. 00:20:41.000 --> 00:20:46.000 And here, to be completely truthful, I have to say defined 00:20:46.000 --> 00:20:50.000 in a simply connected region. Otherwise, we might have the 00:20:50.000 --> 00:20:53.000 same kind of strange things happening as before. 00:20:53.000 --> 00:21:03.000 Let's not worry too much about it. 00:21:03.000 --> 00:21:07.000 For accuracy we need our vector field to be defined in a simply 00:21:07.000 --> 00:21:10.000 connected region. And example is just if it is 00:21:10.000 --> 00:21:14.000 defined everywhere. If you don't have any evil 00:21:14.000 --> 00:21:21.000 eliminators then you can just go ahead and there is no problem. 00:21:21.000 --> 00:21:36.000 It is a gradient field. We need three conditions. 00:21:36.000 --> 00:21:43.000 Let's do it in order. P sub y equals Q sub x. 00:21:43.000 --> 00:21:52.000 And we have P sub z equals R sub x and Q sub z equals R sub 00:21:52.000 --> 00:21:55.000 y. How do you remember these three 00:21:55.000 --> 00:21:56.000 conditions? Well, it is pretty easy. 00:21:56.000 --> 00:22:01.000 You pick any two components, say the x and the z component, 00:22:01.000 --> 00:22:04.000 and you take the partial of the x component with respect to z, 00:22:04.000 --> 00:22:07.000 the partial of the z component with respect to x and you must 00:22:07.000 --> 00:22:12.000 make them equal. And the same with every pair of 00:22:12.000 --> 00:22:15.000 variables. In fact, if you had a function 00:22:15.000 --> 00:22:17.000 of many more variables the criterion would still look 00:22:17.000 --> 00:22:21.000 exactly like that. For every pair of components 00:22:21.000 --> 00:22:24.000 the mixed partials must be the same. 00:22:24.000 --> 00:22:27.000 But we are not going to go beyond three variables so you 00:22:27.000 --> 00:22:32.000 don't need to know that. This you need to know so let me 00:22:32.000 --> 00:22:33.000 box it. 00:22:51.000 --> 00:23:02.000 That is pretty straightforward. Let's do an example just to see 00:23:02.000 --> 00:23:11.000 how it goes. By the way, we can also think 00:23:11.000 --> 00:23:19.000 of it in terms of differentials. Before I do the example, 00:23:19.000 --> 00:23:22.000 let me just say in a different language. 00:23:22.000 --> 00:23:29.000 If we have a differential given to us of a form Pdx Qdy Rdz is 00:23:29.000 --> 00:23:33.000 going to be an exact differential, 00:23:33.000 --> 00:23:40.000 which means it is equal to df for some function F exactly and 00:23:40.000 --> 00:23:43.000 of the same conditions. That is the same thing. 00:23:43.000 --> 00:23:51.000 Just in the language of differentials. 00:23:51.000 --> 00:23:57.000 The example that I promised. Of course, I could do again the 00:23:57.000 --> 00:24:00.000 same one over there and check that it satisfies the condition, 00:24:00.000 --> 00:24:02.000 but then it wouldn't be much fun. 00:24:02.000 --> 00:24:10.000 So let's do a better one. Actually, let's do it in a way 00:24:10.000 --> 00:24:18.000 that looks like an exam problem. Let's say for which a and b is 00:24:18.000 --> 00:24:29.000 a xy dx plus -- Oh, it is not going to fit here. 00:24:29.000 --> 00:24:39.000 But it will fit here. a xy dx ( x^2 z^3) dy (byz^2 - 00:24:39.000 --> 00:24:51.000 4z^3) dz, an exact differential. Or, if you don't like exact 00:24:51.000 --> 00:24:54.000 differentials, for which a and b is the 00:24:54.000 --> 00:24:58.000 corresponding vector field with i, j and k instead, 00:24:58.000 --> 00:25:02.000 a gradient field. Let's just apply the criterion. 