1 00:00:01,000 --> 00:00:03,000 2 00:00:01,000 --> 00:00:03,000 The following content is provided under a Creative 3 00:00:03,000 --> 00:00:05,000 Commons license. Your support will help MIT 4 00:00:05,000 --> 00:00:08,000 OpenCourseWare continue to offer high quality educational 5 00:00:08,000 --> 00:00:13,000 resources for free. To make a donation or to view 6 00:00:13,000 --> 00:00:18,000 additional materials from hundreds of MIT courses, 7 00:00:18,000 --> 00:00:23,000 visit MIT OpenCourseWare at ocw.mit.edu. 8 00:00:23,000 --> 00:00:25,000 So, so far, we've seen things about vectors, 9 00:00:25,000 --> 00:00:27,000 equation of planes, motions in space, 10 00:00:27,000 --> 00:00:30,000 and so on. Basically we've done geometry 11 00:00:30,000 --> 00:00:31,000 in space. But, calculus, 12 00:00:31,000 --> 00:00:33,000 really, is about studying functions. 13 00:00:33,000 --> 00:00:37,000 Now, we're going to actually move on to studying functions of 14 00:00:37,000 --> 00:00:40,000 several variables. So, this new unit, 15 00:00:40,000 --> 00:00:45,000 what we'll do over the next three weeks or so will be about 16 00:00:45,000 --> 00:00:50,000 functions of several variables and their derivatives. 17 00:00:50,000 --> 00:00:55,000 OK, so first of all, we should try to figure out how 18 00:00:55,000 --> 00:00:58,000 we are going to think about functions. 19 00:00:58,000 --> 00:01:02,000 So, remember, if you have a function of one 20 00:01:02,000 --> 00:01:08,000 variable, that means you have a quantity that depends on one 21 00:01:08,000 --> 00:01:12,000 parameter. Maybe f depends on the variable 22 00:01:12,000 --> 00:01:14,000 x. And, for example, 23 00:01:14,000 --> 00:01:19,000 a function that you all know is f of x equals sin(x). 24 00:01:19,000 --> 00:01:22,000 And, the way we would represent that is maybe just by plotting 25 00:01:22,000 --> 00:01:28,000 the graph of the function. So, the graph of a function, 26 00:01:28,000 --> 00:01:34,000 we plot y = f(x). And, the graph of a sine 27 00:01:34,000 --> 00:01:45,000 function that looks like this. OK, so now, let's say that we 28 00:01:45,000 --> 00:01:53,000 had, actually, a function of two variables. 29 00:01:53,000 --> 00:01:56,000 So, that means that the value of F depends actually on two 30 00:01:56,000 --> 00:01:59,000 different parameters, say, if the variables are x and 31 00:01:59,000 --> 00:02:03,000 y, or they can have any names you 32 00:02:03,000 --> 00:02:07,000 want. So, given values of the two 33 00:02:07,000 --> 00:02:13,000 parameters, x and y, the function will give us a 34 00:02:13,000 --> 00:02:17,000 number that we'll call f(x, y). 35 00:02:17,000 --> 00:02:21,000 That depends on x and y according to some formula, 36 00:02:21,000 --> 00:02:29,000 OK, not very surprising so far. So, for example, 37 00:02:29,000 --> 00:02:44,000 I can give you the function f(x, y) = x^2 y^2. 38 00:02:44,000 --> 00:02:47,000 And, of course, as with functions of one 39 00:02:47,000 --> 00:02:50,000 variable, we don't need things to be defined everywhere. 40 00:02:50,000 --> 00:02:53,000 Sometimes there is the domain of definition. 41 00:02:53,000 --> 00:02:57,000 So, this one is defined all the time. 42 00:02:57,000 --> 00:03:01,000 But, if I tell you, say, f of x, 43 00:03:01,000 --> 00:03:08,000 y equals square root of y, well, this is only defined if y 44 00:03:08,000 --> 00:03:14,000 is nonnegative. If I tell you f(x, 45 00:03:14,000 --> 00:03:23,000 y) equals one over x y, that's only defined if x y is 46 00:03:23,000 --> 00:03:30,000 not zero, and so on. Now, so these are mathematical 47 00:03:30,000 --> 00:03:33,000 examples given by explicit formulas. 48 00:03:33,000 --> 00:03:35,000 And, of course, there's physical examples. 49 00:03:35,000 --> 00:03:38,000 For example, so examples coming from real 50 00:03:38,000 --> 00:03:40,000 life, so for example, you can look at the temperature 51 00:03:40,000 --> 00:03:43,000 at the certain point on the surface of the earth. 52 00:03:43,000 --> 00:03:46,000 So, you use maybe longitude and latitude; that's x and y. 53 00:03:46,000 --> 00:03:49,000 And then you have f(x, y) equals the temperature at 54 00:03:49,000 --> 00:03:50,000 that point. 55 00:03:50,000 --> 00:04:12,000 56 00:04:12,000 --> 00:04:17,000 In fact, because temperature depends also may be on how high 57 00:04:17,000 --> 00:04:18,000 up you are. It depends on elevation. 58 00:04:18,000 --> 00:04:21,000 So, it's actually a function of maybe x, y, z. 59 00:04:21,000 --> 00:04:24,000 And, it also depends on time. So, in fact, 60 00:04:24,000 --> 00:04:28,000 maybe it's a function of t in x y z coordinates in space. 61 00:04:28,000 --> 00:04:31,000 So, you see that real-world functions can depends on a lot 62 00:04:31,000 --> 00:04:33,000 of variables. So, our goal will be to 63 00:04:33,000 --> 00:04:35,000 understand how to deal with that. 64 00:04:35,000 --> 00:04:57,000 65 00:04:57,000 --> 00:05:01,000 OK, so now you will see very soon, but actually it's already 66 00:05:01,000 --> 00:05:05,000 tricky enough to picture a function of two variables. 67 00:05:05,000 --> 00:05:08,000 So, we are going to focus on the case of functions of two 68 00:05:08,000 --> 00:05:10,000 variables. And then, we'll see that if we 69 00:05:10,000 --> 00:05:12,000 have more than two variables, then it's harder to plot the 70 00:05:12,000 --> 00:05:14,000 function. We cannot draw with the graph 71 00:05:14,000 --> 00:05:17,000 looks like anymore. But, the tools are the same, 72 00:05:17,000 --> 00:05:20,000 the notion of partial derivatives, grade and vector, 73 00:05:20,000 --> 00:05:23,000 and so on, all the tools that we will 74 00:05:23,000 --> 00:05:27,000 develop work exactly the same way no matter how many variables 75 00:05:27,000 --> 00:05:30,000 you have. So, I'll say, 76 00:05:30,000 --> 00:05:41,000 for simplicity -- -- we'll focus mostly on two or sometimes 77 00:05:41,000 --> 00:05:48,000 three variables. But, it works the same in any 78 00:05:48,000 --> 00:05:56,000 number of variables. OK, so the first question is 79 00:05:56,000 --> 00:06:05,000 how to visualize a function of two variables. 