WEBVTT 00:00:00.000 --> 00:00:02.150 PROFESSOR TODA: Any questions so far? 00:00:02.150 --> 00:00:06.450 I mean, conceptual, theoretical questions first, 00:00:06.450 --> 00:00:08.754 and then we will do the second part 00:00:08.754 --> 00:00:10.185 of [INAUDIBLE] applications. 00:00:10.185 --> 00:00:14.478 Then you can ask for more questions. 00:00:14.478 --> 00:00:15.909 No questions so far? 00:00:15.909 --> 00:00:18.676 I have not finished 11-4. 00:00:18.676 --> 00:00:25.745 I still owe you a long explanation about 11-4. 00:00:25.745 --> 00:00:28.270 Hopefully it's going to make more sense today 00:00:28.270 --> 00:00:30.850 than it made last time. 00:00:30.850 --> 00:00:34.220 I was just saying that I'm doing 11-4. 00:00:34.220 --> 00:00:36.350 This is a lot of chapter. 00:00:36.350 --> 00:00:47.958 So second part of 11-4 today-- tangent plane and applications. 00:00:47.958 --> 00:00:50.870 00:00:50.870 --> 00:00:53.739 Now, we don't say what those applications are 00:00:53.739 --> 00:00:58.970 from the start, but these are some very important concepts 00:00:58.970 --> 00:01:00.838 called the total differential. 00:01:00.838 --> 00:01:07.156 00:01:07.156 --> 00:01:13.960 And the linear approximation number 00:01:13.960 --> 00:01:15.495 is going under the [INAUDIBLE]. 00:01:15.495 --> 00:01:17.180 Thank you, sir. 00:01:17.180 --> 00:01:23.880 Linear approximation for functions of the type z 00:01:23.880 --> 00:01:29.480 equals f of xy, which means graphs of two variables. 00:01:29.480 --> 00:01:33.700 At the end of the chapter, I'll take the notes copy from you. 00:01:33.700 --> 00:01:37.130 So don't give me anything until it's over. 00:01:37.130 --> 00:01:38.910 When is that going to be over? 00:01:38.910 --> 00:01:41.862 We have four more sections to go. 00:01:41.862 --> 00:01:47.140 So I guess right before spring break you give me 00:01:47.140 --> 00:01:50.330 the notes for chapter 11. 00:01:50.330 --> 00:01:52.470 All right, and then I'm thinking of making 00:01:52.470 --> 00:01:54.894 copies of both chapters. 00:01:54.894 --> 00:02:00.260 You get the-- I'm distributing them to you. 00:02:00.260 --> 00:02:02.660 I haven't started and yet go ahead. 00:02:02.660 --> 00:02:09.600 Could anybody tell me what the equation 00:02:09.600 --> 00:02:13.520 that we used last time-- we proved it, actually. 00:02:13.520 --> 00:02:16.260 00:02:16.260 --> 00:02:20.982 What is the equation of the tangent plane 00:02:20.982 --> 00:02:27.266 to a smooth surface or a patch of a surface at the point 00:02:27.266 --> 00:02:33.784 m of coordinates x0, y0, z0, where the graph is 00:02:33.784 --> 00:02:37.480 given by z equals f of x and y. 00:02:37.480 --> 00:02:40.860 I'm going to label it on the patch of a surface. 00:02:40.860 --> 00:02:44.030 OK, imagine it labeled brown there. 00:02:44.030 --> 00:02:52.540 And can somebody tell me the equation of the other plane? 00:02:52.540 --> 00:02:54.450 But because you have better memory, 00:02:54.450 --> 00:03:00.300 being much younger, about 25 years younger than me or so. 00:03:00.300 --> 00:03:05.650 So could you-- could anybody tell me what the tangent 00:03:05.650 --> 00:03:08.620 planes equation-- I'll start. 00:03:08.620 --> 00:03:10.292 And it's going to come to you. 00:03:10.292 --> 00:03:14.880 z minus z0 equals. 00:03:14.880 --> 00:03:15.943 And now let's see. 00:03:15.943 --> 00:03:18.382 I'll pick a nice color. 00:03:18.382 --> 00:03:19.105 I'll wait. 00:03:19.105 --> 00:03:21.845 00:03:21.845 --> 00:03:23.650 STUDENT: fx of x. 00:03:23.650 --> 00:03:27.220 PROFESSOR TODA: f sub x, the partial derivative measured 00:03:27.220 --> 00:03:35.780 at f0 i0 times the quantity x minus x0 plus-- 00:03:35.780 --> 00:03:36.980 STUDENT: f sub y. 00:03:36.980 --> 00:03:38.730 PROFESSOR TODA: f sub y, excellent. 00:03:38.730 --> 00:03:40.676 f sub y. 00:03:40.676 --> 00:03:41.924 STUDENT: x0, y0. 00:03:41.924 --> 00:03:44.930 PROFESSOR TODA: x0, y0 times y minus y0. 00:03:44.930 --> 00:03:49.259 00:03:49.259 --> 00:03:50.221 OK. 00:03:50.221 --> 00:03:51.630 All right. 00:03:51.630 --> 00:03:59.110 Now thinking of what those quantities mean, x minus x0, y 00:03:59.110 --> 00:04:03.730 minus y0, z minus z0, what are they? 00:04:03.730 --> 00:04:06.830 They are small displacements, aren't they? 00:04:06.830 --> 00:04:10.380 I mean, what does it mean small displacement? 00:04:10.380 --> 00:04:20.860 Imagine that you are near the point on both surfaces. 00:04:20.860 --> 00:04:23.500 So what is a small neighborhood-- 00:04:23.500 --> 00:04:27.940 what's a typical small neighborhood [INAUDIBLE]? 00:04:27.940 --> 00:04:30.280 It's a disk, right? 00:04:30.280 --> 00:04:32.672 There are many kinds of neighborhoods, but one of them, 00:04:32.672 --> 00:04:36.956 I'd say, would be this open disk, OK? 00:04:36.956 --> 00:04:38.860 I'll draw that. 00:04:38.860 --> 00:04:44.712 Now, if I have a red point-- I don't 00:04:44.712 --> 00:04:53.190 know how to do that pink point-- somewhere nearby in planes-- 00:04:53.190 --> 00:04:54.550 this is the plane. 00:04:54.550 --> 00:04:59.356 In plane, I have this point that is close. 00:04:59.356 --> 00:05:01.030 And that point is xyz. 00:05:01.030 --> 00:05:04.350 00:05:04.350 --> 00:05:08.750 And you think, OK, can I visualize that better? 00:05:08.750 --> 00:05:11.790 Well, guys, it's hard to visualize that better. 00:05:11.790 --> 00:05:14.692 But I'll draw a triangle [? doing ?] a better job. 00:05:14.692 --> 00:05:17.437 00:05:17.437 --> 00:05:18.145 That's the frame. 00:05:18.145 --> 00:05:22.404 00:05:22.404 --> 00:05:24.869 This is a surface. 00:05:24.869 --> 00:05:27.450 Imagine it's a surface, OK? 00:05:27.450 --> 00:05:32.060 That's the point of x0, y0. 00:05:32.060 --> 00:05:34.720 [? It's ?] the 0 and that. 00:05:34.720 --> 00:05:36.690 Where is the point xyz again? 00:05:36.690 --> 00:05:40.640 The point xyz is not on the pink stuff. 00:05:40.640 --> 00:05:41.790 This is a pink surface. 00:05:41.790 --> 00:05:45.330 It looks like Pepto Bismol or something. 00:05:45.330 --> 00:05:46.310 You shaded it. 00:05:46.310 --> 00:05:47.065 No. 00:05:47.065 --> 00:05:48.230 That's not what I want. 00:05:48.230 --> 00:05:55.560 I want the close enough point on the blue plane. 00:05:55.560 --> 00:06:01.180 It's actually in the blue plane pie and this guy would be xyz. 00:06:01.180 --> 00:06:05.190 So now say, OK, how far I x be from x0? 00:06:05.190 --> 00:06:06.090 Well, I don't know. 00:06:06.090 --> 00:06:13.510 We would have to check the points, the set 0, 00:06:13.510 --> 00:06:15.948 check the blue point. 00:06:15.948 --> 00:06:18.470 This is x. 00:06:18.470 --> 00:06:23.940 So between x and x0, I have this difference, 00:06:23.940 --> 00:06:34.070 which is delta x displacement, displacement along the x-axis, 00:06:34.070 --> 00:06:38.915 away from the point, fixed point. 00:06:38.915 --> 00:06:41.825 00:06:41.825 --> 00:06:44.735 This is the fixed point, this point. 00:06:44.735 --> 00:06:47.160 This point is p. 00:06:47.160 --> 00:06:48.150 OK. 00:06:48.150 --> 00:06:51.233 y minus y0, let's call that delta y, which 00:06:51.233 --> 00:06:53.426 is the displacement along the y-axis. 00:06:53.426 --> 00:06:56.425 00:06:56.425 --> 00:07:02.020 And then the z minus z0 can be. 00:07:02.020 --> 00:07:05.700 Just because I'm a mathematician and I don't like writing down 00:07:05.700 --> 00:07:11.280 a lot, I would use s batch as I can, 00:07:11.280 --> 00:07:16.780 compact symbols, to speed up my computation. 00:07:16.780 --> 00:07:19.200 So I can rewrite this whole thing 00:07:19.200 --> 00:07:27.560 as a delta z equals f sub x, x0 y0, which is a number. 00:07:27.560 --> 00:07:28.520 It's a slope. 00:07:28.520 --> 00:07:31.570 We discussed about that last time. 00:07:31.570 --> 00:07:33.875 We even went skiing last time, when 00:07:33.875 --> 00:07:38.290 we said that's like the slope in-- what's the x direction? 00:07:38.290 --> 00:07:41.970 Slope in the x direction and slope in the y direction 00:07:41.970 --> 00:07:49.000 on the graph that was the white covered with snow hill. 00:07:49.000 --> 00:07:50.886 That was what we had last time. 00:07:50.886 --> 00:07:54.546 Delta x plus f sub 0, another slope 00:07:54.546 --> 00:07:56.742 in the y direction, delta y. 00:07:56.742 --> 00:08:01.622 00:08:01.622 --> 00:08:07.890 And fortunately-- OK, the book is a very good book, obviously, 00:08:07.890 --> 00:08:09.250 right? 00:08:09.250 --> 00:08:15.630 But I wish we could've done certain things better in terms 00:08:15.630 --> 00:08:21.800 of comparisons between this notion in Calc III 00:08:21.800 --> 00:08:27.260 and some corresponding notion in Calc I. 00:08:27.260 --> 00:08:29.610 So you're probably thinking, what the heck 00:08:29.610 --> 00:08:31.240 is this witch thinking about? 00:08:31.240 --> 00:08:34.590 Well, I'm thinking of something that you 00:08:34.590 --> 00:08:39.730 may want to remember from Calc I. 00:08:39.730 --> 00:08:42.880 And that's going to come into place beautifully 00:08:42.880 --> 00:08:47.810 right now because you have the Calc I, Calc III comparison. 00:08:47.810 --> 00:08:52.799 And that's why it would be great-- the books don't even 00:08:52.799 --> 00:08:55.270 talk about this comparison. 00:08:55.270 --> 00:08:59.810 In Calc I, I reminded you about Mr. Leibniz. 00:08:59.810 --> 00:09:01.110 He was a very nice guy. 00:09:01.110 --> 00:09:02.570 I have no idea, right? 00:09:02.570 --> 00:09:04.130 Never met him. 00:09:04.130 --> 00:09:06.660 One of the fathers of calculus. 00:09:06.660 --> 00:09:10.630 And he introduced the so-called Leibniz notation. 00:09:10.630 --> 00:09:15.720 And one of you in office hours last Wednesday 00:09:15.720 --> 00:09:19.280 told me, so the Leibnitz notation 00:09:19.280 --> 00:09:23.455 for a function g of x-- I'm intentionally 00:09:23.455 --> 00:09:26.273 changing notation-- is what? 00:09:26.273 --> 00:09:31.627 Well, this is just the derivative 00:09:31.627 --> 00:09:34.000 which is the limit of the different quotients 00:09:34.000 --> 00:09:38.480 of your delta g over delta x-- as done by some 00:09:38.480 --> 00:09:43.180 blutches-- 0, right, which would be the same as lim 00:09:43.180 --> 00:09:50.840 of g of x minus g of x0 over x minus x0 as x approaches x0, 00:09:50.840 --> 00:09:52.070 right? 00:09:52.070 --> 00:09:52.570 Right. 00:09:52.570 --> 00:09:57.720 So we've done that in Calc I. But it was a long time ago. 00:09:57.720 --> 00:10:00.630 My mission is to teach you all Calc III, 00:10:00.630 --> 00:10:03.835 but I feel that my mission is also 00:10:03.835 --> 00:10:08.635 to teach you what you may not remember very well from Calc I, 00:10:08.635 --> 00:10:11.640 because everything is related. 00:10:11.640 --> 00:10:17.690 So what was the way we could have written this, 00:10:17.690 --> 00:10:21.260 not delta g over delta x equals g prime. 00:10:21.260 --> 00:10:22.486 No. 00:10:22.486 --> 00:10:29.040 But it's an approximation of g prime around a very small 00:10:29.040 --> 00:10:33.745 [INAUDIBLE], very close to x0. 00:10:33.745 --> 00:10:36.550 00:10:36.550 --> 00:10:39.676 So if you wanted to rewrite this approximation, 00:10:39.676 --> 00:10:42.091 how would you have rewritten it? 00:10:42.091 --> 00:10:47.410 00:10:47.410 --> 00:10:48.140 Delta g-- 00:10:48.140 --> 00:10:54.866 00:10:54.866 --> 00:10:57.310 STUDENT: g prime sub x. 00:10:57.310 --> 00:11:02.490 PROFESSOR TODA: g prime of x0 times delta x. 00:11:02.490 --> 00:11:03.930 OK? 00:11:03.930 --> 00:11:08.280 Now, why this approximation? 00:11:08.280 --> 00:11:11.710 What if I had put equal? 00:11:11.710 --> 00:11:14.120 If I had put equal, it would be all nonsense. 00:11:14.120 --> 00:11:15.405 Why? 00:11:15.405 --> 00:11:19.210 Well, say, Magdalena, if you put equal, it's another object. 00:11:19.210 --> 00:11:19.770 What object? 00:11:19.770 --> 00:11:20.330 OK. 00:11:20.330 --> 00:11:22.120 Let's look at the objects. 00:11:22.120 --> 00:11:22.995 Let's draw a picture. 00:11:22.995 --> 00:11:25.730 00:11:25.730 --> 00:11:27.176 This is g. 00:11:27.176 --> 00:11:28.622 This is x0. 00:11:28.622 --> 00:11:30.560 This is g of x. 00:11:30.560 --> 00:11:32.440 What's g prime? 00:11:32.440 --> 00:11:39.420 g prime-- thank god-- is the slope of g prime x0 over here. 00:11:39.420 --> 00:11:46.610 So if I want to write the line, the line is exactly this. 00:11:46.610 --> 00:11:50.170 The red object is the line. 00:11:50.170 --> 00:11:52.770 So what is the red object again? 00:11:52.770 --> 00:11:58.350 It's y minus y over x minus x0 equals m, which 00:11:58.350 --> 00:12:00.290 is g prime number 0. 00:12:00.290 --> 00:12:01.750 m is the slope. 00:12:01.750 --> 00:12:05.210 That's the point slope formula, thank you very much. 00:12:05.210 --> 00:12:06.770 So the red object is this. 00:12:06.770 --> 00:12:08.990 This is the line. 00:12:08.990 --> 00:12:10.770 Attention is not the same. 00:12:10.770 --> 00:12:15.625 The blue thing is my curve, more precisely 00:12:15.625 --> 00:12:17.600 a tiny portion of my curve. 00:12:17.600 --> 00:12:21.610 This neighborhood around the point is what I have here. 00:12:21.610 --> 00:12:22.805 What I'm actually-- what? 00:12:22.805 --> 00:12:26.010 00:12:26.010 --> 00:12:29.510 I'm trying to approximate my curve 00:12:29.510 --> 00:12:32.310 function with a little line. 00:12:32.310 --> 00:12:36.420 And I say, I would rather approximate with a red line 00:12:36.420 --> 00:12:38.582 because this is the best approximation 00:12:38.582 --> 00:12:44.200 to the blue arc of a curve which is on the curve, right? 00:12:44.200 --> 00:12:46.985 So this is what it is is just an approximation 00:12:46.985 --> 00:12:54.620 of a curve, approximation of a curve of an arc of a curve. 00:12:54.620 --> 00:12:57.590 But Magdalena's lazy today-- approximation 00:12:57.590 --> 00:13:03.550 of an arc of a curve with a segment of a line, 00:13:03.550 --> 00:13:07.102 with a segment of the tangent line 00:13:07.102 --> 00:13:10.735 of the tangent [INAUDIBLE]. 00:13:10.735 --> 00:13:13.360 How do we call such a phenomenon? 00:13:13.360 --> 00:13:17.650 An approximation of an arc of a circle 00:13:17.650 --> 00:13:23.115 with a little segment of a tangent line 00:13:23.115 --> 00:13:26.040 is like a discretization, right? 00:13:26.040 --> 00:13:29.416 But we call it linear approximation. 00:13:29.416 --> 00:13:32.460 It's called a linear approximation. 00:13:32.460 --> 00:13:36.590 00:13:36.590 --> 00:13:40.220 A-P-P, approx. 00:13:40.220 --> 00:13:42.460 Have you ever seen a linear approximation 00:13:42.460 --> 00:13:46.880 before coming from Calc II? 00:13:46.880 --> 00:13:49.700 Well, in Calc II you've seen the Taylor's formula. 00:13:49.700 --> 00:13:51.510 What is the Taylor's formula? 00:13:51.510 --> 00:13:55.246 It's a beautiful thing that said what? 00:13:55.246 --> 00:13:55.990 I don't know. 00:13:55.990 --> 00:13:56.990 Let's remember together. 00:13:56.990 --> 00:14:00.206 So relationship with Calc II, I'm 00:14:00.206 --> 00:14:04.670 going to go and make an arrow-- relationship with Calc II, 00:14:04.670 --> 00:14:08.160 because everything is actually related. 00:14:08.160 --> 00:14:13.750 In Calc II-- how did we introduce Taylor's formula? 00:14:13.750 --> 00:14:16.930 Well, instead of little a that you're so used to in Calc II, 00:14:16.930 --> 00:14:21.170 we are going to put x0 is the same thing, right? 00:14:21.170 --> 00:14:23.550 So what was Taylor's formula saying? 00:14:23.550 --> 00:14:28.150 You have this kind of smooth, beautiful curve. 00:14:28.150 --> 00:14:30.860 But being smooth is not enough. 00:14:30.860 --> 00:14:33.860 You have that real analytic. 00:14:33.860 --> 00:14:36.070 Real analytic means that the function can be 00:14:36.070 --> 00:14:41.100 expanded in Taylor's formula. 00:14:41.100 --> 00:14:42.250 So what does it mean? 00:14:42.250 --> 00:14:53.000 It means that we have f of x prime is f of x0 equals-- or g. 00:14:53.000 --> 00:14:54.920 You want-- it doesn't matter. 00:14:54.920 --> 00:15:01.160 f prime of x0 times x minus x0 plus 00:15:01.160 --> 00:15:06.010 dot, dot, dot, dot something that I'm going to put. 00:15:06.010 --> 00:15:09.300 This is [? O. ?] It's a small quantity that's maybe not 00:15:09.300 --> 00:15:12.898 so small, but I declare it to be negligible. 00:15:12.898 --> 00:15:14.690 And so they're going to be negligible. 00:15:14.690 --> 00:15:18.920 I have to make a face, a smiley face and eyes, 00:15:18.920 --> 00:15:23.530 meaning that it's OK to neglect the second order 00:15:23.530 --> 00:15:25.420 term, the third order term. 00:15:25.420 --> 00:15:28.370 So what happens, that little h, when I square it, 00:15:28.370 --> 00:15:29.336 say the heck with it. 00:15:29.336 --> 00:15:30.800 It's going to be very small. 00:15:30.800 --> 00:15:36.700 Like if h is 0.1 and then h squared will be 0.0001. 00:15:36.700 --> 00:15:40.445 And I have a certain range of error that I allow, 00:15:40.445 --> 00:15:41.540 a threshold. 00:15:41.540 --> 00:15:43.470 I say that's negligible. 00:15:43.470 --> 00:15:47.430 If h squared and h cubed and h to the fourth are negligible, 00:15:47.430 --> 00:15:49.930 then I'm fine. 00:15:49.930 --> 00:15:53.440 If I take all the other spot, that's 00:15:53.440 --> 00:15:55.960 the linear approximation. 00:15:55.960 --> 00:15:59.730 And that's exactly what I wrote here 00:15:59.730 --> 00:16:02.140 with little g instead of f. 00:16:02.140 --> 00:16:05.120 The only difference is this is little f and this is little g. 00:16:05.120 --> 00:16:09.340 But it's the same exact formula, linear approximation. 00:16:09.340 --> 00:16:14.596 Do you guys remember then next terms of the Taylor's formula? 00:16:14.596 --> 00:16:15.310 STUDENT: fw-- 00:16:15.310 --> 00:16:16.437 PROFESSOR TODA: fw-- 00:16:16.437 --> 00:16:19.920 STUDENT: w over-- 00:16:19.920 --> 00:16:23.430 PROFESSOR TODA: So fw prime at x0 over-- 00:16:23.430 --> 00:16:24.384 STUDENT: 1 factorial. 00:16:24.384 --> 00:16:25.550 PROFESSOR TODA: 2 factorial. 00:16:25.550 --> 00:16:26.625 This was 1 factorial. 00:16:26.625 --> 00:16:28.950 This was over 1 factorial. 00:16:28.950 --> 00:16:30.573 But I don't write it because it's one. 00:16:30.573 --> 00:16:31.197 STUDENT: Right. 00:16:31.197 --> 00:16:35.823 PROFESSOR TODA: Here I would have f double prime of blah, 00:16:35.823 --> 00:16:41.100 blah, blah over-- what did you say-- 2 factorial times x 00:16:41.100 --> 00:16:44.376 minus x0 squared plus, plus, plus, the cubic [INAUDIBLE] 00:16:44.376 --> 00:16:49.730 of the-- this is the quadratic term that I neglect, right? 