1 00:00:00,000 --> 00:00:02,020 PROFESSOR: I'll go over the exam. 2 00:00:02,020 --> 00:00:04,170 It's good review for the final, and it's 3 00:00:04,170 --> 00:00:09,530 a good feedback for you in case you have questions. 4 00:00:09,530 --> 00:00:15,074 I do not change grades, I do not curve your exam. 5 00:00:15,074 --> 00:00:19,493 I do not make adjustments after I give you the grade. 6 00:00:19,493 --> 00:00:22,193 Therefore, it's very important for me 7 00:00:22,193 --> 00:00:25,385 to explain why you got what you got. 8 00:00:25,385 --> 00:00:30,300 Not everybody did well on this exam. 9 00:00:30,300 --> 00:00:35,060 Most people did pretty good, and I'm 10 00:00:35,060 --> 00:00:41,800 quite happy with what I see as a average for the class. 11 00:00:41,800 --> 00:00:44,120 However, there are many open questions 12 00:00:44,120 --> 00:00:47,920 from many people, things they didn't quite understand, 13 00:00:47,920 --> 00:00:51,180 and I would like to discuss those. 14 00:00:51,180 --> 00:00:54,420 15 00:00:54,420 --> 00:01:09,165 First of all, the midterm exam was 11 questions. 16 00:01:09,165 --> 00:01:16,480 10 were mandatory, so the maximum possible 17 00:01:16,480 --> 00:01:20,800 percentage-wise was 110%. 18 00:01:20,800 --> 00:01:25,400 So for somebody who did perfectly fine, 19 00:01:25,400 --> 00:01:26,430 they would have 110%. 20 00:01:26,430 --> 00:01:29,130 21 00:01:29,130 --> 00:01:33,810 There is one person only who got the high. 22 00:01:33,810 --> 00:01:36,750 I didn't disclose his name, but I would 23 00:01:36,750 --> 00:01:39,576 like to say congratulations. 24 00:01:39,576 --> 00:01:46,827 And I'm going to go ahead and solve each problem with you, 25 00:01:46,827 --> 00:01:48,276 for you. 26 00:01:48,276 --> 00:01:53,600 So you have the function f of x, y, 27 00:01:53,600 --> 00:01:55,954 to be x squared minus y squared. 28 00:01:55,954 --> 00:02:01,800 And the differential was f sub x dx plus f sub y dy, 29 00:02:01,800 --> 00:02:06,240 whihc is 2x dx, minus 2y dy. 30 00:02:06,240 --> 00:02:09,070 31 00:02:09,070 --> 00:02:11,190 That was something very easy. 32 00:02:11,190 --> 00:02:18,324 It was not supposed to give you any headache, and most of you 33 00:02:18,324 --> 00:02:21,726 did a fine job on this one. 34 00:02:21,726 --> 00:02:24,440 What created some problems to most students 35 00:02:24,440 --> 00:02:26,650 was the second problem, though. 36 00:02:26,650 --> 00:02:29,990 And I sorry to hear that, sorry to see that. 37 00:02:29,990 --> 00:02:33,455 Find the directional derivative of a function, 38 00:02:33,455 --> 00:02:35,930 the same function as before. 39 00:02:35,930 --> 00:02:39,890 40 00:02:39,890 --> 00:02:45,520 So I have taken advantage of the previous problem, 41 00:02:45,520 --> 00:02:50,492 in order to make your do time shorter. 42 00:02:50,492 --> 00:02:56,444 At the point p of coordinates x equals 0, y equals 1, 43 00:02:56,444 --> 00:02:59,420 you will have a direction given by the vector. 44 00:02:59,420 --> 00:03:07,090 What does it mean direction given? 45 00:03:07,090 --> 00:03:14,702 Analyze that direction given by the vector means what? 46 00:03:14,702 --> 00:03:20,840 Not the vector i minus j is y, because it's not a unit vector. 47 00:03:20,840 --> 00:03:24,690 What is the corresponding direction given by it? 48 00:03:24,690 --> 00:03:27,285 A corresponding direction given by it 49 00:03:27,285 --> 00:03:32,600 is 1 over square root of 2i minus 1 over square root of 2j. 50 00:03:32,600 --> 00:03:36,900 So it's a collinear vector-- that is, unit varies. 51 00:03:36,900 --> 00:03:38,400 Say it again, [INAUDIBLE]? 52 00:03:38,400 --> 00:03:43,800 The direction u represents a collinear vector. 53 00:03:43,800 --> 00:03:47,209 So pointing in the same direction as v, 54 00:03:47,209 --> 00:03:49,144 but it has to be unitary. 55 00:03:49,144 --> 00:03:49,644 Why? 56 00:03:49,644 --> 00:03:54,514 Because the definition of the directional derivative 57 00:03:54,514 --> 00:03:59,850 is a function along the direction u at the point p, 58 00:03:59,850 --> 00:04:06,256 was given by the formula partial derivative at p 59 00:04:06,256 --> 00:04:10,350 and at 1, plus partial derivative at p times 2. 60 00:04:10,350 --> 00:04:12,420 Did I expect to write all this down? 61 00:04:12,420 --> 00:04:15,350 Yes, I did, as I showed you last time. 62 00:04:15,350 --> 00:04:19,470 So you have 2x evaluated at-- what is x? 63 00:04:19,470 --> 00:04:20,240 0. 64 00:04:20,240 --> 00:04:24,488 1 times u1. 65 00:04:24,488 --> 00:04:31,440 This is u1 minus 2y, evaluated at 0, 1 times 66 00:04:31,440 --> 00:04:36,232 minus 1 over root 2, which is u2. 67 00:04:36,232 --> 00:04:45,540 68 00:04:45,540 --> 00:04:48,120 Well that means the first term goes away, 69 00:04:48,120 --> 00:04:50,420 because this is going to be 0. 70 00:04:50,420 --> 00:04:54,250 And after the second term, you have a plus. 71 00:04:54,250 --> 00:04:57,570 y is 1, thank god, that's easy. 72 00:04:57,570 --> 00:05:01,670 2 over square root of 2, the answer is root 2. 73 00:05:01,670 --> 00:05:05,050 So any other answer would normally receiving a 0. 74 00:05:05,050 --> 00:05:10,610 The answer was b, square root 2. 75 00:05:10,610 --> 00:05:17,790 Now, on number three, the function given is different. 76 00:05:17,790 --> 00:05:24,875 f of x, y equals e to the xy. 77 00:05:24,875 --> 00:05:29,330 78 00:05:29,330 --> 00:05:34,600 And they say, the gradient of this function 79 00:05:34,600 --> 00:05:37,010 is at an arbitrary point. 80 00:05:37,010 --> 00:05:38,461 Say is again? 81 00:05:38,461 --> 00:05:42,150 The gradient of this function is at an arbitrary point. 82 00:05:42,150 --> 00:05:44,625 That was only part of the problem. 83 00:05:44,625 --> 00:05:48,585 A little bit of credit for just writing the gradient. 84 00:05:48,585 --> 00:05:52,050 This is actually easy, a piece of cake. 85 00:05:52,050 --> 00:05:53,040 y is that. 86 00:05:53,040 --> 00:05:57,495 You have f sub x i plus f sub y j. 87 00:05:57,495 --> 00:06:11,130 It equals y to the xy i plus x to the xyj. 88 00:06:11,130 --> 00:06:12,546 That's very good. 89 00:06:12,546 --> 00:06:18,690 90 00:06:18,690 --> 00:06:20,668 Alright, OK? 91 00:06:20,668 --> 00:06:39,660 Then, which direction-- it's just the gradient right? 92 00:06:39,660 --> 00:06:41,410 The direction corresponds to the gradient. 93 00:06:41,410 --> 00:06:43,710 They don't ask you for the u. 94 00:06:43,710 --> 00:06:45,580 Actually, you don't need the u. 95 00:06:45,580 --> 00:06:49,039 You just need the tangent plane in this case. 96 00:06:49,039 --> 00:06:53,260 And if you know the equation of the tangent plane, 97 00:06:53,260 --> 00:06:56,980 as I told you to remember that, that would be very helpful. 98 00:06:56,980 --> 00:06:58,960 Write your answer in the space provided. 99 00:06:58,960 --> 00:07:01,326 So what did I expect you to do? 100 00:07:01,326 --> 00:07:09,220 First this, and then write the equation z minus z0 101 00:07:09,220 --> 00:07:17,110 equals f sub x times x minus x0, plus f sub y, y minus y0. 102 00:07:17,110 --> 00:07:22,274 Here at the point p, evaluate it at the point p. 103 00:07:22,274 --> 00:07:24,470 But attention, what is the point p? 104 00:07:24,470 --> 00:07:29,210 Well, p is the origin, because we say at the origin. 105 00:07:29,210 --> 00:07:32,250 Oh, so that makes things easier. 106 00:07:32,250 --> 00:07:33,429 I'm not done. 107 00:07:33,429 --> 00:07:37,101 Half of the problem is still coming. 108 00:07:37,101 --> 00:07:40,830 If you did until this point, I can only give you 5 out of 10 109 00:07:40,830 --> 00:07:41,823 or something like that. 110 00:07:41,823 --> 00:07:44,538 111 00:07:44,538 --> 00:07:48,620 Many people made a mistake at z0. 112 00:07:48,620 --> 00:07:53,060 Attention guys, you plus in that 0, you don't get 0. 113 00:07:53,060 --> 00:07:55,810 For god's sake, it's 1, right? 114 00:07:55,810 --> 00:07:58,720 So z0 is 1. 115 00:07:58,720 --> 00:08:01,890 Now you're getting the sense that you 116 00:08:01,890 --> 00:08:07,395 have z minus 1 equals f sub x, computed as 0, 117 00:08:07,395 --> 00:08:10,650 0 will be 0, lucky you. 118 00:08:10,650 --> 00:08:18,380 f sub y computed at 0, 0, you were expected to say that. 119 00:08:18,380 --> 00:08:23,540 So the answer for this problem was z equals 1. 120 00:08:23,540 --> 00:08:28,450 Still, if you messed up, I gave some partial credit, 121 00:08:28,450 --> 00:08:32,539 because I didn't want to punish you too much, too harshly. 122 00:08:32,539 --> 00:08:36,380 123 00:08:36,380 --> 00:08:42,870 On number four, find the direction u-- now 124 00:08:42,870 --> 00:08:45,868 you're using number three, so I should not 125 00:08:45,868 --> 00:08:50,050 erase number three completely. 126 00:08:50,050 --> 00:08:52,260 On number four, you use number three, so 127 00:08:52,260 --> 00:08:54,628 the same type of function. 128 00:08:54,628 --> 00:08:58,300 But it says find the direction in which 129 00:08:58,300 --> 00:09:05,820 this function increases most rapidly, at the point 1, 1. 130 00:09:05,820 --> 00:09:06,740 OK. 131 00:09:06,740 --> 00:09:16,320 So what do you do? you compute the gradient at the point 1,1, 132 00:09:16,320 --> 00:09:20,006 and you say, this is a piece of cake. 133 00:09:20,006 --> 00:09:23,408 It's going to be ei plus ej, ee. 134 00:09:23,408 --> 00:09:33,130 135 00:09:33,130 --> 00:09:35,680 Wonderful. 136 00:09:35,680 --> 00:09:40,155 So what you do is a u is the gradient f over the length 137 00:09:40,155 --> 00:09:43,340 of the gradient of f at p. 138 00:09:43,340 --> 00:09:48,730 Which is ee divided by the length of it. 139 00:09:48,730 --> 00:09:52,120 But you say, I don't have to compute the length of it. 140 00:09:52,120 --> 00:09:56,670 I know what is pulling your two e's is what? 141 00:09:56,670 --> 00:10:00,600 And no matter what you have here you get the same unique result. 142 00:10:00,600 --> 00:10:02,840 Remember we talked about that uniqueness? 143 00:10:02,840 --> 00:10:04,780 This is what I tried to emphasize, 144 00:10:04,780 --> 00:10:10,074 that you can have 77, ee, 99, 55, 100 and 100. 145 00:10:10,074 --> 00:10:13,768 If you divide by the norm, you still get the same answer. 146 00:10:13,768 --> 00:10:17,620 Not 11, but 1 over root 2, 1 over root 2. 147 00:10:17,620 --> 00:10:19,490 So no matter what you had there-- 148 00:10:19,490 --> 00:10:22,470 it could have had a million, or something instead of e, 149 00:10:22,470 --> 00:10:24,419 you still have the same u. 150 00:10:24,419 --> 00:10:29,580 151 00:10:29,580 --> 00:10:31,300 Yes, put it back. 152 00:10:31,300 --> 00:10:32,940 Give yourself points, modify that. 153 00:10:32,940 --> 00:10:35,840 154 00:10:35,840 --> 00:10:37,626 OK, so let me tell you. 155 00:10:37,626 --> 00:10:40,800 Normally I should penalize, because I say write the answer 156 00:10:40,800 --> 00:10:41,950 in the space provided. 157 00:10:41,950 --> 00:10:45,160 And thank god you had enough space, right? 158 00:10:45,160 --> 00:10:48,326 Look, this person wrote-- I shouldn't show you who 159 00:10:48,326 --> 00:10:50,160 he is, he's not in here anyway. 160 00:10:50,160 --> 00:10:52,660 He has space and he provided last year 161 00:10:52,660 --> 00:10:54,710 with no square root of 2, because only two 162 00:10:54,710 --> 00:10:57,220 rows are enough to write that. 163 00:10:57,220 --> 00:10:59,710 It's OK, I understand you forgot to copy. 164 00:10:59,710 --> 00:11:02,040 My son did the same thing. 165 00:11:02,040 --> 00:11:03,990 He got a scantron at the UIL. 166 00:11:03,990 --> 00:11:07,520 Come to visit my son, I wanted to kill him, but it's OK. 167 00:11:07,520 --> 00:11:10,322 He got all the answers right, and then 168 00:11:10,322 --> 00:11:14,530 the teacher-- that reminds me of a movie with Mr. Bean. 169 00:11:14,530 --> 00:11:17,260 So the teacher comes to him and says, 170 00:11:17,260 --> 00:11:19,810 wow your scantron is blank. 171 00:11:19,810 --> 00:11:21,750 So what was I supposed to do? 172 00:11:21,750 --> 00:11:24,489 Adjust for all the answers you got in the box, 173 00:11:24,489 --> 00:11:25,805 put them in the scantron. 174 00:11:25,805 --> 00:11:26,305 Oh, really? 175 00:11:26,305 --> 00:11:27,564 So he goes quickly. 176 00:11:27,564 --> 00:11:30,735 And then he got only 75% of them transferred. 177 00:11:30,735 --> 00:11:32,950 The rest of them were not transferred. 178 00:11:32,950 --> 00:11:34,295 I don't know what they did. 179 00:11:34,295 --> 00:11:36,690 I have no idea. 180 00:11:36,690 --> 00:11:40,144 But the professor would have given full credit, 181 00:11:40,144 --> 00:11:43,476 even for the answers that he had in the box. 182 00:11:43,476 --> 00:11:45,860 From what I understood, the rule for scantrons, 183 00:11:45,860 --> 00:11:49,110 exams like you I only say, if you don't have them 184 00:11:49,110 --> 00:11:52,430 on the scantron, they don't count. 185 00:11:52,430 --> 00:11:56,180 This is very harsh, because we don't do that. 186 00:11:56,180 --> 00:11:58,710 For example, the final-- if you-- 187 00:11:58,710 --> 00:12:02,940 that's why I'm trying to read everything. 188 00:12:02,940 --> 00:12:07,975 Suppose you box you answer and it's 1 over square root of 2. 189 00:12:07,975 --> 00:12:09,454 If that's the right answer. 190 00:12:09,454 --> 00:12:10,933 Then if have the multiple choice, 191 00:12:10,933 --> 00:12:14,384 and they forgot to circle 1 over square root of 2. 192 00:12:14,384 --> 00:12:17,342 I still give you 100% on that problem. 193 00:12:17,342 --> 00:12:20,300 Some professor do not. 194 00:12:20,300 --> 00:12:24,244 So this is at the latitude at whoever makes the rules, 195 00:12:24,244 --> 00:12:29,167 or whoever writes the exam. 196 00:12:29,167 --> 00:12:29,667 OK. 197 00:12:29,667 --> 00:12:32,180 198 00:12:32,180 --> 00:12:39,400 So again for the final, even for the multiple choice problems, 199 00:12:39,400 --> 00:12:41,580 I still need the solutions. 200 00:12:41,580 --> 00:12:46,002 I'm going to ask you to use a bluebook. 201 00:12:46,002 --> 00:12:50,880 Some professors do not ask you to use a bluebook. 202 00:12:50,880 --> 00:12:54,770 They say, as long as you can write on the sheet, circle 203 00:12:54,770 --> 00:12:56,690 the answer, I'm fine. 204 00:12:56,690 --> 00:12:57,440 I'm not fine. 205 00:12:57,440 --> 00:13:00,010 I want to keep what's in the bluebook. 206 00:13:00,010 --> 00:13:01,850 So buy-- how much is it? 207 00:13:01,850 --> 00:13:03,320 Like a dollar? 208 00:13:03,320 --> 00:13:06,260 Buy the books ahead of time, make sure you have them. 209 00:13:06,260 --> 00:13:13,120 Now, number five was a piece of cake once you did number four. 210 00:13:13,120 --> 00:13:15,570 You have a question? 211 00:13:15,570 --> 00:13:18,330 STUDENT: What size bluebook do you need for the final? 212 00:13:18,330 --> 00:13:20,020 PROFESSOR: The big one. 213 00:13:20,020 --> 00:13:22,960 Bigger than that, right? 214 00:13:22,960 --> 00:13:27,913 The direction u for five. 215 00:13:27,913 --> 00:13:33,314 With the problem four was i plus j over root 2, right? 216 00:13:33,314 --> 00:13:36,751 This is what you remember that you did in problem four. 217 00:13:36,751 --> 00:13:40,690 If you didn't do problem four, you cannot do problem five. 218 00:13:40,690 --> 00:13:44,272 Problem five says, this is parallel to one line. 219 00:13:44,272 --> 00:13:49,706 This is parallel to-- what is i plus j? 220 00:13:49,706 --> 00:13:51,372 Of course, you don't have to draw that. 221 00:13:51,372 --> 00:13:52,830 I'm not expecting you to draw that. 222 00:13:52,830 --> 00:13:54,670 y equals x is the first bisection. 223 00:13:54,670 --> 00:13:59,460 224 00:13:59,460 --> 00:14:01,830 All you had to do was circle C, and that 225 00:14:01,830 --> 00:14:04,740 was-- once you circled C, you get full credit. 226 00:14:04,740 --> 00:14:09,220 If you don't do that, you don't get credit for anything. 227 00:14:09,220 --> 00:14:11,690 Now six. 228 00:14:11,690 --> 00:14:16,043 What is the maximum rate of increase of the function 229 00:14:16,043 --> 00:14:22,031 z the same of your friend, your fellow z equals 230 00:14:22,031 --> 00:14:27,520 e to the xy at p0, coordinates 1, 1? 231 00:14:27,520 --> 00:14:33,520 Then the value of the maximum rate of change is? 232 00:14:33,520 --> 00:14:34,060 A noun. 233 00:14:34,060 --> 00:14:36,862 234 00:14:36,862 --> 00:14:38,600 What's the simplest way to do it? 235 00:14:38,600 --> 00:14:40,600 There are two ways to do it. 236 00:14:40,600 --> 00:14:42,955 One is the long way, one is the short way. 237 00:14:42,955 --> 00:14:45,800 What's the short way, guys? 238 00:14:45,800 --> 00:14:51,070 Just compute the length of the gradient. 239 00:14:51,070 --> 00:14:54,620 The length of the gradient at the point P. 240 00:14:54,620 --> 00:15:03,210 So you have whatever that was, ee in length. 241 00:15:03,210 --> 00:15:06,380 So the answer was e root 2. 242 00:15:06,380 --> 00:15:07,900 Am I right? 243 00:15:07,900 --> 00:15:09,290 What was the long way? 244 00:15:09,290 --> 00:15:11,177 I saw somebody do it. 245 00:15:11,177 --> 00:15:14,110 This is a lot more work, but of course, 246 00:15:14,110 --> 00:15:18,110 would be to compute the directional derivative 247 00:15:18,110 --> 00:15:20,510 at the point p for this function. 248 00:15:20,510 --> 00:15:24,030 In the direction of u, where u is the gradient 249 00:15:24,030 --> 00:15:27,980 divided by the length. 250 00:15:27,980 --> 00:15:28,780 at the point p. 251 00:15:28,780 --> 00:15:30,745 And you get, of course, the same answer. 252 00:15:30,745 --> 00:15:31,245 Why? 253 00:15:31,245 --> 00:15:35,700 Because we proved that actually the maximum rate of change 254 00:15:35,700 --> 00:15:38,670 represented directional derivative exactly 255 00:15:38,670 --> 00:15:41,640 in the direction given by the gradient. 256 00:15:41,640 --> 00:15:43,620 This is something we proved. 