn a lot more. 00:10:06.520 --> 00:10:11.020 If I were to be right above my [INAUDIBLE] like that. 00:10:11.020 --> 00:10:17.771 So this is the spacial curve in our three imaginary trajectory 00:10:17.771 --> 00:10:20.206 run of a bug or a particle. 00:10:20.206 --> 00:10:24.030 As we said, this is the planar curve-- planar, 00:10:24.030 --> 00:10:28.706 parametrized curve in r2. 00:10:28.706 --> 00:10:29.550 What's different? 00:10:29.550 --> 00:10:31.390 What do we know about them? 00:10:31.390 --> 00:10:35.470 We clearly know section 10.2. 00:10:35.470 --> 00:10:38.900 What I hate in general about processors 00:10:38.900 --> 00:10:43.230 is if they are way too structured. 00:10:43.230 --> 00:10:47.220 Mathematics cannot be talking sections where you say, oh, 00:10:47.220 --> 00:10:51.786 section 10.2 is only about velocity and acceleration. 00:10:51.786 --> 00:10:55.130 But section 10.4 is about tangent unit vector 00:10:55.130 --> 00:10:56.740 and principle normal. 00:10:56.740 --> 00:10:58.840 Well, they are related. 00:10:58.840 --> 00:11:03.740 So it's only natural when we talk about section 10.2 00:11:03.740 --> 00:11:11.470 acceleration and velocity that from acceleration, you 00:11:11.470 --> 00:11:22.290 have a induced line to tangent unit vector-- tangent unit 00:11:22.290 --> 00:11:23.400 vector. 00:11:23.400 --> 00:11:28.450 And later on, you're going to compare acceleration 00:11:28.450 --> 00:11:30.610 with a normal principal vector. 00:11:30.610 --> 00:11:32.490 Sometimes, they are the same thing. 00:11:32.490 --> 00:11:35.020 Sometimes, they are not the same thing. 00:11:35.020 --> 00:11:38.560 It's a good idea to see when they are the same thing 00:11:38.560 --> 00:11:40.600 and when they are not. 00:11:40.600 --> 00:11:44.890 So in section 10.4, we will focus practically 00:11:44.890 --> 00:11:48.430 or t, n, and v, the Frenet frame and its consequences 00:11:48.430 --> 00:11:51.820 on curvature, we already talked about that a little bit. 00:11:51.820 --> 00:11:56.300 In 10.2, practically, we didn't cover much. 00:11:56.300 --> 00:11:59.380 I only told you about velocity, acceleration. 00:11:59.380 --> 00:12:03.150 However, I would like to review that for you. 00:12:03.150 --> 00:12:05.990 Because I don't want to risk losing you. 00:12:05.990 --> 00:12:07.900 I'm going to lose some of you anyway. 00:12:07.900 --> 00:12:10.120 Two people said this course is too hard for me. 00:12:10.120 --> 00:12:11.950 I'm going to drop. 00:12:11.950 --> 00:12:14.270 You are free to drop and I think it's better for you 00:12:14.270 --> 00:12:16.360 to drop than struggle. 00:12:16.360 --> 00:12:21.442 But as long as you can still learn and you can follow, 00:12:21.442 --> 00:12:22.660 you shouldn't drop. 00:12:22.660 --> 00:12:26.870 So try to see exactly how much you can handle. 00:12:26.870 --> 00:12:30.400 If you can handle just the regular section of calc three, 00:12:30.400 --> 00:12:32.150 go to that regular section. 00:12:32.150 --> 00:12:36.430 If you can handle more, if you are good at mathematics, 00:12:36.430 --> 00:12:39.089 if you have always been considered bright 00:12:39.089 --> 00:12:41.780 in mathematics in high school, let us stay here. 00:12:41.780 --> 00:12:43.030 Otherwise, go. 00:12:43.030 --> 00:12:44.280 Don't stay. 00:12:44.280 --> 00:12:48.790 All right, so the velocities are prime of t. 00:12:48.790 --> 00:12:51.680 The acceleration is our double prime of t. 00:12:51.680 --> 00:12:53.610 We have done that last time. 00:12:53.610 --> 00:12:55.330 We were very happy. 00:12:55.330 --> 00:12:58.430 What would happen in a planar curve seen on 2? 00:12:58.430 --> 00:13:02.460 The same thing, of course, except the last component 00:13:02.460 --> 00:13:03.800 is not there. 00:13:03.800 --> 00:13:06.820 It's part of ti plus y prime of tj. 00:13:06.820 --> 00:13:10.780 And there is a 0k in both cases. 00:13:10.780 --> 00:13:12.870 So all these are factors. 00:13:12.870 --> 00:13:15.295 At times, I'm not going to point that out anymore. 00:13:15.295 --> 00:13:18.000 00:13:18.000 --> 00:13:20.320 The derivation goes component-wise. 00:13:20.320 --> 00:13:24.890 So if you forgot how to derive or you want to drink and derive 00:13:24.890 --> 00:13:28.370 or something, then you don't belong in this class. 00:13:28.370 --> 00:13:32.250 So again, make sure you know the derivations and integrations 00:13:32.250 --> 00:13:33.779 really well. 00:13:33.779 --> 00:13:35.445 I'm going to work some examples out just 00:13:35.445 --> 00:13:36.720 to refresh your memory. 00:13:36.720 --> 00:13:40.450 But if you have struggled with differentiation and integration 00:13:40.450 --> 00:13:45.170 in Calc 1, then you do not do belong in this class. 00:13:45.170 --> 00:13:53.136 All right, let's see about speed. 00:13:53.136 --> 00:13:54.620 It's about speed. 00:13:54.620 --> 00:13:56.030 It's about time. 00:13:56.030 --> 00:14:00.090 It's about time to remember what the speed was. 00:14:00.090 --> 00:14:04.320 The speed was the absolute value or the magnitude. 00:14:04.320 --> 00:14:07.015 It's not an absolute value, but it's a magnitude 00:14:07.015 --> 00:14:08.590 of the velocity factor. 00:14:08.590 --> 00:14:11.100 This is the speed. 00:14:11.100 --> 00:14:13.552 And the same in this case. 00:14:13.552 --> 00:14:17.600 If I want to write an explicit formula because somebody 00:14:17.600 --> 00:14:21.130 asked me by email, can I write an explicit formula, of course. 00:14:21.130 --> 00:14:24.246 That's a piece of cake and you should know that from before. 00:14:24.246 --> 00:14:29.780 X prime of t squared plus y prime of t squared plus z 00:14:29.780 --> 00:14:34.110 prime of t squared under the square root. 00:14:34.110 --> 00:14:37.265 I was not going to insist on the planar curve. 00:14:37.265 --> 00:14:41.240 Of course the planar curve will have a speed that all of you 00:14:41.240 --> 00:14:42.420 know about. 00:14:42.420 --> 00:14:44.950 And that's going to be square root of x prime of t 00:14:44.950 --> 00:14:49.070 squared plus y root prime of t squared. 00:14:49.070 --> 00:14:53.340 You should do your own thinking to see what the particular case 00:14:53.340 --> 00:14:55.710 will become. 00:14:55.710 --> 00:14:58.430 However, I want to see if you understood 00:14:58.430 --> 00:15:01.780 what derives from that in the sense 00:15:01.780 --> 00:15:06.494 that you should know the length of a arc of a curve. 00:15:06.494 --> 00:15:09.890 What is the length of an arc of a curve? 00:15:09.890 --> 00:15:15.240 Well, we have to look back at Calculus 2 a little bit 00:15:15.240 --> 00:15:20.900 and remember that the length of an arc of a curve in Calculus 2 00:15:20.900 --> 00:15:24.390 was given by, what? 00:15:24.390 --> 00:15:30.050 So you say, well, yeah. 00:15:30.050 --> 00:15:31.410 That was a long time ago. 00:15:31.410 --> 00:15:33.400 Well, some of you already don't even 00:15:33.400 --> 00:15:39.650 remember that as being integral from a to b of square root of 1 00:15:39.650 --> 00:15:43.480 plus 1 prime of x squared dx. 00:15:43.480 --> 00:15:46.630 And you were freaking out thinking, oh my god, 00:15:46.630 --> 00:15:51.556 I don't see how this formula from Calc 2, 00:15:51.556 --> 00:15:55.170 the arc of a curve, had you travel between time 00:15:55.170 --> 00:16:01.740 equals a and time equals b will relate to this formula. 00:16:01.740 --> 00:16:03.440 So what happened in Calc 2? 00:16:03.440 --> 00:16:07.060 In Calc 2, hopefully, you have a good teacher. 