WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.320 Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.560 continue to offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.300 To make a donation, or view additional materials 00:00:13.300 --> 00:00:17.116 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.116 --> 00:00:17.740 at ocw.mit.edu. 00:00:34.750 --> 00:00:35.820 PROFESSOR: Hi. 00:00:35.820 --> 00:00:40.370 Today, we would like to begin our study of calculus 00:00:40.370 --> 00:00:42.360 of several real variables. 00:00:42.360 --> 00:00:45.890 And as I mentioned in our last lecture, 00:00:45.890 --> 00:00:50.400 I would like to begin it from the traditional point of view. 00:00:50.400 --> 00:00:54.140 And I guess I'd like to make a little bit of a short speech 00:00:54.140 --> 00:00:55.700 even before we begin. 00:00:55.700 --> 00:00:58.360 There may be a little bit of danger 00:00:58.360 --> 00:01:01.320 in the way I've been emphasizing the modern approach 00:01:01.320 --> 00:01:04.190 to mathematics: that one may actually believe 00:01:04.190 --> 00:01:07.400 that the traditional approach to mathematics 00:01:07.400 --> 00:01:10.290 left very, very much to be desired. 00:01:10.290 --> 00:01:12.150 The answer is, it didn't really. 00:01:12.150 --> 00:01:15.870 That the role of logic in good traditional mathematics 00:01:15.870 --> 00:01:19.410 was about the same as it is in good modern mathematics. 00:01:19.410 --> 00:01:22.820 That in fact, if we define new mathematics 00:01:22.820 --> 00:01:25.240 as meaning meaningful mathematics, 00:01:25.240 --> 00:01:28.970 there really was no difference between the two approaches. 00:01:28.970 --> 00:01:30.520 Except for the fact that after you 00:01:30.520 --> 00:01:33.040 have a couple of hundred years of hindsight 00:01:33.040 --> 00:01:36.740 and can take a subject apart from a different point of view, 00:01:36.740 --> 00:01:39.080 then obviously it's no great surprise 00:01:39.080 --> 00:01:43.280 that you can find cleaner ways of really presenting the topic. 00:01:43.280 --> 00:01:47.050 It's the old cliche of hindsight being better than foresight 00:01:47.050 --> 00:01:49.280 by a darn sight, or some such thing. 00:01:49.280 --> 00:01:51.650 But at any rate, without further ado, 00:01:51.650 --> 00:01:53.760 let's actually look at the introduction 00:01:53.760 --> 00:01:56.250 to calculus of several real variables 00:01:56.250 --> 00:01:58.780 from a traditional point of view. 00:01:58.780 --> 00:02:00.900 And I've abbreviated that title to simply 00:02:00.900 --> 00:02:03.070 say, "Calculus of Several Variables." 00:02:03.070 --> 00:02:05.610 Whereas I say that, think of two things. 00:02:05.610 --> 00:02:07.430 First of all, it's an introduction. 00:02:07.430 --> 00:02:09.900 And second of all, we're going to be doing this 00:02:09.900 --> 00:02:12.330 from the traditional point of view. 00:02:12.330 --> 00:02:14.060 Now, what we're assuming is-- remember, 00:02:14.060 --> 00:02:16.380 last time we talked about functions 00:02:16.380 --> 00:02:17.780 of several real variables. 00:02:17.780 --> 00:02:20.050 We talked about the limit concept, 00:02:20.050 --> 00:02:24.730 gave you exercises that drilled you on what limits meant, 00:02:24.730 --> 00:02:25.960 and the like. 00:02:25.960 --> 00:02:28.130 And as a result, it now makes sense 00:02:28.130 --> 00:02:32.229 to talk about the calculus, the instantaneous rate of change. 00:02:32.229 --> 00:02:34.520 Except that you have an awful lot of variables in here. 00:02:34.520 --> 00:02:37.590 In other words, the general case that we'd like to tackle 00:02:37.590 --> 00:02:41.920 is the case where we have w being some real number, 00:02:41.920 --> 00:02:46.050 being a function of the n independent variables 00:02:46.050 --> 00:02:48.300 x_1 up to x_n. 00:02:48.300 --> 00:02:51.230 And of course, you see, the calculus applies to the f. 00:02:51.230 --> 00:02:54.920 The w used here is to indicate the fact that the output is 00:02:54.920 --> 00:02:56.640 a real number. 00:02:56.640 --> 00:03:00.960 It's like talking about f of x versus y equals f of x. 00:03:00.960 --> 00:03:04.400 In a manner of speaking, it's the f of x that's important. 00:03:04.400 --> 00:03:06.180 We won't get into that right now. 00:03:06.180 --> 00:03:08.890 What was important, though, about y equals f of x? 00:03:08.890 --> 00:03:12.410 It indicated a graph-- graph. 00:03:12.410 --> 00:03:15.610 And because it was a graph, we could visualize things 00:03:15.610 --> 00:03:17.590 pictorially that might have been harder 00:03:17.590 --> 00:03:19.610 to understand analytically. 00:03:19.610 --> 00:03:23.810 For this reason, even though n may be any positive integer 00:03:23.810 --> 00:03:26.570 whatsoever-- preferably, of course, greater than 1, 00:03:26.570 --> 00:03:29.117 otherwise we wouldn't have a function of several variables. 00:03:29.117 --> 00:03:30.950 In fact, that's the definition of a function 00:03:30.950 --> 00:03:32.970 of several variables-- it's an expression 00:03:32.970 --> 00:03:35.290 of this type where n is greater than 1. 00:03:35.290 --> 00:03:37.580 But in general, and you'll notice this in the text 00:03:37.580 --> 00:03:41.470 as well, we begin with the special case that n equals 2. 00:03:41.470 --> 00:03:44.790 Because, you see, if n is 2, we need what? 00:03:44.790 --> 00:03:50.250 Two degrees of freedom to plot the independent variables. 00:03:50.250 --> 00:03:52.730 That would be the xy-plane, for example. 00:03:52.730 --> 00:03:56.360 A third degree of freedom, say, the z-axis, 00:03:56.360 --> 00:03:59.490 often called w-axis because of the symbolism. 00:03:59.490 --> 00:04:01.210 In other words, in the same way that I 00:04:01.210 --> 00:04:03.230 could graph a real-valued function 00:04:03.230 --> 00:04:06.700 of a single real variable as a curve in the plane, 00:04:06.700 --> 00:04:10.620 I can graph a real-valued function of two real variables 00:04:10.620 --> 00:04:13.370 as a surface in three-dimensional space. 00:04:13.370 --> 00:04:17.329 And as a result, when one is adjusting to new language, 00:04:17.329 --> 00:04:20.760 it's nice to take the case n equals 2 simply because you 00:04:20.760 --> 00:04:22.880 can view things pictorially. 00:04:22.880 --> 00:04:24.960 The danger, of course, with n equals 2, 00:04:24.960 --> 00:04:27.970 is you get so used to the picture that after a while 00:04:27.970 --> 00:04:30.980 you forget that n could have been more than 2. 