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 --> 00:00:19.650 at ocw.mit.edu. 00:00:33.930 --> 00:00:35.140 PROFESSOR: Hi. 00:00:35.140 --> 00:00:37.780 Today's lecture starts off with what 00:00:37.780 --> 00:00:41.260 probably is the single most important topic 00:00:41.260 --> 00:00:44.100 in the entire course, not just to date, 00:00:44.100 --> 00:00:45.920 but in the entire course. 00:00:45.920 --> 00:00:49.720 And it's a rather sneaky thing in the sense that 00:00:49.720 --> 00:00:51.640 takes a while to grow into it. 00:00:51.640 --> 00:00:55.390 Things have been going so smoothly so far, that perhaps 00:00:55.390 --> 00:00:58.250 this point is going to be a little bit subtle to make, 00:00:58.250 --> 00:01:00.770 and we'll try to lead into it gradually. 00:01:00.770 --> 00:01:05.620 There are probably ten different ways to introduce this topic. 00:01:05.620 --> 00:01:08.150 And whichever one you pick, I think, 00:01:08.150 --> 00:01:10.680 any one of the other nine would've been better. 00:01:10.680 --> 00:01:15.510 But without any further ado, let me talk about today's lesson 00:01:15.510 --> 00:01:19.770 in terms of something called the directional derivative 00:01:19.770 --> 00:01:22.360 and indicate, in a manner of speaking, 00:01:22.360 --> 00:01:25.590 that when we deal with functions of several variables, 00:01:25.590 --> 00:01:28.020 especially in the case that n equals 2, 00:01:28.020 --> 00:01:32.150 there is a very obvious geometrical interpretation, one 00:01:32.150 --> 00:01:35.380 that we made some use of in the previous lecture, 00:01:35.380 --> 00:01:37.010 but which I hope to make more use 00:01:37.010 --> 00:01:39.430 of in this particular lecture. 00:01:39.430 --> 00:01:42.300 Let's take a look and see what the situation is. 00:01:42.300 --> 00:01:45.120 Let's suppose I'm given that w is a function of the two 00:01:45.120 --> 00:01:49.300 independent variables x and y, and I have that w is therefore 00:01:49.300 --> 00:01:51.260 some function of x and y. 00:01:51.260 --> 00:01:54.310 And I can now talk about the graph of w, which 00:01:54.310 --> 00:01:57.870 we assume is some surface. 00:01:57.870 --> 00:02:00.900 And I've drawn it in this particular way. 00:02:00.900 --> 00:02:04.960 I take some point a comma b in the xy-plane 00:02:04.960 --> 00:02:07.170 on which f is defined. 00:02:07.170 --> 00:02:11.820 And at that particular point, I go to the corresponding point 00:02:11.820 --> 00:02:13.960 on the surface, which I call P_0, 00:02:13.960 --> 00:02:15.530 which has coordinates what? 00:02:15.530 --> 00:02:20.820 a, b, and c, where c is simply f of a comma b. 00:02:20.820 --> 00:02:23.850 That, of course, is how you graph a function 00:02:23.850 --> 00:02:25.910 of two independent variables. 00:02:25.910 --> 00:02:30.710 The w coordinate, so to speak, is simply the functional value 00:02:30.710 --> 00:02:34.050 of the x- and y-coordinates. 00:02:34.050 --> 00:02:38.140 Now, what we did last time was we essentially said, lookit, 00:02:38.140 --> 00:02:42.470 if we were to slice this surface by a plane which 00:02:42.470 --> 00:02:47.240 was either parallel to the wy-plane or to the wx-plane, 00:02:47.240 --> 00:02:49.980 we get special curves of intersection, 00:02:49.980 --> 00:02:54.810 and we use this to indicate the idea of partial derivatives. 00:02:54.810 --> 00:02:56.997 Now, without too much imagination, 00:02:56.997 --> 00:02:58.580 I think it should be very easy for you 00:02:58.580 --> 00:03:03.390 to visualize an arbitrary surface over your heads. 00:03:03.390 --> 00:03:06.230 You're standing in a particular point on the floor. 00:03:06.230 --> 00:03:10.290 At that point on the floor, you visualize a coordinate axis, 00:03:10.290 --> 00:03:12.670 which we'll call the x- and y-axis. 00:03:12.670 --> 00:03:15.970 Now, obviously, you can move out from the point at which you're 00:03:15.970 --> 00:03:18.940 standing along the x-axis. 00:03:18.940 --> 00:03:22.260 You can move out in the direction along the y-axis. 00:03:22.260 --> 00:03:24.350 You can visualize that how rapidly 00:03:24.350 --> 00:03:28.190 the height above your head is changing as you move out 00:03:28.190 --> 00:03:30.420 certainly will depend on whether you're moving 00:03:30.420 --> 00:03:33.170 along the x-axis or the y-axis. 00:03:33.170 --> 00:03:35.840 But the key point-- at least, the key point 00:03:35.840 --> 00:03:38.150 from the point of view of today's lecture-- 00:03:38.150 --> 00:03:41.070 is the fact that why were you restricted to either 00:03:41.070 --> 00:03:42.370 of these two directions? 00:03:42.370 --> 00:03:44.620 Why couldn't you have moved in any one 00:03:44.620 --> 00:03:47.020 of infinitely many different directions, 00:03:47.020 --> 00:03:50.030 namely from the point at which you're originating, 00:03:50.030 --> 00:03:53.040 you can move in any direction at all, because we're assuming, 00:03:53.040 --> 00:03:55.200 at least, that the floor on which you're standing 00:03:55.200 --> 00:03:56.440 is continuous. 00:03:56.440 --> 00:03:57.310 It's unbroken. 00:03:57.310 --> 00:03:59.230 You can move in these directions. 00:03:59.230 --> 00:04:02.610 And the idea that you get can be shown pictorially 00:04:02.610 --> 00:04:04.900 by saying, lookit. 00:04:04.900 --> 00:04:07.570 Let's assume that we're at the point a comma b. 00:04:07.570 --> 00:04:11.250 We have this surface over our head just as we did before. 00:04:11.250 --> 00:04:15.620 But now, instead of picking a direction either parallel 00:04:15.620 --> 00:04:19.459 to the wx-plane or to the wy-plane, 00:04:19.459 --> 00:04:24.230 let's pick some arbitrary direction s in which to move. 00:04:24.230 --> 00:04:26.620 And now, what I'm saying is if I now 00:04:26.620 --> 00:04:32.470 take the plane that passes through this direction 00:04:32.470 --> 00:04:37.370 s perpendicular to the xy-plane, that plane will also 00:04:37.370 --> 00:04:41.580 intersect this surface, passing through the point P_0. 