1 00:00:00,000 --> 00:00:00,499 2 00:00:00,499 --> 00:00:04,934 PROFESSOR: We will pick up from where we left. 3 00:00:04,934 --> 00:00:08,406 I hope the attendance will get a little bit better today. 4 00:00:08,406 --> 00:00:13,862 It's not even Friday, it's Thursday night. 5 00:00:13,862 --> 00:00:17,334 So last time we talked a little bit about chapter 13, 6 00:00:17,334 --> 00:00:21,302 we started 13-1. 7 00:00:21,302 --> 00:00:27,750 I wanted to remind you that we revisited the notion of work. 8 00:00:27,750 --> 00:00:40,700 9 00:00:40,700 --> 00:00:44,226 Now, if you notice what the book does, 10 00:00:44,226 --> 00:00:45,980 it doesn't give you any specifics 11 00:00:45,980 --> 00:00:48,860 about the force field. 12 00:00:48,860 --> 00:00:50,550 May the force be with you. 13 00:00:50,550 --> 00:00:56,031 They don't say what kind of animal this f is. 14 00:00:56,031 --> 00:01:02,570 We sort of informally said I'm going to have 15 00:01:02,570 --> 00:01:06,290 some sort of path integral. 16 00:01:06,290 --> 00:01:10,880 And I didn't say what conditions I was assuming about f. 17 00:01:10,880 --> 00:01:15,030 And I just said that r is the position vector. 18 00:01:15,030 --> 00:01:18,170 19 00:01:18,170 --> 00:01:22,825 It's important for us to imagine that is plus c1, what 20 00:01:22,825 --> 00:01:24,290 does that mean, c1? 21 00:01:24,290 --> 00:01:30,200 It means that this function, let's write it R of t, equals. 22 00:01:30,200 --> 00:01:35,360 Let's say we are implying not in space, so we have x of t, 23 00:01:35,360 --> 00:01:40,450 y of t, the parametrization of this position vector. 24 00:01:40,450 --> 00:01:42,900 Of course we wrote that last time as well, we 25 00:01:42,900 --> 00:01:46,280 said x is x of t, y is y of t. 26 00:01:46,280 --> 00:01:51,640 But why I took c1 and not continuous? 27 00:01:51,640 --> 00:01:53,240 Could anybody tell me? 28 00:01:53,240 --> 00:01:56,290 If I'm going to go ahead and differentiate it, 29 00:01:56,290 --> 00:01:59,290 of course I'd like it to be differentiable. 30 00:01:59,290 --> 00:02:03,520 And its derivatives should be continuous. 31 00:02:03,520 --> 00:02:07,660 But that's actually not enough for my purposes. 32 00:02:07,660 --> 00:02:11,170 So if I want R of t to be c1, that's good, 33 00:02:11,170 --> 00:02:13,150 I'm going to smile. 34 00:02:13,150 --> 00:02:18,082 But when we did that in chapter-- was it chapter 10? 35 00:02:18,082 --> 00:02:20,560 It was chapter 10, Erin, am I right? 36 00:02:20,560 --> 00:02:23,615 We assumed this was a regular curve. 37 00:02:23,615 --> 00:02:27,650 A regular curve is not just as differentiable 38 00:02:27,650 --> 00:02:31,390 with the derivative's continuous with respect to time. 39 00:02:31,390 --> 00:02:34,380 x prime of t, y prime of t, both must exist 40 00:02:34,380 --> 00:02:36,030 and must be continuous. 41 00:02:36,030 --> 00:02:39,850 We wanted something else about the velocity. 42 00:02:39,850 --> 00:02:42,200 Do you remember the drunken bug? 43 00:02:42,200 --> 00:02:46,520 The drunken but was fine and he was flying. 44 00:02:46,520 --> 00:02:49,820 As long as he was flying, everything was fine. 45 00:02:49,820 --> 00:02:54,080 When did the drunken bug have a problem? 46 00:02:54,080 --> 00:02:56,895 When the velocity field became 0, 47 00:02:56,895 --> 00:03:02,860 at the instant where the bug lost his velocity, right? 48 00:03:02,860 --> 00:03:10,740 So we said regular means c1 and R prime of t at any value of t 49 00:03:10,740 --> 00:03:13,520 should be different from 0. 50 00:03:13,520 --> 00:03:16,620 We do not allow the particle to stop on it's way. 51 00:03:16,620 --> 00:03:20,680 We don't allow it, whether it is a photon, a drunken bug, 52 00:03:20,680 --> 00:03:23,690 an airplane, or whatever it is. 53 00:03:23,690 --> 00:03:27,570 We don't want it to stop in it's trajectory. 54 00:03:27,570 --> 00:03:31,990 Is that good for other reasons as well? 55 00:03:31,990 --> 00:03:35,260 Very good for the reason that we want 56 00:03:35,260 --> 00:03:39,150 to think later in arc length. 57 00:03:39,150 --> 00:03:43,320 [INAUDIBLE] came up with this idea last time. 58 00:03:43,320 --> 00:03:46,480 I didn't want to tell you the truth, but he was right. 59 00:03:46,480 --> 00:03:53,010 One can define certain path integrals with respect 60 00:03:53,010 --> 00:03:56,540 to s, with respect to arc length parameter. 61 00:03:56,540 --> 00:03:58,400 But as you remember very well, he 62 00:03:58,400 --> 00:04:03,260 had this correspondence between an arbitrary parameter type t 63 00:04:03,260 --> 00:04:07,641 and s, and this is s of t. 64 00:04:07,641 --> 00:04:10,200 And also going back and forth, that 65 00:04:10,200 --> 00:04:15,370 means from s you head back to t. 66 00:04:15,370 --> 00:04:19,820 So here's s of t and this is t of s, right? 67 00:04:19,820 --> 00:04:26,500 So we have this correspondence and everything worked fine 68 00:04:26,500 --> 00:04:31,450 in terms of being able to invert that. 69 00:04:31,450 --> 00:04:34,320 And having some sort of equal morphisms 70 00:04:34,320 --> 00:04:40,620 as long as the velocity was non-0. 71 00:04:40,620 --> 00:04:43,720 OK, do you remember who s of t was? 72 00:04:43,720 --> 00:04:47,900 S of t was defined-- It was a long time ago. 73 00:04:47,900 --> 00:04:50,710 So I'm reminding you s of t was integral 74 00:04:50,710 --> 00:04:55,350 from 0 to t-- or from t0 to t. 75 00:04:55,350 --> 00:04:59,200 Your favorite initial moment in time. 76 00:04:59,200 --> 00:05:03,570 Of the speed, uh-huh, and what the heck was the speed? 77 00:05:03,570 --> 00:05:10,660 The speed was the norm or the length of the R prime of t. 78 00:05:10,660 --> 00:05:16,646 This is called speed, that we assume different from 0, 79 00:05:16,646 --> 00:05:19,160 for a good purpose. 80 00:05:19,160 --> 00:05:24,600 We can go back and forth between t and s, t to s, 81 00:05:24,600 --> 00:05:30,440 s to t, with differentiable functions. 82 00:05:30,440 --> 00:05:35,300 Good, so now we can apply the inverse mapping through them. 83 00:05:35,300 --> 00:05:38,170 We can do all sorts of stuff with that. 84 00:05:38,170 --> 00:05:42,430 On this one we did not quite define it rigorously. 85 00:05:42,430 --> 00:05:43,690 What did they say is? 86 00:05:43,690 --> 00:05:46,490 We said f would be a good enough function, 87 00:05:46,490 --> 00:05:50,330 but know that I do not need f to be c1. 88 00:05:50,330 --> 00:05:54,424 This is too strong, too strong. 89 00:05:54,424 --> 00:05:58,870 So in Calc 1 when you had to integrate a function of one 90 00:05:58,870 --> 00:06:02,972 variable you just assumed that- in Calc 1 91 00:06:02,972 --> 00:06:04,700 I remember- you assume that continuous. 92 00:06:04,700 --> 00:06:07,580 It doesn't even have to be continuous 93 00:06:07,580 --> 00:06:09,862 but let's assume that f would be continuous. 94 00:06:09,862 --> 00:06:15,650 95 00:06:15,650 --> 00:06:21,105 OK, so you have, in one sense, that the composition with R, 96 00:06:21,105 --> 00:06:27,006 if you have f of x of t, y of t, z of t. 97 00:06:27,006 --> 00:06:33,600 In terms of time will be a functions of one variable, 98 00:06:33,600 --> 00:06:35,580 and this will be continuous. 99 00:06:35,580 --> 00:06:39,060 100 00:06:39,060 --> 00:06:41,050 All right? 101 00:06:41,050 --> 00:06:44,610 OK, now what if it's not continuous? 102 00:06:44,610 --> 00:06:47,850 Can't I have a piecewise, continuous function? 103 00:06:47,850 --> 00:06:51,425 Like in Calc 1, do you guys remember we had some of this? 104 00:06:51,425 --> 00:06:54,170 And from here like that and from here like this 105 00:06:54,170 --> 00:06:55,170 and from here like that. 106 00:06:55,170 --> 00:06:58,000 And we had these continuities, and this was piecewise 107 00:06:58,000 --> 00:06:59,860 continuous. 108 00:06:59,860 --> 00:07:02,660 Yeah, for god sake, I can integrate that. 109 00:07:02,660 --> 00:07:06,530 Why do we assume integral of a continuous function? 110 00:07:06,530 --> 00:07:08,680 Just to make our lives easier and also 111 00:07:08,680 --> 00:07:12,500 because we are in freshman and sophomore level Calculus. 112 00:07:12,500 --> 00:07:16,690 If we were in advanced Calculus we would say, 113 00:07:16,690 --> 00:07:21,650 I want this function to be integrable. 114 00:07:21,650 --> 00:07:24,540 This is a lot weaker than continuous, 115 00:07:24,540 --> 00:07:28,360 maybe the set of discontinuities is also very large. 116 00:07:28,360 --> 00:07:31,510 Who told you that you have finitely many jumps 117 00:07:31,510 --> 00:07:32,590 these continuities? 118 00:07:32,590 --> 00:07:34,530 Maybe you have a much larger set. 119 00:07:34,530 --> 00:07:39,070 And this is what you learn in advanced Calculus. 120 00:07:39,070 --> 00:07:41,710 But you are not at the level of a senior 121 00:07:41,710 --> 00:07:44,880 yet so we'll just assume, for the time being, 122 00:07:44,880 --> 00:07:47,070 that f is continuous. 123 00:07:47,070 --> 00:07:49,480 All right, and we say, what is this animal? 124 00:07:49,480 --> 00:07:52,330 We called it w and be baptised it. 125 00:07:52,330 --> 00:07:56,880 We said, just give it some sort of name 126 00:07:56,880 --> 00:07:58,550 and we say that is work. 127 00:07:58,550 --> 00:08:02,600 And by definition, by definition, 128 00:08:02,600 --> 00:08:09,375 this is going to be integral from-- Now, 129 00:08:09,375 --> 00:08:14,890 the thing is, we define this as a simple integral with respect 130 00:08:14,890 --> 00:08:17,900 to time as a definition. 131 00:08:17,900 --> 00:08:19,830 That doesn't mean that I introduced 132 00:08:19,830 --> 00:08:24,660 the notion of path integral the way I should, 133 00:08:24,660 --> 00:08:26,340 I was cheating on that. 134 00:08:26,340 --> 00:08:28,550 So the way we introduced it was like, 135 00:08:28,550 --> 00:08:33,100 let f be a function of the spatial coordinates 136 00:08:33,100 --> 00:08:34,650 in terms of time. 137 00:08:34,650 --> 00:08:37,789 x, y, z are space coordinates, t is time. 138 00:08:37,789 --> 00:08:42,068 So I have f of R of t here. 139 00:08:42,068 --> 00:08:45,443 Dot Product, who the heck is the R? 140 00:08:45,443 --> 00:08:48,280 This is nothing but a vector art drawing. 141 00:08:48,280 --> 00:08:50,510 These are both vectors, sometimes 142 00:08:50,510 --> 00:08:53,440 I should put them in bold like they do in the book. 143 00:08:53,440 --> 00:08:56,460 To make it clear I can put a bar on top of them, 144 00:08:56,460 --> 00:08:57,890 they are free vectors. 145 00:08:57,890 --> 00:09:02,990 So, f of R times R prime of t dt. 146 00:09:02,990 --> 00:09:07,180 And your favorite moments of time are-- let's say on my arc 147 00:09:07,180 --> 00:09:12,270 that I'm describing from time, t, equals a, to time equals b. 148 00:09:12,270 --> 00:09:15,220 Therefore, I'm going to take time for a to b. 149 00:09:15,220 --> 00:09:19,535 And this is how we define the work of a force. 