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 --> 00:00:29.000 OK, so remember, we've seen Stokes theorem, 00:00:29.000 --> 00:00:37.000 which says if I have a closed curve bounding some surface, 00:00:37.000 --> 00:00:40.000 S, and I orient the curve and the 00:00:40.000 --> 00:00:44.000 surface compatible with each other, 00:00:44.000 --> 00:00:53.000 then I can compute the line integral along C along my curve 00:00:53.000 --> 00:00:57.000 in terms of, instead, 00:00:57.000 --> 00:01:04.000 surface integral for flux of a different vector field, 00:01:04.000 --> 00:01:12.000 namely, curl f dot n dS. OK, so that's the statement. 00:01:12.000 --> 00:01:18.000 And, just to clarify a little bit, so, again, 00:01:18.000 --> 00:01:24.000 we've seen various kinds of integrals. 00:01:24.000 --> 00:01:26.000 So, line integrals we know how to evaluate. 00:01:26.000 --> 00:01:30.000 They take place in a curve. You express everything in terms 00:01:30.000 --> 00:01:32.000 of one variable, and after substituting, 00:01:32.000 --> 00:01:36.000 you end up with a usual one variable integral that you know 00:01:36.000 --> 00:01:40.000 how to evaluate. And, surface integrals, 00:01:40.000 --> 00:01:44.000 we know also how to evaluate. Namely, we've seen various 00:01:44.000 --> 00:01:47.000 formulas for ndS. Once you have such a formula, 00:01:47.000 --> 00:01:50.000 due to the dot product with this vector field, 00:01:50.000 --> 00:01:52.000 which is not the same as that one. 00:01:52.000 --> 00:01:56.000 But it's a new vector field that you can build out of f. 00:01:56.000 --> 00:02:00.000 You do the dot product. You express everything in terms 00:02:00.000 --> 00:02:03.000 of your two integration variables, and then you 00:02:03.000 --> 00:02:06.000 evaluate. So, now, what does this have to 00:02:06.000 --> 00:02:12.000 do with various other things? So, one thing I want to say has 00:02:12.000 --> 00:02:18.000 to do with how Stokes helps us understand path independence, 00:02:18.000 --> 00:02:24.000 so, how it actually motivates our criterion for gradient 00:02:24.000 --> 00:02:30.000 fields, independence. 00:02:30.000 --> 00:02:35.000 OK, so, we've seen that if we have a 00:02:35.000 --> 00:02:40.000 vector field defined in a simply connected region, 00:02:40.000 --> 00:02:43.000 and its curl is zero, then it's a gradient field, 00:02:43.000 --> 00:02:47.000 and the line integral is path independent. 00:02:47.000 --> 00:02:53.000 So, let me first define for you when a simply connected region 00:02:53.000 --> 00:03:01.000 is. So, we say that a region in 00:03:01.000 --> 00:03:17.000 space is simply connected -- -- if every closed loop inside this 00:03:17.000 --> 00:03:31.000 region bounds some surface again inside this region. 00:03:31.000 --> 00:03:39.000 OK, so let me just give you some examples just to clarify. 00:03:39.000 --> 00:03:46.000 So, for example, let's say that I have a region 00:03:46.000 --> 00:03:52.000 that's the entire space with the origin removed. 00:03:52.000 --> 00:04:00.000 OK, so space with the origin removed, OK, you think it's 00:04:00.000 --> 00:04:06.000 simply connected? Who thinks it's simply 00:04:06.000 --> 00:04:09.000 connected? Who thinks it's not simply 00:04:09.000 --> 00:04:14.000 connected? Let's think a little bit harder. 00:04:14.000 --> 00:04:17.000 Let's say that I take a loop like this one, 00:04:17.000 --> 00:04:20.000 OK, it doesn't go through the origin. 00:04:20.000 --> 00:04:24.000 Can I find a surface that's bounded by this loop and that 00:04:24.000 --> 00:04:26.000 does not pass through the origin? 00:04:26.000 --> 00:04:30.000 Yeah, I can take the sphere, you know, for example, 00:04:30.000 --> 00:04:34.000 or anything that's just not quite the disk? 00:04:34.000 --> 00:04:36.000 So, and similarly, 00:04:36.000 --> 00:04:39.000 if I take any other loop that avoids the origin, 00:04:39.000 --> 00:04:42.000 I can find, actually, a surface bounded by it that 00:04:42.000 --> 00:04:44.000 does not pass through the origin. 00:04:44.000 --> 00:04:47.000 So, actually, that's kind of a not so obvious 00:04:47.000 --> 00:04:49.000 theorem to prove, but maybe intuitively, 00:04:49.000 --> 00:04:52.000 start by finding any surface. Well, if that surface passes 00:04:52.000 --> 00:04:54.000 through the origin, just wiggle it a little bit, 00:04:54.000 --> 00:04:56.000 you can make sure it doesn't pass through the origin anymore. 00:04:56.000 --> 00:05:00.000 Just push it a little bit. So, in fact, 00:05:00.000 --> 00:05:08.000 this is simply connected. That was a trick question. 00:05:08.000 --> 00:05:13.000 OK, now on the other hand, a good example of something 00:05:13.000 --> 00:05:16.000 that is not simply connected is if I take space, 00:05:16.000 --> 00:05:36.000 and I remove the z axis -- -- that is not simply connected. 00:05:36.000 --> 00:05:39.000 And, see, the reason is, if I look again, say, 00:05:39.000 --> 00:05:44.000 at the unit circle in the x axis, 00:05:44.000 --> 00:05:47.000 sorry, unit circle in the xy plane, 00:05:47.000 --> 00:05:52.000 I mean, in the xy plane, so, if I try to find a surface 00:05:52.000 --> 00:05:58.000 whose boundary is this disk, well, it has to actually cross 00:05:58.000 --> 00:06:03.000 the z axis somewhere. There's no way that I can find 00:06:03.000 --> 00:06:09.000 a surface whose only boundary is this curve, which doesn't hit 00:06:09.000 --> 00:06:13.000 the z axis anywhere. Of course, you could try to use 00:06:13.000 --> 00:06:17.000 the same trick as there, say, maybe we want to go up, 00:06:17.000 --> 00:06:19.000 up, up. You know, let's start with a 00:06:19.000 --> 00:06:21.000 cylinder. Well, the problem is you have 00:06:21.000 --> 00:06:24.000 to go infinitely far because the z axis goes infinitely far. 00:06:24.000 --> 00:06:27.000 And, you'll never be able to actually close your surface. 00:06:27.000 --> 00:06:30.000 So, the matter what kind of trick you might want to use, 00:06:30.000 --> 00:06:34.000 it's actually a theorem in topology that you cannot find a 00:06:34.000 --> 00:06:39.000 surface bounded by this disk without intersecting the z axis. 00:06:39.000 --> 00:06:44.000 Yes? Well, a doughnut shape 00:06:44.000 --> 00:06:47.000 certainly would stay away from the z axis, but it wouldn't be a 00:06:47.000 --> 00:06:50.000 surface with boundary just this guy. 00:06:50.000 --> 00:06:53.000 Right, it would have to have either some other boundary. 