WEBVTT 00:00:01.000 --> 00:00:03.000 The following content is provided under a Creative 00:00:03.000 --> 00:00:05.000 Commons license. Your support will help MIT 00:00:05.000 --> 00:00:08.000 OpenCourseWare continue to offer high quality educational 00:00:08.000 --> 00:00:13.000 resources for free. To make a donation or to view 00:00:13.000 --> 00:00:18.000 additional materials from hundreds of MIT courses, 00:00:18.000 --> 00:00:23.000 visit MIT OpenCourseWare at ocw.mit.edu. 00:00:23.000 --> 00:00:25.000 So, so far, we've seen things about vectors, 00:00:25.000 --> 00:00:27.000 equation of planes, motions in space, 00:00:27.000 --> 00:00:30.000 and so on. Basically we've done geometry 00:00:30.000 --> 00:00:31.000 in space. But, calculus, 00:00:31.000 --> 00:00:33.000 really, is about studying functions. 00:00:33.000 --> 00:00:37.000 Now, we're going to actually move on to studying functions of 00:00:37.000 --> 00:00:40.000 several variables. So, this new unit, 00:00:40.000 --> 00:00:45.000 what we'll do over the next three weeks or so will be about 00:00:45.000 --> 00:00:50.000 functions of several variables and their derivatives. 00:00:50.000 --> 00:00:55.000 OK, so first of all, we should try to figure out how 00:00:55.000 --> 00:00:58.000 we are going to think about functions. 00:00:58.000 --> 00:01:02.000 So, remember, if you have a function of one 00:01:02.000 --> 00:01:08.000 variable, that means you have a quantity that depends on one 00:01:08.000 --> 00:01:12.000 parameter. Maybe f depends on the variable 00:01:12.000 --> 00:01:14.000 x. And, for example, 00:01:14.000 --> 00:01:19.000 a function that you all know is f of x equals sin(x). 00:01:19.000 --> 00:01:22.000 And, the way we would represent that is maybe just by plotting 00:01:22.000 --> 00:01:28.000 the graph of the function. So, the graph of a function, 00:01:28.000 --> 00:01:34.000 we plot y = f(x). And, the graph of a sine 00:01:34.000 --> 00:01:45.000 function that looks like this. OK, so now, let's say that we 00:01:45.000 --> 00:01:53.000 had, actually, a function of two variables. 00:01:53.000 --> 00:01:56.000 So, that means that the value of F depends actually on two 00:01:56.000 --> 00:01:59.000 different parameters, say, if the variables are x and 00:01:59.000 --> 00:02:03.000 y, or they can have any names you 00:02:03.000 --> 00:02:07.000 want. So, given values of the two 00:02:07.000 --> 00:02:13.000 parameters, x and y, the function will give us a 00:02:13.000 --> 00:02:17.000 number that we'll call f(x, y). 00:02:17.000 --> 00:02:21.000 That depends on x and y according to some formula, 00:02:21.000 --> 00:02:29.000 OK, not very surprising so far. So, for example, 00:02:29.000 --> 00:02:44.000 I can give you the function f(x, y) = x^2 y^2. 00:02:44.000 --> 00:02:47.000 And, of course, as with functions of one 00:02:47.000 --> 00:02:50.000 variable, we don't need things to be defined everywhere. 00:02:50.000 --> 00:02:53.000 Sometimes there is the domain of definition. 00:02:53.000 --> 00:02:57.000 So, this one is defined all the time. 00:02:57.000 --> 00:03:01.000 But, if I tell you, say, f of x, 00:03:01.000 --> 00:03:08.000 y equals square root of y, well, this is only defined if y 00:03:08.000 --> 00:03:14.000 is nonnegative. If I tell you f(x, 00:03:14.000 --> 00:03:23.000 y) equals one over x y, that's only defined if x y is 00:03:23.000 --> 00:03:30.000 not zero, and so on. Now, so these are mathematical 00:03:30.000 --> 00:03:33.000 examples given by explicit formulas. 00:03:33.000 --> 00:03:35.000 And, of course, there's physical examples. 00:03:35.000 --> 00:03:38.000 For example, so examples coming from real 00:03:38.000 --> 00:03:40.000 life, so for example, you can look at the temperature 00:03:40.000 --> 00:03:43.000 at the certain point on the surface of the earth. 00:03:43.000 --> 00:03:46.000 So, you use maybe longitude and latitude; that's x and y. 00:03:46.000 --> 00:03:49.000 And then you have f(x, y) equals the temperature at 00:03:49.000 --> 00:03:50.000 that point. 00:04:12.000 --> 00:04:17.000 In fact, because temperature depends also may be on how high 00:04:17.000 --> 00:04:18.000 up you are. It depends on elevation. 00:04:18.000 --> 00:04:21.000 So, it's actually a function of maybe x, y, z. 00:04:21.000 --> 00:04:24.000 And, it also depends on time. So, in fact, 00:04:24.000 --> 00:04:28.000 maybe it's a function of t in x y z coordinates in space. 00:04:28.000 --> 00:04:31.000 So, you see that real-world functions can depends on a lot 00:04:31.000 --> 00:04:33.000 of variables. So, our goal will be to 00:04:33.000 --> 00:04:35.000 understand how to deal with that. 00:04:57.000 --> 00:05:01.000 OK, so now you will see very soon, but actually it's already 00:05:01.000 --> 00:05:05.000 tricky enough to picture a function of two variables. 00:05:05.000 --> 00:05:08.000 So, we are going to focus on the case of functions of two 00:05:08.000 --> 00:05:10.000 variables. And then, we'll see that if we 00:05:10.000 --> 00:05:12.000 have more than two variables, then it's harder to plot the 00:05:12.000 --> 00:05:14.000 function. We cannot draw with the graph 00:05:14.000 --> 00:05:17.000 looks like anymore. But, the tools are the same, 00:05:17.000 --> 00:05:20.000 the notion of partial derivatives, grade and vector, 00:05:20.000 --> 00:05:23.000 and so on, all the tools that we will 00:05:23.000 --> 00:05:27.000 develop work exactly the same way no matter how many variables 00:05:27.000 --> 00:05:30.000 you have. So, I'll say, 00:05:30.000 --> 00:05:41.000 for simplicity -- -- we'll focus mostly on two or sometimes 00:05:41.000 --> 00:05:48.000 three variables. But, it works the same in any 00:05:48.000 --> 00:05:56.000 number of variables. OK, so the first question is 00:05:56.000 --> 00:06:05.000 how to visualize a function of two variables. 