WEBVTT 00:00:00.000 --> 00:00:00.930 00:00:00.930 --> 00:00:05.670 PROFESSOR: So let's forget about this example 00:00:05.670 --> 00:00:10.910 and review what we learned in 11.3, chapter 11. 00:00:10.910 --> 00:00:17.160 Chapter 11, again, was functions of several variables. 00:00:17.160 --> 00:00:21.028 In our case, I'll say functions of two variables. 00:00:21.028 --> 00:00:27.010 00:00:27.010 --> 00:00:30.400 11.3 taught you, what? 00:00:30.400 --> 00:00:32.100 Taught you some beautiful things. 00:00:32.100 --> 00:00:34.980 Practically, if you understand this picture, 00:00:34.980 --> 00:00:37.502 you will remember everything. 00:00:37.502 --> 00:00:43.630 This picture is going to try and [INAUDIBLE] a graph 00:00:43.630 --> 00:00:45.540 that's sitting above here somewhere 00:00:45.540 --> 00:00:50.870 in Euclidean free space, dimensional space. 00:00:50.870 --> 00:00:53.180 You have the origin. 00:00:53.180 --> 00:00:57.154 And you say I want markers. 00:00:57.154 --> 00:00:59.118 No, you don't say I want markers. 00:00:59.118 --> 00:01:02.740 I say I want markers. 00:01:02.740 --> 00:01:09.380 We want to fix a point x0, y0 on the surface, 00:01:09.380 --> 00:01:11.160 assuming the surface is smooth. 00:01:11.160 --> 00:01:14.030 00:01:14.030 --> 00:01:17.260 That x0 of mine should be projected. 00:01:17.260 --> 00:01:20.250 I'm going to try to draw better than I did last time. 00:01:20.250 --> 00:01:23.410 X0, y0 corresponds to a certain altitude 00:01:23.410 --> 00:01:26.565 z0 that is projected like that. 00:01:26.565 --> 00:01:29.050 And this is my [INAUDIBLE] 0 here. 00:01:29.050 --> 00:01:31.310 But I don't care much about that right now. 00:01:31.310 --> 00:01:34.850 I care about the fact that locally, I 00:01:34.850 --> 00:01:40.760 represent the function as a graph-- z of f-- f 00:01:40.760 --> 00:01:43.650 of x and y defined over a domain. 00:01:43.650 --> 00:01:47.630 I have a domain that is an open set. 00:01:47.630 --> 00:01:50.992 And you connect to-- that's more than you need to know. 00:01:50.992 --> 00:01:52.045 Could be anything. 00:01:52.045 --> 00:01:56.340 Could be a square, could be a-- this could be something, 00:01:56.340 --> 00:01:58.343 a nice patch of them like. 00:01:58.343 --> 00:02:01.040 00:02:01.040 --> 00:02:05.350 So the projection of my point here is x0, y0. 00:02:05.350 --> 00:02:08.149 I'm going to draw these parallels as well as I can. 00:02:08.149 --> 00:02:10.660 But I cannot draw very well. 00:02:10.660 --> 00:02:12.130 But I'm trying. 00:02:12.130 --> 00:02:18.220 X0 and y0-- and remember from last time. 00:02:18.220 --> 00:02:19.890 What did we say? 00:02:19.890 --> 00:02:25.125 I'm going to draw a plane of equation x equals x0. 00:02:25.125 --> 00:02:27.730 00:02:27.730 --> 00:02:30.202 All right, I'll try. 00:02:30.202 --> 00:02:33.746 I'll try and do a good job-- x equals x0 is this plane. 00:02:33.746 --> 00:02:35.245 STUDENT: Don't you have the x amount 00:02:35.245 --> 00:02:36.712 and the y amounts backward? 00:02:36.712 --> 00:02:38.882 Or [INAUDIBLE] 00:02:38.882 --> 00:02:39.465 PROFESSOR: No. 00:02:39.465 --> 00:02:40.430 STUDENT: [INAUDIBLE] 00:02:40.430 --> 00:02:44.280 PROFESSOR: x is this 1 coming towards you like that. 00:02:44.280 --> 00:02:47.280 And I also think about that always, Ryan. 00:02:47.280 --> 00:02:48.730 Do I have them backward? 00:02:48.730 --> 00:02:49.930 This time, I was lucky. 00:02:49.930 --> 00:02:51.370 I didn't have them backward. 00:02:51.370 --> 00:02:52.910 So y goes this way. 00:02:52.910 --> 00:02:57.920 For y0, let me pick another color, a more beautiful color. 00:02:57.920 --> 00:03:01.175 00:03:01.175 --> 00:03:07.180 For y0, my video is not going to see the y0. 00:03:07.180 --> 00:03:10.763 But hopefully, it's going to see it, this beautiful line. 00:03:10.763 --> 00:03:12.470 Spring is coming. 00:03:12.470 --> 00:03:15.486 So this is going to be the plane. 00:03:15.486 --> 00:03:19.860 00:03:19.860 --> 00:03:24.160 Label it [INAUDIBLE] y equals 0y. 00:03:24.160 --> 00:03:30.690 Now, the green plane cuts the surface into a plane curve, 00:03:30.690 --> 00:03:33.030 of course, because teasing the plane 00:03:33.030 --> 00:03:35.730 that I drew with the line. 00:03:35.730 --> 00:03:41.080 And in the plane that I drew with red-- was it red or pink? 00:03:41.080 --> 00:03:42.430 It's Valentine's Day. 00:03:42.430 --> 00:03:43.040 It's pink. 00:03:43.040 --> 00:03:47.160 OK, so I have it like that. 00:03:47.160 --> 00:03:48.630 So what is the pink curve? 00:03:48.630 --> 00:03:53.680 The pink curve is the intersection between z 00:03:53.680 --> 00:03:58.080 equals-- x equals x0 plane with my surface. 00:03:58.080 --> 00:04:00.110 My surface is black. 00:04:00.110 --> 00:04:03.800 I'm going to say s on surface. 00:04:03.800 --> 00:04:05.770 And then I have a pink curve. 00:04:05.770 --> 00:04:06.850 Let's call it c1. 00:04:06.850 --> 00:04:11.100 Because you cannot see pink on your notes. 00:04:11.100 --> 00:04:14.590 You only can imagine that it's not the same thing. 00:04:14.590 --> 00:04:24.398 C2 is y equals y0 plane intersected with s. 00:04:24.398 --> 00:04:26.382 And what have we learned last time? 00:04:26.382 --> 00:04:32.720 Last time, we learned that we introduce some derivatives 00:04:32.720 --> 00:04:37.240 at the point at 0, y is 0, so that they represent 00:04:37.240 --> 00:04:40.400 those partial derivatives of the function z with respect 00:04:40.400 --> 00:04:42.010 to x and y. 00:04:42.010 --> 00:04:56.550 So we have the partial z sub x at x0, y0 and the partial and z 00:04:56.550 --> 00:05:00.955 sub y at x0, y0. 00:05:00.955 --> 00:05:04.360 Do we have a more elegant definition? 00:05:04.360 --> 00:05:06.860 That's elegant enough for me, thank you very much. 00:05:06.860 --> 00:05:09.810 But if I wanted to give the original definition, what 00:05:09.810 --> 00:05:11.180 was that? 00:05:11.180 --> 00:05:19.741 That is d of bx at x0, y0, which is a limit of the difference 00:05:19.741 --> 00:05:20.240 quotient. 00:05:20.240 --> 00:05:23.000 And this time, we're going to-- not going to do the x of y. 00:05:23.000 --> 00:05:25.020 I I'm different today. 00:05:25.020 --> 00:05:27.470 So I do h goes to 0. 