150 00:09:19,535 --> 00:09:22,786 151 00:09:22,786 --> 00:09:25,610 The work of a force that's acting 152 00:09:25,610 --> 00:09:32,770 on a particle that is moving between time, a, and time, b, 153 00:09:32,770 --> 00:09:36,890 on this arc of a curve which is called c. 154 00:09:36,890 --> 00:09:38,273 Do you like this c? 155 00:09:38,273 --> 00:09:41,500 Okay, and the force is different. 156 00:09:41,500 --> 00:09:44,055 So we have a force field. 157 00:09:44,055 --> 00:09:46,360 So I cheated, I knew a lot in the sense 158 00:09:46,360 --> 00:09:48,510 that I didn't tell you how you actually 159 00:09:48,510 --> 00:09:55,280 introduce the path integral. 160 00:09:55,280 --> 00:10:01,220 Now this is more or less where I stop and [INAUDIBLE]. 161 00:10:01,220 --> 00:10:04,650 But couldn't we actually introduce this integral 162 00:10:04,650 --> 00:10:11,320 and even define it with respect to some arc length grammar? 163 00:10:11,320 --> 00:10:14,580 Maybe if everything goes fine in terms of theory? 164 00:10:14,580 --> 00:10:16,250 And the answer is yes. 165 00:10:16,250 --> 00:10:20,100 And I'm going to show you how one can do that. 166 00:10:20,100 --> 00:10:25,670 I'm going to go ahead and clean here a little bit. 167 00:10:25,670 --> 00:10:29,170 I'm going to leave this on by comparison for awhile. 168 00:10:29,170 --> 00:10:33,510 And then I will assume something that we have not 169 00:10:33,510 --> 00:10:40,044 defined whatsoever, which is an animal called path integral. 170 00:10:40,044 --> 00:10:54,260 So the path integral of a vector field along a trajectory, c. 171 00:10:54,260 --> 00:10:55,790 I don't know how to draw. 172 00:10:55,790 --> 00:10:58,630 I will draw some skewed curve, how about that? 173 00:10:58,630 --> 00:11:04,130 Some pretty skewed curve, c, it's not self intersecting, 174 00:11:04,130 --> 00:11:05,360 not necessarily. 175 00:11:05,360 --> 00:11:08,770 You guys have to imagine this is like the trajectory 176 00:11:08,770 --> 00:11:13,220 of an airplane in the sky, right? 177 00:11:13,220 --> 00:11:17,740 OK, and I have it on d equals a, to d equals b. 178 00:11:17,740 --> 00:11:21,150 But I said forget about the time, t, 179 00:11:21,150 --> 00:11:24,520 maybe I can do everything in arc length forever. 180 00:11:24,520 --> 00:11:26,320 So if that particle, or airplane, 181 00:11:26,320 --> 00:11:31,120 or whatever it is has a continuous motion, 182 00:11:31,120 --> 00:11:32,750 that's also differentiable. 183 00:11:32,750 --> 00:11:35,710 And the velocity never becomes zero. 184 00:11:35,710 --> 00:11:38,650 Then I can parametrize an arc length 185 00:11:38,650 --> 00:11:44,160 and I can say, forget about it, I have integral over c. 186 00:11:44,160 --> 00:11:46,970 See, this is c, it's not f, okay? 187 00:11:46,970 --> 00:11:54,752 But f of x of s, y of s, z of s, okay? 188 00:11:54,752 --> 00:11:57,570 And this is going to be a ds. 189 00:11:57,570 --> 00:12:00,690 And you'll say, yes Magdelina-- this is little s, I'm sorry. 190 00:12:00,690 --> 00:12:03,310 Yes, Magdelina, but what the heck is this animal, 191 00:12:03,310 --> 00:12:05,040 you've never introduced it. 192 00:12:05,040 --> 00:12:09,000 I have not introduced it because I have to discuss about it. 193 00:12:09,000 --> 00:12:14,740 When we introduced Riemann sums, then we took the limit. 194 00:12:14,740 --> 00:12:19,890 We always have to think how to partition our domains. 195 00:12:19,890 --> 00:12:30,620 So this curve can be partitioned in as many as n, this is s k. 196 00:12:30,620 --> 00:12:36,220 S k, this is s1, and this is s n, the last of the Mohicans. 197 00:12:36,220 --> 00:12:42,170 I have n sub intervals, pieces of the art. 198 00:12:42,170 --> 00:12:44,410 And how am I going to introduce this? 199 00:12:44,410 --> 00:12:47,220 As the limit, if it exists. 200 00:12:47,220 --> 00:12:49,480 Because I can be in trouble, maybe this limit 201 00:12:49,480 --> 00:12:51,580 is not going to exist. 202 00:12:51,580 --> 00:12:54,220 The sum of what? 203 00:12:54,220 --> 00:12:58,760 For every [? seg ?] partition I will take a little arbitrary 204 00:12:58,760 --> 00:13:00,190 point inside the subarc. 205 00:13:00,190 --> 00:13:03,150 206 00:13:03,150 --> 00:13:03,783 Subarc? 207 00:13:03,783 --> 00:13:04,550 STUDENT: Yeah. 208 00:13:04,550 --> 00:13:06,320 PROFESSOR: Subarc, it's a little arc. 209 00:13:06,320 --> 00:13:09,450 Contains a-- let's take it here. 210 00:13:09,450 --> 00:13:13,000 What am I going to define in terms of wind? 211 00:13:13,000 --> 00:13:20,726 s k, y k, and z k, some people put a star on it 212 00:13:20,726 --> 00:13:23,890 to make it obvious. 213 00:13:23,890 --> 00:13:27,040 But I'm going to go ahead and say 214 00:13:27,040 --> 00:13:35,600 x star k, y star k, z star k, is my arbitrary point in the k 215 00:13:35,600 --> 00:13:38,140 subarc. 216 00:13:38,140 --> 00:13:42,095 Times, what shall I multiply by? 217 00:13:42,095 --> 00:13:48,110 A delta sk, and then I take k from one to n 218 00:13:48,110 --> 00:13:52,140 and I press to the limit with respect n. 219 00:13:52,140 --> 00:13:57,150 But actually I could also say in some other ways 220 00:13:57,150 --> 00:14:05,170 that the partitions length goes to 0, delta s goes to zero. 221 00:14:05,170 --> 00:14:07,050 And you say but, now wait a minute, 222 00:14:07,050 --> 00:14:11,600 you have s1, s2 s3, s4, s k, little tiny subarc, 223 00:14:11,600 --> 00:14:13,416 what the heck is delta s? 224 00:14:13,416 --> 00:14:21,820 Delta s is the largest subarc. 225 00:14:21,820 --> 00:14:25,980 So the length of the largest subarc, length of the largest 226 00:14:25,980 --> 00:14:29,180 subarc in the partition. 227 00:14:29,180 --> 00:14:34,274 So the more points I take, the more I refine this. 228 00:14:34,274 --> 00:14:36,190 I take the points closer and closer and closer 229 00:14:36,190 --> 00:14:37,600 in this partition. 230 00:14:37,600 --> 00:14:41,010 What happens to the length of this partition? 231 00:14:41,010 --> 00:14:43,850 It shrinks to-- it goes to 0. 232 00:14:43,850 --> 00:14:45,960 Assuming that this would be the largest 233 00:14:45,960 --> 00:14:48,690 one, well if the largest one goes to 0, 234 00:14:48,690 --> 00:14:51,830 everybody else goes to 0. 235 00:14:51,830 --> 00:14:55,750 So this is a Riemann sum, can we know for sure 236 00:14:55,750 --> 00:14:57,970 that this limit exists? 237 00:14:57,970 --> 00:15:02,690 No, we hope to god that this limit exists. 238 00:15:02,690 --> 00:15:08,150 And if the limit exists then I will introduce this notion 239 00:15:08,150 --> 00:15:09,773 of integral around the back. 240 00:15:09,773 --> 00:15:15,450 241 00:15:15,450 --> 00:15:18,100 And you said, OK I believe you, but look, 242 00:15:18,100 --> 00:15:22,120 what is the connection between the work- the way 243 00:15:22,120 --> 00:15:25,850 you introduced it as a simple Calculus 1 integral here- 244 00:15:25,850 --> 00:15:30,500 and this animal that looks like an alien coming from the sky. 245 00:15:30,500 --> 00:15:33,280 We don't know how to look at it. 246 00:15:33,280 --> 00:15:37,830 Actually guys it's not so bad, you do the same thing 247 00:15:37,830 --> 00:15:40,240 as you did before. 248 00:15:40,240 --> 00:15:44,180 In a sense that, s is connected to any time parameter. 249 00:15:44,180 --> 00:15:47,750 So Mr. ds says, I'm your old friend, 250 00:15:47,750 --> 00:15:55,584 trust me, I know who I am. ds was the speed times dt. 251 00:15:55,584 --> 00:16:00,640 Who can tell me if we are in R three, and we are drunken bugs, 252 00:16:00,640 --> 00:16:02,880 ds will become what? 253 00:16:02,880 --> 00:16:07,550 A long square root times dt, and what's inside here? 254 00:16:07,550 --> 00:16:10,220 I want to see if you guys are awake. 255 00:16:10,220 --> 00:16:11,654 [INTERPOSING VOICES] 256 00:16:11,654 --> 00:16:17,180 PROFESSOR: Very good, x prime of t squared, I'm so lazy 257 00:16:17,180 --> 00:16:20,875 but I'll write it down. y prime of t squared plus z prime of t 258 00:16:20,875 --> 00:16:21,685 squared. 259 00:16:21,685 --> 00:16:24,210 And this is going to be the speed. 260 00:16:24,210 --> 00:16:28,980 So I can always do that, and in this case 261 00:16:28,980 --> 00:16:33,200 this is going to become always some-- let's say from time, t0, 262 00:16:33,200 --> 00:16:36,240 to time t1. 263 00:16:36,240 --> 00:16:39,245 Some in the integrals of-- some of the limit 264 00:16:39,245 --> 00:16:40,580 points for the time. 265 00:16:40,580 --> 00:16:45,460 I'm going to have f of R of s of t, 266 00:16:45,460 --> 00:16:47,670 in the end everything will depend on t. 267 00:16:47,670 --> 00:16:49,900 And this is my face being happy. 268 00:16:49,900 --> 00:16:51,970 It's not part of the integral. 269 00:16:51,970 --> 00:16:52,710 Saying what? 270 00:16:52,710 --> 00:16:55,290 Saying that, guys, if I plug in everything 271 00:16:55,290 --> 00:16:58,565 back in terms of t- I'm more familiar to that type 272 00:16:58,565 --> 00:17:00,770 of integral- then I have what? 273 00:17:00,770 --> 00:17:04,540 Square root of-- that's the arc length element 274 00:17:04,540 --> 00:17:07,450 x prime then t squared, plus y prime then t 275 00:17:07,450 --> 00:17:12,510 squared, plus z prime then t squared, dt. 276 00:17:12,510 --> 00:17:17,790 So in the end it is-- I think the video doesn't see me 277 00:17:17,790 --> 00:17:21,569 but it heard me, presumably. 278 00:17:21,569 --> 00:17:24,220 This is our old friend from Calc 1, 279 00:17:24,220 --> 00:17:29,990 which is the simple integral with respect to t from a to b. 280 00:17:29,990 --> 00:17:34,370 OK, all right, and we believe that the work 281 00:17:34,370 --> 00:17:36,870 can be expressed like that. 282 00:17:36,870 --> 00:17:39,610 I introduced it last time, I even 283 00:17:39,610 --> 00:17:41,860 proved it on some particular cases 284 00:17:41,860 --> 00:17:45,470 last time when Alex wasn't here because, I know why. 285 00:17:45,470 --> 00:17:46,927 Were you sick? 286 00:17:46,927 --> 00:17:48,510 ALEX: I'll talk to you about it later. 287 00:17:48,510 --> 00:17:49,850 I'm a bad person. 288 00:17:49,850 --> 00:17:56,915 PROFESSOR: All right, then I'm dragging an object like, 289 00:17:56,915 --> 00:18:02,610 the f was parallel to the direction of displacement. 290 00:18:02,610 --> 00:18:04,800 And then I said the work would be 291 00:18:04,800 --> 00:18:09,150 the magnitude of f times the magnitude of the displacement. 292 00:18:09,150 --> 00:18:13,450 And then we proved that is just a particular case of this, 293 00:18:13,450 --> 00:18:15,950 we proved that last time, it was a piece of cake. 294 00:18:15,950 --> 00:18:17,860 Actually, we proved the other one. 295 00:18:17,860 --> 00:18:23,280 It proved that if force is going to be oblique and at an angle 296 00:18:23,280 --> 00:18:27,180 theta with the displacement direction, then 297 00:18:27,180 --> 00:18:31,030 the work will be the magnitude of the force times 298 00:18:31,030 --> 00:18:35,796 cosine of theta, times the magnitude of displacement, 299 00:18:35,796 --> 00:18:36,730 all right? 