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 --> 00:00:17.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.210 --> 00:00:18.793 at ocw.mit.edu. 00:00:35.860 --> 00:00:36.850 HERBERT GROSS: Hi. 00:00:36.850 --> 00:00:41.060 Today we do a somewhat computational bit. 00:00:41.060 --> 00:00:45.240 And actually, the lecture for today 00:00:45.240 --> 00:00:47.680 is not nearly as difficult, once you 00:00:47.680 --> 00:00:50.320 get through the maze of symbolism, 00:00:50.320 --> 00:00:53.940 as it is to apply the material. 00:00:53.940 --> 00:00:57.320 In other words, we're going to devote the next two 00:00:57.320 --> 00:01:01.190 units of our course to this particular topic, which 00:01:01.190 --> 00:01:03.060 is known as the chain rule. 00:01:03.060 --> 00:01:07.870 But we'll give one lecture to cover both units. 00:01:07.870 --> 00:01:11.100 And again, the idea is that it's not so much 00:01:11.100 --> 00:01:14.780 that the concept becomes more difficult as much as it 00:01:14.780 --> 00:01:17.820 is that you must develop a certain amount 00:01:17.820 --> 00:01:22.650 of dexterity keeping track of the various partial derivatives 00:01:22.650 --> 00:01:25.090 and the like. 00:01:25.090 --> 00:01:27.330 At any rate, maybe I think the best way 00:01:27.330 --> 00:01:31.120 is to just barge into a hypothetical situation 00:01:31.120 --> 00:01:34.060 and see what the situation really is. 00:01:34.060 --> 00:01:37.610 The idea is essentially the following. 00:01:37.610 --> 00:01:40.150 We're given some function, say, w. 00:01:40.150 --> 00:01:43.100 w is a function of, say, the three independent variables 00:01:43.100 --> 00:01:45.010 x, y, and z. 00:01:45.010 --> 00:01:47.020 Now, for some reason or other, which 00:01:47.020 --> 00:01:49.670 we won't worry about right now, it 00:01:49.670 --> 00:01:53.880 turns out that x, y, and z are, in turn, 00:01:53.880 --> 00:01:57.720 conveniently expressible in terms of the two variables 00:01:57.720 --> 00:01:58.915 r and s. 00:01:58.915 --> 00:02:01.980 In fact, if you want a physical interpretation of this, 00:02:01.980 --> 00:02:06.260 you can think of if x, y, and z are functions 00:02:06.260 --> 00:02:09.050 of the two independent variables r and s, 00:02:09.050 --> 00:02:11.780 that means that we have two degrees of freedom. 00:02:11.780 --> 00:02:15.450 So we may think of this as parametrically representing 00:02:15.450 --> 00:02:17.930 the equation of a surface. 00:02:17.930 --> 00:02:19.840 And what we're talking about here 00:02:19.840 --> 00:02:24.010 is w being a function of something in space and asking, 00:02:24.010 --> 00:02:27.900 what does w look like when you restrict your space 00:02:27.900 --> 00:02:29.300 to a particular surface? 00:02:29.300 --> 00:02:31.700 I mean, that's just a geometrical interpretation 00:02:31.700 --> 00:02:33.050 that one could talk about. 00:02:33.050 --> 00:02:34.350 But the idea is the following. 00:02:34.350 --> 00:02:38.730 After all, if w depends on x, y, and z, and x, y, 00:02:38.730 --> 00:02:43.100 and z each depend on r and s, in particular then, 00:02:43.100 --> 00:02:46.660 it's clear that w itself is some function of r and s, 00:02:46.660 --> 00:02:49.500 where, again, I use the usual notation 00:02:49.500 --> 00:02:52.090 of using a g here rather than the f 00:02:52.090 --> 00:02:57.090 up here to indicate that the relationship between r and s 00:02:57.090 --> 00:02:59.900 which specifies w, may very well be 00:02:59.900 --> 00:03:02.930 a different algebraic relationship than that which 00:03:02.930 --> 00:03:05.820 relates x, y, and z to give w. 00:03:05.820 --> 00:03:08.750 But the point that we have in mind is the following. 00:03:08.750 --> 00:03:10.750 Given that w is a function of x, y, 00:03:10.750 --> 00:03:15.170 and z, given that x, y, and z are functions of r and s, 00:03:15.170 --> 00:03:17.560 hence w is a function of r and s. 00:03:17.560 --> 00:03:20.930 The question that we ask in calculus of several variables 00:03:20.930 --> 00:03:24.910 is, first of all, if we can be sure that these were all 00:03:24.910 --> 00:03:28.060 continuously differentiable functions, 00:03:28.060 --> 00:03:32.190 can we be sure that w will be a continuously differentiable 00:03:32.190 --> 00:03:33.310 function of r and s? 00:03:33.310 --> 00:03:35.290 That's the first question. 00:03:35.290 --> 00:03:38.470 And the second question is, OK, assuming 00:03:38.470 --> 00:03:41.490 that the answer to the first question is in the affirmative, 00:03:41.490 --> 00:03:44.890 that w is a continuously differentiable function of r 00:03:44.890 --> 00:03:49.110 and s, how could we compute, for example, the partial of w 00:03:49.110 --> 00:03:53.940 with respect to r, knowing all of the so-called obvious 00:03:53.940 --> 00:03:55.330 other partial derivatives? 00:03:55.330 --> 00:03:56.590 What do I mean by that? 00:03:56.590 --> 00:04:00.240 Well, what I mean is if you were to look just at this equation, 00:04:00.240 --> 00:04:01.960 just looking at this equation, what 00:04:01.960 --> 00:04:04.060 are the obvious partial derivatives to take? 00:04:04.060 --> 00:04:05.810 You say, well, we'll take the partial of w 00:04:05.810 --> 00:04:09.190 with respect to x, the partial of w with respect to y, 00:04:09.190 --> 00:04:11.790 and the partial of w with respect to z. 00:04:11.790 --> 00:04:14.180 And if you were to look, say, at this equation, 00:04:14.180 --> 00:04:17.050 the natural thing to ask is, what is the partial 00:04:17.050 --> 00:04:18.350 of x with respect to r? 00:04:18.350 --> 00:04:21.029 What is the partial of x with respect to s? 00:04:21.029 --> 00:04:22.010 Et cetera. 00:04:22.010 --> 00:04:23.830 In other words, what we're saying is, 00:04:23.830 --> 00:04:27.580 in this particular problem, we would like to figure out how, 00:04:27.580 --> 00:04:30.410 for example, to compute the partial of w with respect 00:04:30.410 --> 00:04:34.010 to r, knowing that we have at our disposal 00:04:34.010 --> 00:04:38.390 the partials of w with respect to x, y, and z; 00:04:38.390 --> 00:04:43.810 the partials of x, y, and z with respect to r; et cetera; 00:04:43.810 --> 00:04:45.950 meaning we also have the partials of x, y, 00:04:45.950 --> 00:04:48.310 and z with respect to theta. 00:04:48.310 --> 00:04:51.370 Before I go any further, notice, by the way, 00:04:51.370 --> 00:04:53.930 that if I left out that phrase that we were talking 00:04:53.930 --> 00:04:57.120 about in our last lecture, "continuously differentiable," 00:04:57.120 --> 00:04:59.370 notice that all of this would make sense, 00:04:59.370 --> 00:05:02.330 provided that the derivatives existed. 00:05:02.330 --> 00:05:05.010 There was no place here do I make any statement 00:05:05.010 --> 00:05:09.090 that the partials have to not only exist but be continuous. 