WEBVTT 00:00:00.000 --> 00:00:00.499 00:00:00.499 --> 00:00:04.934 PROFESSOR: We will pick up from where we left. 00:00:04.934 --> 00:00:08.406 I hope the attendance will get a little bit better today. 00:00:08.406 --> 00:00:13.862 It's not even Friday, it's Thursday night. 00:00:13.862 --> 00:00:17.334 So last time we talked a little bit about chapter 13, 00:00:17.334 --> 00:00:21.302 we started 13-1. 00:00:21.302 --> 00:00:27.750 I wanted to remind you that we revisited the notion of work. 00:00:27.750 --> 00:00:40.700 00:00:40.700 --> 00:00:44.226 Now, if you notice what the book does, 00:00:44.226 --> 00:00:45.980 it doesn't give you any specifics 00:00:45.980 --> 00:00:48.860 about the force field. 00:00:48.860 --> 00:00:50.550 May the force be with you. 00:00:50.550 --> 00:00:56.031 They don't say what kind of animal this f is. 00:00:56.031 --> 00:01:02.570 We sort of informally said I'm going to have 00:01:02.570 --> 00:01:06.290 some sort of path integral. 00:01:06.290 --> 00:01:10.880 And I didn't say what conditions I was assuming about f. 00:01:10.880 --> 00:01:15.030 And I just said that r is the position vector. 00:01:15.030 --> 00:01:18.170 00:01:18.170 --> 00:01:22.825 It's important for us to imagine that is plus c1, what 00:01:22.825 --> 00:01:24.290 does that mean, c1? 00:01:24.290 --> 00:01:30.200 It means that this function, let's write it R of t, equals. 00:01:30.200 --> 00:01:35.360 Let's say we are implying not in space, so we have x of t, 00:01:35.360 --> 00:01:40.450 y of t, the parametrization of this position vector. 00:01:40.450 --> 00:01:42.900 Of course we wrote that last time as well, we 00:01:42.900 --> 00:01:46.280 said x is x of t, y is y of t. 00:01:46.280 --> 00:01:51.640 But why I took c1 and not continuous? 00:01:51.640 --> 00:01:53.240 Could anybody tell me? 00:01:53.240 --> 00:01:56.290 If I'm going to go ahead and differentiate it, 00:01:56.290 --> 00:01:59.290 of course I'd like it to be differentiable. 00:01:59.290 --> 00:02:03.520 And its derivatives should be continuous. 00:02:03.520 --> 00:02:07.660 But that's actually not enough for my purposes. 00:02:07.660 --> 00:02:11.170 So if I want R of t to be c1, that's good, 00:02:11.170 --> 00:02:13.150 I'm going to smile. 00:02:13.150 --> 00:02:18.082 But when we did that in chapter-- was it chapter 10? 00:02:18.082 --> 00:02:20.560 It was chapter 10, Erin, am I right? 00:02:20.560 --> 00:02:23.615 We assumed this was a regular curve. 00:02:23.615 --> 00:02:27.650 A regular curve is not just as differentiable 00:02:27.650 --> 00:02:31.390 with the derivative's continuous with respect to time. 00:02:31.390 --> 00:02:34.380 x prime of t, y prime of t, both must exist 00:02:34.380 --> 00:02:36.030 and must be continuous. 00:02:36.030 --> 00:02:39.850 We wanted something else about the velocity. 00:02:39.850 --> 00:02:42.200 Do you remember the drunken bug? 00:02:42.200 --> 00:02:46.520 The drunken but was fine and he was flying. 00:02:46.520 --> 00:02:49.820 As long as he was flying, everything was fine. 00:02:49.820 --> 00:02:54.080 When did the drunken bug have a problem? 00:02:54.080 --> 00:02:56.895 When the velocity field became 0, 00:02:56.895 --> 00:03:02.860 at the instant where the bug lost his velocity, right? 00:03:02.860 --> 00:03:10.740 So we said regular means c1 and R prime of t at any value of t 00:03:10.740 --> 00:03:13.520 should be different from 0. 00:03:13.520 --> 00:03:16.620 We do not allow the particle to stop on it's way. 00:03:16.620 --> 00:03:20.680 We don't allow it, whether it is a photon, a drunken bug, 00:03:20.680 --> 00:03:23.690 an airplane, or whatever it is. 00:03:23.690 --> 00:03:27.570 We don't want it to stop in it's trajectory. 00:03:27.570 --> 00:03:31.990 Is that good for other reasons as well? 00:03:31.990 --> 00:03:35.260 Very good for the reason that we want 00:03:35.260 --> 00:03:39.150 to think later in arc length. 00:03:39.150 --> 00:03:43.320 [INAUDIBLE] came up with this idea last time. 00:03:43.320 --> 00:03:46.480 I didn't want to tell you the truth, but he was right. 00:03:46.480 --> 00:03:53.010 One can define certain path integrals with respect 00:03:53.010 --> 00:03:56.540 to s, with respect to arc length parameter. 00:03:56.540 --> 00:03:58.400 But as you remember very well, he 00:03:58.400 --> 00:04:03.260 had this correspondence between an arbitrary parameter type t 00:04:03.260 --> 00:04:07.641 and s, and this is s of t. 00:04:07.641 --> 00:04:10.200 And also going back and forth, that 00:04:10.200 --> 00:04:15.370 means from s you head back to t. 00:04:15.370 --> 00:04:19.820 So here's s of t and this is t of s, right? 00:04:19.820 --> 00:04:26.500 So we have this correspondence and everything worked fine 00:04:26.500 --> 00:04:31.450 in terms of being able to invert that. 00:04:31.450 --> 00:04:34.320 And having some sort of equal morphisms 00:04:34.320 --> 00:04:40.620 as long as the velocity was non-0. 00:04:40.620 --> 00:04:43.720 OK, do you remember who s of t was? 00:04:43.720 --> 00:04:47.900 S of t was defined-- It was a long time ago. 00:04:47.900 --> 00:04:50.710 So I'm reminding you s of t was integral 00:04:50.710 --> 00:04:55.350 from 0 to t-- or from t0 to t. 00:04:55.350 --> 00:04:59.200 Your favorite initial moment in time. 00:04:59.200 --> 00:05:03.570 Of the speed, uh-huh, and what the heck was the speed? 00:05:03.570 --> 00:05:10.660 The speed was the norm or the length of the R prime of t. 00:05:10.660 --> 00:05:16.646 This is called speed, that we assume different from 0, 00:05:16.646 --> 00:05:19.160 for a good purpose. 00:05:19.160 --> 00:05:24.600 We can go back and forth between t and s, t to s, 00:05:24.600 --> 00:05:30.440 s to t, with differentiable functions. 00:05:30.440 --> 00:05:35.300 Good, so now we can apply the inverse mapping through them. 00:05:35.300 --> 00:05:38.170 We can do all sorts of stuff with that. 00:05:38.170 --> 00:05:42.430 On this one we did not quite define it rigorously. 00:05:42.430 --> 00:05:43.690 What did they say is? 00:05:43.690 --> 00:05:46.490 We said f would be a good enough function, 00:05:46.490 --> 00:05:50.330 but know that I do not need f to be c1. 00:05:50.330 --> 00:05:54.424 This is too strong, too strong. 00:05:54.424 --> 00:05:58.870 So in Calc 1 when you had to integrate a function of one 00:05:58.870 --> 00:06:02.972 variable you just assumed that- in Calc 1 00:06:02.972 --> 00:06:04.700 I remember- you assume that continuous. 00:06:04.700 --> 00:06:07.580 It doesn't even have to be continuous 00:06:07.580 --> 00:06:09.862 but let's assume that f would be continuous. 00:06:09.862 --> 00:06:15.650 00:06:15.650 --> 00:06:21.105 OK, so you have, in one sense, that the composition with R, 00:06:21.105 --> 00:06:27.006 if you have f of x of t, y of t, z of t. 00:06:27.006 --> 00:06:33.600 In terms of time will be a functions of one variable, 00:06:33.600 --> 00:06:35.580 and this will be continuous. 00:06:35.580 --> 00:06:39.060 00:06:39.060 --> 00:06:41.050 All right? 00:06:41.050 --> 00:06:44.610 OK, now what if it's not continuous? 00:06:44.610 --> 00:06:47.850 Can't I have a piecewise, continuous function? 00:06:47.850 --> 00:06:51.425 Like in Calc 1, do you guys remember we had some of this? 00:06:51.425 --> 00:06:54.170 And from here like that and from here like this 00:06:54.170 --> 00:06:55.170 and from here like that. 00:06:55.170 --> 00:06:58.000 And we had these continuities, and this was piecewise 00:06:58.000 --> 00:06:59.860 continuous. 00:06:59.860 --> 00:07:02.660 Yeah, for god sake, I can integrate that. 00:07:02.660 --> 00:07:06.530 Why do we assume integral of a continuous function? 00:07:06.530 --> 00:07:08.680 Just to make our lives easier and also 00:07:08.680 --> 00:07:12.500 because we are in freshman and sophomore level Calculus. 00:07:12.500 --> 00:07:16.690 If we were in advanced Calculus we would say, 00:07:16.690 --> 00:07:21.650 I want this function to be integrable. 00:07:21.650 --> 00:07:24.540 This is a lot weaker than continuous, 00:07:24.540 --> 00:07:28.360 maybe the set of discontinuities is also very large. 00:07:28.360 --> 00:07:31.510 Who told you that you have finitely many jumps 00:07:31.510 --> 00:07:32.590 these continuities? 00:07:32.590 --> 00:07:34.530 Maybe you have a much larger set. 00:07:34.530 --> 00:07:39.070 And this is what you learn in advanced Calculus. 00:07:39.070 --> 00:07:41.710 But you are not at the level of a senior 00:07:41.710 --> 00:07:44.880 yet so we'll just assume, for the time being, 00:07:44.880 --> 00:07:47.070 that f is continuous. 00:07:47.070 --> 00:07:49.480 All right, and we say, what is this animal? 00:07:49.480 --> 00:07:52.330 We called it w and be baptised it. 00:07:52.330 --> 00:07:56.880 We said, just give it some sort of name 00:07:56.880 --> 00:07:58.550 and we say that is work. 00:07:58.550 --> 00:08:02.600 And by definition, by definition, 00:08:02.600 --> 00:08:09.375 this is going to be integral from-- Now, 00:08:09.375 --> 00:08:14.890 the thing is, we define this as a simple integral with respect 00:08:14.890 --> 00:08:17.900 to time as a definition. 00:08:17.900 --> 00:08:19.830 That doesn't mean that I introduced 00:08:19.830 --> 00:08:24.660 the notion of path integral the way I should, 00:08:24.660 --> 00:08:26.340 I was cheating on that. 00:08:26.340 --> 00:08:28.550 So the way we introduced it was like, 00:08:28.550 --> 00:08:33.100 let f be a function of the spatial coordinates 00:08:33.100 --> 00:08:34.650 in terms of time. 00:08:34.650 --> 00:08:37.789 x, y, z are space coordinates, t is time. 00:08:37.789 --> 00:08:42.068 So I have f of R of t here. 00:08:42.068 --> 00:08:45.443 Dot Product, who the heck is the R? 00:08:45.443 --> 00:08:48.280 This is nothing but a vector art drawing. 00:08:48.280 --> 00:08:50.510 These are both vectors, sometimes 00:08:50.510 --> 00:08:53.440 I should put them in bold like they do in the book. 00:08:53.440 --> 00:08:56.460 To make it clear I can put a bar on top of them, 00:08:56.460 --> 00:08:57.890 they are free vectors. 