00:06:53.000 --> 00:06:57.000 So, maybe what you have in mind is some sort of doughnut shape 00:06:57.000 --> 00:07:01.000 like this that curves on itself, and maybe comes back. 00:07:01.000 --> 00:07:05.000 Well, if you don't quite close it all the way around, 00:07:05.000 --> 00:07:07.000 so I can try to, indeed, draw some sort of 00:07:07.000 --> 00:07:10.000 doughnut here. Well, if I don't quite close 00:07:10.000 --> 00:07:13.000 it, that it will have another edge at the other end wherever I 00:07:13.000 --> 00:07:15.000 started. If I close it completely, 00:07:15.000 --> 00:07:18.000 then this curve is no longer its boundary because my surface 00:07:18.000 --> 00:07:20.000 lives on both sides of this curve. 00:07:20.000 --> 00:07:22.000 See, I want a surface that stops on this curve, 00:07:22.000 --> 00:07:25.000 and doesn't go beyond it. And, nowhere else does it have 00:07:25.000 --> 00:07:28.000 that kind of behavior. Everywhere else, 00:07:28.000 --> 00:07:33.000 it keeps going on. So, 00:07:33.000 --> 00:07:36.000 actually, I mean, maybe actually another way to 00:07:36.000 --> 00:07:40.000 convince yourself is to find a counter example to the statement 00:07:40.000 --> 00:07:44.000 I'm going to make about vector fields with curl zero and simply 00:07:44.000 --> 00:07:48.000 connected regions always being conservative. 00:07:48.000 --> 00:07:52.000 So, what you can do is you can take the example that we had in 00:07:52.000 --> 00:07:55.000 one of our older problem sets. That was a vector field in the 00:07:55.000 --> 00:07:58.000 plane. But, you can also use it to 00:07:58.000 --> 00:08:02.000 define a vector field in space just with no z component. 00:08:02.000 --> 00:08:05.000 That vector field is actually defined everywhere except on the 00:08:05.000 --> 00:08:08.000 z axis, and it violates the usual theorem that we would 00:08:08.000 --> 00:08:12.000 expect. So, that's one way to check 00:08:12.000 --> 00:08:20.000 just for sure that this thing is not simply connected. 00:08:20.000 --> 00:08:26.000 So, what's the statement I want to make? 00:08:26.000 --> 00:08:40.000 So, recall we've seen if F is a gradient field -- -- then its 00:08:40.000 --> 00:08:46.000 curl is zero. That's just the fact that the 00:08:46.000 --> 00:08:49.000 mixed second partial derivatives are equal. 00:08:49.000 --> 00:08:53.000 So, now, the converse is the following theorem. 00:08:53.000 --> 00:09:01.000 It says if the curl of F equals zero in, sorry, 00:09:01.000 --> 00:09:09.000 and F is defined -- No, is not the logical in which to 00:09:09.000 --> 00:09:15.000 say it. So, if F is defined in a simply 00:09:15.000 --> 00:09:30.000 connected region, and curl F is zero -- -- then F 00:09:30.000 --> 00:09:45.000 is a gradient field, and the line integral for F is 00:09:45.000 --> 00:09:53.000 path independent -- -- F is conservative, 00:09:53.000 --> 00:09:55.000 and so on, all the usual consequences. 00:09:55.000 --> 00:09:58.000 Remember, these are all equivalent to each other, 00:09:58.000 --> 00:10:01.000 for example, because you can use path 00:10:01.000 --> 00:10:05.000 independence to define the potential by doing the line 00:10:05.000 --> 00:10:08.000 integral of F. OK, so where do we use the 00:10:08.000 --> 00:10:12.000 assumption of being defined in a simply connected region? 00:10:12.000 --> 00:10:17.000 Well, the way which we will prove this is to use Stokes 00:10:17.000 --> 00:10:20.000 theorem. OK, so the proof, 00:10:20.000 --> 00:10:25.000 so just going to prove that the line integral is path 00:10:25.000 --> 00:10:29.000 independent; the others work the same way. 00:10:29.000 --> 00:10:34.000 OK, so let's assume that we have a vector field whose curl 00:10:34.000 --> 00:10:38.000 is zero. And, let's say that we have two 00:10:38.000 --> 00:10:44.000 curves, C1 and C2, that go from some point P0 to 00:10:44.000 --> 00:10:49.000 some point P1, the same point to the same 00:10:49.000 --> 00:10:54.000 point. Well, we'd like to understand 00:10:54.000 --> 00:11:00.000 the line integral along C1, say, minus the line integral 00:11:00.000 --> 00:11:04.000 along C2 to show that this is zero. 00:11:04.000 --> 00:11:06.000 That's what we are trying to prove. 00:11:06.000 --> 00:11:12.000 So, how will we compute that? Well, the line integral along 00:11:12.000 --> 00:11:17.000 C1 minus C2, well, let's just form a closed curve 00:11:17.000 --> 00:11:24.000 that is C1 minus C2. OK, so let's call C, 00:11:24.000 --> 00:11:36.000 woops -- So that's equal to the integral along C of f dot dr 00:11:36.000 --> 00:11:44.000 where C is C1 followed by C2 backwards. 00:11:44.000 --> 00:11:50.000 Now, C is a closed curve. So, I can use Stokes theorem. 00:11:50.000 --> 00:11:52.000 Well, to be able to use Stokes theorem, I need, 00:11:52.000 --> 00:11:54.000 actually, to find a surface to apply it to. 00:11:54.000 --> 00:11:57.000 And, that's where the assumption of simply connected 00:11:57.000 --> 00:12:00.000 is useful. I know in advance that any 00:12:00.000 --> 00:12:02.000 closed curve, so, C in particular, 00:12:02.000 --> 00:12:10.000 has to bound some surface. OK, so we can find S, 00:12:10.000 --> 00:12:21.000 a surface, S, that bounds C because the 00:12:21.000 --> 00:12:32.000 region is simply connected. So, now that tells us we can 00:12:32.000 --> 00:12:38.000 actually apply Stokes theorem, except it won't fit here. 00:12:38.000 --> 00:12:40.000 So, instead, I will do that on the next 00:12:40.000 --> 00:12:45.000 line. That's equal by Stokes to the 00:12:45.000 --> 00:12:51.000 double integral over S of curl F dot vector dS, 00:12:51.000 --> 00:12:54.000 or ndS. But now, the curl is zero. 00:12:54.000 --> 00:12:58.000 So, if I integrate zero, I will get zero. 00:12:58.000 --> 00:13:02.000 OK, so I proved that my two line integrals along C1 and C2 00:13:02.000 --> 00:13:04.000 are equal. But for that, 00:13:04.000 --> 00:13:08.000 I needed to be able to find a surface which to apply Stokes 00:13:08.000 --> 00:13:11.000 theorem. And that required my region to 00:13:11.000 --> 00:13:14.000 be simply connected. If I had a vector field that 00:13:14.000 --> 00:13:17.000 was defined only outside of the z axis and I took two paths that 00:13:17.000 --> 00:13:20.000 went on one side and the other side of the z axis, 00:13:20.000 --> 00:13:21.000 I might have obtained, actually, 00:13:21.000 --> 00:13:27.000 different values of the line integral. 