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. 00:00:25.000 --> 00:00:32.000 Recall that yesterday we saw, no, two days ago we learned 00:00:32.000 --> 00:00:37.000 about the curl of a vector field in space. 00:00:37.000 --> 00:00:45.000 And we said the curl of F is defined by taking a cross 00:00:45.000 --> 00:00:52.000 product between the symbol dell and the vector F. 00:00:52.000 --> 00:00:58.000 Concretely, the way we would compute this would be by putting 00:00:58.000 --> 00:01:04.000 the components of F into this determinant and expanding and 00:01:04.000 --> 00:01:09.000 then getting a vector with components Ry minus Qz, 00:01:09.000 --> 00:01:21.000 Pz minus Rx and Qx minus Py. I think I also tried to explain 00:01:21.000 --> 00:01:25.000 very quickly what the significance of a curl is. 00:01:25.000 --> 00:01:28.000 Just to tell you again very quickly, 00:01:28.000 --> 00:01:34.000 basically curl measures, if you mention that your vector 00:01:34.000 --> 00:01:40.000 field measures the velocity in some fluid then the curl 00:01:40.000 --> 00:01:47.000 measures how much rotation is taking place in that fluid. 00:01:47.000 --> 00:02:05.000 Measures the rotation part of a velocity field. 00:02:05.000 --> 00:02:13.000 More precisely the direction corresponds to the axis of 00:02:13.000 --> 00:02:22.000 rotation and the magnitude corresponds to twice the angular 00:02:22.000 --> 00:02:24.000 velocity. 00:02:47.000 --> 00:02:50.000 Just to give you a few quick examples. 00:02:50.000 --> 00:02:53.000 If I take a constant vector field, 00:02:53.000 --> 00:03:01.000 so everything translates at the same speed, 00:03:01.000 --> 00:03:06.000 then obviously when you take the partial derivatives you will 00:03:06.000 --> 00:03:10.000 just get a bunch of zeros so the curl will be zero. 00:03:10.000 --> 00:03:15.000 If you take a vector field that stretches things, 00:03:15.000 --> 00:03:17.000 let's say, for example, we are going to stretch things 00:03:17.000 --> 00:03:23.000 along the x-axis, that would be a vector field 00:03:23.000 --> 00:03:30.000 that goes parallel to the x direction but maybe, 00:03:30.000 --> 00:03:33.000 say, x times i. So that when you are in front 00:03:33.000 --> 00:03:35.000 of a plane of a blackboard you are moving forward, 00:03:35.000 --> 00:03:36.000 when you are behind you are moving backwards, 00:03:36.000 --> 00:03:40.000 things are getting expanded in the x direction. 00:03:40.000 --> 00:03:48.000 If you compute the curl, you can check each of these. 00:03:48.000 --> 00:03:49.000 Again, they are going to be zero. 00:03:49.000 --> 00:03:53.000 There is no curl. This is not what curl measures. 00:03:53.000 --> 00:03:58.000 I mean, actually, what measures expansion, 00:03:58.000 --> 00:04:03.000 stretching is actually divergence. 00:04:03.000 --> 00:04:05.000 If you take the divergence of this field, 00:04:05.000 --> 00:04:07.000 you would get one plus zero plus zero, 00:04:07.000 --> 00:04:10.000 it looks like it will be one, so in case you don't remember, 00:04:10.000 --> 00:04:15.000 I mean divergence precisely measures this stretching effect 00:04:15.000 --> 00:04:18.000 in your field. And, on the other hand, 00:04:18.000 --> 00:04:22.000 if you take something that corresponds to, 00:04:22.000 --> 00:04:26.000 say, rotation about the z-axis at 00:04:26.000 --> 00:04:34.000 unit angular velocity -- That means they are going to moving 00:04:34.000 --> 00:04:41.000 in circles around the z-axis. One way to write down this 00:04:41.000 --> 00:04:46.000 field, let's see, the z component is zero because 00:04:46.000 --> 00:04:50.000 everything is moving horizontally. 00:04:50.000 --> 00:04:54.000 And in the x and y directions, if you look at it from above, 00:04:54.000 --> 00:04:59.000 well, it is just going to be our good old friend the vector 00:04:59.000 --> 00:05:02.000 field that rotates everything [at unit speed?]. 00:05:02.000 --> 00:05:05.000 And we have seen the formula for this one many times. 00:05:05.000 --> 00:05:09.000 The first component is minus y, the second one is x. 00:05:09.000 --> 00:05:17.000 Now, if you compute the curl of this guy, you will get zero, 00:05:17.000 --> 00:05:21.000 zero, two, two k. And so k is the axis of 00:05:21.000 --> 00:05:24.000 rotation, two is twice the angular velocity. 00:05:24.000 --> 00:05:26.000 And now, of course, you can imagine much more 00:05:26.000 --> 00:05:29.000 complicated motions where you will have -- For example, 00:05:29.000 --> 00:05:32.000 if you look at the Charles River very carefully then you 00:05:32.000 --> 00:05:34.000 will see that water is flowing, generally speaking, 00:05:34.000 --> 00:05:38.000 towards the ocean. But, at the same time, 00:05:38.000 --> 00:05:43.000 there are actually a few eddies in there and with water 00:05:43.000 --> 00:05:47.000 swirling. Those are the places where 00:05:47.000 --> 00:05:51.000 there is actually curl in the flow. 00:05:51.000 --> 00:05:56.000 Yes. I don't know how to turn out 00:05:56.000 --> 00:06:00.000 the lights a bit, but I'm sure there is a way. 00:06:00.000 --> 00:06:10.000 Does this do it? Is it working? 00:06:10.000 --> 00:06:23.000 OK. You're welcome. 00:06:23.000 --> 00:06:31.000 Hopefully it is easier to see now. 00:06:31.000 --> 00:06:35.000 That was about curl. Now, why do we care about curl 00:06:35.000 --> 00:06:40.000 besides this motivation of understanding motions? 00:06:40.000 --> 00:06:43.000 One place where it comes up is when we try to understand 00:06:43.000 --> 00:06:45.000 whether a vector field is conservative. 