00:08:57.890 --> 00:09:02.990 So, f of R times R prime of t dt. 00:09:02.990 --> 00:09:07.180 And your favorite moments of time are-- let's say on my arc 00:09:07.180 --> 00:09:12.270 that I'm describing from time, t, equals a, to time equals b. 00:09:12.270 --> 00:09:15.220 Therefore, I'm going to take time for a to b. 00:09:15.220 --> 00:09:19.535 And this is how we define the work of a force. 00:09:19.535 --> 00:09:22.786 00:09:22.786 --> 00:09:25.610 The work of a force that's acting 00:09:25.610 --> 00:09:32.770 on a particle that is moving between time, a, and time, b, 00:09:32.770 --> 00:09:36.890 on this arc of a curve which is called c. 00:09:36.890 --> 00:09:38.273 Do you like this c? 00:09:38.273 --> 00:09:41.500 Okay, and the force is different. 00:09:41.500 --> 00:09:44.055 So we have a force field. 00:09:44.055 --> 00:09:46.360 So I cheated, I knew a lot in the sense 00:09:46.360 --> 00:09:48.510 that I didn't tell you how you actually 00:09:48.510 --> 00:09:55.280 introduce the path integral. 00:09:55.280 --> 00:10:01.220 Now this is more or less where I stop and [INAUDIBLE]. 00:10:01.220 --> 00:10:04.650 But couldn't we actually introduce this integral 00:10:04.650 --> 00:10:11.320 and even define it with respect to some arc length grammar? 00:10:11.320 --> 00:10:14.580 Maybe if everything goes fine in terms of theory? 00:10:14.580 --> 00:10:16.250 And the answer is yes. 00:10:16.250 --> 00:10:20.100 And I'm going to show you how one can do that. 00:10:20.100 --> 00:10:25.670 I'm going to go ahead and clean here a little bit. 00:10:25.670 --> 00:10:29.170 I'm going to leave this on by comparison for awhile. 00:10:29.170 --> 00:10:33.510 And then I will assume something that we have not 00:10:33.510 --> 00:10:40.044 defined whatsoever, which is an animal called path integral. 00:10:40.044 --> 00:10:54.260 So the path integral of a vector field along a trajectory, c. 00:10:54.260 --> 00:10:55.790 I don't know how to draw. 00:10:55.790 --> 00:10:58.630 I will draw some skewed curve, how about that? 00:10:58.630 --> 00:11:04.130 Some pretty skewed curve, c, it's not self intersecting, 00:11:04.130 --> 00:11:05.360 not necessarily. 00:11:05.360 --> 00:11:08.770 You guys have to imagine this is like the trajectory 00:11:08.770 --> 00:11:13.220 of an airplane in the sky, right? 00:11:13.220 --> 00:11:17.740 OK, and I have it on d equals a, to d equals b. 00:11:17.740 --> 00:11:21.150 But I said forget about the time, t, 00:11:21.150 --> 00:11:24.520 maybe I can do everything in arc length forever. 00:11:24.520 --> 00:11:26.320 So if that particle, or airplane, 00:11:26.320 --> 00:11:31.120 or whatever it is has a continuous motion, 00:11:31.120 --> 00:11:32.750 that's also differentiable. 00:11:32.750 --> 00:11:35.710 And the velocity never becomes zero. 00:11:35.710 --> 00:11:38.650 Then I can parametrize an arc length 00:11:38.650 --> 00:11:44.160 and I can say, forget about it, I have integral over c. 00:11:44.160 --> 00:11:46.970 See, this is c, it's not f, okay? 00:11:46.970 --> 00:11:54.752 But f of x of s, y of s, z of s, okay? 00:11:54.752 --> 00:11:57.570 And this is going to be a ds. 00:11:57.570 --> 00:12:00.690 And you'll say, yes Magdelina-- this is little s, I'm sorry. 00:12:00.690 --> 00:12:03.310 Yes, Magdelina, but what the heck is this animal, 00:12:03.310 --> 00:12:05.040 you've never introduced it. 00:12:05.040 --> 00:12:09.000 I have not introduced it because I have to discuss about it. 00:12:09.000 --> 00:12:14.740 When we introduced Riemann sums, then we took the limit. 00:12:14.740 --> 00:12:19.890 We always have to think how to partition our domains. 00:12:19.890 --> 00:12:30.620 So this curve can be partitioned in as many as n, this is s k. 00:12:30.620 --> 00:12:36.220 S k, this is s1, and this is s n, the last of the Mohicans. 00:12:36.220 --> 00:12:42.170 I have n sub intervals, pieces of the art. 00:12:42.170 --> 00:12:44.410 And how am I going to introduce this? 00:12:44.410 --> 00:12:47.220 As the limit, if it exists. 00:12:47.220 --> 00:12:49.480 Because I can be in trouble, maybe this limit 00:12:49.480 --> 00:12:51.580 is not going to exist. 00:12:51.580 --> 00:12:54.220 The sum of what? 00:12:54.220 --> 00:12:58.760 For every [? seg ?] partition I will take a little arbitrary 00:12:58.760 --> 00:13:00.190 point inside the subarc. 00:13:00.190 --> 00:13:03.150 00:13:03.150 --> 00:13:03.783 Subarc? 00:13:03.783 --> 00:13:04.550 STUDENT: Yeah. 00:13:04.550 --> 00:13:06.320 PROFESSOR: Subarc, it's a little arc. 00:13:06.320 --> 00:13:09.450 Contains a-- let's take it here. 00:13:09.450 --> 00:13:13.000 What am I going to define in terms of wind? 00:13:13.000 --> 00:13:20.726 s k, y k, and z k, some people put a star on it 00:13:20.726 --> 00:13:23.890 to make it obvious. 00:13:23.890 --> 00:13:27.040 But I'm going to go ahead and say 00:13:27.040 --> 00:13:35.600 x star k, y star k, z star k, is my arbitrary point in the k 00:13:35.600 --> 00:13:38.140 subarc. 00:13:38.140 --> 00:13:42.095 Times, what shall I multiply by? 00:13:42.095 --> 00:13:48.110 A delta sk, and then I take k from one to n 00:13:48.110 --> 00:13:52.140 and I press to the limit with respect n. 00:13:52.140 --> 00:13:57.150 But actually I could also say in some other ways 00:13:57.150 --> 00:14:05.170 that the partitions length goes to 0, delta s goes to zero. 00:14:05.170 --> 00:14:07.050 And you say but, now wait a minute, 00:14:07.050 --> 00:14:11.600 you have s1, s2 s3, s4, s k, little tiny subarc, 00:14:11.600 --> 00:14:13.416 what the heck is delta s? 00:14:13.416 --> 00:14:21.820 Delta s is the largest subarc. 00:14:21.820 --> 00:14:25.980 So the length of the largest subarc, length of the largest 00:14:25.980 --> 00:14:29.180 subarc in the partition. 00:14:29.180 --> 00:14:34.274 So the more points I take, the more I refine this. 00:14:34.274 --> 00:14:36.190 I take the points closer and closer and closer 00:14:36.190 --> 00:14:37.600 in this partition. 00:14:37.600 --> 00:14:41.010 What happens to the length of this partition? 00:14:41.010 --> 00:14:43.850 It shrinks to-- it goes to 0. 00:14:43.850 --> 00:14:45.960 Assuming that this would be the largest 00:14:45.960 --> 00:14:48.690 one, well if the largest one goes to 0, 00:14:48.690 --> 00:14:51.830 everybody else goes to 0. 00:14:51.830 --> 00:14:55.750 So this is a Riemann sum, can we know for sure 00:14:55.750 --> 00:14:57.970 that this limit exists? 00:14:57.970 --> 00:15:02.690 No, we hope to god that this limit exists. 00:15:02.690 --> 00:15:08.150 And if the limit exists then I will introduce this notion 00:15:08.150 --> 00:15:09.773 of integral around the back. 00:15:09.773 --> 00:15:15.450 00:15:15.450 --> 00:15:18.100 And you said, OK I believe you, but look, 00:15:18.100 --> 00:15:22.120 what is the connection between the work- the way 00:15:22.120 --> 00:15:25.850 you introduced it as a simple Calculus 1 integral here- 00:15:25.850 --> 00:15:30.500 and this animal that looks like an alien coming from the sky. 00:15:30.500 --> 00:15:33.280 We don't know how to look at it. 00:15:33.280 --> 00:15:37.830 Actually guys it's not so bad, you do the same thing 00:15:37.830 --> 00:15:40.240 as you did before. 00:15:40.240 --> 00:15:44.180 In a sense that, s is connected to any time parameter. 00:15:44.180 --> 00:15:47.750 So Mr. ds says, I'm your old friend, 00:15:47.750 --> 00:15:55.584 trust me, I know who I am. ds was the speed times dt. 00:15:55.584 --> 00:16:00.640 Who can tell me if we are in R three, and we are drunken bugs, 00:16:00.640 --> 00:16:02.880 ds will become what? 00:16:02.880 --> 00:16:07.550 A long square root times dt, and what's inside here? 00:16:07.550 --> 00:16:10.220 I want to see if you guys are awake. 00:16:10.220 --> 00:16:11.654 [INTERPOSING VOICES] 00:16:11.654 --> 00:16:17.180 PROFESSOR: Very good, x prime of t squared, I'm so lazy 00:16:17.180 --> 00:16:20.875 but I'll write it down. y prime of t squared plus z prime of t 00:16:20.875 --> 00:16:21.685 squared. 00:16:21.685 --> 00:16:24.210 And this is going to be the speed. 00:16:24.210 --> 00:16:28.980 So I can always do that, and in this case 00:16:28.980 --> 00:16:33.200 this is going to become always some-- let's say from time, t0, 00:16:33.200 --> 00:16:36.240 to time t1. 00:16:36.240 --> 00:16:39.245 Some in the integrals of-- some of the limit 00:16:39.245 --> 00:16:40.580 points for the time. 00:16:40.580 --> 00:16:45.460 I'm going to have f of R of s of t, 00:16:45.460 --> 00:16:47.670 in the end everything will depend on t. 00:16:47.670 --> 00:16:49.900 And this is my face being happy. 00:16:49.900 --> 00:16:51.970 It's not part of the integral. 00:16:51.970 --> 00:16:52.710 Saying what? 00:16:52.710 --> 00:16:55.290 Saying that, guys, if I plug in everything 00:16:55.290 --> 00:16:58.565 back in terms of t- I'm more familiar to that type 00:16:58.565 --> 00:17:00.770 of integral- then I have what? 00:17:00.770 --> 00:17:04.540 Square root of-- that's the arc length element 00:17:04.540 --> 00:17:07.450 x prime then t squared, plus y prime then t 00:17:07.450 --> 00:17:12.510 squared, plus z prime then t squared, dt. 00:17:12.510 --> 00:17:17.790 So in the end it is-- I think the video doesn't see me 00:17:17.790 --> 00:17:21.569 but it heard me, presumably. 00:17:21.569 --> 00:17:24.220 This is our old friend from Calc 1, 00:17:24.220 --> 00:17:29.990 which is the simple integral with respect to t from a to b. 00:17:29.990 --> 00:17:34.370 OK, all right, and we believe that the work 00:17:34.370 --> 00:17:36.870 can be expressed like that. 00:17:36.870 --> 00:17:39.610 I introduced it last time, I even 00:17:39.610 --> 00:17:41.860 proved it on some particular cases 00:17:41.860 --> 00:17:45.470 last time when Alex wasn't here because, I know why. 00:17:45.470 --> 00:17:46.927 Were you sick? 00:17:46.927 --> 00:17:48.510 ALEX: I'll talk to you about it later. 00:17:48.510 --> 00:17:49.850 I'm a bad person. 00:17:49.850 --> 00:17:56.915 PROFESSOR: All right, then I'm dragging an object like, 00:17:56.915 --> 00:18:02.610 the f was parallel to the direction of displacement. 00:18:02.610 --> 00:18:04.800 And then I said the work would be 00:18:04.800 --> 00:18:09.150 the magnitude of f times the magnitude of the displacement. 