00:04:41.580 --> 00:04:43.790 I get a curve of intersection, and I 00:04:43.790 --> 00:04:47.242 can talk about the slope of that particular curve. 00:04:47.242 --> 00:04:48.700 See, ultimately, this is what we're 00:04:48.700 --> 00:04:49.783 going to be talking about. 00:04:49.783 --> 00:04:52.180 Now, what is the slope of that particular curve? 00:04:52.180 --> 00:04:54.500 Well it's a derivative. 00:04:54.500 --> 00:04:58.060 It's a derivative of w with respect to s, 00:04:58.060 --> 00:05:01.090 where all of a sudden we've now chosen the s direction 00:05:01.090 --> 00:05:03.120 over here to be what we're going to take 00:05:03.120 --> 00:05:05.110 our derivative with respect to. 00:05:05.110 --> 00:05:07.720 In other words, it seems to make sense 00:05:07.720 --> 00:05:12.890 to talk about the derivative of f in the direction of s, 00:05:12.890 --> 00:05:15.320 evaluated at the point a comma b. 00:05:15.320 --> 00:05:18.260 In other words, just like we talked about f sub x of a comma 00:05:18.260 --> 00:05:21.160 b and f sub y of a comma b, why can't we 00:05:21.160 --> 00:05:26.220 talk about the derivative of f evaluated at a comma b, 00:05:26.220 --> 00:05:28.050 in the direction of s? 00:05:28.050 --> 00:05:31.620 And if we want to correlate this with the notation in the text, 00:05:31.620 --> 00:05:35.420 observe that we call that dw/ds. 00:05:35.420 --> 00:05:39.440 By the way, notice-- not the partial of w with respect to s, 00:05:39.440 --> 00:05:43.570 but the derivative of w with respect to s. 00:05:43.570 --> 00:05:46.780 Remember, the partials were essentially 00:05:46.780 --> 00:05:51.190 defined in terms of holding either x or y constant. 00:05:51.190 --> 00:05:55.820 Notice that in our general directional derivative idea, 00:05:55.820 --> 00:05:59.470 neither x nor y is being held constant. 00:05:59.470 --> 00:06:04.910 Notice that x and y are varying as we move along s. 00:06:04.910 --> 00:06:08.730 Notice, also by the way, that once x and y are restricted 00:06:08.730 --> 00:06:12.930 to move along s, they are no longer independent variables. 00:06:12.930 --> 00:06:14.650 It's a rather interesting point. 00:06:14.650 --> 00:06:17.410 We talk about w being a function of the two 00:06:17.410 --> 00:06:19.990 independent variables x and y, but as soon 00:06:19.990 --> 00:06:23.710 as we pick that direction s in the xy-plane, 00:06:23.710 --> 00:06:26.709 x and y have to be very specially related-- namely, 00:06:26.709 --> 00:06:28.750 according to the equation of a straight line that 00:06:28.750 --> 00:06:32.450 determines s-- for that point to be on the line. 00:06:32.450 --> 00:06:35.016 And that's why we talk about dw/ds. 00:06:35.016 --> 00:06:38.040 w is a function of a single variable 00:06:38.040 --> 00:06:41.110 once you restrict yourself to the particular direction s. 00:06:41.110 --> 00:06:43.970 At any rate, the question that we want to come to grips with 00:06:43.970 --> 00:06:46.950 is, how do you find dw/ds? 00:06:46.950 --> 00:06:50.640 Well, here again is the beauty of our logical structure. 00:06:50.640 --> 00:06:53.890 By definition-- this is the same definition we knew early 00:06:53.890 --> 00:06:55.605 in part one of our course. 00:06:55.605 --> 00:06:59.890 dw/ds, by definition, is the limit as delta 00:06:59.890 --> 00:07:05.450 s approaches 0, delta w divided by delta s. 00:07:05.450 --> 00:07:08.790 Now, the thing is, what's very difficult to compute 00:07:08.790 --> 00:07:10.420 in real life is delta w. 00:07:10.420 --> 00:07:13.465 After all, this w, being f of x, y, 00:07:13.465 --> 00:07:17.030 f can be a very, very complicated surface. 00:07:17.030 --> 00:07:20.360 And to actually find the true change in w-- well, heck. 00:07:20.360 --> 00:07:25.060 We already saw this in part one when we compared this delta y 00:07:25.060 --> 00:07:26.540 with delta y tan. 00:07:26.540 --> 00:07:30.800 To actually find the change in y was a much more difficult job 00:07:30.800 --> 00:07:33.470 than to find the change in y to the tangent line. 00:07:33.470 --> 00:07:35.490 What we're saying here is, lookit. 00:07:35.490 --> 00:07:39.340 We already know how to find the change in w, 00:07:39.340 --> 00:07:42.080 not to the surface, but to the plane which 00:07:42.080 --> 00:07:45.650 is tangent to the surface at our point P sub 0. 00:07:45.650 --> 00:07:47.820 In other words, we have already discussed 00:07:47.820 --> 00:07:51.100 that delta w tan is the partial of f 00:07:51.100 --> 00:07:54.830 with respect to x evaluated at a comma b times delta x, 00:07:54.830 --> 00:07:58.270 plus the partial of f with respect to y at the point 00:07:58.270 --> 00:08:01.530 a comma b times delta y. 00:08:01.530 --> 00:08:03.780 And now, what we say is-- and I've 00:08:03.780 --> 00:08:06.260 written this to accentuate it, because I 00:08:06.260 --> 00:08:08.010 have to talk very strongly about this. 00:08:08.010 --> 00:08:11.960 It's a point that, if I don't make, most of you, at least, 00:08:11.960 --> 00:08:14.680 will allow me to slip over this and not even notice 00:08:14.680 --> 00:08:16.940 that I've missed something very, very crucial. 00:08:16.940 --> 00:08:21.780 But let me assume that I can approximate delta w by delta w 00:08:21.780 --> 00:08:22.280 tan. 00:08:22.280 --> 00:08:26.850 In other words, let's suppose that delta w 00:08:26.850 --> 00:08:30.440 tan is a reasonable approximation for delta w. 00:08:30.440 --> 00:08:33.549 You say, well, what's such a big assumption about that? 00:08:33.549 --> 00:08:35.309 And I'm going to save that for the very 00:08:35.309 --> 00:08:38.650 last part of the lecture, because my belief is 00:08:38.650 --> 00:08:41.950 that the subtlety is so great that I would like to leave that 00:08:41.950 --> 00:08:45.630 for the very end, and go through as if the subtlety didn't exist 00:08:45.630 --> 00:08:47.430 so that you get the computational feeling 00:08:47.430 --> 00:08:48.790 as to what's going on here. 00:08:48.790 --> 00:08:50.660 But here's the interesting point. 