WEBVTT 00:00:00.000 --> 00:00:01.894 00:00:01.894 --> 00:00:03.810 MAGDALENA TODA: So what's your general feeling 00:00:03.810 --> 00:00:05.423 about Chapter 11? 00:00:05.423 --> 00:00:06.355 STUDENT: It's OK. 00:00:06.355 --> 00:00:07.355 MAGDALENA TODA: It's OK. 00:00:07.355 --> 00:00:12.185 So functions of two variables are 00:00:12.185 --> 00:00:15.566 to be compared all the time with the functions of one variable. 00:00:15.566 --> 00:00:18.810 Every nothing you have seen in Calc 1 00:00:18.810 --> 00:00:23.983 has a corresponding the motion in Calc 3. 00:00:23.983 --> 00:00:27.780 00:00:27.780 --> 00:00:33.020 So really no questions about theory, concepts, Chapter 11 00:00:33.020 --> 00:00:36.880 concepts, previous concepts? 00:00:36.880 --> 00:00:39.760 Feel free to email me this weekend. 00:00:39.760 --> 00:00:43.370 Don't think it's the weekend because we 00:00:43.370 --> 00:00:47.140 are on a 24/7 availability. 00:00:47.140 --> 00:00:48.970 People, we use WeBWork. 00:00:48.970 --> 00:00:50.520 Not just me, but everybody who uses 00:00:50.520 --> 00:00:54.310 WeBWork is on a 24/7 availability, 00:00:54.310 --> 00:00:59.130 answering questions about WeBWork problems. 00:00:59.130 --> 00:01:02.320 Saturday and Sunday is when most of you do the homework. 00:01:02.320 --> 00:01:05.010 00:01:05.010 --> 00:01:09.462 It's convenient for us as well because we are with the family, 00:01:09.462 --> 00:01:11.797 but we don't have many meetings to attend. 00:01:11.797 --> 00:01:14.810 So I'll be happy to answer your questions. 00:01:14.810 --> 00:01:19.384 Last time, we discussed a little bit about preparation 00:01:19.384 --> 00:01:21.598 for The Chain Rule. 00:01:21.598 --> 00:01:23.566 In Calc 3. 00:01:23.566 --> 00:01:34.880 So the chain rule in Calc 3 was something really-- 00:01:34.880 --> 00:01:40.126 this is section 11.5. 00:01:40.126 --> 00:01:45.549 The preparation was done last time, 00:01:45.549 --> 00:01:50.479 but I'm going to review it a little bit. 00:01:50.479 --> 00:01:54.423 Let's see what we discussed. 00:01:54.423 --> 00:01:57.381 I'm going to split, again, the board in two. 00:01:57.381 --> 00:02:05.842 And I'll say, can we review the notions of The Chain Rule. 00:02:05.842 --> 00:02:11.360 When you start with a variable-- let's say it's time. 00:02:11.360 --> 00:02:20.795 Time going to f of t, which goes into g of f of t by something 00:02:20.795 --> 00:02:22.340 called composition. 00:02:22.340 --> 00:02:26.284 We've done that since we were kids in college algebra. 00:02:26.284 --> 00:02:27.170 What? 00:02:27.170 --> 00:02:28.680 You never took college algebra? 00:02:28.680 --> 00:02:30.890 Except in high school, you took high school algebra, 00:02:30.890 --> 00:02:33.210 most of you. 00:02:33.210 --> 00:02:35.324 So what did you do in high school algebra? 00:02:35.324 --> 00:02:38.280 We said g composed with l. 00:02:38.280 --> 00:02:40.760 This is a composition of two functions. 00:02:40.760 --> 00:02:46.360 What I'm skipping here is the theory that you learned then 00:02:46.360 --> 00:02:56.250 that to a compose well, F of t has to be in the domain of g. 00:02:56.250 --> 00:02:59.340 So the image F of t, whatever you get from this image, 00:02:59.340 --> 00:03:01.620 has to be in the domain of g. 00:03:01.620 --> 00:03:05.100 Otherwise, the composition could not exist. 00:03:05.100 --> 00:03:07.870 Now if you have differentiability, 00:03:07.870 --> 00:03:10.390 assuming that this is g composed with F, 00:03:10.390 --> 00:03:16.332 assuming to be c1-- c1 meaning differentiable 00:03:16.332 --> 00:03:24.470 and derivatives are continuous-- assuming both of them are c1, 00:03:24.470 --> 00:03:26.320 they compose well. 00:03:26.320 --> 00:03:28.260 What am I going to do next? 00:03:28.260 --> 00:03:35.660 I'm going to say the d, dt g of F of t. 00:03:35.660 --> 00:03:39.600 And we said last time, we get The Chain Rule 00:03:39.600 --> 00:03:44.463 from the last function we applied, g prime. 00:03:44.463 --> 00:03:50.860 And so you have dg, [? d2 ?] at F of t. 00:03:50.860 --> 00:03:57.590 I'm calling this guy u variable just for my own enjoyment. 00:03:57.590 --> 00:04:01.660 And then I go du, dt. 00:04:01.660 --> 00:04:06.020 But du, dt would be nothing but a prime of t, 00:04:06.020 --> 00:04:09.330 so remember the cowboys shooting at each other? 00:04:09.330 --> 00:04:11.060 The du and du. 00:04:11.060 --> 00:04:16.120 I will replace the u by prime of t, just like you did in Calc 1. 00:04:16.120 --> 00:04:16.620 Why? 00:04:16.620 --> 00:04:23.320 Because I want to a mixture of notations according to Calc 1 00:04:23.320 --> 00:04:24.871 you took here. 00:04:24.871 --> 00:04:30.580 The idea for Calc 3 is the same with [INAUDIBLE] time, 00:04:30.580 --> 00:04:33.370 assuming everything composes well, 00:04:33.370 --> 00:04:37.910 and has differentiability, and the derivatives are continuous. 00:04:37.910 --> 00:04:39.800 Just to make your life easier. 00:04:39.800 --> 00:04:43.700 We have x of t, y of t. 00:04:43.700 --> 00:04:47.175 Two nice functions and a function 00:04:47.175 --> 00:04:54.690 of these variables, F of x and y. 00:04:54.690 --> 00:04:57.710 So I'm going to have to say, how about x 00:04:57.710 --> 00:05:01.330 is a function of t and y is a function of t? 00:05:01.330 --> 00:05:05.290 So I should be able to go ahead and differentiate 00:05:05.290 --> 00:05:07.620 with respect to the t. 00:05:07.620 --> 00:05:10.850 00:05:10.850 --> 00:05:13.354 And how did it go? 00:05:13.354 --> 00:05:14.770 Now that I prepared you last time, 00:05:14.770 --> 00:05:21.370 a little bit, for this kind of new picture, new diagram, 00:05:21.370 --> 00:05:27.365 you should be able to tell me, without looking at the notes 00:05:27.365 --> 00:05:31.530 from last time, how this goes. 00:05:31.530 --> 00:05:36.730 So I'll take the function F of x of t, y of t. 00:05:36.730 --> 00:05:41.721 And when I view it like that, I understand it's ultimately 00:05:41.721 --> 00:05:45.380 a big function, F of t. 00:05:45.380 --> 00:05:49.610 It's a real valued function of t, 00:05:49.610 --> 00:05:52.780 ultimately, as the composition. 00:05:52.780 --> 00:05:56.257 This big F. 00:05:56.257 --> 00:05:58.370 00:05:58.370 --> 00:06:05.290 So does anybody remember how this went? 00:06:05.290 --> 00:06:07.750 Let's see. 00:06:07.750 --> 00:06:10.210 The derivative, with respect to t, 00:06:10.210 --> 00:06:14.965 of this whole thing, F of x of t, y of t? 00:06:14.965 --> 00:06:19.070 00:06:19.070 --> 00:06:19.570 Thoughts? 00:06:19.570 --> 00:06:23.160 00:06:23.160 --> 00:06:25.270 The partial derivative of F with respect 00:06:25.270 --> 00:06:32.704 to x, evaluated at x of t and y to t. 00:06:32.704 --> 00:06:35.620 So everything has to be replaced in terms of t 00:06:35.620 --> 00:06:38.536 because it's going to be y. 00:06:38.536 --> 00:06:43.750 We assume that this derivative exists and it's continuous. 00:06:43.750 --> 00:06:44.270 Why? 00:06:44.270 --> 00:06:46.365 Just to make your life a little bit easier. 00:06:46.365 --> 00:06:49.550 00:06:49.550 --> 00:06:53.990 From the beginning, we had dx, dt, 00:06:53.990 --> 00:06:57.300 which was also defined everywhere 00:06:57.300 --> 00:07:09.000 and continuous, plus df, 2y at the same point times dy, dt. 00:07:09.000 --> 00:07:12.310 00:07:12.310 --> 00:07:20.750 Notice what happens here with these guys looking diagonally, 00:07:20.750 --> 00:07:21.745 staring at each other. 00:07:21.745 --> 00:07:24.810 00:07:24.810 --> 00:07:26.940 Keep in mind the plus sign. 00:07:26.940 --> 00:07:30.670 And of course, some of you told me, well, is that OK? 00:07:30.670 --> 00:07:31.960 You know favorite, right? 00:07:31.960 --> 00:07:35.450 F of x at x of dy of t. 00:07:35.450 --> 00:07:37.120 That's fine. 00:07:37.120 --> 00:07:38.500 I saw that. 00:07:38.500 --> 00:07:39.780 In engineering you use it. 00:07:39.780 --> 00:07:53.358 Physics majors also use a lot of this notation 00:07:53.358 --> 00:07:57.250 as sub [INAUDIBLE] Fs of t. 00:07:57.250 --> 00:07:58.700 We've seen that. 00:07:58.700 --> 00:07:59.600 We've seen that. 00:07:59.600 --> 00:08:03.150 It comes as no surprise to us, but we 00:08:03.150 --> 00:08:06.879 would like to see if there are any other cases we 00:08:06.879 --> 00:08:07.670 should worry about. 00:08:07.670 --> 00:08:18.062 00:08:18.062 --> 00:08:22.710 Now I don't want to jump to the next example 00:08:22.710 --> 00:08:25.840 until I give you something that you 00:08:25.840 --> 00:08:32.110 know very well from Calculus 1. 00:08:32.110 --> 00:08:38.970 It's an example that you saw before that was a melting ice 00:08:38.970 --> 00:08:41.682 sphere. 00:08:41.682 --> 00:08:46.995 It appears a lot in problems, like final exam problems 00:08:46.995 --> 00:08:48.450 and stuff. 00:08:48.450 --> 00:08:53.120 What is the material of this ball? 00:08:53.120 --> 00:08:54.925 It's melting ice. 00:08:54.925 --> 00:08:59.118 00:08:59.118 --> 00:09:08.470 And if you remember, it says that at the moment t0, 00:09:08.470 --> 00:09:13.820 assume the radius was 5 inches. 00:09:13.820 --> 00:09:16.706 00:09:16.706 --> 00:09:42.508 We also know that the rate of change of the radius in time 00:09:42.508 --> 00:09:49.490 will be minus 5. 00:09:49.490 --> 00:09:54.920 But let's suppose that we say that inches per-- meaning, 00:09:54.920 --> 00:09:56.810 it's really hot in the room. 00:09:56.810 --> 00:10:00.270 Not this room, but the hypothetic room 00:10:00.270 --> 00:10:05.760 where the ice ball is melting. 00:10:05.760 --> 00:10:08.890 So imagine, in 1 minute, the radius 00:10:08.890 --> 00:10:13.550 will go down by 5 inches. 00:10:13.550 --> 00:10:16.786 Yes, it must be really hot. 00:10:16.786 --> 00:10:25.980 I want to know the derivative, dv, dt at the time 0. 00:10:25.980 --> 00:10:29.600 So you go, oh my god, I don't remember doing this, actually. 00:10:29.600 --> 00:10:31.330 It is a Calc 1 type of problem. 00:10:31.330 --> 00:10:34.580 00:10:34.580 --> 00:10:37.620 Why am I even discussing it again? 00:10:37.620 --> 00:10:41.280 Because I want to fool you a little bit into remembering 00:10:41.280 --> 00:10:44.953 the elementary formulas for the volume of a sphere, volume 00:10:44.953 --> 00:10:47.720 of a cone, volume of a cylinder. 00:10:47.720 --> 00:10:48.910 That was a long time ago. 00:10:48.910 --> 00:10:53.570 When you ask you teachers in K12 if you should memorize them, 00:10:53.570 --> 00:10:55.950 they said, by all means, memorize them. 00:10:55.950 --> 00:10:59.350 That was elementary geometry, but some of you know them 00:10:59.350 --> 00:11:01.054 by heart, some of you don't. 00:11:01.054 --> 00:11:03.180 Do you remember the volume formula 00:11:03.180 --> 00:11:05.860 for a ball with radius r? 00:11:05.860 --> 00:11:08.161 [INTERPOSING VOICES] 00:11:08.161 --> 00:11:09.095 00:11:09.095 --> 00:11:10.029 What? 00:11:10.029 --> 00:11:11.430 [? STUDENT: High RQ. ?] 00:11:11.430 --> 00:11:12.491 STUDENT: 4/3rds. 00:11:12.491 --> 00:11:13.366 MAGDALENA TODA: Good. 00:11:13.366 --> 00:11:14.860 I'm proud of you guys. 00:11:14.860 --> 00:11:18.496 I've discovered lots of people who are engineering majors 00:11:18.496 --> 00:11:19.870 and they don't know this formula. 00:11:19.870 --> 00:11:23.790 So how are we going to think of this problem? 00:11:23.790 --> 00:11:26.390 We have to think, Chain Rule. 00:11:26.390 --> 00:11:30.570 And Chain Rule means that you view this radius as a shrinking 00:11:30.570 --> 00:11:32.280 thing because that's why you have 00:11:32.280 --> 00:11:34.740 the grade of change negative. 00:11:34.740 --> 00:11:37.120 The radius is shrinking, it's decreasing, 00:11:37.120 --> 00:11:40.940 so you view r as a function of t. 00:11:40.940 --> 00:11:42.530 And of course, you made me cube it. 00:11:42.530 --> 00:11:44.960 I had to cube it. 00:11:44.960 --> 00:11:48.260 And then v will be a function of t ultimately, but you see, 00:11:48.260 --> 00:11:54.400 guys, t goes to r of t, r of t goes to v of t. 00:11:54.400 --> 00:11:56.675 What's the formula for this function? 