1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation, or view additional materials 6 00:00:13,300 --> 00:00:17,116 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,116 --> 00:00:17,740 at ocw.mit.edu. 8 00:00:34,750 --> 00:00:35,820 PROFESSOR: Hi. 9 00:00:35,820 --> 00:00:40,370 Today, we would like to begin our study of calculus 10 00:00:40,370 --> 00:00:42,360 of several real variables. 11 00:00:42,360 --> 00:00:45,890 And as I mentioned in our last lecture, 12 00:00:45,890 --> 00:00:50,400 I would like to begin it from the traditional point of view. 13 00:00:50,400 --> 00:00:54,140 And I guess I'd like to make a little bit of a short speech 14 00:00:54,140 --> 00:00:55,700 even before we begin. 15 00:00:55,700 --> 00:00:58,360 There may be a little bit of danger 16 00:00:58,360 --> 00:01:01,320 in the way I've been emphasizing the modern approach 17 00:01:01,320 --> 00:01:04,190 to mathematics: that one may actually believe 18 00:01:04,190 --> 00:01:07,400 that the traditional approach to mathematics 19 00:01:07,400 --> 00:01:10,290 left very, very much to be desired. 20 00:01:10,290 --> 00:01:12,150 The answer is, it didn't really. 21 00:01:12,150 --> 00:01:15,870 That the role of logic in good traditional mathematics 22 00:01:15,870 --> 00:01:19,410 was about the same as it is in good modern mathematics. 23 00:01:19,410 --> 00:01:22,820 That in fact, if we define new mathematics 24 00:01:22,820 --> 00:01:25,240 as meaning meaningful mathematics, 25 00:01:25,240 --> 00:01:28,970 there really was no difference between the two approaches. 26 00:01:28,970 --> 00:01:30,520 Except for the fact that after you 27 00:01:30,520 --> 00:01:33,040 have a couple of hundred years of hindsight 28 00:01:33,040 --> 00:01:36,740 and can take a subject apart from a different point of view, 29 00:01:36,740 --> 00:01:39,080 then obviously it's no great surprise 30 00:01:39,080 --> 00:01:43,280 that you can find cleaner ways of really presenting the topic. 31 00:01:43,280 --> 00:01:47,050 It's the old cliche of hindsight being better than foresight 32 00:01:47,050 --> 00:01:49,280 by a darn sight, or some such thing. 33 00:01:49,280 --> 00:01:51,650 But at any rate, without further ado, 34 00:01:51,650 --> 00:01:53,760 let's actually look at the introduction 35 00:01:53,760 --> 00:01:56,250 to calculus of several real variables 36 00:01:56,250 --> 00:01:58,780 from a traditional point of view. 37 00:01:58,780 --> 00:02:00,900 And I've abbreviated that title to simply 38 00:02:00,900 --> 00:02:03,070 say, "Calculus of Several Variables." 39 00:02:03,070 --> 00:02:05,610 Whereas I say that, think of two things. 40 00:02:05,610 --> 00:02:07,430 First of all, it's an introduction. 41 00:02:07,430 --> 00:02:09,900 And second of all, we're going to be doing this 42 00:02:09,900 --> 00:02:12,330 from the traditional point of view. 43 00:02:12,330 --> 00:02:14,060 Now, what we're assuming is-- remember, 44 00:02:14,060 --> 00:02:16,380 last time we talked about functions 45 00:02:16,380 --> 00:02:17,780 of several real variables. 46 00:02:17,780 --> 00:02:20,050 We talked about the limit concept, 47 00:02:20,050 --> 00:02:24,730 gave you exercises that drilled you on what limits meant, 48 00:02:24,730 --> 00:02:25,960 and the like. 49 00:02:25,960 --> 00:02:28,130 And as a result, it now makes sense 50 00:02:28,130 --> 00:02:32,229 to talk about the calculus, the instantaneous rate of change. 51 00:02:32,229 --> 00:02:34,520 Except that you have an awful lot of variables in here. 52 00:02:34,520 --> 00:02:37,590 In other words, the general case that we'd like to tackle 53 00:02:37,590 --> 00:02:41,920 is the case where we have w being some real number, 54 00:02:41,920 --> 00:02:46,050 being a function of the n independent variables 55 00:02:46,050 --> 00:02:48,300 x_1 up to x_n. 56 00:02:48,300 --> 00:02:51,230 And of course, you see, the calculus applies to the f. 57 00:02:51,230 --> 00:02:54,920 The w used here is to indicate the fact that the output is 58 00:02:54,920 --> 00:02:56,640 a real number. 59 00:02:56,640 --> 00:03:00,960 It's like talking about f of x versus y equals f of x. 60 00:03:00,960 --> 00:03:04,400 In a manner of speaking, it's the f of x that's important. 61 00:03:04,400 --> 00:03:06,180 We won't get into that right now. 62 00:03:06,180 --> 00:03:08,890 What was important, though, about y equals f of x? 63 00:03:08,890 --> 00:03:12,410 It indicated a graph-- graph. 64 00:03:12,410 --> 00:03:15,610 And because it was a graph, we could visualize things 65 00:03:15,610 --> 00:03:17,590 pictorially that might have been harder 66 00:03:17,590 --> 00:03:19,610 to understand analytically. 67 00:03:19,610 --> 00:03:23,810 For this reason, even though n may be any positive integer 68 00:03:23,810 --> 00:03:26,570 whatsoever-- preferably, of course, greater than 1, 69 00:03:26,570 --> 00:03:29,117 otherwise we wouldn't have a function of several variables. 70 00:03:29,117 --> 00:03:30,950 In fact, that's the definition of a function 71 00:03:30,950 --> 00:03:32,970 of several variables-- it's an expression 72 00:03:32,970 --> 00:03:35,290 of this type where n is greater than 1. 73 00:03:35,290 --> 00:03:37,580 But in general, and you'll notice this in the text 74 00:03:37,580 --> 00:03:41,470 as well, we begin with the special case that n equals 2. 75 00:03:41,470 --> 00:03:44,790 Because, you see, if n is 2, we need what? 76 00:03:44,790 --> 00:03:50,250 Two degrees of freedom to plot the independent variables. 77 00:03:50,250 --> 00:03:52,730 That would be the xy-plane, for example. 78 00:03:52,730 --> 00:03:56,360 A third degree of freedom, say, the z-axis, 79 00:03:56,360 --> 00:03:59,490 often called w-axis because of the symbolism. 80 00:03:59,490 --> 00:04:01,210 In other words, in the same way that I 81 00:04:01,210 --> 00:04:03,230 could graph a real-valued function 82 00:04:03,230 --> 00:04:06,700 of a single real variable as a curve in the plane, 83 00:04:06,700 --> 00:04:10,620 I can graph a real-valued function of two real variables 84 00:04:10,620 --> 00:04:13,370 as a surface in three-dimensional space. 85 00:04:13,370 --> 00:04:17,329 And as a result, when one is adjusting to new language, 86 00:04:17,329 --> 00:04:20,760 it's nice to take the case n equals 2 simply because you 87 00:04:20,760 --> 00:04:22,880 can view things pictorially. 88 00:04:22,880 --> 00:04:24,960 The danger, of course, with n equals 2, 89 00:04:24,960 --> 00:04:27,970 is you get so used to the picture that after a while 90 00:04:27,970 --> 00:04:30,980 you forget that n could have been more than 2. 91 00:04:30,980 --> 00:04:35,950 So I will try to pay homage to both points of view. 92 00:04:35,950 --> 00:04:39,310 I will simply mention here that we will often let n equal 2 93 00:04:39,310 --> 00:04:41,400 to take advantage of geometry. 