00:18:09.150 --> 00:18:13.450 And then we proved that is just a particular case of this, 00:18:13.450 --> 00:18:15.950 we proved that last time, it was a piece of cake. 00:18:15.950 --> 00:18:17.860 Actually, we proved the other one. 00:18:17.860 --> 00:18:23.280 It proved that if force is going to be oblique and at an angle 00:18:23.280 --> 00:18:27.180 theta with the displacement direction, then 00:18:27.180 --> 00:18:31.030 the work will be the magnitude of the force times 00:18:31.030 --> 00:18:35.796 cosine of theta, times the magnitude of displacement, 00:18:35.796 --> 00:18:36.730 all right? 00:18:36.730 --> 00:18:42.190 And that was all an application of this beautiful warp formula. 00:18:42.190 --> 00:18:46.215 Let's see something more interesting from an application 00:18:46.215 --> 00:18:48.000 viewpoint. 00:18:48.000 --> 00:18:53.030 Assume that you are looking at the washer, 00:18:53.030 --> 00:18:56.760 you are just doing laundry. 00:18:56.760 --> 00:19:01.460 And you are looking at this centrifugal force. 00:19:01.460 --> 00:19:08.170 We have two forces, one is centripetal towards the center 00:19:08.170 --> 00:19:11.590 of the motion, circular motion, one is centrifugal. 00:19:11.590 --> 00:19:15.220 I will take a centrifugal force f, 00:19:15.220 --> 00:19:19.440 and I will say I want to measure at the work 00:19:19.440 --> 00:19:25.310 that this force is producing in the circular motion 00:19:25.310 --> 00:19:27.700 of my dryer. 00:19:27.700 --> 00:19:31.710 My poor dryer died so I had to buy another one 00:19:31.710 --> 00:19:33.135 and it cost me a lot of money. 00:19:33.135 --> 00:19:36.180 And I was thinking, such a simple thing, 00:19:36.180 --> 00:19:38.630 you pay hundreds of dollars on it 00:19:38.630 --> 00:19:41.570 but, anyway, we take some things for granted. 00:19:41.570 --> 00:19:45.990 00:19:45.990 --> 00:19:54.610 I will take the washer because the washer is a simpler 00:19:54.610 --> 00:19:59.780 case in the sense that the motion-- I can assume it's 00:19:59.780 --> 00:20:04.180 a circular motion of constant velocity. 00:20:04.180 --> 00:20:07.490 And let's say this is the washer. 00:20:07.490 --> 00:20:10.840 00:20:10.840 --> 00:20:16.940 And centrifugal force is acting here. 00:20:16.940 --> 00:20:21.060 Let's call that-- what should it be? 00:20:21.060 --> 00:20:28.710 Well, it's continuing the position vector 00:20:28.710 --> 00:20:36.440 so let's call that lambda x I, plus lambda y j. 00:20:36.440 --> 00:20:42.360 In the sense that it's collinear to the vector that 00:20:42.360 --> 00:20:46.580 starts at origin, and here is got to be x of t, y. 00:20:46.580 --> 00:20:50.040 X and y are the special components 00:20:50.040 --> 00:20:52.560 at any point on my circular motion. 00:20:52.560 --> 00:20:55.460 If it's a circular motion I have x 00:20:55.460 --> 00:20:57.550 squared plus y squared equals r squared, 00:20:57.550 --> 00:21:01.660 where the radius is the radius of my washer. 00:21:01.660 --> 00:21:06.520 You have to compute the work produced 00:21:06.520 --> 00:21:10.960 by the centrifugal force in one full rotation. 00:21:10.960 --> 00:21:14.306 It doesn't matter, I can have infinitely many rotations. 00:21:14.306 --> 00:21:19.030 I can have a hundred rotations, I couldn't care less. 00:21:19.030 --> 00:21:23.310 But assume that the motion has constant speed. 00:21:23.310 --> 00:21:26.650 So if I wanted, I could parametrize in our things 00:21:26.650 --> 00:21:30.270 but it doesn't bring a difference. 00:21:30.270 --> 00:21:33.710 Because guys, when this speed is already constant, 00:21:33.710 --> 00:21:36.330 like for the circular motion you are familiar with. 00:21:36.330 --> 00:21:39.760 Or the helicoidal case you are familiar with, 00:21:39.760 --> 00:21:44.120 you also saw the case when the speed was constant. 00:21:44.120 --> 00:21:47.810 Practically, you were just rescaling the time 00:21:47.810 --> 00:21:52.750 to get to your speed, to your time parameter s, arc length. 00:21:52.750 --> 00:21:54.840 So whether you work with t, or you work with s, 00:21:54.840 --> 00:21:58.230 it's the same thing if the speed is a constant. 00:21:58.230 --> 00:22:02.030 So I'm not going to use my imagination to go and do it 00:22:02.030 --> 00:22:02.900 with respect to s. 00:22:02.900 --> 00:22:05.500 I could, but I couldn't give a damn 00:22:05.500 --> 00:22:09.565 because I'm going to have a beautiful t that you 00:22:09.565 --> 00:22:13.000 are going to help me recover. 00:22:13.000 --> 00:22:15.640 From here, what is the parametrization 00:22:15.640 --> 00:22:18.696 that comes to mind? 00:22:18.696 --> 00:22:19.570 Can you guys help me? 00:22:19.570 --> 00:22:23.640 I know you can after all the review of chapter 10 00:22:23.640 --> 00:22:25.860 and-- this is what? 00:22:25.860 --> 00:22:29.470 00:22:29.470 --> 00:22:32.970 R what? 00:22:32.970 --> 00:22:34.970 You should whisper cosine t. 00:22:34.970 --> 00:22:38.680 Say it out loud, be proud of what you know. 00:22:38.680 --> 00:22:40.640 This is R sine t. 00:22:40.640 --> 00:22:43.080 And let's take t between 0 and 2 pi. 00:22:43.080 --> 00:22:50.730 One, the revolution only, and then I say, good. 00:22:50.730 --> 00:22:53.996 The speed is what? 00:22:53.996 --> 00:23:00.930 Speed, square root of x prime the t squared 00:23:00.930 --> 00:23:03.430 plus y prime the t squared. 00:23:03.430 --> 00:23:08.260 Which is the same as writing R prime of the t in magnitude. 00:23:08.260 --> 00:23:09.480 Thank God we know that. 00:23:09.480 --> 00:23:11.550 How much is this? 00:23:11.550 --> 00:23:14.266 R, very good, this is R, very good. 00:23:14.266 --> 00:23:18.790 So life is not so hard, it's-- hopefully I'll be able to do 00:23:18.790 --> 00:23:21.660 the w. 00:23:21.660 --> 00:23:22.410 What is the w? 00:23:22.410 --> 00:23:25.910 It's the path integral all over the circle 00:23:25.910 --> 00:23:30.380 I have here, that I traveled counterclockwise from any point 00:23:30.380 --> 00:23:31.270 to any point. 00:23:31.270 --> 00:23:34.559 Let's say this would be the origin of my motion, 00:23:34.559 --> 00:23:37.340 then I go back. 00:23:37.340 --> 00:23:43.320 And I have this force, F, that I have 00:23:43.320 --> 00:23:47.540 to redistribute in terms of R. So this notation 00:23:47.540 --> 00:23:49.222 is giving me a little bit of a headache, 00:23:49.222 --> 00:23:51.410 but in reality it's going to be very simple. 00:23:51.410 --> 00:23:59.740 This is the dot product, R prime dt, which was the R. 00:23:59.740 --> 00:24:05.390 Which some other people asked me, how can you write that? 00:24:05.390 --> 00:24:12.090 Well, read the review, R of x equals 00:24:12.090 --> 00:24:16.810 x of t, i plus y of t, j. 00:24:16.810 --> 00:24:19.990 Also, read the next side plus y j. 00:24:19.990 --> 00:24:26.590 It short, the dR differential out of t, 00:24:26.590 --> 00:24:32.920 sorry, I'll put R. dR is dx i plus dy j. 00:24:32.920 --> 00:24:35.950 And if somebody wants to be expressing 00:24:35.950 --> 00:24:40.390 this in terms of speeds, we'll say this is x prime dt, 00:24:40.390 --> 00:24:43.010 this is y prime dt. 00:24:43.010 --> 00:24:48.130 So we can rewrite this x prime then t i, plus y prime then t 00:24:48.130 --> 00:24:49.836 j, dt. 00:24:49.836 --> 00:24:54.620 00:24:54.620 --> 00:24:55.800 All right? 00:24:55.800 --> 00:25:01.350 OK, which is the same thing as R prime of t, dt. 00:25:01.350 --> 00:25:04.203 00:25:04.203 --> 00:25:04.703 [INAUDIBLE] 00:25:04.703 --> 00:25:07.580 00:25:07.580 --> 00:25:11.760 This looks awfully theoretical. 00:25:11.760 --> 00:25:16.550 I say, I don't like it, I want to put my favorite guys 00:25:16.550 --> 00:25:18.290 in the picture. 00:25:18.290 --> 00:25:22.140 So I have to think, when I do the dot product 00:25:22.140 --> 00:25:26.380 I have the dot product between the vector that 00:25:26.380 --> 00:25:29.890 has components f1 and f2. 00:25:29.890 --> 00:25:31.870 How am I going to do that? 00:25:31.870 --> 00:25:38.700 Well, if I multiply with this guy, dot product, the boss guy, 00:25:38.700 --> 00:25:39.800 with this boss guy. 00:25:39.800 --> 00:25:43.180 Are you guys with me? 00:25:43.180 --> 00:25:44.810 What am I going to do? 00:25:44.810 --> 00:25:48.560 First component times first component, 00:25:48.560 --> 00:25:53.740 plus second component times second component of a vector. 00:25:53.740 --> 00:25:58.830 So I have to be smart and understand how I do that. 00:25:58.830 --> 00:26:01.060 Lambda is a constant. 00:26:01.060 --> 00:26:04.110 Lambda, you're my friend, you stay there. 00:26:04.110 --> 00:26:09.570 x is x of t, x of t, but I multiplied 00:26:09.570 --> 00:26:11.490 with the first component here, so I 00:26:11.490 --> 00:26:14.860 multiplied by x prime of t. 00:26:14.860 --> 00:26:19.600 Plus lambda times y of t, times y prime of t. 00:26:19.600 --> 00:26:23.750 And who gets out of the picture is dt at the end. 00:26:23.750 --> 00:26:27.530 I have integrate with respect to that dt. 00:26:27.530 --> 00:26:29.570 This would be incorrect, why? 00:26:29.570 --> 00:26:34.600 Because t has to move between some specific limits 00:26:34.600 --> 00:26:38.180 when I specify what a path integral is. 00:26:38.180 --> 00:26:41.870 I cannot leave a c-- very good, from 0 to 2 pi, excellent. 00:26:41.870 --> 00:26:45.300 00:26:45.300 --> 00:26:46.240 Is this hard? 00:26:46.240 --> 00:26:48.670 No, It's going to be a piece of cake. 00:26:48.670 --> 00:26:50.336 Why is that a piece of cake? 00:26:50.336 --> 00:26:54.190 Because I can keep writing. 00:26:54.190 --> 00:26:58.519 You actually are faster than me. 00:26:58.519 --> 00:27:00.560 STUDENT: Your chain rule is already done for you. 00:27:00.560 --> 00:27:03.930 PROFESSOR: Right, and then lambda 00:27:03.930 --> 00:27:06.920 gets out just because-- well you remember 00:27:06.920 --> 00:27:09.600 you kick the lambda out, right? 00:27:09.600 --> 00:27:20.140 And then I've put R cosine t times minus R sine t. 00:27:20.140 --> 00:27:21.310 I'm done with who? 00:27:21.310 --> 00:27:23.625 I'm done with this fellow and that fellow. 00:27:23.625 --> 00:27:27.340 00:27:27.340 --> 00:27:37.550 And plus y, R sine t, what is R prime? 00:27:37.550 --> 00:27:48.580 R cosine, thank you guys, dt. 00:27:48.580 --> 00:27:50.910 And now I'm going to ask you, what is this animal? 00:27:50.910 --> 00:27:53.660 00:27:53.660 --> 00:27:56.020 Stare at that, what is the integrand? 