00:16:07.060 --> 00:16:09.660 And hopefully, you've learned a lot. 00:16:09.660 --> 00:16:12.540 This is between a and b, right? 00:16:12.540 --> 00:16:14.210 What did they teach you in Calc 2? 00:16:14.210 --> 00:16:16.600 They taught you that you have to take 00:16:16.600 --> 00:16:18.600 integral from a to b of square root of 1 00:16:18.600 --> 00:16:21.020 plus y prime of x squared ds. 00:16:21.020 --> 00:16:21.810 Why? 00:16:21.810 --> 00:16:23.730 You never asked your teacher why. 00:16:23.730 --> 00:16:24.230 That's bad. 00:16:24.230 --> 00:16:25.922 You should do that. 00:16:25.922 --> 00:16:29.060 You should ask why every time. 00:16:29.060 --> 00:16:32.790 They make you swallow a formula via memorization 00:16:32.790 --> 00:16:35.472 without understanding this is the speed. 00:16:35.472 --> 00:16:37.710 And now I'm coming with the good news. 00:16:37.710 --> 00:16:39.990 I have a proof of that. 00:16:39.990 --> 00:16:42.320 I know what speed means when I'm moving 00:16:42.320 --> 00:16:46.810 along the arc of a curve in plane. 00:16:46.810 --> 00:16:51.145 OK, so what is the distance travelled between time equals A 00:16:51.145 --> 00:16:52.500 and time equals B? 00:16:52.500 --> 00:16:57.070 It's going to be integral form a to be of the speed, right? 00:16:57.070 --> 00:16:59.450 This is the same one I'm driving from-- level two-- 00:16:59.450 --> 00:17:01.711 Amarillo or anywhere else. 00:17:01.711 --> 00:17:02.210 There. 00:17:02.210 --> 00:17:05.069 Now, what they showed you and they fooled you 00:17:05.069 --> 00:17:10.618 into memorizing that is just a consequence of this formula 00:17:10.618 --> 00:17:12.530 because of what he said. 00:17:12.530 --> 00:17:13.480 Why? 00:17:13.480 --> 00:17:16.785 The most usual parameterization is 00:17:16.785 --> 00:17:22.680 going to be y of t equals t-- I'm sorry, x of t equals vxst 00:17:22.680 --> 00:17:25.910 and y of t equals y of t. 00:17:25.910 --> 00:17:27.940 So, again x is time. 00:17:27.940 --> 00:17:33.130 In many linear curves, you can take x to be time, thank God. 00:17:33.130 --> 00:17:38.560 And then your parametrization will be t comma y of t. 00:17:38.560 --> 00:17:40.640 Because x is t. 00:17:40.640 --> 00:17:43.150 And x prime of t will be 1. 00:17:43.150 --> 00:17:45.930 Y prime of t will be y prime of t. 00:17:45.930 --> 00:17:50.040 When you take them, squish them, square them, sum them up, 00:17:50.040 --> 00:17:51.990 you get exactly this one. 00:17:51.990 --> 00:17:54.402 But you notice this is the speed. 00:17:54.402 --> 00:17:56.070 What is this the speed? 00:17:56.070 --> 00:18:03.288 Of some value over prime of t, which is speed. 00:18:03.288 --> 00:18:07.250 You see that what they forced you to memorize in Calc 2 00:18:07.250 --> 00:18:10.920 is nothing but the speed. 00:18:10.920 --> 00:18:12.920 And I could change the parameterization 00:18:12.920 --> 00:18:14.980 to something more general. 00:18:14.980 --> 00:18:19.560 Now, can I do this parameterization for a circle? 00:18:19.560 --> 00:18:20.230 No. 00:18:20.230 --> 00:18:22.460 Why not? 00:18:22.460 --> 00:18:25.000 I could, but then I'd have to split 00:18:25.000 --> 00:18:26.760 into the upper part and lower part 00:18:26.760 --> 00:18:29.040 because the circle is not a graph. 00:18:29.040 --> 00:18:31.210 So I take t between this and that 00:18:31.210 --> 00:18:35.920 and then I have square root of 1 minus t squared on top. 00:18:35.920 --> 00:18:38.800 And underneath, I have minus square root of 1 00:18:38.800 --> 00:18:39.570 minus t squared. 00:18:39.570 --> 00:18:43.980 So I split the poor circle into a graph and another graph. 00:18:43.980 --> 00:18:45.230 And I do it separately. 00:18:45.230 --> 00:18:47.310 And I can still apply that. 00:18:47.310 --> 00:18:49.430 But only a fool would do that, right? 00:18:49.430 --> 00:18:52.900 So what does a smart mathematician do? 00:18:52.900 --> 00:18:54.670 A smart mathematician will say, OK, 00:18:54.670 --> 00:18:59.740 for the circle, x is cosine t, y is sine t. 00:18:59.740 --> 00:19:01.830 And that is the parameterization I'm 00:19:01.830 --> 00:19:04.170 going to use for this formula. 00:19:04.170 --> 00:19:05.900 And I get speed 1. 00:19:05.900 --> 00:19:08.620 And I'm going to be happy, right? 00:19:08.620 --> 00:19:10.970 So it's a lot easier to understand what 00:19:10.970 --> 00:19:13.280 a general parameterization is. 00:19:13.280 --> 00:19:19.490 What is the length of an arc of a curve for a curving space? 00:19:19.490 --> 00:19:20.910 There's the bug. 00:19:20.910 --> 00:19:22.110 Time equals t0. 00:19:22.110 --> 00:19:23.970 He's buzzing. 00:19:23.970 --> 00:19:26.180 And after 10 seconds, he will be at the end. 00:19:26.180 --> 00:19:30.620 So it goes, [BUZZING] jump. 00:19:30.620 --> 00:19:35.120 OK, how much did he travel? 00:19:35.120 --> 00:19:41.700 Integral from a to b of square root of x prime of t squared 00:19:41.700 --> 00:19:44.430 plus y prime of t squared plus z prime of t 00:19:44.430 --> 00:19:50.430 squared-- no matter what that position vector x of ty of t0 00:19:50.430 --> 00:19:51.360 give us. 00:19:51.360 --> 00:19:56.200 So you take the coordinates of the velocity vector. 00:19:56.200 --> 00:19:57.150 You look at them. 00:19:57.150 --> 00:19:57.900 You square them. 00:19:57.900 --> 00:19:59.280 You add them together. 00:19:59.280 --> 00:20:00.740 You put them under the square root. 00:20:00.740 --> 00:20:02.510 That's going to be the speed. 00:20:02.510 --> 00:20:06.370 And displacement is integral of speed. 00:20:06.370 --> 00:20:09.100 When you guys learned in school, your teacher 00:20:09.100 --> 00:20:10.935 oversimplified the things. 00:20:10.935 --> 00:20:12.950 What did your teacher say in physics? 00:20:12.950 --> 00:20:15.700 Space equals speed times time. 00:20:15.700 --> 00:20:16.680 Say it again. 00:20:16.680 --> 00:20:19.935 He said space traveled is speed times time. 00:20:19.935 --> 00:20:23.635 But he assumed the speed is constant or constant 00:20:23.635 --> 00:20:26.570 on portions-- like, speedswise constant. 00:20:26.570 --> 00:20:28.940 Well, if it's a constant, the speed 00:20:28.940 --> 00:20:30.790 will get the heck out of here. 00:20:30.790 --> 00:20:35.300 And then the space will be speed times b minus a. 00:20:35.300 --> 00:20:37.840 But b minus a is delta t. 00:20:37.840 --> 00:20:41.200 In mathematics, in physics, we say b minus a is delta t. 00:20:41.200 --> 00:20:44.720 That's the interval of time that the bug travels or the particle 00:20:44.720 --> 00:20:45.730 travels. 00:20:45.730 --> 00:20:47.870 So he or she was right. 00:20:47.870 --> 00:20:51.060 Space is speed times time, but it's not like 00:20:51.060 --> 00:20:53.670 that unless the speed is constant. 00:20:53.670 --> 00:20:55.830 So he oversimplified your knowledge 00:20:55.830 --> 00:20:57.520 of mathematics and physics. 00:20:57.520 --> 00:20:59.040 Now you see the truth. 00:20:59.040 --> 00:21:04.500 Space is integral of speed. 00:21:04.500 --> 00:21:06.340 OK, now we understand. 00:21:06.340 --> 00:21:09.830 And I promised you last time that after reviewing, 00:21:09.830 --> 00:21:13.730 I didn't even say I would review anything from 10.2 and 10.4. 00:21:13.730 --> 00:21:14.850 I promised you more. 00:21:14.850 --> 00:21:17.580 I promised you that I'm going to compute something that's 00:21:17.580 --> 00:21:23.960 out of 10.4 which is called a curvature of a helix 00:21:23.960 --> 00:21:25.230 in particular. 