94 00:04:41,400 --> 00:04:44,070 When we do this, the usual notation 95 00:04:44,070 --> 00:04:49,370 is to let w equal f of x, y where x and y are independent. 96 00:04:49,370 --> 00:04:52,280 And I should mention why it's so important to talk 97 00:04:52,280 --> 00:04:54,590 about independent variables. 98 00:04:54,590 --> 00:04:56,130 It'll become clear in a moment. 99 00:04:56,130 --> 00:04:58,540 Let me just tell you what independent means, and then 100 00:04:58,540 --> 00:05:01,340 I'll show you why it becomes clear in a moment. 101 00:05:01,340 --> 00:05:03,370 To say that x and y are independent 102 00:05:03,370 --> 00:05:06,880 simply means that you can choose one of the two variables 103 00:05:06,880 --> 00:05:08,760 without determining the other. 104 00:05:08,760 --> 00:05:12,150 For example, if I were to say let y equal 2x, 105 00:05:12,150 --> 00:05:14,620 then certainly y and x are not independent, 106 00:05:14,620 --> 00:05:18,790 because the choice of x determines the choice of y. 107 00:05:18,790 --> 00:05:20,930 But to say that x and y are independent-- 108 00:05:20,930 --> 00:05:23,110 and that's what we mean by two degrees of freedom-- 109 00:05:23,110 --> 00:05:25,670 is I can take any value of x that I want 110 00:05:25,670 --> 00:05:29,170 and not be impeded as to the choice of y that I want. 111 00:05:29,170 --> 00:05:32,615 And that's why we say the domain is the xy-plane. 112 00:05:32,615 --> 00:05:33,865 I have two degrees of freedom. 113 00:05:33,865 --> 00:05:36,390 I can roam anywhere through the plane this way. 114 00:05:36,390 --> 00:05:38,720 But the reason that we want this is as follows. 115 00:05:38,720 --> 00:05:41,130 And this is where the traditional mathematics 116 00:05:41,130 --> 00:05:45,220 was every bit as logical as the modern mathematics. 117 00:05:45,220 --> 00:05:48,350 Let's keep in mind that logic is the art 118 00:05:48,350 --> 00:05:53,630 of taking known situations that we know how to handle, you see, 119 00:05:53,630 --> 00:05:57,390 and reducing unfamiliar situations to these more 120 00:05:57,390 --> 00:05:58,920 familiar situations. 121 00:05:58,920 --> 00:06:01,590 The point is that even in traditional mathematics, 122 00:06:01,590 --> 00:06:04,330 one knew how to handle the calculus 123 00:06:04,330 --> 00:06:06,210 of a single real variable. 124 00:06:06,210 --> 00:06:09,060 So what one said was this, given that w 125 00:06:09,060 --> 00:06:13,000 is a function of both x and y, let's hold y fixed, 126 00:06:13,000 --> 00:06:16,050 for example, and let x vary. 127 00:06:16,050 --> 00:06:18,940 You see, if I fix y at some particular value 128 00:06:18,940 --> 00:06:23,230 and let x vary, once I've made that fixed choice of y, 129 00:06:23,230 --> 00:06:26,090 I have w as a function of x alone. 130 00:06:26,090 --> 00:06:28,880 And as long as w is a function of x alone, 131 00:06:28,880 --> 00:06:31,020 I already know how to take the derivative of w 132 00:06:31,020 --> 00:06:32,300 with respect to x. 133 00:06:32,300 --> 00:06:33,890 And by the way, there's no favoritism 134 00:06:33,890 --> 00:06:35,360 here between x and y. 135 00:06:35,360 --> 00:06:39,330 In a similar way, if I hold x constant and let y vary, 136 00:06:39,330 --> 00:06:41,940 then w is a function of y alone. 137 00:06:41,940 --> 00:06:44,780 And I can then find the derivative of w 138 00:06:44,780 --> 00:06:46,240 with respect to y. 139 00:06:46,240 --> 00:06:49,360 And these things were called partial derivatives, you see, 140 00:06:49,360 --> 00:06:51,480 because it involved holding what? 141 00:06:51,480 --> 00:06:53,750 All but one of the variables constant. 142 00:06:53,750 --> 00:06:57,190 We'll go into that in more detail as we go along. 143 00:06:57,190 --> 00:06:58,500 But the whole idea is what? 144 00:06:58,500 --> 00:07:01,090 By choosing the variables to be independent, 145 00:07:01,090 --> 00:07:04,820 we can hold all of them but one at a time constant. 146 00:07:04,820 --> 00:07:07,990 And as long as we're doing that, the resulting function 147 00:07:07,990 --> 00:07:10,350 is a function of a single real variable. 148 00:07:10,350 --> 00:07:13,980 And this reduces us to part one of our course. 149 00:07:13,980 --> 00:07:15,730 So at any rate, what we're saying is, 150 00:07:15,730 --> 00:07:19,790 fix y at some constant value, say, y equals y sub 0. 151 00:07:19,790 --> 00:07:23,360 And we then let x vary as in ordinary one-dimensional 152 00:07:23,360 --> 00:07:29,110 calculus between x_0 minus the magnitude of delta x and x_0 153 00:07:29,110 --> 00:07:30,760 plus the magnitude of delta x. 154 00:07:30,760 --> 00:07:34,670 In other words, we mark off an interval of magnitude delta x, 155 00:07:34,670 --> 00:07:38,540 on either side of x_0 and let x be any place in here. 156 00:07:38,540 --> 00:07:42,760 Then what we do, is remembering that y0 is a constant, 157 00:07:42,760 --> 00:07:45,760 and that this function, no matter how complicated it looks 158 00:07:45,760 --> 00:07:48,720 like, is a function of just delta x alone, we 159 00:07:48,720 --> 00:07:51,880 mimic the definition of an ordinary derivative. 160 00:07:51,880 --> 00:07:53,740 In other words, we take the limit as delta 161 00:07:53,740 --> 00:07:58,900 x approaches 0 of f of x_0 plus delta x comma y_0, 162 00:07:58,900 --> 00:08:04,340 minus f of x_0 comma y_0, and divide that by delta x. 163 00:08:04,340 --> 00:08:07,490 In other words, with y_0 being treated as a constant, 164 00:08:07,490 --> 00:08:09,960 notice that essentially, this is the change 165 00:08:09,960 --> 00:08:14,250 in f as x varies between x_0 and x_0 plus delta 166 00:08:14,250 --> 00:08:16,630 x, divided by the change in x. 167 00:08:16,630 --> 00:08:19,610 In other words, it's the limit of an average rate 168 00:08:19,610 --> 00:08:24,590 of change of f of x with respect to x for a fixed value of y_0. 169 00:08:24,590 --> 00:08:34,070 And to indicate that, we write this analogously, 170 00:08:34,070 --> 00:08:36,320 I would say, to how we write functions 171 00:08:36,320 --> 00:08:37,870 of a single real variable. 172 00:08:37,870 --> 00:08:40,299 In other words, we write f prime. 173 00:08:40,299 --> 00:08:42,914 Instead of f prime, we indicate-- 174 00:08:42,914 --> 00:08:45,780 let me put it this way, when you had only one real variable, 175 00:08:45,780 --> 00:08:48,575 was there any danger of misinterpreting the prime? 176 00:08:48,575 --> 00:08:50,700 After all, with only one independent variable, when 177 00:08:50,700 --> 00:08:53,180 you said derivative, it was obvious what 178 00:08:53,180 --> 00:08:54,740 variable you were differentiating 179 00:08:54,740 --> 00:08:55,890 with respect to. 180 00:08:55,890 --> 00:08:58,190 Here, there are two variables, x and y. 181 00:08:58,190 --> 00:09:00,360 So instead of the prime as a subscript, 182 00:09:00,360 --> 00:09:02,430 we write down the variable with respect to which 183 00:09:02,430 --> 00:09:03,699 we're taking the derivative. 