00:05:27.470 --> 00:05:31.040 H is my smallest displacement of [INAUDIBLE]. 00:05:31.040 --> 00:05:34.330 Here, I have f of-- now, who is the variable? 00:05:34.330 --> 00:05:34.850 X. 00:05:34.850 --> 00:05:38.940 So who is going to say fixed? 00:05:38.940 --> 00:05:47.730 Y. So I'm going to say I'm displacing mister x0 with an h. 00:05:47.730 --> 00:05:55.320 And y0 will be fixed minus f of x0, y0, all over h. 00:05:55.320 --> 00:06:02.310 So again, instead of-- instead of a delta x, I call the h. 00:06:02.310 --> 00:06:06.010 And the derivative with respect to y 00:06:06.010 --> 00:06:11.320 will assume that x0 is a constant. 00:06:11.320 --> 00:06:13.750 I saw how well [INAUDIBLE] explained that 00:06:13.750 --> 00:06:15.880 and I'm ambitious. 00:06:15.880 --> 00:06:18.040 I want to do an even better job than [INAUDIBLE]. 00:06:18.040 --> 00:06:20.370 Hopefully, I might manage. 00:06:20.370 --> 00:06:32.820 D of ty equals [INAUDIBLE] h going to 0 of that CF of-- now, 00:06:32.820 --> 00:06:34.910 who's telling me what we have? 00:06:34.910 --> 00:06:37.970 Of course, mister x, y and y, yy. 00:06:37.970 --> 00:06:43.620 F of x0, y0 is their constant waiting for his turn. 00:06:43.620 --> 00:06:46.660 H is your parameter. 00:06:46.660 --> 00:06:50.940 And then you'll have, what? 00:06:50.940 --> 00:06:51.990 H0 is fixed, right? 00:06:51.990 --> 00:06:53.280 STUDENT: So h0 is-- 00:06:53.280 --> 00:06:54.280 PROFESSOR: --fixed. 00:06:54.280 --> 00:06:56.260 Y is the variable. 00:06:56.260 --> 00:07:01.910 So I go into the direction of y starting from y0. 00:07:01.910 --> 00:07:05.650 And I displace that with a small quantity, right? 00:07:05.650 --> 00:07:09.010 So these are my partial velocity-- 00:07:09.010 --> 00:07:11.890 my partial derivatives, I'm sorry, not partial velocities. 00:07:11.890 --> 00:07:12.765 Forget what I said. 00:07:12.765 --> 00:07:17.656 I said something that you will learn later. 00:07:17.656 --> 00:07:18.910 What are those? 00:07:18.910 --> 00:07:29.700 Those are the slopes at x0, y0 of the tangents at the point 00:07:29.700 --> 00:07:31.330 here, OK? 00:07:31.330 --> 00:07:36.870 The tangents to the two curves, the pink one-- the pink one 00:07:36.870 --> 00:07:42.700 and the green one, all right? 00:07:42.700 --> 00:07:48.670 For the pink one, for the pink curve, what is the variable? 00:07:48.670 --> 00:07:51.110 The variable is the y, right? 00:07:51.110 --> 00:07:56.880 So this is c1 is a curve that depends on y. 00:07:56.880 --> 00:07:59.670 And c2 is a curve that depends on x. 00:07:59.670 --> 00:08:03.530 So this comes with x0 fixed. 00:08:03.530 --> 00:08:05.560 I better write it like that. 00:08:05.560 --> 00:08:08.390 F of x is 0. 00:08:08.390 --> 00:08:15.200 Y, instead of c2 of x, I'll say f of y-- f of x and yz. 00:08:15.200 --> 00:08:17.970 00:08:17.970 --> 00:08:22.297 So, which slope is which? 00:08:22.297 --> 00:08:26.280 00:08:26.280 --> 00:08:29.890 The d of dy at this point is the slope to this one. 00:08:29.890 --> 00:08:31.335 Are you guys with me? 00:08:31.335 --> 00:08:33.610 The slope of that tangent. 00:08:33.610 --> 00:08:36.760 Considered in the plane where it is. 00:08:36.760 --> 00:08:38.740 How about the other one? 00:08:38.740 --> 00:08:47.630 S of x will be the slope of this line in the green plane, OK? 00:08:47.630 --> 00:08:52.180 That is considered as a plane of axis of coordinates. 00:08:52.180 --> 00:08:57.260 Good, good-- so we know what they are. 00:08:57.260 --> 00:09:04.660 A quick example to review-- I've given you some really ugly, 00:09:04.660 --> 00:09:06.450 nasty functions today. 00:09:06.450 --> 00:09:08.900 The last time, you did a good job. 00:09:08.900 --> 00:09:11.730 So today, I'm not challenging you anymore. 00:09:11.730 --> 00:09:16.165 I'm just going to give you one simple example. 00:09:16.165 --> 00:09:19.370 And I'm asked you, what does this guy look like 00:09:19.370 --> 00:09:25.280 and what will the meanings of z sub x and z sub y be? 00:09:25.280 --> 00:09:26.520 What will they be at? 00:09:26.520 --> 00:09:33.130 Let's say I think I know what I want to take at the point 0, 0. 00:09:33.130 --> 00:09:40.400 And maybe you're going to tell me what else it will be. 00:09:40.400 --> 00:09:42.832 And eventually at another point like z 00:09:42.832 --> 00:09:55.438 sub a, so coordinates, 1 over square root of 2 and 1 00:09:55.438 --> 00:09:58.390 over square root of 2. 00:09:58.390 --> 00:10:02.818 And v sub y is same-- 1 over square root of 2, 00:10:02.818 --> 00:10:05.000 1 over square root of 2. 00:10:05.000 --> 00:10:10.150 Can one draw them and have a geometric explanation 00:10:10.150 --> 00:10:13.580 of what's going on? 00:10:13.580 --> 00:10:16.590 Well, I don't want you to forget the definitions, 00:10:16.590 --> 00:10:19.990 but since you absorbed them with your mind hopefully 00:10:19.990 --> 00:10:23.760 and with your eyes, you're not going to need them anymore. 00:10:23.760 --> 00:10:27.050 We should be able to draw this quadric that you love. 00:10:27.050 --> 00:10:29.250 I'm sure you love it. 00:10:29.250 --> 00:10:31.630 When it's-- what does it look like? 00:10:31.630 --> 00:10:35.995 00:10:35.995 --> 00:10:37.450 STUDENT: [INAUDIBLE] 00:10:37.450 --> 00:10:40.660 PROFESSOR: Wait a minute, you're not awake or I'm not awake. 00:10:40.660 --> 00:10:44.150 So if you do x squared plus y squared, 00:10:44.150 --> 00:10:46.880 don't write it down please. 00:10:46.880 --> 00:10:49.180 It would be that. 00:10:49.180 --> 00:10:50.685 And what is this? 00:10:50.685 --> 00:10:51.960 STUDENT: That's a [INAUDIBLE]. 00:10:51.960 --> 00:10:54.830 PROFESSOR: A circular paraboloid-- you are correct. 00:10:54.830 --> 00:10:57.420 We've done that before. 00:10:57.420 --> 00:10:59.660 I'd say it looks like an egg shell, 00:10:59.660 --> 00:11:02.515 but it's actually-- this is a parabola 00:11:02.515 --> 00:11:04.815 if it's going to infinity. 00:11:04.815 --> 00:11:06.270 And you said a bunch of circles. 00:11:06.270 --> 00:11:06.770 Yes, sir. 00:11:06.770 --> 00:11:08.770 STUDENT: So is it an upside down graph? 00:11:08.770 --> 00:11:10.827 PROFESSOR: It's an upside down paraboloid. 