00:21:25.230 --> 00:21:29.680 Because we looked at curvature of a parametric curve 00:21:29.680 --> 00:21:31.190 in general. 00:21:31.190 --> 00:21:36.700 I want to organize the material of review from 10.2 and 10.4 00:21:36.700 --> 00:21:40.260 in a big problem just like you will have in the exams, 00:21:40.260 --> 00:21:42.347 in the midterm, and in the final. 00:21:42.347 --> 00:21:43.430 I don't want to scare you. 00:21:43.430 --> 00:21:45.920 I just want to prepare you better 00:21:45.920 --> 00:21:49.844 for the kind of multiple questions we are going to have. 00:21:49.844 --> 00:21:55.240 So let me give you a funny looking curve. 00:21:55.240 --> 00:21:59.430 I want you to think about it and tell me what it is. 00:21:59.430 --> 00:22:01.935 a and b are positive numbers. 00:22:01.935 --> 00:22:07.140 a cosine ba sine t bt will be some sort of funny trajectory. 00:22:07.140 --> 00:22:09.930 You are already familiar to that. 00:22:09.930 --> 00:22:13.414 Last time, I gave you an example where a was 4-- oh my god, 00:22:13.414 --> 00:22:14.550 I don't even remember. 00:22:14.550 --> 00:22:16.460 You'll need to help me. 00:22:16.460 --> 00:22:18.684 [INAUDIBLE] 00:22:18.684 --> 00:22:21.120 STUDENT: 4, 4, 3. 00:22:21.120 --> 00:22:24.760 PROFESSOR: I took those because they are Pythagorean numbers. 00:22:24.760 --> 00:22:26.360 So what does it mean? 00:22:26.360 --> 00:22:28.970 3 squared plus 4 squared equals 5 squared. 00:22:28.970 --> 00:22:31.920 I wanted the sum of them to be a perfect square. 00:22:31.920 --> 00:22:33.230 So I was playing games. 00:22:33.230 --> 00:22:36.590 You can do that for any a and b. 00:22:36.590 --> 00:22:37.710 Now, what do I want? 00:22:37.710 --> 00:22:43.620 A-- like in 10.2 where you write r prime of t, 00:22:43.620 --> 00:22:46.540 rewrite that double prime of t. 00:22:46.540 --> 00:22:49.730 So it's a complex problem. 00:22:49.730 --> 00:22:53.210 In b, I want you to find t and r prime 00:22:53.210 --> 00:22:55.750 of t over-- who remembers the formula? 00:22:55.750 --> 00:22:57.700 I shouldn't have spoon-fed you that. 00:22:57.700 --> 00:22:58.620 STUDENT: Absolute-- 00:22:58.620 --> 00:23:00.755 PROFESSOR: Absolute magnitude, actually. 00:23:00.755 --> 00:23:03.520 It's more correct to say magnitude, right? 00:23:03.520 --> 00:23:04.300 Very good. 00:23:04.300 --> 00:23:08.636 And what else did I spoon-feed you last name? 00:23:08.636 --> 00:23:10.280 I spoon-fed you n. 00:23:10.280 --> 00:23:13.970 Let's compute n as well as part of the problem 00:23:13.970 --> 00:23:21.200 t prime t over t prime of t magnitude. 00:23:21.200 --> 00:23:24.140 STUDENT: So you're looking for the tangent unit vector. 00:23:24.140 --> 00:23:25.497 PROFESSOR: Tangent unit vector? 00:23:25.497 --> 00:23:27.080 STUDENT: And then you're looking for-- 00:23:27.080 --> 00:23:27.913 PROFESSOR: Yes, sir. 00:23:27.913 --> 00:23:30.830 And-- OK, don't you like me to also give you 00:23:30.830 --> 00:23:34.050 something like a grading grid, how much everything 00:23:34.050 --> 00:23:35.320 would be worth. 00:23:35.320 --> 00:23:36.570 Imagine you're taking an exam. 00:23:36.570 --> 00:23:39.710 Why not put yourself in an exam mode 00:23:39.710 --> 00:23:44.130 so you don't freak out during the actual exam? 00:23:44.130 --> 00:23:47.680 C will be another question, something smart. 00:23:47.680 --> 00:24:02.480 Let's see-- reparameterize an arc length to a plane, a curve, 00:24:02.480 --> 00:24:05.380 rho of s. 00:24:05.380 --> 00:24:08.510 Why not r of s like some people call-- use it 00:24:08.510 --> 00:24:10.160 and some books use it? 00:24:10.160 --> 00:24:11.759 Because if you're reparameterizing s, 00:24:11.759 --> 00:24:13.467 it's going to be the same physical limits 00:24:13.467 --> 00:24:15.870 but a different function. 00:24:15.870 --> 00:24:19.640 So if you remember the diagram I wrote before, 00:24:19.640 --> 00:24:24.480 little r is a function that comes from integral i time 00:24:24.480 --> 00:24:29.450 integral 2r3 and rho would be coming from a j to r3. 00:24:29.450 --> 00:24:32.610 And what is the relationship between them? 00:24:32.610 --> 00:24:36.360 This is t goes to s and this is s goes to t. 00:24:36.360 --> 00:24:39.450 What is d I'm asking you? 00:24:39.450 --> 00:24:41.380 Well, if you're d and c, of course 00:24:41.380 --> 00:24:44.530 you know what the arc length parameter will be. 00:24:44.530 --> 00:24:49.630 It's going to be integral from 0 to t or any t0 here 00:24:49.630 --> 00:24:54.962 of the speed-- of the speed of the original function here 00:24:54.962 --> 00:24:56.310 of t. 00:24:56.310 --> 00:25:01.820 The tau-- maybe tau is better than the dummy variable t. 00:25:01.820 --> 00:25:05.242 And e I want. 00:25:05.242 --> 00:25:06.750 You say, how much more do you want? 00:25:06.750 --> 00:25:07.650 I want a lot. 00:25:07.650 --> 00:25:09.132 I'm a greedy person. 00:25:09.132 --> 00:25:13.840 I want the curvature of the curve. 00:25:13.840 --> 00:25:17.550 And you have to remind me. 00:25:17.550 --> 00:25:19.990 Some of you are very good students, better than me. 00:25:19.990 --> 00:25:23.622 I mean, I'm still behind with a research course 00:25:23.622 --> 00:25:25.080 that I have-- research paper i have 00:25:25.080 --> 00:25:29.786 to read in two days in biology. 00:25:29.786 --> 00:25:35.500 But this curvature of the curve had a very simple formula 00:25:35.500 --> 00:25:36.980 that we all love. 00:25:36.980 --> 00:25:40.120 For mathematicians, it's a piece of cake to remember it. 00:25:40.120 --> 00:25:43.310 K-- that's what I like about being a mathematician. 00:25:43.310 --> 00:25:45.350 I don't need a good memory. 00:25:45.350 --> 00:25:47.920 Now I remember why I didn't go to medical school-- 00:25:47.920 --> 00:25:51.010 because my father told me, well, you 00:25:51.010 --> 00:25:53.810 should be able to remember all the bones in a person's body. 00:25:53.810 --> 00:25:55.890 And I said, dad, do you know all these names? 00:25:55.890 --> 00:25:56.210 Yes, of course. 00:25:56.210 --> 00:25:57.293 And he started telling me. 00:25:57.293 --> 00:26:01.410 Well, I realized that I would never remember those. 00:26:01.410 --> 00:26:07.030 But I remember this formula which is r rho. 00:26:07.030 --> 00:26:10.330 In this case, if our r is Greek rho, 00:26:10.330 --> 00:26:13.090 it's got to be rho double prime of what? 00:26:13.090 --> 00:26:15.970 of S. Is this correct, what I wrote? 00:26:15.970 --> 00:26:16.470 No. 00:26:16.470 --> 00:26:18.050 What's missing? 00:26:18.050 --> 00:26:22.520 The acceleration and arc length but in magnitude because that's 00:26:22.520 --> 00:26:23.905 a vector, of course. 00:26:23.905 --> 00:26:26.870 This is the scalar function. 00:26:26.870 --> 00:26:28.660 Anything else you want, Magdalena? 00:26:28.660 --> 00:26:30.190 Oh, that's enough. 00:26:30.190 --> 00:26:34.060 All right, so I want to know everything 00:26:34.060 --> 00:26:38.260 that's possible to know about this curve from 10.2 and 10.4 00:26:38.260 --> 00:26:39.890 sections. 00:26:39.890 --> 00:26:41.840 10.3-- skip 10.5. 00:26:41.840 --> 00:26:44.330 Skip-- you're happy about it. 00:26:44.330 --> 00:26:44.940 Yes sir. 00:26:44.940 --> 00:26:48.426 STUDENT: For the parameter on v, is it a t? 00:26:48.426 --> 00:26:49.920 And what's the integral? 00:26:49.920 --> 00:26:51.010 What's on the bottom. 00:26:51.010 --> 00:26:54.230 PROFESSOR: Ah, that value erased when I wrote that one. 00:26:54.230 --> 00:26:56.200 It was there-- t0. 