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. 00:00:24.000 --> 00:00:34.000 We were looking at vector fields last time. 00:00:34.000 --> 00:00:45.000 Last time we saw that if a vector field happens to be a 00:00:45.000 --> 00:00:56.000 gradient field -- -- then the line integral can be computed 00:00:56.000 --> 00:01:08.000 actually by taking the change in value of the potential between 00:01:08.000 --> 00:01:19.000 the end point and the starting point of the curve. 00:01:19.000 --> 00:01:24.000 If we have a curve c, from a point p0 to a point p1 00:01:24.000 --> 00:01:29.000 then the line integral for work depends only on the end points 00:01:29.000 --> 00:01:32.000 and not on the actual path we chose. 00:01:32.000 --> 00:01:43.000 We say that the line integral is path independent. 00:01:43.000 --> 00:01:49.000 And we also said that the vector field is conservative 00:01:49.000 --> 00:01:55.000 because of conservation of energy which tells you if you 00:01:55.000 --> 00:02:02.000 start at a point and you come back to the same point then you 00:02:02.000 --> 00:02:07.000 haven't gotten any work out of that force. 00:02:07.000 --> 00:02:15.000 If we have a closed curve then the line integral for work is 00:02:15.000 --> 00:02:18.000 just zero. And, basically, 00:02:18.000 --> 00:02:23.000 we say that these properties are equivalent being a gradient 00:02:23.000 --> 00:02:28.000 field or being path independent or being conservative. 00:02:28.000 --> 00:02:31.000 And what I promised to you is that today we would see a 00:02:31.000 --> 00:02:35.000 criterion to decide whether a vector field is a gradient field 00:02:35.000 --> 00:02:38.000 or not and how to find the potential function if it is a 00:02:38.000 --> 00:02:47.000 gradient field. So, that is the topic for today. 00:02:47.000 --> 00:03:00.000 The question is testing whether a given vector field, 00:03:00.000 --> 00:03:14.000 let's say M and N compliments, is a gradient field. 00:03:14.000 --> 00:03:16.000 For that, well, let's start with an 00:03:16.000 --> 00:03:26.000 observation. Say that it is a gradient field. 00:03:26.000 --> 00:03:31.000 That means that the first component of a field is just the 00:03:31.000 --> 00:03:35.000 partial of f with respect to some variable x and the second 00:03:35.000 --> 00:03:40.000 component is the partial of f with respect to y. 00:03:40.000 --> 00:03:43.000 Now we have seen an interesting property of the second partial 00:03:43.000 --> 00:03:46.000 derivatives of the function, which is if you take the 00:03:46.000 --> 00:03:49.000 partial derivative first with respect to x, 00:03:49.000 --> 00:03:52.000 then with respect to y, or first with respect to y, 00:03:52.000 --> 00:03:58.000 then with respect to x you get the same thing. 00:03:58.000 --> 00:04:07.000 We know f sub xy equals f sub yx, and that means M sub y 00:04:07.000 --> 00:04:12.000 equals N sub x. If you have a gradient field 00:04:12.000 --> 00:04:14.000 then it should have this property. 00:04:14.000 --> 00:04:17.000 You take the y component, take the derivative with 00:04:17.000 --> 00:04:19.000 respect to x, take the x component, 00:04:19.000 --> 00:04:20.000 differentiate with respect to y, 00:04:20.000 --> 00:04:31.000 you should get the same answer. And that is important to know. 00:04:31.000 --> 00:04:37.000 So, I am going to put that in a box. 00:04:37.000 --> 00:04:43.000 It is a broken box. The claim that I want to make 00:04:43.000 --> 00:04:45.000 is that there is a converse of sorts. 00:04:45.000 --> 00:04:47.000 This is actually basically all we need to check. 00:05:06.000 --> 00:05:18.000 Conversely, if, and I am going to put here a 00:05:18.000 --> 00:05:33.000 condition, My equals Nx, then F is a gradient field. 00:05:33.000 --> 00:05:35.000 What is the condition that I need to put here? 00:05:35.000 --> 00:05:37.000 Well, we will see a more precise version of that next 00:05:37.000 --> 00:05:44.000 week. But for now let's just say if 00:05:44.000 --> 00:05:59.000 our vector field is defined and differentiable everywhere in the 00:05:59.000 --> 00:06:01.000 plane. We need, actually, 00:06:01.000 --> 00:06:04.000 a vector field that is well-defined everywhere. 00:06:04.000 --> 00:06:07.000 You are not allowed to have somehow places where it is not 00:06:07.000 --> 00:06:09.000 well-defined. Otherwise, actually, 00:06:09.000 --> 00:06:13.000 you have a counter example on your problem set this week. 00:06:13.000 --> 00:06:16.000 If you look at the last problem on the problem set this week, 00:06:16.000 --> 00:06:20.000 it gives you a vector field that satisfies this condition 00:06:20.000 --> 00:06:22.000 everywhere where it is defined. But, actually, 00:06:22.000 --> 00:06:24.000 there is a point where it is not defined. 00:06:24.000 --> 00:06:28.000 And that causes it, actually, to somehow -- I mean 00:06:28.000 --> 00:06:33.000 everything that I am going to say today breaks down for that 00:06:33.000 --> 00:06:36.000 example because of that. I mean, we will shed more light 00:06:36.000 --> 00:06:39.000 on this a bit later with the notion of simply connected 00:06:39.000 --> 00:06:42.000 regions and so on. But for now let's just say if 00:06:42.000 --> 00:06:47.000 it is defined everywhere and it satisfies this criterion then it 00:06:47.000 --> 00:06:52.000 is a gradient field. If you ignore the technical 00:06:52.000 --> 00:06:57.000 condition, being a gradient field means essentially the same 00:06:57.000 --> 00:07:11.000 thing as having this property. That is what we need to check. 