00:06:05.000 --> 00:06:10.000 So, the first thing we can do is try to draw the graph of f. 00:06:10.000 --> 00:06:19.000 So, maybe I should say f -- which is a function of two 00:06:19.000 --> 00:06:23.000 variables. So, the first answer will be, 00:06:23.000 --> 00:06:26.000 we can try to look at it's graph. 00:06:26.000 --> 00:06:29.000 And, the idea is the same as with one variable, 00:06:29.000 --> 00:06:31.000 namely, we look at all the possible values of the 00:06:31.000 --> 00:06:34.000 parameters, x and y, and for each of them, 00:06:34.000 --> 00:06:40.000 we plot a point whose height is the value of a function at these 00:06:40.000 --> 00:06:43.000 parameters. So, we'll plot, 00:06:43.000 --> 00:06:47.000 let's say, z equals f(x, y). 00:06:47.000 --> 00:06:52.000 And, now that will become, actually, a surface in space. 00:06:52.000 --> 00:06:57.000 OK, so for each value of x and y, yeah, we have x, 00:06:57.000 --> 00:07:02.000 y in the x, y plane, then we'll plot the point in 00:07:02.000 --> 00:07:05.000 space at position x, y. 00:07:05.000 --> 00:07:13.000 And, z equals f of x, y. OK, and if we take all of these 00:07:13.000 --> 00:07:17.000 points together, they will give us some surface 00:07:17.000 --> 00:07:24.000 that sits in space. Yes? 00:07:24.000 --> 00:07:29.000 Oh, a function of two variables, shorthand. 00:07:29.000 --> 00:07:36.000 Well, let's say how to visualize a function of two 00:07:36.000 --> 00:07:39.000 variables. OK, so, how do we do that 00:07:39.000 --> 00:07:41.000 concretely? Say that I give you a formula 00:07:41.000 --> 00:07:46.000 for f. How do we try to represent it? 00:07:46.000 --> 00:07:57.000 So, let's do our first example. Let's say I give you a function 00:07:57.000 --> 00:08:02.000 f(x, y) = -y. OK, so it looks a little bit 00:08:02.000 --> 00:08:04.000 silly because it doesn't depend on x. 00:08:04.000 --> 00:08:10.000 But, that's not the problem. It's still a valid function of 00:08:10.000 --> 00:08:13.000 x and y. It just happens to be constant 00:08:13.000 --> 00:08:17.000 with respect to x. So, to draw the graph we look 00:08:17.000 --> 00:08:22.000 at the surface in space defined by z equals y. 00:08:22.000 --> 00:08:26.000 What kind of surface is that? It's a plane, OK? 00:08:26.000 --> 00:08:32.000 And, if we want to draw it, z equals minus y will look, 00:08:32.000 --> 00:08:36.000 well, let's put y axis. Let's put x axis. 00:08:36.000 --> 00:08:39.000 Let's put z axis. If I look at what happens in 00:08:39.000 --> 00:08:43.000 the y, z plane in the plane of a blackboard, it will just look 00:08:43.000 --> 00:08:45.000 like a line that goes downward with slope one. 00:08:45.000 --> 00:08:51.000 OK, so it will be this. And, what happens if I change x? 00:08:51.000 --> 00:08:54.000 Well, if I change x, nothing happens because x 00:08:54.000 --> 00:08:57.000 doesn't appear in this equation. So, in fact, 00:08:57.000 --> 00:09:01.000 if instead of setting x equal to zero I set x equal to one, 00:09:01.000 --> 00:09:04.000 I'm in front of the blackboard, or minus one at the back. 00:09:04.000 --> 00:09:07.000 Well, it still looks exactly the same. 00:09:07.000 --> 00:09:15.000 So, I have this plane that actually contains the x axis and 00:09:15.000 --> 00:09:22.000 slopes downwards with slope one. It's kind of hard to draw. 00:09:22.000 --> 00:09:25.000 Now, you see immediately what the big problem with graphs will 00:09:25.000 --> 00:09:29.000 be. But, these pictures are hard to 00:09:29.000 --> 00:09:34.000 read. But that's our first graph. 00:09:34.000 --> 00:09:41.000 OK, a question so far? OK, so let's say that we have a 00:09:41.000 --> 00:09:43.000 slightly more complicated function. 00:09:43.000 --> 00:09:50.000 How do we see it? So, let's draw another example. 00:09:50.000 --> 00:09:58.000 Let's say I give you f(x, y) = 1 - x^2-y^2. 00:09:58.000 --> 00:10:06.000 So, we should try to picture what the surface z=1- x^2-y^2 00:10:06.000 --> 00:10:10.000 looks like. So, how do we do that? 00:10:10.000 --> 00:10:15.000 Well, maybe you are very fast and figured out what it looks 00:10:15.000 --> 00:10:17.000 like. But, if not, 00:10:17.000 --> 00:10:21.000 then we need to work piece by piece. 00:10:21.000 --> 00:10:27.000 So, maybe it will help if we understand first what it does in 00:10:27.000 --> 00:10:35.000 the plane of the blackboard. So, if we look at it in the y, 00:10:35.000 --> 00:10:42.000 z plane, that means we set x equal to zero. 00:10:42.000 --> 00:10:48.000 And then, z becomes 1 - y^2. What is that? 00:10:48.000 --> 00:10:54.000 It's a parabola pointing downwards, and starting at one. 00:10:54.000 --> 00:11:00.000 So, we should draw maybe this downward parabola. 00:11:00.000 --> 00:11:08.000 It starts at one and it cuts the y axis at one. 00:11:08.000 --> 00:11:11.000 When y is one, that gives us zero. 00:11:11.000 --> 00:11:15.000 So, we might have an idea of what it might look like, 00:11:15.000 --> 00:11:19.000 or maybe not. Let's get more slices. 00:11:19.000 --> 00:11:27.000 Let's see what it does in the x, z plane, this other vertical 00:11:27.000 --> 00:11:33.000 plane that's coming towards us. So, in the x, 00:11:33.000 --> 00:11:38.000 z plane, if we set y equal to zero, we get z equals one minus 00:11:38.000 --> 00:11:40.000 x^2. It's, again, 00:11:40.000 --> 00:11:46.000 a parabola going downwards. OK, so I'm going to try to draw 00:11:46.000 --> 00:11:51.000 a parabola that goes downward, but now to the front and to the 00:11:51.000 --> 00:11:54.000 back. So, we are starting to have a 00:11:54.000 --> 00:11:57.000 slightly better idea but we still don't know whether the 00:11:57.000 --> 00:11:59.000 cross section of this thing might be round, 00:11:59.000 --> 00:12:04.000 square, something else. So, it wants more confirmation. 