1 00:00:00,000 --> 00:00:01,894 2 00:00:01,894 --> 00:00:03,810 MAGDALENA TODA: So what's your general feeling 3 00:00:03,810 --> 00:00:05,423 about Chapter 11? 4 00:00:05,423 --> 00:00:06,355 STUDENT: It's OK. 5 00:00:06,355 --> 00:00:07,355 MAGDALENA TODA: It's OK. 6 00:00:07,355 --> 00:00:12,185 So functions of two variables are 7 00:00:12,185 --> 00:00:15,566 to be compared all the time with the functions of one variable. 8 00:00:15,566 --> 00:00:18,810 Every nothing you have seen in Calc 1 9 00:00:18,810 --> 00:00:23,983 has a corresponding the motion in Calc 3. 10 00:00:23,983 --> 00:00:27,780 11 00:00:27,780 --> 00:00:33,020 So really no questions about theory, concepts, Chapter 11 12 00:00:33,020 --> 00:00:36,880 concepts, previous concepts? 13 00:00:36,880 --> 00:00:39,760 Feel free to email me this weekend. 14 00:00:39,760 --> 00:00:43,370 Don't think it's the weekend because we 15 00:00:43,370 --> 00:00:47,140 are on a 24/7 availability. 16 00:00:47,140 --> 00:00:48,970 People, we use WeBWork. 17 00:00:48,970 --> 00:00:50,520 Not just me, but everybody who uses 18 00:00:50,520 --> 00:00:54,310 WeBWork is on a 24/7 availability, 19 00:00:54,310 --> 00:00:59,130 answering questions about WeBWork problems. 20 00:00:59,130 --> 00:01:02,320 Saturday and Sunday is when most of you do the homework. 21 00:01:02,320 --> 00:01:05,010 22 00:01:05,010 --> 00:01:09,462 It's convenient for us as well because we are with the family, 23 00:01:09,462 --> 00:01:11,797 but we don't have many meetings to attend. 24 00:01:11,797 --> 00:01:14,810 So I'll be happy to answer your questions. 25 00:01:14,810 --> 00:01:19,384 Last time, we discussed a little bit about preparation 26 00:01:19,384 --> 00:01:21,598 for The Chain Rule. 27 00:01:21,598 --> 00:01:23,566 In Calc 3. 28 00:01:23,566 --> 00:01:34,880 So the chain rule in Calc 3 was something really-- 29 00:01:34,880 --> 00:01:40,126 this is section 11.5. 30 00:01:40,126 --> 00:01:45,549 The preparation was done last time, 31 00:01:45,549 --> 00:01:50,479 but I'm going to review it a little bit. 32 00:01:50,479 --> 00:01:54,423 Let's see what we discussed. 33 00:01:54,423 --> 00:01:57,381 I'm going to split, again, the board in two. 34 00:01:57,381 --> 00:02:05,842 And I'll say, can we review the notions of The Chain Rule. 35 00:02:05,842 --> 00:02:11,360 When you start with a variable-- let's say it's time. 36 00:02:11,360 --> 00:02:20,795 Time going to f of t, which goes into g of f of t by something 37 00:02:20,795 --> 00:02:22,340 called composition. 38 00:02:22,340 --> 00:02:26,284 We've done that since we were kids in college algebra. 39 00:02:26,284 --> 00:02:27,170 What? 40 00:02:27,170 --> 00:02:28,680 You never took college algebra? 41 00:02:28,680 --> 00:02:30,890 Except in high school, you took high school algebra, 42 00:02:30,890 --> 00:02:33,210 most of you. 43 00:02:33,210 --> 00:02:35,324 So what did you do in high school algebra? 44 00:02:35,324 --> 00:02:38,280 We said g composed with l. 45 00:02:38,280 --> 00:02:40,760 This is a composition of two functions. 46 00:02:40,760 --> 00:02:46,360 What I'm skipping here is the theory that you learned then 47 00:02:46,360 --> 00:02:56,250 that to a compose well, F of t has to be in the domain of g. 48 00:02:56,250 --> 00:02:59,340 So the image F of t, whatever you get from this image, 49 00:02:59,340 --> 00:03:01,620 has to be in the domain of g. 50 00:03:01,620 --> 00:03:05,100 Otherwise, the composition could not exist. 51 00:03:05,100 --> 00:03:07,870 Now if you have differentiability, 52 00:03:07,870 --> 00:03:10,390 assuming that this is g composed with F, 53 00:03:10,390 --> 00:03:16,332 assuming to be c1-- c1 meaning differentiable 54 00:03:16,332 --> 00:03:24,470 and derivatives are continuous-- assuming both of them are c1, 55 00:03:24,470 --> 00:03:26,320 they compose well. 56 00:03:26,320 --> 00:03:28,260 What am I going to do next? 57 00:03:28,260 --> 00:03:35,660 I'm going to say the d, dt g of F of t. 58 00:03:35,660 --> 00:03:39,600 And we said last time, we get The Chain Rule 59 00:03:39,600 --> 00:03:44,463 from the last function we applied, g prime. 60 00:03:44,463 --> 00:03:50,860 And so you have dg, [? d2 ?] at F of t. 61 00:03:50,860 --> 00:03:57,590 I'm calling this guy u variable just for my own enjoyment. 62 00:03:57,590 --> 00:04:01,660 And then I go du, dt. 63 00:04:01,660 --> 00:04:06,020 But du, dt would be nothing but a prime of t, 64 00:04:06,020 --> 00:04:09,330 so remember the cowboys shooting at each other? 65 00:04:09,330 --> 00:04:11,060 The du and du. 66 00:04:11,060 --> 00:04:16,120 I will replace the u by prime of t, just like you did in Calc 1. 67 00:04:16,120 --> 00:04:16,620 Why? 68 00:04:16,620 --> 00:04:23,320 Because I want to a mixture of notations according to Calc 1 69 00:04:23,320 --> 00:04:24,871 you took here. 70 00:04:24,871 --> 00:04:30,580 The idea for Calc 3 is the same with [INAUDIBLE] time, 71 00:04:30,580 --> 00:04:33,370 assuming everything composes well, 72 00:04:33,370 --> 00:04:37,910 and has differentiability, and the derivatives are continuous. 73 00:04:37,910 --> 00:04:39,800 Just to make your life easier. 74 00:04:39,800 --> 00:04:43,700 We have x of t, y of t. 75 00:04:43,700 --> 00:04:47,175 Two nice functions and a function 76 00:04:47,175 --> 00:04:54,690 of these variables, F of x and y. 77 00:04:54,690 --> 00:04:57,710 So I'm going to have to say, how about x 78 00:04:57,710 --> 00:05:01,330 is a function of t and y is a function of t? 79 00:05:01,330 --> 00:05:05,290 So I should be able to go ahead and differentiate 80 00:05:05,290 --> 00:05:07,620 with respect to the t. 81 00:05:07,620 --> 00:05:10,850 82 00:05:10,850 --> 00:05:13,354 And how did it go? 83 00:05:13,354 --> 00:05:14,770 Now that I prepared you last time, 84 00:05:14,770 --> 00:05:21,370 a little bit, for this kind of new picture, new diagram, 85 00:05:21,370 --> 00:05:27,365 you should be able to tell me, without looking at the notes 86 00:05:27,365 --> 00:05:31,530 from last time, how this goes. 87 00:05:31,530 --> 00:05:36,730 So I'll take the function F of x of t, y of t. 88 00:05:36,730 --> 00:05:41,721 And when I view it like that, I understand it's ultimately 89 00:05:41,721 --> 00:05:45,380 a big function, F of t. 90 00:05:45,380 --> 00:05:49,610 It's a real valued function of t, 91 00:05:49,610 --> 00:05:52,780 ultimately, as the composition. 92 00:05:52,780 --> 00:05:56,257 This big F. 93 00:05:56,257 --> 00:05:58,370 94 00:05:58,370 --> 00:06:05,290 So does anybody remember how this went? 95 00:06:05,290 --> 00:06:07,750 Let's see. 96 00:06:07,750 --> 00:06:10,210 The derivative, with respect to t, 97 00:06:10,210 --> 00:06:14,965 of this whole thing, F of x of t, y of t? 98 00:06:14,965 --> 00:06:19,070 99 00:06:19,070 --> 00:06:19,570 Thoughts? 100 00:06:19,570 --> 00:06:23,160 101 00:06:23,160 --> 00:06:25,270 The partial derivative of F with respect 102 00:06:25,270 --> 00:06:32,704 to x, evaluated at x of t and y to t. 103 00:06:32,704 --> 00:06:35,620 So everything has to be replaced in terms of t 104 00:06:35,620 --> 00:06:38,536 because it's going to be y. 105 00:06:38,536 --> 00:06:43,750 We assume that this derivative exists and it's continuous. 106 00:06:43,750 --> 00:06:44,270 Why? 107 00:06:44,270 --> 00:06:46,365 Just to make your life a little bit easier. 108 00:06:46,365 --> 00:06:49,550 109 00:06:49,550 --> 00:06:53,990 From the beginning, we had dx, dt, 110 00:06:53,990 --> 00:06:57,300 which was also defined everywhere 111 00:06:57,300 --> 00:07:09,000 and continuous, plus df, 2y at the same point times dy, dt. 112 00:07:09,000 --> 00:07:12,310 113 00:07:12,310 --> 00:07:20,750 Notice what happens here with these guys looking diagonally, 114 00:07:20,750 --> 00:07:21,745 staring at each other. 115 00:07:21,745 --> 00:07:24,810 116 00:07:24,810 --> 00:07:26,940 Keep in mind the plus sign. 117 00:07:26,940 --> 00:07:30,670 And of course, some of you told me, well, is that OK? 118 00:07:30,670 --> 00:07:31,960 You know favorite, right? 119 00:07:31,960 --> 00:07:35,450 F of x at x of dy of t. 120 00:07:35,450 --> 00:07:37,120 That's fine. 121 00:07:37,120 --> 00:07:38,500 I saw that. 122 00:07:38,500 --> 00:07:39,780 In engineering you use it. 123 00:07:39,780 --> 00:07:53,358 Physics majors also use a lot of this notation 124 00:07:53,358 --> 00:07:57,250 as sub [INAUDIBLE] Fs of t. 125 00:07:57,250 --> 00:07:58,700 We've seen that. 126 00:07:58,700 --> 00:07:59,600 We've seen that. 127 00:07:59,600 --> 00:08:03,150 It comes as no surprise to us, but we 128 00:08:03,150 --> 00:08:06,879 would like to see if there are any other cases we 129 00:08:06,879 --> 00:08:07,670 should worry about. 130 00:08:07,670 --> 00:08:18,062 131 00:08:18,062 --> 00:08:22,710 Now I don't want to jump to the next example 132 00:08:22,710 --> 00:08:25,840 until I give you something that you 133 00:08:25,840 --> 00:08:32,110 know very well from Calculus 1. 134 00:08:32,110 --> 00:08:38,970 It's an example that you saw before that was a melting ice 135 00:08:38,970 --> 00:08:41,682 sphere. 136 00:08:41,682 --> 00:08:46,995 It appears a lot in problems, like final exam problems 137 00:08:46,995 --> 00:08:48,450 and stuff. 138 00:08:48,450 --> 00:08:53,120 What is the material of this ball? 139 00:08:53,120 --> 00:08:54,925 It's melting ice. 140 00:08:54,925 --> 00:08:59,118 141 00:08:59,118 --> 00:09:08,470 And if you remember, it says that at the moment t0, 142 00:09:08,470 --> 00:09:13,820 assume the radius was 5 inches. 143 00:09:13,820 --> 00:09:16,706 144 00:09:16,706 --> 00:09:42,508 We also know that the rate of change of the radius in time 145 00:09:42,508 --> 00:09:49,490 will be minus 5. 146 00:09:49,490 --> 00:09:54,920 But let's suppose that we say that inches per-- meaning, 147 00:09:54,920 --> 00:09:56,810 it's really hot in the room. 148 00:09:56,810 --> 00:10:00,270 Not this room, but the hypothetic room 149 00:10:00,270 --> 00:10:05,760 where the ice ball is melting. 150 00:10:05,760 --> 00:10:08,890 So imagine, in 1 minute, the radius 151 00:10:08,890 --> 00:10:13,550 will go down by 5 inches. 152 00:10:13,550 --> 00:10:16,786 Yes, it must be really hot. 153 00:10:16,786 --> 00:10:25,980 I want to know the derivative, dv, dt at the time 0. 154 00:10:25,980 --> 00:10:29,600 So you go, oh my god, I don't remember doing this, actually. 155 00:10:29,600 --> 00:10:31,330 It is a Calc 1 type of problem. 156 00:10:31,330 --> 00:10:34,580 157 00:10:34,580 --> 00:10:37,620 Why am I even discussing it again? 158 00:10:37,620 --> 00:10:41,280 Because I want to fool you a little bit into remembering 159 00:10:41,280 --> 00:10:44,953 the elementary formulas for the volume of a sphere, volume 160 00:10:44,953 --> 00:10:47,720 of a cone, volume of a cylinder. 161 00:10:47,720 --> 00:10:48,910 That was a long time ago. 162 00:10:48,910 --> 00:10:53,570 When you ask you teachers in K12 if you should memorize them, 163 00:10:53,570 --> 00:10:55,950 they said, by all means, memorize them. 164 00:10:55,950 --> 00:10:59,350 That was elementary geometry, but some of you know them 165 00:10:59,350 --> 00:11:01,054 by heart, some of you don't. 166 00:11:01,054 --> 00:11:03,180 Do you remember the volume formula 167 00:11:03,180 --> 00:11:05,860 for a ball with radius r? 168 00:11:05,860 --> 00:11:08,161 [INTERPOSING VOICES] 169 00:11:08,161 --> 00:11:09,095 170 00:11:09,095 --> 00:11:10,029 What? 171 00:11:10,029 --> 00:11:11,430 [? STUDENT: High RQ. ?] 172 00:11:11,430 --> 00:11:12,491 STUDENT: 4/3rds. 173 00:11:12,491 --> 00:11:13,366 MAGDALENA TODA: Good. 174 00:11:13,366 --> 00:11:14,860 I'm proud of you guys. 175 00:11:14,860 --> 00:11:18,496 I've discovered lots of people who are engineering majors 176 00:11:18,496 --> 00:11:19,870 and they don't know this formula. 177 00:11:19,870 --> 00:11:23,790 So how are we going to think of this problem? 178 00:11:23,790 --> 00:11:26,390 We have to think, Chain Rule. 179 00:11:26,390 --> 00:11:30,570 And Chain Rule means that you view this radius as a shrinking 180 00:11:30,570 --> 00:11:32,280 thing because that's why you have 181 00:11:32,280 --> 00:11:34,740 the grade of change negative. 182 00:11:34,740 --> 00:11:37,120 The radius is shrinking, it's decreasing, 183 00:11:37,120 --> 00:11:40,940 so you view r as a function of t. 184 00:11:40,940 --> 00:11:42,530 And of course, you made me cube it. 185 00:11:42,530 --> 00:11:44,960 I had to cube it. 186 00:11:44,960 --> 00:11:48,260 And then v will be a function of t ultimately, but you see, 187 00:11:48,260 --> 00:11:54,400 guys, t goes to r of t, r of t goes to v of t. 188 00:11:54,400 --> 00:11:56,675 What's the formula for this function? 