00:06:45.000 --> 00:06:49.000 Remember we have seen that a vector field is conservative if 00:06:49.000 --> 00:06:53.000 and only if its curl is zero. That is the situation in which 00:06:53.000 --> 00:06:56.000 we are allowed to try to look for a potential function and 00:06:56.000 --> 00:06:57.000 then use the fundamental theorem. 00:06:57.000 --> 00:07:00.000 But another place where this comes up, 00:07:00.000 --> 00:07:02.000 if you remember what we did in the plane, 00:07:02.000 --> 00:07:06.000 curl also came up when we tried to convert nine integrals into 00:07:06.000 --> 00:07:10.000 double integrals. That was Greene's theorem. 00:07:10.000 --> 00:07:19.000 Well, it turns out we can do the same thing in space and that 00:07:19.000 --> 00:07:28.000 is called Stokes' theorem. What does Stokes' theorem say? 00:07:28.000 --> 00:07:36.000 It says that the work done by a vector field along a closed 00:07:36.000 --> 00:07:44.000 curve can be replaced by a double integral of curl F. 00:07:44.000 --> 00:07:47.000 Let me write it using the dell notation. 00:07:47.000 --> 00:07:54.000 That is curl F. Dot ndS on a suitably chosen 00:07:54.000 --> 00:07:58.000 surface. That is a very strange kind of 00:07:58.000 --> 00:08:01.000 statement. But actually it is not much 00:08:01.000 --> 00:08:04.000 more strange than things we have seen before. 00:08:04.000 --> 00:08:09.000 I should clarify what this means. 00:08:09.000 --> 00:08:17.000 C has to be a closed curve in space. 00:08:17.000 --> 00:08:32.000 And S can be any surface bounded by C. 00:08:32.000 --> 00:08:36.000 For example, what Stokes' theorem tells me 00:08:36.000 --> 00:08:41.000 is that let us say that I have to compute some line integral on 00:08:41.000 --> 00:08:48.000 maybe, say, the unit circle in the x, 00:08:48.000 --> 00:08:52.000 y plane. Of course I can set a line 00:08:52.000 --> 00:08:57.000 integral directly and compute it by setting x equals cosine T, 00:08:57.000 --> 00:09:00.000 y equals sine T, z equals zero. 00:09:00.000 --> 00:09:03.000 But maybe sometimes I don't want to do that because my 00:09:03.000 --> 00:09:06.000 vector field is really complicated. 00:09:06.000 --> 00:09:11.000 And instead I will want to reduce things to a surface 00:09:11.000 --> 00:09:13.000 integral. Now, I know that you guys are 00:09:13.000 --> 00:09:16.000 not necessarily fond of computing flux of vector fields 00:09:16.000 --> 00:09:19.000 for surfaces so maybe you don't really see the point. 00:09:19.000 --> 00:09:22.000 But sometimes it is useful. Sometimes it is also useful 00:09:22.000 --> 00:09:25.000 backwards because, actually, you have a surface 00:09:25.000 --> 00:09:29.000 integral that you would like to turn into a line integral. 00:09:29.000 --> 00:09:34.000 What Stokes' theorem says is that I can choose my favorite 00:09:34.000 --> 00:09:38.000 surface whose boundary is this circle. 00:09:38.000 --> 00:09:42.000 I could choose, for example, 00:09:42.000 --> 00:09:50.000 a half sphere if I want or I can choose, let's call that s1, 00:09:50.000 --> 00:09:54.000 I don't know, a pointy thing, 00:09:54.000 --> 00:09:57.000 s2. Probably the most logical one, 00:09:57.000 --> 00:10:00.000 actually, would be just to choose a disk in the x, 00:10:00.000 --> 00:10:02.000 y plane. That would probably be the 00:10:02.000 --> 00:10:04.000 easiest one to set up for calculating flux. 00:10:04.000 --> 00:10:07.000 Anyway, what Stokes' theorem tells me 00:10:07.000 --> 00:10:09.000 is I can choose any of these surfaces, 00:10:09.000 --> 00:10:14.000 whichever one I want, and I can compute the flux of 00:10:14.000 --> 00:10:18.000 curl F through this surface. Curl F is a new vector field 00:10:18.000 --> 00:10:22.000 when you have this formula that gives you a vector field you 00:10:22.000 --> 00:10:25.000 compute its flux through your favorite surface, 00:10:25.000 --> 00:10:31.000 and you should get the same thing as if you had done the 00:10:31.000 --> 00:10:37.000 line integral for F. That is the statement. 00:10:37.000 --> 00:10:43.000 Now, there is a catch here. What is the catch? 00:10:43.000 --> 00:10:47.000 Well, the catch is we have to figure out what conventions to 00:10:47.000 --> 00:10:51.000 use because remember when we have a surface there are two 00:10:51.000 --> 00:10:54.000 possible orientations. We have to decide which way we 00:10:54.000 --> 00:10:58.000 will counter flux positively, which way we will counter flux 00:10:58.000 --> 00:11:01.000 negatively. And, if we change our choice, 00:11:01.000 --> 00:11:05.000 then of course the flux will become the opposite. 00:11:05.000 --> 00:11:08.000 Well, similarly to define the work, I need to choose which way 00:11:08.000 --> 00:11:12.000 I am going to run my curve. If I change which way I go 00:11:12.000 --> 00:11:16.000 around the curve then my work will become the opposite. 00:11:16.000 --> 00:11:21.000 What happens is I have to orient the curve C and the 00:11:21.000 --> 00:11:28.000 surface S in compatible ways. We have to figure out what the 00:11:28.000 --> 00:11:36.000 rule is for how the orientation of S and that of C relate to 00:11:36.000 --> 00:11:41.000 each other. What about orientation? 