00:16:49.730 --> 00:16:51.180 So that was Taylor's formula. 00:16:51.180 --> 00:16:54.790 Do I mention anything about it now? 00:16:54.790 --> 00:16:55.905 We should. 00:16:55.905 --> 00:16:58.250 But practically, the authors of the book 00:16:58.250 --> 00:17:00.400 thought, well, everything is in the book. 00:17:00.400 --> 00:17:02.120 You can go back and forth. 00:17:02.120 --> 00:17:05.300 It's not like that unless somebody opens your eyes. 00:17:05.300 --> 00:17:09.930 For example, I didn't see that when I was 21. 00:17:09.930 --> 00:17:13.040 I couldn't make any connection between these Calc I, 00:17:13.040 --> 00:17:14.920 Calc II, Calc III notions. 00:17:14.920 --> 00:17:17.886 Because nobody told me, hey, Magdalena, open your eyes 00:17:17.886 --> 00:17:20.118 and look at that in perspective and make 00:17:20.118 --> 00:17:24.720 a comparison between what you learned in different chapters. 00:17:24.720 --> 00:17:26.220 I had to grow. 00:17:26.220 --> 00:17:29.030 After 20 years, I said, oh, I finally 00:17:29.030 --> 00:17:33.680 see the picture of linearization of a function of, let's say, 00:17:33.680 --> 00:17:35.390 n variables. 00:17:35.390 --> 00:17:38.480 So all these total differentials will come in place 00:17:38.480 --> 00:17:41.050 when time comes. 00:17:41.050 --> 00:17:46.410 You have a so-called differential in Calc I. 00:17:46.410 --> 00:17:47.920 And that's not delta g. 00:17:47.920 --> 00:17:49.890 Some people say, OK, no, that's delta g. 00:17:49.890 --> 00:17:52.000 No, no, no, no. 00:17:52.000 --> 00:17:53.610 The delta x is a displacement. 00:17:53.610 --> 00:17:57.305 The delta g is the induced displacement. 00:17:57.305 --> 00:17:59.985 If you want this to be come a differential, 00:17:59.985 --> 00:18:02.840 then you shrink that displacement 00:18:02.840 --> 00:18:05.640 to infinitesimally small. 00:18:05.640 --> 00:18:06.230 OK? 00:18:06.230 --> 00:18:09.684 So it's like going from a molecule to an atom 00:18:09.684 --> 00:18:13.990 to an electron to subatomic particles but even more, 00:18:13.990 --> 00:18:16.060 something infinitesimally small. 00:18:16.060 --> 00:18:17.070 So what do we do? 00:18:17.070 --> 00:18:22.810 We shrink delta x into dx which is infinitesimally small. 00:18:22.810 --> 00:18:26.390 00:18:26.390 --> 00:18:28.932 It's like the notion of God but microscopically 00:18:28.932 --> 00:18:33.920 or like microbiology compared to the universe, OK? 00:18:33.920 --> 00:18:42.210 So dx is multiplied by g prime of x0. 00:18:42.210 --> 00:18:46.430 And instead of delta g, I'm going to have a so-called dg, 00:18:46.430 --> 00:18:49.060 and that's a form. 00:18:49.060 --> 00:18:53.260 In mathematics, this is called a form or a one form. 00:18:53.260 --> 00:18:58.520 And it's a special kind of object, OK? 00:18:58.520 --> 00:19:01.550 So Mr. Leibniz was very smart. 00:19:01.550 --> 00:19:09.720 He said, but I can rewrite this form like dg dx equals g prime. 00:19:09.720 --> 00:19:13.450 So if you ever forget about this form which 00:19:13.450 --> 00:19:18.169 is called differential, differential form, 00:19:18.169 --> 00:19:20.780 you remember Mr. Leibniz, he taught you 00:19:20.780 --> 00:19:25.322 how to write the derivative in two different ways, dg dx or g 00:19:25.322 --> 00:19:26.630 prime. 00:19:26.630 --> 00:19:30.220 What you do is just formally multiply g prime by dx 00:19:30.220 --> 00:19:31.670 and you get dg. 00:19:31.670 --> 00:19:34.700 Say it again, Magdalena-- multiply g prime by dx 00:19:34.700 --> 00:19:35.880 and you get dg. 00:19:35.880 --> 00:19:38.890 And that's your so-called differential. 00:19:38.890 --> 00:19:42.500 Now, why do you say total differential-- total 00:19:42.500 --> 00:19:46.870 differential, my god, like complete differentiation? 00:19:46.870 --> 00:19:52.280 In 11.4, we deal with functions of two variables. 00:19:52.280 --> 00:19:54.750 So can we say differentials? 00:19:54.750 --> 00:19:57.290 Mmm, it's a little bit like a differential 00:19:57.290 --> 00:20:00.030 with respect to what variable? 00:20:00.030 --> 00:20:02.590 If you say with respect to all the variables, 00:20:02.590 --> 00:20:08.960 then you have to be thinking to be smart and event, 00:20:08.960 --> 00:20:11.690 create this new object. 00:20:11.690 --> 00:20:17.312 If one would write Taylor's formula, 00:20:17.312 --> 00:20:22.720 there is a Taylor's formula that we don't give. 00:20:22.720 --> 00:20:23.260 OK. 00:20:23.260 --> 00:20:26.210 Now, you guys are looking at me with excitement. 00:20:26.210 --> 00:20:30.740 For one point extra credit, on the internet, 00:20:30.740 --> 00:20:35.310 find Taylor's formula for n variables, functions 00:20:35.310 --> 00:20:38.590 of n variables or at least two variables, 00:20:38.590 --> 00:20:43.720 which was going to look like z minus z0 equals 00:20:43.720 --> 00:20:49.140 f sub x at the point x0 at 0 times x minus x0 plus 00:20:49.140 --> 00:21:00.200 f sub y at x0 y0 times x minus x0 plus second order terms 00:21:00.200 --> 00:21:04.010 plus third order terms plus fourth order terms. 00:21:04.010 --> 00:21:06.720 And the video cannot see me. 00:21:06.720 --> 00:21:08.850 So what do we do? 00:21:08.850 --> 00:21:13.830 We just truncate this part of Taylor's I say, 00:21:13.830 --> 00:21:18.170 I already take the Taylor polynomial of degree one. 00:21:18.170 --> 00:21:21.470 And the quadratic terms and everything else, the heck 00:21:21.470 --> 00:21:22.850 with that. 00:21:22.850 --> 00:21:25.020 And I call that a linear approximation, 00:21:25.020 --> 00:21:28.330 but it's actually Taylor's formula being discussed. 00:21:28.330 --> 00:21:30.680 We don't tell you in the book because we 00:21:30.680 --> 00:21:31.740 don't want to scare you. 00:21:31.740 --> 00:21:34.865 I think we would better tell you at some point, 00:21:34.865 --> 00:21:38.010 so I decided to tell you now. 00:21:38.010 --> 00:21:38.850 All right. 00:21:38.850 --> 00:21:42.440 So this is Taylor's formula for functions of two variables. 00:21:42.440 --> 00:21:45.630 We have to create not out of nothing 00:21:45.630 --> 00:21:49.810 but out of this the total differential. 00:21:49.810 --> 00:21:51.190 Who tells me? 00:21:51.190 --> 00:21:54.033 Shrink the displacement, Magdalena. 00:21:54.033 --> 00:21:58.141 The delta x shrunk to an infinitesimally small 00:21:58.141 --> 00:21:58.640 will be dx. 00:21:58.640 --> 00:22:01.110 Delta y will become dy. 00:22:01.110 --> 00:22:06.390 The line is a smiley from the skies, just looking at us. 00:22:06.390 --> 00:22:08.040 He loves our notations. 00:22:08.040 --> 00:22:10.896 And this is dz. 00:22:10.896 --> 00:22:18.970 So I'm going to write dz or df's the same thing equals f sub x. 00:22:18.970 --> 00:22:22.420 At the point, you could be at any point 00:22:22.420 --> 00:22:29.780 you are taking in particular, dx plus f sub y xy dy. 00:22:29.780 --> 00:22:34.010 So this is at any point at the arbitrary point xy 00:22:34.010 --> 00:22:39.310 in the domain where your function e is at least c1. 00:22:39.310 --> 00:22:40.730 What does it mean, c1? 00:22:40.730 --> 00:22:43.280 It means the function is differentiable 00:22:43.280 --> 00:22:47.400 and the partial derivatives are continuous. 00:22:47.400 --> 00:22:50.850 I said several times, I want even more than that. 00:22:50.850 --> 00:22:56.790 I want it maybe second order derivatives 00:22:56.790 --> 00:23:02.868 to exist and be continuous and so on and so forth. 00:23:02.868 --> 00:23:08.465 And I will assume that the function can 00:23:08.465 --> 00:23:11.585 be expanded [INAUDIBLE] series. 00:23:11.585 --> 00:23:14.445 00:23:14.445 --> 00:23:17.440 All right, now example of a final problem 00:23:17.440 --> 00:23:22.260 that was my first problem on the final many times 00:23:22.260 --> 00:23:26.310 and also on the common final departmental final. 00:23:26.310 --> 00:23:28.320 And many students screwed up, and I 00:23:28.320 --> 00:23:32.380 don't want you to ever make such a mistake. 00:23:32.380 --> 00:23:37.322 So this is a mistake not to make, OK, mistake not 00:23:37.322 --> 00:23:43.730 to make because after 20 something years of teaching, 00:23:43.730 --> 00:23:46.010 I'm quite familiar with the mistakes students 00:23:46.010 --> 00:23:49.230 make in general and I don't want you to make them. 00:23:49.230 --> 00:23:50.664 You are too good to do this. 00:23:50.664 --> 00:23:52.098 So problem 1. 00:23:52.098 --> 00:23:56.600 On the final, I said-- we said-- the only difference was 00:23:56.600 --> 00:24:00.900 on some departmental finals, we gave a more sophisticated 00:24:00.900 --> 00:24:02.470 function. 00:24:02.470 --> 00:24:06.580 I'm going to give only some simple function 00:24:06.580 --> 00:24:07.820 for this polynomial. 00:24:07.820 --> 00:24:09.770 That's beautiful. 00:24:09.770 --> 00:24:18.930 And then I said we said write the differential 00:24:18.930 --> 00:24:28.090 of this function at an arbitrary point x, y. 00:24:28.090 --> 00:24:28.610 And done. 00:24:28.610 --> 00:24:31.080 And [INAUDIBLE]. 00:24:31.080 --> 00:24:34.642 Well, let me tell you what some of my students-- some 00:24:34.642 --> 00:24:36.350 of my studentss-- don't do that. 00:24:36.350 --> 00:24:38.302 I'm going to cross it with red. 00:24:38.302 --> 00:24:41.770 And some of my students wrote me very beautifully df 00:24:41.770 --> 00:24:44.390 equals 2x plus 2y. 00:24:44.390 --> 00:24:47.550 And that can send me to the hospital. 00:24:47.550 --> 00:24:53.320 If you want to go to the ER soon, do this on the exam 00:24:53.320 --> 00:24:55.960 because this is nonsense. 00:24:55.960 --> 00:24:57.480 Why is this nonsense? 00:24:57.480 --> 00:24:58.360 This is not-- 00:24:58.360 --> 00:24:59.840 STUDENT: [INAUDIBLE] dx or dy. 00:24:59.840 --> 00:25:00.840 PROFESSOR TODA: Exactly. 00:25:00.840 --> 00:25:06.980 So the most important thing is that the df is like-- OK, 00:25:06.980 --> 00:25:09.060 let me come back to driving. 00:25:09.060 --> 00:25:14.480 I'm driving to Amarillo-- and I give this example to my calc 1 00:25:14.480 --> 00:25:18.201 students all the time because it's a linear motion in terms 00:25:18.201 --> 00:25:18.700 of time. 00:25:18.700 --> 00:25:21.090 And let's say I'm on cruise control or not. 00:25:21.090 --> 00:25:22.780 It doesn't matter. 00:25:22.780 --> 00:25:30.190 When we drive and I'm looking at the speedometer and I see 60-- 00:25:30.190 --> 00:25:37.000 I didn't want to say more, but let's say 80, 80 miles an hour. 00:25:37.000 --> 00:25:38.620 That is a miles an hour. 00:25:38.620 --> 00:25:43.344 That means the hour is a huge chunk delta h or delta t. 00:25:43.344 --> 00:25:45.010 Let's call it delta t because it's time. 00:25:45.010 --> 00:25:45.640 I'm silly. 00:25:45.640 --> 00:25:47.660 Delta t is 1. 00:25:47.660 --> 00:25:51.310 Delta s, the space, the space, is going 00:25:51.310 --> 00:25:54.970 to be the chunk of 60 miles. 00:25:54.970 --> 00:26:00.360 But then that is the average speed that I had. 00:26:00.360 --> 00:26:02.130 So that's why I said 60. 00:26:02.130 --> 00:26:04.806 That's the average speed I had in my trip, 00:26:04.806 --> 00:26:05.930 during my trip [INAUDIBLE]. 00:26:05.930 --> 00:26:10.600 There were moments when my speed was 0 or close to 0. 00:26:10.600 --> 00:26:12.390 Let's assume it was never 0. 00:26:12.390 --> 00:26:14.931 But that means there were many moments when my speed could've 00:26:14.931 --> 00:26:18.990 been 100, and nobody knows because they didn't catch me. 00:26:18.990 --> 00:26:21.450 So I was just lucky. 00:26:21.450 --> 00:26:26.300 So in average, if somebody is asking you what is the average, 00:26:26.300 --> 00:26:30.440 that doesn't tell them anything. 00:26:30.440 --> 00:26:34.090 That reminds me of that joke-- overall I'm good, 00:26:34.090 --> 00:26:38.190 the statistician joke who was, are you cold? 00:26:38.190 --> 00:26:39.000 Are you warm? 00:26:39.000 --> 00:26:44.142 And he was actually sitting on with one half of him 00:26:44.142 --> 00:26:47.090 on a block of ice and the other half on the stove, 00:26:47.090 --> 00:26:49.172 and he says, in average, I'm fine. 00:26:49.172 --> 00:26:52.400 But he was dying. 00:26:52.400 --> 00:26:53.910 This is the same kind of thing. 00:26:53.910 --> 00:26:58.360 My average was 60 miles an hour, but I almost 00:26:58.360 --> 00:27:02.110 got caught when I was driving almost 100. 00:27:02.110 --> 00:27:06.250 But nobody knows because I'm not giving you that information. 00:27:06.250 --> 00:27:12.440 That's the infinitesimally small information that I have not 00:27:12.440 --> 00:27:16.610 put correctly here means that what is 00:27:16.610 --> 00:27:18.990 what I see on the speedometer? 00:27:18.990 --> 00:27:21.060 It's the instantaneous rate of change 00:27:21.060 --> 00:27:23.880 that I see that fraction of second. 00:27:23.880 --> 00:27:30.940 So that means maybe a few feet per a fraction of a second. 00:27:30.940 --> 00:27:33.920 It means how many feet did I travel 00:27:33.920 --> 00:27:36.470 in that fraction of a second? 00:27:36.470 --> 00:27:41.240 And if that fraction of a second is very tiny that I cannot even 00:27:41.240 --> 00:27:44.000 express it properly, that's what I'm going to have-- 00:27:44.000 --> 00:27:46.610 df equals f prime dx. 00:27:46.610 --> 00:27:52.010 So df and dx have to be small because their ratio will be 00:27:52.010 --> 00:27:56.180 a good number, like 60, like 80, but [? them in ?] themselves 00:27:56.180 --> 00:27:58.635 delta m delta [? srv, ?] very tiny things. 00:27:58.635 --> 00:28:03.420 It's the ratio that matters in the end to be 60, or 80, 00:28:03.420 --> 00:28:04.470 or whatever. 00:28:04.470 --> 00:28:08.520 So I have 2x dx plus 2y dy. 00:28:08.520 --> 00:28:10.920 Never say that the differential, which 00:28:10.920 --> 00:28:13.160 is something infinitesimally small, 00:28:13.160 --> 00:28:17.376 is equal to this scalar function that it doesn't even 00:28:17.376 --> 00:28:18.160 make any sense. 00:28:18.160 --> 00:28:20.060 Don't do that because you get 0 points 00:28:20.060 --> 00:28:21.900 and then we argue, and I don't want 00:28:21.900 --> 00:28:25.450 you to get 0 points on this problem, right. 00:28:25.450 --> 00:28:27.250 So it's a very simple problem. 00:28:27.250 --> 00:28:31.080 All I want to test you on would be this definition. 00:28:31.080 --> 00:28:36.000 Remember, you're going to see that again on the midterm 00:28:36.000 --> 00:28:39.020 and on the final, or just on the final. 00:28:39.020 --> 00:28:41.650 Any questions about that? 00:28:41.650 --> 00:28:42.250 All right. 00:28:42.250 --> 00:28:53.978 So I want to give you the following homework out 00:28:53.978 --> 00:29:00.680 of section 11.4 on top of the web work. 00:29:00.680 --> 00:29:07.250 00:29:07.250 --> 00:29:16.640 Read all the solved examples of the section. 00:29:16.640 --> 00:29:23.530 00:29:23.530 --> 00:29:24.030 OK. 00:29:24.030 --> 00:29:30.470 So for example, somebody tells you 00:29:30.470 --> 00:29:40.110 I have to apply this knowing that I have 00:29:40.110 --> 00:29:44.610 an error of measurement of some sort in the s direction 00:29:44.610 --> 00:29:48.210 and an error of measurement of some sort in the y direction. 00:29:48.210 --> 00:29:51.010 There are two or three examples like that. 00:29:51.010 --> 00:29:54.910 They will give you all this data, including the error 00:29:54.910 --> 00:29:55.640 measurement. 00:29:55.640 --> 00:29:58.490 For delta, it should be 0.1. 00:29:58.490 --> 00:30:04.240 Don't confuse the 0.1 with dx. dx is not a quantity. 00:30:04.240 --> 00:30:08.608 dx is something like micro cosmic thing. 00:30:08.608 --> 00:30:14.134 It's like infinitely [? small ?]. 00:30:14.134 --> 00:30:15.050 Infinitesimally small. 00:30:15.050 --> 00:30:19.560 So saying that dx should be 0.1 doesn't make any sense, 00:30:19.560 --> 00:30:22.880 but delta x being 0.1 make sense. 00:30:22.880 --> 00:30:26.350 Delta y being 0.3 makes sense. 00:30:26.350 --> 00:30:29.560 And they ask you to plug it in and find 00:30:29.560 --> 00:30:32.130 the general difference. 00:30:32.130 --> 00:30:33.730 For example, where could that happen? 00:30:33.730 --> 00:30:35.760 And you see examples in the book. 00:30:35.760 --> 00:30:40.910 Somebody measures something-- an area of a rectangle 00:30:40.910 --> 00:30:42.970 or a volume of a cube. 00:30:42.970 --> 00:30:46.110 But when you measure, you make mistakes. 00:30:46.110 --> 00:30:48.270 You have measurement errors. 00:30:48.270 --> 00:30:53.250 In the delta x, you have an error of plus minus 0.1. 00:30:53.250 --> 00:31:00.870 In the y direction, you have displacement error 0.2 or 0.3, 00:31:00.870 --> 00:31:02.220 something like that. 00:31:02.220 --> 00:31:05.090 What is the overall error you are 00:31:05.090 --> 00:31:08.100 going to make when you measure that function of two variables? 00:31:08.100 --> 00:31:09.730 That's what you have. 00:31:09.730 --> 00:31:12.140 So you plug in all those displacements 00:31:12.140 --> 00:31:14.790 and you come up with the computational problem. 00:31:14.790 --> 00:31:20.200 Several of you Wednesday we discussed in my office already 00:31:20.200 --> 00:31:24.700 solved those problems through web work and came to me, 00:31:24.700 --> 00:31:27.510 and I said, how did you know to plug in those [? numbers ?]? 00:31:27.510 --> 00:31:28.900 Well, it's not so hard. 00:31:28.900 --> 00:31:30.120 It's sort of common sense. 00:31:30.120 --> 00:31:32.990 Plus, I looked in the book and that gave me the idea 00:31:32.990 --> 00:31:34.517 to remind you to look in the book 00:31:34.517 --> 00:31:37.250 for those numerical examples. 00:31:37.250 --> 00:31:40.370 You will have to use your calculator. 00:31:40.370 --> 00:31:42.990 So you don't have it with you, you generally, we 00:31:42.990 --> 00:31:45.000 don't use in the classroom, but it's very easy. 00:31:45.000 --> 00:31:48.392 All you have to do is use the calculator and [INAUDIBLE] 00:31:48.392 --> 00:31:51.310 examples and see how it goes. 00:31:51.310 --> 00:31:57.430 I wanted to show you something more interesting 00:31:57.430 --> 00:32:09.410 even, more beautiful regarding something 00:32:09.410 --> 00:32:12.930 we don't show in the book until later on, 00:32:12.930 --> 00:32:18.240 and I'm uncomfortable with the idea of not showing this to you 00:32:18.240 --> 00:32:19.610 now. 00:32:19.610 --> 00:32:26.560 An alternate way, or more advanced way, 00:32:26.560 --> 00:32:38.390 more advanced way, to define the tangent plane-- 00:32:38.390 --> 00:32:49.190 the tangent plane-- to a surface S at the point p. 00:32:49.190 --> 00:32:51.690 And I'll draw again. 