80 00:06:05,000 --> 00:06:10,000 So, the first thing we can do is try to draw the graph of f. 81 00:06:10,000 --> 00:06:19,000 So, maybe I should say f -- which is a function of two 82 00:06:19,000 --> 00:06:23,000 variables. So, the first answer will be, 83 00:06:23,000 --> 00:06:26,000 we can try to look at it's graph. 84 00:06:26,000 --> 00:06:29,000 And, the idea is the same as with one variable, 85 00:06:29,000 --> 00:06:31,000 namely, we look at all the possible values of the 86 00:06:31,000 --> 00:06:34,000 parameters, x and y, and for each of them, 87 00:06:34,000 --> 00:06:40,000 we plot a point whose height is the value of a function at these 88 00:06:40,000 --> 00:06:43,000 parameters. So, we'll plot, 89 00:06:43,000 --> 00:06:47,000 let's say, z equals f(x, y). 90 00:06:47,000 --> 00:06:52,000 And, now that will become, actually, a surface in space. 91 00:06:52,000 --> 00:06:57,000 OK, so for each value of x and y, yeah, we have x, 92 00:06:57,000 --> 00:07:02,000 y in the x, y plane, then we'll plot the point in 93 00:07:02,000 --> 00:07:05,000 space at position x, y. 94 00:07:05,000 --> 00:07:13,000 And, z equals f of x, y. OK, and if we take all of these 95 00:07:13,000 --> 00:07:17,000 points together, they will give us some surface 96 00:07:17,000 --> 00:07:24,000 that sits in space. Yes? 97 00:07:24,000 --> 00:07:29,000 Oh, a function of two variables, shorthand. 98 00:07:29,000 --> 00:07:36,000 Well, let's say how to visualize a function of two 99 00:07:36,000 --> 00:07:39,000 variables. OK, so, how do we do that 100 00:07:39,000 --> 00:07:41,000 concretely? Say that I give you a formula 101 00:07:41,000 --> 00:07:46,000 for f. How do we try to represent it? 102 00:07:46,000 --> 00:07:57,000 So, let's do our first example. Let's say I give you a function 103 00:07:57,000 --> 00:08:02,000 f(x, y) = -y. OK, so it looks a little bit 104 00:08:02,000 --> 00:08:04,000 silly because it doesn't depend on x. 105 00:08:04,000 --> 00:08:10,000 But, that's not the problem. It's still a valid function of 106 00:08:10,000 --> 00:08:13,000 x and y. It just happens to be constant 107 00:08:13,000 --> 00:08:17,000 with respect to x. So, to draw the graph we look 108 00:08:17,000 --> 00:08:22,000 at the surface in space defined by z equals y. 109 00:08:22,000 --> 00:08:26,000 What kind of surface is that? It's a plane, OK? 110 00:08:26,000 --> 00:08:32,000 And, if we want to draw it, z equals minus y will look, 111 00:08:32,000 --> 00:08:36,000 well, let's put y axis. Let's put x axis. 112 00:08:36,000 --> 00:08:39,000 Let's put z axis. If I look at what happens in 113 00:08:39,000 --> 00:08:43,000 the y, z plane in the plane of a blackboard, it will just look 114 00:08:43,000 --> 00:08:45,000 like a line that goes downward with slope one. 115 00:08:45,000 --> 00:08:51,000 OK, so it will be this. And, what happens if I change x? 116 00:08:51,000 --> 00:08:54,000 Well, if I change x, nothing happens because x 117 00:08:54,000 --> 00:08:57,000 doesn't appear in this equation. So, in fact, 118 00:08:57,000 --> 00:09:01,000 if instead of setting x equal to zero I set x equal to one, 119 00:09:01,000 --> 00:09:04,000 I'm in front of the blackboard, or minus one at the back. 120 00:09:04,000 --> 00:09:07,000 Well, it still looks exactly the same. 121 00:09:07,000 --> 00:09:15,000 So, I have this plane that actually contains the x axis and 122 00:09:15,000 --> 00:09:22,000 slopes downwards with slope one. It's kind of hard to draw. 123 00:09:22,000 --> 00:09:25,000 Now, you see immediately what the big problem with graphs will 124 00:09:25,000 --> 00:09:29,000 be. But, these pictures are hard to 125 00:09:29,000 --> 00:09:34,000 read. But that's our first graph. 126 00:09:34,000 --> 00:09:41,000 OK, a question so far? OK, so let's say that we have a 127 00:09:41,000 --> 00:09:43,000 slightly more complicated function. 128 00:09:43,000 --> 00:09:50,000 How do we see it? So, let's draw another example. 129 00:09:50,000 --> 00:09:58,000 Let's say I give you f(x, y) = 1 - x^2-y^2. 130 00:09:58,000 --> 00:10:06,000 So, we should try to picture what the surface z=1- x^2-y^2 131 00:10:06,000 --> 00:10:10,000 looks like. So, how do we do that? 132 00:10:10,000 --> 00:10:15,000 Well, maybe you are very fast and figured out what it looks 133 00:10:15,000 --> 00:10:17,000 like. But, if not, 134 00:10:17,000 --> 00:10:21,000 then we need to work piece by piece. 135 00:10:21,000 --> 00:10:27,000 So, maybe it will help if we understand first what it does in 136 00:10:27,000 --> 00:10:35,000 the plane of the blackboard. So, if we look at it in the y, 137 00:10:35,000 --> 00:10:42,000 z plane, that means we set x equal to zero. 138 00:10:42,000 --> 00:10:48,000 And then, z becomes 1 - y^2. What is that? 139 00:10:48,000 --> 00:10:54,000 It's a parabola pointing downwards, and starting at one. 140 00:10:54,000 --> 00:11:00,000 So, we should draw maybe this downward parabola. 141 00:11:00,000 --> 00:11:08,000 It starts at one and it cuts the y axis at one. 142 00:11:08,000 --> 00:11:11,000 When y is one, that gives us zero. 143 00:11:11,000 --> 00:11:15,000 So, we might have an idea of what it might look like, 144 00:11:15,000 --> 00:11:19,000 or maybe not. Let's get more slices. 145 00:11:19,000 --> 00:11:27,000 Let's see what it does in the x, z plane, this other vertical 146 00:11:27,000 --> 00:11:33,000 plane that's coming towards us. So, in the x, 147 00:11:33,000 --> 00:11:38,000 z plane, if we set y equal to zero, we get z equals one minus 148 00:11:38,000 --> 00:11:40,000 x^2. It's, again, 149 00:11:40,000 --> 00:11:46,000 a parabola going downwards. OK, so I'm going to try to draw 150 00:11:46,000 --> 00:11:51,000 a parabola that goes downward, but now to the front and to the 151 00:11:51,000 --> 00:11:54,000 back. So, we are starting to have a 152 00:11:54,000 --> 00:11:57,000 slightly better idea but we still don't know whether the 153 00:11:57,000 --> 00:11:59,000 cross section of this thing might be round, 154 00:11:59,000 --> 00:12:04,000 square, something else. So, it wants more confirmation. 