00:04:30.980 --> 00:04:35.950 So I will try to pay homage to both points of view. 00:04:35.950 --> 00:04:39.310 I will simply mention here that we will often let n equal 2 00:04:39.310 --> 00:04:41.400 to take advantage of geometry. 00:04:41.400 --> 00:04:44.070 When we do this, the usual notation 00:04:44.070 --> 00:04:49.370 is to let w equal f of x, y where x and y are independent. 00:04:49.370 --> 00:04:52.280 And I should mention why it's so important to talk 00:04:52.280 --> 00:04:54.590 about independent variables. 00:04:54.590 --> 00:04:56.130 It'll become clear in a moment. 00:04:56.130 --> 00:04:58.540 Let me just tell you what independent means, and then 00:04:58.540 --> 00:05:01.340 I'll show you why it becomes clear in a moment. 00:05:01.340 --> 00:05:03.370 To say that x and y are independent 00:05:03.370 --> 00:05:06.880 simply means that you can choose one of the two variables 00:05:06.880 --> 00:05:08.760 without determining the other. 00:05:08.760 --> 00:05:12.150 For example, if I were to say let y equal 2x, 00:05:12.150 --> 00:05:14.620 then certainly y and x are not independent, 00:05:14.620 --> 00:05:18.790 because the choice of x determines the choice of y. 00:05:18.790 --> 00:05:20.930 But to say that x and y are independent-- 00:05:20.930 --> 00:05:23.110 and that's what we mean by two degrees of freedom-- 00:05:23.110 --> 00:05:25.670 is I can take any value of x that I want 00:05:25.670 --> 00:05:29.170 and not be impeded as to the choice of y that I want. 00:05:29.170 --> 00:05:32.615 And that's why we say the domain is the xy-plane. 00:05:32.615 --> 00:05:33.865 I have two degrees of freedom. 00:05:33.865 --> 00:05:36.390 I can roam anywhere through the plane this way. 00:05:36.390 --> 00:05:38.720 But the reason that we want this is as follows. 00:05:38.720 --> 00:05:41.130 And this is where the traditional mathematics 00:05:41.130 --> 00:05:45.220 was every bit as logical as the modern mathematics. 00:05:45.220 --> 00:05:48.350 Let's keep in mind that logic is the art 00:05:48.350 --> 00:05:53.630 of taking known situations that we know how to handle, you see, 00:05:53.630 --> 00:05:57.390 and reducing unfamiliar situations to these more 00:05:57.390 --> 00:05:58.920 familiar situations. 00:05:58.920 --> 00:06:01.590 The point is that even in traditional mathematics, 00:06:01.590 --> 00:06:04.330 one knew how to handle the calculus 00:06:04.330 --> 00:06:06.210 of a single real variable. 00:06:06.210 --> 00:06:09.060 So what one said was this, given that w 00:06:09.060 --> 00:06:13.000 is a function of both x and y, let's hold y fixed, 00:06:13.000 --> 00:06:16.050 for example, and let x vary. 00:06:16.050 --> 00:06:18.940 You see, if I fix y at some particular value 00:06:18.940 --> 00:06:23.230 and let x vary, once I've made that fixed choice of y, 00:06:23.230 --> 00:06:26.090 I have w as a function of x alone. 00:06:26.090 --> 00:06:28.880 And as long as w is a function of x alone, 00:06:28.880 --> 00:06:31.020 I already know how to take the derivative of w 00:06:31.020 --> 00:06:32.300 with respect to x. 00:06:32.300 --> 00:06:33.890 And by the way, there's no favoritism 00:06:33.890 --> 00:06:35.360 here between x and y. 00:06:35.360 --> 00:06:39.330 In a similar way, if I hold x constant and let y vary, 00:06:39.330 --> 00:06:41.940 then w is a function of y alone. 00:06:41.940 --> 00:06:44.780 And I can then find the derivative of w 00:06:44.780 --> 00:06:46.240 with respect to y. 00:06:46.240 --> 00:06:49.360 And these things were called partial derivatives, you see, 00:06:49.360 --> 00:06:51.480 because it involved holding what? 00:06:51.480 --> 00:06:53.750 All but one of the variables constant. 00:06:53.750 --> 00:06:57.190 We'll go into that in more detail as we go along. 00:06:57.190 --> 00:06:58.500 But the whole idea is what? 00:06:58.500 --> 00:07:01.090 By choosing the variables to be independent, 00:07:01.090 --> 00:07:04.820 we can hold all of them but one at a time constant. 00:07:04.820 --> 00:07:07.990 And as long as we're doing that, the resulting function 00:07:07.990 --> 00:07:10.350 is a function of a single real variable. 00:07:10.350 --> 00:07:13.980 And this reduces us to part one of our course. 00:07:13.980 --> 00:07:15.730 So at any rate, what we're saying is, 00:07:15.730 --> 00:07:19.790 fix y at some constant value, say, y equals y sub 0. 00:07:19.790 --> 00:07:23.360 And we then let x vary as in ordinary one-dimensional 00:07:23.360 --> 00:07:29.110 calculus between x_0 minus the magnitude of delta x and x_0 00:07:29.110 --> 00:07:30.760 plus the magnitude of delta x. 00:07:30.760 --> 00:07:34.670 In other words, we mark off an interval of magnitude delta x, 00:07:34.670 --> 00:07:38.540 on either side of x_0 and let x be any place in here. 00:07:38.540 --> 00:07:42.760 Then what we do, is remembering that y0 is a constant, 00:07:42.760 --> 00:07:45.760 and that this function, no matter how complicated it looks 00:07:45.760 --> 00:07:48.720 like, is a function of just delta x alone, we 00:07:48.720 --> 00:07:51.880 mimic the definition of an ordinary derivative. 00:07:51.880 --> 00:07:53.740 In other words, we take the limit as delta 00:07:53.740 --> 00:07:58.900 x approaches 0 of f of x_0 plus delta x comma y_0, 00:07:58.900 --> 00:08:04.340 minus f of x_0 comma y_0, and divide that by delta x. 00:08:04.340 --> 00:08:07.490 In other words, with y_0 being treated as a constant, 00:08:07.490 --> 00:08:09.960 notice that essentially, this is the change 00:08:09.960 --> 00:08:14.250 in f as x varies between x_0 and x_0 plus delta 00:08:14.250 --> 00:08:16.630 x, divided by the change in x. 00:08:16.630 --> 00:08:19.610 In other words, it's the limit of an average rate 00:08:19.610 --> 00:08:24.590 of change of f of x with respect to x for a fixed value of y_0. 00:08:24.590 --> 00:08:34.070 And to indicate that, we write this analogously, 00:08:34.070 --> 00:08:36.320 I would say, to how we write functions 00:08:36.320 --> 00:08:37.870 of a single real variable. 00:08:37.870 --> 00:08:40.299 In other words, we write f prime. 00:08:40.299 --> 00:08:42.914 Instead of f prime, we indicate-- 00:08:42.914 --> 00:08:45.780 let me put it this way, when you had only one real variable, 00:08:45.780 --> 00:08:48.575 was there any danger of misinterpreting the prime? 