00:05:09.090 --> 00:05:10.270 I never say that at all. 00:05:10.270 --> 00:05:11.720 This is what the problem is. 00:05:11.720 --> 00:05:13.490 I would like to use the chain rule. 00:05:13.490 --> 00:05:15.880 Do you see why it's called the chain rule? 00:05:15.880 --> 00:05:18.410 w is a function of x, y, and z. 00:05:18.410 --> 00:05:21.400 x, y, and z are each functions of r and s. 00:05:21.400 --> 00:05:24.510 Now, what I claim is that not only is it possible 00:05:24.510 --> 00:05:27.830 to do this but the recipe for doing 00:05:27.830 --> 00:05:32.130 this is a very, very suggestive thing, one which is very, very 00:05:32.130 --> 00:05:35.930 easy to remember, once you see how it's put together. 00:05:35.930 --> 00:05:37.590 If you don't see how it's put together, 00:05:37.590 --> 00:05:39.580 the thing is just a mess-- namely, 00:05:39.580 --> 00:05:42.351 the claim is that the partial of w with respect to r 00:05:42.351 --> 00:05:44.850 is the partial-- I'll just read it to you-- the partial of w 00:05:44.850 --> 00:05:48.210 with respect to x times the partial of x with respect to r, 00:05:48.210 --> 00:05:49.960 plus the partial of w with respect 00:05:49.960 --> 00:05:53.120 to y times the partial of y with respect to r, 00:05:53.120 --> 00:05:54.820 plus the partial of w with respect 00:05:54.820 --> 00:05:57.520 to z times the partial of z with respect to r. 00:05:57.520 --> 00:05:59.640 And as I say, if you try to memorize that, 00:05:59.640 --> 00:06:01.670 it's a very, very nasty business. 00:06:01.670 --> 00:06:03.905 But let's look at this in three separate pieces. 00:06:09.910 --> 00:06:13.390 In a way, can you sense that this 00:06:13.390 --> 00:06:17.260 is nothing more than the change in w with respect 00:06:17.260 --> 00:06:20.410 to r due to the change in x alone? 00:06:20.410 --> 00:06:22.110 In other words, you're taking here what? 00:06:22.110 --> 00:06:25.970 The change in w due to x and multiplying that 00:06:25.970 --> 00:06:28.490 by the change in x with respect to r. 00:06:28.490 --> 00:06:32.270 So this is the contribution of the change in w with respect 00:06:32.270 --> 00:06:35.490 to r due to x alone. 00:06:35.490 --> 00:06:38.590 On the other hand, this is the partial of w with respect 00:06:38.590 --> 00:06:41.840 to r due to the change in y alone. 00:06:41.840 --> 00:06:43.760 And this is the partial of w with respect 00:06:43.760 --> 00:06:46.670 to r due to the change in z alone. 00:06:46.670 --> 00:06:49.630 And since x, y, and z are independent, 00:06:49.630 --> 00:06:52.990 the change in x, the change in y, and the change in z 00:06:52.990 --> 00:06:54.900 are also independent variables. 00:06:54.900 --> 00:06:58.150 Consequently, it seems reasonable to assume 00:06:58.150 --> 00:07:01.840 that to find the total change of w with respect to r, 00:07:01.840 --> 00:07:04.990 we just add up all of the partial contributions. 00:07:04.990 --> 00:07:07.700 Namely, we take the partial of w with respect 00:07:07.700 --> 00:07:11.570 to r due to x alone, add on to that the partial of w 00:07:11.570 --> 00:07:14.070 with respect to r due to y alone, 00:07:14.070 --> 00:07:16.140 add on to that the partial of w with respect 00:07:16.140 --> 00:07:19.940 to r due to z alone, and that that sum should 00:07:19.940 --> 00:07:23.810 be the total change in w with respect r, 00:07:23.810 --> 00:07:25.120 treating s as a constant. 00:07:25.120 --> 00:07:29.000 And by the way, let me point out a pitfall with this notation. 00:07:29.000 --> 00:07:31.980 We're so used to using fractional notation here. 00:07:31.980 --> 00:07:34.350 Have you noticed that if you're not careful here, 00:07:34.350 --> 00:07:37.400 you're almost tempted to cancel-- 00:07:37.400 --> 00:07:40.077 I don't want to write this, because you'll think that it's 00:07:40.077 --> 00:07:41.160 the right way of doing it. 00:07:41.160 --> 00:07:43.620 But see if we say, let's cancel the partials with respect 00:07:43.620 --> 00:07:46.300 to x here, let's cancel the partials with respect 00:07:46.300 --> 00:07:49.180 to y here, and let's cancel the partials with respect 00:07:49.180 --> 00:07:50.200 to z here. 00:07:50.200 --> 00:07:53.450 By the way, if you did that, notice what you would get 00:07:53.450 --> 00:07:57.590 is the contradiction that the partial of w with respect to r 00:07:57.590 --> 00:07:59.750 is equal to the partial of w with respect 00:07:59.750 --> 00:08:02.320 to r, plus the partial of w with respect 00:08:02.320 --> 00:08:04.935 to r, plus the partial of w with respect to r. 00:08:04.935 --> 00:08:06.310 In other words, it seems that you 00:08:06.310 --> 00:08:09.340 would get that the partial of w with respect to r 00:08:09.340 --> 00:08:12.150 is always three times itself, which 00:08:12.150 --> 00:08:15.080 is, I hope, a glaring enough contradiction so I don't 00:08:15.080 --> 00:08:18.500 have to go into any more detail about the contradiction part. 00:08:18.500 --> 00:08:26.460 Notice again, though, why I have made such a fetish 00:08:26.460 --> 00:08:29.090 over labeling the variables. 00:08:29.090 --> 00:08:31.440 Notice that when you're taking the partial of w 00:08:31.440 --> 00:08:35.919 with respect to x, you're assuming that y and z 00:08:35.919 --> 00:08:38.490 are the variables that are being held constant. 00:08:38.490 --> 00:08:41.620 And when you're taking the partial of x with respect to r, 00:08:41.620 --> 00:08:44.900 it's s that you're assuming is being held constant. 00:08:44.900 --> 00:08:47.080 And as soon as you look at these subscripts here, 00:08:47.080 --> 00:08:50.120 somehow or other that should put you on your guard 00:08:50.120 --> 00:08:53.070 to be careful about crossing out because, after all, 00:08:53.070 --> 00:08:55.620 the changes are being made with respect 00:08:55.620 --> 00:08:57.660 to different sets of variables. 00:08:57.660 --> 00:09:00.710 At any rate, this is the statement. 00:09:00.710 --> 00:09:04.990 And my other claim is that the proof follows immediately 00:09:04.990 --> 00:09:09.940 from the main, key theorem that we stressed last time, even 00:09:09.940 --> 00:09:11.380 though we didn't prove it. 00:09:11.380 --> 00:09:14.280 But we've had ample exercises using this. 00:09:14.280 --> 00:09:16.870 Namely, notice that we have already 00:09:16.870 --> 00:09:21.630 seen that if w does happen to be a continuously differentiable 00:09:21.630 --> 00:09:26.840 function of x, y, and z, then delta w is the partial of w 00:09:26.840 --> 00:09:30.290 with respect to x times delta x, plus the partial of w 00:09:30.290 --> 00:09:33.660 with respect to y times delta y, plus the partial of w 00:09:33.660 --> 00:09:37.790 with respect to z times delta z, plus an error term. 00:09:37.790 --> 00:09:39.200 And what is that error term? 00:09:39.200 --> 00:09:43.740 It's k_1 delta x, plus k_2 delta y, plus k_3 delta 00:09:43.740 --> 00:09:48.200 z, where k_1, k_2, and k_3 all approach 00:09:48.200 --> 00:09:52.220 0 as delta x, delta y, and delta z approach 0. 