300 00:18:36,730 --> 00:18:42,190 And that was all an application of this beautiful warp formula. 301 00:18:42,190 --> 00:18:46,215 Let's see something more interesting from an application 302 00:18:46,215 --> 00:18:48,000 viewpoint. 303 00:18:48,000 --> 00:18:53,030 Assume that you are looking at the washer, 304 00:18:53,030 --> 00:18:56,760 you are just doing laundry. 305 00:18:56,760 --> 00:19:01,460 And you are looking at this centrifugal force. 306 00:19:01,460 --> 00:19:08,170 We have two forces, one is centripetal towards the center 307 00:19:08,170 --> 00:19:11,590 of the motion, circular motion, one is centrifugal. 308 00:19:11,590 --> 00:19:15,220 I will take a centrifugal force f, 309 00:19:15,220 --> 00:19:19,440 and I will say I want to measure at the work 310 00:19:19,440 --> 00:19:25,310 that this force is producing in the circular motion 311 00:19:25,310 --> 00:19:27,700 of my dryer. 312 00:19:27,700 --> 00:19:31,710 My poor dryer died so I had to buy another one 313 00:19:31,710 --> 00:19:33,135 and it cost me a lot of money. 314 00:19:33,135 --> 00:19:36,180 And I was thinking, such a simple thing, 315 00:19:36,180 --> 00:19:38,630 you pay hundreds of dollars on it 316 00:19:38,630 --> 00:19:41,570 but, anyway, we take some things for granted. 317 00:19:41,570 --> 00:19:45,990 318 00:19:45,990 --> 00:19:54,610 I will take the washer because the washer is a simpler 319 00:19:54,610 --> 00:19:59,780 case in the sense that the motion-- I can assume it's 320 00:19:59,780 --> 00:20:04,180 a circular motion of constant velocity. 321 00:20:04,180 --> 00:20:07,490 And let's say this is the washer. 322 00:20:07,490 --> 00:20:10,840 323 00:20:10,840 --> 00:20:16,940 And centrifugal force is acting here. 324 00:20:16,940 --> 00:20:21,060 Let's call that-- what should it be? 325 00:20:21,060 --> 00:20:28,710 Well, it's continuing the position vector 326 00:20:28,710 --> 00:20:36,440 so let's call that lambda x I, plus lambda y j. 327 00:20:36,440 --> 00:20:42,360 In the sense that it's collinear to the vector that 328 00:20:42,360 --> 00:20:46,580 starts at origin, and here is got to be x of t, y. 329 00:20:46,580 --> 00:20:50,040 X and y are the special components 330 00:20:50,040 --> 00:20:52,560 at any point on my circular motion. 331 00:20:52,560 --> 00:20:55,460 If it's a circular motion I have x 332 00:20:55,460 --> 00:20:57,550 squared plus y squared equals r squared, 333 00:20:57,550 --> 00:21:01,660 where the radius is the radius of my washer. 334 00:21:01,660 --> 00:21:06,520 You have to compute the work produced 335 00:21:06,520 --> 00:21:10,960 by the centrifugal force in one full rotation. 336 00:21:10,960 --> 00:21:14,306 It doesn't matter, I can have infinitely many rotations. 337 00:21:14,306 --> 00:21:19,030 I can have a hundred rotations, I couldn't care less. 338 00:21:19,030 --> 00:21:23,310 But assume that the motion has constant speed. 339 00:21:23,310 --> 00:21:26,650 So if I wanted, I could parametrize in our things 340 00:21:26,650 --> 00:21:30,270 but it doesn't bring a difference. 341 00:21:30,270 --> 00:21:33,710 Because guys, when this speed is already constant, 342 00:21:33,710 --> 00:21:36,330 like for the circular motion you are familiar with. 343 00:21:36,330 --> 00:21:39,760 Or the helicoidal case you are familiar with, 344 00:21:39,760 --> 00:21:44,120 you also saw the case when the speed was constant. 345 00:21:44,120 --> 00:21:47,810 Practically, you were just rescaling the time 346 00:21:47,810 --> 00:21:52,750 to get to your speed, to your time parameter s, arc length. 347 00:21:52,750 --> 00:21:54,840 So whether you work with t, or you work with s, 348 00:21:54,840 --> 00:21:58,230 it's the same thing if the speed is a constant. 349 00:21:58,230 --> 00:22:02,030 So I'm not going to use my imagination to go and do it 350 00:22:02,030 --> 00:22:02,900 with respect to s. 351 00:22:02,900 --> 00:22:05,500 I could, but I couldn't give a damn 352 00:22:05,500 --> 00:22:09,565 because I'm going to have a beautiful t that you 353 00:22:09,565 --> 00:22:13,000 are going to help me recover. 354 00:22:13,000 --> 00:22:15,640 From here, what is the parametrization 355 00:22:15,640 --> 00:22:18,696 that comes to mind? 356 00:22:18,696 --> 00:22:19,570 Can you guys help me? 357 00:22:19,570 --> 00:22:23,640 I know you can after all the review of chapter 10 358 00:22:23,640 --> 00:22:25,860 and-- this is what? 359 00:22:25,860 --> 00:22:29,470 360 00:22:29,470 --> 00:22:32,970 R what? 361 00:22:32,970 --> 00:22:34,970 You should whisper cosine t. 362 00:22:34,970 --> 00:22:38,680 Say it out loud, be proud of what you know. 363 00:22:38,680 --> 00:22:40,640 This is R sine t. 364 00:22:40,640 --> 00:22:43,080 And let's take t between 0 and 2 pi. 365 00:22:43,080 --> 00:22:50,730 One, the revolution only, and then I say, good. 366 00:22:50,730 --> 00:22:53,996 The speed is what? 367 00:22:53,996 --> 00:23:00,930 Speed, square root of x prime the t squared 368 00:23:00,930 --> 00:23:03,430 plus y prime the t squared. 369 00:23:03,430 --> 00:23:08,260 Which is the same as writing R prime of the t in magnitude. 370 00:23:08,260 --> 00:23:09,480 Thank God we know that. 371 00:23:09,480 --> 00:23:11,550 How much is this? 372 00:23:11,550 --> 00:23:14,266 R, very good, this is R, very good. 373 00:23:14,266 --> 00:23:18,790 So life is not so hard, it's-- hopefully I'll be able to do 374 00:23:18,790 --> 00:23:21,660 the w. 375 00:23:21,660 --> 00:23:22,410 What is the w? 376 00:23:22,410 --> 00:23:25,910 It's the path integral all over the circle 377 00:23:25,910 --> 00:23:30,380 I have here, that I traveled counterclockwise from any point 378 00:23:30,380 --> 00:23:31,270 to any point. 379 00:23:31,270 --> 00:23:34,559 Let's say this would be the origin of my motion, 380 00:23:34,559 --> 00:23:37,340 then I go back. 381 00:23:37,340 --> 00:23:43,320 And I have this force, F, that I have 382 00:23:43,320 --> 00:23:47,540 to redistribute in terms of R. So this notation 383 00:23:47,540 --> 00:23:49,222 is giving me a little bit of a headache, 384 00:23:49,222 --> 00:23:51,410 but in reality it's going to be very simple. 385 00:23:51,410 --> 00:23:59,740 This is the dot product, R prime dt, which was the R. 386 00:23:59,740 --> 00:24:05,390 Which some other people asked me, how can you write that? 387 00:24:05,390 --> 00:24:12,090 Well, read the review, R of x equals 388 00:24:12,090 --> 00:24:16,810 x of t, i plus y of t, j. 389 00:24:16,810 --> 00:24:19,990 Also, read the next side plus y j. 390 00:24:19,990 --> 00:24:26,590 It short, the dR differential out of t, 391 00:24:26,590 --> 00:24:32,920 sorry, I'll put R. dR is dx i plus dy j. 392 00:24:32,920 --> 00:24:35,950 And if somebody wants to be expressing 393 00:24:35,950 --> 00:24:40,390 this in terms of speeds, we'll say this is x prime dt, 394 00:24:40,390 --> 00:24:43,010 this is y prime dt. 395 00:24:43,010 --> 00:24:48,130 So we can rewrite this x prime then t i, plus y prime then t 396 00:24:48,130 --> 00:24:49,836 j, dt. 397 00:24:49,836 --> 00:24:54,620 398 00:24:54,620 --> 00:24:55,800 All right? 399 00:24:55,800 --> 00:25:01,350 OK, which is the same thing as R prime of t, dt. 400 00:25:01,350 --> 00:25:04,203 401 00:25:04,203 --> 00:25:04,703 [INAUDIBLE] 402 00:25:04,703 --> 00:25:07,580 403 00:25:07,580 --> 00:25:11,760 This looks awfully theoretical. 404 00:25:11,760 --> 00:25:16,550 I say, I don't like it, I want to put my favorite guys 405 00:25:16,550 --> 00:25:18,290 in the picture. 406 00:25:18,290 --> 00:25:22,140 So I have to think, when I do the dot product 407 00:25:22,140 --> 00:25:26,380 I have the dot product between the vector that 408 00:25:26,380 --> 00:25:29,890 has components f1 and f2. 409 00:25:29,890 --> 00:25:31,870 How am I going to do that? 410 00:25:31,870 --> 00:25:38,700 Well, if I multiply with this guy, dot product, the boss guy, 411 00:25:38,700 --> 00:25:39,800 with this boss guy. 412 00:25:39,800 --> 00:25:43,180 Are you guys with me? 413 00:25:43,180 --> 00:25:44,810 What am I going to do? 414 00:25:44,810 --> 00:25:48,560 First component times first component, 415 00:25:48,560 --> 00:25:53,740 plus second component times second component of a vector. 416 00:25:53,740 --> 00:25:58,830 So I have to be smart and understand how I do that. 417 00:25:58,830 --> 00:26:01,060 Lambda is a constant. 418 00:26:01,060 --> 00:26:04,110 Lambda, you're my friend, you stay there. 419 00:26:04,110 --> 00:26:09,570 x is x of t, x of t, but I multiplied 420 00:26:09,570 --> 00:26:11,490 with the first component here, so I 421 00:26:11,490 --> 00:26:14,860 multiplied by x prime of t. 422 00:26:14,860 --> 00:26:19,600 Plus lambda times y of t, times y prime of t. 423 00:26:19,600 --> 00:26:23,750 And who gets out of the picture is dt at the end. 424 00:26:23,750 --> 00:26:27,530 I have integrate with respect to that dt. 425 00:26:27,530 --> 00:26:29,570 This would be incorrect, why? 426 00:26:29,570 --> 00:26:34,600 Because t has to move between some specific limits 427 00:26:34,600 --> 00:26:38,180 when I specify what a path integral is. 428 00:26:38,180 --> 00:26:41,870 I cannot leave a c-- very good, from 0 to 2 pi, excellent. 429 00:26:41,870 --> 00:26:45,300 430 00:26:45,300 --> 00:26:46,240 Is this hard? 431 00:26:46,240 --> 00:26:48,670 No, It's going to be a piece of cake. 432 00:26:48,670 --> 00:26:50,336 Why is that a piece of cake? 433 00:26:50,336 --> 00:26:54,190 Because I can keep writing. 434 00:26:54,190 --> 00:26:58,519 You actually are faster than me. 435 00:26:58,519 --> 00:27:00,560 STUDENT: Your chain rule is already done for you. 436 00:27:00,560 --> 00:27:03,930 PROFESSOR: Right, and then lambda 437 00:27:03,930 --> 00:27:06,920 gets out just because-- well you remember 438 00:27:06,920 --> 00:27:09,600 you kick the lambda out, right? 439 00:27:09,600 --> 00:27:20,140 And then I've put R cosine t times minus R sine t. 440 00:27:20,140 --> 00:27:21,310 I'm done with who? 441 00:27:21,310 --> 00:27:23,625 I'm done with this fellow and that fellow. 442 00:27:23,625 --> 00:27:27,340 443 00:27:27,340 --> 00:27:37,550 And plus y, R sine t, what is R prime? 444 00:27:37,550 --> 00:27:48,580 R cosine, thank you guys, dt. 445 00:27:48,580 --> 00:27:50,910 And now I'm going to ask you, what is this animal? 446 00:27:50,910 --> 00:27:53,660 447 00:27:53,660 --> 00:27:56,020 Stare at that, what is the integrand? 448 00:27:56,020 --> 00:27:59,130 Is a friend of yours, he's so cute. 449 00:27:59,130 --> 00:28:01,540 He's staring at you and saying you are done. 450 00:28:01,540 --> 00:28:04,895 Why are you done? 451 00:28:04,895 --> 00:28:08,346 What happens to the integrand? 