WEBVTT 00:00:01.000 --> 00:00:03.000 The following content is provided under a Creative 00:00:03.000 --> 00:00:05.000 Commons license. Your support will help MIT 00:00:05.000 --> 00:00:08.000 OpenCourseWare continue to offer high quality educational 00:00:08.000 --> 00:00:13.000 resources for free. To make a donation or to view 00:00:13.000 --> 00:00:18.000 additional materials from hundreds of MIT courses, 00:00:18.000 --> 00:00:23.000 visit MIT OpenCourseWare at ocw.mit.edu. 00:00:23.000 --> 00:00:28.000 So, let me remind you, yesterday we've defined and 00:00:28.000 --> 00:00:34.000 started to compute line integrals for work as a vector 00:00:34.000 --> 00:00:41.000 field along a curve. So, we have a curve in the 00:00:41.000 --> 00:00:48.000 plane, C. We have a vector field that 00:00:48.000 --> 00:00:55.000 gives us a vector at every point. 00:00:55.000 --> 00:01:04.000 And, we want to find the work done along the curve. 00:01:04.000 --> 00:01:11.000 So, that's the line integral along C of F dr, 00:01:11.000 --> 00:01:16.000 or more geometrically, line integral along C of F.T ds 00:01:16.000 --> 00:01:19.000 where T is the unit tangent vector, 00:01:19.000 --> 00:01:23.000 and ds is the arc length element. 00:01:23.000 --> 00:01:30.000 Or, in coordinates, that they integral of M dx N dy 00:01:30.000 --> 00:01:38.000 where M and N are the components of the vector field. 00:01:38.000 --> 00:01:46.000 OK, so -- Let's do an example that will just summarize what we 00:01:46.000 --> 00:01:51.000 did yesterday, and then we will move on to 00:01:51.000 --> 00:01:57.000 interesting observations about these things. 00:01:57.000 --> 00:02:03.000 So, here's an example we are going to look at now. 00:02:03.000 --> 00:02:11.000 Let's say I give you the vector field yi plus xj. 00:02:11.000 --> 00:02:13.000 So, it's not completely obvious what it looks like, 00:02:13.000 --> 00:02:16.000 but here is a computer plot of that vector field. 00:02:16.000 --> 00:02:21.000 So, that tells you a bit what it does. 00:02:21.000 --> 00:02:24.000 It points in all sorts of directions. 00:02:24.000 --> 00:02:31.000 And, let's say we want to find the work done by this vector 00:02:31.000 --> 00:02:35.000 field. If I move along this closed 00:02:35.000 --> 00:02:40.000 curve, I start at the origin. But, I moved along the x-axis 00:02:40.000 --> 00:02:43.000 to one. That move along the unit circle 00:02:43.000 --> 00:02:46.000 to the diagonal, and then I move back to the 00:02:46.000 --> 00:02:54.000 origin in a straight line. OK, so C consists of three 00:02:54.000 --> 00:03:06.000 parts -- -- so that you enclose a sector of a unit disk -- -- 00:03:06.000 --> 00:03:17.000 corresponding to angles between zero and 45�. 00:03:17.000 --> 00:03:24.000 So, to compute this line integral, all we have to do is 00:03:24.000 --> 00:03:33.000 we have set up three different integrals and add that together. 00:03:33.000 --> 00:03:44.000 OK, so we need to set up the integral of y dx plus x dy for 00:03:44.000 --> 00:03:54.000 each of these pieces. So, let's do the first one on 00:03:54.000 --> 00:03:59.000 the x-axis. Well, one way to parameterize 00:03:59.000 --> 00:04:06.000 that is just use the x variable. And, say that because we are on 00:04:06.000 --> 00:04:12.000 the, let's see, sorry, we are going from the 00:04:12.000 --> 00:04:17.000 origin to (1,0). Well, we know we are on the 00:04:17.000 --> 00:04:22.000 x-axis. So, y there is actually just 00:04:22.000 --> 00:04:25.000 zero. And, the variable will be x 00:04:25.000 --> 00:04:28.000 from zero to one. Or, if you prefer, 00:04:28.000 --> 00:04:31.000 you can parameterize things, say, x equals t for t from zero 00:04:31.000 --> 00:04:36.000 to one, and y equals zero. What doesn't change is y is 00:04:36.000 --> 00:04:41.000 zero, and therefore, dy is also zero. 00:04:41.000 --> 00:04:48.000 So, in fact, we are integrating y dx x dy, 00:04:48.000 --> 00:04:53.000 but that becomes, well, zero dx 0, 00:04:53.000 --> 00:04:59.000 and that's just going to give you zero. 00:04:59.000 --> 00:05:03.000 OK, so there's the line integral. 00:05:03.000 --> 00:05:07.000 Here, it's very easy to compute. Of course, you can also do it 00:05:07.000 --> 00:05:10.000 geometrically because geometrically, 00:05:10.000 --> 00:05:12.000 you can see in the picture along the x-axis, 00:05:12.000 --> 00:05:15.000 the vector field is pointing vertically. 00:05:15.000 --> 00:05:19.000 If I'm on the x-axis, my vector field is actually in 00:05:19.000 --> 00:05:23.000 the y direction. So, it's perpendicular to my 00:05:23.000 --> 00:05:24.000 curve. So, the work done is going to 00:05:24.000 --> 00:05:26.000 be zero. F dot T will be zero. 00:05:48.000 --> 00:05:51.000 OK, so F dot T is zero, so the integral is zero. 00:05:51.000 --> 00:05:57.000 OK, any questions about this first part of the calculation? 00:05:57.000 --> 00:06:04.000 No? It's OK? OK, let's move on to more 00:06:04.000 --> 00:06:11.000 interesting part of it. Let's do the second part, 00:06:11.000 --> 00:06:18.000 which is a portion of the unit circle. 00:06:18.000 --> 00:06:24.000 OK, so I should have drawn my picture. 00:06:24.000 --> 00:06:37.000 And so now we are moving on this part of the curve that's 00:06:37.000 --> 00:06:40.000 C2. And, of course we have to 00:06:40.000 --> 00:06:43.000 choose how to express x and y in terms of a single variable. 