00:09:52.220 --> 00:09:56.350 Now again, the key step in all of this 00:09:56.350 --> 00:10:01.940 is that this amount here I could always call delta w tan, 00:10:01.940 --> 00:10:04.810 or as Professor Thomas calls it for more than two variables, 00:10:04.810 --> 00:10:10.530 delta w sub lin, l-i-n, meaning that this is a linear equation. 00:10:10.530 --> 00:10:12.760 Remember-- I've made an abbreviation here-- 00:10:12.760 --> 00:10:17.340 these partials are assumed to be evaluated at a particular point 00:10:17.340 --> 00:10:19.040 that we're interested in. 00:10:19.040 --> 00:10:24.060 But the idea is, granted that I can always call this delta w 00:10:24.060 --> 00:10:30.070 tan, to say that the error has this small a magnitude depends 00:10:30.070 --> 00:10:33.110 on the fact that w is a continuously differentiable 00:10:33.110 --> 00:10:35.290 function of x, y, and z. 00:10:35.290 --> 00:10:37.310 That's why the theory is so important. 00:10:37.310 --> 00:10:40.380 What happens in real life is that most examples 00:10:40.380 --> 00:10:44.850 you encounter in real-life engineering, the functions 00:10:44.850 --> 00:10:48.170 that you're dealing with are continuously differentiable. 00:10:48.170 --> 00:10:51.260 So it seems like we're making a big issue over nothing. 00:10:51.260 --> 00:10:54.100 I should point out that on the frontiers of knowledge, 00:10:54.100 --> 00:10:57.000 enough situations occur where the functions that we're 00:10:57.000 --> 00:10:59.800 dealing with are not continuously differentiable, 00:10:59.800 --> 00:11:03.610 that some horrible mistakes can be made by assuming that you 00:11:03.610 --> 00:11:09.090 can replace delta w by this, without any significant error. 00:11:09.090 --> 00:11:11.660 But as long as this is the case, we can do this. 00:11:11.660 --> 00:11:14.990 And now notice, what does the partial of w with respect to r 00:11:14.990 --> 00:11:15.680 mean? 00:11:15.680 --> 00:11:20.050 It means you take delta w divided by delta r. 00:11:20.050 --> 00:11:21.820 And let me just do that here. 00:11:21.820 --> 00:11:24.890 I'll just divide every term by delta r. 00:11:32.360 --> 00:11:34.610 And now, what do I have to do next to get the partial? 00:11:34.610 --> 00:11:37.970 I have to take the limit as delta r approaches 0. 00:11:37.970 --> 00:11:41.720 Now the interesting point is as delta approaches 0, 00:11:41.720 --> 00:11:45.300 holding s fixed, this term obviously becomes 00:11:45.300 --> 00:11:48.820 the partial of x with respect to r, by definition. 00:11:48.820 --> 00:11:51.940 This term becomes the partial of y with respect to r, 00:11:51.940 --> 00:11:53.180 by definition. 00:11:53.180 --> 00:11:56.650 And this term becomes a partial of z with respect to r, 00:11:56.650 --> 00:11:57.860 by definition. 00:11:57.860 --> 00:12:01.070 By the way, notice that even though delta x, delta y, 00:12:01.070 --> 00:12:05.020 and delta z are all going to 0 as delta r goes to 0, 00:12:05.020 --> 00:12:08.620 you can not immediately conclude that these terms drop out. 00:12:08.620 --> 00:12:11.800 Because after all, delta r is also approaching 0. 00:12:11.800 --> 00:12:15.380 So delta x over delta r is that 0 over 0 form. 00:12:15.380 --> 00:12:18.890 In fact, that's precisely the partial 00:12:18.890 --> 00:12:21.560 of x with respect to r term that we're talking about. 00:12:21.560 --> 00:12:23.270 The beauty is what? 00:12:23.270 --> 00:12:25.780 That as delta x, delta y, and delta z 00:12:25.780 --> 00:12:28.840 approach 0, each of the k's approach 0. 00:12:31.490 --> 00:12:35.530 You see, the reason that the error term becomes negligible, 00:12:35.530 --> 00:12:39.180 becomes 0 in the limit, isn't because delta x, delta y, 00:12:39.180 --> 00:12:41.350 and delta z are becoming small. 00:12:41.350 --> 00:12:43.550 Because these small numbers are being divided 00:12:43.550 --> 00:12:44.850 by another small number. 00:12:44.850 --> 00:12:49.010 It's because the k_1, k_2, and k_3 are getting small. 00:12:49.010 --> 00:12:52.010 At any rate, putting this all together, 00:12:52.010 --> 00:12:54.260 notice that now, in a manner completely 00:12:54.260 --> 00:12:57.380 analogous to our part-one treatment of the chain rule, 00:12:57.380 --> 00:12:59.680 except that we're now dealing with several variables, 00:12:59.680 --> 00:13:01.530 these three terms drop out. 00:13:01.530 --> 00:13:06.410 And these three terms become the claim that we made before. 00:13:06.410 --> 00:13:10.040 In other words, this is how the partial of w with respect to r 00:13:10.040 --> 00:13:10.640 is computed. 00:13:10.640 --> 00:13:14.600 And again, the theory is the easiest part of this. 00:13:14.600 --> 00:13:15.860 That's the easy part. 00:13:15.860 --> 00:13:17.785 The hard part is getting familiarity 00:13:17.785 --> 00:13:18.910 with how to work with this. 00:13:18.910 --> 00:13:21.270 And I think the best way to get some familiarity 00:13:21.270 --> 00:13:24.470 for working with this is to pick particularly simple problems 00:13:24.470 --> 00:13:26.300 for the lecture, problems where it's 00:13:26.300 --> 00:13:28.620 so easy to do the problem both ways 00:13:28.620 --> 00:13:31.100 that no hangup can possibly occur. 00:13:31.100 --> 00:13:32.900 Let's take a very simple example. 00:13:32.900 --> 00:13:34.830 Let's suppose that w equals x squared 00:13:34.830 --> 00:13:37.260 plus y squared plus z squared. 00:13:37.260 --> 00:13:42.690 Suppose we also know that x is r plus s, y is r minus s, 00:13:42.690 --> 00:13:44.990 and z happens to be 2r. 00:13:44.990 --> 00:13:47.410 In this particular case, notice that we 00:13:47.410 --> 00:13:52.330 would find the partial of w with respect to r very conveniently 00:13:52.330 --> 00:13:53.980 by direct substitution. 00:13:53.980 --> 00:13:57.270 Namely, we simply replace x by r plus s, 00:13:57.270 --> 00:14:02.620 we replace y by r minus s, we replace z by 2r. 00:14:02.620 --> 00:14:06.700 And then w simply becomes this expression here, 00:14:06.700 --> 00:14:09.260 which when we collect terms, becomes 00:14:09.260 --> 00:14:12.460 6 r squared plus 2 s squared. 00:14:12.460 --> 00:14:14.460 And again, the arithmetic there is simple enough 00:14:14.460 --> 00:14:16.543 so I'm not even going to bother worrying about how 00:14:16.543 --> 00:14:18.600 we justify these steps. 00:14:18.600 --> 00:14:22.690 At which stage, to take the partial of w with respect to r, 00:14:22.690 --> 00:14:25.956 holding s constant, this is simply what? 00:14:25.956 --> 00:14:27.070 12r. 00:14:27.070 --> 00:14:29.110 Because s is being treated as a constant, 00:14:29.110 --> 00:14:31.990 its derivative with respect to r is 0. 00:14:31.990 --> 00:14:34.360 You see, in an example like this, 00:14:34.360 --> 00:14:37.530 one would not really be tempted to use the chain rule. 00:14:37.530 --> 00:14:40.680 The chain rule is used in many cases 00:14:40.680 --> 00:14:43.980 not just for convenience, but in cases of great theory 00:14:43.980 --> 00:14:47.950 where you're only given that w is some function of x, y, 00:14:47.950 --> 00:14:50.570 and z, and you're not told explicitly 00:14:50.570 --> 00:14:51.740 what the function is. 00:14:51.740 --> 00:14:53.690 You're just given f of x, y, z. 00:14:53.690 --> 00:14:57.110 In the case where the function is given explicitly, 00:14:57.110 --> 00:15:00.380 it's sometimes very easy to substitute directly. 