257 00:15:43,620 --> 00:15:48,075 One of the few things we proved in this class. 258 00:15:48,075 --> 00:15:50,055 Alright. 259 00:15:50,055 --> 00:15:54,015 So the answer was e root 2. 260 00:15:54,015 --> 00:15:55,500 Let's move on to number seven. 261 00:15:55,500 --> 00:16:01,935 Number seven-- and remind me of your five points. 262 00:16:01,935 --> 00:16:07,380 Can you email me, so I have an Excel sheet, 263 00:16:07,380 --> 00:16:09,220 and I'll put it in. 264 00:16:09,220 --> 00:16:15,510 Consider the function f of x, y e to the negative x squared, 265 00:16:15,510 --> 00:16:17,750 y squared. 266 00:16:17,750 --> 00:16:20,990 What can you tell me about this type of function? 267 00:16:20,990 --> 00:16:22,740 It's the headache function. 268 00:16:22,740 --> 00:16:25,620 If I would ask you to do an anti-derivative of each 269 00:16:25,620 --> 00:16:28,620 of the negative squares, you would say Magdalene, 270 00:16:28,620 --> 00:16:31,580 didn't you say that this is impossible? 271 00:16:31,580 --> 00:16:38,980 While the anti-derivative exists, it cannot be expressed. 272 00:16:38,980 --> 00:16:42,205 It cannot be expressed as an elementary function. 273 00:16:42,205 --> 00:16:44,145 And that's a big headache. 274 00:16:44,145 --> 00:16:47,055 This problem is beautiful, why is it beautiful? 275 00:16:47,055 --> 00:16:50,935 Because in the end, it becomes magic. 276 00:16:50,935 --> 00:16:53,845 So it's a positive function. 277 00:16:53,845 --> 00:16:56,990 It's like a bell on top of the church something. 278 00:16:56,990 --> 00:17:00,930 And then, you have to compute double integral 279 00:17:00,930 --> 00:17:06,368 over the unit disk of centers of 0 and radius 1. 280 00:17:06,368 --> 00:17:09,770 Of e to the negative x squared minus y squared dx/dy. 281 00:17:09,770 --> 00:17:12,280 282 00:17:12,280 --> 00:17:15,618 Well then you say, well I've done this kind of thing 283 00:17:15,618 --> 00:17:19,829 before, but not with Cartesian coordinates. 284 00:17:19,829 --> 00:17:24,880 We did it with the Jacobian r, that changes everything 285 00:17:24,880 --> 00:17:28,790 into polar coordinates. 286 00:17:28,790 --> 00:17:34,130 So this guy becomes e to the minus r squared. 287 00:17:34,130 --> 00:17:39,140 Each of the numbers are squared dr, d theta. 288 00:17:39,140 --> 00:17:42,000 D on the unit [INAUDIBLE] disk means 289 00:17:42,000 --> 00:17:44,730 the radius goes from 0 to 1. 290 00:17:44,730 --> 00:17:48,540 This is a blessing for us, because it's easy data. 291 00:17:48,540 --> 00:17:51,030 Then we have 0 to 2 pi. 292 00:17:51,030 --> 00:17:53,790 You could have put it in any order. 293 00:17:53,790 --> 00:17:58,870 For u, it's easier to close your eyes when it comes to theta. 294 00:17:58,870 --> 00:18:01,530 Say, theta is independent. 295 00:18:01,530 --> 00:18:05,350 He is like a partition that has to do nothing 296 00:18:05,350 --> 00:18:07,440 with what's inside here. 297 00:18:07,440 --> 00:18:10,710 So let's pull him out of this picture. 298 00:18:10,710 --> 00:18:14,132 And he wants to live by himself. 299 00:18:14,132 --> 00:18:18,614 An integral from 0 to 2 pi of d theta was of course 2 pi. 300 00:18:18,614 --> 00:18:22,100 He's happy to go out, having fun. 301 00:18:22,100 --> 00:18:26,590 This guy inside has to be thoroughly computed. 302 00:18:26,590 --> 00:18:30,100 In the sense that you perform the substitution. 303 00:18:30,100 --> 00:18:39,590 I was actually amused that half of you did u equals r squared, 304 00:18:39,590 --> 00:18:42,840 and half of you did u equals minus r squared. 305 00:18:42,840 --> 00:18:44,600 It really doesn't matter which one. 306 00:18:44,600 --> 00:18:46,940 But the problem is that some of you 307 00:18:46,940 --> 00:18:51,670 made a mess when you put the limit points back in place, 308 00:18:51,670 --> 00:18:53,640 and you made mistakes. 309 00:18:53,640 --> 00:18:56,490 Somebody even got negative answers, 310 00:18:56,490 --> 00:18:59,860 I was about to fall off the chair. 311 00:18:59,860 --> 00:19:04,130 Of course, I was in a good mood because it was a holiday, 312 00:19:04,130 --> 00:19:05,270 I graded them. 313 00:19:05,270 --> 00:19:09,040 Fortunately, I graded them over the break. 314 00:19:09,040 --> 00:19:12,330 So after I came back from Georgia. 315 00:19:12,330 --> 00:19:18,680 I have minus r dr. rdr was minus a half du. 316 00:19:18,680 --> 00:19:21,360 317 00:19:21,360 --> 00:19:24,656 This fellow is just into the u, and he's 318 00:19:24,656 --> 00:19:31,104 a blessing because the [INAUDIBLE] 319 00:19:31,104 --> 00:19:36,560 So into the u, however, take it between 1 and what? 320 00:19:36,560 --> 00:19:38,048 Not 0 and 1. 321 00:19:38,048 --> 00:19:41,520 But when you have 0 here, you have 0 here. 322 00:19:41,520 --> 00:19:44,496 When you have 1, you have minus 1. 323 00:19:44,496 --> 00:19:48,470 So pay attention to that, otherwise, you 324 00:19:48,470 --> 00:19:52,440 get something that makes no sense. 325 00:19:52,440 --> 00:19:55,010 Times minus a half. 326 00:19:55,010 --> 00:19:59,475 That, you will have to be careful about. 327 00:19:59,475 --> 00:19:59,975 Why? 328 00:19:59,975 --> 00:20:04,097 Because there will be a minus from here and here, in the end, 329 00:20:04,097 --> 00:20:05,310 the answer will be positive. 330 00:20:05,310 --> 00:20:08,270 And that's reminding me of that city plumber joke 331 00:20:08,270 --> 00:20:11,926 when he doesn't pay attention to the limits of integration. 332 00:20:11,926 --> 00:20:16,910 And you can get a minus volume, or a minus area. 333 00:20:16,910 --> 00:20:24,010 So e to the minus 1 minus 1. 334 00:20:24,010 --> 00:20:25,800 But that leaves a negative number, 335 00:20:25,800 --> 00:20:34,270 but when you multiply it by a minus, you have 1 minus 1/e. 336 00:20:34,270 --> 00:20:35,060 1 minus 1/e. 337 00:20:35,060 --> 00:20:37,850 338 00:20:37,850 --> 00:20:38,540 Good, thank god. 339 00:20:38,540 --> 00:20:41,130 This is a nice guy, less than 1. 340 00:20:41,130 --> 00:20:45,085 And this is key to your answer, because 2 goes away 341 00:20:45,085 --> 00:20:48,373 and pi stays in place and this is less than pi. 342 00:20:48,373 --> 00:20:52,480 So the answer to this question was an answer less than pi. 343 00:20:52,480 --> 00:20:55,070 And if you didn't get it, I'm very sorry, 344 00:20:55,070 --> 00:21:01,130 if you didn't get less than pi, you didn't get any points. 345 00:21:01,130 --> 00:21:04,520 But, there are enough chances for you to get another point. 346 00:21:04,520 --> 00:21:14,476 I was brokenhearted for 10 people or more out of 25 347 00:21:14,476 --> 00:21:24,094 did not remember what I taught in class about the area 348 00:21:24,094 --> 00:21:28,486 of a collateral triangle. 349 00:21:28,486 --> 00:21:31,902 And it broke my heart, and I was about to cry, 350 00:21:31,902 --> 00:21:35,318 but I said, c'mon, they'll do it better in the final. 351 00:21:35,318 --> 00:21:39,222 Honestly, I was so brokenhearted. 352 00:21:39,222 --> 00:21:41,700 So this is 1, 0, 0. 353 00:21:41,700 --> 00:21:42,620 This was 0, 1, 0. 354 00:21:42,620 --> 00:21:44,945 This was 0, 0, 1. 355 00:21:44,945 --> 00:21:49,332 On number eight. 356 00:21:49,332 --> 00:21:50,316 Thank you. 357 00:21:50,316 --> 00:22:03,108 358 00:22:03,108 --> 00:22:04,092 Beautiful. 359 00:22:04,092 --> 00:22:10,380 It's an equilateral triangle, and the l side 360 00:22:10,380 --> 00:22:14,594 of that equilateral triangle is the square root of 2. 361 00:22:14,594 --> 00:22:17,106 I even taught you how to cheat. 362 00:22:17,106 --> 00:22:17,980 That's why I was mad. 363 00:22:17,980 --> 00:22:22,120 I taught you how to cheat, and you didn't take advantage. 364 00:22:22,120 --> 00:22:29,550 So the area was l squared, square root of 2, 4. 365 00:22:29,550 --> 00:22:33,840 Which we did this together in fifth or sixth grade 366 00:22:33,840 --> 00:22:40,616 by multiplying that height and the width and divided by 2. 367 00:22:40,616 --> 00:22:43,520 And then we came up with this formula with the Pythagorean 368 00:22:43,520 --> 00:22:45,456 theorem in the classroom. 369 00:22:45,456 --> 00:22:49,131 If eligible to, you can very quickly get an answer. 370 00:22:49,131 --> 00:22:54,382 So that's going to be 2 root 2 over 4, just root 3 over 2. 371 00:22:54,382 --> 00:22:56,840 And when I saw that people got something else except root 3 372 00:22:56,840 --> 00:22:59,400 over 2, that broke my heart. 373 00:22:59,400 --> 00:23:01,240 Really. 374 00:23:01,240 --> 00:23:06,200 You have plenty of time to catch up 375 00:23:06,200 --> 00:23:07,665 with that being on your final. 376 00:23:07,665 --> 00:23:10,320 377 00:23:10,320 --> 00:23:14,230 Did I expect you to really do the surface integral? 378 00:23:14,230 --> 00:23:20,220 Some people again, need to write integral over the shaded domain 379 00:23:20,220 --> 00:23:24,213 d a square root of f sub x squared 380 00:23:24,213 --> 00:23:28,440 plus f sub y squared plus 1. 381 00:23:28,440 --> 00:23:31,100 That was the right track, because this is root 3. 382 00:23:31,100 --> 00:23:37,611 And then the area, you get the area of the 1 times 1 over 2, 383 00:23:37,611 --> 00:23:38,110 right? 384 00:23:38,110 --> 00:23:40,438 1/3 is the area. 385 00:23:40,438 --> 00:23:45,398 Root 3 gets out of this, so you have-- when you integrate, 386 00:23:45,398 --> 00:23:49,366 you have the area of the shaded base that I have. 387 00:23:49,366 --> 00:23:51,350 And you get the same answer. 388 00:23:51,350 --> 00:23:54,326 No matter how you do it, with calculators 389 00:23:54,326 --> 00:23:58,790 or without calculators, you still could have passed. 390 00:23:58,790 --> 00:24:02,540 Am I if you didn't get the answer? 391 00:24:02,540 --> 00:24:03,080 No. 392 00:24:03,080 --> 00:24:04,844 Absolutely. 393 00:24:04,844 --> 00:24:08,990 But it hurts me as if, I don't know, a relative of mine 394 00:24:08,990 --> 00:24:13,730 messed up some task. 395 00:24:13,730 --> 00:24:16,930 That's why it's better that you don't know your students, 396 00:24:16,930 --> 00:24:21,010 because when you know your students, 397 00:24:21,010 --> 00:24:23,000 you know that they could have done better, 398 00:24:23,000 --> 00:24:24,130 because you know them. 399 00:24:24,130 --> 00:24:27,050 So we can say, OK, it really hurts 400 00:24:27,050 --> 00:24:30,010 when you know that they messed up, not because they are not 401 00:24:30,010 --> 00:24:34,580 smart or educated, but because they just either didn't 402 00:24:34,580 --> 00:24:37,622 pay attention or they were stressed out. 403 00:24:37,622 --> 00:24:41,840 However, my substitute, the guy who came here, 404 00:24:41,840 --> 00:24:43,220 was my Ph.D. student. 405 00:24:43,220 --> 00:24:47,564 He got a doctoral degree mathematics with me last year. 406 00:24:47,564 --> 00:24:50,115 And he told me you were not stressed out at all. 407 00:24:50,115 --> 00:24:51,625 And I said, thank god. 408 00:24:51,625 --> 00:24:55,100 I'm glad that they were calm. 409 00:24:55,100 --> 00:24:58,235 And he said, I didn't look at the exam, 410 00:24:58,235 --> 00:25:01,205 but it seemed like they did very well and they were comfortable. 411 00:25:01,205 --> 00:25:02,690 And I was so happy. 412 00:25:02,690 --> 00:25:04,670 I was in Athens, Georgia. 413 00:25:04,670 --> 00:25:06,155 And reading this email I said, yay! 414 00:25:06,155 --> 00:25:07,640 Everybody's going to get an A! 415 00:25:07,640 --> 00:25:10,610 So I come home and I start grading it. 416 00:25:10,610 --> 00:25:16,055 I was sad to see that my prediction was not correct. 417 00:25:16,055 --> 00:25:20,026 But anyway, [INAUDIBLE] with an average of B. 418 00:25:20,026 --> 00:25:21,930 For an honors class, it's OK. 419 00:25:21,930 --> 00:25:24,786 I just expected a lot better. 420 00:25:24,786 --> 00:25:29,560 And I know it's going to be a lot better in the final. 421 00:25:29,560 --> 00:25:32,430 Number nine. 422 00:25:32,430 --> 00:25:34,744 This was done by almost everybody, 423 00:25:34,744 --> 00:25:36,910 except for a few people who messed up on the limits. 424 00:25:36,910 --> 00:25:38,365 I don't know why. 425 00:25:38,365 --> 00:25:44,185 When they compute-- when they drew, they drew x squared, 426 00:25:44,185 --> 00:25:46,125 and they drew square root of xn. 427 00:25:46,125 --> 00:25:49,520 Of course, you were supposed-- the answer was 0 to 1, 428 00:25:49,520 --> 00:25:50,980 integral of. 429 00:25:50,980 --> 00:25:56,140 Now, if you do first x, you have x from y 430 00:25:56,140 --> 00:25:57,564 squared to square root of y. 431 00:25:57,564 --> 00:25:58,460 You guys with me? 432 00:25:58,460 --> 00:26:02,170 Because this is smaller than that. 433 00:26:02,170 --> 00:26:02,930 OK? 434 00:26:02,930 --> 00:26:08,190 So you have 1 and dx dy equals to integral from 0 to 1, 435 00:26:08,190 --> 00:26:11,690 integral x squared to square root of x1 dy dx. 436 00:26:11,690 --> 00:26:17,190 437 00:26:17,190 --> 00:26:24,270 Now, what a few people did-- and I just forgave them. 438 00:26:24,270 --> 00:26:29,216 They just-- one put this like that. 439 00:26:29,216 --> 00:26:31,790 And here, he put root 2. 440 00:26:31,790 --> 00:26:33,380 Root y and y squared. 441 00:26:33,380 --> 00:26:34,600 Don't do that. 442 00:26:34,600 --> 00:26:37,495 It's like chasing that a positive number equals 443 00:26:37,495 --> 00:26:41,880 a negative number, which is all complete nonsense. 444 00:26:41,880 --> 00:26:45,770 So the correct answer was we put y squared down, 445 00:26:45,770 --> 00:26:48,805 and square root of y because this guy is 446 00:26:48,805 --> 00:26:54,000 bigger than this guy for something between 0 and 1. 447 00:26:54,000 --> 00:26:54,960 Because I told you. 448 00:26:54,960 --> 00:27:04,130 Square root of 0.04 is bigger than the square of that. 449 00:27:04,130 --> 00:27:06,040 OK. 450 00:27:06,040 --> 00:27:15,170 Now am I happy with that? 451 00:27:15,170 --> 00:27:16,450 I'm quite happy. 452 00:27:16,450 --> 00:27:21,578 In general, people understood the vertical strip method 453 00:27:21,578 --> 00:27:23,940 compared to the horizontal strip method. 454 00:27:23,940 --> 00:27:25,350 And why am I happy? 455 00:27:25,350 --> 00:27:30,110 Because I was asked by three people from other classes 456 00:27:30,110 --> 00:27:33,610 to help them, over there, on the corridor. 457 00:27:33,610 --> 00:27:35,225 And I asked them, who is your teacher? 458 00:27:35,225 --> 00:27:36,190 This and that. 459 00:27:36,190 --> 00:27:40,820 But we did not understand reversing the order 460 00:27:40,820 --> 00:27:42,158 of integration in class. 461 00:27:42,158 --> 00:27:43,652 And I said, how come? 462 00:27:43,652 --> 00:27:45,644 Well, they didn't explain it very well. 463 00:27:45,644 --> 00:27:47,636 So I started explaining it to them. 464 00:27:47,636 --> 00:27:50,126 And then I realized that it's a conflict of interest. 465 00:27:50,126 --> 00:27:52,616 I'm not allowed to do that. 466 00:27:52,616 --> 00:27:55,604 And then I go, oh my god, I cannot do the homework for you. 467 00:27:55,604 --> 00:27:56,600 I'm not allowed. 468 00:27:56,600 --> 00:27:59,610 But I was already talking. 469 00:27:59,610 --> 00:28:04,350 So I said, guys, can you do it? 470 00:28:04,350 --> 00:28:05,120 I don't know. 471 00:28:05,120 --> 00:28:07,060 I said, do you draw? 472 00:28:07,060 --> 00:28:07,950 Why would we draw? 473 00:28:07,950 --> 00:28:10,050 They didn't teach us how to draw. 474 00:28:10,050 --> 00:28:13,235 I said, but how do you know about vertical strips 475 00:28:13,235 --> 00:28:14,460 and horizontal strips? 476 00:28:14,460 --> 00:28:15,440 No. 477 00:28:15,440 --> 00:28:17,890 And how do you do this? 478 00:28:17,890 --> 00:28:18,870 We don't know. 479 00:28:18,870 --> 00:28:21,830 We felt like we have to figure it out. 480 00:28:21,830 --> 00:28:25,446 Without drawing, without understanding how the vertical 481 00:28:25,446 --> 00:28:27,816 strips are drawn between two functions, 482 00:28:27,816 --> 00:28:31,070 and how you switch the horizontal strips, 483 00:28:31,070 --> 00:28:33,340 you cannot do this problem, period. 484 00:28:33,340 --> 00:28:36,318 So if you don't have-- maybe some people have 485 00:28:36,318 --> 00:28:39,180 enough imagination-- but that's very rare-- 486 00:28:39,180 --> 00:28:40,937 That they can close their eyes and they 487 00:28:40,937 --> 00:28:44,960 can see a picture with their eyes closed 488 00:28:44,960 --> 00:28:46,050 and they can solve that. 489 00:28:46,050 --> 00:28:48,175 But that's not the way to learn. 490 00:28:48,175 --> 00:28:51,790 The way to learn is a very visual learning thing. 491 00:28:51,790 --> 00:28:54,459 So that's why we draw all the time. 492 00:28:54,459 --> 00:28:57,324 493 00:28:57,324 --> 00:28:59,449 STUDENT: Professor, you can cheat these with Cal 2. 494 00:28:59,449 --> 00:29:00,447 PROFESSOR: Yes. 495 00:29:00,447 --> 00:29:02,443 You can do that with Cal 2. 496 00:29:02,443 --> 00:29:03,441 What's the problem? 497 00:29:03,441 --> 00:29:05,936 You have integral from 0 to 1. 498 00:29:05,936 --> 00:29:09,380 Square root of y minus y squared. 499 00:29:09,380 --> 00:29:17,440 Well, they learn to do the other one. 500 00:29:17,440 --> 00:29:21,750 The one with square root x minus x squared, 0,1 and so on. 501 00:29:21,750 --> 00:29:26,246 But they were told explicitly to write-- the professor even 502 00:29:26,246 --> 00:29:29,772 left these empty and put spaces, fill in the spaces. 503 00:29:29,772 --> 00:29:32,022 And they say, how the heck do we fill in those spaces? 504 00:29:32,022 --> 00:29:35,004 Plus the whiteboard problems have the empty spaces. 505 00:29:35,004 --> 00:29:36,992 And they couldn't believe that at all. 506 00:29:36,992 --> 00:29:40,970 And one of them went to the tutoring center and was lucky. 