00:11:10.827 --> 00:11:11.660 STUDENT: [INAUDIBLE] 00:11:11.660 --> 00:11:13.580 PROFESSOR: So, very good-- how do we do that? 00:11:13.580 --> 00:11:16.286 We make this guy look in the mirror. 00:11:16.286 --> 00:11:17.265 This is the lake. 00:11:17.265 --> 00:11:18.455 The lake is xy plane. 00:11:18.455 --> 00:11:21.470 So this guy is looking in the mirror. 00:11:21.470 --> 00:11:25.160 Take his image and shift it just like he 00:11:25.160 --> 00:11:29.270 said-- shift it one unit up. 00:11:29.270 --> 00:11:30.750 This is one. 00:11:30.750 --> 00:11:34.024 You're going to have another paraboloid. 00:11:34.024 --> 00:11:39.370 So from this construction, I'm going to draw. 00:11:39.370 --> 00:11:42.045 And he's going to look like you took a cup 00:11:42.045 --> 00:11:44.160 and you put it upside down. 00:11:44.160 --> 00:11:45.770 But it's more like an eggshell, right? 00:11:45.770 --> 00:11:48.950 It's not a cup because a cup is supposed 00:11:48.950 --> 00:11:50.770 to have a flat bottom, right? 00:11:50.770 --> 00:11:54.690 But this is like an eggshell. 00:11:54.690 --> 00:11:58.090 And I'll draw. 00:11:58.090 --> 00:12:01.800 And for this fellow, we have a beautiful picture 00:12:01.800 --> 00:12:09.945 that looks like this hopefully But I'm going to try and draw. 00:12:09.945 --> 00:12:12.560 STUDENT: Are you looking from a top to bottom? 00:12:12.560 --> 00:12:16.470 PROFESSOR: We can look it whatever you want to look. 00:12:16.470 --> 00:12:19.300 That's a very good thing. 00:12:19.300 --> 00:12:23.052 You're getting too close to what I wanted to go. 00:12:23.052 --> 00:12:25.380 We'll discuss in one minute. 00:12:25.380 --> 00:12:29.800 So you can imagine this is a hill full of snow. 00:12:29.800 --> 00:12:33.272 Although in two days, we have Valentine's Day 00:12:33.272 --> 00:12:35.310 and there is no snow. 00:12:35.310 --> 00:12:38.610 But assume that we go to New Mexico 00:12:38.610 --> 00:12:41.380 and we find a hill that more or less looks 00:12:41.380 --> 00:12:44.510 like a perfect hill like that. 00:12:44.510 --> 00:12:50.430 And we start thinking of skiing down the hill. 00:12:50.430 --> 00:12:52.570 Where am I at 0, 0? 00:12:52.570 --> 00:12:55.087 I am on top of the hill. 00:12:55.087 --> 00:12:59.420 I'm on top of the hill and I decide 00:12:59.420 --> 00:13:03.740 to analyze the slope of the tangents 00:13:03.740 --> 00:13:08.100 to the surface in the direction of-- who is this? 00:13:08.100 --> 00:13:10.290 Like now, and you make me nervous. 00:13:10.290 --> 00:13:13.550 So in the direction of y, I have one slope. 00:13:13.550 --> 00:13:17.620 In the direction of x, I have another slope in general. 00:13:17.620 --> 00:13:21.050 Only in this case, they are the same slope. 00:13:21.050 --> 00:13:25.490 And what is that same slope if I'm here on top of the hill? 00:13:25.490 --> 00:13:30.180 This is me-- well, I don't know, one of you guys. 00:13:30.180 --> 00:13:34.004 00:13:34.004 --> 00:13:37.282 That looks horrible. 00:13:37.282 --> 00:13:38.240 What's going to happen? 00:13:38.240 --> 00:13:39.820 We don't want to think about it. 00:13:39.820 --> 00:13:42.350 But it definitely is too steep. 00:13:42.350 --> 00:13:46.030 So this will be the slope of the line in the direction 00:13:46.030 --> 00:13:46.950 with respect to y. 00:13:46.950 --> 00:13:50.080 So I'm going to think f sub y and f sub 00:13:50.080 --> 00:13:55.930 x if I change my skis go this direction and I go down. 00:13:55.930 --> 00:14:01.726 So I could go down this way and break my neck. 00:14:01.726 --> 00:14:07.530 Or I could go down this way and break my neck as well. 00:14:07.530 --> 00:14:15.670 OK, it has to go like-- right? 00:14:15.670 --> 00:14:17.937 Can you tell me what these guys will be? 00:14:17.937 --> 00:14:20.270 I'm going to put them in pink because they're beautiful. 00:14:20.270 --> 00:14:21.120 STUDENT: 0 [INAUDIBLE]. 00:14:21.120 --> 00:14:22.828 PROFESSOR: Thank God, they are beautiful. 00:14:22.828 --> 00:14:23.900 Larry, what does it mean? 00:14:23.900 --> 00:14:28.070 That means that the two tangents, the tangents 00:14:28.070 --> 00:14:33.710 to the curves, are horizontal. 00:14:33.710 --> 00:14:38.100 And if I were to draw the plane between those two tangents-- 00:14:38.100 --> 00:14:44.566 one tangent is in pink pen, our is in green. 00:14:44.566 --> 00:14:46.112 Today, I'm all about colors. 00:14:46.112 --> 00:14:47.090 I'm in a good mood. 00:14:47.090 --> 00:14:50.690 00:14:50.690 --> 00:14:54.970 And that's going to be the so-called tangent plane-- 00:14:54.970 --> 00:15:06.980 tangent plane to the surface at x0, y0, which is the origin. 00:15:06.980 --> 00:15:09.290 That was a nice point. 00:15:09.290 --> 00:15:10.680 That is a nice point. 00:15:10.680 --> 00:15:12.550 Not all the points will be [INAUDIBLE] 00:15:12.550 --> 00:15:14.680 and nice but beautiful. 00:15:14.680 --> 00:15:18.304 [INAUDIBLE] I take the nice-- well, not so nice, 00:15:18.304 --> 00:15:19.150 I don't know. 00:15:19.150 --> 00:15:21.534 You'll have to figure it out. 00:15:21.534 --> 00:15:26.530 How do I get-- well, first of all, where is this point? 00:15:26.530 --> 00:15:31.720 If I take x to be 1 over square 2 and y to be 1 over square 2 00:15:31.720 --> 00:15:33.870 and I plug them in, what's z going to be? 00:15:33.870 --> 00:15:35.010 STUDENT: 0. 00:15:35.010 --> 00:15:37.641 PROFESSOR: 0, and I did that on purpose. 00:15:37.641 --> 00:15:41.020 Because in that case, I'm going to be on flat line again. 00:15:41.020 --> 00:15:42.250 This look like [INAUDIBLE]. 00:15:42.250 --> 00:15:45.090 Except [INAUDIBLE] is not z equal 0. 00:15:45.090 --> 00:15:47.490 What is [INAUDIBLE] like? 00:15:47.490 --> 00:15:48.390 z equals-- 00:15:48.390 --> 00:15:49.322 STUDENT: [INAUDIBLE] 00:15:49.322 --> 00:15:49.946 PROFESSOR: Huh? 00:15:49.946 --> 00:15:51.485 STUDENT: I don't know. 00:15:51.485 --> 00:15:54.080 PROFESSOR: Do you want to go in meters or in feet? 00:15:54.080 --> 00:15:56.280 STUDENT: [INAUDIBLE]. 00:15:56.280 --> 00:15:57.600 It's about a mile. 00:15:57.600 --> 00:15:59.080 PROFESSOR: Yes, I don't know. 00:15:59.080 --> 00:16:02.840 I thought it's about one kilometer, 1,000 something 00:16:02.840 --> 00:16:03.820 meters. 