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 --> 00:00:17.862 at ocw.mit.edu. 00:00:31.360 --> 00:00:32.240 PROFESSOR: Hi. 00:00:32.240 --> 00:00:35.930 Our main aim today is to successfully 00:00:35.930 --> 00:00:39.320 conclude block three, and to set the stage 00:00:39.320 --> 00:00:41.530 for introducing block four. 00:00:41.530 --> 00:00:45.730 Now by way of brief review, we may have become so involved 00:00:45.730 --> 00:00:48.680 with our discussion of the last few lessons 00:00:48.680 --> 00:00:52.000 where we've done the chain rule, that we may have forgotten 00:00:52.000 --> 00:00:56.360 that the basic ingredient that we had arrived at, 00:00:56.360 --> 00:00:58.880 that set up our proof of the chain rule-- 00:00:58.880 --> 00:01:00.441 and then after that, we got involved 00:01:00.441 --> 00:01:02.440 in the computation of how to use the chain rule. 00:01:02.440 --> 00:01:04.849 But the basic ingredient was the idea 00:01:04.849 --> 00:01:10.140 that we had obtained a linear approximation, a delta w tan 00:01:10.140 --> 00:01:11.190 idea. 00:01:11.190 --> 00:01:14.680 And this concept leads to the idea of a differential, which 00:01:14.680 --> 00:01:17.170 is an extension of the idea of differentials 00:01:17.170 --> 00:01:19.510 as we knew them in part one of our course. 00:01:19.510 --> 00:01:22.860 And what I would like to do is to talk somewhat briefly 00:01:22.860 --> 00:01:26.570 about this topic, to show how it introduces what 00:01:26.570 --> 00:01:29.200 will be the main body of material 00:01:29.200 --> 00:01:32.080 in block four and an important application 00:01:32.080 --> 00:01:35.780 that we can handle now that will show why it's worth 00:01:35.780 --> 00:01:38.086 understanding certain things about differentials, even 00:01:38.086 --> 00:01:40.130 on a very elementary level. 00:01:40.130 --> 00:01:42.840 At any rate, the topic that I've chosen for today 00:01:42.840 --> 00:01:45.330 is called "Exact Differentials." 00:01:45.330 --> 00:01:47.650 And the idea, again, is completely 00:01:47.650 --> 00:01:50.110 analogous to what we did in calculus of a single variable. 00:01:50.110 --> 00:01:52.870 We start with w equals f of x, y. 00:01:52.870 --> 00:01:55.950 And we showed that if w was a continuously differentiable 00:01:55.950 --> 00:01:58.390 function of x and y, it made sense 00:01:58.390 --> 00:02:02.030 to emphasize the expression called delta w tan, which 00:02:02.030 --> 00:02:03.650 was the partial of f with respect 00:02:03.650 --> 00:02:05.930 to x evaluated at a given point times delta 00:02:05.930 --> 00:02:08.080 x, plus the partial of f with respect 00:02:08.080 --> 00:02:11.090 to y at that same point times delta y. 00:02:11.090 --> 00:02:14.040 And what we do is the same thing that we did, 00:02:14.040 --> 00:02:16.260 as I say, in calculus of a single variable. 00:02:16.260 --> 00:02:18.540 We introduce a new language. 00:02:18.540 --> 00:02:23.160 We write delta w tan now as dw. 00:02:23.160 --> 00:02:29.010 We write delta x as dx, delta y as dy. 00:02:29.010 --> 00:02:34.650 And what we say is that dw is the total differential of w. 00:02:34.650 --> 00:02:37.300 To see how this is analogous to what happened in calculus 00:02:37.300 --> 00:02:40.450 of a single variable, notice, for the sake of argument, 00:02:40.450 --> 00:02:43.890 that if f happens to be a function of x alone, 00:02:43.890 --> 00:02:45.440 so that there is no second variable. 00:02:45.440 --> 00:02:47.740 In other words, if f is independent of y, 00:02:47.740 --> 00:02:51.450 notice that the partial of f with respect to y will be 0. 00:02:51.450 --> 00:02:55.170 And we will then have that w is a function of x alone. 00:02:55.170 --> 00:02:56.830 dw, then, will be what? 00:02:56.830 --> 00:02:59.590 Well in that case, the partial of f with respect to x 00:02:59.590 --> 00:03:03.470 is the ordinary derivative of f with respect to x times dx. 00:03:03.470 --> 00:03:08.780 And we're back to our old recipe that dw is dw/dx times dx. 00:03:08.780 --> 00:03:12.040 That this is a natural extension of what happened in calculus 00:03:12.040 --> 00:03:13.370 of a single variable. 00:03:13.370 --> 00:03:14.090 All right? 00:03:14.090 --> 00:03:15.700 Now what leads into the next block 00:03:15.700 --> 00:03:19.200 and why we're going to drop this for the time being but to go up 00:03:19.200 --> 00:03:21.290 to a different aspect of this, is 00:03:21.290 --> 00:03:26.130 notice that if we can assume that dw can replace 00:03:26.130 --> 00:03:28.665 delta w-- if we can assume, as we talked 00:03:28.665 --> 00:03:30.040 about in the case of continuously 00:03:30.040 --> 00:03:33.430 differentiable functions, that the difference between delta w 00:03:33.430 --> 00:03:37.010 and delta w tan is negligible for sufficiently small delta 00:03:37.010 --> 00:03:42.780 x and delta y, notice that the equation dw equals f sub x dx 00:03:42.780 --> 00:03:46.265 plus f sub y dy, since f sub x and f sub y are computed 00:03:46.265 --> 00:03:48.500 at a given point, these become what? 00:03:48.500 --> 00:03:52.900 Linear combinations of dx and dy, a constant times dx 00:03:52.900 --> 00:03:54.760 plus a constant times dy. 00:03:54.760 --> 00:03:57.480 And we're into the subject of linear equations, which 00:03:57.480 --> 00:03:59.640 we will talk about next time. 00:03:59.640 --> 00:04:01.490 For the time being, all I want to point out 00:04:01.490 --> 00:04:05.920 is that dw is called the total differential of w. 00:04:05.920 --> 00:04:10.330 And that conversely, starting with any expression of the form 00:04:10.330 --> 00:04:14.430 some function M of x and y times dx plus some function 00:04:14.430 --> 00:04:16.600 N of x and y times dy-- in other words, 00:04:16.600 --> 00:04:19.800 M of x, y dx plus N of x, y dy, no matter 00:04:19.800 --> 00:04:22.200 what functions of x and y M and N are, 00:04:22.200 --> 00:04:26.140 any expression of this form is called a differential. 