00:12:04.000 --> 00:12:16.000 We might want to also figure out where the surface intersects 00:12:16.000 --> 00:12:22.000 the x, y plane. So, we hit the x, 00:12:22.000 --> 00:12:29.000 y plane when z equals zero. That means 1-x^2-y^2 should be 00:12:29.000 --> 00:12:38.000 0, that becomes x^2 y^2 = 1. That is a circle of radius 1. 00:12:38.000 --> 00:12:46.000 That's the unit size. So, that means that here, 00:12:46.000 --> 00:12:50.000 we actually have the unit circle. 00:12:50.000 --> 00:12:54.000 And now, you should imagine that you have this thing that 00:12:54.000 --> 00:12:57.000 when you slice it by a vertical plane, looks like a downwards 00:12:57.000 --> 00:13:00.000 parabola. And, it's actually a surface of 00:13:00.000 --> 00:13:03.000 revolution. You can rotate it around the z 00:13:03.000 --> 00:13:06.000 axis, OK? Now, if you stare long enough 00:13:06.000 --> 00:13:09.000 at that equation, you'll actually see that, 00:13:09.000 --> 00:13:12.000 yes, we know that it had to be like that. 00:13:12.000 --> 00:13:17.000 But, see, so these are useful ways of trying to guess what the 00:13:17.000 --> 00:13:19.000 graph looks like. Of course, the other way is to 00:13:19.000 --> 00:13:24.000 just ask your computer to do it. And then, you know, 00:13:24.000 --> 00:13:30.000 you will get that kind of formula. 00:13:30.000 --> 00:13:36.000 OK, well, I can leave it on if you want. 00:13:36.000 --> 00:13:40.000 No, because I plotted a different function that I will 00:13:40.000 --> 00:13:43.000 show you later. So, it goes this way. 00:13:43.000 --> 00:13:46.000 I mean, if you want, it's really going downward. 00:13:46.000 --> 00:13:49.000 Yes, I agree that the sheet is upside down. 00:13:49.000 --> 00:13:52.000 That's because I plotted something else. 00:13:52.000 --> 00:13:58.000 OK, so, in fact, so I plotted in my computer was 00:13:58.000 --> 00:14:03.000 actually x^2 y^2 that looks like that. 00:14:03.000 --> 00:14:08.000 See, it's the same with a parabola going upwards. 00:14:08.000 --> 00:14:14.000 If you want to see more examples, I have various 00:14:14.000 --> 00:14:19.000 examples to show, well, here's the graph, 00:14:19.000 --> 00:14:21.000 y^2-x^2. See, so that one is kind of 00:14:21.000 --> 00:14:23.000 interesting. It looks like a saddle. 00:14:23.000 --> 00:14:29.000 If you look at it in the y, z plane, then it's a parabola 00:14:29.000 --> 00:14:34.000 going up, z = y^2. And, that's what we see to the 00:14:34.000 --> 00:14:38.000 left and to the right. But, if you put it in the x, 00:14:38.000 --> 00:14:42.000 z plane, then that's a parabola going downwards, 00:14:42.000 --> 00:14:45.000 z = - x^2. So, we have a parabola going 00:14:45.000 --> 00:14:48.000 downwards in one direction, upwards in the other one. 00:14:48.000 --> 00:14:53.000 And together, they form this surface. 00:14:53.000 --> 00:14:55.000 And of course, you can plot much more 00:14:55.000 --> 00:14:58.000 complicated functions. So, this one, 00:14:58.000 --> 00:15:00.000 well, if you can read very small things, 00:15:00.000 --> 00:15:03.000 you can see the formula. It doesn't matter, 00:15:03.000 --> 00:15:09.000 just to show you that you can put a formula into a computer: 00:15:09.000 --> 00:15:18.000 it will show you a picture. OK, so that's pretty good. 00:15:18.000 --> 00:15:20.000 I mean, you can see that it can get a bit cluttered because 00:15:20.000 --> 00:15:22.000 maybe those features that are hidden behind, 00:15:22.000 --> 00:15:25.000 or maybe we have trouble seeing if we don't have a computer, 00:15:25.000 --> 00:15:29.000 that looks very readable. But, this is kind of hard to 00:15:29.000 --> 00:15:33.000 visualize sometimes. So, there is another way to 00:15:33.000 --> 00:15:36.000 plot the functions of two variables. 00:15:36.000 --> 00:15:47.000 And, let's call it the contour plot. 00:15:47.000 --> 00:15:51.000 So, the contour plot is a very elegant solution to the problem 00:15:51.000 --> 00:15:55.000 that it's difficult to draw to space pictures on a sheet of 00:15:55.000 --> 00:15:58.000 paper or on a blackboard. So, instead, 00:15:58.000 --> 00:16:02.000 let's try to represent the function of two variables by 00:16:02.000 --> 00:16:04.000 just the map, you know, the same way that 00:16:04.000 --> 00:16:07.000 when you walk around, you have actually geographical 00:16:07.000 --> 00:16:11.000 maps that fit on a piece of paper that tell you about what 00:16:11.000 --> 00:16:17.000 the real world looks like. So, what contour plot looks 00:16:17.000 --> 00:16:22.000 something like this? So, it's an x, y plot. 00:16:22.000 --> 00:16:25.000 And, that, you have a bunch of curves. 00:16:25.000 --> 00:16:32.000 And, what the curves represent are the elevations on the graph. 00:16:32.000 --> 00:16:35.000 So, for example, this curve might correspond to 00:16:35.000 --> 00:16:37.000 all the points where f(x, y) = 1. 