00:16:03.820 --> 00:16:05.870 But somebody said it's more so. 00:16:05.870 --> 00:16:10.040 It's flat land, and I'd say about a mile above the sea 00:16:10.040 --> 00:16:11.090 level. 00:16:11.090 --> 00:16:20.410 All right, now, I am going to be in flat land right here, 00:16:20.410 --> 00:16:24.770 1 over a root 2, 1 over a root 2, and 0. 00:16:24.770 --> 00:16:26.060 What happened here? 00:16:26.060 --> 00:16:28.986 Here, I just already broke my neck, you know. 00:16:28.986 --> 00:16:32.290 Well, if I came in this direction, 00:16:32.290 --> 00:16:36.616 I would need to draw a prospective trajectory that 00:16:36.616 --> 00:16:38.880 was hopefully not mine. 00:16:38.880 --> 00:16:43.699 And the tangent would-- the tangent, the slope 00:16:43.699 --> 00:16:45.650 of the tangent, would be funny. 00:16:45.650 --> 00:16:47.740 Let's see what you need to do. 00:16:47.740 --> 00:16:52.540 You need to say, OK, prime with respect to x, minus 2x. 00:16:52.540 --> 00:16:56.990 And then at the point x equals 1 over [INAUDIBLE] 2 y 00:16:56.990 --> 00:16:59.948 equals 1 over 2, you just plug in. 00:16:59.948 --> 00:17:00.920 And what do you have? 00:17:00.920 --> 00:17:02.420 STUDENT: Square root of [INAUDIBLE]. 00:17:02.420 --> 00:17:05.000 PROFESSOR: Negative square root of 2-- my god 00:17:05.000 --> 00:17:08.940 that is really bad as a slope. 00:17:08.940 --> 00:17:11.002 It's a steep slope. 00:17:11.002 --> 00:17:14.010 And this one-- how about this one? 00:17:14.010 --> 00:17:16.098 Same idea, symmetric function. 00:17:16.098 --> 00:17:20.339 And it's going to be exactly the same-- very steep slope. 00:17:20.339 --> 00:17:22.819 Why are they negative numbers? 00:17:22.819 --> 00:17:27.470 Because the slope is going down, right? 00:17:27.470 --> 00:17:31.990 That's the kind of slope I have in both directions-- 00:17:31.990 --> 00:17:38.570 one and-- all right. 00:17:38.570 --> 00:17:48.310 If I were to draw this thing continuing, 00:17:48.310 --> 00:17:51.800 how would I represent those slopes? 00:17:51.800 --> 00:17:56.682 This circle-- this circle is just making my life harder. 00:17:56.682 --> 00:17:59.600 But I would need to imagine those slopes as being 00:17:59.600 --> 00:18:02.330 like I'm here, all right? 00:18:02.330 --> 00:18:03.700 Are you guys with me? 00:18:03.700 --> 00:18:12.360 And I will need to draw x0-- well, what is that? 00:18:12.360 --> 00:18:24.782 1 over root 2 and 1 over root 2 And I would draw two planes. 00:18:24.782 --> 00:18:30.084 And I would have two curves. 00:18:30.084 --> 00:18:33.220 And when you slice up, imagine this 00:18:33.220 --> 00:18:34.487 would be a piece of cheese. 00:18:34.487 --> 00:18:35.320 STUDENT: [INAUDIBLE] 00:18:35.320 --> 00:18:35.955 PROFESSOR: And you cut-- 00:18:35.955 --> 00:18:36.788 STUDENT: [INAUDIBLE] 00:18:36.788 --> 00:18:41.782 00:18:41.782 --> 00:18:42.490 PROFESSOR: Right? 00:18:42.490 --> 00:18:43.073 STUDENT: Yeah. 00:18:43.073 --> 00:18:44.970 PROFESSOR: And you cut in this other side. 00:18:44.970 --> 00:18:48.806 Well, this is the one that's facing you. 00:18:48.806 --> 00:18:49.652 You cut like that. 00:18:49.652 --> 00:18:51.360 And when you cut like this, it's facing-- 00:18:51.360 --> 00:18:52.193 STUDENT: [INAUDIBLE] 00:18:52.193 --> 00:18:54.853 00:18:54.853 --> 00:18:56.849 PROFESSOR: Hm? 00:18:56.849 --> 00:18:59.344 But anyway, let's not draw the other one. 00:18:59.344 --> 00:19:00.841 It's hard, right? 00:19:00.841 --> 00:19:04.084 STUDENT: [INAUDIBLE] angle like this-- just the piece 00:19:04.084 --> 00:19:05.622 of the corner of the cheese. 00:19:05.622 --> 00:19:06.330 PROFESSOR: Right. 00:19:06.330 --> 00:19:08.350 STUDENT: The corner is facing you. 00:19:08.350 --> 00:19:13.248 PROFESSOR: So yeah, so it's-- the corner is facing you. 00:19:13.248 --> 00:19:17.000 STUDENT: So basically, you [INAUDIBLE] this. 00:19:17.000 --> 00:19:18.600 PROFESSOR: But-- exactly, but-- 00:19:18.600 --> 00:19:20.020 STUDENT: Like this. 00:19:20.020 --> 00:19:22.890 PROFESSOR: Yeah, well-- yeah, it's hard to draw. 00:19:22.890 --> 00:19:25.710 So practically, this is what you're 00:19:25.710 --> 00:19:29.450 looking at it is slope that's negative in both directions. 00:19:29.450 --> 00:19:34.760 So you're going to go this way or this way. 00:19:34.760 --> 00:19:39.140 And it's much steeper than you imagine [INAUDIBLE]. 00:19:39.140 --> 00:19:43.000 OK, they are equal. 00:19:43.000 --> 00:19:44.525 I'm trying to draw them equal. 00:19:44.525 --> 00:19:48.130 I don't know how equal they can be. 00:19:48.130 --> 00:19:55.780 One belongs to one plane just like you said. 00:19:55.780 --> 00:19:58.323 This belongs to this plane. 00:19:58.323 --> 00:20:07.476 And the green one belongs to the plane that's facing you. 00:20:07.476 --> 00:20:08.784 So the slope goes this way. 00:20:08.784 --> 00:20:11.170 But the two slopes are equal. 00:20:11.170 --> 00:20:13.480 You have to have a little bit of imagination. 00:20:13.480 --> 00:20:16.650 We would need some cheese to make a mountain of cheese 00:20:16.650 --> 00:20:18.400 and cut them and slice them. 00:20:18.400 --> 00:20:21.640 We'll eat everything after, yeah. 00:20:21.640 --> 00:20:29.030 All right, let's move on to something more challenging 00:20:29.030 --> 00:20:31.780 now that we got to the tangent plane. 00:20:31.780 --> 00:20:34.200 So if somebody would say, wait a minute, 00:20:34.200 --> 00:20:38.230 you said this is the tangent plane to the surface. 00:20:38.230 --> 00:20:40.370 You just introduced a new notion. 00:20:40.370 --> 00:20:41.450 You were fooling us. 00:20:41.450 --> 00:20:43.820 I'm fooling you guys. 00:20:43.820 --> 00:20:48.590 It's not April 1, but this kind of a not a neat thing. 00:20:48.590 --> 00:20:58.504 I just tried to introduce you into the section 11.4. 00:20:58.504 --> 00:21:02.710 So if you have a piece of a curve that's smooth 00:21:02.710 --> 00:21:08.010 and you have a point x0, y0, can you 00:21:08.010 --> 00:21:12.860 find out the equation of the tangent plane? 00:21:12.860 --> 00:21:16.610 Pi, and this is s form surface. 00:21:16.610 --> 00:21:20.820 How can I find the equation of the tangent plane? 00:21:20.820 --> 00:21:27.180 00:21:27.180 --> 00:21:33.210 That x0, y0-- 12 is going to be also z0. 