507 00:29:40,970 --> 00:29:43,130 Because he got-- this is like when 508 00:29:43,130 --> 00:29:45,827 you go to a medical doctor, sometimes you 509 00:29:45,827 --> 00:29:48,607 are lucky and get a good doctor who takes care of you, 510 00:29:48,607 --> 00:29:49,982 figures out what your problem is. 511 00:29:49,982 --> 00:29:54,140 And sometimes, they give you the wrong medicine. 512 00:29:54,140 --> 00:29:58,010 So one of them got the right tutor who knew how to explain 513 00:29:58,010 --> 00:29:59,860 and sort of knew something. 514 00:29:59,860 --> 00:30:04,176 But the other one got a tutor who never took Calculus 3 515 00:30:04,176 --> 00:30:09,040 and said, I don't know what the heck these multiple snakes are. 516 00:30:09,040 --> 00:30:11,652 So I'm not going to be able to help you. 517 00:30:11,652 --> 00:30:14,991 So he was very disappointed. 518 00:30:14,991 --> 00:30:15,490 OK. 519 00:30:15,490 --> 00:30:20,110 Compute the area of the domain D from the previous problem. 520 00:30:20,110 --> 00:30:23,160 This was something that nobody's telling you, 521 00:30:23,160 --> 00:30:26,270 hey, you have to do it with the double snakes. 522 00:30:26,270 --> 00:30:28,715 You can do it with just with a simple snake 523 00:30:28,715 --> 00:30:31,110 and you're still fine. 524 00:30:31,110 --> 00:30:37,020 So in Calc 1-- this Calc 1, whatever it is. 525 00:30:37,020 --> 00:30:41,842 In Calc 1, you learn that you have to integrate this 526 00:30:41,842 --> 00:30:49,000 and you'll get 2/3 x to the 3/2 minus 1/3 x 527 00:30:49,000 --> 00:30:56,007 cubed at x equals 1 minus whatever you have with 0. 528 00:30:56,007 --> 00:30:59,430 But at 0, you have 0, so you say, forget about it. 529 00:30:59,430 --> 00:31:05,290 And you have 2/3 minus 1/3 equals 1/3, then you're done. 530 00:31:05,290 --> 00:31:05,790 OK? 531 00:31:05,790 --> 00:31:08,130 Did I expect you to show me work? 532 00:31:08,130 --> 00:31:09,380 No. 533 00:31:09,380 --> 00:31:11,890 For everybody who wrote 1.3-- and there 534 00:31:11,890 --> 00:31:13,990 were many people who did this mentally, 535 00:31:13,990 --> 00:31:17,800 and they came up with 1/3. 536 00:31:17,800 --> 00:31:20,900 They got 10 pionts on the problem. 537 00:31:20,900 --> 00:31:25,990 Finally, number 11. 538 00:31:25,990 --> 00:31:29,130 Without computing the volume inside the sphere, 539 00:31:29,130 --> 00:31:35,094 x squared plus y squared plus z squared equals 2. 540 00:31:35,094 --> 00:31:37,850 541 00:31:37,850 --> 00:31:42,860 Set up a triple integral corresponding to it 542 00:31:42,860 --> 00:31:44,748 in the space provided below. 543 00:31:44,748 --> 00:31:48,241 544 00:31:48,241 --> 00:31:52,689 Some people, a few people, messed up. 545 00:31:52,689 --> 00:31:53,730 They forgot the Jacobian. 546 00:31:53,730 --> 00:31:57,223 So they put the 1 instead of r squared [? side-side. ?] 547 00:31:57,223 --> 00:32:01,340 When you work in three components, 548 00:32:01,340 --> 00:32:04,290 they do fine setting up the limits. 549 00:32:04,290 --> 00:32:05,990 [INAUDIBLE] 1 here. 550 00:32:05,990 --> 00:32:07,380 Don't look at it in the final. 551 00:32:07,380 --> 00:32:09,370 You can ruin your life this way. 552 00:32:09,370 --> 00:32:11,950 So we have r squared sine phi. 553 00:32:11,950 --> 00:32:16,150 Phi was the latitude from the North Pole. 554 00:32:16,150 --> 00:32:18,510 it doesn't matter in which order you do it. 555 00:32:18,510 --> 00:32:21,920 But I would do to er b phi b theta. 556 00:32:21,920 --> 00:32:25,380 You tell me what the end points are, and we are done. 557 00:32:25,380 --> 00:32:26,370 STUDENT: From 0 to 5. 558 00:32:26,370 --> 00:32:27,360 PROFESSOR: 0 to-- 559 00:32:27,360 --> 00:32:29,340 STUDENT: No, on the first one. 560 00:32:29,340 --> 00:32:31,315 PROFESSOR: 0 to-- 561 00:32:31,315 --> 00:32:31,815 STUDENT: Dr? 562 00:32:31,815 --> 00:32:33,300 It's the square root of 2. 563 00:32:33,300 --> 00:32:34,290 PROFESSOR: Mm-hmm. 564 00:32:34,290 --> 00:32:35,775 STUDENT: And b theta-- 565 00:32:35,775 --> 00:32:37,260 PROFESSOR: 0. 566 00:32:37,260 --> 00:32:38,745 2pi. 567 00:32:38,745 --> 00:32:41,540 And theta, all around. 568 00:32:41,540 --> 00:32:42,508 STUDENT: 2pi. 569 00:32:42,508 --> 00:32:45,880 PROFESSOR: Longitude 360 meridian degrees. 570 00:32:45,880 --> 00:32:46,380 OK. 571 00:32:46,380 --> 00:32:47,832 0 to 2pi. 572 00:32:47,832 --> 00:32:49,284 So good. 573 00:32:49,284 --> 00:32:50,330 So we are done. 574 00:32:50,330 --> 00:32:52,000 Did I expect you to write it down? 575 00:32:52,000 --> 00:32:52,560 No. 576 00:32:52,560 --> 00:32:57,130 I had three people who were nice and wrote down 4. 577 00:32:57,130 --> 00:32:58,850 I mean, they actually did the work. 578 00:32:58,850 --> 00:33:00,810 Maybe they had nothing better to do. 579 00:33:00,810 --> 00:33:02,400 I have no idea why. 580 00:33:02,400 --> 00:33:04,820 4pi i cubed over 3, right? 581 00:33:04,820 --> 00:33:09,808 And then they proved the formula in general using the Jacobian. 582 00:33:09,808 --> 00:33:14,638 Using the formula, they got the correct formula for r 583 00:33:14,638 --> 00:33:16,087 equals square root of 2. 584 00:33:16,087 --> 00:33:18,030 And I was very happy. 585 00:33:18,030 --> 00:33:19,731 But did I ask you to do that? 586 00:33:19,731 --> 00:33:20,230 No. 587 00:33:20,230 --> 00:33:22,070 Did I give you extra credit. 588 00:33:22,070 --> 00:33:22,990 No. 589 00:33:22,990 --> 00:33:26,810 So all the extra credit was just one problem to 590 00:33:26,810 --> 00:33:30,121 asked to do exactly what you were told to do. 591 00:33:30,121 --> 00:33:32,960 592 00:33:32,960 --> 00:33:36,490 I don't know about how you feel about this exam, 593 00:33:36,490 --> 00:33:39,150 but it wasn't a hard exam. 594 00:33:39,150 --> 00:33:41,138 It was not an easy exam. 595 00:33:41,138 --> 00:33:45,106 It was an exam that was supposed to test 596 00:33:45,106 --> 00:33:49,570 what you learned until now all through the course. 597 00:33:49,570 --> 00:33:53,538 And that was the whole idea. 598 00:33:53,538 --> 00:33:55,890 I think you've learned very much, 599 00:33:55,890 --> 00:33:58,920 and I think you did fine, the majority of you. 600 00:33:58,920 --> 00:34:02,920 And that should ease the pressure on you 601 00:34:02,920 --> 00:34:05,420 when it comes to preparing for the final. 602 00:34:05,420 --> 00:34:10,080 I was thinking last night, I'm going to send you, probably 603 00:34:10,080 --> 00:34:13,121 by email or in-person in class, two 604 00:34:13,121 --> 00:34:17,049 or three samples of the final from old finals 605 00:34:17,049 --> 00:34:22,449 that inspire us when we write the final. 606 00:34:22,449 --> 00:34:25,887 A few of us will provide problems and comments 607 00:34:25,887 --> 00:34:29,324 and suggestions when we write out the departmental final. 608 00:34:29,324 --> 00:34:33,252 But the final will be departmental for all sections. 609 00:34:33,252 --> 00:34:37,670 I don't expect more than 15 problems on the final. 610 00:34:37,670 --> 00:34:44,264 I have yet to think and decide if I want to [? lift ?] 611 00:34:44,264 --> 00:34:46,045 probably the same policy. 612 00:34:46,045 --> 00:34:47,889 I mean, the final is the same for everybody. 613 00:34:47,889 --> 00:34:51,538 But the policy about how to give partial credit 614 00:34:51,538 --> 00:34:54,090 or not give partial credit. [INAUDIBLE]. 615 00:34:54,090 --> 00:34:57,609 And I already decided that I'm going to read everything, 616 00:34:57,609 --> 00:35:00,794 so in case that you mess up at the end with your miracle 617 00:35:00,794 --> 00:35:02,509 answer, you still get partial credit 618 00:35:02,509 --> 00:35:05,939 for your integrals [INAUDIBLE] shown. 619 00:35:05,939 --> 00:35:10,360 Also, one of those 15 problems. might be for extra credit. 620 00:35:10,360 --> 00:35:12,550 I have to think a little bit better 621 00:35:12,550 --> 00:35:17,580 how-- what is the maximum weight I want to put. 622 00:35:17,580 --> 00:35:22,600 What I would say, since I never [INAUDIBLE] open a homework, 623 00:35:22,600 --> 00:35:27,030 and I never curve exams, I would think 624 00:35:27,030 --> 00:35:33,714 I could make 110% as the possible maximum. 625 00:35:33,714 --> 00:35:37,102 In this case, you have some cushion 626 00:35:37,102 --> 00:35:42,360 to make a mistake or two and still get a perfect score. 627 00:35:42,360 --> 00:35:43,670 OK. 628 00:35:43,670 --> 00:35:46,841 I'm going to move on to a new chapter. 629 00:35:46,841 --> 00:35:49,296 I have actually moved on already, 630 00:35:49,296 --> 00:35:51,751 but nobody believed me. 631 00:35:51,751 --> 00:35:56,170 Last time, I started Chapter 13. 632 00:35:56,170 --> 00:36:04,200 Chapter 13 is a mixture of mathematics and physics. 633 00:36:04,200 --> 00:36:07,083 You will be surprised how many things 634 00:36:07,083 --> 00:36:10,751 are coming from solid mechanics, fluid mechanics. 635 00:36:10,751 --> 00:36:12,218 Yes, Regan. 636 00:36:12,218 --> 00:36:14,670 STUDENT: [INAUDIBLE] 637 00:36:14,670 --> 00:36:16,330 PROFESSOR: For a job? 638 00:36:16,330 --> 00:36:18,236 You want me to come with you? 639 00:36:18,236 --> 00:36:19,670 [LAUGHTER] 640 00:36:19,670 --> 00:36:22,060 STUDENT: Because I tried to talk to you [INAUDIBLE]. 641 00:36:22,060 --> 00:36:24,928 PROFESSOR: Yes, yes. 642 00:36:24,928 --> 00:36:26,840 Yes. 643 00:36:26,840 --> 00:36:28,290 Yeah. 644 00:36:28,290 --> 00:36:29,865 And you have to sign up. 645 00:36:29,865 --> 00:36:32,131 Start a [? sheet, ?] attend [? the sheet, ?] and sign 646 00:36:32,131 --> 00:36:34,777 your name and good luck with the interview. 647 00:36:34,777 --> 00:36:37,182 You should have told me before! 648 00:36:37,182 --> 00:36:39,106 I could have said a prayer for you. 649 00:36:39,106 --> 00:36:41,030 This things are very stressful! 650 00:36:41,030 --> 00:36:43,429 I remember my own interviews. 651 00:36:43,429 --> 00:36:44,220 There were several. 652 00:36:44,220 --> 00:36:47,671 I didn't know anything about it, and my hands were all sweaty. 653 00:36:47,671 --> 00:36:49,962 And you know you should never shake hands with somebody 654 00:36:49,962 --> 00:36:51,954 when your hands are sweaty. 655 00:36:51,954 --> 00:36:54,942 You have to do like this first. 656 00:36:54,942 --> 00:36:57,432 Be confident and don't be nervous. 657 00:36:57,432 --> 00:36:59,424 Don't sweat or anything. 658 00:36:59,424 --> 00:37:01,414 Because they can see that. 659 00:37:01,414 --> 00:37:01,914 All right. 660 00:37:01,914 --> 00:37:05,610 You just be yourself. 661 00:37:05,610 --> 00:37:06,830 Do you have earrings? 662 00:37:06,830 --> 00:37:10,074 Because after my several job interviews-- 663 00:37:10,074 --> 00:37:13,500 those are good earrings-- I was told that I should never 664 00:37:13,500 --> 00:37:17,630 wear dangling earrings at the interviews, which I did not, 665 00:37:17,630 --> 00:37:18,930 because I didn't have any. 666 00:37:18,930 --> 00:37:20,990 But I love dangling earrings. 667 00:37:20,990 --> 00:37:25,972 And I was asking some academics why that was [? our ?] problem. 668 00:37:25,972 --> 00:37:27,780 And they say they are distracting. 669 00:37:27,780 --> 00:37:30,666 Because mathematicians are like cats. 670 00:37:30,666 --> 00:37:32,292 [LAUGHTER] 671 00:37:32,292 --> 00:37:33,792 PROFESSOR: --pendulum, and then they 672 00:37:33,792 --> 00:37:36,438 get hypnotized by the dangling. 673 00:37:36,438 --> 00:37:37,320 So I don't know. 674 00:37:37,320 --> 00:37:40,914 I think most of the interviewers have some problems 675 00:37:40,914 --> 00:37:45,306 and they find some things distracting or annoying. 676 00:37:45,306 --> 00:37:46,770 Otherwise, I think you are fine. 677 00:37:46,770 --> 00:37:49,686 You're dressed fine for an interview. 678 00:37:49,686 --> 00:37:50,186 OK. 679 00:37:50,186 --> 00:37:52,990 So now serious job. 680 00:37:52,990 --> 00:37:57,886 We have to remember some of the things we don't remember. 681 00:37:57,886 --> 00:38:03,742 Which are the gradient for a function of let's say 682 00:38:03,742 --> 00:38:05,206 three variables. 683 00:38:05,206 --> 00:38:08,134 Let's grow up a little bit. 684 00:38:08,134 --> 00:38:13,290 And that was what the vector field 685 00:38:13,290 --> 00:38:19,630 F sub xi plus F sub [? I j ?] plus F sub z k. 686 00:38:19,630 --> 00:38:20,170 Right? 687 00:38:20,170 --> 00:38:24,354 At an arbitrary point xyz in your domain. 688 00:38:24,354 --> 00:38:29,880 So where xyz is in some domain, you are in a potato. 689 00:38:29,880 --> 00:38:34,940 And the meaning of the gradient, the geometric meaning of this, 690 00:38:34,940 --> 00:38:36,910 doesn't look like a theta [INAUDIBLE]. 691 00:38:36,910 --> 00:38:41,170 It's some sort of solid that it corresponds 692 00:38:41,170 --> 00:38:42,674 to a closed surface. 693 00:38:42,674 --> 00:38:46,530 And this closed surface that closes up on its own 694 00:38:46,530 --> 00:38:49,660 is having a hard time [INAUDIBLE]. 695 00:38:49,660 --> 00:38:51,900 It has a normal. 696 00:38:51,900 --> 00:38:56,650 And this normal is given by the gradient of this function, 697 00:38:56,650 --> 00:38:59,530 we can increase [? it ?] like that. 698 00:38:59,530 --> 00:39:00,970 You remember that. 699 00:39:00,970 --> 00:39:02,970 And that was a long time ago. 700 00:39:02,970 --> 00:39:06,810 But you should still master that. 701 00:39:06,810 --> 00:39:11,980 Last time, I gave you the z equals f of xy, 702 00:39:11,980 --> 00:39:14,490 z equals little f of xy, as a graph 703 00:39:14,490 --> 00:39:18,240 of the function of two variables over a domain in plane. 704 00:39:18,240 --> 00:39:19,980 We computed the gradient of that. 705 00:39:19,980 --> 00:39:22,860 But that's what we did all through the [? meter ?]. 706 00:39:22,860 --> 00:39:24,620 So that's no fun. 707 00:39:24,620 --> 00:39:27,135 We know that too well. 708 00:39:27,135 --> 00:39:33,677 On this problem, I gave you some new piece 709 00:39:33,677 --> 00:39:34,830 of information last time. 710 00:39:34,830 --> 00:39:38,192 So I said, if you have a vector field that 711 00:39:38,192 --> 00:39:44,480 looks F 1i plus F 2j plus F 3k, where 712 00:39:44,480 --> 00:39:50,444 Fi is C1, that means that the differentiable 713 00:39:50,444 --> 00:39:52,924 and the derivatives are continuous, 714 00:39:52,924 --> 00:39:56,892 what was the divergence of it? 715 00:39:56,892 --> 00:39:59,372 Well, that was before the Easter break. 716 00:39:59,372 --> 00:40:01,356 And I know we had a long break. 717 00:40:01,356 --> 00:40:06,316 I cannot recover from this break so easily, because it was long. 718 00:40:06,316 --> 00:40:08,300 And I also traveled last week. 719 00:40:08,300 --> 00:40:13,700 But before I traveled, I remember that I gave you this. 720 00:40:13,700 --> 00:40:15,850 And you memorized it. 721 00:40:15,850 --> 00:40:17,670 Most of you memorised it. 722 00:40:17,670 --> 00:40:19,859 How was it? 723 00:40:19,859 --> 00:40:23,310 The first component differentiated with respect 724 00:40:23,310 --> 00:40:28,733 to the first variable plus the second component 725 00:40:28,733 --> 00:40:33,170 differentiated with respect to the second variable. 726 00:40:33,170 --> 00:40:37,607 Plus the third component differentiated with respect 727 00:40:37,607 --> 00:40:41,058 to the third variable. 728 00:40:41,058 --> 00:40:46,270 So I'm asking you, as an exercise, like I 729 00:40:46,270 --> 00:40:49,390 did last time, the same thing. 730 00:40:49,390 --> 00:40:54,852 Exercise one for this section. 731 00:40:54,852 --> 00:40:57,317 Compute divergence of the gradient 732 00:40:57,317 --> 00:41:04,519 of F, where F is a C1 function of xyz. 733 00:41:04,519 --> 00:41:07,004 That means F is [? like this ?] differentiable 734 00:41:07,004 --> 00:41:08,992 and with continuous derivatives. 735 00:41:08,992 --> 00:41:10,483 What does it mean? 736 00:41:10,483 --> 00:41:15,453 It means that you have to compute divergence of F sub xi 737 00:41:15,453 --> 00:41:20,423 plus F sub yj plus F sub zk. 738 00:41:20,423 --> 00:41:24,010 And you're thinking, I can do that! 739 00:41:24,010 --> 00:41:29,752 By definition, I take the first component-- who was that? 740 00:41:29,752 --> 00:41:30,748 Hmm? 741 00:41:30,748 --> 00:41:32,242 STUDENT: Brian. 742 00:41:32,242 --> 00:41:33,238 PROFESSOR: Oh, right. 743 00:41:33,238 --> 00:41:35,230 I thought that somebody wanted to come in 744 00:41:35,230 --> 00:41:37,720 and then he heard me and changed his mind. 745 00:41:37,720 --> 00:41:39,214 [LAUGHTER] 746 00:41:39,214 --> 00:41:41,020 PROFESSOR: F sub x parentheses [INAUDIBLE] 747 00:41:41,020 --> 00:41:45,010 x plus F sub-- like when you go on a blind date 748 00:41:45,010 --> 00:41:47,290 and you see, change your mind. 749 00:41:47,290 --> 00:41:47,910 OK. 750 00:41:47,910 --> 00:41:53,780 F sub y y plus F sub z z. 751 00:41:53,780 --> 00:41:57,570 Do you remember that I gave away 95 cents 752 00:41:57,570 --> 00:41:59,990 for this type of question? 753 00:41:59,990 --> 00:42:03,230 So what was this operator? 754 00:42:03,230 --> 00:42:05,450 We can write it better. 755 00:42:05,450 --> 00:42:09,342 We can write it using the second partial derivatives 756 00:42:09,342 --> 00:42:12,357 with respect to z, y, and z. 757 00:42:12,357 --> 00:42:15,766 And we gave a name to this one. 758 00:42:15,766 --> 00:42:17,410 We called this names-- 759 00:42:17,410 --> 00:42:18,201 STUDENT: Laplacian. 760 00:42:18,201 --> 00:42:19,175 PROFESSOR: Laplacian. 761 00:42:19,175 --> 00:42:21,123 Laplace operator. 762 00:42:21,123 --> 00:42:22,584 Laplace. 763 00:42:22,584 --> 00:42:25,506 Laplace. 764 00:42:25,506 --> 00:42:26,480 Laplacian. 765 00:42:26,480 --> 00:42:29,060 That's how you spell it. 766 00:42:29,060 --> 00:42:30,826 Laplac-ian. 767 00:42:30,826 --> 00:42:33,260 OK? 768 00:42:33,260 --> 00:42:44,060 Of F. And then what do you have? 769 00:42:44,060 --> 00:42:48,910 770 00:42:48,910 --> 00:42:51,862 You have to introduce a new notation in. 771 00:42:51,862 --> 00:42:54,020 When you see this triangle that looks 772 00:42:54,020 --> 00:42:56,375 like an equilateral triangle, this 773 00:42:56,375 --> 00:42:58,650 means Laplacian of something. 