00:16:37.000 --> 00:16:46.000 And, that curve might be all the points where f=2 and f=3 and 00:16:46.000 --> 00:16:49.000 so on, OK? So, when you see you this kind 00:16:49.000 --> 00:16:53.000 of plot, you're supposed to think that the graph of the 00:16:53.000 --> 00:16:56.000 function sits somewhere in space above that. 00:16:56.000 --> 00:17:00.000 It's like a map telling you how high things are. 00:17:00.000 --> 00:17:03.000 And, what you would want to do with the function, 00:17:03.000 --> 00:17:06.000 really, is be able to tell quickly what's the value at a 00:17:06.000 --> 00:17:08.000 given point? Well, let's say I want to look 00:17:08.000 --> 00:17:11.000 at that point. I know that f is somewhere 00:17:11.000 --> 00:17:14.000 between 1 and 2. Actually, it's much faster to 00:17:14.000 --> 00:17:17.000 read than the graph. On the graph I might have to 00:17:17.000 --> 00:17:18.000 look carefully, and then measure things, 00:17:18.000 --> 00:17:22.000 and so on. Here, I can just raise the 00:17:22.000 --> 00:17:27.000 value of f by comparing with the nearby lines. 00:17:27.000 --> 00:17:31.000 OK, so let me try to make that more precise. 00:17:31.000 --> 00:17:41.000 So, it shows all the points -- -- where f(x, 00:17:41.000 --> 00:17:53.000 y) equals some fixed values, some fixed constants. 00:17:53.000 --> 00:18:05.000 And, these constants typically are chosen at regular intervals. 00:18:05.000 --> 00:18:07.000 For example, here I chose one, 00:18:07.000 --> 00:18:11.000 two, three, and they could continue with zero minus one, 00:18:11.000 --> 00:18:16.000 and so on. So, one way to think about it, 00:18:16.000 --> 00:18:21.000 how does this relate to the graph? 00:18:21.000 --> 00:18:31.000 Well, that's the same thing as cutting, I mean, 00:18:31.000 --> 00:18:40.000 we slice the graph by horizontal planes. 00:18:40.000 --> 00:18:44.000 So, horizontal planes have equations of a form z equals 00:18:44.000 --> 00:18:46.000 some constant, z equals zero, 00:18:46.000 --> 00:18:48.000 z equals one, z equals two, 00:18:48.000 --> 00:18:51.000 and so on. So, maybe the graph of my 00:18:51.000 --> 00:18:55.000 function will be some sort of plot out there. 00:18:55.000 --> 00:19:00.000 And, if I slice it by the plane z equals one, 00:19:00.000 --> 00:19:04.000 then I will get the level curve, 00:19:04.000 --> 00:19:14.000 which is the point where f(x, y) = 1, 00:19:14.000 --> 00:19:24.000 and so, that's called a level curve of f. 00:19:24.000 --> 00:19:32.000 OK, and so we repeat the process with maybe z equals two, 00:19:32.000 --> 00:19:38.000 and we get another level curve, and so on. 00:19:38.000 --> 00:19:44.000 And, then we squish all of them up, and that's how we get the 00:19:44.000 --> 00:19:47.000 contour plot. OK, so each of these lines, 00:19:47.000 --> 00:19:50.000 imagine that this is like some mountain or something that you 00:19:50.000 --> 00:19:52.000 are hiking on. Each of these lines tells you 00:19:52.000 --> 00:19:55.000 how you could move to stay at a constant height if you want to 00:19:55.000 --> 00:19:58.000 get to the other side of the mountain but without ever going 00:19:58.000 --> 00:20:03.000 up or down. You would just walk along that 00:20:03.000 --> 00:20:08.000 path, and it will get you there without effort. 00:20:08.000 --> 00:20:11.000 So, in fact, if you have been talking about 00:20:11.000 --> 00:20:15.000 hiking on mountains, well, that's exactly what a 00:20:15.000 --> 00:20:21.000 topographical map is about. So, I need to zoom a bit. 00:20:21.000 --> 00:20:27.000 So, a topographic map, this one from the US geological 00:20:27.000 --> 00:20:32.000 survey shows you, basically, all the level curves 00:20:32.000 --> 00:20:37.000 of an altitude function on a piece of land. 00:20:37.000 --> 00:20:40.000 So, you know that if you walk right along these curves, 00:20:40.000 --> 00:20:42.000 you will stay along the same height. 00:20:42.000 --> 00:20:46.000 And you know that if you walk towards, these don't have 00:20:46.000 --> 00:20:48.000 numbers. Let me find a place with 00:20:48.000 --> 00:20:53.000 numbers. Here, there is a 500 in the 00:20:53.000 --> 00:20:56.000 middle. So, you know that if you walk 00:20:56.000 --> 00:20:59.000 on the line that says 500, you stay always at 500 meters 00:20:59.000 --> 00:21:02.000 in elevation. If you walk towards the 00:21:02.000 --> 00:21:05.000 mountain that I think is below it, then you will go up, 00:21:05.000 --> 00:21:07.000 and so on. So, you can see, 00:21:07.000 --> 00:21:10.000 for example, here there's a peak, 00:21:10.000 --> 00:21:13.000 and here there is a valley with the river in it, 00:21:13.000 --> 00:21:17.000 and the altitudes go down, and then back up again on the 00:21:17.000 --> 00:21:19.000 other side. OK, so that's an example of a 00:21:19.000 --> 00:21:22.000 contour plot of a function. Of course, we don't have a 00:21:22.000 --> 00:21:25.000 formula for that function, but we have a contour plot, 00:21:25.000 --> 00:21:29.000 and that's what we need actually to understand what's 00:21:29.000 --> 00:21:36.000 going on there. OK, any questions? 00:21:36.000 --> 00:21:39.000 No? OK, so another example of 00:21:39.000 --> 00:21:42.000 contour plots, well, you've probably seen 00:21:42.000 --> 00:21:46.000 various versions of these temperature maps. 00:21:46.000 --> 00:21:51.000 So, that's supposed to be how warm it is right now. 00:21:51.000 --> 00:21:55.000 So, this one is color-coded. Instead of having curves, 00:21:55.000 --> 00:21:58.000 it has all these colors. But, the effect is the same. 00:21:58.000 --> 00:22:01.000 If you look at the separations between consecutive colors, 00:22:01.000 --> 00:22:05.000 these are the level curves of a function that tells you the 00:22:05.000 --> 00:22:12.000 temperature at a given point. OK, so these are examples of 00:22:12.000 --> 00:22:24.000 contour plots in real life. OK, no questions? 