184 00:09:03,699 --> 00:09:05,990 And then we put in the point at which the derivative is 185 00:09:05,990 --> 00:09:07,520 being evaluated. 186 00:09:07,520 --> 00:09:09,560 You see, in a similar way, to have 187 00:09:09,560 --> 00:09:14,360 defined f sub y of x_0, y_0, that would have meant what? 188 00:09:14,360 --> 00:09:18,440 We hold x constant at some value x_0, 189 00:09:18,440 --> 00:09:23,110 let y vary between y_0 and y_0 plus delta y, 190 00:09:23,110 --> 00:09:26,870 and take the limit of this expression, f of x_0 comma y_0 191 00:09:26,870 --> 00:09:31,770 plus delta y minus f of x_0, y_0, over delta y. 192 00:09:31,770 --> 00:09:33,300 Take the limit of that expression 193 00:09:33,300 --> 00:09:35,460 as delta y approaches 0, you see. 194 00:09:35,460 --> 00:09:37,840 And we're going to show this pictorially in a while. 195 00:09:37,840 --> 00:09:40,960 After all, that's why we chose the case n equals 2, 196 00:09:40,960 --> 00:09:43,240 was so that we could eventually illustrate this thing 197 00:09:43,240 --> 00:09:44,560 pictorially. 198 00:09:44,560 --> 00:09:47,870 But for the time being, I prefer not to introduce the picture 199 00:09:47,870 --> 00:09:49,460 because I want you to see that even 200 00:09:49,460 --> 00:09:53,830 in the case where my input consists of an n-tuple where 201 00:09:53,830 --> 00:09:57,000 n is greater than 2, or 3, or 4-- 202 00:09:57,000 --> 00:09:59,560 whatever you want-- that this thing still makes sense. 203 00:09:59,560 --> 00:10:02,560 In other words, in general, suppose w 204 00:10:02,560 --> 00:10:05,430 is a function of the n independent variables 205 00:10:05,430 --> 00:10:07,670 x sub 1 up to x sub n. 206 00:10:07,670 --> 00:10:12,060 And I want to compute f of x sub 1 at the point x_1 207 00:10:12,060 --> 00:10:15,600 equals a_1, et cetera, x_n equals a_n. 208 00:10:15,600 --> 00:10:22,650 By definition, what I do is I hold a_2 up to a_n constant. 209 00:10:22,650 --> 00:10:27,920 I let x_1 vary from a_1 to a_1 plus delta x. 210 00:10:27,920 --> 00:10:29,120 I compute what? 211 00:10:29,120 --> 00:10:33,700 f of a_1 plus delta x comma a_2, et cetera, comma a_n, 212 00:10:33,700 --> 00:10:36,370 minus f of a_1 up to a_n. 213 00:10:36,370 --> 00:10:43,700 Noticing that that numerator is simply the change in f as x_1 214 00:10:43,700 --> 00:10:46,690 varies from a_1 to a_1 plus delta x_1 215 00:10:46,690 --> 00:10:49,230 while all the other variables are being held constant. 216 00:10:49,230 --> 00:10:52,580 Which of course I can do if the variables are independent. 217 00:10:52,580 --> 00:10:54,740 I then divide that by delta x_1. 218 00:10:54,740 --> 00:10:56,490 That's the average rate of change. 219 00:10:56,490 --> 00:10:59,660 Then I take the limit as delta x_1 approaches 0. 220 00:10:59,660 --> 00:11:02,230 And this I can do for any number of variables, 221 00:11:02,230 --> 00:11:04,100 provided that they're independent. 222 00:11:04,100 --> 00:11:08,190 Keep that in mind, if x sub n could be expressed in terms 223 00:11:08,190 --> 00:11:13,900 of the remaining x's, how can I hold a sub n constant-- 224 00:11:13,900 --> 00:11:18,880 or, how can I vary a sub n while I hold the other ones constant? 225 00:11:18,880 --> 00:11:21,920 In other words, if the a_n's-- if one of variables depends 226 00:11:21,920 --> 00:11:25,160 on the others, they are related in terms of their motion. 227 00:11:25,160 --> 00:11:28,010 Keep in mind here, I just chose delta x_1. 228 00:11:28,010 --> 00:11:32,090 Analogous definition is held for f sub x sub 2, 229 00:11:32,090 --> 00:11:34,870 f sub x sub 3, et cetera. 230 00:11:34,870 --> 00:11:36,290 OK? 231 00:11:36,290 --> 00:11:39,800 I think again that once you see a few examples, 232 00:11:39,800 --> 00:11:41,220 the mystery vanishes. 233 00:11:41,220 --> 00:11:43,560 What this thing says in plain English 234 00:11:43,560 --> 00:11:46,500 is that when you have a function of several variables 235 00:11:46,500 --> 00:11:48,500 and you want to take the derivative with respect 236 00:11:48,500 --> 00:11:51,105 to one of those variables, pretend 237 00:11:51,105 --> 00:11:52,830 that every one of the other variables 238 00:11:52,830 --> 00:11:55,030 was being held constant. 239 00:11:55,030 --> 00:11:56,160 Let me give an example. 240 00:11:56,160 --> 00:11:59,280 Let w be f of x_1, x_2, x_3, where 241 00:11:59,280 --> 00:12:02,690 the specific f that I have in mind is obtained as follows. 242 00:12:02,690 --> 00:12:08,020 It's x_1 times x_2 times x_3 plus e to the x_1 power. 243 00:12:08,020 --> 00:12:11,310 Now, to take the derivative of this with respect to x_1, 244 00:12:11,310 --> 00:12:14,150 all that this says is treat x_2 and x_3 245 00:12:14,150 --> 00:12:15,600 as if they were constants. 246 00:12:15,600 --> 00:12:16,740 Well, look at it. 247 00:12:16,740 --> 00:12:20,370 If I treat x_2 and x_3 as if they were constants, 248 00:12:20,370 --> 00:12:23,980 if I differentiate this with respect to x_1, all I have left 249 00:12:23,980 --> 00:12:25,000 is what? 250 00:12:25,000 --> 00:12:28,350 This particular constant, which is x_2 times x_3. 251 00:12:28,350 --> 00:12:31,250 And the derivative of e to the x_1, with respect to x_1, 252 00:12:31,250 --> 00:12:33,250 is e to the x_1. 253 00:12:33,250 --> 00:12:38,060 Therefore, f sub x_1 in this case is x_2*x_3 plus e 254 00:12:38,060 --> 00:12:39,220 to the x_1. 255 00:12:39,220 --> 00:12:44,310 In particular, if I evaluate f sub x sub 1 at the three-tuple 256 00:12:44,310 --> 00:12:49,740 (1, 2, 3)-- namely, I replace x_1 by 1, x_2 by 2, x_3 by 3-- 257 00:12:49,740 --> 00:12:50,950 I obtain what? 258 00:12:50,950 --> 00:12:54,400 2 times 3 plus e to the first power. 259 00:12:54,400 --> 00:12:58,010 In other words, this is just a number, 6 plus e. 260 00:12:58,010 --> 00:13:00,160 On the other hand, if I had decided 261 00:13:00,160 --> 00:13:02,860 to differentiate this with respect to x_2, 262 00:13:02,860 --> 00:13:05,780 I would've treated x_1 and x_3 as constants. 263 00:13:05,780 --> 00:13:09,060 With x_1 and x_3 as constants, the derivative of this with 264 00:13:09,060 --> 00:13:13,370 respect to x_2 is just x_1*x_3. 265 00:13:13,370 --> 00:13:16,110 Treating x_1 as a constant, the derivative of e 266 00:13:16,110 --> 00:13:20,160 to the x_1 with respect to the variable x_3 is 0. 267 00:13:20,160 --> 00:13:22,220 Because the derivative of any constant is 0. 268 00:13:22,220 --> 00:13:25,450 Notice that this is not really a constant. 