189 00:11:56,675 --> 00:11:59,774 v equals 4 pi i cubed over 3. 190 00:11:59,774 --> 00:12:02,210 191 00:12:02,210 --> 00:12:04,180 So this is how the diagram goes. 192 00:12:04,180 --> 00:12:10,280 You look at that composition and you have dv, dt. 193 00:12:10,280 --> 00:12:14,161 And I remember teaching as a graduate student, that 194 00:12:14,161 --> 00:12:18,320 was a long time ago, in '97 or something, 195 00:12:18,320 --> 00:12:23,020 with this kind of diagram with compositions of functions. 196 00:12:23,020 --> 00:12:25,720 And my students had told me, nobody showed us 197 00:12:25,720 --> 00:12:28,716 this kind of diagram before. 198 00:12:28,716 --> 00:12:29,600 Well, I do. 199 00:12:29,600 --> 00:12:32,120 200 00:12:32,120 --> 00:12:35,290 I think they are very useful for understanding 201 00:12:35,290 --> 00:12:38,220 how a composition will go. 202 00:12:38,220 --> 00:12:42,490 Now I would just going ahead and say v prime because I'm lazy. 203 00:12:42,490 --> 00:12:45,850 And I go v prime of t is 0. 204 00:12:45,850 --> 00:12:51,012 Meaning, that this is the dv, dt at t0. 205 00:12:51,012 --> 00:12:55,440 And somebody has to help me remember how we did The Chain 206 00:12:55,440 --> 00:12:57,852 Rule in Calc 1. 207 00:12:57,852 --> 00:12:59,470 It was ages ago. 208 00:12:59,470 --> 00:13:05,490 4 pi over 3 constant times. 209 00:13:05,490 --> 00:13:07,380 Who jumps down? 210 00:13:07,380 --> 00:13:11,110 The 3 jumps down and he's very happy to do that. 211 00:13:11,110 --> 00:13:12,540 3, r squared. 212 00:13:12,540 --> 00:13:15,320 But r squared is not an independent variable. 213 00:13:15,320 --> 00:13:18,650 He or she depends on t. 214 00:13:18,650 --> 00:13:22,090 So I'll be very happy to say 3 times that times. 215 00:13:22,090 --> 00:13:24,020 And that's the essential part. 216 00:13:24,020 --> 00:13:25,653 I'm not done. 217 00:13:25,653 --> 00:13:26,872 STUDENT: It's dr over dt. 218 00:13:26,872 --> 00:13:27,830 MAGDALENA TODA: dr, dt. 219 00:13:27,830 --> 00:13:31,430 So I have finally applied The Chain Rule. 220 00:13:31,430 --> 00:13:35,440 And how do I plug in the data in order 221 00:13:35,440 --> 00:13:38,900 to get this as the final answer? 222 00:13:38,900 --> 00:13:46,450 I just go 4 pi over 3 times what? 223 00:13:46,450 --> 00:13:48,980 224 00:13:48,980 --> 00:13:56,420 3 times r-- who is r at the time to 0, 225 00:13:56,420 --> 00:14:00,112 where I want to view the whole situation? 226 00:14:00,112 --> 00:14:03,520 r squared at time to 0 would be 25. 227 00:14:03,520 --> 00:14:04,880 Are you guys with me? 228 00:14:04,880 --> 00:14:09,112 dr, dt at time to 0 is negative 5. 229 00:14:09,112 --> 00:14:10,090 All right. 230 00:14:10,090 --> 00:14:12,180 I'm done. 231 00:14:12,180 --> 00:14:15,195 So you are going to ask me, if I'm taking the examine, 232 00:14:15,195 --> 00:14:17,340 do I need this in the exam like that? 233 00:14:17,340 --> 00:14:18,860 Easy. 234 00:14:18,860 --> 00:14:20,820 Oh, it depends on the exam. 235 00:14:20,820 --> 00:14:23,330 If you have a multiple choice where this is simplified, 236 00:14:23,330 --> 00:14:27,440 obviously, it's not the right thing to forget about it, 237 00:14:27,440 --> 00:14:33,372 but I will accept answers like that. 238 00:14:33,372 --> 00:14:37,260 I don't care about the numerical part very much. 239 00:14:37,260 --> 00:14:41,335 If you want to do more, 4 times 25 is hundred times 5. 240 00:14:41,335 --> 00:14:43,534 So I have minus what? 241 00:14:43,534 --> 00:14:44,940 STUDENT: 500 pi. 242 00:14:44,940 --> 00:14:47,074 MAGDALENA TODA: 500 pi. 243 00:14:47,074 --> 00:14:49,050 How do we get the unit of that? 244 00:14:49,050 --> 00:14:50,630 I'm wondering. 245 00:14:50,630 --> 00:14:52,410 STUDENT: Cubic inches per minute. 246 00:14:52,410 --> 00:14:54,350 MAGDALENA TODA: Cubic inches per minute. 247 00:14:54,350 --> 00:14:55,320 Very good. 248 00:14:55,320 --> 00:14:56,650 Cubic inches per minute. 249 00:14:56,650 --> 00:14:59,020 Why don't I write it down? 250 00:14:59,020 --> 00:15:01,440 Because I couldn't care less. 251 00:15:01,440 --> 00:15:02,380 I'm a mathematician. 252 00:15:02,380 --> 00:15:06,709 If I were a physicist, I would definitely write it down. 253 00:15:06,709 --> 00:15:10,100 And he was right. 254 00:15:10,100 --> 00:15:15,990 Now you are going to find this weird. 255 00:15:15,990 --> 00:15:20,190 Why is she doing this review of this kind of melting ice 256 00:15:20,190 --> 00:15:22,360 problem from Calc 1? 257 00:15:22,360 --> 00:15:25,790 Because today I'm being sneaky and mean. 258 00:15:25,790 --> 00:15:28,760 And I want to give you a little challenge 259 00:15:28,760 --> 00:15:30,940 for 1 point of extra credit. 260 00:15:30,940 --> 00:15:33,410 You will have to compose your own problem, 261 00:15:33,410 --> 00:15:37,330 in Calculus 3, that is like that. 262 00:15:37,330 --> 00:15:49,910 So you have to compose a problem about a solid cylinder made 263 00:15:49,910 --> 00:15:51,840 of ice. 264 00:15:51,840 --> 00:15:53,010 Say what, Magdalena? 265 00:15:53,010 --> 00:15:53,730 OK. 266 00:15:53,730 --> 00:15:57,450 So I'll write it down. 267 00:15:57,450 --> 00:16:01,850 Solid cylinder made of ice that's melting in time. 268 00:16:01,850 --> 00:16:04,615 269 00:16:04,615 --> 00:16:07,110 So compose your own problem. 270 00:16:07,110 --> 00:16:10,020 Do you have to solve your own problem? 271 00:16:10,020 --> 00:16:12,530 Yes, I guess so. 272 00:16:12,530 --> 00:16:14,310 Once you compose your own problem, 273 00:16:14,310 --> 00:16:16,545 solve your own problem For extra credit, 1 point. 274 00:16:16,545 --> 00:16:20,680 275 00:16:20,680 --> 00:16:28,770 Compose, write, and solve-- you are the problem author. 276 00:16:28,770 --> 00:16:36,004 Write and solve your own problem, 277 00:16:36,004 --> 00:16:40,396 so that the story includes-- 278 00:16:40,396 --> 00:16:42,836 STUDENT: A solid cylinder. 279 00:16:42,836 --> 00:16:43,812 MAGDALENA TODA: Yes. 280 00:16:43,812 --> 00:16:48,670 Includes-- instead of a nice ball, a solid cylinder. 281 00:16:48,670 --> 00:16:54,300 282 00:16:54,300 --> 00:16:59,240 And necessarily, you cannot write it just a story-- 283 00:16:59,240 --> 00:17:02,860 I once had an ice cylinder, and it was melting, 284 00:17:02,860 --> 00:17:05,990 and I went to watch a movie, and by the time I came back, 285 00:17:05,990 --> 00:17:07,098 it was all melted. 286 00:17:07,098 --> 00:17:08,910 That's not what I want. 287 00:17:08,910 --> 00:17:24,450 I want it so that the problem is an example of applying 288 00:17:24,450 --> 00:17:35,130 The Chain Rule in Calc 3. 289 00:17:35,130 --> 00:17:37,730 And I won't say more. 290 00:17:37,730 --> 00:17:40,688 So maybe somebody can help with a hint. 291 00:17:40,688 --> 00:17:43,033 Maybe I shouldn't give too many hits, 292 00:17:43,033 --> 00:17:46,420 but let's talk as if we were chatting in a cafe, 293 00:17:46,420 --> 00:17:49,290 without me writing too much down. 294 00:17:49,290 --> 00:17:51,380 Of course, you can take notes of our discussion, 295 00:17:51,380 --> 00:17:54,290 but I don't want have it documented. 296 00:17:54,290 --> 00:17:55,759 So we have a cylinder right. 297 00:17:55,759 --> 00:17:58,753 298 00:17:58,753 --> 00:18:01,747 There is the cylinder. 299 00:18:01,747 --> 00:18:02,745 Forget about this. 300 00:18:02,745 --> 00:18:04,250 So there's the cylinder. 301 00:18:04,250 --> 00:18:09,620 It's made of ice and it's melting. 302 00:18:09,620 --> 00:18:14,010 And the volume should be a function of two variables 303 00:18:14,010 --> 00:18:16,725 because otherwise, you don't have it in Calc 3. 304 00:18:16,725 --> 00:18:18,426 So a function of two variables. 305 00:18:18,426 --> 00:18:21,807 306 00:18:21,807 --> 00:18:25,182 What other two variables am I talking about? 307 00:18:25,182 --> 00:18:26,640 STUDENT: The radius and the height. 308 00:18:26,640 --> 00:18:28,640 MAGDALENA TODA: The radius would be one of them. 309 00:18:28,640 --> 00:18:30,120 You don't have to say x and y. 310 00:18:30,120 --> 00:18:33,840 This is r and h. 311 00:18:33,840 --> 00:18:39,002 So h and r are in that formula. 312 00:18:39,002 --> 00:18:40,460 I'm not going to say which formula, 313 00:18:40,460 --> 00:18:44,740 you guys should know of the volume of the cylinder. 314 00:18:44,740 --> 00:18:48,974 But both h and r, what do they have in common in the story? 315 00:18:48,974 --> 00:18:49,940 STUDENT: Time. 316 00:18:49,940 --> 00:18:52,150 MAGDALENA TODA: They are both functions of time. 317 00:18:52,150 --> 00:18:53,771 They are melting in time. 318 00:18:53,771 --> 00:18:55,270 STUDENT: Can I ask a quick question? 319 00:18:55,270 --> 00:18:55,870 MAGDALENA TODA: Yes, sir. 320 00:18:55,870 --> 00:18:58,430 STUDENT: What if we solve for-- what is the negative 500 321 00:18:58,430 --> 00:18:59,786 [? path? ?] 322 00:18:59,786 --> 00:19:04,290 MAGDALENA TODA: This is the speed with which the volume is 323 00:19:04,290 --> 00:19:05,638 shrinking at time to 0. 324 00:19:05,638 --> 00:19:08,850 325 00:19:08,850 --> 00:19:13,090 So the rate of change of the volume at time to o. 326 00:19:13,090 --> 00:19:15,170 And this is something-- by the way, 327 00:19:15,170 --> 00:19:20,260 that's how I would like you to state it. 328 00:19:20,260 --> 00:19:25,805 Find the rate of change of the volume of the ice-- 329 00:19:25,805 --> 00:19:28,780 wasn't that a good cylinder? 330 00:19:28,780 --> 00:19:34,850 At time to 0, if you know that at time to 0 331 00:19:34,850 --> 00:19:37,680 something happened. 332 00:19:37,680 --> 00:19:41,120 Maybe r is given, h is given. 333 00:19:41,120 --> 00:19:44,256 The derivatives are given. 334 00:19:44,256 --> 00:19:47,370 You only have one derivative given here, 335 00:19:47,370 --> 00:19:50,460 which was our prime of t minus 5. 336 00:19:50,460 --> 00:19:52,160 Now I leave it to you. 337 00:19:52,160 --> 00:19:57,120 I ask it to you, and I'll leave it to you, and don't tell me. 338 00:19:57,120 --> 00:20:01,820 When we have a piece of ice-- well, 339 00:20:01,820 --> 00:20:05,590 there was something in the news, but I'm not going to say. 340 00:20:05,590 --> 00:20:08,090 There was some nice, ice sculpture in the news there. 341 00:20:08,090 --> 00:20:10,880 342 00:20:10,880 --> 00:20:18,980 So do the dimensions decrease at the same rate, do you think? 343 00:20:18,980 --> 00:20:20,560 I mean, I don't know. 344 00:20:20,560 --> 00:20:22,020 It's all up to you. 345 00:20:22,020 --> 00:20:24,830 Think of a case when the radius and the height 346 00:20:24,830 --> 00:20:27,800 would shrink at the same speed. 347 00:20:27,800 --> 00:20:31,316 And think of a case when the radius and the height 348 00:20:31,316 --> 00:20:34,030 of the cylinder made of ice would not 349 00:20:34,030 --> 00:20:38,610 change at the same rate for some reason. 350 00:20:38,610 --> 00:20:40,700 I don't know, but the simplest case 351 00:20:40,700 --> 00:20:43,040 would be to assume that all of the dimensions 352 00:20:43,040 --> 00:20:49,520 shrink at the same speed, at the same rate of change. 353 00:20:49,520 --> 00:20:52,000 So you write your own problem, you make up your own data. 354 00:20:52,000 --> 00:20:55,335 Now you will appreciate how much work people 355 00:20:55,335 --> 00:20:57,370 put into that work book. 356 00:20:57,370 --> 00:21:00,680 I mean, if there is a bug, it's one in a thousand, 357 00:21:00,680 --> 00:21:04,210 but for a programmer to be able to write those problems, 358 00:21:04,210 --> 00:21:08,715 he has to know calculus, he has to know C++ or Java, 359 00:21:08,715 --> 00:21:12,520 he has to be good-- that's not a problem, right? 360 00:21:12,520 --> 00:21:13,440 STUDENT: No. 361 00:21:13,440 --> 00:21:14,450 That's fine. 362 00:21:14,450 --> 00:21:20,360 MAGDALENA TODA: He or she has to know how to write a problem, 363 00:21:20,360 --> 00:21:22,600 so that you guys, no matter how you 364 00:21:22,600 --> 00:21:29,130 input your answer, as long as it is correct, you'll get the OK. 365 00:21:29,130 --> 00:21:32,990 Because you can put answers in many equivalent forms 366 00:21:32,990 --> 00:21:36,127 and all of them have to be-- 367 00:21:36,127 --> 00:21:37,210 STUDENT: The right answer. 368 00:21:37,210 --> 00:21:38,043 MAGDALENA TODA: Yes. 369 00:21:38,043 --> 00:21:40,780 To get the right answer. 370 00:21:40,780 --> 00:21:43,950 So since I have new people who just came-- 371 00:21:43,950 --> 00:21:47,023 And I understand you guys come from different buildings 372 00:21:47,023 --> 00:21:52,435 and I'm not mad for people who are coming late because I know 373 00:21:52,435 --> 00:21:55,280 you come from other classes, I wanted 374 00:21:55,280 --> 00:22:03,400 to say we started from a melting ice sphere example in Calc 1 375 00:22:03,400 --> 00:22:07,260 that was on many finals in here, at Texas Tech. 376 00:22:07,260 --> 00:22:14,470 And I want you to compose your own problem based on that. 377 00:22:14,470 --> 00:22:17,490 This time, involving a cylinder made 378 00:22:17,490 --> 00:22:23,650 of ice whose dimensions are doing something special. 