00:08:48.575 --> 00:08:50.700 After all, with only one independent variable, when 00:08:50.700 --> 00:08:53.180 you said derivative, it was obvious what 00:08:53.180 --> 00:08:54.740 variable you were differentiating 00:08:54.740 --> 00:08:55.890 with respect to. 00:08:55.890 --> 00:08:58.190 Here, there are two variables, x and y. 00:08:58.190 --> 00:09:00.360 So instead of the prime as a subscript, 00:09:00.360 --> 00:09:02.430 we write down the variable with respect to which 00:09:02.430 --> 00:09:03.699 we're taking the derivative. 00:09:03.699 --> 00:09:05.990 And then we put in the point at which the derivative is 00:09:05.990 --> 00:09:07.520 being evaluated. 00:09:07.520 --> 00:09:09.560 You see, in a similar way, to have 00:09:09.560 --> 00:09:14.360 defined f sub y of x_0, y_0, that would have meant what? 00:09:14.360 --> 00:09:18.440 We hold x constant at some value x_0, 00:09:18.440 --> 00:09:23.110 let y vary between y_0 and y_0 plus delta y, 00:09:23.110 --> 00:09:26.870 and take the limit of this expression, f of x_0 comma y_0 00:09:26.870 --> 00:09:31.770 plus delta y minus f of x_0, y_0, over delta y. 00:09:31.770 --> 00:09:33.300 Take the limit of that expression 00:09:33.300 --> 00:09:35.460 as delta y approaches 0, you see. 00:09:35.460 --> 00:09:37.840 And we're going to show this pictorially in a while. 00:09:37.840 --> 00:09:40.960 After all, that's why we chose the case n equals 2, 00:09:40.960 --> 00:09:43.240 was so that we could eventually illustrate this thing 00:09:43.240 --> 00:09:44.560 pictorially. 00:09:44.560 --> 00:09:47.870 But for the time being, I prefer not to introduce the picture 00:09:47.870 --> 00:09:49.460 because I want you to see that even 00:09:49.460 --> 00:09:53.830 in the case where my input consists of an n-tuple where 00:09:53.830 --> 00:09:57.000 n is greater than 2, or 3, or 4-- 00:09:57.000 --> 00:09:59.560 whatever you want-- that this thing still makes sense. 00:09:59.560 --> 00:10:02.560 In other words, in general, suppose w 00:10:02.560 --> 00:10:05.430 is a function of the n independent variables 00:10:05.430 --> 00:10:07.670 x sub 1 up to x sub n. 00:10:07.670 --> 00:10:12.060 And I want to compute f of x sub 1 at the point x_1 00:10:12.060 --> 00:10:15.600 equals a_1, et cetera, x_n equals a_n. 00:10:15.600 --> 00:10:22.650 By definition, what I do is I hold a_2 up to a_n constant. 00:10:22.650 --> 00:10:27.920 I let x_1 vary from a_1 to a_1 plus delta x. 00:10:27.920 --> 00:10:29.120 I compute what? 00:10:29.120 --> 00:10:33.700 f of a_1 plus delta x comma a_2, et cetera, comma a_n, 00:10:33.700 --> 00:10:36.370 minus f of a_1 up to a_n. 00:10:36.370 --> 00:10:43.700 Noticing that that numerator is simply the change in f as x_1 00:10:43.700 --> 00:10:46.690 varies from a_1 to a_1 plus delta x_1 00:10:46.690 --> 00:10:49.230 while all the other variables are being held constant. 00:10:49.230 --> 00:10:52.580 Which of course I can do if the variables are independent. 00:10:52.580 --> 00:10:54.740 I then divide that by delta x_1. 00:10:54.740 --> 00:10:56.490 That's the average rate of change. 00:10:56.490 --> 00:10:59.660 Then I take the limit as delta x_1 approaches 0. 00:10:59.660 --> 00:11:02.230 And this I can do for any number of variables, 00:11:02.230 --> 00:11:04.100 provided that they're independent. 00:11:04.100 --> 00:11:08.190 Keep that in mind, if x sub n could be expressed in terms 00:11:08.190 --> 00:11:13.900 of the remaining x's, how can I hold a sub n constant-- 00:11:13.900 --> 00:11:18.880 or, how can I vary a sub n while I hold the other ones constant? 00:11:18.880 --> 00:11:21.920 In other words, if the a_n's-- if one of variables depends 00:11:21.920 --> 00:11:25.160 on the others, they are related in terms of their motion. 00:11:25.160 --> 00:11:28.010 Keep in mind here, I just chose delta x_1. 00:11:28.010 --> 00:11:32.090 Analogous definition is held for f sub x sub 2, 00:11:32.090 --> 00:11:34.870 f sub x sub 3, et cetera. 00:11:34.870 --> 00:11:36.290 OK? 00:11:36.290 --> 00:11:39.800 I think again that once you see a few examples, 00:11:39.800 --> 00:11:41.220 the mystery vanishes. 00:11:41.220 --> 00:11:43.560 What this thing says in plain English 00:11:43.560 --> 00:11:46.500 is that when you have a function of several variables 00:11:46.500 --> 00:11:48.500 and you want to take the derivative with respect 00:11:48.500 --> 00:11:51.105 to one of those variables, pretend 00:11:51.105 --> 00:11:52.830 that every one of the other variables 00:11:52.830 --> 00:11:55.030 was being held constant. 00:11:55.030 --> 00:11:56.160 Let me give an example. 00:11:56.160 --> 00:11:59.280 Let w be f of x_1, x_2, x_3, where 00:11:59.280 --> 00:12:02.690 the specific f that I have in mind is obtained as follows. 00:12:02.690 --> 00:12:08.020 It's x_1 times x_2 times x_3 plus e to the x_1 power. 00:12:08.020 --> 00:12:11.310 Now, to take the derivative of this with respect to x_1, 00:12:11.310 --> 00:12:14.150 all that this says is treat x_2 and x_3 00:12:14.150 --> 00:12:15.600 as if they were constants. 00:12:15.600 --> 00:12:16.740 Well, look at it. 00:12:16.740 --> 00:12:20.370 If I treat x_2 and x_3 as if they were constants, 00:12:20.370 --> 00:12:23.980 if I differentiate this with respect to x_1, all I have left 00:12:23.980 --> 00:12:25.000 is what? 00:12:25.000 --> 00:12:28.350 This particular constant, which is x_2 times x_3. 00:12:28.350 --> 00:12:31.250 And the derivative of e to the x_1, with respect to x_1, 00:12:31.250 --> 00:12:33.250 is e to the x_1. 00:12:33.250 --> 00:12:38.060 Therefore, f sub x_1 in this case is x_2*x_3 plus e 00:12:38.060 --> 00:12:39.220 to the x_1. 00:12:39.220 --> 00:12:44.310 In particular, if I evaluate f sub x sub 1 at the three-tuple 00:12:44.310 --> 00:12:49.740 (1, 2, 3)-- namely, I replace x_1 by 1, x_2 by 2, x_3 by 3-- 00:12:49.740 --> 00:12:50.950 I obtain what? 00:12:50.950 --> 00:12:54.400 2 times 3 plus e to the first power. 00:12:54.400 --> 00:12:58.010 In other words, this is just a number, 6 plus e. 00:12:58.010 --> 00:13:00.160 On the other hand, if I had decided 00:13:00.160 --> 00:13:02.860 to differentiate this with respect to x_2, 00:13:02.860 --> 00:13:05.780 I would've treated x_1 and x_3 as constants. 00:13:05.780 --> 00:13:09.060 With x_1 and x_3 as constants, the derivative of this with 00:13:09.060 --> 00:13:13.370 respect to x_2 is just x_1*x_3. 00:13:13.370 --> 00:13:16.110 Treating x_1 as a constant, the derivative of e 00:13:16.110 --> 00:13:20.160 to the x_1 with respect to the variable x_3 is 0. 00:13:20.160 --> 00:13:22.220 Because the derivative of any constant is 0. 