269 00:13:25,450 --> 00:13:28,150 It's a constant once I've fixed x_1 270 00:13:28,150 --> 00:13:31,120 at a particular value, which is what this particular definition 271 00:13:31,120 --> 00:13:33,180 says. 272 00:13:33,180 --> 00:13:39,360 In particular, f sub x sub 2 of (1, 2, 3) is 1 times 3, or 3. 273 00:13:39,360 --> 00:13:41,830 I should also mention that sometimes instead 274 00:13:41,830 --> 00:13:45,060 of writing f of x sub 1, we write something 275 00:13:45,060 --> 00:13:46,850 that looks like the regular derivative, 276 00:13:46,850 --> 00:13:49,510 only we make a funny kind of script d instead 277 00:13:49,510 --> 00:13:50,870 of an ordinary d. 278 00:13:50,870 --> 00:13:52,700 This is read "the partial derivative 279 00:13:52,700 --> 00:13:55,330 of w with respect to x_1." 280 00:13:55,330 --> 00:13:58,140 And the relationship between these two notations 281 00:13:58,140 --> 00:14:00,990 is rather similar to the relationship that 282 00:14:00,990 --> 00:14:05,900 exists between the notations dy/dx and f prime of x. 283 00:14:05,900 --> 00:14:10,680 At any rate, if we have the n independent variables x_1 up 284 00:14:10,680 --> 00:14:14,040 to x_n, and f is a function of those variables, 285 00:14:14,040 --> 00:14:17,770 f sub x sub 1, et cetera, f sub x sub n 286 00:14:17,770 --> 00:14:23,010 are called the partial derivatives of f with respect 287 00:14:23,010 --> 00:14:25,950 to x_1 up to x_n, respectively. 288 00:14:25,950 --> 00:14:27,860 In other words, this means what? 289 00:14:27,860 --> 00:14:30,630 The partial derivative of f with respect to x_1. 290 00:14:30,630 --> 00:14:31,560 That means what? 291 00:14:31,560 --> 00:14:35,660 Take the ordinary derivative as if x_1 were the only variable. 292 00:14:35,660 --> 00:14:38,580 This is the partial of f with respect to x sub n. 293 00:14:38,580 --> 00:14:41,090 That means take the ordinary derivative of f 294 00:14:41,090 --> 00:14:43,460 as if x sub n were the only variable. 295 00:14:43,460 --> 00:14:45,960 And you can always do this, provided that your variables 296 00:14:45,960 --> 00:14:48,660 are independent. 297 00:14:48,660 --> 00:14:52,490 Because our definitions so closely 298 00:14:52,490 --> 00:14:55,169 parallel the structure for what happened 299 00:14:55,169 --> 00:14:57,710 in the case of one independent variable-- including the limit 300 00:14:57,710 --> 00:15:01,090 theorems, the distance formulas, and what have you-- 301 00:15:01,090 --> 00:15:03,800 it turns out that the usual derivative properties still 302 00:15:03,800 --> 00:15:04,550 hold. 303 00:15:04,550 --> 00:15:08,100 For example, if w happens to be the function of the two 304 00:15:08,100 --> 00:15:13,020 independent variables e to the 3x plus y times sine 2x 305 00:15:13,020 --> 00:15:16,620 minus y, and I want to take the derivative of w with respect 306 00:15:16,620 --> 00:15:19,400 to x-- meaning differentiate this treating y 307 00:15:19,400 --> 00:15:22,930 as a constant-- notice the treating y as a constant 308 00:15:22,930 --> 00:15:27,490 gives me two functions of what? 309 00:15:27,490 --> 00:15:27,990 x. 310 00:15:27,990 --> 00:15:29,800 In other words, my function is a product 311 00:15:29,800 --> 00:15:32,650 of two functions, each of which depends on x. 312 00:15:32,650 --> 00:15:35,710 So I use the ordinary product rule to differentiate this. 313 00:15:35,710 --> 00:15:38,740 Namely, how do I use the product rule to take the partial of w 314 00:15:38,740 --> 00:15:39,840 with respect to x? 315 00:15:39,840 --> 00:15:42,820 I treat y as a constant and I differentiate 316 00:15:42,820 --> 00:15:44,440 as if x is the only variable. 317 00:15:44,440 --> 00:15:45,320 I say what? 318 00:15:45,320 --> 00:15:49,070 It's the first factor, e to the 3x plus y times 319 00:15:49,070 --> 00:15:52,280 the derivative of the second with respect to x, 320 00:15:52,280 --> 00:15:54,020 treating y as a constant. 321 00:15:54,020 --> 00:15:55,960 If I treat y as a constant, the derivative 322 00:15:55,960 --> 00:16:01,080 of sine of 2x minus y is cosine 2x minus y times 323 00:16:01,080 --> 00:16:03,820 the derivative of what's inside with respect to x. 324 00:16:03,820 --> 00:16:06,370 And that's just 2. 325 00:16:06,370 --> 00:16:08,340 Plus what? 326 00:16:08,340 --> 00:16:12,290 The second term times the derivative of the first. 327 00:16:12,290 --> 00:16:14,900 And the derivative of e to the 3x plus y 328 00:16:14,900 --> 00:16:18,070 with respect to x, treating y as a constant, 329 00:16:18,070 --> 00:16:22,610 is e to the 3x plus y times the derivative of 3x plus y, 330 00:16:22,610 --> 00:16:25,130 with respect to x, which is just 3. 331 00:16:25,130 --> 00:16:27,990 And so I have the partial of w with respect to x. 332 00:16:27,990 --> 00:16:30,410 In a similar way, I could have found the partial of w 333 00:16:30,410 --> 00:16:33,590 with respect to y, et cetera. 334 00:16:33,590 --> 00:16:37,870 I should now introduce one little problem 335 00:16:37,870 --> 00:16:41,671 that will bother you, maybe even after I help you with this. 336 00:16:41,671 --> 00:16:43,170 In fact, I guess what really crushes 337 00:16:43,170 --> 00:16:46,251 me is the student who comes up in a live class, after class, 338 00:16:46,251 --> 00:16:48,750 and says, "I understood it until I heard you lecture on it." 339 00:16:48,750 --> 00:16:50,980 I hope this doesn't cause that problem. 340 00:16:50,980 --> 00:16:53,590 But the danger is, I tell you, lookit, everything 341 00:16:53,590 --> 00:16:55,990 that happened for calculus of a single variable 342 00:16:55,990 --> 00:16:58,750 happens for calculus of several variables. 343 00:16:58,750 --> 00:17:00,880 Then all of a sudden, you'll find in our textbook, 344 00:17:00,880 --> 00:17:02,970 in every textbook, just about, they'll 345 00:17:02,970 --> 00:17:06,849 say things like, if you take the partial of the variable u 346 00:17:06,849 --> 00:17:09,230 with respect to x, you do not get 347 00:17:09,230 --> 00:17:12,470 the reciprocal of the partial of x with respect to u. 348 00:17:12,470 --> 00:17:17,869 In other words, remember, for one variable, dy/dx, right, 349 00:17:17,869 --> 00:17:21,220 was the reciprocal of dx/dy. 350 00:17:21,220 --> 00:17:23,369 They say it's not true in several variables. 351 00:17:23,369 --> 00:17:25,980 I have written in an accentuated question mark 352 00:17:25,980 --> 00:17:28,050 here because I want to explain something 353 00:17:28,050 --> 00:17:29,400 very important to you. 354 00:17:29,400 --> 00:17:31,370 And by the way, if you understand this, 355 00:17:31,370 --> 00:17:34,380 you're 90% of the way home free as far as 356 00:17:34,380 --> 00:17:37,850 understanding how calculus of several variables 357 00:17:37,850 --> 00:17:41,410 is used in most important applications. 