00:13:27.000 --> 00:13:35.000 OK, so anyway, that's the customary warning 00:13:35.000 --> 00:13:43.000 about simply connected things. OK, let me just mention very 00:13:43.000 --> 00:13:46.000 quickly that there's a lot of interesting topology you can do, 00:13:46.000 --> 00:13:48.000 actually in space. So, for example, 00:13:48.000 --> 00:13:50.000 this concept of being simply connected or not, 00:13:50.000 --> 00:13:55.000 and studying which loops bound surfaces or not can be used to 00:13:55.000 --> 00:13:58.000 classify shapes of things inside space. 00:13:58.000 --> 00:14:07.000 So, for example, one of the founding 00:14:07.000 --> 00:14:15.000 achievements of topology in the 19th century was to classify 00:14:15.000 --> 00:14:24.000 surfaces in space -- -- by trying to look at loops on them. 00:14:24.000 --> 00:14:33.000 So, what I mean by that is that if I take the surface of a 00:14:33.000 --> 00:14:39.000 sphere, well, I claim the surface of a sphere 00:14:39.000 --> 00:14:44.000 -- -- is simply connected. Why is that? 00:14:44.000 --> 00:14:49.000 Well, let's take my favorite closed curve on the surface of a 00:14:49.000 --> 00:14:53.000 sphere. I can always find a portion of 00:14:53.000 --> 00:14:59.000 the sphere that's bounded by it. OK, so that's the definition of 00:14:59.000 --> 00:15:03.000 the surface of a sphere being simply connected. 00:15:03.000 --> 00:15:06.000 On the other hand, if I take what's called a 00:15:06.000 --> 00:15:07.000 torus, or if you prefer, 00:15:07.000 --> 00:15:10.000 the surface of a doughnut, that's more, 00:15:10.000 --> 00:15:21.000 it's a less technical term, but it's -- -- well, 00:15:21.000 --> 00:15:24.000 that's not simply connected. And, in fact, 00:15:24.000 --> 00:15:26.000 for example, if you look at this loop here 00:15:26.000 --> 00:15:29.000 that goes around it, well, of course it bounds a 00:15:29.000 --> 00:15:32.000 surface in space. But, that surface cannot be 00:15:32.000 --> 00:15:35.000 made to be just a piece of the donut. 00:15:35.000 --> 00:15:39.000 You have to go through the hole. You have to leave the surface 00:15:39.000 --> 00:15:41.000 of a torus. In fact, there's another one. 00:15:41.000 --> 00:15:47.000 See, this one also does not bound anything that's completely 00:15:47.000 --> 00:15:50.000 contained in the torus. And, of course, 00:15:50.000 --> 00:15:53.000 it bounds this disc, but inside of a torus. 00:15:53.000 --> 00:15:56.000 But, that's not a part of the surface itself. 00:15:56.000 --> 00:16:02.000 So, in fact, there's, and topologists would 00:16:02.000 --> 00:16:09.000 say, there's two independent -- -- loops that don't bound 00:16:09.000 --> 00:16:15.000 surfaces, that don't bound anything. 00:16:15.000 --> 00:16:18.000 And, so this number two is somehow an invariant that you 00:16:18.000 --> 00:16:20.000 can associate to this kind of shape. 00:16:20.000 --> 00:16:23.000 And then, if you consider more complicated surfaces with more 00:16:23.000 --> 00:16:24.000 holes in them, you can try, somehow, 00:16:24.000 --> 00:16:27.000 to count independent loops on them, 00:16:27.000 --> 00:16:33.000 and that's the beginning of the classification of surfaces. 00:16:33.000 --> 00:16:39.000 Anyway, that's not really an 18.02 topic, but I thought I 00:16:39.000 --> 00:16:45.000 would mentioned it because it's kind of a cool idea. 00:16:45.000 --> 00:16:55.000 OK, let me say a bit more in the way of fun remarks like 00:16:55.000 --> 00:16:59.000 that. So, food for thought: 00:16:59.000 --> 00:17:05.000 let's say that I want to apply Stokes theorem to simplify a 00:17:05.000 --> 00:17:08.000 line integral along the curve here. 00:17:08.000 --> 00:17:11.000 So, this curve is maybe not easy to see in the picture. 00:17:11.000 --> 00:17:17.000 It kind of goes twice around the z axis, but spirals up and 00:17:17.000 --> 00:17:20.000 then down. OK, so one way to find a 00:17:20.000 --> 00:17:25.000 surface that's bounded by this curve is to take what's called 00:17:25.000 --> 00:17:29.000 the Mobius strip. OK, so the Mobius strip, 00:17:29.000 --> 00:17:32.000 it's a one sided strip where when you go around, 00:17:32.000 --> 00:17:35.000 you flip one side becomes the other. 00:17:35.000 --> 00:17:38.000 So, you just, if you want to take a band of 00:17:38.000 --> 00:17:41.000 paper and glue the two sides with a twist, 00:17:41.000 --> 00:17:44.000 so, it's a one sided surface. And, that gives us, 00:17:44.000 --> 00:17:49.000 actually, serious trouble if we try to orient it to apply Stokes 00:17:49.000 --> 00:17:53.000 theorem. So, see, for example, 00:17:53.000 --> 00:17:58.000 if I take this Mobius strip, and I try to find an 00:17:58.000 --> 00:18:04.000 orientation, so here it looks like that, 00:18:04.000 --> 00:18:08.000 well, let's say that I've oriented my curve going in this 00:18:08.000 --> 00:18:11.000 direction. So, I go around, 00:18:11.000 --> 00:18:13.000 around, around, still going this direction. 00:18:13.000 --> 00:18:19.000 Well, the orientation I should have for Stokes theorem is that 00:18:19.000 --> 00:18:22.000 when I, so, curve continues here. 00:18:22.000 --> 00:18:26.000 Well, if you look at the convention around here, 00:18:26.000 --> 00:18:31.000 it tells us that the normal vector should be going this way. 00:18:31.000 --> 00:18:35.000 OK, if we look at it near here, if we walk along this way, 00:18:35.000 --> 00:18:37.000 the surface is to our right . So, we should actually be 00:18:37.000 --> 00:18:40.000 flipping things upside down. The normal vector should be 00:18:40.000 --> 00:18:41.000 going down. And, in fact, 00:18:41.000 --> 00:18:44.000 if you try to follow your normal vector that's pointing 00:18:44.000 --> 00:18:45.000 up, it's pointing up, up, up. 00:18:45.000 --> 00:18:49.000 It will have to go into things, into, into, down. 00:18:49.000 --> 00:18:53.000 There's no way to choose consistently a normal vector for 00:18:53.000 --> 00:19:01.000 the Mobius strip. So, that's what we call a 00:19:01.000 --> 00:19:08.000 non-orientable surface. And, that just means it has 00:19:08.000 --> 00:19:10.000 only one side. And, if it has only one side, 00:19:10.000 --> 00:19:14.000 that we cannot speak of flux for it because we have no way of 00:19:14.000 --> 00:19:17.000 saying that we'll be counting things positively one way, 00:19:17.000 --> 00:19:19.000 negatively the other way, because there's only one, 00:19:19.000 --> 00:19:22.000 you know, there's no notion of sides. 