00:27:56.020 --> 00:27:59.130 Is a friend of yours, he's so cute. 00:27:59.130 --> 00:28:01.540 He's staring at you and saying you are done. 00:28:01.540 --> 00:28:04.895 Why are you done? 00:28:04.895 --> 00:28:08.346 What happens to the integrand? 00:28:08.346 --> 00:28:12.365 It's zero, it's a blessing, it's zero. 00:28:12.365 --> 00:28:14.000 How come it's zero? 00:28:14.000 --> 00:28:21.390 Because these two terms simplify, they cancel out. 00:28:21.390 --> 00:28:26.290 They cancel out, thank god they cancel out, I got a zero. 00:28:26.290 --> 00:28:28.530 So we discover something that a physicist 00:28:28.530 --> 00:28:32.152 or a mechanical engineer would have told you already. 00:28:32.152 --> 00:28:34.110 And do you think he would have actually plugged 00:28:34.110 --> 00:28:36.230 in the path integral? 00:28:36.230 --> 00:28:39.700 No, they wouldn't think like this. 00:28:39.700 --> 00:28:45.540 He has a simpler explanation for that because he's experienced 00:28:45.540 --> 00:28:47.260 with linear experiments. 00:28:47.260 --> 00:28:51.740 And says, if I drag this like that I know what to work with. 00:28:51.740 --> 00:28:54.770 If I drag in, like at an angle, I 00:28:54.770 --> 00:28:57.950 know that I have the magnitude of this, 00:28:57.950 --> 00:28:59.390 cosine theta, the angle. 00:28:59.390 --> 00:29:03.740 So, he knows for linear cases what we have. 00:29:03.740 --> 00:29:10.060 For a circular case he can smell the result 00:29:10.060 --> 00:29:12.880 without doing the path integral. 00:29:12.880 --> 00:29:18.080 So how do you think the guy, if he's a mechanical engineer, 00:29:18.080 --> 00:29:20.040 would think in a second? 00:29:20.040 --> 00:29:24.315 Say well, think of your trajectory, right? 00:29:24.315 --> 00:29:27.360 It's a circle. 00:29:27.360 --> 00:29:31.620 The problem is that centrifugal force being perpendicular 00:29:31.620 --> 00:29:33.560 to the circle all the time. 00:29:33.560 --> 00:29:37.140 And you say, how can a line be perpendicular to a circle? 00:29:37.140 --> 00:29:39.570 It is, it's the normal to the circle. 00:29:39.570 --> 00:29:43.054 So when you say this is normal to the circle 00:29:43.054 --> 00:29:45.470 you mean it's normal to the tangent of the circle. 00:29:45.470 --> 00:29:50.840 So if you measure the angle made by the normal at every point 00:29:50.840 --> 00:29:53.800 to the trajectory of a circle, it's always lines. 00:29:53.800 --> 00:30:01.520 So he goes, gosh, I got cosine of 90, that's zero. 00:30:01.520 --> 00:30:04.270 So if you have some sort of work produced 00:30:04.270 --> 00:30:06.470 by something perpendicular to your trajectory, 00:30:06.470 --> 00:30:07.800 that must be zero. 00:30:07.800 --> 00:30:12.140 So he or she has very good intuition. 00:30:12.140 --> 00:30:13.875 Of course, how do we prove it? 00:30:13.875 --> 00:30:16.535 We are mathematicians, we prove the path integral, 00:30:16.535 --> 00:30:19.460 we got zero for the work, all right? 00:30:19.460 --> 00:30:25.200 But he could sense that kind of stuff from the beginning. 00:30:25.200 --> 00:30:34.740 Now, there is another example where maybe you don't have 00:30:34.740 --> 00:30:38.360 90 degrees for your trajectory. 00:30:38.360 --> 00:30:47.970 Well, I'm going to just take-- what if I change the force 00:30:47.970 --> 00:30:50.570 and I make a difference problem? 00:30:50.570 --> 00:30:52.844 Make it into a different problem. 00:30:52.844 --> 00:31:01.293 00:31:01.293 --> 00:31:04.772 I will do that later, I won't go and erase it. 00:31:04.772 --> 00:31:15.710 00:31:15.710 --> 00:31:19.620 Last time we did one that was, compute 00:31:19.620 --> 00:31:25.330 the work along a parabola from something to something. 00:31:25.330 --> 00:31:26.830 Let's do that again. 00:31:26.830 --> 00:31:30.668 00:31:30.668 --> 00:31:35.585 For some sort of a nice force field 00:31:35.585 --> 00:31:39.740 I'll take your vector valued function to be nice to you. 00:31:39.740 --> 00:31:42.380 I'll change it, y i plus x j. 00:31:42.380 --> 00:31:45.330 00:31:45.330 --> 00:31:48.550 And then we are in plane and we move 00:31:48.550 --> 00:31:55.020 along this parabola between 0, 0 and-- what is this guys? 00:31:55.020 --> 00:31:59.348 1, 1-- well let make it into a one, it's cute. 00:31:59.348 --> 00:32:04.690 And I'd like you to measure the work along the parabola 00:32:04.690 --> 00:32:11.410 and also along the arc of a-- along the segment of a line 00:32:11.410 --> 00:32:13.440 between the two points. 00:32:13.440 --> 00:32:16.728 So I want you to compute w1 along the parabola, 00:32:16.728 --> 00:32:22.990 and w2 along this thingy, the segment. 00:32:22.990 --> 00:32:24.070 Should it be hard? 00:32:24.070 --> 00:32:27.230 No, this was old session for many finals. 00:32:27.230 --> 00:32:33.180 I remember, I think it was 2003, 2006, 2008, 00:32:33.180 --> 00:32:36.320 and very recently, I think a year and 1/2 ago. 00:32:36.320 --> 00:32:38.020 A problem like that was given. 00:32:38.020 --> 00:32:40.750 Compute the path integrals correspond 00:32:40.750 --> 00:32:45.000 to work for both parametrization and compare them. 00:32:45.000 --> 00:32:45.650 Is it hard? 00:32:45.650 --> 00:32:48.861 I have no idea, let me think. 00:32:48.861 --> 00:32:53.260 For the first one we have parametrization 00:32:53.260 --> 00:32:55.600 that we need to distinguish from the other one. 00:32:55.600 --> 00:32:58.706 The first parametrization for a parabola, 00:32:58.706 --> 00:33:01.430 we discussed it last time, was of course 00:33:01.430 --> 00:33:04.080 the simplest possible one you can think of. 00:33:04.080 --> 00:33:05.830 And we did this last time but I'm 00:33:05.830 --> 00:33:10.360 repeating this because I didn't want Alex to miss that. 00:33:10.360 --> 00:33:18.250 And I'm going to say integral from some time to some time. 00:33:18.250 --> 00:33:21.090 Now, if I'm between 0 and 1, time of course 00:33:21.090 --> 00:33:25.780 will be between 0 and 1 because x is time. 00:33:25.780 --> 00:33:28.554 All right, good, that means what else? 00:33:28.554 --> 00:33:32.140 This Is f1 and this is f2. 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, 00:33:39.720 --> 00:33:46.310 y prime of t, all the [? sausage ?] times dt. 00:33:46.310 --> 00:33:47.930 Is this going to be hard? 00:33:47.930 --> 00:33:52.830 Hopefully not, I'm going to have to identify everybody. 00:33:52.830 --> 00:33:55.970 Identify this guys prime of t with respect to t 00:33:55.970 --> 00:33:58.110 is 1, piece of cake, right? 00:33:58.110 --> 00:34:02.850 This fellow is-- you told me last time you got 2t 00:34:02.850 --> 00:34:05.360 and you got it right. 00:34:05.360 --> 00:34:08.219 This guy, I have to be a little bit careful 00:34:08.219 --> 00:34:10.679 because y is the fourth guy. 00:34:10.679 --> 00:34:15.920 This is t squared and this is t. 00:34:15.920 --> 00:34:19.889 00:34:19.889 --> 00:34:22.960 So my integral will be a joke. 00:34:22.960 --> 00:34:29.630 0 to 1, 2t squared plus t squared equals 3t squared. 00:34:29.630 --> 00:34:30.980 Is it hard to integrate? 00:34:30.980 --> 00:34:33.110 No, for God's sake, this is integral-- this 00:34:33.110 --> 00:34:39.270 is t cubed between 0 and 1, right, right? 00:34:39.270 --> 00:34:44.080 So I should get 1, and if I get, I 00:34:44.080 --> 00:34:48.600 think I did it right, if I get the other parametrization you 00:34:48.600 --> 00:34:52.540 have to help me write it again. 00:34:52.540 --> 00:34:57.190 The parametrization of this straight line 00:34:57.190 --> 00:34:59.380 between 0, 0 and 1, 1. 00:34:59.380 --> 00:35:03.300 Now on the actual exam, I'm never 00:35:03.300 --> 00:35:08.500 going to forgive you if you don't know how to parametrize. 00:35:08.500 --> 00:35:13.140 Now you know it but two months ago you didn't, many of you 00:35:13.140 --> 00:35:14.100 didn't. 00:35:14.100 --> 00:35:21.510 So if somebody gives you 2 points, OK, in plane I ask you, 00:35:21.510 --> 00:35:27.160 how do you write that symmetric equation of the line between? 00:35:27.160 --> 00:35:29.030 You were a little bit hesitant, now 00:35:29.030 --> 00:35:34.030 you shouldn't be hesitant because it's a serious thing. 00:35:34.030 --> 00:35:36.290 So how did we write that? 00:35:36.290 --> 00:35:40.466 We memorized it. x minus x1, over x2 minus x1 00:35:40.466 --> 00:35:44.440 equals y minus y1, over y2 minus y1. 00:35:44.440 --> 00:35:47.300 This can also be written as-- you know, guys, 00:35:47.300 --> 00:35:49.840 that this over that is the actual slope. 00:35:49.840 --> 00:35:52.275 This over that, so it can be written 00:35:52.275 --> 00:35:53.560 as a [INAUDIBLE] formula. 00:35:53.560 --> 00:35:55.250 It can be written in many ways. 00:35:55.250 --> 00:35:59.920 And if we put a, t, we transform it into a parametric equation. 00:35:59.920 --> 00:36:02.140 So you should be able, on the final, 00:36:02.140 --> 00:36:05.690 to do that for any segment of a line with your eyes closed. 00:36:05.690 --> 00:36:07.550 Like, you see the numbers, you plug them in, 00:36:07.550 --> 00:36:10.640 you get the parametric equations. 00:36:10.640 --> 00:36:14.820 We are nice on the exams because we usually give you 00:36:14.820 --> 00:36:18.750 a line that's easy to write. 00:36:18.750 --> 00:36:25.830 Like in this case you would have x equals t and y equal, 00:36:25.830 --> 00:36:29.940 let's see if you are asleep yet, t. 00:36:29.940 --> 00:36:31.500 Why is that? 00:36:31.500 --> 00:36:37.290 Because the line that joins 0, 0 and 1, 1 is y equals x. 00:36:37.290 --> 00:36:40.095 So y equals x is called also, first bicycle. 00:36:40.095 --> 00:36:43.100 It's the old friend of yours from trigonometry, 00:36:43.100 --> 00:36:46.300 from Pre-Calc, from algebra, I don't know where, college 00:36:46.300 --> 00:36:47.820 algebra. 00:36:47.820 --> 00:36:49.160 Alrighty, is this hard to do? 00:36:49.160 --> 00:36:55.850 No, it's easier than before. w2 is integral from 0 to 1, 00:36:55.850 --> 00:37:00.380 this is t and this is t, this is t and this is t, good. 00:37:00.380 --> 00:37:07.990 So we have t times 1 plus t times 1, 00:37:07.990 --> 00:37:15.362 it's like a funny, nice, game that's too simple, 2t, 2t. 00:37:15.362 --> 00:37:18.340 00:37:18.340 --> 00:37:23.100 So the fundamental theorem of Calc 00:37:23.100 --> 00:37:27.042 says t squared between 0 and 1, the answer is 1. 