155 00:12:04,000 --> 00:12:16,000 We might want to also figure out where the surface intersects 156 00:12:16,000 --> 00:12:22,000 the x, y plane. So, we hit the x, 157 00:12:22,000 --> 00:12:29,000 y plane when z equals zero. That means 1-x^2-y^2 should be 158 00:12:29,000 --> 00:12:38,000 0, that becomes x^2 y^2 = 1. That is a circle of radius 1. 159 00:12:38,000 --> 00:12:46,000 That's the unit size. So, that means that here, 160 00:12:46,000 --> 00:12:50,000 we actually have the unit circle. 161 00:12:50,000 --> 00:12:54,000 And now, you should imagine that you have this thing that 162 00:12:54,000 --> 00:12:57,000 when you slice it by a vertical plane, looks like a downwards 163 00:12:57,000 --> 00:13:00,000 parabola. And, it's actually a surface of 164 00:13:00,000 --> 00:13:03,000 revolution. You can rotate it around the z 165 00:13:03,000 --> 00:13:06,000 axis, OK? Now, if you stare long enough 166 00:13:06,000 --> 00:13:09,000 at that equation, you'll actually see that, 167 00:13:09,000 --> 00:13:12,000 yes, we know that it had to be like that. 168 00:13:12,000 --> 00:13:17,000 But, see, so these are useful ways of trying to guess what the 169 00:13:17,000 --> 00:13:19,000 graph looks like. Of course, the other way is to 170 00:13:19,000 --> 00:13:24,000 just ask your computer to do it. And then, you know, 171 00:13:24,000 --> 00:13:30,000 you will get that kind of formula. 172 00:13:30,000 --> 00:13:36,000 OK, well, I can leave it on if you want. 173 00:13:36,000 --> 00:13:40,000 No, because I plotted a different function that I will 174 00:13:40,000 --> 00:13:43,000 show you later. So, it goes this way. 175 00:13:43,000 --> 00:13:46,000 I mean, if you want, it's really going downward. 176 00:13:46,000 --> 00:13:49,000 Yes, I agree that the sheet is upside down. 177 00:13:49,000 --> 00:13:52,000 That's because I plotted something else. 178 00:13:52,000 --> 00:13:58,000 OK, so, in fact, so I plotted in my computer was 179 00:13:58,000 --> 00:14:03,000 actually x^2 y^2 that looks like that. 180 00:14:03,000 --> 00:14:08,000 See, it's the same with a parabola going upwards. 181 00:14:08,000 --> 00:14:14,000 If you want to see more examples, I have various 182 00:14:14,000 --> 00:14:19,000 examples to show, well, here's the graph, 183 00:14:19,000 --> 00:14:21,000 y^2-x^2. See, so that one is kind of 184 00:14:21,000 --> 00:14:23,000 interesting. It looks like a saddle. 185 00:14:23,000 --> 00:14:29,000 If you look at it in the y, z plane, then it's a parabola 186 00:14:29,000 --> 00:14:34,000 going up, z = y^2. And, that's what we see to the 187 00:14:34,000 --> 00:14:38,000 left and to the right. But, if you put it in the x, 188 00:14:38,000 --> 00:14:42,000 z plane, then that's a parabola going downwards, 189 00:14:42,000 --> 00:14:45,000 z = - x^2. So, we have a parabola going 190 00:14:45,000 --> 00:14:48,000 downwards in one direction, upwards in the other one. 191 00:14:48,000 --> 00:14:53,000 And together, they form this surface. 192 00:14:53,000 --> 00:14:55,000 And of course, you can plot much more 193 00:14:55,000 --> 00:14:58,000 complicated functions. So, this one, 194 00:14:58,000 --> 00:15:00,000 well, if you can read very small things, 195 00:15:00,000 --> 00:15:03,000 you can see the formula. It doesn't matter, 196 00:15:03,000 --> 00:15:09,000 just to show you that you can put a formula into a computer: 197 00:15:09,000 --> 00:15:18,000 it will show you a picture. OK, so that's pretty good. 198 00:15:18,000 --> 00:15:20,000 I mean, you can see that it can get a bit cluttered because 199 00:15:20,000 --> 00:15:22,000 maybe those features that are hidden behind, 200 00:15:22,000 --> 00:15:25,000 or maybe we have trouble seeing if we don't have a computer, 201 00:15:25,000 --> 00:15:29,000 that looks very readable. But, this is kind of hard to 202 00:15:29,000 --> 00:15:33,000 visualize sometimes. So, there is another way to 203 00:15:33,000 --> 00:15:36,000 plot the functions of two variables. 204 00:15:36,000 --> 00:15:47,000 And, let's call it the contour plot. 205 00:15:47,000 --> 00:15:51,000 So, the contour plot is a very elegant solution to the problem 206 00:15:51,000 --> 00:15:55,000 that it's difficult to draw to space pictures on a sheet of 207 00:15:55,000 --> 00:15:58,000 paper or on a blackboard. So, instead, 208 00:15:58,000 --> 00:16:02,000 let's try to represent the function of two variables by 209 00:16:02,000 --> 00:16:04,000 just the map, you know, the same way that 210 00:16:04,000 --> 00:16:07,000 when you walk around, you have actually geographical 211 00:16:07,000 --> 00:16:11,000 maps that fit on a piece of paper that tell you about what 212 00:16:11,000 --> 00:16:17,000 the real world looks like. So, what contour plot looks 213 00:16:17,000 --> 00:16:22,000 something like this? So, it's an x, y plot. 214 00:16:22,000 --> 00:16:25,000 And, that, you have a bunch of curves. 215 00:16:25,000 --> 00:16:32,000 And, what the curves represent are the elevations on the graph. 216 00:16:32,000 --> 00:16:35,000 So, for example, this curve might correspond to 217 00:16:35,000 --> 00:16:37,000 all the points where f(x, y) = 1. 