358 00:17:41,410 --> 00:17:43,180 Let me take a very simple case. 359 00:17:43,180 --> 00:17:46,790 Let u equal x plus y, and let v equal x minus y, 360 00:17:46,790 --> 00:17:50,280 Now obviously, if u equals x plus y, the partial of u 361 00:17:50,280 --> 00:17:52,300 with respect to x, if we hold y constant-- 362 00:17:52,300 --> 00:17:54,940 we treat x as the only variable-- is 1. 363 00:17:54,940 --> 00:17:55,440 Right? 364 00:17:55,440 --> 00:17:58,160 The partial of u with respect to x is 1. 365 00:17:58,160 --> 00:18:02,030 On the other hand, notice that if I add these two equations, 366 00:18:02,030 --> 00:18:03,930 the y term drops out. 367 00:18:03,930 --> 00:18:05,220 I get what? 368 00:18:05,220 --> 00:18:10,110 2x equals u plus v, divide through by 2. 369 00:18:10,110 --> 00:18:15,320 That says x is equal to 1/2*u plus 1/2*v. 370 00:18:15,320 --> 00:18:17,820 Let me now take the partial of x with respect 371 00:18:17,820 --> 00:18:21,110 to u, holding v constant, treating u 372 00:18:21,110 --> 00:18:22,500 as the only variable. 373 00:18:22,500 --> 00:18:25,710 It's easy for me to see that the partial of x with respect to u 374 00:18:25,710 --> 00:18:26,880 is 1/2. 375 00:18:26,880 --> 00:18:29,840 And certainly, if I now compare these two, 376 00:18:29,840 --> 00:18:34,490 it should be clear that this is not the reciprocal of this, 377 00:18:34,490 --> 00:18:36,322 or vice versa. 378 00:18:36,322 --> 00:18:37,780 The point that I wanted to mention, 379 00:18:37,780 --> 00:18:39,120 though, was the following. 380 00:18:39,120 --> 00:18:41,900 That when, here, we said, take the partial of u 381 00:18:41,900 --> 00:18:44,830 with respect to x, we were assuming 382 00:18:44,830 --> 00:18:50,320 that the independent variables were x and y. 383 00:18:50,320 --> 00:18:53,700 And when we said, here, take the partial of x with respect 384 00:18:53,700 --> 00:18:55,760 to u, what did we hold constant? 385 00:18:55,760 --> 00:18:58,210 We held v constant. 386 00:18:58,210 --> 00:19:01,300 In other words-- I'm not going to keep this habit up very 387 00:19:01,300 --> 00:19:01,910 long. 388 00:19:01,910 --> 00:19:03,970 But just for the time being, I would 389 00:19:03,970 --> 00:19:07,790 like you to get used to the idea of putting in as a subscript 390 00:19:07,790 --> 00:19:10,410 the variables that are being held constant. 391 00:19:10,410 --> 00:19:12,950 You see, in the textbook when they 392 00:19:12,950 --> 00:19:14,990 say this is the case, what they really 393 00:19:14,990 --> 00:19:18,040 mean is, if you differentiate u with respect 394 00:19:18,040 --> 00:19:22,200 to x, holding y constant, that in general will not 395 00:19:22,200 --> 00:19:25,520 be the reciprocal of the derivative of x with respect 396 00:19:25,520 --> 00:19:27,870 to u, holding v constant. 397 00:19:27,870 --> 00:19:29,769 So I can cross out the question mark now. 398 00:19:29,769 --> 00:19:32,060 But the point is, notice that the variables that you're 399 00:19:32,060 --> 00:19:34,560 holding constant here are different. 400 00:19:34,560 --> 00:19:38,710 You see, suppose instead I took the partial of x with respect 401 00:19:38,710 --> 00:19:40,570 to u, holding y constant. 402 00:19:40,570 --> 00:19:44,920 In other words, suppose I solve for x in terms of u and y. 403 00:19:44,920 --> 00:19:47,580 That happens to be very easy to do here. 404 00:19:47,580 --> 00:19:50,970 Given that u equals x plus y, it follows immediately 405 00:19:50,970 --> 00:19:53,550 that x is equal to u minus y. 406 00:19:53,550 --> 00:19:56,870 Now, if I take the partial of x with respect to u, 407 00:19:56,870 --> 00:20:00,420 treating y as a constant-- so let 408 00:20:00,420 --> 00:20:03,760 me do that-- what do I get for an answer? 409 00:20:03,760 --> 00:20:06,330 I get 1 for an answer. 410 00:20:06,330 --> 00:20:08,900 In other words, the reciprocal of the partial 411 00:20:08,900 --> 00:20:13,360 of x with respect to u, treating y as a constant-- 412 00:20:13,360 --> 00:20:16,180 as the other independent variable-- 413 00:20:16,180 --> 00:20:18,120 is equal to the partial of u with respect 414 00:20:18,120 --> 00:20:21,820 to x when it's also y that's being considered 415 00:20:21,820 --> 00:20:23,320 as the other variable. 416 00:20:23,320 --> 00:20:27,110 When the variables outside here match up, the recipes work. 417 00:20:27,110 --> 00:20:28,710 Of course that leads to the question, 418 00:20:28,710 --> 00:20:31,740 then why do the textbooks tell you that this isn't true? 419 00:20:31,740 --> 00:20:34,910 Why do they pick this particular representation rather than 420 00:20:34,910 --> 00:20:35,990 this one? 421 00:20:35,990 --> 00:20:40,000 And I have a combination example and explanation here. 422 00:20:40,000 --> 00:20:41,160 And it's simply this. 423 00:20:41,160 --> 00:20:45,920 Let's suppose I was given that w equals e to the x plus y times 424 00:20:45,920 --> 00:20:48,260 cosine x minus y. 425 00:20:48,260 --> 00:20:50,860 It seems that since the only way the variable appears 426 00:20:50,860 --> 00:20:53,900 in the first factor is as x plus y, 427 00:20:53,900 --> 00:20:56,010 and the only way it appears in the second factor 428 00:20:56,010 --> 00:20:59,630 is in the form x minus y, a very natural substitution 429 00:20:59,630 --> 00:21:05,200 might be to say that u equal x plus y and v equal x minus y. 430 00:21:05,200 --> 00:21:08,410 In other words, I could either visualize w 431 00:21:08,410 --> 00:21:11,350 as being a function of x and y-- in which case, 432 00:21:11,350 --> 00:21:15,060 it would be e to the x plus y times cosine x minus y-- 433 00:21:15,060 --> 00:21:19,825 or, I could visualize w as being a simpler function of the two 434 00:21:19,825 --> 00:21:24,355 new variables u and v, where w is then just what? e 435 00:21:24,355 --> 00:21:26,370 to the u times cosine v. 436 00:21:26,370 --> 00:21:29,000 Notice again, that this formula here 437 00:21:29,000 --> 00:21:31,300 is much simpler than this one. 438 00:21:31,300 --> 00:21:32,510 This one says what? 439 00:21:32,510 --> 00:21:34,750 Take e to of the sum of these two, 440 00:21:34,750 --> 00:21:38,010 multiplied by the cosine of the difference of these two. 441 00:21:38,010 --> 00:21:39,540 And this one just says what? 442 00:21:39,540 --> 00:21:43,990 Take e to the first times the cosine of the second. 443 00:21:43,990 --> 00:21:47,440 But the point is, that you might make this change of variables 444 00:21:47,440 --> 00:21:49,660 to simplify your computations. 