379 00:22:23,650 --> 00:22:26,130 That shouldn't be hard. 380 00:22:26,130 --> 00:22:29,500 I'm going to erase this part because it's not 381 00:22:29,500 --> 00:22:30,600 the relevant one. 382 00:22:30,600 --> 00:22:32,840 I'm going to keep this one a little bit more 383 00:22:32,840 --> 00:22:35,960 for people who want to take notes. 384 00:22:35,960 --> 00:22:37,271 And I'm going to move on. 385 00:22:37,271 --> 00:22:42,460 386 00:22:42,460 --> 00:22:47,000 Another example we give you in the book 387 00:22:47,000 --> 00:22:54,700 is that one where x and y, the variables the function f, 388 00:22:54,700 --> 00:22:58,820 are not just functions of time, t. 389 00:22:58,820 --> 00:23:03,570 They, themselves, are functions of other two variables. 390 00:23:03,570 --> 00:23:07,990 Is that a lot more different from what I gave you already? 391 00:23:07,990 --> 00:23:08,490 No. 392 00:23:08,490 --> 00:23:10,730 The idea is the same. 393 00:23:10,730 --> 00:23:13,930 And you are imaginative. 394 00:23:13,930 --> 00:23:20,500 You are able to come up with your own answers. 395 00:23:20,500 --> 00:23:26,950 I'm going to ask you to think about what I'll have to write. 396 00:23:26,950 --> 00:23:28,330 This is finished. 397 00:23:28,330 --> 00:23:32,250 398 00:23:32,250 --> 00:23:37,796 So assume that you have function z equals F of x,y. 399 00:23:37,796 --> 00:23:42,470 400 00:23:42,470 --> 00:23:48,840 As we had it before, this is example 2 401 00:23:48,840 --> 00:23:58,330 where x is a function of u and v itself. 402 00:23:58,330 --> 00:24:02,510 And y is a function of u and v itself. 403 00:24:02,510 --> 00:24:07,201 And we assume that all the partial derivatives 404 00:24:07,201 --> 00:24:09,536 are defined and continuous. 405 00:24:09,536 --> 00:24:12,030 And we make the problem really nice. 406 00:24:12,030 --> 00:24:21,490 And now we'll come up with some example 407 00:24:21,490 --> 00:24:38,630 you know from before where x equals x of uv equals uv. 408 00:24:38,630 --> 00:24:48,760 And y equals y of uv equals u plus v. 409 00:24:48,760 --> 00:24:51,860 So these functions are the sum and the product 410 00:24:51,860 --> 00:24:53,195 of other variables. 411 00:24:53,195 --> 00:24:56,130 412 00:24:56,130 --> 00:25:06,780 Can you tell me how I am going to compute the derivative of 0, 413 00:25:06,780 --> 00:25:17,320 or of f, with the script of u at x of uv, y of uv? 414 00:25:17,320 --> 00:25:18,756 Is this hard? 415 00:25:18,756 --> 00:25:19,502 STUDENT: It is. 416 00:25:19,502 --> 00:25:20,710 MAGDALENA TODA: I don't know. 417 00:25:20,710 --> 00:25:27,512 You have to help me because-- why don't I put d here? 418 00:25:27,512 --> 00:25:29,000 STUDENT: Because [INAUDIBLE]. 419 00:25:29,000 --> 00:25:30,630 MAGDALENA TODA: Because you have 2. 420 00:25:30,630 --> 00:25:32,592 So the composition in itself will 421 00:25:32,592 --> 00:25:35,510 be a function of two variables. 422 00:25:35,510 --> 00:25:38,610 So of course, I have [INAUDIBLE]. 423 00:25:38,610 --> 00:25:48,270 I'm going to go ahead and do it as you say without rushing. 424 00:25:48,270 --> 00:25:51,360 Of course, I know you are watching. 425 00:25:51,360 --> 00:25:53,250 What will happen? 426 00:25:53,250 --> 00:25:54,159 STUDENT: 2x and 2y. 427 00:25:54,159 --> 00:25:55,450 MAGDALENA TODA: No, in general. 428 00:25:55,450 --> 00:25:58,470 Over here, I know you want to do it right away, 429 00:25:58,470 --> 00:26:01,790 but I would like you to give me a general formula mimicking 430 00:26:01,790 --> 00:26:06,530 the same thing you had before when you had one parameter, t. 431 00:26:06,530 --> 00:26:08,120 Now you have u and d separately. 432 00:26:08,120 --> 00:26:10,376 You want it to do it straight. 433 00:26:10,376 --> 00:26:19,088 So we have df, dx at x of uv, y of uv. 434 00:26:19,088 --> 00:26:20,540 Shut up, Magdalene. 435 00:26:20,540 --> 00:26:24,450 Let people talk and help you because you're tired. 436 00:26:24,450 --> 00:26:26,880 It's a Thursday. 437 00:26:26,880 --> 00:26:28,366 df, dx. 438 00:26:28,366 --> 00:26:29,259 STUDENT: [INAUDIBLE]. 439 00:26:29,259 --> 00:26:30,050 MAGDALENA TODA: dx. 440 00:26:30,050 --> 00:26:34,250 Again, [INAUDIBLE] notation, partial with respect 441 00:26:34,250 --> 00:26:42,890 to u, plus df, dy. 442 00:26:42,890 --> 00:26:46,430 So the second argument-- so I prime in respect 443 00:26:46,430 --> 00:26:50,610 to the second argument, computing everything 444 00:26:50,610 --> 00:26:55,180 in the end, which means in terms of u and v times, 445 00:26:55,180 --> 00:27:00,350 again, the dy with respect to u. 446 00:27:00,350 --> 00:27:01,550 You are saying that. 447 00:27:01,550 --> 00:27:04,367 Now I'd like you to see the pattern. 448 00:27:04,367 --> 00:27:06,450 Of course, you see the pattern here, smart people, 449 00:27:06,450 --> 00:27:11,660 but I want to emphasize the cowboys. 450 00:27:11,660 --> 00:27:15,021 And green for the other cowboy. 451 00:27:15,021 --> 00:27:16,854 I'm trying to match the college beautifully. 452 00:27:16,854 --> 00:27:22,636 453 00:27:22,636 --> 00:27:26,116 And the independent variable, Mr. u. 454 00:27:26,116 --> 00:27:27,610 Not u, but Mr. u. 455 00:27:27,610 --> 00:27:28,606 Yes, ma'am? 456 00:27:28,606 --> 00:27:31,096 STUDENT: Is it the partial of dx, du? 457 00:27:31,096 --> 00:27:37,771 Or is it-- like you did the partial for the-- 458 00:27:37,771 --> 00:27:39,562 MAGDALENA TODA: So did I do anything wrong? 459 00:27:39,562 --> 00:27:41,554 I don't think I did anything wrong. 460 00:27:41,554 --> 00:27:44,400 STUDENT: So it is the partial for dx over du? 461 00:27:44,400 --> 00:27:48,663 MAGDALENA TODA: So I go du with respect to the first variable, 462 00:27:48,663 --> 00:27:50,865 times that variable with respect to u. 463 00:27:50,865 --> 00:27:52,156 STUDENT: But is it the partial? 464 00:27:52,156 --> 00:27:53,660 That's my question. 465 00:27:53,660 --> 00:27:57,180 MAGDALENA TODA: But it has to be a partial because x is 466 00:27:57,180 --> 00:28:02,995 a function of u and v, so I cannot put d. 467 00:28:02,995 --> 00:28:07,146 And then the same plus the same idea as before. 468 00:28:07,146 --> 00:28:10,110 df with respect to the second argument 469 00:28:10,110 --> 00:28:15,650 times that second argument with respect to the u. 470 00:28:15,650 --> 00:28:20,420 You see, Mr. u is replacing Mr. t. 471 00:28:20,420 --> 00:28:21,485 He is independent. 472 00:28:21,485 --> 00:28:24,170 473 00:28:24,170 --> 00:28:27,310 He's the guy who is moving. 474 00:28:27,310 --> 00:28:29,010 We don't care about anybody else, 475 00:28:29,010 --> 00:28:32,030 but he replaces the time in this kind of problem. 476 00:28:32,030 --> 00:28:35,500 477 00:28:35,500 --> 00:28:38,590 Now the other one. 478 00:28:38,590 --> 00:28:41,064 I will let you speak. 479 00:28:41,064 --> 00:28:43,926 Df, dv. 480 00:28:43,926 --> 00:28:50,136 The same idea, but somebody else is going to talk. 481 00:28:50,136 --> 00:28:52,580 STUDENT: It would be del f, del y. 482 00:28:52,580 --> 00:28:54,450 MAGDALENA TODA: Del f, del x? 483 00:28:54,450 --> 00:28:56,890 Well, let's try to start in order. 484 00:28:56,890 --> 00:29:01,940 485 00:29:01,940 --> 00:29:05,200 And I tried to be organized and write neatly 486 00:29:05,200 --> 00:29:12,700 because I looked at-- so these videos are new and in progress. 487 00:29:12,700 --> 00:29:16,670 And I'm trying to see what I did well and I didn't. 488 00:29:16,670 --> 00:29:18,490 And at times, I wrote neatly. 489 00:29:18,490 --> 00:29:20,785 At times, I wrote not so neatly. 490 00:29:20,785 --> 00:29:23,260 I'm just learning about myself. 491 00:29:23,260 --> 00:29:28,070 It's one thing, what you think about yourself from the inside 492 00:29:28,070 --> 00:29:30,610 and to you see yourself the way other people 493 00:29:30,610 --> 00:29:33,350 see from the outside. 494 00:29:33,350 --> 00:29:34,420 It's not fun. 495 00:29:34,420 --> 00:29:35,670 STUDENT: Can you say it again? 496 00:29:35,670 --> 00:29:41,900 MAGDALENA TODA: This is v. So I'll say it again. 497 00:29:41,900 --> 00:29:46,370 We all have a certain impression about ourselves, 498 00:29:46,370 --> 00:29:49,100 but when you see a movie of yourself, 499 00:29:49,100 --> 00:29:51,740 you see the way other people see you. 500 00:29:51,740 --> 00:29:53,069 And it's not fun. 501 00:29:53,069 --> 00:29:56,010 STUDENT: So what-- 502 00:29:56,010 --> 00:29:58,750 MAGDALENA TODA: So let's see the cowboys. 503 00:29:58,750 --> 00:30:06,360 Ryan is looking at the [? man. ?] He is all [? man. ?] 504 00:30:06,360 --> 00:30:10,480 And y is here, right? 505 00:30:10,480 --> 00:30:15,830 And who is the time variable, kind of, this time? 506 00:30:15,830 --> 00:30:18,205 This time, which one is the time? 507 00:30:18,205 --> 00:30:28,456 v. And v is the only ultimate variable that we care about. 508 00:30:28,456 --> 00:30:31,858 So everything you did before with respect to t, 509 00:30:31,858 --> 00:30:35,307 you do now with respect to u, you 510 00:30:35,307 --> 00:30:37,130 do now with respect to v. It shouldn't 511 00:30:37,130 --> 00:30:38,970 be hard to understand. 512 00:30:38,970 --> 00:30:42,040 I want to work the example, of course. 513 00:30:42,040 --> 00:30:44,440 With your help, I will work it. 514 00:30:44,440 --> 00:30:49,270 Now remember how my students cheated on this one? 515 00:30:49,270 --> 00:30:57,090 So I told my colleague, he did not say, five or six years ago, 516 00:30:57,090 --> 00:31:00,810 by first writing The Chain Rule for functions of two variables, 517 00:31:00,810 --> 00:31:08,430 express all the df, du, df, dv, but he said by any method. 518 00:31:08,430 --> 00:31:12,970 Of course, what they did-- they were sneaky. 519 00:31:12,970 --> 00:31:15,962 They took something like x equals uv 520 00:31:15,962 --> 00:31:17,926 and they plugged it in here. 521 00:31:17,926 --> 00:31:19,872 They took the function [? u and v, ?] 522 00:31:19,872 --> 00:31:20,872 they plugged it in here. 523 00:31:20,872 --> 00:31:22,836 They computed everything in terms of u and v 524 00:31:22,836 --> 00:31:24,309 and took the partials. 525 00:31:24,309 --> 00:31:26,756 STUDENT: Why don't you [INAUDIBLE]? 526 00:31:26,756 --> 00:31:29,130 MAGDALENA TODA: It depends how the problem is formulated. 527 00:31:29,130 --> 00:31:30,455 STUDENT: So if you make it [INAUDIBLE], 528 00:31:30,455 --> 00:31:31,626 then it's [INAUDIBLE]. 529 00:31:31,626 --> 00:31:35,716 530 00:31:35,716 --> 00:31:39,360 MAGDALENA TODA: So when they give you the precise functions, 531 00:31:39,360 --> 00:31:39,919 you're right. 532 00:31:39,919 --> 00:31:41,710 But if they don't give you those functions, 533 00:31:41,710 --> 00:31:44,320 if they keep them a secret, then you still 534 00:31:44,320 --> 00:31:47,230 have to write the general formula. 535 00:31:47,230 --> 00:31:51,450 If they don't give you the functions, all of them 536 00:31:51,450 --> 00:31:54,060 explicitly. 537 00:31:54,060 --> 00:31:57,296 So let's see what to do in this case. 538 00:31:57,296 --> 00:32:09,300 df, du at x of u, vy of uv will be what? 539 00:32:09,300 --> 00:32:11,520 Now people, help me, please. 540 00:32:11,520 --> 00:32:15,710 541 00:32:15,710 --> 00:32:21,980 I want to teach you how engineers and physicists very, 542 00:32:21,980 --> 00:32:26,610 very often express those at x and y. 543 00:32:26,610 --> 00:32:29,080 And many of you know because we talked 544 00:32:29,080 --> 00:32:31,280 about that in office hours. 545 00:32:31,280 --> 00:32:36,661 2x, I might write, but evaluated at-- 546 00:32:36,661 --> 00:32:38,160 and this is a very frequent notation 547 00:32:38,160 --> 00:32:42,030 image in the engineering and physicist world. 548 00:32:42,030 --> 00:32:45,170 So 2x evaluated at where? 549 00:32:45,170 --> 00:32:51,530 At the point where x is uv and y is u plus v. 550 00:32:51,530 --> 00:32:59,670 So I say x of uv, y of uv. 551 00:32:59,670 --> 00:33:05,040 And I'll replace later because I'm not in a hurry. 552 00:33:05,040 --> 00:33:06,970 dx, du. 553 00:33:06,970 --> 00:33:09,300 Who is dx, du? 554 00:33:09,300 --> 00:33:11,403 The derivative of x or with respect to u? 555 00:33:11,403 --> 00:33:12,690 Are you guys awake? 