218 00:16:37,000 --> 00:16:46,000 And, that curve might be all the points where f=2 and f=3 and 219 00:16:46,000 --> 00:16:49,000 so on, OK? So, when you see you this kind 220 00:16:49,000 --> 00:16:53,000 of plot, you're supposed to think that the graph of the 221 00:16:53,000 --> 00:16:56,000 function sits somewhere in space above that. 222 00:16:56,000 --> 00:17:00,000 It's like a map telling you how high things are. 223 00:17:00,000 --> 00:17:03,000 And, what you would want to do with the function, 224 00:17:03,000 --> 00:17:06,000 really, is be able to tell quickly what's the value at a 225 00:17:06,000 --> 00:17:08,000 given point? Well, let's say I want to look 226 00:17:08,000 --> 00:17:11,000 at that point. I know that f is somewhere 227 00:17:11,000 --> 00:17:14,000 between 1 and 2. Actually, it's much faster to 228 00:17:14,000 --> 00:17:17,000 read than the graph. On the graph I might have to 229 00:17:17,000 --> 00:17:18,000 look carefully, and then measure things, 230 00:17:18,000 --> 00:17:22,000 and so on. Here, I can just raise the 231 00:17:22,000 --> 00:17:27,000 value of f by comparing with the nearby lines. 232 00:17:27,000 --> 00:17:31,000 OK, so let me try to make that more precise. 233 00:17:31,000 --> 00:17:41,000 So, it shows all the points -- -- where f(x, 234 00:17:41,000 --> 00:17:53,000 y) equals some fixed values, some fixed constants. 235 00:17:53,000 --> 00:18:05,000 And, these constants typically are chosen at regular intervals. 236 00:18:05,000 --> 00:18:07,000 For example, here I chose one, 237 00:18:07,000 --> 00:18:11,000 two, three, and they could continue with zero minus one, 238 00:18:11,000 --> 00:18:16,000 and so on. So, one way to think about it, 239 00:18:16,000 --> 00:18:21,000 how does this relate to the graph? 240 00:18:21,000 --> 00:18:31,000 Well, that's the same thing as cutting, I mean, 241 00:18:31,000 --> 00:18:40,000 we slice the graph by horizontal planes. 242 00:18:40,000 --> 00:18:44,000 So, horizontal planes have equations of a form z equals 243 00:18:44,000 --> 00:18:46,000 some constant, z equals zero, 244 00:18:46,000 --> 00:18:48,000 z equals one, z equals two, 245 00:18:48,000 --> 00:18:51,000 and so on. So, maybe the graph of my 246 00:18:51,000 --> 00:18:55,000 function will be some sort of plot out there. 247 00:18:55,000 --> 00:19:00,000 And, if I slice it by the plane z equals one, 248 00:19:00,000 --> 00:19:04,000 then I will get the level curve, 249 00:19:04,000 --> 00:19:14,000 which is the point where f(x, y) = 1, 250 00:19:14,000 --> 00:19:24,000 and so, that's called a level curve of f. 251 00:19:24,000 --> 00:19:32,000 OK, and so we repeat the process with maybe z equals two, 252 00:19:32,000 --> 00:19:38,000 and we get another level curve, and so on. 253 00:19:38,000 --> 00:19:44,000 And, then we squish all of them up, and that's how we get the 254 00:19:44,000 --> 00:19:47,000 contour plot. OK, so each of these lines, 255 00:19:47,000 --> 00:19:50,000 imagine that this is like some mountain or something that you 256 00:19:50,000 --> 00:19:52,000 are hiking on. Each of these lines tells you 257 00:19:52,000 --> 00:19:55,000 how you could move to stay at a constant height if you want to 258 00:19:55,000 --> 00:19:58,000 get to the other side of the mountain but without ever going 259 00:19:58,000 --> 00:20:03,000 up or down. You would just walk along that 260 00:20:03,000 --> 00:20:08,000 path, and it will get you there without effort. 261 00:20:08,000 --> 00:20:11,000 So, in fact, if you have been talking about 262 00:20:11,000 --> 00:20:15,000 hiking on mountains, well, that's exactly what a 263 00:20:15,000 --> 00:20:21,000 topographical map is about. So, I need to zoom a bit. 264 00:20:21,000 --> 00:20:27,000 So, a topographic map, this one from the US geological 265 00:20:27,000 --> 00:20:32,000 survey shows you, basically, all the level curves 266 00:20:32,000 --> 00:20:37,000 of an altitude function on a piece of land. 267 00:20:37,000 --> 00:20:40,000 So, you know that if you walk right along these curves, 268 00:20:40,000 --> 00:20:42,000 you will stay along the same height. 269 00:20:42,000 --> 00:20:46,000 And you know that if you walk towards, these don't have 270 00:20:46,000 --> 00:20:48,000 numbers. Let me find a place with 271 00:20:48,000 --> 00:20:53,000 numbers. Here, there is a 500 in the 272 00:20:53,000 --> 00:20:56,000 middle. So, you know that if you walk 273 00:20:56,000 --> 00:20:59,000 on the line that says 500, you stay always at 500 meters 274 00:20:59,000 --> 00:21:02,000 in elevation. If you walk towards the 275 00:21:02,000 --> 00:21:05,000 mountain that I think is below it, then you will go up, 276 00:21:05,000 --> 00:21:07,000 and so on. So, you can see, 277 00:21:07,000 --> 00:21:10,000 for example, here there's a peak, 278 00:21:10,000 --> 00:21:13,000 and here there is a valley with the river in it, 279 00:21:13,000 --> 00:21:17,000 and the altitudes go down, and then back up again on the 280 00:21:17,000 --> 00:21:19,000 other side. OK, so that's an example of a 281 00:21:19,000 --> 00:21:22,000 contour plot of a function. Of course, we don't have a 282 00:21:22,000 --> 00:21:25,000 formula for that function, but we have a contour plot, 283 00:21:25,000 --> 00:21:29,000 and that's what we need actually to understand what's 284 00:21:29,000 --> 00:21:36,000 going on there. OK, any questions? 285 00:21:36,000 --> 00:21:39,000 No? OK, so another example of 286 00:21:39,000 --> 00:21:42,000 contour plots, well, you've probably seen 287 00:21:42,000 --> 00:21:46,000 various versions of these temperature maps. 288 00:21:46,000 --> 00:21:51,000 So, that's supposed to be how warm it is right now. 289 00:21:51,000 --> 00:21:55,000 So, this one is color-coded. Instead of having curves, 290 00:21:55,000 --> 00:21:58,000 it has all these colors. But, the effect is the same. 291 00:21:58,000 --> 00:22:01,000 If you look at the separations between consecutive colors, 292 00:22:01,000 --> 00:22:05,000 these are the level curves of a function that tells you the 293 00:22:05,000 --> 00:22:12,000 temperature at a given point. OK, so these are examples of 294 00:22:12,000 --> 00:22:24,000 contour plots in real life. OK, no questions? 