00:06:43.000 --> 00:06:46.000 Well, most likely, when you are moving on a 00:06:46.000 --> 00:06:49.000 circle, you are going to use the angle along the circle to tell 00:06:49.000 --> 00:06:53.000 you where you are. OK, so we're going to use the 00:06:53.000 --> 00:06:56.000 angle theta as a parameter. And we will say, 00:06:56.000 --> 00:07:02.000 we are on the unit circle. So, x is cosine theta and y is 00:07:02.000 --> 00:07:05.000 sine theta. What's the range of theta? 00:07:05.000 --> 00:07:11.000 Theta goes from zero to pi over four, OK? 00:07:11.000 --> 00:07:15.000 So, whenever I see dx, I will replace it by, 00:07:15.000 --> 00:07:19.000 well, the derivative of cosine is negative sine. 00:07:19.000 --> 00:07:24.000 So, minus sine theta d theta, and dy, the derivative of sine 00:07:24.000 --> 00:07:29.000 is cosine. So, it will become cosine theta 00:07:29.000 --> 00:07:34.000 d theta. OK, so I'm computing the 00:07:34.000 --> 00:07:41.000 integral of y dx x dy. That means -- -- I'll be 00:07:41.000 --> 00:07:52.000 actually computing the integral of, so, y is sine theta. 00:07:52.000 --> 00:08:01.000 dx, that's negative sine theta d theta plus x cosine. 00:08:01.000 --> 00:08:08.000 dy is cosine theta d theta from zero to pi/4. 00:08:08.000 --> 00:08:17.000 OK, so that's integral from zero to pi / 4 of cosine squared 00:08:17.000 --> 00:08:22.000 minus sine squared. And, if you know your trig, 00:08:22.000 --> 00:08:27.000 then you should recognize this as cosine of two theta. 00:08:27.000 --> 00:08:33.000 OK, so that will integrate to one half of sine two theta from 00:08:33.000 --> 00:08:36.000 zero to pi over four, sorry. 00:08:36.000 --> 00:08:44.000 And, sine pi over two is one. So, you will get one half. 00:08:44.000 --> 00:08:49.000 OK, any questions about this one? 00:08:49.000 --> 00:09:07.000 No? OK, then let's do the third one. 00:09:07.000 --> 00:09:15.000 So, the third guy is when we come back to the origin along 00:09:15.000 --> 00:09:18.000 the diagonal. OK, so we go in a straight line 00:09:18.000 --> 00:09:20.000 from this point. Where's this point? 00:09:20.000 --> 00:09:25.000 Well, this point is one over root two, one over root two. 00:09:25.000 --> 00:09:32.000 And, we go back to the origin. OK, so we need to figure out a 00:09:32.000 --> 00:09:38.000 way to express x and y in terms of the same parameter. 00:09:38.000 --> 00:09:43.000 So, one way which is very natural would be to just say, 00:09:43.000 --> 00:09:47.000 well, let's say we move from here to here over time. 00:09:47.000 --> 00:09:50.000 And, at time zero, we are here. At time one, we are here. 00:09:50.000 --> 00:09:54.000 We know how to parameterize this line. 00:09:54.000 --> 00:10:02.000 So, what we could do is say, let's parameterize this line. 00:10:02.000 --> 00:10:09.000 So, we start at one over root two, and we go down by one over 00:10:09.000 --> 00:10:19.000 root two in time one. And, same with y. 00:10:19.000 --> 00:10:24.000 That's actually perfectly fine. But that's unnecessarily 00:10:24.000 --> 00:10:27.000 complicated. OK, why is a complicated? 00:10:27.000 --> 00:10:30.000 Because we will get all of these expressions. 00:10:30.000 --> 00:10:34.000 It would be easier to actually just look at motion in this 00:10:34.000 --> 00:10:38.000 direction and then say, well, if we have a certain work 00:10:38.000 --> 00:10:42.000 if we move from here to here, then the work done moving from 00:10:42.000 --> 00:10:46.000 here to here is just going to be the opposite, 00:10:46.000 --> 00:10:48.000 OK? So, in fact, 00:10:48.000 --> 00:10:55.000 we can do slightly better by just saying, well, 00:10:55.000 --> 00:10:59.000 we'll take x = t, y = t. 00:10:59.000 --> 00:11:07.000 t from zero to one over root two, and take, 00:11:07.000 --> 00:11:15.000 well, sorry, that gives us what I will call 00:11:15.000 --> 00:11:25.000 minus C3, which is C3 backwards. And then we can say the 00:11:25.000 --> 00:11:31.000 integral for work along minus C3 is the opposite of the work 00:11:31.000 --> 00:11:35.000 along C3. Or, if you're comfortable with 00:11:35.000 --> 00:11:39.000 integration where variables go down, 00:11:39.000 --> 00:11:42.000 then you could also say that t just goes from one over square 00:11:42.000 --> 00:11:45.000 root of two down to zero. And, when you set up your 00:11:45.000 --> 00:11:49.000 integral, it will go from one over root two to zero. 00:11:49.000 --> 00:11:50.000 And, of course, that will be the negative of 00:11:50.000 --> 00:11:52.000 the one from zero to one over root two. 00:11:52.000 --> 00:11:59.000 So, it's the same thing. OK, so if we do it with this 00:11:59.000 --> 00:12:03.000 parameterization, we'll get that, 00:12:03.000 --> 00:12:08.000 well of course, dx is dt, dy is dt. 00:12:08.000 --> 00:12:16.000 So, the integral along minus C3 of y dx plus x dy is just the 00:12:16.000 --> 00:12:24.000 integral from zero to one over root two of t dt plus t dt. 00:12:24.000 --> 00:12:30.000 Sorry, I'm messing up my blackboard, OK, 00:12:30.000 --> 00:12:37.000 which is going to be, well, the integral of 2t dt, 00:12:37.000 --> 00:12:46.000 which is t2 between these bounds, which is one half. 00:12:46.000 --> 00:12:51.000 That's the integral along minus C3, along the reversed path. 00:12:51.000 --> 00:13:03.000 And, if I want to do it along C3 instead, then I just take the 00:13:03.000 --> 00:13:07.000 negative. Or, if you prefer, 00:13:07.000 --> 00:13:11.000 you could have done it directly with integral from one over root 00:13:11.000 --> 00:13:14.000 two, two zero, which gives you immediately the 00:13:14.000 --> 00:13:19.000 negative one half. OK, so at the end, 00:13:19.000 --> 00:13:28.000 we get that the total work -- -- was the sum of the three line 00:13:28.000 --> 00:13:32.000 integrals. I'm not writing after dr just 00:13:32.000 --> 00:13:36.000 to save space. But, zero plus one half minus 00:13:36.000 --> 00:13:39.000 one half, and that comes out to zero. 00:13:39.000 --> 00:13:44.000 So, a lot of calculations for nothing. 