00:07:11.000 --> 00:07:20.000 Let's look at an example. Well, one vector field that we 00:07:20.000 --> 00:07:24.000 have been looking at a lot was - yi xj. 00:07:24.000 --> 00:07:30.000 Remember that was the vector field that looked like a 00:07:30.000 --> 00:07:35.000 rotation at the unit speed. I think last time we already 00:07:35.000 --> 00:07:39.000 decided that this guy should not be allowed to be a gradient 00:07:39.000 --> 00:07:42.000 field and should not be conservative because if we 00:07:42.000 --> 00:07:45.000 integrate on the unit circle then we would get a positive 00:07:45.000 --> 00:07:49.000 answer. But let's check that indeed it 00:07:49.000 --> 00:07:55.000 fails our test. Well, let's call this M and 00:07:55.000 --> 00:08:01.000 let's call this guy N. If you look at partial M, 00:08:01.000 --> 00:08:07.000 partial y, that is going to be a negative one. 00:08:07.000 --> 00:08:11.000 If you take partial N, partial x, that is going to be 00:08:11.000 --> 00:08:12.000 one. These are not the same. 00:08:12.000 --> 00:08:17.000 So, indeed, this is not a gradient field. 00:08:32.000 --> 00:08:53.000 Any questions about that? Yes? 00:08:53.000 --> 00:08:58.000 Your question is if I have the property M sub y equals N sub x 00:08:58.000 --> 00:09:03.000 only in a certain part of a plane for some values of x and 00:09:03.000 --> 00:09:06.000 y, can I conclude these things? 00:09:06.000 --> 00:09:09.000 And it is a gradient field in that part of the plane and 00:09:09.000 --> 00:09:13.000 conservative and so on. The answer for now is, 00:09:13.000 --> 00:09:17.000 in general, no. And when we spend a bit more 00:09:17.000 --> 00:09:20.000 time on it, actually, maybe I should move that up. 00:09:20.000 --> 00:09:24.000 Maybe we will talk about it later this week instead of when 00:09:24.000 --> 00:09:28.000 I had planned. There is a notion what it means 00:09:28.000 --> 00:09:30.000 for a region to be without holes. 00:09:30.000 --> 00:09:34.000 Basically, if you have that kind of property in a region 00:09:34.000 --> 00:09:38.000 that doesn't have any holes inside it then things will work. 00:09:38.000 --> 00:09:42.000 The problem comes from a vector field satisfying this criterion 00:09:42.000 --> 00:09:44.000 in a region but it has a hole in it. 00:09:44.000 --> 00:09:47.000 Because what you don't know is whether your potential is 00:09:47.000 --> 00:09:51.000 actually well-defined and takes the same value when you move all 00:09:51.000 --> 00:09:53.000 around the hole. It might come back to take a 00:09:53.000 --> 00:09:56.000 different value. If you look carefully and think 00:09:56.000 --> 00:10:00.000 hard about the example in the problem sets that is exactly 00:10:00.000 --> 00:10:04.000 what happens there. Again, I will say more about 00:10:04.000 --> 00:10:08.000 that later. For now we basically need our 00:10:08.000 --> 00:10:11.000 function to be, I mean, 00:10:11.000 --> 00:10:14.000 I should still say if you have this property for a vector field 00:10:14.000 --> 00:10:16.000 that is not quite defined everywhere, 00:10:16.000 --> 00:10:17.000 you are more than welcome, you know, 00:10:17.000 --> 00:10:20.000 you should probably still try to look for a potential using 00:10:20.000 --> 00:10:23.000 methods that we will see. But something might go wrong 00:10:23.000 --> 00:10:30.000 later. You might end up with a 00:10:30.000 --> 00:10:39.000 potential that is not well-defined. 00:10:39.000 --> 00:10:53.000 Let's do another example. Let's say that I give you this 00:10:53.000 --> 00:11:03.000 vector field. And this a here is a number. 00:11:03.000 --> 00:11:08.000 The question is for which value of a is this going to be 00:11:08.000 --> 00:11:13.000 possibly a gradient? If you have your flashcards 00:11:13.000 --> 00:11:17.000 then that is a good time to use them to vote, 00:11:17.000 --> 00:11:23.000 assuming that the number is small enough to be made with. 00:11:23.000 --> 00:11:27.000 Let's try to think about it. We want to call this guy M. 00:11:27.000 --> 00:11:35.000 We want to call that guy N. And we want to test M sub y 00:11:35.000 --> 00:11:42.000 versus N sub x. I don't see anyone. 00:11:42.000 --> 00:11:46.000 I see people doing it with their hands, and that works very 00:11:46.000 --> 00:11:48.000 well. OK. 00:11:48.000 --> 00:12:04.000 The question is for which value of a is this a gradient? 00:12:04.000 --> 00:12:10.000 I see various people with the correct answer. 00:12:10.000 --> 00:12:15.000 OK. That a strange answer. 00:12:15.000 --> 00:12:20.000 That is a good answer. OK. 00:12:20.000 --> 00:12:28.000 The vote seems to be for a equals eight. 00:12:28.000 --> 00:12:35.000 Let's see. What if I take M sub y? 00:12:35.000 --> 00:12:41.000 That is going to be just ax. And N sub x? 00:12:41.000 --> 00:12:47.000 That is 8x. I would like a equals eight. 00:12:47.000 --> 00:12:50.000 By the way, when you set these two equal to each other, 00:12:50.000 --> 00:12:52.000 they really have to be equal everywhere. 00:12:52.000 --> 00:12:55.000 You don't want to somehow solve for x or anything like that. 00:12:55.000 --> 00:12:59.000 You just want these expressions, in terms of x and 00:12:59.000 --> 00:13:02.000 y, to be the same quantities. I mean you cannot say if x 00:13:02.000 --> 00:13:07.000 equals z they are always equal. Yeah, that is true. 00:13:07.000 --> 00:13:13.000 But that is not what we are asking. 