00:22:24.000 --> 00:22:26.000 No? OK, so basically, 00:22:26.000 --> 00:22:31.000 one of the goals that one should try to achieve at this 00:22:31.000 --> 00:22:35.000 point is becoming familiar with the contour plot, 00:22:35.000 --> 00:22:38.000 the graph, and how to view and deal with 00:22:38.000 --> 00:22:39.000 functions. 00:22:54.000 --> 00:23:02.000 [APPLAUSE] OK, so -- Let's do an example. 00:23:02.000 --> 00:23:04.000 Well, let's do a couple of examples. 00:23:04.000 --> 00:23:08.000 So, let's start with f(x,y) = - y. 00:23:08.000 --> 00:23:12.000 And, I'm going to take the same two examples as there to start 00:23:12.000 --> 00:23:16.000 with so that we see the relation between the graph and the 00:23:16.000 --> 00:23:23.000 contour plots. So, let's try to plot it. 00:23:23.000 --> 00:23:30.000 So, we are asked for the level curve, f equals 0 for this one? 00:23:30.000 --> 00:23:38.000 Well, f is zero when y is zero. So, that's the x axis. 00:23:38.000 --> 00:23:44.000 OK, so that's the level, zero. Where's the level one? 00:23:44.000 --> 00:23:48.000 Well, f is one when negative y is one. 00:23:48.000 --> 00:23:51.000 That means when y is negative one. 00:23:51.000 --> 00:23:57.000 So, you go to minus one, and that will be where my level 00:23:57.000 --> 00:24:02.000 one is, and so on. f is two when y is negative 00:24:02.000 --> 00:24:06.000 two. F is negative one when y is 00:24:06.000 --> 00:24:10.000 one, and so on. Is that convincing? 00:24:10.000 --> 00:24:15.000 Do you see how we got that? OK, let me do it again. 00:24:15.000 --> 00:24:18.000 I don't see anybody nodding, so that's kind of bad news. 00:24:18.000 --> 00:24:22.000 So, if I want to know, where is the level curve, 00:24:22.000 --> 00:24:26.000 say, one, I try to set f equals to one. 00:24:26.000 --> 00:24:31.000 Let's do this one. f equals one means that 00:24:31.000 --> 00:24:36.000 negative y is one means that y is minus one, 00:24:36.000 --> 00:24:43.000 and y equals minus one is this horizontal line on this chart. 00:24:43.000 --> 00:24:47.000 OK, and same with the others. So, you can see on the map that 00:24:47.000 --> 00:24:49.000 the value of a function doesn't depend on x. 00:24:49.000 --> 00:24:52.000 If you move parallel to the x axis, nothing happens. 00:24:52.000 --> 00:24:56.000 If you move in the y direction, it decreases at a constant 00:24:56.000 --> 00:24:59.000 rate. That's why the contours are 00:24:59.000 --> 00:25:03.000 evenly spaced. How spaced out they are tells 00:25:03.000 --> 00:25:06.000 you, actually, how steep things are. 00:25:06.000 --> 00:25:08.000 So, that corresponds exactly to that picture, 00:25:08.000 --> 00:25:11.000 except that here we draw x coming to the front, 00:25:11.000 --> 00:25:14.000 and y to the right. So, you have to rotate the map 00:25:14.000 --> 00:25:19.000 by 90� to get to that. It's an unfortunate consequence 00:25:19.000 --> 00:25:24.000 of the usual way of plotting things in space. 00:25:24.000 --> 00:25:32.000 OK, so these horizontal lines here correspond actually to 00:25:32.000 --> 00:25:35.000 horizontal lines here. 00:25:43.000 --> 00:25:54.000 OK, second example. Let's do 1-x^2-y^2. 00:25:54.000 --> 00:26:00.000 OK, or maybe I will write it as 1 - (x^2 y^2). 00:26:00.000 --> 00:26:06.000 It's really the same thing. So, x, y, let's see, 00:26:06.000 --> 00:26:12.000 where is this function equal to zero? 00:26:12.000 --> 00:26:21.000 Well, we said f is zero in the unit circle. 00:26:21.000 --> 00:26:32.000 OK, so, the zero level, well, let's say that this is my 00:26:32.000 --> 00:26:36.000 unit. That's where it's at zero. 00:26:36.000 --> 00:26:48.000 What about f equals one? Well, that's when x^2 y^2 = 0. 00:26:48.000 --> 00:26:49.000 Well, that's only going to be here. 00:26:49.000 --> 00:27:00.000 So, that's just a single point. What about f equals minus one? 00:27:00.000 --> 00:27:07.000 That's when x^2 y^2 =2. That's a circle of radius 00:27:07.000 --> 00:27:10.000 square root of two, which is about 1.4. 00:27:10.000 --> 00:27:17.000 So, it's somewhere here. Then, minus two, 00:27:17.000 --> 00:27:24.000 similarly, will be x^2 y^2 = 3. Square root of three is about 00:27:24.000 --> 00:27:27.000 1.7. And then, minus three will be 00:27:27.000 --> 00:27:30.000 of radius two, and so on. 00:27:30.000 --> 00:27:38.000 So, what I want to show here is that they are getting closer and 00:27:38.000 --> 00:27:41.000 closer apart, OK? 00:27:41.000 --> 00:27:44.000 So, first it's concentric circles that tells us that we 00:27:44.000 --> 00:27:47.000 have a shape that's a solid of the graph is going to be a 00:27:47.000 --> 00:27:52.000 surface of revolution. Things don't change if I rotate. 00:27:52.000 --> 00:27:56.000 And second, the level curves are getting closer and closer to 00:27:56.000 --> 00:27:59.000 each other. That means it's getting steeper 00:27:59.000 --> 00:28:03.000 and steeper because I have to travel a shorter distance if I 00:28:03.000 --> 00:28:06.000 want my altitude to change by one. 00:28:06.000 --> 00:28:09.000 OK, so, this top here is kind of flat. 00:28:09.000 --> 00:28:11.000 And then it gets steeper and steeper. 00:28:11.000 --> 00:28:16.000 And, that's what we observe on that picture there. 