295 00:22:24,000 --> 00:22:26,000 No? OK, so basically, 296 00:22:26,000 --> 00:22:31,000 one of the goals that one should try to achieve at this 297 00:22:31,000 --> 00:22:35,000 point is becoming familiar with the contour plot, 298 00:22:35,000 --> 00:22:38,000 the graph, and how to view and deal with 299 00:22:38,000 --> 00:22:39,000 functions. 300 00:22:39,000 --> 00:22:54,000 301 00:22:54,000 --> 00:23:02,000 [APPLAUSE] OK, so -- Let's do an example. 302 00:23:02,000 --> 00:23:04,000 Well, let's do a couple of examples. 303 00:23:04,000 --> 00:23:08,000 So, let's start with f(x,y) = - y. 304 00:23:08,000 --> 00:23:12,000 And, I'm going to take the same two examples as there to start 305 00:23:12,000 --> 00:23:16,000 with so that we see the relation between the graph and the 306 00:23:16,000 --> 00:23:23,000 contour plots. So, let's try to plot it. 307 00:23:23,000 --> 00:23:30,000 So, we are asked for the level curve, f equals 0 for this one? 308 00:23:30,000 --> 00:23:38,000 Well, f is zero when y is zero. So, that's the x axis. 309 00:23:38,000 --> 00:23:44,000 OK, so that's the level, zero. Where's the level one? 310 00:23:44,000 --> 00:23:48,000 Well, f is one when negative y is one. 311 00:23:48,000 --> 00:23:51,000 That means when y is negative one. 312 00:23:51,000 --> 00:23:57,000 So, you go to minus one, and that will be where my level 313 00:23:57,000 --> 00:24:02,000 one is, and so on. f is two when y is negative 314 00:24:02,000 --> 00:24:06,000 two. F is negative one when y is 315 00:24:06,000 --> 00:24:10,000 one, and so on. Is that convincing? 316 00:24:10,000 --> 00:24:15,000 Do you see how we got that? OK, let me do it again. 317 00:24:15,000 --> 00:24:18,000 I don't see anybody nodding, so that's kind of bad news. 318 00:24:18,000 --> 00:24:22,000 So, if I want to know, where is the level curve, 319 00:24:22,000 --> 00:24:26,000 say, one, I try to set f equals to one. 320 00:24:26,000 --> 00:24:31,000 Let's do this one. f equals one means that 321 00:24:31,000 --> 00:24:36,000 negative y is one means that y is minus one, 322 00:24:36,000 --> 00:24:43,000 and y equals minus one is this horizontal line on this chart. 323 00:24:43,000 --> 00:24:47,000 OK, and same with the others. So, you can see on the map that 324 00:24:47,000 --> 00:24:49,000 the value of a function doesn't depend on x. 325 00:24:49,000 --> 00:24:52,000 If you move parallel to the x axis, nothing happens. 326 00:24:52,000 --> 00:24:56,000 If you move in the y direction, it decreases at a constant 327 00:24:56,000 --> 00:24:59,000 rate. That's why the contours are 328 00:24:59,000 --> 00:25:03,000 evenly spaced. How spaced out they are tells 329 00:25:03,000 --> 00:25:06,000 you, actually, how steep things are. 330 00:25:06,000 --> 00:25:08,000 So, that corresponds exactly to that picture, 331 00:25:08,000 --> 00:25:11,000 except that here we draw x coming to the front, 332 00:25:11,000 --> 00:25:14,000 and y to the right. So, you have to rotate the map 333 00:25:14,000 --> 00:25:19,000 by 90� to get to that. It's an unfortunate consequence 334 00:25:19,000 --> 00:25:24,000 of the usual way of plotting things in space. 335 00:25:24,000 --> 00:25:32,000 OK, so these horizontal lines here correspond actually to 336 00:25:32,000 --> 00:25:35,000 horizontal lines here. 337 00:25:35,000 --> 00:25:43,000 338 00:25:43,000 --> 00:25:54,000 OK, second example. Let's do 1-x^2-y^2. 339 00:25:54,000 --> 00:26:00,000 OK, or maybe I will write it as 1 - (x^2 y^2). 340 00:26:00,000 --> 00:26:06,000 It's really the same thing. So, x, y, let's see, 341 00:26:06,000 --> 00:26:12,000 where is this function equal to zero? 342 00:26:12,000 --> 00:26:21,000 Well, we said f is zero in the unit circle. 343 00:26:21,000 --> 00:26:32,000 OK, so, the zero level, well, let's say that this is my 344 00:26:32,000 --> 00:26:36,000 unit. That's where it's at zero. 345 00:26:36,000 --> 00:26:48,000 What about f equals one? Well, that's when x^2 y^2 = 0. 346 00:26:48,000 --> 00:26:49,000 Well, that's only going to be here. 347 00:26:49,000 --> 00:27:00,000 So, that's just a single point. What about f equals minus one? 348 00:27:00,000 --> 00:27:07,000 That's when x^2 y^2 =2. That's a circle of radius 349 00:27:07,000 --> 00:27:10,000 square root of two, which is about 1.4. 350 00:27:10,000 --> 00:27:17,000 So, it's somewhere here. Then, minus two, 351 00:27:17,000 --> 00:27:24,000 similarly, will be x^2 y^2 = 3. Square root of three is about 352 00:27:24,000 --> 00:27:27,000 1.7. And then, minus three will be 353 00:27:27,000 --> 00:27:30,000 of radius two, and so on. 354 00:27:30,000 --> 00:27:38,000 So, what I want to show here is that they are getting closer and 355 00:27:38,000 --> 00:27:41,000 closer apart, OK? 356 00:27:41,000 --> 00:27:44,000 So, first it's concentric circles that tells us that we 357 00:27:44,000 --> 00:27:47,000 have a shape that's a solid of the graph is going to be a 358 00:27:47,000 --> 00:27:52,000 surface of revolution. Things don't change if I rotate. 359 00:27:52,000 --> 00:27:56,000 And second, the level curves are getting closer and closer to 360 00:27:56,000 --> 00:27:59,000 each other. That means it's getting steeper 361 00:27:59,000 --> 00:28:03,000 and steeper because I have to travel a shorter distance if I 362 00:28:03,000 --> 00:28:06,000 want my altitude to change by one. 363 00:28:06,000 --> 00:28:09,000 OK, so, this top here is kind of flat. 364 00:28:09,000 --> 00:28:11,000 And then it gets steeper and steeper. 365 00:28:11,000 --> 00:28:16,000 And, that's what we observe on that picture there. 366 00:28:16,000 --> 00:28:24,000 OK, so just to show you a few more, where did I put my, 367 00:28:24,000 --> 00:28:30,000 so, for x^2 y^2, the contour plot looks like 368 00:28:30,000 --> 00:28:37,000 this. Maybe actually I'll make it. 369 00:28:37,000 --> 00:28:41,000 OK, so it looks exactly the same as this one. 370 00:28:41,000 --> 00:28:44,000 But, the difference is if you can see the numbers which are 371 00:28:44,000 --> 00:28:45,000 not there, so you can see them, 372 00:28:45,000 --> 00:28:49,000 then you would know that instead of decreasing as we move 373 00:28:49,000 --> 00:28:52,000 out, this one is increasing as we go 374 00:28:52,000 --> 00:28:54,000 out. OK, so that's where we use, 375 00:28:54,000 --> 00:28:57,000 actually, the labels on the level curves that tell us 376 00:28:57,000 --> 00:29:00,000 whether things are going up or down. 377 00:29:00,000 --> 00:29:04,000 But, the contour plots look exactly the same. 378 00:29:04,000 --> 00:29:14,000 For the next one I had, I think, was y^2-x^2. 