00:26:56.200 --> 00:27:00.510 So I can start with any fixed t0 as my initial moment in time. 00:27:00.510 --> 00:27:02.560 I would like my initial moment in time 00:27:02.560 --> 00:27:05.980 to be 0 just to make my things easier. 00:27:05.980 --> 00:27:07.940 Are we ready to solve this problem together? 00:27:07.940 --> 00:27:11.570 I think we have just about the exact time 00:27:11.570 --> 00:27:14.070 we need to do everything. 00:27:14.070 --> 00:27:17.610 First of all, you have to tell me what kind of curve this is. 00:27:17.610 --> 00:27:20.020 Of course you know because you were here last time. 00:27:20.020 --> 00:27:23.250 Don't skip classes because you are missing everything out 00:27:23.250 --> 00:27:25.380 and then you will have to drop or withdraw. 00:27:25.380 --> 00:27:27.230 So don't skip class. 00:27:27.230 --> 00:27:31.160 What was that from last time? 00:27:31.160 --> 00:27:33.510 It was a helix. 00:27:33.510 --> 00:27:35.280 I'm going to try and redraw it. 00:27:35.280 --> 00:27:38.010 I know I'm wasting my time, but I would 00:27:38.010 --> 00:27:43.750 try to draw a better curve. 00:27:43.750 --> 00:27:46.325 Ah, what's the equation of the cylinder? 00:27:46.325 --> 00:27:49.938 [CLASS MURMURS] 00:27:49.938 --> 00:27:51.327 PROFESSOR: Huh? 00:27:51.327 --> 00:27:53.382 What's the equation of the cylinder? 00:27:53.382 --> 00:27:55.610 That's a quadratic that you are all 00:27:55.610 --> 00:28:01.252 familiar with on which on my beautiful helix is sitting on. 00:28:01.252 --> 00:28:02.980 I taught you the trick last time. 00:28:02.980 --> 00:28:04.350 Don't forget it. 00:28:04.350 --> 00:28:10.100 STUDENT: a over 4 cosine of t squared plus 8 over 4 sine 00:28:10.100 --> 00:28:10.850 of t squared. 00:28:10.850 --> 00:28:13.850 00:28:13.850 --> 00:28:16.250 PROFESSOR: So we do that-- very good. 00:28:16.250 --> 00:28:19.040 X is going to be-- let me right that down. 00:28:19.040 --> 00:28:20.425 X is cosine. 00:28:20.425 --> 00:28:22.940 Y is a sine t. 00:28:22.940 --> 00:28:24.610 And that's exactly what you asked me. 00:28:24.610 --> 00:28:26.060 And z is bt. 00:28:26.060 --> 00:28:29.740 And then what I need to do is square these guys out 00:28:29.740 --> 00:28:31.565 as you said very well. 00:28:31.565 --> 00:28:33.196 I don't care about this 2z. 00:28:33.196 --> 00:28:34.750 He's not in the picture here. 00:28:34.750 --> 00:28:38.820 X squared plus y squared will be a squared, which means I better 00:28:38.820 --> 00:28:42.900 go ahead and draw a circle of radius a on the bottom 00:28:42.900 --> 00:28:44.920 and then build my-- oh my god, it 00:28:44.920 --> 00:28:49.820 looks horrible-- the cylinder based on that circle. 00:28:49.820 --> 00:28:51.050 Guys, it's now straight. 00:28:51.050 --> 00:28:51.780 I'm sorry. 00:28:51.780 --> 00:28:55.380 I mean, I can do better than that. 00:28:55.380 --> 00:28:58.710 OK, good. 00:28:58.710 --> 00:29:02.760 So I'm starting at what point? 00:29:02.760 --> 00:29:06.334 I'm starting at a0 0 time t equals 0. 00:29:06.334 --> 00:29:07.500 We discussed that last time. 00:29:07.500 --> 00:29:09.300 I'm not going to repeat. 00:29:09.300 --> 00:29:12.300 I'm starting here, and two of you 00:29:12.300 --> 00:29:14.090 told me that if t equals phi over two, 00:29:14.090 --> 00:29:17.550 I'm going to be here and so on and so forth. 00:29:17.550 --> 00:29:21.842 If I ask you one more thing for extra credit, what 00:29:21.842 --> 00:29:30.970 is the length of the trajectory traveled by the bug, whatever 00:29:30.970 --> 00:29:38.380 that is, between time t equals 0 and time t equals phi over 2. 00:29:38.380 --> 00:29:40.080 I'd say that's extra credit. 00:29:40.080 --> 00:29:52.400 So, oh my god, 20%, 20%, 20%, 20%, 20%, and 10% for this one. 00:29:52.400 --> 00:29:56.570 And if you think why does she care about the percentages 00:29:56.570 --> 00:29:59.030 and points, you will care and I care. 00:29:59.030 --> 00:30:02.700 Because I want you to see how you are going to be assessed. 00:30:02.700 --> 00:30:05.460 If you have no idea how you're going to assessed, 00:30:05.460 --> 00:30:08.750 then you're going to be happy and i will be unhappy. 00:30:08.750 --> 00:30:12.030 All right, so for 20% credit on this problem, 00:30:12.030 --> 00:30:15.540 we want to see r prime of t will be, r double prime of t 00:30:15.540 --> 00:30:16.040 will be. 00:30:16.040 --> 00:30:18.095 That's going to be a piece of cake. 00:30:18.095 --> 00:30:21.420 And of course, it's maybe the reward is too big for that, 00:30:21.420 --> 00:30:23.200 but that's life. 00:30:23.200 --> 00:30:31.670 Minus a sine t a equals time t and d, d as in infinity. 00:30:31.670 --> 00:30:34.320 So I have an infinite cylinder on which 00:30:34.320 --> 00:30:37.230 I draw an infinite helix coming from hell 00:30:37.230 --> 00:30:39.370 and going to paradise. 00:30:39.370 --> 00:30:44.220 So between minus infinity and plus infinity, there's a guy. 00:30:44.220 --> 00:30:47.790 I'm going to draw a beautiful infinite helix. 00:30:47.790 --> 00:30:50.460 And this is what I posted here. 00:30:50.460 --> 00:30:53.260 What's the acceleration of this helix? 00:30:53.260 --> 00:30:59.600 Minus a cosine t minus 5 sine t and 0. 00:30:59.600 --> 00:31:03.280 Question, quick question for you. 00:31:03.280 --> 00:31:06.840 Will-- you guys are fast. 00:31:06.840 --> 00:31:10.640 Maybe I shouldn't go ahead of myself. 00:31:10.640 --> 00:31:14.530 Nobody's asking me what the speed is right now. 00:31:14.530 --> 00:31:17.760 So why would I do something that's not on the final, right? 00:31:17.760 --> 00:31:19.980 So let's see. 00:31:19.980 --> 00:31:23.130 T, you will have to compute the speed when you get to here. 00:31:23.130 --> 00:31:26.150 But wait until we get there. 00:31:26.150 --> 00:31:27.420 What is mister t? 00:31:27.420 --> 00:31:29.560 Mister t will be the tangent vector. 00:31:29.560 --> 00:31:34.860 So the velocity is going like a crazy guy, long vector. 00:31:34.860 --> 00:31:39.160 The normal unit vector says, I'm the tangent unit vector. 00:31:39.160 --> 00:31:42.750 I'm always perpendicular to the direction. 00:31:42.750 --> 00:31:43.590 I'm of length 1. 00:31:43.590 --> 00:31:47.491 STUDENT: I thought the tangent was parallel to the direction. 00:31:47.491 --> 00:31:49.490 PROFESSOR: Yes, the direction of motion is this. 00:31:49.490 --> 00:31:51.040 Look at me. 00:31:51.040 --> 00:31:53.220 This is my direction of motion. 00:31:53.220 --> 00:31:54.077 And the tangent is-- 00:31:54.077 --> 00:31:55.160 STUDENT: You said it was-- 00:31:55.160 --> 00:31:57.030 PROFESSOR: --in the direction of motion. 00:31:57.030 --> 00:31:57.690 STUDENT: But you said it was perpendicular. 00:31:57.690 --> 00:31:59.130 PROFESSOR: I said perpendicular? 00:31:59.130 --> 00:32:03.030 Because I was thinking ahead of myself and n. 00:32:03.030 --> 00:32:04.250 And I apologize. 00:32:04.250 --> 00:32:06.255 So thank you for correcting me. 00:32:06.255 --> 00:32:08.342 So t is the tangent unit vector. 00:32:08.342 --> 00:32:12.770 00:32:12.770 --> 00:32:15.230 I'm going along the direction of motion. 00:32:15.230 --> 00:32:17.690 And it's going to be perpendicular to t. 00:32:17.690 --> 00:32:22.263 And that's the principal normal unit vector-- 00:32:22.263 --> 00:32:24.227 principal normal unit vector. 00:32:24.227 --> 00:32:26.630 And you're going to tell me what I'm having here. 00:32:26.630 --> 00:32:27.560 Because I don't know. 00:32:27.560 --> 00:32:30.970 00:32:30.970 --> 00:32:37.410 T is minus a sine t a equals sine t 00:32:37.410 --> 00:32:41.120 and v divided by the speed. 