452 00:28:08,346 --> 00:28:12,365 It's zero, it's a blessing, it's zero. 453 00:28:12,365 --> 00:28:14,000 How come it's zero? 454 00:28:14,000 --> 00:28:21,390 Because these two terms simplify, they cancel out. 455 00:28:21,390 --> 00:28:26,290 They cancel out, thank god they cancel out, I got a zero. 456 00:28:26,290 --> 00:28:28,530 So we discover something that a physicist 457 00:28:28,530 --> 00:28:32,152 or a mechanical engineer would have told you already. 458 00:28:32,152 --> 00:28:34,110 And do you think he would have actually plugged 459 00:28:34,110 --> 00:28:36,230 in the path integral? 460 00:28:36,230 --> 00:28:39,700 No, they wouldn't think like this. 461 00:28:39,700 --> 00:28:45,540 He has a simpler explanation for that because he's experienced 462 00:28:45,540 --> 00:28:47,260 with linear experiments. 463 00:28:47,260 --> 00:28:51,740 And says, if I drag this like that I know what to work with. 464 00:28:51,740 --> 00:28:54,770 If I drag in, like at an angle, I 465 00:28:54,770 --> 00:28:57,950 know that I have the magnitude of this, 466 00:28:57,950 --> 00:28:59,390 cosine theta, the angle. 467 00:28:59,390 --> 00:29:03,740 So, he knows for linear cases what we have. 468 00:29:03,740 --> 00:29:10,060 For a circular case he can smell the result 469 00:29:10,060 --> 00:29:12,880 without doing the path integral. 470 00:29:12,880 --> 00:29:18,080 So how do you think the guy, if he's a mechanical engineer, 471 00:29:18,080 --> 00:29:20,040 would think in a second? 472 00:29:20,040 --> 00:29:24,315 Say well, think of your trajectory, right? 473 00:29:24,315 --> 00:29:27,360 It's a circle. 474 00:29:27,360 --> 00:29:31,620 The problem is that centrifugal force being perpendicular 475 00:29:31,620 --> 00:29:33,560 to the circle all the time. 476 00:29:33,560 --> 00:29:37,140 And you say, how can a line be perpendicular to a circle? 477 00:29:37,140 --> 00:29:39,570 It is, it's the normal to the circle. 478 00:29:39,570 --> 00:29:43,054 So when you say this is normal to the circle 479 00:29:43,054 --> 00:29:45,470 you mean it's normal to the tangent of the circle. 480 00:29:45,470 --> 00:29:50,840 So if you measure the angle made by the normal at every point 481 00:29:50,840 --> 00:29:53,800 to the trajectory of a circle, it's always lines. 482 00:29:53,800 --> 00:30:01,520 So he goes, gosh, I got cosine of 90, that's zero. 483 00:30:01,520 --> 00:30:04,270 So if you have some sort of work produced 484 00:30:04,270 --> 00:30:06,470 by something perpendicular to your trajectory, 485 00:30:06,470 --> 00:30:07,800 that must be zero. 486 00:30:07,800 --> 00:30:12,140 So he or she has very good intuition. 487 00:30:12,140 --> 00:30:13,875 Of course, how do we prove it? 488 00:30:13,875 --> 00:30:16,535 We are mathematicians, we prove the path integral, 489 00:30:16,535 --> 00:30:19,460 we got zero for the work, all right? 490 00:30:19,460 --> 00:30:25,200 But he could sense that kind of stuff from the beginning. 491 00:30:25,200 --> 00:30:34,740 Now, there is another example where maybe you don't have 492 00:30:34,740 --> 00:30:38,360 90 degrees for your trajectory. 493 00:30:38,360 --> 00:30:47,970 Well, I'm going to just take-- what if I change the force 494 00:30:47,970 --> 00:30:50,570 and I make a difference problem? 495 00:30:50,570 --> 00:30:52,844 Make it into a different problem. 496 00:30:52,844 --> 00:31:01,293 497 00:31:01,293 --> 00:31:04,772 I will do that later, I won't go and erase it. 498 00:31:04,772 --> 00:31:15,710 499 00:31:15,710 --> 00:31:19,620 Last time we did one that was, compute 500 00:31:19,620 --> 00:31:25,330 the work along a parabola from something to something. 501 00:31:25,330 --> 00:31:26,830 Let's do that again. 502 00:31:26,830 --> 00:31:30,668 503 00:31:30,668 --> 00:31:35,585 For some sort of a nice force field 504 00:31:35,585 --> 00:31:39,740 I'll take your vector valued function to be nice to you. 505 00:31:39,740 --> 00:31:42,380 I'll change it, y i plus x j. 506 00:31:42,380 --> 00:31:45,330 507 00:31:45,330 --> 00:31:48,550 And then we are in plane and we move 508 00:31:48,550 --> 00:31:55,020 along this parabola between 0, 0 and-- what is this guys? 509 00:31:55,020 --> 00:31:59,348 1, 1-- well let make it into a one, it's cute. 510 00:31:59,348 --> 00:32:04,690 And I'd like you to measure the work along the parabola 511 00:32:04,690 --> 00:32:11,410 and also along the arc of a-- along the segment of a line 512 00:32:11,410 --> 00:32:13,440 between the two points. 513 00:32:13,440 --> 00:32:16,728 So I want you to compute w1 along the parabola, 514 00:32:16,728 --> 00:32:22,990 and w2 along this thingy, the segment. 515 00:32:22,990 --> 00:32:24,070 Should it be hard? 516 00:32:24,070 --> 00:32:27,230 No, this was old session for many finals. 517 00:32:27,230 --> 00:32:33,180 I remember, I think it was 2003, 2006, 2008, 518 00:32:33,180 --> 00:32:36,320 and very recently, I think a year and 1/2 ago. 519 00:32:36,320 --> 00:32:38,020 A problem like that was given. 520 00:32:38,020 --> 00:32:40,750 Compute the path integrals correspond 521 00:32:40,750 --> 00:32:45,000 to work for both parametrization and compare them. 522 00:32:45,000 --> 00:32:45,650 Is it hard? 523 00:32:45,650 --> 00:32:48,861 I have no idea, let me think. 524 00:32:48,861 --> 00:32:53,260 For the first one we have parametrization 525 00:32:53,260 --> 00:32:55,600 that we need to distinguish from the other one. 526 00:32:55,600 --> 00:32:58,706 The first parametrization for a parabola, 527 00:32:58,706 --> 00:33:01,430 we discussed it last time, was of course 528 00:33:01,430 --> 00:33:04,080 the simplest possible one you can think of. 529 00:33:04,080 --> 00:33:05,830 And we did this last time but I'm 530 00:33:05,830 --> 00:33:10,360 repeating this because I didn't want Alex to miss that. 531 00:33:10,360 --> 00:33:18,250 And I'm going to say integral from some time to some time. 532 00:33:18,250 --> 00:33:21,090 Now, if I'm between 0 and 1, time of course 533 00:33:21,090 --> 00:33:25,780 will be between 0 and 1 because x is time. 534 00:33:25,780 --> 00:33:28,554 All right, good, that means what else? 535 00:33:28,554 --> 00:33:32,140 This Is f1 and this is f2. 536 00:33:32,140 --> 00:33:39,720 So I'm going to have f1 of t, x prime of t, plus f2 of t, 537 00:33:39,720 --> 00:33:46,310 y prime of t, all the [? sausage ?] times dt. 538 00:33:46,310 --> 00:33:47,930 Is this going to be hard? 539 00:33:47,930 --> 00:33:52,830 Hopefully not, I'm going to have to identify everybody. 540 00:33:52,830 --> 00:33:55,970 Identify this guys prime of t with respect to t 541 00:33:55,970 --> 00:33:58,110 is 1, piece of cake, right? 542 00:33:58,110 --> 00:34:02,850 This fellow is-- you told me last time you got 2t 543 00:34:02,850 --> 00:34:05,360 and you got it right. 544 00:34:05,360 --> 00:34:08,219 This guy, I have to be a little bit careful 545 00:34:08,219 --> 00:34:10,679 because y is the fourth guy. 546 00:34:10,679 --> 00:34:15,920 This is t squared and this is t. 547 00:34:15,920 --> 00:34:19,889 548 00:34:19,889 --> 00:34:22,960 So my integral will be a joke. 549 00:34:22,960 --> 00:34:29,630 0 to 1, 2t squared plus t squared equals 3t squared. 550 00:34:29,630 --> 00:34:30,980 Is it hard to integrate? 551 00:34:30,980 --> 00:34:33,110 No, for God's sake, this is integral-- this 552 00:34:33,110 --> 00:34:39,270 is t cubed between 0 and 1, right, right? 553 00:34:39,270 --> 00:34:44,080 So I should get 1, and if I get, I 554 00:34:44,080 --> 00:34:48,600 think I did it right, if I get the other parametrization you 555 00:34:48,600 --> 00:34:52,540 have to help me write it again. 556 00:34:52,540 --> 00:34:57,190 The parametrization of this straight line 557 00:34:57,190 --> 00:34:59,380 between 0, 0 and 1, 1. 558 00:34:59,380 --> 00:35:03,300 Now on the actual exam, I'm never 559 00:35:03,300 --> 00:35:08,500 going to forgive you if you don't know how to parametrize. 560 00:35:08,500 --> 00:35:13,140 Now you know it but two months ago you didn't, many of you 561 00:35:13,140 --> 00:35:14,100 didn't. 562 00:35:14,100 --> 00:35:21,510 So if somebody gives you 2 points, OK, in plane I ask you, 563 00:35:21,510 --> 00:35:27,160 how do you write that symmetric equation of the line between? 564 00:35:27,160 --> 00:35:29,030 You were a little bit hesitant, now 565 00:35:29,030 --> 00:35:34,030 you shouldn't be hesitant because it's a serious thing. 566 00:35:34,030 --> 00:35:36,290 So how did we write that? 567 00:35:36,290 --> 00:35:40,466 We memorized it. x minus x1, over x2 minus x1 568 00:35:40,466 --> 00:35:44,440 equals y minus y1, over y2 minus y1. 569 00:35:44,440 --> 00:35:47,300 This can also be written as-- you know, guys, 570 00:35:47,300 --> 00:35:49,840 that this over that is the actual slope. 571 00:35:49,840 --> 00:35:52,275 This over that, so it can be written 572 00:35:52,275 --> 00:35:53,560 as a [INAUDIBLE] formula. 573 00:35:53,560 --> 00:35:55,250 It can be written in many ways. 574 00:35:55,250 --> 00:35:59,920 And if we put a, t, we transform it into a parametric equation. 575 00:35:59,920 --> 00:36:02,140 So you should be able, on the final, 576 00:36:02,140 --> 00:36:05,690 to do that for any segment of a line with your eyes closed. 577 00:36:05,690 --> 00:36:07,550 Like, you see the numbers, you plug them in, 578 00:36:07,550 --> 00:36:10,640 you get the parametric equations. 579 00:36:10,640 --> 00:36:14,820 We are nice on the exams because we usually give you 580 00:36:14,820 --> 00:36:18,750 a line that's easy to write. 581 00:36:18,750 --> 00:36:25,830 Like in this case you would have x equals t and y equal, 582 00:36:25,830 --> 00:36:29,940 let's see if you are asleep yet, t. 583 00:36:29,940 --> 00:36:31,500 Why is that? 584 00:36:31,500 --> 00:36:37,290 Because the line that joins 0, 0 and 1, 1 is y equals x. 585 00:36:37,290 --> 00:36:40,095 So y equals x is called also, first bicycle. 586 00:36:40,095 --> 00:36:43,100 It's the old friend of yours from trigonometry, 587 00:36:43,100 --> 00:36:46,300 from Pre-Calc, from algebra, I don't know where, college 588 00:36:46,300 --> 00:36:47,820 algebra. 589 00:36:47,820 --> 00:36:49,160 Alrighty, is this hard to do? 590 00:36:49,160 --> 00:36:55,850 No, it's easier than before. w2 is integral from 0 to 1, 591 00:36:55,850 --> 00:37:00,380 this is t and this is t, this is t and this is t, good. 592 00:37:00,380 --> 00:37:07,990 So we have t times 1 plus t times 1, 593 00:37:07,990 --> 00:37:15,362 it's like a funny, nice, game that's too simple, 2t, 2t. 594 00:37:15,362 --> 00:37:18,340 595 00:37:18,340 --> 00:37:23,100 So the fundamental theorem of Calc 596 00:37:23,100 --> 00:37:27,042 says t squared between 0 and 1, the answer is 1. 597 00:37:27,042 --> 00:37:28,500 Am I surprised? 