00:21:33.210 --> 00:21:44.880 But what I mean that x0, y0 is in on the floor 00:21:44.880 --> 00:21:47.210 as a projection. 00:21:47.210 --> 00:21:50.500 So I'm always looking at the graph. 00:21:50.500 --> 00:21:52.496 And that's why. 00:21:52.496 --> 00:21:54.299 The moment I stop looking at the graph, 00:21:54.299 --> 00:21:55.340 things will be different. 00:21:55.340 --> 00:21:59.770 But I'm looking at the graph of independent variables x, y. 00:21:59.770 --> 00:22:02.505 And that's why those guys are always on the floor. 00:22:02.505 --> 00:22:07.560 A and z would be a function to keep in the variable. 00:22:07.560 --> 00:22:09.300 Now, does anybody know? 00:22:09.300 --> 00:22:12.610 Because I know you guys are reading in advance 00:22:12.610 --> 00:22:16.260 and you have better teachers than me. 00:22:16.260 --> 00:22:17.374 You have the internet. 00:22:17.374 --> 00:22:18.165 You have the links. 00:22:18.165 --> 00:22:18.957 You have YouTube. 00:22:18.957 --> 00:22:20.620 You have Khan Academy. 00:22:20.620 --> 00:22:24.532 I know from a bunch of you that you have already gone 00:22:24.532 --> 00:22:27.240 over half of the chapter 11. 00:22:27.240 --> 00:22:30.840 I just hope that now you can compare what you learned 00:22:30.840 --> 00:22:34.110 with what I'm teaching you, And I'm not 00:22:34.110 --> 00:22:36.840 expecting you to go in advance, but several of you 00:22:36.840 --> 00:22:38.590 already know this formula. 00:22:38.590 --> 00:22:43.620 We talked about it in office hours on yesterday. 00:22:43.620 --> 00:22:45.991 Because Tuesday, I didn't have office hours. 00:22:45.991 --> 00:22:49.790 I had a coordinator meeting. 00:22:49.790 --> 00:22:55.942 So what equation corresponds to the tangent plate? 00:22:55.942 --> 00:22:56.775 STUDENT: [INAUDIBLE] 00:22:56.775 --> 00:23:00.420 00:23:00.420 --> 00:23:02.120 PROFESSOR: Several of you know it. 00:23:02.120 --> 00:23:03.970 You know what I hated? 00:23:03.970 --> 00:23:05.120 It's fine that you know it. 00:23:05.120 --> 00:23:08.250 I'm proud of you guys and I'll write it. 00:23:08.250 --> 00:23:12.320 But when I was a freshman-- or what the heck was I? 00:23:12.320 --> 00:23:17.240 A sophomore I think-- no, I was a freshman when they fed me. 00:23:17.240 --> 00:23:19.365 They spoon-fed me this equation. 00:23:19.365 --> 00:23:22.380 And I didn't understand anything at the time. 00:23:22.380 --> 00:23:25.910 I hated the fact that the Professor painted it 00:23:25.910 --> 00:23:30.860 on the board just like that out of the blue. 00:23:30.860 --> 00:23:33.700 I want to see a proof. 00:23:33.700 --> 00:23:39.630 And he was able to-- I think he could have done a good job. 00:23:39.630 --> 00:23:42.730 But he didn't. 00:23:42.730 --> 00:23:46.790 He showed us a bunch of justifications 00:23:46.790 --> 00:23:53.400 like if you generally have a surface in implicit form, 00:23:53.400 --> 00:23:58.100 I told you that the gradient of F 00:23:58.100 --> 00:24:01.530 represents the normal connection, right? 00:24:01.530 --> 00:24:06.510 And he prepared us pretty good for what could 00:24:06.510 --> 00:24:08.850 have been the proof of that. 00:24:08.850 --> 00:24:10.195 He said, OK, guys. 00:24:10.195 --> 00:24:12.470 You know the duration of the normal 00:24:12.470 --> 00:24:16.000 as even as the gradient over the next of the gradient, 00:24:16.000 --> 00:24:17.954 if you want unit normal. 00:24:17.954 --> 00:24:19.340 How did he do that? 00:24:19.340 --> 00:24:21.900 Well, he had a bunch of examples. 00:24:21.900 --> 00:24:23.440 He had the sphere. 00:24:23.440 --> 00:24:25.805 He showed us that for the sphere, 00:24:25.805 --> 00:24:29.884 you have the normal, which is the continuation 00:24:29.884 --> 00:24:31.400 of the position vector. 00:24:31.400 --> 00:24:34.260 Then he said, OK, you can have approximations 00:24:34.260 --> 00:24:39.370 of a surface that is smooth and round with oscillating spheres 00:24:39.370 --> 00:24:45.292 just the way you have for a curve, a resonating circle, 00:24:45.292 --> 00:24:49.560 a resonating circle-- that's called oscillating circle. 00:24:49.560 --> 00:24:53.070 Resonating circle-- in that case, what will the normal be? 00:24:53.070 --> 00:24:55.570 Well, the normal will have to depend 00:24:55.570 --> 00:24:57.120 on the radius of the circle. 00:24:57.120 --> 00:25:01.945 So you have a principal normal or a normal if it's a plane 00:25:01.945 --> 00:25:02.790 curve. 00:25:02.790 --> 00:25:05.850 And it's easy to understand that's the same 00:25:05.850 --> 00:25:07.210 as the gradient. 00:25:07.210 --> 00:25:12.680 So we have enough justification for the direction 00:25:12.680 --> 00:25:16.230 of the gradient of such a function is always 00:25:16.230 --> 00:25:19.867 normal-- normal to the surface, normal to all 00:25:19.867 --> 00:25:22.730 the curves on the surface. 00:25:22.730 --> 00:25:26.660 If we want to find that without swallowing 00:25:26.660 --> 00:25:32.410 this like I had to when I was a student, it's not hard. 00:25:32.410 --> 00:25:35.320 And let me show you how we do it. 00:25:35.320 --> 00:25:37.330 We start from the graph, right? 00:25:37.330 --> 00:25:40.800 Z equals f of x and y. 00:25:40.800 --> 00:25:44.200 And we say, well, Magdalena, but this is a graph. 00:25:44.200 --> 00:25:47.550 It's not an implicit equation. 00:25:47.550 --> 00:25:50.320 And I'll say, yes it is. 00:25:50.320 --> 00:25:53.980 Let me show you how I make it an implicit equation. 00:25:53.980 --> 00:25:56.302 I move z to the other side. 00:25:56.302 --> 00:26:00.630 I put 0 equals f of xy minus z. 00:26:00.630 --> 00:26:04.272 Now it is an implicit equation. 00:26:04.272 --> 00:26:05.720 So you say you cheated. 00:26:05.720 --> 00:26:07.530 Yes, I did. 00:26:07.530 --> 00:26:08.190 I have cheated. 00:26:08.190 --> 00:26:10.750 00:26:10.750 --> 00:26:14.880 It's funny that whenever somebody gives you a graph, 00:26:14.880 --> 00:26:16.920 you can rewrite that graph immediately 00:26:16.920 --> 00:26:18.710 as an implicit equation. 00:26:18.710 --> 00:26:23.760 So that implicit equation is of the form big F of xyz 00:26:23.760 --> 00:26:27.470 now equals a constant, which is 0. 