1 00:00:00,000 --> 00:00:01,972 PROFESSOR: Questions so far? 2 00:00:01,972 --> 00:00:03,028 STUDENT: [INAUDIBLE] 3 00:00:03,028 --> 00:00:03,944 PROFESSOR: Yes, ma'am. 4 00:00:03,944 --> 00:00:04,882 You are Megan? 5 00:00:04,882 --> 00:00:05,423 STUDENT: Yes. 6 00:00:05,423 --> 00:00:07,888 PROFESSOR: OK. 7 00:00:07,888 --> 00:00:11,092 STUDENT: I was just wondering if we get like a form 8 00:00:11,092 --> 00:00:13,311 of [INAUDIBLE], note cards-- 9 00:00:13,311 --> 00:00:15,710 PROFESSOR: No, you [INAUDIBLE] sheet whatsoever. 10 00:00:15,710 --> 00:00:22,190 So I think it's better that I review some of the formulas 11 00:00:22,190 --> 00:00:25,710 today that you are expected to know by heart, 12 00:00:25,710 --> 00:00:30,280 because they are also-- they require you know to expect you 13 00:00:30,280 --> 00:00:34,440 to know the same formulas by heart for the final 14 00:00:34,440 --> 00:00:36,370 with no cheat sheet. 15 00:00:36,370 --> 00:00:39,435 So the final will have exactly the same policy, 16 00:00:39,435 --> 00:00:40,960 at the end of [INAUDIBLE]. 17 00:00:40,960 --> 00:00:45,860 No calculator, no formula sheet, no cheat sheet, no nothing, 18 00:00:45,860 --> 00:00:50,025 but you know I'm telling you guys this, except what 19 00:00:50,025 --> 00:00:50,760 you remember. 20 00:00:50,760 --> 00:00:58,110 21 00:00:58,110 --> 00:01:01,910 Let me remind you that you are expected 22 00:01:01,910 --> 00:01:06,842 to know the equation of the tangent plane. 23 00:01:06,842 --> 00:01:09,322 I'm not going to give them in chronological order, 24 00:01:09,322 --> 00:01:23,014 but I think it's a good idea to review for midterm and final, 25 00:01:23,014 --> 00:01:30,990 some of the must-know formulas. 26 00:01:30,990 --> 00:01:37,458 27 00:01:37,458 --> 00:01:42,300 One, well, I discussed this before but I didn't remind you, 28 00:01:42,300 --> 00:01:51,018 differential of a function of several variables. 29 00:01:51,018 --> 00:01:55,858 30 00:01:55,858 --> 00:01:59,660 In particular, two variables most likely 31 00:01:59,660 --> 00:02:05,675 are the examples we've worked on a lot this semester. 32 00:02:05,675 --> 00:02:13,250 Number two, the definition and especially formula, 33 00:02:13,250 --> 00:02:27,550 main formula for responding to directional derivatives of F 34 00:02:27,550 --> 00:02:36,426 at P of coordinates X 0, Y 0, in the direction U 1 and U 2, 35 00:02:36,426 --> 00:02:40,790 equals U. 36 00:02:40,790 --> 00:02:44,451 Just to test you, OK, well, I believe 37 00:02:44,451 --> 00:02:49,120 you know the formula of the differential. 38 00:02:49,120 --> 00:02:56,210 But without me reminding you what was that of two variables, 39 00:02:56,210 --> 00:02:59,790 I expect you to say d F equals-- 40 00:02:59,790 --> 00:03:02,670 STUDENT: F [INAUDIBLE]. 41 00:03:02,670 --> 00:03:06,785 D X plus F Y, D Y. 42 00:03:06,785 --> 00:03:08,201 PROFESSOR: How about-- thank you-- 43 00:03:08,201 --> 00:03:11,123 how about the directional derivative 44 00:03:11,123 --> 00:03:18,870 of F at P in the direction of the vector U? 45 00:03:18,870 --> 00:03:21,530 You will need the formula. 46 00:03:21,530 --> 00:03:22,420 Good, [INAUDIBLE]. 47 00:03:22,420 --> 00:03:25,580 Yeah, that's the easiest way to remember it, 48 00:03:25,580 --> 00:03:31,674 but that's not the first thing I want you to say, right? 49 00:03:31,674 --> 00:03:33,943 How did I write this? 50 00:03:33,943 --> 00:03:35,210 [INAUDIBLE] 51 00:03:35,210 --> 00:03:39,000 Of course, F of X-- thank you-- and [INAUDIBLE] 0, 52 00:03:39,000 --> 00:03:43,365 times Z 1 plus derivative inspect Y, 53 00:03:43,365 --> 00:03:46,290 and X 0, Y 0 times U 2. 54 00:03:46,290 --> 00:03:51,380 What do we assume about it, F C 1 on the domain [INAUDIBLE]? 55 00:03:51,380 --> 00:03:54,248 56 00:03:54,248 --> 00:04:00,606 Which means differential goal with continuous derivatives. 57 00:04:00,606 --> 00:04:04,478 This is what we assume through chapter 11. 58 00:04:04,478 --> 00:04:13,923 Number three, I think I told you, but I'm not sure. 59 00:04:13,923 --> 00:04:16,690 But I think I did. 60 00:04:16,690 --> 00:04:24,810 Review the tangent plane formula, 61 00:04:24,810 --> 00:04:27,180 formulas-- how about both? 62 00:04:27,180 --> 00:04:33,968 Well, only one is the one I consider relative for us. 63 00:04:33,968 --> 00:04:41,350 Which is Z equals F of X, Y, will imply that at P 64 00:04:41,350 --> 00:04:46,198 with on the surface, even as a reference, we 65 00:04:46,198 --> 00:04:51,040 have a tangent plane of formula Z minus Z 0, 66 00:04:51,040 --> 00:04:52,610 equals-- who does it? 67 00:04:52,610 --> 00:04:55,800 OK, now you have to remember this. 68 00:04:55,800 --> 00:04:57,907 Of course, it's your midterm. 69 00:04:57,907 --> 00:05:02,880 Review all of these things by [INAUDIBLE] Thursday. 70 00:05:02,880 --> 00:05:09,510 Variable S of X, [INAUDIBLE] 0 at 0, times-- 71 00:05:09,510 --> 00:05:11,590 STUDENT: X minus X 0. 72 00:05:11,590 --> 00:05:13,930 PROFESSOR: Thank you, Roberto, X 0 minus X 0, 73 00:05:13,930 --> 00:05:18,350 plus the same kind o thing in a different color, because I 74 00:05:18,350 --> 00:05:28,000 like to play [INAUDIBLE] orange, S Y, X 0, Y 0, Y minus Y 0. 75 00:05:28,000 --> 00:05:31,350 Don't come to the midterm-- you better not come to the midterm, 76 00:05:31,350 --> 00:05:35,850 and you get a 0 for not knowing the formulas, right? 77 00:05:35,850 --> 00:05:39,530 Now maybe you will see on this midterm, maybe 78 00:05:39,530 --> 00:05:43,350 not, maybe you'll see it on the final-- what 79 00:05:43,350 --> 00:05:47,915 happens when you don't have the graph of a surface? 80 00:05:47,915 --> 00:05:52,290 81 00:05:52,290 --> 00:06:03,740 Maybe you'll have an implicit equation, an implicit equation 82 00:06:03,740 --> 00:06:15,560 where we write F of coordinates, X, Y and Z, equals a constant. 83 00:06:15,560 --> 00:06:18,590 Why is the tangent plane a P? 84 00:06:18,590 --> 00:06:25,480 Tangent plane, tangent plane in both cases should be Y. 85 00:06:25,480 --> 00:06:31,500 Well, if you consider the first formula 86 00:06:31,500 --> 00:06:34,400 as a consequence of the second one, 87 00:06:34,400 --> 00:06:37,752 that would be simply easy, because you 88 00:06:37,752 --> 00:06:41,292 will have to write F of X Y minus Z equals 0. 89 00:06:41,292 --> 00:06:46,638 And there you are, the same kind of formula in this. 90 00:06:46,638 --> 00:06:52,200 So what do you write-- remember the surface, 91 00:06:52,200 --> 00:06:54,450 the implicit formula. 92 00:06:54,450 --> 00:06:57,515 Who gave you the normal to the surface of a point P? 93 00:06:57,515 --> 00:07:00,310 94 00:07:00,310 --> 00:07:01,320 No? 95 00:07:01,320 --> 00:07:04,070 The gradient of who? 96 00:07:04,070 --> 00:07:06,840 Not the gradient of the left, don't confuse-- the gradient 97 00:07:06,840 --> 00:07:09,120 of the big F, right? 98 00:07:09,120 --> 00:07:16,480 OK, at P. And the tangent plane represents a what? 99 00:07:16,480 --> 00:07:20,170 The tangent plane represents exactly the perpendicular plane 100 00:07:20,170 --> 00:07:26,490 that passes through the point P, and is [INAUDIBLE] 101 00:07:26,490 --> 00:07:28,720 to the normal. 102 00:07:28,720 --> 00:07:33,520 So you're going to have your surface, your normal, 103 00:07:33,520 --> 00:07:37,870 and the tangent plane, which is perpendicular to the normal. 104 00:07:37,870 --> 00:07:40,580 Is this easy to remember, maybe for your final? 105 00:07:40,580 --> 00:07:45,655 I want to check if you know-- make a list, this list, 106 00:07:45,655 --> 00:07:49,148 you have to post it in the bathroom or somewhere, 107 00:07:49,148 --> 00:07:52,142 on the wall or a closet. 108 00:07:52,142 --> 00:07:56,040 Because you need to know these things by the final. 109 00:07:56,040 --> 00:08:03,136 S of X at point P becomes S minus X 0 plus what? 110 00:08:03,136 --> 00:08:05,630 The same kind of thing, right? 111 00:08:05,630 --> 00:08:17,970 But it [INAUDIBLE] Y and Z. 112 00:08:17,970 --> 00:08:21,270 So if you have the curiosity to want 113 00:08:21,270 --> 00:08:25,530 to prove that the first colorful formula for the tangent plane, 114 00:08:25,530 --> 00:08:31,550 using the red formula for the tangent plane, it would come, 115 00:08:31,550 --> 00:08:33,270 is an immediate [INAUDIBLE]. 116 00:08:33,270 --> 00:08:35,220 We've actually done that before. 117 00:08:35,220 --> 00:08:38,120 We even did the implicit function theorem. 118 00:08:38,120 --> 00:08:39,950 There are some very nice things you 119 00:08:39,950 --> 00:08:42,499 can do when you have a function of several variables. 120 00:08:42,499 --> 00:08:48,130 And in particular, for a function of two variables, 121 00:08:48,130 --> 00:08:50,465 makes it really easy. 122 00:08:50,465 --> 00:08:59,105 I'm gonna erase one, two, three, and continue. 123 00:08:59,105 --> 00:09:02,370 124 00:09:02,370 --> 00:09:04,920 So I guess when I leave the room, 125 00:09:04,920 --> 00:09:08,336 I have to be careful not to leave the actual midterm 126 00:09:08,336 --> 00:09:10,460 in the room, although I know that you wouldn't even 127 00:09:10,460 --> 00:09:13,005 try to check my papers. 128 00:09:13,005 --> 00:09:16,300 129 00:09:16,300 --> 00:09:20,930 I did also something in this for finding 130 00:09:20,930 --> 00:09:23,876 a direction in which the function increases 131 00:09:23,876 --> 00:09:25,343 most rapidly. 132 00:09:25,343 --> 00:09:27,720 I don't have to write it down, but I 133 00:09:27,720 --> 00:09:29,250 can remind you of the concept. 134 00:09:29,250 --> 00:09:33,410 135 00:09:33,410 --> 00:09:39,082 So it's just the concept now, no formula to actually memorize. 136 00:09:39,082 --> 00:09:43,140 But I'll still say number four, problem number four, 137 00:09:43,140 --> 00:09:52,038 because that's what I set up on the actual exam. 138 00:09:52,038 --> 00:09:59,720 So what is the direction of highest ascend, deepest ascend? 139 00:09:59,720 --> 00:10:01,664 STUDENT: [INAUDIBLE]. 140 00:10:01,664 --> 00:10:03,610 PROFESSOR: Is the direction of the plane. 141 00:10:03,610 --> 00:10:08,914 And what is the direction of the steepest descent? 142 00:10:08,914 --> 00:10:14,480 The opposite of the direction of the grade. 143 00:10:14,480 --> 00:10:28,908 So the direction of steepest ascend and descend 144 00:10:28,908 --> 00:10:48,672 is the direction of for the graph Z equals F of X 1. 145 00:10:48,672 --> 00:10:53,010 This is the function [INAUDIBLE] that I'm talking about. 146 00:10:53,010 --> 00:11:00,547 Five, the direction of U that you 147 00:11:00,547 --> 00:11:07,200 found at the previous problem, I didn't ask if it's unique, OK? 148 00:11:07,200 --> 00:11:09,679 Because that was one-- of course it's unique. 149 00:11:09,679 --> 00:11:12,144 Because we [INAUDIBLE] sizes. 150 00:11:12,144 --> 00:11:15,595 How do you say units of sizing, [INAUDIBLE]? 151 00:11:15,595 --> 00:11:19,539 By deriving with it, [INAUDIBLE] a second, 152 00:11:19,539 --> 00:11:22,497 you have a U and a -U. 153 00:11:22,497 --> 00:11:24,962 STUDENT: So isn't the direction that the actual [INAUDIBLE] 154 00:11:24,962 --> 00:11:27,427 for is the gradient of a normal vector? 155 00:11:27,427 --> 00:11:33,904 PROFESSOR: So yeah, so the way I-- OK, you 156 00:11:33,904 --> 00:11:35,070 want me to read the problem? 157 00:11:35,070 --> 00:11:40,190 I'm going to read the actual function So find the direction 158 00:11:40,190 --> 00:11:47,290 U, in which the function F of X Y, blah, blah, blah, blah, 159 00:11:47,290 --> 00:11:51,340 is here, it increases most rapidly. 160 00:11:51,340 --> 00:11:53,204 So what do you have to do? 161 00:11:53,204 --> 00:11:55,410 So the direction of that is, what is 162 00:11:55,410 --> 00:11:58,370 the direction of that or this? 163 00:11:58,370 --> 00:12:11,860 U equals [INAUDIBLE] respectively minus U at P. 164 00:12:11,860 --> 00:12:21,860 Five, this direction U that you found at the previous problem, 165 00:12:21,860 --> 00:12:25,180 could be perpendicular to a certain line, which 166 00:12:25,180 --> 00:12:26,180 of the following planes? 167 00:12:26,180 --> 00:12:32,260 168 00:12:32,260 --> 00:12:33,950 I may give you multiple choice. 169 00:12:33,950 --> 00:12:37,510 Now what do you have to do when you think 170 00:12:37,510 --> 00:12:44,840 the direction of-- the way it's actually formulated 171 00:12:44,840 --> 00:12:47,980 is zero direction is parallel to one 172 00:12:47,980 --> 00:12:50,880 of the following [INAUDIBLE] planes, which one? 173 00:12:50,880 --> 00:12:52,380 Let me give you an example. 174 00:12:52,380 --> 00:13:01,380 175 00:13:01,380 --> 00:13:07,946 Z equals X [INAUDIBLE] squared, at P coordinates 1, 1. 176 00:13:07,946 --> 00:13:13,650 So that means X 0 is 1, Y 0 is 1, and Z 0 is two. 177 00:13:13,650 --> 00:13:17,190 178 00:13:17,190 --> 00:13:28,680 Find the direction of the gradient of F. 179 00:13:28,680 --> 00:13:36,520 Let me put Z for alpha-- I'm abusing my [INAUDIBLE]-- at P. 180 00:13:36,520 --> 00:13:50,090 And state which of the following lines is parallel for this 181 00:13:50,090 --> 00:13:50,590 direction? 182 00:13:50,590 --> 00:13:58,350 183 00:13:58,350 --> 00:14:02,780 A, lines in plane. 184 00:14:02,780 --> 00:14:07,130 185 00:14:07,130 --> 00:14:08,850 X equals 2. 186 00:14:08,850 --> 00:14:11,655 B, Y equals 3. 187 00:14:11,655 --> 00:14:16,800 Or C, X plus 1 equals 0. 188 00:14:16,800 --> 00:14:20,155 D, these are lines in plane in the plane, 189 00:14:20,155 --> 00:14:24,422 X Y. X plus Y. E, none of the above. 190 00:14:24,422 --> 00:14:29,669 191 00:14:29,669 --> 00:14:31,690 So how are you going to do that quickly? 192 00:14:31,690 --> 00:14:32,730 Well, it's easy, right? 193 00:14:32,730 --> 00:14:34,560 So what do I do? 194 00:14:34,560 --> 00:14:38,730 I say gradient 2 X 2 Y, at the point 195 00:14:38,730 --> 00:14:41,548 1, 1-- you don't have to write down everything. 196 00:14:41,548 --> 00:14:47,030 It's going to be the gradient of F at P, will be 2, 2. 197 00:14:47,030 --> 00:14:53,180 That means U will be normalized 2, 2. 198 00:14:53,180 --> 00:14:54,742 What do you get? 199 00:14:54,742 --> 00:14:56,550 STUDENT: [INAUDIBLE] 200 00:14:56,550 --> 00:15:00,780 PROFESSOR: Well, what do people do normally 201 00:15:00,780 --> 00:15:04,017 if they want to do it by the definition? 202 00:15:04,017 --> 00:15:06,452 They [INAUDIBLE] the vector 2, 2, 203 00:15:06,452 --> 00:15:08,840 by the square root [INAUDIBLE]. 204 00:15:08,840 --> 00:15:11,400 Well, you could be a little bit smarter than that, 205 00:15:11,400 --> 00:15:14,930 and say, F is the same as the direction 1, 206 00:15:14,930 --> 00:15:18,360 1 divided by the square root of the sums, 207 00:15:18,360 --> 00:15:21,450 of a sum of the squares. 208 00:15:21,450 --> 00:15:23,040 It doesn't matter which one you pick. 209 00:15:23,040 --> 00:15:28,570 All the co-linear ones will reveal the unique U. 210 00:15:28,570 --> 00:15:31,980 And that's exactly what I was trying to say, 211 00:15:31,980 --> 00:15:35,420 was this thinking by in just two or three moves ahead of that. 212 00:15:35,420 --> 00:15:38,017 STUDENT: So that's the same as 2, 2 over 4? 213 00:15:38,017 --> 00:15:38,850 PROFESSOR: Yes, sir. 214 00:15:38,850 --> 00:15:43,320 It's the same as 2, 2 over the square root of 4 plus 4. 215 00:15:43,320 --> 00:15:46,880 But it's easier, why it's sort of faster to do it. 216 00:15:46,880 --> 00:15:51,495 So why is that true actually, Ryan is very right? 217 00:15:51,495 --> 00:15:53,865 Why is that true? 218 00:15:53,865 --> 00:15:56,165 Exactly because of that uniqueness that I told you 219 00:15:56,165 --> 00:16:00,510 about last time, when you said, well, [INAUDIBLE], 220 00:16:00,510 --> 00:16:01,720 what is that? 221 00:16:01,720 --> 00:16:06,590 So you get 1 over square root of 2, and 1 over square root of 2, 222 00:16:06,590 --> 00:16:10,640 is that you [INAUDIBLE] vector direction. 223 00:16:10,640 --> 00:16:14,290 Now without doing further work, this 224 00:16:14,290 --> 00:16:16,860 is just a simple multiple question, 225 00:16:16,860 --> 00:16:18,347 of [INAUDIBLE] question. 226 00:16:18,347 --> 00:16:21,840 You are in front of your exam, and you see lines in play. 227 00:16:21,840 --> 00:16:25,910 You close your eyes and see all of-- I will see my what? 228 00:16:25,910 --> 00:16:29,376 You see all the lines in plane. 229 00:16:29,376 --> 00:16:33,400 Of all these lines, your favorite line 230 00:16:33,400 --> 00:16:38,920 has to have the same direction as the vector U. Is X equals 2? 231 00:16:38,920 --> 00:16:40,530 No, that's nothing. 232 00:16:40,530 --> 00:16:41,350 Y equals 3? 233 00:16:41,350 --> 00:16:43,570 Those are horizontal, vertical. 234 00:16:43,570 --> 00:16:46,980 That's the direction of the first [INAUDIBLE]. 235 00:16:46,980 --> 00:16:48,720 So is this true of C? 236 00:16:48,720 --> 00:16:49,386 No. 237 00:16:49,386 --> 00:16:51,010 STUDENT: No, that's for parallel lines. 238 00:16:51,010 --> 00:16:51,814 PROFESSOR: D? 239 00:16:51,814 --> 00:16:52,662 STUDENT: Yes. 240 00:16:52,662 --> 00:16:53,510 PROFESSOR: Right? 241 00:16:53,510 --> 00:16:58,250 So the incline X, Y-- the first bisector 242 00:16:58,250 --> 00:16:59,740 is X equals Y [INAUDIBLE]. 243 00:16:59,740 --> 00:17:05,490 Number C is Y minus X, which is called the second bisector. 244 00:17:05,490 --> 00:17:08,880 You've seen that in college algebra-- high school algebra, 245 00:17:08,880 --> 00:17:10,090 more likely. 246 00:17:10,090 --> 00:17:12,983 So we call this first bisector, second bisector. 247 00:17:12,983 --> 00:17:15,240 All right, so the answer is D. Do you have 248 00:17:15,240 --> 00:17:17,278 the same thing [INAUDIBLE]? 249 00:17:17,278 --> 00:17:19,772 On the two multiple choice things 250 00:17:19,772 --> 00:17:25,262 you have, you see very well, OK, I'm testing you. 251 00:17:25,262 --> 00:17:28,540 I didn't say anything. 