00:25:02.000 --> 00:25:06.000 And, of course, you can guess that what will 00:25:06.000 --> 00:25:11.000 follow is figuring out how to find the potential when there is 00:25:11.000 --> 00:25:12.000 one. 00:25:33.000 --> 00:25:40.000 Let's do it one by one. We want to compare P sub y with 00:25:40.000 --> 00:25:45.000 Q sub x, we want to compare P sub z with 00:25:45.000 --> 00:25:53.000 R sub x and we want to compare Q sub z with R sub y where we call 00:25:53.000 --> 00:25:57.000 P, Q and R these guys. Let's see. 00:25:57.000 --> 00:26:04.000 What is P sub y? That seems to be ax. 00:26:04.000 --> 00:26:09.000 What is Q sub x? 2x. 00:26:09.000 --> 00:26:12.000 Q is this one. Actually, let me write them 00:26:12.000 --> 00:26:16.000 down. Because otherwise I am going to 00:26:16.000 --> 00:26:21.000 get confused myself. This guy here, 00:26:21.000 --> 00:26:32.000 that is P, this guy here, that is Q and that guy here, 00:26:32.000 --> 00:26:38.000 that is R. This one tells us that a should 00:26:38.000 --> 00:26:43.000 be equal to two of the first product that you hold. 00:26:43.000 --> 00:26:49.000 OK. Let's look at P sub z. That is just zero. 00:26:49.000 --> 00:26:51.000 R sub x? Well, R doesn't have any x 00:26:51.000 --> 00:26:56.000 either so that is zero. This one is not a problem. 00:26:56.000 --> 00:27:02.000 Q sub z? Well, that seems to be 3z2. 00:27:02.000 --> 00:27:14.000 R sub y seems to be bz2, so b should be equal to three. 00:27:14.000 --> 00:27:19.000 We need to have a equals two and, this is an and, 00:27:19.000 --> 00:27:23.000 not or, b equals three for this to be exact. 00:27:23.000 --> 00:27:27.000 For those values of a and b, we can look for a potential 00:27:27.000 --> 00:27:30.000 using the method that we are going to see right now. 00:27:30.000 --> 00:27:33.000 For any other values of a and b we cannot. 00:27:33.000 --> 00:27:38.000 If we have to compute a line integral, we have to do it by 00:27:38.000 --> 00:27:42.000 finding a parameter and setting up everything. 00:27:42.000 --> 00:27:57.000 Any questions at this point? Yes? 00:27:57.000 --> 00:28:00.000 I see. Well, if I got the same answer, 00:28:00.000 --> 00:28:05.000 oh, did say bz^2 or 3bz^2? Well, 3bz^2, for example, 00:28:05.000 --> 00:28:09.000 I would need b to be zero because the only time that 3bz2 00:28:09.000 --> 00:28:13.000 equals bz2 as not just at one point but everywhere, 00:28:13.000 --> 00:28:15.000 I need them to be the same function of x, 00:28:15.000 --> 00:28:18.000 y, z. Well, if a coefficient of z2 is 00:28:18.000 --> 00:28:22.000 the same that would be give b equals 3b, that would give me b 00:28:22.000 --> 00:28:25.000 equals zero. If you got bz2 on both sides 00:28:25.000 --> 00:28:29.000 then it would mean for any value of b it works, 00:28:29.000 --> 00:28:35.000 and you wouldn't have to worry about what the value of b is. 00:28:35.000 --> 00:28:41.000 Any other questions? No. 00:28:41.000 --> 00:28:50.000 OK. Now, how do we find the 00:28:50.000 --> 00:28:58.000 potential? Well, there are two methods as 00:28:58.000 --> 00:28:59.000 before. One of them, 00:28:59.000 --> 00:29:02.000 I don't remember if it was the first one or the second one last 00:29:02.000 --> 00:29:04.000 time, but it really doesn't matter. 00:29:04.000 --> 00:29:10.000 One of them was just to say that the value of F at the 00:29:10.000 --> 00:29:15.000 point, let me call that x1, y1, z1, 00:29:15.000 --> 00:29:22.000 is equal to the line integral of my field along a well-chosen 00:29:22.000 --> 00:29:25.000 curve plus, of course, a constant, 00:29:25.000 --> 00:29:30.000 which is going to be the integration constant. 00:29:30.000 --> 00:29:39.000 And the kind of curve that I will take to do this calculation 00:29:39.000 --> 00:29:48.000 will just be my favorite curve going from the origin to the 00:29:48.000 --> 00:29:51.000 point x1, y1, z1. 00:29:51.000 --> 00:29:57.000 And so, typically the most common choice would be to go 00:29:57.000 --> 00:30:04.000 just first along the x-axis, then parallel to the y-axis and 00:30:04.000 --> 00:30:11.000 then parallel to the z-axis all the way to my point x1, 00:30:11.000 --> 00:30:14.000 y1, z1. I would just calculate three 00:30:14.000 --> 00:30:18.000 easy line integrals. Add them together and that 00:30:18.000 --> 00:30:21.000 would give me the value of my function. 