00:11:56.675 --> 00:11:59.774 v equals 4 pi i cubed over 3. 00:11:59.774 --> 00:12:02.210 00:12:02.210 --> 00:12:04.180 So this is how the diagram goes. 00:12:04.180 --> 00:12:10.280 You look at that composition and you have dv, dt. 00:12:10.280 --> 00:12:14.161 And I remember teaching as a graduate student, that 00:12:14.161 --> 00:12:18.320 was a long time ago, in '97 or something, 00:12:18.320 --> 00:12:23.020 with this kind of diagram with compositions of functions. 00:12:23.020 --> 00:12:25.720 And my students had told me, nobody showed us 00:12:25.720 --> 00:12:28.716 this kind of diagram before. 00:12:28.716 --> 00:12:29.600 Well, I do. 00:12:29.600 --> 00:12:32.120 00:12:32.120 --> 00:12:35.290 I think they are very useful for understanding 00:12:35.290 --> 00:12:38.220 how a composition will go. 00:12:38.220 --> 00:12:42.490 Now I would just going ahead and say v prime because I'm lazy. 00:12:42.490 --> 00:12:45.850 And I go v prime of t is 0. 00:12:45.850 --> 00:12:51.012 Meaning, that this is the dv, dt at t0. 00:12:51.012 --> 00:12:55.440 And somebody has to help me remember how we did The Chain 00:12:55.440 --> 00:12:57.852 Rule in Calc 1. 00:12:57.852 --> 00:12:59.470 It was ages ago. 00:12:59.470 --> 00:13:05.490 4 pi over 3 constant times. 00:13:05.490 --> 00:13:07.380 Who jumps down? 00:13:07.380 --> 00:13:11.110 The 3 jumps down and he's very happy to do that. 00:13:11.110 --> 00:13:12.540 3, r squared. 00:13:12.540 --> 00:13:15.320 But r squared is not an independent variable. 00:13:15.320 --> 00:13:18.650 He or she depends on t. 00:13:18.650 --> 00:13:22.090 So I'll be very happy to say 3 times that times. 00:13:22.090 --> 00:13:24.020 And that's the essential part. 00:13:24.020 --> 00:13:25.653 I'm not done. 00:13:25.653 --> 00:13:26.872 STUDENT: It's dr over dt. 00:13:26.872 --> 00:13:27.830 MAGDALENA TODA: dr, dt. 00:13:27.830 --> 00:13:31.430 So I have finally applied The Chain Rule. 00:13:31.430 --> 00:13:35.440 And how do I plug in the data in order 00:13:35.440 --> 00:13:38.900 to get this as the final answer? 00:13:38.900 --> 00:13:46.450 I just go 4 pi over 3 times what? 00:13:46.450 --> 00:13:48.980 00:13:48.980 --> 00:13:56.420 3 times r-- who is r at the time to 0, 00:13:56.420 --> 00:14:00.112 where I want to view the whole situation? 00:14:00.112 --> 00:14:03.520 r squared at time to 0 would be 25. 00:14:03.520 --> 00:14:04.880 Are you guys with me? 00:14:04.880 --> 00:14:09.112 dr, dt at time to 0 is negative 5. 00:14:09.112 --> 00:14:10.090 All right. 00:14:10.090 --> 00:14:12.180 I'm done. 00:14:12.180 --> 00:14:15.195 So you are going to ask me, if I'm taking the examine, 00:14:15.195 --> 00:14:17.340 do I need this in the exam like that? 00:14:17.340 --> 00:14:18.860 Easy. 00:14:18.860 --> 00:14:20.820 Oh, it depends on the exam. 00:14:20.820 --> 00:14:23.330 If you have a multiple choice where this is simplified, 00:14:23.330 --> 00:14:27.440 obviously, it's not the right thing to forget about it, 00:14:27.440 --> 00:14:33.372 but I will accept answers like that. 00:14:33.372 --> 00:14:37.260 I don't care about the numerical part very much. 00:14:37.260 --> 00:14:41.335 If you want to do more, 4 times 25 is hundred times 5. 00:14:41.335 --> 00:14:43.534 So I have minus what? 00:14:43.534 --> 00:14:44.940 STUDENT: 500 pi. 00:14:44.940 --> 00:14:47.074 MAGDALENA TODA: 500 pi. 00:14:47.074 --> 00:14:49.050 How do we get the unit of that? 00:14:49.050 --> 00:14:50.630 I'm wondering. 00:14:50.630 --> 00:14:52.410 STUDENT: Cubic inches per minute. 00:14:52.410 --> 00:14:54.350 MAGDALENA TODA: Cubic inches per minute. 00:14:54.350 --> 00:14:55.320 Very good. 00:14:55.320 --> 00:14:56.650 Cubic inches per minute. 00:14:56.650 --> 00:14:59.020 Why don't I write it down? 00:14:59.020 --> 00:15:01.440 Because I couldn't care less. 00:15:01.440 --> 00:15:02.380 I'm a mathematician. 00:15:02.380 --> 00:15:06.709 If I were a physicist, I would definitely write it down. 00:15:06.709 --> 00:15:10.100 And he was right. 00:15:10.100 --> 00:15:15.990 Now you are going to find this weird. 00:15:15.990 --> 00:15:20.190 Why is she doing this review of this kind of melting ice 00:15:20.190 --> 00:15:22.360 problem from Calc 1? 00:15:22.360 --> 00:15:25.790 Because today I'm being sneaky and mean. 00:15:25.790 --> 00:15:28.760 And I want to give you a little challenge 00:15:28.760 --> 00:15:30.940 for 1 point of extra credit. 00:15:30.940 --> 00:15:33.410 You will have to compose your own problem, 00:15:33.410 --> 00:15:37.330 in Calculus 3, that is like that. 00:15:37.330 --> 00:15:49.910 So you have to compose a problem about a solid cylinder made 00:15:49.910 --> 00:15:51.840 of ice. 00:15:51.840 --> 00:15:53.010 Say what, Magdalena? 00:15:53.010 --> 00:15:53.730 OK. 00:15:53.730 --> 00:15:57.450 So I'll write it down. 00:15:57.450 --> 00:16:01.850 Solid cylinder made of ice that's melting in time. 00:16:01.850 --> 00:16:04.615 00:16:04.615 --> 00:16:07.110 So compose your own problem. 00:16:07.110 --> 00:16:10.020 Do you have to solve your own problem? 00:16:10.020 --> 00:16:12.530 Yes, I guess so. 00:16:12.530 --> 00:16:14.310 Once you compose your own problem, 00:16:14.310 --> 00:16:16.545 solve your own problem For extra credit, 1 point. 00:16:16.545 --> 00:16:20.680 00:16:20.680 --> 00:16:28.770 Compose, write, and solve-- you are the problem author. 00:16:28.770 --> 00:16:36.004 Write and solve your own problem, 00:16:36.004 --> 00:16:40.396 so that the story includes-- 00:16:40.396 --> 00:16:42.836 STUDENT: A solid cylinder. 00:16:42.836 --> 00:16:43.812 MAGDALENA TODA: Yes. 00:16:43.812 --> 00:16:48.670 Includes-- instead of a nice ball, a solid cylinder. 00:16:48.670 --> 00:16:54.300 00:16:54.300 --> 00:16:59.240 And necessarily, you cannot write it just a story-- 00:16:59.240 --> 00:17:02.860 I once had an ice cylinder, and it was melting, 00:17:02.860 --> 00:17:05.990 and I went to watch a movie, and by the time I came back, 00:17:05.990 --> 00:17:07.098 it was all melted. 00:17:07.098 --> 00:17:08.910 That's not what I want. 00:17:08.910 --> 00:17:24.450 I want it so that the problem is an example of applying 00:17:24.450 --> 00:17:35.130 The Chain Rule in Calc 3. 00:17:35.130 --> 00:17:37.730 And I won't say more. 00:17:37.730 --> 00:17:40.688 So maybe somebody can help with a hint. 00:17:40.688 --> 00:17:43.033 Maybe I shouldn't give too many hits, 00:17:43.033 --> 00:17:46.420 but let's talk as if we were chatting in a cafe, 00:17:46.420 --> 00:17:49.290 without me writing too much down. 00:17:49.290 --> 00:17:51.380 Of course, you can take notes of our discussion, 00:17:51.380 --> 00:17:54.290 but I don't want have it documented. 00:17:54.290 --> 00:17:55.759 So we have a cylinder right. 00:17:55.759 --> 00:17:58.753 00:17:58.753 --> 00:18:01.747 There is the cylinder. 00:18:01.747 --> 00:18:02.745 Forget about this. 00:18:02.745 --> 00:18:04.250 So there's the cylinder. 00:18:04.250 --> 00:18:09.620 It's made of ice and it's melting. 00:18:09.620 --> 00:18:14.010 And the volume should be a function of two variables 00:18:14.010 --> 00:18:16.725 because otherwise, you don't have it in Calc 3. 00:18:16.725 --> 00:18:18.426 So a function of two variables. 00:18:18.426 --> 00:18:21.807 00:18:21.807 --> 00:18:25.182 What other two variables am I talking about? 00:18:25.182 --> 00:18:26.640 STUDENT: The radius and the height. 00:18:26.640 --> 00:18:28.640 MAGDALENA TODA: The radius would be one of them. 00:18:28.640 --> 00:18:30.120 You don't have to say x and y. 00:18:30.120 --> 00:18:33.840 This is r and h. 00:18:33.840 --> 00:18:39.002 So h and r are in that formula. 00:18:39.002 --> 00:18:40.460 I'm not going to say which formula, 00:18:40.460 --> 00:18:44.740 you guys should know of the volume of the cylinder. 00:18:44.740 --> 00:18:48.974 But both h and r, what do they have in common in the story? 00:18:48.974 --> 00:18:49.940 STUDENT: Time. 00:18:49.940 --> 00:18:52.150 MAGDALENA TODA: They are both functions of time. 00:18:52.150 --> 00:18:53.771 They are melting in time. 00:18:53.771 --> 00:18:55.270 STUDENT: Can I ask a quick question? 00:18:55.270 --> 00:18:55.870 MAGDALENA TODA: Yes, sir. 00:18:55.870 --> 00:18:58.430 STUDENT: What if we solve for-- what is the negative 500 00:18:58.430 --> 00:18:59.786 [? path? ?] 00:18:59.786 --> 00:19:04.290 MAGDALENA TODA: This is the speed with which the volume is 00:19:04.290 --> 00:19:05.638 shrinking at time to 0. 00:19:05.638 --> 00:19:08.850 00:19:08.850 --> 00:19:13.090 So the rate of change of the volume at time to o. 00:19:13.090 --> 00:19:15.170 And this is something-- by the way, 00:19:15.170 --> 00:19:20.260 that's how I would like you to state it. 00:19:20.260 --> 00:19:25.805 Find the rate of change of the volume of the ice-- 00:19:25.805 --> 00:19:28.780 wasn't that a good cylinder? 00:19:28.780 --> 00:19:34.850 At time to 0, if you know that at time to 0 00:19:34.850 --> 00:19:37.680 something happened. 00:19:37.680 --> 00:19:41.120 Maybe r is given, h is given. 00:19:41.120 --> 00:19:44.256 The derivatives are given. 00:19:44.256 --> 00:19:47.370 You only have one derivative given here, 00:19:47.370 --> 00:19:50.460 which was our prime of t minus 5. 00:19:50.460 --> 00:19:52.160 Now I leave it to you. 00:19:52.160 --> 00:19:57.120 I ask it to you, and I'll leave it to you, and don't tell me. 00:19:57.120 --> 00:20:01.820 When we have a piece of ice-- well, 00:20:01.820 --> 00:20:05.590 there was something in the news, but I'm not going to say. 00:20:05.590 --> 00:20:08.090 There was some nice, ice sculpture in the news there. 00:20:08.090 --> 00:20:10.880 00:20:10.880 --> 00:20:18.980 So do the dimensions decrease at the same rate, do you think? 00:20:18.980 --> 00:20:20.560 I mean, I don't know. 00:20:20.560 --> 00:20:22.020 It's all up to you. 00:20:22.020 --> 00:20:24.830 Think of a case when the radius and the height 00:20:24.830 --> 00:20:27.800 would shrink at the same speed. 00:20:27.800 --> 00:20:31.316 And think of a case when the radius and the height 00:20:31.316 --> 00:20:34.030 of the cylinder made of ice would not 00:20:34.030 --> 00:20:38.610 change at the same rate for some reason. 00:20:38.610 --> 00:20:40.700 I don't know, but the simplest case 00:20:40.700 --> 00:20:43.040 would be to assume that all of the dimensions 00:20:43.040 --> 00:20:49.520 shrink at the same speed, at the same rate of change. 00:20:49.520 --> 00:20:52.000 So you write your own problem, you make up your own data. 00:20:52.000 --> 00:20:55.335 Now you will appreciate how much work people 00:20:55.335 --> 00:20:57.370 put into that work book. 00:20:57.370 --> 00:21:00.680 I mean, if there is a bug, it's one in a thousand, 00:21:00.680 --> 00:21:04.210 but for a programmer to be able to write those problems, 00:21:04.210 --> 00:21:08.715 he has to know calculus, he has to know C++ or Java, 00:21:08.715 --> 00:21:12.520 he has to be good-- that's not a problem, right? 00:21:12.520 --> 00:21:13.440 STUDENT: No. 00:21:13.440 --> 00:21:14.450 That's fine. 00:21:14.450 --> 00:21:20.360 MAGDALENA TODA: He or she has to know how to write a problem, 00:21:20.360 --> 00:21:22.600 so that you guys, no matter how you 00:21:22.600 --> 00:21:29.130 input your answer, as long as it is correct, you'll get the OK. 00:21:29.130 --> 00:21:32.990 Because you can put answers in many equivalent forms 00:21:32.990 --> 00:21:36.127 and all of them have to be-- 00:21:36.127 --> 00:21:37.210 STUDENT: The right answer. 00:21:37.210 --> 00:21:38.043 MAGDALENA TODA: Yes. 00:21:38.043 --> 00:21:40.780 To get the right answer. 