00:08:50.660 --> 00:08:55.590 Notice that delta w is a change in w as you move from the point 00:08:55.590 --> 00:08:58.280 a comma b to some other point in the plane. 00:08:58.280 --> 00:08:59.880 It's a change in height. 00:08:59.880 --> 00:09:02.080 Now, obviously, that change in w is 00:09:02.080 --> 00:09:04.980 going to depend very strongly on what direction 00:09:04.980 --> 00:09:06.060 you're moving in. 00:09:06.060 --> 00:09:09.780 On the other hand, how was delta w tan computed? 00:09:09.780 --> 00:09:17.140 Delta w tan was computed just by knowing 00:09:17.140 --> 00:09:19.200 two special directional derivatives known 00:09:19.200 --> 00:09:21.300 as the partial derivatives. 00:09:21.300 --> 00:09:23.820 You see, notice that to get delta w 00:09:23.820 --> 00:09:27.310 tan, I have made the assumption that all I have to know 00:09:27.310 --> 00:09:29.440 is what's happening in the x direction 00:09:29.440 --> 00:09:32.240 and what's happening in the y direction, everything 00:09:32.240 --> 00:09:36.010 else being determined from the function evaluated at a comma 00:09:36.010 --> 00:09:36.750 b. 00:09:36.750 --> 00:09:39.720 So this is really a very strong assumption, 00:09:39.720 --> 00:09:44.800 that delta w is determined pretty closely by delta w tan. 00:09:44.800 --> 00:09:48.130 And it turns out to be almost universally true 00:09:48.130 --> 00:09:50.990 and we're going to save that part, as I say, for the end. 00:09:50.990 --> 00:09:53.110 But for now, let's suppose that we 00:09:53.110 --> 00:09:57.530 are allowed to replace delta w by delta w tan. 00:09:57.530 --> 00:10:02.550 If we do that, and we go back to our definition for dw/ds-- 00:10:02.550 --> 00:10:05.350 namely, the limit as delta s approaches 0, 00:10:05.350 --> 00:10:08.450 delta w divided by delta s-- we now 00:10:08.450 --> 00:10:14.130 replace delta w by delta w tan and divide through by delta 00:10:14.130 --> 00:10:17.480 s to find delta w over delta s. 00:10:17.480 --> 00:10:22.090 We have, assuming that this is a legal substitution, 00:10:22.090 --> 00:10:27.160 that delta w over delta s is the partial of f with respect to x, 00:10:27.160 --> 00:10:31.450 evaluated at a comma b, times delta x divided by delta 00:10:31.450 --> 00:10:34.250 s plus the partial of f with respect to y 00:10:34.250 --> 00:10:39.180 evaluated at a comma b times the change in y 00:10:39.180 --> 00:10:42.990 divided by the change in s-- delta y divided by delta s. 00:10:42.990 --> 00:10:46.450 Now, remember, in the s direction, x and y 00:10:46.450 --> 00:10:48.330 are not independent variables. 00:10:48.330 --> 00:10:50.560 In fact, how are they related? 00:10:50.560 --> 00:10:53.960 Let's isolate our little diagram here so 00:10:53.960 --> 00:10:56.760 that we see in what direction we're moving here. 00:10:56.760 --> 00:10:58.780 We're starting at the point a comma b. 00:10:58.780 --> 00:11:00.860 And by the way, to get the proper orientation 00:11:00.860 --> 00:11:04.290 here, as I've drawn the xy-plane here, 00:11:04.290 --> 00:11:08.150 imagine the surface coming out from the blackboard. 00:11:08.150 --> 00:11:12.000 In other words, the height is really being measured away 00:11:12.000 --> 00:11:13.330 from the blackboard here. 00:11:13.330 --> 00:11:14.850 That's where my surface is. 00:11:14.850 --> 00:11:16.920 I move in the direction of s. 00:11:16.920 --> 00:11:18.630 Here is a delta x. 00:11:18.630 --> 00:11:20.762 Here is a delta y. 00:11:20.762 --> 00:11:23.740 s has a constant direction, a constant slope. 00:11:23.740 --> 00:11:25.030 It's a straight line. 00:11:25.030 --> 00:11:28.940 Call the angle that it makes with the positive x-axis phi. 00:11:28.940 --> 00:11:32.950 Notice that no matter how big delta x and delta y are, 00:11:32.950 --> 00:11:36.280 they are related by similar triangles to what? 00:11:36.280 --> 00:11:40.720 The delta x divided by delta s will always be cosine phi, 00:11:40.720 --> 00:11:45.300 and delta y divided by delta s will always be sine phi. 00:11:45.300 --> 00:11:48.890 And therefore, if I replace delta x divided by delta 00:11:48.890 --> 00:11:53.640 s by cosine phi, delta y divided by delta s by sine phi, 00:11:53.640 --> 00:11:56.050 I obtain that-- what? 00:11:56.050 --> 00:12:01.110 Delta w divided by delta s is equal to the partial 00:12:01.110 --> 00:12:04.730 of f with respect to x evaluated at point a comma b 00:12:04.730 --> 00:12:08.460 times the cosine of phi, plus the partial of f with respect 00:12:08.460 --> 00:12:12.060 to y evaluated at a comma b times the sine of phi, 00:12:12.060 --> 00:12:13.720 where phi is the what? 00:12:13.720 --> 00:12:18.444 The angle that the direction s makes with the positive x-axis. 00:12:18.444 --> 00:12:20.860 At any rate, notice, by the way, that these things are all 00:12:20.860 --> 00:12:25.710 constants once s is chosen, once the point a comma b is fixed. 00:12:25.710 --> 00:12:29.720 Therefore, when I pass to the limit, nothing really changes. 00:12:29.720 --> 00:12:31.590 In other words, this thing that I'm 00:12:31.590 --> 00:12:34.610 calling the directional derivative of f 00:12:34.610 --> 00:12:38.870 in the direction of s evaluated at a comma b 00:12:38.870 --> 00:12:40.350 turns out to be what? 00:12:40.350 --> 00:12:43.950 The partial of f with respect to x at the point a comma b times 00:12:43.950 --> 00:12:47.970 cosine phi, plus the partial of f with respect to y 00:12:47.970 --> 00:12:51.220 evaluated at a comma b times the sine of phi. 00:12:51.220 --> 00:12:55.130 And what I'd like you to notice is that in these two terms, 00:12:55.130 --> 00:12:58.760 one factor of each term is determined solely 00:12:58.760 --> 00:13:00.750 by the point a comma b. 00:13:00.750 --> 00:13:04.580 In other words, notice that these two factors here 00:13:04.580 --> 00:13:06.880 are just partial derivatives and have nothing 00:13:06.880 --> 00:13:07.830 to do with direction. 