00:04:26.140 --> 00:04:27.670 By the way, you may recall that you 00:04:27.670 --> 00:04:31.180 were used to calling dx a differential back in calculus 00:04:31.180 --> 00:04:32.490 of a single variable. 00:04:32.490 --> 00:04:35.910 Notice that in particular, one choice of capital N 00:04:35.910 --> 00:04:38.540 is to have N of x, y be identically 0. 00:04:38.540 --> 00:04:41.200 We could have M of x, y be identically 1. 00:04:41.200 --> 00:04:46.470 In which case, M*dx plus N*dy would simply be 1*dx plus 0*dy, 00:04:46.470 --> 00:04:48.130 or dx. 00:04:48.130 --> 00:04:51.560 In other words, notice that this extended definition 00:04:51.560 --> 00:04:54.430 of a differential for two or more variables 00:04:54.430 --> 00:04:57.820 includes the definition for a single variable. 00:04:57.820 --> 00:04:59.920 At any rate, any expression of this type 00:04:59.920 --> 00:05:01.740 is called a differential. 00:05:01.740 --> 00:05:03.750 And again, a very natural question that comes up 00:05:03.750 --> 00:05:05.250 is who wants differentials? 00:05:05.250 --> 00:05:06.880 Why do we need them? 00:05:06.880 --> 00:05:10.000 And perhaps the best application comes 00:05:10.000 --> 00:05:12.260 from the clue that a certain subject is 00:05:12.260 --> 00:05:15.370 called "Differential Equations," rather than 00:05:15.370 --> 00:05:17.190 derivative equations. 00:05:17.190 --> 00:05:20.000 Let me illustrate this in terms of what I think 00:05:20.000 --> 00:05:21.780 is a rather nice example, in the sense 00:05:21.780 --> 00:05:23.610 that it will be hard to guess the answer, 00:05:23.610 --> 00:05:26.060 but rather easy to visualize what the problem says, 00:05:26.060 --> 00:05:27.077 at least. 00:05:27.077 --> 00:05:28.785 Let's imagine that I have a certain curve 00:05:28.785 --> 00:05:31.220 S in the xy-plane. 00:05:31.220 --> 00:05:34.030 I don't tell you anything else about that curve 00:05:34.030 --> 00:05:38.160 except that its slope at any point x comma y 00:05:38.160 --> 00:05:41.660 is given by the very interesting relationship 00:05:41.660 --> 00:05:44.680 that it's the quotient of the square of its distance 00:05:44.680 --> 00:05:49.280 from the origin over twice the product of its coordinates. 00:05:49.280 --> 00:05:51.525 In other words, to find the slope of the curve S 00:05:51.525 --> 00:05:56.720 at the point x comma y, I square the distance of the point 00:05:56.720 --> 00:06:00.340 from the origin, take the negative of that, 00:06:00.340 --> 00:06:02.720 divide by twice the product of the coordinates, 00:06:02.720 --> 00:06:04.956 and that's going to be the slope. 00:06:04.956 --> 00:06:06.830 Now certainly that makes sense geometrically. 00:06:06.830 --> 00:06:10.190 But what type of a curve would have that property? 00:06:10.190 --> 00:06:13.090 And even assuming that we knew such a curve, how in the world 00:06:13.090 --> 00:06:15.540 can you express it in a convenient manner? 00:06:15.540 --> 00:06:16.780 Well here's the idea. 00:06:16.780 --> 00:06:18.250 Again, back to the same technique 00:06:18.250 --> 00:06:20.240 that we used in calculus of a single variable. 00:06:20.240 --> 00:06:23.030 Algebraically, this is our equation. 00:06:23.030 --> 00:06:26.260 If we cross-multiply and collect all our terms 00:06:26.260 --> 00:06:28.300 onto one side of the equation, we 00:06:28.300 --> 00:06:31.650 obtain what we call the differential equation, see? 00:06:31.650 --> 00:06:34.900 It's an equation involving a differential. 00:06:34.900 --> 00:06:38.550 In fact, in this case, using our general notation, M of x, y 00:06:38.550 --> 00:06:40.390 is x squared plus y squared. 00:06:40.390 --> 00:06:42.790 And N of x, y is 2x*y. 00:06:42.790 --> 00:06:45.510 We have a differential equal to 0. 00:06:45.510 --> 00:06:47.340 Now without going into this right now, 00:06:47.340 --> 00:06:50.576 suppose I just happen to have a very quick eye. 00:06:50.576 --> 00:06:53.280 Well in fact, before I even say that, let me point this out. 00:06:53.280 --> 00:06:56.020 We've had problems like this in part one of our course. 00:06:56.020 --> 00:06:58.660 The main difference was that we could separate out 00:06:58.660 --> 00:06:59.310 the variables. 00:06:59.310 --> 00:07:01.690 We could get the x's and the y's separated. 00:07:01.690 --> 00:07:04.790 Remember now, y is implicitly a function of x here, 00:07:04.790 --> 00:07:07.240 in this expression, because of the equation. 00:07:07.240 --> 00:07:09.580 See? x and y are no longer independent 00:07:09.580 --> 00:07:11.570 when we equate this to 0. 00:07:11.570 --> 00:07:15.450 The question is though, we cannot separate 00:07:15.450 --> 00:07:17.540 the x's and the y's here. 00:07:17.540 --> 00:07:20.500 But if we were lucky-- that's where luck comes in-- we would 00:07:20.500 --> 00:07:23.790 recognize that treating y as a function of x, 00:07:23.790 --> 00:07:25.910 the differential on the left-hand side 00:07:25.910 --> 00:07:30.650 is precisely the differential of 1/3 x cubed plus x squared y. 00:07:30.650 --> 00:07:33.160 In other words, the partial of this with respect to x is 00:07:33.160 --> 00:07:37.890 simply x squared-- do I have... 00:07:37.890 --> 00:07:40.522 I think I may have this thing in reverse. 