00:19:22.000 --> 00:19:26.000 So, you can't define a side towards which things will be 00:19:26.000 --> 00:19:34.000 going positively. So, that's actually a situation 00:19:34.000 --> 00:19:44.000 where flux cannot be defined. OK, so as much as Mobius strips 00:19:44.000 --> 00:19:48.000 and climb-bottles are exciting and really cool, 00:19:48.000 --> 00:19:51.000 well, we can't use them in this class because we can't define 00:19:51.000 --> 00:19:54.000 flux through them. So, if we really wanted to 00:19:54.000 --> 00:19:57.000 apply Stokes theorem, because I've been telling you 00:19:57.000 --> 00:19:59.000 that space is simply connected, and I will always be able to 00:19:59.000 --> 00:20:01.000 apply Stokes theorem to any curve, 00:20:01.000 --> 00:20:05.000 what would I do? Well, I claim this curve 00:20:05.000 --> 00:20:10.000 actually bounds another surface that is orientable. 00:20:10.000 --> 00:20:11.000 Yeah, that looks counterintuitive. 00:20:11.000 --> 00:20:16.000 Well, let's see it. I claim you can take a 00:20:16.000 --> 00:20:22.000 hemisphere, and you can take a small thing and twist it around. 00:20:22.000 --> 00:20:26.000 So, in case you don't believe me, let me do it again with the 00:20:26.000 --> 00:20:28.000 transparency. Here's my loop, 00:20:28.000 --> 00:20:31.000 and see, well, the scale is not exactly the 00:20:31.000 --> 00:20:33.000 same. So, it doesn't quite match. 00:20:33.000 --> 00:20:35.000 But, and it's getting a bit dark. 00:20:35.000 --> 00:20:41.000 But, that spherical thing with a little slit going twisting 00:20:41.000 --> 00:20:46.000 into it will actually have boundary my loop. 00:20:46.000 --> 00:20:50.000 And, that one is orientable. I mean, I leave it up to you to 00:20:50.000 --> 00:20:56.000 stare at the picture long enough to convince yourselves that 00:20:56.000 --> 00:21:00.000 there's a well-defined up and down. 00:21:00.000 --> 00:21:11.000 OK. So now, I mean, 00:21:11.000 --> 00:21:14.000 in case you are getting really, really worried, 00:21:14.000 --> 00:21:18.000 I mean, there won't be any Mobius strips on the exam on 00:21:18.000 --> 00:21:24.000 Tuesday, OK? It's just to show you some cool 00:21:24.000 --> 00:21:29.000 stuff. OK, questions? 00:21:29.000 --> 00:21:34.000 No? OK, one last thing I want to 00:21:34.000 --> 00:21:38.000 show you before we start reviewing, 00:21:38.000 --> 00:21:41.000 so one question you might have about Stokes theorem is, 00:21:41.000 --> 00:21:44.000 how come we can choose whatever surface we want? 00:21:44.000 --> 00:21:47.000 I mean, sure, it seems to work, 00:21:47.000 --> 00:21:52.000 but why? So, I'm going to say a couple 00:21:52.000 --> 00:22:02.000 of words about surface independence in Stokes theorem. 00:22:02.000 --> 00:22:08.000 So, let's say that I have a curve, C, in space. 00:22:08.000 --> 00:22:11.000 And, let's say that I want to apply Stokes theorem. 00:22:11.000 --> 00:22:16.000 So, then I can choose my favorite surface bounded by C. 00:22:16.000 --> 00:22:18.000 So, in a situation like this, for example, 00:22:18.000 --> 00:22:21.000 I might want to make my first choice be this guy, 00:22:21.000 --> 00:22:25.000 S1, like maybe some sort of upper half sphere. 00:22:25.000 --> 00:22:28.000 And, if you pay attention to the orientation conventions, 00:22:28.000 --> 00:22:31.000 you'll see that you need to take it with normal vector 00:22:31.000 --> 00:22:34.000 pointing up. Maybe actually I would rather 00:22:34.000 --> 00:22:36.000 make a different choice. And actually, 00:22:36.000 --> 00:22:41.000 I will choose another surface, S2, that maybe looks like that. 00:22:41.000 --> 00:22:44.000 And, if I look carefully at the orientation convention, 00:22:44.000 --> 00:22:47.000 Stokes theorem tells me that I have to take the normal vector 00:22:47.000 --> 00:22:52.000 pointing up again. So, that's actually into things. 00:22:52.000 --> 00:22:57.000 So, Stokes says that the line 00:22:57.000 --> 00:23:04.000 integral along C of my favorite vector field can be computed 00:23:04.000 --> 00:23:09.000 either as a flux integral for the curl through S1, 00:23:09.000 --> 00:23:16.000 or as the same integral, but through S2 instead of S1. 00:23:16.000 --> 00:23:21.000 So, that seems to suggest that curl F has some sort of surface 00:23:21.000 --> 00:23:24.000 independence property. It doesn't really matter which 00:23:24.000 --> 00:23:27.000 surface I take, as long as the boundary is this 00:23:27.000 --> 00:23:29.000 given curve, C. Why is that? 00:23:29.000 --> 00:23:31.000 That's a strange property to have. 00:23:31.000 --> 00:23:36.000 Where does it come from? Well, let's think about it for 00:23:36.000 --> 00:23:40.000 a second. So, why are these the same? 00:23:40.000 --> 00:23:42.000 I mean, of course, they have to be the same 00:23:42.000 --> 00:23:44.000 because that's what Stokes tell us. 00:23:44.000 --> 00:23:48.000 But, why is that OK? Well, let's think about 00:23:48.000 --> 00:23:53.000 comparing the flux integral for S1 and the flux integral for S2. 00:23:53.000 --> 00:23:57.000 So, if we want to compare them, we should probably subtract 00:23:57.000 --> 00:24:02.000 them from each other. OK, so let's do the flux 00:24:02.000 --> 00:24:09.000 integral for S1 minus the flux integral for S2 of the same 00:24:09.000 --> 00:24:12.000 thing. Well, let's give a name. 00:24:12.000 --> 00:24:18.000 Let's call S the surface S1 minus S2. 00:24:18.000 --> 00:24:21.000 So, what is S? S is S1 with its given 00:24:21.000 --> 00:24:26.000 orientation together with S2 with the reversed orientation. 00:24:26.000 --> 00:24:32.000 So, S is actually this whole closed surface here. 00:24:32.000 --> 00:24:37.000 And, the normal vector to S seems to be pointing outwards 00:24:37.000 --> 00:24:39.000 everywhere. OK, so now, if we have a closed 00:24:39.000 --> 00:24:41.000 surface with a normal vector pointing outwards, 00:24:41.000 --> 00:24:44.000 and we want to find a flux integral for it, 00:24:44.000 --> 00:24:47.000 well, we can replace that with a 00:24:47.000 --> 00:24:57.000 triple integral. So, that's the divergence 00:24:57.000 --> 00:25:03.000 theorem. So, that's by the divergence 00:25:03.000 --> 00:25:09.000 theorem using the fact that S is a closed surface. 00:25:09.000 --> 00:25:13.000 That's equal to the triple integral over the region inside. 