00:15:00.380 --> 00:15:03.050 At any rate, what the chain rule says is roughly this. 00:15:03.050 --> 00:15:03.920 They say, lookit. 00:15:03.920 --> 00:15:06.440 From this equation, you could immediately 00:15:06.440 --> 00:15:09.590 say the partial of w with respect to x is 2x, 00:15:09.590 --> 00:15:12.470 the partial of w with respect to y is 2y, 00:15:12.470 --> 00:15:15.710 the partial of w with respect to z is 2z. 00:15:15.710 --> 00:15:17.460 From this equation, you could immediately 00:15:17.460 --> 00:15:20.340 say that the partial of x with respect to r is 1, 00:15:20.340 --> 00:15:23.040 the partial of x with respect to s is 1, 00:15:23.040 --> 00:15:25.920 the partial of y with respect to r is 1, 00:15:25.920 --> 00:15:29.850 the partial of y with respect to s is minus 1, 00:15:29.850 --> 00:15:32.910 the partial of z with respect to r is 2, 00:15:32.910 --> 00:15:36.450 and the partial of z with respect to s is 0. 00:15:36.450 --> 00:15:43.380 In particular, summarizing our results, we have these here. 00:15:43.380 --> 00:15:45.300 Now, what the chain rule says is what? 00:15:45.300 --> 00:15:48.190 To find the partial of w with respect to r, 00:15:48.190 --> 00:15:50.410 you just take the partial of w with respect 00:15:50.410 --> 00:15:53.540 to x times the partial of x with respect to r, 00:15:53.540 --> 00:15:55.630 plus the partial of w with respect 00:15:55.630 --> 00:15:58.600 to y times the partial of y with respect to r, 00:15:58.600 --> 00:16:00.320 plus the partial of w with respect 00:16:00.320 --> 00:16:03.100 to z times the partial of z with respect to r. 00:16:03.100 --> 00:16:06.150 And if we do that in this case, we simply get what? 00:16:06.150 --> 00:16:11.380 2x plus 2y plus 4z. 00:16:11.380 --> 00:16:14.170 Now again, I picked, deliberately, 00:16:14.170 --> 00:16:16.260 a very simple problem here. 00:16:16.260 --> 00:16:24.840 Remember, by definition, x is r plus s, y is r minus s, 00:16:24.840 --> 00:16:28.340 and z happens to be 2r. 00:16:28.340 --> 00:16:31.770 And now you can see very quickly here that when I substitute in, 00:16:31.770 --> 00:16:32.380 I get what? 00:16:32.380 --> 00:16:40.360 2r plus 2r is 4r, plus 8r is 12r, and 2s minus 2s is 0. 00:16:40.360 --> 00:16:42.750 The partial of w with respect to r 00:16:42.750 --> 00:16:45.710 is also 12r, also meaning what? 00:16:45.710 --> 00:16:48.390 We found that same answer before. 00:16:48.390 --> 00:16:51.630 At least that's how the chain rule works. 00:16:51.630 --> 00:16:55.220 And again, we have to remember that the chain rule does not 00:16:55.220 --> 00:16:57.390 depend on the number of variables, 00:16:57.390 --> 00:17:00.730 even though this may start to look a little bit sticky. 00:17:00.730 --> 00:17:02.340 Let's word it as follows. 00:17:02.340 --> 00:17:07.109 Suppose w happens to be a continuously differentiable 00:17:07.109 --> 00:17:13.099 function of the n independent variables x_1 up to x_n. 00:17:13.099 --> 00:17:16.190 See, that's what this parenthetical remark means. 00:17:16.190 --> 00:17:20.210 I'm saying that not only do the partials of f with respect 00:17:20.210 --> 00:17:22.880 to x_1 up to x_n exist at a given point, 00:17:22.880 --> 00:17:24.670 but they are continuous there. 00:17:24.670 --> 00:17:26.310 And why do I want that in there? 00:17:26.310 --> 00:17:28.650 So I can say that my error term is never 00:17:28.650 --> 00:17:33.550 any greater than that k_1 delta x_1, plus k_2 delta x_2, 00:17:33.550 --> 00:17:36.680 plus, et cetera, k_n delta x_n, where the k's go 00:17:36.680 --> 00:17:39.310 to 0 as the delta x's go to 0. 00:17:39.310 --> 00:17:41.880 I'm going to spare you the details of proofs. 00:17:41.880 --> 00:17:43.530 But I just want you to keep seeing 00:17:43.530 --> 00:17:45.400 why these things are necessary. 00:17:45.400 --> 00:17:50.170 At any rate, let's suppose now that each of the n variables 00:17:50.170 --> 00:17:56.020 x_1 up to x_n turn out to be functions of the m variables. 00:17:56.020 --> 00:17:58.800 n and m could conceivably be equal. 00:17:58.800 --> 00:18:01.500 But m could even be more than n. 00:18:01.500 --> 00:18:02.637 It can be less than n. 00:18:02.637 --> 00:18:04.470 There's no reason why they have to be equal. 00:18:04.470 --> 00:18:07.630 All we're saying is, speaking in the most general terms, 00:18:07.630 --> 00:18:12.040 suppose each of the n x's is a continuously differentiable 00:18:12.040 --> 00:18:16.870 function of the m independent variables y sub 1 up 00:18:16.870 --> 00:18:17.880 to y sub m. 00:18:17.880 --> 00:18:20.400 In fact, that's what this "et cetera" means here. 00:18:20.400 --> 00:18:23.370 The "et cetera" refers to the parenthetical remark here. 00:18:23.370 --> 00:18:27.590 I mean that not only are the x's functions of y_1 up to y_m, 00:18:27.590 --> 00:18:30.890 but they're continuously differentiable functions. 00:18:30.890 --> 00:18:35.180 Now obviously, what we're saying is that if w can be expressed 00:18:35.180 --> 00:18:38.260 in terms of the x's, the x's can be expressed in terms 00:18:38.260 --> 00:18:42.480 of the y's, obviously then, w can be expressed in terms 00:18:42.480 --> 00:18:43.090 of the y's. 00:18:43.090 --> 00:18:47.590 In other words, w is some function of y_1 up to y_m. 00:18:47.590 --> 00:18:51.040 Now the question that comes up is that just looking at this, 00:18:51.040 --> 00:18:54.810 I can talk about the partial of w with respect to y_1, 00:18:54.810 --> 00:18:57.640 the partial of w with respect to y_2, 00:18:57.640 --> 00:19:02.660 the partial of w with respect to y_3, et cetera, all the way up 00:19:02.660 --> 00:19:05.820 to the partial of w with respect to y sub m. 00:19:05.820 --> 00:19:07.610 And the question is, lookit. 00:19:07.610 --> 00:19:10.900 From the original form of w, it was easy to talk about 00:19:10.900 --> 00:19:14.060 the partials of w with respect to the x's. 00:19:14.060 --> 00:19:17.620 From how the x's are given in terms of the y's, it's easy 00:19:17.620 --> 00:19:20.420 to talk about the derivatives of the x's with respect 00:19:20.420 --> 00:19:21.550 to the y's. 00:19:21.550 --> 00:19:23.170 And so the question is, how do you 00:19:23.170 --> 00:19:26.660 find the partial of w with respect to, say, y sub 00:19:26.660 --> 00:19:29.690 1, given all of these other partial derivatives? 