00:13:44.000 --> 00:13:49.000 OK, so that should give you overview of various ways to 00:13:49.000 --> 00:13:57.000 compute line integrals. Any questions about all that? 00:13:57.000 --> 00:14:03.000 No? OK. So, next, let me tell you about 00:14:03.000 --> 00:14:07.000 how to avoid computing like integrals. 00:14:07.000 --> 00:14:08.000 Well, one is easy: don't take this class. 00:14:08.000 --> 00:14:17.000 But that's not, so here's another way not to do 00:14:17.000 --> 00:14:20.000 it, OK? So, let's look a little bit 00:14:20.000 --> 00:14:24.000 about one kind of vector field that actually we've encountered 00:14:24.000 --> 00:14:26.000 a few weeks ago without saying it. 00:14:26.000 --> 00:14:30.000 So, we said when we have a function of two variables, 00:14:30.000 --> 00:14:32.000 we have the gradient vector. Well, at the time, 00:14:32.000 --> 00:14:35.000 it was just a vector. But, that vector depended on x 00:14:35.000 --> 00:14:36.000 and y. So, in fact, 00:14:36.000 --> 00:14:43.000 it's a vector field. OK, so here's an interesting 00:14:43.000 --> 00:14:48.000 special case. Say that F, our vector field is 00:14:48.000 --> 00:14:52.000 actually the gradient of some function. 00:14:52.000 --> 00:15:01.000 So, it's a gradient field. And, so f is a function of two 00:15:01.000 --> 00:15:08.000 variables, x and y, and that's called the potential 00:15:08.000 --> 00:15:12.000 for the vector field. The reason is, 00:15:12.000 --> 00:15:16.000 of course, from physics. In physics, you call potential, 00:15:16.000 --> 00:15:21.000 electrical potential or gravitational potential, 00:15:21.000 --> 00:15:25.000 the potential energy. This function of position that 00:15:25.000 --> 00:15:29.000 tells you how much actually energy stored somehow by the 00:15:29.000 --> 00:15:33.000 force field, and this gradient gives you the force. 00:15:33.000 --> 00:15:37.000 Actually, not quite. If you are a physicist, 00:15:37.000 --> 00:15:40.000 that the force will be negative the gradient. 00:15:40.000 --> 00:15:44.000 So, that means that physicists' potentials are the opposite of a 00:15:44.000 --> 00:15:46.000 mathematician's potential. Okay? 00:15:46.000 --> 00:15:48.000 So it's just here to confuse you. 00:15:48.000 --> 00:15:50.000 It doesn't really matter all the time. 00:15:50.000 --> 00:15:55.000 So to make things simpler we are using this convention and 00:15:55.000 --> 00:15:59.000 you just put a minus sign if you are doing physics. 00:15:59.000 --> 00:16:13.000 So then I claim that we can simplify the evaluation of the 00:16:13.000 --> 00:16:21.000 line integral for work. Perhaps you've seen in physics, 00:16:21.000 --> 00:16:24.000 the work done by, say, the electrical force, 00:16:24.000 --> 00:16:28.000 is actually given by the change in the value of a potential from 00:16:28.000 --> 00:16:30.000 the starting point of the ending point, 00:16:30.000 --> 00:16:36.000 or same for gravitational force. So, these are special cases of 00:16:36.000 --> 00:16:40.000 what's called the fundamental theorem of calculus for line 00:16:40.000 --> 00:16:43.000 integrals. So, the fundamental theorem of 00:16:43.000 --> 00:16:46.000 calculus, not for line integrals, tells you if you 00:16:46.000 --> 00:16:49.000 integrate a derivative, then you get back the function. 00:16:49.000 --> 00:16:52.000 And here, it's the same thing in multivariable calculus. 00:16:52.000 --> 00:16:54.000 It tells you, if you take the line integral 00:16:54.000 --> 00:16:58.000 of the gradient of a function, what you get back is the 00:16:58.000 --> 00:16:58.000 function. 00:17:23.000 --> 00:17:30.000 OK, so -- -- the fundamental 00:17:30.000 --> 00:17:43.000 theorem of calculus for line integrals -- -- says if you 00:17:43.000 --> 00:17:58.000 integrate a vector field that's the gradient of a function along 00:17:58.000 --> 00:18:03.000 a curve, let's say that you have a curve 00:18:03.000 --> 00:18:06.000 that goes from some starting point, P0, 00:18:06.000 --> 00:18:15.000 to some ending point, P1. All you will get is the value 00:18:15.000 --> 00:18:21.000 of F at P1 minus the value of F at P0. 00:18:21.000 --> 00:18:25.000 OK, so, that's a pretty nifty formula that only works if the 00:18:25.000 --> 00:18:28.000 field that you are integrating is a gradient. 00:18:28.000 --> 00:18:32.000 You know it's a gradient, and you know the function, 00:18:32.000 --> 00:18:35.000 little f. I mean, we can't put just any 00:18:35.000 --> 00:18:38.000 vector field in here. We have to put the gradient of 00:18:38.000 --> 00:18:41.000 F. So, actually on Tuesday we'll 00:18:41.000 --> 00:18:47.000 see how to decide whether a vector field is a gradient or 00:18:47.000 --> 00:18:49.000 not, and if it is a gradient, 00:18:49.000 --> 00:18:52.000 how to find the potential function. 