379 00:29:14,000 --> 00:29:18,000 So, the contour plot, well, let me actually zoom out. 380 00:29:18,000 --> 00:29:20,000 So, the contour plot looks like that. 381 00:29:20,000 --> 00:29:23,000 So, the level curve corresponding to zero is 382 00:29:23,000 --> 00:29:27,000 actually two diagonal lines. And, if you look on the plot, 383 00:29:27,000 --> 00:29:30,000 say that you started at the saddle point in the middle and 384 00:29:30,000 --> 00:29:33,000 you try to stay at the same level. 385 00:29:33,000 --> 00:29:35,000 So, it looks like a mountain pass, right? 386 00:29:35,000 --> 00:29:38,000 Well, there's actually four directions from that point that 387 00:29:38,000 --> 00:29:41,000 you can go staying at the same height. 388 00:29:41,000 --> 00:29:44,000 And actually, on the map, they look exactly 389 00:29:44,000 --> 00:29:46,000 like this, too, these crossing lines. 390 00:29:46,000 --> 00:29:49,000 OK, so, these are things that go on the side of the two 391 00:29:49,000 --> 00:29:53,000 mountains that are to the left and right, and stay at the same 392 00:29:53,000 --> 00:29:57,000 height as the mountain pass. On the other hand, 393 00:29:57,000 --> 00:30:01,000 if you go along the y direction, to the left or to the 394 00:30:01,000 --> 00:30:05,000 right, then you go towards positive values. 395 00:30:05,000 --> 00:30:11,000 And, if you go along the x axis, then you get towards 396 00:30:11,000 --> 00:30:18,000 negative values. OK, the equation for, 397 00:30:18,000 --> 00:30:25,000 the function was y^2-x^2. So, you can try to plot them by 398 00:30:25,000 --> 00:30:27,000 hand and confirmed that it does look like that. 399 00:30:27,000 --> 00:30:33,000 But, I trust my computer. And, finally, 400 00:30:33,000 --> 00:30:39,000 this one, well, so the contour plot looks a bit 401 00:30:39,000 --> 00:30:43,000 complicated. But, you can see two things. 402 00:30:43,000 --> 00:30:45,000 In the middle, you can see these two origins 403 00:30:45,000 --> 00:30:47,000 with these concentric circles which are not really circles, 404 00:30:47,000 --> 00:30:50,000 but, you know, these closed curves that are 405 00:30:50,000 --> 00:30:53,000 concentric. And, they correspond to the two 406 00:30:53,000 --> 00:30:56,000 mountains. And then, at some point in the 407 00:30:56,000 --> 00:31:00,000 middle, we have a mountain pass. And there, we see the two 408 00:31:00,000 --> 00:31:05,000 crossing lines again, like, on the plot of y^2-x^2. 409 00:31:05,000 --> 00:31:11,000 And so, at this saddle point here, if we go north or south, 410 00:31:11,000 --> 00:31:15,000 then we go down on either side to the Valley. 411 00:31:15,000 --> 00:31:17,000 And, if we go east or west, then we go towards the 412 00:31:17,000 --> 00:31:21,000 mountains. We'll go up. 413 00:31:21,000 --> 00:31:26,000 OK, does that make sense a little bit? 414 00:31:26,000 --> 00:31:31,000 OK, so, reading plots is not easy, but hopefully we'll get 415 00:31:31,000 --> 00:31:32,000 used to it very soon. 416 00:31:32,000 --> 00:31:44,000 417 00:31:44,000 --> 00:31:49,000 OK, so actually let's say, well, OK, so, 418 00:31:49,000 --> 00:31:55,000 I want to point out one thing. The contour plot tells us, 419 00:31:55,000 --> 00:32:00,000 actually, what happens when we move, when we change x and y. 420 00:32:00,000 --> 00:32:05,000 So, if I change the value of x and y, that means I'm moving 421 00:32:05,000 --> 00:32:08,000 east-west or north-south on the map. 422 00:32:08,000 --> 00:32:12,000 And then, I can ask myself, is the value of the function 423 00:32:12,000 --> 00:32:15,000 increase or decrease in each of these situations? 424 00:32:15,000 --> 00:32:18,000 Well, that's the kind of thing that the contour plot can tell 425 00:32:18,000 --> 00:32:19,000 me very quickly. 426 00:32:19,000 --> 00:32:54,000 427 00:32:54,000 --> 00:32:56,000 So -- OK, so say, for example, 428 00:32:56,000 --> 00:32:59,000 that I have a piece of contour plot. 429 00:32:59,000 --> 00:33:01,000 That looks, you know, like that. 430 00:33:01,000 --> 00:33:06,000 Maybe this is f equals one, and this is f equals two. 431 00:33:06,000 --> 00:33:13,000 And here, this is f equals zero. And, let's say that I start at 432 00:33:13,000 --> 00:33:17,000 the point, say, at this point. 433 00:33:17,000 --> 00:33:23,000 OK, so here I have (x0, y0). And, the question I might ask 434 00:33:23,000 --> 00:33:26,000 myself is if I change x or y, how does f change? 435 00:33:26,000 --> 00:33:34,000 Well, the contour plot tells me that if x increases, 436 00:33:34,000 --> 00:33:41,000 and I keep y constant, then what happens to f(x, 437 00:33:41,000 --> 00:33:44,000 y)? Well, it will increase because 438 00:33:44,000 --> 00:33:47,000 if I move to the right, then I go from one to a value 439 00:33:47,000 --> 00:33:50,000 bigger than one. I don't know exactly how much, 440 00:33:50,000 --> 00:33:53,000 but I know that somewhere between one and two, 441 00:33:53,000 --> 00:33:57,000 it's more than one. If x decreases, 442 00:33:57,000 --> 00:34:02,000 then f decreases. And, similarly, 443 00:34:02,000 --> 00:34:07,000 I can tell that if y increases, then f, well, 444 00:34:07,000 --> 00:34:14,000 it looks like if I increase y, then f will also increase. 445 00:34:14,000 --> 00:34:20,000 And, if y decreases, then f decreases. 