00:21:40.780 --> 00:21:43.950 So since I have new people who just came-- 00:21:43.950 --> 00:21:47.023 And I understand you guys come from different buildings 00:21:47.023 --> 00:21:52.435 and I'm not mad for people who are coming late because I know 00:21:52.435 --> 00:21:55.280 you come from other classes, I wanted 00:21:55.280 --> 00:22:03.400 to say we started from a melting ice sphere example in Calc 1 00:22:03.400 --> 00:22:07.260 that was on many finals in here, at Texas Tech. 00:22:07.260 --> 00:22:14.470 And I want you to compose your own problem based on that. 00:22:14.470 --> 00:22:17.490 This time, involving a cylinder made 00:22:17.490 --> 00:22:23.650 of ice whose dimensions are doing something special. 00:22:23.650 --> 00:22:26.130 That shouldn't be hard. 00:22:26.130 --> 00:22:29.500 I'm going to erase this part because it's not 00:22:29.500 --> 00:22:30.600 the relevant one. 00:22:30.600 --> 00:22:32.840 I'm going to keep this one a little bit more 00:22:32.840 --> 00:22:35.960 for people who want to take notes. 00:22:35.960 --> 00:22:37.271 And I'm going to move on. 00:22:37.271 --> 00:22:42.460 00:22:42.460 --> 00:22:47.000 Another example we give you in the book 00:22:47.000 --> 00:22:54.700 is that one where x and y, the variables the function f, 00:22:54.700 --> 00:22:58.820 are not just functions of time, t. 00:22:58.820 --> 00:23:03.570 They, themselves, are functions of other two variables. 00:23:03.570 --> 00:23:07.990 Is that a lot more different from what I gave you already? 00:23:07.990 --> 00:23:08.490 No. 00:23:08.490 --> 00:23:10.730 The idea is the same. 00:23:10.730 --> 00:23:13.930 And you are imaginative. 00:23:13.930 --> 00:23:20.500 You are able to come up with your own answers. 00:23:20.500 --> 00:23:26.950 I'm going to ask you to think about what I'll have to write. 00:23:26.950 --> 00:23:28.330 This is finished. 00:23:28.330 --> 00:23:32.250 00:23:32.250 --> 00:23:37.796 So assume that you have function z equals F of x,y. 00:23:37.796 --> 00:23:42.470 00:23:42.470 --> 00:23:48.840 As we had it before, this is example 2 00:23:48.840 --> 00:23:58.330 where x is a function of u and v itself. 00:23:58.330 --> 00:24:02.510 And y is a function of u and v itself. 00:24:02.510 --> 00:24:07.201 And we assume that all the partial derivatives 00:24:07.201 --> 00:24:09.536 are defined and continuous. 00:24:09.536 --> 00:24:12.030 And we make the problem really nice. 00:24:12.030 --> 00:24:21.490 And now we'll come up with some example 00:24:21.490 --> 00:24:38.630 you know from before where x equals x of uv equals uv. 00:24:38.630 --> 00:24:48.760 And y equals y of uv equals u plus v. 00:24:48.760 --> 00:24:51.860 So these functions are the sum and the product 00:24:51.860 --> 00:24:53.195 of other variables. 00:24:53.195 --> 00:24:56.130 00:24:56.130 --> 00:25:06.780 Can you tell me how I am going to compute the derivative of 0, 00:25:06.780 --> 00:25:17.320 or of f, with the script of u at x of uv, y of uv? 00:25:17.320 --> 00:25:18.756 Is this hard? 00:25:18.756 --> 00:25:19.502 STUDENT: It is. 00:25:19.502 --> 00:25:20.710 MAGDALENA TODA: I don't know. 00:25:20.710 --> 00:25:27.512 You have to help me because-- why don't I put d here? 00:25:27.512 --> 00:25:29.000 STUDENT: Because [INAUDIBLE]. 00:25:29.000 --> 00:25:30.630 MAGDALENA TODA: Because you have 2. 00:25:30.630 --> 00:25:32.592 So the composition in itself will 00:25:32.592 --> 00:25:35.510 be a function of two variables. 00:25:35.510 --> 00:25:38.610 So of course, I have [INAUDIBLE]. 00:25:38.610 --> 00:25:48.270 I'm going to go ahead and do it as you say without rushing. 00:25:48.270 --> 00:25:51.360 Of course, I know you are watching. 00:25:51.360 --> 00:25:53.250 What will happen? 00:25:53.250 --> 00:25:54.159 STUDENT: 2x and 2y. 00:25:54.159 --> 00:25:55.450 MAGDALENA TODA: No, in general. 00:25:55.450 --> 00:25:58.470 Over here, I know you want to do it right away, 00:25:58.470 --> 00:26:01.790 but I would like you to give me a general formula mimicking 00:26:01.790 --> 00:26:06.530 the same thing you had before when you had one parameter, t. 00:26:06.530 --> 00:26:08.120 Now you have u and d separately. 00:26:08.120 --> 00:26:10.376 You want it to do it straight. 00:26:10.376 --> 00:26:19.088 So we have df, dx at x of uv, y of uv. 00:26:19.088 --> 00:26:20.540 Shut up, Magdalene. 00:26:20.540 --> 00:26:24.450 Let people talk and help you because you're tired. 00:26:24.450 --> 00:26:26.880 It's a Thursday. 00:26:26.880 --> 00:26:28.366 df, dx. 00:26:28.366 --> 00:26:29.259 STUDENT: [INAUDIBLE]. 00:26:29.259 --> 00:26:30.050 MAGDALENA TODA: dx. 00:26:30.050 --> 00:26:34.250 Again, [INAUDIBLE] notation, partial with respect 00:26:34.250 --> 00:26:42.890 to u, plus df, dy. 00:26:42.890 --> 00:26:46.430 So the second argument-- so I prime in respect 00:26:46.430 --> 00:26:50.610 to the second argument, computing everything 00:26:50.610 --> 00:26:55.180 in the end, which means in terms of u and v times, 00:26:55.180 --> 00:27:00.350 again, the dy with respect to u. 00:27:00.350 --> 00:27:01.550 You are saying that. 00:27:01.550 --> 00:27:04.367 Now I'd like you to see the pattern. 00:27:04.367 --> 00:27:06.450 Of course, you see the pattern here, smart people, 00:27:06.450 --> 00:27:11.660 but I want to emphasize the cowboys. 00:27:11.660 --> 00:27:15.021 And green for the other cowboy. 00:27:15.021 --> 00:27:16.854 I'm trying to match the college beautifully. 00:27:16.854 --> 00:27:22.636 00:27:22.636 --> 00:27:26.116 And the independent variable, Mr. u. 00:27:26.116 --> 00:27:27.610 Not u, but Mr. u. 00:27:27.610 --> 00:27:28.606 Yes, ma'am? 00:27:28.606 --> 00:27:31.096 STUDENT: Is it the partial of dx, du? 00:27:31.096 --> 00:27:37.771 Or is it-- like you did the partial for the-- 00:27:37.771 --> 00:27:39.562 MAGDALENA TODA: So did I do anything wrong? 00:27:39.562 --> 00:27:41.554 I don't think I did anything wrong. 00:27:41.554 --> 00:27:44.400 STUDENT: So it is the partial for dx over du? 00:27:44.400 --> 00:27:48.663 MAGDALENA TODA: So I go du with respect to the first variable, 00:27:48.663 --> 00:27:50.865 times that variable with respect to u. 00:27:50.865 --> 00:27:52.156 STUDENT: But is it the partial? 00:27:52.156 --> 00:27:53.660 That's my question. 00:27:53.660 --> 00:27:57.180 MAGDALENA TODA: But it has to be a partial because x is 00:27:57.180 --> 00:28:02.995 a function of u and v, so I cannot put d. 00:28:02.995 --> 00:28:07.146 And then the same plus the same idea as before. 00:28:07.146 --> 00:28:10.110 df with respect to the second argument 00:28:10.110 --> 00:28:15.650 times that second argument with respect to the u. 00:28:15.650 --> 00:28:20.420 You see, Mr. u is replacing Mr. t. 00:28:20.420 --> 00:28:21.485 He is independent. 00:28:21.485 --> 00:28:24.170 00:28:24.170 --> 00:28:27.310 He's the guy who is moving. 00:28:27.310 --> 00:28:29.010 We don't care about anybody else, 00:28:29.010 --> 00:28:32.030 but he replaces the time in this kind of problem. 00:28:32.030 --> 00:28:35.500 00:28:35.500 --> 00:28:38.590 Now the other one. 00:28:38.590 --> 00:28:41.064 I will let you speak. 00:28:41.064 --> 00:28:43.926 Df, dv. 00:28:43.926 --> 00:28:50.136 The same idea, but somebody else is going to talk. 00:28:50.136 --> 00:28:52.580 STUDENT: It would be del f, del y. 00:28:52.580 --> 00:28:54.450 MAGDALENA TODA: Del f, del x? 00:28:54.450 --> 00:28:56.890 Well, let's try to start in order. 00:28:56.890 --> 00:29:01.940 00:29:01.940 --> 00:29:05.200 And I tried to be organized and write neatly 00:29:05.200 --> 00:29:12.700 because I looked at-- so these videos are new and in progress. 00:29:12.700 --> 00:29:16.670 And I'm trying to see what I did well and I didn't. 00:29:16.670 --> 00:29:18.490 And at times, I wrote neatly. 00:29:18.490 --> 00:29:20.785 At times, I wrote not so neatly. 00:29:20.785 --> 00:29:23.260 I'm just learning about myself. 00:29:23.260 --> 00:29:28.070 It's one thing, what you think about yourself from the inside 00:29:28.070 --> 00:29:30.610 and to you see yourself the way other people 00:29:30.610 --> 00:29:33.350 see from the outside. 00:29:33.350 --> 00:29:34.420 It's not fun. 00:29:34.420 --> 00:29:35.670 STUDENT: Can you say it again? 00:29:35.670 --> 00:29:41.900 MAGDALENA TODA: This is v. So I'll say it again. 00:29:41.900 --> 00:29:46.370 We all have a certain impression about ourselves, 00:29:46.370 --> 00:29:49.100 but when you see a movie of yourself, 00:29:49.100 --> 00:29:51.740 you see the way other people see you. 00:29:51.740 --> 00:29:53.069 And it's not fun. 00:29:53.069 --> 00:29:56.010 STUDENT: So what-- 00:29:56.010 --> 00:29:58.750 MAGDALENA TODA: So let's see the cowboys. 00:29:58.750 --> 00:30:06.360 Ryan is looking at the [? man. ?] He is all [? man. ?] 00:30:06.360 --> 00:30:10.480 And y is here, right? 00:30:10.480 --> 00:30:15.830 And who is the time variable, kind of, this time? 00:30:15.830 --> 00:30:18.205 This time, which one is the time? 00:30:18.205 --> 00:30:28.456 v. And v is the only ultimate variable that we care about. 00:30:28.456 --> 00:30:31.858 So everything you did before with respect to t, 00:30:31.858 --> 00:30:35.307 you do now with respect to u, you 00:30:35.307 --> 00:30:37.130 do now with respect to v. It shouldn't 00:30:37.130 --> 00:30:38.970 be hard to understand. 00:30:38.970 --> 00:30:42.040 I want to work the example, of course. 00:30:42.040 --> 00:30:44.440 With your help, I will work it. 00:30:44.440 --> 00:30:49.270 Now remember how my students cheated on this one? 00:30:49.270 --> 00:30:57.090 So I told my colleague, he did not say, five or six years ago, 00:30:57.090 --> 00:31:00.810 by first writing The Chain Rule for functions of two variables, 00:31:00.810 --> 00:31:08.430 express all the df, du, df, dv, but he said by any method. 00:31:08.430 --> 00:31:12.970 Of course, what they did-- they were sneaky. 00:31:12.970 --> 00:31:15.962 They took something like x equals uv 00:31:15.962 --> 00:31:17.926 and they plugged it in here. 00:31:17.926 --> 00:31:19.872 They took the function [? u and v, ?] 00:31:19.872 --> 00:31:20.872 they plugged it in here. 00:31:20.872 --> 00:31:22.836 They computed everything in terms of u and v 00:31:22.836 --> 00:31:24.309 and took the partials. 00:31:24.309 --> 00:31:26.756 STUDENT: Why don't you [INAUDIBLE]? 00:31:26.756 --> 00:31:29.130 MAGDALENA TODA: It depends how the problem is formulated. 00:31:29.130 --> 00:31:30.455 STUDENT: So if you make it [INAUDIBLE], 00:31:30.455 --> 00:31:31.626 then it's [INAUDIBLE]. 00:31:31.626 --> 00:31:35.716 00:31:35.716 --> 00:31:39.360 MAGDALENA TODA: So when they give you the precise functions, 00:31:39.360 --> 00:31:39.919 you're right. 00:31:39.919 --> 00:31:41.710 But if they don't give you those functions, 00:31:41.710 --> 00:31:44.320 if they keep them a secret, then you still 00:31:44.320 --> 00:31:47.230 have to write the general formula. 00:31:47.230 --> 00:31:51.450 If they don't give you the functions, all of them 00:31:51.450 --> 00:31:54.060 explicitly. 00:31:54.060 --> 00:31:57.296 So let's see what to do in this case. 00:31:57.296 --> 00:32:09.300 df, du at x of u, vy of uv will be what? 00:32:09.300 --> 00:32:11.520 Now people, help me, please. 00:32:11.520 --> 00:32:15.710 00:32:15.710 --> 00:32:21.980 I want to teach you how engineers and physicists very, 00:32:21.980 --> 00:32:26.610 very often express those at x and y. 00:32:26.610 --> 00:32:29.080 And many of you know because we talked 00:32:29.080 --> 00:32:31.280 about that in office hours. 00:32:31.280 --> 00:32:36.661 2x, I might write, but evaluated at-- 00:32:36.661 --> 00:32:38.160 and this is a very frequent notation 00:32:38.160 --> 00:32:42.030 image in the engineering and physicist world. 00:32:42.030 --> 00:32:45.170 So 2x evaluated at where? 00:32:45.170 --> 00:32:51.530 At the point where x is uv and y is u plus v. 00:32:51.530 --> 00:32:59.670 So I say x of uv, y of uv. 00:32:59.670 --> 00:33:05.040 And I'll replace later because I'm not in a hurry. 00:33:05.040 --> 00:33:06.970 dx, du. 00:33:06.970 --> 00:33:09.300 Who is dx, du? 00:33:09.300 --> 00:33:11.403 The derivative of x or with respect to u? 00:33:11.403 --> 00:33:12.690 Are you guys awake? 