445 00:21:49,660 --> 00:21:51,970 The point is that either we would treat w 446 00:21:51,970 --> 00:21:56,620 as a function of x and y, or we would have treated w 447 00:21:56,620 --> 00:21:58,280 as a function of u and v. 448 00:21:58,280 --> 00:22:02,420 And by the way, notice here my use of f and g. 449 00:22:02,420 --> 00:22:06,570 Notice that w is a different function of x and y 450 00:22:06,570 --> 00:22:09,770 than it is of u and v. But this notation says what? 451 00:22:09,770 --> 00:22:12,730 We can consider w either in terms of x and y, 452 00:22:12,730 --> 00:22:18,320 or in terms of u and v. Hardly ever would we consider w 453 00:22:18,320 --> 00:22:19,630 as a function of u and y. 454 00:22:19,630 --> 00:22:22,770 In other words, either we would change the variables or we 455 00:22:22,770 --> 00:22:23,610 don't. 456 00:22:23,610 --> 00:22:25,890 Or, for example, in terms of polar versus Cartesian 457 00:22:25,890 --> 00:22:29,740 coordinates, either we use x and y or we use r and theta. 458 00:22:29,740 --> 00:22:32,040 We don't usually use r and x. 459 00:22:32,040 --> 00:22:35,240 In other words, going back to the textbook example again, 460 00:22:35,240 --> 00:22:37,510 if we look at this particular situation here, 461 00:22:37,510 --> 00:22:39,600 all the book is saying is, when you're 462 00:22:39,600 --> 00:22:41,480 differentiating with respect to n, 463 00:22:41,480 --> 00:22:44,310 you usually mean that y is the variable being held constant. 464 00:22:44,310 --> 00:22:46,275 When we differentiate with respect to u, 465 00:22:46,275 --> 00:22:49,120 we usually means that v is the variable that's 466 00:22:49,120 --> 00:22:50,370 being held constant. 467 00:22:50,370 --> 00:22:54,020 Again, more of this is said in our exercises. 468 00:22:54,020 --> 00:22:55,790 But for the time being, I want you 469 00:22:55,790 --> 00:22:59,930 to see how important it is, when you have several variables, 470 00:22:59,930 --> 00:23:02,820 to keep track of which are the dependent variables, which 471 00:23:02,820 --> 00:23:06,500 are the independent variables, and how they're coupled. 472 00:23:06,500 --> 00:23:09,490 I said earlier that one of the nice things about picking n 473 00:23:09,490 --> 00:23:13,760 equals 2 is that you can draw a nice picture of the situation. 474 00:23:13,760 --> 00:23:16,340 And what I thought you might like to see is the following. 475 00:23:16,340 --> 00:23:20,310 Let's suppose you have that w is some function of x and y. 476 00:23:20,310 --> 00:23:23,420 What that means is, we can locate-- for a given value of x 477 00:23:23,420 --> 00:23:24,580 and a given value of y. 478 00:23:24,580 --> 00:23:28,170 In other words, for x equals x_0, y equals y_0, 479 00:23:28,170 --> 00:23:32,800 we can think of the point (x_0, y_0) as being in the xy-plane. 480 00:23:32,800 --> 00:23:38,050 w, which is f of x_0, y_0, is just the height 481 00:23:38,050 --> 00:23:39,930 to this particular surface. 482 00:23:39,930 --> 00:23:46,200 In other words, if I let f of x_0, y_0 be called w_0, 483 00:23:46,200 --> 00:23:51,670 notice that the function f of x, y, at the point x equals x_0, 484 00:23:51,670 --> 00:23:55,940 y equals y_0, graphically corresponds to the point 485 00:23:55,940 --> 00:24:01,060 whose coordinates are x_0, y_0, and w_0. 486 00:24:01,060 --> 00:24:02,530 OK? 487 00:24:02,530 --> 00:24:03,790 This is now a surface. 488 00:24:03,790 --> 00:24:08,600 And in the same way that we use tangent lines to replace curves 489 00:24:08,600 --> 00:24:11,260 when we had one independent variable, in two 490 00:24:11,260 --> 00:24:15,340 independent variables, we use tangent planes 491 00:24:15,340 --> 00:24:16,920 to replace surfaces. 492 00:24:16,920 --> 00:24:20,190 The question that comes up is, how do you get a tangent plane? 493 00:24:20,190 --> 00:24:23,000 And what does this have to do with the partial derivatives? 494 00:24:23,000 --> 00:24:25,000 And what I would like show you is the following. 495 00:24:25,000 --> 00:24:28,640 First of all, one very natural way 496 00:24:28,640 --> 00:24:32,390 of intersecting this surface with a plane-- in other words, 497 00:24:32,390 --> 00:24:35,330 when you take the partial of w with respect 498 00:24:35,330 --> 00:24:40,330 to y, you're holding x constant, you're saying, let x equal x_0. 499 00:24:40,330 --> 00:24:44,340 Notice that in three-dimensional space, x equals x_0 500 00:24:44,340 --> 00:24:47,550 is the equation of this plane. 501 00:24:47,550 --> 00:24:51,380 And this plane-- see, what is it the plane x equals x_0? 502 00:24:51,380 --> 00:24:53,710 It's the plane that goes through the line x equals 503 00:24:53,710 --> 00:24:57,540 x_0 parallel to the xy-plane. 504 00:24:57,540 --> 00:25:02,327 This plane intersects my surface in a particular curve. 505 00:25:02,327 --> 00:25:06,150 That curve passes through P_0. 506 00:25:06,150 --> 00:25:09,340 I can talk about the slope of the curve 507 00:25:09,340 --> 00:25:11,620 at that particular point. 508 00:25:11,620 --> 00:25:12,500 OK? 509 00:25:12,500 --> 00:25:16,620 In a similar way, I could've sliced the surface 510 00:25:16,620 --> 00:25:19,540 by the plane y equals y_0. 511 00:25:19,540 --> 00:25:23,620 In other words, hold y constant. 512 00:25:23,620 --> 00:25:26,710 Well, you see, to hold y constant, that 513 00:25:26,710 --> 00:25:29,850 means I, again, draw my plane. 514 00:25:29,850 --> 00:25:32,330 It intersects the surface in a different curve, 515 00:25:32,330 --> 00:25:35,630 but that curve must also go through the point P_0. 516 00:25:35,630 --> 00:25:37,930 That's the same P_0 that's over here, 517 00:25:37,930 --> 00:25:41,880 because after all, the point on the surface that is directly 518 00:25:41,880 --> 00:25:44,704 above (x_0, y_0) is P_0. 519 00:25:44,704 --> 00:25:46,370 And that point doesn't change, no matter 520 00:25:46,370 --> 00:25:48,460 what plane you slice this thing with. 521 00:25:48,460 --> 00:25:49,530 OK? 522 00:25:49,530 --> 00:25:53,710 And so again, I can talk about the slope of the tangent 523 00:25:53,710 --> 00:25:56,160 to this particular curve of intersection. 524 00:25:56,160 --> 00:26:00,070 By the way, one very brief aside before I continue on with this. 525 00:26:00,070 --> 00:26:03,180 One of the things that makes functions of several variables 526 00:26:03,180 --> 00:26:06,170 so difficult is-- I would like you to observe, 527 00:26:06,170 --> 00:26:08,190 because this will become the backbone 528 00:26:08,190 --> 00:26:11,870 of our future investigations-- that these two 529 00:26:11,870 --> 00:26:15,740 particular planes that I drew, to intersect my surface, 530 00:26:15,740 --> 00:26:17,570 were very special planes. 531 00:26:17,570 --> 00:26:19,720 One of them was parallel-- they both 532 00:26:19,720 --> 00:26:21,890 went through the point (x_0, y_0), 533 00:26:21,890 --> 00:26:25,470 but one happened to be parallel to the wy-plane 534 00:26:25,470 --> 00:26:29,300 and the other one happened to be parallel to the wx-plane. 535 00:26:29,300 --> 00:26:32,310 Notice that, in general, I could have 536 00:26:32,310 --> 00:26:37,530 passed infinitely many different planes through (x_0, y_0). 