00:13:22.220 --> 00:13:25.450 Notice that this is not really a constant. 00:13:25.450 --> 00:13:28.150 It's a constant once I've fixed x_1 00:13:28.150 --> 00:13:31.120 at a particular value, which is what this particular definition 00:13:31.120 --> 00:13:33.180 says. 00:13:33.180 --> 00:13:39.360 In particular, f sub x sub 2 of (1, 2, 3) is 1 times 3, or 3. 00:13:39.360 --> 00:13:41.830 I should also mention that sometimes instead 00:13:41.830 --> 00:13:45.060 of writing f of x sub 1, we write something 00:13:45.060 --> 00:13:46.850 that looks like the regular derivative, 00:13:46.850 --> 00:13:49.510 only we make a funny kind of script d instead 00:13:49.510 --> 00:13:50.870 of an ordinary d. 00:13:50.870 --> 00:13:52.700 This is read "the partial derivative 00:13:52.700 --> 00:13:55.330 of w with respect to x_1." 00:13:55.330 --> 00:13:58.140 And the relationship between these two notations 00:13:58.140 --> 00:14:00.990 is rather similar to the relationship that 00:14:00.990 --> 00:14:05.900 exists between the notations dy/dx and f prime of x. 00:14:05.900 --> 00:14:10.680 At any rate, if we have the n independent variables x_1 up 00:14:10.680 --> 00:14:14.040 to x_n, and f is a function of those variables, 00:14:14.040 --> 00:14:17.770 f sub x sub 1, et cetera, f sub x sub n 00:14:17.770 --> 00:14:23.010 are called the partial derivatives of f with respect 00:14:23.010 --> 00:14:25.950 to x_1 up to x_n, respectively. 00:14:25.950 --> 00:14:27.860 In other words, this means what? 00:14:27.860 --> 00:14:30.630 The partial derivative of f with respect to x_1. 00:14:30.630 --> 00:14:31.560 That means what? 00:14:31.560 --> 00:14:35.660 Take the ordinary derivative as if x_1 were the only variable. 00:14:35.660 --> 00:14:38.580 This is the partial of f with respect to x sub n. 00:14:38.580 --> 00:14:41.090 That means take the ordinary derivative of f 00:14:41.090 --> 00:14:43.460 as if x sub n were the only variable. 00:14:43.460 --> 00:14:45.960 And you can always do this, provided that your variables 00:14:45.960 --> 00:14:48.660 are independent. 00:14:48.660 --> 00:14:52.490 Because our definitions so closely 00:14:52.490 --> 00:14:55.169 parallel the structure for what happened 00:14:55.169 --> 00:14:57.710 in the case of one independent variable-- including the limit 00:14:57.710 --> 00:15:01.090 theorems, the distance formulas, and what have you-- 00:15:01.090 --> 00:15:03.800 it turns out that the usual derivative properties still 00:15:03.800 --> 00:15:04.550 hold. 00:15:04.550 --> 00:15:08.100 For example, if w happens to be the function of the two 00:15:08.100 --> 00:15:13.020 independent variables e to the 3x plus y times sine 2x 00:15:13.020 --> 00:15:16.620 minus y, and I want to take the derivative of w with respect 00:15:16.620 --> 00:15:19.400 to x-- meaning differentiate this treating y 00:15:19.400 --> 00:15:22.930 as a constant-- notice the treating y as a constant 00:15:22.930 --> 00:15:27.490 gives me two functions of what? 00:15:27.490 --> 00:15:27.990 x. 00:15:27.990 --> 00:15:29.800 In other words, my function is a product 00:15:29.800 --> 00:15:32.650 of two functions, each of which depends on x. 00:15:32.650 --> 00:15:35.710 So I use the ordinary product rule to differentiate this. 00:15:35.710 --> 00:15:38.740 Namely, how do I use the product rule to take the partial of w 00:15:38.740 --> 00:15:39.840 with respect to x? 00:15:39.840 --> 00:15:42.820 I treat y as a constant and I differentiate 00:15:42.820 --> 00:15:44.440 as if x is the only variable. 00:15:44.440 --> 00:15:45.320 I say what? 00:15:45.320 --> 00:15:49.070 It's the first factor, e to the 3x plus y times 00:15:49.070 --> 00:15:52.280 the derivative of the second with respect to x, 00:15:52.280 --> 00:15:54.020 treating y as a constant. 00:15:54.020 --> 00:15:55.960 If I treat y as a constant, the derivative 00:15:55.960 --> 00:16:01.080 of sine of 2x minus y is cosine 2x minus y times 00:16:01.080 --> 00:16:03.820 the derivative of what's inside with respect to x. 00:16:03.820 --> 00:16:06.370 And that's just 2. 00:16:06.370 --> 00:16:08.340 Plus what? 00:16:08.340 --> 00:16:12.290 The second term times the derivative of the first. 00:16:12.290 --> 00:16:14.900 And the derivative of e to the 3x plus y 00:16:14.900 --> 00:16:18.070 with respect to x, treating y as a constant, 00:16:18.070 --> 00:16:22.610 is e to the 3x plus y times the derivative of 3x plus y, 00:16:22.610 --> 00:16:25.130 with respect to x, which is just 3. 00:16:25.130 --> 00:16:27.990 And so I have the partial of w with respect to x. 00:16:27.990 --> 00:16:30.410 In a similar way, I could have found the partial of w 00:16:30.410 --> 00:16:33.590 with respect to y, et cetera. 00:16:33.590 --> 00:16:37.870 I should now introduce one little problem 00:16:37.870 --> 00:16:41.671 that will bother you, maybe even after I help you with this. 00:16:41.671 --> 00:16:43.170 In fact, I guess what really crushes 00:16:43.170 --> 00:16:46.251 me is the student who comes up in a live class, after class, 00:16:46.251 --> 00:16:48.750 and says, "I understood it until I heard you lecture on it." 00:16:48.750 --> 00:16:50.980 I hope this doesn't cause that problem. 00:16:50.980 --> 00:16:53.590 But the danger is, I tell you, lookit, everything 00:16:53.590 --> 00:16:55.990 that happened for calculus of a single variable 00:16:55.990 --> 00:16:58.750 happens for calculus of several variables. 00:16:58.750 --> 00:17:00.880 Then all of a sudden, you'll find in our textbook, 00:17:00.880 --> 00:17:02.970 in every textbook, just about, they'll 00:17:02.970 --> 00:17:06.849 say things like, if you take the partial of the variable u 00:17:06.849 --> 00:17:09.230 with respect to x, you do not get 00:17:09.230 --> 00:17:12.470 the reciprocal of the partial of x with respect to u. 00:17:12.470 --> 00:17:17.869 In other words, remember, for one variable, dy/dx, right, 00:17:17.869 --> 00:17:21.220 was the reciprocal of dx/dy. 00:17:21.220 --> 00:17:23.369 They say it's not true in several variables. 00:17:23.369 --> 00:17:25.980 I have written in an accentuated question mark 00:17:25.980 --> 00:17:28.050 here because I want to explain something 00:17:28.050 --> 00:17:29.400 very important to you. 00:17:29.400 --> 00:17:31.370 And by the way, if you understand this, 00:17:31.370 --> 00:17:34.380 you're 90% of the way home free as far as 00:17:34.380 --> 00:17:37.850 understanding how calculus of several variables 00:17:37.850 --> 00:17:41.410 is used in most important applications. 