598 00:37:28,500 --> 00:37:31,360 Look at me, do I look surprised at all 599 00:37:31,360 --> 00:37:34,240 that I got the same answer? 600 00:37:34,240 --> 00:37:37,224 No, I told you a secret last time. 601 00:37:37,224 --> 00:37:39,390 I didn't prove it. 602 00:37:39,390 --> 00:37:43,100 I said that R times happy times. 603 00:37:43,100 --> 00:37:46,740 When depending on the force that is with you, 604 00:37:46,740 --> 00:37:51,300 you have the same work no matter what path 605 00:37:51,300 --> 00:37:53,830 you are taking between a and b. 606 00:37:53,830 --> 00:37:57,720 Between the origin and finish line. 607 00:37:57,720 --> 00:38:01,030 So I'm claiming that if I give you this zig-zag line 608 00:38:01,030 --> 00:38:08,010 and I asked you what-- look, it could be any crazy path 609 00:38:08,010 --> 00:38:11,090 but it has to be a nice differentiable path. 610 00:38:11,090 --> 00:38:14,630 Along this differentiable path, no matter 611 00:38:14,630 --> 00:38:17,420 how you compute the work, that's your business, 612 00:38:17,420 --> 00:38:19,040 I claim I still get 1. 613 00:38:19,040 --> 00:38:21,840 614 00:38:21,840 --> 00:38:26,440 Can you even think why, some of you remember maybe, 615 00:38:26,440 --> 00:38:28,986 the force was key? 616 00:38:28,986 --> 00:38:30,444 STUDENT: It's a conservative force. 617 00:38:30,444 --> 00:38:32,780 PROFESSOR: It had to be good conservative. 618 00:38:32,780 --> 00:38:37,320 Now this is conservative but why is that conservative? 619 00:38:37,320 --> 00:38:39,900 What the heck is a conservative force? 620 00:38:39,900 --> 00:38:46,710 So let's write it down on the-- we 621 00:38:46,710 --> 00:39:01,650 say that the vector valued function, f, 622 00:39:01,650 --> 00:39:19,084 valued in R2 or R3, is conservative if there exists 623 00:39:19,084 --> 00:39:25,460 a smooth function. 624 00:39:25,460 --> 00:39:28,990 Little f, it actually has to be just c1, 625 00:39:28,990 --> 00:39:31,660 called scalar potential. 626 00:39:31,660 --> 00:39:45,380 Called scalar potential, such that big F as a vector 627 00:39:45,380 --> 00:39:48,100 field will be not little f. 628 00:39:48,100 --> 00:39:52,830 That means it will be the gradient of the scalar 629 00:39:52,830 --> 00:39:54,838 potential. 630 00:39:54,838 --> 00:39:57,370 Definition, that was the definition, 631 00:39:57,370 --> 00:40:08,440 and then criterion for a f in R2 to be conservative. 632 00:40:08,440 --> 00:40:13,390 633 00:40:13,390 --> 00:40:21,370 I claim that f equals f1 i, plus f two eyes, no, f2 j. 634 00:40:21,370 --> 00:40:23,370 I'm just making silly puns, I don't 635 00:40:23,370 --> 00:40:27,880 know if you guys follow me. 636 00:40:27,880 --> 00:40:33,470 If and only if f sub 1 prime, with respect to y, 637 00:40:33,470 --> 00:40:37,890 is f sub 2 prime, with respect to x. 638 00:40:37,890 --> 00:40:40,166 Can I prove this? 639 00:40:40,166 --> 00:40:45,560 Prove, prove Magdelina, don't just stare at it, prove. 640 00:40:45,560 --> 00:40:48,030 Why would that be necessary and sufficient? 641 00:40:48,030 --> 00:40:52,020 642 00:40:52,020 --> 00:40:58,890 Well, for big F to be conservative 643 00:40:58,890 --> 00:41:02,180 it means that it has to be the gradient 644 00:41:02,180 --> 00:41:06,970 of some little function, little f, some scalar potential. 645 00:41:06,970 --> 00:41:13,204 Alrighty, so let me write it down, proof. 646 00:41:13,204 --> 00:41:20,584 f conservative if and only if there 647 00:41:20,584 --> 00:41:27,964 exists f, such that gradient of f is F. If and only 648 00:41:27,964 --> 00:41:32,990 if-- what does it mean about f1 and f2? 649 00:41:32,990 --> 00:41:36,540 f1 and f2 are the f sub of x and the f sub 650 00:41:36,540 --> 00:41:39,460 y of some scalar potential. 651 00:41:39,460 --> 00:41:44,300 So if f is the gradient, that means that the first component 652 00:41:44,300 --> 00:41:48,600 has to be little f sub x And the second component should 653 00:41:48,600 --> 00:41:51,906 have to be little f sub y. 654 00:41:51,906 --> 00:41:58,240 But that is if and only if f sub 1 prime, with respect to y, 655 00:41:58,240 --> 00:42:01,710 is the same as f sub 2 prime, with respect to x. 656 00:42:01,710 --> 00:42:04,144 What is that? 657 00:42:04,144 --> 00:42:14,052 The red thing here is called a compatibility condition 658 00:42:14,052 --> 00:42:16,440 of this system. 659 00:42:16,440 --> 00:42:22,660 This is a system of two OD's. 660 00:42:22,660 --> 00:42:27,426 You are going to study ODEs in 3350. 661 00:42:27,426 --> 00:42:29,730 And you are going to remember this 662 00:42:29,730 --> 00:42:33,020 and say, Oh, I know that because she taught me that in Calc 3. 663 00:42:33,020 --> 00:42:36,310 Not all instructors will teach you this in Calc 3. 664 00:42:36,310 --> 00:42:38,880 Some of them fool you and skip this material 665 00:42:38,880 --> 00:42:44,410 that's very important to understand in 3350. 666 00:42:44,410 --> 00:42:49,490 So guys, what's going to happened when you prime this 667 00:42:49,490 --> 00:42:51,580 with respect to y? 668 00:42:51,580 --> 00:42:55,440 You get f sub x prime, with respect to y. 669 00:42:55,440 --> 00:42:57,110 When you prime this with respect to x 670 00:42:57,110 --> 00:42:59,860 you get f sub y prime, with respect to x. 671 00:42:59,860 --> 00:43:01,680 Why are they the same thing? 672 00:43:01,680 --> 00:43:04,716 I'm going to remind you that they 673 00:43:04,716 --> 00:43:07,220 are the same thing for a smooth function. 674 00:43:07,220 --> 00:43:09,250 Who said that? 675 00:43:09,250 --> 00:43:19,060 A crazy German mathematician whose name was Schwartz. 676 00:43:19,060 --> 00:43:22,134 Which means black, that's what I'm painting it in black. 677 00:43:22,134 --> 00:43:29,090 Because is the Schwartz guy, the first criterion 678 00:43:29,090 --> 00:43:31,750 saying that no matter in what order 679 00:43:31,750 --> 00:43:34,320 you differentiate the smooth function you 680 00:43:34,320 --> 00:43:37,800 get the same answer for the mixed derivative. 681 00:43:37,800 --> 00:43:41,620 So you see we prove if and only if that you 682 00:43:41,620 --> 00:43:43,900 have to have this criterion, otherwise 683 00:43:43,900 --> 00:43:46,210 it's not going to be conservative. 684 00:43:46,210 --> 00:43:49,630 So I'm asking you, for your old friend, f 685 00:43:49,630 --> 00:43:52,840 equals- example one or example two, 686 00:43:52,840 --> 00:43:56,200 I don't know- y i plus x j. 687 00:43:56,200 --> 00:43:58,510 Is the conservative? 688 00:43:58,510 --> 00:44:00,440 You can prove it in two ways. 689 00:44:00,440 --> 00:44:06,685 Prove in two differently ways that it is conservative. 690 00:44:06,685 --> 00:44:13,210 691 00:44:13,210 --> 00:44:16,716 a, find the criteria. 692 00:44:16,716 --> 00:44:20,450 What does this criteria say? 693 00:44:20,450 --> 00:44:25,430 Take your first component, prime it with respect to y. 694 00:44:25,430 --> 00:44:27,780 So y prime with respect to y. 695 00:44:27,780 --> 00:44:32,870 Take your second component, x, prime it with respect to x. 696 00:44:32,870 --> 00:44:34,750 Is this true? 697 00:44:34,750 --> 00:44:37,460 Yes, and this is me, happy that it's true. 698 00:44:37,460 --> 00:44:40,740 So this is 1 equals 1, so it's true. 699 00:44:40,740 --> 00:44:44,780 So it must be conservative, so it must be conservative. 700 00:44:44,780 --> 00:44:46,780 Could I have done it another way? 701 00:44:46,780 --> 00:44:49,610 702 00:44:49,610 --> 00:44:57,530 By definition, by definition, to prove that a force field 703 00:44:57,530 --> 00:45:01,640 is conservative by definition, that it a matter 704 00:45:01,640 --> 00:45:04,075 of the smart people. 705 00:45:04,075 --> 00:45:07,100 There are people who- unlike me when 706 00:45:07,100 --> 00:45:11,670 I was 18- are able to see the scalar potential in just 707 00:45:11,670 --> 00:45:13,852 about any problem I give them. 708 00:45:13,852 --> 00:45:18,070 I'm not going to make this experiment with a bunch of you 709 00:45:18,070 --> 00:45:22,320 and I'm going to reward you for the correct answers. 710 00:45:22,320 --> 00:45:25,430 But, could anybody see the existence 711 00:45:25,430 --> 00:45:27,710 of the scalar potential? 712 00:45:27,710 --> 00:45:30,750 So these there, this exists. 713 00:45:30,750 --> 00:45:37,590 That's there exists a little f scalar potential such 714 00:45:37,590 --> 00:45:43,081 that nabla f equals F. 715 00:45:43,081 --> 00:45:47,680 And some of you may see it and say, I see it. 716 00:45:47,680 --> 00:45:50,070 So, can you see a little function 717 00:45:50,070 --> 00:45:54,660 f scalar function so that f sub of x i is y 718 00:45:54,660 --> 00:46:00,810 and f sub- Magdelena-- x of y j is this x j? 719 00:46:00,810 --> 00:46:02,220 STUDENT: [INAUDIBLE]? 720 00:46:02,220 --> 00:46:07,330 PROFESSOR: No, you need to drink some coffee first. 721 00:46:07,330 --> 00:46:12,990 You can get this, x times what? 722 00:46:12,990 --> 00:46:14,165 Why is that? 723 00:46:14,165 --> 00:46:17,270 I'll teach you how to get it. 724 00:46:17,270 --> 00:46:18,880 Nevertheless, there are some people 725 00:46:18,880 --> 00:46:21,140 who can do it with their naked eye 726 00:46:21,140 --> 00:46:26,066 because they have a little computer in their head. 727 00:46:26,066 --> 00:46:28,710 But how did I do it? 728 00:46:28,710 --> 00:46:30,970 It's just a matter of experience, I said, 729 00:46:30,970 --> 00:46:36,690 if I take f to be x y, I sort of guessed it. 730 00:46:36,690 --> 00:46:42,000 f sub x would be y and f sub y will be x so this should be it, 731 00:46:42,000 --> 00:46:44,440 and this is going to do. 732 00:46:44,440 --> 00:46:49,040 And f is a nice function, polynomial in two variables, 733 00:46:49,040 --> 00:46:50,420 it's a smooth function. 734 00:46:50,420 --> 00:46:55,200 I'm very happy I'm over the domain, 735 00:46:55,200 --> 00:46:57,770 open this or whatever, open domain in plane. 736 00:46:57,770 --> 00:47:00,160 I'm very happy, I have no problem with it. 737 00:47:00,160 --> 00:47:04,756 So I can know that this is conservative in two ways. 738 00:47:04,756 --> 00:47:07,750 Either I get to the source of the problem 739 00:47:07,750 --> 00:47:10,380 and I find the little scalar potential whose 740 00:47:10,380 --> 00:47:13,480 gradient is my force field. 741 00:47:13,480 --> 00:47:16,630 Or I can verify the criterion and I say, 742 00:47:16,630 --> 00:47:19,100 the derivative of this with respect to y 743 00:47:19,100 --> 00:47:20,972 is the derivative of this with respect 744 00:47:20,972 --> 00:47:23,140 to x is-- one is the same. 745 00:47:23,140 --> 00:47:29,145 The same thing, you're going to see it again in math 3350. 