00:33:12.690 --> 00:33:13.370 STUDENT: Yes. 00:33:13.370 --> 00:33:14.730 So it's v. 00:33:14.730 --> 00:33:19.470 MAGDALENA TODA: v. Very good. v plus-- the next term, who's 00:33:19.470 --> 00:33:22.610 going to tell me what we have? 00:33:22.610 --> 00:33:24.070 STUDENT: 2y evaluated at-- 00:33:24.070 --> 00:33:28.340 MAGDALENA TODA: 2y evaluated at-- look how lazy I am. 00:33:28.340 --> 00:33:37.540 Times the derivative of y with respect to u. 00:33:37.540 --> 00:33:39.640 So you were right because of 2y. 00:33:39.640 --> 00:33:42.740 00:33:42.740 --> 00:33:44.200 Attention, right? 00:33:44.200 --> 00:33:48.140 So it's dy, du is 1. 00:33:48.140 --> 00:33:49.910 It's very easy to make a mistake. 00:33:49.910 --> 00:33:52.430 I've had mistakes who made mistakes in the final 00:33:52.430 --> 00:33:55.740 from just miscalculating because when 00:33:55.740 --> 00:33:57.700 you are close to some formula, you 00:33:57.700 --> 00:34:00.040 don't see the whole picture. 00:34:00.040 --> 00:34:01.038 What do you do? 00:34:01.038 --> 00:34:05.030 At the end of your exams, go back and rather 00:34:05.030 --> 00:34:08.620 than quickly turning in a paper, never do that, 00:34:08.620 --> 00:34:11.510 go back and check all your problems. 00:34:11.510 --> 00:34:13.489 It's a good habit. 00:34:13.489 --> 00:34:20.880 2 times x, which is uv, I plug it as a function of u and v, 00:34:20.880 --> 00:34:23.350 right? 00:34:23.350 --> 00:34:27.370 Times a v plus-- who is 2y? 00:34:27.370 --> 00:34:28.870 That's the last of the Mohicans. 00:34:28.870 --> 00:34:30.310 One is out. 00:34:30.310 --> 00:34:31.100 STUDENT: 2. 00:34:31.100 --> 00:34:37.130 MAGDALENA TODA: 2y 2 times replace y in terms of u and v. 00:34:37.130 --> 00:34:38.050 And you're done. 00:34:38.050 --> 00:34:40.489 So do you like it? 00:34:40.489 --> 00:34:42.020 I don't. 00:34:42.020 --> 00:34:44.440 And how would you write it? 00:34:44.440 --> 00:34:48.510 Not much better than that, but at least let's try. 00:34:48.510 --> 00:34:53.380 2uv squared plus 2u plus 2v. 00:34:53.380 --> 00:34:55.004 You can do a little bit more than that, 00:34:55.004 --> 00:35:01.130 but if you want to list it in the order of the degrees 00:35:01.130 --> 00:35:04.190 of the polynomials, that's OK. 00:35:04.190 --> 00:35:06.220 Now next one. 00:35:06.220 --> 00:35:10.015 df, dv, x of uv, y of uv. 00:35:10.015 --> 00:35:12.790 00:35:12.790 --> 00:35:15.861 Such examples are in the book. 00:35:15.861 --> 00:35:17.860 Many things are in the book and out of the book. 00:35:17.860 --> 00:35:21.400 I mean, on the white board. 00:35:21.400 --> 00:35:25.180 I don't know why it gives you so many combinations of this type, 00:35:25.180 --> 00:35:31.130 u plus v, u minus-- 2u plus 2v, 2u you minus 2v. 00:35:31.130 --> 00:35:31.960 Well, I know why. 00:35:31.960 --> 00:35:35.210 Because that's a rotation and rescaling. 00:35:35.210 --> 00:35:37.150 So there is a reason behind that, 00:35:37.150 --> 00:35:42.110 but I thought of something different for df, dv. 00:35:42.110 --> 00:35:44.800 Now what do I do? 00:35:44.800 --> 00:35:45.355 df, dx. 00:35:45.355 --> 00:35:46.022 STUDENT: You [? have to find something symmetrical to that. 00:35:46.022 --> 00:35:46.860 ?] 00:35:46.860 --> 00:35:48.443 MAGDALENA TODA: Again, the same thing. 00:35:48.443 --> 00:35:52.781 2x evaluated at whoever times-- 00:35:52.781 --> 00:35:53.280 STUDENT: u. 00:35:53.280 --> 00:35:55.831 00:35:55.831 --> 00:35:58.080 MAGDALENA TODA: Because you have dx with respect to v, 00:35:58.080 --> 00:36:02.090 so you have u plus-- 00:36:02.090 --> 00:36:03.770 STUDENT: df, dy. 00:36:03.770 --> 00:36:07.060 MAGDALENA TODA: df, dy, which is 2y, evaluated 00:36:07.060 --> 00:36:10.002 at the same kind of guy. 00:36:10.002 --> 00:36:13.360 So all you have to do is replace with respect to u and v. 00:36:13.360 --> 00:36:16.240 And finally, multiplied by- 00:36:16.240 --> 00:36:16.870 STUDENT: dy. 00:36:16.870 --> 00:36:18.180 MAGDALENA TODA: dy, dv. 00:36:18.180 --> 00:36:21.650 dy, dv is 1 again. 00:36:21.650 --> 00:36:23.723 Just pay attention when you plug in 00:36:23.723 --> 00:36:25.722 because you realize you can know these very well 00:36:25.722 --> 00:36:28.780 and understand it as a process, but if you make an algebra 00:36:28.780 --> 00:36:31.134 and everything is out. 00:36:31.134 --> 00:36:32.800 And then you send me an email that says, 00:36:32.800 --> 00:36:34.960 I've tried this problem 15 times. 00:36:34.960 --> 00:36:37.820 And I don't even hold you responsible for that 00:36:37.820 --> 00:36:41.770 because I can make algebra mistakes anytime. 00:36:41.770 --> 00:36:54.400 So 2uv times u plus 2 times u plus v. So what did I do here? 00:36:54.400 --> 00:37:00.906 I simply replaced the given functions in terms of u and v. 00:37:00.906 --> 00:37:03.200 And I'm done. 00:37:03.200 --> 00:37:03.900 Do I like it? 00:37:03.900 --> 00:37:08.532 No, but I'd like you to notice something as soon as I'm done. 00:37:08.532 --> 00:37:11.899 2u squared v plus 2u plus 2v. 00:37:11.899 --> 00:37:17.610 00:37:17.610 --> 00:37:20.040 Could I have expected that? 00:37:20.040 --> 00:37:21.540 Look at the beauty of the functions. 00:37:21.540 --> 00:37:24.490 00:37:24.490 --> 00:37:27.550 Z is a symmetric function. 00:37:27.550 --> 00:37:31.515 x and y have some of the symmetry as well. 00:37:31.515 --> 00:37:34.550 If you swap u and v, these are symmetric polynomials 00:37:34.550 --> 00:37:38.615 of order 2 and 1. 00:37:38.615 --> 00:37:40.070 [INAUDIBLE] 00:37:40.070 --> 00:37:42.620 Swap the variables, you still get the same thing. 00:37:42.620 --> 00:37:45.615 Swap the variables u and v, you get the same thing. 00:37:45.615 --> 00:37:48.380 So how could I have imagined that I'm 00:37:48.380 --> 00:37:54.470 going to get-- if I were smart, without doing all the work, 00:37:54.470 --> 00:37:57.920 I could figure out this by just swapping 00:37:57.920 --> 00:38:01.760 the u and v, the rows of u and v. I would have said, 00:38:01.760 --> 00:38:07.980 2vu squared, dv plus 2u and it's the same thing I got here. 00:38:07.980 --> 00:38:13.520 But not always are you so lucky to be given nice data. 00:38:13.520 --> 00:38:15.690 Well, in real life, it's a mess. 00:38:15.690 --> 00:38:21.940 If you are, let's say, working with geophysics real data, 00:38:21.940 --> 00:38:27.560 you two parameters and for each parameter, x and y, 00:38:27.560 --> 00:38:28.810 you have other parameters. 00:38:28.810 --> 00:38:31.565 You will never have anything that nice. 00:38:31.565 --> 00:38:35.940 You may have nasty truncations of polynomials 00:38:35.940 --> 00:38:39.057 with many, many terms that you work 00:38:39.057 --> 00:38:41.390 with approximating polynomials all the time. [INAUDIBLE] 00:38:41.390 --> 00:38:43.350 or something like that. 00:38:43.350 --> 00:38:46.870 So don't expect these miracles to happen with real data, 00:38:46.870 --> 00:38:49.740 but the process is the same. 00:38:49.740 --> 00:38:52.960 And, of course, there are programs 00:38:52.960 --> 00:38:56.410 that incorporate all of the Calculus 3 notions 00:38:56.410 --> 00:38:59.380 that we went over. 00:38:59.380 --> 00:39:03.670 There were people who already wrote 00:39:03.670 --> 00:39:08.310 lots of programs that enable you to compute derivatives 00:39:08.310 --> 00:39:11.105 of function of several variables. 00:39:11.105 --> 00:39:23.690 00:39:23.690 --> 00:39:27.490 Now let me take your temperature again. 00:39:27.490 --> 00:39:29.520 Is this hard? 00:39:29.520 --> 00:39:30.420 No. 00:39:30.420 --> 00:39:33.770 It's sort of logical you just have to pay attention to what? 00:39:33.770 --> 00:39:36.450 00:39:36.450 --> 00:39:41.523 Pay attention to not making too many algebra mistakes, right? 00:39:41.523 --> 00:39:42.522 That's kind of the idea. 00:39:42.522 --> 00:39:45.330 00:39:45.330 --> 00:39:48.701 More things that I wanted to-- there 00:39:48.701 --> 00:39:50.950 are many more things I wanted to share with you today, 00:39:50.950 --> 00:39:56.230 but I'm glad we reached some consensus in the sense 00:39:56.230 --> 00:40:01.718 that you feel there is logic and order in this type of problem. 00:40:01.718 --> 00:40:21.900 00:40:21.900 --> 00:40:35.170 I tried to give you a little bit of an introduction to why 00:40:35.170 --> 00:40:39.645 the gradient is so important last time. 00:40:39.645 --> 00:40:41.220 And I'm going to come back to that 00:40:41.220 --> 00:40:45.239 again, so I'm not going to leave you in the air. 00:40:45.239 --> 00:40:49.071 But before then, I would like to do 00:40:49.071 --> 00:40:50.687 the directional derivative, which 00:40:50.687 --> 00:40:52.914 is a very important section. 00:40:52.914 --> 00:40:57.378 So I'm going to start again. 00:40:57.378 --> 00:41:07.890 And I'll also do, at the same time, some review of 11.5. 00:41:07.890 --> 00:41:10.060 So I will combine them. 00:41:10.060 --> 00:41:11.844 And I want to introduce the notion 00:41:11.844 --> 00:41:14.528 of directional derivatives because it's 00:41:14.528 --> 00:41:15.992 right there for us to grab it. 00:41:15.992 --> 00:41:26.240 00:41:26.240 --> 00:41:28.850 And you say, well, that sounds familiar. 00:41:28.850 --> 00:41:32.290 It sounds like I dealt with direction before, 00:41:32.290 --> 00:41:35.780 but I didn't what that was. 00:41:35.780 --> 00:41:37.500 That's exactly true. 00:41:37.500 --> 00:41:41.360 You dealt with it before, you just didn't know what it was. 00:41:41.360 --> 00:41:44.060 And I'll give you the general definition, 00:41:44.060 --> 00:41:49.120 but then I would like you to think about if you have ever 00:41:49.120 --> 00:41:50.260 seen that before. 00:41:50.260 --> 00:41:53.040 00:41:53.040 --> 00:41:59.420 I'm going to say I have the derivative of a function, f, 00:41:59.420 --> 00:42:01.440 in the direction, u. 00:42:01.440 --> 00:42:04.290 And I'm going put u bar as if you were free, 00:42:04.290 --> 00:42:05.480 not a married man. 00:42:05.480 --> 00:42:09.180 But u as a direction as always a unit vector. 00:42:09.180 --> 00:42:10.380 STUDENT: [INAUDIBLE]. 00:42:10.380 --> 00:42:11.921 MAGDALENA TODA: I told you last time, 00:42:11.921 --> 00:42:18.188 just to prepare you, direction, u, is always a unit vector. 00:42:18.188 --> 00:42:23.544 00:42:23.544 --> 00:42:24.044 Always. 00:42:24.044 --> 00:42:28.450 00:42:28.450 --> 00:42:29.390 Computed at x0y0. 00:42:29.390 --> 00:42:35.556 But x0y0 is a given view point. 00:42:35.556 --> 00:42:40.860 00:42:40.860 --> 00:42:46.490 And I'm going to say what that's going to be. 00:42:46.490 --> 00:42:49.100 I have a limit. 00:42:49.100 --> 00:42:51.300 I'm going to use the h. 00:42:51.300 --> 00:42:53.760 And you say, why in the world is she using h? 00:42:53.760 --> 00:42:57.426 You will see in a second-- h goes to 0-- because we 00:42:57.426 --> 00:42:59.130 haven't used h in awhile. 00:42:59.130 --> 00:43:02.480 h is like a small displacement that shrinks to 0. 00:43:02.480 --> 00:43:07.480 00:43:07.480 --> 00:43:25.818 And I put here, f of x0 plus hu1, y0 plus hu2, 00:43:25.818 --> 00:43:32.300 close, minus f of x0y0. 00:43:32.300 --> 00:43:34.912 So you say, wait a minute, Magdalena, oh my god, I've 00:43:34.912 --> 00:43:37.986 got a headache. 00:43:37.986 --> 00:43:38.970 I'm not here. 00:43:38.970 --> 00:43:42.530 Z0 is easy to understand for everybody, right? 00:43:42.530 --> 00:43:47.040 That's going to be altitude at the point x0y0. 00:43:47.040 --> 00:43:48.620 It shouldn't be hard. 00:43:48.620 --> 00:43:51.560 00:43:51.560 --> 00:43:54.530 On the other hand, what am I doing? 00:43:54.530 --> 00:44:00.090 I have to look at a real graph, in the real world. 00:44:00.090 --> 00:44:04.860 And that's going to be a patch of a smooth surface. 00:44:04.860 --> 00:44:08.640 And I say, OK, this is my favorite point. 00:44:08.640 --> 00:44:12.330 I have x0y0 on the ground. 