00:17:41.410 --> 00:17:43.180 Let me take a very simple case. 00:17:43.180 --> 00:17:46.790 Let u equal x plus y, and let v equal x minus y, 00:17:46.790 --> 00:17:50.280 Now obviously, if u equals x plus y, the partial of u 00:17:50.280 --> 00:17:52.300 with respect to x, if we hold y constant-- 00:17:52.300 --> 00:17:54.940 we treat x as the only variable-- is 1. 00:17:54.940 --> 00:17:55.440 Right? 00:17:55.440 --> 00:17:58.160 The partial of u with respect to x is 1. 00:17:58.160 --> 00:18:02.030 On the other hand, notice that if I add these two equations, 00:18:02.030 --> 00:18:03.930 the y term drops out. 00:18:03.930 --> 00:18:05.220 I get what? 00:18:05.220 --> 00:18:10.110 2x equals u plus v, divide through by 2. 00:18:10.110 --> 00:18:15.320 That says x is equal to 1/2*u plus 1/2*v. 00:18:15.320 --> 00:18:17.820 Let me now take the partial of x with respect 00:18:17.820 --> 00:18:21.110 to u, holding v constant, treating u 00:18:21.110 --> 00:18:22.500 as the only variable. 00:18:22.500 --> 00:18:25.710 It's easy for me to see that the partial of x with respect to u 00:18:25.710 --> 00:18:26.880 is 1/2. 00:18:26.880 --> 00:18:29.840 And certainly, if I now compare these two, 00:18:29.840 --> 00:18:34.490 it should be clear that this is not the reciprocal of this, 00:18:34.490 --> 00:18:36.322 or vice versa. 00:18:36.322 --> 00:18:37.780 The point that I wanted to mention, 00:18:37.780 --> 00:18:39.120 though, was the following. 00:18:39.120 --> 00:18:41.900 That when, here, we said, take the partial of u 00:18:41.900 --> 00:18:44.830 with respect to x, we were assuming 00:18:44.830 --> 00:18:50.320 that the independent variables were x and y. 00:18:50.320 --> 00:18:53.700 And when we said, here, take the partial of x with respect 00:18:53.700 --> 00:18:55.760 to u, what did we hold constant? 00:18:55.760 --> 00:18:58.210 We held v constant. 00:18:58.210 --> 00:19:01.300 In other words-- I'm not going to keep this habit up very 00:19:01.300 --> 00:19:01.910 long. 00:19:01.910 --> 00:19:03.970 But just for the time being, I would 00:19:03.970 --> 00:19:07.790 like you to get used to the idea of putting in as a subscript 00:19:07.790 --> 00:19:10.410 the variables that are being held constant. 00:19:10.410 --> 00:19:12.950 You see, in the textbook when they 00:19:12.950 --> 00:19:14.990 say this is the case, what they really 00:19:14.990 --> 00:19:18.040 mean is, if you differentiate u with respect 00:19:18.040 --> 00:19:22.200 to x, holding y constant, that in general will not 00:19:22.200 --> 00:19:25.520 be the reciprocal of the derivative of x with respect 00:19:25.520 --> 00:19:27.870 to u, holding v constant. 00:19:27.870 --> 00:19:29.769 So I can cross out the question mark now. 00:19:29.769 --> 00:19:32.060 But the point is, notice that the variables that you're 00:19:32.060 --> 00:19:34.560 holding constant here are different. 00:19:34.560 --> 00:19:38.710 You see, suppose instead I took the partial of x with respect 00:19:38.710 --> 00:19:40.570 to u, holding y constant. 00:19:40.570 --> 00:19:44.920 In other words, suppose I solve for x in terms of u and y. 00:19:44.920 --> 00:19:47.580 That happens to be very easy to do here. 00:19:47.580 --> 00:19:50.970 Given that u equals x plus y, it follows immediately 00:19:50.970 --> 00:19:53.550 that x is equal to u minus y. 00:19:53.550 --> 00:19:56.870 Now, if I take the partial of x with respect to u, 00:19:56.870 --> 00:20:00.420 treating y as a constant-- so let 00:20:00.420 --> 00:20:03.760 me do that-- what do I get for an answer? 00:20:03.760 --> 00:20:06.330 I get 1 for an answer. 00:20:06.330 --> 00:20:08.900 In other words, the reciprocal of the partial 00:20:08.900 --> 00:20:13.360 of x with respect to u, treating y as a constant-- 00:20:13.360 --> 00:20:16.180 as the other independent variable-- 00:20:16.180 --> 00:20:18.120 is equal to the partial of u with respect 00:20:18.120 --> 00:20:21.820 to x when it's also y that's being considered 00:20:21.820 --> 00:20:23.320 as the other variable. 00:20:23.320 --> 00:20:27.110 When the variables outside here match up, the recipes work. 00:20:27.110 --> 00:20:28.710 Of course that leads to the question, 00:20:28.710 --> 00:20:31.740 then why do the textbooks tell you that this isn't true? 00:20:31.740 --> 00:20:34.910 Why do they pick this particular representation rather than 00:20:34.910 --> 00:20:35.990 this one? 00:20:35.990 --> 00:20:40.000 And I have a combination example and explanation here. 00:20:40.000 --> 00:20:41.160 And it's simply this. 00:20:41.160 --> 00:20:45.920 Let's suppose I was given that w equals e to the x plus y times 00:20:45.920 --> 00:20:48.260 cosine x minus y. 00:20:48.260 --> 00:20:50.860 It seems that since the only way the variable appears 00:20:50.860 --> 00:20:53.900 in the first factor is as x plus y, 00:20:53.900 --> 00:20:56.010 and the only way it appears in the second factor 00:20:56.010 --> 00:20:59.630 is in the form x minus y, a very natural substitution 00:20:59.630 --> 00:21:05.200 might be to say that u equal x plus y and v equal x minus y. 00:21:05.200 --> 00:21:08.410 In other words, I could either visualize w 00:21:08.410 --> 00:21:11.350 as being a function of x and y-- in which case, 00:21:11.350 --> 00:21:15.060 it would be e to the x plus y times cosine x minus y-- 00:21:15.060 --> 00:21:19.825 or, I could visualize w as being a simpler function of the two 00:21:19.825 --> 00:21:24.355 new variables u and v, where w is then just what? e 00:21:24.355 --> 00:21:26.370 to the u times cosine v. 00:21:26.370 --> 00:21:29.000 Notice again, that this formula here 00:21:29.000 --> 00:21:31.300 is much simpler than this one. 00:21:31.300 --> 00:21:32.510 This one says what? 00:21:32.510 --> 00:21:34.750 Take e to of the sum of these two, 00:21:34.750 --> 00:21:38.010 multiplied by the cosine of the difference of these two. 00:21:38.010 --> 00:21:39.540 And this one just says what? 00:21:39.540 --> 00:21:43.990 Take e to the first times the cosine of the second. 00:21:43.990 --> 00:21:47.440 But the point is, that you might make this change of variables 00:21:47.440 --> 00:21:49.660 to simplify your computations. 