774 00:42:58,650 --> 00:43:01,085 So if you have a function of two variables-- so 775 00:43:01,085 --> 00:43:03,377 let's say z equals F of xy. 776 00:43:03,377 --> 00:43:07,796 What is the Laplacian of this little f? 777 00:43:07,796 --> 00:43:11,724 Little f x x plus little f y y. 778 00:43:11,724 --> 00:43:14,670 So we could be second partial with respect 779 00:43:14,670 --> 00:43:17,460 to x plus the second partial with respect to y. 780 00:43:17,460 --> 00:43:20,530 What if I have something else? 781 00:43:20,530 --> 00:43:26,202 Like let me give you a more general function. 782 00:43:26,202 --> 00:43:28,886 Let's say I have a differentiable function 783 00:43:28,886 --> 00:43:31,570 of N variables with continuous derivatives. 784 00:43:31,570 --> 00:43:33,522 And it looks like crazy. 785 00:43:33,522 --> 00:43:35,474 It looks like that. 786 00:43:35,474 --> 00:43:39,480 x1, x2, x n minus what? 787 00:43:39,480 --> 00:43:46,743 Well, the Laplace operator in this case will be F sub x1 x1 788 00:43:46,743 --> 00:43:48,430 plus [? A of ?] sub x2 x2. 789 00:43:48,430 --> 00:43:53,260 Which means the partial of F, the second derivative 790 00:43:53,260 --> 00:43:55,240 with respect to x2. 791 00:43:55,240 --> 00:44:03,160 And plus the last derivative with respect-- two [INAUDIBLE] 792 00:44:03,160 --> 00:44:04,645 with respect to the same variable. 793 00:44:04,645 --> 00:44:07,120 The last variable is xm minus 1. 794 00:44:07,120 --> 00:44:09,100 This could be one million and 1. 795 00:44:09,100 --> 00:44:10,620 I don't know. 796 00:44:10,620 --> 00:44:14,270 You can have this as many variables as you want. 797 00:44:14,270 --> 00:44:17,480 Now, actually in engineering, there 798 00:44:17,480 --> 00:44:20,440 are functions that have many parameters. 799 00:44:20,440 --> 00:44:22,090 You have three special opponents. 800 00:44:22,090 --> 00:44:22,900 Then you have time. 801 00:44:22,900 --> 00:44:25,080 Then you have temperature, then you have pressure, 802 00:44:25,080 --> 00:44:27,020 then you have god knows what. 803 00:44:27,020 --> 00:44:29,780 The surface tension of the membrane. 804 00:44:29,780 --> 00:44:32,070 Many things. 805 00:44:32,070 --> 00:44:34,350 You really have a million parameters. 806 00:44:34,350 --> 00:44:35,700 Actually, it's impossible. 807 00:44:35,700 --> 00:44:38,560 It's even hard to work with 10 parameters. 808 00:44:38,560 --> 00:44:41,600 Imagine always working with equations 809 00:44:41,600 --> 00:44:47,960 that have lots of variables and having do deal with that. 810 00:44:47,960 --> 00:44:52,690 In fluid flows, hydrodynamical problems, 811 00:44:52,690 --> 00:44:56,180 most the time in 3D turbulent flows, 812 00:44:56,180 --> 00:44:58,930 for example, then you have xyz spatial coordinates 813 00:44:58,930 --> 00:45:02,530 and time T. So even with four variables, 814 00:45:02,530 --> 00:45:07,110 once you get those operators, you could have something like F 815 00:45:07,110 --> 00:45:14,254 sub x x x t plus g sub x x t plus and so on. 816 00:45:14,254 --> 00:45:17,982 All sorts of ugly components. 817 00:45:17,982 --> 00:45:21,780 Sometimes you'll have equations of fluid flows 818 00:45:21,780 --> 00:45:24,320 in dynamic software. 819 00:45:24,320 --> 00:45:26,170 Fluid flows with turbulence are really 820 00:45:26,170 --> 00:45:30,680 an area of mathematics in itself, 821 00:45:30,680 --> 00:45:36,566 of really complicated equations with most of the operators. 822 00:45:36,566 --> 00:45:38,554 I was looking at them in Georgia, 823 00:45:38,554 --> 00:45:40,045 where I went to this conference. 824 00:45:40,045 --> 00:45:43,027 Most of those equations were order 4. 825 00:45:43,027 --> 00:45:47,003 Of course, most of them you cannot even think about solving 826 00:45:47,003 --> 00:45:50,000 by hand, or with any known methods. 827 00:45:50,000 --> 00:45:54,230 You can solve them numerically with computational software. 828 00:45:54,230 --> 00:45:58,100 That is the only [INAUDIBLE] that modern mathematics 829 00:45:58,100 --> 00:46:01,392 has in some areas right now. 830 00:46:01,392 --> 00:46:04,270 The right software, in order to find solutions 831 00:46:04,270 --> 00:46:06,866 to a fluid flow with turbulence. 832 00:46:06,866 --> 00:46:09,306 That is the solution to this type of equation. 833 00:46:09,306 --> 00:46:13,700 Like [INAUDIBLE], for example. 834 00:46:13,700 --> 00:46:18,712 Now we are going to see-- well, you are going to see. 835 00:46:18,712 --> 00:46:20,876 I'm too old and I saw that 20 years ago. 836 00:46:20,876 --> 00:46:23,654 When you're going 3350 [INAUDIBLE] 837 00:46:23,654 --> 00:46:25,097 differential equations. 838 00:46:25,097 --> 00:46:29,907 And then, if you do PD 3350 one in engineering, 839 00:46:29,907 --> 00:46:32,312 You're going to see lots of equations 840 00:46:32,312 --> 00:46:34,430 that are hard to solve. 841 00:46:34,430 --> 00:46:37,050 But in many of them, you're going to see partials, 842 00:46:37,050 --> 00:46:38,120 like that. 843 00:46:38,120 --> 00:46:40,145 And you're going to say, oh, thank god 844 00:46:40,145 --> 00:46:42,180 that I like partials in Calc Three 845 00:46:42,180 --> 00:46:43,530 so they became my friends. 846 00:46:43,530 --> 00:46:47,350 And you'll never have headaches-- [? you know what ?] 847 00:46:47,350 --> 00:46:50,740 would be easy, if you understood that notion of differential 848 00:46:50,740 --> 00:46:55,380 well, the notion of partial derivatives very well. 849 00:46:55,380 --> 00:46:59,376 So I'm going to erase this one. 850 00:46:59,376 --> 00:47:13,640 851 00:47:13,640 --> 00:47:14,460 OK. 852 00:47:14,460 --> 00:47:17,880 And then I'll say, I don't how many of you-- 853 00:47:17,880 --> 00:47:21,688 I'll try to make this formula more visible. 854 00:47:21,688 --> 00:47:25,010 Some of you maybe, who are engineering majors 855 00:47:25,010 --> 00:47:27,991 know about curl. 856 00:47:27,991 --> 00:47:30,510 Have you heard about curl? 857 00:47:30,510 --> 00:47:32,831 Curl of a vector value function. 858 00:47:32,831 --> 00:47:33,331 No. 859 00:47:33,331 --> 00:47:34,822 You haven't. 860 00:47:34,822 --> 00:47:38,301 Suppose that you have a vector value function. 861 00:47:38,301 --> 00:47:44,762 862 00:47:44,762 --> 00:47:49,732 That is F of coordinates x, y, z, the coordinates. 863 00:47:49,732 --> 00:47:53,470 The C1 of over seven domain omega. 864 00:47:53,470 --> 00:47:57,530 Omega is the domain that your special coordinates live in. 865 00:47:57,530 --> 00:47:59,878 Xyz living some potato. 866 00:47:59,878 --> 00:48:02,070 That's it. 867 00:48:02,070 --> 00:48:06,830 Whose solid body enclosed by a closed surface. 868 00:48:06,830 --> 00:48:11,150 In that potato, F is a differentiable function 869 00:48:11,150 --> 00:48:16,200 with respect to xyz, and the derivatives are continuous. 870 00:48:16,200 --> 00:48:19,478 Now, in most cases, if you work with Laplacian, 871 00:48:19,478 --> 00:48:21,609 this is not enough C1. 872 00:48:21,609 --> 00:48:24,104 If you work with Laplacian, what do you want? 873 00:48:24,104 --> 00:48:25,102 What do you need? 874 00:48:25,102 --> 00:48:28,096 You have F sub x x plus F sub y1. 875 00:48:28,096 --> 00:48:29,593 So you need C2. 876 00:48:29,593 --> 00:48:32,587 You work with at least C2. 877 00:48:32,587 --> 00:48:35,082 Many examples have C infinity. 878 00:48:35,082 --> 00:48:37,826 That means you're having really beautiful functions that 879 00:48:37,826 --> 00:48:39,074 are elementary. 880 00:48:39,074 --> 00:48:41,070 Some of them even polynomial approximations. 881 00:48:41,070 --> 00:48:43,930 And then you really can differentiate 882 00:48:43,930 --> 00:48:47,812 them ad infinitum and all the derivatives [INAUDIBLE], 883 00:48:47,812 --> 00:48:50,292 and then you can call yourself lucky. 884 00:48:50,292 --> 00:48:54,260 How do you introduce the notion of curl of it? 885 00:48:54,260 --> 00:48:57,732 And it sounds funny, and this is why they made this fun. 886 00:48:57,732 --> 00:49:01,204 And my hair used to be curly, but I shaved my head 887 00:49:01,204 --> 00:49:04,340 over the holiday, and now it's between. 888 00:49:04,340 --> 00:49:09,180 So curl of F is something that looks horrible 889 00:49:09,180 --> 00:49:12,145 when you try to memorize it. 890 00:49:12,145 --> 00:49:15,510 So you say, OK, if I'm going to get this on the final, 891 00:49:15,510 --> 00:49:18,030 you better wear this T-shirt. 892 00:49:18,030 --> 00:49:21,510 No, there is something better than that. 893 00:49:21,510 --> 00:49:24,890 One time I was the wearing-- OK. 894 00:49:24,890 --> 00:49:29,930 My students got no permission from the [INAUDIBLE] 895 00:49:29,930 --> 00:49:33,374 to come in with a cheat sheet. 896 00:49:33,374 --> 00:49:36,265 But I was wearing a T-shirt that had Green's theorem. 897 00:49:36,265 --> 00:49:37,640 I don't know how many of you have 898 00:49:37,640 --> 00:49:39,062 heard about Green's theorem. 899 00:49:39,062 --> 00:49:41,050 We are going to learn it in two weeks. 900 00:49:41,050 --> 00:49:44,262 And I was wearing that T-shirt. 901 00:49:44,262 --> 00:49:46,732 And it was by accident, OK? 902 00:49:46,732 --> 00:49:49,696 I didn't do it on purpose to help my students cheat. 903 00:49:49,696 --> 00:49:53,318 So one student at some point goes like, well, I 904 00:49:53,318 --> 00:49:54,636 don't remember Green's theorem. 905 00:49:54,636 --> 00:49:56,029 And then he looked my T-shirt. 906 00:49:56,029 --> 00:49:56,612 Oh, all right. 907 00:49:56,612 --> 00:49:58,094 Never mind. 908 00:49:58,094 --> 00:50:01,780 So I had Green's theorem on my shirt, [INAUDIBLE]. 909 00:50:01,780 --> 00:50:04,334 910 00:50:04,334 --> 00:50:08,473 But it's hard to wear like 10 T-shirts, one for the-- I 911 00:50:08,473 --> 00:50:12,126 have one for the formula of the curvature of a curve in space. 912 00:50:12,126 --> 00:50:14,561 Remember that one, how it is so nasty? 913 00:50:14,561 --> 00:50:16,022 OK, I have this one. 914 00:50:16,022 --> 00:50:16,996 I have Green's theorem. 915 00:50:16,996 --> 00:50:19,410 I have [INAUDIBLE], all the important formulas actually. 916 00:50:19,410 --> 00:50:20,753 I have 10 T-shirts. 917 00:50:20,753 --> 00:50:23,218 And then I was thinking, how will I 918 00:50:23,218 --> 00:50:27,162 be if I were like taking ten T-shirts on top of the other 919 00:50:27,162 --> 00:50:31,120 and taking them one off at a time during the final. 920 00:50:31,120 --> 00:50:32,780 There is no cheat sheet. 921 00:50:32,780 --> 00:50:35,506 There are no formula sheets, no nothing. 922 00:50:35,506 --> 00:50:38,060 But I would look like Joey from "Friends." 923 00:50:38,060 --> 00:50:41,950 Remember Joey, when he was dressed in many layers. 924 00:50:41,950 --> 00:50:47,410 So rather than that, I say ask me. 925 00:50:47,410 --> 00:50:49,970 Say oh, you know, I'm freaking out. 926 00:50:49,970 --> 00:50:55,100 I'm taking this final, and I forgot curl. 927 00:50:55,100 --> 00:50:59,320 Rather than not attempting the complex problem at all, 928 00:50:59,320 --> 00:51:03,780 ask me before the exam, and I will remind everybody 929 00:51:03,780 --> 00:51:07,700 how to set up the curl formula. 930 00:51:07,700 --> 00:51:11,376 So you simply have to think in terms 931 00:51:11,376 --> 00:51:15,420 of operators-- ddx, ddy, ddz. 932 00:51:15,420 --> 00:51:16,386 What are these? 933 00:51:16,386 --> 00:51:22,092 These are derivative operators. 934 00:51:22,092 --> 00:51:29,510 So if you take this and multiply it by a function, 935 00:51:29,510 --> 00:51:34,260 that means df, ds-- [INAUDIBLE]. 936 00:51:34,260 --> 00:51:39,714 All right, so in this case, if F is-- I'll 937 00:51:39,714 --> 00:51:54,050 go by my T-shirt-- PI plus QJ plus RK, where PQ and R are all 938 00:51:54,050 --> 00:52:00,720 scalar functions of xyz. 939 00:52:00,720 --> 00:52:07,920 940 00:52:07,920 --> 00:52:09,840 STUDENT: Then we will not forget it. 941 00:52:09,840 --> 00:52:11,760 PROFESSOR: Then we are no longer forget it, 942 00:52:11,760 --> 00:52:15,490 and you'll no longer need my T-shirt. 943 00:52:15,490 --> 00:52:18,580 All right, so how do you do that? 944 00:52:18,580 --> 00:52:22,430 You go expand along your first row, 945 00:52:22,430 --> 00:52:28,780 I times whoever the minor will be, which is this guy. 946 00:52:28,780 --> 00:52:31,490 How do you do the [? cowboy ?] problem? 947 00:52:31,490 --> 00:52:34,167 These guys multiply each other. 948 00:52:34,167 --> 00:52:38,530 So you go dr, dy. 949 00:52:38,530 --> 00:52:39,220 Plus or minus? 950 00:52:39,220 --> 00:52:45,410 Minus dq, dz. 951 00:52:45,410 --> 00:52:55,497 Close times I. So the I is the corresponding element 952 00:52:55,497 --> 00:52:58,395 to the minor that I just completed. 953 00:52:58,395 --> 00:53:03,650 This minor is the determinant, which is exactly this guy. 954 00:53:03,650 --> 00:53:07,030 And this is exactly what my T-shirt says. 955 00:53:07,030 --> 00:53:08,130 Right, precisely. 956 00:53:08,130 --> 00:53:09,380 OK. 957 00:53:09,380 --> 00:53:12,895 The second term, if we put the minus-- no, 958 00:53:12,895 --> 00:53:14,840 they changed the signs. 959 00:53:14,840 --> 00:53:15,780 That's the thing. 960 00:53:15,780 --> 00:53:23,970 I would put minus, because I am expanding along the first row. 961 00:53:23,970 --> 00:53:28,010 And the second that I'm in minus something minor times 962 00:53:28,010 --> 00:53:29,790 J. Which minor? 963 00:53:29,790 --> 00:53:34,160 Let me make in the lime. 964 00:53:34,160 --> 00:53:35,520 Lime is a nice color. 965 00:53:35,520 --> 00:53:50,990 And then I'll take this, this, this, and that-- dr, 966 00:53:50,990 --> 00:53:58,210 dx shooting [? cowboys ?] there-- minus dq, dz. 967 00:53:58,210 --> 00:54:10,070 And of course they wrote dq, dz minus dr, dx. 968 00:54:10,070 --> 00:54:12,684 So I would leave it like that. 969 00:54:12,684 --> 00:54:13,676 It doesn't matter. 970 00:54:13,676 --> 00:54:16,160 You can put the minus in if you want. 971 00:54:16,160 --> 00:54:19,620 Plus the k dot. 972 00:54:19,620 --> 00:54:22,308 k goes at the end. 973 00:54:22,308 --> 00:54:24,292 All right, now k goes at the end. 974 00:54:24,292 --> 00:54:27,280 975 00:54:27,280 --> 00:54:38,544 And then k multiplies this determinant-- dq, dx minus dp, 976 00:54:38,544 --> 00:54:39,044 dy. 977 00:54:39,044 --> 00:54:43,970 978 00:54:43,970 --> 00:54:45,620 dq, dx minus dp, dy. 979 00:54:45,620 --> 00:54:48,434 980 00:54:48,434 --> 00:54:49,341 Is it hard? 981 00:54:49,341 --> 00:54:49,841 No. 982 00:54:49,841 --> 00:54:51,700 It is not going to be hard to memorize. 983 00:54:51,700 --> 00:54:54,014 So then how did we do that? 984 00:54:54,014 --> 00:54:57,592 We set up the first row to be I, J, K, the second row 985 00:54:57,592 --> 00:54:59,870 to be ddx, ddy, and ddz. 986 00:54:59,870 --> 00:55:04,262 And then all in order the components of your vector value 987 00:55:04,262 --> 00:55:07,190 function in the exact order they are with respect 988 00:55:07,190 --> 00:55:11,120 to the standard basis i j k. 989 00:55:11,120 --> 00:55:13,860 All right, now there are other names 990 00:55:13,860 --> 00:55:18,490 and other symbols for curl of F. They use 991 00:55:18,490 --> 00:55:21,766 curl because it's in English. 992 00:55:21,766 --> 00:55:24,110 Well actually, in Great Britain I 993 00:55:24,110 --> 00:55:30,293 saw that they used [INAUDIBLE], or else they use both. 994 00:55:30,293 --> 00:55:34,450 In my language, in Romanian, we call it [? rotore. ?] 995 00:55:34,450 --> 00:55:38,390 And I saw that in French it's very similar. 996 00:55:38,390 --> 00:55:40,070 They use the same. 997 00:55:40,070 --> 00:55:45,200 Now in the mechanical engineering notation 998 00:55:45,200 --> 00:55:46,625 it's funny. 999 00:55:46,625 --> 00:55:53,644 They use another symbol and a cross [? broad dot ?] symbol F. 1000 00:55:53,644 --> 00:56:00,390 And by that they mean curl F. So if you 1001 00:56:00,390 --> 00:56:03,694 talk to a professor who's in mechanical engineering, 1002 00:56:03,694 --> 00:56:06,260 or fluid mechanics, or something, 1003 00:56:06,260 --> 00:56:10,680 when they talk about curl, they will use this notation. 1004 00:56:10,680 --> 00:56:13,526 When they use this other notation, 1005 00:56:13,526 --> 00:56:16,558 what do you think this is again? 1006 00:56:16,558 --> 00:56:17,743 Divergence, yes. 1007 00:56:17,743 --> 00:56:21,300 I told you last time that is divergence of F. 1008 00:56:21,300 --> 00:56:26,090 So make the distinction between-- again, 1009 00:56:26,090 --> 00:56:28,846 when are you leaving? 1010 00:56:28,846 --> 00:56:29,824 Huh? 1011 00:56:29,824 --> 00:56:34,050 OK, so you have been in [INAUDIBLE]. 1012 00:56:34,050 --> 00:56:38,620 And then we have this distinction 1013 00:56:38,620 --> 00:56:40,272 we use here, like for dot product 1014 00:56:40,272 --> 00:56:42,860 and you use here as a cross product. 1015 00:56:42,860 --> 00:56:46,990 Now you have to understand the conceptual difference is 1016 00:56:46,990 --> 00:56:49,280 huge between these guys. 1017 00:56:49,280 --> 00:56:52,210 This is a scalar function. 1018 00:56:52,210 --> 00:56:55,760 This is a vector function-- vector, scalar-- vector, 1019 00:56:55,760 --> 00:56:58,220 scalar, vector scalar. 1020 00:56:58,220 --> 00:57:00,188 Because I've had to do it on so [INAUDIBLE]. 1021 00:57:00,188 --> 00:57:01,664 It makes [INAUDIBLE]. 1022 00:57:01,664 --> 00:57:06,584 And I heard of colleagues complaining 1023 00:57:06,584 --> 00:57:10,028 while grading the final that the students did not 1024 00:57:10,028 --> 00:57:14,990 understand that this is a vector, and this is a scalar. 1025 00:57:14,990 --> 00:57:18,800 OK, a few simple exercises-- I'm going to go ahead and do 1026 00:57:18,800 --> 00:57:20,510 some of them. 1027 00:57:20,510 --> 00:57:24,450 We tried to make the data on the final exam 1028 00:57:24,450 --> 00:57:30,314 very accessible and very easy to apply in problems. 