252 00:17:28,540 --> 00:17:33,705 It was three feet away, OK? 253 00:17:33,705 --> 00:17:37,330 254 00:17:37,330 --> 00:17:42,190 We have just a quick answer, and it's 255 00:17:42,190 --> 00:17:44,455 going to be easy, without algebra, 256 00:17:44,455 --> 00:17:45,700 without computational stuff. 257 00:17:45,700 --> 00:17:48,550 258 00:17:48,550 --> 00:17:50,900 Just from the first glance, you'll be able to answer. 259 00:17:50,900 --> 00:17:53,806 Number six, what is the maximum rate 260 00:17:53,806 --> 00:17:58,800 of increase in the same case as in problem five? 261 00:17:58,800 --> 00:18:05,976 You say [INAUDIBLE], what is the maximum, maximal rate 262 00:18:05,976 --> 00:18:12,189 of increase of a [INAUDIBLE]? 263 00:18:12,189 --> 00:18:14,674 And we all know what I'm talking about, 264 00:18:14,674 --> 00:18:16,662 although maybe not everybody. 265 00:18:16,662 --> 00:18:19,147 But this is the gradient. 266 00:18:19,147 --> 00:18:23,290 Who is giving you the maximum rate of increase? 267 00:18:23,290 --> 00:18:27,430 As I said last time in the review, 268 00:18:27,430 --> 00:18:29,500 that's actually the directional derivative 269 00:18:29,500 --> 00:18:31,200 in the direction of the gradient. 270 00:18:31,200 --> 00:18:35,240 But you are supposed to know without proving again 271 00:18:35,240 --> 00:18:37,490 that the directional derivative and the direction 272 00:18:37,490 --> 00:18:41,140 of the gradient will give you that what? 273 00:18:41,140 --> 00:18:43,580 Gradient of norm of-- 274 00:18:43,580 --> 00:18:44,470 STUDENT: [INAUDIBLE]. 275 00:18:44,470 --> 00:18:50,840 PROFESSOR: Exactly, the magnitude of this F. So 276 00:18:50,840 --> 00:18:59,554 what does that-- what would that be in my case? 277 00:18:59,554 --> 00:19:02,010 [INAUDIBLE] pay attention, please. 278 00:19:02,010 --> 00:19:04,620 Don't look at this, if it's confusing you. 279 00:19:04,620 --> 00:19:06,420 Look at that, right? 280 00:19:06,420 --> 00:19:07,630 How much is that? 281 00:19:07,630 --> 00:19:11,520 282 00:19:11,520 --> 00:19:14,120 All right, [INAUDIBLE]. 283 00:19:14,120 --> 00:19:19,105 So you can put this as [INAUDIBLE]. 284 00:19:19,105 --> 00:19:23,820 So your multiple choice-- how many multiple choices do you 285 00:19:23,820 --> 00:19:24,320 have? 286 00:19:24,320 --> 00:19:25,320 Only two. 287 00:19:25,320 --> 00:19:28,500 It may seem like what is the maximum rate? 288 00:19:28,500 --> 00:19:33,721 1, 0, 0, is telling-- that means you have no increase. 289 00:19:33,721 --> 00:19:34,699 You're not moving. 290 00:19:34,699 --> 00:19:37,090 You're just lying there on the plane. 291 00:19:37,090 --> 00:19:37,900 OK. 292 00:19:37,900 --> 00:19:40,050 What else? 293 00:19:40,050 --> 00:19:42,580 2 root 2, 2 [INAUDIBLE] to infinity. 294 00:19:42,580 --> 00:19:43,320 I don't know. 295 00:19:43,320 --> 00:19:45,430 I'm giving some nonsensical choices. 296 00:19:45,430 --> 00:19:48,382 But one of them would be 2 root 2. 297 00:19:48,382 --> 00:19:53,750 So you would see, it would jump in front of your eyes. 298 00:19:53,750 --> 00:20:01,155 Number seven, I think I'm going-- I thought about this, 299 00:20:01,155 --> 00:20:05,610 and I said one of you guys asked me, 300 00:20:05,610 --> 00:20:08,100 can you re-open any homework? 301 00:20:08,100 --> 00:20:10,260 And I said, nope. 302 00:20:10,260 --> 00:20:10,800 Why? 303 00:20:10,800 --> 00:20:15,300 Because once the homework closes, automatically 304 00:20:15,300 --> 00:20:21,110 a few seconds later, all the answers are gonna be up. 305 00:20:21,110 --> 00:20:23,680 Do I have other problems handy to create 306 00:20:23,680 --> 00:20:29,470 a make-up for that individual person who had the problem? 307 00:20:29,470 --> 00:20:32,280 My cat almost died this week, but she said, 308 00:20:32,280 --> 00:20:35,450 but I have a treatment, and hopefully she's going to live. 309 00:20:35,450 --> 00:20:40,040 So in situations of [INAUDIBLE], like an accident, a problem, 310 00:20:40,040 --> 00:20:44,220 [INAUDIBLE] hospitalization, and so on, I'm sorry, 311 00:20:44,220 --> 00:20:46,480 I cannot re-open the homework. 312 00:20:46,480 --> 00:20:49,760 The homework is already up there with all the answers. 313 00:20:49,760 --> 00:20:54,430 When I extend homework, it's still doing that interval when 314 00:20:54,430 --> 00:20:56,200 you cannot see the answers. 315 00:20:56,200 --> 00:20:59,100 So I can extend it by there [INAUDIBLE], 316 00:20:59,100 --> 00:21:00,170 that was an exception. 317 00:21:00,170 --> 00:21:03,440 So you have until the fourth-- is the the fourth? 318 00:21:03,440 --> 00:21:05,110 OK. 319 00:21:05,110 --> 00:21:07,990 But once that closes, I cannot re-open it. 320 00:21:07,990 --> 00:21:14,440 However, I thought of giving you a compensation midterm exam, 321 00:21:14,440 --> 00:21:16,952 contains an extra credit problem. 322 00:21:16,952 --> 00:21:20,810 Because once you told me that, I started 323 00:21:20,810 --> 00:21:27,470 feeling bad for the two people who have problems. 324 00:21:27,470 --> 00:21:31,285 There were two or three people who had very serious problems 325 00:21:31,285 --> 00:21:33,500 this past weekend. 326 00:21:33,500 --> 00:21:36,825 So in the midterm, you have that extra credit problem, 327 00:21:36,825 --> 00:21:41,310 that is meant to touch up a little bit of let's say 328 00:21:41,310 --> 00:21:44,720 if you missed a few problems from the homework, 329 00:21:44,720 --> 00:21:48,440 you had some bad day, whatever. 330 00:21:48,440 --> 00:21:51,360 So you have ten problems plus one. 331 00:21:51,360 --> 00:21:56,876 Seven, you've seen that before I told you about it. 332 00:21:56,876 --> 00:21:59,774 It's an easy problem. 333 00:21:59,774 --> 00:22:06,720 You have Z equals F over X Y. And I'm 334 00:22:06,720 --> 00:22:28,140 saying compute the volume of the body that lies below the graph 335 00:22:28,140 --> 00:22:40,260 and above the unit [INAUDIBLE] D. Fine. 336 00:22:40,260 --> 00:22:45,675 Eight, unfortunately eight have [INAUDIBLE] 337 00:22:45,675 --> 00:22:49,120 was disclosed because Ryan was dreaming 338 00:22:49,120 --> 00:22:50,936 of the problems in the midterm. 339 00:22:50,936 --> 00:22:55,630 But it was something like that, very good information. 340 00:22:55,630 --> 00:23:01,860 So I would say a problem like that, maybe a plane that 341 00:23:01,860 --> 00:23:04,640 is cut in what? 342 00:23:04,640 --> 00:23:08,730 The plane's coordinates form something like a tetrahedron, 343 00:23:08,730 --> 00:23:13,655 find the volume, something like that. 344 00:23:13,655 --> 00:23:21,340 Nine, again without giving you the exact values, 345 00:23:21,340 --> 00:23:25,690 you will have a function F of X, and U of X, 346 00:23:25,690 --> 00:23:29,780 I'd say positive over a certain interval. 347 00:23:29,780 --> 00:23:35,010 348 00:23:35,010 --> 00:23:44,610 Set up the double integral, set up a double integral 349 00:23:44,610 --> 00:23:51,898 for the area of the domain between F and G contained. 350 00:23:51,898 --> 00:24:01,360 351 00:24:01,360 --> 00:24:15,400 Compute that, and also reverse the order of integration 352 00:24:15,400 --> 00:24:16,510 to check your work. 353 00:24:16,510 --> 00:24:19,378 354 00:24:19,378 --> 00:24:24,992 Your answer, because here, a multiple choice answer, 355 00:24:24,992 --> 00:24:27,845 [INAUDIBLE] answer, no guesses. 356 00:24:27,845 --> 00:24:30,270 It's going to not be hard at all-- very 357 00:24:30,270 --> 00:24:32,870 nice, friendly functions, very nice, friendly [INAUDIBLE] 358 00:24:32,870 --> 00:24:34,330 functions. 359 00:24:34,330 --> 00:24:37,927 Just I have done this before, but I'm not 360 00:24:37,927 --> 00:24:39,530 going to repeat what it was. 361 00:24:39,530 --> 00:24:45,021 I did it in the-- it's like the one I did last week. 362 00:24:45,021 --> 00:24:48,870 363 00:24:48,870 --> 00:24:53,460 All right, so remember you write the vertical strip thing, 364 00:24:53,460 --> 00:24:56,400 integration with respect to Y first, and then with respect 365 00:24:56,400 --> 00:24:58,574 to X. You switch to the horizontal strip 366 00:24:58,574 --> 00:25:01,580 method of integration with respect 367 00:25:01,580 --> 00:25:07,596 to X first, and then with respect to Y. Okie-doke? 368 00:25:07,596 --> 00:25:13,030 And the actual algebra here will be [INAUDIBLE] 369 00:25:13,030 --> 00:25:16,480 expect to be done in one line. 370 00:25:16,480 --> 00:25:19,760 So you will have something extremely simple. 371 00:25:19,760 --> 00:25:24,290 Ten-- it's another long exam. 372 00:25:24,290 --> 00:25:27,610 So I have to try to test everything 373 00:25:27,610 --> 00:25:31,060 you know without you spending more than one minute 374 00:25:31,060 --> 00:25:36,770 per problem, just to conceive the result. Formally, 375 00:25:36,770 --> 00:25:38,990 hold on-- now nine, I split it. 376 00:25:38,990 --> 00:25:41,490 Because I felt pity for you. 377 00:25:41,490 --> 00:25:45,750 So I put [INAUDIBLE], I put just set up the level integral, 378 00:25:45,750 --> 00:25:50,066 and reverse the order of integration. 379 00:25:50,066 --> 00:25:53,546 So you have to write integral integral equals integral 380 00:25:53,546 --> 00:25:57,210 integral, nothing else, no answer, no number. 381 00:25:57,210 --> 00:26:07,028 And ten, actually compute any of the two integrals 382 00:26:07,028 --> 00:26:14,875 at number nine to find the area of the domain. 383 00:26:14,875 --> 00:26:18,340 384 00:26:18,340 --> 00:26:22,156 Just like we did last time, and you 385 00:26:22,156 --> 00:26:24,460 don't have a calculator, OK? 386 00:26:24,460 --> 00:26:27,780 Suppose your answer will be-- what was last time? 387 00:26:27,780 --> 00:26:30,440 One over six, I don't know. 388 00:26:30,440 --> 00:26:33,250 If you give me decimals, I will be very upset. 389 00:26:33,250 --> 00:26:36,760 You have to give me the precise answer for that problem, 390 00:26:36,760 --> 00:26:40,650 because it's so easy to compute that you would have 391 00:26:40,650 --> 00:26:47,276 no need for using a calculator or software, or any kind 392 00:26:47,276 --> 00:26:50,657 of electronic device. 393 00:26:50,657 --> 00:26:56,692 And finally number 11-- and number 11, I 394 00:26:56,692 --> 00:27:01,140 shouldn't say what it is, because it's extra credit. 395 00:27:01,140 --> 00:27:03,374 But I'll still say what it is. 396 00:27:03,374 --> 00:27:05,480 It's some simple integral where you 397 00:27:05,480 --> 00:27:08,296 are going to have to use spherical coordinates. 398 00:27:08,296 --> 00:27:09,837 And shut up, [INAUDIBLE], because you 399 00:27:09,837 --> 00:27:11,745 are talking too much. 400 00:27:11,745 --> 00:27:16,350 So again, number 11 will be a triple integral 401 00:27:16,350 --> 00:27:17,855 that is easy to compute. 402 00:27:17,855 --> 00:27:22,610 And when you're going-- well, you don't have to use vehicle. 403 00:27:22,610 --> 00:27:26,110 You can still do it with cylindrical coordinates, 404 00:27:26,110 --> 00:27:27,350 for example. 405 00:27:27,350 --> 00:27:33,300 But it's a big pain doing the cylindrical coordinates 406 00:27:33,300 --> 00:27:35,120 for that kind of problem. 407 00:27:35,120 --> 00:27:41,076 So imagine maybe I'm looking at the domain to be a sphere. 408 00:27:41,076 --> 00:27:43,980 409 00:27:43,980 --> 00:27:49,180 The problems we worked as a training in class, 410 00:27:49,180 --> 00:27:53,326 are actually harder than the ones I put on the exam. 411 00:27:53,326 --> 00:27:58,100 I have a professor who's a grad student, and he used to say, 412 00:27:58,100 --> 00:28:00,522 the easy problems are the professor 413 00:28:00,522 --> 00:28:02,927 to work in classes examples. 414 00:28:02,927 --> 00:28:07,960 The hard problems are the students who have on the exam. 415 00:28:07,960 --> 00:28:10,390 I think exactly the opposite. 416 00:28:10,390 --> 00:28:17,555 Because when you will in training for any kind of sports 417 00:28:17,555 --> 00:28:21,310 or a skill or music, you have to train yourself 418 00:28:21,310 --> 00:28:26,320 above the level of your competition. 419 00:28:26,320 --> 00:28:29,045 Otherwise your competition will be bored. 420 00:28:29,045 --> 00:28:30,620 So what you're doing with training 421 00:28:30,620 --> 00:28:32,620 should not always be how whether you 422 00:28:32,620 --> 00:28:37,030 are an athlete or a mathematician or a violinist 423 00:28:37,030 --> 00:28:39,420 or whatever. 424 00:28:39,420 --> 00:28:44,150 So you're not going to see something like intersector 425 00:28:44,150 --> 00:28:48,700 cylinders, passing one through the other, one the cone, 426 00:28:48,700 --> 00:28:51,240 ice cream cone will be doing a parabola, 427 00:28:51,240 --> 00:28:55,277 then the cone is full of ice cream-- nothing like that. 428 00:28:55,277 --> 00:28:57,730 Something simpler. 429 00:28:57,730 --> 00:29:02,201 And you may guess what it is, but keep in mind that force 430 00:29:02,201 --> 00:29:06,290 of speed components, you have to know the Jacobian, 431 00:29:06,290 --> 00:29:07,470 don't hesitate. 432 00:29:07,470 --> 00:29:09,575 That's assumed to be memorized. 433 00:29:09,575 --> 00:29:11,560 Don't ask me in the middle of the exam. 434 00:29:11,560 --> 00:29:14,104 Why was the Jacobian [INAUDIBLE] components? 435 00:29:14,104 --> 00:29:17,170 You are supposed to know that as being what? 436 00:29:17,170 --> 00:29:17,885 What was that? 437 00:29:17,885 --> 00:29:18,867 Roberto knows. 438 00:29:18,867 --> 00:29:19,700 [INTERPOSING VOICES] 439 00:29:19,700 --> 00:29:21,770 PROFESSOR: [INAUDIBLE] assign by, and by 440 00:29:21,770 --> 00:29:24,860 was what, for a friend of yours? 441 00:29:24,860 --> 00:29:28,820 The latitude from Santa Claus, measured down all the way 442 00:29:28,820 --> 00:29:31,090 to [INAUDIBLE]. 443 00:29:31,090 --> 00:29:35,214 Theta is the longitude from 0 to 2 pi. 444 00:29:35,214 --> 00:29:40,450 You are going to have some very nice domain. 445 00:29:40,450 --> 00:29:42,034 All right. 446 00:29:42,034 --> 00:29:42,700 That's it, guys. 447 00:29:42,700 --> 00:29:45,390 That's what the exam will say. 448 00:29:45,390 --> 00:29:47,570 I'm asking you for a few things. 449 00:29:47,570 --> 00:29:51,610 First of all, you are already prepared, I guarantee it. 450 00:29:51,610 --> 00:29:55,010 Do not stay up late at night. 451 00:29:55,010 --> 00:29:57,905 The biggest mistake students make 452 00:29:57,905 --> 00:30:02,400 is staying up the night before a midterm or a final because they 453 00:30:02,400 --> 00:30:04,450 want to study everything. 454 00:30:04,450 --> 00:30:06,200 That's bad. 455 00:30:06,200 --> 00:30:10,406 The next day you will be tired and you won't perform as well. 456 00:30:10,406 --> 00:30:15,698 Second of all, do not be nervous at all. 457 00:30:15,698 --> 00:30:17,150 You have no reason to be nervous. 458 00:30:17,150 --> 00:30:19,446 You have plenty of time. 459 00:30:19,446 --> 00:30:24,360 You have plenty of things to write down. 460 00:30:24,360 --> 00:30:28,750 OK, about the way I grade. 461 00:30:28,750 --> 00:30:31,180 If you leave the problem completely blank, 462 00:30:31,180 --> 00:30:33,300 yes, that's a zero. 463 00:30:33,300 --> 00:30:39,235 But if you provide me with at least a hint, a formula that 464 00:30:39,235 --> 00:30:41,580 serves you-- not just an arbitrary formula 465 00:30:41,580 --> 00:30:43,020 that has nothing to do with it. 466 00:30:43,020 --> 00:30:46,380 But a formula that's in the regulation, 467 00:30:46,380 --> 00:30:49,740 I give you partial credit for everything. 468 00:30:49,740 --> 00:30:52,960 So you have no reason to freak out. 469 00:30:52,960 --> 00:30:57,320 Even if you mess up, let's say, one or two problems, 470 00:30:57,320 --> 00:31:00,838 your algebra at the end, you should still 471 00:31:00,838 --> 00:31:04,150 gather together lots of credit. 472 00:31:04,150 --> 00:31:11,540 I wrote the exam especially because I, myself, 473 00:31:11,540 --> 00:31:14,530 hate some medical answers from web work. 474 00:31:14,530 --> 00:31:18,400 I made the answers to be fat and sassy, not 475 00:31:18,400 --> 00:31:20,610 like the ones in the web works. 476 00:31:20,610 --> 00:31:24,940 So something that you can do even mentally, 477 00:31:24,940 --> 00:31:28,600 not have to struggle with the answer. 478 00:31:28,600 --> 00:31:31,340 Several people have asked me to go over 479 00:31:31,340 --> 00:31:38,584 the last two problems of the homework, and I'll go forward. 480 00:31:38,584 --> 00:31:40,894 I'll give you an example of a problem that bothered 481 00:31:40,894 --> 00:31:45,234 a few bothered a few people. 482 00:31:45,234 --> 00:31:51,850 And it's somewhat interesting because, of course, 483 00:31:51,850 --> 00:31:57,580 I make the algebra easier than it is. 484 00:31:57,580 --> 00:32:01,170 You have 2, 3, 7, 9, I don't know. 485 00:32:01,170 --> 00:32:06,960 You have a function x and y that are both functions of u and v. 486 00:32:06,960 --> 00:32:10,620 And instead of asking you for the determinant-- well, 487 00:32:10,620 --> 00:32:14,565 one of them may have asked you for the determinant, 488 00:32:14,565 --> 00:32:19,330 the functions x, derivative of x, y with [INAUDIBLE] u, 489 00:32:19,330 --> 00:32:20,358 v as Jacobian. 490 00:32:20,358 --> 00:32:25,400 But the other one was asking you for just the opposite. 491 00:32:25,400 --> 00:32:27,390 Well, several people didn't see that. 492 00:32:27,390 --> 00:32:30,720 And they kept asking me, so I answered some 20 questions 493 00:32:30,720 --> 00:32:34,370 during the weekend exactly about problems like that. 494 00:32:34,370 --> 00:32:36,230 Which doesn't bother me. 495 00:32:36,230 --> 00:32:40,270 I think I would have had the same problem when 496 00:32:40,270 --> 00:32:41,730 I was like you. 497 00:32:41,730 --> 00:32:44,690 The easy part on the first Jacobian 498 00:32:44,690 --> 00:32:49,850 is that you have 1, 1, 1 minus 1, whatever that is. 499 00:32:49,850 --> 00:32:54,480 The definition is x sub u, x sub v, 500 00:32:54,480 --> 00:32:57,850 y sub u, y sub v. These are the partial derivatives 501 00:32:57,850 --> 00:32:59,670 and that's called Jacobian. 