00:18:52.000 --> 00:18:58.000 So, we'll cover that. But, for now we need to try to 00:18:58.000 --> 00:19:05.000 figure out a bit more about this, what it says, 00:19:05.000 --> 00:19:11.000 what it means physically, how to think of it 00:19:11.000 --> 00:19:15.000 geometrically, and so on. 00:19:15.000 --> 00:19:18.000 So, maybe I should say, if you're trying to write this 00:19:18.000 --> 00:19:21.000 in coordinates, because that's also a useful 00:19:21.000 --> 00:19:24.000 way to think about it, if I give you the line integral 00:19:24.000 --> 00:19:27.000 along C, so, the gradient field, 00:19:27.000 --> 00:19:29.000 the components are f sub x and f sub y. 00:19:29.000 --> 00:19:36.000 So, it means I'm actually integrating f sub x dx plus f 00:19:36.000 --> 00:19:38.000 sub y dy. Or, if you prefer, 00:19:38.000 --> 00:19:42.000 that's the same thing as actually integrating df. 00:19:42.000 --> 00:19:46.000 So, I'm integrating the differential of a function, 00:19:46.000 --> 00:19:54.000 f. Well then, that's the change in 00:19:54.000 --> 00:19:56.000 F. And, of course, 00:19:56.000 --> 00:20:02.000 if you write it in this form, then probably it's quite 00:20:02.000 --> 00:20:06.000 obvious to you that this should be true. 00:20:06.000 --> 00:20:11.000 I mean, in this form, actually it's the same 00:20:11.000 --> 00:20:15.000 statement as in single variable calculus. 00:20:15.000 --> 00:20:17.000 OK, and actually that's how we prove the theorem. 00:20:17.000 --> 00:20:27.000 So, let's prove this theorem. How do we prove it? 00:20:27.000 --> 00:20:31.000 Well, let's say I give you a curve and I ask you to compute 00:20:31.000 --> 00:20:34.000 this integral. How will you do that? 00:20:34.000 --> 00:20:38.000 Well, the way you compute the integral actually is by choosing 00:20:38.000 --> 00:20:41.000 a parameter, and expressing everything in terms of that 00:20:41.000 --> 00:20:46.000 parameter. So, we'll set, 00:20:46.000 --> 00:20:57.000 well, so we know it's f sub x dx plus f sub y dy. 00:20:57.000 --> 00:21:03.000 And, we'll want to parameterize C in the form x equals x of t. 00:21:03.000 --> 00:21:09.000 y equals y of t. So, if we do that, 00:21:09.000 --> 00:21:12.000 then dx becomes x prime of t dt. 00:21:12.000 --> 00:21:25.000 dy becomes y prime of t dt. So, we know x is x of t. 00:21:25.000 --> 00:21:31.000 That tells us dx is x prime of t dt. 00:21:31.000 --> 00:21:38.000 y is y of t gives us dy is y prime of t dt. 00:21:38.000 --> 00:21:52.000 So, now what we are integrating actually becomes the integral of 00:21:52.000 --> 00:22:05.000 f sub x times dx dt plus f sub y times dy dt times dt. 00:22:05.000 --> 00:22:09.000 OK, but now, here I recognize a familiar 00:22:09.000 --> 00:22:13.000 guy. I've seen this one before in 00:22:13.000 --> 00:22:15.000 the chain rule. OK, this guy, 00:22:15.000 --> 00:22:19.000 by the chain rule, is the rate of change of f if I 00:22:19.000 --> 00:22:22.000 take x and y to be functions of t. 00:22:22.000 --> 00:22:26.000 And, I plug those into f. So, in fact, 00:22:26.000 --> 00:22:34.000 what I'm integrating is df dt when I think of f as a function 00:22:34.000 --> 00:22:42.000 of t by just plugging x and y as functions of t. 00:22:42.000 --> 00:22:51.000 And so maybe actually I should now say I have sometimes t goes 00:22:51.000 --> 00:22:59.000 from some initial time, let's say, t zero to t one. 00:22:59.000 --> 00:23:03.000 And now, by the usual fundamental theorem of calculus, 00:23:03.000 --> 00:23:07.000 I know that this will be just the change in the value of f 00:23:07.000 --> 00:23:09.000 between t zero and t one. 00:23:36.000 --> 00:23:44.000 OK, so integral from t zero to one of (df /dt) dt, 00:23:44.000 --> 00:23:52.000 well, that becomes f between t zero and t one. 00:23:52.000 --> 00:23:55.000 f of what? We just have to be a little bit 00:23:55.000 --> 00:23:58.000 careful here. Well, it's not quite f of t. 00:23:58.000 --> 00:24:02.000 It's f seen as a function of t by putting x of t and y of t 00:24:02.000 --> 00:24:07.000 into it. So, let me read that carefully. 00:24:07.000 --> 00:24:15.000 What I'm integrating to is f of x of t and y of t. 00:24:15.000 --> 00:24:19.000 Does that sound fair? Yeah, and so, 00:24:19.000 --> 00:24:23.000 when I plug in t1, I get the point where I am at 00:24:23.000 --> 00:24:26.000 time t1. That's the endpoint of my curve. 00:24:26.000 --> 00:24:31.000 When I plug t0, I will get the starting point 00:24:31.000 --> 00:24:37.000 of my curve, p0. And, that's the end of the 00:24:37.000 --> 00:24:43.000 proof. It wasn't that hard, see? 00:24:43.000 --> 00:24:56.000 OK, so let's see an example. Well, let's look at that 00:24:56.000 --> 00:25:00.000 example again. So, we have this curve. 00:25:00.000 --> 00:25:03.000 We have this vector field. Could it be that, 00:25:03.000 --> 00:25:05.000 by accident, that vector field was a 00:25:05.000 --> 00:25:08.000 gradient field? So, remember, 00:25:08.000 --> 00:25:12.000 our vector field was y, x. 00:25:12.000 --> 00:25:16.000 Can we think of a function whose derivative with respect to 00:25:16.000 --> 00:25:19.000 x is y, and derivative with respect to y is x? 00:25:19.000 --> 00:25:27.000 Yeah, x times y sounds like a good candidate where f( x, 00:25:27.000 --> 00:25:30.000 y) is xy. OK, 00:25:30.000 --> 00:25:34.000 so that means that the line integrals that we computed along 00:25:34.000 --> 00:25:39.000 these things can be just evaluated from just finding out 00:25:39.000 --> 00:25:44.000 the values of f at the endpoint? So, here's version two of my 00:25:44.000 --> 00:25:50.000 plot where I've added the contour plot of a function, 00:25:50.000 --> 00:25:53.000 x, y on top of the vector field. 