1029 00:57:30,314 --> 00:57:36,736 And one of the problems that-- we'll start with example 2-- 1030 00:57:36,736 --> 00:57:40,700 would be this one. 1031 00:57:40,700 --> 00:57:43,251 And you may think, why? 1032 00:57:43,251 --> 00:57:45,736 Sometimes we put it in disguise. 1033 00:57:45,736 --> 00:57:50,457 And we said assume you have a sphere-- that's 1034 00:57:50,457 --> 00:58:07,590 the unit sphere-- of origin O. And say compute. 1035 00:58:07,590 --> 00:58:10,500 1036 00:58:10,500 --> 00:58:13,842 What is the equation of the unit sphere, guys? 1037 00:58:13,842 --> 00:58:17,720 X squared plus y squared plus z squared equals one, right? 1038 00:58:17,720 --> 00:58:23,749 From [INAUDIBLE], F equals normal-- external 1039 00:58:23,749 --> 00:58:35,270 normal-- to the unit sphere pointing out, [? through ?] 1040 00:58:35,270 --> 00:58:49,770 than N is the same at a different point as the position 1041 00:58:49,770 --> 00:58:50,270 vector. 1042 00:58:50,270 --> 00:58:55,210 1043 00:58:55,210 --> 00:58:57,085 Then compute. 1044 00:58:57,085 --> 00:59:00,963 1045 00:59:00,963 --> 00:59:11,786 [? Now follow. ?] Gradient of F, divergence of F, 1046 00:59:11,786 --> 00:59:15,782 and curl of F. Now that should be a piece of cake. 1047 00:59:15,782 --> 00:59:19,275 Now one is not [INAUDIBLE] so much of a piece of cake 1048 00:59:19,275 --> 00:59:23,220 if you don't understand what the problem wants from you. 1049 00:59:23,220 --> 00:59:28,240 It is to actually graph the expression of this one. 1050 00:59:28,240 --> 00:59:32,462 So you're going to say what is the normal 1051 00:59:32,462 --> 00:59:34,910 to a function like that? 1052 00:59:34,910 --> 00:59:37,360 First of all, we just talked today about it. 1053 00:59:37,360 --> 00:59:42,220 If you have a function, even if it's implicitly as F of x, 1054 00:59:42,220 --> 00:59:52,530 y, z equals c, in that case N is your friend from the past. 1055 00:59:52,530 --> 01:00:00,832 If it's a unit normal, unit normal to a surface 1056 01:00:00,832 --> 01:00:03,796 happens all the time in engineering. 1057 01:00:03,796 --> 01:00:06,760 Whether you do solid mechanics or fluid mechanics, 1058 01:00:06,760 --> 01:00:09,970 you always have to complete these things. 1059 01:00:09,970 --> 01:00:11,757 This is going to be hard. 1060 01:00:11,757 --> 01:00:24,177 The gradient of F divided by the length 1061 01:00:24,177 --> 01:00:27,099 of-- but here I have a problem. 1062 01:00:27,099 --> 01:00:32,943 I have to put G here, because G will be my position vector. 1063 01:00:32,943 --> 01:00:35,890 This is the point x,y,z. 1064 01:00:35,890 --> 01:00:38,890 Or you prefer big R. But I think I prefer big G, 1065 01:00:38,890 --> 01:00:41,900 because big R looks like a scalar radius, 1066 01:00:41,900 --> 01:00:43,289 and I don't like that. 1067 01:00:43,289 --> 01:00:47,260 So the position vector will be the circle middle 1068 01:00:47,260 --> 01:00:52,180 that starts at the origin and whose N is on the surface, 1069 01:00:52,180 --> 01:00:53,160 right? 1070 01:00:53,160 --> 01:00:57,080 And this is the equation, xy equals yj plus [? ek1. ?] 1071 01:00:57,080 --> 01:01:02,430 Because my point x,y,z has a corresponding vector xi plus yj 1072 01:01:02,430 --> 01:01:04,550 plus zk-- big deal. 1073 01:01:04,550 --> 01:01:08,130 Now I'm trying to convince you that, for the unit 1074 01:01:08,130 --> 01:01:12,570 normal for the sphere, I have the same kind of thing. 1075 01:01:12,570 --> 01:01:17,330 So how do we compute this normally? 1076 01:01:17,330 --> 01:01:23,092 I take the function F that implicitly defines the surface. 1077 01:01:23,092 --> 01:01:27,020 All right, so in my case F is something else. 1078 01:01:27,020 --> 01:01:29,966 What is it? x squared plus y squared plus z squared. 1079 01:01:29,966 --> 01:01:34,385 1080 01:01:34,385 --> 01:01:37,331 Let's compute it. 1081 01:01:37,331 --> 01:01:40,440 N is going to be [INAUDIBLE]. 1082 01:01:40,440 --> 01:01:41,680 It's very nice. 1083 01:01:41,680 --> 01:01:50,740 2x comma 2y comma 2z divided by the square root of the sums. 1084 01:01:50,740 --> 01:01:52,405 Do I like this? 1085 01:01:52,405 --> 01:01:56,285 Uh, no, but I'll have to do it whether I like it or not. 1086 01:01:56,285 --> 01:02:02,590 1087 01:02:02,590 --> 01:02:05,710 I want to simplify up and down via 2. 1088 01:02:05,710 --> 01:02:07,264 Can I do that? 1089 01:02:07,264 --> 01:02:08,080 Of course I can. 1090 01:02:08,080 --> 01:02:15,950 I'm going to get x,y,z divided by square root of x squared 1091 01:02:15,950 --> 01:02:18,110 plus y squared plus z squared. 1092 01:02:18,110 --> 01:02:22,054 1093 01:02:22,054 --> 01:02:23,040 And this was 1. 1094 01:02:23,040 --> 01:02:28,195 1095 01:02:28,195 --> 01:02:29,695 STUDENT: Wouldn't there still be a 2 1096 01:02:29,695 --> 01:02:32,407 there, because it's 2 squared [INAUDIBLE]? 1097 01:02:32,407 --> 01:02:33,886 PROFESSOR: No, I pulled it out. 1098 01:02:33,886 --> 01:02:35,650 That's exactly what I said. 1099 01:02:35,650 --> 01:02:37,237 There was a 4 inside. 1100 01:02:37,237 --> 01:02:39,185 I pulled out with the forceps. 1101 01:02:39,185 --> 01:02:41,133 I put it up here, square root of 4. 1102 01:02:41,133 --> 01:02:45,010 And I have a 2 here, and that cancels out. 1103 01:02:45,010 --> 01:02:48,200 So I got something much simpler than you guys 1104 01:02:48,200 --> 01:02:50,370 expected at first. 1105 01:02:50,370 --> 01:02:59,325 I got xi plus yj plus zk as being the normal. 1106 01:02:59,325 --> 01:03:01,273 Did you expect this? 1107 01:03:01,273 --> 01:03:03,708 And you were supposed to expect that this is y, 1108 01:03:03,708 --> 01:03:07,117 because this is the position vector that has one length. 1109 01:03:07,117 --> 01:03:09,990 The length of a root vector is 1, 1110 01:03:09,990 --> 01:03:11,445 and the point is on the sphere. 1111 01:03:11,445 --> 01:03:14,700 The normal will be exactly the continuation. 1112 01:03:14,700 --> 01:03:16,900 Take your root vector, and continue 1113 01:03:16,900 --> 01:03:20,080 in the same direction-- this is the beauty 1114 01:03:20,080 --> 01:03:23,840 of the normal to a surface, that it continues the radius. 1115 01:03:23,840 --> 01:03:27,750 It continues the radius of the sphere in the same direction. 1116 01:03:27,750 --> 01:03:30,160 So you copy and paste your vector here. 1117 01:03:30,160 --> 01:03:34,870 Position vector G will be the same as the normal N. 1118 01:03:34,870 --> 01:03:38,056 All you do is you shift, but it's the same vector 1119 01:03:38,056 --> 01:03:40,350 at the different point. 1120 01:03:40,350 --> 01:03:43,530 Instead of starting at O, it starts at P. So 1121 01:03:43,530 --> 01:03:45,546 [? that ?] is the same vector. 1122 01:03:45,546 --> 01:03:48,580 So you take the radius vector from inside the sphere-- 1123 01:03:48,580 --> 01:03:51,315 the position vector-- and you shift it out, 1124 01:03:51,315 --> 01:03:54,790 and that's the normal to the sphere. 1125 01:03:54,790 --> 01:03:59,290 So the equation is still xi plus yj plus zk. 1126 01:03:59,290 --> 01:03:59,790 Yes, sir. 1127 01:03:59,790 --> 01:04:02,225 STUDENT: Does it remain the same for any other functions, 1128 01:04:02,225 --> 01:04:03,199 like [INAUDIBLE]? 1129 01:04:03,199 --> 01:04:07,810 1130 01:04:07,810 --> 01:04:09,800 PROFESSOR: For the unit sphere, yes it is. 1131 01:04:09,800 --> 01:04:12,800 But for a general sphere, no. 1132 01:04:12,800 --> 01:04:15,760 For example, what if my sphere will be 1133 01:04:15,760 --> 01:04:19,360 of center origin and radius R? 1134 01:04:19,360 --> 01:04:22,610 1135 01:04:22,610 --> 01:04:28,750 And its position vector v is x,y,z-- like that. 1136 01:04:28,750 --> 01:04:31,635 1137 01:04:31,635 --> 01:04:33,380 [INAUDIBLE] I don't know. 1138 01:04:33,380 --> 01:04:34,090 G, right? 1139 01:04:34,090 --> 01:04:35,610 That's the position normal. 1140 01:04:35,610 --> 01:04:40,557 STUDENT: [INAUDIBLE] just divide them by the R. 1141 01:04:40,557 --> 01:04:44,350 PROFESSOR: You just divide by the R. 1142 01:04:44,350 --> 01:04:47,960 So instead of radius being big R, 1143 01:04:47,960 --> 01:04:50,580 your unit vector will be this one. 1144 01:04:50,580 --> 01:04:52,660 And you take this one and shift it here, 1145 01:04:52,660 --> 01:04:53,940 and that's all you have. 1146 01:04:53,940 --> 01:04:55,380 For the sphere, it's beautiful. 1147 01:04:55,380 --> 01:04:58,077 For any surface in general, no. 1148 01:04:58,077 --> 01:04:59,985 Let me show you. 1149 01:04:59,985 --> 01:05:04,278 You have a bunch of [INAUDIBLE], and your position vectors 1150 01:05:04,278 --> 01:05:06,663 look like crazies like that. 1151 01:05:06,663 --> 01:05:11,990 And the normals could be-- they don't have 1152 01:05:11,990 --> 01:05:13,240 to continue their position. 1153 01:05:13,240 --> 01:05:23,220 They could be-- it depends how the tangent planes look like. 1154 01:05:23,220 --> 01:05:31,204 And the tangent plane at the point 1155 01:05:31,204 --> 01:05:33,699 has to be perpendicular to the normal. 1156 01:05:33,699 --> 01:05:37,810 So the normal field is the N of [INAUDIBLE] vectors. 1157 01:05:37,810 --> 01:05:41,200 But the little thingies that look like rectangles 1158 01:05:41,200 --> 01:05:44,024 or whatever they are-- those are the tangent planes 1159 01:05:44,024 --> 01:05:45,680 of those points. 1160 01:05:45,680 --> 01:05:48,585 So in general there is no obvious relationship 1161 01:05:48,585 --> 01:05:53,240 between the position and the normal for the surface. 1162 01:05:53,240 --> 01:05:55,430 You are really lucky for this [? field. ?] 1163 01:05:55,430 --> 01:06:00,166 And for many reasons, like how beautiful the sphere is, 1164 01:06:00,166 --> 01:06:04,118 these functions will be easy to compute. 1165 01:06:04,118 --> 01:06:06,588 Can you tell me what they are without computing? 1166 01:06:06,588 --> 01:06:11,034 Because that should be a piece of cake. 1167 01:06:11,034 --> 01:06:14,492 What is the gradient field? 1168 01:06:14,492 --> 01:06:19,432 STUDENT: [INAUDIBLE] to that one? 1169 01:06:19,432 --> 01:06:22,235 That's the x, y, and z. 1170 01:06:22,235 --> 01:06:24,890 PROFESSOR: For the sphere. 1171 01:06:24,890 --> 01:06:26,710 STUDENT: 2x, 2y-- 1172 01:06:26,710 --> 01:06:36,795 PROFESSOR: Actually, let's do it for both divergence G and curl 1173 01:06:36,795 --> 01:06:46,840 G. And you say wait, they will be-- so gradient-- no, 1174 01:06:46,840 --> 01:06:47,630 I meant here. 1175 01:06:47,630 --> 01:06:50,408 You don't have gradient. 1176 01:06:50,408 --> 01:06:54,840 When F is a scalar function, then you have gradient. 1177 01:06:54,840 --> 01:07:01,270 Then for that gradient you're going to have divergence. 1178 01:07:01,270 --> 01:07:05,330 And for that-- I changed notations, that's shy 1179 01:07:05,330 --> 01:07:06,860 I have to fix it. 1180 01:07:06,860 --> 01:07:10,820 Because F used to be that, and it's not a vector anymore. 1181 01:07:10,820 --> 01:07:12,800 So big F is not a vector anymore. 1182 01:07:12,800 --> 01:07:16,760 It's a scalar function, and now I have to change the problem. 1183 01:07:16,760 --> 01:07:18,245 What is the gradient there? 1184 01:07:18,245 --> 01:07:19,730 What's divergence of the gradient? 1185 01:07:19,730 --> 01:07:24,185 [INAUDIBLE] gradient of F. And for the G that I gave you, 1186 01:07:24,185 --> 01:07:27,660 I want the divergence in the curve? 1187 01:07:27,660 --> 01:07:31,320 So I made the problem fluffier that it was before. 1188 01:07:31,320 --> 01:07:34,590 More things to confuse for practice. 1189 01:07:34,590 --> 01:07:35,480 What's the gradient? 1190 01:07:35,480 --> 01:07:36,600 We did it before. 1191 01:07:36,600 --> 01:07:37,100 2x-- 1192 01:07:37,100 --> 01:07:38,260 STUDENT: 2xi, 2-- 1193 01:07:38,260 --> 01:07:42,470 PROFESSOR: 2y, 2z-- we are at a [? 93 ?] point p on the sphere. 1194 01:07:42,470 --> 01:07:46,195 It could be anywhere-- anywhere in space. 1195 01:07:46,195 --> 01:07:49,165 What's the divergence of this individual? 1196 01:07:49,165 --> 01:07:51,640 So remember guys, what I told you? 1197 01:07:51,640 --> 01:07:54,610 First component differentiated with a straight 2x 1198 01:07:54,610 --> 01:07:57,332 plus second component differentiated with respect 1199 01:07:57,332 --> 01:08:01,045 to y plus third component differentiated 1200 01:08:01,045 --> 01:08:04,015 with respect to z. 1201 01:08:04,015 --> 01:08:11,950 2 plus 2 plus 2 equals 6-- piece of cake. 1202 01:08:11,950 --> 01:08:18,023 And curl of the gradient of F-- is that hard? 1203 01:08:18,023 --> 01:08:18,856 [? STUDENT: Yeah. ?] 1204 01:08:18,856 --> 01:08:22,307 PROFESSOR: No, but we have to know the definition. 1205 01:08:22,307 --> 01:08:26,743 And without looking at the T-shirt, how do we do that? 1206 01:08:26,743 --> 01:08:30,194 1207 01:08:30,194 --> 01:08:36,899 The determinant-- I, J, K. Operators-- ddx, ddy, and ddz. 1208 01:08:36,899 --> 01:08:40,875 1209 01:08:40,875 --> 01:08:44,850 STUDENT: [INAUDIBLE] 2x, 2y, 2z, correct? 1210 01:08:44,850 --> 01:08:48,827 PROFESSOR: And we copy and paste the three components. 1211 01:08:48,827 --> 01:08:50,814 [INAUDIBLE] in the trash. 1212 01:08:50,814 --> 01:08:53,830 I'll take the blue. 1213 01:08:53,830 --> 01:08:57,957 So we put 2x, 2y, 2z. 1214 01:08:57,957 --> 01:09:00,790 Do you think it's going to be easy or hard? 1215 01:09:00,790 --> 01:09:02,130 Do you see the answer? 1216 01:09:02,130 --> 01:09:06,189 Some of are very sharp, and you may see the answer. 1217 01:09:06,189 --> 01:09:10,254 For example, when the cowboys shoot at each other 1218 01:09:10,254 --> 01:09:14,413 like this, dz, dy is here. 1219 01:09:14,413 --> 01:09:16,341 dy, dz is here. 1220 01:09:16,341 --> 01:09:23,970 So this, as a minor, is 0-- 0I, an eye for an eye. 1221 01:09:23,970 --> 01:09:25,759 And what else? 1222 01:09:25,759 --> 01:09:29,660 dz, dx-- dx dz, same thing, minus 0j. 1223 01:09:29,660 --> 01:09:32,395 Is this meant to say minus 0j? 1224 01:09:32,395 --> 01:09:33,210 Yes it is. 1225 01:09:33,210 --> 01:09:37,715 But I did it because I want you to have the good habit 1226 01:09:37,715 --> 01:09:40,189 of saying plus minus plus. 1227 01:09:40,189 --> 01:09:42,910 And that's finally the same kind of thing 1228 01:09:42,910 --> 01:09:46,883 that'll give you 0k if you think that when 1229 01:09:46,883 --> 01:09:49,313 you do partial derivative of y with respect to [? f ?], 1230 01:09:49,313 --> 01:09:50,770 you get 0. 1231 01:09:50,770 --> 01:09:52,229 You have 0. 1232 01:09:52,229 --> 01:09:57,580 So some student of mine asked, so this is the 0 vector, 1233 01:09:57,580 --> 01:10:01,820 how in the world do I write a 0 vector on short? 1234 01:10:01,820 --> 01:10:03,014 Let me show you how. 1235 01:10:03,014 --> 01:10:04,180 You're going to laugh at me. 1236 01:10:04,180 --> 01:10:08,990 Some people write 0 bar, which means the 0 vector. 1237 01:10:08,990 --> 01:10:11,561 Some other people don't like it, it's silly. 1238 01:10:11,561 --> 01:10:17,642 Some people write O with double like that, meaning that, 1239 01:10:17,642 --> 01:10:21,498 hey, this is a vector element, the vector with its components 1240 01:10:21,498 --> 01:10:25,836 of 0, 0, 0-- to distinguish that vector from the number 0, 1241 01:10:25,836 --> 01:10:33,618 which is not in bold-- So the notations for the vector are 0. 1242 01:10:33,618 --> 01:10:37,546 So I'm going to write here 0, 0, 0. 1243 01:10:37,546 --> 01:10:39,510 How about Mr. G? 1244 01:10:39,510 --> 01:10:42,456 Mr. G will act similarly. 1245 01:10:42,456 --> 01:10:47,250 When you do the divergence it's going to be-- 1 1246 01:10:47,250 --> 01:10:50,315 plus 1 plus 1 equals 3. 1247 01:10:50,315 --> 01:10:53,522 1248 01:10:53,522 --> 01:10:55,982 You should remember this thing. 1249 01:10:55,982 --> 01:10:59,180 We are going to do the divergence 3, 1250 01:10:59,180 --> 01:11:02,870 and they will ask you to do a triple integral of a divergence 1251 01:11:02,870 --> 01:11:04,346 of a vector field. 1252 01:11:04,346 --> 01:11:06,642 And when you do that, you are going 1253 01:11:06,642 --> 01:11:08,780 to get a triple integer of something 1254 01:11:08,780 --> 01:11:13,330 like 3, which is a custom, which will make your life very easy. 1255 01:11:13,330 --> 01:11:16,430 So you will very easily compute those triple integrals 1256 01:11:16,430 --> 01:11:18,150 of constants. 1257 01:11:18,150 --> 01:11:22,635 Curl of G, G being of [? a. ?] OK? 1258 01:11:22,635 --> 01:11:26,274 I should make the distinction between a scalar function 1259 01:11:26,274 --> 01:11:30,603 and a vector function by putting a G bar on the vector function. 1260 01:11:30,603 --> 01:11:34,932 How about this? 1261 01:11:34,932 --> 01:11:35,750 Is it hard? 1262 01:11:35,750 --> 01:11:38,156 No, because it's the same fellow. 1263 01:11:38,156 --> 01:11:40,920 Instead of that, I have just x, y, z. 1264 01:11:40,920 --> 01:11:42,770 The answer will be the same. 1265 01:11:42,770 --> 01:11:48,830 So I still want to get 0, 0, 0-- the vector 0. 1266 01:11:48,830 --> 01:11:52,670 So the point was that we will give you enough. 1267 01:11:52,670 --> 01:11:54,848 You may expect them to be very hard, 1268 01:11:54,848 --> 01:11:57,842 but they are not going to be very hard. 1269 01:11:57,842 --> 01:12:02,333 Let's do one more like the ones we have in the book. 1270 01:12:02,333 --> 01:12:06,325 What do you think this one will be? 1271 01:12:06,325 --> 01:12:10,816 I'm making you a new vector value function. 1272 01:12:10,816 --> 01:12:16,305 That's maybe two little exercises 1273 01:12:16,305 --> 01:12:19,922 we can do just working exercise three, four, 1274 01:12:19,922 --> 01:12:21,295 I don't know what they are. 1275 01:12:21,295 --> 01:12:23,902 1276 01:12:23,902 --> 01:12:28,276 Let me give you R vector of x, y, z 1277 01:12:28,276 --> 01:12:38,470 equals yzI plus xzj plus xyk. 1278 01:12:38,470 --> 01:12:41,170 Compute the curl. 1279 01:12:41,170 --> 01:12:47,375 Let me write it like engineers do just for fun-- [INAUDIBLE] 1280 01:12:47,375 --> 01:12:48,373 cross. 