00:21:49.660 --> 00:21:51.970 The point is that either we would treat w 00:21:51.970 --> 00:21:56.620 as a function of x and y, or we would have treated w 00:21:56.620 --> 00:21:58.280 as a function of u and v. 00:21:58.280 --> 00:22:02.420 And by the way, notice here my use of f and g. 00:22:02.420 --> 00:22:06.570 Notice that w is a different function of x and y 00:22:06.570 --> 00:22:09.770 than it is of u and v. But this notation says what? 00:22:09.770 --> 00:22:12.730 We can consider w either in terms of x and y, 00:22:12.730 --> 00:22:18.320 or in terms of u and v. Hardly ever would we consider w 00:22:18.320 --> 00:22:19.630 as a function of u and y. 00:22:19.630 --> 00:22:22.770 In other words, either we would change the variables or we 00:22:22.770 --> 00:22:23.610 don't. 00:22:23.610 --> 00:22:25.890 Or, for example, in terms of polar versus Cartesian 00:22:25.890 --> 00:22:29.740 coordinates, either we use x and y or we use r and theta. 00:22:29.740 --> 00:22:32.040 We don't usually use r and x. 00:22:32.040 --> 00:22:35.240 In other words, going back to the textbook example again, 00:22:35.240 --> 00:22:37.510 if we look at this particular situation here, 00:22:37.510 --> 00:22:39.600 all the book is saying is, when you're 00:22:39.600 --> 00:22:41.480 differentiating with respect to n, 00:22:41.480 --> 00:22:44.310 you usually mean that y is the variable being held constant. 00:22:44.310 --> 00:22:46.275 When we differentiate with respect to u, 00:22:46.275 --> 00:22:49.120 we usually means that v is the variable that's 00:22:49.120 --> 00:22:50.370 being held constant. 00:22:50.370 --> 00:22:54.020 Again, more of this is said in our exercises. 00:22:54.020 --> 00:22:55.790 But for the time being, I want you 00:22:55.790 --> 00:22:59.930 to see how important it is, when you have several variables, 00:22:59.930 --> 00:23:02.820 to keep track of which are the dependent variables, which 00:23:02.820 --> 00:23:06.500 are the independent variables, and how they're coupled. 00:23:06.500 --> 00:23:09.490 I said earlier that one of the nice things about picking n 00:23:09.490 --> 00:23:13.760 equals 2 is that you can draw a nice picture of the situation. 00:23:13.760 --> 00:23:16.340 And what I thought you might like to see is the following. 00:23:16.340 --> 00:23:20.310 Let's suppose you have that w is some function of x and y. 00:23:20.310 --> 00:23:23.420 What that means is, we can locate-- for a given value of x 00:23:23.420 --> 00:23:24.580 and a given value of y. 00:23:24.580 --> 00:23:28.170 In other words, for x equals x_0, y equals y_0, 00:23:28.170 --> 00:23:32.800 we can think of the point (x_0, y_0) as being in the xy-plane. 00:23:32.800 --> 00:23:38.050 w, which is f of x_0, y_0, is just the height 00:23:38.050 --> 00:23:39.930 to this particular surface. 00:23:39.930 --> 00:23:46.200 In other words, if I let f of x_0, y_0 be called w_0, 00:23:46.200 --> 00:23:51.670 notice that the function f of x, y, at the point x equals x_0, 00:23:51.670 --> 00:23:55.940 y equals y_0, graphically corresponds to the point 00:23:55.940 --> 00:24:01.060 whose coordinates are x_0, y_0, and w_0. 00:24:01.060 --> 00:24:02.530 OK? 00:24:02.530 --> 00:24:03.790 This is now a surface. 00:24:03.790 --> 00:24:08.600 And in the same way that we use tangent lines to replace curves 00:24:08.600 --> 00:24:11.260 when we had one independent variable, in two 00:24:11.260 --> 00:24:15.340 independent variables, we use tangent planes 00:24:15.340 --> 00:24:16.920 to replace surfaces. 00:24:16.920 --> 00:24:20.190 The question that comes up is, how do you get a tangent plane? 00:24:20.190 --> 00:24:23.000 And what does this have to do with the partial derivatives? 00:24:23.000 --> 00:24:25.000 And what I would like show you is the following. 00:24:25.000 --> 00:24:28.640 First of all, one very natural way 00:24:28.640 --> 00:24:32.390 of intersecting this surface with a plane-- in other words, 00:24:32.390 --> 00:24:35.330 when you take the partial of w with respect 00:24:35.330 --> 00:24:40.330 to y, you're holding x constant, you're saying, let x equal x_0. 00:24:40.330 --> 00:24:44.340 Notice that in three-dimensional space, x equals x_0 00:24:44.340 --> 00:24:47.550 is the equation of this plane. 00:24:47.550 --> 00:24:51.380 And this plane-- see, what is it the plane x equals x_0? 00:24:51.380 --> 00:24:53.710 It's the plane that goes through the line x equals 00:24:53.710 --> 00:24:57.540 x_0 parallel to the xy-plane. 00:24:57.540 --> 00:25:02.327 This plane intersects my surface in a particular curve. 00:25:02.327 --> 00:25:06.150 That curve passes through P_0. 00:25:06.150 --> 00:25:09.340 I can talk about the slope of the curve 00:25:09.340 --> 00:25:11.620 at that particular point. 00:25:11.620 --> 00:25:12.500 OK? 00:25:12.500 --> 00:25:16.620 In a similar way, I could've sliced the surface 00:25:16.620 --> 00:25:19.540 by the plane y equals y_0. 00:25:19.540 --> 00:25:23.620 In other words, hold y constant. 00:25:23.620 --> 00:25:26.710 Well, you see, to hold y constant, that 00:25:26.710 --> 00:25:29.850 means I, again, draw my plane. 00:25:29.850 --> 00:25:32.330 It intersects the surface in a different curve, 00:25:32.330 --> 00:25:35.630 but that curve must also go through the point P_0. 00:25:35.630 --> 00:25:37.930 That's the same P_0 that's over here, 00:25:37.930 --> 00:25:41.880 because after all, the point on the surface that is directly 00:25:41.880 --> 00:25:44.704 above (x_0, y_0) is P_0. 00:25:44.704 --> 00:25:46.370 And that point doesn't change, no matter 00:25:46.370 --> 00:25:48.460 what plane you slice this thing with. 00:25:48.460 --> 00:25:49.530 OK? 00:25:49.530 --> 00:25:53.710 And so again, I can talk about the slope of the tangent 00:25:53.710 --> 00:25:56.160 to this particular curve of intersection. 00:25:56.160 --> 00:26:00.070 By the way, one very brief aside before I continue on with this. 00:26:00.070 --> 00:26:03.180 One of the things that makes functions of several variables 00:26:03.180 --> 00:26:06.170 so difficult is-- I would like you to observe, 00:26:06.170 --> 00:26:08.190 because this will become the backbone 00:26:08.190 --> 00:26:11.870 of our future investigations-- that these two 00:26:11.870 --> 00:26:15.740 particular planes that I drew, to intersect my surface, 00:26:15.740 --> 00:26:17.570 were very special planes. 00:26:17.570 --> 00:26:19.720 One of them was parallel-- they both 00:26:19.720 --> 00:26:21.890 went through the point (x_0, y_0), 00:26:21.890 --> 00:26:25.470 but one happened to be parallel to the wy-plane 00:26:25.470 --> 00:26:29.300 and the other one happened to be parallel to the wx-plane. 00:26:29.300 --> 00:26:32.310 Notice that, in general, I could have 00:26:32.310 --> 00:26:37.530 passed infinitely many different planes through (x_0, y_0). 