00:19:29.690 --> 00:19:32.460 And the answer, again, is something that you just 00:19:32.460 --> 00:19:33.640 have to get used to. 00:19:33.640 --> 00:19:36.600 The proof goes through for n and m the same way 00:19:36.600 --> 00:19:38.780 as it did for the lower-dimensional case. 00:19:38.780 --> 00:19:41.410 And the intuitive interpretation is the same. 00:19:41.410 --> 00:19:45.380 Namely, to find the partial of w with respect to y_1, 00:19:45.380 --> 00:19:48.590 we simply see how much w changed with respect 00:19:48.590 --> 00:19:52.300 to y_1 due to the change in x_1 alone, 00:19:52.300 --> 00:19:54.940 add on to that the change in w with respect 00:19:54.940 --> 00:19:59.720 to y_1 due to the change in x_2 alone, et cetera, 00:19:59.720 --> 00:20:07.450 add on to that, finally, the change in w with respect to y 00:20:07.450 --> 00:20:10.970 sub 1 due to the change in x sub n alone. 00:20:10.970 --> 00:20:12.580 In other words, again, if you think 00:20:12.580 --> 00:20:15.740 of this in terms of cancellation, 00:20:15.740 --> 00:20:19.210 if you cross these things out, don't think of adding them, 00:20:19.210 --> 00:20:20.710 but think of them as what? 00:20:20.710 --> 00:20:24.130 Giving you the individual components 00:20:24.130 --> 00:20:27.450 that tell you how the partial of w with respect to y_1 00:20:27.450 --> 00:20:28.350 is made up. 00:20:28.350 --> 00:20:30.422 By the way, there is one parenthetical remark 00:20:30.422 --> 00:20:31.880 that I haven't written on the board 00:20:31.880 --> 00:20:33.920 that I would like to make at this time. 00:20:33.920 --> 00:20:36.440 In Professor Thomas's text, he has 00:20:36.440 --> 00:20:39.360 elected to introduce matrix algebra prior 00:20:39.360 --> 00:20:41.420 to this particular chapter. 00:20:41.420 --> 00:20:43.800 It again turns out that one does not 00:20:43.800 --> 00:20:46.340 need matrices to talk about the chain rule 00:20:46.340 --> 00:20:49.180 but that if one had matrix notation, 00:20:49.180 --> 00:20:51.580 the matrix notation is particularly 00:20:51.580 --> 00:20:54.680 convenient for summarizing the chain rule. 00:20:54.680 --> 00:20:59.010 I have elected to hold off on matrix algebra 00:20:59.010 --> 00:21:01.210 till the near future because it comes up 00:21:01.210 --> 00:21:03.940 in a much better motivated way, I 00:21:03.940 --> 00:21:07.570 think, in terms of these linear approximations. 00:21:07.570 --> 00:21:10.410 But the point is if, as you're reading the text, 00:21:10.410 --> 00:21:12.980 you see the matrix notation, and you are not 00:21:12.980 --> 00:21:15.510 familiar with the matrices, forget it. 00:21:15.510 --> 00:21:20.300 All the matrix is, is a shortcut notation for saying this. 00:21:20.300 --> 00:21:22.470 And if I want a shortcut notation here, 00:21:22.470 --> 00:21:24.530 I don't need matrices for saying this. 00:21:24.530 --> 00:21:27.510 I can say this in terms of our sigma notation. 00:21:27.510 --> 00:21:29.780 Notice that one other way of writing 00:21:29.780 --> 00:21:33.560 this thing very compactly that may be more suggestive 00:21:33.560 --> 00:21:34.810 is the following. 00:21:34.810 --> 00:21:39.440 Notice that I'm adding up n terms here. 00:21:39.440 --> 00:21:43.680 Each term consists of two factors, each of which 00:21:43.680 --> 00:21:45.430 looks like a fraction. 00:21:45.430 --> 00:21:47.260 The numerator of the first fraction 00:21:47.260 --> 00:21:50.960 is always a partial of w. 00:21:50.960 --> 00:21:52.790 The denominator of the second fraction 00:21:52.790 --> 00:21:55.980 is always the partial y_1. 00:21:55.980 --> 00:21:59.330 And it appears that the denominator of the first 00:21:59.330 --> 00:22:03.600 and the numerator of the second always have the same subscript, 00:22:03.600 --> 00:22:07.910 but they seem to vary consecutively from 1 to n. 00:22:07.910 --> 00:22:10.850 And that's precisely where the sigma notation comes in handy. 00:22:10.850 --> 00:22:13.130 Why don't we just write, therefore, 00:22:13.130 --> 00:22:16.610 that this is the sum, partial of w with respect 00:22:16.610 --> 00:22:20.200 to x sub k, plus the partial of x sub k 00:22:20.200 --> 00:22:23.020 with respect to y_1, as the subscript 00:22:23.020 --> 00:22:28.170 k ranges through all integral values from 1 to n? 00:22:28.170 --> 00:22:31.270 In other words, notice that in this particular form, 00:22:31.270 --> 00:22:34.310 we have simply rewritten this thing compactly. 00:22:34.310 --> 00:22:36.830 But if you look at this and look at this, 00:22:36.830 --> 00:22:40.220 I think it's very suggestive to see how the chain rule works. 00:22:40.220 --> 00:22:43.940 You see, here's your partial of w with respect to y_1. 00:22:43.940 --> 00:22:45.350 And these are what? 00:22:45.350 --> 00:22:51.040 The contributions due to each of the changes of the n variables. 00:22:51.040 --> 00:22:54.350 This is the change due to the x sub k variable. 00:22:54.350 --> 00:22:57.400 And you add these all up because they're independent variables, 00:22:57.400 --> 00:22:59.620 as k goes from 1 to n. 00:22:59.620 --> 00:23:02.280 And I write "et cetera" here simply to point out 00:23:02.280 --> 00:23:04.610 that I could have computed the partial of w 00:23:04.610 --> 00:23:07.670 with respect to y_2 instead of y sub 1. 00:23:07.670 --> 00:23:10.560 By the way, the recipe would've looked exactly the same, 00:23:10.560 --> 00:23:12.560 except that if there was a 2 here, there 00:23:12.560 --> 00:23:13.940 would have been a 2 here. 00:23:13.940 --> 00:23:17.160 If there were a 3 here, they would've been a 3 here. 00:23:17.160 --> 00:23:21.020 If there were an m here, there would have been an m here. 00:23:21.020 --> 00:23:22.800 OK, now lookit. 00:23:22.800 --> 00:23:25.630 At this particular stage of our lecture 00:23:25.630 --> 00:23:30.080 today, this could end with the idea 00:23:30.080 --> 00:23:32.440 that for the unit that's now assigned, 00:23:32.440 --> 00:23:34.450 this is as far as you have to go. 00:23:34.450 --> 00:23:36.870 In other words, for the exercises 00:23:36.870 --> 00:23:39.890 that I've given you in this particular unit, 00:23:39.890 --> 00:23:43.460 we do nothing higher than using the chain rule 00:23:43.460 --> 00:23:46.250 for first-order derivatives. 00:23:46.250 --> 00:23:49.670 The point is that in many applications in real life, 00:23:49.670 --> 00:23:52.840 we must take higher-order derivatives. 00:23:52.840 --> 00:23:55.480 In other words, there are many differential equations, 00:23:55.480 --> 00:23:58.360 partial differential equations, where we must work 00:23:58.360 --> 00:24:00.100 with higher-order derivatives. 00:24:00.100 --> 00:24:03.670 And for that reason, it becomes very important, sometimes, 00:24:03.670 --> 00:24:06.360 to be able to take a second derivative 00:24:06.360 --> 00:24:08.700 or a third derivative or a fourth derivative 00:24:08.700 --> 00:24:10.480 by means of the chain rule. 00:24:10.480 --> 00:24:14.140 Now the interesting point is that the theory that we've used 00:24:14.140 --> 00:24:16.420 so far doesn't change at all. 00:24:16.420 --> 00:24:19.070 What does happen is that the average student, 00:24:19.070 --> 00:24:21.960 in learning this material for the first time, 00:24:21.960 --> 00:24:24.080 gets swamped by the notation. 