446 00:34:20,000 --> 00:34:23,000 And, that's the kind of qualitative analysis that we can 447 00:34:23,000 --> 00:34:27,000 do easily from the contour plot. But, maybe we'd like to 448 00:34:27,000 --> 00:34:30,000 actually be more precise in that, and tell how fast f 449 00:34:30,000 --> 00:34:34,000 changes if I change x or y. OK, so to find the rate of 450 00:34:34,000 --> 00:34:39,000 change, that's exactly where we use derivatives. 451 00:34:39,000 --> 00:34:47,000 So -- So, we are going to have to deal with partial 452 00:34:47,000 --> 00:34:58,000 derivatives. So, I will explain to you soon 453 00:34:58,000 --> 00:35:05,000 why partial. So, let me just remind you 454 00:35:05,000 --> 00:35:12,000 first, if you have a function of one variable, 455 00:35:12,000 --> 00:35:18,000 then so let's say f of x, then you have a derivative, 456 00:35:18,000 --> 00:35:22,000 f prime of x is also called df/dx. 457 00:35:22,000 --> 00:35:31,000 And, it's defined as a limit when delta x goes to zero of the 458 00:35:31,000 --> 00:35:35,000 change in f. Sorry, it's not going to fit. 459 00:35:35,000 --> 00:35:42,000 I have to go to the next line. It's going to be the limit as 460 00:35:42,000 --> 00:35:47,000 delta x goes to zero of the rate of change. 461 00:35:47,000 --> 00:35:52,000 So, the change in f between x and x plus delta x divided by 462 00:35:52,000 --> 00:35:56,000 delta x. Sometimes you write delta f for 463 00:35:56,000 --> 00:35:59,000 the change in f divided by delta x. 464 00:35:59,000 --> 00:36:04,000 And then, you take the limit of this rate of increase as delta x 465 00:36:04,000 --> 00:36:05,000 goes to zero. Now, of course, 466 00:36:05,000 --> 00:36:08,000 if you have a formula for f, then you know, 467 00:36:08,000 --> 00:36:12,000 at least you should know, I suspect most of you know how 468 00:36:12,000 --> 00:36:19,000 to actually take the derivative of a function from its formula. 469 00:36:19,000 --> 00:36:30,000 So -- Now, how do we do that? Sorry, and I should also tell 470 00:36:30,000 --> 00:36:32,000 you what this means on the graph. 471 00:36:32,000 --> 00:36:36,000 So, if I plot the graph of a function, and to have my point, 472 00:36:36,000 --> 00:36:41,000 x, and here I have f of x, how do I see the derivative? 473 00:36:41,000 --> 00:36:48,000 Well, I look at the tangent line to the graph, 474 00:36:48,000 --> 00:36:55,000 and the slope of the tangent line is f prime of x, 475 00:36:55,000 --> 00:36:59,000 OK? And, not every function has a 476 00:36:59,000 --> 00:37:03,000 derivative. You have functions that are not 477 00:37:03,000 --> 00:37:05,000 regular enough to actually have a derivative. 478 00:37:05,000 --> 00:37:08,000 So, in this class, we are not going to actually 479 00:37:08,000 --> 00:37:11,000 worry too much about differentiability. 480 00:37:11,000 --> 00:37:16,000 But, it's good, at least, to be aware that you 481 00:37:16,000 --> 00:37:19,000 can't always take the derivative. 482 00:37:19,000 --> 00:37:24,000 So, yes, and what else do I want to remind you of? 483 00:37:24,000 --> 00:37:32,000 Well, they also have an approximation formula -- -- 484 00:37:32,000 --> 00:37:39,000 which says that, you know, if we have the value 485 00:37:39,000 --> 00:37:41,000 of f at some point, x0, 486 00:37:41,000 --> 00:37:47,000 and that we want to find the value at a nearby point, 487 00:37:47,000 --> 00:37:51,000 x close to x0, then our best guess is that 488 00:37:51,000 --> 00:37:58,000 it's f of x0 plus the derivative f prime at x0 times delta x, 489 00:37:58,000 --> 00:38:02,000 or if you want, x minus x0, OK? 490 00:38:02,000 --> 00:38:06,000 Is this kind of familiar to you? Yeah, I mean, 491 00:38:06,000 --> 00:38:09,000 maybe with different notations. Maybe you called that delta x 492 00:38:09,000 --> 00:38:12,000 or something. Maybe you called that x0 plus h 493 00:38:12,000 --> 00:38:14,000 or something. But, it's the usual 494 00:38:14,000 --> 00:38:18,000 approximation formula using the derivative. 495 00:38:18,000 --> 00:38:21,000 If you put more terms, then you get the dreaded Taylor 496 00:38:21,000 --> 00:38:24,000 approximation that I know you guys don't like. 497 00:38:24,000 --> 00:38:36,000 So, the question is how do we do the same for a function of 498 00:38:36,000 --> 00:38:41,000 two variables, f(x, y)? 499 00:38:41,000 --> 00:38:45,000 So, the difficulty there is we can change x, 500 00:38:45,000 --> 00:38:49,000 or we can change y, or we can change both. 501 00:38:49,000 --> 00:38:52,000 And, presumably, the manner in which f changes 502 00:38:52,000 --> 00:38:56,000 will be different depending on whether we change x or y. 503 00:38:56,000 --> 00:39:00,000 So, that's why we have several different notions of derivative. 504 00:39:00,000 --> 00:39:24,000 505 00:39:24,000 --> 00:39:37,000 So, OK, we have a notation. OK, so this is a curly d, 506 00:39:37,000 --> 00:39:41,000 and it is not a straight d, and it is not a delta. 507 00:39:41,000 --> 00:39:44,000 It's a d that kind of curves backwards like that. 508 00:39:44,000 --> 00:39:50,000 And, this symbol is partial. OK, so it's a special notation 509 00:39:50,000 --> 00:39:54,000 for partial derivatives. And, essentially what it means 510 00:39:54,000 --> 00:39:56,000 is we are going to do a derivative where we care about 511 00:39:56,000 --> 00:39:59,000 only one variable at a time. That's why it's partial. 512 00:39:59,000 --> 00:40:02,000 It's missing the other variables. 