00:28:16.000 --> 00:28:24.000 OK, so just to show you a few more, where did I put my, 00:28:24.000 --> 00:28:30.000 so, for x^2 y^2, the contour plot looks like 00:28:30.000 --> 00:28:37.000 this. Maybe actually I'll make it. 00:28:37.000 --> 00:28:41.000 OK, so it looks exactly the same as this one. 00:28:41.000 --> 00:28:44.000 But, the difference is if you can see the numbers which are 00:28:44.000 --> 00:28:45.000 not there, so you can see them, 00:28:45.000 --> 00:28:49.000 then you would know that instead of decreasing as we move 00:28:49.000 --> 00:28:52.000 out, this one is increasing as we go 00:28:52.000 --> 00:28:54.000 out. OK, so that's where we use, 00:28:54.000 --> 00:28:57.000 actually, the labels on the level curves that tell us 00:28:57.000 --> 00:29:00.000 whether things are going up or down. 00:29:00.000 --> 00:29:04.000 But, the contour plots look exactly the same. 00:29:04.000 --> 00:29:14.000 For the next one I had, I think, was y^2-x^2. 00:29:14.000 --> 00:29:18.000 So, the contour plot, well, let me actually zoom out. 00:29:18.000 --> 00:29:20.000 So, the contour plot looks like that. 00:29:20.000 --> 00:29:23.000 So, the level curve corresponding to zero is 00:29:23.000 --> 00:29:27.000 actually two diagonal lines. And, if you look on the plot, 00:29:27.000 --> 00:29:30.000 say that you started at the saddle point in the middle and 00:29:30.000 --> 00:29:33.000 you try to stay at the same level. 00:29:33.000 --> 00:29:35.000 So, it looks like a mountain pass, right? 00:29:35.000 --> 00:29:38.000 Well, there's actually four directions from that point that 00:29:38.000 --> 00:29:41.000 you can go staying at the same height. 00:29:41.000 --> 00:29:44.000 And actually, on the map, they look exactly 00:29:44.000 --> 00:29:46.000 like this, too, these crossing lines. 00:29:46.000 --> 00:29:49.000 OK, so, these are things that go on the side of the two 00:29:49.000 --> 00:29:53.000 mountains that are to the left and right, and stay at the same 00:29:53.000 --> 00:29:57.000 height as the mountain pass. On the other hand, 00:29:57.000 --> 00:30:01.000 if you go along the y direction, to the left or to the 00:30:01.000 --> 00:30:05.000 right, then you go towards positive values. 00:30:05.000 --> 00:30:11.000 And, if you go along the x axis, then you get towards 00:30:11.000 --> 00:30:18.000 negative values. OK, the equation for, 00:30:18.000 --> 00:30:25.000 the function was y^2-x^2. So, you can try to plot them by 00:30:25.000 --> 00:30:27.000 hand and confirmed that it does look like that. 00:30:27.000 --> 00:30:33.000 But, I trust my computer. And, finally, 00:30:33.000 --> 00:30:39.000 this one, well, so the contour plot looks a bit 00:30:39.000 --> 00:30:43.000 complicated. But, you can see two things. 00:30:43.000 --> 00:30:45.000 In the middle, you can see these two origins 00:30:45.000 --> 00:30:47.000 with these concentric circles which are not really circles, 00:30:47.000 --> 00:30:50.000 but, you know, these closed curves that are 00:30:50.000 --> 00:30:53.000 concentric. And, they correspond to the two 00:30:53.000 --> 00:30:56.000 mountains. And then, at some point in the 00:30:56.000 --> 00:31:00.000 middle, we have a mountain pass. And there, we see the two 00:31:00.000 --> 00:31:05.000 crossing lines again, like, on the plot of y^2-x^2. 00:31:05.000 --> 00:31:11.000 And so, at this saddle point here, if we go north or south, 00:31:11.000 --> 00:31:15.000 then we go down on either side to the Valley. 00:31:15.000 --> 00:31:17.000 And, if we go east or west, then we go towards the 00:31:17.000 --> 00:31:21.000 mountains. We'll go up. 00:31:21.000 --> 00:31:26.000 OK, does that make sense a little bit? 00:31:26.000 --> 00:31:31.000 OK, so, reading plots is not easy, but hopefully we'll get 00:31:31.000 --> 00:31:32.000 used to it very soon. 00:31:44.000 --> 00:31:49.000 OK, so actually let's say, well, OK, so, 00:31:49.000 --> 00:31:55.000 I want to point out one thing. The contour plot tells us, 00:31:55.000 --> 00:32:00.000 actually, what happens when we move, when we change x and y. 00:32:00.000 --> 00:32:05.000 So, if I change the value of x and y, that means I'm moving 00:32:05.000 --> 00:32:08.000 east-west or north-south on the map. 00:32:08.000 --> 00:32:12.000 And then, I can ask myself, is the value of the function 00:32:12.000 --> 00:32:15.000 increase or decrease in each of these situations? 00:32:15.000 --> 00:32:18.000 Well, that's the kind of thing that the contour plot can tell 00:32:18.000 --> 00:32:19.000 me very quickly. 00:32:54.000 --> 00:32:56.000 So -- OK, so say, for example, 00:32:56.000 --> 00:32:59.000 that I have a piece of contour plot. 00:32:59.000 --> 00:33:01.000 That looks, you know, like that. 00:33:01.000 --> 00:33:06.000 Maybe this is f equals one, and this is f equals two. 00:33:06.000 --> 00:33:13.000 And here, this is f equals zero. And, let's say that I start at 00:33:13.000 --> 00:33:17.000 the point, say, at this point. 00:33:17.000 --> 00:33:23.000 OK, so here I have (x0, y0). And, the question I might ask 00:33:23.000 --> 00:33:26.000 myself is if I change x or y, how does f change? 00:33:26.000 --> 00:33:34.000 Well, the contour plot tells me that if x increases, 00:33:34.000 --> 00:33:41.000 and I keep y constant, then what happens to f(x, 00:33:41.000 --> 00:33:44.000 y)? Well, it will increase because 00:33:44.000 --> 00:33:47.000 if I move to the right, then I go from one to a value 00:33:47.000 --> 00:33:50.000 bigger than one. I don't know exactly how much, 00:33:50.000 --> 00:33:53.000 but I know that somewhere between one and two, 00:33:53.000 --> 00:33:57.000 it's more than one. If x decreases, 00:33:57.000 --> 00:34:02.000 then f decreases. And, similarly, 00:34:02.000 --> 00:34:07.000 I can tell that if y increases, then f, well, 00:34:07.000 --> 00:34:14.000 it looks like if I increase y, then f will also increase. 00:34:14.000 --> 00:34:20.000 And, if y decreases, then f decreases. 