746 00:47:29,145 --> 00:47:30,710 All right, that I taught many times, 747 00:47:30,710 --> 00:47:33,020 I'm not going to teach that in the fall. 748 00:47:33,020 --> 00:47:34,820 But I know of some very good people 749 00:47:34,820 --> 00:47:36,065 who teach that in the fall. 750 00:47:36,065 --> 00:47:38,670 In any case, they would reteach it to you 751 00:47:38,670 --> 00:47:42,060 because good teachers don't assume that you know much. 752 00:47:42,060 --> 00:47:45,970 But when you will see it you'll remember me. 753 00:47:45,970 --> 00:47:49,970 Hopefully fondly, not cursing me or anything, right? 754 00:47:49,970 --> 00:47:54,890 OK, how do we actually get to compute f by hand 755 00:47:54,890 --> 00:47:59,150 if we're not experienced enough to guess it like I was 756 00:47:59,150 --> 00:48:01,740 experienced enough to guess? 757 00:48:01,740 --> 00:48:08,240 So let me show you how you solve a system of two differential 758 00:48:08,240 --> 00:48:11,290 equations like that. 759 00:48:11,290 --> 00:48:15,625 So how I got-- how you are supposed 760 00:48:15,625 --> 00:48:23,010 to get the scalar potential. 761 00:48:23,010 --> 00:48:28,790 f sub x equals F1, f sub y equals F2. 762 00:48:28,790 --> 00:48:35,358 So by integration, 1 and 2. 763 00:48:35,358 --> 00:48:38,740 764 00:48:38,740 --> 00:48:40,910 And you say, what you mean 1 and 2? 765 00:48:40,910 --> 00:48:42,670 I'll show you in a second. 766 00:48:42,670 --> 00:48:47,820 So for my case, example 2, I'll take 767 00:48:47,820 --> 00:48:50,570 my f sub x must be y, right? 768 00:48:50,570 --> 00:48:51,070 Good. 769 00:48:51,070 --> 00:48:54,025 My f sub y must be x, right? 770 00:48:54,025 --> 00:48:55,270 Right. 771 00:48:55,270 --> 00:48:58,160 Who is f? 772 00:48:58,160 --> 00:49:00,690 Solve this property. 773 00:49:00,690 --> 00:49:05,580 Oh, I have to start integrating from the first guy. 774 00:49:05,580 --> 00:49:08,390 What kind of information am I going to squeeze? 775 00:49:08,390 --> 00:49:11,572 I'm going to say I have to go backwards, 776 00:49:11,572 --> 00:49:17,581 I have to get-- f is going to be what? 777 00:49:17,581 --> 00:49:23,156 Integral of y with respect to x, say it again, Magdelina. 778 00:49:23,156 --> 00:49:27,440 Integral of y with respect to x, but attention, 779 00:49:27,440 --> 00:49:31,620 this may come because, for me, the variable is x here, 780 00:49:31,620 --> 00:49:35,480 and y is like, you cannot stay in this picture. 781 00:49:35,480 --> 00:49:40,590 So I have a constant c that depends on y. 782 00:49:40,590 --> 00:49:41,510 Say what? 783 00:49:41,510 --> 00:49:46,290 Yes, because if you go backwards and prime this with respect 784 00:49:46,290 --> 00:49:49,658 to x, what do you get? f sub x will 785 00:49:49,658 --> 00:49:51,840 be y because this is the anti-derivative. 786 00:49:51,840 --> 00:49:55,460 Plus this prime with respect to x, zero. 787 00:49:55,460 --> 00:49:59,520 So this c of y may, a little bit, ruin your plans. 788 00:49:59,520 --> 00:50:02,630 I've had students who forgot about it 789 00:50:02,630 --> 00:50:04,950 and then they got in trouble because they couldn't get 790 00:50:04,950 --> 00:50:08,130 the scalar potential correctly. 791 00:50:08,130 --> 00:50:08,630 All right? 792 00:50:08,630 --> 00:50:14,340 OK, so from this one you say, OK I have some-- 793 00:50:14,340 --> 00:50:16,980 what is the integral of y dx? 794 00:50:16,980 --> 00:50:22,246 xy, plus some guy c constant that depends on y. 795 00:50:22,246 --> 00:50:25,610 From this fellow I go, but I have 796 00:50:25,610 --> 00:50:29,520 to verify the second condition, if I don't I'm dead meat. 797 00:50:29,520 --> 00:50:32,320 There are two coupled equations, these 798 00:50:32,320 --> 00:50:33,980 are coupled equations that have to be 799 00:50:33,980 --> 00:50:36,320 verified at the same time. 800 00:50:36,320 --> 00:50:45,120 So f sub y will be prime with respect to y. x plus prime 801 00:50:45,120 --> 00:50:52,750 with respect to y. c prime of y, God gave me x here. 802 00:50:52,750 --> 00:50:56,760 So I'm really lucky in that sense that c prime of y 803 00:50:56,760 --> 00:51:01,300 will be 0 because I have an x here and an x here. 804 00:51:01,300 --> 00:51:05,540 So c of y will simply be any constant k. c of y 805 00:51:05,540 --> 00:51:09,487 is just a constant k, it's not going to depend on y, 806 00:51:09,487 --> 00:51:10,980 it's a constant k. 807 00:51:10,980 --> 00:51:14,510 So my answer was not correct. 808 00:51:14,510 --> 00:51:21,320 The best answer would have been f of xy must be xy plus k. 809 00:51:21,320 --> 00:51:26,650 But any function like xy will work, I just need one to work. 810 00:51:26,650 --> 00:51:29,330 I just need a scalar potential, not all of them. 811 00:51:29,330 --> 00:51:33,846 This will work, x2, xy plus 7 will work, xy plus 3 will work, 812 00:51:33,846 --> 00:51:38,860 xy minus 1,033,045 will work. 813 00:51:38,860 --> 00:51:43,830 But I only need one so I'll take xy. 814 00:51:43,830 --> 00:51:46,220 Now that I trained your mind a little bit, 815 00:51:46,220 --> 00:51:48,370 maybe you don't need to actually solve 816 00:51:48,370 --> 00:51:53,720 the system because your brain wasn't ready before. 817 00:51:53,720 --> 00:51:57,300 But you'd be amazed, we are very trainable people. 818 00:51:57,300 --> 00:52:03,550 And in the process of doing something completely new, 819 00:52:03,550 --> 00:52:05,400 we are learning. 820 00:52:05,400 --> 00:52:10,305 And your brain next, will say, I think 821 00:52:10,305 --> 00:52:15,330 I know how to function a little bit backwards. 822 00:52:15,330 --> 00:52:20,460 And try to integrate and see and guess a potential 823 00:52:20,460 --> 00:52:24,080 because it's not so hard. 824 00:52:24,080 --> 00:52:25,580 So let me give you example 3. 825 00:52:25,580 --> 00:52:29,650 826 00:52:29,650 --> 00:52:36,240 Somebody give you over a domain in plane x i plus y j, 827 00:52:36,240 --> 00:52:40,605 and says, over D, simply connected domain in plane, 828 00:52:40,605 --> 00:52:44,000 open, doesn't matter. 829 00:52:44,000 --> 00:52:45,455 Is this conservative? 830 00:52:45,455 --> 00:52:48,370 831 00:52:48,370 --> 00:52:51,880 Find a scalar potential. 832 00:52:51,880 --> 00:53:00,710 833 00:53:00,710 --> 00:53:03,960 This is again, we do section 13-2, 834 00:53:03,960 --> 00:53:07,650 so today we did 13-1 and 13-2 jointly. 835 00:53:07,650 --> 00:53:10,630 836 00:53:10,630 --> 00:53:11,930 Find the scalar potential. 837 00:53:11,930 --> 00:53:14,994 Do you see it now? 838 00:53:14,994 --> 00:53:16,392 STUDENT: [INAUDIBLE]? 839 00:53:16,392 --> 00:53:19,560 PROFESSOR: Excellent, we teach now, got it. 840 00:53:19,560 --> 00:53:28,450 He says, I know where this comes from. 841 00:53:28,450 --> 00:53:31,650 I've got it, x squared plus y squared over 2. 842 00:53:31,650 --> 00:53:32,850 How did he do it? 843 00:53:32,850 --> 00:53:33,900 He's a genius. 844 00:53:33,900 --> 00:53:39,560 No he's not, he's just learning from the first time 845 00:53:39,560 --> 00:53:40,760 when he failed. 846 00:53:40,760 --> 00:53:44,180 And now he knows what he has to do and his brain says, 847 00:53:44,180 --> 00:53:47,230 oh, I got it. 848 00:53:47,230 --> 00:53:50,110 Now, [INAUDIBLE] could have applied this method 849 00:53:50,110 --> 00:53:54,640 and solved the coupled system and do it slowly 850 00:53:54,640 --> 00:53:56,820 and it would have taken him another 10 minutes. 851 00:53:56,820 --> 00:54:00,100 And he's in the final, he doesn't have time to spare. 852 00:54:00,100 --> 00:54:06,030 If he can guess the potential and then verify that, 853 00:54:06,030 --> 00:54:07,720 it's going to be easy for him. 854 00:54:07,720 --> 00:54:08,315 Why is that? 855 00:54:08,315 --> 00:54:15,490 This is going to be 2x over 2 x i, and this is 2y over 2 y j. 856 00:54:15,490 --> 00:54:17,838 So yeah, he was right. 857 00:54:17,838 --> 00:54:21,806 858 00:54:21,806 --> 00:54:26,634 All right, let me give you another one. 859 00:54:26,634 --> 00:54:29,104 Let's see who gets this one. 860 00:54:29,104 --> 00:54:34,080 F is a vector valued function, maybe a force field, 861 00:54:34,080 --> 00:54:35,341 that is this. 862 00:54:35,341 --> 00:54:38,047 863 00:54:38,047 --> 00:54:41,640 Of course there are many ways-- maybe somebody's 864 00:54:41,640 --> 00:54:44,180 going to ask you to prove this is 865 00:54:44,180 --> 00:54:49,320 conservative by the criterion, but they shouldn't tell you 866 00:54:49,320 --> 00:54:51,550 how to do it. 867 00:54:51,550 --> 00:54:53,050 So show this is conservative. 868 00:54:53,050 --> 00:54:56,550 If somebody doesn't want the scalar potential because they 869 00:54:56,550 --> 00:54:57,890 don't need it, let's say. 870 00:54:57,890 --> 00:55:01,945 Well, prime f1 with respect to y, I'll prime this with respect 871 00:55:01,945 --> 00:55:03,400 to x. 872 00:55:03,400 --> 00:55:07,560 f1 prime with respect to y equals 2x is the same 873 00:55:07,560 --> 00:55:09,720 as f2 prime with respect with. 874 00:55:09,720 --> 00:55:13,600 Yeah, it is conservative, I know it from the criterion. 875 00:55:13,600 --> 00:55:16,570 But [INAUDIBLE] knows that later I will ask him 876 00:55:16,570 --> 00:55:19,150 for the scalar potential. 877 00:55:19,150 --> 00:55:25,610 And I wonder if he can find it for me without computing it 878 00:55:25,610 --> 00:55:27,030 by solving the system. 879 00:55:27,030 --> 00:55:30,265 Just from his mathematical intuition 880 00:55:30,265 --> 00:55:33,036 that is running in the background of your-- 881 00:55:33,036 --> 00:55:35,880 STUDENT: x squared, multiply y [INAUDIBLE]. 882 00:55:35,880 --> 00:55:38,405 PROFESSOR: x squared y, excellent. 883 00:55:38,405 --> 00:55:42,770 884 00:55:42,770 --> 00:55:47,140 Zach came up with it and anybody else? 885 00:55:47,140 --> 00:55:49,090 Alex? 886 00:55:49,090 --> 00:55:51,610 So all three of you, OK? 887 00:55:51,610 --> 00:55:54,060 Squared y, very good. 888 00:55:54,060 --> 00:55:56,030 Was it hard? 889 00:55:56,030 --> 00:55:59,480 Yeah, it's hard for most people. 890 00:55:59,480 --> 00:56:02,720 It was hard for me when I first saw 891 00:56:02,720 --> 00:56:05,530 that in the first 30 minutes of becoming familiar 892 00:56:05,530 --> 00:56:09,283 with the scalar potential, I was 18 or 19. 893 00:56:09,283 --> 00:56:11,928 But then I got it in about half an hour 894 00:56:11,928 --> 00:56:15,536 and I was able to do them mentally. 895 00:56:15,536 --> 00:56:20,430 Most of the examples I got were really nice. 896 00:56:20,430 --> 00:56:28,730 Were on purpose made nice for us for the exam to work fast. 897 00:56:28,730 --> 00:56:35,790 And now let's see why would the work really not 898 00:56:35,790 --> 00:56:38,620 depend on the trajectory you are taking 899 00:56:38,620 --> 00:56:41,070 if your force is conservative. 900 00:56:41,070 --> 00:56:42,955 If the force is conservative there 901 00:56:42,955 --> 00:56:46,440 is something magic that's going to happen. 902 00:56:46,440 --> 00:56:50,600 And we really don't know what that is, 903 00:56:50,600 --> 00:56:53,160 but we should be able to prove. 