00:44:12.330 --> 00:44:16.030 And the corresponding point in three dimensions, 00:44:16.030 --> 00:44:21.990 would be x0y0 and z0, which is the f of x0y0. 00:44:21.990 --> 00:44:24.210 And you say, wait a minute, what do you mean 00:44:24.210 --> 00:44:25.860 I can't move in a direction? 00:44:25.860 --> 00:44:33.430 Is it like when took a sleigh and we went 00:44:33.430 --> 00:44:35.680 to have fun on the hill? 00:44:35.680 --> 00:44:38.430 Yes, but I said that would be the last time 00:44:38.430 --> 00:44:42.620 we talked about the hilly area with snow on it. 00:44:42.620 --> 00:44:47.545 It was a good preparation for today in the sense that-- 00:44:47.545 --> 00:44:51.522 Remember, we went somewhere when I picked your direction north, 00:44:51.522 --> 00:44:52.830 east? 00:44:52.830 --> 00:44:54.590 i plus j? 00:44:54.590 --> 00:44:57.150 And in the direction of i plus j, 00:44:57.150 --> 00:44:59.104 which is not quite the direction and I'll 00:44:59.104 --> 00:45:04.530 ask you why in a second, I was going down along a meridian. 00:45:04.530 --> 00:45:05.570 Remember last time? 00:45:05.570 --> 00:45:11.820 And then that was the direction of the steepest descent. 00:45:11.820 --> 00:45:12.830 I was sliding down. 00:45:12.830 --> 00:45:16.650 If I wanted the direction of the steepest ascent, 00:45:16.650 --> 00:45:19.680 that would have been minus i minus j. 00:45:19.680 --> 00:45:23.460 So I had plus i plus j, minus i minus j. 00:45:23.460 --> 00:45:26.087 And I told you last time, why are those not quite directions? 00:45:26.087 --> 00:45:27.670 STUDENT: Because they are not unitary. 00:45:27.670 --> 00:45:29.211 MAGDALENA TODA: They are not unitary. 00:45:29.211 --> 00:45:31.840 So to make them like this u, I should 00:45:31.840 --> 00:45:34.270 have said, in the direction i plus 00:45:34.270 --> 00:45:37.140 j, that was one minus x squared minus y 00:45:37.140 --> 00:45:41.430 squared, the parabola way, that was the hill full of snow. 00:45:41.430 --> 00:45:45.220 So in the direction i plus j, I go down 00:45:45.220 --> 00:45:47.140 the fastest possible way. 00:45:47.140 --> 00:45:50.760 In the direction i plus j over square root of 2, 00:45:50.760 --> 00:45:53.580 I would be fine with a unit vector. 00:45:53.580 --> 00:45:57.660 In the opposite direction, I go up the fastest way possible, 00:45:57.660 --> 00:46:01.600 but you don't want to because it's-- can you imagine hiking 00:46:01.600 --> 00:46:09.021 the steepest possible direction in the steepest way? 00:46:09.021 --> 00:46:14.750 00:46:14.750 --> 00:46:16.360 Now with my direction. 00:46:16.360 --> 00:46:22.340 My direction in plane should be the i vector. 00:46:22.340 --> 00:46:25.600 And that magic vector should have length 1 from here 00:46:25.600 --> 00:46:26.100 to here. 00:46:26.100 --> 00:46:29.500 And when you measure this guy, he has to have length 1. 00:46:29.500 --> 00:46:35.466 And if you decompose, you have to decompose him along the-- 00:46:35.466 --> 00:46:36.940 what is this? 00:46:36.940 --> 00:46:40.090 The x direction and the y direction, right? 00:46:40.090 --> 00:46:47.450 How do you split a vector in such a decomposition? 00:46:47.450 --> 00:46:54.110 Well, Mr. u will be u1i plus 1i. 00:46:54.110 --> 00:46:55.710 It sounds funny. 00:46:55.710 --> 00:46:58.550 Plus u2j. 00:46:58.550 --> 00:47:02.220 So you have u1 from here to here. 00:47:02.220 --> 00:47:04.070 I don't well you can draw. 00:47:04.070 --> 00:47:06.790 I think some of you can draw really well, especially 00:47:06.790 --> 00:47:10.680 better than me because you took technical drawing. 00:47:10.680 --> 00:47:13.952 How many of you took technical drawing in this glass? 00:47:13.952 --> 00:47:15.860 STUDENT: Only in this class? 00:47:15.860 --> 00:47:17.270 MAGDALENA TODA: In anything. 00:47:17.270 --> 00:47:17.630 STUDENT: In high school. 00:47:17.630 --> 00:47:19.254 MAGDALENA TODA: High school or college. 00:47:19.254 --> 00:47:21.700 STUDENT: I went to it in middle school. 00:47:21.700 --> 00:47:23.660 So it gives you so that [INAUDIBLE] 00:47:23.660 --> 00:47:24.740 and you'd have to draw it. [INAUDIBLE]. 00:47:24.740 --> 00:47:26.364 MAGDALENA TODA: It's really helping you 00:47:26.364 --> 00:47:31.090 with the perspective view, 3D view, from an angle. 00:47:31.090 --> 00:47:33.490 So now you're looking at this u direction 00:47:33.490 --> 00:47:35.930 as being u1i plus u2j. 00:47:35.930 --> 00:47:39.810 And you say, OK, I think I know what's going on. 00:47:39.810 --> 00:47:47.085 You have a displacement in the direction of the x 00:47:47.085 --> 00:47:51.740 coordinate by 1 times h. 00:47:51.740 --> 00:47:54.570 So it's a small displacement that you're talking about. 00:47:54.570 --> 00:47:56.890 And-- yes? 00:47:56.890 --> 00:47:58.326 STUDENT: Why 1 [INAUDIBLE]? 00:47:58.326 --> 00:48:01.917 00:48:01.917 --> 00:48:03.000 MAGDALENA TODA: Which one? 00:48:03.000 --> 00:48:04.319 STUDENT: You said 1 times H. 00:48:04.319 --> 00:48:05.110 MAGDALENA TODA: u1. 00:48:05.110 --> 00:48:08.719 00:48:08.719 --> 00:48:09.760 You will see in a second. 00:48:09.760 --> 00:48:12.416 That's the way you define it. 00:48:12.416 --> 00:48:15.080 This is adjusted information. 00:48:15.080 --> 00:48:18.940 I would like you to tell me what the whole animal is, if I 00:48:18.940 --> 00:48:21.240 want to represent it later. 00:48:21.240 --> 00:48:23.980 And if you can give me some examples. 00:48:23.980 --> 00:48:29.450 And if I go in a y direction with a small displacement, 00:48:29.450 --> 00:48:33.600 from y0, I have to leave and go. 00:48:33.600 --> 00:48:38.030 So I am here at x0y0. 00:48:38.030 --> 00:48:41.960 And this is the x direction and this is the y direction. 00:48:41.960 --> 00:48:47.660 And when I displace a little bit, I displace with the green. 00:48:47.660 --> 00:48:49.890 I displace in this direction. 00:48:49.890 --> 00:48:54.455 I will have to displace and see what happens here. 00:48:54.455 --> 00:48:58.100 00:48:58.100 --> 00:49:02.182 And then in this direction-- I'm not going to write it yet. 00:49:02.182 --> 00:49:04.190 So I'm displacing in this direction 00:49:04.190 --> 00:49:06.530 and in that direction. 00:49:06.530 --> 00:49:08.490 Why am I keeping it h? 00:49:08.490 --> 00:49:12.940 Well, because I have the coordinates x0y0 plus-- 00:49:12.940 --> 00:49:21.125 how do you give me a collinear vector to u, but a small one? 00:49:21.125 --> 00:49:23.480 You say, wait a minute, I know what you mean. 00:49:23.480 --> 00:49:28.410 I start from the point x0, this is p, plus a small multiple 00:49:28.410 --> 00:49:31.990 of the direction you give me. 00:49:31.990 --> 00:49:34.550 So here, you had it before in Calc 2. 00:49:34.550 --> 00:49:40.640 You had t times uru2, which is my vector, u. 00:49:40.640 --> 00:49:55.112 So give me a very small displacement vector 00:49:55.112 --> 00:50:04.665 in the direction u, which is u1u2, u2 as a vector. 00:50:04.665 --> 00:50:06.162 You like angular graphics. 00:50:06.162 --> 00:50:07.660 I don't, but it doesn't matter. 00:50:07.660 --> 00:50:09.829 STUDENT: So basically, h. 00:50:09.829 --> 00:50:11.245 MAGDALENA TODA: So basically, this 00:50:11.245 --> 00:50:16.060 is x0 plus-- you want t or h? 00:50:16.060 --> 00:50:17.640 t or h, it doesn't matter. 00:50:17.640 --> 00:50:23.470 hu1, ui0 plus hu2. 00:50:23.470 --> 00:50:24.430 Why not t? 00:50:24.430 --> 00:50:27.310 Why did I take h? 00:50:27.310 --> 00:50:30.470 It is like time parameter that I'm doing with h, 00:50:30.470 --> 00:50:33.990 but h is a very small time parameter. 00:50:33.990 --> 00:50:36.190 It's an infinitesimally small time. 00:50:36.190 --> 00:50:41.050 It's just a fraction of a second after I start. 00:50:41.050 --> 00:50:43.800 That's why I use little h and not little t. 00:50:43.800 --> 00:50:46.500 00:50:46.500 --> 00:50:52.230 H, in general, indicates a very small time displacement. 00:50:52.230 --> 00:50:58.180 So tried to say, where am I here? 00:50:58.180 --> 00:51:02.480 I'm here, just one step further with a small displacement. 00:51:02.480 --> 00:51:06.150 And that's going to p at this whole thing. 00:51:06.150 --> 00:51:11.030 00:51:11.030 --> 00:51:17.322 Let's call this F of-- the blue one is F of x0y0. 00:51:17.322 --> 00:51:23.140 00:51:23.140 --> 00:51:27.940 And the green altitude, or the altitude of the green point, 00:51:27.940 --> 00:51:29.780 will be what? 00:51:29.780 --> 00:51:31.850 Well, this is something, something, 00:51:31.850 --> 00:51:44.002 and the altitude would be F of x0 plus hu1, y0 plus hu2. 00:51:44.002 --> 00:51:49.630 And I measure how far away the altitudes are. 00:51:49.630 --> 00:51:50.750 They are very close. 00:51:50.750 --> 00:51:53.380 The blue altitude and the green altitude 00:51:53.380 --> 00:51:55.185 varies the displacement. 00:51:55.185 --> 00:51:57.640 And how can I draw that? 00:51:57.640 --> 00:51:58.630 Here. 00:51:58.630 --> 00:52:00.134 You see this one? 00:52:00.134 --> 00:52:02.062 This is the delta z. 00:52:02.062 --> 00:52:05.920 So this thing is like a delta z kind of guy. 00:52:05.920 --> 00:52:06.504 Any questions? 00:52:06.504 --> 00:52:08.711 It's a little bit hard, but you will see in a second. 00:52:08.711 --> 00:52:09.330 Yes, sir? 00:52:09.330 --> 00:52:11.460 STUDENT: Is it like a small displacement 00:52:11.460 --> 00:52:17.290 that has to be perpendicular to the [INAUDIBLE]? 00:52:17.290 --> 00:52:18.202 MAGDALENA TODA: No. 00:52:18.202 --> 00:52:19.785 STUDENT: It's a result of [INAUDIBLE]? 00:52:19.785 --> 00:52:21.282 MAGDALENA TODA: It is in the direction. 00:52:21.282 --> 00:52:22.365 STUDENT: In the direction? 00:52:22.365 --> 00:52:24.290 MAGDALENA TODA: So let's model it better. 00:52:24.290 --> 00:52:27.170 I don't have a three dimensional-- they sent me 00:52:27.170 --> 00:52:29.310 an email this morning from the library saying, 00:52:29.310 --> 00:52:31.780 do you want your three dimensional print-- 00:52:31.780 --> 00:52:35.400 do you want to support the idea of Texas Tech having a three 00:52:35.400 --> 00:52:39.691 dimensional printer available for educational purposes? 00:52:39.691 --> 00:52:41.232 STUDENT: Did you say, of course, yes? 00:52:41.232 --> 00:52:43.235 MAGDALENA TODA: Of course, I would. 00:52:43.235 --> 00:52:45.110 But I don't have a three dimensional printer, 00:52:45.110 --> 00:52:47.590 but you have imagination and imagine 00:52:47.590 --> 00:52:50.450 we have a surface that, again, looks like a hill. 00:52:50.450 --> 00:52:52.710 That's my hand. 00:52:52.710 --> 00:52:58.354 And this engagement ring that I have is actually p0, 00:52:58.354 --> 00:52:59.020 which is x0y0zz. 00:52:59.020 --> 00:53:03.590 00:53:03.590 --> 00:53:09.246 And I'm going in a direction of somebody. 00:53:09.246 --> 00:53:10.246 It doesn't have to be u. 00:53:10.246 --> 00:53:12.240 No, [INAUDIBLE]. 00:53:12.240 --> 00:53:14.450 So I'm going in the direction of u-- yu2, 00:53:14.450 --> 00:53:18.000 is that horizontal thing. 00:53:18.000 --> 00:53:20.340 I'm going in that direction. 00:53:20.340 --> 00:53:22.570 So this is the direction I'm going in 00:53:22.570 --> 00:53:25.430 and I say, OK, where do I go? 00:53:25.430 --> 00:53:29.030 We'll do a small displacement, an infinitesimally small 00:53:29.030 --> 00:53:32.340 displacement in that direction here. 00:53:32.340 --> 00:53:37.970 So the two points are related to one another. 00:53:37.970 --> 00:53:42.480 And you say, but there's such a small difference in altitudes 00:53:42.480 --> 00:53:44.980 because you have an infinitesimally small 00:53:44.980 --> 00:53:47.080 displacement in that direction. 00:53:47.080 --> 00:53:47.580 Yes, I know. 00:53:47.580 --> 00:53:50.955 But when you make the ratio between that small delta 00:53:50.955 --> 00:53:57.670 z and the small h, the ratio could be 65 or 120 minus 32. 