556 00:33:12,690 --> 00:33:13,370 STUDENT: Yes. 557 00:33:13,370 --> 00:33:14,730 So it's v. 558 00:33:14,730 --> 00:33:19,470 MAGDALENA TODA: v. Very good. v plus-- the next term, who's 559 00:33:19,470 --> 00:33:22,610 going to tell me what we have? 560 00:33:22,610 --> 00:33:24,070 STUDENT: 2y evaluated at-- 561 00:33:24,070 --> 00:33:28,340 MAGDALENA TODA: 2y evaluated at-- look how lazy I am. 562 00:33:28,340 --> 00:33:37,540 Times the derivative of y with respect to u. 563 00:33:37,540 --> 00:33:39,640 So you were right because of 2y. 564 00:33:39,640 --> 00:33:42,740 565 00:33:42,740 --> 00:33:44,200 Attention, right? 566 00:33:44,200 --> 00:33:48,140 So it's dy, du is 1. 567 00:33:48,140 --> 00:33:49,910 It's very easy to make a mistake. 568 00:33:49,910 --> 00:33:52,430 I've had mistakes who made mistakes in the final 569 00:33:52,430 --> 00:33:55,740 from just miscalculating because when 570 00:33:55,740 --> 00:33:57,700 you are close to some formula, you 571 00:33:57,700 --> 00:34:00,040 don't see the whole picture. 572 00:34:00,040 --> 00:34:01,038 What do you do? 573 00:34:01,038 --> 00:34:05,030 At the end of your exams, go back and rather 574 00:34:05,030 --> 00:34:08,620 than quickly turning in a paper, never do that, 575 00:34:08,620 --> 00:34:11,510 go back and check all your problems. 576 00:34:11,510 --> 00:34:13,489 It's a good habit. 577 00:34:13,489 --> 00:34:20,880 2 times x, which is uv, I plug it as a function of u and v, 578 00:34:20,880 --> 00:34:23,350 right? 579 00:34:23,350 --> 00:34:27,370 Times a v plus-- who is 2y? 580 00:34:27,370 --> 00:34:28,870 That's the last of the Mohicans. 581 00:34:28,870 --> 00:34:30,310 One is out. 582 00:34:30,310 --> 00:34:31,100 STUDENT: 2. 583 00:34:31,100 --> 00:34:37,130 MAGDALENA TODA: 2y 2 times replace y in terms of u and v. 584 00:34:37,130 --> 00:34:38,050 And you're done. 585 00:34:38,050 --> 00:34:40,489 So do you like it? 586 00:34:40,489 --> 00:34:42,020 I don't. 587 00:34:42,020 --> 00:34:44,440 And how would you write it? 588 00:34:44,440 --> 00:34:48,510 Not much better than that, but at least let's try. 589 00:34:48,510 --> 00:34:53,380 2uv squared plus 2u plus 2v. 590 00:34:53,380 --> 00:34:55,004 You can do a little bit more than that, 591 00:34:55,004 --> 00:35:01,130 but if you want to list it in the order of the degrees 592 00:35:01,130 --> 00:35:04,190 of the polynomials, that's OK. 593 00:35:04,190 --> 00:35:06,220 Now next one. 594 00:35:06,220 --> 00:35:10,015 df, dv, x of uv, y of uv. 595 00:35:10,015 --> 00:35:12,790 596 00:35:12,790 --> 00:35:15,861 Such examples are in the book. 597 00:35:15,861 --> 00:35:17,860 Many things are in the book and out of the book. 598 00:35:17,860 --> 00:35:21,400 I mean, on the white board. 599 00:35:21,400 --> 00:35:25,180 I don't know why it gives you so many combinations of this type, 600 00:35:25,180 --> 00:35:31,130 u plus v, u minus-- 2u plus 2v, 2u you minus 2v. 601 00:35:31,130 --> 00:35:31,960 Well, I know why. 602 00:35:31,960 --> 00:35:35,210 Because that's a rotation and rescaling. 603 00:35:35,210 --> 00:35:37,150 So there is a reason behind that, 604 00:35:37,150 --> 00:35:42,110 but I thought of something different for df, dv. 605 00:35:42,110 --> 00:35:44,800 Now what do I do? 606 00:35:44,800 --> 00:35:45,355 df, dx. 607 00:35:45,355 --> 00:35:46,022 STUDENT: You [? have to find something symmetrical to that. 608 00:35:46,022 --> 00:35:46,860 ?] 609 00:35:46,860 --> 00:35:48,443 MAGDALENA TODA: Again, the same thing. 610 00:35:48,443 --> 00:35:52,781 2x evaluated at whoever times-- 611 00:35:52,781 --> 00:35:53,280 STUDENT: u. 612 00:35:53,280 --> 00:35:55,831 613 00:35:55,831 --> 00:35:58,080 MAGDALENA TODA: Because you have dx with respect to v, 614 00:35:58,080 --> 00:36:02,090 so you have u plus-- 615 00:36:02,090 --> 00:36:03,770 STUDENT: df, dy. 616 00:36:03,770 --> 00:36:07,060 MAGDALENA TODA: df, dy, which is 2y, evaluated 617 00:36:07,060 --> 00:36:10,002 at the same kind of guy. 618 00:36:10,002 --> 00:36:13,360 So all you have to do is replace with respect to u and v. 619 00:36:13,360 --> 00:36:16,240 And finally, multiplied by- 620 00:36:16,240 --> 00:36:16,870 STUDENT: dy. 621 00:36:16,870 --> 00:36:18,180 MAGDALENA TODA: dy, dv. 622 00:36:18,180 --> 00:36:21,650 dy, dv is 1 again. 623 00:36:21,650 --> 00:36:23,723 Just pay attention when you plug in 624 00:36:23,723 --> 00:36:25,722 because you realize you can know these very well 625 00:36:25,722 --> 00:36:28,780 and understand it as a process, but if you make an algebra 626 00:36:28,780 --> 00:36:31,134 and everything is out. 627 00:36:31,134 --> 00:36:32,800 And then you send me an email that says, 628 00:36:32,800 --> 00:36:34,960 I've tried this problem 15 times. 629 00:36:34,960 --> 00:36:37,820 And I don't even hold you responsible for that 630 00:36:37,820 --> 00:36:41,770 because I can make algebra mistakes anytime. 631 00:36:41,770 --> 00:36:54,400 So 2uv times u plus 2 times u plus v. So what did I do here? 632 00:36:54,400 --> 00:37:00,906 I simply replaced the given functions in terms of u and v. 633 00:37:00,906 --> 00:37:03,200 And I'm done. 634 00:37:03,200 --> 00:37:03,900 Do I like it? 635 00:37:03,900 --> 00:37:08,532 No, but I'd like you to notice something as soon as I'm done. 636 00:37:08,532 --> 00:37:11,899 2u squared v plus 2u plus 2v. 637 00:37:11,899 --> 00:37:17,610 638 00:37:17,610 --> 00:37:20,040 Could I have expected that? 639 00:37:20,040 --> 00:37:21,540 Look at the beauty of the functions. 640 00:37:21,540 --> 00:37:24,490 641 00:37:24,490 --> 00:37:27,550 Z is a symmetric function. 642 00:37:27,550 --> 00:37:31,515 x and y have some of the symmetry as well. 643 00:37:31,515 --> 00:37:34,550 If you swap u and v, these are symmetric polynomials 644 00:37:34,550 --> 00:37:38,615 of order 2 and 1. 645 00:37:38,615 --> 00:37:40,070 [INAUDIBLE] 646 00:37:40,070 --> 00:37:42,620 Swap the variables, you still get the same thing. 647 00:37:42,620 --> 00:37:45,615 Swap the variables u and v, you get the same thing. 648 00:37:45,615 --> 00:37:48,380 So how could I have imagined that I'm 649 00:37:48,380 --> 00:37:54,470 going to get-- if I were smart, without doing all the work, 650 00:37:54,470 --> 00:37:57,920 I could figure out this by just swapping 651 00:37:57,920 --> 00:38:01,760 the u and v, the rows of u and v. I would have said, 652 00:38:01,760 --> 00:38:07,980 2vu squared, dv plus 2u and it's the same thing I got here. 653 00:38:07,980 --> 00:38:13,520 But not always are you so lucky to be given nice data. 654 00:38:13,520 --> 00:38:15,690 Well, in real life, it's a mess. 655 00:38:15,690 --> 00:38:21,940 If you are, let's say, working with geophysics real data, 656 00:38:21,940 --> 00:38:27,560 you two parameters and for each parameter, x and y, 657 00:38:27,560 --> 00:38:28,810 you have other parameters. 658 00:38:28,810 --> 00:38:31,565 You will never have anything that nice. 659 00:38:31,565 --> 00:38:35,940 You may have nasty truncations of polynomials 660 00:38:35,940 --> 00:38:39,057 with many, many terms that you work 661 00:38:39,057 --> 00:38:41,390 with approximating polynomials all the time. [INAUDIBLE] 662 00:38:41,390 --> 00:38:43,350 or something like that. 663 00:38:43,350 --> 00:38:46,870 So don't expect these miracles to happen with real data, 664 00:38:46,870 --> 00:38:49,740 but the process is the same. 665 00:38:49,740 --> 00:38:52,960 And, of course, there are programs 666 00:38:52,960 --> 00:38:56,410 that incorporate all of the Calculus 3 notions 667 00:38:56,410 --> 00:38:59,380 that we went over. 668 00:38:59,380 --> 00:39:03,670 There were people who already wrote 669 00:39:03,670 --> 00:39:08,310 lots of programs that enable you to compute derivatives 670 00:39:08,310 --> 00:39:11,105 of function of several variables. 671 00:39:11,105 --> 00:39:23,690 672 00:39:23,690 --> 00:39:27,490 Now let me take your temperature again. 673 00:39:27,490 --> 00:39:29,520 Is this hard? 674 00:39:29,520 --> 00:39:30,420 No. 675 00:39:30,420 --> 00:39:33,770 It's sort of logical you just have to pay attention to what? 676 00:39:33,770 --> 00:39:36,450 677 00:39:36,450 --> 00:39:41,523 Pay attention to not making too many algebra mistakes, right? 678 00:39:41,523 --> 00:39:42,522 That's kind of the idea. 679 00:39:42,522 --> 00:39:45,330 680 00:39:45,330 --> 00:39:48,701 More things that I wanted to-- there 681 00:39:48,701 --> 00:39:50,950 are many more things I wanted to share with you today, 682 00:39:50,950 --> 00:39:56,230 but I'm glad we reached some consensus in the sense 683 00:39:56,230 --> 00:40:01,718 that you feel there is logic and order in this type of problem. 684 00:40:01,718 --> 00:40:21,900 685 00:40:21,900 --> 00:40:35,170 I tried to give you a little bit of an introduction to why 686 00:40:35,170 --> 00:40:39,645 the gradient is so important last time. 687 00:40:39,645 --> 00:40:41,220 And I'm going to come back to that 688 00:40:41,220 --> 00:40:45,239 again, so I'm not going to leave you in the air. 689 00:40:45,239 --> 00:40:49,071 But before then, I would like to do 690 00:40:49,071 --> 00:40:50,687 the directional derivative, which 691 00:40:50,687 --> 00:40:52,914 is a very important section. 692 00:40:52,914 --> 00:40:57,378 So I'm going to start again. 693 00:40:57,378 --> 00:41:07,890 And I'll also do, at the same time, some review of 11.5. 694 00:41:07,890 --> 00:41:10,060 So I will combine them. 695 00:41:10,060 --> 00:41:11,844 And I want to introduce the notion 696 00:41:11,844 --> 00:41:14,528 of directional derivatives because it's 697 00:41:14,528 --> 00:41:15,992 right there for us to grab it. 698 00:41:15,992 --> 00:41:26,240 699 00:41:26,240 --> 00:41:28,850 And you say, well, that sounds familiar. 700 00:41:28,850 --> 00:41:32,290 It sounds like I dealt with direction before, 701 00:41:32,290 --> 00:41:35,780 but I didn't what that was. 702 00:41:35,780 --> 00:41:37,500 That's exactly true. 703 00:41:37,500 --> 00:41:41,360 You dealt with it before, you just didn't know what it was. 704 00:41:41,360 --> 00:41:44,060 And I'll give you the general definition, 705 00:41:44,060 --> 00:41:49,120 but then I would like you to think about if you have ever 706 00:41:49,120 --> 00:41:50,260 seen that before. 707 00:41:50,260 --> 00:41:53,040 708 00:41:53,040 --> 00:41:59,420 I'm going to say I have the derivative of a function, f, 709 00:41:59,420 --> 00:42:01,440 in the direction, u. 710 00:42:01,440 --> 00:42:04,290 And I'm going put u bar as if you were free, 711 00:42:04,290 --> 00:42:05,480 not a married man. 712 00:42:05,480 --> 00:42:09,180 But u as a direction as always a unit vector. 713 00:42:09,180 --> 00:42:10,380 STUDENT: [INAUDIBLE]. 714 00:42:10,380 --> 00:42:11,921 MAGDALENA TODA: I told you last time, 715 00:42:11,921 --> 00:42:18,188 just to prepare you, direction, u, is always a unit vector. 716 00:42:18,188 --> 00:42:23,544 717 00:42:23,544 --> 00:42:24,044 Always. 718 00:42:24,044 --> 00:42:28,450 719 00:42:28,450 --> 00:42:29,390 Computed at x0y0. 720 00:42:29,390 --> 00:42:35,556 But x0y0 is a given view point. 721 00:42:35,556 --> 00:42:40,860 722 00:42:40,860 --> 00:42:46,490 And I'm going to say what that's going to be. 723 00:42:46,490 --> 00:42:49,100 I have a limit. 724 00:42:49,100 --> 00:42:51,300 I'm going to use the h. 725 00:42:51,300 --> 00:42:53,760 And you say, why in the world is she using h? 726 00:42:53,760 --> 00:42:57,426 You will see in a second-- h goes to 0-- because we 727 00:42:57,426 --> 00:42:59,130 haven't used h in awhile. 728 00:42:59,130 --> 00:43:02,480 h is like a small displacement that shrinks to 0. 729 00:43:02,480 --> 00:43:07,480 730 00:43:07,480 --> 00:43:25,818 And I put here, f of x0 plus hu1, y0 plus hu2, 731 00:43:25,818 --> 00:43:32,300 close, minus f of x0y0. 732 00:43:32,300 --> 00:43:34,912 So you say, wait a minute, Magdalena, oh my god, I've 733 00:43:34,912 --> 00:43:37,986 got a headache. 734 00:43:37,986 --> 00:43:38,970 I'm not here. 735 00:43:38,970 --> 00:43:42,530 Z0 is easy to understand for everybody, right? 736 00:43:42,530 --> 00:43:47,040 That's going to be altitude at the point x0y0. 737 00:43:47,040 --> 00:43:48,620 It shouldn't be hard. 738 00:43:48,620 --> 00:43:51,560 739 00:43:51,560 --> 00:43:54,530 On the other hand, what am I doing? 740 00:43:54,530 --> 00:44:00,090 I have to look at a real graph, in the real world. 741 00:44:00,090 --> 00:44:04,860 And that's going to be a patch of a smooth surface. 742 00:44:04,860 --> 00:44:08,640 And I say, OK, this is my favorite point. 743 00:44:08,640 --> 00:44:12,330 I have x0y0 on the ground. 744 00:44:12,330 --> 00:44:16,030 And the corresponding point in three dimensions, 745 00:44:16,030 --> 00:44:21,990 would be x0y0 and z0, which is the f of x0y0. 746 00:44:21,990 --> 00:44:24,210 And you say, wait a minute, what do you mean 747 00:44:24,210 --> 00:44:25,860 I can't move in a direction? 