00:07:40.522 --> 00:07:41.230 Let me make sure. 00:07:41.230 --> 00:07:44.060 If I take the partial of this with respect to x, 00:07:44.060 --> 00:07:48.170 I have 1/3 x cubed plus 2x*y, right? 00:07:48.170 --> 00:07:51.290 And if I take the partial with respect to y, 00:07:51.290 --> 00:07:53.455 I simply have x squared. 00:07:53.455 --> 00:07:55.670 At any rate-- what I'm saying is, 00:07:55.670 --> 00:07:58.837 if we knew a function whose-- see, I have this in reverse. 00:07:58.837 --> 00:08:00.170 That's what's bothering me here. 00:08:00.170 --> 00:08:03.051 I think the square should go this way. 00:08:03.051 --> 00:08:03.925 That's not important. 00:08:03.925 --> 00:08:05.780 All I'm saying is if I've done this thing correctly, 00:08:05.780 --> 00:08:07.710 if you take the partial of this with respect 00:08:07.710 --> 00:08:11.220 to y-- with respect x, you should get this. 00:08:11.220 --> 00:08:13.887 If you take the partial of this with respect to y, 00:08:13.887 --> 00:08:14.720 you should get this. 00:08:14.720 --> 00:08:16.350 And that comes out correctly now. 00:08:16.350 --> 00:08:19.220 In other words, the partial of this with respect to x 00:08:19.220 --> 00:08:21.350 is x squared plus y squared. 00:08:21.350 --> 00:08:25.250 The partial of this with respect to y-- this term drops out, 00:08:25.250 --> 00:08:26.670 and there's just 2x*y. 00:08:26.670 --> 00:08:28.330 I'm sorry for that little mistake. 00:08:28.330 --> 00:08:31.000 But I won't really apologize for it in the sense 00:08:31.000 --> 00:08:34.559 that once we've found it-- or even if we didn't find it, 00:08:34.559 --> 00:08:35.820 the key theory is the same. 00:08:35.820 --> 00:08:39.850 The point is that if we knew a function whose differential was 00:08:39.850 --> 00:08:43.110 this, then the fact that this differential is 0 00:08:43.110 --> 00:08:46.760 means the expression itself must be a constant. 00:08:46.760 --> 00:08:54.570 In other words, 1/3 x cubed plus x y squared must be a constant. 00:08:54.570 --> 00:08:58.520 And therefore, what we're saying is if we now 00:08:58.520 --> 00:09:01.525 solve the equation-- I really am sorry to have 00:09:01.525 --> 00:09:02.400 botched this for you. 00:09:02.400 --> 00:09:05.060 But again, as I say, the idea basically 00:09:05.060 --> 00:09:06.440 comes through unimpeded. 00:09:06.440 --> 00:09:10.080 I now solve for y in terms of x, and I get what? 00:09:10.080 --> 00:09:13.460 That y is equal to-- transposing here-- 00:09:13.460 --> 00:09:16.720 plus or minus the square root of c minus 1/3 x 00:09:16.720 --> 00:09:18.960 cubed, all over x. 00:09:18.960 --> 00:09:21.330 In other words, the family of curves 00:09:21.330 --> 00:09:24.900 that satisfies this differential equation 00:09:24.900 --> 00:09:27.714 is precisely this family here, whatever that may look like. 00:09:27.714 --> 00:09:29.130 In other words, this is the family 00:09:29.130 --> 00:09:32.110 of curves that has the interesting slope property 00:09:32.110 --> 00:09:33.495 that we just discussed. 00:09:33.495 --> 00:09:35.350 Now you see, again, the reason I say 00:09:35.350 --> 00:09:37.340 that I wasn't too concerned about what 00:09:37.340 --> 00:09:39.720 this function actually was is, notice 00:09:39.720 --> 00:09:42.580 that I was just pulling it out of the hat for you here anyway. 00:09:42.580 --> 00:09:45.620 I said suppose we could find a function w such 00:09:45.620 --> 00:09:47.800 that dw was this. 00:09:47.800 --> 00:09:51.950 The major question that comes up in how one uses differentials 00:09:51.950 --> 00:09:54.990 and the like is as follows. 00:09:54.990 --> 00:09:58.440 Suppose all I told you was that we had the differential x 00:09:58.440 --> 00:10:02.620 squared plus y squared dx plus 2x*y*dy, and I said to you, 00:10:02.620 --> 00:10:07.390 find a function w such that dw will be x squared plus y 00:10:07.390 --> 00:10:10.590 squared dx plus 2x*y*dy. 00:10:10.590 --> 00:10:14.740 The key computational aid in doing this involves the fact 00:10:14.740 --> 00:10:16.670 that if u and v are independent variables-- 00:10:16.670 --> 00:10:18.550 and it's crucial that they be independent-- 00:10:18.550 --> 00:10:22.580 that if a*u plus b*v equals c*u plus d*v, 00:10:22.580 --> 00:10:27.780 it must be that a equals c, and b equals d. 00:10:27.780 --> 00:10:31.360 In other words, the only way that two linear combinations 00:10:31.360 --> 00:10:33.860 of independent variables can be equal 00:10:33.860 --> 00:10:37.310 is if they're equal coefficient by coefficient. 00:10:37.310 --> 00:10:39.150 You see, the proof is very simple. 00:10:39.150 --> 00:10:42.540 Namely, after all, if u and v are independent variables, 00:10:42.540 --> 00:10:47.660 why can't I pick v to be 0, let u be any non-zero number. 00:10:47.660 --> 00:10:50.500 Then if v is 0, when I equate these two, 00:10:50.500 --> 00:10:52.740 it says a*u equals c*u. 00:10:52.740 --> 00:10:56.550 Since u is not 0, I can cancel, and obtain that a equals c. 00:10:56.550 --> 00:11:00.690 And in a similar way letting u equal 0, I have b*v equals d*v. 00:11:00.690 --> 00:11:03.620 I can cancel the v's and get that b equals d. 00:11:03.620 --> 00:11:06.410 But it's crucial that u and v be independent. 00:11:06.410 --> 00:11:08.440 Because if u and v are not independent, 00:11:08.440 --> 00:11:13.320 how do I know that I can have u equal to 0 when v is not 0? 00:11:13.320 --> 00:11:16.470 For example, let's suppose that v and u were dependent. 