00:53:57.670 --> 00:53:59.420 You don't know what you get. 00:53:59.420 --> 00:54:04.290 So just like in general limit of the difference quotient 00:54:04.290 --> 00:54:10.260 being the derivative, you'll get the ratio between some things 00:54:10.260 --> 00:54:12.110 that are very small. 00:54:12.110 --> 00:54:15.050 But in the end, you can get something unexpected. 00:54:15.050 --> 00:54:16.360 Finite or anything. 00:54:16.360 --> 00:54:21.340 Now what do you think this guy-- according 00:54:21.340 --> 00:54:27.280 to your previous Chain Rule preparation. 00:54:27.280 --> 00:54:31.330 I taught you about Chain Rule. 00:54:31.330 --> 00:54:36.440 What will this be if we compute them? 00:54:36.440 --> 00:54:37.830 There is a proof for this. 00:54:37.830 --> 00:54:41.430 It would be like a page or a 2 page proof 00:54:41.430 --> 00:54:44.070 for what I'm claiming to have. 00:54:44.070 --> 00:54:45.870 Or how do you think I'm going to get 00:54:45.870 --> 00:54:49.750 to this without doing the limit of a difference quotient? 00:54:49.750 --> 00:54:51.540 Because if I give you functions and you 00:54:51.540 --> 00:54:53.206 do the limit of the difference quotients 00:54:53.206 --> 00:54:56.970 for some nasty functions, you'll never finish. 00:54:56.970 --> 00:55:01.820 So what do you think we ought to do? 00:55:01.820 --> 00:55:06.152 This is going to be some sort of derivative, right? 00:55:06.152 --> 00:55:09.144 And it's going to be a derivative of what? 00:55:09.144 --> 00:55:11.040 Yes, sir. 00:55:11.040 --> 00:55:14.440 STUDENT: Well, it's going to be like a partial derivative, 00:55:14.440 --> 00:55:20.006 except the plane you're using to cut the surface 00:55:20.006 --> 00:55:22.660 is not going to be in the x direction or the y direction. 00:55:22.660 --> 00:55:24.324 It's going to be along the [? uz. ?] 00:55:24.324 --> 00:55:25.240 MAGDALENA TODA: Right. 00:55:25.240 --> 00:55:28.190 So that is a very good observation. 00:55:28.190 --> 00:55:31.600 And it would be like I would the partial not in this direction, 00:55:31.600 --> 00:55:34.080 not in that direction, but in this direction. 00:55:34.080 --> 00:55:35.330 Let me tell you what this is. 00:55:35.330 --> 00:55:38.030 So according to a theorem, this would 00:55:38.030 --> 00:55:42.620 be df, dx, exactly like The Chain Rule, 00:55:42.620 --> 00:55:49.670 at my favorite point here, x0y0 [INAUDIBLE] 00:55:49.670 --> 00:55:55.070 p times-- now you say, oh, Magdalena, I understand. 00:55:55.070 --> 00:55:56.730 You're doing some sort of derivation. 00:55:56.730 --> 00:56:02.380 The derivative of that with respect to h would be u1. 00:56:02.380 --> 00:56:02.880 Yes. 00:56:02.880 --> 00:56:04.090 It's a Chain Rule. 00:56:04.090 --> 00:56:12.946 So then I go times u1 plus df, dy at the point times u2. 00:56:12.946 --> 00:56:14.790 00:56:14.790 --> 00:56:18.370 And you say, OK, but can I prove that? 00:56:18.370 --> 00:56:21.130 Yes, you could, but to prove that you 00:56:21.130 --> 00:56:26.136 would need to play a game. 00:56:26.136 --> 00:56:30.530 The proof will involve that you multiply up and down 00:56:30.530 --> 00:56:32.770 by an additional expression. 00:56:32.770 --> 00:56:35.364 And then you take limit of a product. 00:56:35.364 --> 00:56:37.320 If you take product, the product of limits, 00:56:37.320 --> 00:56:43.090 and you study them separately until you get to this Actually, 00:56:43.090 --> 00:56:47.360 this is an application of The Chain. 00:56:47.360 --> 00:56:54.440 But I want to come back to what Alexander just notice. 00:56:54.440 --> 00:56:57.550 I can explain this much better if we only 00:56:57.550 --> 00:57:01.864 think of derivative in the direction of i and derivative 00:57:01.864 --> 00:57:02.780 in the direction of j. 00:57:02.780 --> 00:57:04.680 What the heck are those? 00:57:04.680 --> 00:57:07.360 What are they going to be? 00:57:07.360 --> 00:57:13.690 The direction of deritivie-- if I have i instead of u, that 00:57:13.690 --> 00:57:17.179 will make you understand the whole notion much better. 00:57:17.179 --> 00:57:18.970 So what would be the directional derivative 00:57:18.970 --> 00:57:22.650 of in the direction of i only? 00:57:22.650 --> 00:57:23.850 Well, i for an i. 00:57:23.850 --> 00:57:25.132 It goes this way. 00:57:25.132 --> 00:57:27.020 This is a hard lesson. 00:57:27.020 --> 00:57:29.852 And it's advanced calculus rather than Calc 3, 00:57:29.852 --> 00:57:32.220 but you're going to get it. 00:57:32.220 --> 00:57:36.800 So if I go in the direction of i, 00:57:36.800 --> 00:57:40.720 I should have the df, dx, right? 00:57:40.720 --> 00:57:41.890 That should be it. 00:57:41.890 --> 00:57:42.837 Do I? 00:57:42.837 --> 00:57:44.170 STUDENT: Yes, but [INAUDIBLE] 0. 00:57:44.170 --> 00:57:45.170 MAGDALENA TODA: Exactly. 00:57:45.170 --> 00:57:47.620 Was I able to invent something so 00:57:47.620 --> 00:57:53.210 when I come back to what I already know, I recreate df, dx 00:57:53.210 --> 00:57:56.060 and nothing else? 00:57:56.060 --> 00:58:04.752 Precisely because for i as being u, what will be u1 and u2? 00:58:04.752 --> 00:58:06.680 STUDENT: [INAUDIBLE]. 00:58:06.680 --> 00:58:09.110 MAGDALENA TODA: u1 is 1. 00:58:09.110 --> 00:58:11.621 u2 is 0. 00:58:11.621 --> 00:58:12.120 Right? 00:58:12.120 --> 00:58:16.300 Because when we write i as a function of i and j, 00:58:16.300 --> 00:58:19.070 that's 1 times i plus 0 times j. 00:58:19.070 --> 00:58:22.946 So u1 is 1, u2 is zero. 00:58:22.946 --> 00:58:24.110 Thank god. 00:58:24.110 --> 00:58:26.950 According to the anything, this difference quotient 00:58:26.950 --> 00:58:32.155 or the simpler way to define it from the theorem would 00:58:32.155 --> 00:58:34.620 be simply the second goes away. 00:58:34.620 --> 00:58:36.240 It vanishes. 00:58:36.240 --> 00:58:41.540 u1 would be 1 and what I'm left with is df, dx. 00:58:41.540 --> 00:58:45.071 And that's exactly what Alex noticed. 00:58:45.071 --> 00:58:49.350 So the directional derivative is defined, 00:58:49.350 --> 00:58:53.770 as a combination of vectors, such that you recreate 00:58:53.770 --> 00:58:56.350 the directional derivative in the direction of i 00:58:56.350 --> 00:58:59.190 being the partial, df, dx. 00:58:59.190 --> 00:59:02.810 Exactly like you learned before in 11.3. 00:59:02.810 --> 00:59:06.050 And what do I have if I try to recreate 00:59:06.050 --> 00:59:10.141 the directional derivative in the direct of j? 00:59:10.141 --> 00:59:10.640 x0y0. 00:59:10.640 --> 00:59:14.500 We don't explain this much in the book. 00:59:14.500 --> 00:59:17.510 I think on this one, I'm doing a better job than the book. 00:59:17.510 --> 00:59:21.750 So what is df in the direction of j? 00:59:21.750 --> 00:59:23.890 j is this way. 00:59:23.890 --> 00:59:27.130 Well, [INAUDIBLE] is that 1j-- you 00:59:27.130 --> 00:59:32.170 let me write it down-- is 0i plus 1j. 00:59:32.170 --> 00:59:33.550 0 is u1. 00:59:33.550 --> 00:59:36.250 1 is u2. 00:59:36.250 --> 00:59:41.980 So by this formula, I simply should 00:59:41.980 --> 00:59:47.970 get the directional deritive-- I mean, 00:59:47.970 --> 00:59:50.975 directional derivative is the partial deritive-- with respect 00:59:50.975 --> 00:59:56.890 to y at my point times a 1 that I'm not going to write. 00:59:56.890 --> 01:00:07.330 So it's a concoction, so that in the directions of i and j, 01:00:07.330 --> 01:00:10.600 you actually get the partial deritives. 01:00:10.600 --> 01:00:13.020 And everything else is linear algebra. 01:00:13.020 --> 01:00:19.550 So if you have a problem understanding the composition 01:00:19.550 --> 01:00:21.640 of vectors, the sum of vectors, this 01:00:21.640 --> 01:00:25.568 is because-- u1 and u2 are [INAUDIBLE], 01:00:25.568 --> 01:00:28.250 I'm sorry-- this is because you haven't taken 01:00:28.250 --> 01:00:33.348 the linear algebra yet, which teaches you a lot about how 01:00:33.348 --> 01:00:36.330 a vector decomposes in two different directions 01:00:36.330 --> 01:00:39.312 or along the standard canonical bases. 01:00:39.312 --> 01:00:41.860 01:00:41.860 --> 01:00:44.940 Let's see some problems of the type 01:00:44.940 --> 01:00:49.452 that I've always put in the midterm and the same kind 01:00:49.452 --> 01:00:54.642 of problems like we have seen in the final. 01:00:54.642 --> 01:00:57.559 For example 3, is it, guys? 01:00:57.559 --> 01:00:58.100 I don't know. 01:00:58.100 --> 01:00:59.641 Example 3, 4, or something like that? 01:00:59.641 --> 01:01:00.460 STUDENT: 3. 01:01:00.460 --> 01:01:03.130 MAGDALENA TODA: Given z equals F of xy-- 01:01:03.130 --> 01:01:06.430 what do you like best, the value or the hill? 01:01:06.430 --> 01:01:09.250 This appeared in most of my exams. 01:01:09.250 --> 01:01:12.510 x squared plus y squared, circular [INAUDIBLE] 01:01:12.510 --> 01:01:14.470 was one of my favorite examples. 01:01:14.470 --> 01:01:16.470 1 minus x squared minus y squared 01:01:16.470 --> 01:01:22.930 was the circular parabola upside down. 01:01:22.930 --> 01:01:24.330 Which one do you prefer? 01:01:24.330 --> 01:01:25.470 I don't care. 01:01:25.470 --> 01:01:26.495 Which one? 01:01:26.495 --> 01:01:27.370 STUDENT: [INAUDIBLE]. 01:01:27.370 --> 01:01:27.680 MAGDALENA TODA: The [INAUDIBLE]? 01:01:27.680 --> 01:01:29.240 The first one. 01:01:29.240 --> 01:01:30.060 It's easier. 01:01:30.060 --> 01:01:34.870 01:01:34.870 --> 01:01:36.690 And a typical problem. 01:01:36.690 --> 01:01:50.060 Compute the directional derivative of z 01:01:50.060 --> 01:02:00.050 equals F of x and y at the point p of coordinates 1, 1, 2 01:02:00.050 --> 01:02:14.050 in the following directions-- A, i. 01:02:14.050 --> 01:02:15.220 B, j. 01:02:15.220 --> 01:02:18.060 C, i plus j. 01:02:18.060 --> 01:02:24.000 01:02:24.000 --> 01:02:29.180 D, the opposite, minus i, minus j over square 2. 01:02:29.180 --> 01:02:31.960 And E-- 01:02:31.960 --> 01:02:33.460 STUDENT: That's a square root 3. 01:02:33.460 --> 01:02:34.460 MAGDALENA TODA: What? 01:02:34.460 --> 01:02:36.040 STUDENT: You wrote a square root 3. 01:02:36.040 --> 01:02:37.080 MAGDALENA TODA: I wrote square root of 3. 01:02:37.080 --> 01:02:37.705 Thank you guys. 01:02:37.705 --> 01:02:38.880 Thanks for being vigilant. 01:02:38.880 --> 01:02:43.264 So always keep an eye on me because I'm full of surprises, 01:02:43.264 --> 01:02:43.805 good and bad. 01:02:43.805 --> 01:02:46.210 No, just kidding. 01:02:46.210 --> 01:02:47.807 So let's see. 01:02:47.807 --> 01:02:48.932 What do I want to put here? 01:02:48.932 --> 01:02:51.402 Something. 01:02:51.402 --> 01:02:52.390 How about this? 01:02:52.390 --> 01:03:01.282 01:03:01.282 --> 01:03:06.716 3 over root 5, pi plus [? y ?] over 5j. 01:03:06.716 --> 01:03:10.668 Is this a unit vector or not? 01:03:10.668 --> 01:03:12.150 STUDENT: No. 01:03:12.150 --> 01:03:13.474 STUDENT: Yes, it is. 01:03:13.474 --> 01:03:15.140 So you're going to drag the [INAUDIBLE]. 01:03:15.140 --> 01:03:17.031 MAGDALENA TODA: Why is that a unit vector? 01:03:17.031 --> 01:03:18.761 STUDENT: It's missing-- no, it's not. 01:03:18.761 --> 01:03:20.927 MAGDALENA TODA: Then how do I make it a unit vector? 01:03:20.927 --> 01:03:22.875 STUDENT: [INAUDIBLE]. 01:03:22.875 --> 01:03:24.840 STUDENT: [INAUDIBLE]. 01:03:24.840 --> 01:03:28.362 STUDENT: I have to take down-- there's a 3 that has to be 1. 01:03:28.362 --> 01:03:29.346 [INAUDIBLE] 01:03:29.346 --> 01:03:32.298 And the second one has to be 1, on the top, 01:03:32.298 --> 01:03:34.266 to make it a unit vector. 01:03:34.266 --> 01:03:39.200 01:03:39.200 --> 01:03:41.560 MAGDALENA TODA: Give me a unit vector. 01:03:41.560 --> 01:03:46.668 Another one then these easy ones. 01:03:46.668 --> 01:03:48.233 STUDENT: 3 over 5 by 4 or 5. 01:03:48.233 --> 01:03:49.108 MAGDALENA TODA: What? 01:03:49.108 --> 01:03:52.050 STUDENT: 3 over 5 by 4 over 5j. 01:03:52.050 --> 01:03:53.810 MAGDALENA TODA: 3 over-- I cannot hear. 01:03:53.810 --> 01:03:54.080 STUDENT: 3 over 5-- 01:03:54.080 --> 01:03:55.420 MAGDALENA TODA: 3 over 5. 