00:32:41.120 --> 00:32:43.630 That's why I was getting ahead of myself 00:32:43.630 --> 00:32:46.920 thinking about the speed that you'll need later on anyway. 00:32:46.920 --> 00:32:50.240 But you already need it here, right? 00:32:50.240 --> 00:32:55.180 Because the denominator of this expression will be the speed. 00:32:55.180 --> 00:32:58.290 Magnitude of r prime-- what is that? 00:32:58.290 --> 00:33:01.470 Piece of cake-- square root of the sum 00:33:01.470 --> 00:33:05.747 of the squares of square root of a squared plus b squared. 00:33:05.747 --> 00:33:06.330 Piece of cake. 00:33:06.330 --> 00:33:07.290 I love it. 00:33:07.290 --> 00:33:09.004 So what do I notice? 00:33:09.004 --> 00:33:11.830 That although I'm going on a funny curve which 00:33:11.830 --> 00:33:15.650 is a parametrized helix, I expect some-- maybe 00:33:15.650 --> 00:33:18.210 I expected something wild in terms of speed. 00:33:18.210 --> 00:33:19.500 Well, the speed is constant. 00:33:19.500 --> 00:33:26.850 STUDENT: [INAUDIBLE] the square root of negative a sine t 00:33:26.850 --> 00:33:27.350 squared-- 00:33:27.350 --> 00:33:29.570 PROFESSOR: And what are those? 00:33:29.570 --> 00:33:33.470 A squared sine squared plus c squared cosine squared plus b 00:33:33.470 --> 00:33:35.512 squared, right? 00:33:35.512 --> 00:33:37.220 And what sine squared plus cosine squared 00:33:37.220 --> 00:33:38.450 is 1 [INAUDIBLE]. 00:33:38.450 --> 00:33:41.160 So you get a squared plus b squared. 00:33:41.160 --> 00:33:46.485 Good-- now let's go on and do the n. 00:33:46.485 --> 00:33:53.020 The n will be t prime over magnitude of t prime. 00:33:53.020 --> 00:33:56.350 When you do t prime, you'll say, wait a minute. 00:33:56.350 --> 00:33:59.752 I have square root of a squared plus b squared on the bottom. 00:33:59.752 --> 00:34:04.930 On the top, I have minus equals sine t minus a sine t and 0. 00:34:04.930 --> 00:34:06.310 We have time to finish? 00:34:06.310 --> 00:34:07.036 I think. 00:34:07.036 --> 00:34:08.590 I hope so. 00:34:08.590 --> 00:34:18.110 Divided by-- divided by the magnitude of this fellow. 00:34:18.110 --> 00:34:20.560 I will say, oh, wait a minute. 00:34:20.560 --> 00:34:24.371 The magnitude of this fellow is simply the magnitude 00:34:24.371 --> 00:34:26.364 of this over this magnitude. 00:34:26.364 --> 00:34:29.860 00:34:29.860 --> 00:34:34.449 And we've seen last time this is the magnitude of this vector a, 00:34:34.449 --> 00:34:35.190 right? 00:34:35.190 --> 00:34:35.840 Good. 00:34:35.840 --> 00:34:39.049 Now, so the answer will be n is going to be a unit 00:34:39.049 --> 00:34:41.920 vector, very nice friend of yours, minus cosine t 00:34:41.920 --> 00:34:44.146 minus sine t0. 00:34:44.146 --> 00:34:49.520 Can you draw a conclusion about how I should draw this vector? 00:34:49.520 --> 00:34:51.609 You see the component in k is 0. 00:34:51.609 --> 00:34:55.610 So this vector cannot be like that-- 00:34:55.610 --> 00:34:57.721 cannot be inclined with respect to the horizontal. 00:34:57.721 --> 00:34:58.220 Yes sir. 00:34:58.220 --> 00:35:00.487 STUDENT: So what happens to-- down there-- square root 00:35:00.487 --> 00:35:01.854 of a squared plus b squared? 00:35:01.854 --> 00:35:02.895 PROFESSOR: They simplify. 00:35:02.895 --> 00:35:04.414 This is division. 00:35:04.414 --> 00:35:05.080 STUDENT: Oh, OK. 00:35:05.080 --> 00:35:07.730 PROFESSOR: So this simplifies with that and a simplifies 00:35:07.730 --> 00:35:10.000 with a. 00:35:10.000 --> 00:35:12.090 I should leave some things as an exercise, 00:35:12.090 --> 00:35:15.600 but this is an obvious one so I don't have to explain anything. 00:35:15.600 --> 00:35:18.950 Minus cosine t minus sine t-- if do 00:35:18.950 --> 00:35:22.290 you guys imagine what that is? 00:35:22.290 --> 00:35:25.760 Remember your washer and dryer. 00:35:25.760 --> 00:35:32.500 So if you have an acceleration that's pointing inside 00:35:32.500 --> 00:35:36.170 like from a centrifugal force, the corresponding acceleration 00:35:36.170 --> 00:35:39.480 would go pointing inside, not outside. 00:35:39.480 --> 00:35:43.780 That's going to be exactly minus cosine t minus sine t0. 00:35:43.780 --> 00:35:47.520 So the way I should draw the n would not be just any n, 00:35:47.520 --> 00:35:52.660 but should be at every point a beautiful vector 00:35:52.660 --> 00:35:55.770 that's horizontal and is moving along the helix. 00:35:55.770 --> 00:35:57.840 My elbow is moving along the helix. 00:35:57.840 --> 00:35:58.630 See my elbow? 00:35:58.630 --> 00:35:59.965 Where's my elbow moving? 00:35:59.965 --> 00:36:01.200 I'm trying. 00:36:01.200 --> 00:36:03.070 I swear, I won't do it that way. 00:36:03.070 --> 00:36:06.910 So this is the helix and this is the acceleration, which 00:36:06.910 --> 00:36:12.525 is acceleration and the normal unit vector in this case 00:36:12.525 --> 00:36:13.210 are co-linear. 00:36:13.210 --> 00:36:14.980 They are not co-linear in general. 00:36:14.980 --> 00:36:18.970 But if the speed is a constant, they are co-linear. 00:36:18.970 --> 00:36:20.816 The n and the acceleration. 00:36:20.816 --> 00:36:21.316 Yes, sir? 00:36:21.316 --> 00:36:24.774 STUDENT: How do you know it's pointing in the central axis? 00:36:24.774 --> 00:36:25.645 I thought it was-- 00:36:25.645 --> 00:36:26.686 PROFESSOR: Good question. 00:36:26.686 --> 00:36:27.956 Good question. 00:36:27.956 --> 00:36:28.890 Well, yeah. 00:36:28.890 --> 00:36:29.590 Let's see now. 00:36:29.590 --> 00:36:31.160 Plug in t equals 0. 00:36:31.160 --> 00:36:32.330 What do you have? 00:36:32.330 --> 00:36:35.820 Minus cosine 0 minus 1 0, 0. 00:36:35.820 --> 00:36:40.380 So you guys would have to draw the vector minus 1, 0, 0. 00:36:40.380 --> 00:36:42.310 That's minus i, right? 00:36:42.310 --> 00:36:47.980 So when I start here, this is my n-- from here to here, 00:36:47.980 --> 00:36:50.910 from the particle to the insid. 00:36:50.910 --> 00:36:52.720 So I go on that. 00:36:52.720 --> 00:36:55.100 All right, so this is the normal principal vector. 00:36:55.100 --> 00:36:57.066 I'm very happy about it. 00:36:57.066 --> 00:36:59.820 STUDENT: Isn't the normal principal vector is the-- is it 00:36:59.820 --> 00:37:01.440 the derivative of t, or is just-- 00:37:01.440 --> 00:37:02.815 PROFESSOR: It was by definition-- 00:37:02.815 --> 00:37:05.490 it's in your notes-- t prime over the magnitude of the-- 00:37:05.490 --> 00:37:09.100 STUDENT: So then did you-- why didn't you 00:37:09.100 --> 00:37:10.982 take a derivative of t prime? 00:37:10.982 --> 00:37:11.690 PROFESSOR: I did. 00:37:11.690 --> 00:37:12.606 STUDENT: Yeah, I know. 00:37:12.606 --> 00:37:15.880 I see you took a derivative of t of-- 00:37:15.880 --> 00:37:19.172 PROFESSOR: This is t prime. 00:37:19.172 --> 00:37:20.380 STUDENT: OK. 00:37:20.380 --> 00:37:23.570 PROFESSOR: And this is magnitude of t prime. 00:37:23.570 --> 00:37:26.360 Why don't you try this at home, like, 00:37:26.360 --> 00:37:30.020 slowly until you're sure this is what yo got? 00:37:30.020 --> 00:37:32.410 So I did-- I did the derivative of i. 00:37:32.410 --> 00:37:33.880 STUDENT: I saw that. 00:37:33.880 --> 00:37:36.060 PROFESSOR: This is a [INAUDIBLE]. 00:37:36.060 --> 00:37:37.950 STUDENT: You said you were-- 00:37:37.950 --> 00:37:40.340 PROFESSOR: So when we have t times a function 00:37:40.340 --> 00:37:43.460 and we prime the product, k goes out. 00:37:43.460 --> 00:37:45.380 Lucky for us-- imagine how life would 00:37:45.380 --> 00:37:46.970 be if it weren't like that. 00:37:46.970 --> 00:37:49.180 So the constant that falls out is 00:37:49.180 --> 00:37:51.890 1 over square root of what I derived. 