00:25:53.000 --> 00:25:57.000 Actually, they have a vector field is still pointing 00:25:57.000 --> 00:26:00.000 perpendicular to the level curves that we have seen, 00:26:00.000 --> 00:26:02.000 just to remind you. And, so now, 00:26:02.000 --> 00:26:05.000 when we move, now when we move, 00:26:05.000 --> 00:26:09.000 the origin is on the level curve, f equals zero. 00:26:09.000 --> 00:26:14.000 And, when we start going along C1, we stay on f equals zero. 00:26:14.000 --> 00:26:17.000 So, there's no work. The potential doesn't change. 00:26:17.000 --> 00:26:21.000 Then on C2, the potential increases from zero to one half. 00:26:21.000 --> 00:26:24.000 The work is one half. And then, on C3, 00:26:24.000 --> 00:26:27.000 we go back down from one half to zero. 00:26:27.000 --> 00:26:40.000 The work is negative one half. See, that was much easier than 00:26:40.000 --> 00:26:47.000 computing. So, for example, 00:26:47.000 --> 00:26:53.000 the integral along C2 is actually just, 00:26:53.000 --> 00:27:00.000 so, C2 goes from one zero to one over root two, 00:27:00.000 --> 00:27:09.000 one over root two. So, that's one half minus zero, 00:27:09.000 --> 00:27:18.000 and that's one half, OK, because C2 was going here. 00:27:18.000 --> 00:27:26.000 And, at this point, f is zero. At that point, f is one half. 00:27:26.000 --> 00:27:29.000 And, similarly for the others, and of course when you sum, 00:27:29.000 --> 00:27:33.000 you get zero because the total change in f when you go from 00:27:33.000 --> 00:27:35.000 here, to here, to here, to here, 00:27:35.000 --> 00:27:37.000 eventually you are back at the same place. 00:27:37.000 --> 00:27:44.000 So, f hasn't changed. OK, so that's a neat trick. 00:27:44.000 --> 00:27:48.000 And it's important conceptually because a lot of the forces are 00:27:48.000 --> 00:27:53.000 gradients of potentials, namely, gravitational force, 00:27:53.000 --> 00:27:57.000 electric force. The problem is not every vector 00:27:57.000 --> 00:28:00.000 field is a gradient. A lot of vector fields are not 00:28:00.000 --> 00:28:02.000 gradients. For example, 00:28:02.000 --> 00:28:08.000 magnetic fields certainly are not gradients. 00:28:08.000 --> 00:28:33.000 So -- -- a big warning: everything today only applies 00:28:33.000 --> 00:28:48.000 if F is a gradient field. OK, it's not true otherwise. 00:29:07.000 --> 00:29:19.000 OK, still, let's see, what are the consequences of 00:29:19.000 --> 00:29:30.000 the fundamental theorem? So, just to put one more time 00:29:30.000 --> 00:29:39.000 this disclaimer, if F is a gradient field -- -- 00:29:39.000 --> 00:29:44.000 then what do we have? Well, there's various nice 00:29:44.000 --> 00:29:47.000 features of work done by gradient fields that are not too 00:29:47.000 --> 00:29:53.000 far off the vector fields. So, one of them is this 00:29:53.000 --> 00:30:02.000 property of path independence. OK, so the claim is if I have a 00:30:02.000 --> 00:30:05.000 line integral to compute, that it doesn't matter which 00:30:05.000 --> 00:30:09.000 path I take as long as it goes from point a to point b. 00:30:09.000 --> 00:30:14.000 It just depends on the point where I start and the point 00:30:14.000 --> 00:30:18.000 where I end. And, that's certainly false in 00:30:18.000 --> 00:30:22.000 general, but for a gradient field that works. 00:30:22.000 --> 00:30:25.000 So if I have a point, P0, 00:30:25.000 --> 00:30:28.000 a point, P1, and I have two different paths 00:30:28.000 --> 00:30:33.000 that go there, say, C1 and C2, 00:30:33.000 --> 00:30:39.000 so they go from the same point to the same point but in 00:30:39.000 --> 00:30:44.000 different ways, then in this situation, 00:30:44.000 --> 00:30:55.000 the line integral along C1 is equal to the line integral along 00:30:55.000 --> 00:30:56.000 C2. Well, actually, 00:30:56.000 --> 00:31:02.000 let me insist that this is only for gradient fields by putting 00:31:02.000 --> 00:31:09.000 gradient F in here, just so you don't get tempted 00:31:09.000 --> 00:31:19.000 to ever use this for a field that's not a gradient field -- 00:31:19.000 --> 00:31:28.000 -- if C1 and C2 have the same start and end point. 00:31:28.000 --> 00:31:30.000 OK, how do you prove that? Well, it's very easy. 00:31:30.000 --> 00:31:32.000 We just use the fundamental theorem. 00:31:32.000 --> 00:31:35.000 It tells us, if you compute the line 00:31:35.000 --> 00:31:38.000 integral along C1, it's just F at this point minus 00:31:38.000 --> 00:31:41.000 F at this point. If you do it for C2, 00:31:41.000 --> 00:31:45.000 well, the same. So, they are the same. 00:31:45.000 --> 00:31:48.000 And for that you don't actually even need to know what little f 00:31:48.000 --> 00:31:50.000 is. You know in advance that it's 00:31:50.000 --> 00:31:53.000 going to be the same. So, if I give you a vector 00:31:53.000 --> 00:31:56.000 field and I tell you it's the gradient of mysterious function 00:31:56.000 --> 00:31:58.000 but I don't tell you what the function is and you don't want 00:31:58.000 --> 00:32:00.000 to find out, you can still use path 00:32:00.000 --> 00:32:03.000 independence, but only if you know it's a 00:32:03.000 --> 00:32:03.000 gradient. 