748 00:44:25,860 --> 00:44:33,430 Is it like when took a sleigh and we went 749 00:44:33,430 --> 00:44:35,680 to have fun on the hill? 750 00:44:35,680 --> 00:44:38,430 Yes, but I said that would be the last time 751 00:44:38,430 --> 00:44:42,620 we talked about the hilly area with snow on it. 752 00:44:42,620 --> 00:44:47,545 It was a good preparation for today in the sense that-- 753 00:44:47,545 --> 00:44:51,522 Remember, we went somewhere when I picked your direction north, 754 00:44:51,522 --> 00:44:52,830 east? 755 00:44:52,830 --> 00:44:54,590 i plus j? 756 00:44:54,590 --> 00:44:57,150 And in the direction of i plus j, 757 00:44:57,150 --> 00:44:59,104 which is not quite the direction and I'll 758 00:44:59,104 --> 00:45:04,530 ask you why in a second, I was going down along a meridian. 759 00:45:04,530 --> 00:45:05,570 Remember last time? 760 00:45:05,570 --> 00:45:11,820 And then that was the direction of the steepest descent. 761 00:45:11,820 --> 00:45:12,830 I was sliding down. 762 00:45:12,830 --> 00:45:16,650 If I wanted the direction of the steepest ascent, 763 00:45:16,650 --> 00:45:19,680 that would have been minus i minus j. 764 00:45:19,680 --> 00:45:23,460 So I had plus i plus j, minus i minus j. 765 00:45:23,460 --> 00:45:26,087 And I told you last time, why are those not quite directions? 766 00:45:26,087 --> 00:45:27,670 STUDENT: Because they are not unitary. 767 00:45:27,670 --> 00:45:29,211 MAGDALENA TODA: They are not unitary. 768 00:45:29,211 --> 00:45:31,840 So to make them like this u, I should 769 00:45:31,840 --> 00:45:34,270 have said, in the direction i plus 770 00:45:34,270 --> 00:45:37,140 j, that was one minus x squared minus y 771 00:45:37,140 --> 00:45:41,430 squared, the parabola way, that was the hill full of snow. 772 00:45:41,430 --> 00:45:45,220 So in the direction i plus j, I go down 773 00:45:45,220 --> 00:45:47,140 the fastest possible way. 774 00:45:47,140 --> 00:45:50,760 In the direction i plus j over square root of 2, 775 00:45:50,760 --> 00:45:53,580 I would be fine with a unit vector. 776 00:45:53,580 --> 00:45:57,660 In the opposite direction, I go up the fastest way possible, 777 00:45:57,660 --> 00:46:01,600 but you don't want to because it's-- can you imagine hiking 778 00:46:01,600 --> 00:46:09,021 the steepest possible direction in the steepest way? 779 00:46:09,021 --> 00:46:14,750 780 00:46:14,750 --> 00:46:16,360 Now with my direction. 781 00:46:16,360 --> 00:46:22,340 My direction in plane should be the i vector. 782 00:46:22,340 --> 00:46:25,600 And that magic vector should have length 1 from here 783 00:46:25,600 --> 00:46:26,100 to here. 784 00:46:26,100 --> 00:46:29,500 And when you measure this guy, he has to have length 1. 785 00:46:29,500 --> 00:46:35,466 And if you decompose, you have to decompose him along the-- 786 00:46:35,466 --> 00:46:36,940 what is this? 787 00:46:36,940 --> 00:46:40,090 The x direction and the y direction, right? 788 00:46:40,090 --> 00:46:47,450 How do you split a vector in such a decomposition? 789 00:46:47,450 --> 00:46:54,110 Well, Mr. u will be u1i plus 1i. 790 00:46:54,110 --> 00:46:55,710 It sounds funny. 791 00:46:55,710 --> 00:46:58,550 Plus u2j. 792 00:46:58,550 --> 00:47:02,220 So you have u1 from here to here. 793 00:47:02,220 --> 00:47:04,070 I don't well you can draw. 794 00:47:04,070 --> 00:47:06,790 I think some of you can draw really well, especially 795 00:47:06,790 --> 00:47:10,680 better than me because you took technical drawing. 796 00:47:10,680 --> 00:47:13,952 How many of you took technical drawing in this glass? 797 00:47:13,952 --> 00:47:15,860 STUDENT: Only in this class? 798 00:47:15,860 --> 00:47:17,270 MAGDALENA TODA: In anything. 799 00:47:17,270 --> 00:47:17,630 STUDENT: In high school. 800 00:47:17,630 --> 00:47:19,254 MAGDALENA TODA: High school or college. 801 00:47:19,254 --> 00:47:21,700 STUDENT: I went to it in middle school. 802 00:47:21,700 --> 00:47:23,660 So it gives you so that [INAUDIBLE] 803 00:47:23,660 --> 00:47:24,740 and you'd have to draw it. [INAUDIBLE]. 804 00:47:24,740 --> 00:47:26,364 MAGDALENA TODA: It's really helping you 805 00:47:26,364 --> 00:47:31,090 with the perspective view, 3D view, from an angle. 806 00:47:31,090 --> 00:47:33,490 So now you're looking at this u direction 807 00:47:33,490 --> 00:47:35,930 as being u1i plus u2j. 808 00:47:35,930 --> 00:47:39,810 And you say, OK, I think I know what's going on. 809 00:47:39,810 --> 00:47:47,085 You have a displacement in the direction of the x 810 00:47:47,085 --> 00:47:51,740 coordinate by 1 times h. 811 00:47:51,740 --> 00:47:54,570 So it's a small displacement that you're talking about. 812 00:47:54,570 --> 00:47:56,890 And-- yes? 813 00:47:56,890 --> 00:47:58,326 STUDENT: Why 1 [INAUDIBLE]? 814 00:47:58,326 --> 00:48:01,917 815 00:48:01,917 --> 00:48:03,000 MAGDALENA TODA: Which one? 816 00:48:03,000 --> 00:48:04,319 STUDENT: You said 1 times H. 817 00:48:04,319 --> 00:48:05,110 MAGDALENA TODA: u1. 818 00:48:05,110 --> 00:48:08,719 819 00:48:08,719 --> 00:48:09,760 You will see in a second. 820 00:48:09,760 --> 00:48:12,416 That's the way you define it. 821 00:48:12,416 --> 00:48:15,080 This is adjusted information. 822 00:48:15,080 --> 00:48:18,940 I would like you to tell me what the whole animal is, if I 823 00:48:18,940 --> 00:48:21,240 want to represent it later. 824 00:48:21,240 --> 00:48:23,980 And if you can give me some examples. 825 00:48:23,980 --> 00:48:29,450 And if I go in a y direction with a small displacement, 826 00:48:29,450 --> 00:48:33,600 from y0, I have to leave and go. 827 00:48:33,600 --> 00:48:38,030 So I am here at x0y0. 828 00:48:38,030 --> 00:48:41,960 And this is the x direction and this is the y direction. 829 00:48:41,960 --> 00:48:47,660 And when I displace a little bit, I displace with the green. 830 00:48:47,660 --> 00:48:49,890 I displace in this direction. 831 00:48:49,890 --> 00:48:54,455 I will have to displace and see what happens here. 832 00:48:54,455 --> 00:48:58,100 833 00:48:58,100 --> 00:49:02,182 And then in this direction-- I'm not going to write it yet. 834 00:49:02,182 --> 00:49:04,190 So I'm displacing in this direction 835 00:49:04,190 --> 00:49:06,530 and in that direction. 836 00:49:06,530 --> 00:49:08,490 Why am I keeping it h? 837 00:49:08,490 --> 00:49:12,940 Well, because I have the coordinates x0y0 plus-- 838 00:49:12,940 --> 00:49:21,125 how do you give me a collinear vector to u, but a small one? 839 00:49:21,125 --> 00:49:23,480 You say, wait a minute, I know what you mean. 840 00:49:23,480 --> 00:49:28,410 I start from the point x0, this is p, plus a small multiple 841 00:49:28,410 --> 00:49:31,990 of the direction you give me. 842 00:49:31,990 --> 00:49:34,550 So here, you had it before in Calc 2. 843 00:49:34,550 --> 00:49:40,640 You had t times uru2, which is my vector, u. 844 00:49:40,640 --> 00:49:55,112 So give me a very small displacement vector 845 00:49:55,112 --> 00:50:04,665 in the direction u, which is u1u2, u2 as a vector. 846 00:50:04,665 --> 00:50:06,162 You like angular graphics. 847 00:50:06,162 --> 00:50:07,660 I don't, but it doesn't matter. 848 00:50:07,660 --> 00:50:09,829 STUDENT: So basically, h. 849 00:50:09,829 --> 00:50:11,245 MAGDALENA TODA: So basically, this 850 00:50:11,245 --> 00:50:16,060 is x0 plus-- you want t or h? 851 00:50:16,060 --> 00:50:17,640 t or h, it doesn't matter. 852 00:50:17,640 --> 00:50:23,470 hu1, ui0 plus hu2. 853 00:50:23,470 --> 00:50:24,430 Why not t? 854 00:50:24,430 --> 00:50:27,310 Why did I take h? 855 00:50:27,310 --> 00:50:30,470 It is like time parameter that I'm doing with h, 856 00:50:30,470 --> 00:50:33,990 but h is a very small time parameter. 857 00:50:33,990 --> 00:50:36,190 It's an infinitesimally small time. 858 00:50:36,190 --> 00:50:41,050 It's just a fraction of a second after I start. 859 00:50:41,050 --> 00:50:43,800 That's why I use little h and not little t. 860 00:50:43,800 --> 00:50:46,500 861 00:50:46,500 --> 00:50:52,230 H, in general, indicates a very small time displacement. 862 00:50:52,230 --> 00:50:58,180 So tried to say, where am I here? 863 00:50:58,180 --> 00:51:02,480 I'm here, just one step further with a small displacement. 864 00:51:02,480 --> 00:51:06,150 And that's going to p at this whole thing. 865 00:51:06,150 --> 00:51:11,030 866 00:51:11,030 --> 00:51:17,322 Let's call this F of-- the blue one is F of x0y0. 867 00:51:17,322 --> 00:51:23,140 868 00:51:23,140 --> 00:51:27,940 And the green altitude, or the altitude of the green point, 869 00:51:27,940 --> 00:51:29,780 will be what? 870 00:51:29,780 --> 00:51:31,850 Well, this is something, something, 871 00:51:31,850 --> 00:51:44,002 and the altitude would be F of x0 plus hu1, y0 plus hu2. 872 00:51:44,002 --> 00:51:49,630 And I measure how far away the altitudes are. 873 00:51:49,630 --> 00:51:50,750 They are very close. 874 00:51:50,750 --> 00:51:53,380 The blue altitude and the green altitude 875 00:51:53,380 --> 00:51:55,185 varies the displacement. 876 00:51:55,185 --> 00:51:57,640 And how can I draw that? 877 00:51:57,640 --> 00:51:58,630 Here. 878 00:51:58,630 --> 00:52:00,134 You see this one? 879 00:52:00,134 --> 00:52:02,062 This is the delta z. 880 00:52:02,062 --> 00:52:05,920 So this thing is like a delta z kind of guy. 881 00:52:05,920 --> 00:52:06,504 Any questions? 882 00:52:06,504 --> 00:52:08,711 It's a little bit hard, but you will see in a second. 883 00:52:08,711 --> 00:52:09,330 Yes, sir? 884 00:52:09,330 --> 00:52:11,460 STUDENT: Is it like a small displacement 885 00:52:11,460 --> 00:52:17,290 that has to be perpendicular to the [INAUDIBLE]? 886 00:52:17,290 --> 00:52:18,202 MAGDALENA TODA: No. 887 00:52:18,202 --> 00:52:19,785 STUDENT: It's a result of [INAUDIBLE]? 888 00:52:19,785 --> 00:52:21,282 MAGDALENA TODA: It is in the direction. 889 00:52:21,282 --> 00:52:22,365 STUDENT: In the direction? 890 00:52:22,365 --> 00:52:24,290 MAGDALENA TODA: So let's model it better. 891 00:52:24,290 --> 00:52:27,170 I don't have a three dimensional-- they sent me 892 00:52:27,170 --> 00:52:29,310 an email this morning from the library saying, 893 00:52:29,310 --> 00:52:31,780 do you want your three dimensional print-- 894 00:52:31,780 --> 00:52:35,400 do you want to support the idea of Texas Tech having a three 895 00:52:35,400 --> 00:52:39,691 dimensional printer available for educational purposes? 896 00:52:39,691 --> 00:52:41,232 STUDENT: Did you say, of course, yes? 897 00:52:41,232 --> 00:52:43,235 MAGDALENA TODA: Of course, I would. 898 00:52:43,235 --> 00:52:45,110 But I don't have a three dimensional printer, 899 00:52:45,110 --> 00:52:47,590 but you have imagination and imagine 900 00:52:47,590 --> 00:52:50,450 we have a surface that, again, looks like a hill. 901 00:52:50,450 --> 00:52:52,710 That's my hand. 902 00:52:52,710 --> 00:52:58,354 And this engagement ring that I have is actually p0, 903 00:52:58,354 --> 00:52:59,020 which is x0y0zz. 904 00:52:59,020 --> 00:53:03,590 905 00:53:03,590 --> 00:53:09,246 And I'm going in a direction of somebody. 906 00:53:09,246 --> 00:53:10,246 It doesn't have to be u. 907 00:53:10,246 --> 00:53:12,240 No, [INAUDIBLE]. 908 00:53:12,240 --> 00:53:14,450 So I'm going in the direction of u-- yu2, 909 00:53:14,450 --> 00:53:18,000 is that horizontal thing. 910 00:53:18,000 --> 00:53:20,340 I'm going in that direction. 911 00:53:20,340 --> 00:53:22,570 So this is the direction I'm going in 912 00:53:22,570 --> 00:53:25,430 and I say, OK, where do I go? 913 00:53:25,430 --> 00:53:29,030 We'll do a small displacement, an infinitesimally small 914 00:53:29,030 --> 00:53:32,340 displacement in that direction here. 915 00:53:32,340 --> 00:53:37,970 So the two points are related to one another. 916 00:53:37,970 --> 00:53:42,480 And you say, but there's such a small difference in altitudes 917 00:53:42,480 --> 00:53:44,980 because you have an infinitesimally small 918 00:53:44,980 --> 00:53:47,080 displacement in that direction. 919 00:53:47,080 --> 00:53:47,580 Yes, I know. 920 00:53:47,580 --> 00:53:50,955 But when you make the ratio between that small delta 921 00:53:50,955 --> 00:53:57,670 z and the small h, the ratio could be 65 or 120 minus 32. 922 00:53:57,670 --> 00:53:59,420 You don't know what you get. 923 00:53:59,420 --> 00:54:04,290 So just like in general limit of the difference quotient 924 00:54:04,290 --> 00:54:10,260 being the derivative, you'll get the ratio between some things 925 00:54:10,260 --> 00:54:12,110 that are very small. 926 00:54:12,110 --> 00:54:15,050 But in the end, you can get something unexpected. 927 00:54:15,050 --> 00:54:16,360 Finite or anything. 