00:13:07.830 --> 00:13:10.630 They're determined solely by the choice of the function 00:13:10.630 --> 00:13:13.340 f and the point a comma b. 00:13:13.340 --> 00:13:17.330 On the other hand, notice that these two factors, cosine phi 00:13:17.330 --> 00:13:20.300 and sine phi, have nothing to do with f 00:13:20.300 --> 00:13:23.270 and have to do only with the direction 00:13:23.270 --> 00:13:27.000 s itself, which, again, should make intuitive sense to you. 00:13:27.000 --> 00:13:29.470 That if you're asking, in terms of the surface 00:13:29.470 --> 00:13:32.560 being up over your head, for a directional derivative, 00:13:32.560 --> 00:13:35.070 obviously, once the surface is given, 00:13:35.070 --> 00:13:36.460 the directional derivative should 00:13:36.460 --> 00:13:40.880 depend on two things: one, what point you start at, 00:13:40.880 --> 00:13:44.950 and the other, what direction you move in once you've started 00:13:44.950 --> 00:13:46.340 at that particular point. 00:13:46.340 --> 00:13:47.972 And what direction you move in has 00:13:47.972 --> 00:13:49.680 nothing to do with what the surface looks 00:13:49.680 --> 00:13:51.410 like over your head. 00:13:51.410 --> 00:13:55.600 At any rate, what we now do is invoke our dot product 00:13:55.600 --> 00:13:57.990 notation-- in other words, this little trick 00:13:57.990 --> 00:14:00.420 that we talked about when we learned dot products. 00:14:00.420 --> 00:14:06.400 This particular sum can be written very suggestively 00:14:06.400 --> 00:14:09.070 as the dot product of two vectors. 00:14:09.070 --> 00:14:11.550 Namely, I will write this as what? 00:14:11.550 --> 00:14:13.710 The vector whose components are f 00:14:13.710 --> 00:14:18.210 sub x and f sub y dotted with the vector whose components are 00:14:18.210 --> 00:14:21.080 cosine phi and sine phi, because remember, 00:14:21.080 --> 00:14:24.230 when you dot two vectors in Cartesian coordinates, 00:14:24.230 --> 00:14:27.670 you multiply them coefficient by coefficient. 00:14:27.670 --> 00:14:31.650 To make a long story short, dw/ds is what? 00:14:31.650 --> 00:14:34.960 It's this vector, whose i component 00:14:34.960 --> 00:14:39.040 is f sub x evaluated at a comma b, whose j complement is 00:14:39.040 --> 00:14:41.810 f sub y evaluated at a comma b. 00:14:41.810 --> 00:14:45.450 I call that vector g of a comma b. 00:14:45.450 --> 00:14:48.010 I'm going to give that a special name a little bit later. 00:14:48.010 --> 00:14:50.590 But for now, it's very important to notice 00:14:50.590 --> 00:14:52.460 that this is not a number. 00:14:52.460 --> 00:14:54.990 It's an ordered pair of numbers. 00:14:54.990 --> 00:14:57.134 In other words, it's a vector, and if you 00:14:57.134 --> 00:14:59.300 want to think of this as a vector, what we're saying 00:14:59.300 --> 00:15:00.000 is what? 00:15:00.000 --> 00:15:02.790 Think of the vector whose i component 00:15:02.790 --> 00:15:06.130 is the partial of f with respect to x evaluated at a comma 00:15:06.130 --> 00:15:11.370 b, whose j complement is f sub y evaluated at a comma b. 00:15:11.370 --> 00:15:13.450 Notice that these are numbers, because a 00:15:13.450 --> 00:15:15.530 and b are fixed constants here. 00:15:15.530 --> 00:15:20.210 Therefore, this vector, g of a comma b, is a constant vector, 00:15:20.210 --> 00:15:22.300 and it's in the xy-plane. 00:15:22.300 --> 00:15:24.000 It's a 2-tuple. 00:15:24.000 --> 00:15:27.490 On the other hand, the other vector, (cosine phi, sine phi), 00:15:27.490 --> 00:15:31.830 hopefully, you recognize by now is nothing more than the unit 00:15:31.830 --> 00:15:34.250 vector in the direction of s. 00:15:34.250 --> 00:15:36.760 You see, this is the unit vector in the direction of s. 00:15:36.760 --> 00:15:42.530 So if I now use my abbreviation, dw/ds evaluated at a comma 00:15:42.530 --> 00:15:47.120 b-- in other words, the directional derivative of w 00:15:47.120 --> 00:15:51.200 in the direction of s evaluated at a comma b-- is just 00:15:51.200 --> 00:15:55.580 my vector g of a comma b dotted with the unit vector 00:15:55.580 --> 00:15:57.110 in the direction of s. 00:15:57.110 --> 00:15:58.880 Now, observe have two things. 00:15:58.880 --> 00:16:01.920 First of all, when you dot, this is a constant. 00:16:01.920 --> 00:16:04.780 I can't change this once a and b are given. 00:16:04.780 --> 00:16:06.510 This is fixed. 00:16:06.510 --> 00:16:09.220 So all I can vary is u sub s. 00:16:09.220 --> 00:16:12.950 But u sub s is a unit vector, so the only way I can vary u 00:16:12.950 --> 00:16:15.240 sub s is to change its direction. 00:16:15.240 --> 00:16:19.010 Notice that for two vectors of constant magnitude, 00:16:19.010 --> 00:16:21.270 their dot product is maximum when 00:16:21.270 --> 00:16:23.210 the two vectors are parallel. 00:16:23.210 --> 00:16:26.610 In other words, this will be as big as possible 00:16:26.610 --> 00:16:32.310 when u sub s is chosen to be in the direction of my vector g. 00:16:32.310 --> 00:16:35.960 In other words, dw/ds at a comma b 00:16:35.960 --> 00:16:40.810 is maximum in the direction of g of a comma b. 00:16:40.810 --> 00:16:42.600 That's the first thing to observe. 00:16:42.600 --> 00:16:45.690 The second thing is that when they are parallel, 00:16:45.690 --> 00:16:48.200 the cosine of the angle between them is 1. 00:16:48.200 --> 00:16:50.760 So the magnitude of this vector will just 00:16:50.760 --> 00:16:53.180 be the product of these two magnitudes. 