502 00:32:59,670 --> 00:33:06,150 And what you have, you have an easy answer. 503 00:33:06,150 --> 00:33:09,380 In this case, you have the answer negative 2. 504 00:33:09,380 --> 00:33:15,710 And if you, however, are asked by the author of the problem, 505 00:33:15,710 --> 00:33:17,900 whoever created the problem of this. 506 00:33:17,900 --> 00:33:19,615 And you put negative 2, it's going 507 00:33:19,615 --> 00:33:21,820 to say no, this is not correct. 508 00:33:21,820 --> 00:33:24,090 And this is what happened to several people. 509 00:33:24,090 --> 00:33:26,830 Now, there are two ways around it. 510 00:33:26,830 --> 00:33:30,988 There are two ways you can solve that. 511 00:33:30,988 --> 00:33:34,327 STUDENT: On the work, it says the reverse [INAUDIBLE]. 512 00:33:34,327 --> 00:33:37,189 That Jacobian times the reverse Jacobian [INAUDIBLE]. 513 00:33:37,189 --> 00:33:40,600 514 00:33:40,600 --> 00:33:45,720 PROFESSOR: I want to say why that is. 515 00:33:45,720 --> 00:33:48,925 For a student who doesn't know why this Jacobian is exactly 516 00:33:48,925 --> 00:33:53,580 j inverse, there are still chances the student can say, 517 00:33:53,580 --> 00:33:56,870 well, here's how smart I am. 518 00:33:56,870 --> 00:34:01,340 I'm going to say u out, v out in terms of x and y. 519 00:34:01,340 --> 00:34:03,560 I inverse the functions because they 520 00:34:03,560 --> 00:34:06,510 are linear functions [INAUDIBLE] linear system. 521 00:34:06,510 --> 00:34:08,853 So I say x plus y. 522 00:34:08,853 --> 00:34:12,060 This is elimination called-- when we were little, 523 00:34:12,060 --> 00:34:15,889 this was called elimination 2u. 524 00:34:15,889 --> 00:34:22,280 x minus y equals 2v. 525 00:34:22,280 --> 00:34:27,989 So u is x plus y over 2. 526 00:34:27,989 --> 00:34:30,070 That means 1/2 x, 1/2 y. 527 00:34:30,070 --> 00:34:30,780 Right, guys? 528 00:34:30,780 --> 00:34:31,280 I'm right? 529 00:34:31,280 --> 00:34:32,320 STUDENT: Mm-hmm. 530 00:34:32,320 --> 00:34:33,420 PROFESSOR: OK. 531 00:34:33,420 --> 00:34:38,320 And 1/2 of x and minus 1/2 of y. 532 00:34:38,320 --> 00:34:40,159 And then, what does the student say? 533 00:34:40,159 --> 00:34:42,530 I know what I'm going to do. 534 00:34:42,530 --> 00:34:48,025 Just by the same definition, I say the du, v dx, 535 00:34:48,025 --> 00:34:51,295 y as I have an inverse function. 536 00:34:51,295 --> 00:34:53,739 And I knew how to invert the system. 537 00:34:53,739 --> 00:34:56,150 I get 1/2. 538 00:34:56,150 --> 00:34:59,710 Not matrix, Magdalena, now, determinant. 539 00:34:59,710 --> 00:35:04,220 1/2, 1/2 and 1/2, minus 1/2. 540 00:35:04,220 --> 00:35:07,160 541 00:35:07,160 --> 00:35:09,130 And guess what? 542 00:35:09,130 --> 00:35:09,900 What do I get? 543 00:35:09,900 --> 00:35:14,770 Exactly what it was saying, but I did it the long way. 544 00:35:14,770 --> 00:35:21,170 I got minus 1 over 4 minus 1 over 4, which is minus 1/2. 545 00:35:21,170 --> 00:35:21,671 Which is-- 546 00:35:21,671 --> 00:35:22,378 STUDENT: Inverse. 547 00:35:22,378 --> 00:35:23,992 PROFESSOR: The inverse of that. 548 00:35:23,992 --> 00:35:26,700 549 00:35:26,700 --> 00:35:33,080 And you are going to ask me, OK, I don't understand why. 550 00:35:33,080 --> 00:35:35,652 That's why I want to tell you a story that I 551 00:35:35,652 --> 00:35:37,390 think is beautiful. 552 00:35:37,390 --> 00:35:40,724 The book doesn't start like that, because the book doesn't 553 00:35:40,724 --> 00:35:45,372 necessarily have enough space to remind you 554 00:35:45,372 --> 00:35:50,180 everything you learned in Calc 1 when you are in Calc 3. 555 00:35:50,180 --> 00:35:54,630 But if you think of what you learned in Calc 1, in Calc 1 556 00:35:54,630 --> 00:35:58,500 your professor-- I'm sure that he or she showed you this. 557 00:35:58,500 --> 00:36:01,815 If you have a function y equals f of x, 558 00:36:01,815 --> 00:36:06,100 assume this is a c1 function and everything is nice. 559 00:36:06,100 --> 00:36:14,039 And then you have that f prime of x exists 560 00:36:14,039 --> 00:36:15,330 and it's continuous everywhere. 561 00:36:15,330 --> 00:36:17,570 That's what it means c1. 562 00:36:17,570 --> 00:36:20,970 And you want to invert this function. 563 00:36:20,970 --> 00:36:25,030 You want to invert this function around the point x0. 564 00:36:25,030 --> 00:36:38,030 So you know that at least for some interval 565 00:36:38,030 --> 00:36:41,030 that f is one-to-one. 566 00:36:41,030 --> 00:36:42,494 So it's invertible. 567 00:36:42,494 --> 00:36:48,960 568 00:36:48,960 --> 00:36:53,590 What is the derivative of [INAUDIBLE]? 569 00:36:53,590 --> 00:36:56,420 570 00:36:56,420 --> 00:37:00,270 Somebody asks you, so what is the-- the derivative 571 00:37:00,270 --> 00:37:02,030 of the inverse function is a function 572 00:37:02,030 --> 00:37:06,400 of x with respect to x. 573 00:37:06,400 --> 00:37:08,563 [INAUDIBLE]? 574 00:37:08,563 --> 00:37:10,000 I don't know. 575 00:37:10,000 --> 00:37:15,680 576 00:37:15,680 --> 00:37:17,770 Remind yourself how you did that. 577 00:37:17,770 --> 00:37:18,970 Was this hard? 578 00:37:18,970 --> 00:37:23,470 579 00:37:23,470 --> 00:37:31,450 Anybody remembers the formula for the inverse function? 580 00:37:31,450 --> 00:37:32,350 STUDENT: [INAUDIBLE]. 581 00:37:32,350 --> 00:37:35,214 PROFESSOR: 1 over f prime of x. 582 00:37:35,214 --> 00:37:38,651 583 00:37:38,651 --> 00:37:42,088 So assume that you do that at the next 0, 584 00:37:42,088 --> 00:37:45,525 assume that f prime of x0 is different from 0. 585 00:37:45,525 --> 00:37:49,250 Now, how would you prove that and how-- 586 00:37:49,250 --> 00:37:51,520 well, too much memorization. 587 00:37:51,520 --> 00:38:03,120 This is what we are doing in-- the derivative of e 588 00:38:03,120 --> 00:38:06,680 to the x was what? 589 00:38:06,680 --> 00:38:10,350 What was the derivative of natural log of this? 590 00:38:10,350 --> 00:38:11,735 STUDENT: [INAUDIBLE]. 591 00:38:11,735 --> 00:38:12,360 PROFESSOR: 1/x. 592 00:38:12,360 --> 00:38:15,120 593 00:38:15,120 --> 00:38:18,838 Now, when you have an arbitrary function f 594 00:38:18,838 --> 00:38:24,020 and you compose with inverse, what is it by definition? 595 00:38:24,020 --> 00:38:25,156 X equals x. 596 00:38:25,156 --> 00:38:26,530 So this is the identity function. 597 00:38:26,530 --> 00:38:32,200 598 00:38:32,200 --> 00:38:35,070 Chain rule tells you, wait a minute. 599 00:38:35,070 --> 00:38:41,200 Chain rule tell you how to prime the whole thing. 600 00:38:41,200 --> 00:38:45,575 So I prime-- what is this animal? 601 00:38:45,575 --> 00:38:49,630 F of f inverse of x is the composition of functions, 602 00:38:49,630 --> 00:38:50,260 right? 603 00:38:50,260 --> 00:38:56,425 So apply chain rule to f of f inverse of x, all prime, 604 00:38:56,425 --> 00:38:57,670 with respect to x. 605 00:38:57,670 --> 00:39:00,600 What is x prime with respect to x? 606 00:39:00,600 --> 00:39:03,356 x prime with respect to x is 1. 607 00:39:03,356 --> 00:39:05,160 All right. 608 00:39:05,160 --> 00:39:05,820 Chain rule. 609 00:39:05,820 --> 00:39:07,220 What does the chain rule say? 610 00:39:07,220 --> 00:39:11,477 Chain rule says f prime of f inverse 611 00:39:11,477 --> 00:39:16,850 of x times a function on the outside prime first. 612 00:39:16,850 --> 00:39:21,178 We go from the outside to the inside one step at a time. 613 00:39:21,178 --> 00:39:24,482 Derivative of the guy-- you cover f with your hand. 614 00:39:24,482 --> 00:39:27,140 Derivative of the guy inside, the core function inside, 615 00:39:27,140 --> 00:39:33,025 that will simply be f inverse of x prime with respect to x 616 00:39:33,025 --> 00:39:35,060 equals 1. 617 00:39:35,060 --> 00:39:37,020 That was x prime. 618 00:39:37,020 --> 00:39:41,966 So the derivative of the inverse function-- all right. 619 00:39:41,966 --> 00:39:51,800 f inverse prime is 1 over f prime of f inverse of x. 620 00:39:51,800 --> 00:39:56,180 621 00:39:56,180 --> 00:39:59,520 So if you think of this being the y, 622 00:39:59,520 --> 00:40:08,980 you have f inverse prime at y equals 1 over-- well, yeah. 623 00:40:08,980 --> 00:40:16,230 If you put it at x, it's f prime of f inverse of x. 624 00:40:16,230 --> 00:40:20,416 Because the f inverse-- this is x. 625 00:40:20,416 --> 00:40:22,304 This is f of x. 626 00:40:22,304 --> 00:40:24,650 This is the y [INAUDIBLE]. 627 00:40:24,650 --> 00:40:28,620 When you have f inverse, x is the image of y. 628 00:40:28,620 --> 00:40:30,530 So f inverse has an input. 629 00:40:30,530 --> 00:40:33,270 630 00:40:33,270 --> 00:40:35,150 How is this called? 631 00:40:35,150 --> 00:40:37,240 In the domain of f inverse. 632 00:40:37,240 --> 00:40:40,040 That means, who is in the domain of f? 633 00:40:40,040 --> 00:40:41,233 f inverse of x. 634 00:40:41,233 --> 00:40:43,437 So one is x, one is y. 635 00:40:43,437 --> 00:40:47,120 636 00:40:47,120 --> 00:40:52,250 So keep that in mind that when you have to invert a function, 637 00:40:52,250 --> 00:40:53,060 what do you do? 638 00:40:53,060 --> 00:41:02,070 You say 1 over the derivative of the initial-- 639 00:41:02,070 --> 00:41:05,500 so the derivative of the inverse function 640 00:41:05,500 --> 00:41:09,920 is 1 over derivative of our initial function 641 00:41:09,920 --> 00:41:13,640 at the corresponding point. 642 00:41:13,640 --> 00:41:19,060 This is how you did the derivative for the Calc 1 643 00:41:19,060 --> 00:41:20,890 people. 644 00:41:20,890 --> 00:41:21,640 All right. 645 00:41:21,640 --> 00:41:34,490 646 00:41:34,490 --> 00:41:38,070 So how do I apply that formula? 647 00:41:38,070 --> 00:41:40,950 Well, I have two functions here. 648 00:41:40,950 --> 00:41:48,585 One is e to the x and one is natural log of x. 649 00:41:48,585 --> 00:41:50,501 How do I know they are inverse to one another? 650 00:41:50,501 --> 00:41:54,640 Their graphs should be symmetric with respect to the? 651 00:41:54,640 --> 00:41:56,313 STUDENT: y equals x. 652 00:41:56,313 --> 00:41:59,151 PROFESSOR: With respect to the first [INAUDIBLE]. 653 00:41:59,151 --> 00:42:02,940 654 00:42:02,940 --> 00:42:09,370 Assume that f of x is e to the x. 655 00:42:09,370 --> 00:42:11,190 OK. 656 00:42:11,190 --> 00:42:18,548 f inverse of-- well, let's say f inverse of y. 657 00:42:18,548 --> 00:42:22,532 That would be natural log of y, right? 658 00:42:22,532 --> 00:42:28,510 659 00:42:28,510 --> 00:42:35,236 So what if you put here, what is f inverse? 660 00:42:35,236 --> 00:42:38,080 661 00:42:38,080 --> 00:42:39,830 Natural log. 662 00:42:39,830 --> 00:42:43,864 Let's say a simple way to write this, simple division. 663 00:42:43,864 --> 00:42:48,238 664 00:42:48,238 --> 00:42:51,960 According to that formula, how would you do the math? 665 00:42:51,960 --> 00:43:01,976 You go f inverse prime of x must be 1 over the derivative of f 666 00:43:01,976 --> 00:43:06,880 with respect of f inverse x. 667 00:43:06,880 --> 00:43:09,520 So you go, wait a minute. 668 00:43:09,520 --> 00:43:12,237 OK, who is f inverse of x? 669 00:43:12,237 --> 00:43:20,690 670 00:43:20,690 --> 00:43:24,700 Sorry, if f of x is e to the x, who is f inverse of x? 671 00:43:24,700 --> 00:43:31,504 672 00:43:31,504 --> 00:43:34,110 You want me to change a letter? 673 00:43:34,110 --> 00:43:36,240 I can put a y here. 674 00:43:36,240 --> 00:43:39,030 But in any case, I want to convince you that this is 1/x. 675 00:43:39,030 --> 00:43:42,180 676 00:43:42,180 --> 00:43:43,820 Why? 677 00:43:43,820 --> 00:43:47,490 Because f prime is e to the x. 678 00:43:47,490 --> 00:43:53,060 This is going to be e to the f inverse of x, 679 00:43:53,060 --> 00:43:58,200 which is e to the natural log of x, which is x. 680 00:43:58,200 --> 00:44:01,090 That's why I have x here. 681 00:44:01,090 --> 00:44:05,460 So again, if f of x equals e to the x, then f 682 00:44:05,460 --> 00:44:11,030 inverse of x is the national log of x. 683 00:44:11,030 --> 00:44:13,212 By this formula, you know that you 684 00:44:13,212 --> 00:44:19,150 have to compute natural log-- this is f inverse. 685 00:44:19,150 --> 00:44:22,540 Natural log of x prime, right? 686 00:44:22,540 --> 00:44:24,778 What is this by that formula? 687 00:44:24,778 --> 00:44:31,240 1 over the derivative of f prime computed at f inverse of x. 688 00:44:31,240 --> 00:44:33,520 Now, who is f inverse of x? 689 00:44:33,520 --> 00:44:35,060 f inverse of x is natural log of x. 690 00:44:35,060 --> 00:44:36,650 So again, let me write. 691 00:44:36,650 --> 00:44:41,400 All this guy here in the orange thing, this [INAUDIBLE] f 692 00:44:41,400 --> 00:44:44,920 inverse of x is ln x. 693 00:44:44,920 --> 00:44:47,740 Who is f prime? 694 00:44:47,740 --> 00:44:50,630 f prime of x is e to the x. 695 00:44:50,630 --> 00:44:53,387 So f prime of natural log of x will 696 00:44:53,387 --> 00:44:56,033 be e to the natural log of x. 697 00:44:56,033 --> 00:44:59,503 Applied to natural log of x, which is x. 698 00:44:59,503 --> 00:45:02,740 So you got it. 699 00:45:02,740 --> 00:45:04,036 All right? 700 00:45:04,036 --> 00:45:10,280 So remember, this formula, professors actually avoid. 701 00:45:10,280 --> 00:45:11,300 They say, oh my god. 702 00:45:11,300 --> 00:45:13,735 My students will never understand this composition 703 00:45:13,735 --> 00:45:15,610 thing, derivative. 704 00:45:15,610 --> 00:45:17,492 So Magdalena, I don't care. 705 00:45:17,492 --> 00:45:18,950 You are the undergraduate director. 706 00:45:18,950 --> 00:45:21,310 I'll never give it like that. 707 00:45:21,310 --> 00:45:23,190 That's a mistake. 708 00:45:23,190 --> 00:45:25,660 They should show it to you like that. 709 00:45:25,660 --> 00:45:31,230 f inverse prime of x equals 1 over f prime of what? 710 00:45:31,230 --> 00:45:33,750 Of the inverse image of x. 711 00:45:33,750 --> 00:45:37,980 Because you act on x. 712 00:45:37,980 --> 00:45:43,590 So if x is acting on-- this is f of x, then 713 00:45:43,590 --> 00:45:47,350 you have to invert by acting on f of x, like this 714 00:45:47,350 --> 00:45:48,526 and like that. 715 00:45:48,526 --> 00:45:54,786 If x is in the domain of f inverse, that means what? 716 00:45:54,786 --> 00:46:00,950 That in the domain of f, you have f inverse of x as input. 717 00:46:00,950 --> 00:46:04,430 So instead of giving you the formula, 718 00:46:04,430 --> 00:46:08,045 they just make you memorize the formulas 719 00:46:08,045 --> 00:46:12,420 for the inverse functions, like-- believe me, 720 00:46:12,420 --> 00:46:16,405 you take e to the x [INAUDIBLE] derivative. 721 00:46:16,405 --> 00:46:19,800 You take natural log, it's 1/x derivative. 722 00:46:19,800 --> 00:46:23,894 Don't worry about the fact that they are inverse to one another 723 00:46:23,894 --> 00:46:26,640 and you an relate the derivatives of two 724 00:46:26,640 --> 00:46:28,140 inverse functions. 725 00:46:28,140 --> 00:46:33,010 They try to stay out of trouble because this is hard to follow. 726 00:46:33,010 --> 00:46:36,494 You could see that you had [INAUDIBLE] a little bit 727 00:46:36,494 --> 00:46:37,160 and concentrate. 728 00:46:37,160 --> 00:46:38,710 What is this woman saying? 729 00:46:38,710 --> 00:46:41,520 This looks hard. 730 00:46:41,520 --> 00:46:46,136 But it's the same process that happens in the Jacobian. 731 00:46:46,136 --> 00:46:52,620 So in the Jacobian of a function of two variables. 732 00:46:52,620 --> 00:46:57,030 733 00:46:57,030 --> 00:47:05,446 Now, remember the signed area that I told you about. 734 00:47:05,446 --> 00:47:08,610 Signed area notion. 735 00:47:08,610 --> 00:47:09,650 What did we say? 736 00:47:09,650 --> 00:47:13,830 We said that dA is dx dy, but it's not 737 00:47:13,830 --> 00:47:17,420 the way they explain in the book because it's 738 00:47:17,420 --> 00:47:20,650 more like a wedge thing. 739 00:47:20,650 --> 00:47:25,640 And that wedge thingy had a meaning in the sense 740 00:47:25,640 --> 00:47:30,856 that if you were to not take the exterior derivative dx dy, 741 00:47:30,856 --> 00:47:35,610 but take dy wedge dx, it would change sign. 742 00:47:35,610 --> 00:47:37,850 So we thought of signed area before. 743 00:47:37,850 --> 00:47:42,160 When we did dx wedge dy, what did we 744 00:47:42,160 --> 00:47:44,720 get in terms of Jacobian? 745 00:47:44,720 --> 00:47:49,280 We get j d r [INAUDIBLE] coordinates. 746 00:47:49,280 --> 00:47:52,647 Do you remember what this j was? 747 00:47:52,647 --> 00:47:53,540 STUDENT: r. 748 00:47:53,540 --> 00:47:56,140 PROFESSOR: Very good, r. 749 00:47:56,140 --> 00:48:00,760 In the case, in the simple case of Cartesian versus polar, 750 00:48:00,760 --> 00:48:06,540 Cartesian going to polar, you have a function f. 751 00:48:06,540 --> 00:48:09,640 Coming back it's called the inverse function. 752 00:48:09,640 --> 00:48:15,240 So I'm asking, this is the Jacobian of which function? 753 00:48:15,240 --> 00:48:18,160 This is the Jacobian of the function that goes 754 00:48:18,160 --> 00:48:20,495 from [INAUDIBLE] theta to x, y. 755 00:48:20,495 --> 00:48:23,090 If I want the Jacobian of the function that 756 00:48:23,090 --> 00:48:25,410 goes from x, y into [INAUDIBLE] theta, 757 00:48:25,410 --> 00:48:30,120 I should write-- well, d r d theta 758 00:48:30,120 --> 00:48:33,890 will be something times dx dy. 759 00:48:33,890 --> 00:48:36,760 And now you understand better what's going on. 760 00:48:36,760 --> 00:48:39,180 1/j. 761 00:48:39,180 --> 00:48:40,160 1/j. 762 00:48:40,160 --> 00:48:42,710 So Matthew was right, in the sense 763 00:48:42,710 --> 00:48:48,390 that he said why are you so clumsy and go ahead and compute 764 00:48:48,390 --> 00:48:50,150 again u, v? 765 00:48:50,150 --> 00:48:52,360 Express u, v in terms of x, y. 766 00:48:52,360 --> 00:48:54,710 You waste your time and get minus a 1/2. 