00:24:24.080 --> 00:24:26.340 Consequently, what I want to do is 00:24:26.340 --> 00:24:30.090 to give you the lecture on this material at the same time 00:24:30.090 --> 00:24:33.850 that I'm lecturing on first-order derivatives, simply 00:24:33.850 --> 00:24:37.640 because the continuity follows smoother this way, 00:24:37.640 --> 00:24:40.380 so that you see what the whole overall picture is, then 00:24:40.380 --> 00:24:42.740 to make sure that you cement these things down. 00:24:42.740 --> 00:24:44.830 The next unit after this will give 00:24:44.830 --> 00:24:47.490 you drill on taking higher-order derivatives. 00:24:47.490 --> 00:24:49.440 What this may mean is that many of you 00:24:49.440 --> 00:24:54.640 may prefer to watch this half of the film 00:24:54.640 --> 00:24:57.920 a second time, after you've already 00:24:57.920 --> 00:24:59.490 tried working some of the problems 00:24:59.490 --> 00:25:01.530 with higher-order derivatives, if you're still 00:25:01.530 --> 00:25:02.490 confused by this. 00:25:02.490 --> 00:25:03.960 But at any rate, let's take a look 00:25:03.960 --> 00:25:05.660 at a hypothetical situation. 00:25:05.660 --> 00:25:08.160 Since we're so used to polar coordinates, 00:25:08.160 --> 00:25:10.910 let's talk in terms of polar coordinates. 00:25:10.910 --> 00:25:13.690 Suppose w happens to be a continuously differentiable 00:25:13.690 --> 00:25:15.875 function of x and y. 00:25:15.875 --> 00:25:20.760 x and y, in turn, are continuously differentiable 00:25:20.760 --> 00:25:23.280 functions of the polar coordinates r and theta. 00:25:23.280 --> 00:25:25.860 In fact, they're x equals r cosine theta, 00:25:25.860 --> 00:25:27.850 y equals r sine theta. 00:25:27.850 --> 00:25:28.540 Now lookit. 00:25:28.540 --> 00:25:32.710 If all I want to do is find the partial of w with respect to r, 00:25:32.710 --> 00:25:35.650 I can do that by the ordinary chain rule. 00:25:35.650 --> 00:25:38.010 Namely, it's the partial of w with respect 00:25:38.010 --> 00:25:40.700 to x times the partial of x with respect to r, 00:25:40.700 --> 00:25:42.360 plus the partial of w with respect 00:25:42.360 --> 00:25:45.680 to y times the partial of y with respect to r. 00:25:45.680 --> 00:25:48.240 Now, knowing what x looks like explicitly 00:25:48.240 --> 00:25:51.480 in terms of r and theta and what y looks like explicitly 00:25:51.480 --> 00:25:53.690 in terms of r and theta, I can certainly 00:25:53.690 --> 00:25:56.300 compute the partials of x and y with respect 00:25:56.300 --> 00:25:59.560 to r, holding theta constant. 00:25:59.560 --> 00:26:02.550 In particular, the partial of x with respect to r 00:26:02.550 --> 00:26:04.200 is simply cosine theta. 00:26:04.200 --> 00:26:06.740 And the partial of y with respect to r 00:26:06.740 --> 00:26:08.170 is simply sine theta. 00:26:08.170 --> 00:26:11.140 So the partial of w with respect to r 00:26:11.140 --> 00:26:13.560 is partial of w with respect to x times 00:26:13.560 --> 00:26:18.010 cosine theta, plus partial of w with respect to y times sine 00:26:18.010 --> 00:26:18.510 theta. 00:26:18.510 --> 00:26:23.410 And by the way, notice that I cannot simplify these terms. 00:26:23.410 --> 00:26:25.790 I cannot simplify these terms in general, 00:26:25.790 --> 00:26:29.570 because all I'm given is that w is some function of x and y. 00:26:29.570 --> 00:26:33.120 I don't know what w looks like explicitly in terms of x and y. 00:26:33.120 --> 00:26:36.660 So all I can do is talk about the partials of w with respect 00:26:36.660 --> 00:26:40.320 to x, partial of w respect to y, without worrying 00:26:40.320 --> 00:26:43.300 any more about this, with the understanding that if I knew 00:26:43.300 --> 00:26:46.440 what w looked like explicitly in terms of x and y, 00:26:46.440 --> 00:26:48.985 I could work out what this thing was. 00:26:48.985 --> 00:26:50.440 Now, here's the key point. 00:26:50.440 --> 00:26:52.340 That's why I've accentuated it. 00:26:52.340 --> 00:26:55.400 In the same way that w is a function 00:26:55.400 --> 00:27:00.120 of both x and y, so also are the partials of w with respect 00:27:00.120 --> 00:27:03.530 to x and the partials of w with respect to y. 00:27:03.530 --> 00:27:06.750 In other words, even though this looks 00:27:06.750 --> 00:27:09.180 like this emphasizes the x, notice 00:27:09.180 --> 00:27:11.290 that when you take the derivative 00:27:11.290 --> 00:27:16.040 of a function of both x and y with respect to x, in general, 00:27:16.040 --> 00:27:17.640 the resulting function will again 00:27:17.640 --> 00:27:20.300 be a function of both x and y. 00:27:20.300 --> 00:27:23.070 And so what we're saying is that if the partials of w 00:27:23.070 --> 00:27:27.310 with respect to x and the partials of w with respect to y 00:27:27.310 --> 00:27:30.580 also happen to be continuously differentiable functions of x 00:27:30.580 --> 00:27:34.905 and y, we could, if we wished, use the chain rule again. 00:27:34.905 --> 00:27:37.120 In other words, suppose in the particular problem 00:27:37.120 --> 00:27:38.890 that I was dealing with, it wasn't 00:27:38.890 --> 00:27:42.320 enough to know the partial of w with respect to r. 00:27:42.320 --> 00:27:45.670 Suppose, for example, I wanted the second partial of w 00:27:45.670 --> 00:27:46.980 with respect to r. 00:27:46.980 --> 00:27:50.220 Well obviously, that simply means what? 00:27:50.220 --> 00:27:54.360 Take the partial of this with respect to r. 00:27:54.360 --> 00:27:57.590 In other words, the second partial of w with respect to r 00:27:57.590 --> 00:28:02.070 is just the partial of the partial of w with respect to r, 00:28:02.070 --> 00:28:03.770 with respect r. 00:28:03.770 --> 00:28:07.510 I'm just going to differentiate this thing with respect to r. 00:28:07.510 --> 00:28:10.720 In other words, writing this out more succinctly for you, 00:28:10.720 --> 00:28:13.860 the second partial derivative of w with respect to r 00:28:13.860 --> 00:28:17.290 is the partial with respect to r of cosine theta 00:28:17.290 --> 00:28:21.400 partial of w with respect to x, plus sine theta partial 00:28:21.400 --> 00:28:23.640 of w with respect to y. 00:28:23.640 --> 00:28:24.750 Now, here's the key point. 00:28:24.750 --> 00:28:26.890 When we differentiate here, we're 00:28:26.890 --> 00:28:29.570 assuming that theta is being held constant. 00:28:29.570 --> 00:28:30.850 Isn't that right? 00:28:30.850 --> 00:28:33.720 So consequently, when I'm differentiating with respect 00:28:33.720 --> 00:28:37.180 to r, cosine theta is a constant. 