00:16:53.180 --> 00:16:56.730 But the magnitude of u sub s, u sub s being a unit vector, 00:16:56.730 --> 00:16:57.740 is 1. 00:16:57.740 --> 00:17:00.720 Therefore, the maximum magnitude not only 00:17:00.720 --> 00:17:05.490 occurs in the direction of g, but it is also numerically 00:17:05.490 --> 00:17:09.220 equal to the magnitude of g. 00:17:09.220 --> 00:17:13.660 In other words, the maximum value of dw/ds 00:17:13.660 --> 00:17:17.140 evaluated at a comma b not only occurs 00:17:17.140 --> 00:17:19.119 in the direction of the vector g, 00:17:19.119 --> 00:17:23.910 but that maximum magnitude is the magnitude of g evaluated 00:17:23.910 --> 00:17:25.400 at a comma b. 00:17:25.400 --> 00:17:30.600 For that reason, g of a comma b is given a very important name. 00:17:30.600 --> 00:17:34.790 And I decided to hold off on the name until as late as possible 00:17:34.790 --> 00:17:36.820 so that the name wouldn't frighten you. 00:17:36.820 --> 00:17:39.380 But the name is the gradient vector. 00:17:39.380 --> 00:17:42.280 In other words, the vector g of a comma b 00:17:42.280 --> 00:17:45.990 is called the gradient of f at a comma b. 00:17:45.990 --> 00:17:48.640 And it's usually written in this notation. 00:17:48.640 --> 00:17:51.250 An upside down delta. 00:17:51.250 --> 00:17:54.200 It's called del, usually, with an arrow over it, 00:17:54.200 --> 00:17:56.570 or in boldface print in the text. 00:17:56.570 --> 00:17:58.990 And it's written this way, and it's read what? 00:17:58.990 --> 00:18:02.750 The gradient of f evaluated at a comma b. 00:18:02.750 --> 00:18:06.060 What is the gradient of f evaluated at a comma b? 00:18:06.060 --> 00:18:09.660 It's the vector, which gives you the hint as to how 00:18:09.660 --> 00:18:12.620 to compute the directional derivative in any direction 00:18:12.620 --> 00:18:13.650 that you wish. 00:18:13.650 --> 00:18:16.280 Namely, in terms of the gradient vector, 00:18:16.280 --> 00:18:20.420 the directional derivative of f in the direction of s, 00:18:20.420 --> 00:18:23.410 evaluated at a, b, is the gradient 00:18:23.410 --> 00:18:28.260 of f evaluated at a comma b dotted with the unit vector 00:18:28.260 --> 00:18:29.520 in the direction of s. 00:18:29.520 --> 00:18:31.830 By the way, you may recall that when 00:18:31.830 --> 00:18:35.180 you dot a vector with a unit vector, 00:18:35.180 --> 00:18:39.310 you get the projection of that vector in the direction 00:18:39.310 --> 00:18:40.420 of the unit vector. 00:18:40.420 --> 00:18:42.990 In other words, the directional derivative-- another way 00:18:42.990 --> 00:18:45.960 of looking at this physically is nothing more 00:18:45.960 --> 00:18:49.840 than the projection of the gradient vector 00:18:49.840 --> 00:18:53.700 onto the given direction in which you're moving. 00:18:53.700 --> 00:18:57.390 And the important point is that this particular definition 00:18:57.390 --> 00:19:00.570 does not depend on our coordinate system. 00:19:00.570 --> 00:19:04.380 What is interesting is that, in Cartesian coordinates, 00:19:04.380 --> 00:19:07.970 there is a very simple way of computing the gradient vector. 00:19:07.970 --> 00:19:11.030 Namely, the i component of the gradient vector 00:19:11.030 --> 00:19:12.900 is just the partial of f with respect 00:19:12.900 --> 00:19:16.010 to x and the j component is just the partial 00:19:16.010 --> 00:19:17.700 of f with respect to y. 00:19:17.700 --> 00:19:20.840 But that was a very special case, because, you see, 00:19:20.840 --> 00:19:26.430 i and x happen to have the same direction, as do y and j. 00:19:26.430 --> 00:19:30.450 For arbitrary coordinate systems, this need not be true. 00:19:30.450 --> 00:19:33.032 And I'm going to drill you on that in the exercises. 00:19:33.032 --> 00:19:34.490 But in other words, what I'm saying 00:19:34.490 --> 00:19:36.640 is remember the gradient vector in terms 00:19:36.640 --> 00:19:38.890 of a maximum directional derivative. 00:19:38.890 --> 00:19:42.310 Don't memorize it as a formula, because if you do, 00:19:42.310 --> 00:19:43.700 you're going to get in trouble. 00:19:43.700 --> 00:19:46.830 For example, if I were to give my surface 00:19:46.830 --> 00:19:51.060 in polar coordinates, say w of some function of r and theta, 00:19:51.060 --> 00:19:53.060 then it turns out-- and there's an exercise 00:19:53.060 --> 00:19:55.750 on this in the notes-- that the gradient of f 00:19:55.750 --> 00:20:01.550 is the partial of w with respect to r times u sub r plus-- 00:20:01.550 --> 00:20:03.440 and here's the big difference-- 1 00:20:03.440 --> 00:20:08.040 over r times the partial of w with respect to theta times 00:20:08.040 --> 00:20:09.140 u sub theta. 00:20:09.140 --> 00:20:11.020 In other words, the gradient vector 00:20:11.020 --> 00:20:14.580 is not the partial of w with respect to r times 00:20:14.580 --> 00:20:18.870 u sub r plus the partial of w with respect to theta times 00:20:18.870 --> 00:20:19.580 u sub theta. 00:20:19.580 --> 00:20:22.660 In other words, you don't just mechanically differentiate 00:20:22.660 --> 00:20:24.230 with respect to these variables. 00:20:24.230 --> 00:20:27.500 And the key reason that you can't do this-- well, lookit. 00:20:27.500 --> 00:20:29.290 Let's just look at this little diagram. 00:20:29.290 --> 00:20:31.830 And I think the whole idea will become very clear. 00:20:31.830 --> 00:20:34.000 Remember that in polar coordinates 00:20:34.000 --> 00:20:36.280 r is denoted this way. 00:20:36.280 --> 00:20:39.730 u sub theta is at right angles to r. 00:20:39.730 --> 00:20:44.920 Notice that u sub theta is not in the direction of theta. 00:20:44.920 --> 00:20:46.920 Notice that the direction of theta 00:20:46.920 --> 00:20:50.390 is sort of the tangent to this circle of radius 00:20:50.390 --> 00:20:51.750 r at this point. 