00:32:51.690 --> 00:32:56.470 Half of my job is drawing in this class, which I like. 00:32:56.470 --> 00:32:59.910 I mean, I was having an argument with one of my colleagues who 00:32:59.910 --> 00:33:03.480 said, I hate when they are giving me to teach calculus 3 00:33:03.480 --> 00:33:07.660 because I cannot draw. 00:33:07.660 --> 00:33:09.910 I think that the most beautiful part 00:33:09.910 --> 00:33:15.450 is that we can represent things visually, 00:33:15.450 --> 00:33:20.262 and this is just pi, the tangent plane I'm after, 00:33:20.262 --> 00:33:24.880 and p will be a coordinate 0 by 0, z0. 00:33:24.880 --> 00:33:26.900 And what was the label? 00:33:26.900 --> 00:33:27.790 Oh, the label. 00:33:27.790 --> 00:33:28.365 The label. 00:33:28.365 --> 00:33:34.330 The label was internal where z equals f of xy. 00:33:34.330 --> 00:33:40.160 But more generally, I'll say this time plus more generally, 00:33:40.160 --> 00:33:58.970 what if you have f of xyz equals c for that surface. 00:33:58.970 --> 00:34:00.560 Let's call it [INAUDIBLE]. 00:34:00.560 --> 00:34:04.800 F of xy is [INAUDIBLE]. 00:34:04.800 --> 00:34:08.210 And somebody even said, can you have a parametrization? 00:34:08.210 --> 00:34:10.440 And this is where I wanted to go. 00:34:10.440 --> 00:34:14.469 00:34:14.469 --> 00:34:16.230 Ryan was the first one who asked me, 00:34:16.230 --> 00:34:18.870 but then there were three more of you 00:34:18.870 --> 00:34:21.159 who have restless minds plus you-- 00:34:21.159 --> 00:34:25.670 because that's the essence of being active here. 00:34:25.670 --> 00:34:29.840 We don't lose our connections. 00:34:29.840 --> 00:34:34.300 We lose neurons anyway, but we don't lose our connections 00:34:34.300 --> 00:34:37.949 if we think, and anticipate things, 00:34:37.949 --> 00:34:40.080 and try to relate concepts. 00:34:40.080 --> 00:34:42.590 So if you don't want to get Alzheimer's, just 00:34:42.590 --> 00:34:45.730 think about the parametrization. 00:34:45.730 --> 00:34:49.699 So can I have a parametrization for a surface? 00:34:49.699 --> 00:34:52.179 All righty, what do you mean? 00:34:52.179 --> 00:34:58.240 What if somebody says for a curve, we have r of t, right, 00:34:58.240 --> 00:34:59.075 which was what? 00:34:59.075 --> 00:35:06.500 It was x of ti plus y of tj plus z of tk, and we were so happy 00:35:06.500 --> 00:35:09.825 and we were happy because we were traveling 00:35:09.825 --> 00:35:12.320 in time with respect to the origin, 00:35:12.320 --> 00:35:15.640 and this was r of t at time t. 00:35:15.640 --> 00:35:18.330 [INAUDIBLE] 00:35:18.330 --> 00:35:20.210 But somebody asked me, [INAUDIBLE], 00:35:20.210 --> 00:35:27.010 can you have such a position vector moving on a surface? 00:35:27.010 --> 00:35:30.240 Like look, it's a rigid motion. 00:35:30.240 --> 00:35:32.770 If you went to the robotics science 00:35:32.770 --> 00:35:36.340 fair, Texas Tech, or something like that, you know about that. 00:35:36.340 --> 00:35:37.180 Yeah, cities. 00:35:37.180 --> 00:35:39.977 So how do we introduce such a parametrization? 00:35:39.977 --> 00:35:44.470 We have an origin of course. 00:35:44.470 --> 00:35:46.390 An origin is always important. 00:35:46.390 --> 00:35:48.326 Everybody has an origin. 00:35:48.326 --> 00:35:53.170 00:35:53.170 --> 00:35:57.610 And I take that position vector, and where does it start? 00:35:57.610 --> 00:36:02.120 It starts at the origin, and the tip of it is on the surface, 00:36:02.120 --> 00:36:05.382 And it's gliding on the surface, the tip of it. 00:36:05.382 --> 00:36:10.500 And that's going to be r, but it's not going to be r of t. 00:36:10.500 --> 00:36:12.930 It's going to be r of longitude and latitude. 00:36:12.930 --> 00:36:16.110 Like imagine, that would be the radius coming 00:36:16.110 --> 00:36:18.360 from the center of the earth. 00:36:18.360 --> 00:36:20.980 And it depends on two parameters. 00:36:20.980 --> 00:36:24.780 One of them would be latitude. 00:36:24.780 --> 00:36:26.140 Am I drawing this right? 00:36:26.140 --> 00:36:26.640 Latitude-- 00:36:26.640 --> 00:36:28.730 STUDENT: [INAUDIBLE] longitude. 00:36:28.730 --> 00:36:30.870 PROFESSOR TODA: --from a latitude 0. 00:36:30.870 --> 00:36:32.010 I'm at the equator. 00:36:32.010 --> 00:36:33.760 Then latitude 90 degrees. 00:36:33.760 --> 00:36:35.970 I'm at the North Pole. 00:36:35.970 --> 00:36:37.760 In mathematics, we are funny. 00:36:37.760 --> 00:36:40.880 We say latitude 0, latitude 90 North Pole, 00:36:40.880 --> 00:36:45.165 latitude negative 90, which is South Pole. 00:36:45.165 --> 00:36:49.290 And longitude from 0 to 2 pi. 00:36:49.290 --> 00:36:53.740 Meridian 0 to all around. 00:36:53.740 --> 00:36:58.200 So r will be not a function of t but a function of u and b, 00:36:58.200 --> 00:37:02.240 thank god, because u and b are the latitude and longitude 00:37:02.240 --> 00:37:03.320 sort of. 00:37:03.320 --> 00:37:12.324 So we have x of uv i plus y of uv j plus z of uv k. 00:37:12.324 --> 00:37:20.620 00:37:20.620 --> 00:37:23.030 You can do that. 00:37:23.030 --> 00:37:26.010 And you say, but can you give us an example, because this 00:37:26.010 --> 00:37:28.210 looks so abstract for god sake. 00:37:28.210 --> 00:37:31.830 If you give me the graph the way you gave it to me 00:37:31.830 --> 00:37:37.307 before z equals f of xy, please parametrize this for me. 00:37:37.307 --> 00:37:41.880 00:37:41.880 --> 00:37:44.640 Parametrize it for me because I'm lost. 00:37:44.640 --> 00:37:45.610 You are not lost. 00:37:45.610 --> 00:37:47.530 We can do this together. 00:37:47.530 --> 00:37:51.480 Now what's the simplest way to parametrize 00:37:51.480 --> 00:37:57.260 a graph of the type z equals f of xy? 00:37:57.260 --> 00:38:01.970 Take the xy to be u and v. Take x 00:38:01.970 --> 00:38:05.360 and y to be your independent variables 00:38:05.360 --> 00:38:07.850 and take z to be the dependent variable. 00:38:07.850 --> 00:38:12.700 00:38:12.700 --> 00:38:16.930 I'm again expressing these things in terms of variables 00:38:16.930 --> 00:38:18.340 like I did last time. 00:38:18.340 --> 00:38:23.370 Then I say, let's take this kind of parametrization. [INAUDIBLE] 00:38:23.370 --> 00:38:24.380 vu, right. 00:38:24.380 --> 00:38:33.080 y would be v. Then I'm going to write r of x and y 00:38:33.080 --> 00:38:36.710 just like that guy will be [INAUDIBLE] of xn. 00:38:36.710 --> 00:38:38.770 [? y ?] will say, wait a minute. 00:38:38.770 --> 00:38:42.884 I will have to re-denote everybody with capitals. 00:38:42.884 --> 00:38:46.300 Then my life will become better because you 00:38:46.300 --> 00:38:47.300 don't have to erase. 00:38:47.300 --> 00:38:50.670 You just make little x big, little y bigs, 00:38:50.670 --> 00:38:53.890 bigs, big, capitalized XYZ. 00:38:53.890 --> 00:39:02.150 And then I'll say OK, XYZ will be my setting here in 3D. 00:39:02.150 --> 00:39:07.020 00:39:07.020 --> 00:39:07.560 All right. 00:39:07.560 --> 00:39:10.290 So how am I going to re-parametrize 00:39:10.290 --> 00:39:12.576 the whole surface? 00:39:12.576 --> 00:39:22.220 Whole surface will be r of xy equals in this case, well, 00:39:22.220 --> 00:39:23.280 let's think about it. 00:39:23.280 --> 00:39:29.020 In this case, I'm going to have xy. 00:39:29.020 --> 00:39:31.350 And where's the little f? 00:39:31.350 --> 00:39:32.550 I just erased it. 00:39:32.550 --> 00:39:35.085 I was smart, right, that I erased f of xy. 00:39:35.085 --> 00:39:37.830 00:39:37.830 --> 00:39:46.010 So I have x, y, and z, which is f of xy. 00:39:46.010 --> 00:39:53.240 00:39:53.240 --> 00:40:01.430 And this is the generic point p of coordinates xy f of xy. 00:40:01.430 --> 00:40:04.980 00:40:04.980 --> 00:40:07.580 So I say, OK, what does it mean? 00:40:07.580 --> 00:40:10.100 I will project this point. 00:40:10.100 --> 00:40:13.175 And this is the point when big x becomes little 00:40:13.175 --> 00:40:17.860 x, when big y becomes-- where is my y-axis? 00:40:17.860 --> 00:40:20.090 Somebody ate my y axis. 00:40:20.090 --> 00:40:22.190 [INAUDIBLE] 00:40:22.190 --> 00:40:28.400 So when big Y becomes little y, little y 00:40:28.400 --> 00:40:33.830 is just an instance of big Y. And big Z will take what value? 00:40:33.830 --> 00:40:35.630 Well, I need to project that. 00:40:35.630 --> 00:40:39.120 How do you project from a point to the z-axis? 00:40:39.120 --> 00:40:42.680 You have to take the parallel from the point 00:40:42.680 --> 00:40:47.630 to the horizontal plane until you 00:40:47.630 --> 00:40:52.940 hit the-- [INAUDIBLE] the whole plane parallel to the floor 00:40:52.940 --> 00:40:54.210 through the point p. 00:40:54.210 --> 00:40:55.450 And what do I get here? 00:40:55.450 --> 00:40:56.410 STUDENT: [INAUDIBLE]. 00:40:56.410 --> 00:40:58.670 PROFESSOR TODA: Not z0, but it's little z 00:40:58.670 --> 00:41:03.120 equals f of xy, which is an instance of the variable xz. 00:41:03.120 --> 00:41:06.460 For you programmers, you know that big z will be a variable 00:41:06.460 --> 00:41:11.640 and little z will be [INAUDIBLE] a variable. 00:41:11.640 --> 00:41:12.140 OK. 00:41:12.140 --> 00:41:16.610 So I parametrized my graph in a more general way, 00:41:16.610 --> 00:41:18.578 general parametrization for a graph. 00:41:18.578 --> 00:41:25.960 00:41:25.960 --> 00:41:33.420 And now, what are-- what's the meaning of r sub x and r sub y? 00:41:33.420 --> 00:41:34.489 What are they? 00:41:34.489 --> 00:41:35.364 STUDENT: [INAUDIBLE]. 00:41:35.364 --> 00:41:38.180 00:41:38.180 --> 00:41:41.660 PROFESSOR TODA: Now, we don't say that in the book. 00:41:41.660 --> 00:41:42.990 Shame on us. 00:41:42.990 --> 00:41:43.630 Shame on us. 00:41:43.630 --> 00:41:47.480 We should have because I was browsing through the projects 00:41:47.480 --> 00:41:49.900 about a year and a half ago. 00:41:49.900 --> 00:41:52.970 The senior projects of a few of my students 00:41:52.970 --> 00:41:56.340 who are-- two of them were in mechanical engineering. 00:41:56.340 --> 00:42:00.660 One of them was in petroleum engineering. 00:42:00.660 --> 00:42:03.965 And he actually showed me that they were doing this. 00:42:03.965 --> 00:42:07.830 They were taking vectors that depend on parameters-- 00:42:07.830 --> 00:42:11.250 this is a vector [INAUDIBLE]-- and differentiated them with 00:42:11.250 --> 00:42:13.720 respect to those parameters. 00:42:13.720 --> 00:42:17.215 And I was thinking OK, did we do the partial derivatives r sub 00:42:17.215 --> 00:42:17.960 x, r sub y? 00:42:17.960 --> 00:42:19.340 Not so much. 00:42:19.340 --> 00:42:22.380 But now I want to do it because I think that prepares 00:42:22.380 --> 00:42:24.640 you better as engineers. 00:42:24.640 --> 00:42:29.070 So what is r sub x and what is r sub y? 00:42:29.070 --> 00:42:31.250 And you say, well, OK. [INAUDIBLE], 00:42:31.250 --> 00:42:34.860 I think I know how to do that in my sleep, right. 00:42:34.860 --> 00:42:36.780 If you want me to do that theoretically 00:42:36.780 --> 00:42:39.720 from this formula, but on the picture, 00:42:39.720 --> 00:42:42.450 I really don't know what it is. 00:42:42.450 --> 00:42:45.590 So I'm asking you what I'm going to have in terms 00:42:45.590 --> 00:42:47.240 of r sub x and r sub y. 00:42:47.240 --> 00:42:48.950 They will be vectors. 00:42:48.950 --> 00:42:51.880 This should be a vector as well, right. 00:42:51.880 --> 00:42:56.620 And for me, vector triple means the identification 00:42:56.620 --> 00:42:59.930 between the three coordinates and the physical vector. 00:42:59.930 --> 00:43:01.960 So this is the physical vector. 00:43:01.960 --> 00:43:06.032 Go ahead and write x prime with respect to x is 1. 00:43:06.032 --> 00:43:08.684 00:43:08.684 --> 00:43:13.776 y prime with respect to x is 0. 00:43:13.776 --> 00:43:15.970 The third [INAUDIBLE] prime with respect 00:43:15.970 --> 00:43:20.190 to x is just whatever this little f is, 00:43:20.190 --> 00:43:21.984 it's not any of my business. 00:43:21.984 --> 00:43:24.786 It's a [INAUDIBLE] function f sub x. 00:43:24.786 --> 00:43:28.290 00:43:28.290 --> 00:43:30.590 Well, what is the second vector? 00:43:30.590 --> 00:43:32.285 STUDENT: 0, 1, f sub y. 00:43:32.285 --> 00:43:34.810 PROFESSOR TODA: 0, 1, f sub y. 00:43:34.810 --> 00:43:36.596 Now, are they slopes? 00:43:36.596 --> 00:43:37.096 No. 00:43:37.096 --> 00:43:38.010 These are slopes. 00:43:38.010 --> 00:43:40.770 That's a slope and that's a slope. 00:43:40.770 --> 00:43:44.950 And we learned about those in 11.3, 00:43:44.950 --> 00:43:49.530 and we understood that those are ski slopes, they were. 00:43:49.530 --> 00:43:52.312 In the direction of x and the direction of y, 00:43:52.312 --> 00:44:00.030 the slopes of the tangents to the coordinate lines. 00:44:00.030 --> 00:44:04.980 But this looks like I have a direction of a line, 00:44:04.980 --> 00:44:08.610 and this would be the lope, and that's the direction of a line, 00:44:08.610 --> 00:44:10.310 and that would be the slope. 00:44:10.310 --> 00:44:12.700 What are those lines? 00:44:12.700 --> 00:44:16.314 STUDENT: [INAUDIBLE] to the function [INAUDIBLE]. 00:44:16.314 --> 00:44:17.480 PROFESSOR TODA: Let me draw. 00:44:17.480 --> 00:44:19.440 Then shall I erase the whole thing? 00:44:19.440 --> 00:44:20.160 No. 00:44:20.160 --> 00:44:23.948 I'm just going to keep-- I'll erase the tangent. 00:44:23.948 --> 00:44:27.470 Don't erase anything on your notebooks. 00:44:27.470 --> 00:44:28.922 So this is the point p. 00:44:28.922 --> 00:44:29.630 It's still there. 00:44:29.630 --> 00:44:30.570 This is the surface. 00:44:30.570 --> 00:44:33.060 It's still there. 00:44:33.060 --> 00:44:38.200 So my surface will be x, slices of x, [? S ?] constant 00:44:38.200 --> 00:44:39.590 are coming towards you. 00:44:39.590 --> 00:44:45.800 They are these [? walls ?] like that, like this, yes. 00:44:45.800 --> 00:44:47.606 It's like the CT scan. 00:44:47.606 --> 00:44:52.190 I think that when they slice up your body, 00:44:52.190 --> 00:44:54.260 tch tch tch tch tch tch, take pictures 00:44:54.260 --> 00:44:57.590 of the slices of your body, that's the same kind of thing. 00:44:57.590 --> 00:44:59.508 So x0, x0, x0, x0. 00:44:59.508 --> 00:45:05.414 I'm going to [INAUDIBLE] planes and I had x equals x0. 00:45:05.414 --> 00:45:12.402 And in the other direction, I cut and I get, what do I get? 00:45:12.402 --> 00:45:18.400 00:45:18.400 --> 00:45:20.226 Well, I started bad. 00:45:20.226 --> 00:45:23.650 00:45:23.650 --> 00:45:25.195 Great, Magdalena, this is-- 00:45:25.195 --> 00:45:27.226 What is this pink? 00:45:27.226 --> 00:45:32.350 It's not Valentine's Day anymore. y equals [INAUDIBLE]. 00:45:32.350 --> 00:45:34.810 And this is the point. 00:45:34.810 --> 00:45:39.320 So, as Alex was trying to tell you, 00:45:39.320 --> 00:45:44.980 our sub x would represent the vector, the physical vector 00:45:44.980 --> 00:45:52.260 in 3D, that is originating at p and tangent to which 00:45:52.260 --> 00:45:55.760 of the two, to the purple one or to the red one? 00:45:55.760 --> 00:45:57.185 STUDENT: Red. 00:45:57.185 --> 00:45:58.135 Uh, purple. 00:45:58.135 --> 00:45:59.560 PROFESSOR TODA: Make up your mind. 00:45:59.560 --> 00:46:01.494 STUDENT: The purple one. 00:46:01.494 --> 00:46:03.660 PROFESSOR TODA: [INAUDIBLE] constant and [INAUDIBLE] 00:46:03.660 --> 00:46:06.770 constant in the red one, y equals y0, right? 00:46:06.770 --> 00:46:08.915 So, this depends on x. 00:46:08.915 --> 00:46:11.010 So this has r sub x. 00:46:11.010 --> 00:46:14.800 00:46:14.800 --> 00:46:18.830 This is the velocity with respect to the variable x. 00:46:18.830 --> 00:46:23.200 And the other one, the blue one, x equals x0, 00:46:23.200 --> 00:46:27.640 means x0 is held fixed and y is the variable. 00:46:27.640 --> 00:46:30.505 So I have to do r sub y, and what am I gonna get? 00:46:30.505 --> 00:46:32.696 I'm gonna get the blue vector. 00:46:32.696 --> 00:46:34.880 What's the property of the blue vector? 00:46:34.880 --> 00:46:37.830 It's tangent to the purple line. 00:46:37.830 --> 00:46:44.160 So r sub y has to be tangent to the curve. 00:46:44.160 --> 00:46:47.440 00:46:47.440 --> 00:46:55.310 x0, y, f of x0 and y is the curve. 00:46:55.310 --> 00:46:59.770 And r sub x is tangent to which curve? 00:46:59.770 --> 00:47:02.400 Who is telling me which curve? 00:47:02.400 --> 00:47:12.020 x, y0 sub constant, f of x and y0. 00:47:12.020 --> 00:47:14.489 So that's a curve that depends only on y, 00:47:14.489 --> 00:47:16.854 y is the time in this case. 00:47:16.854 --> 00:47:19.000 And that's the curve that depends only on x. 00:47:19.000 --> 00:47:21.210 x is the time in this case. 00:47:21.210 --> 00:47:24.580 r sub x and r sub y are the tangent vectors. 00:47:24.580 --> 00:47:26.830 What's magical about them? 00:47:26.830 --> 00:47:30.540 If I shape this triangle between them, 00:47:30.540 --> 00:47:32.172 that will be the tangent plane. 00:47:32.172 --> 00:47:35.950 00:47:35.950 --> 00:47:39.170 And I make a smile because I discovered the tangent plane 00:47:39.170 --> 00:47:43.230 in a different way than we did it last time. 00:47:43.230 --> 00:47:51.005 So the tangent plane represents the plane of the vector r sub 00:47:51.005 --> 00:47:54.532 x and r sub y. 00:47:54.532 --> 00:48:02.290 The tangent plane represents the plane 00:48:02.290 --> 00:48:13.080 given by vectors r sub x and r sub y with what conditions? 00:48:13.080 --> 00:48:14.025 It's a conditional. 00:48:14.025 --> 00:48:17.010 00:48:17.010 --> 00:48:20.630 r sub x and r sub y shouldn't be 0. 00:48:20.630 --> 00:48:24.850 r sub x different from 0, r sub y different from 0, 00:48:24.850 --> 00:48:27.455 and r sub x and r sub y are not collinear. 00:48:27.455 --> 00:48:32.160 00:48:32.160 --> 00:48:35.050 What's gonna happen if they are collinear? 00:48:35.050 --> 00:48:36.880 Well, they're gonna collapse; they are not 00:48:36.880 --> 00:48:38.190 gonna determine a plane. 00:48:38.190 --> 00:48:40.770 So there will be no tangent planes. 00:48:40.770 --> 00:48:43.720 So they have to be linearly independent. 00:48:43.720 --> 00:48:47.940 For the people who are taking now linear algebra, I'm saying. 00:48:47.940 --> 00:48:50.940 So we have no other choice, we have 00:48:50.940 --> 00:48:54.820 to assume that these vectors, called partial velocities, 00:48:54.820 --> 00:49:04.120 by the way, for the motion across the surface. 