00:25:13.000 --> 00:25:26.000 Let me call that region D of divergence, of curl F dV. 00:25:26.000 --> 00:25:34.000 OK, and what I'm going to claim now is that we can actually 00:25:34.000 --> 00:25:41.000 check that if you take the divergence of the curl of a 00:25:41.000 --> 00:25:47.000 vector field, you always get zero. 00:25:47.000 --> 00:25:50.000 OK, and so that will tell you that this integral will always 00:25:50.000 --> 00:25:53.000 be zero. And that's why the flux for S1, 00:25:53.000 --> 00:25:58.000 and the flux for S2 were the same a priori and we didn't have 00:25:58.000 --> 00:26:01.000 to worry about which one we chose when we did Stokes 00:26:01.000 --> 00:26:06.000 theorem. OK, so let's just check quickly 00:26:06.000 --> 00:26:10.000 that divergence of a curve is zero. 00:26:10.000 --> 00:26:12.000 OK, in case you're wondering why I'm doing all this, 00:26:12.000 --> 00:26:13.000 well, first I think it's kind of interesting, 00:26:13.000 --> 00:26:17.000 and second, it reminds you of a statement of all these theorems, 00:26:17.000 --> 00:26:19.000 and all these definitions. So, in a way, 00:26:19.000 --> 00:26:25.000 we are already reviewing. OK, so let's see. 00:26:25.000 --> 00:26:30.000 If my vector field has components P, 00:26:30.000 --> 00:26:39.000 Q, and R, remember that the curl was defined by this cross 00:26:39.000 --> 00:26:47.000 product between del and our given vector field. 00:26:47.000 --> 00:27:04.000 So, that's Ry - Qz followed by Pz - Rx, and Qx - Py. 00:27:04.000 --> 00:27:14.000 So, now, we want to take the divergence of this. 00:27:14.000 --> 00:27:19.000 Well, so we have to take the first component, 00:27:19.000 --> 00:27:23.000 Ry minus Qz, and take its partial with 00:27:23.000 --> 00:27:28.000 respect to x. Then, take the y component, 00:27:28.000 --> 00:27:35.000 Pz minus Rx partial with respect to y plus Qx minus Py 00:27:35.000 --> 00:27:41.000 partial with respect to z. And, well, now we should expand 00:27:41.000 --> 00:27:43.000 this. But I claim it will always 00:27:43.000 --> 00:27:44.000 simplify to zero. 00:28:11.000 --> 00:28:24.000 OK, so I think we have over there, becomes R sub yx minus Q 00:28:24.000 --> 00:28:39.000 sub zx plus P sub zy minus R sub xy plus Q sub xz minus P sub yz. 00:28:39.000 --> 00:28:47.000 Well, let's see. We have P sub zy minus P sub yz. 00:28:47.000 --> 00:28:54.000 These two cancel out. We have R sub yx minus R sub xy. 00:28:54.000 --> 00:28:58.000 These cancel out. Q sub zx and Q sub xz, 00:28:58.000 --> 00:29:04.000 these two also cancel out. So, indeed, the divergence of a 00:29:04.000 --> 00:29:10.000 curl is always zero. OK, so the claim is divergence 00:29:10.000 --> 00:29:17.000 of curl is always zero. Del cross F is always zero, 00:29:17.000 --> 00:29:26.000 and just a small remark, if we had actually real vectors 00:29:26.000 --> 00:29:30.000 rather than this strange del guy, 00:29:30.000 --> 00:29:33.000 indeed we know that if we have two vectors, 00:29:33.000 --> 00:29:37.000 U and V, and we do u dot u cross v, 00:29:37.000 --> 00:29:40.000 what is that? Well, one way to say it is it's 00:29:40.000 --> 00:29:43.000 the determinant of u, u, and v, which is the volume 00:29:43.000 --> 00:29:45.000 of the box. But, it's completely flat 00:29:45.000 --> 00:29:47.000 because u, u, and v are all in the plane 00:29:47.000 --> 00:29:50.000 defined by u and v. The other way to say it is that 00:29:50.000 --> 00:29:53.000 u cross v is perpendicular to u and v. 00:29:53.000 --> 00:29:56.000 Well, if it's perpendicular u, then its dot product with u 00:29:56.000 --> 00:29:59.000 will be zero. So, no matter how you say it, 00:29:59.000 --> 00:30:02.000 this is always zero. So, in a way, 00:30:02.000 --> 00:30:09.000 this reinforces our intuition that del, even though it's not 00:30:09.000 --> 00:30:15.000 at all an actual vector sometimes can be manipulated in 00:30:15.000 --> 00:30:20.000 the same way. OK, I think that's it for new 00:30:20.000 --> 00:30:26.000 topics for today. And, 00:30:26.000 --> 00:30:30.000 so, now I should maybe try to recap quickly what we've learned 00:30:30.000 --> 00:30:34.000 in these past three weeks so that you know, 00:30:34.000 --> 00:30:39.000 so, the exam is probably going to be similar in difficulty to 00:30:39.000 --> 00:30:42.000 the practice exams. That's my goal. 00:30:42.000 --> 00:30:45.000 I don't know if I will have reached that goal or not. 00:30:45.000 --> 00:30:48.000 We'll only know that after you've taken the test. 00:30:48.000 --> 00:30:53.000 But, the idea is it's meant to be more or less the same level 00:30:53.000 --> 00:30:58.000 of difficulty. So, at this point, 00:30:58.000 --> 00:31:06.000 we've learned about three kinds of beasts in space. 00:31:06.000 --> 00:31:12.000 OK, so I'm going to divide my blackboard into three pieces, 00:31:12.000 --> 00:31:16.000 and here I will write triple integrals. 00:31:16.000 --> 00:31:20.000 We've learned about double integrals, and we've learned 00:31:20.000 --> 00:31:26.000 about line integrals. OK, so triple integrals over a 00:31:26.000 --> 00:31:33.000 region in space, we integrate a scalar quantity, 00:31:33.000 --> 00:31:35.000 dV. How do we do that? 00:31:35.000 --> 00:31:41.000 Well, we can do that in rectangular coordinates where dV 00:31:41.000 --> 00:31:46.000 becomes something like, maybe, dz dx dy, 00:31:46.000 --> 00:31:52.000 or any permutation of these. We've seen how to do it also in 00:31:52.000 --> 00:31:59.000 cylindrical coordinates where dV is maybe dz times r dr d theta 00:31:59.000 --> 00:32:02.000 or more commonly r dr d theta dz. 00:32:02.000 --> 00:32:06.000 But, what I want to emphasize in this way is that both of 00:32:06.000 --> 00:32:09.000 these you set up pretty much in the same way. 00:32:09.000 --> 00:32:12.000 So, remember, the main trick here is to find 00:32:12.000 --> 00:32:15.000 the bounds of integration. So, when you do it, 00:32:15.000 --> 00:32:18.000 say, with dz first, that means for fixed xy, 00:32:18.000 --> 00:32:23.000 so, for a fixed point in the xy plane, you have to look at the 00:32:23.000 --> 00:32:25.000 bounds for z. So, that means you have to 00:32:25.000 --> 00:32:28.000 figure out what's the bottom surface of your solid, 00:32:28.000 --> 00:32:31.000 and what's the top surface of your solid? 