00:37:27.042 --> 00:37:28.500 Am I surprised? 00:37:28.500 --> 00:37:31.360 Look at me, do I look surprised at all 00:37:31.360 --> 00:37:34.240 that I got the same answer? 00:37:34.240 --> 00:37:37.224 No, I told you a secret last time. 00:37:37.224 --> 00:37:39.390 I didn't prove it. 00:37:39.390 --> 00:37:43.100 I said that R times happy times. 00:37:43.100 --> 00:37:46.740 When depending on the force that is with you, 00:37:46.740 --> 00:37:51.300 you have the same work no matter what path 00:37:51.300 --> 00:37:53.830 you are taking between a and b. 00:37:53.830 --> 00:37:57.720 Between the origin and finish line. 00:37:57.720 --> 00:38:01.030 So I'm claiming that if I give you this zig-zag line 00:38:01.030 --> 00:38:08.010 and I asked you what-- look, it could be any crazy path 00:38:08.010 --> 00:38:11.090 but it has to be a nice differentiable path. 00:38:11.090 --> 00:38:14.630 Along this differentiable path, no matter 00:38:14.630 --> 00:38:17.420 how you compute the work, that's your business, 00:38:17.420 --> 00:38:19.040 I claim I still get 1. 00:38:19.040 --> 00:38:21.840 00:38:21.840 --> 00:38:26.440 Can you even think why, some of you remember maybe, 00:38:26.440 --> 00:38:28.986 the force was key? 00:38:28.986 --> 00:38:30.444 STUDENT: It's a conservative force. 00:38:30.444 --> 00:38:32.780 PROFESSOR: It had to be good conservative. 00:38:32.780 --> 00:38:37.320 Now this is conservative but why is that conservative? 00:38:37.320 --> 00:38:39.900 What the heck is a conservative force? 00:38:39.900 --> 00:38:46.710 So let's write it down on the-- we 00:38:46.710 --> 00:39:01.650 say that the vector valued function, f, 00:39:01.650 --> 00:39:19.084 valued in R2 or R3, is conservative if there exists 00:39:19.084 --> 00:39:25.460 a smooth function. 00:39:25.460 --> 00:39:28.990 Little f, it actually has to be just c1, 00:39:28.990 --> 00:39:31.660 called scalar potential. 00:39:31.660 --> 00:39:45.380 Called scalar potential, such that big F as a vector 00:39:45.380 --> 00:39:48.100 field will be not little f. 00:39:48.100 --> 00:39:52.830 That means it will be the gradient of the scalar 00:39:52.830 --> 00:39:54.838 potential. 00:39:54.838 --> 00:39:57.370 Definition, that was the definition, 00:39:57.370 --> 00:40:08.440 and then criterion for a f in R2 to be conservative. 00:40:08.440 --> 00:40:13.390 00:40:13.390 --> 00:40:21.370 I claim that f equals f1 i, plus f two eyes, no, f2 j. 00:40:21.370 --> 00:40:23.370 I'm just making silly puns, I don't 00:40:23.370 --> 00:40:27.880 know if you guys follow me. 00:40:27.880 --> 00:40:33.470 If and only if f sub 1 prime, with respect to y, 00:40:33.470 --> 00:40:37.890 is f sub 2 prime, with respect to x. 00:40:37.890 --> 00:40:40.166 Can I prove this? 00:40:40.166 --> 00:40:45.560 Prove, prove Magdelina, don't just stare at it, prove. 00:40:45.560 --> 00:40:48.030 Why would that be necessary and sufficient? 00:40:48.030 --> 00:40:52.020 00:40:52.020 --> 00:40:58.890 Well, for big F to be conservative 00:40:58.890 --> 00:41:02.180 it means that it has to be the gradient 00:41:02.180 --> 00:41:06.970 of some little function, little f, some scalar potential. 00:41:06.970 --> 00:41:13.204 Alrighty, so let me write it down, proof. 00:41:13.204 --> 00:41:20.584 f conservative if and only if there 00:41:20.584 --> 00:41:27.964 exists f, such that gradient of f is F. If and only 00:41:27.964 --> 00:41:32.990 if-- what does it mean about f1 and f2? 00:41:32.990 --> 00:41:36.540 f1 and f2 are the f sub of x and the f sub 00:41:36.540 --> 00:41:39.460 y of some scalar potential. 00:41:39.460 --> 00:41:44.300 So if f is the gradient, that means that the first component 00:41:44.300 --> 00:41:48.600 has to be little f sub x And the second component should 00:41:48.600 --> 00:41:51.906 have to be little f sub y. 00:41:51.906 --> 00:41:58.240 But that is if and only if f sub 1 prime, with respect to y, 00:41:58.240 --> 00:42:01.710 is the same as f sub 2 prime, with respect to x. 00:42:01.710 --> 00:42:04.144 What is that? 00:42:04.144 --> 00:42:14.052 The red thing here is called a compatibility condition 00:42:14.052 --> 00:42:16.440 of this system. 00:42:16.440 --> 00:42:22.660 This is a system of two OD's. 00:42:22.660 --> 00:42:27.426 You are going to study ODEs in 3350. 00:42:27.426 --> 00:42:29.730 And you are going to remember this 00:42:29.730 --> 00:42:33.020 and say, Oh, I know that because she taught me that in Calc 3. 00:42:33.020 --> 00:42:36.310 Not all instructors will teach you this in Calc 3. 00:42:36.310 --> 00:42:38.880 Some of them fool you and skip this material 00:42:38.880 --> 00:42:44.410 that's very important to understand in 3350. 00:42:44.410 --> 00:42:49.490 So guys, what's going to happened when you prime this 00:42:49.490 --> 00:42:51.580 with respect to y? 00:42:51.580 --> 00:42:55.440 You get f sub x prime, with respect to y. 00:42:55.440 --> 00:42:57.110 When you prime this with respect to x 00:42:57.110 --> 00:42:59.860 you get f sub y prime, with respect to x. 00:42:59.860 --> 00:43:01.680 Why are they the same thing? 00:43:01.680 --> 00:43:04.716 I'm going to remind you that they 00:43:04.716 --> 00:43:07.220 are the same thing for a smooth function. 00:43:07.220 --> 00:43:09.250 Who said that? 00:43:09.250 --> 00:43:19.060 A crazy German mathematician whose name was Schwartz. 00:43:19.060 --> 00:43:22.134 Which means black, that's what I'm painting it in black. 00:43:22.134 --> 00:43:29.090 Because is the Schwartz guy, the first criterion 00:43:29.090 --> 00:43:31.750 saying that no matter in what order 00:43:31.750 --> 00:43:34.320 you differentiate the smooth function you 00:43:34.320 --> 00:43:37.800 get the same answer for the mixed derivative. 00:43:37.800 --> 00:43:41.620 So you see we prove if and only if that you 00:43:41.620 --> 00:43:43.900 have to have this criterion, otherwise 00:43:43.900 --> 00:43:46.210 it's not going to be conservative. 00:43:46.210 --> 00:43:49.630 So I'm asking you, for your old friend, f 00:43:49.630 --> 00:43:52.840 equals- example one or example two, 00:43:52.840 --> 00:43:56.200 I don't know- y i plus x j. 00:43:56.200 --> 00:43:58.510 Is the conservative? 00:43:58.510 --> 00:44:00.440 You can prove it in two ways. 00:44:00.440 --> 00:44:06.685 Prove in two differently ways that it is conservative. 00:44:06.685 --> 00:44:13.210 00:44:13.210 --> 00:44:16.716 a, find the criteria. 00:44:16.716 --> 00:44:20.450 What does this criteria say? 00:44:20.450 --> 00:44:25.430 Take your first component, prime it with respect to y. 00:44:25.430 --> 00:44:27.780 So y prime with respect to y. 00:44:27.780 --> 00:44:32.870 Take your second component, x, prime it with respect to x. 00:44:32.870 --> 00:44:34.750 Is this true? 00:44:34.750 --> 00:44:37.460 Yes, and this is me, happy that it's true. 00:44:37.460 --> 00:44:40.740 So this is 1 equals 1, so it's true. 00:44:40.740 --> 00:44:44.780 So it must be conservative, so it must be conservative. 00:44:44.780 --> 00:44:46.780 Could I have done it another way? 00:44:46.780 --> 00:44:49.610 00:44:49.610 --> 00:44:57.530 By definition, by definition, to prove that a force field 00:44:57.530 --> 00:45:01.640 is conservative by definition, that it a matter 00:45:01.640 --> 00:45:04.075 of the smart people. 00:45:04.075 --> 00:45:07.100 There are people who- unlike me when 00:45:07.100 --> 00:45:11.670 I was 18- are able to see the scalar potential in just 00:45:11.670 --> 00:45:13.852 about any problem I give them. 00:45:13.852 --> 00:45:18.070 I'm not going to make this experiment with a bunch of you 00:45:18.070 --> 00:45:22.320 and I'm going to reward you for the correct answers. 00:45:22.320 --> 00:45:25.430 But, could anybody see the existence 00:45:25.430 --> 00:45:27.710 of the scalar potential? 00:45:27.710 --> 00:45:30.750 So these there, this exists. 00:45:30.750 --> 00:45:37.590 That's there exists a little f scalar potential such 00:45:37.590 --> 00:45:43.081 that nabla f equals F. 00:45:43.081 --> 00:45:47.680 And some of you may see it and say, I see it. 00:45:47.680 --> 00:45:50.070 So, can you see a little function 00:45:50.070 --> 00:45:54.660 f scalar function so that f sub of x i is y 00:45:54.660 --> 00:46:00.810 and f sub- Magdelena-- x of y j is this x j? 00:46:00.810 --> 00:46:02.220 STUDENT: [INAUDIBLE]? 00:46:02.220 --> 00:46:07.330 PROFESSOR: No, you need to drink some coffee first. 00:46:07.330 --> 00:46:12.990 You can get this, x times what? 00:46:12.990 --> 00:46:14.165 Why is that? 00:46:14.165 --> 00:46:17.270 I'll teach you how to get it. 00:46:17.270 --> 00:46:18.880 Nevertheless, there are some people 00:46:18.880 --> 00:46:21.140 who can do it with their naked eye 00:46:21.140 --> 00:46:26.066 because they have a little computer in their head. 00:46:26.066 --> 00:46:28.710 But how did I do it? 00:46:28.710 --> 00:46:30.970 It's just a matter of experience, I said, 00:46:30.970 --> 00:46:36.690 if I take f to be x y, I sort of guessed it. 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, 00:46:42.000 --> 00:46:44.440 and this is going to do. 00:46:44.440 --> 00:46:49.040 And f is a nice function, polynomial in two variables, 00:46:49.040 --> 00:46:50.420 it's a smooth function. 00:46:50.420 --> 00:46:55.200 I'm very happy I'm over the domain, 00:46:55.200 --> 00:46:57.770 open this or whatever, open domain in plane. 00:46:57.770 --> 00:47:00.160 I'm very happy, I have no problem with it. 00:47:00.160 --> 00:47:04.756 So I can know that this is conservative in two ways. 00:47:04.756 --> 00:47:07.750 Either I get to the source of the problem 00:47:07.750 --> 00:47:10.380 and I find the little scalar potential whose 00:47:10.380 --> 00:47:13.480 gradient is my force field. 00:47:13.480 --> 00:47:16.630 Or I can verify the criterion and I say, 00:47:16.630 --> 00:47:19.100 the derivative of this with respect to y 00:47:19.100 --> 00:47:20.972 is the derivative of this with respect 00:47:20.972 --> 00:47:23.140 to x is-- one is the same. 00:47:23.140 --> 00:47:29.145 The same thing, you're going to see it again in math 3350. 