00:32:25.000 --> 00:32:35.000 OK, I guess this one is dead. So, that will stay here forever 00:32:35.000 --> 00:32:40.000 because nobody is tall enough to erase it. 00:32:40.000 --> 00:32:49.000 When you come back next year and you still see that formula, 00:32:49.000 --> 00:32:53.000 you'll see. Yes, but there's no useful 00:32:53.000 --> 00:32:59.000 information here. That's a good point. 00:32:59.000 --> 00:33:06.000 OK, so what's another consequence? 00:33:06.000 --> 00:33:14.000 So, if you have a gradient field, it's what's called 00:33:14.000 --> 00:33:19.000 conservative. OK, so what a conservative 00:33:19.000 --> 00:33:21.000 field? Well, the word conservative 00:33:21.000 --> 00:33:26.000 comes from the idea in physics; if the conservation of energy. 00:33:26.000 --> 00:33:31.000 It tells you that you cannot get energy for free out of your 00:33:31.000 --> 00:33:33.000 force field. So, 00:33:33.000 --> 00:33:36.000 what it means is that in particular, 00:33:36.000 --> 00:33:39.000 if you take a closed trajectory, 00:33:39.000 --> 00:33:44.000 so a trajectory that goes from some point back to the same 00:33:44.000 --> 00:33:52.000 point, so, if C is a closed curve, 00:33:52.000 --> 00:34:03.000 then the work done along C -- -- is zero. 00:34:03.000 --> 00:34:06.000 OK, that's the definition of what it means to be 00:34:06.000 --> 00:34:09.000 conservative. If I take any closed curve, 00:34:09.000 --> 00:34:13.000 the work will always be zero. On the contrary, 00:34:13.000 --> 00:34:17.000 not conservative means somewhere there is a curve along 00:34:17.000 --> 00:34:21.000 which the work is not zero. If you find a curve where the 00:34:21.000 --> 00:34:23.000 work is zero, that's not enough to say it's 00:34:23.000 --> 00:34:26.000 conservative. You have show that no matter 00:34:26.000 --> 00:34:30.000 what curve I give you, if it's a closed curve, 00:34:30.000 --> 00:34:34.000 it will always be zero. So, what that means concretely 00:34:34.000 --> 00:34:37.000 is if you have a force field that conservative, 00:34:37.000 --> 00:34:42.000 then you cannot build somehow some perpetual motion out of it. 00:34:42.000 --> 00:34:44.000 You can't build something that will just keep going just 00:34:44.000 --> 00:34:47.000 powered by that force because that force is actually not 00:34:47.000 --> 00:34:50.000 providing any energy. After you've gone one loop 00:34:50.000 --> 00:34:53.000 around, nothings happened from the point of view of the energy 00:34:53.000 --> 00:34:57.000 provided by that force. There's no work coming from the 00:34:57.000 --> 00:34:59.000 force, while if you have a force field 00:34:59.000 --> 00:35:02.000 that's not conservative than you can try to actually maybe find a 00:35:02.000 --> 00:35:04.000 loop where the work would be positive. 00:35:04.000 --> 00:35:07.000 And then, you know, that thing will just keep 00:35:07.000 --> 00:35:08.000 running. So actually, 00:35:08.000 --> 00:35:13.000 if you just look at magnetic fields and transformers or power 00:35:13.000 --> 00:35:15.000 adapters, and things like that, 00:35:15.000 --> 00:35:18.000 you precisely extract energy from the magnetic field. 00:35:18.000 --> 00:35:20.000 Of course, I mean, you actually have to take some 00:35:20.000 --> 00:35:22.000 power supply to maintain the magnetic fields. 00:35:22.000 --> 00:35:25.000 But, so a magnetic field, you could actually try to get 00:35:25.000 --> 00:35:31.000 energy from it almost for free. A gravitational field or an 00:35:31.000 --> 00:35:35.000 electric field, you can't. 00:35:35.000 --> 00:35:41.000 OK, so and now why does that hold? 00:35:41.000 --> 00:35:43.000 Well, if I have a gradient field, 00:35:43.000 --> 00:35:47.000 then if I try to compute this line integral, 00:35:47.000 --> 00:35:50.000 I know it will be the value of the function at the end point 00:35:50.000 --> 00:35:52.000 minus the value at the starting point. 00:35:52.000 --> 00:36:04.000 But, they are the same. So, the value is the same. 00:36:04.000 --> 00:36:09.000 So, if I have a gradient field, and I do the line integral, 00:36:09.000 --> 00:36:13.000 then I will get f at the endpoint minus f at the starting 00:36:13.000 --> 00:36:23.000 point. But, they're the same point, 00:36:23.000 --> 00:36:32.000 so that's zero. OK, so just to reinforce my 00:36:32.000 --> 00:36:38.000 warning that not every field is a gradient field, 00:36:38.000 --> 00:36:45.000 let's look again at our favorite vector field from 00:36:45.000 --> 00:36:49.000 yesterday. So, our favorite vector field 00:36:49.000 --> 00:36:54.000 yesterday was negative y and x. It's a vector field that just 00:36:54.000 --> 00:36:59.000 rotates around the origin counterclockwise. 00:36:59.000 --> 00:37:07.000 Well, we said, say you take just the unit 00:37:07.000 --> 00:37:17.000 circle -- -- for example, counterclockwise. 00:37:17.000 --> 00:37:23.000 Well, remember we said yesterday that the line integral 00:37:23.000 --> 00:37:28.000 of F dr, maybe I should say F dot T ds now, 00:37:28.000 --> 00:37:34.000 because the vector field is tangent to the circle. 00:37:34.000 --> 00:37:43.000 So, on the unit circle, F is tangent to the curve. 