00:11:16.470 --> 00:11:19.500 Suppose, for example, that v equals twice u. 00:11:19.500 --> 00:11:25.410 Observe, in this case, that 9u plus 4v-- since v is 2u-- 00:11:25.410 --> 00:11:27.540 turns out to be 17u. 00:11:27.540 --> 00:11:34.570 On the other hand, 7u plus 5v is 7u plus 10u, which is also 17u. 00:11:34.570 --> 00:11:39.410 Notice that 9u plus 4v equals 7u plus 5v, 00:11:39.410 --> 00:11:42.780 even though the coefficients don't line up as far 00:11:42.780 --> 00:11:45.540 as being equal is concerned. 00:11:45.540 --> 00:11:46.200 You see? 00:11:46.200 --> 00:11:49.130 In other words, notice that being able to equate things 00:11:49.130 --> 00:11:51.860 coefficient by coefficient hinges on the fact 00:11:51.860 --> 00:11:54.740 that the variables that we're dealing with are independent. 00:11:54.740 --> 00:11:56.810 And the key thing going back to this problem 00:11:56.810 --> 00:11:58.780 is, what were dx and dy? 00:11:58.780 --> 00:12:00.800 They were delta x and delta y. 00:12:00.800 --> 00:12:03.620 And as long as x and y are independent variables, 00:12:03.620 --> 00:12:07.185 certainly the change in x can be done independently 00:12:07.185 --> 00:12:08.670 of the change in y. 00:12:08.670 --> 00:12:12.540 In other words, dx and dy are independent variables. 00:12:12.540 --> 00:12:17.550 So if we now say, let's find that function w such that dw is 00:12:17.550 --> 00:12:23.550 x squared plus y squared dx plus 2x*y*dy, 00:12:23.550 --> 00:12:26.650 the one thing we know about dw, if it exists, 00:12:26.650 --> 00:12:30.070 by our basic definition of this section of this lecture, 00:12:30.070 --> 00:12:34.530 is that dw is the partial of f with respect to x times dx plus 00:12:34.530 --> 00:12:37.540 the partial of f with respect to y times dy. 00:12:37.540 --> 00:12:40.420 Since dx and dy are independent variables, 00:12:40.420 --> 00:12:43.170 the only way these two expressions can be identical 00:12:43.170 --> 00:12:45.170 is coefficient by coefficient. 00:12:45.170 --> 00:12:47.230 That means in particular, therefore, 00:12:47.230 --> 00:12:48.740 that the partial of f with respect 00:12:48.740 --> 00:12:51.750 to x, which is the coefficient of dx here, 00:12:51.750 --> 00:12:53.670 must be x squared plus y squared, which 00:12:53.670 --> 00:12:55.350 is the coefficient of dx here. 00:12:55.350 --> 00:12:58.990 And similarly, the partial of f with respect to y must equal 00:12:58.990 --> 00:13:00.800 2x*y. 00:13:00.800 --> 00:13:03.030 Now, armed with this information, 00:13:03.030 --> 00:13:05.750 I can actually go out and construct f. 00:13:05.750 --> 00:13:07.420 And the way I do this is remember-- 00:13:07.420 --> 00:13:10.170 as soon as I see the partial of f with respect to x, 00:13:10.170 --> 00:13:12.460 it means I'm treating y as a constant. 00:13:12.460 --> 00:13:16.170 If I'm treating y as a constant, I just integrate this thing 00:13:16.170 --> 00:13:18.080 as if x were the only variable. 00:13:18.080 --> 00:13:20.820 Treating x as a variable and y as a constant, 00:13:20.820 --> 00:13:23.680 the integral of x squared is 1/3 x cubed. 00:13:23.680 --> 00:13:25.830 The integral of y squared is simply 00:13:25.830 --> 00:13:29.160 y squared x, because y is being treated as a constant. 00:13:29.160 --> 00:13:33.180 And finally, there must be a constant of integration, 00:13:33.180 --> 00:13:36.460 but my constant now is any function of y. 00:13:36.460 --> 00:13:38.700 In other words, if I have any function 00:13:38.700 --> 00:13:42.360 of y, its partial with respect to x, by definition of x and y 00:13:42.360 --> 00:13:44.890 being independent-- the partial of any function 00:13:44.890 --> 00:13:46.890 of y with respect to x is 0. 00:13:46.890 --> 00:13:49.640 Therefore my constant of integration, in this case, 00:13:49.640 --> 00:13:51.349 is a function of y alone. 00:13:51.349 --> 00:13:53.640 Again, to look at this in a different perspective, what 00:13:53.640 --> 00:13:55.920 I'm saying is the most general function 00:13:55.920 --> 00:13:57.910 of two variables in the whole world 00:13:57.910 --> 00:13:59.760 whose derivative with respect to x 00:13:59.760 --> 00:14:04.020 is x squared plus y squared is 1/3 x cubed plus y squared 00:14:04.020 --> 00:14:07.410 x plus a function of y alone. 00:14:07.410 --> 00:14:11.920 Now all I don't know so far is what specific function of y 00:14:11.920 --> 00:14:13.400 is g? 00:14:13.400 --> 00:14:15.620 In other words, I've determined f now up 00:14:15.620 --> 00:14:20.940 to this particular arbitrary constant of integration. 00:14:20.940 --> 00:14:22.850 How do I get a hold of this thing? 00:14:22.850 --> 00:14:26.340 Notice that as yet, I have not used a piece of information 00:14:26.340 --> 00:14:30.020 that tells me the partial of f with respect to y is 2x*y. 00:14:30.020 --> 00:14:32.320 And to utilize that piece of information, what I do 00:14:32.320 --> 00:14:34.570 is I come to this equation, and I say, lookit. 00:14:34.570 --> 00:14:37.720 Let me, from here, just differentiate this with respect 00:14:37.720 --> 00:14:39.880 to y, treating x as a constant. 00:14:39.880 --> 00:14:42.070 If I do that, this term drops out. 00:14:42.070 --> 00:14:44.250 This term becomes 2x*y. 00:14:44.250 --> 00:14:46.910 And this, being a function of y alone, 00:14:46.910 --> 00:14:49.940 its derivative with respect to y is just g prime of y. 00:14:49.940 --> 00:14:55.810 So whatever f is, the derivative of f with respect to y is 2x*y 00:14:55.810 --> 00:14:57.640 plus g prime of y. 00:14:57.640 --> 00:15:00.310 On the other hand, we also know that the partial of f with 00:15:00.310 --> 00:15:02.860 respect to y is 2x*y. 00:15:02.860 --> 00:15:05.640 Consequently, since these are two different expressions 00:15:05.640 --> 00:15:09.040 for the same quantity, they must be equal. 00:15:09.040 --> 00:15:12.210 Equating these two, the 2x*y's cancel. 