00:13:13.000 --> 00:13:18.000 Now we come to the next logical question. 00:13:18.000 --> 00:13:20.000 Let's say that we have passed the test. 00:13:20.000 --> 00:13:23.000 We have put a equals eight in here. 00:13:23.000 --> 00:13:26.000 Now it should be a gradient field. 00:13:26.000 --> 00:13:30.000 The question is how do we find the potential? 00:13:30.000 --> 00:13:36.000 That becomes eight from now on. The question is how do we find 00:13:36.000 --> 00:13:39.000 the function which has this as gradient? 00:13:39.000 --> 00:13:43.000 One option is to try to guess. Actually, quite often you will 00:13:43.000 --> 00:13:47.000 succeed that way. But that is not a valid method 00:13:47.000 --> 00:13:50.000 on next week's test. We are going to see two 00:13:50.000 --> 00:13:55.000 different systematic methods. And you should be using one of 00:13:55.000 --> 00:14:00.000 these because guessing doesn't always work. 00:14:00.000 --> 00:14:03.000 And, actually, I can come up with examples 00:14:03.000 --> 00:14:07.000 where if you try to guess you will surely fail. 00:14:07.000 --> 00:14:15.000 I can come up with trick ones, but I don't want to put that on 00:14:15.000 --> 00:14:24.000 the test. The next stage is finding the 00:14:24.000 --> 00:14:30.000 potential. And let me just emphasize that 00:14:30.000 --> 00:14:36.000 we can only do that if step one was successful. 00:14:36.000 --> 00:14:41.000 If we have a vector field that cannot possibly be a gradient 00:14:41.000 --> 00:14:45.000 then we shouldn't try to look for a potential. 00:14:45.000 --> 00:14:52.000 It is kind of obvious but is probably worth pointing out. 00:14:52.000 --> 00:15:00.000 There are two methods. The first method that we will 00:15:00.000 --> 00:15:16.000 see is computing line integrals. Let's see how that works. 00:15:16.000 --> 00:15:25.000 Let's say that I take some path that starts at the origin. 00:15:25.000 --> 00:15:26.000 Or, actually, anywhere you want, 00:15:26.000 --> 00:15:29.000 but let's take the origin. That is my favorite point. 00:15:29.000 --> 00:15:36.000 And let's go to a point with coordinates (x1, 00:15:36.000 --> 00:15:40.000 y1). And let's take my favorite 00:15:40.000 --> 00:15:45.000 curve and compute the line integral of that field, 00:15:45.000 --> 00:15:49.000 you know, the work done along the curve. 00:15:49.000 --> 00:15:55.000 Well, by the fundamental theorem, that should be equal to 00:15:55.000 --> 00:16:02.000 the value of the potential at the end point minus the value at 00:16:02.000 --> 00:16:09.000 the origin. That means I can actually write 00:16:09.000 --> 00:16:19.000 f of (x1, y1) equals -- -- that line integral plus the value at 00:16:19.000 --> 00:16:26.000 the origin. And that is just a constant. 00:16:26.000 --> 00:16:27.000 We don't know what it is. And, actually, 00:16:27.000 --> 00:16:30.000 we can choose what it is. Because if you have a 00:16:30.000 --> 00:16:33.000 potential, say that you have some potential function. 00:16:33.000 --> 00:16:34.000 And let's say that you add one to it. 00:16:34.000 --> 00:16:36.000 It is still a potential function. 00:16:36.000 --> 00:16:38.000 Adding one doesn't change the gradient. 00:16:38.000 --> 00:16:41.000 You can even add 18 or any number that you want. 00:16:41.000 --> 00:16:44.000 This is just going to be an integration constant. 00:16:44.000 --> 00:16:47.000 It is the same thing as, in one variable calculus, 00:16:47.000 --> 00:16:49.000 when you take the anti-derivative of a function it 00:16:49.000 --> 00:16:52.000 is only defined up to adding the constant. 00:16:52.000 --> 00:16:56.000 We have this integration constant, but apart from that we 00:16:56.000 --> 00:16:59.000 know that we should be able to get a potential from this. 00:16:59.000 --> 00:17:03.000 And this we can compute using the definition of the line 00:17:03.000 --> 00:17:06.000 integral. And we don't know what little f 00:17:06.000 --> 00:17:11.000 is, but we know what the vector field is so we can compute that. 00:17:11.000 --> 00:17:14.000 Of course, to do the calculation we probably don't 00:17:14.000 --> 00:17:18.000 want to use this kind of path. I mean if that is your favorite 00:17:18.000 --> 00:17:21.000 path then that is fine, but it is not very easy to 00:17:21.000 --> 00:17:24.000 compute the line integral along this, 00:17:24.000 --> 00:17:28.000 especially since I didn't tell you what the definition is. 00:17:28.000 --> 00:17:31.000 There are easier favorite paths to have. 00:17:31.000 --> 00:17:33.000 For example, you can go on a straight line 00:17:33.000 --> 00:17:37.000 from the origin to that point. That would be slightly easier. 00:17:37.000 --> 00:17:40.000 But then there is one easier. The easiest of all, 00:17:40.000 --> 00:17:47.000 probably, is to just go first along the x-axis to (x1,0) and 00:17:47.000 --> 00:17:51.000 then go up parallel to the y-axis. 00:17:51.000 --> 00:17:54.000 Why is that easy? Well, that is because when we 00:17:54.000 --> 00:17:57.000 do the line integral it becomes M dx N dy. 00:17:57.000 --> 00:18:05.000 And then, on each of these pieces, one-half just goes away 00:18:05.000 --> 00:18:11.000 because x, y is constant. Let's try to use that method in 00:18:11.000 --> 00:18:12.000 our example. 