00:11:41.000 --> 00:11:55.000 Well, we need the orientations of C and S to be compatible and 00:11:55.000 --> 00:12:05.000 they have to explain to you what the rule is. 00:12:05.000 --> 00:12:15.000 Let me show you a picture. The rule is if I walk along C 00:12:15.000 --> 00:12:23.000 with S to my left then the normal vector is pointing up for 00:12:23.000 --> 00:12:29.000 me. Let me write that. 00:12:29.000 --> 00:12:37.000 If I walk along C, I should say in the positive 00:12:37.000 --> 00:12:48.000 direction, in the direction that I have chosen to orient C. 00:12:48.000 --> 00:13:06.000 With S to my left then n is pointing up for me. 00:13:06.000 --> 00:13:10.000 Here is the example. If I am walking on this curve, 00:13:10.000 --> 00:13:12.000 it looks like the surface is to my left. 00:13:12.000 --> 00:13:19.000 And so the normal vector is going towards what is up for me. 00:13:19.000 --> 00:13:26.000 Any questions about that? I see some people using their 00:13:26.000 --> 00:13:28.000 right hands. That is also right-handable 00:13:28.000 --> 00:13:31.000 which I am going to say in just a few moments. 00:13:31.000 --> 00:13:32.000 That is another way to remember this. 00:13:32.000 --> 00:13:38.000 Before I tell you about the right-handable version, 00:13:38.000 --> 00:13:43.000 let me just try something. Actually, I am not happy with 00:13:43.000 --> 00:13:47.000 this orientation of C and I want to orient my curve C going 00:13:47.000 --> 00:13:51.000 clockwise on the picture. So the other orientation. 00:13:51.000 --> 00:13:55.000 Then, if I walk on it this way, the surface would be to my 00:13:55.000 --> 00:13:56.000 right. You can remember, 00:13:56.000 --> 00:13:59.000 if it helps you, that if a surface is to your 00:13:59.000 --> 00:14:01.000 right then the normal vector will go down. 00:14:01.000 --> 00:14:04.000 The other way to think about this rule is enough because if 00:14:04.000 --> 00:14:07.000 you are walking clockwise, well, you can change that to 00:14:07.000 --> 00:14:10.000 counterclockwise just by walking upside down. 00:14:10.000 --> 00:14:14.000 This guy is walking clockwise on C. 00:14:14.000 --> 00:14:21.000 And while for him, if you look carefully at the 00:14:21.000 --> 00:14:31.000 picture, the surface is actually to his left when you flip upside 00:14:31.000 --> 00:14:34.000 down. Yeah, it is kind of confusing. 00:14:34.000 --> 00:14:38.000 But, anyway, maybe it's easier if you 00:14:38.000 --> 00:14:44.000 actually rotate in the picture. And now it is getting actually 00:14:44.000 --> 00:14:50.000 really confusing because his walking upside up with, 00:14:50.000 --> 00:14:54.000 actually, the surface is to his left. 00:14:54.000 --> 00:14:58.000 I mean where he is at here is actually at the front and this 00:14:58.000 --> 00:15:01.000 is the back, but that is kind of hard to see. 00:15:01.000 --> 00:15:05.000 Anyway, whichever method will work best for you. 00:15:05.000 --> 00:15:07.000 Perhaps it is easiest to first do it with the other 00:15:07.000 --> 00:15:09.000 orientation, this one, 00:15:09.000 --> 00:15:13.000 and this side, if you want the opposite one, 00:15:13.000 --> 00:15:21.000 then you will just flip everything. 00:15:21.000 --> 00:15:25.000 Now, what is the other way of remembering this with the 00:15:25.000 --> 00:15:27.000 right-hand rule? First of all, 00:15:27.000 --> 00:15:29.000 take your right hand, not your left. 00:15:29.000 --> 00:15:32.000 Even if your right hand is actually using a pen or 00:15:32.000 --> 00:15:34.000 something like that in your right hand do this. 00:15:34.000 --> 00:15:37.000 And let's take your fingers in order. 00:15:37.000 --> 00:15:40.000 First your thumb. Let's make your thumb go along 00:15:40.000 --> 00:15:42.000 the object that has only one dimension in there. 00:15:42.000 --> 00:15:47.000 That is the curve. Well, let's look at the top 00:15:47.000 --> 00:15:52.000 picture up there. I want my thumb to go along the 00:15:52.000 --> 00:15:56.000 curve so that is kind of towards the right. 00:15:56.000 --> 00:16:06.000 Then I want to make my index finger point towards the 00:16:06.000 --> 00:16:09.000 surface. Towards the surface I mean 00:16:09.000 --> 00:16:12.000 towards the interior of the surface from the curve. 00:16:12.000 --> 00:16:15.000 And when I am on the curve I am on the boundary of the surface, 00:16:15.000 --> 00:16:18.000 so there is a direction along the surface that is the curve 00:16:18.000 --> 00:16:20.000 and the other one is pointing into the surface. 00:16:20.000 --> 00:16:24.000 That one would be pointing kind of to the back slightly up 00:16:24.000 --> 00:16:27.000 maybe, so like that. And now your middle finger is 00:16:27.000 --> 00:16:30.000 going to point in the direction of the normal vector. 00:16:30.000 --> 00:16:37.000 That is up, at least if you have the same kind of right hand 00:16:37.000 --> 00:16:49.000 as I do. The other way of doing it is 00:16:49.000 --> 00:17:08.000 using the right-hand rule along C positively. 00:17:08.000 --> 00:17:18.000 The index finger towards the interior of S. 00:17:18.000 --> 00:17:27.000 Sorry, I shouldn't say interior. I should say tangent to S 00:17:27.000 --> 00:17:34.000 towards the interior of S. What I mean by that is really 00:17:34.000 --> 00:17:39.000 the part of S that is not its boundary, so the rest of the 00:17:39.000 --> 00:17:49.000 surface. Then the middle finger points 00:17:49.000 --> 00:18:00.000 parallel to n. Let's practice. 00:18:00.000 --> 00:18:09.000 Let's say that I gave you this curve bounding this surface. 00:18:09.000 --> 00:18:13.000 Which way do you think the normal vector will be going? 