00:15:12.210 --> 00:15:16.060 We obtain that g prime of y is 0, which means, in this case, 00:15:16.060 --> 00:15:18.960 that g of y was a bona fide constant, that not 00:15:18.960 --> 00:15:22.702 only is g independent of x, it's also independent of y. 00:15:22.702 --> 00:15:24.410 And we see that the most general function 00:15:24.410 --> 00:15:27.410 f which has the desired property is what? 00:15:27.410 --> 00:15:30.560 All we have to do is now go back to this expression here, 00:15:30.560 --> 00:15:34.140 which gave us the answer up to g of y, 00:15:34.140 --> 00:15:37.780 put this value of g of y in, and we see that the most general 00:15:37.780 --> 00:15:42.530 function whose total differential is x squared plus 00:15:42.530 --> 00:15:48.070 y squared dx plus 2x*y*dy is 1/3 x cubed plus y squared x plus 00:15:48.070 --> 00:15:48.660 c. 00:15:48.660 --> 00:15:51.700 And I guess there's a moral to this story, after all. 00:15:51.700 --> 00:15:52.200 You know? 00:15:52.200 --> 00:15:53.690 When I did it the quick way, I made 00:15:53.690 --> 00:15:55.910 a careless computational mistake earlier 00:15:55.910 --> 00:15:58.220 in the lecture that sort of threw me off a bit 00:15:58.220 --> 00:15:59.640 and perplexed me. 00:15:59.640 --> 00:16:03.520 And it seems it was poetic justice. 00:16:03.520 --> 00:16:05.360 Because, you know, when I took this method 00:16:05.360 --> 00:16:07.900 where we pretended we didn't know the answer in advance 00:16:07.900 --> 00:16:10.270 and just worked the thing out systematically, 00:16:10.270 --> 00:16:12.150 we came up with the right term. 00:16:12.150 --> 00:16:15.810 In other words, it should be y squared x, not y x squared. 00:16:15.810 --> 00:16:19.417 And that was what we saw should be the correct answer. 00:16:19.417 --> 00:16:21.750 So maybe there's something to be said about doing things 00:16:21.750 --> 00:16:23.870 systematically, after all. 00:16:23.870 --> 00:16:24.370 You see? 00:16:24.370 --> 00:16:25.810 If I can't win them all, at least 00:16:25.810 --> 00:16:29.780 I have the gift of rationalizing to think that I win them all. 00:16:29.780 --> 00:16:34.320 Time after time, I snatch defeat from the jaws of victory. 00:16:34.320 --> 00:16:34.820 No? 00:16:34.820 --> 00:16:35.330 All right. 00:16:35.330 --> 00:16:36.240 You know what I mean. 00:16:36.240 --> 00:16:36.890 Lookit. 00:16:36.890 --> 00:16:40.200 Anyway, the question that we've now solved is what? 00:16:40.200 --> 00:16:42.180 We have worked out this routine. 00:16:42.180 --> 00:16:44.560 Notice that what we are really doing 00:16:44.560 --> 00:16:48.220 is the counterpart of calculus of a single variable, that 00:16:48.220 --> 00:16:51.680 knowing what a partial is, we can integrate with respect 00:16:51.680 --> 00:16:54.440 to that variable, treating all the other variables 00:16:54.440 --> 00:16:55.650 as a constant. 00:16:55.650 --> 00:16:57.690 And this is the general technique. 00:16:57.690 --> 00:17:00.250 Now let's summarize this more generally. 00:17:00.250 --> 00:17:01.760 The general definition is this. 00:17:01.760 --> 00:17:05.579 And by the way, for the sake of conserving space, I will now 00:17:05.579 --> 00:17:09.619 abbreviate M of x, y and N of x, y just by M and N. 00:17:09.619 --> 00:17:11.270 But whenever I write M and N here, 00:17:11.270 --> 00:17:14.339 it's assumed that M and N are functions of x and y. 00:17:14.339 --> 00:17:15.420 Well what we say is this. 00:17:15.420 --> 00:17:18.170 M*dx plus N*dy, which by the way, 00:17:18.170 --> 00:17:21.510 by a previous definition is always a differential. 00:17:21.510 --> 00:17:25.380 But this differential is called an exact differential 00:17:25.380 --> 00:17:30.010 if and only if there exists a function w-- f of x, y-- 00:17:30.010 --> 00:17:33.720 such that dw is M*dx plus N*dy. 00:17:33.720 --> 00:17:37.340 Or in terms of our previous discussion, 00:17:37.340 --> 00:17:38.790 what this means is what? 00:17:38.790 --> 00:17:45.330 dw is also equal to f sub x dx plus f sub y dy. 00:17:45.330 --> 00:17:47.410 Since x and y are independent variables, 00:17:47.410 --> 00:17:49.520 dx and dy are independent variables. 00:17:49.520 --> 00:17:51.870 So an alternative definition is what? 00:17:51.870 --> 00:17:55.940 That M*dx plus N*dy is an exact differential if there exists 00:17:55.940 --> 00:17:59.950 a function f such that the partial of f with respect to x 00:17:59.950 --> 00:18:04.310 is M, and the partial of f with respect to y is N. 00:18:04.310 --> 00:18:07.190 Now, by the way, let's just pause here for a moment. 00:18:07.190 --> 00:18:09.700 You might suspect that with all the functions 00:18:09.700 --> 00:18:12.920 to choose from, that no matter how M and N were given, 00:18:12.920 --> 00:18:17.235 we're bound to find at least one function which satisfies-- 00:18:17.235 --> 00:18:18.610 I should put this in, because you 00:18:18.610 --> 00:18:21.340 want both of these conditions fulfilled at the same time. 00:18:21.340 --> 00:18:24.260 The amazing thing is that it is not always 00:18:24.260 --> 00:18:27.240 possible to find a function which 00:18:27.240 --> 00:18:31.040 has a given differential as its total differential. 00:18:31.040 --> 00:18:33.570 In fact, let me show you why, if we can just do something 00:18:33.570 --> 00:18:34.990 a little bit tricky here. 00:18:34.990 --> 00:18:37.630 Let's suppose that the function f exists such 00:18:37.630 --> 00:18:39.100 that its partial with respect to x 00:18:39.100 --> 00:18:41.190 is M and its partial with respect to y 00:18:41.190 --> 00:18:45.640 is N. Suppose that both of these happen to be differentiable. 