00:32:31.000 --> 00:32:34.000 And, you have to find the value of z at the bottom, 00:32:34.000 --> 00:32:37.000 the value of z at the top as functions of x and y. 00:32:37.000 --> 00:32:40.000 And then, you will put that as bounds for z. 00:32:40.000 --> 00:32:43.000 Once you've done that, you are left with the question 00:32:43.000 --> 00:32:45.000 of finding bounds for x and y. Well, for that, 00:32:45.000 --> 00:32:49.000 you just rotate the picture, look at your solid from above, 00:32:49.000 --> 00:32:52.000 so, look at its projection to the xy plane, 00:32:52.000 --> 00:32:56.000 and you set up a double integral either in rectangular 00:32:56.000 --> 00:33:02.000 xy coordinates, or in polar coordinates for x 00:33:02.000 --> 00:33:04.000 and y. Of course, you can always do it 00:33:04.000 --> 00:33:08.000 a different orders. And, I'll let you figure out 00:33:08.000 --> 00:33:11.000 again how that goes. But, if you do dz first, 00:33:11.000 --> 00:33:15.000 then the inner bounds are given by bottom and top, 00:33:15.000 --> 00:33:20.000 and the outer ones are given by looking at the shadow of the 00:33:20.000 --> 00:33:23.000 region. Now, there's also spherical 00:33:23.000 --> 00:33:28.000 coordinates. And there, we've seen that dV 00:33:28.000 --> 00:33:32.000 is rho squared sine phi d rho d phi d theta. 00:33:32.000 --> 00:33:35.000 So now, of course, if this orgy of Greek letters 00:33:35.000 --> 00:33:39.000 is confusing you at this point, then you probably need to first 00:33:39.000 --> 00:33:41.000 review spherical coordinates for themselves. 00:33:41.000 --> 00:33:44.000 Remember that rho is the distance from the origin. 00:33:44.000 --> 00:33:47.000 Phi is the angle down from the z axis. 00:33:47.000 --> 00:33:49.000 So, it's zero, and the positive z axis, 00:33:49.000 --> 00:33:53.000 pi over two in the xy plane, and increases all the way to pi 00:33:53.000 --> 00:33:59.000 on the negative z axis. And, theta is the angle around 00:33:59.000 --> 00:34:02.000 the z axis. So, now, when we set up bounds 00:34:02.000 --> 00:34:04.000 here, it will look a lot like what 00:34:04.000 --> 00:34:07.000 you've done in polar coordinates in the plane because when you 00:34:07.000 --> 00:34:09.000 look at the inner bound down on rho, 00:34:09.000 --> 00:34:12.000 for a fixed phi and theta, that means you're shooting a 00:34:12.000 --> 00:34:15.000 straight ray from the origin in some direction in space. 00:34:15.000 --> 00:34:17.000 So, you know, you're sending a laser beam, 00:34:17.000 --> 00:34:20.000 and you want to know what part of your beam is going to be in 00:34:20.000 --> 00:34:23.000 your given solid. You want to solve for the value 00:34:23.000 --> 00:34:26.000 of rho when you enter the solid and when you leave it. 00:34:26.000 --> 00:34:29.000 I mean, very often, if the origin is in your solid, 00:34:29.000 --> 00:34:33.000 then rho will start at zero. Then you want to know when you 00:34:33.000 --> 00:34:34.000 exit. And, I mean, 00:34:34.000 --> 00:34:38.000 there's a fairly small list of kinds of surfaces that we've 00:34:38.000 --> 00:34:41.000 seen how to set up in spherical coordinates. 00:34:41.000 --> 00:34:44.000 So, if you're really upset by this, go over the problems in 00:34:44.000 --> 00:34:47.000 the notes. That will give you a good idea 00:34:47.000 --> 00:34:53.000 of what kinds of things we've seen in spherical coordinates. 00:34:53.000 --> 00:34:56.000 OK, and then evaluation is the usual way. 00:34:56.000 --> 00:35:01.000 Questions about this? No? 00:35:01.000 --> 00:35:08.000 OK, so, I should say we can do something bad, 00:35:08.000 --> 00:35:15.000 but so we've seen, of course, applications of 00:35:15.000 --> 00:35:19.000 this. So, we should know how to use a 00:35:19.000 --> 00:35:24.000 triple integral to evaluate things like a mass of a solid, 00:35:24.000 --> 00:35:29.000 the average value of a function, 00:35:29.000 --> 00:35:37.000 the moment of inertia about one of the coordinate axes, 00:35:37.000 --> 00:35:54.000 or the gravitational attraction on a mass at the origin. 00:35:54.000 --> 00:35:58.000 OK, so these are just formulas to remember for examples of 00:35:58.000 --> 00:36:01.000 triple integrals. It doesn't change conceptually. 00:36:01.000 --> 00:36:04.000 You always set them up and evaluate them the same way. 00:36:04.000 --> 00:36:11.000 It just tells you what to put there for the integrand. 00:36:11.000 --> 00:36:15.000 Now, double integrals: so, 00:36:15.000 --> 00:36:18.000 when we have a surface in space, 00:36:18.000 --> 00:36:21.000 well, what we will integrate on it, 00:36:21.000 --> 00:36:26.000 at least what we've seen how to integrate is a vector field 00:36:26.000 --> 00:36:31.000 dotted with the unit normal vector times the area element. 00:36:31.000 --> 00:36:38.000 OK, and this is sometimes called vector dS. 00:36:38.000 --> 00:36:48.000 Now, how do we evaluate that? Well, we've seen formulas for 00:36:48.000 --> 00:36:55.000 ndS in various settings. And, once you have a formula 00:36:55.000 --> 00:37:01.000 for ndS, that will relate ndS to maybe dx dy, or something else. 00:37:01.000 --> 00:37:07.000 And then, you will express, so, for example, 00:37:07.000 --> 00:37:15.000 ndS equals something dx dy. And then, it becomes a double 00:37:15.000 --> 00:37:21.000 integral of something dx dy. Now, in the integrand, 00:37:21.000 --> 00:37:23.000 you want to express everything in terms of x and y. 00:37:23.000 --> 00:37:26.000 So, if you had a z, maybe you have a formula for z 00:37:26.000 --> 00:37:28.000 in terms of x and y. And, when you set up the 00:37:28.000 --> 00:37:30.000 bounds, well, you try to figure out what are 00:37:30.000 --> 00:37:33.000 the bounds for x and y? That would be just looking at 00:37:33.000 --> 00:37:35.000 it from above. Of course, if you are using 00:37:35.000 --> 00:37:37.000 other variables, figure out the bounds for those 00:37:37.000 --> 00:37:40.000 variables. And, when you've done that, 00:37:40.000 --> 00:37:44.000 it becomes just a double integral in the usual sense. 00:37:44.000 --> 00:37:46.000 OK, so maybe I should be a bit more explicit about formulas 00:37:46.000 --> 00:37:52.000 because there have been a lot. So, let me tell you about a few 00:37:52.000 --> 00:37:56.000 of them. Let me actually do that over 00:37:56.000 --> 00:38:02.000 here because I don't want to make this too crowded. 00:38:24.000 --> 00:38:28.000 OK, so what kinds of formulas for ndS have we seen? 00:38:28.000 --> 00:38:32.000 Well, we've seen a formula, for example, 00:38:32.000 --> 00:38:37.000 for a horizontal plane, or for something that's 00:38:37.000 --> 00:38:42.000 parallel to the yz plane or the xz plane. 00:38:42.000 --> 00:38:47.000 Well, let's do just the yz plane for a quick reminder. 