00:26:27.470 --> 00:26:32.330 F of xy is your old friend and minus z. 00:26:32.330 --> 00:26:36.340 Now, can you tell me what is the normal to this surface? 00:26:36.340 --> 00:26:40.010 Yeah, give me a splash in a minute like that. 00:26:40.010 --> 00:26:43.192 So what is the gradient of f? 00:26:43.192 --> 00:26:44.995 Gradient of f will be the normal. 00:26:44.995 --> 00:26:47.110 I don't care if it's unit or not. 00:26:47.110 --> 00:26:49.460 To heck with the unit or normal. 00:26:49.460 --> 00:26:54.180 I'm going to say I wanted prime with respect to x, y, 00:26:54.180 --> 00:26:58.030 and z respectively. 00:26:58.030 --> 00:26:59.760 And what is the gradient? 00:26:59.760 --> 00:27:01.790 Is the vector. 00:27:01.790 --> 00:27:06.920 Big F sub x comma big F sub y comma big F sub z. 00:27:06.920 --> 00:27:08.900 We see that last time. 00:27:08.900 --> 00:27:13.620 So the gradient of a function is the vector 00:27:13.620 --> 00:27:17.130 whose coordinates are the partial velocity-- 00:27:17.130 --> 00:27:19.560 your friends form last time. 00:27:19.560 --> 00:27:22.090 Can we represent this again? 00:27:22.090 --> 00:27:22.650 I don't know. 00:27:22.650 --> 00:27:24.300 You need to help me. 00:27:24.300 --> 00:27:29.339 Who is big F prime with respect to x? 00:27:29.339 --> 00:27:30.130 There is no x here. 00:27:30.130 --> 00:27:32.620 Thank God that's like a constant. 00:27:32.620 --> 00:27:36.170 I just have to take this little one, f, and prime it 00:27:36.170 --> 00:27:36.960 with respect to x. 00:27:36.960 --> 00:27:41.050 And that's exactly what that's going to be-- little f sub x. 00:27:41.050 --> 00:27:44.800 What is big F with respect to y? 00:27:44.800 --> 00:27:46.040 STUDENT: [INAUDIBLE] 00:27:46.040 --> 00:27:49.205 PROFESSOR: Little f sub y prime with respect 00:27:49.205 --> 00:27:51.940 to y-- differentiated with respect to y. 00:27:51.940 --> 00:27:56.210 And finally, if I differentiated with respect to z, 00:27:56.210 --> 00:27:57.520 there is no z here, right? 00:27:57.520 --> 00:27:58.490 There is no z. 00:27:58.490 --> 00:28:00.000 So that's like a constant. 00:28:00.000 --> 00:28:04.540 Prime [INAUDIBLE] 0 and minus 1. 00:28:04.540 --> 00:28:05.770 So I know the gradient. 00:28:05.770 --> 00:28:06.880 I know the normal. 00:28:06.880 --> 00:28:09.140 This is the normal. 00:28:09.140 --> 00:28:14.180 Now, if somebody gives you the normal, there you are. 00:28:14.180 --> 00:28:20.230 You have the normal to the surface-- normal to surface. 00:28:20.230 --> 00:28:21.282 What does it mean? 00:28:21.282 --> 00:28:26.480 Equals normal to the tangent plane to the surface. 00:28:26.480 --> 00:28:30.054 Normal or perpendicular to the tangent plane- 00:28:30.054 --> 00:28:37.520 to the plane-- of the surface. 00:28:37.520 --> 00:28:41.470 At that point-- point is the point p. 00:28:41.470 --> 00:28:44.500 00:28:44.500 --> 00:28:52.530 All right, so if you were to study a surface that's-- do you 00:28:52.530 --> 00:28:53.570 have a [INAUDIBLE]? 00:28:53.570 --> 00:28:54.700 STUDENT: Uh, no. 00:28:54.700 --> 00:28:55.517 Do you? 00:28:55.517 --> 00:28:56.100 PROFESSOR: OK. 00:28:56.100 --> 00:28:59.300 00:28:59.300 --> 00:29:03.520 OK, I want to study the tangent plane 00:29:03.520 --> 00:29:05.270 at this point to the surface. 00:29:05.270 --> 00:29:06.670 Well, that's flat, Magdalena. 00:29:06.670 --> 00:29:08.460 You have no imagination. 00:29:08.460 --> 00:29:13.770 The tangent plane is this plane, is the same as the surface. 00:29:13.770 --> 00:29:16.490 So, no fun-- no fun. 00:29:16.490 --> 00:29:19.880 How about I pick my favorite plane here 00:29:19.880 --> 00:29:24.690 and I take-- what is-- OK. 00:29:24.690 --> 00:29:26.765 I have-- this is Children Internationals. 00:29:26.765 --> 00:29:30.300 I have a little girl abroad that I'm sponsoring. 00:29:30.300 --> 00:29:34.500 So you have a point here and a plane 00:29:34.500 --> 00:29:38.780 that passes through that point. 00:29:38.780 --> 00:29:40.590 This is the tangent plane. 00:29:40.590 --> 00:29:43.610 And my finger is the normal. 00:29:43.610 --> 00:29:46.910 And the normal, we call that normal to the surface 00:29:46.910 --> 00:29:49.316 when it's normal to the tangent plane. 00:29:49.316 --> 00:29:52.972 At every point, this is what the normal is. 00:29:52.972 --> 00:29:55.490 All right, can we write that based on chapter nine? 00:29:55.490 --> 00:29:59.179 Now I will see what you remember from chapter nine if anything 00:29:59.179 --> 00:30:00.157 at all. 00:30:00.157 --> 00:30:03.580 00:30:03.580 --> 00:30:09.430 All right, how do we write the tangent plane 00:30:09.430 --> 00:30:11.826 if we know the normal? 00:30:11.826 --> 00:30:21.550 OK, review-- if the normal vector is ai plus bj plus ck, 00:30:21.550 --> 00:30:28.060 that means the plane that is perpendicular to it 00:30:28.060 --> 00:30:30.650 is of what form? 00:30:30.650 --> 00:30:37.548 Ax plus by plus cz plus d equals 0, right? 00:30:37.548 --> 00:30:39.540 You've learned that in chapter nine. 00:30:39.540 --> 00:30:43.370 Most of you learned that last semester in Calculus 2 00:30:43.370 --> 00:30:44.590 at the end. 00:30:44.590 --> 00:30:51.290 Now, if my normal is f sub x, f sub y, and minus 1, 00:30:51.290 --> 00:30:52.810 those are ABC for God's sake. 00:30:52.810 --> 00:30:53.930 Well, good. 00:30:53.930 --> 00:30:59.400 Big A, big B, big C at the given point. 00:30:59.400 --> 00:31:09.810 So I'm going to have f sub x at the given point d times 00:31:09.810 --> 00:31:17.760 x plus f sub y at any given point d times y. 00:31:17.760 --> 00:31:18.980 Who is c? 00:31:18.980 --> 00:31:20.420 C is minus 1. 00:31:20.420 --> 00:31:24.180 Minus 1 times z is-- say you're being silly. 00:31:24.180 --> 00:31:26.552 Magdalena, why do you write minus 1? 00:31:26.552 --> 00:31:28.890 Just because I'm having fun. 00:31:28.890 --> 00:31:32.780 And plus, d equals 0. 00:31:32.780 --> 00:31:34.590 And you say, well, wait, wait, wait. 00:31:34.590 --> 00:31:39.880 This starts looking like that but it's not the same thing. 00:31:39.880 --> 00:31:42.830 All right, what? 00:31:42.830 --> 00:31:44.243 How do you get to d? 00:31:44.243 --> 00:31:47.700 00:31:47.700 --> 00:31:50.510 Now, actually, the plane perpendicular 00:31:50.510 --> 00:31:55.180 to n that passes through a given point 00:31:55.180 --> 00:31:58.920 can be written much faster, right? 