1281 01:12:48,373 --> 01:12:54,361 R is the same as curl R, which is I, 1282 01:12:54,361 --> 01:12:59,840 J, K-- oh my god-- ddx, ddy, ddz. 1283 01:12:59,840 --> 01:13:02,500 1284 01:13:02,500 --> 01:13:07,820 Why is z-- xz-- xy. 1285 01:13:07,820 --> 01:13:13,490 Are you saying oh, that's not so easy anymore. 1286 01:13:13,490 --> 01:13:17,356 You-- you will see that it becomes easy, 1287 01:13:17,356 --> 01:13:22,160 OK? i times what is the minor? 1288 01:13:22,160 --> 01:13:24,880 This times-- x, right? 1289 01:13:24,880 --> 01:13:31,877 Minus x plus minus j times 1 minus what? 1290 01:13:31,877 --> 01:13:34,811 Minor will be the red thingie. 1291 01:13:34,811 --> 01:13:39,212 And the red thingie is beautiful, 1292 01:13:39,212 --> 01:13:45,430 because it's gonna be y minus y plus k times-- 1293 01:13:45,430 --> 01:13:49,810 who do you think it's gonna be? z z minus. 1294 01:13:49,810 --> 01:13:53,220 So it's still 0. 1295 01:13:53,220 --> 01:13:57,460 Do we expect something like that on the final? 1296 01:13:57,460 --> 01:13:58,900 An easy computation. 1297 01:13:58,900 --> 01:14:03,494 Somebody says, find me the curve of this function. 1298 01:14:03,494 --> 01:14:04,868 And the functions usually we give 1299 01:14:04,868 --> 01:14:06,460 you are nice and significant. 1300 01:14:06,460 --> 01:14:11,880 Something where the result will be pretty. 1301 01:14:11,880 --> 01:14:12,380 OK. 1302 01:14:12,380 --> 01:14:15,836 1303 01:14:15,836 --> 01:14:18,954 Let me see what else I wanted. 1304 01:14:18,954 --> 01:14:25,898 I'm gonna-- I have space here. 1305 01:14:25,898 --> 01:14:45,738 So compute the curl and Laplace operator of f of xyz 1306 01:14:45,738 --> 01:14:56,238 equals x squared yzi plus x y squared zj plus xy z squared k. 1307 01:14:56,238 --> 01:15:02,530 1308 01:15:02,530 --> 01:15:04,466 Of divergence. 1309 01:15:04,466 --> 01:15:05,930 Sorry, guys. 1310 01:15:05,930 --> 01:15:09,328 This is not a-- it's not a scalar function. 1311 01:15:09,328 --> 01:15:11,240 I want the divergence and the curl. 1312 01:15:11,240 --> 01:15:13,152 The curl will be a vector. 1313 01:15:13,152 --> 01:15:15,550 The divergence will be a scalar function. 1314 01:15:15,550 --> 01:15:18,190 Later on I'll give you a nice function where you can 1315 01:15:18,190 --> 01:15:20,300 compute the Laplace operator. 1316 01:15:20,300 --> 01:15:23,800 That's gonna have to be a scalar function. 1317 01:15:23,800 --> 01:15:27,302 And although the Laplace operator 1318 01:15:27,302 --> 01:15:29,787 can be generalized to vector functions, 1319 01:15:29,787 --> 01:15:31,775 and I'll tell you later how-- what that is. 1320 01:15:31,775 --> 01:15:32,769 It's very easy. 1321 01:15:32,769 --> 01:15:37,620 It's practically the Laplace operators in every direction. 1322 01:15:37,620 --> 01:15:38,450 OK. 1323 01:15:38,450 --> 01:15:42,276 So let's see the curl. 1324 01:15:42,276 --> 01:15:48,500 1325 01:15:48,500 --> 01:15:56,390 i j k, d dx, d dy, d dz. 1326 01:15:56,390 --> 01:15:58,740 Today I'm gonna cook up the homework. 1327 01:15:58,740 --> 01:16:01,120 And with all the practice that we are doing now, 1328 01:16:01,120 --> 01:16:04,880 you should have absolutely no problem doing the homework 1329 01:16:04,880 --> 01:16:06,840 for the first two sections. 1330 01:16:06,840 --> 01:16:10,330 At least for the section-- today's section, 13.1. 1331 01:16:10,330 --> 01:16:15,340 x squared yz, x y squared z, xy z squared. 1332 01:16:15,340 --> 01:16:17,393 You see there is some sort of symmetry. 1333 01:16:17,393 --> 01:16:18,800 I'm playing a game here. 1334 01:16:18,800 --> 01:16:23,940 1335 01:16:23,940 --> 01:16:29,840 So I have i. 1336 01:16:29,840 --> 01:16:33,930 I want you to tell me [INAUDIBLE], see now, 1337 01:16:33,930 --> 01:16:35,615 I don't work much in groups. 1338 01:16:35,615 --> 01:16:38,116 I don't make you work in groups, but I 1339 01:16:38,116 --> 01:16:40,690 want you to answer my question. 1340 01:16:40,690 --> 01:16:46,925 So what is going to be this minor that I-- the first thing 1341 01:16:46,925 --> 01:16:47,425 is gonna be? 1342 01:16:47,425 --> 01:16:48,910 STUDENT: x z squared. 1343 01:16:48,910 --> 01:16:49,900 PROFESSOR: Very good. 1344 01:16:49,900 --> 01:16:52,375 STUDENT: Minus x y squared. 1345 01:16:52,375 --> 01:16:57,325 1346 01:16:57,325 --> 01:16:58,315 PROFESSOR: Minus j. 1347 01:16:58,315 --> 01:17:00,295 Potential [? plus or ?] minus. 1348 01:17:00,295 --> 01:17:01,780 OK, what is the next guy? 1349 01:17:01,780 --> 01:17:03,070 STUDENT: y z squared. 1350 01:17:03,070 --> 01:17:07,970 PROFESSOR: y z squared, thank you. 1351 01:17:07,970 --> 01:17:08,950 STUDENT: x squared. 1352 01:17:08,950 --> 01:17:12,870 PROFESSOR: x squared, right? 1353 01:17:12,870 --> 01:17:16,790 Plus k times-- 1354 01:17:16,790 --> 01:17:20,710 STUDENT: y squared z. 1355 01:17:20,710 --> 01:17:23,160 PROFESSOR: So, you see what I'm doing? 1356 01:17:23,160 --> 01:17:27,504 I'm doing this from respect to x. y squared z. 1357 01:17:27,504 --> 01:17:28,971 You said it, right. 1358 01:17:28,971 --> 01:17:32,883 Minus this guy, x squared z. 1359 01:17:32,883 --> 01:17:37,630 Can I write it more-- I don't really like the way I wrote it. 1360 01:17:37,630 --> 01:17:38,750 But I'll write like that. 1361 01:17:38,750 --> 01:17:45,402 How about x times z squared minus y squared i minus j. 1362 01:17:45,402 --> 01:17:47,390 Or maybe better plus j. 1363 01:17:47,390 --> 01:17:49,378 I'll change this up. 1364 01:17:49,378 --> 01:17:50,869 Plus j at the end. 1365 01:17:50,869 --> 01:17:55,342 Because it's the vector. y times x 1366 01:17:55,342 --> 01:18:02,797 squared minus z squared j plus-- who gets out? 1367 01:18:02,797 --> 01:18:09,258 z-- z times y squared minus x squared k. 1368 01:18:09,258 --> 01:18:12,737 1369 01:18:12,737 --> 01:18:13,731 OK. 1370 01:18:13,731 --> 01:18:16,713 Good There is some symmetry in there. 1371 01:18:16,713 --> 01:18:19,695 The break in the symmetry is in the middle. 1372 01:18:19,695 --> 01:18:23,174 Because, as you see, x is separate, 1373 01:18:23,174 --> 01:18:26,618 and then z is followed by y, and then 1374 01:18:26,618 --> 01:18:31,770 x squared-- x is followed by z and y is followed by x. 1375 01:18:31,770 --> 01:18:37,098 So I have some-- some symmetry of some sort. 1376 01:18:37,098 --> 01:18:40,080 1377 01:18:40,080 --> 01:18:42,565 What else did I want? 1378 01:18:42,565 --> 01:18:44,056 Divergence operator. 1379 01:18:44,056 --> 01:18:47,535 And that will be the last example of the kind. 1380 01:18:47,535 --> 01:18:50,020 [INAUDIBLE] 1381 01:18:50,020 --> 01:18:51,964 How do you write the divergence? 1382 01:18:51,964 --> 01:18:52,505 Is this hard? 1383 01:18:52,505 --> 01:18:53,005 Very easy? 1384 01:18:53,005 --> 01:18:55,510 1385 01:18:55,510 --> 01:18:58,771 I'm going go ask you to simplify because I don't like it, 1386 01:18:58,771 --> 01:19:01,226 like, as a sum. 1387 01:19:01,226 --> 01:19:02,208 2xyz-- 1388 01:19:02,208 --> 01:19:10,560 STUDENT: 2xyz plus 2xyz plus 2xyz. 1389 01:19:10,560 --> 01:19:13,411 PROFESSOR: And now you see why I don't like it as a sum. 1390 01:19:13,411 --> 01:19:17,387 Because it's 6xyz and it's very pretty like that. 1391 01:19:17,387 --> 01:19:20,120 I'd like you-- on the exam, I'd like 1392 01:19:20,120 --> 01:19:24,345 you to take the function and box the answer, 1393 01:19:24,345 --> 01:19:28,815 and that's all I want you to do. 1394 01:19:28,815 --> 01:19:29,315 All right. 1395 01:19:29,315 --> 01:19:31,303 I'm gonna go ahead and erase. 1396 01:19:31,303 --> 01:19:49,692 1397 01:19:49,692 --> 01:19:53,171 I'm going to move on to 13.2, but I'd 1398 01:19:53,171 --> 01:19:59,848 like to review some physics a little bit with you 1399 01:19:59,848 --> 01:20:02,800 and see what you remember from physics. 1400 01:20:02,800 --> 01:20:21,496 1401 01:20:21,496 --> 01:20:23,464 It's a little bit messy. 1402 01:20:23,464 --> 01:20:27,892 I'll use this instead because I like the board to be clean. 1403 01:20:27,892 --> 01:20:34,260 If I were to ask you to remember work in physics, 1404 01:20:34,260 --> 01:20:41,708 I would say-- I'm changing a little bit the order in 13.2. 1405 01:20:41,708 --> 01:20:46,688 I'd like you to go back in time and see 1406 01:20:46,688 --> 01:20:48,680 what work was in physics class. 1407 01:20:48,680 --> 01:20:53,660 1408 01:20:53,660 --> 01:20:56,150 STUDENT: [INAUDIBLE] force dx. 1409 01:20:56,150 --> 01:20:59,670 1410 01:20:59,670 --> 01:21:02,090 PROFESSOR: What if you didn't know any calculus? 1411 01:21:02,090 --> 01:21:04,318 Let's go a long time. 1412 01:21:04,318 --> 01:21:10,780 The section is 13.2 preliminaries. 1413 01:21:10,780 --> 01:21:13,472 STUDENT: Force multiplied distance. 1414 01:21:13,472 --> 01:21:15,424 PROFESSOR: Very good. 1415 01:21:15,424 --> 01:21:17,376 Preliminary work. 1416 01:21:17,376 --> 01:21:25,498 [INAUDIBLE] The notion of work from physics-- hey, come on. 1417 01:21:25,498 --> 01:21:34,960 1418 01:21:34,960 --> 01:21:39,130 Physics, or engineering, mechanics, whatever you study, 1419 01:21:39,130 --> 01:21:39,800 work. 1420 01:21:39,800 --> 01:21:43,922 Imagine that you're taking a-- this 1421 01:21:43,922 --> 01:21:47,393 is your body that you're playing with-- not your own body, 1422 01:21:47,393 --> 01:21:50,470 but the body you are acting on in physics. 1423 01:21:50,470 --> 01:21:58,480 And you are dragging this object from a place A 1424 01:21:58,480 --> 01:22:00,890 to a place B, another position. 1425 01:22:00,890 --> 01:22:04,264 A B is the distance. 1426 01:22:04,264 --> 01:22:09,190 And the force is parallel to the trajectory. 1427 01:22:09,190 --> 01:22:10,400 This is very important. 1428 01:22:10,400 --> 01:22:12,349 This is a simpler case. 1429 01:22:12,349 --> 01:22:14,225 In general, it's not so simple. 1430 01:22:14,225 --> 01:22:17,840 So this force is acting, and it's a constant force. 1431 01:22:17,840 --> 01:22:21,910 And you pull the object from one place to another. 1432 01:22:21,910 --> 01:22:24,040 That's case one. 1433 01:22:24,040 --> 01:22:28,010 In case two, life is harder. 1434 01:22:28,010 --> 01:22:35,825 You actually pull the poor object 1435 01:22:35,825 --> 01:22:38,272 with the force in this direction. 1436 01:22:38,272 --> 01:22:40,682 Actually, most of us do that, right? 1437 01:22:40,682 --> 01:22:45,090 If I were to have a gliding object on the surface, 1438 01:22:45,090 --> 01:22:48,500 I would actually act on that object in the direction 1439 01:22:48,500 --> 01:22:51,100 of my arm by pulling it. 1440 01:22:51,100 --> 01:22:56,520 So when I displace this body from point a to point b, 1441 01:22:56,520 --> 01:23:01,140 I still travel the distance d, but-- so d 1442 01:23:01,140 --> 01:23:04,230 is a displacement vector that can be written like-- 1443 01:23:04,230 --> 01:23:07,200 or it can be drawn like that. 1444 01:23:07,200 --> 01:23:09,550 I have to be smart in both cases, 1445 01:23:09,550 --> 01:23:11,320 figure out what I want from life, 1446 01:23:11,320 --> 01:23:13,530 because it's not so clear. 1447 01:23:13,530 --> 01:23:15,610 When they taught me I think the first time 1448 01:23:15,610 --> 01:23:18,460 I was in-- oh my God-- eighth grade, 1449 01:23:18,460 --> 01:23:20,770 and they-- that was a long time ago. 1450 01:23:20,770 --> 01:23:26,640 In this case, w is gonna be a scalar 1451 01:23:26,640 --> 01:23:32,260 and I'm gonna have the magnitude of the force F. F is a vector, 1452 01:23:32,260 --> 01:23:36,895 but to indicate it in Newtons or whatever I measure it in, 1453 01:23:36,895 --> 01:23:40,850 it's gonna be in magnitude. 1454 01:23:40,850 --> 01:23:44,552 Times the little d, but instead of little d 1455 01:23:44,552 --> 01:23:48,180 I should be a little smarter and say, 1456 01:23:48,180 --> 01:23:51,859 Magdalena, this is the magnitude of the vector A B, which 1457 01:23:51,859 --> 01:23:52,900 is a displacement vector. 1458 01:23:52,900 --> 01:23:55,520 1459 01:23:55,520 --> 01:23:58,590 So this is called work. 1460 01:23:58,590 --> 01:24:03,151 And if the force is 10 Newtons, and the distance 1461 01:24:03,151 --> 01:24:07,480 is 10 meters, because we want to go international. 1462 01:24:07,480 --> 01:24:10,972 We want to be global, right, at Texas Tech, so good. 1463 01:24:10,972 --> 01:24:12,180 So we have 100 Newton-meters. 1464 01:24:12,180 --> 01:24:17,450 1465 01:24:17,450 --> 01:24:23,034 Now you can measure-- well, you can have another example. 1466 01:24:23,034 --> 01:24:27,914 I'm thinking gravity and then you can say it in pounds, 1467 01:24:27,914 --> 01:24:29,866 and that measures force. 1468 01:24:29,866 --> 01:24:32,306 And you have other units that are not international. 1469 01:24:32,306 --> 01:24:34,260 I'm not gonna mess up. 1470 01:24:34,260 --> 01:24:38,310 When you have the work in this case, though, 1471 01:24:38,310 --> 01:24:41,430 it's more complicated. 1472 01:24:41,430 --> 01:24:44,100 And I'm not gonna be mad at you. [INAUDIBLE] is 1473 01:24:44,100 --> 01:24:45,805 trying to tell me what it is. 1474 01:24:45,805 --> 01:24:48,940 I'm not going to be mad at the people who don't know 1475 01:24:48,940 --> 01:24:51,340 what the work is in this case. 1476 01:24:51,340 --> 01:24:56,638 Although, I was looking at-- I am the person 1477 01:24:56,638 --> 01:25:01,558 who has run a committee to oversee 1478 01:25:01,558 --> 01:25:04,264 the finals for different math classes, all the math 1479 01:25:04,264 --> 01:25:05,520 classes we offer here. 1480 01:25:05,520 --> 01:25:09,950 And every semester I see the [? streak ?] pre-calculus, 1481 01:25:09,950 --> 01:25:11,976 calc 1, calc 2, calc 3. 1482 01:25:11,976 --> 01:25:13,350 In trig and pre-calculus, they'll 1483 01:25:13,350 --> 01:25:15,880 always have a work there. 1484 01:25:15,880 --> 01:25:19,930 And I was wondering how many of you took pre-calculus, 1485 01:25:19,930 --> 01:25:22,520 and how many of you remember that you 1486 01:25:22,520 --> 01:25:26,158 studied this in pre-calculus. 1487 01:25:26,158 --> 01:25:27,820 It's a little bit awkward. 1488 01:25:27,820 --> 01:25:31,400 I'm thinking, how do they do it, but I gave you the formula. 1489 01:25:31,400 --> 01:25:35,770 And they say the force in itself as a vector dot product 1490 01:25:35,770 --> 01:25:38,680 the displacement vector. 1491 01:25:38,680 --> 01:25:41,674 So they are both forces in dot product. 1492 01:25:41,674 --> 01:25:45,482 And I was surprised to see that they gave you [INAUDIBLE] 1493 01:25:45,482 --> 01:25:51,420 If I were to express it, how would I express it? 1494 01:25:51,420 --> 01:25:55,340 I'll say the magnitude of F, of course, 1495 01:25:55,340 --> 01:25:57,790 in Newtons, whatever it is, times the magnitude 1496 01:25:57,790 --> 01:26:00,240 of the displacement vector-- 1497 01:26:00,240 --> 01:26:01,720 STUDENT: Multiply those cosines. 1498 01:26:01,720 --> 01:26:04,580 PROFESSOR: Cosine of the angle between. 1499 01:26:04,580 --> 01:26:06,580 And I'm too lazy, I don't know, theta. 1500 01:26:06,580 --> 01:26:08,980 Let's call it angle theta. 1501 01:26:08,980 --> 01:26:16,200 Because I don't want to include that in locations, OK? 1502 01:26:16,200 --> 01:26:18,460 It really doesn't matter in which direction 1503 01:26:18,460 --> 01:26:23,340 I'm going, because cosine theta, thank God, is an even function. 1504 01:26:23,340 --> 01:26:25,790 It's equal to cosine of minus theta. 1505 01:26:25,790 --> 01:26:29,710 So whether I go this way or that way, it's the same cosine. 1506 01:26:29,710 --> 01:26:30,690 All right. 1507 01:26:30,690 --> 01:26:33,630 So the cosine of the angle between the two. 1508 01:26:33,630 --> 01:26:37,060 It's very easy when you don't need calculus. 1509 01:26:37,060 --> 01:26:41,960 But when you use calculus, because your trajectory is 1510 01:26:41,960 --> 01:26:45,880 no longer a line, life is becoming more complicated. 1511 01:26:45,880 --> 01:26:49,733 So we have to come up with a different formula, 1512 01:26:49,733 --> 01:26:53,581 with a different notion of work. 1513 01:26:53,581 --> 01:26:56,510 I'm gonna erase-- are you guys done with that? 1514 01:26:56,510 --> 01:26:58,200 Is it visible? 1515 01:26:58,200 --> 01:26:59,192 You're done. 1516 01:26:59,192 --> 01:27:08,116 1517 01:27:08,116 --> 01:27:08,616 OK. 1518 01:27:08,616 --> 01:27:14,100 1519 01:27:14,100 --> 01:27:18,680 So again, life is not so easy in reality anymore. 1520 01:27:18,680 --> 01:27:28,500 I have a particle in physics-- a photon enters a four-star hotel 1521 01:27:28,500 --> 01:27:32,838 and says-- talks to the bellboy, and the bellboy, 1522 01:27:32,838 --> 01:27:34,780 can I help you with your luggage. 1523 01:27:34,780 --> 01:27:36,732 No, I'm traveling light. 1524 01:27:36,732 --> 01:27:37,694 [LAUGHTER] 1525 01:27:37,694 --> 01:27:41,390 So the particle, the photon-- whatever. 1526 01:27:41,390 --> 01:27:45,190 A particle is moving-- is moving on a trajectory. 1527 01:27:45,190 --> 01:27:47,790 Suppose that trajectory is planar, 1528 01:27:47,790 --> 01:27:50,690 just to make your life easier at first. 1529 01:27:50,690 --> 01:27:53,426 It's in the plane x y. 1530 01:27:53,426 --> 01:27:56,842 And this is the little particle that's moving. 1531 01:27:56,842 --> 01:28:04,162 And this is R. And that is x i plus yj. 1532 01:28:04,162 --> 01:28:08,554 1533 01:28:08,554 --> 01:28:11,482 Good. 1534 01:28:11,482 --> 01:28:13,900 And this is the point x, y. 1535 01:28:13,900 --> 01:28:16,350 And that's the position, the current position 1536 01:28:16,350 --> 01:28:18,320 of my particle, right now. 1537 01:28:18,320 --> 01:28:20,650 Not in the past, not in the future. 