00:37:43.000 --> 00:37:49.000 And so, F dot T is length F times, well, length T. 00:37:49.000 --> 00:37:53.000 But, T is a unit vector. So, it's length F. 00:37:53.000 --> 00:37:57.000 And, the length of F on the unit circle was just one. 00:37:57.000 --> 00:38:02.000 So, that's the integral of 1 ds. So, it's just the length of the 00:38:02.000 --> 00:38:07.000 circle that's 2 pi. And 2 pi is definitely not zero. 00:38:07.000 --> 00:38:13.000 So, this vector field is not conservative. 00:38:13.000 --> 00:38:17.000 And so, now we know actually it's not the gradient of 00:38:17.000 --> 00:38:22.000 anything because if it were a gradient, then it would be 00:38:22.000 --> 00:38:28.000 conservative and it's not. So, it's an example of a vector 00:38:28.000 --> 00:38:33.000 field that is not conservative. It's not path independent 00:38:33.000 --> 00:38:38.000 either by the way because, see, if I go from here to here 00:38:38.000 --> 00:38:43.000 along the upper half circle or along the lower half circle, 00:38:43.000 --> 00:38:46.000 in one case I will get pi. In the other case I will get 00:38:46.000 --> 00:38:49.000 negative pi. I don't get the same answer, 00:38:49.000 --> 00:38:54.000 and so on, and so on. It just fails to have all of 00:38:54.000 --> 00:38:59.000 these properties. So, maybe I will write that 00:38:59.000 --> 00:39:06.000 down. It's not conservative, 00:39:06.000 --> 00:39:21.000 not path independent. It's not a gradient. 00:39:21.000 --> 00:39:29.000 It doesn't have any of these properties. 00:39:29.000 --> 00:39:38.000 OK, any questions? Yes? 00:39:38.000 --> 00:39:40.000 How do you determine whether something is a gradient or not? 00:39:40.000 --> 00:39:44.000 Well, that's what we will see on Tuesday. 00:39:44.000 --> 00:39:50.000 Yes? Is it possible that it's 00:39:50.000 --> 00:39:52.000 conservative and not path independent, or vice versa? 00:39:52.000 --> 00:39:54.000 The answer is no; these two properties are 00:39:54.000 --> 00:39:57.000 equivalent, and we are going to see that right now. 00:39:57.000 --> 00:40:12.000 At least that's the plan. OK, yes? 00:40:12.000 --> 00:40:15.000 Let's see, so you said if it's not path independent, 00:40:15.000 --> 00:40:19.000 then we cannot draw level curves that are perpendicular to 00:40:19.000 --> 00:40:23.000 it at every point. I wouldn't necessarily go that 00:40:23.000 --> 00:40:25.000 far. You might be able to draw 00:40:25.000 --> 00:40:27.000 curves that are perpendicular to it. 00:40:27.000 --> 00:40:32.000 But they won't be the level curves of a function for which 00:40:32.000 --> 00:40:34.000 this is the gradient. I mean, you might still have, 00:40:34.000 --> 00:40:36.000 you know, if you take, say, 00:40:36.000 --> 00:40:40.000 take his gradient field and scale it that in strange ways, 00:40:40.000 --> 00:40:42.000 you know, multiply by two in some places, 00:40:42.000 --> 00:40:45.000 by one in other places, by five and some other places, 00:40:45.000 --> 00:40:48.000 you will get something that won't be conservative anymore. 00:40:48.000 --> 00:40:51.000 And it will still be perpendicular to the curves. 00:40:51.000 --> 00:40:56.000 So, it's more subtle than that, but certainly if it's not 00:40:56.000 --> 00:41:01.000 conservative then it's not a gradient, and you cannot do what 00:41:01.000 --> 00:41:04.000 we said. And how to decide whether it is 00:41:04.000 --> 00:41:06.000 or not, they'll be Tuesday's topic. 00:41:06.000 --> 00:41:17.000 So, for now, I just want to figure out again 00:41:17.000 --> 00:41:31.000 actually, let's now state all these properties -- Actually, 00:41:31.000 --> 00:41:41.000 let me first do one minute of physics. 00:41:41.000 --> 00:41:48.000 So, let me just tell you again what's the physics in here. 00:41:48.000 --> 00:42:00.000 So, it's a force field is the gradient of a potential -- -- 00:42:00.000 --> 00:42:07.000 so, I'll still keep my plus signs. 00:42:07.000 --> 00:42:13.000 So, maybe I should say this is minus physics. 00:42:13.000 --> 00:42:20.000 [LAUGHTER] So, the work of F is the change 00:42:20.000 --> 00:42:31.000 in value of potential from one endpoint to the other endpoint. 00:42:31.000 --> 00:42:45.000 [PAUSE ] And -- -- so, you know, you might know about 00:42:45.000 --> 00:42:58.000 gravitational fields, or electrical -- -- fields 00:42:58.000 --> 00:43:13.000 versus gravitational -- -- or electrical potential. 00:43:13.000 --> 00:43:16.000 And, in case you haven't done any 8.02 yet, 00:43:16.000 --> 00:43:19.000 electrical potential is also commonly known as voltage. 00:43:19.000 --> 00:43:27.000 It's the one that makes it hurt when you stick your fingers into 00:43:27.000 --> 00:43:33.000 the socket. [LAUGHTER] Don't try it. 00:43:33.000 --> 00:43:45.000 OK, and so now, conservativeness means no 00:43:45.000 --> 00:44:01.000 energy can be extracted for free -- -- from the field. 00:44:01.000 --> 00:44:05.000 You can't just have, you know, a particle moving in that field 00:44:05.000 --> 00:44:08.000 and going on in definitely, faster and faster, 00:44:08.000 --> 00:44:11.000 or if there's actually friction, 00:44:11.000 --> 00:44:25.000 then keep moving. So, total energy is conserved. 00:44:25.000 --> 00:44:29.000 And, I guess, that's why we call that 00:44:29.000 --> 00:44:43.000 conservative. OK, so let's end with the recap 00:44:43.000 --> 00:44:57.000 of various equivalent properties. 