00:18:45.640 --> 00:18:48.520 Take the partial of f sub x with respect to y. 00:18:48.520 --> 00:18:51.980 That tells me that f sub x, y is the partial of M 00:18:51.980 --> 00:18:54.010 with respect to y. 00:18:54.010 --> 00:18:57.860 Let's take the partial of f sub y with respect to x. 00:18:57.860 --> 00:19:00.160 That tells me that the partial of f sub y with respect 00:19:00.160 --> 00:19:02.070 to x is N sub x. 00:19:02.070 --> 00:19:04.820 In other words, f sub y, x equals N sub x. 00:19:04.820 --> 00:19:08.100 Now if f is continuous-- see, if we 00:19:08.100 --> 00:19:09.605 have the right amount of continuity 00:19:09.605 --> 00:19:11.230 and differentiability that we've talked 00:19:11.230 --> 00:19:13.670 about in previous lectures, notice 00:19:13.670 --> 00:19:16.690 that for most well-defined functions-- in particular, 00:19:16.690 --> 00:19:19.430 if these two partials-- see, we don't know what f is. 00:19:19.430 --> 00:19:20.790 We're trying to find f. 00:19:20.790 --> 00:19:22.034 And we're assuming it exists. 00:19:22.034 --> 00:19:23.700 The thing that's given in our definition 00:19:23.700 --> 00:19:26.900 are M and N. So to word this more succinctly, if it turns 00:19:26.900 --> 00:19:29.570 out that the partial of M with respect to y 00:19:29.570 --> 00:19:31.750 and the partial of N with respect to x 00:19:31.750 --> 00:19:34.550 happen to be continuous functions, 00:19:34.550 --> 00:19:37.820 notice that these two things are equal. 00:19:37.820 --> 00:19:40.690 And in particular, that means that the partial of M 00:19:40.690 --> 00:19:43.440 with respect to y has to equal the partial of N 00:19:43.440 --> 00:19:44.940 with respect to x. 00:19:44.940 --> 00:19:47.490 Coming back to the original definition here, 00:19:47.490 --> 00:19:49.290 this is a rather amazing thing. 00:19:49.290 --> 00:19:52.060 It says that if this differential is exact, 00:19:52.060 --> 00:19:55.450 if you take the coefficient of dx-- 00:19:55.450 --> 00:19:58.265 see, think of M as being the coefficient of dx and N 00:19:58.265 --> 00:20:00.450 as being the coefficient of dy. 00:20:00.450 --> 00:20:02.870 If you take the coefficient of dx 00:20:02.870 --> 00:20:05.390 and differentiate that with respect to y, 00:20:05.390 --> 00:20:08.360 you must get the same answer as if you 00:20:08.360 --> 00:20:11.790 took the coefficient of dy and differentiated that 00:20:11.790 --> 00:20:13.160 with respect to x. 00:20:13.160 --> 00:20:15.440 That's an amazing coincidence if that happens. 00:20:15.440 --> 00:20:19.020 Given two arbitrary functions, this indeed won't happen. 00:20:19.020 --> 00:20:21.560 Now before I show you that, let's summarize this 00:20:21.560 --> 00:20:23.900 in words written down so that I make sure 00:20:23.900 --> 00:20:26.260 that you understand what I'm saying. 00:20:26.260 --> 00:20:28.880 What I'm saying is, if the partial of M with respect to y 00:20:28.880 --> 00:20:32.220 and the partial of N with respect to x are continuous, 00:20:32.220 --> 00:20:37.470 then if M*dx plus N*dy is exact, the implication is that 00:20:37.470 --> 00:20:40.820 the partial of M with respect to y must equal the partial of N 00:20:40.820 --> 00:20:42.100 with respect to x. 00:20:42.100 --> 00:20:47.130 And to invert the emphasis here, another way of saying this is 00:20:47.130 --> 00:20:50.290 that if the partial of M with respect to y is not equal 00:20:50.290 --> 00:20:52.310 to the partial of N with respect to x, 00:20:52.310 --> 00:20:55.860 then M*dx plus N*dy is not exact. 00:20:55.860 --> 00:20:58.690 And let me show you that by means of an example. 00:20:58.690 --> 00:21:01.890 Let's look at the differential-- a very simple-looking one. 00:21:01.890 --> 00:21:05.600 In fact, it looks far less complicated than the expression 00:21:05.600 --> 00:21:07.610 that we worked with previously. 00:21:07.610 --> 00:21:10.720 Let's look simply at y*dx minus x*dy. 00:21:10.720 --> 00:21:14.070 In this problem, the coefficient of dx is y. 00:21:14.070 --> 00:21:19.050 That plays the role of M. The coefficient of dy is minus x. 00:21:19.050 --> 00:21:21.450 That's what we're calling N, you see. 00:21:21.450 --> 00:21:24.060 And the partial of M with respect to y 00:21:24.060 --> 00:21:27.040 is just the partial of y with respect to y, which is 1. 00:21:27.040 --> 00:21:29.400 The partial of N with respect to x 00:21:29.400 --> 00:21:33.120 is the partial of minus x with respect to x, which is minus 1. 00:21:33.120 --> 00:21:36.542 Since 1 is not equal to minus 1, this is not exact. 00:21:36.542 --> 00:21:38.375 Because if it were exact, these two partials 00:21:38.375 --> 00:21:39.600 would have to be equal. 00:21:39.600 --> 00:21:41.800 What does it mean to say that this is not exact? 00:21:41.800 --> 00:21:44.130 What it means is that there is not 00:21:44.130 --> 00:21:46.550 a single function in the whole world that 00:21:46.550 --> 00:21:50.250 has the property that its partial with respect to x is y, 00:21:50.250 --> 00:21:53.280 and it's partial with respect to y is minus x. 00:21:53.280 --> 00:21:56.040 You can look forever. 00:21:56.040 --> 00:21:58.800 And not only won't you find one, there isn't one. 00:21:58.800 --> 00:22:01.470 And by the way, let me make one very quick aside here. 00:22:01.470 --> 00:22:04.080 Notice the difference between saying you can't find one, 00:22:04.080 --> 00:22:05.250 and there isn't one. 00:22:05.250 --> 00:22:06.940 There may be one, and you're just not 00:22:06.940 --> 00:22:08.444 lucky enough to find it. 00:22:08.444 --> 00:22:10.860 I'm saying that the reason you don't find an answer here-- 00:22:10.860 --> 00:22:13.750 unless you make a mistake and think you've found an answer-- 00:22:13.750 --> 00:22:16.320 the reason you don't find an answer is there is none. 00:22:16.320 --> 00:22:18.310 And how can I show you that there is none? 00:22:18.310 --> 00:22:20.320 Well the best way is to do the same thing 00:22:20.320 --> 00:22:22.220 that we did in the previous example, 00:22:22.220 --> 00:22:23.970 and see something go wrong. 