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