00:18:45.000 --> 00:18:56.000 Let's say that I want to go along this path from the origin, 00:18:56.000 --> 00:19:06.000 first along the x-axis to (x1,0) and then vertically to 00:19:06.000 --> 00:19:14.000 (x1, y1). And so I want to compute for 00:19:14.000 --> 00:19:21.000 the line integral along that curve. 00:19:21.000 --> 00:19:24.000 Let's say I want to do it for this vector field. 00:19:24.000 --> 00:19:33.000 I want to find the potential for this vector field. 00:19:33.000 --> 00:19:37.000 Let me copy it because I will have to erase at some point. 00:19:37.000 --> 00:19:50.000 4x squared plus 8xy and 3y squared plus 4x squared. 00:19:50.000 --> 00:19:59.000 That will become the integral of 4x squared plus 8 xy times dx 00:19:59.000 --> 00:20:05.000 plus 3y squared plus 4x squared times dy. 00:20:05.000 --> 00:20:08.000 To evaluate on this broken line, I will, 00:20:08.000 --> 00:20:13.000 of course, evaluate separately on each of the two segments. 00:20:13.000 --> 00:20:20.000 I will start with this segment that I will call c1 and then I 00:20:20.000 --> 00:20:25.000 will do this one that I will call c2. 00:20:25.000 --> 00:20:30.000 On c1, how do I evaluate my integral? 00:20:30.000 --> 00:20:38.000 Well, if I am on c1 then x varies from zero to x1. 00:20:38.000 --> 00:20:40.000 Well, actually, I don't know if x1 is positive 00:20:40.000 --> 00:20:41.000 or not so I shouldn't write this. 00:20:41.000 --> 00:20:48.000 I really should say just x goes from zero to x1. 00:20:48.000 --> 00:20:54.000 And what about y? y is just 0. 00:20:54.000 --> 00:21:00.000 I will set y equal to zero and also dy equal to zero. 00:21:00.000 --> 00:21:08.000 I get that the line integral on c1 -- Well, a lot of stuff goes 00:21:08.000 --> 00:21:11.000 away. The entire second term with dy 00:21:11.000 --> 00:21:15.000 goes away because dy is zero. And, in the first term, 00:21:15.000 --> 00:21:18.000 8xy goes away because y is zero as well. 00:21:18.000 --> 00:21:27.000 I just have an integral of 4x squared dx from zero to x1. 00:21:27.000 --> 00:21:31.000 By the way, now you see why I have been using an x1 and a y1 00:21:31.000 --> 00:21:33.000 for my point and not just x and y. 00:21:33.000 --> 00:21:36.000 It is to avoid confusion. I am using x and y as my 00:21:36.000 --> 00:21:41.000 integration variables and x1, y1 as constants that are 00:21:41.000 --> 00:21:45.000 representing the end point of my path. 00:21:45.000 --> 00:21:51.000 And so, if I integrate this, I should get four-thirds x1 00:21:51.000 --> 00:21:54.000 cubed. That is the first part. 00:21:54.000 --> 00:22:01.000 Next I need to do the second segment. 00:22:01.000 --> 00:22:09.000 If I am on c2, y goes from zero to y1. 00:22:09.000 --> 00:22:16.000 And what about x? x is constant equal to x1 so dx 00:22:16.000 --> 00:22:22.000 becomes just zero. It is a constant. 00:22:22.000 --> 00:22:30.000 If I take the line integral of c2, F dot dr then I will get the 00:22:30.000 --> 00:22:37.000 integral from zero to y1. The entire first term with dx 00:22:37.000 --> 00:22:47.000 goes away and then I have 3y squared plus 4x1 squared times 00:22:47.000 --> 00:22:52.000 dy. That integrates to y cubed plus 00:22:52.000 --> 00:23:01.000 4x1 squared y from zero to y1. Or, if you prefer, 00:23:01.000 --> 00:23:11.000 that is y1 cubed plus 4x1 squared y1. 00:23:11.000 --> 00:23:15.000 Now that we have done both of them we can just add them 00:23:15.000 --> 00:23:19.000 together, and that will give us the formula for the potential. 00:23:40.000 --> 00:23:50.000 F of x1 and y1 is four-thirds x1 cubed plus y1 cubed plus 4x1 00:23:50.000 --> 00:23:57.000 squared y1 plus a constant. That constant is just the 00:23:57.000 --> 00:24:03.000 integration constant that we had from the beginning. 00:24:03.000 --> 00:24:05.000 Now you can drop the subscripts if you prefer. 00:24:05.000 --> 00:24:14.000 You can just say f is four-thirds x cubed plus y cubed 00:24:14.000 --> 00:24:20.000 plus 4x squared y plus constant. And you can check. 00:24:20.000 --> 00:24:25.000 If you take the gradient of this, you should get again this 00:24:25.000 --> 00:24:29.000 vector field over there. Any questions about this method? 00:24:29.000 --> 00:24:33.000 Yes? No. 00:24:33.000 --> 00:24:35.000 Well, it depends whether you are just trying to find one 00:24:35.000 --> 00:24:38.000 potential or if you are trying to find all the possible 00:24:38.000 --> 00:24:40.000 potentials. If a problem just says find a 00:24:40.000 --> 00:24:43.000 potential then you don't have to use the constant. 00:24:43.000 --> 00:24:47.000 This guy without the constant is a valid potential. 00:24:47.000 --> 00:24:52.000 You just have others. If your neighbor comes to you 00:24:52.000 --> 00:24:58.000 and say your answer must be wrong because I got this plus 00:24:58.000 --> 00:25:01.000 18, well, both answers are correct. 00:25:01.000 --> 00:25:05.000 By the way. Instead of going first along 00:25:05.000 --> 00:25:08.000 the x-axis vertically, you could do it the other way 00:25:08.000 --> 00:25:11.000 around. Of course, start along the 00:25:11.000 --> 00:25:15.000 y-axis and then horizontally. That is the same level of 00:25:15.000 --> 00:25:19.000 difficulty. You just exchange roles of x 00:25:19.000 --> 00:25:21.000 and y. In some cases, 00:25:21.000 --> 00:25:26.000 it is actually even making more sense maybe to go radially, 00:25:26.000 --> 00:25:30.000 start out from the origin to your end point. 