00:18:13.000 --> 00:18:16.000 Up. Yes. Everyone is voting up. Imaging that I am walking 00:18:16.000 --> 00:18:18.000 around C. That is to my left. 00:18:18.000 --> 00:18:24.000 Normal vector points up. Imagine that you put your thumb 00:18:24.000 --> 00:18:32.000 along C, your index towards S and then your middle finger 00:18:32.000 --> 00:18:36.000 points up. Very good. 00:18:36.000 --> 00:18:43.000 N points up. Another one. 00:19:05.000 --> 00:19:07.000 It is interesting to watch you guys. 00:19:07.000 --> 00:19:13.000 I think mostly it is going up. The correct answer is it goes 00:19:13.000 --> 00:19:20.000 up and into the cone. How do we see that? 00:19:20.000 --> 00:19:24.000 Well, one way to think about it is imagine that you are walking 00:19:24.000 --> 00:19:27.000 on C, on the rim of this cone. You have two options. 00:19:27.000 --> 00:19:30.000 Imagine that you are walking kind of inside or imagine that 00:19:30.000 --> 00:19:33.000 you are walking kind of outside. If you are walking outside then 00:19:33.000 --> 00:19:35.000 S is to your right, but it does not sound good. 00:19:35.000 --> 00:19:39.000 Let's say instead that you are walking on the inside of a cone 00:19:39.000 --> 00:19:43.000 following the boundary. Well, then the surface is to 00:19:43.000 --> 00:19:45.000 your left. And so the normal vector will 00:19:45.000 --> 00:19:50.000 be up for you which means it will be pointing slightly up and 00:19:50.000 --> 00:19:52.000 into the cone. Another way to think about it, 00:19:52.000 --> 00:19:56.000 through the right-hand rule, from this way index going kind 00:19:56.000 --> 00:20:00.000 of down because the surface goes down and a bit to the back. 00:20:00.000 --> 00:20:04.000 And then the normal vector points up and in. 00:20:04.000 --> 00:20:08.000 Yet another way, if you deform continuously your 00:20:08.000 --> 00:20:12.000 surface then the conventions will not change. 00:20:12.000 --> 00:20:15.000 See, this is kind of [UNINTELLIGIBLE] 00:20:15.000 --> 00:20:17.000 in a way. You can deform things and 00:20:17.000 --> 00:20:21.000 nothing will change. So what if we somehow flatten 00:20:21.000 --> 00:20:27.000 our cone, push it a bit up so that it becomes completely flat? 00:20:27.000 --> 00:20:30.000 Then, if you had a flat disk with the curve going 00:20:30.000 --> 00:20:33.000 counterclockwise, the normal vector would go up. 00:20:33.000 --> 00:20:36.000 Now take your disk with its normal vector sticking up. 00:20:36.000 --> 00:20:39.000 If you want to paint the face a different color so that you can 00:20:39.000 --> 00:20:43.000 remember that was beside with a normal vector and then push it 00:20:43.000 --> 00:20:45.000 back down to the cone, you will see that the painted 00:20:45.000 --> 00:20:48.000 face, the one with the normal vector 00:20:48.000 --> 00:20:52.000 on that side is the one that is inside and up. 00:20:52.000 --> 00:20:59.000 Does that make sense? Anyway, I think you have just 00:20:59.000 --> 00:21:03.000 to play with these examples for long enough and get it. 00:21:03.000 --> 00:21:07.000 OK. The last one. Let's say that I have a 00:21:07.000 --> 00:21:10.000 cylinder. So now this guy has actually 00:21:10.000 --> 00:21:12.000 two boundary curves, C and C prime. 00:21:12.000 --> 00:21:16.000 And let's say I want to orient my cylinder so that the normal 00:21:16.000 --> 00:21:20.000 vector sticks out. How should I choose the 00:21:20.000 --> 00:21:30.000 orientation of my curves? Let's start with, 00:21:30.000 --> 00:21:40.000 say, the bottom one. Would the bottom one be going 00:21:40.000 --> 00:21:44.000 clockwise or counterclockwise. Most people seem to say 00:21:44.000 --> 00:21:47.000 counterclockwise, and I agree with that. 00:21:47.000 --> 00:21:51.000 Let me write that down and claim C prime should go 00:21:51.000 --> 00:21:55.000 counterclockwise. One way to think about it, 00:21:55.000 --> 00:21:58.000 actually, it's quite easy, you mentioned that you're 00:21:58.000 --> 00:22:02.000 walking on the outside of the cylinder along C prime. 00:22:02.000 --> 00:22:06.000 If you want to walk along C prime so that the cylinder is to 00:22:06.000 --> 00:22:10.000 your left, that means you have to actually go counterclockwise 00:22:10.000 --> 00:22:14.000 around it. The other way is use your right 00:22:14.000 --> 00:22:17.000 hand. Say when you're at the front of 00:22:17.000 --> 00:22:19.000 C prime, your thumb points to the right, 00:22:19.000 --> 00:22:23.000 your index points up because that's where the surface is, 00:22:23.000 --> 00:22:28.000 and then your middle finger will point out. 00:22:28.000 --> 00:22:39.000 What about C? Well, C I claim we should be 00:22:39.000 --> 00:22:43.000 doing clockwise. I mean think about just walking 00:22:43.000 --> 00:22:46.000 again on the surface of the cylinder along C. 00:22:46.000 --> 00:22:52.000 If you walk clockwise, you will see that the surface 00:22:52.000 --> 00:22:57.000 is to your left or use the right-hand rule. 00:22:57.000 --> 00:23:00.000 Now, if a problem gives you neither the orientation of a 00:23:00.000 --> 00:23:04.000 curve nor that of the surface then it's up to you to make them 00:23:04.000 --> 00:23:05.000 up. But you have to make them up in 00:23:05.000 --> 00:23:09.000 a consistent way. You cannot choose them both at 00:23:09.000 --> 00:23:13.000 random. All right. 00:23:13.000 --> 00:23:30.000 Now we're all set to try to use Stokes' theorem. 00:23:30.000 --> 00:23:35.000 Well, let me do an example first. 00:23:35.000 --> 00:23:43.000 The first example that I will do is actually a comparison. 