537 00:26:37,530 --> 00:26:41,610 Each of which would have intersected the surface 538 00:26:41,610 --> 00:26:43,020 in a different curve. 539 00:26:43,020 --> 00:26:46,540 And this is what leads later to the more generalized concept 540 00:26:46,540 --> 00:26:49,530 of directional derivatives that we will talk about in more 541 00:26:49,530 --> 00:26:51,800 detail in our future lectures. 542 00:26:51,800 --> 00:26:53,930 But I simply mention the word now 543 00:26:53,930 --> 00:26:56,340 because in the reading material in this assignment, 544 00:26:56,340 --> 00:26:59,310 some mention is made of directional derivatives. 545 00:26:59,310 --> 00:27:01,690 In other words, I wanted to point out 546 00:27:01,690 --> 00:27:03,670 that to take partial derivatives, 547 00:27:03,670 --> 00:27:06,630 we are interested in two special directions, 548 00:27:06,630 --> 00:27:08,557 even though there are other directions. 549 00:27:08,557 --> 00:27:10,140 The question that comes up, of course, 550 00:27:10,140 --> 00:27:12,180 is where do the partials come in here? 551 00:27:12,180 --> 00:27:19,370 And secondly, knowing where partials come up in here, how 552 00:27:19,370 --> 00:27:21,590 do we find the equation of a tangent plane? 553 00:27:21,590 --> 00:27:23,300 And all I would like you to see from here 554 00:27:23,300 --> 00:27:26,820 is that I'm going to use the same old technique as always. 555 00:27:26,820 --> 00:27:30,820 Given these two tangent lines, I'm going to vectorize them. 556 00:27:30,820 --> 00:27:34,460 I am then going to find the equation of a plane that passes 557 00:27:34,460 --> 00:27:37,130 through those two vectors. 558 00:27:37,130 --> 00:27:39,310 To do that, I'm going to have to find 559 00:27:39,310 --> 00:27:41,971 the normal to that plane, et cetera. 560 00:27:41,971 --> 00:27:43,720 And I'm going to wind up with the equation 561 00:27:43,720 --> 00:27:45,030 of a tangent plane. 562 00:27:45,030 --> 00:27:46,900 In more slow motion, all I'm saying 563 00:27:46,900 --> 00:27:50,690 is if I now take this picture and draw it over here-- notice 564 00:27:50,690 --> 00:27:54,070 the representation-- it looks as if the curve that I get 565 00:27:54,070 --> 00:27:55,950 is in the wy-plane. 566 00:27:55,950 --> 00:27:57,710 It's actually in a plane parallel 567 00:27:57,710 --> 00:28:02,240 to the wy-plane characterized by x equals x sub 0. 568 00:28:02,240 --> 00:28:04,610 And by the way, notice even though x 569 00:28:04,610 --> 00:28:09,060 equals x_0 was chosen so that x_0 was fixed, 570 00:28:09,060 --> 00:28:13,970 I hope it's clear to you that if I let x sub 0 vary and I picked 571 00:28:13,970 --> 00:28:17,420 different values of x sub 0, the curve that I get here 572 00:28:17,420 --> 00:28:19,680 will, in general, differ. 573 00:28:19,680 --> 00:28:21,400 You see, what I'm saying is, if I 574 00:28:21,400 --> 00:28:24,740 take slices parallel to the wy-plane 575 00:28:24,740 --> 00:28:26,970 for a particular surface, the slices 576 00:28:26,970 --> 00:28:29,930 that I get-- the shape-- will depend on what 577 00:28:29,930 --> 00:28:32,380 the particular value of x_0 is. 578 00:28:32,380 --> 00:28:34,990 But that's not the crucial issue right now. 579 00:28:34,990 --> 00:28:36,760 The point that I'm driving at is, 580 00:28:36,760 --> 00:28:41,650 if I if I vectorize the tangent line and call that v_1, 581 00:28:41,650 --> 00:28:44,610 what does that tangent line look like? 582 00:28:44,610 --> 00:28:47,405 First of all, let's see what its slope is. 583 00:28:47,405 --> 00:28:50,910 Now again, if you didn't see this over here 584 00:28:50,910 --> 00:28:53,310 and all I showed was this diagram, 585 00:28:53,310 --> 00:28:55,840 you would say, hey, that slope is just the derivative 586 00:28:55,840 --> 00:28:58,240 of w with respect to y. 587 00:28:58,240 --> 00:29:01,050 But we write that the partial of w with respect 588 00:29:01,050 --> 00:29:05,020 to y in deference to the fact that w is not a function of y 589 00:29:05,020 --> 00:29:08,320 alone-- that w depends on both x and y, 590 00:29:08,320 --> 00:29:11,350 and the reason that we got this curve was that we fixed 591 00:29:11,350 --> 00:29:13,770 x at some particular value. 592 00:29:13,770 --> 00:29:16,060 So this is really what? 593 00:29:16,060 --> 00:29:19,510 Not the derivative of w with respect to y evaluated at y 594 00:29:19,510 --> 00:29:20,890 equals y_0. 595 00:29:20,890 --> 00:29:25,795 It's the partial of w with respect to y evaluated at what? 596 00:29:25,795 --> 00:29:28,980 x_0 comma y_0. 597 00:29:28,980 --> 00:29:32,340 All right, at any rate, now that we have the slope, 598 00:29:32,340 --> 00:29:34,240 notice that the slope of a vector 599 00:29:34,240 --> 00:29:37,700 can always be viewed as the k component-- 600 00:29:37,700 --> 00:29:41,890 in this case, the k component divided by the j component. 601 00:29:41,890 --> 00:29:44,760 If I make the j component 1, the slope 602 00:29:44,760 --> 00:29:46,720 will just be the k component. 603 00:29:46,720 --> 00:29:49,120 In other words, a vector which is 604 00:29:49,120 --> 00:29:51,930 tangent to the curve-- and one that's easy to write down-- 605 00:29:51,930 --> 00:29:56,770 is j plus the partial of w with respect to y, evaluated 606 00:29:56,770 --> 00:29:59,340 at (x_0, y_0), times k. 607 00:29:59,340 --> 00:30:04,270 And notice, by the way, that this is a number. 608 00:30:04,270 --> 00:30:06,010 That once I take the partial derivative 609 00:30:06,010 --> 00:30:08,224 and evaluate it at a point, I have a number. 610 00:30:08,224 --> 00:30:08,890 So this is what? 611 00:30:08,890 --> 00:30:12,470 A constant vector once x_0 and y_0 are fixed. 612 00:30:12,470 --> 00:30:17,740 In a similar way, my vector v_2, which is for my curve parallel 613 00:30:17,740 --> 00:30:23,690 to the wx-plane-- the slope of that curve at the point P_0 614 00:30:23,690 --> 00:30:27,300 is not dw/dx, it's the partial of w with respect 615 00:30:27,300 --> 00:30:30,340 to x, evaluated (x_0, y_0). 616 00:30:30,340 --> 00:30:31,440 Again, why? 617 00:30:31,440 --> 00:30:34,830 Because, even though this looks like w is a function of x alone 618 00:30:34,830 --> 00:30:37,450 here, it was only that way because we 619 00:30:37,450 --> 00:30:39,500 took the second variable and, in a sense, 620 00:30:39,500 --> 00:30:41,990 froze it at the value y_0. 621 00:30:41,990 --> 00:30:44,450 At any rate, in an analogous manner, 622 00:30:44,450 --> 00:30:47,080 v_2 turns out to be what vector? 623 00:30:47,080 --> 00:30:50,450 It's i plus the partial of w with respect 624 00:30:50,450 --> 00:30:53,760 to x evaluated (x_0, y_0) times k. 625 00:30:53,760 --> 00:30:57,090 In other words, the slope is the partial of w with respect to x. 626 00:30:57,090 --> 00:30:59,005 And it's in the ik-plane. 627 00:30:59,005 --> 00:31:01,460 In other words, the xw-plane. 