00:28:37.180 --> 00:28:40.530 I can skip over that, see, and differentiate 00:28:40.530 --> 00:28:42.764 what's left with respect to r. 00:28:42.764 --> 00:28:43.930 In other words, that's what? 00:28:43.930 --> 00:28:47.440 It's the partial of w with respect to x differentiated 00:28:47.440 --> 00:28:49.430 with respect to r. 00:28:49.430 --> 00:28:54.090 See, I'm using the ordinary rule for the derivative of a sum. 00:28:54.090 --> 00:28:55.920 Now, the derivative of sine theta-- 00:28:55.920 --> 00:28:58.860 see, sine theta is a constant with respect to r. 00:28:58.860 --> 00:29:00.590 So the derivative of this term is just 00:29:00.590 --> 00:29:03.800 sine theta times the derivative of the partial 00:29:03.800 --> 00:29:09.730 of w with respect to y, with respect to r, written this way. 00:29:09.730 --> 00:29:13.320 Now, the key point is that both of these functions 00:29:13.320 --> 00:29:18.712 here, both of these are functions of x and y. 00:29:18.712 --> 00:29:22.240 x and y, in turn, are functions of r and theta. 00:29:22.240 --> 00:29:24.920 So in other words, to differentiate 00:29:24.920 --> 00:29:30.120 this thing with respect to r, I must use the chain rule again. 00:29:30.120 --> 00:29:33.000 Now, because this may seem difficult for you, 00:29:33.000 --> 00:29:34.920 all I'm really saying is, lookit. 00:29:34.920 --> 00:29:37.670 If this term here looks messy, since we 00:29:37.670 --> 00:29:40.000 know that the partial of w with respect to x 00:29:40.000 --> 00:29:44.480 is some function of x and y, let's call that h of x, y. 00:29:44.480 --> 00:29:50.570 Then all we're saying is that the partial of the partial of w 00:29:50.570 --> 00:29:53.820 with respect to x, with respect to r, 00:29:53.820 --> 00:29:56.860 is just the partial of h with respect to r. 00:29:56.860 --> 00:29:59.540 But to find the partial of h with respect to r, 00:29:59.540 --> 00:30:01.640 we know how to use the chain rule there. 00:30:01.640 --> 00:30:02.580 It's just what? 00:30:02.580 --> 00:30:04.350 It's the partial of h with respect 00:30:04.350 --> 00:30:07.720 to x times the partial of x with respect to r, 00:30:07.720 --> 00:30:10.040 plus the partial of h with respect 00:30:10.040 --> 00:30:13.780 to y times the partial of y with respect to r. 00:30:13.780 --> 00:30:17.655 Of course, if we now remember what h is-- see, 00:30:17.655 --> 00:30:21.210 h is the partial of w with respect to x. 00:30:21.210 --> 00:30:25.110 So if I differentiate again with respect to x, 00:30:25.110 --> 00:30:28.210 I get the second partial of w with respect to x. 00:30:28.210 --> 00:30:30.760 We've already seen that the partial of x with respect to r 00:30:30.760 --> 00:30:34.220 is cosine theta, so I have this term. 00:30:34.220 --> 00:30:37.950 The partial of h with respect to y really says what? 00:30:37.950 --> 00:30:41.310 Differentiate the partial of w with respect 00:30:41.310 --> 00:30:44.470 to x, with respect to y. 00:30:44.470 --> 00:30:46.820 And the usual way of abbreviating that 00:30:46.820 --> 00:30:49.590 is like this, which, again, is explained in the reading 00:30:49.590 --> 00:30:51.100 material. 00:30:51.100 --> 00:30:55.540 And we now multiply that by the partial of y with respect 00:30:55.540 --> 00:30:58.580 to r, which happens to be sine theta. 00:30:58.580 --> 00:31:03.870 Now, look at this a few times in your spare time, 00:31:03.870 --> 00:31:05.060 if it's bothering you. 00:31:05.060 --> 00:31:07.390 It is not really that difficult. It 00:31:07.390 --> 00:31:10.260 is messy notation in the sense that you're not 00:31:10.260 --> 00:31:12.804 used to notation that's quite that messy. 00:31:12.804 --> 00:31:13.720 That's why it's messy. 00:31:13.720 --> 00:31:15.150 Once you get used to it, it is not 00:31:15.150 --> 00:31:18.550 any tougher than the chain rule for one independent variable. 00:31:18.550 --> 00:31:21.910 In fact, to take the partial of the partial of w 00:31:21.910 --> 00:31:26.270 with respect to y, with respect to r, I'll do that in one step 00:31:26.270 --> 00:31:27.830 without using a substitution. 00:31:27.830 --> 00:31:30.220 All I'm saying is that this function 00:31:30.220 --> 00:31:32.200 depends on both x and y. 00:31:32.200 --> 00:31:35.710 So to see what its derivative is with respect to r, 00:31:35.710 --> 00:31:38.790 I'll see what the contribution of its derivative with respect 00:31:38.790 --> 00:31:41.780 to r is due to just x alone. 00:31:41.780 --> 00:31:44.120 Then I'll see what contribution of its derivative 00:31:44.120 --> 00:31:46.940 with respect to r is due to just y alone. 00:31:46.940 --> 00:31:48.780 And by the way, when I say it that way, 00:31:48.780 --> 00:31:51.070 notice how quick it is to write this thing down. 00:31:51.070 --> 00:31:52.490 I differentiate this with respect 00:31:52.490 --> 00:31:56.120 to x multiplied by the partial of x with respect to r. 00:31:56.120 --> 00:31:59.130 Add on to that the partial of this with respect to y. 00:31:59.130 --> 00:32:02.390 Multiply that by the partial of y with respect to r. 00:32:02.390 --> 00:32:05.400 If I do this, notice now I have what? 00:32:05.400 --> 00:32:07.710 I have the partial with respect to y. 00:32:07.710 --> 00:32:10.370 And I differentiate that with respect to x. 00:32:10.370 --> 00:32:12.350 That's written this way. 00:32:12.350 --> 00:32:16.960 And by the way, notice that this is the reverse order 00:32:16.960 --> 00:32:19.020 of what we did over here. 00:32:19.020 --> 00:32:22.060 Namely, in one case, we first differentiated with respect 00:32:22.060 --> 00:32:24.410 to x and then with respect to y. 00:32:24.410 --> 00:32:26.810 In the other case, we differentiated first 00:32:26.810 --> 00:32:29.770 with respect to y and then with respect to x. 00:32:29.770 --> 00:32:32.510 So that actually, conceptually there is a difference. 00:32:32.510 --> 00:32:35.980 That's why we write these things differently. 00:32:35.980 --> 00:32:40.010 It does, again, turn out that in most cases, 00:32:40.010 --> 00:32:42.680 the answer that you get-- thank goodness-- 00:32:42.680 --> 00:32:44.820 doesn't depend on the order in which you 00:32:44.820 --> 00:32:46.120 perform the derivatives. 00:32:46.120 --> 00:32:48.070 But this is not at all self-evident, 00:32:48.070 --> 00:32:50.170 even though you'd like to believe that it is. 00:32:50.170 --> 00:32:54.050 But we'll talk about that more in the exercises and the like. 00:32:54.050 --> 00:32:57.320 But all I'm saying now is that if we put everything together 00:32:57.320 --> 00:33:00.270 of what we've had before, we can obtain, 00:33:00.270 --> 00:33:03.390 in this particular case, that the second partial of w 00:33:03.390 --> 00:33:08.700 with respect to r is this somewhat messy but nonetheless 00:33:08.700 --> 00:33:10.770 straightforward expression. 00:33:10.770 --> 00:33:12.280 And see, I've circled these things 00:33:12.280 --> 00:33:16.120 to sort of tell you that if it is permissible to interchange 00:33:16.120 --> 00:33:20.670 the order of differentiation, we could combine these two terms. 00:33:20.670 --> 00:33:24.650 On the other hand, if you couldn't interchange the order, 00:33:24.650 --> 00:33:27.040 this would be a rather dangerous thing 00:33:27.040 --> 00:33:31.090 to do over here because these might be different answers. 