00:23:43.000 --> 00:23:52.000 Stokes' versus Green. I want to show you how Green's 00:23:52.000 --> 00:23:55.000 theorem for work that we saw in the plane, 00:23:55.000 --> 00:23:58.000 but also involved work and curl and so on, 00:23:58.000 --> 00:24:04.000 is actually a special case of this. 00:24:04.000 --> 00:24:11.000 Let's say that we will look at the special case where our curve 00:24:11.000 --> 00:24:16.000 C is actually a curve in the x, y plane. 00:24:16.000 --> 00:24:19.000 And let's make it go counterclockwise in the x, 00:24:19.000 --> 00:24:23.000 y plane because that's what we did for Green's theorem. 00:24:23.000 --> 00:24:25.000 Now let's choose a surface bounded by this curve. 00:24:25.000 --> 00:24:28.000 Well, as I said, I could make up any surface 00:24:28.000 --> 00:24:32.000 that comes to my mind. But, if I want to relate to 00:24:32.000 --> 00:24:35.000 this stuff, I should probably stay in the x, 00:24:35.000 --> 00:24:38.000 y plane. So I am just going to take my 00:24:38.000 --> 00:24:43.000 surface to be the piece of the x, y plane that is inside my 00:24:43.000 --> 00:24:52.000 curve. So let's say S is going to be a 00:24:52.000 --> 00:25:02.000 portion of x, y plane bounded by a curve C, 00:25:02.000 --> 00:25:11.000 and the curve C goes counterclockwise. 00:25:11.000 --> 00:25:17.000 Well, then I should look at [the table?]. 00:25:17.000 --> 00:25:24.000 For work along C of my favorite vector field F dot dr. 00:25:24.000 --> 00:25:31.000 So that will be the line integral of Pdx plus Qdy. 00:25:31.000 --> 00:25:34.000 Like I said, if I call the components of my 00:25:34.000 --> 00:25:38.000 field P, Q and R, it will be Pdx plus Qdy plus 00:25:38.000 --> 00:25:42.000 Rdz, but I don't have any Z here. 00:25:42.000 --> 00:25:49.000 Dz is zero on C. If I evaluate for line 00:25:49.000 --> 00:25:53.000 integral, I don't have any term involving dz. 00:25:53.000 --> 00:25:59.000 Z is zero. Now, let's see what Strokes 00:25:59.000 --> 00:26:05.000 says. Stokes says instead I can 00:26:05.000 --> 00:26:13.000 compute for flux through S of curve F. 00:26:13.000 --> 00:26:17.000 But now what's the normal vector to my surface? 00:26:17.000 --> 00:26:19.000 Well, it's going to be either k or negative k. 00:26:19.000 --> 00:26:22.000 I just have to figure out which one it is. 00:26:22.000 --> 00:26:25.000 Well, if you followed what we've done there, 00:26:25.000 --> 00:26:30.000 you know that the normal vector compatible with this choice for 00:26:30.000 --> 00:26:32.000 the curve C is the one that points up. 00:26:32.000 --> 00:26:43.000 My normal vector is just going to be k hat, so I am going to 00:26:43.000 --> 00:26:49.000 replace my normal vector by k hat. 00:26:49.000 --> 00:26:53.000 That means, actually, I will be integrating curl dot 00:26:53.000 --> 00:26:56.000 k. That means I am integrating the 00:26:56.000 --> 00:27:04.000 z component of curl. Let's look at curl F dot k. 00:27:04.000 --> 00:27:14.000 That's the z component of curl F. 00:27:14.000 --> 00:27:18.000 And what's the z component of curl? 00:27:18.000 --> 00:27:21.000 Well, I conveniently still have the values up there. 00:27:21.000 --> 00:27:36.000 It's Q sub x minus P sub y. My double integral becomes 00:27:36.000 --> 00:27:42.000 double integral of Q sub x minus P sub y. 00:27:42.000 --> 00:27:46.000 What about dS? Well, I am in a piece of the x, 00:27:46.000 --> 00:27:52.000 y plane, so dS is just dxdy or your favorite combination that 00:27:52.000 --> 00:27:56.000 does the same thing. Now, see, if you look at this 00:27:56.000 --> 00:28:02.000 equality, integral of Pdx plus Qdy along a closed curve equals 00:28:02.000 --> 00:28:05.000 double integral of Qx minus Py dxdy. 00:28:05.000 --> 00:28:11.000 That is exactly the statement of Green's theorem. 00:28:11.000 --> 00:28:17.000 I mean except at that time we called things m and n, 00:28:17.000 --> 00:28:20.000 but really that shouldn't matter. 00:28:20.000 --> 00:28:35.000 This tells you that, in fact, Green's theorem is 00:28:35.000 --> 00:28:51.000 just a special case of Stokes' in the x, y plane. 00:28:51.000 --> 00:28:55.000 Now, another small remark I want to make right away before I 00:28:55.000 --> 00:28:57.000 forget, you might think that these 00:28:57.000 --> 00:29:01.000 rules that we've made up about compatibility of orientations 00:29:01.000 --> 00:29:05.000 are completely arbitrary. Well, they are literally in the 00:29:05.000 --> 00:29:10.000 same way as our convention for which we guy curl is arbitrary. 00:29:10.000 --> 00:29:14.000 We chose to make the curl be this thing and not the opposite 00:29:14.000 --> 00:29:17.000 which would have been pretty much just as sensible. 00:29:17.000 --> 00:29:21.000 And, ultimately, that comes from our choice of 00:29:21.000 --> 00:29:25.000 making the cross-product be what it is but of the opposite. 00:29:25.000 --> 00:29:30.000 Ultimately, it all comes from our preference for right-handed 00:29:30.000 --> 00:29:33.000 coordinate systems. If we had been on the planet 00:29:33.000 --> 00:29:36.000 with left-handed coordinate systems then actually our 00:29:36.000 --> 00:29:40.000 conventions would be all the other way around, 00:29:40.000 --> 00:30:05.000 but they are this way. Any other questions? 00:30:05.000 --> 00:30:08.000 A surface that you use in Stokes' theorem is usually not 00:30:08.000 --> 00:30:12.000 going to be closed because its boundary needs to be the curve 00:30:12.000 --> 00:30:14.000 C. So if you had a closed surface 00:30:14.000 --> 00:30:17.000 you wouldn't know where to put your curve. 