00:49:04.120 --> 00:49:04.620 OK? 00:49:04.620 --> 00:49:06.970 These are the partial velocities, or partial velocity 00:49:06.970 --> 00:49:08.630 vectors. 00:49:08.630 --> 00:49:12.860 Partial velocity vectors have to determine a plane, 00:49:12.860 --> 00:49:16.560 so I have to assume that they are non-zero, 00:49:16.560 --> 00:49:20.120 they never become 0, and they are not collinear. 00:49:20.120 --> 00:49:23.270 If they are collinear, life is over for you. 00:49:23.270 --> 00:49:24.140 OK? 00:49:24.140 --> 00:49:29.390 So I have to assume that I throw away all the points where 00:49:29.390 --> 00:49:35.100 the velocities become 0, and all the points where--those are 00:49:35.100 --> 00:49:39.710 singularity points--where my velocity vectors are 0. 00:49:39.710 --> 00:49:43.710 00:49:43.710 --> 00:49:45.820 Have you ever studied design? 00:49:45.820 --> 00:49:47.350 Any kind of experimental design. 00:49:47.350 --> 00:49:52.310 Like, how do you design a car, the coordinate lines on a car? 00:49:52.310 --> 00:49:53.280 I'm just dreaming. 00:49:53.280 --> 00:50:00.200 You have a car, a beautiful car, and then you have-- Well, 00:50:00.200 --> 00:50:04.790 I cannot draw really well, but anyway. 00:50:04.790 --> 00:50:08.730 I have these coordinate lines on this car. 00:50:08.730 --> 00:50:12.060 It's a mesh what I have there. 00:50:12.060 --> 00:50:15.510 Actually, we do that in animation all the time. 00:50:15.510 --> 00:50:21.030 We have meshes over the models we have in animation. 00:50:21.030 --> 00:50:22.660 Think Avatar. 00:50:22.660 --> 00:50:27.210 Now, those are all coordinate lines. 00:50:27.210 --> 00:50:33.650 Those coordinate lines would be, even your singularities, where? 00:50:33.650 --> 00:50:38.510 For example, if you take a body in a mesh like that, in a net, 00:50:38.510 --> 00:50:43.190 in, like, a fishnet, then you pull from the fishnet, 00:50:43.190 --> 00:50:52.980 all the coordinate lines will come together, 00:50:52.980 --> 00:50:55.310 and this would be a singularity. 00:50:55.310 --> 00:50:57.890 We avoid this kind of singularity. 00:50:57.890 --> 00:51:00.430 So these are points where something bad happened. 00:51:00.430 --> 00:51:05.380 Either the velocity vectors become collinear. 00:51:05.380 --> 00:51:07.430 You see what I'm talking about? 00:51:07.430 --> 00:51:11.260 Or the velocity vectors shrank to 0. 00:51:11.260 --> 00:51:14.190 So that's a bad point; that's a singularity point. 00:51:14.190 --> 00:51:16.870 They have this problem when meshing. 00:51:16.870 --> 00:51:20.670 So when they make these models that 00:51:20.670 --> 00:51:26.850 involve two-dimensional meshing and three-dimensional ambient 00:51:26.850 --> 00:51:31.490 space, like it is in animation, the mesh 00:51:31.490 --> 00:51:34.630 is called regular if we don't have 00:51:34.630 --> 00:51:39.770 this kind of singularity, where the velocity vectors become 0, 00:51:39.770 --> 00:51:42.000 or collinear. 00:51:42.000 --> 00:51:45.795 It's very important for a person who programs in animation 00:51:45.795 --> 00:51:47.220 to know mathematics. 00:51:47.220 --> 00:51:50.100 If they don't understand these things, it's over. 00:51:50.100 --> 00:51:55.910 Because you write the matrix, and you will know the vectors 00:51:55.910 --> 00:51:59.954 will become collinear when the two vectors--let's say two rows 00:51:59.954 --> 00:52:00.495 of a matrix-- 00:52:00.495 --> 00:52:00.810 STUDENT: Parallel. 00:52:00.810 --> 00:52:01.880 PROFESSOR TODA: Are proportional. 00:52:01.880 --> 00:52:02.510 Or parallel. 00:52:02.510 --> 00:52:03.870 Or proportional. 00:52:03.870 --> 00:52:07.550 So, everything is numerical in terms of those matrices, 00:52:07.550 --> 00:52:12.890 but it's just a discretization of a continuous phenomenon, 00:52:12.890 --> 00:52:14.010 which is this one. 00:52:14.010 --> 00:52:17.690 00:52:17.690 --> 00:52:19.970 Do you remember Toy Story? 00:52:19.970 --> 00:52:20.815 OK. 00:52:20.815 --> 00:52:24.300 The Toy Story people, the renderers, 00:52:24.300 --> 00:52:27.010 the ones who did the rendering techniques for Toy Story, 00:52:27.010 --> 00:52:30.410 both have their master's in mathematics. 00:52:30.410 --> 00:52:33.970 And you realize why now to do that you 00:52:33.970 --> 00:52:38.860 have to know calc I, calc II, calc III, linear algebra, 00:52:38.860 --> 00:52:41.120 be able to deal with matrices. 00:52:41.120 --> 00:52:45.610 Have a programming course or two; that's essential. 00:52:45.610 --> 00:52:50.042 They took advanced calculus because some people 00:52:50.042 --> 00:52:55.420 don't cover thi-- I was about to skip it right now in calc III. 00:52:55.420 --> 00:53:00.110 But they teach that in advanced calculus 4350, 4351. 00:53:00.110 --> 00:53:02.672 So that's about as far as you can get, 00:53:02.672 --> 00:53:05.870 and differential equation's also very important. 00:53:05.870 --> 00:53:09.510 So, if you master those and you go into something else, 00:53:09.510 --> 00:53:12.320 like programming, electrical engineering, 00:53:12.320 --> 00:53:14.250 you're ready for animation. 00:53:14.250 --> 00:53:16.970 [INAUDIBLE] If you went I want to be a rendering 00:53:16.970 --> 00:53:20.140 guy for the next movie, then they'll say no, 00:53:20.140 --> 00:53:21.600 we won't take you. 00:53:21.600 --> 00:53:23.920 I have a friend who works for Disney. 00:53:23.920 --> 00:53:26.780 She wanted to get a PhD. 00:53:26.780 --> 00:53:29.384 At some point, she changed her mind 00:53:29.384 --> 00:53:31.967 and ended up just with a master's in mathematics 00:53:31.967 --> 00:53:33.800 while I was in Kansas, University of Kansas, 00:53:33.800 --> 00:53:36.565 and she said, "You know what? 00:53:36.565 --> 00:53:41.620 Disney's just giving me $65,000 as an intern." 00:53:41.620 --> 00:53:45.582 And I was like OK and probably asked [INAUDIBLE] $40,000 as 00:53:45.582 --> 00:53:46.676 a postdoc. 00:53:46.676 --> 00:53:48.050 And she said, "Good luck to you." 00:53:48.050 --> 00:53:49.420 Good luck to you, too. 00:53:49.420 --> 00:53:52.520 But we stayed in touch, and right now she's 00:53:52.520 --> 00:53:57.460 making twice as much as I'm making, for Disney. 00:53:57.460 --> 00:53:58.676 Is she happy? 00:53:58.676 --> 00:53:59.568 Yeah. 00:53:59.568 --> 00:54:00.460 Would I be happy? 00:54:00.460 --> 00:54:01.406 No. 00:54:01.406 --> 00:54:05.764 Because she works for 11 hours a day. 00:54:05.764 --> 00:54:08.120 11 hours a day, on a chair. 00:54:08.120 --> 00:54:09.090 That would kill me. 00:54:09.090 --> 00:54:15.070 I mean, I spend about six hours sitting on a chair every day 00:54:15.070 --> 00:54:19.160 of the week, but it's still too much. 00:54:19.160 --> 00:54:20.800 She's a hard worker, though. 00:54:20.800 --> 00:54:22.820 She loves what she's doing. 00:54:22.820 --> 00:54:24.060 The problem is your eyes. 00:54:24.060 --> 00:54:27.420 After a while, your eyes are going bad. 00:54:27.420 --> 00:54:33.600 So, what is the normal for the plane in this case? 00:54:33.600 --> 00:54:37.298 I'll try my best ability to draw normal. 00:54:37.298 --> 00:54:38.714 The normal has to be perpendicular 00:54:38.714 --> 00:54:41.950 to the tangent space, right? 00:54:41.950 --> 00:54:43.700 Tangent plane. 00:54:43.700 --> 00:54:46.230 So, n has to be perpendicular to our sub 00:54:46.230 --> 00:54:49.790 x and has to be perpendicular to our sub y. 00:54:49.790 --> 00:54:53.045 00:54:53.045 --> 00:54:56.240 So, can you have any guess how in the world 00:54:56.240 --> 00:54:59.470 I'm gonna get n vector? 00:54:59.470 --> 00:55:01.454 STUDENT: [INAUDIBLE] 00:55:01.454 --> 00:55:02.870 PROFESSOR TODA: [INAUDIBLE] That's 00:55:02.870 --> 00:55:05.070 why you need to know linear algebra 00:55:05.070 --> 00:55:09.040 sort of at the same time, but you guys are making it fine. 00:55:09.040 --> 00:55:10.460 It's not a big deal. 00:55:10.460 --> 00:55:16.450 You have a matrix, i, j, k in the front row vectors, 00:55:16.450 --> 00:55:21.570 and then you have r sub x that you gave me, and I erased it. 00:55:21.570 --> 00:55:23.605 1, 0, f sub x. 00:55:23.605 --> 00:55:26.590 00:55:26.590 --> 00:55:29.150 0, 1, f sub y. 00:55:29.150 --> 00:55:40.604 And you have exactly 18 seconds to compute this vector. 00:55:40.604 --> 00:55:47.627 00:55:47.627 --> 00:55:48.460 STUDENT: [INAUDIBLE] 00:55:48.460 --> 00:55:52.890 00:55:52.890 --> 00:55:55.690 PROFESSOR TODA: You want k, but I want to leave k at the end 00:55:55.690 --> 00:55:58.540 because I always order my vectors. 00:55:58.540 --> 00:56:02.137 Something i plus something j plus something k. 00:56:02.137 --> 00:56:02.970 [INTERPOSING VOICES] 00:56:02.970 --> 00:56:05.276 00:56:05.276 --> 00:56:06.400 PROFESSOR TODA: Am I right? 00:56:06.400 --> 00:56:07.025 Minus f sub x-- 00:56:07.025 --> 00:56:09.910 STUDENT: Minus f of x plus k. 00:56:09.910 --> 00:56:11.892 PROFESSOR TODA: --times i. 00:56:11.892 --> 00:56:14.262 For j, do I have to change sign? 00:56:14.262 --> 00:56:18.370 Yeah, because 1 plus 2 is odd. 00:56:18.370 --> 00:56:21.273 So I go minus 1. 00:56:21.273 --> 00:56:22.600 And do it slowly. 00:56:22.600 --> 00:56:25.740 You're not gonna make fun of me; I gotta make fun of you, OK? 00:56:25.740 --> 00:56:28.100 And minus 1 times-- 00:56:28.100 --> 00:56:29.440 STUDENT: Did you forget f y? 00:56:29.440 --> 00:56:37.150 PROFESSOR TODA: --f sub y--I go like that--sub y times j plus 00:56:37.150 --> 00:56:39.228 k. 00:56:39.228 --> 00:56:42.120 As you said very well in the most elegant way 00:56:42.120 --> 00:56:45.750 without being like yours, but I say it like this. 00:56:45.750 --> 00:56:49.870 So you have minus f sub x, minus f sub y, 00:56:49.870 --> 00:56:54.580 and 1 as a triple with angular brackets--You love that. 00:56:54.580 --> 00:57:00.250 I don't; I like it parentheses [INAUDIBLE]--equals n. 00:57:00.250 --> 00:57:03.485 But n is non-unitary, but I don't care. 00:57:03.485 --> 00:57:04.730 Why don't I care? 00:57:04.730 --> 00:57:08.270 I can write the tangent plane very well 00:57:08.270 --> 00:57:13.216 without that n being unitary, right? 00:57:13.216 --> 00:57:14.540 It doesn't matter in the end. 00:57:14.540 --> 00:57:17.680 These would be my a, b, c. 00:57:17.680 --> 00:57:18.860 Now I know my ABC. 00:57:18.860 --> 00:57:20.400 I know my ABC. 00:57:20.400 --> 00:57:26.315 So, the tangent plane is your next guess. 00:57:26.315 --> 00:57:30.140 The tangent plane would be perpendicular to n. 00:57:30.140 --> 00:57:32.150 So this is n. 00:57:32.150 --> 00:57:35.515 The tangent plane passes through the point p 00:57:35.515 --> 00:57:37.350 and is perpendicular to n. 00:57:37.350 --> 00:57:43.147 So, what is the equation of the tangent plane? 00:57:43.147 --> 00:57:44.730 STUDENT: Do you want scalar equations? 00:57:44.730 --> 00:57:49.160 PROFESSOR TODA: A by x minus 0. 00:57:49.160 --> 00:57:50.220 Very good. 00:57:50.220 --> 00:57:56.330 That's exactly what I wanted you to write. 00:57:56.330 --> 00:58:01.390 All right, so, does it look familiar? 00:58:01.390 --> 00:58:01.920 Not yet. 00:58:01.920 --> 00:58:02.387 [STUDENT SNEEZES] 00:58:02.387 --> 00:58:02.854 STUDENT: Bless you. 00:58:02.854 --> 00:58:03.757 STUDENT: Bless you. 00:58:03.757 --> 00:58:04.840 PROFESSOR TODA: Bless you. 00:58:04.840 --> 00:58:05.994 Who sneezed? 00:58:05.994 --> 00:58:08.800 OK. 00:58:08.800 --> 00:58:10.370 Am I almost done? 00:58:10.370 --> 00:58:11.700 Well, I am almost done. 00:58:11.700 --> 00:58:14.930 I have to go backwards, and whatever I get 00:58:14.930 --> 00:58:17.760 I'll put it big here in a big formula on top. 00:58:17.760 --> 00:58:22.370 I'm gonna say oh, my God. 00:58:22.370 --> 00:58:24.020 No, that's not what I'm gonna say. 00:58:24.020 --> 00:58:33.330 I'm gonna say minus f sub x at my point p--that is a, right? 00:58:33.330 --> 00:58:37.089 Times x minus x0. 00:58:37.089 --> 00:58:45.956 Plus minus f sub y at the point p; that's b. 00:58:45.956 --> 00:58:54.660 y minus y0 plus--c is 1, right? 00:58:54.660 --> 00:58:55.360 c is 1. 00:58:55.360 --> 00:58:58.020 I'm not gonna write it because if I write 00:58:58.020 --> 00:59:03.990 it you'll want to make fun of me. z minus z0 equals 0. 00:59:03.990 --> 00:59:08.560 Now it starts looking like something familiar, finally. 00:59:08.560 --> 00:59:14.961 Now we discovered that the tangent plane 00:59:14.961 --> 00:59:20.630 can be written as z minus z0. 00:59:20.630 --> 00:59:24.630 I'm keeping the guys z minus z0 on the left-hand side. 00:59:24.630 --> 00:59:28.630 And these guys are gonna move to the right-hand side. 00:59:28.630 --> 00:59:33.570 So, I'm gonna have again, my friend, 00:59:33.570 --> 00:59:45.460 the equation of the tangent plane for the graph z equals f 00:59:45.460 --> 00:59:46.180 of x,y. 00:59:46.180 --> 00:59:51.940 00:59:51.940 --> 00:59:54.870 But you will say OK, I think by now 00:59:54.870 --> 00:59:57.340 we've learned these by heart, we know 00:59:57.340 --> 01:00:00.480 the equation of the tangent plane, and now we're asleep. 01:00:00.480 --> 01:00:06.160 But what if your surface would be implicit the way 01:00:06.160 --> 01:00:08.760 you gave it to us at first. 01:00:08.760 --> 01:00:11.840 Maybe you remember the sphere that was an implicit equation, 01:00:11.840 --> 01:00:14.720 x squared plus x squared plus x squared equals-- 01:00:14.720 --> 01:00:16.030 What do you want it to be? 01:00:16.030 --> 01:00:16.777 STUDENT: 16. 01:00:16.777 --> 01:00:17.610 PROFESSOR TODA: Huh? 01:00:17.610 --> 01:00:18.800 STUDENT: 16. 01:00:18.800 --> 01:00:20.920 PROFESSOR TODA: 16. 01:00:20.920 --> 01:00:22.385 So, radius should be 4. 01:00:22.385 --> 01:00:26.798 01:00:26.798 --> 01:00:31.060 And in such a case, the equation is of the type f of x, y, z 01:00:31.060 --> 01:00:33.190 equals constant. 01:00:33.190 --> 01:00:35.740 Can we write again the equation [INAUDIBLE]? 01:00:35.740 --> 01:00:39.770 01:00:39.770 --> 01:00:42.240 Well, you say well, you just taught 01:00:42.240 --> 01:00:51.240 us some theory that says I have to think of u and v, but not x 01:00:51.240 --> 01:00:51.850 and y. 01:00:51.850 --> 01:00:55.190 Because if I think of x and y, what would they be? 01:00:55.190 --> 01:00:57.960 I think the sphere as being an apple. 01:00:57.960 --> 01:01:01.880 Not an apple, something you can cut easily. 01:01:01.880 --> 01:01:05.480 Well, an apple, an orange, something. 01:01:05.480 --> 01:01:07.080 A round piece of soft cheese. 01:01:07.080 --> 01:01:09.510 I started being hungry, and I'm dreaming. 01:01:09.510 --> 01:01:14.190 So, this is a huge something you're gonna slice up. 01:01:14.190 --> 01:01:19.100 If you are gonna do it with x and y, 01:01:19.100 --> 01:01:21.580 the slices would be like this. 01:01:21.580 --> 01:01:24.630 Like that and like this, right? 01:01:24.630 --> 01:01:27.120 And in that case, your coordinate curves 01:01:27.120 --> 01:01:30.540 are sort of weird. 01:01:30.540 --> 01:01:33.610 If you want to do it in different coordinates, 01:01:33.610 --> 01:01:35.080 so we want to change coordinates, 01:01:35.080 --> 01:01:39.810 and those coordinates should be plotted to the longitude, 01:01:39.810 --> 01:01:43.628 then we cannot use x and y. 01:01:43.628 --> 01:01:44.990 Am I right? 01:01:44.990 --> 01:01:46.590 We cannot use x and y. 01:01:46.590 --> 01:01:50.630 So those u and v will be different coordinates, 01:01:50.630 --> 01:01:55.160 and then we can do it like that, latitude. 01:01:55.160 --> 01:01:57.790 01:01:57.790 --> 01:02:00.010 [INAUDIBLE] minus [INAUDIBLE]. 01:02:00.010 --> 01:02:00.805 And longitude. 01:02:00.805 --> 01:02:03.080 We are gonna talk about spherical coordinates 01:02:03.080 --> 01:02:05.202 later, not today. 01:02:05.202 --> 01:02:06.160 Latitude and longitude. 01:02:06.160 --> 01:02:10.340 01:02:10.340 --> 01:02:12.890 1 point extra credit, because eventually we 01:02:12.890 --> 01:02:16.884 are gonna get there, chapter 12.7. 01:02:16.884 --> 01:02:20.650 12.7 comes way after spring break. 01:02:20.650 --> 01:02:27.390 But before we get there, who is in mechanical engineering 01:02:27.390 --> 01:02:28.830 again? 01:02:28.830 --> 01:02:32.710 You know about Euler's angles, and stuff like that. 01:02:32.710 --> 01:02:33.550 OK. 01:02:33.550 --> 01:02:40.330 Can you write me the equations of x 01:02:40.330 --> 01:02:47.850 and y and z of the sphere with respect to u and v, 01:02:47.850 --> 01:02:51.200 u being latitude and v being longitude? 01:02:51.200 --> 01:02:53.980 01:02:53.980 --> 01:02:58.641 These have to be trigonometric functions. 01:02:58.641 --> 01:03:03.860 01:03:03.860 --> 01:03:10.770 In terms of u and v, when u is latitude and v is longitude. 01:03:10.770 --> 01:03:15.310 1 point extra credit until a week from today. 01:03:15.310 --> 01:03:16.280 How about that? 01:03:16.280 --> 01:03:20.650 01:03:20.650 --> 01:03:23.850 U and v are latitude and longitude. 01:03:23.850 --> 01:03:33.800 And express the xyz point in the ambient space on the sphere. 01:03:33.800 --> 01:03:36.460 x squared plus x squared plus x squared would be 16. 01:03:36.460 --> 01:03:40.020 So you'll have lots of cosines and sines [INAUDIBLE] 01:03:40.020 --> 01:03:46.024 of those angles, the latitude angle and the longitude angle. 01:03:46.024 --> 01:03:49.800 And I would suggest to you that you take--for the extra credit 01:03:49.800 --> 01:03:54.910 thing--you take the longitude angle to be from 0 to 2pi, 01:03:54.910 --> 01:04:00.150 from the Greenwich 0 meridian going back to himself, 01:04:00.150 --> 01:04:07.725 and--well, there are two ways we do this in mathematics 01:04:07.725 --> 01:04:09.810 because mathematicians are so diverse. 01:04:09.810 --> 01:04:14.850 Some of us, say, for me, I measure the latitude 01:04:14.850 --> 01:04:17.100 starting from the North Pole. 01:04:17.100 --> 01:04:20.270 I think that's because we all believe in Santa or something. 01:04:20.270 --> 01:04:23.440 So, we start measuring always from the North Pole 01:04:23.440 --> 01:04:27.030 because that's the most important place on Earth. 01:04:27.030 --> 01:04:35.633 They go 0, pi over 2, and then-- what is our lat--shame on me. 01:04:35.633 --> 01:04:36.480 STUDENT: It's 33. 01:04:36.480 --> 01:04:37.271 PROFESSOR TODA: 33? 01:04:37.271 --> 01:04:39.220 OK. 01:04:39.220 --> 01:04:44.060 Then pi would be the equator, and then pi 01:04:44.060 --> 01:04:45.834 would be the South Pole. 01:04:45.834 --> 01:04:50.625 But some other mathematicians, especially biologists 01:04:50.625 --> 01:04:54.530 and differential geometry people, I'm one of them, 01:04:54.530 --> 01:04:56.090 we go like that. 01:04:56.090 --> 01:05:01.620 Minus pi over 2, South Pole 0, pi over 2 North Pole. 01:05:01.620 --> 01:05:06.820 So we shift that kind of interval. 