00:25:30.000 --> 00:25:37.000 But usually this setting is easier just because each of 00:25:37.000 --> 00:25:43.000 these two guys were very easy to compute. 00:25:43.000 --> 00:25:46.000 But somehow maybe if you suspect that polar coordinates 00:25:46.000 --> 00:25:49.000 will be involved somehow in the answer then maybe it makes sense 00:25:49.000 --> 00:26:01.000 to choose different paths. Maybe a straight line is better. 00:26:01.000 --> 00:26:13.000 Now we have another method to look at which is using 00:26:13.000 --> 00:26:19.000 anti-derivatives. The goal is the same, 00:26:19.000 --> 00:26:21.000 still to find the potential function. 00:26:21.000 --> 00:26:26.000 And you see that finding the potential is really the 00:26:26.000 --> 00:26:31.000 multivariable analog of finding the anti-derivative in the one 00:26:31.000 --> 00:26:34.000 variable. Here we did it basically by 00:26:34.000 --> 00:26:38.000 hand by computing the integral. The other thing you could try 00:26:38.000 --> 00:26:39.000 to say is, wait, I already know how to take 00:26:39.000 --> 00:26:42.000 anti-derivatives. Let's use that instead of 00:26:42.000 --> 00:26:45.000 computing integrals. And it works but you have to be 00:26:45.000 --> 00:26:51.000 careful about how you do it. Let's see how that works. 00:26:51.000 --> 00:26:53.000 Let's still do it with the same example. 00:26:53.000 --> 00:27:02.000 We want to solve the equations. We want a function such that f 00:27:02.000 --> 00:27:13.000 sub x is 4x squared plus 8xy and f sub y is 3y squared plus 4x 00:27:13.000 --> 00:27:16.000 squared. Let's just look at one of these 00:27:16.000 --> 00:27:20.000 at a time. If we look at this one, 00:27:20.000 --> 00:27:28.000 well, we know how to solve this because it is just telling us we 00:27:28.000 --> 00:27:33.000 have to integrate this with respect to x. 00:27:33.000 --> 00:27:38.000 Well, let's call them one and two because I will have to refer 00:27:38.000 --> 00:27:43.000 to them again. Let's start with equation one 00:27:43.000 --> 00:27:48.000 and lets integrate with respect to x. 00:27:48.000 --> 00:27:51.000 Well, it tells us that f should be, 00:27:51.000 --> 00:27:55.000 what do I get when I integrate this with respect to x, 00:27:55.000 --> 00:28:02.000 four-thirds x cubed plus, when I integrate 8xy, 00:28:02.000 --> 00:28:08.000 y is just a constant, so I will get 4x squared y. 00:28:08.000 --> 00:28:11.000 And that is not quite the end to it because there is an 00:28:11.000 --> 00:28:15.000 integration constant. And here, when I say there is 00:28:15.000 --> 00:28:18.000 an integration constant, it just means the extra term 00:28:18.000 --> 00:28:21.000 does not depend on x. That is what it means to be a 00:28:21.000 --> 00:28:25.000 constant in this setting. But maybe my constant still 00:28:25.000 --> 00:28:28.000 depends on y so it is not actually a true constant. 00:28:28.000 --> 00:28:30.000 A constant that depends on y is not really a constant. 00:28:30.000 --> 00:28:38.000 It is actually a function of y. The good news that we have is 00:28:38.000 --> 00:28:40.000 that this function normally depends on x. 00:28:40.000 --> 00:28:46.000 We have made some progress. We have part of the answer and 00:28:46.000 --> 00:28:53.000 we have simplified the problem. If we have anything that looks 00:28:53.000 --> 00:28:56.000 like this, it will satisfy the first condition. 00:28:56.000 --> 00:28:59.000 Now we need to look at the second condition. 00:28:59.000 --> 00:29:12.000 We want f sub y to be that. But we know what f is, 00:29:12.000 --> 00:29:15.000 so let's compute f sub y from this. 00:29:15.000 --> 00:29:20.000 From this I get f sub y. What do I get if I 00:29:20.000 --> 00:29:22.000 differentiate this with respect to y? 00:29:22.000 --> 00:29:37.000 Well, I get zero plus 4x squared plus the derivative of 00:29:37.000 --> 00:29:46.000 g. I would like to match this with 00:29:46.000 --> 00:29:51.000 what I had. If I match this with equation 00:29:51.000 --> 00:29:55.000 two then that will tell me what the derivative of g should be. 00:30:15.000 --> 00:30:20.000 If we compare the two things there, we get 4x squared plus g 00:30:20.000 --> 00:30:26.000 prime of y should be equal to 3y squared by 4x squared. 00:30:26.000 --> 00:30:31.000 And, of course, the 4x squares go away. 00:30:31.000 --> 00:30:35.000 That tells you g prime is 3y squared. 00:30:35.000 --> 00:30:42.000 And that integrates to y cubed plus constant. 00:30:42.000 --> 00:30:46.000 Now, this time the constant is a true constant because g did 00:30:46.000 --> 00:30:48.000 not depend on anything other than y. 00:30:48.000 --> 00:30:54.000 And the constant does not depend on y so it is a real 00:30:54.000 --> 00:30:58.000 constant now. Now we just plug this back into 00:30:58.000 --> 00:31:05.000 this guy. Let's call him star. 00:31:05.000 --> 00:31:13.000 If we plug this into star, we get f equals four-thirds x 00:31:13.000 --> 00:31:21.000 cubed plus 4x squared y plus y cubed plus constant. 00:31:21.000 --> 00:31:30.000 I mean, of course, again, now this constant is 00:31:30.000 --> 00:31:33.000 optional. The advantage of this method is 00:31:33.000 --> 00:31:35.000 you don't have to write any integrals. 