767 00:48:54,710 --> 00:48:55,660 What was that, guys? 768 00:48:55,660 --> 00:48:56,850 Minus 1/2. 769 00:48:56,850 --> 00:49:01,750 When I'm telling you that for the inverse mapping, 770 00:49:01,750 --> 00:49:05,380 the Jacobian you get is the inverse of a Jacobian. 771 00:49:05,380 --> 00:49:06,760 It's very simple. 772 00:49:06,760 --> 00:49:08,600 It's a very simple relationship. 773 00:49:08,600 --> 00:49:10,530 I could observe that. 774 00:49:10,530 --> 00:49:11,750 And he was right. 775 00:49:11,750 --> 00:49:19,150 So keep in mind that when you have Jacobian of the map where 776 00:49:19,150 --> 00:49:24,900 x, y are functions of u, v, this is 1 over the Jacobian 777 00:49:24,900 --> 00:49:30,040 where you have u, v as functions of x, y. 778 00:49:30,040 --> 00:49:32,980 So you have inverse mapping. 779 00:49:32,980 --> 00:49:36,439 In Advanced Calculus, you may learn a little bit more 780 00:49:36,439 --> 00:49:37,855 about the inverse mapping theorem. 781 00:49:37,855 --> 00:49:40,825 This is what I'm talking about. 782 00:49:40,825 --> 00:49:42,420 For the inverse mapping theorem, you 783 00:49:42,420 --> 00:49:47,937 go, well, if the derivative of these two with respect 784 00:49:47,937 --> 00:49:50,431 to these two are done as j Jacobian, the derivative 785 00:49:50,431 --> 00:49:53,798 of these two with respect to these two 786 00:49:53,798 --> 00:49:58,610 in a Jacobian [INAUDIBLE] exactly j inverse, or 1/j. 787 00:49:58,610 --> 00:50:01,700 j is a real number. 788 00:50:01,700 --> 00:50:05,680 So for a real number, whether I write 1/j or j inverse, 789 00:50:05,680 --> 00:50:09,050 it's the same. 790 00:50:09,050 --> 00:50:12,190 So as an application, do you have to know all this? 791 00:50:12,190 --> 00:50:13,936 No, you don't. 792 00:50:13,936 --> 00:50:21,480 But as an application, let me ask you the following. 793 00:50:21,480 --> 00:50:33,710 794 00:50:33,710 --> 00:50:37,690 Something harder than [INAUDIBLE] in the book. 795 00:50:37,690 --> 00:50:41,096 In the book, you have simple transformations. 796 00:50:41,096 --> 00:50:45,732 What is the Jacobian of r theta-- theta, phi 797 00:50:45,732 --> 00:50:46,315 or phi, theta. 798 00:50:46,315 --> 00:50:47,474 It doesn't matter. 799 00:50:47,474 --> 00:50:49,894 If I swap the two, I still have the same thing. 800 00:50:49,894 --> 00:50:53,766 801 00:50:53,766 --> 00:50:56,642 If a determinant swaps two rows or two columns, 802 00:50:56,642 --> 00:50:59,500 do you guys know what happens? 803 00:50:59,500 --> 00:51:01,880 You took linear algebra. 804 00:51:01,880 --> 00:51:02,640 STUDENT: Swap. 805 00:51:02,640 --> 00:51:04,715 PROFESSOR: You swap two rows or two columns. 806 00:51:04,715 --> 00:51:05,590 STUDENT: [INAUDIBLE]. 807 00:51:05,590 --> 00:51:07,990 PROFESSOR: It's going to pick up a minus sign, very good. 808 00:51:07,990 --> 00:51:10,390 But only three people in this class figured it out. 809 00:51:10,390 --> 00:51:13,269 810 00:51:13,269 --> 00:51:14,060 How shall I denote? 811 00:51:14,060 --> 00:51:16,890 Not j, but the notation was [INAUDIBLE]. 812 00:51:16,890 --> 00:51:19,740 And this is j. 813 00:51:19,740 --> 00:51:23,832 So [INAUDIBLE] phi theta over [INAUDIBLE] x, y, z. 814 00:51:23,832 --> 00:51:25,660 How do you compute them? 815 00:51:25,660 --> 00:51:28,670 You say, no, I'm not going to compute it 816 00:51:28,670 --> 00:51:32,530 by hand because until tomorrow I'm not going to finish it. 817 00:51:32,530 --> 00:51:34,810 STUDENT: Does that need a 3 by 3 matrix? 818 00:51:34,810 --> 00:51:36,640 PROFESSOR: It's a determinant. 819 00:51:36,640 --> 00:51:39,750 So when you were to write this, you're 820 00:51:39,750 --> 00:51:45,800 not going to do it because it's a killer for somebody to work 821 00:51:45,800 --> 00:51:49,310 like that in spherical coordinates with only 822 00:51:49,310 --> 00:51:50,930 those inverse functions. 823 00:51:50,930 --> 00:51:56,760 Do you remember as a review what spherical coordinates were? 824 00:51:56,760 --> 00:52:00,890 x, y, z versus r, theta, phi. 825 00:52:00,890 --> 00:52:02,080 We reviewed that. 826 00:52:02,080 --> 00:52:03,290 Theta was the longitude. 827 00:52:03,290 --> 00:52:05,710 Phi was the latitude from the North Pole. 828 00:52:05,710 --> 00:52:08,147 So x was-- who remembers that? 829 00:52:08,147 --> 00:52:08,980 [INTERPOSING VOICES] 830 00:52:08,980 --> 00:52:13,955 831 00:52:13,955 --> 00:52:18,960 PROFESSOR: Cosine theta r sine phi sine theta. 832 00:52:18,960 --> 00:52:21,130 And z was the adjacent guy. 833 00:52:21,130 --> 00:52:24,120 Remember, this was the thingy? 834 00:52:24,120 --> 00:52:27,220 And this was the phi. 835 00:52:27,220 --> 00:52:31,622 And to express x, the phi was adjacent to it. 836 00:52:31,622 --> 00:52:33,080 And that's why you have cosine phi. 837 00:52:33,080 --> 00:52:36,140 838 00:52:36,140 --> 00:52:41,310 It's a killer if somebody wants to pull out the r, phi, theta. 839 00:52:41,310 --> 00:52:43,140 First of all, r will be easy. 840 00:52:43,140 --> 00:52:46,160 But the other ones are a little bit of a headache. 841 00:52:46,160 --> 00:52:47,680 And with all those big functions, 842 00:52:47,680 --> 00:52:51,190 you would waste a lot of time to compute the determinant. 843 00:52:51,190 --> 00:52:52,150 What do you do? 844 00:52:52,150 --> 00:52:55,070 You say, well, didn't you say that if I 845 00:52:55,070 --> 00:52:58,450 take the inverse mapping, the Jacobian would be 846 00:52:58,450 --> 00:53:01,170 1 over the original Jacobian? 847 00:53:01,170 --> 00:53:03,960 Yes, I just said that. 848 00:53:03,960 --> 00:53:09,010 So go ahead and remember what the original Jacobian was 849 00:53:09,010 --> 00:53:13,430 and leave us alone you're going to say. 850 00:53:13,430 --> 00:53:15,940 And you're right. 851 00:53:15,940 --> 00:53:21,655 What was that I just said the other Jacobian was? 852 00:53:21,655 --> 00:53:22,650 STUDENT: [INAUDIBLE]. 853 00:53:22,650 --> 00:53:24,634 PROFESSOR: You told me. 854 00:53:24,634 --> 00:53:26,830 I forgot it already. 855 00:53:26,830 --> 00:53:29,330 r squared sine phi, right? 856 00:53:29,330 --> 00:53:33,320 So if somebody's asking you to solve this problem, 857 00:53:33,320 --> 00:53:36,600 you don't need to write out anything. 858 00:53:36,600 --> 00:53:38,080 Just 1 over [INAUDIBLE]. 859 00:53:38,080 --> 00:53:39,960 I'm done. 860 00:53:39,960 --> 00:53:41,550 But I'm not going to ask you. 861 00:53:41,550 --> 00:53:44,930 Of course, I saw this problem exactly. 862 00:53:44,930 --> 00:53:47,900 Find the Jacobian of the inverse mapping 863 00:53:47,900 --> 00:53:49,830 for the spherical coordinates. 864 00:53:49,830 --> 00:53:54,220 That was given at Princeton in Advanced Calculus. 865 00:53:54,220 --> 00:53:59,280 There were three variables, and then there was a generalization 866 00:53:59,280 --> 00:54:00,880 to [INAUDIBLE] variables. 867 00:54:00,880 --> 00:54:02,660 But based on this at Princeton, I'm 868 00:54:02,660 --> 00:54:04,350 not going to give you anything like that 869 00:54:04,350 --> 00:54:05,418 to compute in the exam. 870 00:54:05,418 --> 00:54:08,340 871 00:54:08,340 --> 00:54:12,389 And I just expect that you know your basics about how 872 00:54:12,389 --> 00:54:15,742 to compute triple integrals. 873 00:54:15,742 --> 00:54:20,420 Use the Jacobians and be successful with it. 874 00:54:20,420 --> 00:54:25,320 875 00:54:25,320 --> 00:54:28,630 Let's do one last problem about the review. 876 00:54:28,630 --> 00:54:30,380 Although, it's not in the midterm, 877 00:54:30,380 --> 00:54:36,530 but I would like to-- I'd like to see how you solve it. 878 00:54:36,530 --> 00:55:01,274 879 00:55:01,274 --> 00:55:05,756 A student from another class, Calc 3, came to me. 880 00:55:05,756 --> 00:55:10,238 And I was hesitant about even helping him on the homework 881 00:55:10,238 --> 00:55:16,070 because we're not supposed to help our college students. 882 00:55:16,070 --> 00:55:19,040 So I told him, did you go to the tutoring center? 883 00:55:19,040 --> 00:55:22,490 And he said yes, but they couldn't help him much. 884 00:55:22,490 --> 00:55:24,474 So I said, OK. 885 00:55:24,474 --> 00:55:27,397 So let me see the problem. 886 00:55:27,397 --> 00:55:28,938 He showed me the problem and I wanted 887 00:55:28,938 --> 00:55:32,590 to talk about this problem with you. 888 00:55:32,590 --> 00:55:36,910 This is not a hard problem, OK? 889 00:55:36,910 --> 00:55:41,566 You just have to see what this is about. 890 00:55:41,566 --> 00:55:43,054 Understand what this is about. 891 00:55:43,054 --> 00:55:49,502 892 00:55:49,502 --> 00:55:56,198 So you have the z equals x squared plus y squared, which 893 00:55:56,198 --> 00:55:59,918 is the [INAUDIBLE]. 894 00:55:59,918 --> 00:56:01,406 Sorry about my typos. 895 00:56:01,406 --> 00:56:07,854 896 00:56:07,854 --> 00:56:10,680 We didn't write this problem in the book. 897 00:56:10,680 --> 00:56:16,330 So I suspect that his instructor came up with this problem. 898 00:56:16,330 --> 00:56:19,580 This is a cone. 899 00:56:19,580 --> 00:56:21,435 We only look at the upper halves. 900 00:56:21,435 --> 00:56:24,030 901 00:56:24,030 --> 00:56:25,555 Do these surfaces intersect? 902 00:56:25,555 --> 00:56:34,430 903 00:56:34,430 --> 00:56:43,480 Draw the body between them if the case. 904 00:56:43,480 --> 00:56:48,980 905 00:56:48,980 --> 00:56:51,386 And compute the volume of that body. 906 00:56:51,386 --> 00:57:01,530 907 00:57:01,530 --> 00:57:03,485 And what do you think my reaction was? 908 00:57:03,485 --> 00:57:05,170 Oh, this is a piece of cake. 909 00:57:05,170 --> 00:57:07,290 And it is a piece of cake. 910 00:57:07,290 --> 00:57:10,530 But you need to learn Calc 3 first 911 00:57:10,530 --> 00:57:14,985 in order to help other people do Calc 3 problems. 912 00:57:14,985 --> 00:57:16,610 Especially if they are not in the book. 913 00:57:16,610 --> 00:57:21,910 So one has to have a very good understanding of the theory 914 00:57:21,910 --> 00:57:26,310 and of geometry, analytic geometry, and conics 915 00:57:26,310 --> 00:57:32,250 before they move onto triple integrals and so on. 916 00:57:32,250 --> 00:57:36,470 Can you imagine these with the eyes of your imagination? 917 00:57:36,470 --> 00:57:37,740 Can we draw them? 918 00:57:37,740 --> 00:57:38,790 Yeah. 919 00:57:38,790 --> 00:57:42,648 We better draw them because they are not nasty to draw. 920 00:57:42,648 --> 00:57:46,860 Of course this looks like the Tower of Pisa. 921 00:57:46,860 --> 00:57:49,590 Let me do it again. 922 00:57:49,590 --> 00:57:50,090 Better. 923 00:57:50,090 --> 00:57:53,410 x, y, and z. 924 00:57:53,410 --> 00:57:56,030 And then I'll take the cone. 925 00:57:56,030 --> 00:58:01,690 Well, let me draw the paraboloid first. 926 00:58:01,690 --> 00:58:04,180 Kind of sort of. 927 00:58:04,180 --> 00:58:07,076 And then the cone. 928 00:58:07,076 --> 00:58:10,070 I hate myself when I cannot draw. 929 00:58:10,070 --> 00:58:14,062 930 00:58:14,062 --> 00:58:20,380 If you were to cut, slice up, it could be this. 931 00:58:20,380 --> 00:58:24,516 And who asked me last time, was it Alex, or Ryan, 932 00:58:24,516 --> 00:58:29,530 or maybe somebody else, who said maybe we could do that even 933 00:58:29,530 --> 00:58:31,007 in Calc 2 by-- 934 00:58:31,007 --> 00:58:31,590 STUDENT: Yeah. 935 00:58:31,590 --> 00:58:32,270 PROFESSOR: You asked me. 936 00:58:32,270 --> 00:58:32,930 STUDENT: [INAUDIBLE]. 937 00:58:32,930 --> 00:58:34,680 PROFESSOR: If you take a leaf like that 938 00:58:34,680 --> 00:58:37,340 and you rotate it around the body, 939 00:58:37,340 --> 00:58:41,130 like in-- using one of the two methods 940 00:58:41,130 --> 00:58:44,400 that you learned in Calc 2. 941 00:58:44,400 --> 00:58:45,630 Well, we can do that. 942 00:58:45,630 --> 00:58:49,340 But you see we have in Calc 3. 943 00:58:49,340 --> 00:58:52,815 So I would like to write that in terms 944 00:58:52,815 --> 00:58:58,600 of the volume of the body faster with knowledge I have. 945 00:58:58,600 --> 00:58:59,430 Do they intersect? 946 00:58:59,430 --> 00:59:00,596 And where do they intersect? 947 00:59:00,596 --> 00:59:02,530 And how do I find this out? 948 00:59:02,530 --> 00:59:04,350 STUDENT: [INAUDIBLE]. 949 00:59:04,350 --> 00:59:06,080 PROFESSOR: Yes. 950 00:59:06,080 --> 00:59:16,740 I have to make them equal and solve for z, and then the rest. 951 00:59:16,740 --> 00:59:18,588 How do I solve for z? 952 00:59:18,588 --> 00:59:23,730 Well, z equals z0 gives me two possibilities. 953 00:59:23,730 --> 00:59:28,246 One is z equals 0 and 1 is z equals 1 954 00:59:28,246 --> 00:59:33,710 because this is the same as writing z times z minus 1 955 00:59:33,710 --> 00:59:35,060 equals 0. 956 00:59:35,060 --> 00:59:36,310 So where do they intersect? 957 00:59:36,310 --> 00:59:37,720 They intersect here at the origin 958 00:59:37,720 --> 00:59:42,060 and they intersect where z equals 1. 959 00:59:42,060 --> 00:59:46,985 And where z equals 1, I'm going to have what circle? 960 00:59:46,985 --> 00:59:49,190 The unit circle. 961 00:59:49,190 --> 00:59:52,604 I'll draw over-- I'll make it in red. 962 00:59:52,604 --> 00:59:58,190 This is x squared plus y squared equals 1 at the altitude 1, 963 00:59:58,190 --> 00:59:59,750 z equals 1. 964 00:59:59,750 --> 01:00:02,411 This is the plane z equals 1. 965 01:00:02,411 --> 01:00:07,560 966 01:00:07,560 --> 01:00:15,070 OK, so how many ways to do this are there? 967 01:00:15,070 --> 01:00:21,970 When we were in Chapter 12, we said the triple integral 968 01:00:21,970 --> 01:00:23,564 will give me the volume. 969 01:00:23,564 --> 01:00:26,050 So the volume will be triple integral 970 01:00:26,050 --> 01:00:31,280 of a certain body-- of 1 over a certain body 971 01:00:31,280 --> 01:00:40,950 dv, where the body is the body of revolution 972 01:00:40,950 --> 01:00:50,480 created by the motion of-- what is this thing? 973 01:00:50,480 --> 01:00:51,480 What shall we call it? 974 01:00:51,480 --> 01:00:53,480 A wing. 975 01:00:53,480 --> 01:00:53,980 [INAUDIBLE] 976 01:00:53,980 --> 01:00:57,480 977 01:00:57,480 --> 01:01:03,480 Domain D. No, domain D is usually what's on [INAUDIBLE]. 978 01:01:03,480 --> 01:01:09,480 979 01:01:09,480 --> 01:01:10,480 I don't know. 980 01:01:10,480 --> 01:01:13,980 STUDENT: L for leaf? 981 01:01:13,980 --> 01:01:14,980 PROFESSOR: L for leaf. 982 01:01:14,980 --> 01:01:15,480 Wonderful. 983 01:01:15,480 --> 01:01:15,980 I like that. 984 01:01:15,980 --> 01:01:21,480 L. OK. 985 01:01:21,480 --> 01:01:26,730 So I can write it up as a triple integral how? 986 01:01:26,730 --> 01:01:29,580 Is it easy to use it in spherical coordinates? 987 01:01:29,580 --> 01:01:30,080 No. 988 01:01:30,080 --> 01:01:32,820 That's not a spherical coordinate problem. 989 01:01:32,820 --> 01:01:35,410 That's a cylindrical coordinate problem. 990 01:01:35,410 --> 01:01:36,180 Why is that? 991 01:01:36,180 --> 01:01:38,535 I'm going to have to think where I live. 992 01:01:38,535 --> 01:01:44,816 I live above a beautiful disk, which is the shadowy plane. 993 01:01:44,816 --> 01:01:48,074 And that beautiful disk has exactly radius 1. 994 01:01:48,074 --> 01:01:48,740 So we are lucky. 995 01:01:48,740 --> 01:01:53,330 That's the unit disk, x squared plus y squared less than 1 996 01:01:53,330 --> 01:01:55,610 and greater than 0. 997 01:01:55,610 --> 01:02:00,230 So when I revolve, I'm using polar coordinates. 998 01:02:00,230 --> 01:02:02,830 And that means I'm using cylindrical coordinates, which 999 01:02:02,830 --> 01:02:05,170 is practically the same thing. 1000 01:02:05,170 --> 01:02:09,715 r will be between what and what? 1001 01:02:09,715 --> 01:02:10,590 STUDENT: [INAUDIBLE]. 1002 01:02:10,590 --> 01:02:13,412 PROFESSOR: 0 to 1, very good. 1003 01:02:13,412 --> 01:02:14,304 Theta? 1004 01:02:14,304 --> 01:02:15,179 STUDENT: [INAUDIBLE]. 1005 01:02:15,179 --> 01:02:17,450 PROFESSOR: 0 to 2 pi. 1006 01:02:17,450 --> 01:02:19,210 How about z? 1007 01:02:19,210 --> 01:02:21,320 z is the z from cylindrical coordinates. 1008 01:02:21,320 --> 01:02:24,680 STUDENT: Square root x squared plus y squared-- 1009 01:02:24,680 --> 01:02:27,080 PROFESSOR: Who is on the bottom? 1010 01:02:27,080 --> 01:02:29,490 STUDENT: 0. 1011 01:02:29,490 --> 01:02:34,820 PROFESSOR: So the z is between-- let me write it in x first, 1012 01:02:34,820 --> 01:02:36,710 and then switch to polar. 1013 01:02:36,710 --> 01:02:37,609 Is that OK? 1014 01:02:37,609 --> 01:02:38,150 STUDENT: Yeah 1015 01:02:38,150 --> 01:02:39,025 PROFESSOR: All right. 1016 01:02:39,025 --> 01:02:42,185 So what do I write on the left-hand side? 1017 01:02:42,185 --> 01:02:43,226 I need water. 1018 01:02:43,226 --> 01:02:44,101 STUDENT: [INAUDIBLE]. 1019 01:02:44,101 --> 01:02:50,820 1020 01:02:50,820 --> 01:02:52,050 PROFESSOR: Who is smaller? 1021 01:02:52,050 --> 01:02:53,966 Who is smaller? 1022 01:02:53,966 --> 01:02:56,870 Square root of x squared y squared or x 1023 01:02:56,870 --> 01:02:59,075 squared plus y squared? 1024 01:02:59,075 --> 01:03:00,776 STUDENT: Square root over. 1025 01:03:00,776 --> 01:03:01,900 PROFESSOR: This is smaller. 1026 01:03:01,900 --> 01:03:02,780 Why? 1027 01:03:02,780 --> 01:03:04,704 STUDENT: Because [INAUDIBLE]. 1028 01:03:04,704 --> 01:03:07,110 PROFESSOR: So it's less than 1. 1029 01:03:07,110 --> 01:03:08,250 I mean, less than 1. 1030 01:03:08,250 --> 01:03:09,640 This is less than 1. 1031 01:03:09,640 --> 01:03:11,110 It's between 0 and 1. 1032 01:03:11,110 --> 01:03:14,580 1033 01:03:14,580 --> 01:03:18,750 So I was trying to explain this to my son, but I couldn't. 1034 01:03:18,750 --> 01:03:20,210 But he's 10. 1035 01:03:20,210 --> 01:03:21,685 It's so hard. 1036 01:03:21,685 --> 01:03:32,687 So I said compare square root of 0.04 with 0.04. 1037 01:03:32,687 --> 01:03:35,200 This is smaller, obviously. 1038 01:03:35,200 --> 01:03:36,852 This is 0.2. 1039 01:03:36,852 --> 01:03:37,710 He can understand. 