01:03:55.420 --> 01:03:56.970 STUDENT: And 4 over 5j. 01:03:56.970 --> 01:03:58.710 MAGDALENA TODA: And 4 over 5j. 01:03:58.710 --> 01:04:00.560 And why is that a unit vector? 01:04:00.560 --> 01:04:05.160 STUDENT: Because 3 squared is [INAUDIBLE]. 01:04:05.160 --> 01:04:07.284 MAGDALENA TODA: And what do we call these numbers? 01:04:07.284 --> 01:04:08.200 You say, what is that? 01:04:08.200 --> 01:04:10.711 And interview? 01:04:10.711 --> 01:04:12.920 Yes, it is an interview. 01:04:12.920 --> 01:04:13.890 Pythagorean numbers. 01:04:13.890 --> 01:04:16.162 3, 4, and 5 are Pythagorean numbers. 01:04:16.162 --> 01:04:19.180 01:04:19.180 --> 01:04:23.840 So let me think a little bit where I should write. 01:04:23.840 --> 01:04:26.200 Is this seen by the-- yes, it's seen 01:04:26.200 --> 01:04:34.438 by the-- I'll just leave what's important for me 01:04:34.438 --> 01:04:35.875 to solve this problem. 01:04:35.875 --> 01:04:44.990 01:04:44.990 --> 01:04:48.160 A. So what do we do? 01:04:48.160 --> 01:04:55.580 The same thing. i is 1.i plus u, or 1 times i plus u times j. 01:04:55.580 --> 01:04:58.675 So simply, you can write the formula or you can say, 01:04:58.675 --> 01:05:01.430 the heck with the formula. 01:05:01.430 --> 01:05:03.920 You know that df is df, dx. 01:05:03.920 --> 01:05:07.810 The derivative of this at the point p. 01:05:07.810 --> 01:05:14.022 So what you want to do is say, 2x-- are you guys with me? 01:05:14.022 --> 01:05:15.160 STUDENT: Yes. 01:05:15.160 --> 01:05:23.090 MAGDALENA TODA: At the value 1, 1, 2, which is 2. 01:05:23.090 --> 01:05:24.940 And at the end of this exercise, I'm 01:05:24.940 --> 01:05:28.430 going to ask you if there's any connection between-- 01:05:28.430 --> 01:05:30.610 or maybe I will ask you next time. 01:05:30.610 --> 01:05:34.590 Oh, we have time. 01:05:34.590 --> 01:05:37.540 What is d in the direction of j? 01:05:37.540 --> 01:05:41.070 The partial derivative with respect to y. 01:05:41.070 --> 01:05:43.600 Nothing else, but our old friend. 01:05:43.600 --> 01:05:47.680 And our old friend says, I have 2y 01:05:47.680 --> 01:05:51.780 computed for the point p, 1, 1, 2. 01:05:51.780 --> 01:05:52.980 What does it mean? 01:05:52.980 --> 01:05:58.794 Y is 1, so just plug this 1 into the thingy. 01:05:58.794 --> 01:05:59.738 It's 2. 01:05:59.738 --> 01:06:03.990 01:06:03.990 --> 01:06:07.110 Now do I see some-- I'm a scientist. 01:06:07.110 --> 01:06:09.200 I have to find interpretations when 01:06:09.200 --> 01:06:11.210 I get results that coincide. 01:06:11.210 --> 01:06:12.645 It's a pattern. 01:06:12.645 --> 01:06:14.014 Why do I get the same answer? 01:06:14.014 --> 01:06:15.930 STUDENT: Because your functions are symmetric. 01:06:15.930 --> 01:06:16.846 MAGDALENA TODA: Right. 01:06:16.846 --> 01:06:20.070 And more than that, because the function is symmetric, 01:06:20.070 --> 01:06:24.621 it's a quadric that I love, it's just a circular problem. 01:06:24.621 --> 01:06:27.850 It's rotation is symmetric. 01:06:27.850 --> 01:06:33.500 So I just take one parabola, one branch of a parabola, 01:06:33.500 --> 01:06:38.400 and I rotate it by 360 degrees. 01:06:38.400 --> 01:06:45.760 So the slope will be the same in both directions, i and j, 01:06:45.760 --> 01:06:47.255 at the point that I have. 01:06:47.255 --> 01:06:49.860 01:06:49.860 --> 01:06:52.750 Well, it depends on the point. 01:06:52.750 --> 01:06:55.145 If the point is, itself, symmetric 01:06:55.145 --> 01:06:58.490 like that, x and y are the same, one in one, 01:06:58.490 --> 01:07:03.750 I did it on purpose-- if you didn't have one and one, 01:07:03.750 --> 01:07:07.520 you had an x variable and y variable to plug in. 01:07:07.520 --> 01:07:10.960 But your magic point is where? 01:07:10.960 --> 01:07:11.780 Oh my god. 01:07:11.780 --> 01:07:15.370 I don't know how to explain with my hands. 01:07:15.370 --> 01:07:16.650 Here I am, the frame. 01:07:16.650 --> 01:07:19.910 I am the frame. x, y, and z. 01:07:19.910 --> 01:07:21.820 1, 1. 01:07:21.820 --> 01:07:23.220 Go up. 01:07:23.220 --> 01:07:25.060 Where do you meet the vase? 01:07:25.060 --> 01:07:26.900 At c equals 2. 01:07:26.900 --> 01:07:30.356 So it's really symmetric and really beautiful. 01:07:30.356 --> 01:07:34.140 01:07:34.140 --> 01:07:37.800 Next I say, oh, in the direction i plus 01:07:37.800 --> 01:07:43.590 j, which is exactly the direction of this meridian 01:07:43.590 --> 01:07:47.782 that I was talking about, i plus j over square root 2. 01:07:47.782 --> 01:07:50.780 Now I've had students-- that's where I was broken hearted. 01:07:50.780 --> 01:07:53.330 Really, I didn't know what to do, 01:07:53.330 --> 01:07:56.500 how much partial credit to give. 01:07:56.500 --> 01:08:00.680 The definition of direction derivative is very strict. 01:08:00.680 --> 01:08:04.750 It says you cannot take whatever 1 and 2 that you want. 01:08:04.750 --> 01:08:09.490 You cannot multiply them by proportionality. 01:08:09.490 --> 01:08:14.410 You have to have u to be a unit vector. 01:08:14.410 --> 01:08:18.279 And then the directional derivative will be unique. 01:08:18.279 --> 01:08:24.319 If I take 1 and 1 for u1 and u2, then I can take 2 and 2, 01:08:24.319 --> 01:08:26.189 and 7 and 7, and 9 and 9. 01:08:26.189 --> 01:08:28.399 And that's going to be a mess because 01:08:28.399 --> 01:08:32.040 the directional derivative wouldn't be unique anymore. 01:08:32.040 --> 01:08:36.020 And that's why whoever gave this definition, 01:08:36.020 --> 01:08:39.104 I think Euler-- I tried to see in the history who 01:08:39.104 --> 01:08:42.779 was the first mathematician who gave 01:08:42.779 --> 01:08:46.950 the definition of the directional derivative. 01:08:46.950 --> 01:08:49.710 And some people said it was Gateaux 01:08:49.710 --> 01:08:53.196 because that's a french mathematician who first talked 01:08:53.196 --> 01:08:55.231 about the Gateaux derivative, which 01:08:55.231 --> 01:08:56.689 is like the directional derivative, 01:08:56.689 --> 01:08:58.859 but other people said, no, look at Euler's work. 01:08:58.859 --> 01:09:00.290 He was a genius. 01:09:00.290 --> 01:09:04.710 He's the guy who discovered the transcendental number 01:09:04.710 --> 01:09:06.740 e and many other things. 01:09:06.740 --> 01:09:09.080 And the exponential e to the x is also 01:09:09.080 --> 01:09:10.510 from Euler and everything. 01:09:10.510 --> 01:09:12.569 He was one of the fathers of calculus. 01:09:12.569 --> 01:09:19.060 Apparently, he knew the first 32 decimals of the number e. 01:09:19.060 --> 01:09:22.910 And how he got to them is by hand. 01:09:22.910 --> 01:09:24.090 Do you guys know of them? 01:09:24.090 --> 01:09:29.620 2.71828-- and that's all I know. 01:09:29.620 --> 01:09:32.000 The first five decimals. 01:09:32.000 --> 01:09:35.729 Well, he knew 32 of them and he got to them by hand. 01:09:35.729 --> 01:09:39.200 And they are non-repeating, infinitely remaining decimals. 01:09:39.200 --> 01:09:40.460 It's a transcendental number. 01:09:40.460 --> 01:09:41.858 STUDENT: And his 32 are correct? 01:09:41.858 --> 01:09:42.733 MAGDALENA TODA: What? 01:09:42.733 --> 01:09:44.180 STUDENT: His 32 are correct? 01:09:44.180 --> 01:09:46.960 MAGDALENA TODA: His first 32 decimals were correct. 01:09:46.960 --> 01:09:49.790 I don't know what-- I mean, the guy 01:09:49.790 --> 01:09:53.260 was something like-- he was working at night. 01:09:53.260 --> 01:09:56.690 And he would fill out, in one night, hundreds 01:09:56.690 --> 01:10:04.270 of pages, computations, both by hand formulas and numerical. 01:10:04.270 --> 01:10:07.170 So imagine-- of course, he would never make a WeBWork mistake. 01:10:07.170 --> 01:10:10.866 I mean, if we built a time machine, 01:10:10.866 --> 01:10:13.240 and we bring Euler back, and he's at Texas Tech, 01:10:13.240 --> 01:10:16.160 and we make him solve our WeBWork problems, 01:10:16.160 --> 01:10:17.910 I think he would take a thousand problems 01:10:17.910 --> 01:10:19.880 and solve them in one night. 01:10:19.880 --> 01:10:21.850 He need to know how to type, so we 01:10:21.850 --> 01:10:24.250 have to teach him how to type. 01:10:24.250 --> 01:10:27.650 But he would be able to compute what you guys have, 01:10:27.650 --> 01:10:31.790 all those numerical answers, in his head. 01:10:31.790 --> 01:10:35.250 He was a scary fellow. 01:10:35.250 --> 01:10:41.380 So u has to be [INAUDIBLE] in some way, made unique. 01:10:41.380 --> 01:10:43.350 u1 and u2. 01:10:43.350 --> 01:10:45.920 I have students-- that's where the story started-- 01:10:45.920 --> 01:10:49.990 who were very good, very smart, both honors and non-honors, who 01:10:49.990 --> 01:10:54.350 took u1 to be 1, u2 to be 2 because they thought direction 01:10:54.350 --> 01:11:00.640 1 and 1, which is not made unique as a direction, unitary. 01:11:00.640 --> 01:11:03.420 And they plugged in here 1, they plugged in here 1, 01:11:03.420 --> 01:11:08.210 they got these correctly, what was I supposed to give them, as 01:11:08.210 --> 01:11:09.066 a [? friend? ?] 01:11:09.066 --> 01:11:09.941 STUDENT: [INAUDIBLE]. 01:11:09.941 --> 01:11:10.430 MAGDALENA TODA: What? 01:11:10.430 --> 01:11:11.530 STUDENT: [INAUDIBLE]. 01:11:11.530 --> 01:11:12.696 MAGDALENA TODA: I gave them. 01:11:12.696 --> 01:11:14.413 How much do you think? 01:11:14.413 --> 01:11:15.204 You should know me. 01:11:15.204 --> 01:11:16.198 STUDENT: [INAUDIBLE]. 01:11:16.198 --> 01:11:17.192 STUDENT: Full. 01:11:17.192 --> 01:11:18.186 MAGDALENA TODA: 60%. 01:11:18.186 --> 01:11:19.504 No. 01:11:19.504 --> 01:11:20.920 Some people don't give any credit, 01:11:20.920 --> 01:11:22.720 so pay attention to this. 01:11:22.720 --> 01:11:31.498 In this case, this has to be 1 over square root 01:11:31.498 --> 01:11:41.602 of 2 times the derivative of f at x, which is computed before 01:11:41.602 --> 01:11:50.570 at the point, plus 1 over square root of 2 times the derivative 01:11:50.570 --> 01:11:52.025 of the function. 01:11:52.025 --> 01:11:54.450 Again, compute it at the same place. 01:11:54.450 --> 01:12:02.514 Which is, oh my god, square root of 2 plus square root of 2, 01:12:02.514 --> 01:12:04.470 which is 2 square root of 2. 01:12:04.470 --> 01:12:20.630 01:12:20.630 --> 01:12:29.746 And finally, the derivative of F at the same point-- I 01:12:29.746 --> 01:12:31.222 should have put at the point. 01:12:31.222 --> 01:12:35.160 Like a physicist would say, at p. 01:12:35.160 --> 01:12:38.290 That would make you familiar with this notation. 01:12:38.290 --> 01:12:40.330 And then measured at what? 01:12:40.330 --> 01:12:43.540 The opposite direction, minus i minus j. 01:12:43.540 --> 01:12:46.060 And now I'm getting lazy and I'm going to ask you 01:12:46.060 --> 01:12:48.624 what the answer will be. 01:12:48.624 --> 01:12:50.040 STUDENT: 2 minus square root of 2. 01:12:50.040 --> 01:12:53.045 MAGDALENA TODA: So you see, there is another pattern. 01:12:53.045 --> 01:12:55.190 In the opposite direction, the direction 01:12:55.190 --> 01:12:59.500 of the derivative in this case would just be the negative one. 01:12:59.500 --> 01:13:03.190 What if we took this directional derivative in absolute value? 01:13:03.190 --> 01:13:05.374 Because you see, in this direction, 01:13:05.374 --> 01:13:07.610 there's a positive directional derivaty. 01:13:07.610 --> 01:13:11.930 In the other direction, it's like it's because-- I know why. 01:13:11.930 --> 01:13:13.600 I'm a vase. 01:13:13.600 --> 01:13:18.132 So in the direction i plus j over square root of 2, 01:13:18.132 --> 01:13:20.305 the directional derivative will be positive. 01:13:20.305 --> 01:13:21.780 It goes up. 01:13:21.780 --> 01:13:24.260 But in the direction minus i minus 01:13:24.260 --> 01:13:28.290 j, which is the opposite, over square root of 2, it goes down. 01:13:28.290 --> 01:13:30.620 So the slope is negative. 01:13:30.620 --> 01:13:32.200 So that's why we have negative. 01:13:32.200 --> 01:13:34.770 Everything you get in life or in math, 01:13:34.770 --> 01:13:36.410 you have to find an interpretation. 