00:26:37.530 --> 00:26:41.610 Each of which would have intersected the surface 00:26:41.610 --> 00:26:43.020 in a different curve. 00:26:43.020 --> 00:26:46.540 And this is what leads later to the more generalized concept 00:26:46.540 --> 00:26:49.530 of directional derivatives that we will talk about in more 00:26:49.530 --> 00:26:51.800 detail in our future lectures. 00:26:51.800 --> 00:26:53.930 But I simply mention the word now 00:26:53.930 --> 00:26:56.340 because in the reading material in this assignment, 00:26:56.340 --> 00:26:59.310 some mention is made of directional derivatives. 00:26:59.310 --> 00:27:01.690 In other words, I wanted to point out 00:27:01.690 --> 00:27:03.670 that to take partial derivatives, 00:27:03.670 --> 00:27:06.630 we are interested in two special directions, 00:27:06.630 --> 00:27:08.557 even though there are other directions. 00:27:08.557 --> 00:27:10.140 The question that comes up, of course, 00:27:10.140 --> 00:27:12.180 is where do the partials come in here? 00:27:12.180 --> 00:27:19.370 And secondly, knowing where partials come up in here, how 00:27:19.370 --> 00:27:21.590 do we find the equation of a tangent plane? 00:27:21.590 --> 00:27:23.300 And all I would like you to see from here 00:27:23.300 --> 00:27:26.820 is that I'm going to use the same old technique as always. 00:27:26.820 --> 00:27:30.820 Given these two tangent lines, I'm going to vectorize them. 00:27:30.820 --> 00:27:34.460 I am then going to find the equation of a plane that passes 00:27:34.460 --> 00:27:37.130 through those two vectors. 00:27:37.130 --> 00:27:39.310 To do that, I'm going to have to find 00:27:39.310 --> 00:27:41.971 the normal to that plane, et cetera. 00:27:41.971 --> 00:27:43.720 And I'm going to wind up with the equation 00:27:43.720 --> 00:27:45.030 of a tangent plane. 00:27:45.030 --> 00:27:46.900 In more slow motion, all I'm saying 00:27:46.900 --> 00:27:50.690 is if I now take this picture and draw it over here-- notice 00:27:50.690 --> 00:27:54.070 the representation-- it looks as if the curve that I get 00:27:54.070 --> 00:27:55.950 is in the wy-plane. 00:27:55.950 --> 00:27:57.710 It's actually in a plane parallel 00:27:57.710 --> 00:28:02.240 to the wy-plane characterized by x equals x sub 0. 00:28:02.240 --> 00:28:04.610 And by the way, notice even though x 00:28:04.610 --> 00:28:09.060 equals x_0 was chosen so that x_0 was fixed, 00:28:09.060 --> 00:28:13.970 I hope it's clear to you that if I let x sub 0 vary and I picked 00:28:13.970 --> 00:28:17.420 different values of x sub 0, the curve that I get here 00:28:17.420 --> 00:28:19.680 will, in general, differ. 00:28:19.680 --> 00:28:21.400 You see, what I'm saying is, if I 00:28:21.400 --> 00:28:24.740 take slices parallel to the wy-plane 00:28:24.740 --> 00:28:26.970 for a particular surface, the slices 00:28:26.970 --> 00:28:29.930 that I get-- the shape-- will depend on what 00:28:29.930 --> 00:28:32.380 the particular value of x_0 is. 00:28:32.380 --> 00:28:34.990 But that's not the crucial issue right now. 00:28:34.990 --> 00:28:36.760 The point that I'm driving at is, 00:28:36.760 --> 00:28:41.650 if I if I vectorize the tangent line and call that v_1, 00:28:41.650 --> 00:28:44.610 what does that tangent line look like? 00:28:44.610 --> 00:28:47.405 First of all, let's see what its slope is. 00:28:47.405 --> 00:28:50.910 Now again, if you didn't see this over here 00:28:50.910 --> 00:28:53.310 and all I showed was this diagram, 00:28:53.310 --> 00:28:55.840 you would say, hey, that slope is just the derivative 00:28:55.840 --> 00:28:58.240 of w with respect to y. 00:28:58.240 --> 00:29:01.050 But we write that the partial of w with respect 00:29:01.050 --> 00:29:05.020 to y in deference to the fact that w is not a function of y 00:29:05.020 --> 00:29:08.320 alone-- that w depends on both x and y, 00:29:08.320 --> 00:29:11.350 and the reason that we got this curve was that we fixed 00:29:11.350 --> 00:29:13.770 x at some particular value. 00:29:13.770 --> 00:29:16.060 So this is really what? 00:29:16.060 --> 00:29:19.510 Not the derivative of w with respect to y evaluated at y 00:29:19.510 --> 00:29:20.890 equals y_0. 00:29:20.890 --> 00:29:25.795 It's the partial of w with respect to y evaluated at what? 00:29:25.795 --> 00:29:28.980 x_0 comma y_0. 00:29:28.980 --> 00:29:32.340 All right, at any rate, now that we have the slope, 00:29:32.340 --> 00:29:34.240 notice that the slope of a vector 00:29:34.240 --> 00:29:37.700 can always be viewed as the k component-- 00:29:37.700 --> 00:29:41.890 in this case, the k component divided by the j component. 00:29:41.890 --> 00:29:44.760 If I make the j component 1, the slope 00:29:44.760 --> 00:29:46.720 will just be the k component. 00:29:46.720 --> 00:29:49.120 In other words, a vector which is 00:29:49.120 --> 00:29:51.930 tangent to the curve-- and one that's easy to write down-- 00:29:51.930 --> 00:29:56.770 is j plus the partial of w with respect to y, evaluated 00:29:56.770 --> 00:29:59.340 at (x_0, y_0), times k. 00:29:59.340 --> 00:30:04.270 And notice, by the way, that this is a number. 00:30:04.270 --> 00:30:06.010 That once I take the partial derivative 00:30:06.010 --> 00:30:08.224 and evaluate it at a point, I have a number. 00:30:08.224 --> 00:30:08.890 So this is what? 00:30:08.890 --> 00:30:12.470 A constant vector once x_0 and y_0 are fixed. 00:30:12.470 --> 00:30:17.740 In a similar way, my vector v_2, which is for my curve parallel 00:30:17.740 --> 00:30:23.690 to the wx-plane-- the slope of that curve at the point P_0 00:30:23.690 --> 00:30:27.300 is not dw/dx, it's the partial of w with respect 00:30:27.300 --> 00:30:30.340 to x, evaluated (x_0, y_0). 00:30:30.340 --> 00:30:31.440 Again, why? 00:30:31.440 --> 00:30:34.830 Because, even though this looks like w is a function of x alone 00:30:34.830 --> 00:30:37.450 here, it was only that way because we 00:30:37.450 --> 00:30:39.500 took the second variable and, in a sense, 00:30:39.500 --> 00:30:41.990 froze it at the value y_0. 00:30:41.990 --> 00:30:44.450 At any rate, in an analogous manner, 00:30:44.450 --> 00:30:47.080 v_2 turns out to be what vector? 00:30:47.080 --> 00:30:50.450 It's i plus the partial of w with respect 00:30:50.450 --> 00:30:53.760 to x evaluated (x_0, y_0) times k. 00:30:53.760 --> 00:30:57.090 In other words, the slope is the partial of w with respect to x. 00:30:57.090 --> 00:30:59.005 And it's in the ik-plane. 00:30:59.005 --> 00:31:01.460 In other words, the xw-plane. 