1538 01:28:20,650 --> 01:28:23,697 My particle is moving, and this is now. 1539 01:28:23,697 --> 01:28:26,132 Suppose time doesn't even exist. 1540 01:28:26,132 --> 01:28:30,220 We think of the movies that we saw lately, 1541 01:28:30,220 --> 01:28:33,670 in The Theory of Everything. 1542 01:28:33,670 --> 01:28:38,740 So then, they say OK, we only care about now. x, y is now 1543 01:28:38,740 --> 01:28:43,740 and that is the current position vector. 1544 01:28:43,740 --> 01:28:49,600 Well, what would be the work between now-- 1545 01:28:49,600 --> 01:28:55,480 whatever now-- and the next, let's say, this is gonna be x1, 1546 01:28:55,480 --> 01:28:57,254 y1. 1547 01:28:57,254 --> 01:28:59,222 And this is x0, y0. 1548 01:28:59,222 --> 01:29:04,634 1549 01:29:04,634 --> 01:29:14,000 That's the general formula, will be x i plus yj. 1550 01:29:14,000 --> 01:29:19,150 So I actually cannot forget about time. 1551 01:29:19,150 --> 01:29:21,070 Not as much as I want. 1552 01:29:21,070 --> 01:29:27,500 So x and y-- x and y are both changing in time. 1553 01:29:27,500 --> 01:29:30,965 We're gonna have x equals x sub t, y equals y sub t. 1554 01:29:30,965 --> 01:29:34,925 Do you guys remember what we call that kind of equation 1555 01:29:34,925 --> 01:29:39,352 for a curve from here to here? 1556 01:29:39,352 --> 01:29:40,275 Para-- 1557 01:29:40,275 --> 01:29:41,316 STUDENT: Parametrization. 1558 01:29:41,316 --> 01:29:45,735 PROFESSOR: Parametrization, or parametric equations. 1559 01:29:45,735 --> 01:29:50,154 Parametric equations. 1560 01:29:50,154 --> 01:29:57,028 1561 01:29:57,028 --> 01:29:59,020 Good. 1562 01:29:59,020 --> 01:30:04,330 So I have may the force be with you. 1563 01:30:04,330 --> 01:30:06,780 I have a force. 1564 01:30:06,780 --> 01:30:10,890 I have a force, and I have some sort of displacement. 1565 01:30:10,890 --> 01:30:15,330 But I cannot express that displacement linearly anymore. 1566 01:30:15,330 --> 01:30:18,630 I'm moving along a [INAUDIBLE] of a curve. 1567 01:30:18,630 --> 01:30:22,572 So I have to think differently. 1568 01:30:22,572 --> 01:30:27,810 And the work will be defined, whether you like it or not, 1569 01:30:27,810 --> 01:30:35,760 as F vector field, dot product with dR over c. 1570 01:30:35,760 --> 01:30:40,080 And you say, what in the world is that? 1571 01:30:40,080 --> 01:30:43,535 What would c be? 1572 01:30:43,535 --> 01:30:46,430 How do I integrate along a path? 1573 01:30:46,430 --> 01:30:50,336 And I will tell you in a second what we mean by that. 1574 01:30:50,336 --> 01:30:53,230 1575 01:30:53,230 --> 01:30:58,596 Meaning that-- this is by definition if you want. 1576 01:30:58,596 --> 01:31:03,490 This is like an application of calculus 1. 1577 01:31:03,490 --> 01:31:07,699 It can be proved, but we don't-- we do a rigorous job 1578 01:31:07,699 --> 01:31:12,316 in the book about introducing and proving that along 1579 01:31:12,316 --> 01:31:13,774 a curvilinear path. 1580 01:31:13,774 --> 01:31:17,662 This is gonna be-- where am I here, at time t0? 1581 01:31:17,662 --> 01:31:19,710 And this is at time t1. 1582 01:31:19,710 --> 01:31:23,370 That means x0 is x of t0. 1583 01:31:23,370 --> 01:31:26,580 y0 is y of t0. 1584 01:31:26,580 --> 01:31:31,530 And at the finish point, I'm at x1, which is x of t1, 1585 01:31:31,530 --> 01:31:34,880 and y1 equals y of t1. 1586 01:31:34,880 --> 01:31:40,170 So between t0 and t1, I have traveled 1587 01:31:40,170 --> 01:31:46,130 and I have F where measure at x of t y of t, 1588 01:31:46,130 --> 01:31:50,262 where t is between-- moving between t0 and t1. 1589 01:31:50,262 --> 01:31:53,040 I'm done with this is the F part. 1590 01:31:53,040 --> 01:31:54,981 What is the dR? 1591 01:31:54,981 --> 01:31:57,869 Now you guys know about differential. 1592 01:31:57,869 --> 01:32:00,364 Thank God you know about differential, 1593 01:32:00,364 --> 01:32:06,851 because this is gonna help you very much. 1594 01:32:06,851 --> 01:32:07,849 OK. 1595 01:32:07,849 --> 01:32:18,100 So instead of dR, I'm going to write dot, and let's 1596 01:32:18,100 --> 01:32:24,295 see how I write-- what I write in terms of dR. You may say, 1597 01:32:24,295 --> 01:32:28,020 well, what does she mean? 1598 01:32:28,020 --> 01:32:34,617 dR was dxi plus dyj. 1599 01:32:34,617 --> 01:32:36,281 And you say, why is that? 1600 01:32:36,281 --> 01:32:37,072 I don't understand. 1601 01:32:37,072 --> 01:32:41,491 Because R itself is x i plus yj. 1602 01:32:41,491 --> 01:32:46,638 And x is a function of t, and y is a function of t. 1603 01:32:46,638 --> 01:32:49,910 That means that when you apply the differential, 1604 01:32:49,910 --> 01:32:53,566 you are gonna apply the differentials to dx and dy, 1605 01:32:53,566 --> 01:32:57,003 and these are gonna be infinitesimal displacement. 1606 01:32:57,003 --> 01:33:00,440 Infinitesimal displacement. 1607 01:33:00,440 --> 01:33:07,805 1608 01:33:07,805 --> 01:33:09,769 Infinitesimal displacement. 1609 01:33:09,769 --> 01:33:14,188 What is an infinitesimal displacement in terms of time? 1610 01:33:14,188 --> 01:33:16,590 Well, we have our parametric equations. 1611 01:33:16,590 --> 01:33:21,280 So Mr. dx as a differential is just x prime dt. 1612 01:33:21,280 --> 01:33:24,060 It's like in the [INAUDIBLE] substitution. 1613 01:33:24,060 --> 01:33:26,162 dy is just y prime dt. 1614 01:33:26,162 --> 01:33:28,617 So let me write this down again. 1615 01:33:28,617 --> 01:33:44,554 This is x prime of t i plus y prime of t j times dt. 1616 01:33:44,554 --> 01:33:48,033 So Mr. dt is like a common factor. 1617 01:33:48,033 --> 01:33:50,021 If he wants to go out for a walk, 1618 01:33:50,021 --> 01:33:52,177 he says, I'm gonna go out for a walk. 1619 01:33:52,177 --> 01:33:53,010 I go out for a walk. 1620 01:33:53,010 --> 01:33:58,420 So dR is actually x prime of t times i 1621 01:33:58,420 --> 01:34:02,972 plus y prime of t times j dt. 1622 01:34:02,972 --> 01:34:06,850 1623 01:34:06,850 --> 01:34:14,650 And this will represent the derivative of R 1624 01:34:14,650 --> 01:34:16,054 with respect to pi. 1625 01:34:16,054 --> 01:34:18,030 So that will be what? 1626 01:34:18,030 --> 01:34:19,512 The differential. 1627 01:34:19,512 --> 01:34:26,428 Differential of R with respect to pi. 1628 01:34:26,428 --> 01:34:31,370 1629 01:34:31,370 --> 01:34:34,550 This is the same as writing dx i plus dy j. 1630 01:34:34,550 --> 01:34:37,690 1631 01:34:37,690 --> 01:34:44,112 And it's the same as writing dR. Why is this happening? 1632 01:34:44,112 --> 01:34:47,330 Because it's [INAUDIBLE] Because x and y 1633 01:34:47,330 --> 01:34:52,536 themselves are functions of one variable only, which is time. 1634 01:34:52,536 --> 01:34:56,488 This is why it happens. 1635 01:34:56,488 --> 01:34:59,946 Oh, so we will simply have to do-- 1636 01:34:59,946 --> 01:35:02,910 to learn new things, right? 1637 01:35:02,910 --> 01:35:05,380 We are gonna have to learn new things, 1638 01:35:05,380 --> 01:35:10,080 like integral from a time-- fixed time 0 to t1, which 1639 01:35:10,080 --> 01:35:15,457 is 10 seconds, of a dot product between a certain vector that 1640 01:35:15,457 --> 01:35:18,610 depends on time and another vector that depends on time, 1641 01:35:18,610 --> 01:35:20,552 and dt. 1642 01:35:20,552 --> 01:35:23,468 So we are gonna have to learn how 1643 01:35:23,468 --> 01:35:28,000 to compute the work through this type of curvilinear integral. 1644 01:35:28,000 --> 01:35:32,492 And this is-- this is called either path integral-- path 1645 01:35:32,492 --> 01:35:50,907 integral along the curve c, or curvilinear integral along c. 1646 01:35:50,907 --> 01:35:53,853 Yes. 1647 01:35:53,853 --> 01:35:56,799 STUDENT: Let's say if I move the force this is 1648 01:35:56,799 --> 01:35:58,763 [INAUDIBLE] function, correct? 1649 01:35:58,763 --> 01:36:01,954 So if I can find [? arc length ?] that 1650 01:36:01,954 --> 01:36:03,720 is between the x-- 1651 01:36:03,720 --> 01:36:04,570 PROFESSOR: Yeah-- 1652 01:36:04,570 --> 01:36:05,736 STUDENT: [INAUDIBLE] points. 1653 01:36:05,736 --> 01:36:08,870 PROFESSOR: Yeah, we will do the one with that length next. 1654 01:36:08,870 --> 01:36:10,914 The [? reason ?] so-- 1655 01:36:10,914 --> 01:36:11,830 STUDENT: Is it harder? 1656 01:36:11,830 --> 01:36:12,790 PROFESSOR: No. 1657 01:36:12,790 --> 01:36:15,064 No, you can pass through a plane. 1658 01:36:15,064 --> 01:36:17,499 And you can-- we'll do that next time. 1659 01:36:17,499 --> 01:36:20,908 You will have an integral with respect to S. 1660 01:36:20,908 --> 01:36:23,343 So the integration will be with respect to dS, 1661 01:36:23,343 --> 01:36:24,317 they are correct. 1662 01:36:24,317 --> 01:36:27,726 And then you will have a function that depends on S 1663 01:36:27,726 --> 01:36:31,340 [INAUDIBLE] So I'll-- for today, I'll only teach you that. 1664 01:36:31,340 --> 01:36:33,360 Next time I'll teach you that with respect 1665 01:36:33,360 --> 01:36:36,766 to arc length, which is also very-- it's not hard at all. 1666 01:36:36,766 --> 01:36:37,266 STUDENT: OK. 1667 01:36:37,266 --> 01:36:41,532 PROFESSOR: So I will work with you on [INAUDIBLE] 1668 01:36:41,532 --> 01:36:48,773 Now assume that we have-- I will spray all this thing. 1669 01:36:48,773 --> 01:36:55,156 1670 01:36:55,156 --> 01:36:58,102 Assume that I have a problem. 1671 01:36:58,102 --> 01:37:02,670 I have a parabola-- arc of a parabola, all right? 1672 01:37:02,670 --> 01:37:08,504 Between-- let's say the parabola is 1673 01:37:08,504 --> 01:37:16,440 y equals x squared between two points. 1674 01:37:16,440 --> 01:37:19,416 1675 01:37:19,416 --> 01:37:21,896 And I'll ask you to compute some work, 1676 01:37:21,896 --> 01:37:25,368 and I'll tell you in a second what [INAUDIBLE] to do. 1677 01:37:25,368 --> 01:37:38,264 1678 01:37:38,264 --> 01:37:52,648 So exercise-- assume the parabola y 1679 01:37:52,648 --> 01:38:03,340 equals x squared between points A of coordinates 0, 0 1680 01:38:03,340 --> 01:38:07,040 and point B of coordinates 1, 1. 1681 01:38:07,040 --> 01:38:15,826 a, Parametrized this parabola in the simplest way you can. 1682 01:38:15,826 --> 01:38:31,070 1683 01:38:31,070 --> 01:38:41,360 And b, compute the work along this arc of a parabola, 1684 01:38:41,360 --> 01:38:44,316 arc AB of this parabola. 1685 01:38:44,316 --> 01:38:50,268 1686 01:38:50,268 --> 01:39:04,246 For [INAUDIBLE] the function big F of t, 1687 01:39:04,246 --> 01:39:09,922 you see that-- I'm going to say, no, big F of the point x, y, 1688 01:39:09,922 --> 01:39:15,131 because you haven't parametrized that yet, big F of x, 1689 01:39:15,131 --> 01:39:22,310 y being xi plus yg. 1690 01:39:22,310 --> 01:39:27,722 1691 01:39:27,722 --> 01:39:30,182 So you say, OK, wait a minute. 1692 01:39:30,182 --> 01:39:34,610 W will be integral over the arc of a parabola. 1693 01:39:34,610 --> 01:39:37,040 Do you want to draw that first? 1694 01:39:37,040 --> 01:39:39,160 Yes, I need to draw that first. 1695 01:39:39,160 --> 01:39:43,998 So I have this parabola from A to B. A is of coordinates 0, 0. 1696 01:39:43,998 --> 01:39:45,962 B is of coordinates 1, 1. 1697 01:39:45,962 --> 01:39:48,908 And this is y equals x squared. 1698 01:39:48,908 --> 01:39:52,099 So what kind of parametrization is 1699 01:39:52,099 --> 01:39:55,291 the simplest one, the regular one that people take? 1700 01:39:55,291 --> 01:39:57,255 Take x to be t? 1701 01:39:57,255 --> 01:40:00,390 And of course, take y in that case. y will be t squared. 1702 01:40:00,390 --> 01:40:01,553 And for 1 you have 1. 1703 01:40:01,553 --> 01:40:04,018 For 0 you have 0. 1704 01:40:04,018 --> 01:40:07,469 When you have that work by definition, what was that? 1705 01:40:07,469 --> 01:40:12,247 It was written as integral or on the graph C. Let's call 1706 01:40:12,247 --> 01:40:14,227 this path C a curvilinear path. 1707 01:40:14,227 --> 01:40:16,455 Look, script C-- so beautiful. 1708 01:40:16,455 --> 01:40:25,365 Let me [INAUDIBLE] red and draw the C of what is work? 1709 01:40:25,365 --> 01:40:36,616 F force-- may the force be with us-- dot dR. All righty, that's 1710 01:40:36,616 --> 01:40:39,050 a little bit of a headache. 1711 01:40:39,050 --> 01:40:45,910 This F is going to be-- can I write an alternative formula 1712 01:40:45,910 --> 01:40:50,810 that I have not written yet but I will write in a second? 1713 01:40:50,810 --> 01:40:56,060 dR will be dxi plus dyj. 1714 01:40:56,060 --> 01:41:01,628 So I can also write that F dot dR 1715 01:41:01,628 --> 01:41:04,556 as the dot product will seem to be-- what was the dot product 1716 01:41:04,556 --> 01:41:06,020 guys, do you remember? 1717 01:41:06,020 --> 01:41:10,430 First component times first component, F1dx 1718 01:41:10,430 --> 01:41:14,931 plus second scalar component times second scalar component, 1719 01:41:14,931 --> 01:41:15,431 F2dy. 1720 01:41:15,431 --> 01:41:20,170 1721 01:41:20,170 --> 01:41:21,510 I'll write it down. 1722 01:41:21,510 --> 01:41:26,382 Along the path C I'll have F1dx plus F2dy. 1723 01:41:26,382 --> 01:41:29,890 But god knows what it's going to be in terms of time. 1724 01:41:29,890 --> 01:41:33,140 So I have to change variable thinking. 1725 01:41:33,140 --> 01:41:36,690 Okey-dokey, Mr. dx by substitution 1726 01:41:36,690 --> 01:41:41,260 was x prime to T. Mr. dy by substitution was y prime dt. 1727 01:41:41,260 --> 01:41:46,010 So I'd rather write this in a simpler way. 1728 01:41:46,010 --> 01:41:50,206 This is a new object, path integral. 1729 01:41:50,206 --> 01:41:52,980 But we know this object from Calc I 1730 01:41:52,980 --> 01:41:56,830 as being a simple integral from time t0-- 1731 01:41:56,830 --> 01:42:04,070 I'll write it down-- time t1, F1x prime of t plus F2y 1732 01:42:04,070 --> 01:42:05,911 prime of t. 1733 01:42:05,911 --> 01:42:09,760 This is the integral dt. 1734 01:42:09,760 --> 01:42:12,927 This would be a piece of cake for us 1735 01:42:12,927 --> 01:42:16,067 to apply in this problem. 1736 01:42:16,067 --> 01:42:19,931 Equals-- now you tell me what I'm supposed to write. 1737 01:42:19,931 --> 01:42:23,260 Because if you don't, I'm going to not write anything. 1738 01:42:23,260 --> 01:42:26,400 t0 for me is what time? 1739 01:42:26,400 --> 01:42:28,370 When did we leave this? 1740 01:42:28,370 --> 01:42:29,346 0. 1741 01:42:29,346 --> 01:42:31,251 And when did we arrive? 1742 01:42:31,251 --> 01:42:32,604 At 1 o'clock. 1743 01:42:32,604 --> 01:42:38,150 We arrived when t is 1, or every one second or whatever 1744 01:42:38,150 --> 01:42:40,287 depending on [INAUDIBLE]. 1745 01:42:40,287 --> 01:42:45,780 OK, from 0 to 1, now who is F1? 1746 01:42:45,780 --> 01:42:48,365 F1 is this. 1747 01:42:48,365 --> 01:42:49,350 But it drives me crazy. 1748 01:42:49,350 --> 01:42:52,800 Because I need this to be expressed in t. 1749 01:42:52,800 --> 01:42:55,810 So I think of x and y as functions of t. 1750 01:42:55,810 --> 01:42:59,524 So if 1 is not x, not [INAUDIBLE] 1751 01:42:59,524 --> 01:43:04,484 right here, but t, which is the same thing in parametrization-- 1752 01:43:04,484 --> 01:43:07,956 this is t, t times. 1753 01:43:07,956 --> 01:43:10,436 Who is x prime? 1754 01:43:10,436 --> 01:43:11,924 1, thank god. 1755 01:43:11,924 --> 01:43:17,380 That is easy, times 1, plus F2. 1756 01:43:17,380 --> 01:43:19,860 Who is F2? 1757 01:43:19,860 --> 01:43:20,360 t squared. 1758 01:43:20,360 --> 01:43:22,640 I'll have to write it down. 1759 01:43:22,640 --> 01:43:25,420 Times who is y prime? 1760 01:43:25,420 --> 01:43:26,230 2t. 1761 01:43:26,230 --> 01:43:27,925 y prime is t2. 1762 01:43:27,925 --> 01:43:34,140 So I write it down-- 2t, dt. 1763 01:43:34,140 --> 01:43:37,980 1764 01:43:37,980 --> 01:43:40,970 So that's how I compute this integral back. 1765 01:43:40,970 --> 01:43:41,820 Is it hard? 1766 01:43:41,820 --> 01:43:47,370 No, because it's just a simple integral from Calculus I. 1767 01:43:47,370 --> 01:43:49,900 So I have to integrate what function? 1768 01:43:49,900 --> 01:43:55,690 A polynomial, 2t cubed plus t with respect 1769 01:43:55,690 --> 01:43:59,720 to t between 0, time 0 and time 1. 1770 01:43:59,720 --> 01:44:02,630 1771 01:44:02,630 --> 01:44:05,060 Good, let's do it. 1772 01:44:05,060 --> 01:44:09,620 Because that's a piece of cake-- 2 times t to the 4 over 4 1773 01:44:09,620 --> 01:44:12,390 plus t squared over 2. 1774 01:44:12,390 --> 01:44:16,782 I take the whole thing between, I apply the fundamental theorem 1775 01:44:16,782 --> 01:44:21,590 of calculus, and I have between t equals 1 up and t equals 0 1776 01:44:21,590 --> 01:44:22,360 down. 1777 01:44:22,360 --> 01:44:26,114 What's the final answer? 1778 01:44:26,114 --> 01:44:27,600 It's a single final answer. 1779 01:44:27,600 --> 01:44:30,430 And again, on the exam, on the final, 1780 01:44:30,430 --> 01:44:32,576 do not expect a headache computation. 1781 01:44:32,576 --> 01:44:34,730 Do expect something simple like that 1782 01:44:34,730 --> 01:44:36,706 where you don't need a calculator. 1783 01:44:36,706 --> 01:44:40,510 You just have either integers only or simple fractions 1784 01:44:40,510 --> 01:44:42,130 to add, and you should get the answer. 1785 01:44:42,130 --> 01:44:44,340 What is the answer, guys? 1786 01:44:44,340 --> 01:44:47,760 1-- 1/2 plus 1/2 equals 1. 1787 01:44:47,760 --> 01:44:52,390 So 1 is the value of the work in what? 1788 01:44:52,390 --> 01:44:54,940 Measured in newtons times meters, 1789 01:44:54,940 --> 01:44:57,815 whatever your units are. 1790 01:44:57,815 --> 01:45:01,630 When you drag the object from this point 1791 01:45:01,630 --> 01:45:06,050 to this point, on which the acting force is the only 1792 01:45:06,050 --> 01:45:08,030 acting force-- it could be the result 1793 01:45:08,030 --> 01:45:09,880 that there are several forces. 1794 01:45:09,880 --> 01:45:13,318 That is that force that you have here. 1795 01:45:13,318 --> 01:45:15,274 Is it useful? 1796 01:45:15,274 --> 01:45:17,230 It's very useful for engineers. 