00:20:51.750 --> 00:20:54.580 If I call this increment d theta, 00:20:54.580 --> 00:20:57.062 notice that this arc length is r d 00:20:57.062 --> 00:21:00.250 theta, so the vector in the direction of u sub theta 00:21:00.250 --> 00:21:03.480 is r d theta, not d theta. 00:21:03.480 --> 00:21:05.890 I don't know if you noticed that, but coming back up 00:21:05.890 --> 00:21:09.240 here for a moment, notice that this was OK here, 00:21:09.240 --> 00:21:13.190 because r was in the same direction as u sub r. 00:21:13.190 --> 00:21:18.010 Notice, however, that it's r d theta which is in the u sub 00:21:18.010 --> 00:21:19.770 theta direction. 00:21:19.770 --> 00:21:22.630 Again, I leave most of these details for the notes. 00:21:22.630 --> 00:21:25.460 But I feel that if I don't say these things to you, 00:21:25.460 --> 00:21:28.430 it becomes very easy to miss these points when 00:21:28.430 --> 00:21:30.660 we talk about them or write about them, 00:21:30.660 --> 00:21:33.530 but somehow I hope that by you hearing me say this, 00:21:33.530 --> 00:21:35.390 you will be keyed in when you come 00:21:35.390 --> 00:21:39.460 to these concepts in the unit that we're studying. 00:21:39.460 --> 00:21:42.230 But I think the best way to augment what we're doing 00:21:42.230 --> 00:21:45.100 is by means of a specific example. 00:21:45.100 --> 00:21:49.890 Let us suppose that we're given the surface w equals f of x, y, 00:21:49.890 --> 00:21:53.250 where f of x, y is x to the fifth plus x cubed y 00:21:53.250 --> 00:21:54.870 plus y to the fifth. 00:21:54.870 --> 00:21:57.290 And we want to compute the directional derivative 00:21:57.290 --> 00:22:01.540 of f at the point 1 comma 1 in the direction-- 00:22:01.540 --> 00:22:06.940 let's call it s sub 1, where s sub 1 is the direction that 00:22:06.940 --> 00:22:11.212 goes from the point 1 comma 1 to the point 4 comma 5. 00:22:11.212 --> 00:22:13.170 Now, what we're saying is-- and I guess, maybe, 00:22:13.170 --> 00:22:17.570 if we look at these two diagrams concurrently, 00:22:17.570 --> 00:22:19.180 maybe this'll be easier to see. 00:22:19.180 --> 00:22:21.600 Here we are at the point 1 comma 1. 00:22:21.600 --> 00:22:26.650 We want to see how fast the slope over our head-- the w 00:22:26.650 --> 00:22:30.160 value-- is changing in the direction of s_1, where 00:22:30.160 --> 00:22:32.950 s_1 is chosen to be what? 00:22:32.950 --> 00:22:35.540 We're moving from the point 1 comma 1 00:22:35.540 --> 00:22:39.290 in the xy-plane to the point 4 comma 5. 00:22:39.290 --> 00:22:41.100 See, we're moving in this direction 00:22:41.100 --> 00:22:45.450 and we want to see how fast w is changing over our heads, which 00:22:45.450 --> 00:22:49.120 geometrically means you draw this plane, 00:22:49.120 --> 00:22:53.840 intersect it with the particular surface here. 00:22:53.840 --> 00:22:59.090 And this point, P_0-- what we really want geometrically 00:22:59.090 --> 00:22:59.780 is what? 00:22:59.780 --> 00:23:03.030 The slope of the line tangent to this curve 00:23:03.030 --> 00:23:08.440 in the w, s_1 plane tangent to this curve at the point P_0. 00:23:08.440 --> 00:23:11.420 And my claim is that this can be done very, very 00:23:11.420 --> 00:23:13.870 easy from a mechanical point of view 00:23:13.870 --> 00:23:16.540 now that we have our gradient vector behind us. 00:23:16.540 --> 00:23:19.570 Namely, what we do is, given what 00:23:19.570 --> 00:23:25.430 w looks like as a function of x and y, we take the partial of w 00:23:25.430 --> 00:23:28.600 with respect to both x and y, which, hopefully, 00:23:28.600 --> 00:23:30.620 you can all do quite mechanically now 00:23:30.620 --> 00:23:32.570 based on our last unit's work. 00:23:32.570 --> 00:23:35.360 We differentiate, first holding y constant, 00:23:35.360 --> 00:23:37.310 then holding x constant. 00:23:37.310 --> 00:23:39.060 At any rate, we obtain what? 00:23:39.060 --> 00:23:41.800 That the partial of w with respect to x 00:23:41.800 --> 00:23:45.290 is 5 x to the fourth plus 3 x squared y. 00:23:45.290 --> 00:23:47.880 And so if we compute that at the point 1 00:23:47.880 --> 00:23:50.210 comma 1, when x and y are both 1, 00:23:50.210 --> 00:23:52.830 this simply turns out to be 8. 00:23:52.830 --> 00:23:56.840 In a similar way, the partial of w with respect to y 00:23:56.840 --> 00:23:59.710 is x cubed plus 6 y to the fifth. 00:23:59.710 --> 00:24:02.810 So if we compute that at the point 1 comma 1, 00:24:02.810 --> 00:24:04.710 that turns out to be 7. 00:24:04.710 --> 00:24:06.610 In other words, then, by definition 00:24:06.610 --> 00:24:10.900 of our gradient, which is the partial of f with respect 00:24:10.900 --> 00:24:14.200 to x evaluated at 1 comma 1 times i, 00:24:14.200 --> 00:24:16.420 plus the partial of f with respect 00:24:16.420 --> 00:24:19.580 to y evaluated at 1 comma 1 times j, 00:24:19.580 --> 00:24:24.640 the gradient of f at 1 comma 1 is just 8i plus 7j. 00:24:24.640 --> 00:24:26.875 Very easy to write down mechanically when you're 00:24:26.875 --> 00:24:28.650 using Cartesian coordinates. 00:24:28.650 --> 00:24:32.510 Now, let me make a brief aside, an interruption here. 00:24:32.510 --> 00:24:36.600 The idea is to emphasize what the gradient vector means. 00:24:36.600 --> 00:24:41.030 What this tells me is that if I were to leave the point 1 00:24:41.030 --> 00:24:46.820 comma 1 in the direction of the vector 8i plus 4j-- 00:24:46.820 --> 00:24:49.710 if I were to leave in that direction, 00:24:49.710 --> 00:24:51.230 that would be the direction in which 00:24:51.230 --> 00:24:53.610 the directional derivative would be maximum. 00:24:53.610 --> 00:24:56.900 And moreover, that maximum directional derivative 00:24:56.900 --> 00:25:00.260 would just be the magnitude of this gradient vector. 