00:31:01.460 --> 00:31:04.130 Now that I have my two vectors, to find 00:31:04.130 --> 00:31:07.310 the plane that passes through them, all I need is what? 00:31:07.310 --> 00:31:10.460 The normal vector and a point in the plane. 00:31:10.460 --> 00:31:14.240 Well, one of the nice things about having studied 00:31:14.240 --> 00:31:16.500 the cross product here, is that I can now 00:31:16.500 --> 00:31:19.250 take the cross product of v_1 and v_2. 00:31:19.250 --> 00:31:21.710 Sparing you the details, remember what I do here. 00:31:21.710 --> 00:31:26.580 I just write down i, j, k. v_1 has these components, 00:31:26.580 --> 00:31:29.490 v_2 has these components. 00:31:29.490 --> 00:31:34.050 I now expand this determinant in the usual way. 00:31:34.050 --> 00:31:35.490 And I get what? 00:31:35.490 --> 00:31:37.209 This particular vector. 00:31:37.209 --> 00:31:39.125 In other words, the i component is the partial 00:31:39.125 --> 00:31:40.730 of w with respect to x. 00:31:40.730 --> 00:31:43.660 The j component is a partial of w with respect to y. 00:31:43.660 --> 00:31:46.270 Both evaluated at the point (x_0, y_0). 00:31:46.270 --> 00:31:48.640 And the k component is minus 1. 00:31:48.640 --> 00:31:52.310 Now, to find the equation of a tangent plane, all I have to do 00:31:52.310 --> 00:31:53.750 is what? 00:31:53.750 --> 00:31:55.550 Take the standard form of the plane. 00:31:55.550 --> 00:31:58.910 In other words, the coefficients will be these. 00:31:58.910 --> 00:32:03.950 And then I take (x_0, y_0, w_0), which is a point in my plane. 00:32:03.950 --> 00:32:06.910 And then the equation of the tangent plane 00:32:06.910 --> 00:32:10.580 becomes, quite simply, this expression here. 00:32:10.580 --> 00:32:13.290 And in fact, if I rewrite that, notice 00:32:13.290 --> 00:32:17.180 I can transpose the w minus w_0 term. 00:32:17.180 --> 00:32:19.260 That is the change in w. 00:32:19.260 --> 00:32:21.920 But to remind us that we're in the tangent plane, 00:32:21.920 --> 00:32:26.830 we don't write this as delta w, we write it as delta w sub tan. 00:32:26.830 --> 00:32:29.950 And notice that the change in w to the tangent plane 00:32:29.950 --> 00:32:33.380 is just the partial of w with respect to x times delta x, 00:32:33.380 --> 00:32:37.290 plus the partial of w with respect to y times delta y. 00:32:37.290 --> 00:32:40.570 And hopefully, this starts to seem familiar to you. 00:32:40.570 --> 00:32:44.310 This should almost look like the equation of delta y tan, 00:32:44.310 --> 00:32:46.060 only in two dimensions. 00:32:46.060 --> 00:32:49.040 Remember when we had delta y tan in part one of the course? 00:32:49.040 --> 00:32:51.160 See, what this really says is what? 00:32:51.160 --> 00:32:54.390 This is the change in w with respect to x, multiplied 00:32:54.390 --> 00:32:56.050 by the total change in x. 00:32:56.050 --> 00:32:58.150 In other words, this term is what? 00:32:58.150 --> 00:33:02.380 It's the change in w due to the change in x alone. 00:33:02.380 --> 00:33:05.450 This term is the change in w, with respect to y, 00:33:05.450 --> 00:33:07.510 times the total change in y. 00:33:07.510 --> 00:33:09.720 So this term tells you the change 00:33:09.720 --> 00:33:12.630 in w due to the change in y alone. 00:33:12.630 --> 00:33:15.970 And if you add these two up, since x and y are independent, 00:33:15.970 --> 00:33:18.514 this should give you the total change in w. 00:33:18.514 --> 00:33:20.930 The reason, of course, that you don't get the total change 00:33:20.930 --> 00:33:24.320 in w, but rather, delta w tan, is the fact 00:33:24.320 --> 00:33:27.160 that if these expressions here are not 00:33:27.160 --> 00:33:31.380 constant numbers as delta x and delta y ore varying, 00:33:31.380 --> 00:33:33.410 these numbers here are varying. 00:33:33.410 --> 00:33:34.800 These are variables. 00:33:34.800 --> 00:33:38.460 But you see, once you fix these at these values, 00:33:38.460 --> 00:33:42.270 these become numbers, and now you know what delta w tan is. 00:33:42.270 --> 00:33:44.640 Now, in some of the homework problems 00:33:44.640 --> 00:33:46.350 in the text, what we're going to do 00:33:46.350 --> 00:33:48.610 is just practice the geometry itself. 00:33:48.610 --> 00:33:52.280 We're going to find equations of tangent planes. 00:33:52.280 --> 00:33:55.430 The thing that's very important to us in future lectures, 00:33:55.430 --> 00:33:57.810 from a structural point of view, is 00:33:57.810 --> 00:34:00.320 that the equation of the tangent plane-- in other words, 00:34:00.320 --> 00:34:05.040 the equation for delta w tan-- is far more 00:34:05.040 --> 00:34:09.260 simple to handle than the equation for the true delta w. 00:34:09.260 --> 00:34:11.350 You see, the equation for delta w tan 00:34:11.350 --> 00:34:15.020 just has delta x and delta y appearing to the first power. 00:34:15.020 --> 00:34:17.929 Whereas, if you try to find delta w exactly 00:34:17.929 --> 00:34:20.210 for an arbitrary function of x and y, 00:34:20.210 --> 00:34:22.840 this can become a very, very messy thing. 00:34:22.840 --> 00:34:26.980 In short, delta w tan will play for two 00:34:26.980 --> 00:34:31.210 and more independent variables the same role 00:34:31.210 --> 00:34:33.620 that delta y sub tan played for us 00:34:33.620 --> 00:34:37.360 in part one of this course as a differential. 00:34:37.360 --> 00:34:40.280 And what we'll be discussing next, or at least 00:34:40.280 --> 00:34:42.810 trying to get at the root of, is just 00:34:42.810 --> 00:34:45.170 what does play the role of a differential 00:34:45.170 --> 00:34:48.170 when you're dealing with calculus of several variables? 00:34:48.170 --> 00:34:51.889 And what we're going to find is that, again, the computations 00:34:51.889 --> 00:34:55.100 become messy enough so that even though the structure stays 00:34:55.100 --> 00:34:58.070 fairly nice, there are some wrinkles that come up 00:34:58.070 --> 00:35:01.830 that will take us a considerable amount of time to untangle. 00:35:01.830 --> 00:35:04.620 But we'll worry about that when the time comes. 00:35:04.620 --> 00:35:06.560 And so until that time, goodbye. 00:35:11.800 --> 00:35:14.180 Funding for the publication of this video 00:35:14.180 --> 00:35:19.050 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:35:19.050 --> 00:35:23.230 Help OCW continue to provide free and open access to MIT 00:35:23.230 --> 00:35:27.640 courses by making a donation at ocw.mit.edu/donate.