628 00:31:01,460 --> 00:31:04,130 Now that I have my two vectors, to find 629 00:31:04,130 --> 00:31:07,310 the plane that passes through them, all I need is what? 630 00:31:07,310 --> 00:31:10,460 The normal vector and a point in the plane. 631 00:31:10,460 --> 00:31:14,240 Well, one of the nice things about having studied 632 00:31:14,240 --> 00:31:16,500 the cross product here, is that I can now 633 00:31:16,500 --> 00:31:19,250 take the cross product of v_1 and v_2. 634 00:31:19,250 --> 00:31:21,710 Sparing you the details, remember what I do here. 635 00:31:21,710 --> 00:31:26,580 I just write down i, j, k. v_1 has these components, 636 00:31:26,580 --> 00:31:29,490 v_2 has these components. 637 00:31:29,490 --> 00:31:34,050 I now expand this determinant in the usual way. 638 00:31:34,050 --> 00:31:35,490 And I get what? 639 00:31:35,490 --> 00:31:37,209 This particular vector. 640 00:31:37,209 --> 00:31:39,125 In other words, the i component is the partial 641 00:31:39,125 --> 00:31:40,730 of w with respect to x. 642 00:31:40,730 --> 00:31:43,660 The j component is a partial of w with respect to y. 643 00:31:43,660 --> 00:31:46,270 Both evaluated at the point (x_0, y_0). 644 00:31:46,270 --> 00:31:48,640 And the k component is minus 1. 645 00:31:48,640 --> 00:31:52,310 Now, to find the equation of a tangent plane, all I have to do 646 00:31:52,310 --> 00:31:53,750 is what? 647 00:31:53,750 --> 00:31:55,550 Take the standard form of the plane. 648 00:31:55,550 --> 00:31:58,910 In other words, the coefficients will be these. 649 00:31:58,910 --> 00:32:03,950 And then I take (x_0, y_0, w_0), which is a point in my plane. 650 00:32:03,950 --> 00:32:06,910 And then the equation of the tangent plane 651 00:32:06,910 --> 00:32:10,580 becomes, quite simply, this expression here. 652 00:32:10,580 --> 00:32:13,290 And in fact, if I rewrite that, notice 653 00:32:13,290 --> 00:32:17,180 I can transpose the w minus w_0 term. 654 00:32:17,180 --> 00:32:19,260 That is the change in w. 655 00:32:19,260 --> 00:32:21,920 But to remind us that we're in the tangent plane, 656 00:32:21,920 --> 00:32:26,830 we don't write this as delta w, we write it as delta w sub tan. 657 00:32:26,830 --> 00:32:29,950 And notice that the change in w to the tangent plane 658 00:32:29,950 --> 00:32:33,380 is just the partial of w with respect to x times delta x, 659 00:32:33,380 --> 00:32:37,290 plus the partial of w with respect to y times delta y. 660 00:32:37,290 --> 00:32:40,570 And hopefully, this starts to seem familiar to you. 661 00:32:40,570 --> 00:32:44,310 This should almost look like the equation of delta y tan, 662 00:32:44,310 --> 00:32:46,060 only in two dimensions. 663 00:32:46,060 --> 00:32:49,040 Remember when we had delta y tan in part one of the course? 664 00:32:49,040 --> 00:32:51,160 See, what this really says is what? 665 00:32:51,160 --> 00:32:54,390 This is the change in w with respect to x, multiplied 666 00:32:54,390 --> 00:32:56,050 by the total change in x. 667 00:32:56,050 --> 00:32:58,150 In other words, this term is what? 668 00:32:58,150 --> 00:33:02,380 It's the change in w due to the change in x alone. 669 00:33:02,380 --> 00:33:05,450 This term is the change in w, with respect to y, 670 00:33:05,450 --> 00:33:07,510 times the total change in y. 671 00:33:07,510 --> 00:33:09,720 So this term tells you the change 672 00:33:09,720 --> 00:33:12,630 in w due to the change in y alone. 673 00:33:12,630 --> 00:33:15,970 And if you add these two up, since x and y are independent, 674 00:33:15,970 --> 00:33:18,514 this should give you the total change in w. 675 00:33:18,514 --> 00:33:20,930 The reason, of course, that you don't get the total change 676 00:33:20,930 --> 00:33:24,320 in w, but rather, delta w tan, is the fact 677 00:33:24,320 --> 00:33:27,160 that if these expressions here are not 678 00:33:27,160 --> 00:33:31,380 constant numbers as delta x and delta y ore varying, 679 00:33:31,380 --> 00:33:33,410 these numbers here are varying. 680 00:33:33,410 --> 00:33:34,800 These are variables. 681 00:33:34,800 --> 00:33:38,460 But you see, once you fix these at these values, 682 00:33:38,460 --> 00:33:42,270 these become numbers, and now you know what delta w tan is. 683 00:33:42,270 --> 00:33:44,640 Now, in some of the homework problems 684 00:33:44,640 --> 00:33:46,350 in the text, what we're going to do 685 00:33:46,350 --> 00:33:48,610 is just practice the geometry itself. 686 00:33:48,610 --> 00:33:52,280 We're going to find equations of tangent planes. 687 00:33:52,280 --> 00:33:55,430 The thing that's very important to us in future lectures, 688 00:33:55,430 --> 00:33:57,810 from a structural point of view, is 689 00:33:57,810 --> 00:34:00,320 that the equation of the tangent plane-- in other words, 690 00:34:00,320 --> 00:34:05,040 the equation for delta w tan-- is far more 691 00:34:05,040 --> 00:34:09,260 simple to handle than the equation for the true delta w. 692 00:34:09,260 --> 00:34:11,350 You see, the equation for delta w tan 693 00:34:11,350 --> 00:34:15,020 just has delta x and delta y appearing to the first power. 694 00:34:15,020 --> 00:34:17,929 Whereas, if you try to find delta w exactly 695 00:34:17,929 --> 00:34:20,210 for an arbitrary function of x and y, 696 00:34:20,210 --> 00:34:22,840 this can become a very, very messy thing. 697 00:34:22,840 --> 00:34:26,980 In short, delta w tan will play for two 698 00:34:26,980 --> 00:34:31,210 and more independent variables the same role 699 00:34:31,210 --> 00:34:33,620 that delta y sub tan played for us 700 00:34:33,620 --> 00:34:37,360 in part one of this course as a differential. 701 00:34:37,360 --> 00:34:40,280 And what we'll be discussing next, or at least 702 00:34:40,280 --> 00:34:42,810 trying to get at the root of, is just 703 00:34:42,810 --> 00:34:45,170 what does play the role of a differential 704 00:34:45,170 --> 00:34:48,170 when you're dealing with calculus of several variables? 705 00:34:48,170 --> 00:34:51,889 And what we're going to find is that, again, the computations 706 00:34:51,889 --> 00:34:55,100 become messy enough so that even though the structure stays 707 00:34:55,100 --> 00:34:58,070 fairly nice, there are some wrinkles that come up 708 00:34:58,070 --> 00:35:01,830 that will take us a considerable amount of time to untangle. 709 00:35:01,830 --> 00:35:04,620 But we'll worry about that when the time comes. 710 00:35:04,620 --> 00:35:06,560 And so until that time, goodbye. 711 00:35:11,800 --> 00:35:14,180 Funding for the publication of this video 712 00:35:14,180 --> 00:35:19,050 was provided by the Gabriella and Paul Rosenbaum Foundation. 713 00:35:19,050 --> 00:35:23,230 Help OCW continue to provide free and open access to MIT 714 00:35:23,230 --> 00:35:27,640 courses by making a donation at ocw.mit.edu/donate.