00:38:47.000 --> 00:38:52.000 So, if I have a surface that's contained inside the yz plane, 00:38:52.000 --> 00:38:56.000 then obviously I will express ds in terms of, 00:38:56.000 --> 00:39:01.000 well, I will use y and z as my variables. 00:39:01.000 --> 00:39:05.000 So, I will say that ds is dy dz, or dz dy, 00:39:05.000 --> 00:39:11.000 whatever's most convenient. Maybe we will even switch to 00:39:11.000 --> 00:39:15.000 polar coordinates after that if a problem wants us to. 00:39:15.000 --> 00:39:16.000 And, what about the normal vector? 00:39:16.000 --> 00:39:21.000 Well, the normal vector is either coming straight at us, 00:39:21.000 --> 00:39:26.000 or it's maybe going back away from us depending on which 00:39:26.000 --> 00:39:30.000 orientation we've chosen. So, this gives us ndS. 00:39:30.000 --> 00:39:32.000 We dot our favorite vector field with it. 00:39:32.000 --> 00:39:38.000 We integrate, and we get the answer. 00:39:38.000 --> 00:39:48.000 OK, we've seen about spheres and cylinders centered at the 00:39:48.000 --> 00:39:54.000 origin or centered on the z axis. 00:39:54.000 --> 00:40:00.000 So, the normal vector sticks straight out or straight in, 00:40:00.000 --> 00:40:05.000 depending on which direction you do it in. 00:40:05.000 --> 00:40:09.000 So, for a sphere, the normal vector is 00:40:14.000 y, z> divided by the radius of the sphere. 00:40:14.000 --> 00:40:17.000 For a cylinder, it's 00:40:21.000 0>, divided by the radius of a cylinder. 00:40:21.000 --> 00:40:25.000 And, the surface element on a sphere, 00:40:25.000 --> 00:40:28.000 so, see, it's very closely related to the volume element of 00:40:28.000 --> 00:40:31.000 spherical coordinates except you don't have a rho anymore. 00:40:31.000 --> 00:40:37.000 You just plug in a rho equals a. So, you get a squared sine phi 00:40:37.000 --> 00:40:41.000 d phi d theta. And, for a cylinder, 00:40:41.000 --> 00:40:47.000 it would be a dz d theta. So, 00:40:47.000 --> 00:40:51.000 by the way, just a quick check, when you're doing an integral, 00:40:51.000 --> 00:40:55.000 if it's the surface integral, there should be two integral 00:40:55.000 --> 00:40:56.000 signs, and there should be two 00:40:56.000 --> 00:40:59.000 integration variables. And, there should be two d 00:40:59.000 --> 00:41:03.000 somethings. If you end up with a dx, 00:41:03.000 --> 00:41:10.000 dy, dz in the surface integral, something is seriously wrong. 00:41:10.000 --> 00:41:18.000 OK, now, besides these specific formulas, we've seen two general 00:41:18.000 --> 00:41:24.000 formulas that are also useful. So, one is, 00:41:24.000 --> 00:41:29.000 if we know how to express z in terms of x and y, 00:41:29.000 --> 00:41:33.000 and just to change notation to show you that it's not set in 00:41:33.000 --> 00:41:36.000 stone, let's say that z is known as a 00:41:36.000 --> 00:41:42.000 function z of x and y. So, how do I get ndS in that 00:41:42.000 --> 00:41:45.000 case? Well, we've seen a formula that 00:41:45.000 --> 00:41:51.000 says negative partial z partial x, negative partial z partial y, 00:41:51.000 --> 00:41:54.000 one dx dy. So, this formula relates the 00:41:54.000 --> 00:41:57.000 volume, sorry, the surface element on our 00:41:57.000 --> 00:42:01.000 surface to the area element in the xy plane. 00:42:01.000 --> 00:42:08.000 It lets us convert between dS and dx dy. 00:42:08.000 --> 00:42:11.000 OK, so we just plug in this, and we dot with F, 00:42:11.000 --> 00:42:14.000 and then we substitute everything in terms of x and y, 00:42:14.000 --> 00:42:17.000 and we evaluate the integral over x and y. 00:42:17.000 --> 00:42:22.000 If we don't really want to find a way to find z as a function of 00:42:22.000 --> 00:42:29.000 x and y, but we have a normal vector 00:42:29.000 --> 00:42:35.000 given to us, then we have another formula 00:42:35.000 --> 00:42:39.000 which says that ndS is, sorry, I should have said it's 00:42:39.000 --> 00:42:42.000 always up to sign because we have a two orientation 00:42:42.000 --> 00:42:45.000 convention. We have to decide based on what 00:42:45.000 --> 00:42:48.000 we are trying to do, whether we are doing the 00:42:48.000 --> 00:42:51.000 correct convention or the wrong one. 00:42:51.000 --> 00:43:02.000 So, the other formula is n over n dot k dx dy. 00:43:02.000 --> 00:43:09.000 Sorry, are they all the same? Well, if you want, 00:43:09.000 --> 00:43:12.000 you can put an absolute value here. 00:43:12.000 --> 00:43:16.000 But, it doesn't matter because it's up to sign anyway. 00:43:16.000 --> 00:43:22.000 So, I mean, this formula is valid as it is. 00:43:22.000 --> 00:43:24.000 OK, and, I mean, if you're in a situation where 00:43:24.000 --> 00:43:26.000 you can apply more than one formula, 00:43:26.000 --> 00:43:32.000 they will all give you the same answer in the end because it's 00:43:32.000 --> 00:43:37.000 the same flux integral. OK, so anyway, 00:43:37.000 --> 00:43:40.000 so we have various ways of computing surface integrals, 00:43:40.000 --> 00:43:44.000 and probably one of the best possible things you can do to 00:43:44.000 --> 00:43:48.000 prepare for the test is actually to look again at some practice 00:43:48.000 --> 00:43:51.000 problems from the notes that do flux integrals over various 00:43:51.000 --> 00:43:55.000 kinds of surfaces because that's probably one of the hardest 00:43:55.000 --> 00:43:58.000 topics in this unit of the class. 00:43:58.000 --> 00:44:05.000 OK, anyway, let's move on to line integrals. 00:44:05.000 --> 00:44:11.000 So, those are actually a piece of cake in comparison, 00:44:11.000 --> 00:44:17.000 OK, because all that this is, is just integral of P dx Q dy R 00:44:17.000 --> 00:44:24.000 dz. And, then all you have to do is 00:44:24.000 --> 00:44:35.000 parameterize the curve, C, to express everything in 00:44:35.000 --> 00:44:42.000 terms of a single variable. And then, you end up with a 00:44:42.000 --> 00:44:46.000 usual single integral, and you can just compute it. 00:44:46.000 --> 00:44:48.000 So, that one works pretty much as it did in the plane. 00:44:48.000 --> 00:44:52.000 So, if you forgotten what we did in the plane, 00:44:52.000 --> 00:44:56.000 it's really the same thing. OK, so now we have three 00:44:56.000 --> 00:44:58.000 different kinds of integrals, and really, well, 00:44:58.000 --> 00:45:01.000 they certainly have in common that they integrate things 00:45:01.000 --> 00:45:03.000 somehow. But, apart from that, 00:45:03.000 --> 00:45:05.000 they are extremely different in what they do. 00:45:05.000 --> 00:45:08.000 I mean, this one involves a function, a scalar quantity. 