904 00:56:53,160 --> 00:56:57,600 905 00:56:57,600 --> 00:57:19,890 So theorem, actually this is funny. 906 00:57:19,890 --> 00:57:25,511 It's called the fundamental theorem of path integrals 907 00:57:25,511 --> 00:57:28,457 but it's the fundamental theorem of calculus 3. 908 00:57:28,457 --> 00:57:36,313 I'm going to write it like this, the fundamental theorem 909 00:57:36,313 --> 00:57:45,170 of calc 3, path integrals. 910 00:57:45,170 --> 00:57:54,708 It's also called- 13.3, section- Independence of path. 911 00:57:54,708 --> 00:58:01,980 912 00:58:01,980 --> 00:58:11,733 So remember you have a work, w, over a path, c. 913 00:58:11,733 --> 00:58:22,610 F dot dR where there R is the regular parametrized curve 914 00:58:22,610 --> 00:58:23,110 overseen. 915 00:58:23,110 --> 00:58:33,850 916 00:58:33,850 --> 00:58:36,850 This is called a supposition vector. 917 00:58:36,850 --> 00:58:40,850 918 00:58:40,850 --> 00:58:45,730 Regular meaning c1, and never vanishing speed, 919 00:58:45,730 --> 00:58:49,400 the velocity never vanishes. 920 00:58:49,400 --> 00:58:55,025 Velocity times 0 such that f is continuous, 921 00:58:55,025 --> 00:59:01,580 or a nice enough integral. 922 00:59:01,580 --> 00:59:07,950 923 00:59:07,950 --> 00:59:25,920 If F is conservative of scalar potential, little f, 924 00:59:25,920 --> 00:59:39,430 then the work, w, equals little f at the endpoint 925 00:59:39,430 --> 00:59:41,560 minus little f at the origin. 926 00:59:41,560 --> 00:59:44,730 927 00:59:44,730 --> 00:59:56,682 Where, by origin and endpoint are those for the path, 928 00:59:56,682 --> 01:00:00,618 are those for the arc, are those for the curve, c. 929 01:00:00,618 --> 01:00:05,060 930 01:00:05,060 --> 01:00:09,576 So the work, the w, will be independent of time. 931 01:00:09,576 --> 01:00:18,720 So w will be independent of f. 932 01:00:18,720 --> 01:00:21,120 And you saw an example when I took 933 01:00:21,120 --> 01:00:24,260 a conservative function that was really nice, 934 01:00:24,260 --> 01:00:27,620 y times i plus x times j. 935 01:00:27,620 --> 01:00:29,440 That was the force field. 936 01:00:29,440 --> 01:00:33,470 Because that was conservative, we got w being 1 937 01:00:33,470 --> 01:00:34,780 no matter what path we took. 938 01:00:34,780 --> 01:00:37,970 We took a parabola, we took a straight line, 939 01:00:37,970 --> 01:00:39,770 and we could have taken a zig-zag 940 01:00:39,770 --> 01:00:41,860 and we still get w equals 1. 941 01:00:41,860 --> 01:00:44,550 So no matter what path you are taking. 942 01:00:44,550 --> 01:00:46,572 Can we prove this? 943 01:00:46,572 --> 01:00:53,000 Well, regular classes don't prove anything, almost nothing. 944 01:00:53,000 --> 01:00:56,900 But we are honor students so lets see what we can do. 945 01:00:56,900 --> 01:00:59,210 We have to understand what's going on. 946 01:00:59,210 --> 01:01:06,520 Why do we have this fundamental theorem of calculus 3? 947 01:01:06,520 --> 01:01:15,610 The work, w, can be expressed-- assume f is conservative 948 01:01:15,610 --> 01:01:19,700 which means it's going to come from a potential little f. 949 01:01:19,700 --> 01:01:29,077 Where f is [INAUDIBLE] scalar function over my domain, omega. 950 01:01:29,077 --> 01:01:33,280 951 01:01:33,280 --> 01:01:36,670 Now, the curve, c, is part of this omega 952 01:01:36,670 --> 01:01:40,490 so I don't have any problems on the curve. 953 01:01:40,490 --> 01:01:44,680 w will be rewritten beautifully. 954 01:01:44,680 --> 01:01:46,625 So I'm giving you a sketch of a proof. 955 01:01:46,625 --> 01:01:50,335 But you would be able to do this, maybe even better than me 956 01:01:50,335 --> 01:01:56,310 because I have taught you what you need to do. 957 01:01:56,310 --> 01:02:03,580 So this is going to be f1 i, plus f2 j. 958 01:02:03,580 --> 01:02:09,750 And I'm going to write it. f1 times-- what is this guys? 959 01:02:09,750 --> 01:02:14,420 dR, I taught you, you taught me, x prime of t, right? 960 01:02:14,420 --> 01:02:23,010 Plus f2 times y prime of t, all dt, and time from t0 to t1. 961 01:02:23,010 --> 01:02:27,480 I start my motion along the curve at t equals t0 962 01:02:27,480 --> 01:02:30,728 and I finished my motion at t equals t1. 963 01:02:30,728 --> 01:02:33,620 964 01:02:33,620 --> 01:02:36,770 Do I know where f1 and f2 are? 965 01:02:36,770 --> 01:02:38,860 This is the point, that's the whole point, 966 01:02:38,860 --> 01:02:41,780 I know who they are, thank God. 967 01:02:41,780 --> 01:02:47,310 And now I have to again apply some magical think, 968 01:02:47,310 --> 01:02:49,650 I'll ask you in a minute what that is. 969 01:02:49,650 --> 01:02:53,510 So, what is f1? 970 01:02:53,510 --> 01:02:56,100 df dx, or f sub base. 971 01:02:56,100 --> 01:02:59,260 If you don't like f sub base, if you don't like my notation, 972 01:02:59,260 --> 01:03:00,600 you put f sub x, right? 973 01:03:00,600 --> 01:03:02,551 And this df dy. 974 01:03:02,551 --> 01:03:03,050 Why? 975 01:03:03,050 --> 01:03:05,340 Because it's conservative and that 976 01:03:05,340 --> 01:03:08,080 was the gradient of little f. 977 01:03:08,080 --> 01:03:11,060 Of course I'm using the fact that the first component would 978 01:03:11,060 --> 01:03:13,610 be the partial of little f with respect to x. 979 01:03:13,610 --> 01:03:16,032 The second component would be the partial of little 980 01:03:16,032 --> 01:03:16,865 f with respect to y. 981 01:03:16,865 --> 01:03:19,270 Have you seen this formula before? 982 01:03:19,270 --> 01:03:23,599 What in the world is this formula? 983 01:03:23,599 --> 01:03:25,502 STUDENT: It's the chain rule? 984 01:03:25,502 --> 01:03:26,828 PROFESSOR: It's the chain rule. 985 01:03:26,828 --> 01:03:30,630 I don't have a dollar but I will give you a dollar, OK? 986 01:03:30,630 --> 01:03:33,960 Imagine a virtual dollar. 987 01:03:33,960 --> 01:03:35,190 This is the chain rule. 988 01:03:35,190 --> 01:03:38,110 So by the chain rule we can write 989 01:03:38,110 --> 01:03:41,350 this to be the derivative with respect 990 01:03:41,350 --> 01:03:48,840 to t of little f of x of t, and y of t. 991 01:03:48,840 --> 01:03:52,630 Alrighty, so I know what I'm doing. 992 01:03:52,630 --> 01:03:57,860 I know that by chain rule I had little f evaluated at x of t, 993 01:03:57,860 --> 01:04:01,640 y of t, and time t, prime with respect to t. 994 01:04:01,640 --> 01:04:07,750 Now when we take the fundamental theorem of calculus, FTC. 995 01:04:07,750 --> 01:04:13,540 That reminds me, I was teaching calc 1 a few years ago 996 01:04:13,540 --> 01:04:17,390 and I said, that's the Federal Trade Commission. 997 01:04:17,390 --> 01:04:20,755 Federal Trade Commission, fundamental theorem 998 01:04:20,755 --> 01:04:22,450 of calculus. 999 01:04:22,450 --> 01:04:27,988 So coming back to what I have, I prove 1000 01:04:27,988 --> 01:04:36,890 that w is the Federal Trade Commission, no. 1001 01:04:36,890 --> 01:04:43,450 w is the application of something 1002 01:04:43,450 --> 01:04:49,719 that we knew from calc 1, which is beautiful. 1003 01:04:49,719 --> 01:05:00,460 f of xt, y of t dt, this is nothing but what? 1004 01:05:00,460 --> 01:05:05,530 Little f evaluated at-- I'm going to have to write it down, 1005 01:05:05,530 --> 01:05:07,450 this whole sausage. 1006 01:05:07,450 --> 01:05:13,700 f of x of t1, y of t1, minus f of x of t0, y of t0. 1007 01:05:13,700 --> 01:05:17,240 For somebody as lazy as I am, that they effort. 1008 01:05:17,240 --> 01:05:19,490 How can I write It better? 1009 01:05:19,490 --> 01:05:26,060 f at the endpoint minus f at the origin. 1010 01:05:26,060 --> 01:05:29,450 And of course, we are trying to be quite rigorous in the book. 1011 01:05:29,450 --> 01:05:31,480 We would never say that in the book. 1012 01:05:31,480 --> 01:05:34,650 We actually denote the first point 1013 01:05:34,650 --> 01:05:37,690 with p, the origin, and the endpoint with q. 1014 01:05:37,690 --> 01:05:41,808 So we say, f of q minus f of p. 1015 01:05:41,808 --> 01:05:49,920 And we proved q e d, we proved the fundamental theorem 1016 01:05:49,920 --> 01:05:53,000 of path integrals, the independence of that. 1017 01:05:53,000 --> 01:06:00,200 So that means the work is independent of path 1018 01:06:00,200 --> 01:06:03,000 when the force is conservative. 1019 01:06:03,000 --> 01:06:07,355 Now attention, if f is not conservative you are dead meat. 1020 01:06:07,355 --> 01:06:12,500 You cannot say what I just said. 1021 01:06:12,500 --> 01:06:16,390 So I'll give you two separate examples 1022 01:06:16,390 --> 01:06:19,692 and let's see how we solve each of them. 1023 01:06:19,692 --> 01:06:27,400 1024 01:06:27,400 --> 01:06:30,400 A final exam type of problem-- every final exam 1025 01:06:30,400 --> 01:06:36,100 contains an application like that. 1026 01:06:36,100 --> 01:06:41,940 Even the force field, f, or the vector value, f. 1027 01:06:41,940 --> 01:06:44,056 Is it conservative? 1028 01:06:44,056 --> 01:06:51,600 Prove what is proved and after that-- so the path 1029 01:06:51,600 --> 01:06:54,642 integral in any way you can. 1030 01:06:54,642 --> 01:06:58,020 If it's conservative you're really lucky because you're 1031 01:06:58,020 --> 01:06:58,520 in business. 1032 01:06:58,520 --> 01:06:59,900 You don't have to do any work. 1033 01:06:59,900 --> 01:07:03,585 You just find the little scalar potential evaluated 1034 01:07:03,585 --> 01:07:08,500 at the endpoints and subtract, and that's your answer. 1035 01:07:08,500 --> 01:07:13,070 So I'm going to give you an example of a final exam problem 1036 01:07:13,070 --> 01:07:15,545 that happened in the past year. 1037 01:07:15,545 --> 01:07:18,515 So, a final type exam problem. 1038 01:07:18,515 --> 01:07:25,445 1039 01:07:25,445 --> 01:07:37,242 f of xy equals 2xy i plus x squared j, 1040 01:07:37,242 --> 01:07:43,500 over r over r squared. 1041 01:07:43,500 --> 01:07:45,750 STUDENT: Didn't we just do [INAUDIBLE] that? 1042 01:07:45,750 --> 01:07:49,150 PROFESSOR: Well, I just did that but I changed the problem. 1043 01:07:49,150 --> 01:07:51,242 I wanted to keep the same force field. 1044 01:07:51,242 --> 01:07:51,950 STUDENT: Alright. 1045 01:07:51,950 --> 01:07:52,810 PROFESSOR: OK. 1046 01:07:52,810 --> 01:08:18,319 Compute the work, w, performed by f along the arc 1047 01:08:18,319 --> 01:08:23,725 of the circle in the picture. 1048 01:08:23,725 --> 01:08:26,710 1049 01:08:26,710 --> 01:08:27,930 And they draw a picture. 1050 01:08:27,930 --> 01:08:31,649 And they do a picture for you, and you stare at this picture 1051 01:08:31,649 --> 01:08:49,290 and-- So you say, oh my God, if I 1052 01:08:49,290 --> 01:08:53,046 were to parametrize it would be a little bit of-- I 1053 01:08:53,046 --> 01:08:56,259 could, but it would be a little bit of work. 1054 01:08:56,259 --> 01:09:00,340 I would have x equals 2 cosine t, y equals sine t. 1055 01:09:00,340 --> 01:09:05,029 I would have to plug in and do that whole work, definition 1056 01:09:05,029 --> 01:09:06,270 with parametrization. 