00:37:51.890 --> 00:37:56.000 And then I have to derive this whole function also. 00:37:56.000 --> 00:37:59.050 So I would suggest to everybody, not just to yo-- 00:37:59.050 --> 00:38:01.751 go home and see if you can redo this 00:38:01.751 --> 00:38:03.000 without looking in your notes. 00:38:03.000 --> 00:38:05.100 Close the damn notes. 00:38:05.100 --> 00:38:08.780 Open and then you look at-- it's line by line, line by line 00:38:08.780 --> 00:38:10.620 all the derivations. 00:38:10.620 --> 00:38:13.970 Because you guys will have to do that yourselves in the exam, 00:38:13.970 --> 00:38:17.468 either midterm or final anyway. 00:38:17.468 --> 00:38:23.710 Reparameterizing arc lengths to obtain a curve-- I 00:38:23.710 --> 00:38:25.930 still have that to finish the problem. 00:38:25.930 --> 00:38:31.660 Reparameterizing arc length to obtain a curve rho of s. 00:38:31.660 --> 00:38:33.100 How do we do that? 00:38:33.100 --> 00:38:34.070 Who is s? 00:38:34.070 --> 00:38:37.300 First of all, you should start with the s and then 00:38:37.300 --> 00:38:38.660 reparameterize. 00:38:38.660 --> 00:38:39.820 So you say, hey, teacher. 00:38:39.820 --> 00:38:42.050 You try to fool me, right? 00:38:42.050 --> 00:38:45.590 I want s to be grabbed as a parameter first. 00:38:45.590 --> 00:38:49.700 And then I will reparameterize the way you want me to do that. 00:38:49.700 --> 00:38:52.990 So s of t will be integral from 0 00:38:52.990 --> 00:38:56.390 to t square root of a prime a squared times 00:38:56.390 --> 00:38:59.825 b squared b tau-- d tau, yes. 00:38:59.825 --> 00:39:01.205 S of t will be, what? 00:39:01.205 --> 00:39:02.740 Who's helping me on that? 00:39:02.740 --> 00:39:05.040 Because I want you to be awake. 00:39:05.040 --> 00:39:05.985 Are you guys awake? 00:39:05.985 --> 00:39:07.334 [CLASS MURMURS] 00:39:07.334 --> 00:39:08.750 PROFESSOR: The square root of that 00:39:08.750 --> 00:39:14.190 is a constant gets out times t. 00:39:14.190 --> 00:39:18.935 So what did I tell you when it comes to these functions? 00:39:18.935 --> 00:39:21.600 I have to turn my head badly like that. 00:39:21.600 --> 00:39:23.960 This was the alpha t or s of t. 00:39:23.960 --> 00:39:31.685 And this was t of s, which is the inverse function. 00:39:31.685 --> 00:39:33.310 I'm not going to write anything stupid. 00:39:33.310 --> 00:39:36.750 But this is practically the inverse function of s of t. 00:39:36.750 --> 00:39:39.190 I told you it was easiest t do. 00:39:39.190 --> 00:39:40.370 Put it here. 00:39:40.370 --> 00:39:43.480 T has to be replaced by, in terms of s, 00:39:43.480 --> 00:39:45.930 by a certain expression. 00:39:45.930 --> 00:39:47.924 So who is t? 00:39:47.924 --> 00:39:51.860 And you will do that in no time in the exam. 00:39:51.860 --> 00:39:55.910 T pulled out from there will be just 00:39:55.910 --> 00:40:00.720 s over square root a squared plus b squared 00:40:00.720 --> 00:40:04.560 s over square root a squared plus b squared 00:40:04.560 --> 00:40:08.668 and s over square root. 00:40:08.668 --> 00:40:10.600 OK? 00:40:10.600 --> 00:40:13.420 So can I keep the notation out of s? 00:40:13.420 --> 00:40:14.590 No. 00:40:14.590 --> 00:40:18.850 It's not mathematically correct to keep r of s. 00:40:18.850 --> 00:40:21.460 Why do the books sometimes by using 00:40:21.460 --> 00:40:23.040 multiplication keep r of s? 00:40:23.040 --> 00:40:27.390 Because the books are not always rigorous. 00:40:27.390 --> 00:40:28.925 But I'm trying to be rigorous. 00:40:28.925 --> 00:40:30.960 This is an honors class. 00:40:30.960 --> 00:40:34.940 So How do I rewrite the whole thing? 00:40:34.940 --> 00:40:39.961 r of t, who is a function of s, t as a function of s 00:40:39.961 --> 00:40:45.770 was again s over square root a squared plus b squared 00:40:45.770 --> 00:40:49.250 will be renamed rho of s. 00:40:49.250 --> 00:40:51.030 And what is that? 00:40:51.030 --> 00:40:54.810 That is a of cosine of parentheses 00:40:54.810 --> 00:41:00.570 s over square root a squared r b squared, comma, 00:41:00.570 --> 00:41:06.240 a sine of s over square root a squared plus b squared 00:41:06.240 --> 00:41:12.894 and b times s over square root a squared plus b squared. 00:41:12.894 --> 00:41:14.510 So what have I done? 00:41:14.510 --> 00:41:16.221 Did I get my 20%? 00:41:16.221 --> 00:41:16.720 Yes. 00:41:16.720 --> 00:41:17.230 Why? 00:41:17.230 --> 00:41:19.480 Because I reparameterized the curve. 00:41:19.480 --> 00:41:21.920 Did I get my other 20%? 00:41:21.920 --> 00:41:25.680 Yes, because I told people who s of t was. 00:41:25.680 --> 00:41:32.980 So 20% for this box and 20% for this expression. 00:41:32.980 --> 00:41:34.600 So what have I done? 00:41:34.600 --> 00:41:39.100 On the same physical curve, I have slowed down, thank God. 00:41:39.100 --> 00:41:41.860 You say, finally, she's slowing down, right? 00:41:41.860 --> 00:41:42.860 I've changed this speed. 00:41:42.860 --> 00:41:46.270 00:41:46.270 --> 00:41:51.130 On the contrary, if a would be 4 and be would be 3, 00:41:51.130 --> 00:41:56.170 I increase my speed multiple five times, right? 00:41:56.170 --> 00:41:59.500 So you can go back and forth between s and t. 00:41:59.500 --> 00:42:02.820 What does s do compared to t? 00:42:02.820 --> 00:42:04.601 It increases the speed five times. 00:42:04.601 --> 00:42:05.100 Yes sir. 00:42:05.100 --> 00:42:06.992 STUDENT: So when you reparameterize, 00:42:06.992 --> 00:42:08.825 it's just basically the integral from 0 to t 00:42:08.825 --> 00:42:11.850 of whatever [INAUDIBLE] of tau is. 00:42:11.850 --> 00:42:14.210 PROFESSOR: Exactly. 00:42:14.210 --> 00:42:18.680 So my suggestion to all of you-- it took me a year 00:42:18.680 --> 00:42:21.170 to understand how to reparameterize 00:42:21.170 --> 00:42:24.950 because I was not smart enough to get it as a freshman. 00:42:24.950 --> 00:42:26.380 I got an A in that class. 00:42:26.380 --> 00:42:28.390 I didn't understand anything. 00:42:28.390 --> 00:42:31.808 As a sophomore, I really-- because sometimes, you know, 00:42:31.808 --> 00:42:36.180 you can get an A without understanding things in there. 00:42:36.180 --> 00:42:38.607 As a sophomore, I said, OK, what the heck 00:42:38.607 --> 00:42:39.860 was that reparameterization? 00:42:39.860 --> 00:42:42.770 I have to understand that because it bothers me. 00:42:42.770 --> 00:42:43.340 I went back. 00:42:43.340 --> 00:42:45.220 I took the book. 00:42:45.220 --> 00:42:48.080 I learned about reparameterization. 00:42:48.080 --> 00:42:50.540 Our book, I think, does a very good job 00:42:50.540 --> 00:42:52.070 when it comes to reparameterizing. 00:42:52.070 --> 00:42:57.540 So if you open the 10.2 and 10.4, you have to skip-- well, 00:42:57.540 --> 00:42:59.860 am I telling you to skip 10.3? 00:42:59.860 --> 00:43:01.100 That's about ballistics. 00:43:01.100 --> 00:43:03.970 If you're interested in dancing and all sorts of, 00:43:03.970 --> 00:43:08.440 like, how the bullet will be projected 00:43:08.440 --> 00:43:11.346 in this motion or that motion, you can learn that. 00:43:11.346 --> 00:43:14.084 Those are plane curves that are interested in physics 00:43:14.084 --> 00:43:14.750 and mathematics. 00:43:14.750 --> 00:43:18.912 But 10.3 is not part of them and they are required. 00:43:18.912 --> 00:43:20.317 Read 10.2 and 10.4. 