00:25:00.260 --> 00:25:02.510 The magnitude of that gradient vector 00:25:02.510 --> 00:25:06.870 is just the square root of 8 squared plus 7 squared, which 00:25:06.870 --> 00:25:09.020 is the square root of 113. 00:25:09.020 --> 00:25:10.970 In other words, what this tells us 00:25:10.970 --> 00:25:14.550 is that the maximum directional derivative, leaving the point 1 00:25:14.550 --> 00:25:17.900 comma 1, is the square root of 113, 00:25:17.900 --> 00:25:21.630 and it occurs in a direction 8i plus 7j 00:25:21.630 --> 00:25:23.890 as you leave the point 1 comma 1. 00:25:23.890 --> 00:25:25.770 At any rate, getting back to the main stream 00:25:25.770 --> 00:25:29.250 of the problem, what we want is a directional derivative 00:25:29.250 --> 00:25:31.950 in the direction of s_1. 00:25:31.950 --> 00:25:32.810 That means what? 00:25:32.810 --> 00:25:36.140 We take our gradient vector, which is 8 comma 7m 00:25:36.140 --> 00:25:39.860 and dot that with the unit vector in the direction of s_1. 00:25:39.860 --> 00:25:42.430 You may recall from this diagram here 00:25:42.430 --> 00:25:46.320 that the vector in the direction of s_1 00:25:46.320 --> 00:25:50.940 has its i component equal to 3, its j complement equal to 4. 00:25:50.940 --> 00:25:53.980 This makes this a 3, 4, 5 right triangle. 00:25:53.980 --> 00:25:56.270 So the unit vector in this direction 00:25:56.270 --> 00:26:00.610 has as its components 3/5 and 4/5. 00:26:00.610 --> 00:26:04.200 In other words, the directional derivative of f at the point 1 00:26:04.200 --> 00:26:09.960 comma 1 in the given direction s_1 is just the gradient dotted 00:26:09.960 --> 00:26:14.170 with the unit vector 3/5*i plus 4/5*j. 00:26:14.170 --> 00:26:16.810 Just mechanically carrying out this operation 00:26:16.810 --> 00:26:19.210 leads to 52 over 5. 00:26:19.210 --> 00:26:23.160 And by the way, this had better turn out 00:26:23.160 --> 00:26:29.430 to be less than this, because this is what? 00:26:29.430 --> 00:26:32.740 The maximum value that the directional derivative 00:26:32.740 --> 00:26:33.270 can have. 00:26:33.270 --> 00:26:35.850 In other words, if we haven't made a mistake here, 00:26:35.850 --> 00:26:37.390 one of the checkpoints is what? 00:26:37.390 --> 00:26:41.450 That the vector can't project to be any longer 00:26:41.450 --> 00:26:43.920 than what it really is in this. 00:26:43.920 --> 00:26:48.630 It can't be more than the gradient vector. 00:26:48.630 --> 00:26:51.860 But at any rate, let's now conclude the lecture 00:26:51.860 --> 00:26:54.750 by coming to the part which is probably 00:26:54.750 --> 00:26:57.260 the hardest thing that we're going to encounter 00:26:57.260 --> 00:26:58.220 in the whole course. 00:26:58.220 --> 00:27:00.190 In a way, I feel a little bit like a man 00:27:00.190 --> 00:27:01.890 who fell off the Empire State Building. 00:27:01.890 --> 00:27:03.740 And when he went past the 40th floor, 00:27:03.740 --> 00:27:05.198 somebody said, "How are you doing?" 00:27:05.198 --> 00:27:06.650 And he said, "So far so good." 00:27:06.650 --> 00:27:10.010 And that is, we've taken some tremendous liberties here. 00:27:10.010 --> 00:27:12.110 And the biggest liberty that we've taken-- 00:27:12.110 --> 00:27:13.680 and it's not just a liberty. 00:27:13.680 --> 00:27:16.660 It's the kind of a liberty that to solve 00:27:16.660 --> 00:27:19.430 involves the foundations of our entire course. 00:27:19.430 --> 00:27:23.050 We are now at the grassroots of what 00:27:23.050 --> 00:27:25.990 at least the calculus of functions of several variables 00:27:25.990 --> 00:27:27.000 is all about. 00:27:27.000 --> 00:27:28.240 And that's this trouble spot. 00:27:28.240 --> 00:27:31.110 First of all, does delta w tan exist? 00:27:31.110 --> 00:27:32.690 That's the first question. 00:27:32.690 --> 00:27:35.210 Namely, how do you know that there is a tangent plane? 00:27:35.210 --> 00:27:37.640 Just because the surface happens to be smooth 00:27:37.640 --> 00:27:41.730 when you cut it by a plane parallel to the wy-plane 00:27:41.730 --> 00:27:45.910 and smooth when you cut it by a plane parallel to the wx-plane, 00:27:45.910 --> 00:27:50.076 how do you that it's going to be smooth for any given direction? 00:27:50.076 --> 00:27:52.140 See, that's the first intellectual question 00:27:52.140 --> 00:27:54.270 that comes up in the reading assignment that 00:27:54.270 --> 00:27:57.060 has to be solved effectively. 00:27:57.060 --> 00:28:00.150 First of all, does delta w tan exist meaningfully? 00:28:00.150 --> 00:28:04.700 And secondly, if it does exist, how is it related to delta w? 00:28:04.700 --> 00:28:07.970 And now, we come to that key theorem, the proof of which 00:28:07.970 --> 00:28:09.200 is quite hairy. 00:28:09.200 --> 00:28:11.200 It's done in the text. 00:28:11.200 --> 00:28:15.700 It's also done as an optional exercise 00:28:15.700 --> 00:28:19.400 to help you generalize what's done in the text. 00:28:19.400 --> 00:28:21.510 And it's the counterpart of what happens 00:28:21.510 --> 00:28:25.250 with differentials in functions of a single real variable. 00:28:25.250 --> 00:28:28.300 But the key theorem, which I'll state here without proof, 00:28:28.300 --> 00:28:29.710 is simply this. 00:28:29.710 --> 00:28:32.710 Suppose that w is a function of x and y 00:28:32.710 --> 00:28:36.180 and that f sub x and f sub y both happen 00:28:36.180 --> 00:28:39.901 to exist in some neighborhood of the point a comma b. 00:28:39.901 --> 00:28:40.400 All right? 00:28:40.400 --> 00:28:41.960 So far so good. 00:28:41.960 --> 00:28:45.360 Now, here is they key additional hypothesis. 00:28:45.360 --> 00:28:49.570 Suppose also that f sub x and f sub y happen 00:28:49.570 --> 00:28:52.630 to be continuous at a comma b. 00:28:52.630 --> 00:28:55.660 My claim is that if this additional hypothesis is 00:28:55.660 --> 00:28:58.620 obeyed, the tangent plane will exist. 