1057 01:09:06,270 --> 01:09:08,210 Do you have to parametrize? 1058 01:09:08,210 --> 01:09:10,149 Not in this case, why? 1059 01:09:10,149 --> 01:09:12,729 Because the f is conservative. 1060 01:09:12,729 --> 01:09:19,470 If they ask you- some professors give hints, most of them 1061 01:09:19,470 --> 01:09:22,558 are nice and give hints- show f is conservative. 1062 01:09:22,558 --> 01:09:28,859 1063 01:09:28,859 --> 01:09:30,729 So that's a big hint in the sense 1064 01:09:30,729 --> 01:09:33,095 that you see it immediately, how you do it. 1065 01:09:33,095 --> 01:09:36,149 You have 2xy prime with respect to y, 1066 01:09:36,149 --> 01:09:39,710 is 2x, which is x squared prime with respect to x. 1067 01:09:39,710 --> 01:09:41,700 So it is conservative. 1068 01:09:41,700 --> 01:09:44,350 But he or she told you more. 1069 01:09:44,350 --> 01:09:46,859 He said, I'm selling you something here, 1070 01:09:46,859 --> 01:09:50,080 you have to get your own scalar potential. 1071 01:09:50,080 --> 01:09:56,920 And you did, and you got x squared y. 1072 01:09:56,920 --> 01:09:58,970 Now, most of the scalar potentials 1073 01:09:58,970 --> 01:10:01,620 that we are giving you on the exam 1074 01:10:01,620 --> 01:10:04,196 can be seen with naked eyes. 1075 01:10:04,196 --> 01:10:06,770 You wouldn't have to do all the integration of that 1076 01:10:06,770 --> 01:10:09,540 coupled system with respect to x, with respect to y, 1077 01:10:09,540 --> 01:10:14,160 integrate backwards, and things like that. 1078 01:10:14,160 --> 01:10:16,700 What do I need to do in that case guys? 1079 01:10:16,700 --> 01:10:19,018 Say in words. 1080 01:10:19,018 --> 01:10:22,480 Since the force is conservative, just two lines. 1081 01:10:22,480 --> 01:10:25,800 I'm applying the fundamental theorem of calculus. 1082 01:10:25,800 --> 01:10:29,650 I'm applying the fundamental theorem of path integrals. 1083 01:10:29,650 --> 01:10:33,050 I know the work is independent of that. 1084 01:10:33,050 --> 01:10:38,750 So w, in this case is already there, 1085 01:10:38,750 --> 01:10:43,230 is going to be f at the point. 1086 01:10:43,230 --> 01:10:48,006 I didn't say how I'm going to travel it, in what direction. 1087 01:10:48,006 --> 01:10:52,028 f of q minus f of p. 1088 01:10:52,028 --> 01:10:57,360 1089 01:10:57,360 --> 01:11:00,630 And you'll say, well what does it mean? 1090 01:11:00,630 --> 01:11:02,490 How do we do that? 1091 01:11:02,490 --> 01:11:05,890 That means x squared y, evaluated at q, 1092 01:11:05,890 --> 01:11:09,940 who the heck is q? 1093 01:11:09,940 --> 01:11:13,710 Attention, negative 2, 0, Matt, you got it? 1094 01:11:13,710 --> 01:11:14,920 OK, right? 1095 01:11:14,920 --> 01:11:17,770 So, are you guys with me? 1096 01:11:17,770 --> 01:11:19,550 Right? 1097 01:11:19,550 --> 01:11:22,520 And p is 2, 0. 1098 01:11:22,520 --> 01:11:31,690 So at negative 2, 0, minus x squared y at 2, 0. 1099 01:11:31,690 --> 01:11:35,170 So, if you set 0, big, as you knew that you got 0, 1100 01:11:35,170 --> 01:11:38,235 that is the answer. 1101 01:11:38,235 --> 01:11:43,830 Now if somebody would give you a wiggly look that this guy's 1102 01:11:43,830 --> 01:11:49,640 wrong, then he took this past. 1103 01:11:49,640 --> 01:11:52,740 He's going to do exactly the same work 1104 01:11:52,740 --> 01:11:57,800 if he's under influence the same conservative force. 1105 01:11:57,800 --> 01:12:01,254 If the force acting on it is the same. 1106 01:12:01,254 --> 01:12:04,265 No matter what path you are taking-- yes, sir? 1107 01:12:04,265 --> 01:12:07,320 STUDENT: Is it still working for your self-intersecting. 1108 01:12:07,320 --> 01:12:11,420 PROFESSOR: Yeah, because you're not stopping, 1109 01:12:11,420 --> 01:12:13,120 it works for any parametrization. 1110 01:12:13,120 --> 01:12:16,390 So if you're able to parametrize that as a differentiable 1111 01:12:16,390 --> 01:12:20,640 function so that the derivative would never vanish, 1112 01:12:20,640 --> 01:12:23,090 it's going to work, right? 1113 01:12:23,090 --> 01:12:27,412 All right, it can work also for this piecewise contour 1114 01:12:27,412 --> 01:12:29,300 or any other path point. 1115 01:12:29,300 --> 01:12:32,720 As long as it starts and it ends at the same point 1116 01:12:32,720 --> 01:12:38,020 and as long as your conservative force is the same. 1117 01:12:38,020 --> 01:12:39,890 So that force is very [INAUDIBLE], yes? 1118 01:12:39,890 --> 01:12:42,536 STUDENT: Do [INAUDIBLE] graphs but the endpoints the same. 1119 01:12:42,536 --> 01:12:44,840 And if they are conservative then [INTERPOSING VOICES]. 1120 01:12:44,840 --> 01:12:46,506 PROFESSOR: If everything is conservative 1121 01:12:46,506 --> 01:12:49,550 the work along this path is the same. 1122 01:12:49,550 --> 01:12:52,080 There were people who played games 1123 01:12:52,080 --> 01:12:54,254 like that to catch if the student knows what 1124 01:12:54,254 --> 01:12:55,420 they are talking-- yes, sir? 1125 01:12:55,420 --> 01:12:59,067 STUDENT: So, in a problem, if they wanted 1126 01:12:59,067 --> 01:13:00,733 to find the work couldn't we simplify it 1127 01:13:00,733 --> 01:13:02,539 by just saying, we need to find-- because, 1128 01:13:02,539 --> 01:13:05,080 since we're in R2, couldn't we just say it's a straight line? 1129 01:13:05,080 --> 01:13:07,132 Because that's the-- Like, instead of a curve 1130 01:13:07,132 --> 01:13:08,660 we could just set the straight line [INTERPOSING VOICES]. 1131 01:13:08,660 --> 01:13:10,260 PROFESSOR: If it were moving from here 1132 01:13:10,260 --> 01:13:12,270 to here in a straight line you would still 1133 01:13:12,270 --> 01:13:13,940 get the same answer. 1134 01:13:13,940 --> 01:13:17,810 And you could have-- if you did that, actually, 1135 01:13:17,810 --> 01:13:21,170 if you were to compute this kind of work on a straight line 1136 01:13:21,170 --> 01:13:26,120 it has-- let me show you something. 1137 01:13:26,120 --> 01:13:32,470 You see, when you compute dx y dx, plus x squared dy. 1138 01:13:32,470 --> 01:13:38,460 If y is 0 like Matthew said, I'm walking on-- I'm not drunk. 1139 01:13:38,460 --> 01:13:41,350 I'm walking straight. 1140 01:13:41,350 --> 01:13:46,810 y will be 0 here, and 0 here, and the integral will be 0. 1141 01:13:46,810 --> 01:13:50,580 So he would have noticed that from the beginning. 1142 01:13:50,580 --> 01:13:54,240 But unless you know the force is conservative, 1143 01:13:54,240 --> 01:13:57,890 there is no guarantee that on another path 1144 01:13:57,890 --> 01:14:01,490 you don't have a different answer, right? 1145 01:14:01,490 --> 01:14:05,230 So, let me give you another example because now Matthew 1146 01:14:05,230 --> 01:14:06,610 brought this up. 1147 01:14:06,610 --> 01:14:09,230 1148 01:14:09,230 --> 01:14:13,620 A catchy example that a professor 1149 01:14:13,620 --> 01:14:18,470 gave just to make the students life miserable, 1150 01:14:18,470 --> 01:14:22,740 and I'll show it to you in a second. 1151 01:14:22,740 --> 01:14:29,060 1152 01:14:29,060 --> 01:14:36,860 He said, for the picture-- very similar to this one, 1153 01:14:36,860 --> 01:14:39,924 just to make people confused. 1154 01:14:39,924 --> 01:14:43,270 Somebody gives you this arc of a circle 1155 01:14:43,270 --> 01:14:45,660 and you travel from a to b. 1156 01:14:45,660 --> 01:14:49,500 And this the thing. 1157 01:14:49,500 --> 01:15:01,850 And he says, compute w for the force given by y i plus j. 1158 01:15:01,850 --> 01:15:07,760 And the students said OK, I guess 1159 01:15:07,760 --> 01:15:12,440 I'm going to get 0 because I'm going to get something 1160 01:15:12,440 --> 01:15:16,610 like y dx plus x dy. 1161 01:15:16,610 --> 01:15:21,699 And if y is 0, I get 0, and that way you wouldn't be 0 1162 01:15:21,699 --> 01:15:22,240 and I'm done. 1163 01:15:22,240 --> 01:15:27,489 No, it's not how you think because this is not 1164 01:15:27,489 --> 01:15:30,210 conservative. 1165 01:15:30,210 --> 01:15:33,800 So you cannot say I can change my path and it's still going 1166 01:15:33,800 --> 01:15:35,820 to be the same. 1167 01:15:35,820 --> 01:15:38,260 No, why is this not conservative? 1168 01:15:38,260 --> 01:15:39,980 Quickly, this prime with respect to y 1169 01:15:39,980 --> 01:15:42,580 is 1, this prime with respect to x is 0. 1170 01:15:42,580 --> 01:15:46,055 So 1 different from 0 so, oh my God, no. 1171 01:15:46,055 --> 01:15:48,160 In that case, why do we do? 1172 01:15:48,160 --> 01:15:52,480 We have no other choice but say, x equals 2 cosine t, 1173 01:15:52,480 --> 01:15:57,770 y equals 2 sine t, and t between 0 and pi. 1174 01:15:57,770 --> 01:16:04,110 And then I get integral of f1, y, what the heck is y? 1175 01:16:04,110 --> 01:16:08,948 2 sine t, times x prime of t. 1176 01:16:08,948 --> 01:16:15,360 1177 01:16:15,360 --> 01:16:20,850 Yeah, minus 2 sine t, this is x prime. 1178 01:16:20,850 --> 01:16:24,650 Plus 1, are you guys with me? 1179 01:16:24,650 --> 01:16:30,140 Times y prime, which is 2 cosine t, dt. 1180 01:16:30,140 --> 01:16:33,370 And t between 0 and pi. 1181 01:16:33,370 --> 01:16:36,190 1182 01:16:36,190 --> 01:16:42,242 And you get something ugly, you get 0 to pi. 1183 01:16:42,242 --> 01:16:43,660 What is the nice thing? 1184 01:16:43,660 --> 01:16:48,845 When you integrate this with respect to t, you get sine t. 1185 01:16:48,845 --> 01:16:53,480 And thank God, sine, whether you are at 0 or if pi is still 0. 1186 01:16:53,480 --> 01:16:55,600 So this part will disappear. 1187 01:16:55,600 --> 01:17:01,800 So all you have left is minus 4 sine squared dt. 1188 01:17:01,800 --> 01:17:04,760 But you are not done so compute this at home 1189 01:17:04,760 --> 01:17:07,438 because we are out of time. 1190 01:17:07,438 --> 01:17:11,075 So don't jump to conclusions unless you know 1191 01:17:11,075 --> 01:17:12,894 that the force is conservative. 1192 01:17:12,894 --> 01:17:14,878 If your force is not conservative 1193 01:17:14,878 --> 01:17:17,688 then things are going to look very ugly 1194 01:17:17,688 --> 01:17:20,830 and your only chance is to go back to the parametrization, 1195 01:17:20,830 --> 01:17:24,302 to the basics. 1196 01:17:24,302 --> 01:17:27,310 So we are practically done with 13.3 1197 01:17:27,310 --> 01:17:31,320 but I want to watch more examples next time. 1198 01:17:31,320 --> 01:17:33,120 And I'll send you the homework. 1199 01:17:33,120 --> 01:17:38,220 Over the weekend you would be able to start doing homework. 1200 01:17:38,220 --> 01:17:40,620 Now, when shall I grab the homework? 1201 01:17:40,620 --> 01:17:43,320 What if I closed it right before the final, is that? 1202 01:17:43,320 --> 01:17:44,220 STUDENT: Yeah. 1203 01:17:44,220 --> 01:17:46,070 PROFESSOR: Yeah? 1204 01:17:46,070 --> 02:39:17,402