1797 01:45:17,230 --> 01:45:18,697 It's very useful for physicists. 1798 01:45:18,697 --> 01:45:21,300 It's very useful for anybody who works 1799 01:45:21,300 --> 01:45:27,227 in applied mathematics, this notion of work given like that. 1800 01:45:27,227 --> 01:45:29,620 I'm going to go ahead and erase. 1801 01:45:29,620 --> 01:45:34,860 And I'll ask you one thing here that is not 1802 01:45:34,860 --> 01:45:38,812 in the book I think as far as I remember. 1803 01:45:38,812 --> 01:45:45,350 Can you guys prove that this sophisticated formula becomes 1804 01:45:45,350 --> 01:45:48,030 your formula of the one you claimed, 1805 01:45:48,030 --> 01:45:52,060 the first formula you gave me? 1806 01:45:52,060 --> 01:45:53,052 Is it hard? 1807 01:45:53,052 --> 01:45:58,012 Do you think it's hard to prove this? 1808 01:45:58,012 --> 01:46:01,484 OK, what if we have the simplest possible case. 1809 01:46:01,484 --> 01:46:03,468 Let's think of-- 1810 01:46:03,468 --> 01:46:06,940 STUDENT: [INAUDIBLE] 1811 01:46:06,940 --> 01:46:09,916 1812 01:46:09,916 --> 01:46:11,652 PROFESSOR: Yeah, I'm thinking maybe I 1813 01:46:11,652 --> 01:46:22,316 should do, well, A to B, right? 1814 01:46:22,316 --> 01:46:33,228 AB, what kind of expression do I have [INAUDIBLE]? 1815 01:46:33,228 --> 01:46:38,188 If I take this to be-- I could have any line, right? 1816 01:46:38,188 --> 01:46:39,854 I could have any line. 1817 01:46:39,854 --> 01:46:43,500 But if I have any line, I can pick my frame 1818 01:46:43,500 --> 01:46:46,930 according to my preference. 1819 01:46:46,930 --> 01:46:49,420 Nobody's going to tell me, well, you 1820 01:46:49,420 --> 01:46:51,430 have to take the frame like that, 1821 01:46:51,430 --> 01:46:55,990 and then your line will be of the form ax plus by equals. 1822 01:46:55,990 --> 01:47:04,760 No, I'll just take the frame to be this one, where 1823 01:47:04,760 --> 01:47:09,846 AB will be x axis, and A will be of coordinates 0, 0 1824 01:47:09,846 --> 01:47:14,540 and B will be of coordinates B and 0. 1825 01:47:14,540 --> 01:47:18,550 And this is just my line. 1826 01:47:18,550 --> 01:47:28,100 So x will be moving between 0 and B. And y is 0, right? 1827 01:47:28,100 --> 01:47:31,390 It should be, at least. 1828 01:47:31,390 --> 01:47:43,729 And then F, let's say, should be this function, this. 1829 01:47:43,729 --> 01:47:48,530 1830 01:47:48,530 --> 01:47:50,570 I'll assume the angle is constant, 1831 01:47:50,570 --> 01:47:53,398 just like I had it with theta. 1832 01:47:53,398 --> 01:47:57,009 And then it's acting all the way on your object. 1833 01:47:57,009 --> 01:47:58,965 You have the same angle here always. 1834 01:47:58,965 --> 01:48:02,877 1835 01:48:02,877 --> 01:48:07,820 So F is F1i plus F2j. 1836 01:48:07,820 --> 01:48:12,820 1837 01:48:12,820 --> 01:48:22,561 dR will be dxi plus dyj. 1838 01:48:22,561 --> 01:48:29,766 1839 01:48:29,766 --> 01:48:32,390 But then you say, wait a minute, but didn't you say, Magdalena, 1840 01:48:32,390 --> 01:48:34,400 that you are along this line? 1841 01:48:34,400 --> 01:48:36,740 Didn't you say that y is 0? 1842 01:48:36,740 --> 01:48:37,630 So which y? 1843 01:48:37,630 --> 01:48:38,450 So there is no y. 1844 01:48:38,450 --> 01:48:41,690 So this is 0, right? 1845 01:48:41,690 --> 01:48:50,068 OK, Mr. x, I want to parametrize my trajectory. 1846 01:48:50,068 --> 01:48:53,393 How do I parametrize it the simplest way? 1847 01:48:53,393 --> 01:48:55,858 I'll take x to be t. 1848 01:48:55,858 --> 01:49:00,788 And time will be exactly between 0 and d. 1849 01:49:00,788 --> 01:49:02,267 And y will be 0. 1850 01:49:02,267 --> 01:49:06,211 And thank you god, because that's easy. 1851 01:49:06,211 --> 01:49:10,180 And so all you need to give me is 1852 01:49:10,180 --> 01:49:17,024 W is integral of F dR C in that case. 1853 01:49:17,024 --> 01:49:20,398 So what am I going to have in that case? 1854 01:49:20,398 --> 01:49:21,844 I'll have this formula. 1855 01:49:21,844 --> 01:49:26,440 I'll skip a step, and I'll have that formula. 1856 01:49:26,440 --> 01:49:30,280 And that means I have integral from t0 equals 0 to t1 1857 01:49:30,280 --> 01:49:35,736 equals B. 1858 01:49:35,736 --> 01:49:39,208 F1-- now you have to tell me what F1 will be. x prime 1859 01:49:39,208 --> 01:49:40,696 [? noted ?] is 1. 1860 01:49:40,696 --> 01:49:45,656 The second guy is 0, thank you very much, and [INAUDIBLE]. 1861 01:49:45,656 --> 01:49:49,128 1862 01:49:49,128 --> 01:49:52,970 F1 will be what? 1863 01:49:52,970 --> 01:49:56,230 Well, life is nice. 1864 01:49:56,230 --> 01:50:01,480 F1 will be the projection of the vector F on my x-axis. 1865 01:50:01,480 --> 01:50:06,406 So F1 is the length of this blue vector, I'll say. 1866 01:50:06,406 --> 01:50:09,328 So F1 is a scalar. 1867 01:50:09,328 --> 01:50:12,250 Let's say F1 is a scalar component. 1868 01:50:12,250 --> 01:50:16,650 That means it's F length cosine theta. 1869 01:50:16,650 --> 01:50:19,710 Because it's hypotenuse times cosine theta. 1870 01:50:19,710 --> 01:50:20,610 So it's easy. 1871 01:50:20,610 --> 01:50:25,730 So you have length of F, how much it is, 1872 01:50:25,730 --> 01:50:29,890 how big this vector is, times cosine theta, times what 1873 01:50:29,890 --> 01:50:31,060 when you integrate it, guys? 1874 01:50:31,060 --> 01:50:34,676 When you integrate 1 with respect to t, what do you get? 1875 01:50:34,676 --> 01:50:36,940 t between d and 0. 1876 01:50:36,940 --> 01:50:41,170 So you have t between d and 0. 1877 01:50:41,170 --> 01:50:42,924 We got the formula. 1878 01:50:42,924 --> 01:50:48,043 So we got that F length times [INAUDIBLE] 1879 01:50:48,043 --> 01:50:50,550 times cosine theta times d, this is the displacement. 1880 01:50:50,550 --> 01:50:52,478 This is the cosine. 1881 01:50:52,478 --> 01:50:57,418 This is the magnitude of the force that I'm-- look, 1882 01:50:57,418 --> 01:50:59,394 this is the force. 1883 01:50:59,394 --> 01:51:01,864 My force is along my arm. 1884 01:51:01,864 --> 01:51:04,828 I'm just dragging this poor object. 1885 01:51:04,828 --> 01:51:07,298 The force I'm acting with, suppose 1886 01:51:07,298 --> 01:51:11,744 it's always the same parallel to that that I can feel. 1887 01:51:11,744 --> 01:51:16,172 So that's what I have, F cosine theta, and it was easy. 1888 01:51:16,172 --> 01:51:21,100 So as a particular case of this nasty integral, 1889 01:51:21,100 --> 01:51:26,940 I have my old work from school that I had to believe. 1890 01:51:26,940 --> 01:51:30,230 I tell you guys, I did not believe a word. 1891 01:51:30,230 --> 01:51:32,980 Because my teacher in eighth grade 1892 01:51:32,980 --> 01:51:35,660 came up with this out of nothing, 1893 01:51:35,660 --> 01:51:38,810 and we were supposed to be good students preparing 1894 01:51:38,810 --> 01:51:42,970 for a high school like this kind of scientific-- back home, 1895 01:51:42,970 --> 01:51:45,130 there are different kinds of high school. 1896 01:51:45,130 --> 01:51:47,400 There is scientific high school with emphasis 1897 01:51:47,400 --> 01:51:48,233 in math and physics. 1898 01:51:48,233 --> 01:51:49,950 There is one for chemistry/biology. 1899 01:51:49,950 --> 01:51:54,780 There is one for language, linguistics, [INAUDIBLE], 1900 01:51:54,780 --> 01:51:55,746 and so on. 1901 01:51:55,746 --> 01:51:59,610 And I was for the math and physics one. 1902 01:51:59,610 --> 01:52:03,980 And I had to solve this formula without understanding it. 1903 01:52:03,980 --> 01:52:07,980 And it took me many other years to understand 1904 01:52:07,980 --> 01:52:10,930 that it's just a little piece of a big picture, 1905 01:52:10,930 --> 01:52:16,270 and that there's something bigger than what 1906 01:52:16,270 --> 01:52:18,350 we were taught in eighth grade. 1907 01:52:18,350 --> 01:52:28,440 1908 01:52:28,440 --> 01:52:31,736 STUDENT: [INAUDIBLE] 1909 01:52:31,736 --> 01:52:39,672 1910 01:52:39,672 --> 01:52:43,640 PROFESSOR: Yeah, yeah, it's true. 1911 01:52:43,640 --> 01:52:47,112 Now I want to ask you a question. 1912 01:52:47,112 --> 01:52:58,725 So do you think that I would get any kind of conservation laws 1913 01:52:58,725 --> 01:53:03,108 in physics that apply to calculus? 1914 01:53:03,108 --> 01:53:10,360 I mean, how hard is it really to compute the work? 1915 01:53:10,360 --> 01:53:15,012 1916 01:53:15,012 --> 01:53:18,180 And I'm making an announcement now. 1917 01:53:18,180 --> 01:53:22,572 Since I have not given you a break, 1918 01:53:22,572 --> 01:53:26,964 I have to let you go in a few minutes. 1919 01:53:26,964 --> 01:53:29,910 1920 01:53:29,910 --> 01:53:32,390 But I'm making a big announcement 1921 01:53:32,390 --> 01:53:33,845 without proving it. 1922 01:53:33,845 --> 01:53:42,100 1923 01:53:42,100 --> 01:53:46,122 So we will, in about one week at the maximum, 1924 01:53:46,122 --> 01:54:11,529 in maximum one week, study the independence of path of work 1925 01:54:11,529 --> 01:54:24,347 if that work is performed by a conservative force. 1926 01:54:24,347 --> 01:54:31,742 1927 01:54:31,742 --> 01:54:33,730 And you're going to say, wait a minute, 1928 01:54:33,730 --> 01:54:37,618 what the heck is a conservative force and what does she mean? 1929 01:54:37,618 --> 01:54:42,478 Well, I just showed you that the work is a path integral. 1930 01:54:42,478 --> 01:54:44,908 We don't know what that is. 1931 01:54:44,908 --> 01:54:46,360 I'll introduce more. 1932 01:54:46,360 --> 01:54:50,468 I just introduced the definition of a path integral with respect 1933 01:54:50,468 --> 01:54:54,241 to parametrization, general parametrization with respect 1934 01:54:54,241 --> 01:54:54,741 to t. 1935 01:54:54,741 --> 01:54:57,699 So that becomes an integral with respect to dt, 1936 01:54:57,699 --> 01:55:01,020 like the one in Calc I. This is how 1937 01:55:01,020 --> 01:55:02,662 you have to view it at first. 1938 01:55:02,662 --> 01:55:08,185 But guys, if this force is not just any force, 1939 01:55:08,185 --> 01:55:26,841 it's something magic, if F comes from a scalar potential that 1940 01:55:26,841 --> 01:55:33,140 is F represents the gradient of a scalar function F-- 1941 01:55:33,140 --> 01:55:44,352 this is called scalar potential-- then 1942 01:55:44,352 --> 01:55:51,946 F is called-- now let's see how much money I 1943 01:55:51,946 --> 01:55:56,906 have for just the last two or three minutes that I have left. 1944 01:55:56,906 --> 01:55:59,386 I don't have money or I have money? 1945 01:55:59,386 --> 01:56:00,900 Come on, big money. 1946 01:56:00,900 --> 01:56:03,605 1947 01:56:03,605 --> 01:56:07,028 No, I have $5. 1948 01:56:07,028 --> 01:56:10,766 I was looking for $1. 1949 01:56:10,766 --> 01:56:14,682 Here, I'll give you $5 if you give me $4 back 1950 01:56:14,682 --> 01:56:18,168 if you guess-- I don't know. 1951 01:56:18,168 --> 01:56:21,156 So maybe in your engineering courses-- maybe 1952 01:56:21,156 --> 01:56:23,148 I give you some candy instead. 1953 01:56:23,148 --> 01:56:30,640 1954 01:56:30,640 --> 01:56:38,540 So if there is a scalar function little f of coordinates x, 1955 01:56:38,540 --> 01:56:41,180 y, whatever you have in the problem, 1956 01:56:41,180 --> 01:56:44,430 so that big F will be the nabla. 1957 01:56:44,430 --> 01:56:47,070 F nabla means the gradient. 1958 01:56:47,070 --> 01:56:50,030 We say that F comes from a scalar potential. 1959 01:56:50,030 --> 01:57:00,627 But it has also a name, which is called-- god. 1960 01:57:00,627 --> 01:57:06,110 It starts with a C, ends with an E. In that case, 1961 01:57:06,110 --> 01:57:12,100 if this is going to be equal to nabla F, in that case, 1962 01:57:12,100 --> 01:57:14,842 there is a magic theorem that I'm anticipating. 1963 01:57:14,842 --> 01:57:16,660 I'm not proving. 1964 01:57:16,660 --> 01:57:18,550 I'm doing exercises right now. 1965 01:57:18,550 --> 01:57:22,244 We'll see it in two sections, that the work does not depend 1966 01:57:22,244 --> 01:57:23,940 on the path you are taking. 1967 01:57:23,940 --> 01:57:27,241 So you can go from A to B like that, or you can go like this. 1968 01:57:27,241 --> 01:57:28,116 You can go like this. 1969 01:57:28,116 --> 01:57:28,612 You can go like this. 1970 01:57:28,612 --> 01:57:29,487 You can go like that. 1971 01:57:29,487 --> 01:57:33,030 You can go on a parabola, on a line, on anything. 1972 01:57:33,030 --> 01:57:34,749 The result is always the same. 1973 01:57:34,749 --> 01:57:36,290 And it's like the fundamental theorem 1974 01:57:36,290 --> 01:57:39,020 of Calc III in plane for the work. 1975 01:57:39,020 --> 01:57:42,851 So you have little f endpoint. 1976 01:57:42,851 --> 01:57:45,940 STUDENT: Is that [INAUDIBLE]. 1977 01:57:45,940 --> 01:57:47,980 PROFESSOR: Little f of [INAUDIBLE]. 1978 01:57:47,980 --> 01:57:51,660 So all that matters is computing this scalar potential 1979 01:57:51,660 --> 01:57:53,520 here and here, making the difference, 1980 01:57:53,520 --> 01:57:55,008 and that will be your work. 1981 01:57:55,008 --> 01:57:56,496 It's a magic thing. 1982 01:57:56,496 --> 01:57:58,976 In mechanical engineering maybe you 1983 01:57:58,976 --> 01:58:03,440 met it, in physics-- in mechanical engineering, 1984 01:58:03,440 --> 01:58:06,664 because that's where you guys drag all sorts of objects 1985 01:58:06,664 --> 01:58:09,392 around. 1986 01:58:09,392 --> 01:58:10,880 STUDENT: Conservative. 1987 01:58:10,880 --> 01:58:12,864 PROFESSOR: Ah, thank god. 1988 01:58:12,864 --> 01:58:16,336 Rachel, you're a math major I think. 1989 01:58:16,336 --> 01:58:18,320 You're an engineering major. 1990 01:58:18,320 --> 01:58:20,800 STUDENT: [INAUDIBLE] 1991 01:58:20,800 --> 01:58:24,272 PROFESSOR: Wow, OK, and who else said conservative? 1992 01:58:24,272 --> 01:58:28,160 And were there other people who said conservative? 1993 01:58:28,160 --> 01:58:29,926 I'm sorry I don't have. 1994 01:58:29,926 --> 01:58:32,480 Well, next time I'll bring a bunch of dollars, 1995 01:58:32,480 --> 01:58:35,740 and I'll start giving prizes as dollar bills 1996 01:58:35,740 --> 01:58:37,957 like I used to give in differential equations. 1997 01:58:37,957 --> 01:58:39,865 Everybody was so happy in my class. 1998 01:58:39,865 --> 01:58:42,250 Because for everything that they got quickly and right, 1999 01:58:42,250 --> 01:58:44,160 they got $1. 2000 01:58:44,160 --> 01:58:47,026 So conservative-- very good. 2001 01:58:47,026 --> 01:58:52,160 Remember that for the next few lessons. 2002 01:58:52,160 --> 01:58:57,475 We will show that when this f is magical, that is conservative, 2003 01:58:57,475 --> 01:59:02,730 you guys don't have to compute the integral at all. 2004 01:59:02,730 --> 01:59:05,020 There's no parametrization, no nothing. 2005 01:59:05,020 --> 01:59:07,192 It really doesn't depend on what path you take. 2006 01:59:07,192 --> 01:59:11,128 All you would need is to figure who this little f will be, 2007 01:59:11,128 --> 01:59:12,604 this scalar potential. 2008 01:59:12,604 --> 01:59:14,080 Our future work can do that. 2009 01:59:14,080 --> 01:59:18,370 And then you compute the values of that scalar potential here 2010 01:59:18,370 --> 01:59:19,890 and here, make the difference. 2011 01:59:19,890 --> 01:59:23,600 And for sure you'll have such a problem in the final. 2012 01:59:23,600 --> 01:59:26,190 So I'm just anticipating it, because I 2013 01:59:26,190 --> 01:59:32,600 want this to be absorbed in time into your system. 2014 01:59:32,600 --> 01:59:34,780 When we will do the final exam review, 2015 01:59:34,780 --> 01:59:36,813 you should be baptized in this kind of problem 2016 01:59:36,813 --> 01:59:41,766 so that everybody will get 100% on that for the final. 2017 01:59:41,766 --> 01:59:43,740 OK, now I'll let you go. 2018 01:59:43,740 --> 01:59:45,240 Sorry I didn't give you a break. 2019 01:59:45,240 --> 01:59:47,848 But now I give you more time. 2020 01:59:47,848 --> 01:59:49,844 And enjoy the day. 2021 01:59:49,844 --> 01:59:51,341 I'll see you Thursday. 2022 01:59:51,341 --> 01:59:58,824 2023 01:59:58,824 --> 01:59:59,824 I'm moving to my office. 2024 01:59:59,824 --> 02:00:02,818 If you have questions, you can come to my office. 2025 02:00:02,818 --> 02:00:06,311 2026 02:00:06,311 --> 02:00:09,305 Maybe you were getting close. 2027 02:00:09,305 --> 02:00:13,796 How did-- did you know, or it just came to you? 2028 02:00:13,796 --> 02:00:17,788 [BACKGROUND CHATTER] 2029 02:00:17,788 --> 02:01:03,524 2030 02:01:03,524 --> 02:01:05,440 STUDENT: Do you know what section it would be? 2031 02:01:05,440 --> 02:01:09,470 Because I don't even think he's listed or anything. 2032 02:01:09,470 --> 02:01:12,374 PROFESSOR: Send me an email if you don't figure it out. 2033 02:01:12,374 --> 02:01:15,350 But for sure [INAUDIBLE]. 2034 02:01:15,350 --> 02:01:17,830 STUDENT: OK, because I was going to do the honors, 2035 02:01:17,830 --> 02:01:19,318 but it was with [INAUDIBLE]. 2036 02:01:19,318 --> 02:01:20,310 I don't know if he's good, or she's good. 2037 02:01:20,310 --> 02:01:21,226 PROFESSOR: She's good. 2038 02:01:21,226 --> 02:01:25,370 But he's fantastic in the sense that he will help you 2039 02:01:25,370 --> 02:01:27,160 whenever you stumble. 2040 02:01:27,160 --> 02:01:29,860 He's an extremely good teacher. 2041 02:01:29,860 --> 02:01:31,660 He explains really well. 2042 02:01:31,660 --> 02:01:32,860 He has a talent. 2043 02:01:32,860 --> 02:01:36,235 2044 02:01:36,235 --> 02:01:37,360 STUDENT: I'll look for him. 2045 02:01:37,360 --> 02:01:38,260 Thank you. 2046 02:01:38,260 --> 02:01:40,360 PROFESSOR: And if you don't get him, 2047 02:01:40,360 --> 02:01:43,360 she is good as well-- not exceptional like he is. 2048 02:01:43,360 --> 02:01:44,860 He's an exceptional teacher. 2049 02:01:44,860 --> 02:01:48,760 2050 02:01:48,760 --> 02:01:50,560 STUDENT: I'll go to the office. 2051 02:01:50,560 --> 02:01:53,310 PROFESSOR: Yes, yes, [INAUDIBLE]. 2052 02:01:53,310 --> 02:01:54,911