00:34:20.000 --> 00:34:23.000 And, that's the kind of qualitative analysis that we can 00:34:23.000 --> 00:34:27.000 do easily from the contour plot. But, maybe we'd like to 00:34:27.000 --> 00:34:30.000 actually be more precise in that, and tell how fast f 00:34:30.000 --> 00:34:34.000 changes if I change x or y. OK, so to find the rate of 00:34:34.000 --> 00:34:39.000 change, that's exactly where we use derivatives. 00:34:39.000 --> 00:34:47.000 So -- So, we are going to have to deal with partial 00:34:47.000 --> 00:34:58.000 derivatives. So, I will explain to you soon 00:34:58.000 --> 00:35:05.000 why partial. So, let me just remind you 00:35:05.000 --> 00:35:12.000 first, if you have a function of one variable, 00:35:12.000 --> 00:35:18.000 then so let's say f of x, then you have a derivative, 00:35:18.000 --> 00:35:22.000 f prime of x is also called df/dx. 00:35:22.000 --> 00:35:31.000 And, it's defined as a limit when delta x goes to zero of the 00:35:31.000 --> 00:35:35.000 change in f. Sorry, it's not going to fit. 00:35:35.000 --> 00:35:42.000 I have to go to the next line. It's going to be the limit as 00:35:42.000 --> 00:35:47.000 delta x goes to zero of the rate of change. 00:35:47.000 --> 00:35:52.000 So, the change in f between x and x plus delta x divided by 00:35:52.000 --> 00:35:56.000 delta x. Sometimes you write delta f for 00:35:56.000 --> 00:35:59.000 the change in f divided by delta x. 00:35:59.000 --> 00:36:04.000 And then, you take the limit of this rate of increase as delta x 00:36:04.000 --> 00:36:05.000 goes to zero. Now, of course, 00:36:05.000 --> 00:36:08.000 if you have a formula for f, then you know, 00:36:08.000 --> 00:36:12.000 at least you should know, I suspect most of you know how 00:36:12.000 --> 00:36:19.000 to actually take the derivative of a function from its formula. 00:36:19.000 --> 00:36:30.000 So -- Now, how do we do that? Sorry, and I should also tell 00:36:30.000 --> 00:36:32.000 you what this means on the graph. 00:36:32.000 --> 00:36:36.000 So, if I plot the graph of a function, and to have my point, 00:36:36.000 --> 00:36:41.000 x, and here I have f of x, how do I see the derivative? 00:36:41.000 --> 00:36:48.000 Well, I look at the tangent line to the graph, 00:36:48.000 --> 00:36:55.000 and the slope of the tangent line is f prime of x, 00:36:55.000 --> 00:36:59.000 OK? And, not every function has a 00:36:59.000 --> 00:37:03.000 derivative. You have functions that are not 00:37:03.000 --> 00:37:05.000 regular enough to actually have a derivative. 00:37:05.000 --> 00:37:08.000 So, in this class, we are not going to actually 00:37:08.000 --> 00:37:11.000 worry too much about differentiability. 00:37:11.000 --> 00:37:16.000 But, it's good, at least, to be aware that you 00:37:16.000 --> 00:37:19.000 can't always take the derivative. 00:37:19.000 --> 00:37:24.000 So, yes, and what else do I want to remind you of? 00:37:24.000 --> 00:37:32.000 Well, they also have an approximation formula -- -- 00:37:32.000 --> 00:37:39.000 which says that, you know, if we have the value 00:37:39.000 --> 00:37:41.000 of f at some point, x0, 00:37:41.000 --> 00:37:47.000 and that we want to find the value at a nearby point, 00:37:47.000 --> 00:37:51.000 x close to x0, then our best guess is that 00:37:51.000 --> 00:37:58.000 it's f of x0 plus the derivative f prime at x0 times delta x, 00:37:58.000 --> 00:38:02.000 or if you want, x minus x0, OK? 00:38:02.000 --> 00:38:06.000 Is this kind of familiar to you? Yeah, I mean, 00:38:06.000 --> 00:38:09.000 maybe with different notations. Maybe you called that delta x 00:38:09.000 --> 00:38:12.000 or something. Maybe you called that x0 plus h 00:38:12.000 --> 00:38:14.000 or something. But, it's the usual 00:38:14.000 --> 00:38:18.000 approximation formula using the derivative. 00:38:18.000 --> 00:38:21.000 If you put more terms, then you get the dreaded Taylor 00:38:21.000 --> 00:38:24.000 approximation that I know you guys don't like. 00:38:24.000 --> 00:38:36.000 So, the question is how do we do the same for a function of 00:38:36.000 --> 00:38:41.000 two variables, f(x, y)? 00:38:41.000 --> 00:38:45.000 So, the difficulty there is we can change x, 00:38:45.000 --> 00:38:49.000 or we can change y, or we can change both. 00:38:49.000 --> 00:38:52.000 And, presumably, the manner in which f changes 00:38:52.000 --> 00:38:56.000 will be different depending on whether we change x or y. 00:38:56.000 --> 00:39:00.000 So, that's why we have several different notions of derivative. 00:39:24.000 --> 00:39:37.000 So, OK, we have a notation. OK, so this is a curly d, 00:39:37.000 --> 00:39:41.000 and it is not a straight d, and it is not a delta. 00:39:41.000 --> 00:39:44.000 It's a d that kind of curves backwards like that. 00:39:44.000 --> 00:39:50.000 And, this symbol is partial. OK, so it's a special notation 00:39:50.000 --> 00:39:54.000 for partial derivatives. And, essentially what it means 00:39:54.000 --> 00:39:56.000 is we are going to do a derivative where we care about 00:39:56.000 --> 00:39:59.000 only one variable at a time. That's why it's partial. 00:39:59.000 --> 00:40:02.000 It's missing the other variables. 