1040 01:03:37,710 --> 01:03:40,720 1041 01:03:40,720 --> 01:03:43,290 So this is what we're doing. 1042 01:03:43,290 --> 01:03:47,345 We are saying that this is x squared 1043 01:03:47,345 --> 01:03:52,240 plus y squared, the round thing on the bottom. 1044 01:03:52,240 --> 01:04:01,268 And this is going to be on the top, square root of x squared 1045 01:04:01,268 --> 01:04:03,380 plus y squared from the cylinder, 1046 01:04:03,380 --> 01:04:07,124 from the cone-- sorry guys, the upper half. 1047 01:04:07,124 --> 01:04:08,790 Because I only work with the upper half. 1048 01:04:08,790 --> 01:04:12,320 Everything is about the sea level. 1049 01:04:12,320 --> 01:04:15,420 Good, now let's write out the whole thing. 1050 01:04:15,420 --> 01:04:18,905 So I have integral from the polar coordinates, 1051 01:04:18,905 --> 01:04:19,970 from what to what? 1052 01:04:19,970 --> 01:04:23,320 1053 01:04:23,320 --> 01:04:24,460 STUDENT: r squared to r. 1054 01:04:24,460 --> 01:04:30,630 PROFESSOR: r squared to r, 0 to 1, 0 to 2 pi. 1055 01:04:30,630 --> 01:04:35,125 So the order of integration would be dz dr d theta. 1056 01:04:35,125 --> 01:04:38,190 1057 01:04:38,190 --> 01:04:40,384 And what's inside here? 1058 01:04:40,384 --> 01:04:42,976 STUDENT: [INAUDIBLE]. 1059 01:04:42,976 --> 01:04:43,800 PROFESSOR: No. 1060 01:04:43,800 --> 01:04:44,430 STUDENT: r. 1061 01:04:44,430 --> 01:04:47,550 PROFESSOR: r, excellent, r-- why r? 1062 01:04:47,550 --> 01:04:49,900 Because 1 was 1. 1063 01:04:49,900 --> 01:04:58,650 But dv is Jacobian times dr d theta dz-- dz, dr, d theta. 1064 01:04:58,650 --> 01:05:02,102 So this is going to be the r from the change of coordinates, 1065 01:05:02,102 --> 01:05:03,922 the Jacobian. 1066 01:05:03,922 --> 01:05:05,404 Is this hard? 1067 01:05:05,404 --> 01:05:06,392 Well, let's do it. 1068 01:05:06,392 --> 01:05:08,360 Come on, this shouldn't be hard. 1069 01:05:08,360 --> 01:05:13,854 We can even separate the functions. 1070 01:05:13,854 --> 01:05:19,586 And I got you some tricks. 1071 01:05:19,586 --> 01:05:21,890 The first one we have to work it out. 1072 01:05:21,890 --> 01:05:22,890 We have no other choice. 1073 01:05:22,890 --> 01:05:27,910 So I'm going to have the integral from 0 1074 01:05:27,910 --> 01:05:32,060 to 2 pi, integral from 0 to 1. 1075 01:05:32,060 --> 01:05:33,310 And then I go what? 1076 01:05:33,310 --> 01:05:39,730 I go integral of what you see with z, the z between r and r 1077 01:05:39,730 --> 01:05:46,640 squared times r dr d theta. 1078 01:05:46,640 --> 01:05:48,380 Who is going on my nerves? 1079 01:05:48,380 --> 01:05:49,520 Not you guys. 1080 01:05:49,520 --> 01:05:53,840 Here, there is a guy that goes on my nerves-- the theta. 1081 01:05:53,840 --> 01:05:56,700 I can get rid of him, and I say, I don't need the theta. 1082 01:05:56,700 --> 01:05:58,660 I've got things to do with the r. 1083 01:05:58,660 --> 01:06:05,810 So I go 2 pi, which is the integral from 0 to 2 pi of 1 1084 01:06:05,810 --> 01:06:06,520 d theta. 1085 01:06:06,520 --> 01:06:08,710 2 pi goes out. 1086 01:06:08,710 --> 01:06:14,790 Now 2 pi times integral from 0 to 1 of what? 1087 01:06:14,790 --> 01:06:17,494 What's the simplest way to write it? 1088 01:06:17,494 --> 01:06:20,350 STUDENT: [INAUDIBLE]. 1089 01:06:20,350 --> 01:06:22,300 PROFESSOR: r squared in the end. 1090 01:06:22,300 --> 01:06:23,985 I mean, I do the whole thing in the end. 1091 01:06:23,985 --> 01:06:31,340 I have r squared minus r cubed, right guys? 1092 01:06:31,340 --> 01:06:33,112 Are you with me? 1093 01:06:33,112 --> 01:06:36,180 dr, so this is when I did it. 1094 01:06:36,180 --> 01:06:40,320 But I didn't do the anti-derivative, not yet. 1095 01:06:40,320 --> 01:06:42,212 I did not apply the fundamental. 1096 01:06:42,212 --> 01:06:46,666 Now you apply the fundamental [INAUDIBLE] 1097 01:06:46,666 --> 01:06:48,200 and tell me what you get. 1098 01:06:48,200 --> 01:06:50,525 What is this? 1099 01:06:50,525 --> 01:06:53,780 STUDENT: [INAUDIBLE] 1/12. 1100 01:06:53,780 --> 01:06:58,563 PROFESSOR: 1/12, that's very good-- r cubed over 3 minus r 1101 01:06:58,563 --> 01:07:04,530 to the 4 over 4, 1/3 minus 1/4, 1/12, very good. 1102 01:07:04,530 --> 01:07:09,426 So you have 2 pi times 1/12 equals pi over 6. 1103 01:07:09,426 --> 01:07:11,920 Thank god, we got it. 1104 01:07:11,920 --> 01:07:12,690 Was it hard? 1105 01:07:12,690 --> 01:07:15,460 1106 01:07:15,460 --> 01:07:17,980 Would you have spent two days without doing this? 1107 01:07:17,980 --> 01:07:22,220 I think you would have gotten it by yourselves. 1108 01:07:22,220 --> 01:07:25,420 Am I right, with no problem? 1109 01:07:25,420 --> 01:07:26,440 Why is that? 1110 01:07:26,440 --> 01:07:30,090 Because I think you worked enough problems 1111 01:07:30,090 --> 01:07:32,900 to master the material, and you are prepared. 1112 01:07:32,900 --> 01:07:37,170 And this is not a surprise for you 1113 01:07:37,170 --> 01:07:40,591 like it is for many students in other classes. 1114 01:07:40,591 --> 01:07:41,573 Yes, sir. 1115 01:07:41,573 --> 01:07:44,519 STUDENT: Can you put that one in spherical coordinates? 1116 01:07:44,519 --> 01:07:45,840 PROFESSOR: You can. 1117 01:07:45,840 --> 01:07:50,480 That is going to be a hassle. 1118 01:07:50,480 --> 01:07:55,070 I would do one more problem that is not 1119 01:07:55,070 --> 01:07:57,070 quite appropriate for spherical work, 1120 01:07:57,070 --> 01:07:59,640 but I want to do it [INAUDIBLE]. 1121 01:07:59,640 --> 01:08:03,100 Because it looks like the ones I gave you as a homework, 1122 01:08:03,100 --> 01:08:05,680 and several people struggled with that. 1123 01:08:05,680 --> 01:08:13,230 And I want to see how it's done since not everybody finished 1124 01:08:13,230 --> 01:08:13,730 it. 1125 01:08:13,730 --> 01:08:16,510 1126 01:08:16,510 --> 01:08:21,354 Given [INAUDIBLE] numbers, you have a flat plane 1127 01:08:21,354 --> 01:08:26,854 z equals a at some altitude a and a cone 1128 01:08:26,854 --> 01:08:29,178 exactly like the cone I gave you before. 1129 01:08:29,178 --> 01:08:32,600 And of course this is not just like you asked. 1130 01:08:32,600 --> 01:08:36,154 This is not very appropriate for spherical coordinates. 1131 01:08:36,154 --> 01:08:38,950 It's appropriate for cylindrical. 1132 01:08:38,950 --> 01:08:41,602 But they ask you to do it in both. 1133 01:08:41,602 --> 01:08:44,069 Remember that problem, guys? 1134 01:08:44,069 --> 01:08:49,300 So you have the volume of, or some function, or something. 1135 01:08:49,300 --> 01:08:52,609 And they say, put it in both spherical coordinates 1136 01:08:52,609 --> 01:08:57,260 and cylindrical coordinates. 1137 01:08:57,260 --> 01:08:59,390 And let's assume that you don't know what 1138 01:08:59,390 --> 01:09:00,710 function you are integrating. 1139 01:09:00,710 --> 01:09:02,825 I'm working too much with volumes. 1140 01:09:02,825 --> 01:09:04,729 Let's suppose that you are simply 1141 01:09:04,729 --> 01:09:10,130 integrating in function F of x, y, z dV, which 1142 01:09:10,130 --> 01:09:17,100 is dx dy dx over the body of the [INAUDIBLE], of the-- this 1143 01:09:17,100 --> 01:09:23,210 is the flat cone, the flat ice cream cone. 1144 01:09:23,210 --> 01:09:28,367 Then somebody licked your ice cream up to this point. 1145 01:09:28,367 --> 01:09:31,295 And you are left with the ice cream 1146 01:09:31,295 --> 01:09:36,720 only under this at the level of the rim of the waffle. 1147 01:09:36,720 --> 01:09:39,840 1148 01:09:39,840 --> 01:09:43,779 Let's break this into two-- they don't ask you to compute it. 1149 01:09:43,779 --> 01:09:51,210 They ask you to set up cylindrical coordinates 1150 01:09:51,210 --> 01:09:55,120 and set up the spherical coordinates. 1151 01:09:55,120 --> 01:09:56,600 But thank you for the idea. 1152 01:09:56,600 --> 01:09:58,790 That was great. 1153 01:09:58,790 --> 01:10:01,590 So let's see, how hard is it? 1154 01:10:01,590 --> 01:10:06,890 I think it's very easy in cylindrical coordinates. 1155 01:10:06,890 --> 01:10:08,780 What do you do in cylindrical coordinates? 1156 01:10:08,780 --> 01:10:10,010 You say, well, wait a minute. 1157 01:10:10,010 --> 01:10:13,940 If z equals a has to be intersected 1158 01:10:13,940 --> 01:10:17,810 with z squared equals x squared, I 1159 01:10:17,810 --> 01:10:19,640 know the circle that I'm going to get 1160 01:10:19,640 --> 01:10:21,400 is going to be a piece of cake. 1161 01:10:21,400 --> 01:10:25,870 x squared plus y squared equals a squared. 1162 01:10:25,870 --> 01:10:33,880 So really my ice cream cone has the radius a. 1163 01:10:33,880 --> 01:10:35,150 Are you guys with me? 1164 01:10:35,150 --> 01:10:36,070 Is it true? 1165 01:10:36,070 --> 01:10:40,060 Is it true that the radius of this licked ice cream cone 1166 01:10:40,060 --> 01:10:40,560 is a? 1167 01:10:40,560 --> 01:10:41,260 STUDENT: Mhmm. 1168 01:10:41,260 --> 01:10:44,250 PROFESSOR: It is true. 1169 01:10:44,250 --> 01:10:49,255 Whatever that a was-- yours was 43, 34, 37, god knows what, 1170 01:10:49,255 --> 01:10:52,060 doesn't matter. 1171 01:10:52,060 --> 01:10:58,103 I would foresee-- I'm not a prophet or even a witch. 1172 01:10:58,103 --> 01:10:58,660 I am a witch. 1173 01:10:58,660 --> 01:11:03,404 But anyway, I would not foresee somebody 1174 01:11:03,404 --> 01:11:04,820 giving you a hard problem to solve 1175 01:11:04,820 --> 01:11:07,200 like that computationally. 1176 01:11:07,200 --> 01:11:10,600 But on the final, they can make you set up the limits 1177 01:11:10,600 --> 01:11:13,020 and leave it like that. 1178 01:11:13,020 --> 01:11:16,900 So how do we do cylindrical? 1179 01:11:16,900 --> 01:11:18,730 Is this hard? 1180 01:11:18,730 --> 01:11:21,870 So r will be from 0 to a. 1181 01:11:21,870 --> 01:11:24,420 And god, that's easy. 1182 01:11:24,420 --> 01:11:27,235 0 to 2 pi is going to be for the theta. 1183 01:11:27,235 --> 01:11:28,690 First I write dz. 1184 01:11:28,690 --> 01:11:32,278 Then I do dr and d theta. 1185 01:11:32,278 --> 01:11:35,540 1186 01:11:35,540 --> 01:11:40,040 Theta will be between 0 and 2 pi, r between 0 and a. 1187 01:11:40,040 --> 01:11:44,336 z-- you guys have to tell me, because it's 1188 01:11:44,336 --> 01:11:47,822 between a bottom and a top. 1189 01:11:47,822 --> 01:11:51,335 And I was about to take this to drink. 1190 01:11:51,335 --> 01:11:52,210 STUDENT: [INAUDIBLE]. 1191 01:11:52,210 --> 01:12:02,454 1192 01:12:02,454 --> 01:12:04,884 PROFESSOR: r is the one on the bottom, 1193 01:12:04,884 --> 01:12:08,772 and a is the one on the top. 1194 01:12:08,772 --> 01:12:10,716 And I think that's clear to everybody, right? 1195 01:12:10,716 --> 01:12:15,790 Is there anything missing obviously? 1196 01:12:15,790 --> 01:12:19,695 So what do I do when they ask me on the final-- 1197 01:12:19,695 --> 01:12:22,870 when I say this is a mysterious function, what 1198 01:12:22,870 --> 01:12:24,416 do you put in here? 1199 01:12:24,416 --> 01:12:28,980 F of x, y, z, yes, but yes and no. 1200 01:12:28,980 --> 01:12:40,340 Because you say F of x of r z theta, y of r z theta, z of god 1201 01:12:40,340 --> 01:12:41,615 knows z, z. 1202 01:12:41,615 --> 01:12:43,670 z is the same, do you understand? 1203 01:12:43,670 --> 01:12:45,740 So you indicate to the poor people 1204 01:12:45,740 --> 01:12:51,110 that I'm not going to stay in x, y, z, because I'm not stupid. 1205 01:12:51,110 --> 01:12:53,510 I'm going to transform the whole thing 1206 01:12:53,510 --> 01:12:56,370 so it's going to be expressed in terms of these letters-- r 1207 01:12:56,370 --> 01:12:59,890 theta and z. 1208 01:12:59,890 --> 01:13:02,102 Do you have to write all this? 1209 01:13:02,102 --> 01:13:05,100 If you were a professional writing the math paper, yes, 1210 01:13:05,100 --> 01:13:08,800 you have to, or a math book or whatever, you have to. 1211 01:13:08,800 --> 01:13:11,730 But you can also skip it and put the F. I'm not 1212 01:13:11,730 --> 01:13:13,040 going to take off points. 1213 01:13:13,040 --> 01:13:15,530 I will understand. 1214 01:13:15,530 --> 01:13:19,120 Times r-- very good. 1215 01:13:19,120 --> 01:13:21,320 Never forget about your nice Jacobian. 1216 01:13:21,320 --> 01:13:26,350 If you forget the r, this is no good, 0 points, 1217 01:13:26,350 --> 01:13:30,480 even with all the setup you tried to do going into it. 1218 01:13:30,480 --> 01:13:34,410 OK, finally let's see. 1219 01:13:34,410 --> 01:13:37,590 How you do this in spherical is not-- yes, sir. 1220 01:13:37,590 --> 01:13:40,120 STUDENT: When you're finding the volume, 1221 01:13:40,120 --> 01:13:43,665 isn't it with a triple integral, don't you just put a 1? 1222 01:13:43,665 --> 01:13:44,422 PROFESSOR: Hm? 1223 01:13:44,422 --> 01:13:46,005 STUDENT: When you're finding a volume? 1224 01:13:46,005 --> 01:13:48,540 PROFESSOR: No, I didn't say-- I just said, 1225 01:13:48,540 --> 01:13:51,470 but you probably were thinking of [INAUDIBLE]. 1226 01:13:51,470 --> 01:13:53,760 I said, I gave you too many volumes. 1227 01:13:53,760 --> 01:13:56,710 I just said, and I'm tired of saying 1228 01:13:56,710 --> 01:13:59,140 volume of this, volume of that. 1229 01:13:59,140 --> 01:14:01,017 And in the actual problem, they may 1230 01:14:01,017 --> 01:14:04,860 ask you to do triple integral of any function, 1231 01:14:04,860 --> 01:14:07,910 differentiable function or continuous function, 1232 01:14:07,910 --> 01:14:12,700 over a volume, over a body. 1233 01:14:12,700 --> 01:14:15,634 So this could be-- in the next chapter we're 1234 01:14:15,634 --> 01:14:16,925 going to see some applications. 1235 01:14:16,925 --> 01:14:20,010 1236 01:14:20,010 --> 01:14:24,690 I maybe saw some in 12.6 like mass moment, those things. 1237 01:14:24,690 --> 01:14:28,146 But in three coordinates, you have other functions 1238 01:14:28,146 --> 01:14:29,890 that are these functions. 1239 01:14:29,890 --> 01:14:35,150 You'll have that included, row z, x, y, z, and so on. 1240 01:14:35,150 --> 01:14:37,275 OK, good. 1241 01:14:37,275 --> 01:14:39,650 When you would integrate a density function in that case, 1242 01:14:39,650 --> 01:14:42,100 you will have a mass. 1243 01:14:42,100 --> 01:14:45,370 Because you integrate this, d over volume, you'd have a mass. 1244 01:14:45,370 --> 01:14:49,850 OK, in this case, we have to be smart, 1245 01:14:49,850 --> 01:14:56,190 say F times r squared sine phi is the Jacobian. 1246 01:14:56,190 --> 01:15:02,500 This is a function in r phi and theta, right guys? 1247 01:15:02,500 --> 01:15:04,230 We don't care what it is. 1248 01:15:04,230 --> 01:15:05,960 We are going to have the d something, 1249 01:15:05,960 --> 01:15:07,100 d something, d something. 1250 01:15:07,100 --> 01:15:08,745 The question is, which ones? 1251 01:15:08,745 --> 01:15:13,760 Because it's not obvious at all, except for theta. 1252 01:15:13,760 --> 01:15:15,180 Theta is nice. 1253 01:15:15,180 --> 01:15:16,160 He's so nice. 1254 01:15:16,160 --> 01:15:20,135 And we say, OK theta, we are grateful to you. 1255 01:15:20,135 --> 01:15:25,410 We put you at the end, because it's a complete rotation. 1256 01:15:25,410 --> 01:15:31,134 And we know you are between 0 and 2 pi, very reliable guy. 1257 01:15:31,134 --> 01:15:33,996 Phi is not so reliable. 1258 01:15:33,996 --> 01:15:35,770 Well, phi is a nice guy. 1259 01:15:35,770 --> 01:15:39,970 But he puts us through a little bit of work. 1260 01:15:39,970 --> 01:15:41,390 Do we like to work? 1261 01:15:41,390 --> 01:15:45,970 Well, not so much, but we'll try. 1262 01:15:45,970 --> 01:15:52,632 So we need to know a little bit more about this triangle. 1263 01:15:52,632 --> 01:15:56,104 1264 01:15:56,104 --> 01:16:01,064 We need to understand a little bit more about this triangle. 1265 01:16:01,064 --> 01:16:05,280 STUDENT: Well, the angle between the angle at the bottom 1266 01:16:05,280 --> 01:16:06,315 is 45 degrees. 1267 01:16:06,315 --> 01:16:07,490 PROFESSOR: How can you say? 1268 01:16:07,490 --> 01:16:09,365 STUDENT: Because the slope of that line is 1. 1269 01:16:09,365 --> 01:16:12,610 PROFESSOR: Right, so say, now I'm going to observe z 1270 01:16:12,610 --> 01:16:13,661 was a as well. 1271 01:16:13,661 --> 01:16:18,580 1272 01:16:18,580 --> 01:16:23,060 So that means it's a right isosceles triangle. 1273 01:16:23,060 --> 01:16:25,100 If it's a right isosceles triangle, 1274 01:16:25,100 --> 01:16:27,700 this is 45 degree angle. 1275 01:16:27,700 --> 01:16:37,286 So this is from d phi from 0 to pi over 4, excellent. 1276 01:16:37,286 --> 01:16:41,010 Finally, the only one that gives us a little bit of a headache 1277 01:16:41,010 --> 01:16:46,800 but not too much of a headache is the radius r. 1278 01:16:46,800 --> 01:16:48,500 Should I change the color? 1279 01:16:48,500 --> 01:16:51,480 No, I'll leave it r dr. So we have 1280 01:16:51,480 --> 01:17:01,780 to think a little bit of the meaning of our rays. 1281 01:17:01,780 --> 01:17:04,280 Drawing vertical strips or horizontal strips 1282 01:17:04,280 --> 01:17:06,780 or whatever strips is not a good idea. 1283 01:17:06,780 --> 01:17:08,760 When we are in spherical coordinates, 1284 01:17:08,760 --> 01:17:11,820 what do we need to draw? 1285 01:17:11,820 --> 01:17:15,620 Rays, like rays of sun coming from a source. 1286 01:17:15,620 --> 01:17:20,600 The source is here at the origin in spherical coordinates. 1287 01:17:20,600 --> 01:17:25,235 These are like rays of sun that are free to move. 1288 01:17:25,235 --> 01:17:26,590 But they bump. 1289 01:17:26,590 --> 01:17:31,944 They just bump against the plane, the flat roof. 1290 01:17:31,944 --> 01:17:36,240 So they would reflect if this were a physical problem. 1291 01:17:36,240 --> 01:17:38,920 1292 01:17:38,920 --> 01:17:42,966 So definitely all your rays start at 0. 1293 01:17:42,966 --> 01:17:44,730 So you have to put 0 here. 1294 01:17:44,730 --> 01:17:49,311 But this is a question mark. 1295 01:17:49,311 --> 01:17:50,186 STUDENT: [INAUDIBLE]. 