00:31:35.000 --> 00:31:40.000 The small drawback is you have to follow this procedure 00:31:40.000 --> 00:31:45.000 carefully. By the way, one common pitfall 00:31:45.000 --> 00:31:48.000 that is tempting. After you have done this, 00:31:48.000 --> 00:31:51.000 what is very tempting is to just say, well, 00:31:51.000 --> 00:31:53.000 let's do the same with this guy. 00:31:53.000 --> 00:31:55.000 Let's integrate this with respect to y. 00:31:55.000 --> 00:31:58.000 You will get another expression for f up to a constant that 00:31:58.000 --> 00:32:01.000 depends on x. And then let's match them. 00:32:01.000 --> 00:32:04.000 Well, the difficulty is matching is actually quite 00:32:04.000 --> 00:32:09.000 tricky because you don't know in advance whether they will be the 00:32:09.000 --> 00:32:13.000 same expression. It could be you could say let's 00:32:13.000 --> 00:32:16.000 just take the terms that are here and missing there and 00:32:16.000 --> 00:32:20.000 combine the terms, you know, take all the terms 00:32:20.000 --> 00:32:23.000 that appear in either one. That is actually not a good way 00:32:23.000 --> 00:32:25.000 to do it, because if I put sufficiently 00:32:25.000 --> 00:32:28.000 complicated trig functions in there then you might not be able 00:32:28.000 --> 00:32:30.000 to see that two terms are the same. 00:32:30.000 --> 00:32:34.000 Take an easy one. Let's say that here I have one 00:32:34.000 --> 00:32:40.000 plus tangent square and here I have a secan square then you 00:32:40.000 --> 00:32:46.000 might not actually notice that there is a difference. 00:32:46.000 --> 00:32:50.000 But there is no difference. Whatever. 00:32:50.000 --> 00:32:54.000 Anyway, I am saying do it this way, don't do it any other way 00:32:54.000 --> 00:32:57.000 because there is a risk of making a mistake otherwise. 00:32:57.000 --> 00:33:00.000 I mean, on the other hand, you could start with 00:33:00.000 --> 00:33:03.000 integrating with respect to y and then differentiate and match 00:33:03.000 --> 00:33:06.000 with respect to x. But what I am saying is just 00:33:06.000 --> 00:33:09.000 take one of them, integrate, 00:33:09.000 --> 00:33:12.000 get an answer that involves a function of the other variable, 00:33:12.000 --> 00:33:18.000 then differentiate that answer and compare and see what you 00:33:18.000 --> 00:33:21.000 get. By the way, here, 00:33:21.000 --> 00:33:27.000 of course, after we simplified there were only y's here. 00:33:27.000 --> 00:33:29.000 There were no x's. And that is kind of good news. 00:33:29.000 --> 00:33:33.000 I mean, if you had had an x here in this expression that 00:33:33.000 --> 00:33:36.000 would have told you that something is going wrong. 00:33:36.000 --> 00:33:39.000 g is a function of y only. If you get an x here, 00:33:39.000 --> 00:33:42.000 maybe you want to go back and check whether it is really a 00:33:42.000 --> 00:33:47.000 gradient field. Yes? 00:33:47.000 --> 00:33:49.000 Yes, this will work with functions of more than two 00:33:49.000 --> 00:33:51.000 variables. Both methods work with more 00:33:51.000 --> 00:33:53.000 than two variables. We are going to see it in the 00:33:53.000 --> 00:33:56.000 case where more than two means three. 00:33:56.000 --> 00:34:00.000 We are going to see that in two or three weeks from now. 00:34:00.000 --> 00:34:04.000 I mean, basically starting at the end of next week, 00:34:04.000 --> 00:34:08.000 we are going to do triple integrals, line integrals in 00:34:08.000 --> 00:34:10.000 space and so on. The format is first we do 00:34:10.000 --> 00:34:13.000 everything in two variables. Then we will do three variables. 00:34:13.000 --> 00:34:20.000 And then what happens with more than three will be left to your 00:34:20.000 --> 00:34:25.000 imagination. Any other questions about 00:34:25.000 --> 00:34:29.000 either of these methods? A quick poll. 00:34:29.000 --> 00:34:34.000 Who prefers the first method? Who prefers the second method? 00:34:34.000 --> 00:34:41.000 Wow. OK. 00:34:41.000 --> 00:34:45.000 Anyway, you will get to use whichever one you want. 00:34:45.000 --> 00:34:47.000 And I would agree with you, but the second method is 00:34:47.000 --> 00:34:50.000 slightly more effective in that you are writing less stuff. 00:34:50.000 --> 00:34:54.000 You don't have to set up all these line integrals. 00:34:54.000 --> 00:35:03.000 On the other hand, it does require a little bit 00:35:03.000 --> 00:35:19.000 more attention. Let's move on a bit. 00:35:19.000 --> 00:35:24.000 Let me start by actually doing a small recap. 00:35:24.000 --> 00:35:38.000 We said we have various notions. One is to say that the vector 00:35:38.000 --> 00:35:48.000 field is a gradient in a certain region of a plane. 00:35:48.000 --> 00:35:54.000 And we have another notion which is being conservative. 00:35:54.000 --> 00:36:06.000 It says that the line integral is zero along any closed curve. 00:36:06.000 --> 00:36:10.000 Actually, let me introduce a new piece of notation. 00:36:10.000 --> 00:36:14.000 To remind ourselves that we are doing it along a closed curve, 00:36:14.000 --> 00:36:18.000 very often we put just a circle for the integral to tell us this 00:36:18.000 --> 00:36:21.000 is a curve that closes on itself. 00:36:21.000 --> 00:36:25.000 It ends where it started. I mean it doesn't change 00:36:25.000 --> 00:36:28.000 anything concerning the definition or how you compute it 00:36:28.000 --> 00:36:31.000 or anything. It just reminds you that you 00:36:31.000 --> 00:36:34.000 are doing it on a closed curve. It is actually useful for 00:36:34.000 --> 00:36:37.000 various physical applications. And also, when you state 00:36:37.000 --> 00:36:41.000 theorems in that way, it reminds you,oh.. 00:36:41.000 --> 00:36:45.000 I need to be on a closed curve to do it. 00:36:45.000 --> 00:36:51.000 And so we have said these two things are equivalent. 