928 00:54:16,360 --> 00:54:21,340 Now what do you think this guy-- according 929 00:54:21,340 --> 00:54:27,280 to your previous Chain Rule preparation. 930 00:54:27,280 --> 00:54:31,330 I taught you about Chain Rule. 931 00:54:31,330 --> 00:54:36,440 What will this be if we compute them? 932 00:54:36,440 --> 00:54:37,830 There is a proof for this. 933 00:54:37,830 --> 00:54:41,430 It would be like a page or a 2 page proof 934 00:54:41,430 --> 00:54:44,070 for what I'm claiming to have. 935 00:54:44,070 --> 00:54:45,870 Or how do you think I'm going to get 936 00:54:45,870 --> 00:54:49,750 to this without doing the limit of a difference quotient? 937 00:54:49,750 --> 00:54:51,540 Because if I give you functions and you 938 00:54:51,540 --> 00:54:53,206 do the limit of the difference quotients 939 00:54:53,206 --> 00:54:56,970 for some nasty functions, you'll never finish. 940 00:54:56,970 --> 00:55:01,820 So what do you think we ought to do? 941 00:55:01,820 --> 00:55:06,152 This is going to be some sort of derivative, right? 942 00:55:06,152 --> 00:55:09,144 And it's going to be a derivative of what? 943 00:55:09,144 --> 00:55:11,040 Yes, sir. 944 00:55:11,040 --> 00:55:14,440 STUDENT: Well, it's going to be like a partial derivative, 945 00:55:14,440 --> 00:55:20,006 except the plane you're using to cut the surface 946 00:55:20,006 --> 00:55:22,660 is not going to be in the x direction or the y direction. 947 00:55:22,660 --> 00:55:24,324 It's going to be along the [? uz. ?] 948 00:55:24,324 --> 00:55:25,240 MAGDALENA TODA: Right. 949 00:55:25,240 --> 00:55:28,190 So that is a very good observation. 950 00:55:28,190 --> 00:55:31,600 And it would be like I would the partial not in this direction, 951 00:55:31,600 --> 00:55:34,080 not in that direction, but in this direction. 952 00:55:34,080 --> 00:55:35,330 Let me tell you what this is. 953 00:55:35,330 --> 00:55:38,030 So according to a theorem, this would 954 00:55:38,030 --> 00:55:42,620 be df, dx, exactly like The Chain Rule, 955 00:55:42,620 --> 00:55:49,670 at my favorite point here, x0y0 [INAUDIBLE] 956 00:55:49,670 --> 00:55:55,070 p times-- now you say, oh, Magdalena, I understand. 957 00:55:55,070 --> 00:55:56,730 You're doing some sort of derivation. 958 00:55:56,730 --> 00:56:02,380 The derivative of that with respect to h would be u1. 959 00:56:02,380 --> 00:56:02,880 Yes. 960 00:56:02,880 --> 00:56:04,090 It's a Chain Rule. 961 00:56:04,090 --> 00:56:12,946 So then I go times u1 plus df, dy at the point times u2. 962 00:56:12,946 --> 00:56:14,790 963 00:56:14,790 --> 00:56:18,370 And you say, OK, but can I prove that? 964 00:56:18,370 --> 00:56:21,130 Yes, you could, but to prove that you 965 00:56:21,130 --> 00:56:26,136 would need to play a game. 966 00:56:26,136 --> 00:56:30,530 The proof will involve that you multiply up and down 967 00:56:30,530 --> 00:56:32,770 by an additional expression. 968 00:56:32,770 --> 00:56:35,364 And then you take limit of a product. 969 00:56:35,364 --> 00:56:37,320 If you take product, the product of limits, 970 00:56:37,320 --> 00:56:43,090 and you study them separately until you get to this Actually, 971 00:56:43,090 --> 00:56:47,360 this is an application of The Chain. 972 00:56:47,360 --> 00:56:54,440 But I want to come back to what Alexander just notice. 973 00:56:54,440 --> 00:56:57,550 I can explain this much better if we only 974 00:56:57,550 --> 00:57:01,864 think of derivative in the direction of i and derivative 975 00:57:01,864 --> 00:57:02,780 in the direction of j. 976 00:57:02,780 --> 00:57:04,680 What the heck are those? 977 00:57:04,680 --> 00:57:07,360 What are they going to be? 978 00:57:07,360 --> 00:57:13,690 The direction of deritivie-- if I have i instead of u, that 979 00:57:13,690 --> 00:57:17,179 will make you understand the whole notion much better. 980 00:57:17,179 --> 00:57:18,970 So what would be the directional derivative 981 00:57:18,970 --> 00:57:22,650 of in the direction of i only? 982 00:57:22,650 --> 00:57:23,850 Well, i for an i. 983 00:57:23,850 --> 00:57:25,132 It goes this way. 984 00:57:25,132 --> 00:57:27,020 This is a hard lesson. 985 00:57:27,020 --> 00:57:29,852 And it's advanced calculus rather than Calc 3, 986 00:57:29,852 --> 00:57:32,220 but you're going to get it. 987 00:57:32,220 --> 00:57:36,800 So if I go in the direction of i, 988 00:57:36,800 --> 00:57:40,720 I should have the df, dx, right? 989 00:57:40,720 --> 00:57:41,890 That should be it. 990 00:57:41,890 --> 00:57:42,837 Do I? 991 00:57:42,837 --> 00:57:44,170 STUDENT: Yes, but [INAUDIBLE] 0. 992 00:57:44,170 --> 00:57:45,170 MAGDALENA TODA: Exactly. 993 00:57:45,170 --> 00:57:47,620 Was I able to invent something so 994 00:57:47,620 --> 00:57:53,210 when I come back to what I already know, I recreate df, dx 995 00:57:53,210 --> 00:57:56,060 and nothing else? 996 00:57:56,060 --> 00:58:04,752 Precisely because for i as being u, what will be u1 and u2? 997 00:58:04,752 --> 00:58:06,680 STUDENT: [INAUDIBLE]. 998 00:58:06,680 --> 00:58:09,110 MAGDALENA TODA: u1 is 1. 999 00:58:09,110 --> 00:58:11,621 u2 is 0. 1000 00:58:11,621 --> 00:58:12,120 Right? 1001 00:58:12,120 --> 00:58:16,300 Because when we write i as a function of i and j, 1002 00:58:16,300 --> 00:58:19,070 that's 1 times i plus 0 times j. 1003 00:58:19,070 --> 00:58:22,946 So u1 is 1, u2 is zero. 1004 00:58:22,946 --> 00:58:24,110 Thank god. 1005 00:58:24,110 --> 00:58:26,950 According to the anything, this difference quotient 1006 00:58:26,950 --> 00:58:32,155 or the simpler way to define it from the theorem would 1007 00:58:32,155 --> 00:58:34,620 be simply the second goes away. 1008 00:58:34,620 --> 00:58:36,240 It vanishes. 1009 00:58:36,240 --> 00:58:41,540 u1 would be 1 and what I'm left with is df, dx. 1010 00:58:41,540 --> 00:58:45,071 And that's exactly what Alex noticed. 1011 00:58:45,071 --> 00:58:49,350 So the directional derivative is defined, 1012 00:58:49,350 --> 00:58:53,770 as a combination of vectors, such that you recreate 1013 00:58:53,770 --> 00:58:56,350 the directional derivative in the direction of i 1014 00:58:56,350 --> 00:58:59,190 being the partial, df, dx. 1015 00:58:59,190 --> 00:59:02,810 Exactly like you learned before in 11.3. 1016 00:59:02,810 --> 00:59:06,050 And what do I have if I try to recreate 1017 00:59:06,050 --> 00:59:10,141 the directional derivative in the direct of j? 1018 00:59:10,141 --> 00:59:10,640 x0y0. 1019 00:59:10,640 --> 00:59:14,500 We don't explain this much in the book. 1020 00:59:14,500 --> 00:59:17,510 I think on this one, I'm doing a better job than the book. 1021 00:59:17,510 --> 00:59:21,750 So what is df in the direction of j? 1022 00:59:21,750 --> 00:59:23,890 j is this way. 1023 00:59:23,890 --> 00:59:27,130 Well, [INAUDIBLE] is that 1j-- you 1024 00:59:27,130 --> 00:59:32,170 let me write it down-- is 0i plus 1j. 1025 00:59:32,170 --> 00:59:33,550 0 is u1. 1026 00:59:33,550 --> 00:59:36,250 1 is u2. 1027 00:59:36,250 --> 00:59:41,980 So by this formula, I simply should 1028 00:59:41,980 --> 00:59:47,970 get the directional deritive-- I mean, 1029 00:59:47,970 --> 00:59:50,975 directional derivative is the partial deritive-- with respect 1030 00:59:50,975 --> 00:59:56,890 to y at my point times a 1 that I'm not going to write. 1031 00:59:56,890 --> 01:00:07,330 So it's a concoction, so that in the directions of i and j, 1032 01:00:07,330 --> 01:00:10,600 you actually get the partial deritives. 1033 01:00:10,600 --> 01:00:13,020 And everything else is linear algebra. 1034 01:00:13,020 --> 01:00:19,550 So if you have a problem understanding the composition 1035 01:00:19,550 --> 01:00:21,640 of vectors, the sum of vectors, this 1036 01:00:21,640 --> 01:00:25,568 is because-- u1 and u2 are [INAUDIBLE], 1037 01:00:25,568 --> 01:00:28,250 I'm sorry-- this is because you haven't taken 1038 01:00:28,250 --> 01:00:33,348 the linear algebra yet, which teaches you a lot about how 1039 01:00:33,348 --> 01:00:36,330 a vector decomposes in two different directions 1040 01:00:36,330 --> 01:00:39,312 or along the standard canonical bases. 1041 01:00:39,312 --> 01:00:41,860 1042 01:00:41,860 --> 01:00:44,940 Let's see some problems of the type 1043 01:00:44,940 --> 01:00:49,452 that I've always put in the midterm and the same kind 1044 01:00:49,452 --> 01:00:54,642 of problems like we have seen in the final. 1045 01:00:54,642 --> 01:00:57,559 For example 3, is it, guys? 1046 01:00:57,559 --> 01:00:58,100 I don't know. 1047 01:00:58,100 --> 01:00:59,641 Example 3, 4, or something like that? 1048 01:00:59,641 --> 01:01:00,460 STUDENT: 3. 1049 01:01:00,460 --> 01:01:03,130 MAGDALENA TODA: Given z equals F of xy-- 1050 01:01:03,130 --> 01:01:06,430 what do you like best, the value or the hill? 1051 01:01:06,430 --> 01:01:09,250 This appeared in most of my exams. 1052 01:01:09,250 --> 01:01:12,510 x squared plus y squared, circular [INAUDIBLE] 1053 01:01:12,510 --> 01:01:14,470 was one of my favorite examples. 1054 01:01:14,470 --> 01:01:16,470 1 minus x squared minus y squared 1055 01:01:16,470 --> 01:01:22,930 was the circular parabola upside down. 1056 01:01:22,930 --> 01:01:24,330 Which one do you prefer? 1057 01:01:24,330 --> 01:01:25,470 I don't care. 1058 01:01:25,470 --> 01:01:26,495 Which one? 1059 01:01:26,495 --> 01:01:27,370 STUDENT: [INAUDIBLE]. 1060 01:01:27,370 --> 01:01:27,680 MAGDALENA TODA: The [INAUDIBLE]? 1061 01:01:27,680 --> 01:01:29,240 The first one. 1062 01:01:29,240 --> 01:01:30,060 It's easier. 1063 01:01:30,060 --> 01:01:34,870 1064 01:01:34,870 --> 01:01:36,690 And a typical problem. 1065 01:01:36,690 --> 01:01:50,060 Compute the directional derivative of z 1066 01:01:50,060 --> 01:02:00,050 equals F of x and y at the point p of coordinates 1, 1, 2 1067 01:02:00,050 --> 01:02:14,050 in the following directions-- A, i. 1068 01:02:14,050 --> 01:02:15,220 B, j. 1069 01:02:15,220 --> 01:02:18,060 C, i plus j. 1070 01:02:18,060 --> 01:02:24,000 1071 01:02:24,000 --> 01:02:29,180 D, the opposite, minus i, minus j over square 2. 1072 01:02:29,180 --> 01:02:31,960 And E-- 1073 01:02:31,960 --> 01:02:33,460 STUDENT: That's a square root 3. 1074 01:02:33,460 --> 01:02:34,460 MAGDALENA TODA: What? 1075 01:02:34,460 --> 01:02:36,040 STUDENT: You wrote a square root 3. 1076 01:02:36,040 --> 01:02:37,080 MAGDALENA TODA: I wrote square root of 3. 1077 01:02:37,080 --> 01:02:37,705 Thank you guys. 1078 01:02:37,705 --> 01:02:38,880 Thanks for being vigilant. 1079 01:02:38,880 --> 01:02:43,264 So always keep an eye on me because I'm full of surprises, 1080 01:02:43,264 --> 01:02:43,805 good and bad. 1081 01:02:43,805 --> 01:02:46,210 No, just kidding. 1082 01:02:46,210 --> 01:02:47,807 So let's see. 1083 01:02:47,807 --> 01:02:48,932 What do I want to put here? 1084 01:02:48,932 --> 01:02:51,402 Something. 1085 01:02:51,402 --> 01:02:52,390 How about this? 1086 01:02:52,390 --> 01:03:01,282 1087 01:03:01,282 --> 01:03:06,716 3 over root 5, pi plus [? y ?] over 5j. 1088 01:03:06,716 --> 01:03:10,668 Is this a unit vector or not? 1089 01:03:10,668 --> 01:03:12,150 STUDENT: No. 1090 01:03:12,150 --> 01:03:13,474 STUDENT: Yes, it is. 1091 01:03:13,474 --> 01:03:15,140 So you're going to drag the [INAUDIBLE]. 1092 01:03:15,140 --> 01:03:17,031 MAGDALENA TODA: Why is that a unit vector? 1093 01:03:17,031 --> 01:03:18,761 STUDENT: It's missing-- no, it's not. 1094 01:03:18,761 --> 01:03:20,927 MAGDALENA TODA: Then how do I make it a unit vector? 1095 01:03:20,927 --> 01:03:22,875 STUDENT: [INAUDIBLE]. 1096 01:03:22,875 --> 01:03:24,840 STUDENT: [INAUDIBLE]. 1097 01:03:24,840 --> 01:03:28,362 STUDENT: I have to take down-- there's a 3 that has to be 1. 1098 01:03:28,362 --> 01:03:29,346 [INAUDIBLE] 1099 01:03:29,346 --> 01:03:32,298 And the second one has to be 1, on the top, 1100 01:03:32,298 --> 01:03:34,266 to make it a unit vector. 1101 01:03:34,266 --> 01:03:39,200 1102 01:03:39,200 --> 01:03:41,560 MAGDALENA TODA: Give me a unit vector. 1103 01:03:41,560 --> 01:03:46,668 Another one then these easy ones. 1104 01:03:46,668 --> 01:03:48,233 STUDENT: 3 over 5 by 4 or 5. 1105 01:03:48,233 --> 01:03:49,108 MAGDALENA TODA: What? 1106 01:03:49,108 --> 01:03:52,050 STUDENT: 3 over 5 by 4 over 5j. 1107 01:03:52,050 --> 01:03:53,810 MAGDALENA TODA: 3 over-- I cannot hear. 1108 01:03:53,810 --> 01:03:54,080 STUDENT: 3 over 5-- 1109 01:03:54,080 --> 01:03:55,420 MAGDALENA TODA: 3 over 5. 1110 01:03:55,420 --> 01:03:56,970 STUDENT: And 4 over 5j. 1111 01:03:56,970 --> 01:03:58,710 MAGDALENA TODA: And 4 over 5j. 1112 01:03:58,710 --> 01:04:00,560 And why is that a unit vector? 1113 01:04:00,560 --> 01:04:05,160 STUDENT: Because 3 squared is [INAUDIBLE]. 1114 01:04:05,160 --> 01:04:07,284 MAGDALENA TODA: And what do we call these numbers? 