00:45:08.000 --> 00:45:11.000 These involve vector quantities. They don't involve the same 00:45:11.000 --> 00:45:13.000 kinds of shapes over which to integrate. 00:45:13.000 --> 00:45:16.000 Here, you integrate over a three-dimensional region. 00:45:16.000 --> 00:45:19.000 Here, you integrate only over a two-dimensional surface, 00:45:19.000 --> 00:45:21.000 and here, only a one-dimensional curve. 00:45:21.000 --> 00:45:24.000 So, try not to confuse them. That's basically the most 00:45:24.000 --> 00:45:27.000 important advice. Don't get mistaken. 00:45:27.000 --> 00:45:30.000 Each of them has a different way of getting evaluated. 00:45:30.000 --> 00:45:34.000 Eventually, they will all give you numbers, but through 00:45:34.000 --> 00:45:36.000 different processes. So now, well, 00:45:36.000 --> 00:45:38.000 I said these guys are completely different. 00:45:38.000 --> 00:45:40.000 Well, they are, but we still have some bridges 00:45:40.000 --> 00:45:42.000 between them. OK, so we have two, 00:45:42.000 --> 00:45:46.000 maybe I should say three, well, two bridges between these 00:45:46.000 --> 00:45:49.000 guys. OK, so we have somehow a 00:45:49.000 --> 00:45:54.000 connection between these which is the divergence theorem. 00:45:54.000 --> 00:46:02.000 We have a connection between that, which is Stokes theorem. 00:46:02.000 --> 00:46:20.000 So -- Just to write them again, so the divergence theorem says 00:46:20.000 --> 00:46:25.000 if I have a region in space, and I call its boundary S, 00:46:25.000 --> 00:46:27.000 so, it's going to be a closed surface, 00:46:27.000 --> 00:46:31.000 and I orient S with a normal vector pointing outwards, 00:46:31.000 --> 00:46:35.000 then whenever I have a surface integral over S, 00:46:35.000 --> 00:46:40.000 sorry, I can replace it by a triple 00:46:40.000 --> 00:46:47.000 integral over the region inside. OK, so this guy is a vector 00:46:47.000 --> 00:46:49.000 field. And, this guy is a function 00:46:49.000 --> 00:46:52.000 that somehow relates to the vector field. 00:46:52.000 --> 00:46:54.000 I mean, you should know how. You should know the definition 00:46:54.000 --> 00:46:55.000 of divergence, of course. 00:46:55.000 --> 00:46:59.000 But, what I want to point out is if you have to compute the 00:46:59.000 --> 00:47:01.000 two sides separately, well, this is just, 00:47:01.000 --> 00:47:04.000 you know, your standard flux integral. 00:47:04.000 --> 00:47:07.000 This is just your standard triple integral over a region in 00:47:07.000 --> 00:47:09.000 space. Once you have computed what 00:47:09.000 --> 00:47:13.000 this guy is, it's really just a triple integral of the function. 00:47:13.000 --> 00:47:16.000 So, the way in which you compute it doesn't see that it 00:47:16.000 --> 00:47:19.000 came from a divergence. It's just the same way that you 00:47:19.000 --> 00:47:23.000 would compute any other triple integral. 00:47:23.000 --> 00:47:28.000 The way we compute it doesn't depend on what actually we are 00:47:28.000 --> 00:47:32.000 integrating. Stokes theorem says if I have a 00:47:32.000 --> 00:47:35.000 curve that's the boundary of a surface, S, 00:47:35.000 --> 00:47:40.000 and I orient the two in compatible manners, 00:47:40.000 --> 00:47:52.000 then I can replace a line integral on C by a surface 00:47:52.000 --> 00:47:57.000 integral on S. OK, and that surface integral, 00:47:57.000 --> 00:48:00.000 well, it's not for the same vector field. 00:48:00.000 --> 00:48:03.000 This relates a line integral for one field to a surface 00:48:03.000 --> 00:48:06.000 integral from another field. That other field is given from 00:48:06.000 --> 00:48:10.000 the first one just by taking its curl So, after you take the 00:48:10.000 --> 00:48:13.000 curl, you obtain a different vector field. 00:48:13.000 --> 00:48:17.000 And, the way in which you would compute the surface integral is 00:48:17.000 --> 00:48:19.000 just as with any surface integral. 00:48:19.000 --> 00:48:23.000 You just find a formula for ndS dot product, substitute, 00:48:23.000 --> 00:48:26.000 evaluate. The calculation of this thing, 00:48:26.000 --> 00:48:30.000 once you've computed curl does not remember that it was a curl. 00:48:30.000 --> 00:48:33.000 It's the same as with any other flux integral. 00:48:33.000 --> 00:48:35.000 OK, and finally, the last bridge, 00:48:35.000 --> 00:48:38.000 so this was between two and three. 00:48:38.000 --> 00:48:41.000 This was between one and two. Let me just say, 00:48:41.000 --> 00:48:45.000 there's a bridge between zero and one, 00:48:45.000 --> 00:48:53.000 which is that if you have a function in its gradient, 00:48:53.000 --> 00:48:57.000 well, the fundamental theorem of calculus says that the line 00:48:57.000 --> 00:49:01.000 integral for the vector field given by the gradient of a 00:49:01.000 --> 00:49:04.000 function is actually equal to the change in value of a 00:49:04.000 --> 00:49:08.000 function. That's if you have a curve 00:49:08.000 --> 00:49:11.000 bounded by P0 and P1. So in a way, actually, 00:49:11.000 --> 00:49:15.000 each of these three theorems relates a quantity with a 00:49:15.000 --> 00:49:19.000 certain number of integral signs to a quantity with one more 00:49:19.000 --> 00:49:22.000 integral sign. And, that's actually somehow a 00:49:22.000 --> 00:49:24.000 fundamental similarity between them. 00:49:24.000 --> 00:49:27.000 But maybe it's easier to think of them as completely different 00:49:27.000 --> 00:49:30.000 stories. So now, with this one, 00:49:30.000 --> 00:49:36.000 we additionally have to remember another topic is given 00:49:36.000 --> 00:49:41.000 a vector field, F, with curl equal to zero, 00:49:41.000 --> 00:49:45.000 find the potential. And, we've seen two methods for 00:49:45.000 --> 00:49:47.000 that, and I'm sure you remember them. 00:49:47.000 --> 00:49:53.000 So, if not, then try to remember them for Tuesday. 00:49:53.000 --> 00:49:54.000 OK, so anyway, again, conceptually, 00:49:54.000 --> 00:49:56.000 we have, really, three different kinds of 00:49:56.000 --> 00:49:59.000 integrals. We evaluated them in completely 00:49:59.000 --> 00:50:01.000 different ways, and we have a handful of 00:50:01.000 --> 00:50:03.000 theorems, connecting them to each other. 00:50:03.000 --> 00:50:06.000 But, that doesn't have any impact on how we actually 00:50:06.000 --> 00:50:08.000 compute things.