00:30:17.000 --> 00:30:20.000 I mean of course you could make a tiny hole in it and get a tiny 00:30:20.000 --> 00:30:22.000 curve. Actually, what that would say, 00:30:22.000 --> 00:30:26.000 and we are going to see more about that so not very important 00:30:26.000 --> 00:30:28.000 right now, but what we would see is that 00:30:28.000 --> 00:30:31.000 for a close surface we would end up getting zero for the flux. 00:30:31.000 --> 00:30:34.000 And that is actually because divergence of curl is zero, 00:30:34.000 --> 00:30:36.000 but I am getting ahead of myself. 00:30:36.000 --> 00:30:45.000 We are going to see that probably tomorrow in more 00:30:45.000 --> 00:30:49.000 detail. Stokes' theorem only works if 00:30:49.000 --> 00:30:53.000 you can make sense of this. That means you need your vector 00:30:53.000 --> 00:30:58.000 field to be continuous and differentiable everywhere on the 00:30:58.000 --> 00:31:03.000 surface S. Now, why is that relevant? 00:31:03.000 --> 00:31:05.000 Well, say that your vector field was not defined at the 00:31:05.000 --> 00:31:07.000 origin and say that you wanted to do, 00:31:07.000 --> 00:31:11.000 you know, the example that I had first with the unit circling 00:31:11.000 --> 00:31:14.000 the x, y plane. Normally, the most sensible 00:31:14.000 --> 00:31:17.000 choice of surface to apply Stokes' theorem to would be just 00:31:17.000 --> 00:31:19.000 the flat disk in the x, y plane. 00:31:19.000 --> 00:31:23.000 But that assumes that your vector field is well-defined 00:31:23.000 --> 00:31:24.000 there. If your vector field is not 00:31:24.000 --> 00:31:27.000 defined at the origin but defined everywhere else you 00:31:27.000 --> 00:31:29.000 cannot use this guy, but maybe you can still use, 00:31:29.000 --> 00:31:31.000 say, the half-sphere, for example. 00:31:31.000 --> 00:31:35.000 Or, you could use a piece of cylinder plus a flat top or 00:31:35.000 --> 00:31:38.000 whatever you want but not pressing for the origin. 00:31:38.000 --> 00:31:41.000 So you could still use Stokes but you'd have to be careful 00:31:41.000 --> 00:31:44.000 about which surface you choose. Now, if instead your vector 00:31:44.000 --> 00:31:49.000 field is not defined anywhere on the z-axis then you're out of 00:31:49.000 --> 00:31:54.000 luck because there is no way to find a surface bounded by this 00:31:54.000 --> 00:31:59.000 unit circle without crossing the z-axis somewhere. 00:31:59.000 --> 00:32:07.000 Then you wouldn't be able to Stokes' theorem at all or at 00:32:07.000 --> 00:32:16.000 least not directly. Maybe I should write it F 00:32:16.000 --> 00:32:25.000 defines a differentiable everywhere on this. 00:32:25.000 --> 00:32:27.000 But we don't care about what happens outside of this. 00:32:27.000 --> 00:32:35.000 It's really only on the surface that we need it to be OK. 00:32:35.000 --> 00:32:39.000 I mean, again, 99% of the vector fields that 00:32:39.000 --> 00:32:44.000 we see in this class are defined everywhere so that's not an 00:32:44.000 --> 00:32:47.000 urgent concern, but still. 00:32:47.000 --> 00:32:49.000 OK. Should we move on? 00:32:49.000 --> 00:33:01.000 Yes. I have a yes. Let me explain to you quickly 00:33:01.000 --> 00:33:08.000 why Stokes is true. How do we prove a theorem like 00:33:08.000 --> 00:33:10.000 that? Well, 00:33:10.000 --> 00:33:12.000 the strategy, I mean there are other ways, 00:33:12.000 --> 00:33:16.000 but the least painful strategy at this point is to observe what 00:33:16.000 --> 00:33:19.000 we already know is a special case of Stokes's theorem. 00:33:19.000 --> 00:33:22.000 Namely we know the case where the curve is actually in the x, 00:33:22.000 --> 00:33:24.000 y plane and the surface is a flat piece of the x, 00:33:24.000 --> 00:33:34.000 y plane because that's Green's theorem which we proved a while 00:33:34.000 --> 00:33:42.000 ago. We know it for C and S in the 00:33:42.000 --> 00:33:47.000 x, y plane. Now, what if C and S were, 00:33:47.000 --> 00:33:49.000 say, in the y, z plane instead of the x, 00:33:49.000 --> 00:33:51.000 y plane? Well, then it will not quite 00:33:51.000 --> 00:33:55.000 give the same picture because the normal vector would be i hat 00:33:55.000 --> 00:33:58.000 instead of k hat and they would be having different notations 00:33:58.000 --> 00:34:01.000 and it would be integrating with y and z. 00:34:01.000 --> 00:34:02.000 But you see that it would become, again, 00:34:02.000 --> 00:34:05.000 exactly the same formula. We'd know it for any of the 00:34:05.000 --> 00:34:08.000 coordinate planes. In fact, I claim we know it for 00:34:08.000 --> 00:34:13.000 absolutely any plane. And the reason for that is, 00:34:13.000 --> 00:34:15.000 sure, when we write it in coordinates, 00:34:15.000 --> 00:34:19.000 when we write that this line integral is integral of Pdx plus 00:34:19.000 --> 00:34:24.000 Qdy plus Rdz or when we write that the curl is given by this 00:34:24.000 --> 00:34:28.000 formula we use the x, y, z coordinate system. 00:34:28.000 --> 00:34:30.000 But there is something I haven't quite told you about. 00:34:30.000 --> 00:34:33.000 Which is if I switch to any other right-handed coordinate 00:34:33.000 --> 00:34:35.000 system, so I do some sort of rotation 00:34:35.000 --> 00:34:40.000 of my space coordinates, then somehow the line integral, 00:34:40.000 --> 00:34:44.000 the flux integral, the notion of curl makes sense 00:34:44.000 --> 00:34:47.000 in coordinates. And the reason is that they all 00:34:47.000 --> 00:34:50.000 have geometric interpretations. For example, 00:34:50.000 --> 00:34:52.000 when I think of this as the work done by a force, 00:34:52.000 --> 00:34:55.000 well, the force doesn't care whether it's being put in x, 00:34:55.000 --> 00:34:56.000 y coordinates this way or that way. 00:34:56.000 --> 00:35:00.000 It still does the same work because it's the same force. 