00:22:23.970 --> 00:22:25.810 Let's suppose there were an answer to this. 00:22:25.810 --> 00:22:29.350 Well if the partial of f with respect to x equals y, 00:22:29.350 --> 00:22:31.330 let me integrate this with respect 00:22:31.330 --> 00:22:33.470 to x, treating y as a constant. 00:22:33.470 --> 00:22:35.760 It means that the function that I'm looking for 00:22:35.760 --> 00:22:37.140 must have what form? 00:22:37.140 --> 00:22:41.500 It's x times y plus some function of y alone. 00:22:41.500 --> 00:22:43.570 And now we say, gee, we're in pretty good shape. 00:22:43.570 --> 00:22:46.730 All we've got to do is find what g of y is, and we're home free. 00:22:46.730 --> 00:22:49.210 Let's continue to mimic what we did before. 00:22:49.210 --> 00:22:52.090 Knowing this, I can take the partial of this with respect 00:22:52.090 --> 00:22:53.860 to y, OK? 00:22:53.860 --> 00:22:54.860 Which is what? 00:22:54.860 --> 00:22:58.230 If I take the partial of this with respect to y, 00:22:58.230 --> 00:23:04.830 this is f sub y is x plus g prime of y. 00:23:04.830 --> 00:23:06.050 All right? 00:23:06.050 --> 00:23:07.850 Now this is the partial of f with respect 00:23:07.850 --> 00:23:10.010 to y, computed from here. 00:23:10.010 --> 00:23:12.750 I know by hypothesis that the function I'm looking for 00:23:12.750 --> 00:23:15.580 must have its partial with respect to y equal to this. 00:23:15.580 --> 00:23:18.780 Consequently, these two expressions must be equal. 00:23:18.780 --> 00:23:20.840 And equating these two expressions says 00:23:20.840 --> 00:23:24.080 that g prime of y-- see, x plus g prime of y 00:23:24.080 --> 00:23:25.550 must equal minus x. 00:23:25.550 --> 00:23:29.660 Transposing says that g prime of y is minus 2x. 00:23:29.660 --> 00:23:30.880 And this is a contradiction. 00:23:30.880 --> 00:23:33.930 Because lookit, g prime of y is a function of y alone. 00:23:33.930 --> 00:23:36.780 And here we have it equal to some function of x. 00:23:36.780 --> 00:23:38.730 Or another way of looking at it, this 00:23:38.730 --> 00:23:42.590 says that x depends on y. x is some function of y, which 00:23:42.590 --> 00:23:46.330 is a contradiction since we assumed that x and y are 00:23:46.330 --> 00:23:48.400 independent variables. 00:23:48.400 --> 00:23:51.470 By the way, just as a quick aside, 00:23:51.470 --> 00:23:54.910 notice that in the case where we were able to succeed at this-- 00:23:54.910 --> 00:23:59.570 namely the previous example, x squared plus y dx plus 2x*y*dy, 00:23:59.570 --> 00:24:03.010 M was x squared plus y squared, N was 2x*y. 00:24:03.010 --> 00:24:07.270 Notice that the partial of M with respect to y is 2y. 00:24:07.270 --> 00:24:11.800 And the partial of N with respect to x is also 2y. 00:24:11.800 --> 00:24:14.240 So it seems that the deciding factor really 00:24:14.240 --> 00:24:17.530 seems to be that the partial of M with respect to y 00:24:17.530 --> 00:24:20.100 has to equal the partial of N with respect to x. 00:24:20.100 --> 00:24:22.552 And the major result is-- and I didn't write this in, 00:24:22.552 --> 00:24:24.510 because I didn't want to use up too much space. 00:24:24.510 --> 00:24:27.700 But this is emphasized in the text and in the exercises. 00:24:27.700 --> 00:24:34.870 If M, N, M sub y, and N sub x all exist and are continuous, 00:24:34.870 --> 00:24:40.240 then not only can we say that M*dx plus N*dy is exact implies 00:24:40.240 --> 00:24:43.640 this part, but we can do the converse too. 00:24:43.640 --> 00:24:46.860 That in particular, if M sub y equals N sub x, 00:24:46.860 --> 00:24:48.914 we can conclude that this is exact. 00:24:48.914 --> 00:24:50.330 And the reason for this-- and I'll 00:24:50.330 --> 00:24:52.371 go through this very quickly because the proof is 00:24:52.371 --> 00:24:53.464 given in the text. 00:24:53.464 --> 00:24:55.880 I just want to show you the highlights of what happens is, 00:24:55.880 --> 00:24:58.090 what really happened when we tried 00:24:58.090 --> 00:25:01.270 to construct f that made things work in one case 00:25:01.270 --> 00:25:02.480 but not in the other? 00:25:02.480 --> 00:25:05.200 I think the best way to do this is to work abstractly 00:25:05.200 --> 00:25:08.070 here without specifying what M and N are, 00:25:08.070 --> 00:25:09.420 and see what happens. 00:25:09.420 --> 00:25:12.490 What we were trying to solve was a pair of equations-- 00:25:12.490 --> 00:25:15.140 we were trying to find f such that the partial of f 00:25:15.140 --> 00:25:17.970 with respect to x is M, partial of f with respect to y 00:25:17.970 --> 00:25:20.080 is N. The first thing that we did 00:25:20.080 --> 00:25:23.140 was we treated y as a constant and integrated this 00:25:23.140 --> 00:25:24.650 with respect to x. 00:25:24.650 --> 00:25:26.680 Well, I can't do this specifically, 00:25:26.680 --> 00:25:28.050 because I don't know what M is. 00:25:28.050 --> 00:25:31.095 So let me just put the integral sign in here to say what? 00:25:31.095 --> 00:25:32.470 I'm integrating this with respect 00:25:32.470 --> 00:25:34.480 to x, treating y as a constant. 00:25:34.480 --> 00:25:37.150 That's some function of x and y, you see over here. 00:25:37.150 --> 00:25:38.800 See M is a function of x and y. 00:25:38.800 --> 00:25:39.840 I'm holding y constant. 