513 00:40:02,000 --> 00:40:06,000 So, a function of several variables doesn't have the usual 514 00:40:06,000 --> 00:40:10,000 derivative. It has only partial derivatives 515 00:40:10,000 --> 00:40:15,000 for each variable. So, the partial derivative, 516 00:40:15,000 --> 00:40:23,000 the partial f partial x at (x0, y0) is defined to be the limit 517 00:40:23,000 --> 00:40:29,000 when I take a small change in x, delta x, 518 00:40:29,000 --> 00:40:43,000 of the change in f -- -- divided by delta x. 519 00:40:43,000 --> 00:40:47,000 OK, so here I'm actually not changing y at all. 520 00:40:47,000 --> 00:40:51,000 I'm just changing x and looking at the rate of change with 521 00:40:51,000 --> 00:40:54,000 respect to x. And, I have the same with 522 00:40:54,000 --> 00:40:58,000 respect to y. Partial f partial y is the 523 00:40:58,000 --> 00:41:04,000 limit, so I should say, at a point x0 y0 is the limit 524 00:41:04,000 --> 00:41:13,000 as delta y turns to zero. So, this time I keep x the 525 00:41:13,000 --> 00:41:21,000 same, but I change y. OK, so that's the definition of 526 00:41:21,000 --> 00:41:26,000 a partial derivative. And, we say that a function is 527 00:41:26,000 --> 00:41:29,000 differentiable if these things exist. 528 00:41:29,000 --> 00:41:31,000 OK, so most of the functions we'll see are differentiable. 529 00:41:31,000 --> 00:41:34,000 And, we'll actually learn how to compute their partial 530 00:41:34,000 --> 00:41:38,000 derivatives without having to do this because we'll just have the 531 00:41:38,000 --> 00:41:41,000 usual methods for computing derivatives. 532 00:41:41,000 --> 00:41:46,000 So, in fact, I claim you already know how to 533 00:41:46,000 --> 00:41:49,000 take partial derivatives. So, let's see what it means 534 00:41:49,000 --> 00:41:50,000 geometrically. 535 00:41:50,000 --> 00:42:00,000 536 00:42:00,000 --> 00:42:07,000 So, geometrically, my function can be represented 537 00:42:07,000 --> 00:42:12,000 by this graph, and I fix some point, 538 00:42:12,000 --> 00:42:18,000 (x0, y0). And then, I'm going to ask 539 00:42:18,000 --> 00:42:24,000 myself what happens if I change the value of, 540 00:42:24,000 --> 00:42:30,000 well, x, keeping y constant. So, if I keep y constant and 541 00:42:30,000 --> 00:42:33,000 change x, it means that I'm moving forwards or backwards 542 00:42:33,000 --> 00:42:38,000 parallel to the x axis. So, that determines for me the 543 00:42:38,000 --> 00:42:46,000 vertical plane parallel to the x, z plane when I fix y equals 544 00:42:46,000 --> 00:42:51,000 constant. And now, if I slice the graph 545 00:42:51,000 --> 00:42:59,000 by that, I will get some curve that goes, it's a slice of the 546 00:42:59,000 --> 00:43:03,000 graph of f. And now, what I want to find is 547 00:43:03,000 --> 00:43:06,000 how f depends on x if I keep y constant. 548 00:43:06,000 --> 00:43:09,000 That means it's the rate of change if I move along this 549 00:43:09,000 --> 00:43:11,000 curve. So, in fact, 550 00:43:11,000 --> 00:43:17,000 if I look at the slope of this thing. 551 00:43:17,000 --> 00:43:22,000 So, if I draw the tangent line to this slice, 552 00:43:22,000 --> 00:43:28,000 then the slope will be partial f of partial x. 553 00:43:28,000 --> 00:43:32,000 I think I have a better picture of that somewhere. 554 00:43:32,000 --> 00:43:40,000 Yes, here it is. OK, that's the same picture, 555 00:43:40,000 --> 00:43:43,000 just with different colors. So, I take the graph. 556 00:43:43,000 --> 00:43:46,000 I slice it by a vertical plane. I get the curve, 557 00:43:46,000 --> 00:43:50,000 and now I take the slope of that curve, and that gives me 558 00:43:50,000 --> 00:43:54,000 the partial derivative. And, to finish, 559 00:43:54,000 --> 00:43:59,000 let me just tell you how, and I should say, 560 00:43:59,000 --> 00:44:02,000 partial f partial y is the same thing but slicing now by your 561 00:44:02,000 --> 00:44:05,000 plane that goes in the y, z directions. 562 00:44:05,000 --> 00:44:11,000 OK, so I fix x equals constant. That means that I slice by a 563 00:44:11,000 --> 00:44:13,000 plane that's parallel to the blackboard. 564 00:44:13,000 --> 00:44:17,000 I get a curve, and I looked at the slope of 565 00:44:17,000 --> 00:44:20,000 that curve. OK, so it's just a matter of 566 00:44:20,000 --> 00:44:23,000 formatting one variable, setting it constant, 567 00:44:23,000 --> 00:44:27,000 and looking at the other one. So, how to compute these 568 00:44:27,000 --> 00:44:29,000 things, well, we actually, 569 00:44:29,000 --> 00:44:33,000 to find, well, there's a piece of notation I 570 00:44:33,000 --> 00:44:38,000 haven't told you yet. So, another notation you will 571 00:44:38,000 --> 00:44:42,000 see, I think this is what one uses a lot in physics. 572 00:44:42,000 --> 00:44:45,000 And, this is what one uses a lot in applied math, 573 00:44:45,000 --> 00:44:47,000 which is the same thing as physics but with different 574 00:44:47,000 --> 00:44:50,000 notations. OK, so it just too different 575 00:44:50,000 --> 00:44:54,000 notations: partial f partial x, or f subscript x. 576 00:44:54,000 --> 00:45:01,000 And, they are the same thing. Well, we just treat y as a 577 00:45:01,000 --> 00:45:10,000 constant, and x as a variable. And, vice versa if we want to 578 00:45:10,000 --> 00:45:16,000 find partial with aspect to y. So, let me just finish with one 579 00:45:16,000 --> 00:45:22,000 quick example. Let's say that they give you f 580 00:45:22,000 --> 00:45:28,000 of x, y equals x^3y y^2, then partial f partial x. 581 00:45:28,000 --> 00:45:32,000 Well, let's take the derivative. So, here it's x^3 times a 582 00:45:32,000 --> 00:45:37,000 constant. Derivative of x^3 is 3x^2 times 583 00:45:37,000 --> 00:45:42,000 the constant plus what's the derivative of y^2? 584 00:45:42,000 --> 00:45:45,000 Zero, because it's a constant. If you do, instead, 585 00:45:45,000 --> 00:45:48,000 partial f partial y, then this is actually a 586 00:45:48,000 --> 00:45:51,000 constant times y. The derivative of y is one. 587 00:45:51,000 --> 00:45:57,000 So, that's just x^3. And, the derivative of y^2 is 588 00:45:57,000 --> 00:45:59,000 2y. OK, so it's fairly easy. 589 00:45:59,000 --> 00:46:02,000 You just have to keep remembering which one is a 590 00:46:02,000 --> 00:46:06,000 variable, and which one isn't. OK, so more about this next 591 00:46:06,000 --> 00:46:10,000 time, and we will also learn about maxima and minima in 592 00:46:10,000 --> 00:46:13,000 several variables. 593 00:46:13,000 --> 00:46:18,000