00:36:51.000 --> 00:37:00.000 Now we have a third thing which is N sub x equals M sub y at 00:37:00.000 --> 00:37:03.000 every point. Just to summarize the 00:37:03.000 --> 00:37:06.000 discussion. We have said if we have a 00:37:06.000 --> 00:37:09.000 gradient field then we have this. 00:37:09.000 --> 00:37:18.000 And the converse is true in suitable regions. 00:37:18.000 --> 00:37:32.000 We have a converse if F is defined in the entire plane. 00:37:32.000 --> 00:37:43.000 Or, as we will see soon, in a simply connected region. 00:37:43.000 --> 00:37:45.000 I guess some of you cannot see what I am writing here, 00:37:45.000 --> 00:37:48.000 but it doesn't matter because you are not officially supposed 00:37:48.000 --> 00:37:53.000 to know it yet. That will be next week. 00:37:53.000 --> 00:37:57.000 Anyway, I said the fact that Nx equals 00:37:57.000 --> 00:38:01.000 My implies that we have a gradient field and is only if a 00:38:01.000 --> 00:38:06.000 vector field is defined in the entire plane or in a region that 00:38:06.000 --> 00:38:12.000 is called simply connected. And more about that later. 00:38:12.000 --> 00:38:17.000 Now let me just introduce a quantity that probably a lot of 00:38:17.000 --> 00:38:22.000 you have heard about in physics that measures precisely fairly 00:38:22.000 --> 00:38:26.000 ought to be conservative. That is called the curl of a 00:38:26.000 --> 00:38:27.000 vector field. 00:39:06.000 --> 00:39:19.000 For the definition we say that the curl of F is the quantity N 00:39:19.000 --> 00:39:27.000 sub x - M sub y. It is just replicating the 00:39:27.000 --> 00:39:35.000 information we had but in a way that is a single quantity. 00:39:35.000 --> 00:39:43.000 In this new language, the conditions that we had over 00:39:43.000 --> 00:39:50.000 there, this condition says curl F equals zero. 00:39:50.000 --> 00:39:56.000 That is the new version of Nx equals My. 00:39:56.000 --> 00:40:06.000 It measures failure of a vector field to be conservative. 00:40:06.000 --> 00:40:21.000 The test for conservativeness is that the curl of F should be 00:40:21.000 --> 00:40:25.000 zero. I should probably tell you a 00:40:25.000 --> 00:40:29.000 little bit about what the curl is, what it measures and what it 00:40:29.000 --> 00:40:34.000 does because that is something that is probably useful. 00:40:34.000 --> 00:40:37.000 It is a very strange quantity if you put it in that form. 00:40:37.000 --> 00:40:42.000 Yes? I think it is the same as the 00:40:42.000 --> 00:40:45.000 physics one, but I haven't checked the physics textbook. 00:40:45.000 --> 00:40:49.000 I believe it is the same. Yes, I think it is the same as 00:40:49.000 --> 00:40:53.000 the physics one. It is not the opposite this 00:40:53.000 --> 00:40:55.000 time. Of course, in physics maybe you 00:40:55.000 --> 00:40:59.000 have seen curl in space. We are going to see curl in 00:40:59.000 --> 00:41:07.000 space in two or three weeks. Yes? 00:41:07.000 --> 00:41:11.000 Yes. Well, you can also use it. If you fail this test then you 00:41:11.000 --> 00:41:14.000 know for sure that you are not gradient field so you might as 00:41:14.000 --> 00:41:18.000 well do that. If you satisfy the test but you 00:41:18.000 --> 00:41:24.000 are not defined everywhere then there is still a bit of 00:41:24.000 --> 00:41:29.000 ambiguity and you don't know for sure. 00:41:29.000 --> 00:41:40.000 OK. Let's try to see a little bit 00:41:40.000 --> 00:41:48.000 what the curl measures. Just to give you some 00:41:48.000 --> 00:41:55.000 intuition, let's first think about a velocity field. 00:41:55.000 --> 00:42:10.000 The curl measures the rotation component of a motion. 00:42:10.000 --> 00:42:13.000 If you want a fancy word, it measures the vorticity of a 00:42:13.000 --> 00:42:16.000 motion. It tells you how much twisting 00:42:16.000 --> 00:42:19.000 is taking place at a given point. 00:42:19.000 --> 00:42:24.000 For example, if I take a constant vector 00:42:24.000 --> 00:42:32.000 field where my fluid is just all moving in the same direction 00:42:32.000 --> 00:42:37.000 where this is just constants then, 00:42:37.000 --> 00:42:41.000 of course, the curl is zero. Because if you take the 00:42:41.000 --> 00:42:43.000 partials you get zero. And, indeed, 00:42:43.000 --> 00:42:46.000 that is not what you would call swirling. 00:42:46.000 --> 00:42:58.000 There is no vortex in here. Let's do another one where this 00:42:58.000 --> 00:43:02.000 is still nothing going on. Let's say that I take the 00:43:02.000 --> 00:43:06.000 radial vector field where everything just flows away from 00:43:06.000 --> 00:43:11.000 the origin. That is f equals x, y. 00:43:11.000 --> 00:43:16.000 Well, if I take the curl, I have to take partial over 00:43:16.000 --> 00:43:18.000 partial x of the second component, 00:43:18.000 --> 00:43:21.000 which is y, minus partial over partial y of 00:43:21.000 --> 00:43:22.000 the first component, which is x. 00:43:22.000 --> 00:43:25.000 I will get zero. And, indeed, 00:43:25.000 --> 00:43:29.000 if you think about what is going on here, 00:43:29.000 --> 00:43:32.000 there is no rotation involved. On the other hand, 00:43:32.000 --> 00:43:45.000 if you consider our favorite rotation vector field -- -- 00:43:45.000 --> 00:44:00.000 negative y and x then this curl is going to be N sub x minus M 00:44:00.000 --> 00:44:08.000 sub y, one plus one equals two. 00:44:08.000 --> 00:44:13.000 That corresponds to the fact that we are rotating. 