00:44:57.000 --> 00:45:04.000 OK, so the first property that I will have for a vector field 00:45:04.000 --> 00:45:12.000 is that it's conservative. So, to say that a vector field 00:45:12.000 --> 00:45:22.000 with conservative means that the line integral is zero along any 00:45:22.000 --> 00:45:26.000 closed curve. Maybe to clarify, 00:45:26.000 --> 00:45:32.000 sorry, along all closed curves, OK, every closed curve; 00:45:32.000 --> 00:45:36.000 give me any closed curve, I get zero. 00:45:36.000 --> 00:45:44.000 So, now I claim this is the same thing as a second property, 00:45:44.000 --> 00:45:53.000 which is that the line integral of F is path independent. 00:45:53.000 --> 00:45:56.000 OK, so that means if I have two paths with the same endpoint, 00:45:56.000 --> 00:45:58.000 then I will get always the same answer. 00:45:58.000 --> 00:46:03.000 Why is that equivalent? Well, let's say that I am path 00:46:03.000 --> 00:46:06.000 independent. If I am path independent, 00:46:06.000 --> 00:46:10.000 then if I take a closed curve, well, it has the same endpoints 00:46:10.000 --> 00:46:13.000 as just the curve that doesn't move at all. 00:46:13.000 --> 00:46:16.000 So, path independence tells me instead of going all around, 00:46:16.000 --> 00:46:21.000 I could just stay where I am. And then, the work would just 00:46:21.000 --> 00:46:24.000 be zero. So, if I path independent, 00:46:24.000 --> 00:46:28.000 tonight conservative. Conversely, let's say that I'm 00:46:28.000 --> 00:46:32.000 just conservative and I want to check path independence. 00:46:32.000 --> 00:46:37.000 Well, so I have two points, and then I had to paths between 00:46:37.000 --> 00:46:39.000 that. I want to show that the work is 00:46:39.000 --> 00:46:43.000 the same. Well, how I do that? 00:46:43.000 --> 00:46:49.000 C1 and C2, well, I observe that if I do C1 minus 00:46:49.000 --> 00:46:54.000 C2, I get a closed path. If I go first from here to 00:46:54.000 --> 00:46:57.000 here, and then back along that one, I get a closed path. 00:46:57.000 --> 00:47:01.000 So, if I am conservative, I should get zero. 00:47:01.000 --> 00:47:05.000 But, if I get zero on C1 minus C2, it means that the work on C1 00:47:05.000 --> 00:47:10.000 and the work on C2 are the same. See, so it's the same. 00:47:10.000 --> 00:47:19.000 It's just a different way to think about the situation. 00:47:19.000 --> 00:47:24.000 More things that are equivalent, I have two more 00:47:24.000 --> 00:47:29.000 things to say. The third one, 00:47:29.000 --> 00:47:38.000 it's equivalent to F being a gradient field. 00:47:38.000 --> 00:47:46.000 OK, so this is equivalent to the third property. 00:47:46.000 --> 00:47:59.000 F is a gradient field. Why? 00:47:59.000 --> 00:48:02.000 Well, if we know that it's a gradient field, 00:48:02.000 --> 00:48:05.000 that we've seen that we get these properties out of the 00:48:05.000 --> 00:48:08.000 fundamental theorem. The question is, 00:48:08.000 --> 00:48:12.000 if I have a conservative, or path independent vector 00:48:12.000 --> 00:48:15.000 field, why is it the gradient of something? 00:48:15.000 --> 00:48:25.000 OK, so this way is a fundamental theorem. 00:48:25.000 --> 00:48:31.000 That way, well, so that actually, 00:48:31.000 --> 00:48:43.000 let me just say that will be how we find the potential. 00:48:43.000 --> 00:48:46.000 So, how do we find potential? Well, let's say that I know the 00:48:46.000 --> 00:48:49.000 value of my potential here. Actually, I get to choose what 00:48:49.000 --> 00:48:51.000 it is. Remember, in physics, 00:48:51.000 --> 00:48:53.000 the potential is defined up to adding or subtracting a 00:48:53.000 --> 00:48:56.000 constant. What matters is only the change 00:48:56.000 --> 00:48:58.000 in potential. So, let's say I know my 00:48:58.000 --> 00:49:01.000 potential here and I want to know my potential here. 00:49:01.000 --> 00:49:04.000 What do I do? Well, I take my favorite 00:49:04.000 --> 00:49:07.000 particle and I move it from here to here. 00:49:07.000 --> 00:49:10.000 And, I look at the work done. And that tells me how much 00:49:10.000 --> 00:49:14.000 potential has changed. So, that tells me what the 00:49:14.000 --> 00:49:18.000 potential should be here. And, this does not depend on my 00:49:18.000 --> 00:49:21.000 choice of path because I've assumed that I'm path 00:49:21.000 --> 00:49:25.000 independence. So, that's who we will do on 00:49:25.000 --> 00:49:29.000 Tuesday. And, let me just state the 00:49:29.000 --> 00:49:36.000 fourth property that's the same. So, all that stuff is the same 00:49:36.000 --> 00:49:42.000 as also four. If I look at M dx N dy is 00:49:42.000 --> 00:49:48.000 what's called an exact differential. 00:49:48.000 --> 00:49:52.000 So, what that means, an exact differential, 00:49:52.000 --> 00:49:56.000 means that it can be put in the form df for some function, 00:49:56.000 --> 00:49:58.000 f, and just reformulating this 00:49:58.000 --> 00:50:02.000 thing, right, because I'm saying I can just 00:50:02.000 --> 00:50:05.000 put it in the form f sub x dx plus f sub y dy, 00:50:05.000 --> 00:50:08.000 which means my vector field was a gradient field. 00:50:08.000 --> 00:50:13.000 So, these things are really the same. 00:50:13.000 --> 00:50:16.000 OK, so after the weekend, on Tuesday we will actually 00:50:16.000 --> 00:50:19.000 figure out how to decide whether these things hold or not, 00:50:19.000 --> 00:50:22.000 and how to find the potential.