00:43:20.317 --> 00:43:21.650 You understand this much better. 00:43:21.650 --> 00:43:22.683 Yes, ma'am. 00:43:22.683 --> 00:43:24.891 STUDENT: Will the midterm or the final just be, like, 00:43:24.891 --> 00:43:26.682 a series problems, or will it be anything-- 00:43:26.682 --> 00:43:29.730 PROFESSOR: This is going to be like that-- 15 problems 00:43:29.730 --> 00:43:30.255 like that. 00:43:30.255 --> 00:43:31.963 STUDENT: Will it be anything, like, super 00:43:31.963 --> 00:43:33.214 in depth like the extra credit? 00:43:33.214 --> 00:43:35.090 PROFESSOR: That-- isn't that in-depth enough? 00:43:35.090 --> 00:43:37.390 OK, this is going to be like that. 00:43:37.390 --> 00:43:40.610 So I would say at this point, the way I feel, 00:43:40.610 --> 00:43:45.490 I feel that I am ready to put extra credit there. 00:43:45.490 --> 00:43:48.890 My policy is that I read everything. 00:43:48.890 --> 00:43:52.960 So even if at this point, you say extra credit. 00:43:52.960 --> 00:43:54.900 And you put it at the end for me. 00:43:54.900 --> 00:43:57.120 Say, look, I'm doing the extra credit here. 00:43:57.120 --> 00:44:00.350 Then I'll be ready and I'll say, OK, what did she mean? 00:44:00.350 --> 00:44:01.920 Length of the arc? 00:44:01.920 --> 00:44:02.420 Which arc? 00:44:02.420 --> 00:44:05.620 From here to here is ready to be computed. 00:44:05.620 --> 00:44:08.430 00:44:08.430 --> 00:44:11.290 And that's going to be-- you can include your extra credit 00:44:11.290 --> 00:44:13.340 inside the actual problem. 00:44:13.340 --> 00:44:14.259 I see it. 00:44:14.259 --> 00:44:14.800 STUDENT: Yes. 00:44:14.800 --> 00:44:15.670 PROFESSOR: Don't worry. 00:44:15.670 --> 00:44:17.336 STUDENT: Would it just be as like-- just 00:44:17.336 --> 00:44:19.720 like the casual problem on the test or midterm 00:44:19.720 --> 00:44:22.534 or whatever-- would it be, like, an extra credit 00:44:22.534 --> 00:44:23.478 problem in itself? 00:44:23.478 --> 00:44:24.894 I know there will be extra credit, 00:44:24.894 --> 00:44:26.310 but the kind of proving-- 00:44:26.310 --> 00:44:29.620 PROFESSOR: That is-- that is decided together 00:44:29.620 --> 00:44:32.830 with the course coordinator. 00:44:32.830 --> 00:44:35.100 The course coordinator himself said 00:44:35.100 --> 00:44:39.940 that he is encouraging us to set up the scale so that if you 00:44:39.940 --> 00:44:43.150 all the problems that are written on the exam, 00:44:43.150 --> 00:44:46.680 you get something like 120% if everything is perfect. 00:44:46.680 --> 00:44:47.860 STUDENT: OK, if we can-- 00:44:47.860 --> 00:44:49.738 PROFESSOR: So it's sort of in-built in that-- yes. 00:44:49.738 --> 00:44:51.220 STUDENT: If we can do the web work, 00:44:51.220 --> 00:44:53.200 is that a good indication of-- 00:44:53.200 --> 00:44:54.090 PROFESSOR: Wonderful. 00:44:54.090 --> 00:44:56.160 That's exactly-- because the way we 00:44:56.160 --> 00:44:58.290 write those problems for the final, 00:44:58.290 --> 00:45:01.830 we pull them out of the web work problems we do for homework. 00:45:01.830 --> 00:45:03.670 So a square root of a squared times 00:45:03.670 --> 00:45:08.195 b squared times pi over 2-- so what have I discovered? 00:45:08.195 --> 00:45:11.480 If I would take a piece of that paper 00:45:11.480 --> 00:45:13.760 and I would measure from this point to this point 00:45:13.760 --> 00:45:17.600 how much I traveled in inches from here to here, 00:45:17.600 --> 00:45:21.380 that's exactly that square root of- this would be like a 5. 00:45:21.380 --> 00:45:24.580 That's 3.1415 divided by 2. 00:45:24.580 --> 00:45:25.290 Yes, sir. 00:45:25.290 --> 00:45:29.340 STUDENT: So in the interval of a squared plus 00:45:29.340 --> 00:45:31.292 b squared, I know that that's supposed 00:45:31.292 --> 00:45:33.770 to be the interval the magnitude of r-- 00:45:33.770 --> 00:45:35.855 PROFESSOR: The speed-- integral of speed? 00:45:35.855 --> 00:45:36.480 STUDENT: Right. 00:45:36.480 --> 00:45:39.397 So which is the r prime, right? 00:45:39.397 --> 00:45:40.230 PROFESSOR: Yes, sir. 00:45:40.230 --> 00:45:43.667 STUDENT: OK, so r prime was-- 00:45:43.667 --> 00:45:44.500 PROFESSOR: Velocity. 00:45:44.500 --> 00:45:46.780 STUDENT: --a sine-- or negative a sine t, 00:45:46.780 --> 00:45:49.380 a cosine t, and then b? 00:45:49.380 --> 00:45:51.855 So where did the square root of a squared plus b squared 00:45:51.855 --> 00:45:53.487 come from? 00:45:53.487 --> 00:45:54.570 STUDENT: That's from the-- 00:45:54.570 --> 00:45:57.290 PROFESSOR: I just erased it. 00:45:57.290 --> 00:46:02.630 OK, so you have minus i-- minus a sine b equals sine p and d. 00:46:02.630 --> 00:46:04.970 When you squared them, what did you get? 00:46:04.970 --> 00:46:05.938 He has the same thing. 00:46:05.938 --> 00:46:06.979 STUDENT: So that's just-- 00:46:06.979 --> 00:46:08.831 PROFESSOR: The square of that plus the square root of that 00:46:08.831 --> 00:46:10.039 plus the square root of that. 00:46:10.039 --> 00:46:14.717 STUDENT: So it's just like a 2D representation of the top one. 00:46:14.717 --> 00:46:15.550 STUDENT: This side-- 00:46:15.550 --> 00:46:18.980 00:46:18.980 --> 00:46:21.365 PROFESSOR: I just need the magnitude of r prime, which 00:46:21.365 --> 00:46:22.880 is this p, right? 00:46:22.880 --> 00:46:23.657 STUDENT: Right. 00:46:23.657 --> 00:46:25.115 PROFESSOR: The magnitude of this is 00:46:25.115 --> 00:46:29.214 the speed, which is square root of a squared plus b squared. 00:46:29.214 --> 00:46:30.620 Is that clear? 00:46:30.620 --> 00:46:31.230 STUDENT: Yes. 00:46:31.230 --> 00:46:32.870 PROFESSOR: I can go on if you want. 00:46:32.870 --> 00:46:36.835 So a square root of-- the sum of the squares of this, this, 00:46:36.835 --> 00:46:40.510 and that is exactly square of [INAUDIBLE]. 00:46:40.510 --> 00:46:41.850 Keep this in mind as an example. 00:46:41.850 --> 00:46:44.710 It's an extremely important one. 00:46:44.710 --> 00:46:48.710 It appears very frequently in tests-- on tests. 00:46:48.710 --> 00:46:52.560 And it's one of the most beautiful examples 00:46:52.560 --> 00:46:57.900 in applications of mathematics to physics. 00:46:57.900 --> 00:47:02.740 I have something else that was there. 00:47:02.740 --> 00:47:03.570 Yes ma'am 00:47:03.570 --> 00:47:07.866 STUDENT: I was just going to ask if you want to curvature. 00:47:07.866 --> 00:47:08.480 PROFESSOR: Eh? 00:47:08.480 --> 00:47:09.265 STUDENT: The letter-- 00:47:09.265 --> 00:47:10.146 PROFESSOR: Curvature? 00:47:10.146 --> 00:47:11.020 STUDENT: Curvature. 00:47:11.020 --> 00:47:12.603 PROFESSOR: That's exactly what I want. 00:47:12.603 --> 00:47:17.970 And when I said I had something else for 20%, what was k? 00:47:17.970 --> 00:47:22.672 K was rho double prime of s in magnitude. 00:47:22.672 --> 00:47:29.784 So I have to be smart enough to look at that and rho of s. 00:47:29.784 --> 00:47:32.742 And rho of s was the thing that had 00:47:32.742 --> 00:47:35.693 here-- that's going to be probably the end of my lesson 00:47:35.693 --> 00:47:36.193 today. 00:47:36.193 --> 00:47:39.650 00:47:39.650 --> 00:47:46.420 Since you have so many questions, I will continue. 00:47:46.420 --> 00:47:50.360 I should consider-- the chapter is finished 00:47:50.360 --> 00:47:54.030 but I will continue with a deeper review, how about that, 00:47:54.030 --> 00:47:57.300 on Tuesday with more problems. 00:47:57.300 --> 00:48:00.890 Because I have the feeling that although we covered 10.1, 10.2, 00:48:00.890 --> 00:48:03.640 10.4, you need a lot more examples 00:48:03.640 --> 00:48:05.790 until you feel comfortable. 