00:44:13.000 --> 00:44:16.000 Actually, we are rotating at unit angular speed. 00:44:16.000 --> 00:44:20.000 The curl actually measures twice the angular speed of a 00:44:20.000 --> 00:44:24.000 rotation part of a motion at any given point. 00:44:24.000 --> 00:44:26.000 Now, if you have an actual motion, 00:44:26.000 --> 00:44:30.000 a more complicated field than these then no matter where you 00:44:30.000 --> 00:44:34.000 are you can think of a motion as a combination of translation 00:44:34.000 --> 00:44:37.000 effects, maybe dilation effects, 00:44:37.000 --> 00:44:43.000 maybe rotation effects, possibly other things like that. 00:44:43.000 --> 00:44:48.000 And what a curl will measure is how intense the rotation effect 00:44:48.000 --> 00:44:52.000 is at that particular point. I am not going to try to make a 00:44:52.000 --> 00:44:55.000 much more precise statement. A precise statement is what a 00:44:55.000 --> 00:44:58.000 curl measures is really this quantity up there. 00:44:58.000 --> 00:45:01.000 But the intuition you should have is it measures how much 00:45:01.000 --> 00:45:04.000 rotation is taking place at any given point. 00:45:04.000 --> 00:45:06.000 And, of course, in a complicated motion you 00:45:06.000 --> 00:45:09.000 might have more rotation at some point than at some others, 00:45:09.000 --> 00:45:12.000 which is why the curl will depend on x and y. 00:45:12.000 --> 00:45:20.000 It is not just a constant because how much you rotate 00:45:20.000 --> 00:45:26.000 depends on where you are. If you are looking at actual 00:45:26.000 --> 00:45:30.000 wind velocities in weather prediction then the regions with 00:45:30.000 --> 00:45:33.000 high curl tend to be hurricanes or tornadoes or things like 00:45:33.000 --> 00:45:37.000 that. They are not very pleasant 00:45:37.000 --> 00:45:40.000 things. And the sign of a curl tells 00:45:40.000 --> 00:45:43.000 you whether you are going clockwise or counterclockwise. 00:46:09.000 --> 00:46:27.000 Curl measures twice the angular velocity of the rotation 00:46:27.000 --> 00:46:41.000 component of a velocity field. Now, what about a force field? 00:46:41.000 --> 00:46:44.000 Because, after all, how we got to this was coming 00:46:44.000 --> 00:46:47.000 from and trying to understand forces and the work they do. 00:46:47.000 --> 00:46:50.000 So I should tell you what it means for a force. 00:46:50.000 --> 00:47:10.000 Well, the curl of a force field -- -- measures the torque 00:47:10.000 --> 00:47:29.000 exerted on a test object that you put at any point. 00:47:29.000 --> 00:47:36.000 Remember, torque is the rotational analog of the force. 00:47:36.000 --> 00:47:41.000 We had this analogy about velocity versus angular velocity 00:47:41.000 --> 00:47:45.000 and mass versus moment of inertia. 00:47:45.000 --> 00:47:49.000 And then, in that analogy, force divided by the mass is 00:47:49.000 --> 00:47:53.000 what will cause acceleration, which is the derivative of 00:47:53.000 --> 00:47:56.000 velocity. Torque divided by moment of 00:47:56.000 --> 00:47:59.000 inertia is what will cause the angular acceleration, 00:47:59.000 --> 00:48:02.000 namely the derivative of angular velocity. 00:48:02.000 --> 00:48:04.000 Maybe I should write that down. 00:48:18.000 --> 00:48:31.000 Torque divided by moment of inertia is going to be d over dt 00:48:31.000 --> 00:48:38.000 of angular velocity. I leave it up to your physics 00:48:38.000 --> 00:48:41.000 teachers to decide what letters to use for all these things. 00:48:41.000 --> 00:48:49.000 That is the analog of force divided by mass equals 00:48:49.000 --> 00:48:56.000 acceleration, which is d over dt of velocity. 00:48:56.000 --> 00:49:03.000 And so now you see if the curl of a velocity field measure the 00:49:03.000 --> 00:49:07.000 angular velocity of its rotation then, 00:49:07.000 --> 00:49:13.000 by this analogy, the curl of a force field 00:49:13.000 --> 00:49:24.000 should measure the torque it exerts on a mass per unit moment 00:49:24.000 --> 00:49:28.000 of inertia. Concretely, if you imagine that 00:49:28.000 --> 00:49:29.000 you are putting something in there, 00:49:29.000 --> 00:49:32.000 you know, if you are in a velocity field the curl will 00:49:32.000 --> 00:49:35.000 tell you how fast your guy is spinning at a given time. 00:49:35.000 --> 00:49:37.000 If you put something that floats, for example, 00:49:37.000 --> 00:49:40.000 in your fluid, something very light then it is 00:49:40.000 --> 00:49:44.000 going to start spinning. And the curl of a velocity 00:49:44.000 --> 00:49:48.000 field tells you how fast it is spinning at any given time up to 00:49:48.000 --> 00:49:51.000 a factor of two. And the curl of a force field 00:49:51.000 --> 00:49:55.000 tells you how quickly the angular velocity is going to 00:49:55.000 --> 00:50:01.000 increase or decrease. OK. 00:50:01.000 --> 00:50:04.000 Well, next time we are going to see Green's theorem which is 00:50:04.000 --> 00:50:08.000 actually going to tell us a lot more about curl and failure of 00:50:08.000 --> 00:50:11.000 conservativeness.