1296 01:17:50,186 --> 01:17:57,551 1297 01:17:57,551 --> 01:17:59,024 STUDENT: a square root 2. 1298 01:17:59,024 --> 01:18:01,990 1299 01:18:01,990 --> 01:18:03,980 PROFESSOR: No, it's not a fixed answer. 1300 01:18:03,980 --> 01:18:07,190 So you have z will be a fixed. 1301 01:18:07,190 --> 01:18:09,770 But who was z in spherical coordinates? 1302 01:18:09,770 --> 01:18:12,565 That was the only thing you can ask. 1303 01:18:12,565 --> 01:18:16,445 So z equals a is your tradition that is the roof. 1304 01:18:16,445 --> 01:18:18,460 STUDENT: That would be r [INAUDIBLE]. 1305 01:18:18,460 --> 01:18:28,052 PROFESSOR: Very good, r cosine of phi, 1306 01:18:28,052 --> 01:18:32,950 of the latitude from the North Pole. 1307 01:18:32,950 --> 01:18:33,950 This is 45. 1308 01:18:33,950 --> 01:18:38,622 But I mean for a point like this, phi will be this phi. 1309 01:18:38,622 --> 01:18:40,210 Do you guys understand? 1310 01:18:40,210 --> 01:18:44,050 Phi could be any point where the point inside [INAUDIBLE], phi 1311 01:18:44,050 --> 01:18:46,430 will be the latitude from the North Pole. 1312 01:18:46,430 --> 01:18:51,830 OK, so the way you do it is r is between 0 and z 1313 01:18:51,830 --> 01:18:56,650 over cosine phi. 1314 01:18:56,650 --> 01:18:58,210 And that's the hard thing. 1315 01:18:58,210 --> 01:19:01,510 Since z at the roof is a, you have 1316 01:19:01,510 --> 01:19:06,550 to put here a over-- a is fixed, that 43 of yours, 1317 01:19:06,550 --> 01:19:08,820 whatever it was-- cosine phi. 1318 01:19:08,820 --> 01:19:13,940 1319 01:19:13,940 --> 01:19:17,870 So when you guys integrate with respect to r, 1320 01:19:17,870 --> 01:19:24,330 assume this F will be 1, just like you asked me, Alex. 1321 01:19:24,330 --> 01:19:27,470 That would make my life easier and would be good. 1322 01:19:27,470 --> 01:19:29,460 When I integrate with respect to r, 1323 01:19:29,460 --> 01:19:32,030 would it be hard to solve a problem? 1324 01:19:32,030 --> 01:19:33,200 Oh, not so hard. 1325 01:19:33,200 --> 01:19:34,880 Why? 1326 01:19:34,880 --> 01:19:37,650 OK, integrate this with respect to r. 1327 01:19:37,650 --> 01:19:39,550 We have r cubed. 1328 01:19:39,550 --> 01:19:41,590 Integrate r squared. 1329 01:19:41,590 --> 01:19:43,457 We have r cubed over 3, right? 1330 01:19:43,457 --> 01:19:49,421 Let's do this, solve the same problem when F is 1. 1331 01:19:49,421 --> 01:19:54,900 1332 01:19:54,900 --> 01:20:02,564 Solve the same problem when F would be 1, for F equals 1. 1333 01:20:02,564 --> 01:20:06,530 Then you get integral from 0 to 2 pi, 1334 01:20:06,530 --> 01:20:10,940 integral from 0 to pi over 4, integral from 0 1335 01:20:10,940 --> 01:20:21,537 to a over cosine phi, 1 r squared sine phi dr d phi d 1336 01:20:21,537 --> 01:20:22,311 theta. 1337 01:20:22,311 --> 01:20:25,840 The guy that sits on my nerves is again theta. 1338 01:20:25,840 --> 01:20:27,130 He's very nice. 1339 01:20:27,130 --> 01:20:29,880 He can be eliminated from the game. 1340 01:20:29,880 --> 01:20:34,210 So 2 pi out, and I will focus my attention 1341 01:20:34,210 --> 01:20:37,120 to the product of function. 1342 01:20:37,120 --> 01:20:40,610 Well, OK, I have to integrate one at a time. 1343 01:20:40,610 --> 01:20:43,212 So I integrate with respect to what? 1344 01:20:43,212 --> 01:20:45,572 STUDENT: [INAUDIBLE]. 1345 01:20:45,572 --> 01:20:50,180 PROFESSOR: So I get r cubed over 3, all right, the integral 1346 01:20:50,180 --> 01:20:52,845 from 0 to pi over 4. 1347 01:20:52,845 --> 01:20:54,770 STUDENT: [INAUDIBLE]. 1348 01:20:54,770 --> 01:20:59,345 PROFESSOR: r cubed over 3 between-- it's a little bit 1349 01:20:59,345 --> 01:21:03,180 of a headache. r equals a over cosine phi. 1350 01:21:03,180 --> 01:21:05,300 And I bet you my video doesn't see anything, 1351 01:21:05,300 --> 01:21:07,840 so let me change the colors. 1352 01:21:07,840 --> 01:21:10,760 r equals a over cosine phi. 1353 01:21:10,760 --> 01:21:13,020 And r equals 0 down. 1354 01:21:13,020 --> 01:21:14,780 That's the easy part. 1355 01:21:14,780 --> 01:21:19,100 Inside I have r cubed over 3, right? 1356 01:21:19,100 --> 01:21:25,640 All right, and sine phi, and all I'm 1357 01:21:25,640 --> 01:21:31,759 left with is a phi integration, is an integration 1358 01:21:31,759 --> 01:21:33,552 with respect to phi. 1359 01:21:33,552 --> 01:21:35,997 Let's see-- yes, sir. 1360 01:21:35,997 --> 01:21:36,975 STUDENT: [INAUDIBLE]. 1361 01:21:36,975 --> 01:21:40,562 PROFESSOR: Well, I should be able to manage with this guy. 1362 01:21:40,562 --> 01:21:46,575 1363 01:21:46,575 --> 01:21:47,450 STUDENT: [INAUDIBLE]. 1364 01:21:47,450 --> 01:22:05,180 1365 01:22:05,180 --> 01:22:07,293 PROFESSOR: I'm writing just as you said, OK? 1366 01:22:07,293 --> 01:22:13,645 1367 01:22:13,645 --> 01:22:16,150 Now, how much of a headache do you think this is? 1368 01:22:16,150 --> 01:22:18,304 STUDENT: It's not much of one, because it's 1369 01:22:18,304 --> 01:22:20,620 the same as a tangent times the secant squared 1370 01:22:20,620 --> 01:22:22,860 with a constant pulled out. 1371 01:22:22,860 --> 01:22:26,075 STUDENT: So psi and cosine don't [INAUDIBLE]. 1372 01:22:26,075 --> 01:22:34,012 Tangents will give you 1 over cosine-- 1373 01:22:34,012 --> 01:22:36,512 PROFESSOR: What's the simplest way to do it without thinking 1374 01:22:36,512 --> 01:22:39,930 of tangent and cotangent, huh? 1375 01:22:39,930 --> 01:22:41,254 STUDENT: [INAUDIBLE]. 1376 01:22:41,254 --> 01:22:43,520 PROFESSOR: Instead, a u substitution there? 1377 01:22:43,520 --> 01:22:45,246 What is the u substitution? 1378 01:22:45,246 --> 01:22:47,626 STUDENT: [INAUDIBLE]. 1379 01:22:47,626 --> 01:22:49,530 PROFESSOR: Is this good? 1380 01:22:49,530 --> 01:22:50,970 STUDENT: No. 1381 01:22:50,970 --> 01:22:52,168 PROFESSOR: No? 1382 01:22:52,168 --> 01:22:54,995 STUDENT: [INAUDIBLE]. 1383 01:22:54,995 --> 01:22:57,700 PROFESSOR: It's u to the minus 3. 1384 01:22:57,700 --> 01:22:59,880 And that's OK. 1385 01:22:59,880 --> 01:23:05,740 So I have 2 pi a cubed over 3 [INAUDIBLE] 1386 01:23:05,740 --> 01:23:09,780 because they are in my way there making my life miserable, 1387 01:23:09,780 --> 01:23:11,120 integral. 1388 01:23:11,120 --> 01:23:18,100 And then I have u to the minus 3 times-- for du 1389 01:23:18,100 --> 01:23:22,321 I get a minus that that is sort of ugh. 1390 01:23:22,321 --> 01:23:26,040 I have to invent the minus, and I have to invent the minus here 1391 01:23:26,040 --> 01:23:27,990 in front as well. 1392 01:23:27,990 --> 01:23:32,085 So they will compensate for one another. 1393 01:23:32,085 --> 01:23:34,680 And I'll say du. 1394 01:23:34,680 --> 01:23:39,710 But these limit points, of course I can do them by myself. 1395 01:23:39,710 --> 01:23:41,680 I don't need your help. 1396 01:23:41,680 --> 01:23:45,830 But I pretend that I need your help. 1397 01:23:45,830 --> 01:23:48,330 What will be u when phi is 0? 1398 01:23:48,330 --> 01:23:49,150 STUDENT: 1. 1399 01:23:49,150 --> 01:23:51,840 PROFESSOR: 1. 1400 01:23:51,840 --> 01:23:56,824 What will be u when phi is pi over 4? 1401 01:23:56,824 --> 01:23:57,699 STUDENT: [INAUDIBLE]. 1402 01:23:57,699 --> 01:24:02,086 1403 01:24:02,086 --> 01:24:04,460 PROFESSOR: And from now on you should be able to do this. 1404 01:24:04,460 --> 01:24:14,670 So I have minus 2 pi a cubed over 3 times-- I integrate. 1405 01:24:14,670 --> 01:24:17,850 So I add the power, I add the 1, and I add the 1. 1406 01:24:17,850 --> 01:24:20,450 So you have u to the minus 2 over minus 2. 1407 01:24:20,450 --> 01:24:27,262 Are you guys with me-- between u equals 1408 01:24:27,262 --> 01:24:30,630 1 and u equals root 2 over 2. 1409 01:24:30,630 --> 01:24:33,010 I promise you if you have something 1410 01:24:33,010 --> 01:24:35,056 like that in the final and you stop here, 1411 01:24:35,056 --> 01:24:38,030 I'm not going to be blaming you. 1412 01:24:38,030 --> 01:24:41,410 I'll say, very good, leave it there, I don't care. 1413 01:24:41,410 --> 01:24:44,920 Because from this point on, what follows 1414 01:24:44,920 --> 01:24:47,940 is just routine algebra. 1415 01:24:47,940 --> 01:24:52,470 So we have-- I hate this. 1416 01:24:52,470 --> 01:24:55,255 I'm not a calculator. 1417 01:24:55,255 --> 01:24:59,885 But it's better for me to write 1 over root 2, like you said. 1418 01:24:59,885 --> 01:25:03,150 Because in that case, the square will be 1 over 2. 1419 01:25:03,150 --> 01:25:06,180 And when I invert 1 over 2, I get a 2. 1420 01:25:06,180 --> 01:25:10,546 So I have 2 over minus 2. 1421 01:25:10,546 --> 01:25:11,670 Are you guys with me again? 1422 01:25:11,670 --> 01:25:14,220 So I'm thinking the same-- 1 over root 2. 1423 01:25:14,220 --> 01:25:16,685 Square it, you have 1 over 2. 1424 01:25:16,685 --> 01:25:20,567 Take it as a minus, you have exactly 2. 1425 01:25:20,567 --> 01:25:24,130 And you have 2 over minus-- is this a minus? 1426 01:25:24,130 --> 01:25:26,856 I'm so silly, look at me, minus 2. 1427 01:25:26,856 --> 01:25:29,290 STUDENT: It's a cubed over 3, not over 2. 1428 01:25:29,290 --> 01:25:31,590 PROFESSOR: It's going to be-- 1429 01:25:31,590 --> 01:25:33,676 STUDENT: You've got an a cubed over 2 right there. 1430 01:25:33,676 --> 01:25:35,628 And it was-- 1431 01:25:35,628 --> 01:25:36,604 PROFESSOR: Huh? 1432 01:25:36,604 --> 01:25:39,044 STUDENT: You just wrote 2 pi a cubed over 2. 1433 01:25:39,044 --> 01:25:41,000 It's a cubed over 3. 1434 01:25:41,000 --> 01:25:43,370 PROFESSOR: Yes, it's my silliness. 1435 01:25:43,370 --> 01:25:46,420 I looked, and I say this instead of that. 1436 01:25:46,420 --> 01:25:48,000 Thank you so much. 1437 01:25:48,000 --> 01:25:49,790 What do I have here? 1438 01:25:49,790 --> 01:25:51,620 1 over minus 2. 1439 01:25:51,620 --> 01:25:54,844 In the end, what does this mean? 1440 01:25:54,844 --> 01:25:56,245 Let's see, what does this mean? 1441 01:25:56,245 --> 01:26:00,920 When I plug in, I subtract. 1442 01:26:00,920 --> 01:26:02,240 This is what? 1443 01:26:02,240 --> 01:26:07,880 This is minus 1 plus 1/2 is minus 1/2. 1444 01:26:07,880 --> 01:26:12,250 But that minus should not scare me. 1445 01:26:12,250 --> 01:26:14,050 Because of course a minus in a volume 1446 01:26:14,050 --> 01:26:15,865 would be completely wrong. 1447 01:26:15,865 --> 01:26:18,230 But I have a minus from before. 1448 01:26:18,230 --> 01:26:31,940 So it's plus 2 pi times a cubed over 3, and times 1/2. 1449 01:26:31,940 --> 01:26:35,270 So in the end, the answer, the total answer, 1450 01:26:35,270 --> 01:26:37,320 would be answered what? 1451 01:26:37,320 --> 01:26:39,790 STUDENT: [INAUDIBLE]. 1452 01:26:39,790 --> 01:26:45,100 PROFESSOR: Pi a cubed over 3, pi a cubed over 3. 1453 01:26:45,100 --> 01:26:46,560 It looks very-- huh? 1454 01:26:46,560 --> 01:26:48,340 It looks pretty. 1455 01:26:48,340 --> 01:26:53,480 Actually yes, it looks pretty because-- now, 1456 01:26:53,480 --> 01:26:55,900 OK, I'm asking you a question. 1457 01:26:55,900 --> 01:26:59,750 Would we have done that without calculus? 1458 01:26:59,750 --> 01:27:06,200 If somebody told you [INAUDIBLE] it has a volume of some cone, 1459 01:27:06,200 --> 01:27:08,590 what's the volume of a cone? 1460 01:27:08,590 --> 01:27:13,060 Area of the base times the height divided by 3. 1461 01:27:13,060 --> 01:27:17,470 So you could have very nicely cheated on me on the exam 1462 01:27:17,470 --> 01:27:21,010 by saying, you have this cone that 1463 01:27:21,010 --> 01:27:26,190 has pi is squared times a-- pi is squared times a-- divided 1464 01:27:26,190 --> 01:27:29,579 by 3 equals pi cubed over 3. 1465 01:27:29,579 --> 01:27:32,875 When can you not cheat on this problem? 1466 01:27:32,875 --> 01:27:35,000 STUDENT: When you say, you've got to do it with a-- 1467 01:27:35,000 --> 01:27:38,310 PROFESSOR: Exactly, when I say, do it with a-- well, 1468 01:27:38,310 --> 01:27:45,250 I can say, OK, if we say, set up the integral and write it down, 1469 01:27:45,250 --> 01:27:47,100 you set up the integral and write it down. 1470 01:27:47,100 --> 01:27:50,380 If we say, set up the integral and compute it, 1471 01:27:50,380 --> 01:27:53,910 you set up the integral, you fake the computation, 1472 01:27:53,910 --> 01:27:55,260 and you come up with this. 1473 01:27:55,260 --> 01:27:58,400 1474 01:27:58,400 --> 01:28:02,110 If we say, set up the integral and show all your work, 1475 01:28:02,110 --> 01:28:03,490 then you're in trouble. 1476 01:28:03,490 --> 01:28:09,180 But I'm going to try to advocate that for a simple problem, that 1477 01:28:09,180 --> 01:28:11,130 is actually elementary. 1478 01:28:11,130 --> 01:28:15,720 One should not have to show all the work. 1479 01:28:15,720 --> 01:28:19,110 All right, but keep in mind when you have 2 minuses 1480 01:28:19,110 --> 01:28:22,580 like that-- that reminds me. 1481 01:28:22,580 --> 01:28:28,788 So there was a professor whose sink didn't work anymore. 1482 01:28:28,788 --> 01:28:32,064 And he asked for a plumber to come to his house. 1483 01:28:32,064 --> 01:28:33,470 He was a math professor. 1484 01:28:33,470 --> 01:28:36,530 So the plumber comes to his house and fixes this, and says, 1485 01:28:36,530 --> 01:28:37,580 what else is wrong? 1486 01:28:37,580 --> 01:28:40,115 Fixes the toilet, fixes everything in the house, 1487 01:28:40,115 --> 01:28:44,214 and then he shows the professor the bill. 1488 01:28:44,214 --> 01:28:50,058 So the guy said, oh my god, this is 1/3 of my monthly salary. 1489 01:28:50,058 --> 01:28:53,575 So the plumber said, yeah, I mean, really? 1490 01:28:53,575 --> 01:28:54,620 You're a smart guy. 1491 01:28:54,620 --> 01:28:55,411 You're a professor. 1492 01:28:55,411 --> 01:28:57,260 You make that little money? 1493 01:28:57,260 --> 01:28:58,850 Yeah, really. 1494 01:28:58,850 --> 01:28:59,990 I'm so sorry for you. 1495 01:28:59,990 --> 01:29:03,780 Why don't you apply to our company 1496 01:29:03,780 --> 01:29:06,850 and become a plumber if you're interested, if you crave money? 1497 01:29:06,850 --> 01:29:09,004 No, of course, I need money desperately. 1498 01:29:09,004 --> 01:29:12,976 I have five children and a wife [INAUDIBLE]. 1499 01:29:12,976 --> 01:29:15,465 OK, he applies. 1500 01:29:15,465 --> 01:29:16,830 And he says, pay attention. 1501 01:29:16,830 --> 01:29:21,240 Don't write that you are a professor or you have a PhD. 1502 01:29:21,240 --> 01:29:23,820 Just say you just finished high school or say, 1503 01:29:23,820 --> 01:29:25,390 I didn't finish high school. 1504 01:29:25,390 --> 01:29:27,460 So he writes, I didn't finish high school. 1505 01:29:27,460 --> 01:29:28,510 I went to 10th grade. 1506 01:29:28,510 --> 01:29:29,785 They accept him. 1507 01:29:29,785 --> 01:29:32,070 They give him a job. 1508 01:29:32,070 --> 01:29:34,770 And they say, this is your salary. 1509 01:29:34,770 --> 01:29:36,370 But there is something new. 1510 01:29:36,370 --> 01:29:40,150 Everybody has to finish high school. 1511 01:29:40,150 --> 01:29:42,842 So they have to take AP Calculus. 1512 01:29:42,842 --> 01:29:44,630 So he goes, oh my god. 1513 01:29:44,630 --> 01:29:45,630 They all go. 1514 01:29:45,630 --> 01:29:49,640 And there comes a TA from the community college. 1515 01:29:49,640 --> 01:29:51,410 The class was full. 1516 01:29:51,410 --> 01:29:56,260 He tries to solve a problem-- with calculus 1517 01:29:56,260 --> 01:30:00,835 compute the area inside this disc of radius a. 1518 01:30:00,835 --> 01:30:03,750 So the TA-- OK, I did this. 1519 01:30:03,750 --> 01:30:07,390 I got minus pi a squared. 1520 01:30:07,390 --> 01:30:11,670 And the professor says, OK, you cannot get that. 1521 01:30:11,670 --> 01:30:13,854 Let me explain to you. 1522 01:30:13,854 --> 01:30:15,770 He goes, I don't know where he made a mistake. 1523 01:30:15,770 --> 01:30:21,540 Because I still get-- where is minus pi a squared? 1524 01:30:21,540 --> 01:30:22,970 I don't see where the mistake is. 1525 01:30:22,970 --> 01:30:26,631 And then the whole class, 12, 15-- reverse 1526 01:30:26,631 --> 01:30:30,040 the integral limits. 1527 01:30:30,040 --> 01:30:34,423 Change the integral limits and you'll get it right. 1528 01:30:34,423 --> 01:30:41,420 So we can all pretend that we want to do something else 1529 01:30:41,420 --> 01:30:45,920 and we didn't finish high school and we'll get a lot more money. 1530 01:30:45,920 --> 01:30:47,930 The person who came to fix my air conditioner 1531 01:30:47,930 --> 01:30:55,400 said that he actually makes about $100 an hour. 1532 01:30:55,400 --> 01:30:57,370 And I was thinking, wow. 1533 01:30:57,370 --> 01:30:59,910 Wow, I'll never get there. 1534 01:30:59,910 --> 01:31:01,750 But that's impressive. 1535 01:31:01,750 --> 01:31:05,270 Just changing some things and fix, 1536 01:31:05,270 --> 01:31:08,120 press the button, $100 an hour. 1537 01:31:08,120 --> 01:31:11,445 STUDENT: But they don't work full time. [INAUDIBLE]. 1538 01:31:11,445 --> 01:31:15,466 PROFESSOR: Yeah, and I think they are paid by the job. 1539 01:31:15,466 --> 01:31:19,000 But in any case, whether it's a simple job 1540 01:31:19,000 --> 01:31:21,360 and they just-- there is a contact that's 1541 01:31:21,360 --> 01:31:25,814 missing or something trivial, they still 1542 01:31:25,814 --> 01:31:27,758 charge a lot of money. 1543 01:31:27,758 --> 01:31:30,188 STUDENT: [INAUDIBLE]. 1544 01:31:30,188 --> 01:31:31,646 PROFESSOR: A professor? 1545 01:31:31,646 --> 01:31:33,590 STUDENT: [INAUDIBLE]. 1546 01:31:33,590 --> 01:31:34,562 PROFESSOR: What? 1547 01:31:34,562 --> 01:31:37,686 [LAUGHING] No. 1548 01:31:37,686 --> 01:31:38,936 STUDENT: I know the professor. 1549 01:31:38,936 --> 01:31:40,080 I won't tell you who. 1550 01:31:40,080 --> 01:31:41,580 PROFESSOR: OK, I don't want to know. 1551 01:31:41,580 --> 01:31:42,945 I don't want to know. 1552 01:31:42,945 --> 01:31:44,765 But anyway, it's interesting. 1553 01:31:44,765 --> 01:31:46,560 STUDENT: But he doesn't do in the college. 