00:48:05.790 --> 00:48:08.350 Many of you not, maybe 10 people. 00:48:08.350 --> 00:48:09.880 They feel very comfortable. 00:48:09.880 --> 00:48:10.450 They get it. 00:48:10.450 --> 00:48:13.330 But I think nobody will be hurt by more review and more 00:48:13.330 --> 00:48:16.096 examples and more applications. 00:48:16.096 --> 00:48:20.565 Now, who can help me finish my goal for today? 00:48:20.565 --> 00:48:22.640 Is this hard? 00:48:22.640 --> 00:48:25.580 This is rho of s. 00:48:25.580 --> 00:48:30.360 So you have to tell me with the derivation, is it hard? 00:48:30.360 --> 00:48:31.330 No. 00:48:31.330 --> 00:48:37.530 Minus a sine of the whole thing times 1 00:48:37.530 --> 00:48:40.551 over square root of a squared plus b squared because I'm 00:48:40.551 --> 00:48:42.280 applying the chain rule, right? 00:48:42.280 --> 00:48:43.900 Let me change color. 00:48:43.900 --> 00:48:45.070 Who's the next guy? 00:48:45.070 --> 00:48:50.135 A Cosine of s over square root a squared plus b squared. 00:48:50.135 --> 00:48:53.040 I'm now going to leave you this as an exercise 00:48:53.040 --> 00:48:56.250 because you're going to haunt me back ask me why I got this. 00:48:56.250 --> 00:48:58.810 So I want to make it very clear. 00:48:58.810 --> 00:49:03.940 B times 1 over square root a squared by b squared. 00:49:03.940 --> 00:49:06.380 So are we happy with this? 00:49:06.380 --> 00:49:07.460 Is this understood? 00:49:07.460 --> 00:49:10.980 It's a simple derivation of the philosophy. 00:49:10.980 --> 00:49:12.530 We are not done. 00:49:12.530 --> 00:49:15.300 We have to do the acceleration. 00:49:15.300 --> 00:49:18.400 So the acceleration with respect to s 00:49:18.400 --> 00:49:21.610 of this curve where s was the arc length parameter 00:49:21.610 --> 00:49:24.300 is real easy to compute in the same way. 00:49:24.300 --> 00:49:26.140 What is different? 00:49:26.140 --> 00:49:29.880 I'm not going to write more explicitly 00:49:29.880 --> 00:49:32.360 because this should be visible for everybody. 00:49:32.360 --> 00:49:35.450 STUDENT: x [INAUDIBLE]. 00:49:35.450 --> 00:49:38.850 PROFESSOR: Good, minus a over-- I'll 00:49:38.850 --> 00:49:41.640 wait for you to simplify because I don't 00:49:41.640 --> 00:49:43.280 want to pull two roots out. 00:49:43.280 --> 00:49:44.160 STUDENT: A squared-- 00:49:44.160 --> 00:49:45.659 PROFESSOR: A squared plus b squared. 00:49:45.659 --> 00:49:47.215 And why is that, [INAUDIBLE]? 00:49:47.215 --> 00:49:52.180 Because you have once and twice from the chain rule. 00:49:52.180 --> 00:49:55.590 So again, I hope you guys don't have a problem with the chain 00:49:55.590 --> 00:50:00.750 rule so I don't have to send you back to Calculus 1. 00:50:00.750 --> 00:50:05.630 A over a squared times b squared with a minus-- why 00:50:05.630 --> 00:50:06.280 with a minus? 00:50:06.280 --> 00:50:07.040 Somebody explain. 00:50:07.040 --> 00:50:09.190 STUDENT: Use the derivative of cosine. 00:50:09.190 --> 00:50:13.360 PROFESSOR: There's a cosine and there's a minus sine. 00:50:13.360 --> 00:50:15.844 From deriving, I have a minus and a sine. 00:50:15.844 --> 00:50:21.510 00:50:21.510 --> 00:50:25.470 And finally, thank God, the 0-- why 0? 00:50:25.470 --> 00:50:31.310 Because I have a constant that I'm deriving with respect to s. 00:50:31.310 --> 00:50:33.050 Is it hard to see what's up? 00:50:33.050 --> 00:50:36.110 What's going out? 00:50:36.110 --> 00:50:40.814 What is the curvature of the helix? 00:50:40.814 --> 00:50:45.230 A beautiful, beautiful function that 00:50:45.230 --> 00:50:48.560 is known in most of these math, calculus, 00:50:48.560 --> 00:50:54.070 multivariable calculus and differential geometry classes. 00:50:54.070 --> 00:50:56.640 What did you get? 00:50:56.640 --> 00:51:02.755 Square root of sum of the squares of all these guys. 00:51:02.755 --> 00:51:04.115 You process it. 00:51:04.115 --> 00:51:06.440 That's very easy. 00:51:06.440 --> 00:51:07.395 Shall I write it down? 00:51:07.395 --> 00:51:09.790 Let me write it down like a silly girl-- 00:51:09.790 --> 00:51:13.500 square root of a squared, although I hate when I cannot 00:51:13.500 --> 00:51:15.190 go ahead and simplify it. 00:51:15.190 --> 00:51:18.950 But let's say there's this little baby thing. 00:51:18.950 --> 00:51:21.550 00:51:21.550 --> 00:51:25.070 Now I can say it's a over a squared 00:51:25.070 --> 00:51:26.750 plus b squared-- finally. 00:51:26.750 --> 00:51:29.106 So I'm going to ask you a few questions 00:51:29.106 --> 00:51:30.480 and then I'm going to let you go. 00:51:30.480 --> 00:51:32.840 It's a punishment for one minute. 00:51:32.840 --> 00:51:36.580 OK, if I have the curve we had before, 00:51:36.580 --> 00:51:41.590 the beautiful helix with a Pythagorean number 00:51:41.590 --> 00:51:45.440 like 3 cosine t, 3 sine t, and 4t, what 00:51:45.440 --> 00:51:48.498 is the curvature of that helix? 00:51:48.498 --> 00:51:49.800 STUDENT: 3 over 5-- 00:51:49.800 --> 00:51:52.246 PROFESSOR: 3 over 5, excellent. 00:51:52.246 --> 00:51:53.730 How about my helix? 00:51:53.730 --> 00:51:57.540 What if I changed the numbers in web work or on the midterm 00:51:57.540 --> 00:52:00.600 and I say it's going to be-- it could even 00:52:00.600 --> 00:52:02.150 be with a minus, guys. 00:52:02.150 --> 00:52:05.130 It's just the way you travel it would be different. 00:52:05.130 --> 00:52:08.030 So whether I put plus minus here, 00:52:08.030 --> 00:52:09.950 you will try on different examples. 00:52:09.950 --> 00:52:13.150 Sometimes if we put minus here or minus here, 00:52:13.150 --> 00:52:15.380 it really doesn't matter. 00:52:15.380 --> 00:52:17.880 Let's say we have cosine t sine t and t. 00:52:17.880 --> 00:52:22.044 What's the curvature of that parametrized curve? 00:52:22.044 --> 00:52:23.010 1 over-- 00:52:23.010 --> 00:52:23.976 STUDENT: 2. 00:52:23.976 --> 00:52:26.620 PROFESSOR: 1 over 2-- excellent. 00:52:26.620 --> 00:52:27.380 So you got it. 00:52:27.380 --> 00:52:28.740 So I'm proud of you. 00:52:28.740 --> 00:52:32.396 Now, I want to do more examples until you feel confident 00:52:32.396 --> 00:52:32.895 about it. 00:52:32.895 --> 00:52:36.590 I know I got most of you to the point where I want it. 00:52:36.590 --> 00:52:38.690 But you need more reading definitely 00:52:38.690 --> 00:52:41.360 and you need to see more examples. 00:52:41.360 --> 00:52:43.020 Feel free to read the whole chapter. 00:52:43.020 --> 00:52:48.340 I would-- if you don't have time for 10.3, skip it. 00:52:48.340 --> 00:52:50.140 10.5 is not going to be required. 00:52:50.140 --> 00:52:53.090 So if I were a student, I'd go home, open the book, 00:52:53.090 --> 00:52:55.860 read 10.1, 10.2, 10.4, close the book. 00:52:55.860 --> 00:52:59.636 It's actually a lot less than you think it is. 00:52:59.636 --> 00:53:01.510 If you go over the most important formulas, 00:53:01.510 --> 00:53:04.230 then you are ready for the homework. 00:53:04.230 --> 00:53:06.470 The second homework is due when? 00:53:06.470 --> 00:53:08.090 February 11. 00:53:08.090 --> 00:53:10.060 You guys have plenty of time. 00:53:10.060 --> 00:53:14.260 Rather than going to the tutors, ask me for Tuesday. 00:53:14.260 --> 00:53:17.420 On Tuesday, you'll have plenty of time for applications. 00:53:17.420 --> 00:53:20.030 OK, have a wonderful weekend. 00:53:20.030 --> 00:53:23.320 Don't forget to email when you get in trouble, OK? 00:53:23.320 --> 00:53:27.696