00:35:00.000 --> 00:35:03.000 And when I say that the curl measures the rotation in a 00:35:03.000 --> 00:35:06.000 motion, well, that depends on which 00:35:06.000 --> 00:35:09.000 coordinates you use. And the same for interpretation 00:35:09.000 --> 00:35:12.000 of flux. In fact, if I rotated my 00:35:12.000 --> 00:35:17.000 coordinates to fit with any other plane, I could still do 00:35:17.000 --> 00:35:23.000 the same things. What I'm trying to say is, 00:35:23.000 --> 00:35:31.000 in fact, if C and S are in any plane then we can still claim 00:35:31.000 --> 00:35:37.000 that it reduces to Green's theorem. 00:35:37.000 --> 00:35:45.000 It will be Green's theorem not in x, y, z coordinates but in 00:35:45.000 --> 00:35:50.000 some funny rotated coordinate systems. 00:35:50.000 --> 00:35:56.000 What I'm saying is that work, flux and curl makes sense 00:35:56.000 --> 00:35:59.000 independently of coordinates. 00:36:20.000 --> 00:36:23.000 Now, this has to stop somewhere. I can start claiming that I can 00:36:23.000 --> 00:36:26.000 somehow bend my coordinates to a plane, any surface is flat. 00:36:26.000 --> 00:36:29.000 That doesn't really work. But what I can say is if I have 00:36:29.000 --> 00:36:31.000 any surface I can cut it into tiny pieces. 00:36:31.000 --> 00:36:35.000 And these tiny pieces are basically flat. 00:36:35.000 --> 00:36:39.000 So that's basically the idea of a proof. 00:36:39.000 --> 00:36:47.000 I am going to decompose my surface into very small flat 00:36:47.000 --> 00:36:56.000 pieces. Given any S we are just going 00:36:56.000 --> 00:37:08.000 to decompose it into tiny almost flat pieces. 00:37:08.000 --> 00:37:15.000 For example, if I have my surface like this, 00:37:15.000 --> 00:37:23.000 what I will do is I will just cut it into tiles. 00:37:23.000 --> 00:37:28.000 I mean a good example of that is if you look at 00:37:28.000 --> 00:37:31.000 [UNINTELLIGIBLE], for example, 00:37:31.000 --> 00:37:36.000 it's made of all these hexagons and pentagons. 00:37:36.000 --> 00:37:38.000 Well, actually, they're not quite flat in the 00:37:38.000 --> 00:37:41.000 usual rule, but you could make them flat and it would still 00:37:41.000 --> 00:37:45.000 look pretty much like a sphere. Anyway, you're going to cut 00:37:45.000 --> 00:37:49.000 your surface into lots of tiny pieces. 00:37:49.000 --> 00:37:53.000 And then you can use Stokes' theorem on each small piece. 00:37:53.000 --> 00:38:00.000 What it says on each small flat piece -- It says that the line 00:38:00.000 --> 00:38:04.000 integral along say, for example, 00:38:04.000 --> 00:38:08.000 this curve is equal to the flux of a curl through this tiny 00:38:08.000 --> 00:38:12.000 piece of surface. And now I will add all of these 00:38:12.000 --> 00:38:14.000 terms together. If I add all of the small 00:38:14.000 --> 00:38:17.000 contributions to flux I get the total flux. 00:38:17.000 --> 00:38:19.000 What if I add all of the small line integrals? 00:38:19.000 --> 00:38:23.000 Well, I get lots of extra junk because I never asked to compute 00:38:23.000 --> 00:38:26.000 the line integral along this. But this guy will come in twice 00:38:26.000 --> 00:38:30.000 when I do this little plate and when I do that little plate with 00:38:30.000 --> 00:38:34.000 opposite orientations. When I sum all of the little 00:38:34.000 --> 00:38:38.000 line integrals together, all of the inner things cancel 00:38:38.000 --> 00:38:40.000 out, and the only ones that I go 00:38:40.000 --> 00:38:44.000 through only once are those that are at the outer most edges. 00:38:44.000 --> 00:38:50.000 So, when I sum all of my works together, I will get the work 00:38:50.000 --> 00:38:54.000 done just along the outer boundary C. 00:38:54.000 --> 00:39:12.000 Sum of work around each little piece is just actually the work 00:39:12.000 --> 00:39:27.000 along C, the outer curve. And the sum of the flux for 00:39:27.000 --> 00:39:39.000 each piece is going to be the flux through S. 00:39:39.000 --> 00:39:45.000 From Stokes' theorem for flat surfaces, I can get it for any 00:39:45.000 --> 00:39:47.000 surface. I am cheating a little bit 00:39:47.000 --> 00:39:50.000 because you would actually have to check carefully that this 00:39:50.000 --> 00:39:53.000 approximately where you flatten the little pieces that are 00:39:53.000 --> 00:39:56.000 almost flat is [UNINTELLIGIBLE]. But, trust me, 00:39:56.000 --> 00:39:56.000 it actually works. 00:40:13.000 --> 00:40:15.000 Let's do an actual example. I mean I said example, 00:40:15.000 --> 00:40:19.000 but that was more like getting us ready for the proof so 00:40:19.000 --> 00:40:22.000 probably that doesn't count as an actual example. 00:40:22.000 --> 00:40:25.000 I should probably keep these statements for now so I am not 00:40:25.000 --> 00:40:26.000 going to erase this side. 00:41:08.000 --> 00:41:21.000 Let's do an example. Let's try to find the work of 00:41:21.000 --> 00:41:40.000 vector field zi plus xj plus yk around the unit circle in the x, 00:41:40.000 --> 00:41:58.000 y plane counterclockwise. The picture is conveniently 00:41:58.000 --> 00:42:05.000 already there. Just as a quick review, 00:42:05.000 --> 00:42:08.000 let's see how we do that directly. 