1115 01:04:07,284 --> 01:04:08,200 You say, what is that? 1116 01:04:08,200 --> 01:04:10,711 And interview? 1117 01:04:10,711 --> 01:04:12,920 Yes, it is an interview. 1118 01:04:12,920 --> 01:04:13,890 Pythagorean numbers. 1119 01:04:13,890 --> 01:04:16,162 3, 4, and 5 are Pythagorean numbers. 1120 01:04:16,162 --> 01:04:19,180 1121 01:04:19,180 --> 01:04:23,840 So let me think a little bit where I should write. 1122 01:04:23,840 --> 01:04:26,200 Is this seen by the-- yes, it's seen 1123 01:04:26,200 --> 01:04:34,438 by the-- I'll just leave what's important for me 1124 01:04:34,438 --> 01:04:35,875 to solve this problem. 1125 01:04:35,875 --> 01:04:44,990 1126 01:04:44,990 --> 01:04:48,160 A. So what do we do? 1127 01:04:48,160 --> 01:04:55,580 The same thing. i is 1.i plus u, or 1 times i plus u times j. 1128 01:04:55,580 --> 01:04:58,675 So simply, you can write the formula or you can say, 1129 01:04:58,675 --> 01:05:01,430 the heck with the formula. 1130 01:05:01,430 --> 01:05:03,920 You know that df is df, dx. 1131 01:05:03,920 --> 01:05:07,810 The derivative of this at the point p. 1132 01:05:07,810 --> 01:05:14,022 So what you want to do is say, 2x-- are you guys with me? 1133 01:05:14,022 --> 01:05:15,160 STUDENT: Yes. 1134 01:05:15,160 --> 01:05:23,090 MAGDALENA TODA: At the value 1, 1, 2, which is 2. 1135 01:05:23,090 --> 01:05:24,940 And at the end of this exercise, I'm 1136 01:05:24,940 --> 01:05:28,430 going to ask you if there's any connection between-- 1137 01:05:28,430 --> 01:05:30,610 or maybe I will ask you next time. 1138 01:05:30,610 --> 01:05:34,590 Oh, we have time. 1139 01:05:34,590 --> 01:05:37,540 What is d in the direction of j? 1140 01:05:37,540 --> 01:05:41,070 The partial derivative with respect to y. 1141 01:05:41,070 --> 01:05:43,600 Nothing else, but our old friend. 1142 01:05:43,600 --> 01:05:47,680 And our old friend says, I have 2y 1143 01:05:47,680 --> 01:05:51,780 computed for the point p, 1, 1, 2. 1144 01:05:51,780 --> 01:05:52,980 What does it mean? 1145 01:05:52,980 --> 01:05:58,794 Y is 1, so just plug this 1 into the thingy. 1146 01:05:58,794 --> 01:05:59,738 It's 2. 1147 01:05:59,738 --> 01:06:03,990 1148 01:06:03,990 --> 01:06:07,110 Now do I see some-- I'm a scientist. 1149 01:06:07,110 --> 01:06:09,200 I have to find interpretations when 1150 01:06:09,200 --> 01:06:11,210 I get results that coincide. 1151 01:06:11,210 --> 01:06:12,645 It's a pattern. 1152 01:06:12,645 --> 01:06:14,014 Why do I get the same answer? 1153 01:06:14,014 --> 01:06:15,930 STUDENT: Because your functions are symmetric. 1154 01:06:15,930 --> 01:06:16,846 MAGDALENA TODA: Right. 1155 01:06:16,846 --> 01:06:20,070 And more than that, because the function is symmetric, 1156 01:06:20,070 --> 01:06:24,621 it's a quadric that I love, it's just a circular problem. 1157 01:06:24,621 --> 01:06:27,850 It's rotation is symmetric. 1158 01:06:27,850 --> 01:06:33,500 So I just take one parabola, one branch of a parabola, 1159 01:06:33,500 --> 01:06:38,400 and I rotate it by 360 degrees. 1160 01:06:38,400 --> 01:06:45,760 So the slope will be the same in both directions, i and j, 1161 01:06:45,760 --> 01:06:47,255 at the point that I have. 1162 01:06:47,255 --> 01:06:49,860 1163 01:06:49,860 --> 01:06:52,750 Well, it depends on the point. 1164 01:06:52,750 --> 01:06:55,145 If the point is, itself, symmetric 1165 01:06:55,145 --> 01:06:58,490 like that, x and y are the same, one in one, 1166 01:06:58,490 --> 01:07:03,750 I did it on purpose-- if you didn't have one and one, 1167 01:07:03,750 --> 01:07:07,520 you had an x variable and y variable to plug in. 1168 01:07:07,520 --> 01:07:10,960 But your magic point is where? 1169 01:07:10,960 --> 01:07:11,780 Oh my god. 1170 01:07:11,780 --> 01:07:15,370 I don't know how to explain with my hands. 1171 01:07:15,370 --> 01:07:16,650 Here I am, the frame. 1172 01:07:16,650 --> 01:07:19,910 I am the frame. x, y, and z. 1173 01:07:19,910 --> 01:07:21,820 1, 1. 1174 01:07:21,820 --> 01:07:23,220 Go up. 1175 01:07:23,220 --> 01:07:25,060 Where do you meet the vase? 1176 01:07:25,060 --> 01:07:26,900 At c equals 2. 1177 01:07:26,900 --> 01:07:30,356 So it's really symmetric and really beautiful. 1178 01:07:30,356 --> 01:07:34,140 1179 01:07:34,140 --> 01:07:37,800 Next I say, oh, in the direction i plus 1180 01:07:37,800 --> 01:07:43,590 j, which is exactly the direction of this meridian 1181 01:07:43,590 --> 01:07:47,782 that I was talking about, i plus j over square root 2. 1182 01:07:47,782 --> 01:07:50,780 Now I've had students-- that's where I was broken hearted. 1183 01:07:50,780 --> 01:07:53,330 Really, I didn't know what to do, 1184 01:07:53,330 --> 01:07:56,500 how much partial credit to give. 1185 01:07:56,500 --> 01:08:00,680 The definition of direction derivative is very strict. 1186 01:08:00,680 --> 01:08:04,750 It says you cannot take whatever 1 and 2 that you want. 1187 01:08:04,750 --> 01:08:09,490 You cannot multiply them by proportionality. 1188 01:08:09,490 --> 01:08:14,410 You have to have u to be a unit vector. 1189 01:08:14,410 --> 01:08:18,279 And then the directional derivative will be unique. 1190 01:08:18,279 --> 01:08:24,319 If I take 1 and 1 for u1 and u2, then I can take 2 and 2, 1191 01:08:24,319 --> 01:08:26,189 and 7 and 7, and 9 and 9. 1192 01:08:26,189 --> 01:08:28,399 And that's going to be a mess because 1193 01:08:28,399 --> 01:08:32,040 the directional derivative wouldn't be unique anymore. 1194 01:08:32,040 --> 01:08:36,020 And that's why whoever gave this definition, 1195 01:08:36,020 --> 01:08:39,104 I think Euler-- I tried to see in the history who 1196 01:08:39,104 --> 01:08:42,779 was the first mathematician who gave 1197 01:08:42,779 --> 01:08:46,950 the definition of the directional derivative. 1198 01:08:46,950 --> 01:08:49,710 And some people said it was Gateaux 1199 01:08:49,710 --> 01:08:53,196 because that's a french mathematician who first talked 1200 01:08:53,196 --> 01:08:55,231 about the Gateaux derivative, which 1201 01:08:55,231 --> 01:08:56,689 is like the directional derivative, 1202 01:08:56,689 --> 01:08:58,859 but other people said, no, look at Euler's work. 1203 01:08:58,859 --> 01:09:00,290 He was a genius. 1204 01:09:00,290 --> 01:09:04,710 He's the guy who discovered the transcendental number 1205 01:09:04,710 --> 01:09:06,740 e and many other things. 1206 01:09:06,740 --> 01:09:09,080 And the exponential e to the x is also 1207 01:09:09,080 --> 01:09:10,510 from Euler and everything. 1208 01:09:10,510 --> 01:09:12,569 He was one of the fathers of calculus. 1209 01:09:12,569 --> 01:09:19,060 Apparently, he knew the first 32 decimals of the number e. 1210 01:09:19,060 --> 01:09:22,910 And how he got to them is by hand. 1211 01:09:22,910 --> 01:09:24,090 Do you guys know of them? 1212 01:09:24,090 --> 01:09:29,620 2.71828-- and that's all I know. 1213 01:09:29,620 --> 01:09:32,000 The first five decimals. 1214 01:09:32,000 --> 01:09:35,729 Well, he knew 32 of them and he got to them by hand. 1215 01:09:35,729 --> 01:09:39,200 And they are non-repeating, infinitely remaining decimals. 1216 01:09:39,200 --> 01:09:40,460 It's a transcendental number. 1217 01:09:40,460 --> 01:09:41,858 STUDENT: And his 32 are correct? 1218 01:09:41,858 --> 01:09:42,733 MAGDALENA TODA: What? 1219 01:09:42,733 --> 01:09:44,180 STUDENT: His 32 are correct? 1220 01:09:44,180 --> 01:09:46,960 MAGDALENA TODA: His first 32 decimals were correct. 1221 01:09:46,960 --> 01:09:49,790 I don't know what-- I mean, the guy 1222 01:09:49,790 --> 01:09:53,260 was something like-- he was working at night. 1223 01:09:53,260 --> 01:09:56,690 And he would fill out, in one night, hundreds 1224 01:09:56,690 --> 01:10:04,270 of pages, computations, both by hand formulas and numerical. 1225 01:10:04,270 --> 01:10:07,170 So imagine-- of course, he would never make a WeBWork mistake. 1226 01:10:07,170 --> 01:10:10,866 I mean, if we built a time machine, 1227 01:10:10,866 --> 01:10:13,240 and we bring Euler back, and he's at Texas Tech, 1228 01:10:13,240 --> 01:10:16,160 and we make him solve our WeBWork problems, 1229 01:10:16,160 --> 01:10:17,910 I think he would take a thousand problems 1230 01:10:17,910 --> 01:10:19,880 and solve them in one night. 1231 01:10:19,880 --> 01:10:21,850 He need to know how to type, so we 1232 01:10:21,850 --> 01:10:24,250 have to teach him how to type. 1233 01:10:24,250 --> 01:10:27,650 But he would be able to compute what you guys have, 1234 01:10:27,650 --> 01:10:31,790 all those numerical answers, in his head. 1235 01:10:31,790 --> 01:10:35,250 He was a scary fellow. 1236 01:10:35,250 --> 01:10:41,380 So u has to be [INAUDIBLE] in some way, made unique. 1237 01:10:41,380 --> 01:10:43,350 u1 and u2. 1238 01:10:43,350 --> 01:10:45,920 I have students-- that's where the story started-- 1239 01:10:45,920 --> 01:10:49,990 who were very good, very smart, both honors and non-honors, who 1240 01:10:49,990 --> 01:10:54,350 took u1 to be 1, u2 to be 2 because they thought direction 1241 01:10:54,350 --> 01:11:00,640 1 and 1, which is not made unique as a direction, unitary. 1242 01:11:00,640 --> 01:11:03,420 And they plugged in here 1, they plugged in here 1, 1243 01:11:03,420 --> 01:11:08,210 they got these correctly, what was I supposed to give them, as 1244 01:11:08,210 --> 01:11:09,066 a [? friend? ?] 1245 01:11:09,066 --> 01:11:09,941 STUDENT: [INAUDIBLE]. 1246 01:11:09,941 --> 01:11:10,430 MAGDALENA TODA: What? 1247 01:11:10,430 --> 01:11:11,530 STUDENT: [INAUDIBLE]. 1248 01:11:11,530 --> 01:11:12,696 MAGDALENA TODA: I gave them. 1249 01:11:12,696 --> 01:11:14,413 How much do you think? 1250 01:11:14,413 --> 01:11:15,204 You should know me. 1251 01:11:15,204 --> 01:11:16,198 STUDENT: [INAUDIBLE]. 1252 01:11:16,198 --> 01:11:17,192 STUDENT: Full. 1253 01:11:17,192 --> 01:11:18,186 MAGDALENA TODA: 60%. 1254 01:11:18,186 --> 01:11:19,504 No. 1255 01:11:19,504 --> 01:11:20,920 Some people don't give any credit, 1256 01:11:20,920 --> 01:11:22,720 so pay attention to this. 1257 01:11:22,720 --> 01:11:31,498 In this case, this has to be 1 over square root 1258 01:11:31,498 --> 01:11:41,602 of 2 times the derivative of f at x, which is computed before 1259 01:11:41,602 --> 01:11:50,570 at the point, plus 1 over square root of 2 times the derivative 1260 01:11:50,570 --> 01:11:52,025 of the function. 1261 01:11:52,025 --> 01:11:54,450 Again, compute it at the same place. 1262 01:11:54,450 --> 01:12:02,514 Which is, oh my god, square root of 2 plus square root of 2, 1263 01:12:02,514 --> 01:12:04,470 which is 2 square root of 2. 1264 01:12:04,470 --> 01:12:20,630 1265 01:12:20,630 --> 01:12:29,746 And finally, the derivative of F at the same point-- I 1266 01:12:29,746 --> 01:12:31,222 should have put at the point. 1267 01:12:31,222 --> 01:12:35,160 Like a physicist would say, at p. 1268 01:12:35,160 --> 01:12:38,290 That would make you familiar with this notation. 1269 01:12:38,290 --> 01:12:40,330 And then measured at what? 1270 01:12:40,330 --> 01:12:43,540 The opposite direction, minus i minus j. 1271 01:12:43,540 --> 01:12:46,060 And now I'm getting lazy and I'm going to ask you 1272 01:12:46,060 --> 01:12:48,624 what the answer will be. 1273 01:12:48,624 --> 01:12:50,040 STUDENT: 2 minus square root of 2. 1274 01:12:50,040 --> 01:12:53,045 MAGDALENA TODA: So you see, there is another pattern. 1275 01:12:53,045 --> 01:12:55,190 In the opposite direction, the direction 1276 01:12:55,190 --> 01:12:59,500 of the derivative in this case would just be the negative one. 1277 01:12:59,500 --> 01:13:03,190 What if we took this directional derivative in absolute value? 1278 01:13:03,190 --> 01:13:05,374 Because you see, in this direction, 1279 01:13:05,374 --> 01:13:07,610 there's a positive directional derivaty. 1280 01:13:07,610 --> 01:13:11,930 In the other direction, it's like it's because-- I know why. 1281 01:13:11,930 --> 01:13:13,600 I'm a vase. 1282 01:13:13,600 --> 01:13:18,132 So in the direction i plus j over square root of 2, 1283 01:13:18,132 --> 01:13:20,305 the directional derivative will be positive. 1284 01:13:20,305 --> 01:13:21,780 It goes up. 1285 01:13:21,780 --> 01:13:24,260 But in the direction minus i minus 1286 01:13:24,260 --> 01:13:28,290 j, which is the opposite, over square root of 2, it goes down. 1287 01:13:28,290 --> 01:13:30,620 So the slope is negative. 1288 01:13:30,620 --> 01:13:32,200 So that's why we have negative. 1289 01:13:32,200 --> 01:13:34,770 Everything you get in life or in math, 1290 01:13:34,770 --> 01:13:36,410 you have to find an interpretation. 1291 01:13:36,410 --> 01:13:40,354 1292 01:13:40,354 --> 01:13:44,460 Sometimes in life and mathematics, things are subtle. 