00:47:29.145 --> 00:47:30.710 All right, that I taught many times, 00:47:30.710 --> 00:47:33.020 I'm not going to teach that in the fall. 00:47:33.020 --> 00:47:34.820 But I know of some very good people 00:47:34.820 --> 00:47:36.065 who teach that in the fall. 00:47:36.065 --> 00:47:38.670 In any case, they would reteach it to you 00:47:38.670 --> 00:47:42.060 because good teachers don't assume that you know much. 00:47:42.060 --> 00:47:45.970 But when you will see it you'll remember me. 00:47:45.970 --> 00:47:49.970 Hopefully fondly, not cursing me or anything, right? 00:47:49.970 --> 00:47:54.890 OK, how do we actually get to compute f by hand 00:47:54.890 --> 00:47:59.150 if we're not experienced enough to guess it like I was 00:47:59.150 --> 00:48:01.740 experienced enough to guess? 00:48:01.740 --> 00:48:08.240 So let me show you how you solve a system of two differential 00:48:08.240 --> 00:48:11.290 equations like that. 00:48:11.290 --> 00:48:15.625 So how I got-- how you are supposed 00:48:15.625 --> 00:48:23.010 to get the scalar potential. 00:48:23.010 --> 00:48:28.790 f sub x equals F1, f sub y equals F2. 00:48:28.790 --> 00:48:35.358 So by integration, 1 and 2. 00:48:35.358 --> 00:48:38.740 00:48:38.740 --> 00:48:40.910 And you say, what you mean 1 and 2? 00:48:40.910 --> 00:48:42.670 I'll show you in a second. 00:48:42.670 --> 00:48:47.820 So for my case, example 2, I'll take 00:48:47.820 --> 00:48:50.570 my f sub x must be y, right? 00:48:50.570 --> 00:48:51.070 Good. 00:48:51.070 --> 00:48:54.025 My f sub y must be x, right? 00:48:54.025 --> 00:48:55.270 Right. 00:48:55.270 --> 00:48:58.160 Who is f? 00:48:58.160 --> 00:49:00.690 Solve this property. 00:49:00.690 --> 00:49:05.580 Oh, I have to start integrating from the first guy. 00:49:05.580 --> 00:49:08.390 What kind of information am I going to squeeze? 00:49:08.390 --> 00:49:11.572 I'm going to say I have to go backwards, 00:49:11.572 --> 00:49:17.581 I have to get-- f is going to be what? 00:49:17.581 --> 00:49:23.156 Integral of y with respect to x, say it again, Magdelina. 00:49:23.156 --> 00:49:27.440 Integral of y with respect to x, but attention, 00:49:27.440 --> 00:49:31.620 this may come because, for me, the variable is x here, 00:49:31.620 --> 00:49:35.480 and y is like, you cannot stay in this picture. 00:49:35.480 --> 00:49:40.590 So I have a constant c that depends on y. 00:49:40.590 --> 00:49:41.510 Say what? 00:49:41.510 --> 00:49:46.290 Yes, because if you go backwards and prime this with respect 00:49:46.290 --> 00:49:49.658 to x, what do you get? f sub x will 00:49:49.658 --> 00:49:51.840 be y because this is the anti-derivative. 00:49:51.840 --> 00:49:55.460 Plus this prime with respect to x, zero. 00:49:55.460 --> 00:49:59.520 So this c of y may, a little bit, ruin your plans. 00:49:59.520 --> 00:50:02.630 I've had students who forgot about it 00:50:02.630 --> 00:50:04.950 and then they got in trouble because they couldn't get 00:50:04.950 --> 00:50:08.130 the scalar potential correctly. 00:50:08.130 --> 00:50:08.630 All right? 00:50:08.630 --> 00:50:14.340 OK, so from this one you say, OK I have some-- 00:50:14.340 --> 00:50:16.980 what is the integral of y dx? 00:50:16.980 --> 00:50:22.246 xy, plus some guy c constant that depends on y. 00:50:22.246 --> 00:50:25.610 From this fellow I go, but I have 00:50:25.610 --> 00:50:29.520 to verify the second condition, if I don't I'm dead meat. 00:50:29.520 --> 00:50:32.320 There are two coupled equations, these 00:50:32.320 --> 00:50:33.980 are coupled equations that have to be 00:50:33.980 --> 00:50:36.320 verified at the same time. 00:50:36.320 --> 00:50:45.120 So f sub y will be prime with respect to y. x plus prime 00:50:45.120 --> 00:50:52.750 with respect to y. c prime of y, God gave me x here. 00:50:52.750 --> 00:50:56.760 So I'm really lucky in that sense that c prime of y 00:50:56.760 --> 00:51:01.300 will be 0 because I have an x here and an x here. 00:51:01.300 --> 00:51:05.540 So c of y will simply be any constant k. c of y 00:51:05.540 --> 00:51:09.487 is just a constant k, it's not going to depend on y, 00:51:09.487 --> 00:51:10.980 it's a constant k. 00:51:10.980 --> 00:51:14.510 So my answer was not correct. 00:51:14.510 --> 00:51:21.320 The best answer would have been f of xy must be xy plus k. 00:51:21.320 --> 00:51:26.650 But any function like xy will work, I just need one to work. 00:51:26.650 --> 00:51:29.330 I just need a scalar potential, not all of them. 00:51:29.330 --> 00:51:33.846 This will work, x2, xy plus 7 will work, xy plus 3 will work, 00:51:33.846 --> 00:51:38.860 xy minus 1,033,045 will work. 00:51:38.860 --> 00:51:43.830 But I only need one so I'll take xy. 00:51:43.830 --> 00:51:46.220 Now that I trained your mind a little bit, 00:51:46.220 --> 00:51:48.370 maybe you don't need to actually solve 00:51:48.370 --> 00:51:53.720 the system because your brain wasn't ready before. 00:51:53.720 --> 00:51:57.300 But you'd be amazed, we are very trainable people. 00:51:57.300 --> 00:52:03.550 And in the process of doing something completely new, 00:52:03.550 --> 00:52:05.400 we are learning. 00:52:05.400 --> 00:52:10.305 And your brain next, will say, I think 00:52:10.305 --> 00:52:15.330 I know how to function a little bit backwards. 00:52:15.330 --> 00:52:20.460 And try to integrate and see and guess a potential 00:52:20.460 --> 00:52:24.080 because it's not so hard. 00:52:24.080 --> 00:52:25.580 So let me give you example 3. 00:52:25.580 --> 00:52:29.650 00:52:29.650 --> 00:52:36.240 Somebody give you over a domain in plane x i plus y j, 00:52:36.240 --> 00:52:40.605 and says, over D, simply connected domain in plane, 00:52:40.605 --> 00:52:44.000 open, doesn't matter. 00:52:44.000 --> 00:52:45.455 Is this conservative? 00:52:45.455 --> 00:52:48.370 00:52:48.370 --> 00:52:51.880 Find a scalar potential. 00:52:51.880 --> 00:53:00.710 00:53:00.710 --> 00:53:03.960 This is again, we do section 13-2, 00:53:03.960 --> 00:53:07.650 so today we did 13-1 and 13-2 jointly. 00:53:07.650 --> 00:53:10.630 00:53:10.630 --> 00:53:11.930 Find the scalar potential. 00:53:11.930 --> 00:53:14.994 Do you see it now? 00:53:14.994 --> 00:53:16.392 STUDENT: [INAUDIBLE]? 00:53:16.392 --> 00:53:19.560 PROFESSOR: Excellent, we teach now, got it. 00:53:19.560 --> 00:53:28.450 He says, I know where this comes from. 00:53:28.450 --> 00:53:31.650 I've got it, x squared plus y squared over 2. 00:53:31.650 --> 00:53:32.850 How did he do it? 00:53:32.850 --> 00:53:33.900 He's a genius. 00:53:33.900 --> 00:53:39.560 No he's not, he's just learning from the first time 00:53:39.560 --> 00:53:40.760 when he failed. 00:53:40.760 --> 00:53:44.180 And now he knows what he has to do and his brain says, 00:53:44.180 --> 00:53:47.230 oh, I got it. 00:53:47.230 --> 00:53:50.110 Now, [INAUDIBLE] could have applied this method 00:53:50.110 --> 00:53:54.640 and solved the coupled system and do it slowly 00:53:54.640 --> 00:53:56.820 and it would have taken him another 10 minutes. 00:53:56.820 --> 00:54:00.100 And he's in the final, he doesn't have time to spare. 00:54:00.100 --> 00:54:06.030 If he can guess the potential and then verify that, 00:54:06.030 --> 00:54:07.720 it's going to be easy for him. 00:54:07.720 --> 00:54:08.315 Why is that? 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. 00:54:15.490 --> 00:54:17.838 So yeah, he was right. 00:54:17.838 --> 00:54:21.806 00:54:21.806 --> 00:54:26.634 All right, let me give you another one. 00:54:26.634 --> 00:54:29.104 Let's see who gets this one. 00:54:29.104 --> 00:54:34.080 F is a vector valued function, maybe a force field, 00:54:34.080 --> 00:54:35.341 that is this. 00:54:35.341 --> 00:54:38.047 00:54:38.047 --> 00:54:41.640 Of course there are many ways-- maybe somebody's 00:54:41.640 --> 00:54:44.180 going to ask you to prove this is 00:54:44.180 --> 00:54:49.320 conservative by the criterion, but they shouldn't tell you 00:54:49.320 --> 00:54:51.550 how to do it. 00:54:51.550 --> 00:54:53.050 So show this is conservative. 00:54:53.050 --> 00:54:56.550 If somebody doesn't want the scalar potential because they 00:54:56.550 --> 00:54:57.890 don't need it, let's say. 00:54:57.890 --> 00:55:01.945 Well, prime f1 with respect to y, I'll prime this with respect 00:55:01.945 --> 00:55:03.400 to x. 00:55:03.400 --> 00:55:07.560 f1 prime with respect to y equals 2x is the same 00:55:07.560 --> 00:55:09.720 as f2 prime with respect with. 00:55:09.720 --> 00:55:13.600 Yeah, it is conservative, I know it from the criterion. 00:55:13.600 --> 00:55:16.570 But [INAUDIBLE] knows that later I will ask him 00:55:16.570 --> 00:55:19.150 for the scalar potential. 00:55:19.150 --> 00:55:25.610 And I wonder if he can find it for me without computing it 00:55:25.610 --> 00:55:27.030 by solving the system. 00:55:27.030 --> 00:55:30.265 Just from his mathematical intuition 00:55:30.265 --> 00:55:33.036 that is running in the background of your-- 00:55:33.036 --> 00:55:35.880 STUDENT: x squared, multiply y [INAUDIBLE]. 00:55:35.880 --> 00:55:38.405 PROFESSOR: x squared y, excellent. 00:55:38.405 --> 00:55:42.770 00:55:42.770 --> 00:55:47.140 Zach came up with it and anybody else? 00:55:47.140 --> 00:55:49.090 Alex? 00:55:49.090 --> 00:55:51.610 So all three of you, OK? 00:55:51.610 --> 00:55:54.060 Squared y, very good. 00:55:54.060 --> 00:55:56.030 Was it hard? 00:55:56.030 --> 00:55:59.480 Yeah, it's hard for most people. 00:55:59.480 --> 00:56:02.720 It was hard for me when I first saw 00:56:02.720 --> 00:56:05.530 that in the first 30 minutes of becoming familiar 00:56:05.530 --> 00:56:09.283 with the scalar potential, I was 18 or 19. 00:56:09.283 --> 00:56:11.928 But then I got it in about half an hour 00:56:11.928 --> 00:56:15.536 and I was able to do them mentally. 00:56:15.536 --> 00:56:20.430 Most of the examples I got were really nice. 00:56:20.430 --> 00:56:28.730 Were on purpose made nice for us for the exam to work fast. 00:56:28.730 --> 00:56:35.790 And now let's see why would the work really not 00:56:35.790 --> 00:56:38.620 depend on the trajectory you are taking 00:56:38.620 --> 00:56:41.070 if your force is conservative. 00:56:41.070 --> 00:56:42.955 If the force is conservative there 00:56:42.955 --> 00:56:46.440 is something magic that's going to happen. 