1554 01:31:46,560 --> 01:31:50,960 He does outside the college by just advising it. 1555 01:31:50,960 --> 01:31:53,526 PROFESSOR: Oh, you mean like consulting or tutoring 1556 01:31:53,526 --> 01:31:54,498 or stuff like that? 1557 01:31:54,498 --> 01:31:56,928 STUDENT: Not tutoring, consulting for the-- 1558 01:31:56,928 --> 01:31:59,358 PROFESSOR: Consulting. 1559 01:31:59,358 --> 01:32:02,760 Actually, I bet that if we did tutoring, 1560 01:32:02,760 --> 01:32:06,440 which we don't have time for, we would make a lot of money. 1561 01:32:06,440 --> 01:32:08,760 But the nature of my job, for example, 1562 01:32:08,760 --> 01:32:11,812 is that I work about 60 hours a week, 65, 1563 01:32:11,812 --> 01:32:14,788 and I will not have any time left to do other things, 1564 01:32:14,788 --> 01:32:18,260 like consulting, tutoring, and stuff. 1565 01:32:18,260 --> 01:32:20,437 STUDENT: [INAUDIBLE]. 1566 01:32:20,437 --> 01:32:22,228 PROFESSOR: I don't need normally that much. 1567 01:32:22,228 --> 01:32:24,708 I don't crave money that much. 1568 01:32:24,708 --> 01:32:27,049 STUDENT: [INAUDIBLE]. 1569 01:32:27,049 --> 01:32:28,924 PROFESSOR: I have a friend who got a masters. 1570 01:32:28,924 --> 01:32:31,156 She didn't get a PhD. 1571 01:32:31,156 --> 01:32:34,628 She got an offer from this-- I told you about her. 1572 01:32:34,628 --> 01:32:36,116 She moved to California. 1573 01:32:36,116 --> 01:32:38,610 She was a single mom. 1574 01:32:38,610 --> 01:32:42,140 She earns a lot of money working for Pixar. 1575 01:32:42,140 --> 01:32:44,790 And she helped with all the animation things. 1576 01:32:44,790 --> 01:32:47,860 1577 01:32:47,860 --> 01:32:52,310 It was about 15 years ago that she started. 1578 01:32:52,310 --> 01:32:54,190 And it was really hard. 1579 01:32:54,190 --> 01:33:00,776 We were all on Toy Story and that kind of-- what 1580 01:33:00,776 --> 01:33:03,646 was that called? 1581 01:33:03,646 --> 01:33:07,820 There were two rendering algorithms, 1582 01:33:07,820 --> 01:33:09,211 rendering algorithms. 1583 01:33:09,211 --> 01:33:13,860 Two masters students were interested in that. 1584 01:33:13,860 --> 01:33:16,240 They got in immediately. 1585 01:33:16,240 --> 01:33:19,410 To be hired, I think a post-doc with a PhD 1586 01:33:19,410 --> 01:33:23,150 was making about $40,000. 1587 01:33:23,150 --> 01:33:24,800 That was my offer. 1588 01:33:24,800 --> 01:33:31,120 My first offer was a post-doc at Urbana-Champaign for $38,000 1589 01:33:31,120 --> 01:33:34,110 while she was at the hundred and something thousand 1590 01:33:34,110 --> 01:33:36,981 dollars to start with working at Disney. 1591 01:33:36,981 --> 01:33:41,260 Imagine-- with just a masters, no aspiration for a PhD 1592 01:33:41,260 --> 01:33:42,670 whatever. 1593 01:33:42,670 --> 01:33:46,071 So in a way, if you're thinking of doing this, 1594 01:33:46,071 --> 01:33:50,881 a masters in mathematics is probably paying off. 1595 01:33:50,881 --> 01:33:54,340 Because it opens a lot of doors for you. 1596 01:33:54,340 --> 01:33:55,800 And that's just in general. 1597 01:33:55,800 --> 01:33:58,866 I mean, masters in engineering opens a lot of doors. 1598 01:33:58,866 --> 01:34:02,222 But in a way, you pay a price after 1599 01:34:02,222 --> 01:34:05,530 if you want to start even further, get a PhD, 1600 01:34:05,530 --> 01:34:06,570 stay in academia. 1601 01:34:06,570 --> 01:34:08,470 Then you pay a price. 1602 01:34:08,470 --> 01:34:10,540 And if you want to augment your salary, 1603 01:34:10,540 --> 01:34:16,360 you really have to be very good and accomplish some-- 1604 01:34:16,360 --> 01:34:19,160 get [INAUDIBLE] two or three times 1605 01:34:19,160 --> 01:34:22,648 and get higher up each [INAUDIBLE]. 1606 01:34:22,648 --> 01:34:27,500 But we all struggle with these issues. 1607 01:34:27,500 --> 01:34:31,420 It's a lot of work. 1608 01:34:31,420 --> 01:34:34,360 But having a masters in math is not so hard. 1609 01:34:34,360 --> 01:34:39,060 If you like math, it's easy to get it. 1610 01:34:39,060 --> 01:34:39,826 It's a pleasure. 1611 01:34:39,826 --> 01:34:41,540 It's not a lot of hours. 1612 01:34:41,540 --> 01:34:45,980 I think in 36 hours in most schools you can get a masters. 1613 01:34:45,980 --> 01:34:49,598 And it's doable. 1614 01:34:49,598 --> 01:35:02,494 All right, let's go back to review Chapter 11 briefly here. 1615 01:35:02,494 --> 01:35:11,918 1616 01:35:11,918 --> 01:35:13,902 Is this on the midterm? 1617 01:35:13,902 --> 01:35:16,382 No, but it's going to be on the final. 1618 01:35:16,382 --> 01:35:23,326 1619 01:35:23,326 --> 01:35:33,180 Assume you have a x equals u plus v, 1620 01:35:33,180 --> 01:35:47,390 y equals u minus v. Write the following derivative. 1621 01:35:47,390 --> 01:35:57,050 1622 01:35:57,050 --> 01:36:14,012 dx/dv where u of t equals t squared and v equals t. 1623 01:36:14,012 --> 01:36:31,350 Do these both directly and by writing a chain 1624 01:36:31,350 --> 01:36:38,282 rule for the values you have. 1625 01:36:38,282 --> 01:36:43,529 1626 01:36:43,529 --> 01:36:45,910 OK, how do we do this directly? 1627 01:36:45,910 --> 01:36:47,910 It's probably the simplest way. 1628 01:36:47,910 --> 01:36:56,910 1629 01:36:56,910 --> 01:37:02,170 Replace u by t squared, replace v by t and see what you have. 1630 01:37:02,170 --> 01:37:03,219 So 1, directly. 1631 01:37:03,219 --> 01:37:07,809 1632 01:37:07,809 --> 01:37:12,270 X of t equals t squared plus t. 1633 01:37:12,270 --> 01:37:16,414 y of t equals t squared minus t. 1634 01:37:16,414 --> 01:37:23,660 1635 01:37:23,660 --> 01:37:24,930 Good. 1636 01:37:24,930 --> 01:37:32,860 So it's a piece of cake. dx/dt equals 2t plus 1. 1637 01:37:32,860 --> 01:37:37,941 Unfortunately, this is just the first part of the problem. 1638 01:37:37,941 --> 01:37:41,410 And it's actually [INAUDIBLE] to show the chain 1639 01:37:41,410 --> 01:37:43,520 rule for the mappings we have. 1640 01:37:43,520 --> 01:37:45,290 And what mappings do we have? 1641 01:37:45,290 --> 01:37:54,460 We have a map from t to u of t and v of t. 1642 01:37:54,460 --> 01:38:00,340 And then again, from u of t and v of t to x of t and y of t. 1643 01:38:00,340 --> 01:38:04,760 And the transformation is what? x equals u plus v, 1644 01:38:04,760 --> 01:38:10,074 y equals u minus v. And we have another transformation here. 1645 01:38:10,074 --> 01:38:15,138 So how do you write dx/dt? 1646 01:38:15,138 --> 01:38:20,550 x is a function of u and v, right? 1647 01:38:20,550 --> 01:38:25,330 So first you say that dx/dv round, 1648 01:38:25,330 --> 01:38:28,180 which means we do it with the first variable. 1649 01:38:28,180 --> 01:38:31,210 I'll write it for you to see better, 1650 01:38:31,210 --> 01:38:39,120 that initially your x and y were functions of u and v. 1651 01:38:39,120 --> 01:38:51,252 Times-- what is that? dv/dt plus dx/du. 1652 01:38:51,252 --> 01:38:52,830 You can change the order. 1653 01:38:52,830 --> 01:38:55,560 If you didn't like that I started with v, 1654 01:38:55,560 --> 01:39:00,095 I could have started with the u, and the u, and the v, 1655 01:39:00,095 --> 01:39:00,720 and the v here. 1656 01:39:00,720 --> 01:39:03,740 It doesn't matter. 1657 01:39:03,740 --> 01:39:06,130 Guys, do you mind, really? 1658 01:39:06,130 --> 01:39:07,640 v, v. u, u. 1659 01:39:07,640 --> 01:39:08,704 Shooting cowboys? 1660 01:39:08,704 --> 01:39:10,995 Doesn't matter, remember just that they're [INAUDIBLE]. 1661 01:39:10,995 --> 01:39:14,030 1662 01:39:14,030 --> 01:39:15,463 D sorry, d. 1663 01:39:15,463 --> 01:39:18,160 Because where there is no other variable, 1664 01:39:18,160 --> 01:39:22,560 we would put v. So dx/dt? 1665 01:39:22,560 --> 01:39:25,460 Lets see if we get the same answer. 1666 01:39:25,460 --> 01:39:26,320 We should. 1667 01:39:26,320 --> 01:39:27,810 What is dx/dt? 1668 01:39:27,810 --> 01:39:30,908 1, from here. 1669 01:39:30,908 --> 01:39:32,375 What is dv/dt? 1670 01:39:32,375 --> 01:39:37,990 1671 01:39:37,990 --> 01:39:42,640 1 plus dx/du. 1672 01:39:42,640 --> 01:39:46,140 1, du/dt. 1673 01:39:46,140 --> 01:39:47,790 2t. 1674 01:39:47,790 --> 01:39:50,080 If we were to do the same thing-- so 1675 01:39:50,080 --> 01:39:52,100 we got the same answer. 1676 01:39:52,100 --> 01:39:56,140 If you want to do the same thing, 1677 01:39:56,140 --> 01:40:01,910 quickly with respect to say dy/dt, 1678 01:40:01,910 --> 01:40:05,790 suppose that most finals ask you to do both. 1679 01:40:05,790 --> 01:40:08,140 I have students who didn't finish 1680 01:40:08,140 --> 01:40:13,340 because they didn't have the time to finish, 1681 01:40:13,340 --> 01:40:14,915 but that was just my policy. 1682 01:40:14,915 --> 01:40:17,705 When I grade it, I gave them 100%, 1683 01:40:17,705 --> 01:40:20,640 no matter if they stopped here, because I 1684 01:40:20,640 --> 01:40:22,890 said you prove to me that you know the chain rule. 1685 01:40:22,890 --> 01:40:26,730 Why would I punish you further? 1686 01:40:26,730 --> 01:40:27,955 So that's what I do. 1687 01:40:27,955 --> 01:40:32,280 But I want you to do it now, without my help. 1688 01:40:32,280 --> 01:40:34,666 Both ways, dy/dt. 1689 01:40:34,666 --> 01:40:39,496 First you do it with the chain rule. 1690 01:40:39,496 --> 01:40:42,394 First you write those three [INAUDIBLE]. 1691 01:40:42,394 --> 01:40:47,730 1692 01:40:47,730 --> 01:40:48,680 dy del y. 1693 01:40:48,680 --> 01:40:51,575 1694 01:40:51,575 --> 01:40:56,420 del u, du/dt, plus del y. 1695 01:40:56,420 --> 01:40:59,050 del v, dv/dt. 1696 01:40:59,050 --> 01:41:03,772 I'm not going to write it down, you write it down. 1697 01:41:03,772 --> 01:41:05,880 What I'm going to write down is what 1698 01:41:05,880 --> 01:41:07,878 you tell me the numbers are. 1699 01:41:07,878 --> 01:41:13,185 1700 01:41:13,185 --> 01:41:13,810 For everything. 1701 01:41:13,810 --> 01:41:19,724 1702 01:41:19,724 --> 01:41:24,684 STUDENT: Dy divided by d 1703 01:41:24,684 --> 01:41:29,640 PROFESSOR: Or just give me the final answer in terms of 1704 01:41:29,640 --> 01:41:30,140 [INAUDIBLE]. 1705 01:41:30,140 --> 01:41:42,090 1706 01:41:42,090 --> 01:41:47,700 What are the two [INAUDIBLE]? 1707 01:41:47,700 --> 01:41:49,668 Tell me. 1708 01:41:49,668 --> 01:41:56,580 Tell me, this times this, plus this times that. 1709 01:41:56,580 --> 01:41:58,170 What? 1710 01:41:58,170 --> 01:41:59,370 So let's write down. 1711 01:41:59,370 --> 01:42:00,890 Let's write it down together. 1712 01:42:00,890 --> 01:42:16,260 dy/du, du/dt, plus dy/dv dv/dt. 1713 01:42:16,260 --> 01:42:16,760 Alright. 1714 01:42:16,760 --> 01:42:30,480 1715 01:42:30,480 --> 01:42:34,540 1 This is 1. 1716 01:42:34,540 --> 01:42:35,770 How much is dy/dt? 1717 01:42:35,770 --> 01:42:38,550 1718 01:42:38,550 --> 01:42:40,140 Or du/dt, I'm sorry. 1719 01:42:40,140 --> 01:42:44,940 I said dy, it's du/dt. 1720 01:42:44,940 --> 01:42:50,610 Plus minus 1, excellent. 1721 01:42:50,610 --> 01:42:54,032 Times 1. 1722 01:42:54,032 --> 01:42:56,070 Of course you would have done the same thing, 1723 01:42:56,070 --> 01:42:59,640 by plugging in the variables and saying well, 1724 01:42:59,640 --> 01:43:04,770 I have y, which is this is t squared, this is t, 1725 01:43:04,770 --> 01:43:09,270 and I have t squared minus t prime is 2t minus 1. 1726 01:43:09,270 --> 01:43:13,690 That's a simpler way to verify [INAUDIBLE]. 1727 01:43:13,690 --> 01:43:14,470 OK. 1728 01:43:14,470 --> 01:43:20,070 So remember to do that, have this in mind, 1729 01:43:20,070 --> 01:43:25,126 because on the final you may have something like that. 1730 01:43:25,126 --> 01:43:28,619 As we keep going in the month of April, 1731 01:43:28,619 --> 01:43:32,620 I'm going to do as much review as possible for the final. 1732 01:43:32,620 --> 01:43:37,470 Mark a star, or F, not the grade F, but F around for the final, 1733 01:43:37,470 --> 01:43:45,150 put F and circle there to say review this for the final. 1734 01:43:45,150 --> 01:43:48,954 And since we are still in chapter 11 review, 1735 01:43:48,954 --> 01:43:56,870 we'll do another problem of F, final review 1736 01:43:56,870 --> 01:44:02,400 that I didn't put on the midterm but it may be on the final. 1737 01:44:02,400 --> 01:44:06,965 Let's say given the constraint x squared 1738 01:44:06,965 --> 01:44:13,420 plus y squared plus z squared equals 5, compute z sub x 1739 01:44:13,420 --> 01:44:16,658 and z sub y. 1740 01:44:16,658 --> 01:44:18,077 How do you do that? 1741 01:44:18,077 --> 01:44:20,550 What is this called, actually, and why is it 1742 01:44:20,550 --> 01:44:22,720 so important for the final? 1743 01:44:22,720 --> 01:44:24,220 It's called implicit differentiation 1744 01:44:24,220 --> 01:44:28,975 and it appears on almost every final, at least once a year, 1745 01:44:28,975 --> 01:44:31,770 so there is always a big possibility 1746 01:44:31,770 --> 01:44:35,880 that you are going to see something like that. 1747 01:44:35,880 --> 01:44:41,810 I taught you how to think in terms of implicit functions. 1748 01:44:41,810 --> 01:44:49,730 If you think of z as a function of x and y. 1749 01:44:49,730 --> 01:44:52,340 That's a way of changing your perspective. 1750 01:44:52,340 --> 01:44:56,910 So you say, OK, I understand that z 1751 01:44:56,910 --> 01:45:01,250 has to be viewed as a function of x and y. 1752 01:45:01,250 --> 01:45:05,178 I'm just changing my perspective. 1753 01:45:05,178 --> 01:45:10,090 STUDENT: For that one, wouldn't you just solve for z? 1754 01:45:10,090 --> 01:45:11,230 PROFESSOR: No. 1755 01:45:11,230 --> 01:45:15,880 Solving for z would make your life a lot harder. 1756 01:45:15,880 --> 01:45:17,940 The point of implicit functions is 1757 01:45:17,940 --> 01:45:20,470 that you don't separate them. 1758 01:45:20,470 --> 01:45:22,570 If you're going to separate them, 1759 01:45:22,570 --> 01:45:26,040 you have to separately integrate these. 1760 01:45:26,040 --> 01:45:27,918 And it's a headache. 1761 01:45:27,918 --> 01:45:30,852 It's easier-- actually it's a good question. 1762 01:45:30,852 --> 01:45:36,440 It's easier to do z sub x, z sub y without splitting it 1763 01:45:36,440 --> 01:45:37,380 into two cases. 1764 01:45:37,380 --> 01:45:39,960 1765 01:45:39,960 --> 01:45:42,985 Step two. 1766 01:45:42,985 --> 01:45:44,563 Differentiate this with respect to x. 1767 01:45:44,563 --> 01:45:47,520 What do we have? 1768 01:45:47,520 --> 01:45:56,140 2x plus 0 plus the chain rule-- don't write the chain rule. 1769 01:45:56,140 --> 01:46:00,420 2 jumping down, it jumped down. 1770 01:46:00,420 --> 01:46:06,340 2z times-- cover the 2 with your hand. 1771 01:46:06,340 --> 01:46:07,805 z sub x, very good. 1772 01:46:07,805 --> 01:46:11,730 z prime with respect to x equals zero. 1773 01:46:11,730 --> 01:46:13,790 Good. 1774 01:46:13,790 --> 01:46:18,260 So z sub x, step three. 1775 01:46:18,260 --> 01:46:19,400 And the last step. 1776 01:46:19,400 --> 01:46:22,790 See sub x will be what? 1777 01:46:22,790 --> 01:46:23,620 Pull it out. 1778 01:46:23,620 --> 01:46:26,260 Pull this 2 out. 1779 01:46:26,260 --> 01:46:31,595 Minus x over z. 1780 01:46:31,595 --> 01:46:36,250 1781 01:46:36,250 --> 01:46:40,060 The other one is symmetric. 1782 01:46:40,060 --> 01:46:43,510 Alex said let's be smart and not do the whole thing 1783 01:46:43,510 --> 01:46:44,390 all over again. 1784 01:46:44,390 --> 01:46:46,850 Look at beautiful symmetric polynomial. 1785 01:46:46,850 --> 01:46:49,992 You would have to be a little bit careful with when 1786 01:46:49,992 --> 01:46:53,848 you have a 1 here and y would have a 2 here. 1787 01:46:53,848 --> 01:46:56,910 It wouldn't be symmetric in x and y. 1788 01:46:56,910 --> 01:47:00,180 But here, if you reverse the roles of x and y, 1789 01:47:00,180 --> 01:47:02,015 it's not a big deal. 1790 01:47:02,015 --> 01:47:02,890 Are you guys with me? 1791 01:47:02,890 --> 01:47:07,260 Here we are. z sub y equals minus y over z. 1792 01:47:07,260 --> 01:47:10,250 Am I right? 1793 01:47:10,250 --> 01:47:14,090 Keep this in mind for-- I also saw, 1794 01:47:14,090 --> 01:47:18,610 when I was looking at the [INAUDIBLE] library 1795 01:47:18,610 --> 01:47:21,115 files, [INAUDIBLE]. 1796 01:47:21,115 --> 01:47:25,355 I also saw exams, and I was looking at your reviews there. 1797 01:47:25,355 --> 01:47:27,750 I was looking at [INAUDIBLE]. 1798 01:47:27,750 --> 01:47:32,250 The University of Houston has a very beautiful online, free 1799 01:47:32,250 --> 01:47:36,740 library of calculus 1 and calculus 2 exams 1800 01:47:36,740 --> 01:47:39,680 that I found very useful. 1801 01:47:39,680 --> 01:47:44,100 Now, one of them-- listen to me so you 1802 01:47:44,100 --> 01:47:47,330 don't fall through this crack. 1803 01:47:47,330 --> 01:47:54,300 On the Cal 2 exam, they wrote something like that. 1804 01:47:54,300 --> 01:48:00,211 1805 01:48:00,211 --> 01:48:05,850 You don't have to write 1 over x squared, and then compute. 1806 01:48:05,850 --> 01:48:09,120 You just say, OK, if the natural part of the of the argument 1807 01:48:09,120 --> 01:48:13,430 is 5, then the argument is a constant. 1808 01:48:13,430 --> 01:48:15,405 And I don't care what constant it 1809 01:48:15,405 --> 01:48:18,300 is, it it's something that prime will give me 0, 1810 01:48:18,300 --> 01:48:20,090 it's the same problem. 1811 01:48:20,090 --> 01:48:21,390 Are you guys with me? 1812 01:48:21,390 --> 01:48:26,040 So in that case, I'm going have just what kind of change? 1813 01:48:26,040 --> 01:48:28,501 This will be to the 5. 1814 01:48:28,501 --> 01:48:30,740 And I still have 0. 1815 01:48:30,740 --> 01:48:32,850 It's the same answer. 1816 01:48:32,850 --> 01:48:37,220 They just wanted to play games, and you can play games. 1817 01:48:37,220 --> 01:48:40,763 For example, you can make this. 1818 01:48:40,763 --> 01:48:43,450 If you really have a working mind, 1819 01:48:43,450 --> 01:48:50,510 and most mathematicians do, give this to your students. 