00:42:08.000 --> 00:42:14.000 If we do that directly, I have to find the integral 00:42:14.000 --> 00:42:21.000 along C. So F dot dr becomes zdx plus 00:42:21.000 --> 00:42:28.000 xdy plus ydz. But now we actually know that 00:42:28.000 --> 00:42:33.000 on this circle, well, z is zero. 00:42:33.000 --> 00:42:39.000 And we can parameterize x and y, the unit circle in the x, 00:42:39.000 --> 00:42:44.000 y plane, so we can take x equals cosine t, 00:42:44.000 --> 00:42:49.000 y equals sine t. That will just become the 00:42:49.000 --> 00:42:54.000 integral over C. Well, z times dx, 00:42:54.000 --> 00:43:05.000 z is zero so we have nothing, plus x is cosine t times dy is 00:43:05.000 --> 00:43:17.000 -- Well, if y is sine t then dy is cosine tdt plus ydz but z is 00:43:17.000 --> 00:43:22.000 zero. Now, the range of values for t, 00:43:22.000 --> 00:43:26.000 well, we are going counterclockwise around the 00:43:26.000 --> 00:43:31.000 entire circle so that should go from zero to 2pi. 00:43:31.000 --> 00:43:39.000 We will get integral from zero to 2pi of cosine square tdt 00:43:39.000 --> 00:43:45.000 which, if you do the calculation, turns out to be 00:43:45.000 --> 00:43:50.000 just pi. Now, let's instead try to use 00:43:50.000 --> 00:43:55.000 Stokes' theorem to do the calculation. 00:43:55.000 --> 00:44:00.000 Now, of course the smart choice would be to just take the flat 00:44:00.000 --> 00:44:02.000 unit disk. I am not going to do that. 00:44:02.000 --> 00:44:06.000 That would be too boring. Plus we have already kind of 00:44:06.000 --> 00:44:09.000 checked it because we already trust Green's theorem. 00:44:09.000 --> 00:44:11.000 Instead, just to convince you that, 00:44:11.000 --> 00:44:14.000 yes, I can choose really any surface I want, 00:44:14.000 --> 00:44:23.000 let's say that I'm going to choose a piece of paraboloid z 00:44:23.000 --> 00:44:30.000 equals one minus x squared minus y squared. 00:44:30.000 --> 00:44:36.000 Well, to get our conventions straight, we should take the 00:44:36.000 --> 00:44:43.000 normal vector pointing up for compatibility with our choice. 00:44:43.000 --> 00:44:48.000 Well, we will have to compute the flux through S. 00:44:48.000 --> 00:44:50.000 We don't really have to because we could have chosen the disk, 00:44:50.000 --> 00:44:54.000 it would be easier, but if we want to do it this 00:44:54.000 --> 00:45:00.000 way we will compute the flux of curl F through our paraboloid. 00:45:00.000 --> 00:45:03.000 How do we do that? Well, we need to find the curl 00:45:03.000 --> 00:45:10.000 and we need to find ndS. Let's start with the curl. 00:45:10.000 --> 00:45:23.000 Curl F let's take the cross-product between dell and F 00:45:23.000 --> 00:45:28.000 which is zxy. If we compute this, 00:45:28.000 --> 00:45:31.000 the i component will be one minus zero. 00:45:31.000 --> 00:45:37.000 It looks like it is one i. Minus the j component is zero 00:45:37.000 --> 00:45:41.000 minus one. Plus the k component is one 00:45:41.000 --> 00:45:52.000 minus zero. In fact, the curl of the field 00:45:52.000 --> 00:45:59.000 is one, one, one. Now, what about ndS? 00:45:59.000 --> 00:46:03.000 Well, this is a surface for which we know z is a function of 00:46:03.000 --> 00:46:08.000 x and y. ndS we can write as, 00:46:08.000 --> 00:46:14.000 let's call this F of xy, then we can use the formula 00:46:14.000 --> 00:46:19.000 that says ndS equals negative F sub x, negative F sub y, 00:46:19.000 --> 00:46:26.000 one dxdy, which here gives us 2x, 00:46:26.000 --> 00:46:34.000 2y, one dxdy. Now, when we want to compute 00:46:34.000 --> 00:46:41.000 the flux, we will have to do double integral over S of one, 00:46:41.000 --> 00:46:47.000 one, one dot product with 2x, 2y, one dxdy. 00:46:47.000 --> 00:46:55.000 It will become the double integral of 2x plus 2y plus one 00:46:55.000 --> 00:46:58.000 dxdy. And, of course, 00:46:58.000 --> 00:47:01.000 the region which we are integrating, the range of values 00:47:01.000 --> 00:47:04.000 of x and y will be the shadow of our surface. 00:47:04.000 --> 00:47:07.000 That is just going to be, if you look at this paraboloid 00:47:07.000 --> 00:47:11.000 from above, all you will see is the unit 00:47:11.000 --> 00:47:17.000 disk so it will be a double integral of the unit disk. 00:47:17.000 --> 00:47:23.000 And the way we will do that, one way is to switch to polar 00:47:23.000 --> 00:47:28.000 coordinates and do the calculation and then you will 00:47:28.000 --> 00:47:31.000 end up with pi. The other way is to try to do 00:47:31.000 --> 00:47:34.000 it by symmetry. Observe, when you integrate x 00:47:34.000 --> 00:47:37.000 above this, x is as negative on the left as it is positive on 00:47:37.000 --> 00:47:40.000 the right. So the integral of x will be 00:47:40.000 --> 00:47:42.000 zero. The integral of y will be zero 00:47:42.000 --> 00:47:46.000 also by symmetry. Then the integral of one dxdy 00:47:46.000 --> 00:47:52.000 will just be the area of this unit disk which is pi. 00:47:52.000 --> 00:47:54.000 That was our first example. And, of course, 00:47:54.000 --> 00:47:57.000 if you're actually free to choose your favorite surface, 00:47:57.000 --> 00:48:01.000 there is absolutely no reason why you would actually choose 00:48:01.000 --> 00:48:04.000 this paraboloid in this example. I mean it would be much easier 00:48:04.000 --> 00:48:05.000 to choose a flat disk. OK. 00:48:05.000 --> 00:48:09.000 Tomorrow I will tell you a few more things about curl fits in 00:48:09.000 --> 00:48:13.000 with conservativeness and with the divergence theorem, 00:48:13.000 --> 00:48:17.000 Stokes all together, and we will look at Practice 00:48:17.000 --> 00:48:20.000 Exam 4B so please bring the exam with you. 00:48:20.000 --> 00:48:25.000 with you.