00:31:58.920 --> 00:32:06.024 So if a plane is perpendicular to a certain line, 00:32:06.024 --> 00:32:09.535 how do we write if we know a point? 00:32:09.535 --> 00:32:15.190 If we know a point in the normal ABC-- 00:32:15.190 --> 00:32:18.920 I have to go backwards to read it backwards-- then 00:32:18.920 --> 00:32:22.990 the plane is going to be x minus x0 00:32:22.990 --> 00:32:29.530 plus b times y times y0 plus c times z minus c0 equals 0. 00:32:29.530 --> 00:32:33.030 00:32:33.030 --> 00:32:34.680 So who is the d? 00:32:34.680 --> 00:32:39.310 The d is all the constant that gets out of here. 00:32:39.310 --> 00:32:43.671 So the point x0, y0, z0 has to verify the plane. 00:32:43.671 --> 00:32:47.080 And that's why when you plug in x0, y0, z0, 00:32:47.080 --> 00:32:50.420 you get 0 plus 0 plus 0 equals 0. 00:32:50.420 --> 00:32:53.490 That's what it means for a point to verify the plane. 00:32:53.490 --> 00:32:59.035 When you take the x0, y0, z0 and you plug it into the equation, 00:32:59.035 --> 00:33:02.770 you have to have an identity 0 equals 0. 00:33:02.770 --> 00:33:06.800 So this can be rewritten zx plus by plus cz 00:33:06.800 --> 00:33:10.890 just like we did there plus a d. 00:33:10.890 --> 00:33:12.535 And who in the world is the d? 00:33:12.535 --> 00:33:18.730 The d will be exactly minus ax0 minus by0 minus cz0. 00:33:18.730 --> 00:33:22.940 If that makes you uncomfortable, this is in chapter nine. 00:33:22.940 --> 00:33:28.890 Look at the equation of a plane and the normal to it. 00:33:28.890 --> 00:33:32.760 Now I know that I can do better than that if I'm smart. 00:33:32.760 --> 00:33:35.850 So again, I collect the ABC. 00:33:35.850 --> 00:33:37.176 Now I know my ABC. 00:33:37.176 --> 00:33:40.325 00:33:40.325 --> 00:33:42.225 I put them in here. 00:33:42.225 --> 00:33:46.170 So I have f sub x at the point in time. 00:33:46.170 --> 00:33:50.600 Oh, OK, x minus x0 plus, who is my b? 00:33:50.600 --> 00:33:57.120 F sub y computed at the point p times y minus y0. 00:33:57.120 --> 00:33:58.760 And, what? 00:33:58.760 --> 00:33:59.930 Minus, right? 00:33:59.930 --> 00:34:03.500 Minus-- minus 1. 00:34:03.500 --> 00:34:05.040 I'm not going to write minus 1. 00:34:05.040 --> 00:34:07.330 You're going to make fun of me. 00:34:07.330 --> 00:34:10.237 Minus z minus cz. 00:34:10.237 --> 00:34:12.060 And my proof is done. 00:34:12.060 --> 00:34:17.020 QED-- what does it mean, QED? 00:34:17.020 --> 00:34:19.620 In Latin. 00:34:19.620 --> 00:34:22.530 QED means I proved what I wanted to prove. 00:34:22.530 --> 00:34:23.960 Do you know what it stands for? 00:34:23.960 --> 00:34:26.717 Did you take Latin, any of you? 00:34:26.717 --> 00:34:29.090 You took Latin? 00:34:29.090 --> 00:34:33.190 Quod erat demonstrandum. 00:34:33.190 --> 00:34:36.699 00:34:36.699 --> 00:34:39.431 So this was to be proved. 00:34:39.431 --> 00:34:42.050 That's exactly what it was to be proved. 00:34:42.050 --> 00:34:44.647 That, what, that c minus z0, which 00:34:44.647 --> 00:34:50.179 was my fellow over here pretty in pink, is going to be f sub x 00:34:50.179 --> 00:34:54.620 times x minus x0 plus yf sub y times y minus y0. 00:34:54.620 --> 00:35:01.140 So now you know why the equation of the tangent plane is that. 00:35:01.140 --> 00:35:04.525 I proved it more or less, making some assumptions, 00:35:04.525 --> 00:35:06.830 some axioms as assumption. 00:35:06.830 --> 00:35:09.540 But you don't know how to use it. 00:35:09.540 --> 00:35:10.860 So let's use it. 00:35:10.860 --> 00:35:14.255 So for the same valley-- not valley, hill-- 00:35:14.255 --> 00:35:16.170 it was full of snow. 00:35:16.170 --> 00:35:19.375 Z equals 1 minus x squared-- what was you 00:35:19.375 --> 00:35:20.810 guys have forgotten? 00:35:20.810 --> 00:35:24.970 OK, 1 minus x squared minus y squared. 00:35:24.970 --> 00:35:32.570 Find the tangent plane at the following points. 00:35:32.570 --> 00:35:37.045 Ah, x0, y0 to be origin. 00:35:37.045 --> 00:35:39.470 And you say, did you say that that's trivial? 00:35:39.470 --> 00:35:40.480 Yes, it is trivial. 00:35:40.480 --> 00:35:42.950 But I'm going to do it one more time. 00:35:42.950 --> 00:35:47.190 And what was my [INAUDIBLE] point before? 00:35:47.190 --> 00:35:48.630 STUDENT: [INAUDIBLE] 00:35:48.630 --> 00:35:53.430 PROFESSOR: 1 over 2 and 1 over 2. 00:35:53.430 --> 00:35:57.790 OK, and what will be the corresponding point in 3D? 00:35:57.790 --> 00:36:01.500 1 over 2, 1 over 2, I plug in. 00:36:01.500 --> 00:36:03.540 Ah, yes. 00:36:03.540 --> 00:36:07.066 And with this, I hope to finish the day so we 00:36:07.066 --> 00:36:10.236 can go to our other businesses. 00:36:10.236 --> 00:36:11.620 Is this hard? 00:36:11.620 --> 00:36:15.960 Now, I was not able-- I have to be honest with you. 00:36:15.960 --> 00:36:20.670 I was not able to memorize the equation of a tangent plane 00:36:20.670 --> 00:36:27.010 when I was-- when I was young, like a freshman and sophomore. 00:36:27.010 --> 00:36:29.990 I wasn't ready to understand that this 00:36:29.990 --> 00:36:33.220 is a linear approximation of a curved something. 00:36:33.220 --> 00:36:35.530 This practically like the Taylor equation 00:36:35.530 --> 00:36:39.400 for functions of two variables when 00:36:39.400 --> 00:36:42.710 you neglect the quadratic third term and so on. 00:36:42.710 --> 00:36:46.115 You just take the-- I'll teach you 00:36:46.115 --> 00:36:52.150 next time when this is, a first order linear approximation. 00:36:52.150 --> 00:36:54.010 All right, can we do this really quickly? 00:36:54.010 --> 00:36:55.800 It's going to be a piece of cake. 00:36:55.800 --> 00:36:56.630 Let's see. 00:36:56.630 --> 00:36:58.360 Again, how do we do that? 00:36:58.360 --> 00:36:59.820 This is f of x and y. 00:36:59.820 --> 00:37:01.700 We computed that again. 00:37:01.700 --> 00:37:04.440 F of 0, 0 was this 0. 00:37:04.440 --> 00:37:08.490 Guys, if I say something silly, will you stop me? 00:37:08.490 --> 00:37:12.760 F of f sub x-- f of y at 0, 0 is 0. 00:37:12.760 --> 00:37:14.330 So I have two slopes. 