00:56:46.440 --> 00:56:50.600 And we really don't know what that is, 00:56:50.600 --> 00:56:53.160 but we should be able to prove. 00:56:53.160 --> 00:56:57.600 00:56:57.600 --> 00:57:19.890 So theorem, actually this is funny. 00:57:19.890 --> 00:57:25.511 It's called the fundamental theorem of path integrals 00:57:25.511 --> 00:57:28.457 but it's the fundamental theorem of calculus 3. 00:57:28.457 --> 00:57:36.313 I'm going to write it like this, the fundamental theorem 00:57:36.313 --> 00:57:45.170 of calc 3, path integrals. 00:57:45.170 --> 00:57:54.708 It's also called- 13.3, section- Independence of path. 00:57:54.708 --> 00:58:01.980 00:58:01.980 --> 00:58:11.733 So remember you have a work, w, over a path, c. 00:58:11.733 --> 00:58:22.610 F dot dR where there R is the regular parametrized curve 00:58:22.610 --> 00:58:23.110 overseen. 00:58:23.110 --> 00:58:33.850 00:58:33.850 --> 00:58:36.850 This is called a supposition vector. 00:58:36.850 --> 00:58:40.850 00:58:40.850 --> 00:58:45.730 Regular meaning c1, and never vanishing speed, 00:58:45.730 --> 00:58:49.400 the velocity never vanishes. 00:58:49.400 --> 00:58:55.025 Velocity times 0 such that f is continuous, 00:58:55.025 --> 00:59:01.580 or a nice enough integral. 00:59:01.580 --> 00:59:07.950 00:59:07.950 --> 00:59:25.920 If F is conservative of scalar potential, little f, 00:59:25.920 --> 00:59:39.430 then the work, w, equals little f at the endpoint 00:59:39.430 --> 00:59:41.560 minus little f at the origin. 00:59:41.560 --> 00:59:44.730 00:59:44.730 --> 00:59:56.682 Where, by origin and endpoint are those for the path, 00:59:56.682 --> 01:00:00.618 are those for the arc, are those for the curve, c. 01:00:00.618 --> 01:00:05.060 01:00:05.060 --> 01:00:09.576 So the work, the w, will be independent of time. 01:00:09.576 --> 01:00:18.720 So w will be independent of f. 01:00:18.720 --> 01:00:21.120 And you saw an example when I took 01:00:21.120 --> 01:00:24.260 a conservative function that was really nice, 01:00:24.260 --> 01:00:27.620 y times i plus x times j. 01:00:27.620 --> 01:00:29.440 That was the force field. 01:00:29.440 --> 01:00:33.470 Because that was conservative, we got w being 1 01:00:33.470 --> 01:00:34.780 no matter what path we took. 01:00:34.780 --> 01:00:37.970 We took a parabola, we took a straight line, 01:00:37.970 --> 01:00:39.770 and we could have taken a zig-zag 01:00:39.770 --> 01:00:41.860 and we still get w equals 1. 01:00:41.860 --> 01:00:44.550 So no matter what path you are taking. 01:00:44.550 --> 01:00:46.572 Can we prove this? 01:00:46.572 --> 01:00:53.000 Well, regular classes don't prove anything, almost nothing. 01:00:53.000 --> 01:00:56.900 But we are honor students so lets see what we can do. 01:00:56.900 --> 01:00:59.210 We have to understand what's going on. 01:00:59.210 --> 01:01:06.520 Why do we have this fundamental theorem of calculus 3? 01:01:06.520 --> 01:01:15.610 The work, w, can be expressed-- assume f is conservative 01:01:15.610 --> 01:01:19.700 which means it's going to come from a potential little f. 01:01:19.700 --> 01:01:29.077 Where f is [INAUDIBLE] scalar function over my domain, omega. 01:01:29.077 --> 01:01:33.280 01:01:33.280 --> 01:01:36.670 Now, the curve, c, is part of this omega 01:01:36.670 --> 01:01:40.490 so I don't have any problems on the curve. 01:01:40.490 --> 01:01:44.680 w will be rewritten beautifully. 01:01:44.680 --> 01:01:46.625 So I'm giving you a sketch of a proof. 01:01:46.625 --> 01:01:50.335 But you would be able to do this, maybe even better than me 01:01:50.335 --> 01:01:56.310 because I have taught you what you need to do. 01:01:56.310 --> 01:02:03.580 So this is going to be f1 i, plus f2 j. 01:02:03.580 --> 01:02:09.750 And I'm going to write it. f1 times-- what is this guys? 01:02:09.750 --> 01:02:14.420 dR, I taught you, you taught me, x prime of t, right? 01:02:14.420 --> 01:02:23.010 Plus f2 times y prime of t, all dt, and time from t0 to t1. 01:02:23.010 --> 01:02:27.480 I start my motion along the curve at t equals t0 01:02:27.480 --> 01:02:30.728 and I finished my motion at t equals t1. 01:02:30.728 --> 01:02:33.620 01:02:33.620 --> 01:02:36.770 Do I know where f1 and f2 are? 01:02:36.770 --> 01:02:38.860 This is the point, that's the whole point, 01:02:38.860 --> 01:02:41.780 I know who they are, thank God. 01:02:41.780 --> 01:02:47.310 And now I have to again apply some magical think, 01:02:47.310 --> 01:02:49.650 I'll ask you in a minute what that is. 01:02:49.650 --> 01:02:53.510 So, what is f1? 01:02:53.510 --> 01:02:56.100 df dx, or f sub base. 01:02:56.100 --> 01:02:59.260 If you don't like f sub base, if you don't like my notation, 01:02:59.260 --> 01:03:00.600 you put f sub x, right? 01:03:00.600 --> 01:03:02.551 And this df dy. 01:03:02.551 --> 01:03:03.050 Why? 01:03:03.050 --> 01:03:05.340 Because it's conservative and that 01:03:05.340 --> 01:03:08.080 was the gradient of little f. 01:03:08.080 --> 01:03:11.060 Of course I'm using the fact that the first component would 01:03:11.060 --> 01:03:13.610 be the partial of little f with respect to x. 01:03:13.610 --> 01:03:16.032 The second component would be the partial of little 01:03:16.032 --> 01:03:16.865 f with respect to y. 01:03:16.865 --> 01:03:19.270 Have you seen this formula before? 01:03:19.270 --> 01:03:23.599 What in the world is this formula? 01:03:23.599 --> 01:03:25.502 STUDENT: It's the chain rule? 01:03:25.502 --> 01:03:26.828 PROFESSOR: It's the chain rule. 01:03:26.828 --> 01:03:30.630 I don't have a dollar but I will give you a dollar, OK? 01:03:30.630 --> 01:03:33.960 Imagine a virtual dollar. 01:03:33.960 --> 01:03:35.190 This is the chain rule. 01:03:35.190 --> 01:03:38.110 So by the chain rule we can write 01:03:38.110 --> 01:03:41.350 this to be the derivative with respect 01:03:41.350 --> 01:03:48.840 to t of little f of x of t, and y of t. 01:03:48.840 --> 01:03:52.630 Alrighty, so I know what I'm doing. 01:03:52.630 --> 01:03:57.860 I know that by chain rule I had little f evaluated at x of t, 01:03:57.860 --> 01:04:01.640 y of t, and time t, prime with respect to t. 01:04:01.640 --> 01:04:07.750 Now when we take the fundamental theorem of calculus, FTC. 01:04:07.750 --> 01:04:13.540 That reminds me, I was teaching calc 1 a few years ago 01:04:13.540 --> 01:04:17.390 and I said, that's the Federal Trade Commission. 01:04:17.390 --> 01:04:20.755 Federal Trade Commission, fundamental theorem 01:04:20.755 --> 01:04:22.450 of calculus. 01:04:22.450 --> 01:04:27.988 So coming back to what I have, I prove 01:04:27.988 --> 01:04:36.890 that w is the Federal Trade Commission, no. 01:04:36.890 --> 01:04:43.450 w is the application of something 01:04:43.450 --> 01:04:49.719 that we knew from calc 1, which is beautiful. 01:04:49.719 --> 01:05:00.460 f of xt, y of t dt, this is nothing but what? 01:05:00.460 --> 01:05:05.530 Little f evaluated at-- I'm going to have to write it down, 01:05:05.530 --> 01:05:07.450 this whole sausage. 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. 01:05:13.700 --> 01:05:17.240 For somebody as lazy as I am, that they effort. 01:05:17.240 --> 01:05:19.490 How can I write It better? 01:05:19.490 --> 01:05:26.060 f at the endpoint minus f at the origin. 01:05:26.060 --> 01:05:29.450 And of course, we are trying to be quite rigorous in the book. 01:05:29.450 --> 01:05:31.480 We would never say that in the book. 01:05:31.480 --> 01:05:34.650 We actually denote the first point 01:05:34.650 --> 01:05:37.690 with p, the origin, and the endpoint with q. 01:05:37.690 --> 01:05:41.808 So we say, f of q minus f of p. 01:05:41.808 --> 01:05:49.920 And we proved q e d, we proved the fundamental theorem 01:05:49.920 --> 01:05:53.000 of path integrals, the independence of that. 01:05:53.000 --> 01:06:00.200 So that means the work is independent of path 01:06:00.200 --> 01:06:03.000 when the force is conservative. 01:06:03.000 --> 01:06:07.355 Now attention, if f is not conservative you are dead meat. 01:06:07.355 --> 01:06:12.500 You cannot say what I just said. 01:06:12.500 --> 01:06:16.390 So I'll give you two separate examples 01:06:16.390 --> 01:06:19.692 and let's see how we solve each of them. 01:06:19.692 --> 01:06:27.400 01:06:27.400 --> 01:06:30.400 A final exam type of problem-- every final exam 01:06:30.400 --> 01:06:36.100 contains an application like that. 01:06:36.100 --> 01:06:41.940 Even the force field, f, or the vector value, f. 01:06:41.940 --> 01:06:44.056 Is it conservative? 01:06:44.056 --> 01:06:51.600 Prove what is proved and after that-- so the path 01:06:51.600 --> 01:06:54.642 integral in any way you can. 01:06:54.642 --> 01:06:58.020 If it's conservative you're really lucky because you're 01:06:58.020 --> 01:06:58.520 in business. 01:06:58.520 --> 01:06:59.900 You don't have to do any work. 01:06:59.900 --> 01:07:03.585 You just find the little scalar potential evaluated 01:07:03.585 --> 01:07:08.500 at the endpoints and subtract, and that's your answer. 01:07:08.500 --> 01:07:13.070 So I'm going to give you an example of a final exam problem 01:07:13.070 --> 01:07:15.545 that happened in the past year. 01:07:15.545 --> 01:07:18.515 So, a final type exam problem. 01:07:18.515 --> 01:07:25.445 01:07:25.445 --> 01:07:37.242 f of xy equals 2xy i plus x squared j, 01:07:37.242 --> 01:07:43.500 over r over r squared. 01:07:43.500 --> 01:07:45.750 STUDENT: Didn't we just do [INAUDIBLE] that? 01:07:45.750 --> 01:07:49.150 PROFESSOR: Well, I just did that but I changed the problem. 01:07:49.150 --> 01:07:51.242 I wanted to keep the same force field. 01:07:51.242 --> 01:07:51.950 STUDENT: Alright. 01:07:51.950 --> 01:07:52.810 PROFESSOR: OK. 01:07:52.810 --> 01:08:18.319 Compute the work, w, performed by f along the arc 01:08:18.319 --> 01:08:23.725 of the circle in the picture. 01:08:23.725 --> 01:08:26.710 01:08:26.710 --> 01:08:27.930 And they draw a picture. 01:08:27.930 --> 01:08:31.649 And they do a picture for you, and you stare at this picture 01:08:31.649 --> 01:08:49.290 and-- So you say, oh my God, if I 01:08:49.290 --> 01:08:53.046 were to parametrize it would be a little bit of-- I 01:08:53.046 --> 01:08:56.259 could, but it would be a little bit of work. 01:08:56.259 --> 01:09:00.340 I would have x equals 2 cosine t, y equals sine t. 01:09:00.340 --> 01:09:05.029 I would have to plug in and do that whole work, definition 01:09:05.029 --> 01:09:06.270 with parametrization. 