01:05:06.820 --> 01:05:10.280 Then for us, the trigonometric functions of these angles 01:05:10.280 --> 01:05:12.020 would be a little bit different when we 01:05:12.020 --> 01:05:14.395 do the spherical coordinates. 01:05:14.395 --> 01:05:16.335 OK, that's just extra credit. 01:05:16.335 --> 01:05:19.070 It has nothing to do with what I'm gonna do right now. 01:05:19.070 --> 01:05:22.960 What I'm gonna do right now is to pick a point on Earth. 01:05:22.960 --> 01:05:26.000 We have to find Lubbock. 01:05:26.000 --> 01:05:27.210 STUDENT: It's on the left. 01:05:27.210 --> 01:05:28.740 PROFESSOR TODA: Here? 01:05:28.740 --> 01:05:29.870 Is that a good point? 01:05:29.870 --> 01:05:32.400 01:05:32.400 --> 01:05:34.486 This is LBB. 01:05:34.486 --> 01:05:38.430 That's Lubbock International Airport. 01:05:38.430 --> 01:05:47.530 So, for Lubbock--let's call it p as well--draw the r sub u, 01:05:47.530 --> 01:05:52.550 r sub v. So, u was latitude. 01:05:52.550 --> 01:05:55.750 So if I fix the latitude, that means I fix 01:05:55.750 --> 01:05:58.650 the 33 point whatever you said. 01:05:58.650 --> 01:06:00.060 u equals u0. 01:06:00.060 --> 01:06:09.630 It is fixed, so I have u fixed, and v equals v0 is that. 01:06:09.630 --> 01:06:14.345 I fixed the meridian where we are. 01:06:14.345 --> 01:06:15.990 What is this tangent vector? 01:06:15.990 --> 01:06:20.518 01:06:20.518 --> 01:06:22.950 To the pink parallel, the tangent vector 01:06:22.950 --> 01:06:25.660 would be r sub what? 01:06:25.660 --> 01:06:26.160 STUDENT: v. 01:06:26.160 --> 01:06:27.785 PROFESSOR TODA: r sub v. You are right. 01:06:27.785 --> 01:06:28.920 You've got the idea. 01:06:28.920 --> 01:06:33.370 And the blue vector would be the partial velocity. 01:06:33.370 --> 01:06:39.466 That's the tangent vector to the blue meridian, 01:06:39.466 --> 01:06:43.920 which is r sub u. 01:06:43.920 --> 01:06:48.675 And what is n gonna be? n's gonna be r sub u [INAUDIBLE]. 01:06:48.675 --> 01:06:53.370 But is there any other way to do it in a simpler way 01:06:53.370 --> 01:06:55.515 without you guys going oh, man. 01:06:55.515 --> 01:06:58.055 Suppose some of you don't wanna do the extra credit 01:06:58.055 --> 01:07:00.332 and then say the heck with it; I don't 01:07:00.332 --> 01:07:03.610 care about her stinking extra credit until chapter 12, 01:07:03.610 --> 01:07:07.700 when I have to study the spherical coordinates, 01:07:07.700 --> 01:07:11.170 and is there another way to get n. 01:07:11.170 --> 01:07:13.408 I told you another way to get n. 01:07:13.408 --> 01:07:15.384 Well, we are getting there. 01:07:15.384 --> 01:07:21.750 n was the gradient of f over the length of that. 01:07:21.750 --> 01:07:26.490 And if we want it unitary, the length of f was what? 01:07:26.490 --> 01:07:31.720 f sub x, f sub y, f sub z vector, where 01:07:31.720 --> 01:07:36.530 the implicit equation of the surface was f of x, y, z 01:07:36.530 --> 01:07:38.400 equals c. 01:07:38.400 --> 01:07:40.240 So now we've done this before. 01:07:40.240 --> 01:07:42.470 You say Magdalena, you're repeating yourself. 01:07:42.470 --> 01:07:47.210 I know I'm repeating myself, but I want you to learn this twice 01:07:47.210 --> 01:07:49.260 so you can remember it. 01:07:49.260 --> 01:07:52.410 What is f of x, y, z? 01:07:52.410 --> 01:07:56.700 In my case, it's x squared plus y squared plus z squared 01:07:56.700 --> 01:07:59.930 minus 16, or even nothing. 01:07:59.930 --> 01:08:01.850 Because the constant doesn't matter anyway 01:08:01.850 --> 01:08:04.434 when I do the gradient. 01:08:04.434 --> 01:08:05.600 You guys are doing homework. 01:08:05.600 --> 01:08:08.210 You saw how the gradient goes. 01:08:08.210 --> 01:08:13.730 So gradient of f would be 2x times-- and that's 01:08:13.730 --> 01:08:19.384 the partial derivative times i plus 2y times j plus 2z times 01:08:19.384 --> 01:08:22.964 k-- that's very important. 01:08:22.964 --> 01:08:28.270 [? Lovett ?] has some coordinates we plug in. 01:08:28.270 --> 01:08:33.500 Now, can we write-- two things. 01:08:33.500 --> 01:08:35.620 I want two things from you. 01:08:35.620 --> 01:08:41.340 Write me a total differential b tangent plane 01:08:41.340 --> 01:08:46.140 at the point-- so, a, write the total differential. 01:08:46.140 --> 01:08:50.970 01:08:50.970 --> 01:08:53.670 I'm not going to ask you you to do a linear approximation. 01:08:53.670 --> 01:08:55.810 I could. 01:08:55.810 --> 01:09:23.660 B, write the tangent plane to the sphere at the point 01:09:23.660 --> 01:09:25.189 that-- I don't know. 01:09:25.189 --> 01:09:26.870 I don't want one that's trivial. 01:09:26.870 --> 01:09:30.040 01:09:30.040 --> 01:09:37.770 Let's take this 0, square root of 8, and square root of 8. 01:09:37.770 --> 01:09:39.640 I just have to make sure that I don't 01:09:39.640 --> 01:09:41.700 come with some nonsensical point that's 01:09:41.700 --> 01:09:43.290 not going to be on the sphere. 01:09:43.290 --> 01:09:45.863 This will be because I plugged it in in my mind. 01:09:45.863 --> 01:09:50.229 I get 8 plus 8 is 16 last time I checked, right? 01:09:50.229 --> 01:09:54.980 So after we do this we take a break. 01:09:54.980 --> 01:09:58.282 Suppose that this is a problem on your midterm, 01:09:58.282 --> 01:10:00.742 or on your final or on your homework, 01:10:00.742 --> 01:10:04.326 or on somebody [? YouTubed it ?] for a lot of money, 01:10:04.326 --> 01:10:10.010 you asked them, $25 an hour for me to work that problem. 01:10:10.010 --> 01:10:10.570 That's good. 01:10:10.570 --> 01:10:16.730 I mean-- it's-- it's a class that you're taking 01:10:16.730 --> 01:10:20.030 for your general requirement because your school wants you 01:10:20.030 --> 01:10:22.470 to take calc 3. 01:10:22.470 --> 01:10:25.570 But it gives you-- and I know from experience, 01:10:25.570 --> 01:10:27.670 some of my students came back to me and said, 01:10:27.670 --> 01:10:30.160 after I took calc 3, I understood it 01:10:30.160 --> 01:10:33.380 so well that I was able to tutor calc 1, calc 2, calc 3, 01:10:33.380 --> 01:10:35.840 so I got a double job. 01:10:35.840 --> 01:10:38.060 Several hours a week, the tutoring center, 01:10:38.060 --> 01:10:39.557 math department, and several hours 01:10:39.557 --> 01:10:40.640 at the [INAUDIBLE] center. 01:10:40.640 --> 01:10:42.670 You know what I'm talking about? 01:10:42.670 --> 01:10:46.220 So I've had students who did well and ended up liking this, 01:10:46.220 --> 01:10:49.276 and said I can tutor this in my sleep. 01:10:49.276 --> 01:10:53.760 So-- and also private tutoring is always a possibility. 01:10:53.760 --> 01:10:55.200 OK. 01:10:55.200 --> 01:10:58.670 Write total differential. 01:10:58.670 --> 01:11:04.260 df equals, and now I'll say at any point. 01:11:04.260 --> 01:11:06.965 So I don't care what the value will be. 01:11:06.965 --> 01:11:08.820 I didn't say at what point. 01:11:08.820 --> 01:11:09.923 It means in general. 01:11:09.923 --> 01:11:12.010 Why is that? 01:11:12.010 --> 01:11:14.900 You tell me, you know that by now. 01:11:14.900 --> 01:11:18.410 2x times what? 01:11:18.410 --> 01:11:20.305 Now, you learned your lesson, you're 01:11:20.305 --> 01:11:21.980 never gonna make mistakes. 01:11:21.980 --> 01:11:25.490 2y plus 2z dz. 01:11:25.490 --> 01:11:26.450 That is very good. 01:11:26.450 --> 01:11:28.010 That's the total differential. 01:11:28.010 --> 01:11:33.960 Now, what is the equation of the tangent plane? 01:11:33.960 --> 01:11:37.040 It's not gonna be that. 01:11:37.040 --> 01:11:40.670 Because I'm not considering a graph. 01:11:40.670 --> 01:11:44.590 I'm considering an implicitly given surface 01:11:44.590 --> 01:11:52.720 by this implicit equation f of x, y, z, equals c, your friend. 01:11:52.720 --> 01:11:57.734 So what was, in that case, the equation of the plane 01:11:57.734 --> 01:11:59.630 written as? 01:11:59.630 --> 01:12:02.480 STUDENT: [INAUDIBLE] 01:12:02.480 --> 01:12:05.130 PROFESSOR TODA: I'm-- yeah, you guys are smart. 01:12:05.130 --> 01:12:06.440 I mean, you are fast. 01:12:06.440 --> 01:12:07.790 Let's do it in general. 01:12:07.790 --> 01:12:11.635 F sub x-- we did that last time, [INAUDIBLE] times-- 01:12:11.635 --> 01:12:14.260 do you guys remember? 01:12:14.260 --> 01:12:16.470 x minus x0. 01:12:16.470 --> 01:12:20.660 And this is at the point plus big F sub y at the point times 01:12:20.660 --> 01:12:25.590 y minus y0 plus big F sub z at the point z minus z0. 01:12:25.590 --> 01:12:26.810 This is just review. 01:12:26.810 --> 01:12:27.990 Equals 0. 01:12:27.990 --> 01:12:28.490 Stop. 01:12:28.490 --> 01:12:31.466 Where do these guys come from? 01:12:31.466 --> 01:12:32.954 From the gradient. 01:12:32.954 --> 01:12:34.830 From the gradient. 01:12:34.830 --> 01:12:40.150 Which are the a,b,c, now I know my ABCs, from the normal. 01:12:40.150 --> 01:12:41.925 My ABCs from the normal. 01:12:41.925 --> 01:12:46.636 So in this case-- I don't want to erase 01:12:46.636 --> 01:12:49.016 this beautiful picture. 01:12:49.016 --> 01:12:54.910 The last thing I have to do before the break is-- you 01:12:54.910 --> 01:12:56.900 said 0. 01:12:56.900 --> 01:12:58.960 I'm a lazy person by definition. 01:12:58.960 --> 01:13:02.990 Can you tell me why you said 0 times? 01:13:02.990 --> 01:13:05.000 STUDENT: Because the x value is [INAUDIBLE] 01:13:05.000 --> 01:13:07.430 PROFESSOR TODA: You said 2x, plug in and x equals 0 01:13:07.430 --> 01:13:10.050 from your point, Magdalena, so you don't 01:13:10.050 --> 01:13:12.410 have to write down everything. 01:13:12.410 --> 01:13:19.670 But I'm gonna write down 0 times x minus 0 plus-- what's 01:13:19.670 --> 01:13:20.613 next for me? 01:13:20.613 --> 01:13:21.710 STUDENT: 2 square root 8. 01:13:21.710 --> 01:13:23.630 PROFESSOR TODA: 2y, 2 root 8. 01:13:23.630 --> 01:13:26.187 Is root 8 beautiful? 01:13:26.187 --> 01:13:28.095 It looks like heck. 01:13:28.095 --> 01:13:32.870 At the end I'm gonna brush it up a little bit. 01:13:32.870 --> 01:13:39.120 This is the partial-- f sub y of t times y minus-- who is y, z? 01:13:39.120 --> 01:13:40.782 Root 8. 01:13:40.782 --> 01:13:41.734 Do I like it? 01:13:41.734 --> 01:13:43.638 I hate it, but it doesn't matter. 01:13:43.638 --> 01:13:45.542 Because I'm gonna simplify. 01:13:45.542 --> 01:13:52.470 Plus again, 2 root 8, thank you. 01:13:52.470 --> 01:13:56.710 This is my c guy. 01:13:56.710 --> 01:14:02.440 Times z minus root 8 equals 0. 01:14:02.440 --> 01:14:05.430 I picked another example from the one from the book, 01:14:05.430 --> 01:14:08.830 because you are gonna read the book anyway. 01:14:08.830 --> 01:14:11.960 I'm gonna erase that. 01:14:11.960 --> 01:14:14.830 And I'm gonna brush this up because it 01:14:14.830 --> 01:14:17.490 looks horrible to me. 01:14:17.490 --> 01:14:19.890 Thank God this goes away. 01:14:19.890 --> 01:14:21.980 So the plane will simply be a combination 01:14:21.980 --> 01:14:24.250 of my y and z in a constant. 01:14:24.250 --> 01:14:28.000 And if I want to make my life easier, 01:14:28.000 --> 01:14:30.466 I'm gonna divide by what? 01:14:30.466 --> 01:14:32.280 By this. 01:14:32.280 --> 01:14:34.390 So in the end, it doesn't matter. 01:14:34.390 --> 01:14:35.916 Come on. 01:14:35.916 --> 01:14:42.300 I'll get y minus root 8 plus c minus root 8 equals 0. 01:14:42.300 --> 01:14:44.020 Do I like it? 01:14:44.020 --> 01:14:44.770 I hate it. 01:14:44.770 --> 01:14:46.660 No, you know, I don't like it. 01:14:46.660 --> 01:14:49.040 Why don't I like it? 01:14:49.040 --> 01:14:50.430 It's not simplified. 01:14:50.430 --> 01:14:56.000 So in any case, if this were multiple choice, 01:14:56.000 --> 01:14:59.450 it would not be written like that, right? 01:14:59.450 --> 01:15:03.990 So what would be the simplified claim in this case? 01:15:03.990 --> 01:15:09.270 The way I would write it-- a y plus a z minus-- 01:15:09.270 --> 01:15:11.486 think, what is root 8? 01:15:11.486 --> 01:15:12.530 STUDENT: 2 root 2. 01:15:12.530 --> 01:15:13.738 PROFESSOR TODA: And 2 root 2. 01:15:13.738 --> 01:15:20.990 And 2 root 2, how much-- minus 4 root 2. 01:15:20.990 --> 01:15:28.840 And this is how you are expected to leave this answer boxed. 01:15:28.840 --> 01:15:37.812 This is that tangent plane at the point. 01:15:37.812 --> 01:15:41.200 01:15:41.200 --> 01:15:42.652 To the sphere. 01:15:42.652 --> 01:15:45.570 01:15:45.570 --> 01:15:48.570 There are programs-- one time I was teaching 01:15:48.570 --> 01:15:53.970 advance geometry, 4331, and one thing I gave my students to do, 01:15:53.970 --> 01:15:58.920 which was a lot of fun-- using a parametrization, 01:15:58.920 --> 01:16:02.710 plot the entire sphere with MathLab. 01:16:02.710 --> 01:16:04.060 We did it with MathLab. 01:16:04.060 --> 01:16:06.930 Some people said they know [INAUDIBLE] I didn't care. 01:16:06.930 --> 01:16:09.326 So MathLab for me was easier, so we 01:16:09.326 --> 01:16:11.800 plotted the sphere in MathLab. 01:16:11.800 --> 01:16:14.940 We picked a point, and we drew-- well, 01:16:14.940 --> 01:16:21.960 we drew-- with MathLab we drew the tangent plane that 01:16:21.960 --> 01:16:25.950 was tangent to the sphere at that point. 01:16:25.950 --> 01:16:27.220 And they liked it. 01:16:27.220 --> 01:16:29.550 It was-- you know what this class is, 01:16:29.550 --> 01:16:31.750 is-- if you're math majors you take it. 01:16:31.750 --> 01:16:34.250 It's called advanced geometries. 01:16:34.250 --> 01:16:35.630 Mainly it's theoretical. 01:16:35.630 --> 01:16:38.540 It teaches you Euclidian axioms and stuff, 01:16:38.540 --> 01:16:41.540 and then some non-Euclidian geometries. 01:16:41.540 --> 01:16:45.830 But I thought that I would do it into an honors class. 01:16:45.830 --> 01:16:49.270 And I put one third of that last class visualization 01:16:49.270 --> 01:16:50.850 with MathLab of geometry. 01:16:50.850 --> 01:16:54.020 And I think that was what they liked the most, not so 01:16:54.020 --> 01:16:56.070 much the axiomatic part and the proofs, 01:16:56.070 --> 01:17:03.270 but the hands-on computation and visualization in the lab. 01:17:03.270 --> 01:17:04.980 We have this lab, 113. 01:17:04.980 --> 01:17:07.340 We used to have two labs, but now we are poor, 01:17:07.340 --> 01:17:09.090 we only have one. 01:17:09.090 --> 01:17:10.510 No, we lost the lab. 01:17:10.510 --> 01:17:13.660 The undergraduate lab-- 009, next to you, 01:17:13.660 --> 01:17:18.560 is lost because-- I used to each calc 3 there. 01:17:18.560 --> 01:17:21.536 Not because-- that's not why we lost it. 01:17:21.536 --> 01:17:24.830 We lost it because we-- we put some 20 graduate students 01:17:24.830 --> 01:17:25.330 there. 01:17:25.330 --> 01:17:26.686 We have no space. 01:17:26.686 --> 01:17:30.810 And we have 130 graduate students in mathematics. 01:17:30.810 --> 01:17:32.430 Where do you put them? 01:17:32.430 --> 01:17:34.165 We just cram them into cubicles. 01:17:34.165 --> 01:17:37.590 So they made 20 cubicles here, and they put some, 01:17:37.590 --> 01:17:40.010 so we lost the lab. 01:17:40.010 --> 01:17:41.860 It's sad. 01:17:41.860 --> 01:17:42.730 All right. 01:17:42.730 --> 01:17:45.090 So that's it for now. 01:17:45.090 --> 01:17:47.540 We are gonna take a short break, and we 01:17:47.540 --> 01:17:52.120 will continue for one more hour, which is mostly application. 01:17:52.120 --> 01:17:54.660 I'm sort of done with 11.4. 01:17:54.660 --> 01:17:57.838 I'll jump into 11.5 next. 01:17:57.838 --> 01:18:00.832 Take a short break. 01:18:00.832 --> 01:18:02.828 Thanks for the attendance. 01:18:02.828 --> 01:18:04.824 Oh, and you did the calculus. 01:18:04.824 --> 01:18:05.822 Very good. 01:18:05.822 --> 01:19:51.584 01:19:51.584 --> 01:19:55.077 Did this homework give you a lot of headaches, troubles 01:19:55.077 --> 01:19:56.075 or anything, or not? 01:19:56.075 --> 01:19:57.572 Not too much? 01:19:57.572 --> 01:19:59.069 It's a long homework. 01:19:59.069 --> 01:20:00.566 49 problems-- 42 problems. 01:20:00.566 --> 01:20:05.556 01:20:05.556 --> 01:20:07.053 It wasn't bad? 01:20:07.053 --> 01:22:39.080 01:22:39.080 --> 01:22:45.831 OK, questions from the-- what was it, the first part-- mainly 01:22:45.831 --> 01:22:47.620 the first part of chapter 11. 01:22:47.620 --> 01:22:49.510 This is where we are. 01:22:49.510 --> 01:22:56.690 Right now we hit the half point because 11.8 01:22:56.690 --> 01:22:59.250 is the last section. 01:22:59.250 --> 01:23:03.310 And we will do that, that's Lagrange multipliers. 01:23:03.310 --> 01:23:06.900 So, let's do a little bit of a review. 01:23:06.900 --> 01:23:08.836 Questions about homework. 01:23:08.836 --> 01:23:11.160 Do you have them? 01:23:11.160 --> 01:23:13.990 Imagine this would be office hour. 01:23:13.990 --> 01:23:15.116 What would you ask? 01:23:15.116 --> 01:23:17.804 01:23:17.804 --> 01:23:19.510 STUDENT: I know it's a stupid question, 01:23:19.510 --> 01:23:22.210 but my visualization [INAUDIBLE] coming along, and question 01:23:22.210 --> 01:23:26.910 three about the sphere passing the plane and passing the line. 01:23:26.910 --> 01:23:31.590 So you have a 3, 5, and 4 x, y, and z, 01:23:31.590 --> 01:23:34.335 and you have a radius of 5. 01:23:34.335 --> 01:23:36.275 Is it passing the x, y plane? 01:23:36.275 --> 01:23:40.710 Is it passing [INAUDIBLE] x plane and [INAUDIBLE] 01:23:40.710 --> 01:23:42.220 passing the other plane. 01:23:42.220 --> 01:23:44.010 PROFESSOR TODA: So-- say again. 01:23:44.010 --> 01:23:46.050 So you have 3 and 4 and 5-- 01:23:46.050 --> 01:23:47.520 STUDENT: x minus-- yes. 01:23:47.520 --> 01:23:49.380 PROFESSOR TODA: What are the coordinates? 01:23:49.380 --> 01:23:50.691 STUDENT: 3, 4, and 5. 01:23:50.691 --> 01:23:53.410 PROFESSOR TODA: 3, 4, and 5, just as you said them. 01:23:53.410 --> 01:23:54.190 You can-- 01:23:54.190 --> 01:23:55.730 STUDENT: And the radius is 5. 01:23:55.730 --> 01:23:56.840 PROFESSOR TODA: Radius of? 01:23:56.840 --> 01:23:57.340 STUDENT: 5. 01:23:57.340 --> 01:23:59.640 Radius is equal to 5. 01:23:59.640 --> 01:24:00.600 [INAUDIBLE] 01:24:00.600 --> 01:24:02.110 PROFESSOR TODA: Yeah, well, OK. 01:24:02.110 --> 01:24:07.779 So assume you have a sphere of radius 5, which 01:24:07.779 --> 01:24:09.270 means you have 25. 01:24:09.270 --> 01:24:14.710 If you do the 3 squared plus 4 squared plus 5 squared, 01:24:14.710 --> 01:24:16.465 what is that? 01:24:16.465 --> 01:24:17.090 For this point. 