00:40:02.000 --> 00:40:06.000 So, a function of several variables doesn't have the usual 00:40:06.000 --> 00:40:10.000 derivative. It has only partial derivatives 00:40:10.000 --> 00:40:15.000 for each variable. So, the partial derivative, 00:40:15.000 --> 00:40:23.000 the partial f partial x at (x0, y0) is defined to be the limit 00:40:23.000 --> 00:40:29.000 when I take a small change in x, delta x, 00:40:29.000 --> 00:40:43.000 of the change in f -- -- divided by delta x. 00:40:43.000 --> 00:40:47.000 OK, so here I'm actually not changing y at all. 00:40:47.000 --> 00:40:51.000 I'm just changing x and looking at the rate of change with 00:40:51.000 --> 00:40:54.000 respect to x. And, I have the same with 00:40:54.000 --> 00:40:58.000 respect to y. Partial f partial y is the 00:40:58.000 --> 00:41:04.000 limit, so I should say, at a point x0 y0 is the limit 00:41:04.000 --> 00:41:13.000 as delta y turns to zero. So, this time I keep x the 00:41:13.000 --> 00:41:21.000 same, but I change y. OK, so that's the definition of 00:41:21.000 --> 00:41:26.000 a partial derivative. And, we say that a function is 00:41:26.000 --> 00:41:29.000 differentiable if these things exist. 00:41:29.000 --> 00:41:31.000 OK, so most of the functions we'll see are differentiable. 00:41:31.000 --> 00:41:34.000 And, we'll actually learn how to compute their partial 00:41:34.000 --> 00:41:38.000 derivatives without having to do this because we'll just have the 00:41:38.000 --> 00:41:41.000 usual methods for computing derivatives. 00:41:41.000 --> 00:41:46.000 So, in fact, I claim you already know how to 00:41:46.000 --> 00:41:49.000 take partial derivatives. So, let's see what it means 00:41:49.000 --> 00:41:50.000 geometrically. 00:42:00.000 --> 00:42:07.000 So, geometrically, my function can be represented 00:42:07.000 --> 00:42:12.000 by this graph, and I fix some point, 00:42:12.000 --> 00:42:18.000 (x0, y0). And then, I'm going to ask 00:42:18.000 --> 00:42:24.000 myself what happens if I change the value of, 00:42:24.000 --> 00:42:30.000 well, x, keeping y constant. So, if I keep y constant and 00:42:30.000 --> 00:42:33.000 change x, it means that I'm moving forwards or backwards 00:42:33.000 --> 00:42:38.000 parallel to the x axis. So, that determines for me the 00:42:38.000 --> 00:42:46.000 vertical plane parallel to the x, z plane when I fix y equals 00:42:46.000 --> 00:42:51.000 constant. And now, if I slice the graph 00:42:51.000 --> 00:42:59.000 by that, I will get some curve that goes, it's a slice of the 00:42:59.000 --> 00:43:03.000 graph of f. And now, what I want to find is 00:43:03.000 --> 00:43:06.000 how f depends on x if I keep y constant. 00:43:06.000 --> 00:43:09.000 That means it's the rate of change if I move along this 00:43:09.000 --> 00:43:11.000 curve. So, in fact, 00:43:11.000 --> 00:43:17.000 if I look at the slope of this thing. 00:43:17.000 --> 00:43:22.000 So, if I draw the tangent line to this slice, 00:43:22.000 --> 00:43:28.000 then the slope will be partial f of partial x. 00:43:28.000 --> 00:43:32.000 I think I have a better picture of that somewhere. 00:43:32.000 --> 00:43:40.000 Yes, here it is. OK, that's the same picture, 00:43:40.000 --> 00:43:43.000 just with different colors. So, I take the graph. 00:43:43.000 --> 00:43:46.000 I slice it by a vertical plane. I get the curve, 00:43:46.000 --> 00:43:50.000 and now I take the slope of that curve, and that gives me 00:43:50.000 --> 00:43:54.000 the partial derivative. And, to finish, 00:43:54.000 --> 00:43:59.000 let me just tell you how, and I should say, 00:43:59.000 --> 00:44:02.000 partial f partial y is the same thing but slicing now by your 00:44:02.000 --> 00:44:05.000 plane that goes in the y, z directions. 00:44:05.000 --> 00:44:11.000 OK, so I fix x equals constant. That means that I slice by a 00:44:11.000 --> 00:44:13.000 plane that's parallel to the blackboard. 00:44:13.000 --> 00:44:17.000 I get a curve, and I looked at the slope of 00:44:17.000 --> 00:44:20.000 that curve. OK, so it's just a matter of 00:44:20.000 --> 00:44:23.000 formatting one variable, setting it constant, 00:44:23.000 --> 00:44:27.000 and looking at the other one. So, how to compute these 00:44:27.000 --> 00:44:29.000 things, well, we actually, 00:44:29.000 --> 00:44:33.000 to find, well, there's a piece of notation I 00:44:33.000 --> 00:44:38.000 haven't told you yet. So, another notation you will 00:44:38.000 --> 00:44:42.000 see, I think this is what one uses a lot in physics. 00:44:42.000 --> 00:44:45.000 And, this is what one uses a lot in applied math, 00:44:45.000 --> 00:44:47.000 which is the same thing as physics but with different 00:44:47.000 --> 00:44:50.000 notations. OK, so it just too different 00:44:50.000 --> 00:44:54.000 notations: partial f partial x, or f subscript x. 00:44:54.000 --> 00:45:01.000 And, they are the same thing. Well, we just treat y as a 00:45:01.000 --> 00:45:10.000 constant, and x as a variable. And, vice versa if we want to 00:45:10.000 --> 00:45:16.000 find partial with aspect to y. So, let me just finish with one 00:45:16.000 --> 00:45:22.000 quick example. Let's say that they give you f 00:45:22.000 --> 00:45:28.000 of x, y equals x^3y y^2, then partial f partial x. 00:45:28.000 --> 00:45:32.000 Well, let's take the derivative. So, here it's x^3 times a 00:45:32.000 --> 00:45:37.000 constant. Derivative of x^3 is 3x^2 times 00:45:37.000 --> 00:45:42.000 the constant plus what's the derivative of y^2? 00:45:42.000 --> 00:45:45.000 Zero, because it's a constant. If you do, instead, 00:45:45.000 --> 00:45:48.000 partial f partial y, then this is actually a 00:45:48.000 --> 00:45:51.000 constant times y. The derivative of y is one. 00:45:51.000 --> 00:45:57.000 So, that's just x^3. And, the derivative of y^2 is 00:45:57.000 --> 00:45:59.000 2y. OK, so it's fairly easy. 00:45:59.000 --> 00:46:02.000 You just have to keep remembering which one is a 00:46:02.000 --> 00:46:06.000 variable, and which one isn't. OK, so more about this next 00:46:06.000 --> 00:46:10.000 time, and we will also learn about maxima and minima in 00:46:10.000 --> 00:46:13.000 several variables.