00:33:31.090 --> 00:33:34.960 As I say again, if you have enough continuity, 00:33:34.960 --> 00:33:37.514 it turns out that these two factors are the same. 00:33:37.514 --> 00:33:39.180 But that's not the important issue here. 00:33:39.180 --> 00:33:41.160 The important issue here is that I 00:33:41.160 --> 00:33:45.400 can keep using the chain rule to take higher-order derivatives. 00:33:45.400 --> 00:33:47.780 And even though the notation is messier, 00:33:47.780 --> 00:33:51.020 this happened when we dealt with functions of a single variable. 00:33:51.020 --> 00:33:52.750 Remember when we used the chain rule 00:33:52.750 --> 00:33:58.640 to find dy/dx when y and x were given, say, as functions of t? 00:33:58.640 --> 00:34:01.910 We could also use the chain rule to find the second derivative 00:34:01.910 --> 00:34:03.380 of y with respect to x. 00:34:03.380 --> 00:34:06.550 But we had to be a little bit more careful of the computation 00:34:06.550 --> 00:34:10.409 because certain factors crept in that we had to keep track of. 00:34:10.409 --> 00:34:13.870 At any rate, again, to illustrate this idea 00:34:13.870 --> 00:34:15.980 rather than to keep droning on about it, 00:34:15.980 --> 00:34:19.449 let me take a particularly simple computational problem 00:34:19.449 --> 00:34:20.409 to check this thing on. 00:34:20.409 --> 00:34:21.908 In other words, what I'm going to do 00:34:21.908 --> 00:34:25.469 is take this messy formula over here and apply it to a case 00:34:25.469 --> 00:34:29.929 where the arithmetic happens to be very, very simple. 00:34:29.929 --> 00:34:32.070 I'm going to rig this very, very nicely. 00:34:32.070 --> 00:34:35.130 I'm going to let f of x, y just be x squared plus y squared, 00:34:35.130 --> 00:34:36.489 in this case. 00:34:36.489 --> 00:34:38.800 Let w be x squared plus y squared. 00:34:38.800 --> 00:34:41.790 In polar coordinates, notice that x squared plus y squared 00:34:41.790 --> 00:34:45.030 is just r squared. 00:34:45.030 --> 00:34:46.820 So w is just r squared. 00:34:46.820 --> 00:34:49.870 What is the partial of w with respect to r, then? 00:34:49.870 --> 00:34:54.139 The partial of w with respect to r is 2r. 00:34:54.139 --> 00:34:56.675 And if I now differentiate that with respect to r, 00:34:56.675 --> 00:35:00.010 the second partial of w with respect to r is 2. 00:35:00.010 --> 00:35:03.760 Obviously, one would not use the chain rule in real life 00:35:03.760 --> 00:35:06.090 to find the answer to this particular problem. 00:35:06.090 --> 00:35:10.340 We've chosen this problem simply to emphasize how 00:35:10.340 --> 00:35:12.570 the chain rule would work here. 00:35:12.570 --> 00:35:15.350 At any rate, going back here, notice 00:35:15.350 --> 00:35:18.000 that it's very simple to see from this equation 00:35:18.000 --> 00:35:20.910 that the partial of w with respect to x is 2x. 00:35:20.910 --> 00:35:24.480 Therefore, the second partial of w with respect to x is 2. 00:35:24.480 --> 00:35:28.090 The partial of w with respect to y is 2y. 00:35:28.090 --> 00:35:34.400 Therefore, the second partial of w with respect to y is also 2. 00:35:34.400 --> 00:35:37.800 The partial of w with respect to x is a function of x alone, 00:35:37.800 --> 00:35:38.870 in this case. 00:35:38.870 --> 00:35:42.450 Consequently, the derivative with respect to y will be 0. 00:35:42.450 --> 00:35:46.510 Similarly, the partial of w with respect to y is a function of y 00:35:46.510 --> 00:35:47.320 alone. 00:35:47.320 --> 00:35:48.970 Consequently, when I differentiate 00:35:48.970 --> 00:35:51.930 that with respect to x, meaning I'm holding y constant, 00:35:51.930 --> 00:35:54.190 that derivative will also be 0. 00:35:54.190 --> 00:36:00.670 And the interesting point now is if I take these values 00:36:00.670 --> 00:36:06.430 and substitute those into this equation, what happens? 00:36:06.430 --> 00:36:06.990 Look. 00:36:06.990 --> 00:36:10.900 The second partial of w with respect to x is just 2. 00:36:10.900 --> 00:36:15.840 The second partial of w with respect to y is just 2. 00:36:15.840 --> 00:36:19.700 The mixed partials are both 0, regardless of which order 00:36:19.700 --> 00:36:20.860 you did them in. 00:36:20.860 --> 00:36:22.620 That's what we saw over here. 00:36:22.620 --> 00:36:25.100 So consequently, according to this recipe, 00:36:25.100 --> 00:36:27.840 the second partial of w with respect to r 00:36:27.840 --> 00:36:32.580 is 2 cosine squared theta plus 0 plus 2 sine 00:36:32.580 --> 00:36:35.390 squared theta plus 0, where the reason I've 00:36:35.390 --> 00:36:37.326 written these 0's in is simply so 00:36:37.326 --> 00:36:39.200 that when you're looking at your notes later, 00:36:39.200 --> 00:36:42.980 that traces the analog of these terms over here. 00:36:42.980 --> 00:36:47.310 At any rate, notice now that if I add these up, 00:36:47.310 --> 00:36:49.740 2 cosine squared theta plus 2 sine 00:36:49.740 --> 00:36:51.900 squared theta, since sine squared 00:36:51.900 --> 00:36:54.330 theta plus cosine squared theta is 1, 00:36:54.330 --> 00:36:57.570 this sum is just-- I'll write that in white chalk 00:36:57.570 --> 00:37:00.020 just so we don't accentuate it. 00:37:00.020 --> 00:37:01.830 Let it just be part of the answer. 00:37:01.830 --> 00:37:05.180 This is 2 plus 0, which is 2. 00:37:05.180 --> 00:37:09.280 And this certainly does check with the result 00:37:09.280 --> 00:37:11.520 that we got the so-called easier way. 00:37:11.520 --> 00:37:14.990 And again, I don't want to leave you with the idea 00:37:14.990 --> 00:37:18.320 that the second way was just the hard way of doing 00:37:18.320 --> 00:37:22.140 the same problem that we did easily the first way. 00:37:22.140 --> 00:37:25.490 I picked a simple example so you can see how this works. 00:37:25.490 --> 00:37:28.140 I'm going to have a multitude of exercises 00:37:28.140 --> 00:37:31.920 for you to do in the next unit, simply so that you'll pick up 00:37:31.920 --> 00:37:36.160 the kind of know-how that will allow you to change variables 00:37:36.160 --> 00:37:40.750 using the chain rule, with a minimum degree of difficulty. 00:37:40.750 --> 00:37:42.660 In fact, hopefully, I would like to feel 00:37:42.660 --> 00:37:45.930 by the time we're through with the next two units, 00:37:45.930 --> 00:37:49.280 you will be doing this almost as second nature. 00:37:49.280 --> 00:37:51.950 Well, we have other topics to consider 00:37:51.950 --> 00:37:55.780 in terms of our linear approximations and the like. 00:37:55.780 --> 00:37:58.950 We'll talk about that more as the course unfolds. 00:37:58.950 --> 00:38:02.560 For the time being, I would like you to concentrate simply 00:38:02.560 --> 00:38:04.930 on mastering the chain rule. 00:38:04.930 --> 00:38:07.620 And so until we meet next time, good bye. 00:38:12.620 --> 00:38:15.010 Funding for the publication of this video 00:38:15.010 --> 00:38:19.860 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:38:19.860 --> 00:38:24.040 Help OCW continue to provide free and open access to MIT 00:38:24.040 --> 00:38:28.435 courses by making a donation at ocw.mit.edu/donate.