01:24:17.090 --> 01:24:18.750 You have two separate points. 01:24:18.750 --> 01:24:22.919 For this point you have 25 plus 25. 01:24:22.919 --> 01:24:24.875 Are you guys with me? 01:24:24.875 --> 01:24:30.254 So you have the specific x0, y0, z0. 01:24:30.254 --> 01:24:39.060 You do the sum of the squares, and you get 50. 01:24:39.060 --> 01:24:43.900 My question is, is this point outside, inside the sphere 01:24:43.900 --> 01:24:45.370 or on the sphere? 01:24:45.370 --> 01:24:47.010 On the sphere, obviously, it's not, 01:24:47.010 --> 01:24:54.140 because it does not verify the equation of the sphere, right? 01:24:54.140 --> 01:24:59.140 STUDENT: [INAUDIBLE] those the location of the center point. 01:24:59.140 --> 01:25:01.307 STUDENT: Where's the center of the sphere? 01:25:01.307 --> 01:25:02.140 STUDENT: [INAUDIBLE] 01:25:02.140 --> 01:25:05.640 01:25:05.640 --> 01:25:09.140 PROFESSOR TODA: The center of the sphere would be at 0. 01:25:09.140 --> 01:25:11.640 STUDENT: [INAUDIBLE] 01:25:11.640 --> 01:25:13.520 PROFESSOR TODA: We are making up a question. 01:25:13.520 --> 01:25:14.730 So, right? 01:25:14.730 --> 01:25:16.784 So practically, I am making up a question. 01:25:16.784 --> 01:25:17.450 STUDENT: Oh, OK. 01:25:17.450 --> 01:25:22.930 PROFESSOR TODA: So I'm saying if you have a sphere of radius 5, 01:25:22.930 --> 01:25:27.175 and somebody gives you this point of coordinates 3, 4, 01:25:27.175 --> 01:25:29.240 and 5, where is the point? 01:25:29.240 --> 01:25:34.934 Is it inside the sphere, outside the sphere or on the sphere? 01:25:34.934 --> 01:25:37.100 On the sphere it cannot be because it doesn't verify 01:25:37.100 --> 01:25:39.776 the sphere. 01:25:39.776 --> 01:25:44.580 Ah, it looks like a Mr. Egg. 01:25:44.580 --> 01:25:47.280 I don't like it. 01:25:47.280 --> 01:25:50.610 I'm sorry, it's a sphere. 01:25:50.610 --> 01:25:54.880 So a point on a sphere that will have-- that's a hint. 01:25:54.880 --> 01:25:58.470 A point on a sphere that will have coordinates 3 and 4 01:25:58.470 --> 01:26:02.490 would be exactly 3, 4, and 0. 01:26:02.490 --> 01:26:05.960 So it would be where? 01:26:05.960 --> 01:26:07.760 STUDENT: 16, 4. 01:26:07.760 --> 01:26:11.480 PROFESSOR TODA: 3 squared plus 4 squared is 5 squared, right? 01:26:11.480 --> 01:26:13.300 So those are Pythagorean numbers. 01:26:13.300 --> 01:26:15.162 That's the beauty of them. 01:26:15.162 --> 01:26:22.602 01:26:22.602 --> 01:26:27.570 I'm trying to draw well. 01:26:27.570 --> 01:26:28.410 Right. 01:26:28.410 --> 01:26:29.980 This is the point a. 01:26:29.980 --> 01:26:33.235 01:26:33.235 --> 01:26:36.620 You go up how many? 01:26:36.620 --> 01:26:38.940 You shift by 5. 01:26:38.940 --> 01:26:41.329 So are you inside or outside? 01:26:41.329 --> 01:26:42.307 STUDENT: Outside. 01:26:42.307 --> 01:26:43.182 PROFESSOR TODA: Yeah. 01:26:43.182 --> 01:26:50.131 01:26:50.131 --> 01:26:55.020 STUDENT: Are you outside or are you exactly on-- oh. 01:26:55.020 --> 01:26:55.770 Sorry, I thought-- 01:26:55.770 --> 01:26:56.400 PROFESSOR TODA: You go-- 01:26:56.400 --> 01:26:58.286 STUDENT: I thought you were saying point a. 01:26:58.286 --> 01:26:59.960 Point a is like exactly-- [INAUDIBLE] 01:26:59.960 --> 01:27:00.810 PROFESSOR TODA: You are on the equator, 01:27:00.810 --> 01:27:02.260 and from the Equator of the Earth, 01:27:02.260 --> 01:27:05.750 you're going parallel to the z-axis, then you stay outside. 01:27:05.750 --> 01:27:08.570 But the question is more subtle than that. 01:27:08.570 --> 01:27:12.000 This is pretty-- you figured it out. 01:27:12.000 --> 01:27:15.310 1 point-- 0.5 extra credit. 01:27:15.310 --> 01:27:18.580 That we don't have-- I wish we had-- maybe 01:27:18.580 --> 01:27:19.900 we'll find some time. 01:27:19.900 --> 01:27:23.030 When I-- when we rewrite the book, maybe we should do that. 01:27:23.030 --> 01:27:38.636 So express the points outside the sphere, inside the sphere, 01:27:38.636 --> 01:27:50.210 and on the sphere using exclusively 01:27:50.210 --> 01:27:51.636 equalities and inequalities. 01:27:51.636 --> 01:27:57.900 01:27:57.900 --> 01:27:58.900 And that's extra credit. 01:27:58.900 --> 01:28:01.000 So, of course, the [INAUDIBLE] is obvious. 01:28:01.000 --> 01:28:06.800 The sphere is the set of the triples x, y, z in R3. 01:28:06.800 --> 01:28:09.680 01:28:09.680 --> 01:28:13.480 OK, I'm teaching you a little bit of mathematical language. 01:28:13.480 --> 01:28:19.560 x, y, z belongs to R3, R3 being the free space, 01:28:19.560 --> 01:28:23.810 with the property that x squared plus y squared plus z squared 01:28:23.810 --> 01:28:26.720 equals given a squared. 01:28:26.720 --> 01:28:29.840 What if you have less than, what if you have greater than? 01:28:29.840 --> 01:28:31.836 Ah, shut up, Magdalena. 01:28:31.836 --> 01:28:33.400 This is all up to you. 01:28:33.400 --> 01:28:35.910 You will figure out how the points 01:28:35.910 --> 01:28:40.920 on the outside and the points on the inside are characterized. 01:28:40.920 --> 01:28:47.060 And unfortunately we don't emphasize that in the textbook. 01:28:47.060 --> 01:28:49.510 I'll erase. 01:28:49.510 --> 01:28:51.682 You figured it out. 01:28:51.682 --> 01:28:53.265 And now I want to move on to something 01:28:53.265 --> 01:28:57.056 a little bit challenging, but not very challenging. 01:28:57.056 --> 01:29:11.203 01:29:11.203 --> 01:29:12.494 STUDENT: Professor, [INAUDIBLE] 01:29:12.494 --> 01:29:19.830 01:29:19.830 --> 01:29:21.330 PROFESSOR TODA: The last requirement 01:29:21.330 --> 01:29:22.630 on the extra credit? 01:29:22.630 --> 01:29:26.660 So I said the sphere represents the set of all 01:29:26.660 --> 01:29:29.532 triples x, y, z in R3 with the property 01:29:29.532 --> 01:29:31.990 that x squared plus y squared plus y squared plus z squared 01:29:31.990 --> 01:29:33.880 equals a squared. 01:29:33.880 --> 01:29:36.840 With the equality sign. 01:29:36.840 --> 01:29:40.020 Represent the points on the inside of the sphere 01:29:40.020 --> 01:29:44.560 and the outside of the sphere using just inequalities. 01:29:44.560 --> 01:29:45.280 Mathematics. 01:29:45.280 --> 01:29:48.710 No writing, no words, just mathematics. 01:29:48.710 --> 01:29:50.150 In set theory symbols. 01:29:50.150 --> 01:29:54.824 Like, the set of points with braces like that. 01:29:54.824 --> 01:29:57.680 OK. 01:29:57.680 --> 01:30:02.620 I'll help you review a little bit of stuff from the chain 01:30:02.620 --> 01:30:12.150 rule in-- in chapter-- I don't know, guys, 01:30:12.150 --> 01:30:14.770 it was a long time ago. 01:30:14.770 --> 01:30:15.730 Shame on me. 01:30:15.730 --> 01:30:19.320 Chapter 3, calc 1. 01:30:19.320 --> 01:30:38.180 Versus chain rule rules in calc in-- chapter 5 calc 3. 01:30:38.180 --> 01:30:40.550 This is a little bit of a warmup. 01:30:40.550 --> 01:30:42.325 I don't want to [INAUDIBLE] again 01:30:42.325 --> 01:30:44.330 next time when we meet on Thursday. 01:30:44.330 --> 01:30:45.990 Bless you. 01:30:45.990 --> 01:30:48.924 The bless you was out of the context. 01:30:48.924 --> 01:30:51.580 What was the chain rule? 01:30:51.580 --> 01:30:53.670 We did compositions of functions, 01:30:53.670 --> 01:31:01.090 and we had a diagram that we don't show you, but we should. 01:31:01.090 --> 01:31:05.050 There is practically a function that comes from a set A 01:31:05.050 --> 01:31:08.490 to a set B to a set C. These are the sets. 01:31:08.490 --> 01:31:12.760 And we have g and an f. 01:31:12.760 --> 01:31:17.480 And we have g of f of t. 01:31:17.480 --> 01:31:22.450 t is your favorite letter here. 01:31:22.450 --> 01:31:26.790 How do you do the derivative with respect 01:31:26.790 --> 01:31:28.940 to g composed with f? 01:31:28.940 --> 01:31:32.920 01:31:32.920 --> 01:31:36.850 I asked the same question to my Calc 1 and Calc 2 students, 01:31:36.850 --> 01:31:42.470 and they really had a hard time expressing themselves, 01:31:42.470 --> 01:31:44.710 expressing the chain rule. 01:31:44.710 --> 01:31:46.530 And when I gave them an example, they 01:31:46.530 --> 01:31:49.600 said, oh, I know how to do it on the example. 01:31:49.600 --> 01:31:55.090 I just don't know how to do it on the-- I like the numbers, 01:31:55.090 --> 01:31:57.510 but I don't like them letters. 01:31:57.510 --> 01:32:02.345 So how do we do it in an example? 01:32:02.345 --> 01:32:05.340 01:32:05.340 --> 01:32:09.140 I chose natural log, which you find everywhere. 01:32:09.140 --> 01:32:14.442 So how do you do d dt of this animal? 01:32:14.442 --> 01:32:15.888 It's an animal. 01:32:15.888 --> 01:32:18.310 STUDENT: [INAUDIBLE] 01:32:18.310 --> 01:32:21.490 PROFESSOR TODA: So the idea is you go from the outside 01:32:21.490 --> 01:32:23.140 to the inside, one at a time. 01:32:23.140 --> 01:32:24.680 My students know that. 01:32:24.680 --> 01:32:27.480 You prime the function, the outer function, 01:32:27.480 --> 01:32:30.572 the last one you applied, to the function inside. 01:32:30.572 --> 01:32:33.770 And you prime that with respect to the argument. 01:32:33.770 --> 01:32:37.040 This is called the argument in that case. 01:32:37.040 --> 01:32:40.694 Derivative of natural log is 1 over what? 01:32:40.694 --> 01:32:43.610 The argument. 01:32:43.610 --> 01:32:46.315 And you cover up natural log with your hand, 01:32:46.315 --> 01:32:47.170 and you keep going. 01:32:47.170 --> 01:32:51.868 And you say, next I go, times the derivative 01:32:51.868 --> 01:32:55.700 of this square, plus 1, prime with respect to t. 01:32:55.700 --> 01:32:58.460 So I go times 2t. 01:32:58.460 --> 01:33:01.290 And that's what we have. 01:33:01.290 --> 01:33:04.620 And they say, when you explain it like that, they said to me, 01:33:04.620 --> 01:33:06.350 I can understand it. 01:33:06.350 --> 01:33:09.050 But I'm having a problem understanding it 01:33:09.050 --> 01:33:12.960 when you express this diagram-- that it throws me off. 01:33:12.960 --> 01:33:19.242 So in order to avoid that kind of theoretical misconception, 01:33:19.242 --> 01:33:24.990 I'm saying, let us see what the heck this is. 01:33:24.990 --> 01:33:32.805 d dt of g of f of t, because this is what you're doing, 01:33:32.805 --> 01:33:34.680 has to have some understanding. 01:33:34.680 --> 01:33:38.615 The problem is that Mister f of t, that lives here, 01:33:38.615 --> 01:33:40.370 has a different argument. 01:33:40.370 --> 01:33:45.410 The letter in B should be, let's say, u. 01:33:45.410 --> 01:33:48.510 01:33:48.510 --> 01:33:51.684 That doesn't say anything practically. 01:33:51.684 --> 01:33:54.000 How do you differentiate with respect to what? 01:33:54.000 --> 01:33:56.240 You cannot say d dt here. 01:33:56.240 --> 01:34:00.940 So you have to call f of t something generic. 01:34:00.940 --> 01:34:05.210 You have to have a generic variable for that. 01:34:05.210 --> 01:34:13.690 So you have then dg du, at what specific value of u? 01:34:13.690 --> 01:34:18.050 At the specific value of u that we have as f of t. 01:34:18.050 --> 01:34:21.570 Do you understand the specificity of this? 01:34:21.570 --> 01:34:26.700 Times-- that's the chain rule, the product coming 01:34:26.700 --> 01:34:31.764 from the chain rule-- df pt. 01:34:31.764 --> 01:34:33.890 You take du dt or d of dt. 01:34:33.890 --> 01:34:34.940 It is the same thing. 01:34:34.940 --> 01:34:36.884 Say it again, df dt. 01:34:36.884 --> 01:34:41.260 01:34:41.260 --> 01:34:43.540 I had a student ask me, what if I put du dt? 01:34:43.540 --> 01:34:44.790 Would it be wrong? 01:34:44.790 --> 01:34:50.079 No, as long as you understand that u is a-something, 01:34:50.079 --> 01:34:54.867 as the image of this t. 01:34:54.867 --> 01:34:55.950 Do you know what he liked? 01:34:55.950 --> 01:34:58.935 01:34:58.935 --> 01:35:01.790 He said, do you know what I like about that? 01:35:01.790 --> 01:35:07.165 I like that I can imagine that these are two cowboys-- I 01:35:07.165 --> 01:35:09.450 told the same thing to my son. 01:35:09.450 --> 01:35:12.510 He was so excited, not about that, 01:35:12.510 --> 01:35:14.596 but about these two cowboys. 01:35:14.596 --> 01:35:17.329 Of course, he is 10. 01:35:17.329 --> 01:35:18.245 These are the cowboys. 01:35:18.245 --> 01:35:20.420 They are across. 01:35:20.420 --> 01:35:22.640 One is on top of the building there, 01:35:22.640 --> 01:35:24.850 shooting at this guy, who is here 01:35:24.850 --> 01:35:28.480 across the street on the bottom. 01:35:28.480 --> 01:35:31.390 So they are annihilating each other. 01:35:31.390 --> 01:35:33.290 They shoot and they die. 01:35:33.290 --> 01:35:37.080 And they die, and you're left with 1/3. 01:35:37.080 --> 01:35:41.780 The same idea is that, actually, these guys do not simplify. 01:35:41.780 --> 01:35:46.100 du and-- [? du, ?] they're not cowboys who shoot at each other 01:35:46.100 --> 01:35:48.580 at the same time and both die at the same time. 01:35:48.580 --> 01:35:53.220 It is not so romantic. 01:35:53.220 --> 01:35:59.640 But the idea of remembering this formula is the same. 01:35:59.640 --> 01:36:03.700 Because practically, if you want to annihilate the two cowboys 01:36:03.700 --> 01:36:06.330 and put your hands over them so you don't see them anymore, 01:36:06.330 --> 01:36:10.580 du dt, you would have to remember, oh, 01:36:10.580 --> 01:36:12.430 so that was the derivative with respect 01:36:12.430 --> 01:36:15.930 to t that I initially have of the guy on top, 01:36:15.930 --> 01:36:19.110 which was g of f of the composed function. 01:36:19.110 --> 01:36:22.850 So if you view g of f of t as the composed function, 01:36:22.850 --> 01:36:23.838 who is that? 01:36:23.838 --> 01:36:28.980 The composition g composed with f of t 01:36:28.980 --> 01:36:31.900 is the function g of f of t. 01:36:31.900 --> 01:36:34.940 This is the function that you want to differentiate 01:36:34.940 --> 01:36:37.370 with respect to time, t. 01:36:37.370 --> 01:36:40.980 This is this, prime with respect to t. 01:36:40.980 --> 01:36:46.024 It's like they would be killing each other, and you would die. 01:36:46.024 --> 01:36:48.200 And I liked this idea, and I said, 01:36:48.200 --> 01:36:50.390 I should tell that to my students and to my son. 01:36:50.390 --> 01:36:52.925 And, of course, my son started jumping around 01:36:52.925 --> 01:36:56.485 and said that he understands multiplication of fractions 01:36:56.485 --> 01:36:57.890 better now. 01:36:57.890 --> 01:37:01.390 They don't learn about simplifications-- I don't 01:37:01.390 --> 01:37:03.042 know how they teach these kids. 01:37:03.042 --> 01:37:06.320 01:37:06.320 --> 01:37:07.830 It became so complicated. 01:37:07.830 --> 01:37:10.900 It's as if mathematics-- mathematics is the same. 01:37:10.900 --> 01:37:12.110 It hasn't changed. 01:37:12.110 --> 01:37:14.310 It's the people who make the rules 01:37:14.310 --> 01:37:17.415 on how to teach it that change. 01:37:17.415 --> 01:37:21.530 So he simply doesn't see that this simplifies. 01:37:21.530 --> 01:37:24.690 And when I tell him simplify, he's like, what is simplify? 01:37:24.690 --> 01:37:25.850 What is this word simplify? 01:37:25.850 --> 01:37:27.235 My teacher doesn't use it. 01:37:27.235 --> 01:37:31.705 So I feel like sometimes I want to shoot myself. 01:37:31.705 --> 01:37:35.380 But he went over that and he understood about the idea 01:37:35.380 --> 01:37:37.420 of simplification. 01:37:37.420 --> 01:37:39.370 [? He ?] composing something on top 01:37:39.370 --> 01:37:43.190 and the bottom finding the common factors up and down, 01:37:43.190 --> 01:37:44.820 crossing them out, and so on. 01:37:44.820 --> 01:37:47.260 And so now he knows what it means. 01:37:47.260 --> 01:37:50.576 But imagine going to college without having 01:37:50.576 --> 01:37:51.450 this early knowledge. 01:37:51.450 --> 01:37:55.132 You come to college, you were good in school, 01:37:55.132 --> 01:37:57.090 and you've never learned enough simplification. 01:37:57.090 --> 01:38:00.220 And then somebody like me, and tells you simplification. 01:38:00.220 --> 01:38:03.010 You say, she is a foreigner. 01:38:03.010 --> 01:38:07.770 She has a language barrier that is [INAUDIBLE] she has 01:38:07.770 --> 01:38:10.050 that I've never heard before. 01:38:10.050 --> 01:38:15.250 So I wish the people who really re-conceive, re-write 01:38:15.250 --> 01:38:18.820 the curriculum for K12 would be a little bit 01:38:18.820 --> 01:38:21.730 more respectful of the history. 01:38:21.730 --> 01:38:25.590 Imagine that I would teach calculus 01:38:25.590 --> 01:38:28.813 without ever telling you anything about Leibniz, who 01:38:28.813 --> 01:38:31.200 was Leibniz, he doesn't exist. 01:38:31.200 --> 01:38:34.100 Or Euler, or one of these fathers. 01:38:34.100 --> 01:38:37.660 They are the ones who created these notations. 01:38:37.660 --> 01:38:42.630 And if we never tell you about them, that I guess, 01:38:42.630 --> 01:38:47.400 wherever they are, it is an injustice that we are doing. 01:38:47.400 --> 01:38:48.180 All right. 01:38:48.180 --> 01:38:53.520 Chain rule in Chapter 5 of Calc 3. 01:38:53.520 --> 01:38:56.310 This is a little bit more complicated, 01:38:56.310 --> 01:38:59.690 but I'm going to teach it to you because I like it. 01:38:59.690 --> 01:39:05.582 Imagine that you have z equals x squared plus y squared. 01:39:05.582 --> 01:39:06.570 What is that? 01:39:06.570 --> 01:39:08.052 It's an example of a graph. 01:39:08.052 --> 01:39:10.614 And I just taught you what a graph is. 01:39:10.614 --> 01:39:13.490 01:39:13.490 --> 01:39:22.939 But imagine that xy follow a curve. 01:39:22.939 --> 01:39:25.900 01:39:25.900 --> 01:39:27.884 [INAUDIBLE] with respect to time. 01:39:27.884 --> 01:39:37.780 01:39:37.780 --> 01:39:41.330 And you will say, Magdalena, can you draw that? 01:39:41.330 --> 01:39:45.670 What in the world do you mean that x and y follow a curve? 01:39:45.670 --> 01:39:46.770 I'll try to draw it. 01:39:46.770 --> 01:39:48.640 First of all, you are on a walk. 01:39:48.640 --> 01:39:50.480 You are in a beautiful valley. 01:39:50.480 --> 01:39:51.480 It's not a vase. 01:39:51.480 --> 01:39:57.088 It's a circular paraboloid, as an example. 01:39:57.088 --> 01:40:00.944 01:40:00.944 --> 01:40:01.908 It's like an egg shell. 01:40:01.908 --> 01:40:05.290 01:40:05.290 --> 01:40:07.050 You have a curve on that. 01:40:07.050 --> 01:40:08.050 You draw that. 01:40:08.050 --> 01:40:10.340 You have nothing better to do than decorating eggs 01:40:10.340 --> 01:40:10.960 for Easter. 