01:13:36.410 --> 01:13:40.354 01:13:40.354 --> 01:13:44.460 Sometimes in life and mathematics, things are subtle. 01:13:44.460 --> 01:13:46.850 People will say one thing and they mean another thing. 01:13:46.850 --> 01:13:49.824 You have to try to see beyond their words. 01:13:49.824 --> 01:13:50.700 That's sad. 01:13:50.700 --> 01:13:53.760 And in mathematics, you have to try to see beyond the numbers. 01:13:53.760 --> 01:13:55.420 You see a pattern. 01:13:55.420 --> 01:13:58.050 So being in opposite directions, I 01:13:58.050 --> 01:14:02.239 got opposite signs of the directional derivative 01:14:02.239 --> 01:14:03.530 because I have opposite slopes. 01:14:03.530 --> 01:14:07.750 01:14:07.750 --> 01:14:10.992 What else do I want to learn in this example? 01:14:10.992 --> 01:14:12.120 One last thing. 01:14:12.120 --> 01:14:13.286 STUDENT: E. 01:14:13.286 --> 01:14:22.670 MAGDALENA TODA: E. So I have the same thing. 01:14:22.670 --> 01:14:25.242 So it's not going to matter, the direction 01:14:25.242 --> 01:14:26.920 is the only thing that changes. 01:14:26.920 --> 01:14:28.960 These guys are the same. 01:14:28.960 --> 01:14:33.630 The partials are the same at the same point. 01:14:33.630 --> 01:14:35.130 I'm not going to worry about them. 01:14:35.130 --> 01:14:39.200 So I get 2 or both. 01:14:39.200 --> 01:14:41.260 What changes is the blue guys. 01:14:41.260 --> 01:14:47.847 They are going to be 3 over 5 and 4 over 5. 01:14:47.847 --> 01:14:53.620 01:14:53.620 --> 01:14:56.270 And what do I get? 01:14:56.270 --> 01:15:04.885 I get-- right? 01:15:04.885 --> 01:15:09.110 01:15:09.110 --> 01:15:12.890 Now I want to tell you something-- 01:15:12.890 --> 01:15:16.100 I already anticipated something last time. 01:15:16.100 --> 01:15:21.280 And let me tell you what I said last time. 01:15:21.280 --> 01:15:25.970 01:15:25.970 --> 01:15:27.930 Maybe I should not erase-- well, I 01:15:27.930 --> 01:15:30.240 have to erase this whether I like it or not. 01:15:30.240 --> 01:15:33.800 01:15:33.800 --> 01:15:35.775 And now I'll review what this was. 01:15:35.775 --> 01:15:38.180 What was this? d equals x squared plus y squared? 01:15:38.180 --> 01:15:38.960 Yes or no? 01:15:38.960 --> 01:15:41.410 STUDENT: Yes. 01:15:41.410 --> 01:15:46.035 MAGDALENA TODA: So what did I say last time? 01:15:46.035 --> 01:15:52.730 We have no result. We noticed it last time. 01:15:52.730 --> 01:15:55.060 We did not prove it. 01:15:55.060 --> 01:16:08.570 We did not prove it, only found it experimentally 01:16:08.570 --> 01:16:11.810 using our physical common sense. 01:16:11.810 --> 01:16:16.990 When you have a function z equals F of xy, 01:16:16.990 --> 01:16:30.878 we studied the maximum rate of change 01:16:30.878 --> 01:16:39.560 at the point x0y0 in the domain, assuming this is a c1 function. 01:16:39.560 --> 01:16:40.890 I don't know. 01:16:40.890 --> 01:16:44.330 Maximum rate of change was a magic thing. 01:16:44.330 --> 01:16:48.130 And you probably thought, what in the world is that? 01:16:48.130 --> 01:17:01.120 And we also said, this maximum for the rate of change 01:17:01.120 --> 01:17:23.849 is always attained in the direction of the gradient. 01:17:23.849 --> 01:17:31.310 01:17:31.310 --> 01:17:38.050 So you realize that it's the steepest ascent, 01:17:38.050 --> 01:17:40.920 the way it's called in many, many other fields, 01:17:40.920 --> 01:17:42.832 but mathematics. 01:17:42.832 --> 01:17:45.420 Or the steepest descent. 01:17:45.420 --> 01:17:51.530 01:17:51.530 --> 01:17:58.240 Now if it's an ascent, then it's in the direction gradient of F. 01:17:58.240 --> 01:18:00.210 But if it's a descent, it's going 01:18:00.210 --> 01:18:04.630 to be in the opposite direction, minus gradient of F. 01:18:04.630 --> 01:18:07.580 But then I [INAUDIBLE] first of all, 01:18:07.580 --> 01:18:11.890 it's not the same direction, if you have opposites. 01:18:11.890 --> 01:18:14.750 Well, direction is sort of given by one line. 01:18:14.750 --> 01:18:18.840 Whether you take this or the opposite, it's the same thing. 01:18:18.840 --> 01:18:21.280 What this means is that we say direction 01:18:21.280 --> 01:18:25.680 and we didn't [? unitarize ?] it. 01:18:25.680 --> 01:18:31.050 So we could say, or gradient of F 01:18:31.050 --> 01:18:35.980 over length of gradient of F. Or minus gradient of F 01:18:35.980 --> 01:18:39.750 over length of gradient of F. Can this theorem be proved? 01:18:39.750 --> 01:18:41.330 Yes, it can be proved. 01:18:41.330 --> 01:18:45.370 We are going to discuss a little bit more next time about it, 01:18:45.370 --> 01:18:49.360 but I want to tell you a big disclosure today. 01:18:49.360 --> 01:18:55.020 This maximum rate of change is the directional derivative. 01:18:55.020 --> 01:19:07.808 This maximum rate of change is exactly 01:19:07.808 --> 01:19:15.630 the directional derivative in the direction 01:19:15.630 --> 01:19:35.068 of the gradient, which is also the magnitude of the gradient. 01:19:35.068 --> 01:19:43.380 01:19:43.380 --> 01:19:47.230 And you'll say, wait a minute, what? 01:19:47.230 --> 01:19:48.360 What did you say? 01:19:48.360 --> 01:19:51.082 Let's first verify my claim. 01:19:51.082 --> 01:19:53.350 I'm not even sure my claim is true. 01:19:53.350 --> 01:19:55.480 We will see next time. 01:19:55.480 --> 01:19:59.830 Can I verify my claim on one example? 01:19:59.830 --> 01:20:01.690 Well, OK. 01:20:01.690 --> 01:20:04.870 Maximum rate of change would be exactly 01:20:04.870 --> 01:20:07.964 as the directional derivative and the direction 01:20:07.964 --> 01:20:08.630 of the gradient? 01:20:08.630 --> 01:20:10.070 I don't know about that. 01:20:10.070 --> 01:20:11.382 That all sounds crazy. 01:20:11.382 --> 01:20:12.825 So what do I have to compute? 01:20:12.825 --> 01:20:16.673 I have to compute that directional derivative 01:20:16.673 --> 01:20:21.640 of, let's say, my function F in the direction of the gradient-- 01:20:21.640 --> 01:20:22.956 what is the gradient? 01:20:22.956 --> 01:20:26.270 01:20:26.270 --> 01:20:28.815 We have to figure it out. 01:20:28.815 --> 01:20:30.640 We did it last time, but you forgot. 01:20:30.640 --> 01:20:37.360 So for this guy, nabla F, what will be the gradient? 01:20:37.360 --> 01:20:39.740 Where is my function? 01:20:39.740 --> 01:20:47.620 Nabla F will be 2x, 2y, right? 01:20:47.620 --> 01:20:51.750 Which means 2xi plus 2yj, right? 01:20:51.750 --> 01:20:54.676 But if I'm at the point p, what does it mean? 01:20:54.676 --> 01:20:59.020 At the point p, it means that I have 2 times i plus 2 times j, 01:20:59.020 --> 01:21:00.342 right? 01:21:00.342 --> 01:21:06.030 And what is the magnitude of the gradient? 01:21:06.030 --> 01:21:08.140 Yes. 01:21:08.140 --> 01:21:13.292 The magnitude of the gradient is somebody I know, which is what? 01:21:13.292 --> 01:21:18.583 Which is square root of 2 squared plus 2 squared. 01:21:18.583 --> 01:21:20.784 I cannot do that now. 01:21:20.784 --> 01:21:21.950 What's the square root of 8? 01:21:21.950 --> 01:21:22.839 STUDENT: 2 root 2. 01:21:22.839 --> 01:21:23.880 MAGDALENA TODA: 2 root 2. 01:21:23.880 --> 01:21:24.629 This is a pattern. 01:21:24.629 --> 01:21:25.230 2 root 2. 01:21:25.230 --> 01:21:27.240 I've seen this 2 root 2 again somewhere. 01:21:27.240 --> 01:21:28.880 Where the heck have I seen it? 01:21:28.880 --> 01:21:29.922 STUDENT: That was the directional derivative. 01:21:29.922 --> 01:21:31.713 MAGDALENA TODA: The directional derivative. 01:21:31.713 --> 01:21:33.320 So the claim may be right. 01:21:33.320 --> 01:21:36.452 It says it is the directional derivative in the direction 01:21:36.452 --> 01:21:37.810 of the gradient. 01:21:37.810 --> 01:21:40.920 But is this really the direction of the gradient? 01:21:40.920 --> 01:21:42.770 Yes. 01:21:42.770 --> 01:21:45.910 Because when you compote the direction for the gradient, 2y 01:21:45.910 --> 01:21:52.190 plus 2j, you don't mean 2i plus 2j as a twice i plus j, 01:21:52.190 --> 01:21:55.647 you mean the unit vector correspondent to that. 01:21:55.647 --> 01:21:57.230 So what is the direction corresponding 01:21:57.230 --> 01:22:00.550 to the gradient 2i plus 2j? 01:22:00.550 --> 01:22:01.850 STUDENT: i plus j [? over 2. ?] 01:22:01.850 --> 01:22:02.850 MAGDALENA TODA: Exactly. 01:22:02.850 --> 01:22:06.140 U equals i plus j divided by square 2. 01:22:06.140 --> 01:22:09.310 So this is the directional derivative 01:22:09.310 --> 01:22:13.120 in the direction of the gradient at the point p, which is 2 root 01:22:13.120 --> 01:22:13.620 2. 01:22:13.620 --> 01:22:18.250 And it's the same thing-- for some reason that's mysterious 01:22:18.250 --> 01:22:19.860 and we will see next time. 01:22:19.860 --> 01:22:23.340 For some mysterious reason you get exactly the same 01:22:23.340 --> 01:22:27.780 as the length of the gradient vector. 01:22:27.780 --> 01:22:30.460 We will see about this mystery next time. 01:22:30.460 --> 01:22:35.220 I have you enough to torment you until Tuesday. 01:22:35.220 --> 01:22:38.280 What have you promised me besides doing the homework? 01:22:38.280 --> 01:22:39.756 STUDENT: To read the book. 01:22:39.756 --> 01:22:41.130 MAGDALENA TODA: To read the book. 01:22:41.130 --> 01:22:41.950 You're very smart. 01:22:41.950 --> 01:22:43.620 Please, read the book. 01:22:43.620 --> 01:22:45.078 All the examples in the book. 01:22:45.078 --> 01:22:47.070 They are short. 01:22:47.070 --> 01:22:48.066 Thank you so much. 01:22:48.066 --> 01:22:50.556 Have a wonderful weekend and I'll 01:22:50.556 --> 01:22:54.540 talk to you on Tuesday about anything you have trouble with. 01:22:54.540 --> 01:22:57.030 When is the homework due? 01:22:57.030 --> 01:22:59.022 STUDENT: Saturday. 01:22:59.022 --> 01:23:00.514 MAGDALENA TODA: On Saturday. 01:23:00.514 --> 01:23:01.014 I was mean. 01:23:01.014 --> 01:23:04.500 I should have given it you until Sunday night, but-- 01:23:04.500 --> 01:23:05.943 STUDENT: Yes. 01:23:05.943 --> 01:23:08.484 MAGDALENA TODA: Do you want me to make it until Sunday night? 01:23:08.484 --> 01:23:08.982 STUDENT: Yes. 01:23:08.982 --> 01:23:10.148 MAGDALENA TODA: At midnight? 01:23:10.148 --> 01:23:10.974 STUDENT: Yes. 01:23:10.974 --> 01:23:12.966 MAGDALENA TODA: I'll do that. 01:23:12.966 --> 01:23:14.958 I will extend it. 01:23:14.958 --> 01:23:19.440 01:23:19.440 --> 01:23:22.428 STUDENT: She asked, I said yes. 01:23:22.428 --> 01:23:23.922 STUDENT: Why did you do that, dude? 01:23:23.922 --> 01:23:28.238 Come on, my life is ruined now because I have more time 01:23:28.238 --> 01:23:29.987 to work on my homework. 01:23:29.987 --> 01:23:31.820 MAGDALENA TODA: And I've ruined your Sunday. 01:23:31.820 --> 01:23:32.361 STUDENT: Yes. 01:23:32.361 --> 01:23:33.020 No. 01:23:33.020 --> 01:23:33.920 MAGDALENA TODA: No. 01:23:33.920 --> 01:23:36.362 Actually, I know why I did that. 01:23:36.362 --> 01:23:37.820 I thought that the 28th of February 01:23:37.820 --> 01:23:42.620 is the last day of the month, but it's a short month. 01:23:42.620 --> 01:23:45.020 So if we [? try it, ?] we have to extend the months 01:23:45.020 --> 01:23:48.620 a little bit by pulling it by one more day. 01:23:48.620 --> 01:23:49.754 STUDENT: We did? 01:23:49.754 --> 01:23:51.920 MAGDALENA TODA: The first of March is Sunday, right? 01:23:51.920 --> 01:23:53.720 STUDENT: Yes. 01:23:53.720 --> 01:23:55.520 [INTERPOSING VOICES] 01:23:55.520 --> 01:24:05.812 01:24:05.812 --> 01:24:07.520 STUDENT: You're going to miss the speech. 01:24:07.520 --> 01:24:09.320 STUDENT: Oh, we're doing that? 01:24:09.320 --> 01:24:10.520 STUDENT: You're in English? 01:24:10.520 --> 01:24:11.395 STUDENT: [INAUDIBLE]. 01:24:11.395 --> 01:24:13.987 01:24:13.987 --> 01:24:15.320 STUDENT: You don't know English? 01:24:15.320 --> 01:24:15.920 Why are you talking English? 01:24:15.920 --> 01:24:17.720 That's what my father used to say. 01:24:17.720 --> 01:24:19.570 You don't know your own tongue?