1293 01:13:44,460 --> 01:13:46,850 People will say one thing and they mean another thing. 1294 01:13:46,850 --> 01:13:49,824 You have to try to see beyond their words. 1295 01:13:49,824 --> 01:13:50,700 That's sad. 1296 01:13:50,700 --> 01:13:53,760 And in mathematics, you have to try to see beyond the numbers. 1297 01:13:53,760 --> 01:13:55,420 You see a pattern. 1298 01:13:55,420 --> 01:13:58,050 So being in opposite directions, I 1299 01:13:58,050 --> 01:14:02,239 got opposite signs of the directional derivative 1300 01:14:02,239 --> 01:14:03,530 because I have opposite slopes. 1301 01:14:03,530 --> 01:14:07,750 1302 01:14:07,750 --> 01:14:10,992 What else do I want to learn in this example? 1303 01:14:10,992 --> 01:14:12,120 One last thing. 1304 01:14:12,120 --> 01:14:13,286 STUDENT: E. 1305 01:14:13,286 --> 01:14:22,670 MAGDALENA TODA: E. So I have the same thing. 1306 01:14:22,670 --> 01:14:25,242 So it's not going to matter, the direction 1307 01:14:25,242 --> 01:14:26,920 is the only thing that changes. 1308 01:14:26,920 --> 01:14:28,960 These guys are the same. 1309 01:14:28,960 --> 01:14:33,630 The partials are the same at the same point. 1310 01:14:33,630 --> 01:14:35,130 I'm not going to worry about them. 1311 01:14:35,130 --> 01:14:39,200 So I get 2 or both. 1312 01:14:39,200 --> 01:14:41,260 What changes is the blue guys. 1313 01:14:41,260 --> 01:14:47,847 They are going to be 3 over 5 and 4 over 5. 1314 01:14:47,847 --> 01:14:53,620 1315 01:14:53,620 --> 01:14:56,270 And what do I get? 1316 01:14:56,270 --> 01:15:04,885 I get-- right? 1317 01:15:04,885 --> 01:15:09,110 1318 01:15:09,110 --> 01:15:12,890 Now I want to tell you something-- 1319 01:15:12,890 --> 01:15:16,100 I already anticipated something last time. 1320 01:15:16,100 --> 01:15:21,280 And let me tell you what I said last time. 1321 01:15:21,280 --> 01:15:25,970 1322 01:15:25,970 --> 01:15:27,930 Maybe I should not erase-- well, I 1323 01:15:27,930 --> 01:15:30,240 have to erase this whether I like it or not. 1324 01:15:30,240 --> 01:15:33,800 1325 01:15:33,800 --> 01:15:35,775 And now I'll review what this was. 1326 01:15:35,775 --> 01:15:38,180 What was this? d equals x squared plus y squared? 1327 01:15:38,180 --> 01:15:38,960 Yes or no? 1328 01:15:38,960 --> 01:15:41,410 STUDENT: Yes. 1329 01:15:41,410 --> 01:15:46,035 MAGDALENA TODA: So what did I say last time? 1330 01:15:46,035 --> 01:15:52,730 We have no result. We noticed it last time. 1331 01:15:52,730 --> 01:15:55,060 We did not prove it. 1332 01:15:55,060 --> 01:16:08,570 We did not prove it, only found it experimentally 1333 01:16:08,570 --> 01:16:11,810 using our physical common sense. 1334 01:16:11,810 --> 01:16:16,990 When you have a function z equals F of xy, 1335 01:16:16,990 --> 01:16:30,878 we studied the maximum rate of change 1336 01:16:30,878 --> 01:16:39,560 at the point x0y0 in the domain, assuming this is a c1 function. 1337 01:16:39,560 --> 01:16:40,890 I don't know. 1338 01:16:40,890 --> 01:16:44,330 Maximum rate of change was a magic thing. 1339 01:16:44,330 --> 01:16:48,130 And you probably thought, what in the world is that? 1340 01:16:48,130 --> 01:17:01,120 And we also said, this maximum for the rate of change 1341 01:17:01,120 --> 01:17:23,849 is always attained in the direction of the gradient. 1342 01:17:23,849 --> 01:17:31,310 1343 01:17:31,310 --> 01:17:38,050 So you realize that it's the steepest ascent, 1344 01:17:38,050 --> 01:17:40,920 the way it's called in many, many other fields, 1345 01:17:40,920 --> 01:17:42,832 but mathematics. 1346 01:17:42,832 --> 01:17:45,420 Or the steepest descent. 1347 01:17:45,420 --> 01:17:51,530 1348 01:17:51,530 --> 01:17:58,240 Now if it's an ascent, then it's in the direction gradient of F. 1349 01:17:58,240 --> 01:18:00,210 But if it's a descent, it's going 1350 01:18:00,210 --> 01:18:04,630 to be in the opposite direction, minus gradient of F. 1351 01:18:04,630 --> 01:18:07,580 But then I [INAUDIBLE] first of all, 1352 01:18:07,580 --> 01:18:11,890 it's not the same direction, if you have opposites. 1353 01:18:11,890 --> 01:18:14,750 Well, direction is sort of given by one line. 1354 01:18:14,750 --> 01:18:18,840 Whether you take this or the opposite, it's the same thing. 1355 01:18:18,840 --> 01:18:21,280 What this means is that we say direction 1356 01:18:21,280 --> 01:18:25,680 and we didn't [? unitarize ?] it. 1357 01:18:25,680 --> 01:18:31,050 So we could say, or gradient of F 1358 01:18:31,050 --> 01:18:35,980 over length of gradient of F. Or minus gradient of F 1359 01:18:35,980 --> 01:18:39,750 over length of gradient of F. Can this theorem be proved? 1360 01:18:39,750 --> 01:18:41,330 Yes, it can be proved. 1361 01:18:41,330 --> 01:18:45,370 We are going to discuss a little bit more next time about it, 1362 01:18:45,370 --> 01:18:49,360 but I want to tell you a big disclosure today. 1363 01:18:49,360 --> 01:18:55,020 This maximum rate of change is the directional derivative. 1364 01:18:55,020 --> 01:19:07,808 This maximum rate of change is exactly 1365 01:19:07,808 --> 01:19:15,630 the directional derivative in the direction 1366 01:19:15,630 --> 01:19:35,068 of the gradient, which is also the magnitude of the gradient. 1367 01:19:35,068 --> 01:19:43,380 1368 01:19:43,380 --> 01:19:47,230 And you'll say, wait a minute, what? 1369 01:19:47,230 --> 01:19:48,360 What did you say? 1370 01:19:48,360 --> 01:19:51,082 Let's first verify my claim. 1371 01:19:51,082 --> 01:19:53,350 I'm not even sure my claim is true. 1372 01:19:53,350 --> 01:19:55,480 We will see next time. 1373 01:19:55,480 --> 01:19:59,830 Can I verify my claim on one example? 1374 01:19:59,830 --> 01:20:01,690 Well, OK. 1375 01:20:01,690 --> 01:20:04,870 Maximum rate of change would be exactly 1376 01:20:04,870 --> 01:20:07,964 as the directional derivative and the direction 1377 01:20:07,964 --> 01:20:08,630 of the gradient? 1378 01:20:08,630 --> 01:20:10,070 I don't know about that. 1379 01:20:10,070 --> 01:20:11,382 That all sounds crazy. 1380 01:20:11,382 --> 01:20:12,825 So what do I have to compute? 1381 01:20:12,825 --> 01:20:16,673 I have to compute that directional derivative 1382 01:20:16,673 --> 01:20:21,640 of, let's say, my function F in the direction of the gradient-- 1383 01:20:21,640 --> 01:20:22,956 what is the gradient? 1384 01:20:22,956 --> 01:20:26,270 1385 01:20:26,270 --> 01:20:28,815 We have to figure it out. 1386 01:20:28,815 --> 01:20:30,640 We did it last time, but you forgot. 1387 01:20:30,640 --> 01:20:37,360 So for this guy, nabla F, what will be the gradient? 1388 01:20:37,360 --> 01:20:39,740 Where is my function? 1389 01:20:39,740 --> 01:20:47,620 Nabla F will be 2x, 2y, right? 1390 01:20:47,620 --> 01:20:51,750 Which means 2xi plus 2yj, right? 1391 01:20:51,750 --> 01:20:54,676 But if I'm at the point p, what does it mean? 1392 01:20:54,676 --> 01:20:59,020 At the point p, it means that I have 2 times i plus 2 times j, 1393 01:20:59,020 --> 01:21:00,342 right? 1394 01:21:00,342 --> 01:21:06,030 And what is the magnitude of the gradient? 1395 01:21:06,030 --> 01:21:08,140 Yes. 1396 01:21:08,140 --> 01:21:13,292 The magnitude of the gradient is somebody I know, which is what? 1397 01:21:13,292 --> 01:21:18,583 Which is square root of 2 squared plus 2 squared. 1398 01:21:18,583 --> 01:21:20,784 I cannot do that now. 1399 01:21:20,784 --> 01:21:21,950 What's the square root of 8? 1400 01:21:21,950 --> 01:21:22,839 STUDENT: 2 root 2. 1401 01:21:22,839 --> 01:21:23,880 MAGDALENA TODA: 2 root 2. 1402 01:21:23,880 --> 01:21:24,629 This is a pattern. 1403 01:21:24,629 --> 01:21:25,230 2 root 2. 1404 01:21:25,230 --> 01:21:27,240 I've seen this 2 root 2 again somewhere. 1405 01:21:27,240 --> 01:21:28,880 Where the heck have I seen it? 1406 01:21:28,880 --> 01:21:29,922 STUDENT: That was the directional derivative. 1407 01:21:29,922 --> 01:21:31,713 MAGDALENA TODA: The directional derivative. 1408 01:21:31,713 --> 01:21:33,320 So the claim may be right. 1409 01:21:33,320 --> 01:21:36,452 It says it is the directional derivative in the direction 1410 01:21:36,452 --> 01:21:37,810 of the gradient. 1411 01:21:37,810 --> 01:21:40,920 But is this really the direction of the gradient? 1412 01:21:40,920 --> 01:21:42,770 Yes. 1413 01:21:42,770 --> 01:21:45,910 Because when you compote the direction for the gradient, 2y 1414 01:21:45,910 --> 01:21:52,190 plus 2j, you don't mean 2i plus 2j as a twice i plus j, 1415 01:21:52,190 --> 01:21:55,647 you mean the unit vector correspondent to that. 1416 01:21:55,647 --> 01:21:57,230 So what is the direction corresponding 1417 01:21:57,230 --> 01:22:00,550 to the gradient 2i plus 2j? 1418 01:22:00,550 --> 01:22:01,850 STUDENT: i plus j [? over 2. ?] 1419 01:22:01,850 --> 01:22:02,850 MAGDALENA TODA: Exactly. 1420 01:22:02,850 --> 01:22:06,140 U equals i plus j divided by square 2. 1421 01:22:06,140 --> 01:22:09,310 So this is the directional derivative 1422 01:22:09,310 --> 01:22:13,120 in the direction of the gradient at the point p, which is 2 root 1423 01:22:13,120 --> 01:22:13,620 2. 1424 01:22:13,620 --> 01:22:18,250 And it's the same thing-- for some reason that's mysterious 1425 01:22:18,250 --> 01:22:19,860 and we will see next time. 1426 01:22:19,860 --> 01:22:23,340 For some mysterious reason you get exactly the same 1427 01:22:23,340 --> 01:22:27,780 as the length of the gradient vector. 1428 01:22:27,780 --> 01:22:30,460 We will see about this mystery next time. 1429 01:22:30,460 --> 01:22:35,220 I have you enough to torment you until Tuesday. 1430 01:22:35,220 --> 01:22:38,280 What have you promised me besides doing the homework? 1431 01:22:38,280 --> 01:22:39,756 STUDENT: To read the book. 1432 01:22:39,756 --> 01:22:41,130 MAGDALENA TODA: To read the book. 1433 01:22:41,130 --> 01:22:41,950 You're very smart. 1434 01:22:41,950 --> 01:22:43,620 Please, read the book. 1435 01:22:43,620 --> 01:22:45,078 All the examples in the book. 1436 01:22:45,078 --> 01:22:47,070 They are short. 1437 01:22:47,070 --> 01:22:48,066 Thank you so much. 1438 01:22:48,066 --> 01:22:50,556 Have a wonderful weekend and I'll 1439 01:22:50,556 --> 01:22:54,540 talk to you on Tuesday about anything you have trouble with. 1440 01:22:54,540 --> 01:22:57,030 When is the homework due? 1441 01:22:57,030 --> 01:22:59,022 STUDENT: Saturday. 1442 01:22:59,022 --> 01:23:00,514 MAGDALENA TODA: On Saturday. 1443 01:23:00,514 --> 01:23:01,014 I was mean. 1444 01:23:01,014 --> 01:23:04,500 I should have given it you until Sunday night, but-- 1445 01:23:04,500 --> 01:23:05,943 STUDENT: Yes. 1446 01:23:05,943 --> 01:23:08,484 MAGDALENA TODA: Do you want me to make it until Sunday night? 1447 01:23:08,484 --> 01:23:08,982 STUDENT: Yes. 1448 01:23:08,982 --> 01:23:10,148 MAGDALENA TODA: At midnight? 1449 01:23:10,148 --> 01:23:10,974 STUDENT: Yes. 1450 01:23:10,974 --> 01:23:12,966 MAGDALENA TODA: I'll do that. 1451 01:23:12,966 --> 01:23:14,958 I will extend it. 1452 01:23:14,958 --> 01:23:19,440 1453 01:23:19,440 --> 01:23:22,428 STUDENT: She asked, I said yes. 1454 01:23:22,428 --> 01:23:23,922 STUDENT: Why did you do that, dude? 1455 01:23:23,922 --> 01:23:28,238 Come on, my life is ruined now because I have more time 1456 01:23:28,238 --> 01:23:29,987 to work on my homework. 1457 01:23:29,987 --> 01:23:31,820 MAGDALENA TODA: And I've ruined your Sunday. 1458 01:23:31,820 --> 01:23:32,361 STUDENT: Yes. 1459 01:23:32,361 --> 01:23:33,020 No. 1460 01:23:33,020 --> 01:23:33,920 MAGDALENA TODA: No. 1461 01:23:33,920 --> 01:23:36,362 Actually, I know why I did that. 1462 01:23:36,362 --> 01:23:37,820 I thought that the 28th of February 1463 01:23:37,820 --> 01:23:42,620 is the last day of the month, but it's a short month. 1464 01:23:42,620 --> 01:23:45,020 So if we [? try it, ?] we have to extend the months 1465 01:23:45,020 --> 01:23:48,620 a little bit by pulling it by one more day. 1466 01:23:48,620 --> 01:23:49,754 STUDENT: We did? 1467 01:23:49,754 --> 01:23:51,920 MAGDALENA TODA: The first of March is Sunday, right? 1468 01:23:51,920 --> 01:23:53,720 STUDENT: Yes. 1469 01:23:53,720 --> 01:23:55,520 [INTERPOSING VOICES] 1470 01:23:55,520 --> 01:24:05,812 1471 01:24:05,812 --> 01:24:07,520 STUDENT: You're going to miss the speech. 1472 01:24:07,520 --> 01:24:09,320 STUDENT: Oh, we're doing that? 1473 01:24:09,320 --> 01:24:10,520 STUDENT: You're in English? 1474 01:24:10,520 --> 01:24:11,395 STUDENT: [INAUDIBLE]. 1475 01:24:11,395 --> 01:24:13,987 1476 01:24:13,987 --> 01:24:15,320 STUDENT: You don't know English? 1477 01:24:15,320 --> 01:24:15,920 Why are you talking English? 1478 01:24:15,920 --> 01:24:17,720 That's what my father used to say. 1479 01:24:17,720 --> 01:24:19,570 You don't know your own tongue?