01:40:10.960 --> 01:40:12.190 Hey, wait. 01:40:12.190 --> 01:40:14.695 Easter is far, far away. 01:40:14.695 --> 01:40:17.355 But let's say you want to decorate eggs for Easter. 01:40:17.355 --> 01:40:22.800 You take some color of paint and put paint on the egg. 01:40:22.800 --> 01:40:28.400 You are actually describing an arc of a curve. 01:40:28.400 --> 01:40:38.240 And x and y, their projection on the floor 01:40:38.240 --> 01:40:39.605 will be x of t, y of t. 01:40:39.605 --> 01:40:42.790 01:40:42.790 --> 01:40:45.470 Because you paint in time. 01:40:45.470 --> 01:40:46.220 You paint in time. 01:40:46.220 --> 01:40:48.090 You describe this in time. 01:40:48.090 --> 01:40:54.090 Now, if x of ty of t is being projected on the floor. 01:40:54.090 --> 01:40:58.820 Of course, you have a curve here as well, which is what? 01:40:58.820 --> 01:41:05.700 Which it will be x of t, y of t, z of t. 01:41:05.700 --> 01:41:06.620 Oh, my god. 01:41:06.620 --> 01:41:11.910 Yes, because the altitude also depends on the motion in time. 01:41:11.910 --> 01:41:13.810 All right. 01:41:13.810 --> 01:41:16.460 So what's missing here? 01:41:16.460 --> 01:41:18.880 It's missing the third coordinate, duh, that's 01:41:18.880 --> 01:41:21.382 0 because I'm on the floor. 01:41:21.382 --> 01:41:26.500 I'm on the xy plane, which is the floor z equals z. 01:41:26.500 --> 01:41:28.560 But now let's suppose that I want 01:41:28.560 --> 01:41:36.570 to say this is f of x and y, and I want to differentiate 01:41:36.570 --> 01:41:39.400 f with respect to t. 01:41:39.400 --> 01:41:40.730 And you go, say what? 01:41:40.730 --> 01:41:41.440 Oh, my god. 01:41:41.440 --> 01:41:42.480 What is that? 01:41:42.480 --> 01:41:45.840 I differentiate f with respect to t. 01:41:45.840 --> 01:41:48.730 By differentiating f with respect to t, 01:41:48.730 --> 01:41:54.780 I mean that I have f of x and y differentiated 01:41:54.780 --> 01:41:56.145 with respect to t. 01:41:56.145 --> 01:41:58.070 And you say, wait, Magdalena. 01:41:58.070 --> 01:41:59.820 This doesn't make any sense. 01:41:59.820 --> 01:42:03.640 And you would be right to say it doesn't make any sense. 01:42:03.640 --> 01:42:07.280 Can somebody tell me why it doesn't make any sense? 01:42:07.280 --> 01:42:13.580 It's not clear where in the world the variable t is inside. 01:42:13.580 --> 01:42:17.220 So I'm going to say, OK, x are themselves functions 01:42:17.220 --> 01:42:19.500 of t, functions of that. 01:42:19.500 --> 01:42:21.342 x of t, y of t. 01:42:21.342 --> 01:42:23.950 If I don't do that, it's not clear. 01:42:23.950 --> 01:42:27.722 So this is a composed function just like this one. 01:42:27.722 --> 01:42:28.680 Look at the similarity. 01:42:28.680 --> 01:42:31.100 It's really beautiful. 01:42:31.100 --> 01:42:35.670 This is a function of a function, g of f. 01:42:35.670 --> 01:42:38.520 This is a function of two functions. 01:42:38.520 --> 01:42:43.304 Say it again, f is a function of two functions, x and y. 01:42:43.304 --> 01:42:45.232 This was a function of a function of t. 01:42:45.232 --> 01:42:47.642 This was a function of two functions of t. 01:42:47.642 --> 01:42:48.606 Oh, my God. 01:42:48.606 --> 01:42:52.470 01:42:52.470 --> 01:42:55.080 How do we compute this? 01:42:55.080 --> 01:42:56.851 There is a rule. 01:42:56.851 --> 01:42:58.204 It can be proved. 01:42:58.204 --> 01:43:01.950 We will look a little bit into the theoretical justification 01:43:01.950 --> 01:43:03.403 of this proof later. 01:43:03.403 --> 01:43:05.720 But practically what you do, you say, 01:43:05.720 --> 01:43:07.955 I have to have some order in my life. 01:43:07.955 --> 01:43:09.090 OK.? 01:43:09.090 --> 01:43:12.880 So the way we do that, we differentiate first 01:43:12.880 --> 01:43:17.150 with respect to the first location, which is x. 01:43:17.150 --> 01:43:21.515 I go there, but I cannot write df dx because f is a mother 01:43:21.515 --> 01:43:23.110 of two babies. 01:43:23.110 --> 01:43:26.520 f is a function of two variables, x and y. 01:43:26.520 --> 01:43:28.800 She has to be a mother to both of them; 01:43:28.800 --> 01:43:31.620 otherwise, they get jealous of one another. 01:43:31.620 --> 01:43:37.630 So I have to say, partial of f with respect to x, 01:43:37.630 --> 01:43:38.860 I cannot use d. 01:43:38.860 --> 01:43:43.510 Like Leibniz, I have to use del, d of dx. 01:43:43.510 --> 01:43:49.030 At the point x of dy of t, this is the location I have. 01:43:49.030 --> 01:43:50.630 Times what? 01:43:50.630 --> 01:43:51.970 I keep derivation. 01:43:51.970 --> 01:43:55.640 I keep derivating, like don't drink and derive. 01:43:55.640 --> 01:43:56.630 What is that? 01:43:56.630 --> 01:43:58.981 The chain rule. 01:43:58.981 --> 01:44:05.430 Prime again, this guy x with respect to t, dx dt. 01:44:05.430 --> 01:44:09.320 And then you go, plus because she has 01:44:09.320 --> 01:44:11.570 to be a mother to both kids. 01:44:11.570 --> 01:44:14.670 The same thing for the second child. 01:44:14.670 --> 01:44:17.690 So you go, the derivative of f with respect 01:44:17.690 --> 01:44:26.990 to y, add x of ty of t times dy dt. 01:44:26.990 --> 01:44:30.230 01:44:30.230 --> 01:44:35.440 So you see on the surface, x and y are moving according to time. 01:44:35.440 --> 01:44:39.000 And somehow we want to measure the derivative 01:44:39.000 --> 01:44:42.792 of the resulting function, or composition function, 01:44:42.792 --> 01:44:44.610 with respect to time. 01:44:44.610 --> 01:44:46.330 This is a very important chain rule 01:44:46.330 --> 01:44:50.020 that I would like you to memorize. 01:44:50.020 --> 01:44:53.430 A chain rule. 01:44:53.430 --> 01:44:54.060 Chain Rule No. 01:44:54.060 --> 01:44:54.560 1. 01:44:54.560 --> 01:44:58.720 01:44:58.720 --> 01:44:59.920 Is it hard? 01:44:59.920 --> 01:45:01.490 No, but for me it was. 01:45:01.490 --> 01:45:04.690 When I was 21 and I saw that-- and, of course, 01:45:04.690 --> 01:45:06.020 my teacher was good. 01:45:06.020 --> 01:45:10.350 And he told me, Magdalena, imagine that instead of del you 01:45:10.350 --> 01:45:13.530 would have d's. 01:45:13.530 --> 01:45:16.680 So you have d and d and d and d. 01:45:16.680 --> 01:45:21.300 The dx dx here, dy dy here, they should be in your mind. 01:45:21.300 --> 01:45:22.720 They are facing each other. 01:45:22.720 --> 01:45:25.850 They are across on a diagonal. 01:45:25.850 --> 01:45:29.140 And then, of course, I didn't tell my teacher my idea 01:45:29.140 --> 01:45:31.770 with the cowboys, but it was funny. 01:45:31.770 --> 01:45:38.810 So this is the chain rule that re-makes, or generalizes 01:45:38.810 --> 01:45:42.870 this idea to two variables. 01:45:42.870 --> 01:45:47.970 Let's finish the example because we didn't do it. 01:45:47.970 --> 01:45:53.310 What is the derivative of f in our case? 01:45:53.310 --> 01:46:01.660 df dt will be-- oh, my god-- at any point p, how arbitary, 01:46:01.660 --> 01:46:03.848 would be what? 01:46:03.848 --> 01:46:07.640 First, you write with respect to x. 01:46:07.640 --> 01:46:10.501 2x, right? 01:46:10.501 --> 01:46:11.000 2x. 01:46:11.000 --> 01:46:16.900 But then you have to compute this dx, add the pair you give. 01:46:16.900 --> 01:46:19.650 And the pair they gave you has a t. 01:46:19.650 --> 01:46:23.450 So 2x is add x of ty-- if you're going 01:46:23.450 --> 01:46:25.335 to write it first like that, you're 01:46:25.335 --> 01:46:29.730 going to find it weird-- times, I'm done with the first guy. 01:46:29.730 --> 01:46:32.795 Then I'm going to take the second guy in red, 01:46:32.795 --> 01:46:35.310 and I'll put it here. 01:46:35.310 --> 01:46:39.278 dx dt, but dx dt everybody knows. 01:46:39.278 --> 01:46:45.080 [INAUDIBLE] Let me write it like this. 01:46:45.080 --> 01:46:52.186 Plus [INAUDIBLE] that guy again with green-- dy 01:46:52.186 --> 01:46:59.146 computed at the pair x of dy of [? t ?] times, 01:46:59.146 --> 01:47:01.511 again, in red, dy dt. 01:47:01.511 --> 01:47:06.730 01:47:06.730 --> 01:47:08.772 So how do we write the whole thing? 01:47:08.772 --> 01:47:10.990 Could I have written it from the beginning better? 01:47:10.990 --> 01:47:11.490 Yeah. 01:47:11.490 --> 01:47:20.630 2x of t, dx dt plus 2y of t dy. 01:47:20.630 --> 01:47:21.610 Is it hard? 01:47:21.610 --> 01:47:25.150 No, this is the idea. 01:47:25.150 --> 01:47:28.070 Let's have something more specific. 01:47:28.070 --> 01:47:30.230 I'm going to erase the whole thing. 01:47:30.230 --> 01:47:36.230 01:47:36.230 --> 01:47:40.205 I'll give you a problem that we gave on the final 01:47:40.205 --> 01:47:41.630 a few years ago. 01:47:41.630 --> 01:47:44.890 And I'll show you how my students cheated on that. 01:47:44.890 --> 01:47:53.386 And I let them cheat, in a way, because in the end 01:47:53.386 --> 01:47:54.060 they were smart. 01:47:54.060 --> 01:47:59.350 It didn't matter how they did the problem, as long as they 01:47:59.350 --> 01:48:01.790 got the correct answer. 01:48:01.790 --> 01:48:03.330 So the problem was like that. 01:48:03.330 --> 01:48:10.155 And my colleague did that many years ago, several years ago, 01:48:10.155 --> 01:48:11.980 did that several times. 01:48:11.980 --> 01:48:19.700 So he said, let's do f of t, dt squared and g of t. 01:48:19.700 --> 01:48:27.000 I'll I'll do this one, dq plus 1. 01:48:27.000 --> 01:48:42.980 And then let's [INAUDIBLE] the w of u 01:48:42.980 --> 01:48:54.100 and B, exactly the same thing I gave you before, [INAUDIBLE] I 01:48:54.100 --> 01:48:56.040 remember that. 01:48:56.040 --> 01:49:05.708 And he said, compute the derivative of w of f of t, 01:49:05.708 --> 01:49:10.280 and g of t with respect to t. 01:49:10.280 --> 01:49:12.250 And you will ask, wait a minute here. 01:49:12.250 --> 01:49:14.560 Why do you put d and not del? 01:49:14.560 --> 01:49:17.850 Because this is a composed function that in the end 01:49:17.850 --> 01:49:20.580 is a function of t only. 01:49:20.580 --> 01:49:22.680 So if you do it as a composed function, 01:49:22.680 --> 01:49:26.040 because this goes like this. 01:49:26.040 --> 01:49:31.560 t goes to two functions, f of t and u. 01:49:31.560 --> 01:49:34.454 01:49:34.454 --> 01:49:40.850 And there is a function w that takes both of them, that 01:49:40.850 --> 01:49:42.870 is a function of both of them. 01:49:42.870 --> 01:49:46.835 In the end, this composition that's straight from here 01:49:46.835 --> 01:49:50.826 to here, is a function of one variable only. 01:49:50.826 --> 01:49:54.850 01:49:54.850 --> 01:49:58.280 So my students then-- it was in the beginning of the examine, 01:49:58.280 --> 01:49:59.020 I remember. 01:49:59.020 --> 01:50:02.300 And they said, well, I forgot, they said. 01:50:02.300 --> 01:50:03.860 I stayed up almost all night. 01:50:03.860 --> 01:50:05.432 Don't do that. 01:50:05.432 --> 01:50:06.390 Don't do what they did. 01:50:06.390 --> 01:50:08.330 Many of my students stay up all night 01:50:08.330 --> 01:50:11.090 before the final because I think I scare people, 01:50:11.090 --> 01:50:12.700 and that's not what I mean. 01:50:12.700 --> 01:50:15.480 I just want you to study. 01:50:15.480 --> 01:50:18.670 But they stay up before the final and the next day, 01:50:18.670 --> 01:50:19.380 I'm a vegetable. 01:50:19.380 --> 01:50:21.410 I don't even remember the chain rule. 01:50:21.410 --> 01:50:23.450 So they did not remember the chain rule 01:50:23.450 --> 01:50:25.000 that I've just wrote. 01:50:25.000 --> 01:50:28.490 And they said, oh, but I think I know how to do it. 01:50:28.490 --> 01:50:29.990 And I said, shh. 01:50:29.990 --> 01:50:31.980 Just don't say anything. 01:50:31.980 --> 01:50:34.840 Let me show you how the course coordinator wanted 01:50:34.840 --> 01:50:37.170 that done several years ago. 01:50:37.170 --> 01:50:40.165 So he wanted it done by the chain rule. 01:50:40.165 --> 01:50:41.550 He didn't say how you do it. 01:50:41.550 --> 01:50:42.050 OK? 01:50:42.050 --> 01:50:44.160 He said just get to the right answer. 01:50:44.160 --> 01:50:45.614 It doesn't matter. 01:50:45.614 --> 01:50:46.780 He wanted it done like that. 01:50:46.780 --> 01:50:55.700 He said, dw of f of tg of p with respect to t, 01:50:55.700 --> 01:51:06.738 would be dw du, instead of u you have f of t. 01:51:06.738 --> 01:51:16.730 f of tg of t times df dt plus dw with respect 01:51:16.730 --> 01:51:18.940 to the second variable. 01:51:18.940 --> 01:51:25.137 So this would be u, and this would be v with respect 01:51:25.137 --> 01:51:27.300 to the variable v, the second variable 01:51:27.300 --> 01:51:30.726 where [? measure ?] that f of dg of t. 01:51:30.726 --> 01:51:39.410 Evaluate it there times dg dt. 01:51:39.410 --> 01:51:46.136 So it's like dv dt, which is dg dt. [INAUDIBLE] So he did that, 01:51:46.136 --> 01:51:48.045 and he expected people to do what? 01:51:48.045 --> 01:51:51.192 He expected people to take a u squared the same 2 times 01:51:51.192 --> 01:51:54.251 u, just like you did before, 2 times. 01:51:54.251 --> 01:51:57.715 And instead of u, since u is f of t to [INAUDIBLE] puts 01:51:57.715 --> 01:52:13.171 2f of t, this is the first squiggly thing, times v of dt. 01:52:13.171 --> 01:52:19.932 2t is this smiley face. 01:52:19.932 --> 01:52:31.200 This is 2t plus-- what is the f dv? 01:52:31.200 --> 01:52:37.600 Dw with respect to dv is going to be 2v 2 time gf t. 01:52:37.600 --> 01:52:46.794 When I evaluate add gf t, this funny fellow 01:52:46.794 --> 01:52:57.580 with this funny fellow, times qg d, which, with your permission 01:52:57.580 --> 01:53:00.890 I'm going to erase and write 3p squared. 01:53:00.890 --> 01:53:04.030 01:53:04.030 --> 01:53:07.340 And the last row he expected my students to write 01:53:07.340 --> 01:53:22.135 was 2t squared times 2t plus 2pq plus 1, times 3t squared. 01:53:22.135 --> 01:53:27.580 01:53:27.580 --> 01:53:31.540 Are you guys with me? 01:53:31.540 --> 01:53:43.210 So [INAUDIBLE] 2t 2x 2t squared, correct. 01:53:43.210 --> 01:53:49.730 I forgot to identify this as that. 01:53:49.730 --> 01:53:50.230 All right. 01:53:50.230 --> 01:53:52.650 So in the end, the answer is a simplified answer. 01:53:52.650 --> 01:53:53.930 Can you tell me what it is? 01:53:53.930 --> 01:53:55.250 I'm too lazy to write it down. 01:53:55.250 --> 01:53:56.935 You compute it. 01:53:56.935 --> 01:53:58.930 How much is it simplified? 01:53:58.930 --> 01:54:00.421 Find it as a polynomial. 01:54:00.421 --> 01:54:01.415 STUDENT: [INAUDIBLE]. 01:54:01.415 --> 01:54:04.397 01:54:04.397 --> 01:54:08.870 PROFESSOR TODA: So you have 6, 6-- 01:54:08.870 --> 01:54:10.150 STUDENT: 16 cubed plus 3-- 01:54:10.150 --> 01:54:15.300 PROFESSOR TODA: T to the 5th plus-- 01:54:15.300 --> 01:54:17.240 STUDENT: [INAUDIBLE]. 01:54:17.240 --> 01:54:19.350 PROFESSOR TODA: In order, in order. 01:54:19.350 --> 01:54:20.420 What's the next guy? 01:54:20.420 --> 01:54:21.636 STUDENT: [INAUDIBLE]. 01:54:21.636 --> 01:54:22.830 PROFESSOR TODA: 4t cubed. 01:54:22.830 --> 01:54:23.975 And the last guy-- 01:54:23.975 --> 01:54:24.925 STUDENT: 6t squared. 01:54:24.925 --> 01:54:26.050 PROFESSOR TODA: 6t squared. 01:54:26.050 --> 01:54:31.220 01:54:31.220 --> 01:54:31.720 Yes? 01:54:31.720 --> 01:54:33.320 Did you get the same thing? 01:54:33.320 --> 01:54:34.220 OK. 01:54:34.220 --> 01:54:37.145 Now, how did my students do it? 01:54:37.145 --> 01:54:37.644 [INAUDIBLE] 01:54:37.644 --> 01:54:40.171 01:54:40.171 --> 01:54:41.420 Did they apply the chain rule? 01:54:41.420 --> 01:54:41.920 No. 01:54:41.920 --> 01:54:44.100 They said OK, this is how it goes. 01:54:44.100 --> 01:54:46.960 01:54:46.960 --> 01:54:58.210 W of U of T and V of T is U is F. So this guy is T squared, 01:54:58.210 --> 01:55:01.886 T squared squared, plus this guy is T 01:55:01.886 --> 01:55:08.960 cubed plus 1 taken and shaken and squared. 01:55:08.960 --> 01:55:13.710 And then when I do the whole thing, derivative 01:55:13.710 --> 01:55:22.640 of this with respect to T, I get-- 01:55:22.640 --> 01:55:27.570 I'm too lazy-- T to the 4 prime is 40 cubed. 01:55:27.570 --> 01:55:28.870 I'm not going to do on the map. 01:55:28.870 --> 01:55:37.380 2 out T cubed plus 1 times chain rule, 3t squared. 01:55:37.380 --> 01:55:49.610 40 cubed plus 16 to the 5 plus-- [INAUDIBLE] 2 and 6t squared. 01:55:49.610 --> 01:55:56.450 So you realize that I have to give them 100%. 01:55:56.450 --> 01:55:59.465 Although they were very honest and said, we blanked. 01:55:59.465 --> 01:56:01.130 We don't remember the chain rule. 01:56:01.130 --> 01:56:02.926 We don't remember the formula. 01:56:02.926 --> 01:56:03.550 So that's fine. 01:56:03.550 --> 01:56:05.080 Do whatever you can. 01:56:05.080 --> 01:56:06.920 So I gave them 100% for that. 01:56:06.920 --> 01:56:11.280 But realize that the author of the problem 01:56:11.280 --> 01:56:14.080 was a little bit naive. 01:56:14.080 --> 01:56:16.520 Because you could have done this differently. 01:56:16.520 --> 01:56:22.190 I mean if you wanted to actually test the whole thing, 01:56:22.190 --> 01:56:26.170 you wouldn't have given-- let's say you wouldn't have given 01:56:26.170 --> 01:56:32.400 the actual-- yeah, you wouldn't have given the actual functions 01:56:32.400 --> 01:56:37.530 and say write the chain formula symbolically 01:56:37.530 --> 01:56:44.875 for this function applied for F of T and G of T. 01:56:44.875 --> 01:56:49.340 So it was-- they were just lucky. 01:56:49.340 --> 01:56:52.470 Remember that you need to know this chain rule. 01:56:52.470 --> 01:56:53.970 It's going to be one of the problems 01:56:53.970 --> 01:56:56.900 to be emphasized in the exams. 01:56:56.900 --> 01:57:02.408 Maybe one of the top 15 or 16 most important topics. 01:57:02.408 --> 01:57:07.170 01:57:07.170 --> 01:57:07.900 Is that OK? 01:57:07.900 --> 01:57:09.396 Can I erase the whole thing? 01:57:09.396 --> 01:57:09.896 OK. 01:57:09.896 --> 01:57:11.393 Let me erase the whole thing. 01:57:11.393 --> 01:57:44.326 01:57:44.326 --> 01:57:44.826 OK. 01:57:44.826 --> 01:57:45.824 Any other questions? 01:57:45.824 --> 01:58:02.291 01:58:02.291 --> 01:58:03.661 No? 01:58:03.661 --> 01:58:05.285 I'm not going to let you go right away, 01:58:05.285 --> 01:58:07.780 we're going to work one more problem or two more 01:58:07.780 --> 01:58:08.778 simple problems. 01:58:08.778 --> 01:58:10.773 And then we are going to go. 01:58:10.773 --> 01:58:11.273 OK? 01:58:11.273 --> 01:58:22.750 01:58:22.750 --> 01:58:26.480 So question. 01:58:26.480 --> 01:58:27.986 A question. 