1820 01:48:50,510 --> 01:48:54,300 I mean, most people freak out so bad 1821 01:48:54,300 --> 01:48:58,690 when they see that, the won't even touch it. 1822 01:48:58,690 --> 01:49:02,220 It's just all in the head. 1823 01:49:02,220 --> 01:49:12,620 Remember that log in base 17 of a would be what? 1824 01:49:12,620 --> 01:49:16,260 STUDENT: If it's a constant, it's to the 17th. 1825 01:49:16,260 --> 01:49:17,260 PROFESSOR: Who knows? 1826 01:49:17,260 --> 01:49:19,260 STUDENT: What do you mean, you don't do that? 1827 01:49:19,260 --> 01:49:23,440 PROFESSOR: No, no, expressed in terms of natural logs. 1828 01:49:23,440 --> 01:49:24,400 STUDENT: Natural log? 1829 01:49:24,400 --> 01:49:26,885 The natural log of a over natural log of 17. 1830 01:49:26,885 --> 01:49:27,760 PROFESSOR: Very good. 1831 01:49:27,760 --> 01:49:29,680 So what does this matter? 1832 01:49:29,680 --> 01:49:33,380 In the end, you multiply 2, you do the derivative, you still 1833 01:49:33,380 --> 01:49:34,950 get the same answer. 1834 01:49:34,950 --> 01:49:38,140 Some people are trying to make things scarier 1835 01:49:38,140 --> 01:49:40,610 than they are, just to impress. 1836 01:49:40,610 --> 01:49:44,616 When you think of the problem, it's a piece of cake. 1837 01:49:44,616 --> 01:49:47,592 So don't be afraid of it. 1838 01:49:47,592 --> 01:50:05,490 1839 01:50:05,490 --> 01:50:16,570 Oh, by the way, the final exam-- so the midterm would 1840 01:50:16,570 --> 01:50:24,114 be 10 problems pus 1 extra one. 1841 01:50:24,114 --> 01:50:27,090 1842 01:50:27,090 --> 01:50:29,210 And did I tell you how much time? 1843 01:50:29,210 --> 01:50:34,750 It's going to be approximately-- I say, in actual time. 1844 01:50:34,750 --> 01:50:35,380 Needed time. 1845 01:50:35,380 --> 01:50:39,520 1846 01:50:39,520 --> 01:50:43,630 For average student, it'll be about 40 minutes. 1847 01:50:43,630 --> 01:50:53,244 Allowed time one hour and 40 minutes. 1848 01:50:53,244 --> 01:50:57,600 So you have from 12:10 to 1:50. 1849 01:50:57,600 --> 01:51:01,000 1850 01:51:01,000 --> 01:51:06,095 On the final, just a guess, about 15-16 problems. 1851 01:51:06,095 --> 01:51:09,795 1852 01:51:09,795 --> 01:51:11,820 Two hours and a half. 1853 01:51:11,820 --> 01:51:15,982 1854 01:51:15,982 --> 01:51:18,380 STUDENT: Is that allowed time? 1855 01:51:18,380 --> 01:51:21,446 PROFESSOR: Not allowed time. 1856 01:51:21,446 --> 01:51:24,650 If I manage to review very well with you on these concepts 1857 01:51:24,650 --> 01:51:29,710 guys, I guarantee you're not going to need more than 1.5. 1858 01:51:29,710 --> 01:51:31,452 This is the allowed time. 1859 01:51:31,452 --> 01:51:34,405 The allowed time for somebody who hasn't practiced enough. 1860 01:51:34,405 --> 01:51:38,710 1861 01:51:38,710 --> 01:51:41,000 Let me ask you what you think would be good. 1862 01:51:41,000 --> 01:51:44,220 1863 01:51:44,220 --> 01:51:45,894 I have a bunch of finals. 1864 01:51:45,894 --> 01:51:50,410 All the finals for Cal 3 look very similar in nature. 1865 01:51:50,410 --> 01:51:54,250 The same kind of topics as the ones I review. 1866 01:51:54,250 --> 01:51:57,360 I would like to know what you would prefer. 1867 01:51:57,360 --> 01:52:02,988 I would have two or three finals to give you. 1868 01:52:02,988 --> 01:52:05,840 Would you prefer that you try them yourselves first, 1869 01:52:05,840 --> 01:52:08,050 and then I give you the solutions? 1870 01:52:08,050 --> 01:52:08,840 STUDENT: Yes. 1871 01:52:08,840 --> 01:52:11,260 PROFESSOR: Or I give you the solutions from the beginning? 1872 01:52:11,260 --> 01:52:17,397 I'll give you the solutions anyway, but-- 1873 01:52:17,397 --> 01:52:19,230 STUDENT: Can it just be on a separate sheet, 1874 01:52:19,230 --> 01:52:20,958 where we could go through-- 1875 01:52:20,958 --> 01:52:23,770 PROFESSOR: No, no, they are already on a separate sheet. 1876 01:52:23,770 --> 01:52:28,437 For example, I have Fall 2013, or Spring 2012. 1877 01:52:28,437 --> 01:52:30,200 They are from different semesters. 1878 01:52:30,200 --> 01:52:31,480 They are all very similar. 1879 01:52:31,480 --> 01:52:35,290 So I'll give you-- I have two files on this blog. 1880 01:52:35,290 --> 01:52:37,460 The exam itself and the solutions. 1881 01:52:37,460 --> 01:52:43,039 I'll give you the exam, I'll let you work if for two weeks, 1882 01:52:43,039 --> 01:52:44,580 and then I'll give you the solutions. 1883 01:52:44,580 --> 01:52:45,600 How about that? 1884 01:52:45,600 --> 01:52:48,734 Put you'll work on it, you don't cheat on me and any way. 1885 01:52:48,734 --> 01:52:52,192 Because working things yourself, you're learning. 1886 01:52:52,192 --> 01:52:55,650 If you expect other people to feed you the solutions, 1887 01:52:55,650 --> 01:52:57,626 you're not learning as much. 1888 01:52:57,626 --> 01:53:01,578 You are learning some, but you're not learning as much. 1889 01:53:01,578 --> 01:53:03,554 OK, it's getting ready. 1890 01:53:03,554 --> 01:53:05,530 I have a few more things to tell you. 1891 01:53:05,530 --> 01:53:19,856 1892 01:53:19,856 --> 01:53:22,190 Chapter 13, necessary reminders. 1893 01:53:22,190 --> 01:53:30,884 1894 01:53:30,884 --> 01:53:34,760 The gradient is very important. 1895 01:53:34,760 --> 01:53:45,790 Gradient of a function f from r 2 to r. 1896 01:53:45,790 --> 01:53:51,780 We write that as z equals f of x and y, usually. 1897 01:53:51,780 --> 01:53:53,132 And what was the gradient? 1898 01:53:53,132 --> 01:53:55,060 This is good review for the midterm, 1899 01:53:55,060 --> 01:53:58,950 but that's the beginning of section 13.1. 1900 01:53:58,950 --> 01:54:01,390 So I'm actually doing two things, 1901 01:54:01,390 --> 01:54:03,860 I'm giving you the beginning of section 13.1, 1902 01:54:03,860 --> 01:54:06,520 while doing review for the final. 1903 01:54:06,520 --> 01:54:12,020 1904 01:54:12,020 --> 01:54:18,155 You have gradient of f of x, y-- some people are ask me, 1905 01:54:18,155 --> 01:54:22,440 do you prefer that I write on the exams, 1906 01:54:22,440 --> 01:54:27,510 on the midterm, on the final a granular bracket? 1907 01:54:27,510 --> 01:54:34,570 Or do you prefer I write this in this form in the standard base 1908 01:54:34,570 --> 01:54:35,230 i, j. 1909 01:54:35,230 --> 01:54:36,760 Standard [INAUDIBLE]. 1910 01:54:36,760 --> 01:54:38,530 It doesn't make a difference. 1911 01:54:38,530 --> 01:54:40,750 In linear algebra, you would have 1912 01:54:40,750 --> 01:54:43,370 to say what bases you are using. 1913 01:54:43,370 --> 01:54:45,990 But in calculus, we assume that you are using 1914 01:54:45,990 --> 01:54:48,460 the bases which is 1, 0, 0, 1. 1915 01:54:48,460 --> 01:54:51,110 1916 01:54:51,110 --> 01:54:55,920 So you have space in a plane. 1917 01:54:55,920 --> 01:54:58,870 I'm indifferent. 1918 01:54:58,870 --> 01:55:01,730 This is OK, you can use whatever you like. 1919 01:55:01,730 --> 01:55:05,554 If you have a function of three variables, of course 1920 01:55:05,554 --> 01:55:06,710 you have a gradient. 1921 01:55:06,710 --> 01:55:09,760 1922 01:55:09,760 --> 01:55:16,410 But I prefer to write f sub x, i plus f sub y, j plus f sub z, 1923 01:55:16,410 --> 01:55:19,160 the beginning of some ck. 1924 01:55:19,160 --> 01:55:31,446 1925 01:55:31,446 --> 01:55:33,901 Has anybody heard of divergence before? 1926 01:55:33,901 --> 01:55:35,865 What is divergence? 1927 01:55:35,865 --> 01:55:37,829 Gradient is something you've heard before. 1928 01:55:37,829 --> 01:55:40,775 But divergence, have you ever heard of divergence? 1929 01:55:40,775 --> 01:55:47,010 1930 01:55:47,010 --> 01:55:58,950 Maybe in mechanical engineering, have you heard of it before? 1931 01:55:58,950 --> 01:56:00,890 No? 1932 01:56:00,890 --> 01:56:02,256 OK. 1933 01:56:02,256 --> 01:56:05,637 Suppose that you have a function, 1934 01:56:05,637 --> 01:56:08,052 and that is a vector value function. 1935 01:56:08,052 --> 01:56:13,296 1936 01:56:13,296 --> 01:56:14,610 What does it mean? 1937 01:56:14,610 --> 01:56:21,410 A vector in itself will have coordinates at x, y. 1938 01:56:21,410 --> 01:56:29,000 And it's assumed that will be f1 of x, y-- no, 1939 01:56:29,000 --> 01:56:31,475 this is not a vector, that's scalar. 1940 01:56:31,475 --> 01:56:35,120 Times i, plus f2 x, y, j. 1941 01:56:35,120 --> 01:56:38,770 And somebody, one of you actually showed me-- of course 1942 01:56:38,770 --> 01:56:44,860 in mechanics-- you were using divergence in that. 1943 01:56:44,860 --> 01:56:49,355 And I feel bad that I was not the first maybe for some 1944 01:56:49,355 --> 01:56:53,220 of you, I was not the first to tell you what divergence means. 1945 01:56:53,220 --> 01:57:00,300 Divergence f, assuming that f would be missing one function. 1946 01:57:00,300 --> 01:57:01,530 What does this mean? 1947 01:57:01,530 --> 01:57:06,130 It means that it's differential, but its derivatives 1948 01:57:06,130 --> 01:57:07,004 are continuous. 1949 01:57:07,004 --> 01:57:09,670 1950 01:57:09,670 --> 01:57:13,504 We did note that this is the diff of f. 1951 01:57:13,504 --> 01:57:18,470 But in engineering, they denoted most of the time like that. 1952 01:57:18,470 --> 01:57:22,376 There's not a lot of symbols, but you saw the gradient 1953 01:57:22,376 --> 01:57:24,368 with a little dot after that. 1954 01:57:24,368 --> 01:57:29,846 1955 01:57:29,846 --> 01:57:32,475 If you don't put the dot, it doesn't make sense 1956 01:57:32,475 --> 01:57:34,240 with what I'm saying. 1957 01:57:34,240 --> 01:57:37,480 So pay attention to the dot. 1958 01:57:37,480 --> 01:57:38,140 Alright. 1959 01:57:38,140 --> 01:57:39,030 What does this mean? 1960 01:57:39,030 --> 01:57:44,240 It means that you have the derivative 1961 01:57:44,240 --> 01:57:47,564 of the first component with respect to x. 1962 01:57:47,564 --> 01:57:50,840 1963 01:57:50,840 --> 01:57:55,105 Plus it's going to be a value function, 1964 01:57:55,105 --> 01:57:58,070 the derivative of the second component with respect to y. 1965 01:57:58,070 --> 01:58:03,866 1966 01:58:03,866 --> 01:58:08,440 How do you generalize for higher powers? 1967 01:58:08,440 --> 01:58:18,071 What if you have a function-- assume you have a function 1968 01:58:18,071 --> 01:58:20,027 f that looks like that. 1969 01:58:20,027 --> 01:58:30,628 If x1, x2, x n variables, i plus the last one will be 1970 01:58:30,628 --> 01:58:35,226 a [INAUDIBLE] of x1, x2 x n variables times-- 1971 01:58:35,226 --> 01:58:40,374 1972 01:58:40,374 --> 01:58:42,322 eij doesn't make any sense. 1973 01:58:42,322 --> 01:58:46,705 So e1, e2, e n would be the standard bases. 1974 01:58:46,705 --> 01:58:54,510 [INAUDIBLE] doesn't make the [INAUDIBLE] 1975 01:58:54,510 --> 01:59:02,940 for a computer scientist, an ordered set of components 1976 01:59:02,940 --> 01:59:04,810 and values. 1977 01:59:04,810 --> 01:59:08,300 And would be 7, 17, 29. 1978 01:59:08,300 --> 01:59:10,100 Some natural numbers. 1979 01:59:10,100 --> 01:59:18,848 So all these values are taken in r, with every r x is in on. 1980 01:59:18,848 --> 01:59:21,830 What do you think that the divergence of u 1981 01:59:21,830 --> 01:59:24,315 would be in that case? 1982 01:59:24,315 --> 01:59:27,794 If you were to generalize y. 1983 01:59:27,794 --> 01:59:32,764 First component, prime with respect to the first variable. 1984 01:59:32,764 --> 01:59:35,750 Alright. 1985 01:59:35,750 --> 01:59:41,740 Only plus second component with respect to the second variable, 1986 01:59:41,740 --> 01:59:43,350 and so on. 1987 01:59:43,350 --> 01:59:46,010 Last component with respect to the last variable. 1988 01:59:46,010 --> 01:59:49,448 So that would be the general definition. 1989 01:59:49,448 --> 01:59:58,485 And now I'm asking you, assume x that somebody gives you 1990 01:59:58,485 --> 02:00:03,210 a function, f of x, y. 1991 02:00:03,210 --> 02:00:04,805 And with domain in the plane. 1992 02:00:04,805 --> 02:00:10,130 1993 02:00:10,130 --> 02:00:14,180 And f is c1. 1994 02:00:14,180 --> 02:00:18,200 [INAUDIBLE] with continuous radius. 1995 02:00:18,200 --> 02:00:20,180 Actually no, I want more. 1996 02:00:20,180 --> 02:00:23,160 I want c2. 1997 02:00:23,160 --> 02:00:34,130 So twice differential bond, with continuous variables. 1998 02:00:34,130 --> 02:00:39,110 1999 02:00:39,110 --> 02:00:45,230 Compute a. 2000 02:00:45,230 --> 02:00:46,041 Gradient double f. 2001 02:00:46,041 --> 02:00:48,750 2002 02:00:48,750 --> 02:00:53,543 b, divergence of gradient of f, which you can also 2003 02:00:53,543 --> 02:01:00,280 write divergence like engineers do, or gradient to our left. 2004 02:01:00,280 --> 02:01:07,662 Do you know what name that's the last thing we need today, 2005 02:01:07,662 --> 02:01:14,798 the name for this operator. 2006 02:01:14,798 --> 02:01:15,782 Underlined here. 2007 02:01:15,782 --> 02:01:21,686 2008 02:01:21,686 --> 02:01:25,100 So what would be a good name for this kind? 2009 02:01:25,100 --> 02:01:29,160 I'm curious if any of you know if from engineering. 2010 02:01:29,160 --> 02:01:30,615 But we will see. 2011 02:01:30,615 --> 02:01:39,840 2012 02:01:39,840 --> 02:01:44,330 So we are in 13. 2013 02:01:44,330 --> 02:01:50,689 a will be the gradient of f, that's a piece of cake. 2014 02:01:50,689 --> 02:01:52,105 She only wants the definition, let 2015 02:01:52,105 --> 02:01:57,520 me give her the definition of f sub xi, plus f sub y j. 2016 02:01:57,520 --> 02:01:59,020 And if we don't know what those are, 2017 02:01:59,020 --> 02:02:02,401 this is the variable with respect to x. 2018 02:02:02,401 --> 02:02:07,721 And then for dy, df/dy j. 2019 02:02:07,721 --> 02:02:08,220 Good. 2020 02:02:08,220 --> 02:02:11,580 So we know what a gradient is. 2021 02:02:11,580 --> 02:02:15,468 What will this divergence with the gradient be? 2022 02:02:15,468 --> 02:02:17,412 That sounds really weird. 2023 02:02:17,412 --> 02:02:21,790 2024 02:02:21,790 --> 02:02:25,315 According to this definition, we have 2025 02:02:25,315 --> 02:02:30,080 to see what big F1 and big F2 are. 2026 02:02:30,080 --> 02:02:32,564 Or, big F1 and big F2. 2027 02:02:32,564 --> 02:02:35,546 I'm going to take them in breaths. 2028 02:02:35,546 --> 02:02:38,528 Big F1, and big F2. 2029 02:02:38,528 --> 02:02:42,010 2030 02:02:42,010 --> 02:02:48,071 The components of the vector, you apply divergence to it. 2031 02:02:48,071 --> 02:02:50,070 So now that I'm finishing, what do I have to do? 2032 02:02:50,070 --> 02:02:53,270 Somebody tell me. 2033 02:02:53,270 --> 02:02:57,570 So yeah, I can write it f sub x plus f sub y, 2034 02:02:57,570 --> 02:03:03,470 and that shows that you are fast, and very [INAUDIBLE]. 2035 02:03:03,470 --> 02:03:05,600 I can also write it like this, which 2036 02:03:05,600 --> 02:03:11,377 is what I meant-- this is what the book shows first course. 2037 02:03:11,377 --> 02:03:12,830 This is the same thing. 2038 02:03:12,830 --> 02:03:15,380 2039 02:03:15,380 --> 02:03:21,050 Now I really doubt that somebody knows that, 2040 02:03:21,050 --> 02:03:25,332 but I want to give a dollar to the person who 2041 02:03:25,332 --> 02:03:27,065 would know the name of this. 2042 02:03:27,065 --> 02:03:30,035 2043 02:03:30,035 --> 02:03:33,005 Let me see if I have a dollar. 2044 02:03:33,005 --> 02:03:37,960 2045 02:03:37,960 --> 02:03:40,876 Maybe I have $0.35 and a candy. 2046 02:03:40,876 --> 02:03:44,334 Does anybody know the name of this? 2047 02:03:44,334 --> 02:03:55,144 Maybe I can help you a little bit. $0.25 $0.85, $0.95. 2048 02:03:55,144 --> 02:03:57,090 Do you know what this is? 2049 02:03:57,090 --> 02:04:01,460 I'll give you a hint, because I know in mechanical engineering, 2050 02:04:01,460 --> 02:04:03,030 I already introduced this. 2051 02:04:03,030 --> 02:04:08,965 And some physics classes and we would try angle in front, 2052 02:04:08,965 --> 02:04:15,020 and we did all of this triangle operators in the way. 2053 02:04:15,020 --> 02:04:18,972 And we can play a game. 2054 02:04:18,972 --> 02:04:26,890 It's a letter that starts with L. 2055 02:04:26,890 --> 02:04:30,460 But $0.95 we have two more minutes. 2056 02:04:30,460 --> 02:04:32,697 STUDENT: [INAUDIBLE]? 2057 02:04:32,697 --> 02:04:33,280 PROFESSOR: No. 2058 02:04:33,280 --> 02:04:38,262 You are getting close though, because-- [INTERPOSING VOICES] 2059 02:04:38,262 --> 02:04:41,226 What kind of operator is this? 2060 02:04:41,226 --> 02:04:43,980 You're getting close, $0.95. 2061 02:04:43,980 --> 02:04:46,435 Tomorrow, I don't need this. 2062 02:04:46,435 --> 02:04:47,860 When I go to the airports, I don't 2063 02:04:47,860 --> 02:04:49,470 like to have coins with me. 2064 02:04:49,470 --> 02:04:56,990 2065 02:04:56,990 --> 02:04:58,866 STUDENT: Laplace? 2066 02:04:58,866 --> 02:04:59,574 PROFESSOR: $0.95! 2067 02:04:59,574 --> 02:05:02,054 I wish I had a dollar. 2068 02:05:02,054 --> 02:05:04,534 Yes, this is the famous Laplace operator. 2069 02:05:04,534 --> 02:05:06,022 Laplace was a mathematician. 2070 02:05:06,022 --> 02:05:11,500 2071 02:05:11,500 --> 02:05:13,570 And remember it. 2072 02:05:13,570 --> 02:05:16,610 If you take-- how many of you-- you all 2073 02:05:16,610 --> 02:05:19,790 have to take differential equations, right? 2074 02:05:19,790 --> 02:05:21,510 They will kill you with that. 2075 02:05:21,510 --> 02:05:23,940 You're going to see this all the time. 2076 02:05:23,940 --> 02:05:25,907 This Laplace operator is really famous. 2077 02:05:25,907 --> 02:05:28,650 2078 02:05:28,650 --> 02:05:31,340 I will tell you more when I come back. 2079 02:05:31,340 --> 02:05:34,530 I'm going to see you on Tuesday. 2080 02:05:34,530 --> 02:05:36,860 We'll knock out the midterm. 2081 02:05:36,860 --> 02:05:40,600 For you, the people who feel overly prepared for midterm 2082 02:05:40,600 --> 02:05:45,540 can go ahead and read section 13.1 2083 02:05:45,540 --> 02:05:48,840 and see a little bit about Laplace's operator. 2084 02:05:48,840 --> 02:05:51,290 [INTERPOSING VOICES] 2085 02:05:51,290 --> 02:06:45,857