00:28:58.620 --> 00:29:01.940 In other words, it's not enough for the directional derivative 00:29:01.940 --> 00:29:04.380 to exist in the x and y directions 00:29:04.380 --> 00:29:07.000 in order to guarantee that the directional derivative will 00:29:07.000 --> 00:29:08.650 exist in every direction. 00:29:08.650 --> 00:29:12.340 But it is enough provided that these directional derivatives 00:29:12.340 --> 00:29:14.150 happen to be continuous. 00:29:14.150 --> 00:29:17.020 And by the way, if these conditions are met, 00:29:17.020 --> 00:29:21.760 we say that f is a continuously differentiable function 00:29:21.760 --> 00:29:22.790 of x and y. 00:29:22.790 --> 00:29:26.180 But I'll talk about that more next time, or in the notes, 00:29:26.180 --> 00:29:27.940 or in the exercises. 00:29:27.940 --> 00:29:30.550 We're going to make a big issue over this sooner or later. 00:29:30.550 --> 00:29:33.310 But for now what I do want to do is just 00:29:33.310 --> 00:29:35.670 end with what the key theorem is. 00:29:35.670 --> 00:29:37.680 The key theorem says, lookit. 00:29:37.680 --> 00:29:41.020 Just like with one variable, if these conditions are met, 00:29:41.020 --> 00:29:44.590 then there is a very reasonable approximation 00:29:44.590 --> 00:29:47.160 to delta w by delta w tan. 00:29:47.160 --> 00:29:49.920 Namely, what the theorem says is, in this case, 00:29:49.920 --> 00:29:53.300 delta w will be the partial of f with respect 00:29:53.300 --> 00:29:57.170 to x evaluated at a comma b times delta x 00:29:57.170 --> 00:29:59.390 plus the partial of f with respect to y 00:29:59.390 --> 00:30:03.100 evaluated at a comma b times delta y-- and notice, 00:30:03.100 --> 00:30:05.940 of course, that this is a thing that we've been calling delta y 00:30:05.940 --> 00:30:10.880 tan-- plus an arrow. 00:30:10.880 --> 00:30:15.680 And the arrow has the form k_1 delta x plus k_2 delta y, 00:30:15.680 --> 00:30:20.900 where k_1 and k_2 both approach 0 as delta x and delta y 00:30:20.900 --> 00:30:21.930 approach 0. 00:30:21.930 --> 00:30:24.090 And this is very, very crucial. 00:30:24.090 --> 00:30:25.910 It's not enough, as we're going to see 00:30:25.910 --> 00:30:29.570 in the very next lecture, that delta x and delta y approach 0. 00:30:29.570 --> 00:30:34.680 It's that these things go to 0 as delta x and delta y go to 0. 00:30:34.680 --> 00:30:38.620 Consequently, these terms go to 0 faster. 00:30:38.620 --> 00:30:42.030 They go to 0 as a second-order infinitesimal. 00:30:42.030 --> 00:30:44.400 And what this really says is, look, 00:30:44.400 --> 00:30:48.660 for very small values of delta x and delta y, even when you're 00:30:48.660 --> 00:30:53.310 dealing with 0 over 9 forms, if you pick a sufficiently 00:30:53.310 --> 00:30:56.550 small neighborhood of the point a comma b-- 00:30:56.550 --> 00:30:59.400 and that's the key point, a sufficiently small neighborhood 00:30:59.400 --> 00:31:02.590 of the point a comma b-- then delta w 00:31:02.590 --> 00:31:06.510 is approximately equal to delta w tan 00:31:06.510 --> 00:31:10.920 And by the way, this holds also in several variables. 00:31:10.920 --> 00:31:16.070 In other words, I picked the case n equals 2 here simply 00:31:16.070 --> 00:31:18.660 so that we can utilize the geometry. 00:31:18.660 --> 00:31:22.510 Namely, what we're saying is, in terms of the geometry, 00:31:22.510 --> 00:31:25.800 if f happens to be a continuously differentiable 00:31:25.800 --> 00:31:29.800 function of x and y, and we look at the surface w 00:31:29.800 --> 00:31:34.650 equals f of x, y above the point a comma b, what we're saying 00:31:34.650 --> 00:31:38.100 is that in a neighborhood of that point, 00:31:38.100 --> 00:31:40.710 there is-- well, first of all we're saying what? 00:31:40.710 --> 00:31:45.300 A tangent plane exists to the surface above that point 00:31:45.300 --> 00:31:48.400 and that in a neighborhood of that point of tangency, 00:31:48.400 --> 00:31:51.990 the tangent plane is an excellent approximation 00:31:51.990 --> 00:31:54.810 for the true change in w. 00:31:54.810 --> 00:31:57.450 Now, what happens is, if n is greater than 2, 00:31:57.450 --> 00:32:01.890 we can no longer use the geometric interpretation. 00:32:01.890 --> 00:32:05.130 But what is important is that the analytic proof never 00:32:05.130 --> 00:32:06.610 makes use of the picture. 00:32:06.610 --> 00:32:08.300 And the key point is-- and I'm going 00:32:08.300 --> 00:32:10.320 to exploit this in future lectures. 00:32:10.320 --> 00:32:15.720 The really key point is that the delta w tan never gets messy, 00:32:15.720 --> 00:32:18.300 that the variables-- delta x, delta y, 00:32:18.300 --> 00:32:21.880 et cetera-- all occur as linear terms. 00:32:21.880 --> 00:32:25.830 And this is why the so-called linear algebra subject 00:32:25.830 --> 00:32:29.500 becomes so important in the study of functions 00:32:29.500 --> 00:32:30.900 of several variables. 00:32:30.900 --> 00:32:34.160 At any rate, I think this is enough for one lesson. 00:32:34.160 --> 00:32:36.800 And in our next lesson, what we will do 00:32:36.800 --> 00:32:39.980 is show how using this key theorem 00:32:39.980 --> 00:32:43.340 has its analog in something called the chain rule, 00:32:43.340 --> 00:32:46.290 just as it did in the case of part one 00:32:46.290 --> 00:32:49.730 when we studied functions of a single independent variable. 00:32:49.730 --> 00:32:52.070 At any rate, then, until next time. 00:32:52.070 --> 00:32:54.600 Good bye. 00:32:54.600 --> 00:32:56.980 Funding for the publication of this video 00:32:56.980 --> 00:33:01.850 was provided by the Gabrielle and Paul Rosenbaum Foundation. 00:33:01.850 --> 00:33:06.020 Help OCW continue to provide free and open access to MIT 00:33:06.020 --> 00:33:13.730 courses by making a donation at ocw.mit.edu/donate.