01:58:27.986 --> 01:58:32.946 01:58:32.946 --> 01:58:39.890 What do you think the gradient is good at? 01:58:39.890 --> 01:58:49.314 01:58:49.314 --> 01:58:50.980 Two reasons, right. 01:58:50.980 --> 01:58:54.340 Review number one. 01:58:54.340 --> 01:58:59.120 If you have an increasingly defined function, 01:58:59.120 --> 01:59:02.860 then the gradient of F was what? 01:59:02.860 --> 01:59:21.904 Equals direction of the normal to the surface S-- 01:59:21.904 --> 01:59:26.395 let's say S is given increasingly at the point 01:59:26.395 --> 01:59:27.393 with [INAUDIBLE]. 01:59:27.393 --> 01:59:31.884 01:59:31.884 --> 01:59:33.381 But any other reason? 01:59:33.381 --> 02:00:00.327 02:00:00.327 --> 02:00:01.824 Let's take that again. 02:00:01.824 --> 02:00:05.816 Z equals x squared plus y squared. 02:00:05.816 --> 02:00:07.812 Let's compute a few partial derivatives. 02:00:07.812 --> 02:00:09.309 Let's compute the gradient. 02:00:09.309 --> 02:00:21.120 The gradient is Fs of x, Fs of y, where this is F of xy 02:00:21.120 --> 02:00:24.888 or Fs of xi plus Fs of yj. 02:00:24.888 --> 02:00:28.880 02:00:28.880 --> 02:00:31.375 [INAUDIBLE] 02:00:31.375 --> 02:00:34.369 And we drew it. 02:00:34.369 --> 02:00:42.353 I drew this case, and we also drew another related example, 02:00:42.353 --> 02:00:45.846 where we took Z equals 1 minus x squared minus y squared. 02:00:45.846 --> 02:00:46.844 And we went skiing. 02:00:46.844 --> 02:00:52.333 And we were so happy last week to go skiing, because we still 02:00:52.333 --> 02:00:57.650 had snow in New Mexico, and we-- and we 02:00:57.650 --> 02:01:02.546 said now we computed the Z to be minus 2x minus 2y. 02:01:02.546 --> 02:01:06.018 02:01:06.018 --> 02:01:09.597 And we said, I'm looking at the slopes. 02:01:09.597 --> 02:01:12.936 This is the x duration and the y duration. 02:01:12.936 --> 02:01:18.670 And I'm looking at the slopes of the lines of these two curves. 02:01:18.670 --> 02:01:23.630 So one that goes down, like that. 02:01:23.630 --> 02:01:25.010 So this was for what? 02:01:25.010 --> 02:01:27.540 For y equals 0. 02:01:27.540 --> 02:01:32.190 And this was for x equals 0. 02:01:32.190 --> 02:01:36.645 02:01:36.645 --> 02:01:39.580 Curve, x equals 0 curve in plane. 02:01:39.580 --> 02:01:40.360 Right? 02:01:40.360 --> 02:01:42.742 We just cross-section our surface, 02:01:42.742 --> 02:01:43.950 and we have this [INAUDIBLE]. 02:01:43.950 --> 02:01:51.594 And then we have the two tangents, two slopes. 02:01:51.594 --> 02:01:54.064 And we computed them everywhere. 02:01:54.064 --> 02:02:00.486 02:02:00.486 --> 02:02:01.968 At every point. 02:02:01.968 --> 02:02:06.910 02:02:06.910 --> 02:02:10.845 But realize that to go up or down these hills, 02:02:10.845 --> 02:02:15.095 I can go on a curve like that, or I 02:02:15.095 --> 02:02:17.950 can go-- remember the train of Mickey Mouse going 02:02:17.950 --> 02:02:20.182 on the hilly point on the hill? 02:02:20.182 --> 02:02:22.174 We try to take different paths. 02:02:22.174 --> 02:02:24.166 We are going hiking. 02:02:24.166 --> 02:02:28.648 We are going hiking, and we'll take hiking through the pass. 02:02:28.648 --> 02:02:38.608 02:02:38.608 --> 02:02:41.098 OK. 02:02:41.098 --> 02:03:01.420 How do we get the maximum rate of change of the function 02:03:01.420 --> 02:03:03.600 Z equals F of x1? 02:03:03.600 --> 02:03:05.870 So now I'm anticipating something. 02:03:05.870 --> 02:03:10.680 I'd like to see your intuition, your inborn sense of I 02:03:10.680 --> 02:03:12.300 know what's going to happen. 02:03:12.300 --> 02:03:14.092 And you know what that from Mister-- 02:03:14.092 --> 02:03:14.842 STUDENT: Heinrich. 02:03:14.842 --> 02:03:17.590 PROFESSOR TODA: [? Heinrich ?] from high school. 02:03:17.590 --> 02:03:21.280 So I'm asking-- let me rephrase the question 02:03:21.280 --> 02:03:23.130 like a non-mathematician. 02:03:23.130 --> 02:03:24.230 Let's go hiking. 02:03:24.230 --> 02:03:30.274 This is [INAUDIBLE] we go to the lighthouse. 02:03:30.274 --> 02:03:33.790 Which path shall I take on my mountain, my hill, 02:03:33.790 --> 02:03:37.570 my god knows what geography, in order 02:03:37.570 --> 02:03:40.440 to obtain the maximum rate of change? 02:03:40.440 --> 02:03:43.515 That means the highest derivative. 02:03:43.515 --> 02:03:46.470 In what direction do I get the highest derivative? 02:03:46.470 --> 02:03:49.110 STUDENT: In what direction you get the highest derivative-- 02:03:49.110 --> 02:03:50.735 PROFESSOR TODA: So in which direction-- 02:03:50.735 --> 02:03:53.330 in which direction on this hill do 02:03:53.330 --> 02:03:55.098 I get the highest derivative? 02:03:55.098 --> 02:03:57.014 The highest rate of change. 02:03:57.014 --> 02:04:03.740 Rate of change means I want to get the fastest possible way 02:04:03.740 --> 02:04:04.850 somewhere. 02:04:04.850 --> 02:04:08.230 STUDENT: The shortest slope? 02:04:08.230 --> 02:04:09.857 Along just the straight line up. 02:04:09.857 --> 02:04:10.831 PROFESSOR TODA: Along-- 02:04:10.831 --> 02:04:12.292 STUDENT: You don't want to take any [INAUDIBLE]. 02:04:12.292 --> 02:04:13.270 PROFESSOR TODA: Right. 02:04:13.270 --> 02:04:13.910 STUDENT: [INAUDIBLE]. 02:04:13.910 --> 02:04:15.140 It could be along any axis. 02:04:15.140 --> 02:04:17.710 PROFESSOR TODA: So could you see which direction 02:04:17.710 --> 02:04:19.070 those are-- very good. 02:04:19.070 --> 02:04:21.230 Actually you were getting to the same direction. 02:04:21.230 --> 02:04:24.370 So [INAUDIBLE] says Magdalena, don't be silly. 02:04:24.370 --> 02:04:28.295 The actual maximum rate of change for the function Z 02:04:28.295 --> 02:04:31.070 is obviously, because it is common sense, 02:04:31.070 --> 02:04:36.630 it's obviously happening if you take the so-called-- what 02:04:36.630 --> 02:04:37.880 are these guys? 02:04:37.880 --> 02:04:40.810 [INAUDIBLE], not meridians. 02:04:40.810 --> 02:04:42.280 STUDENT: Longtitudes? 02:04:42.280 --> 02:04:43.260 PROFESSOR TODA: OK. 02:04:43.260 --> 02:04:44.730 That is-- OK. 02:04:44.730 --> 02:04:47.810 Suppose that we don't hike, because it's too tiring. 02:04:47.810 --> 02:04:51.170 We go down from the top of the hill. 02:04:51.170 --> 02:04:53.310 Ah, there's also very good idea. 02:04:53.310 --> 02:04:58.870 So when you let yourself go down on a sleigh, 02:04:58.870 --> 02:05:02.560 don't think bobsled or anything-- just a sleigh, 02:05:02.560 --> 02:05:04.110 think of a child's sleigh. 02:05:04.110 --> 02:05:07.680 No, take a plastic bag and put your butt in it 02:05:07.680 --> 02:05:10.590 and let yourself go. 02:05:10.590 --> 02:05:14.140 What is their direction actually? 02:05:14.140 --> 02:05:19.925 Your body will find the fastest way to get down. 02:05:19.925 --> 02:05:23.065 The fastest way to get down will happen exactly 02:05:23.065 --> 02:05:27.710 in the same directions going down 02:05:27.710 --> 02:05:29.600 in the directions of these meridians. 02:05:29.600 --> 02:05:34.100 02:05:34.100 --> 02:05:35.512 OK? 02:05:35.512 --> 02:05:37.000 And now, [INAUDIBLE]. 02:05:37.000 --> 02:05:46.424 02:05:46.424 --> 02:05:58.576 The maximum rate of change will always 02:05:58.576 --> 02:06:07.256 happen in the direction of the gradient. 02:06:07.256 --> 02:06:14.696 02:06:14.696 --> 02:06:18.950 You can get a little bit ahead of time 02:06:18.950 --> 02:06:21.860 by just-- I would like this to [INAUDIBLE] in your heads 02:06:21.860 --> 02:06:23.860 until we get to that section. 02:06:23.860 --> 02:06:26.880 In one section we will be there. 02:06:26.880 --> 02:06:40.430 We also-- it's also reformulated as the highest, the steepest, 02:06:40.430 --> 02:06:42.079 ascent or descent. 02:06:42.079 --> 02:06:44.574 The steepest. 02:06:44.574 --> 02:06:58.546 The steepest ascent or the steepest descent 02:06:58.546 --> 02:07:09.524 always happens in the direction of the gradient. 02:07:09.524 --> 02:07:14.550 02:07:14.550 --> 02:07:17.110 Ascent is when you hike to the top of the hill. 02:07:17.110 --> 02:07:21.450 Descent is when you let yourself go in the plastic [INAUDIBLE] 02:07:21.450 --> 02:07:25.270 bag in the snow. 02:07:25.270 --> 02:07:26.080 Right? 02:07:26.080 --> 02:07:30.030 Can you verify this happens just on this example? 02:07:30.030 --> 02:07:32.540 It's true in general, for any smooth function. 02:07:32.540 --> 02:07:36.010 Our smooth function is a really nice function. 02:07:36.010 --> 02:07:39.816 So what is the gradient? 02:07:39.816 --> 02:07:42.720 Well again, it was 2x 2y, right? 02:07:42.720 --> 02:07:45.850 02:07:45.850 --> 02:07:50.508 And that means at a certain point, x0 y0, whenever you are, 02:07:50.508 --> 02:07:52.300 guys you don't necessarily have to start 02:07:52.300 --> 02:07:54.750 from the top of the hill. 02:07:54.750 --> 02:07:58.870 You can be-- OK, this is your cabin. 02:07:58.870 --> 02:08:01.970 And here you are with friends, or with mom and dad, 02:08:01.970 --> 02:08:05.110 or whoever, on the hill. 02:08:05.110 --> 02:08:09.030 You get out, you take the sleigh, and you go down. 02:08:09.030 --> 02:08:14.320 So no matter where you are, there you go. 02:08:14.320 --> 02:08:22.933 You have 2x0 times i plus 2y0 times j. 02:08:22.933 --> 02:08:31.640 And the direction of the gradient will be 2x0 2y0. 02:08:31.640 --> 02:08:34.520 Do you like this one? 02:08:34.520 --> 02:08:39.240 Well in this case, if you were-- suppose 02:08:39.240 --> 02:08:42.300 you were at the point [INAUDIBLE]. 02:08:42.300 --> 02:08:49.230 02:08:49.230 --> 02:08:53.596 You are at the point of coordinates-- 02:08:53.596 --> 02:08:55.130 do you want to be here? 02:08:55.130 --> 02:08:57.000 You want to be here, right? 02:08:57.000 --> 02:08:58.770 So we've done that before. 02:08:58.770 --> 02:09:02.560 I'll take it as 1 over [? square root of ?] 02:09:02.560 --> 02:09:09.620 2-- I'm trying to be creative today-- [INAUDIBLE] y equals 0, 02:09:09.620 --> 02:09:14.580 and Z equals-- what's left? 02:09:14.580 --> 02:09:16.460 1/2, right? 02:09:16.460 --> 02:09:17.750 Where am I? 02:09:17.750 --> 02:09:20.390 Guys, do you realize where I am? 02:09:20.390 --> 02:09:21.640 I'll [? take a ?] [INAUDIBLE]. 02:09:21.640 --> 02:09:24.340 02:09:24.340 --> 02:09:25.140 y0. 02:09:25.140 --> 02:09:28.656 So I need to be on this meridian on the red thingy. 02:09:28.656 --> 02:09:33.920 02:09:33.920 --> 02:09:37.356 And somewhere here. 02:09:37.356 --> 02:09:40.250 02:09:40.250 --> 02:09:43.385 What's the duration of the gradient here? 02:09:43.385 --> 02:09:45.830 Delta z at this p. 02:09:45.830 --> 02:09:56.610 02:09:56.610 --> 02:09:58.760 Then you say ah, well, I don't get it. 02:09:58.760 --> 02:10:04.040 I have-- the second guy will become 0, because y0 is 0. 02:10:04.040 --> 02:10:06.980 The first guy will become 1 over square root of 2. 02:10:06.980 --> 02:10:15.280 So I have 2 times 1 over square root of 2 times i plus 0j. 02:10:15.280 --> 02:10:29.416 It means in the direction of i-- in the direction of i-- from p, 02:10:29.416 --> 02:10:39.201 I have the fastest-- fastest, Magdalena, fastest-- descent 02:10:39.201 --> 02:10:39.700 possible. 02:10:39.700 --> 02:10:43.010 02:10:43.010 --> 02:10:46.730 But we don't say in the direction of i 02:10:46.730 --> 02:10:49.640 in our everyday life, right? 02:10:49.640 --> 02:10:53.270 Let's say geographic points. 02:10:53.270 --> 02:10:58.610 We are-- I'm a bug, and this is north. 02:10:58.610 --> 02:11:00.086 This is south. 02:11:00.086 --> 02:11:05.006 02:11:05.006 --> 02:11:05.990 This is east. 02:11:05.990 --> 02:11:08.960 02:11:08.960 --> 02:11:11.470 And this is west. 02:11:11.470 --> 02:11:18.140 So if I go east, going east means going in the direction i. 02:11:18.140 --> 02:11:23.040 02:11:23.040 --> 02:11:25.510 Now suppose-- I'm going to finish with this one. 02:11:25.510 --> 02:11:28.870 Suppose that my house is not on the prairie 02:11:28.870 --> 02:11:31.710 but my house is here. 02:11:31.710 --> 02:11:34.400 House, h. 02:11:34.400 --> 02:11:37.834 Find me a wood point to be there. 02:11:37.834 --> 02:11:39.738 STUDENT: Northeast. 02:11:39.738 --> 02:11:41.170 Or to get further down. 02:11:41.170 --> 02:11:45.047 PROFESSOR TODA: Anything, what would look like why I'm here? 02:11:45.047 --> 02:11:48.041 x0, y0, z0. 02:11:48.041 --> 02:11:50.040 Hm. 02:11:50.040 --> 02:11:57.800 1/2, 1/2, and I need the minimum. 02:11:57.800 --> 02:12:02.580 So I want to be on the bisecting plane between the two. 02:12:02.580 --> 02:12:03.420 You understand? 02:12:03.420 --> 02:12:04.400 This is my quarter. 02:12:04.400 --> 02:12:06.870 And I want to be in this bisecting plane. 02:12:06.870 --> 02:12:10.420 So I'll take 1/2, 1/2, and what results from here? 02:12:10.420 --> 02:12:11.540 I have to do math. 02:12:11.540 --> 02:12:16.110 1 minus 1/4 minus 1/4 is 1/2. 02:12:16.110 --> 02:12:17.902 Right? 02:12:17.902 --> 02:12:19.740 1/2, 1/2, 1/2. 02:12:19.740 --> 02:12:22.440 This is where my house is [? and so on. ?] 02:12:22.440 --> 02:12:24.106 And this is full of smoke. 02:12:24.106 --> 02:12:29.940 And what is the maximum rate of change? 02:12:29.940 --> 02:12:34.500 What is the steepest descent is the trajectory 02:12:34.500 --> 02:12:37.920 that my body will take when I let myself go down 02:12:37.920 --> 02:12:39.402 on the sleigh. 02:12:39.402 --> 02:12:40.884 How do I compute that? 02:12:40.884 --> 02:12:43.624 I will just do the same thing. 02:12:43.624 --> 02:12:49.900 Delta z at the point x0 equals 1/2, y0 equals 1/2, 02:12:49.900 --> 02:12:52.100 z0 equals 1/2. 02:12:52.100 --> 02:12:54.060 Well what do I get as direction? 02:12:54.060 --> 02:12:57.490 That will be the direction of the gradient. 02:12:57.490 --> 02:13:02.920 2 times 1/2-- you guys with me still? 02:13:02.920 --> 02:13:09.240 i plus 2 times 1/2 with j. 02:13:09.240 --> 02:13:14.311 And there is no Mr. z0 In the picture. 02:13:14.311 --> 02:13:14.810 Why? 02:13:14.810 --> 02:13:17.100 Because that will give me the direction 02:13:17.100 --> 02:13:22.000 like on-- in a geographic way. 02:13:22.000 --> 02:13:24.420 North, west, east, south. 02:13:24.420 --> 02:13:26.490 These are the direction in plane. 02:13:26.490 --> 02:13:28.120 I'm not talking directions on the hill, 02:13:28.120 --> 02:13:31.496 I'm talking directions on the map. 02:13:31.496 --> 02:13:33.530 These are directions on the map. 02:13:33.530 --> 02:13:35.930 So what is the direction i plus j on the map? 02:13:35.930 --> 02:13:39.820 If you show this to a geography major and say, 02:13:39.820 --> 02:13:43.340 I'm going in the direction i plus j on the map, 02:13:43.340 --> 02:13:45.700 he will say you are crazy. 02:13:45.700 --> 02:13:47.980 He doesn't understand the thing. 02:13:47.980 --> 02:13:50.280 But you know what you mean. 02:13:50.280 --> 02:13:54.100 East for you is the direction of i in the x-axis. 02:13:54.100 --> 02:13:56.210 [INAUDIBLE] 02:13:56.210 --> 02:13:58.430 And this is north. 02:13:58.430 --> 02:13:59.630 Are you guys with me? 02:13:59.630 --> 02:14:01.790 The y direction is north. 02:14:01.790 --> 02:14:06.410 So I'm going perfectly northeast at a 45-degree angle. 02:14:06.410 --> 02:14:08.093 If I tell the geography major I'm 02:14:08.093 --> 02:14:10.725 going northeast perfectly in the middle, he will say I know. 02:14:10.725 --> 02:14:13.560 But you will know that for you, that is i plus j. 02:14:13.560 --> 02:14:15.649 Because you are the mathematician. 02:14:15.649 --> 02:14:17.100 Right? 02:14:17.100 --> 02:14:18.980 So you go down. 02:14:18.980 --> 02:14:20.780 And this is where you are. 02:14:20.780 --> 02:14:22.500 And you're on the meridian. 02:14:22.500 --> 02:14:25.090 This is the direction i plus j. 02:14:25.090 --> 02:14:29.610 So if I want to project my trajectory-- I went down 02:14:29.610 --> 02:14:33.260 with the sleigh, all the way down-- project the trajectory, 02:14:33.260 --> 02:14:36.810 my trajectory is a body on the snow. 02:14:36.810 --> 02:14:39.320 Projecting it on the ground is this one. 02:14:39.320 --> 02:14:43.800 So it is exactly the direction i plus j. 02:14:43.800 --> 02:14:44.320 Right, guys? 02:14:44.320 --> 02:14:48.170 So exactly northeast perfectly at 45-degree angles. 02:14:48.170 --> 02:14:51.150 Now one caveat. 02:14:51.150 --> 02:14:53.455 One caveat, because when we get there, 02:14:53.455 --> 02:14:59.060 you should be ready already, in 11.6 and 11.7. 02:14:59.060 --> 02:15:02.930 When we will say direction, we are also crazy people. 02:15:02.930 --> 02:15:04.920 I told you, mathematicians are not normal. 02:15:04.920 --> 02:15:07.108 You have to be a little bit crazy 02:15:07.108 --> 02:15:11.460 to want to do all the stuff in your head like that. 02:15:11.460 --> 02:15:16.420 i plus j for us is not a direction most of the time. 02:15:16.420 --> 02:15:20.015 When we say direction, we mean we normalize that direction. 02:15:20.015 --> 02:15:23.055 We take the unit vector, which is unique, 02:15:23.055 --> 02:15:25.940 for responding to i plus j. 02:15:25.940 --> 02:15:28.890 So what is that unique unit vector? 02:15:28.890 --> 02:15:32.510 You learned in Chapter 9 everything is connected. 02:15:32.510 --> 02:15:33.890 It's a big circle. 02:15:33.890 --> 02:15:35.340 i plus j, very good. 02:15:35.340 --> 02:15:40.190 So direction is a unit vector for most mathematicians, 02:15:40.190 --> 02:15:45.390 which means you will be i plus j over square root of 2. 02:15:45.390 --> 02:15:51.966 So in Chapter 5, please remember, unlike Chapter 9, 02:15:51.966 --> 02:15:55.613 direction is a unit vector. 02:15:55.613 --> 02:15:59.609 In Chapter 9, Chapter 10, it said direction lmn, 02:15:59.609 --> 02:16:00.650 direction god knows what. 02:16:00.650 --> 02:16:05.900 But in Chapter 11, direction is a vector in plane, 02:16:05.900 --> 02:16:07.860 like this one, i plus [INAUDIBLE] 02:16:07.860 --> 02:16:12.015 has to be a unique normal-- a unique vector. 02:16:12.015 --> 02:16:12.514 OK? 02:16:12.514 --> 02:16:14.318 And we-- keep that in mind. 02:16:14.318 --> 02:16:16.050 Next time, when we meet on Thursday, 02:16:16.050 --> 02:16:19.956 you will understand why we need to normalize it. 02:16:19.956 --> 02:16:23.220 Now can we say goodbye to the snow and everything? 02:16:23.220 --> 02:16:25.810 It's not going to show up much anymore. 02:16:25.810 --> 02:16:27.555 Remember this example. 02:16:27.555 --> 02:16:30.660 But we will start with flowers next time. 02:16:30.660 --> 02:16:31.260 OK. 02:16:31.260 --> 02:16:32.760 Have a nice day. 02:16:32.760 --> 02:16:33.959 Yes, sir? 02:16:33.959 --> 02:16:36.410 Let me stop the video. 02:16:36.410 --> 02:16:37.245