00:37:14.330 --> 00:37:15.520 Those are my hands. 00:37:15.520 --> 00:37:19.430 The slopes of my hands are 0. 00:37:19.430 --> 00:37:27.270 So the tangent plane will be z minus z0 equals 0. 00:37:27.270 --> 00:37:29.050 What is the 0? 00:37:29.050 --> 00:37:29.550 STUDENT: 1 00:37:29.550 --> 00:37:30.550 PROFESSOR: 1, excellent. 00:37:30.550 --> 00:37:31.754 STUDENT: [INAUDIBLE] 00:37:31.754 --> 00:37:32.795 PROFESSOR: Why is that 1? 00:37:32.795 --> 00:37:36.060 0 and 0 give me 1. 00:37:36.060 --> 00:37:39.875 So that was the picture that I had z equals 1 00:37:39.875 --> 00:37:42.845 as the tangent plane at the point corresponding 00:37:42.845 --> 00:37:44.825 to the origin. 00:37:44.825 --> 00:37:48.340 That look like the north pole, 0, 0, 1. 00:37:48.340 --> 00:37:50.052 OK, no. 00:37:50.052 --> 00:37:52.550 It's the top of a hill. 00:37:52.550 --> 00:37:56.330 And finally, one last thing [INAUDIBLE]. 00:37:56.330 --> 00:37:58.200 Maybe you can do this by yourselves, 00:37:58.200 --> 00:38:01.060 but I will shut up if I can. 00:38:01.060 --> 00:38:03.390 I can't in general, but I'll shut up. 00:38:03.390 --> 00:38:09.080 Let's see-- f sub x at 1 over root 2, 1 over root 2. 00:38:09.080 --> 00:38:10.195 Why was that? 00:38:10.195 --> 00:38:11.895 What is f sub x? 00:38:11.895 --> 00:38:14.320 STUDENT: The square root of-- negative square root of 2. 00:38:14.320 --> 00:38:17.370 PROFESSOR: Right, we've done that before. 00:38:17.370 --> 00:38:20.240 And you got exactly what you said-- [INAUDIBLE] 00:38:20.240 --> 00:38:24.760 2 f sub y at the same point. 00:38:24.760 --> 00:38:29.520 I am too lazy to write it down again-- minus root 2. 00:38:29.520 --> 00:38:32.940 And how do we actually express the final answer 00:38:32.940 --> 00:38:37.260 so we can go home and whatever-- to the next class? 00:38:37.260 --> 00:38:39.261 Is it hard? 00:38:39.261 --> 00:38:39.760 No. 00:38:39.760 --> 00:38:40.970 What's the answer? 00:38:40.970 --> 00:38:44.140 Z minus-- now, attention. 00:38:44.140 --> 00:38:45.804 What is z0? 00:38:45.804 --> 00:38:46.730 STUDENT: 0. 00:38:46.730 --> 00:38:48.820 PROFESSOR: 0, right. 00:38:48.820 --> 00:38:49.470 Why is that? 00:38:49.470 --> 00:38:54.280 Because when I plug 1 over a 2, 1 over a 2, I got 0. 00:38:54.280 --> 00:38:56.780 0-- do I have to write it down? 00:38:56.780 --> 00:38:59.300 No, not unless I want to be silly. 00:38:59.300 --> 00:39:02.140 But if you do write down everything 00:39:02.140 --> 00:39:04.980 and you don't simplify the equation of the plane, 00:39:04.980 --> 00:39:08.650 we don't penalize you in any way in the final, OK? 00:39:08.650 --> 00:39:14.140 So if you show your work like that, you're going to be fine. 00:39:14.140 --> 00:39:16.960 What is that 1 over 2? 00:39:16.960 --> 00:39:25.160 Plus minus root 2 times y minus 1 over root 2. 00:39:25.160 --> 00:39:27.780 Is it elegant? 00:39:27.780 --> 00:39:30.820 No, it's not elegant at all. 00:39:30.820 --> 00:39:35.960 So as the last row for today, one final line. 00:39:35.960 --> 00:39:40.060 Can we make it look more elegant? 00:39:40.060 --> 00:39:43.540 Do we care to make it more elegant? 00:39:43.540 --> 00:39:47.610 Definitely some of you care. 00:39:47.610 --> 00:39:52.180 Z will be minus root 2x. 00:39:52.180 --> 00:39:56.660 I want to be consistent and keep the same style in y. 00:39:56.660 --> 00:39:58.930 And yet the constant goes wherever 00:39:58.930 --> 00:40:00.365 it wants to go at the end. 00:40:00.365 --> 00:40:01.994 What's that constant? 00:40:01.994 --> 00:40:03.380 STUDENT: 2 [INAUDIBLE]. 00:40:03.380 --> 00:40:05.111 PROFESSOR: So you see what you have. 00:40:05.111 --> 00:40:06.152 You have this times that. 00:40:06.152 --> 00:40:07.910 It's a 1, this then that is a 1. 00:40:07.910 --> 00:40:10.100 1 plus 1 is 2. 00:40:10.100 --> 00:40:12.490 All right, are you happy with this? 00:40:12.490 --> 00:40:13.910 I'm not. 00:40:13.910 --> 00:40:15.930 I'm happy. 00:40:15.930 --> 00:40:18.260 You-- if this were a multiple choice, 00:40:18.260 --> 00:40:21.500 you would be able to recognize it right away. 00:40:21.500 --> 00:40:25.440 What's the standardized general equation of a plane, though? 00:40:25.440 --> 00:40:29.140 Something x plus something y plus something z plus something 00:40:29.140 --> 00:40:31.050 equals 0. 00:40:31.050 --> 00:40:34.260 So if you wanted to make me very happy, 00:40:34.260 --> 00:40:38.555 you would still move everybody to the left hand side. 00:40:38.555 --> 00:40:41.080 00:40:41.080 --> 00:40:43.220 Do you want equal to or minus 3? 00:40:43.220 --> 00:40:45.100 Yes, it does. 00:40:45.100 --> 00:40:46.040 STUDENT: [INAUDIBLE] 00:40:46.040 --> 00:40:46.980 PROFESSOR: Huh? 00:40:46.980 --> 00:40:49.790 Negative 2-- is that OK? 00:40:49.790 --> 00:40:50.680 Is that fine? 00:40:50.680 --> 00:40:51.610 Are you guys done? 00:40:51.610 --> 00:40:52.400 Is this hard? 00:40:52.400 --> 00:40:53.560 Mm-mm. 00:40:53.560 --> 00:40:55.200 It's hard? 00:40:55.200 --> 00:40:56.280 No. 00:40:56.280 --> 00:40:58.770 Who said it's hard? 00:40:58.770 --> 00:41:05.180 So-- so I would work more tangent planes next time. 00:41:05.180 --> 00:41:08.320 But I think it's something that we can practice on. 00:41:08.320 --> 00:41:12.600 And do expect one exercise like that from one 00:41:12.600 --> 00:41:16.410 of those, God knows, 15, 16 on the final. 00:41:16.410 --> 00:41:18.010 I'm not sure about the midterm. 00:41:18.010 --> 00:41:19.860 I like this type of problem. 00:41:19.860 --> 00:41:23.230 So you might even see something with tangent planes 00:41:23.230 --> 00:41:26.956 on the midterm-- normal to a surface tangent plane. 00:41:26.956 --> 00:41:28.070 It's a good topic. 00:41:28.070 --> 00:41:29.470 It's really pretty. 00:41:29.470 --> 00:41:33.200 For people who like to draw, it's also nice to draw them. 00:41:33.200 --> 00:41:34.590 But do you have to? 00:41:34.590 --> 00:41:35.730 No. 00:41:35.730 --> 00:41:39.360 Some of you don't like to. 00:41:39.360 --> 00:41:43.160 OK, so now I say thank you for the attendance 00:41:43.160 --> 00:41:48.130 and I'll see you next time on Thursday-- on Tuesday. 00:41:48.130 --> 00:41:50.580 Happy Valentine's Day. 00:41:50.580 --> 00:41:52.189