00:25:39.840 --> 00:25:40.890 I'm integrating this. 00:25:40.890 --> 00:25:45.000 Plus a constant of integration which depends only on y. 00:25:45.000 --> 00:25:45.500 All right? 00:25:45.500 --> 00:25:46.760 Same as before. 00:25:46.760 --> 00:25:48.080 What was my next step? 00:25:48.080 --> 00:25:52.470 I took the partial of this with respect to y, 00:25:52.470 --> 00:25:54.500 and I was going to compare that with what 00:25:54.500 --> 00:25:56.420 I knew the partial of f with respect to y 00:25:56.420 --> 00:25:59.440 had to be-- namely, N. So I then took what? 00:25:59.440 --> 00:26:01.530 The partial of this with respect to y. 00:26:01.530 --> 00:26:04.040 That gave me the partial of this integral with respect 00:26:04.040 --> 00:26:06.510 to y plus g prime of y. 00:26:06.510 --> 00:26:10.070 And equating that to N, I get that g prime of y 00:26:10.070 --> 00:26:14.390 was N minus the partial with respect to y integral M dx. 00:26:14.390 --> 00:26:17.200 And again, don't be alarmed by this expression. 00:26:17.200 --> 00:26:19.550 If M had been given explicitly, I 00:26:19.550 --> 00:26:22.530 would have solved explicitly for what this function was. 00:26:22.530 --> 00:26:26.570 The key point is to notice that this side here 00:26:26.570 --> 00:26:29.120 is independent of x. 00:26:29.120 --> 00:26:32.250 Consequently, this equation will be a contradiction 00:26:32.250 --> 00:26:37.500 unless the right-hand side is independent of x. 00:26:37.500 --> 00:26:39.820 In other words, we have determined f up 00:26:39.820 --> 00:26:42.450 to this function g of y. 00:26:42.450 --> 00:26:44.570 And what's going to happen is we are 00:26:44.570 --> 00:26:46.670 going to be able to compute g of y 00:26:46.670 --> 00:26:49.140 if this expression is independent of x, namely just 00:26:49.140 --> 00:26:50.270 by integrating this. 00:26:50.270 --> 00:26:56.070 But if g of y is not independent of x, we are in trouble. 00:26:56.070 --> 00:26:57.880 It means that it's not going to exist. 00:26:57.880 --> 00:27:01.880 The key step is, is this independent of x? 00:27:01.880 --> 00:27:03.710 And the answer is, the best way to find out 00:27:03.710 --> 00:27:06.820 in terms of calculus is to take its derivative with respect 00:27:06.820 --> 00:27:07.570 to x. 00:27:07.570 --> 00:27:11.300 If this depends on y alone, its partial with respect to x 00:27:11.300 --> 00:27:12.600 must be 0. 00:27:12.600 --> 00:27:15.570 In other words, if this partial with respect to x is not 0, 00:27:15.570 --> 00:27:18.080 it means that this expression varies with x. 00:27:18.080 --> 00:27:20.720 At any rate, to see what this derivative is, 00:27:20.720 --> 00:27:23.770 we just differentiate term by term. 00:27:23.770 --> 00:27:24.990 And then we say, you know? 00:27:24.990 --> 00:27:26.540 It's rather interesting. 00:27:26.540 --> 00:27:31.530 If this thing had been first-- notice that if you integrate 00:27:31.530 --> 00:27:33.445 with respect to x and then differentiate 00:27:33.445 --> 00:27:37.150 with respect to x, you wind up with just the integrand. 00:27:37.150 --> 00:27:40.360 Well again, if we have enough continuity, 00:27:40.360 --> 00:27:43.020 the order of differentiation is irrelevant. 00:27:43.020 --> 00:27:45.690 We can reverse the order without changing anything 00:27:45.690 --> 00:27:47.820 if we now interchange this order, 00:27:47.820 --> 00:27:50.230 take the partial first with respect to x. 00:27:50.230 --> 00:27:52.640 That will give us just M. And the partial 00:27:52.640 --> 00:27:54.800 of M with respect to y is just the partial of M 00:27:54.800 --> 00:27:56.180 with respect for y, of course. 00:27:56.180 --> 00:27:58.930 What we're saying is that from this step, we get to this step. 00:27:58.930 --> 00:28:00.920 And this in turn implies this. 00:28:00.920 --> 00:28:04.940 Notice that the only way that this can be 0 00:28:04.940 --> 00:28:07.510 is if the partial of N with respect to x 00:28:07.510 --> 00:28:10.630 equals the partial of M with respect to y. 00:28:10.630 --> 00:28:14.850 So in summary then, this is a very interesting device. 00:28:14.850 --> 00:28:19.000 It tells us how to tell whether a differential is exact. 00:28:19.000 --> 00:28:20.950 And the proof is very nice in this case 00:28:20.950 --> 00:28:24.290 because the proof exactly imitates 00:28:24.290 --> 00:28:28.020 how we construct the function f if it turns out to be exact. 00:28:28.020 --> 00:28:31.880 But the point is, if given M*dx plus N*dy, 00:28:31.880 --> 00:28:34.240 if the partial of M with respect to y is not equal 00:28:34.240 --> 00:28:35.920 to the partial of N with respect to x, 00:28:35.920 --> 00:28:39.330 there's no hope that the function will be exact-- 00:28:39.330 --> 00:28:41.160 the differential be exact. 00:28:41.160 --> 00:28:45.220 At any rate, we will have sufficient exercises 00:28:45.220 --> 00:28:48.670 for drill in how one uses exact differentials. 00:28:48.670 --> 00:28:52.180 This concludes block three of our material. 00:28:52.180 --> 00:28:54.670 Our next block of material which will 00:28:54.670 --> 00:28:57.690 concern linear systems of equations 00:28:57.690 --> 00:28:59.710 will begin with our next lecture. 00:28:59.710 --> 00:29:03.500 And until next time, goodbye. 00:29:03.500 --> 00:29:05.880 Funding for the publication of this video 00:29:05.880 --> 00:29:10.750 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:29:10.750 --> 00:29:14.920 Help OCW continue to provide free and open access to MIT 00:29:14.920 --> 00:29:19.338 courses by making a donation at ocw.mit.edu/donate.