01:09:06.270 --> 01:09:08.210 Do you have to parametrize? 01:09:08.210 --> 01:09:10.149 Not in this case, why? 01:09:10.149 --> 01:09:12.729 Because the f is conservative. 01:09:12.729 --> 01:09:19.470 If they ask you- some professors give hints, most of them 01:09:19.470 --> 01:09:22.558 are nice and give hints- show f is conservative. 01:09:22.558 --> 01:09:28.859 01:09:28.859 --> 01:09:30.729 So that's a big hint in the sense 01:09:30.729 --> 01:09:33.095 that you see it immediately, how you do it. 01:09:33.095 --> 01:09:36.149 You have 2xy prime with respect to y, 01:09:36.149 --> 01:09:39.710 is 2x, which is x squared prime with respect to x. 01:09:39.710 --> 01:09:41.700 So it is conservative. 01:09:41.700 --> 01:09:44.350 But he or she told you more. 01:09:44.350 --> 01:09:46.859 He said, I'm selling you something here, 01:09:46.859 --> 01:09:50.080 you have to get your own scalar potential. 01:09:50.080 --> 01:09:56.920 And you did, and you got x squared y. 01:09:56.920 --> 01:09:58.970 Now, most of the scalar potentials 01:09:58.970 --> 01:10:01.620 that we are giving you on the exam 01:10:01.620 --> 01:10:04.196 can be seen with naked eyes. 01:10:04.196 --> 01:10:06.770 You wouldn't have to do all the integration of that 01:10:06.770 --> 01:10:09.540 coupled system with respect to x, with respect to y, 01:10:09.540 --> 01:10:14.160 integrate backwards, and things like that. 01:10:14.160 --> 01:10:16.700 What do I need to do in that case guys? 01:10:16.700 --> 01:10:19.018 Say in words. 01:10:19.018 --> 01:10:22.480 Since the force is conservative, just two lines. 01:10:22.480 --> 01:10:25.800 I'm applying the fundamental theorem of calculus. 01:10:25.800 --> 01:10:29.650 I'm applying the fundamental theorem of path integrals. 01:10:29.650 --> 01:10:33.050 I know the work is independent of that. 01:10:33.050 --> 01:10:38.750 So w, in this case is already there, 01:10:38.750 --> 01:10:43.230 is going to be f at the point. 01:10:43.230 --> 01:10:48.006 I didn't say how I'm going to travel it, in what direction. 01:10:48.006 --> 01:10:52.028 f of q minus f of p. 01:10:52.028 --> 01:10:57.360 01:10:57.360 --> 01:11:00.630 And you'll say, well what does it mean? 01:11:00.630 --> 01:11:02.490 How do we do that? 01:11:02.490 --> 01:11:05.890 That means x squared y, evaluated at q, 01:11:05.890 --> 01:11:09.940 who the heck is q? 01:11:09.940 --> 01:11:13.710 Attention, negative 2, 0, Matt, you got it? 01:11:13.710 --> 01:11:14.920 OK, right? 01:11:14.920 --> 01:11:17.770 So, are you guys with me? 01:11:17.770 --> 01:11:19.550 Right? 01:11:19.550 --> 01:11:22.520 And p is 2, 0. 01:11:22.520 --> 01:11:31.690 So at negative 2, 0, minus x squared y at 2, 0. 01:11:31.690 --> 01:11:35.170 So, if you set 0, big, as you knew that you got 0, 01:11:35.170 --> 01:11:38.235 that is the answer. 01:11:38.235 --> 01:11:43.830 Now if somebody would give you a wiggly look that this guy's 01:11:43.830 --> 01:11:49.640 wrong, then he took this past. 01:11:49.640 --> 01:11:52.740 He's going to do exactly the same work 01:11:52.740 --> 01:11:57.800 if he's under influence the same conservative force. 01:11:57.800 --> 01:12:01.254 If the force acting on it is the same. 01:12:01.254 --> 01:12:04.265 No matter what path you are taking-- yes, sir? 01:12:04.265 --> 01:12:07.320 STUDENT: Is it still working for your self-intersecting. 01:12:07.320 --> 01:12:11.420 PROFESSOR: Yeah, because you're not stopping, 01:12:11.420 --> 01:12:13.120 it works for any parametrization. 01:12:13.120 --> 01:12:16.390 So if you're able to parametrize that as a differentiable 01:12:16.390 --> 01:12:20.640 function so that the derivative would never vanish, 01:12:20.640 --> 01:12:23.090 it's going to work, right? 01:12:23.090 --> 01:12:27.412 All right, it can work also for this piecewise contour 01:12:27.412 --> 01:12:29.300 or any other path point. 01:12:29.300 --> 01:12:32.720 As long as it starts and it ends at the same point 01:12:32.720 --> 01:12:38.020 and as long as your conservative force is the same. 01:12:38.020 --> 01:12:39.890 So that force is very [INAUDIBLE], yes? 01:12:39.890 --> 01:12:42.536 STUDENT: Do [INAUDIBLE] graphs but the endpoints the same. 01:12:42.536 --> 01:12:44.840 And if they are conservative then [INTERPOSING VOICES]. 01:12:44.840 --> 01:12:46.506 PROFESSOR: If everything is conservative 01:12:46.506 --> 01:12:49.550 the work along this path is the same. 01:12:49.550 --> 01:12:52.080 There were people who played games 01:12:52.080 --> 01:12:54.254 like that to catch if the student knows what 01:12:54.254 --> 01:12:55.420 they are talking-- yes, sir? 01:12:55.420 --> 01:12:59.067 STUDENT: So, in a problem, if they wanted 01:12:59.067 --> 01:13:00.733 to find the work couldn't we simplify it 01:13:00.733 --> 01:13:02.539 by just saying, we need to find-- because, 01:13:02.539 --> 01:13:05.080 since we're in R2, couldn't we just say it's a straight line? 01:13:05.080 --> 01:13:07.132 Because that's the-- Like, instead of a curve 01:13:07.132 --> 01:13:08.660 we could just set the straight line [INTERPOSING VOICES]. 01:13:08.660 --> 01:13:10.260 PROFESSOR: If it were moving from here 01:13:10.260 --> 01:13:12.270 to here in a straight line you would still 01:13:12.270 --> 01:13:13.940 get the same answer. 01:13:13.940 --> 01:13:17.810 And you could have-- if you did that, actually, 01:13:17.810 --> 01:13:21.170 if you were to compute this kind of work on a straight line 01:13:21.170 --> 01:13:26.120 it has-- let me show you something. 01:13:26.120 --> 01:13:32.470 You see, when you compute dx y dx, plus x squared dy. 01:13:32.470 --> 01:13:38.460 If y is 0 like Matthew said, I'm walking on-- I'm not drunk. 01:13:38.460 --> 01:13:41.350 I'm walking straight. 01:13:41.350 --> 01:13:46.810 y will be 0 here, and 0 here, and the integral will be 0. 01:13:46.810 --> 01:13:50.580 So he would have noticed that from the beginning. 01:13:50.580 --> 01:13:54.240 But unless you know the force is conservative, 01:13:54.240 --> 01:13:57.890 there is no guarantee that on another path 01:13:57.890 --> 01:14:01.490 you don't have a different answer, right? 01:14:01.490 --> 01:14:05.230 So, let me give you another example because now Matthew 01:14:05.230 --> 01:14:06.610 brought this up. 01:14:06.610 --> 01:14:09.230 01:14:09.230 --> 01:14:13.620 A catchy example that a professor 01:14:13.620 --> 01:14:18.470 gave just to make the students life miserable, 01:14:18.470 --> 01:14:22.740 and I'll show it to you in a second. 01:14:22.740 --> 01:14:29.060 01:14:29.060 --> 01:14:36.860 He said, for the picture-- very similar to this one, 01:14:36.860 --> 01:14:39.924 just to make people confused. 01:14:39.924 --> 01:14:43.270 Somebody gives you this arc of a circle 01:14:43.270 --> 01:14:45.660 and you travel from a to b. 01:14:45.660 --> 01:14:49.500 And this the thing. 01:14:49.500 --> 01:15:01.850 And he says, compute w for the force given by y i plus j. 01:15:01.850 --> 01:15:07.760 And the students said OK, I guess 01:15:07.760 --> 01:15:12.440 I'm going to get 0 because I'm going to get something 01:15:12.440 --> 01:15:16.610 like y dx plus x dy. 01:15:16.610 --> 01:15:21.699 And if y is 0, I get 0, and that way you wouldn't be 0 01:15:21.699 --> 01:15:22.240 and I'm done. 01:15:22.240 --> 01:15:27.489 No, it's not how you think because this is not 01:15:27.489 --> 01:15:30.210 conservative. 01:15:30.210 --> 01:15:33.800 So you cannot say I can change my path and it's still going 01:15:33.800 --> 01:15:35.820 to be the same. 01:15:35.820 --> 01:15:38.260 No, why is this not conservative? 01:15:38.260 --> 01:15:39.980 Quickly, this prime with respect to y 01:15:39.980 --> 01:15:42.580 is 1, this prime with respect to x is 0. 01:15:42.580 --> 01:15:46.055 So 1 different from 0 so, oh my God, no. 01:15:46.055 --> 01:15:48.160 In that case, why do we do? 01:15:48.160 --> 01:15:52.480 We have no other choice but say, x equals 2 cosine t, 01:15:52.480 --> 01:15:57.770 y equals 2 sine t, and t between 0 and pi. 01:15:57.770 --> 01:16:04.110 And then I get integral of f1, y, what the heck is y? 01:16:04.110 --> 01:16:08.948 2 sine t, times x prime of t. 01:16:08.948 --> 01:16:15.360 01:16:15.360 --> 01:16:20.850 Yeah, minus 2 sine t, this is x prime. 01:16:20.850 --> 01:16:24.650 Plus 1, are you guys with me? 01:16:24.650 --> 01:16:30.140 Times y prime, which is 2 cosine t, dt. 01:16:30.140 --> 01:16:33.370 And t between 0 and pi. 01:16:33.370 --> 01:16:36.190 01:16:36.190 --> 01:16:42.242 And you get something ugly, you get 0 to pi. 01:16:42.242 --> 01:16:43.660 What is the nice thing? 01:16:43.660 --> 01:16:48.845 When you integrate this with respect to t, you get sine t. 01:16:48.845 --> 01:16:53.480 And thank God, sine, whether you are at 0 or if pi is still 0. 01:16:53.480 --> 01:16:55.600 So this part will disappear. 01:16:55.600 --> 01:17:01.800 So all you have left is minus 4 sine squared dt. 01:17:01.800 --> 01:17:04.760 But you are not done so compute this at home 01:17:04.760 --> 01:17:07.438 because we are out of time. 01:17:07.438 --> 01:17:11.075 So don't jump to conclusions unless you know 01:17:11.075 --> 01:17:12.894 that the force is conservative. 01:17:12.894 --> 01:17:14.878 If your force is not conservative 01:17:14.878 --> 01:17:17.688 then things are going to look very ugly 01:17:17.688 --> 01:17:20.830 and your only chance is to go back to the parametrization, 01:17:20.830 --> 01:17:24.302 to the basics. 01:17:24.302 --> 01:17:27.310 So we are practically done with 13.3 01:17:27.310 --> 01:17:31.320 but I want to watch more examples next time. 01:17:31.320 --> 01:17:33.120 And I'll send you the homework. 01:17:33.120 --> 01:17:38.220 Over the weekend you would be able to start doing homework. 01:17:38.220 --> 01:17:40.620 Now, when shall I grab the homework? 01:17:40.620 --> 01:17:43.320 What if I closed it right before the final, is that? 01:17:43.320 --> 01:17:44.220 STUDENT: Yeah. 01:17:44.220 --> 01:17:46.070 PROFESSOR: Yeah? 01:17:46.070 --> 02:39:17.402