00:30:21.000 --> 00:30:27.000 That method works exactly the same way as it did in two 00:30:27.000 --> 00:30:30.000 variables. Now, I seem to recall that you 00:30:30.000 --> 00:30:32.000 guys mostly preferred the other method. 00:30:32.000 --> 00:30:34.000 I am going to tell you about the other method as well, 00:30:34.000 --> 00:30:37.000 but I just want to point out this one actually doesn't become 00:30:37.000 --> 00:30:40.000 more complicated. The other one has actually more 00:30:40.000 --> 00:30:42.000 steps. I mean, of course, 00:30:42.000 --> 00:30:45.000 here there are also a bit more steps because you have three 00:30:45.000 --> 00:30:47.000 parts to your path instead of two. 00:30:47.000 --> 00:30:54.000 You have three line integrals to compute instead of two, 00:30:54.000 --> 00:30:59.000 but conceptually it remains exactly the same idea. 00:30:59.000 --> 00:31:07.000 I should say it works the same way as in 2D. 00:31:07.000 --> 00:31:17.000 Not much changes. Let's look at the other method 00:31:17.000 --> 00:31:24.000 using anti-derivatives. Remember we want to find a 00:31:24.000 --> 00:31:29.000 function little f whose partials are exactly the things we have 00:31:29.000 --> 00:31:31.000 been given. We want to solve, 00:31:31.000 --> 00:31:36.000 well, let me plug in the values of a and b that will work. 00:31:36.000 --> 00:31:47.000 We said a should be two, so f sub x should be 2xy, 00:31:47.000 --> 00:31:59.000 f sub y should be x2 plus z3, and f sub z should be 3yz^2 00:31:59.000 --> 00:32:04.000 minus 4z^3. We are going to look at them 00:32:04.000 --> 00:32:07.000 one at a time and get partial information on the function. 00:32:07.000 --> 00:32:11.000 And then we will compare with the others to get more 00:32:11.000 --> 00:32:15.000 information until we are completely done. 00:32:15.000 --> 00:32:25.000 The first thing we will do, we know that f sub x is 2xy. 00:32:25.000 --> 00:32:27.000 That should tell us something about f. 00:32:27.000 --> 00:32:31.000 Well, let's just integrate that with respect to x. 00:32:31.000 --> 00:32:36.000 Let me write integral dx next to that. 00:32:36.000 --> 00:32:41.000 That tells us that f should be, well, if we integrate that with 00:32:41.000 --> 00:32:44.000 respect to x, 2x integrates to x^2, 00:32:44.000 --> 00:32:47.000 so we should get x2y. Plus, of course, 00:32:47.000 --> 00:32:50.000 an integration constant. Now, what do we mean by 00:32:50.000 --> 00:32:53.000 integration constant. It means that for given values 00:32:53.000 --> 00:32:58.000 of y and z we will get a term that does not depend on x. 00:32:58.000 --> 00:33:02.000 It still depends on y and z. In fact, what we get is a 00:33:02.000 --> 00:33:07.000 function of y and z. See, if you took the derivative 00:33:07.000 --> 00:33:12.000 of this with respect to x you will get 2xy and this guy will 00:33:12.000 --> 00:33:16.000 go away because there is no x in it. 00:33:16.000 --> 00:33:19.000 That is the first step. Now we need to get some 00:33:19.000 --> 00:33:21.000 information on g. How do we do that? 00:33:21.000 --> 00:33:28.000 Well, we look at the other partials. 00:33:28.000 --> 00:33:35.000 F sub y, we want that to be x^2 z^3. 00:33:35.000 --> 00:33:41.000 But we have another way to find it, which is starting from this 00:33:41.000 --> 00:33:50.000 and differentiating. Let me try to use color for 00:33:50.000 --> 00:33:54.000 this. Now, if I take the partial of 00:33:54.000 --> 00:33:58.000 this with respect to y, I am going to get a different 00:33:58.000 --> 00:34:06.000 formula for f sub y. That will be x^2 plus g sub y. 00:34:06.000 --> 00:34:15.000 Well, if I compare these two expressions that tells me that g 00:34:15.000 --> 00:34:23.000 sub y should be z3. Now, if I have this I can 00:34:23.000 --> 00:34:32.000 integrate with respect to y. That will tell me that g is 00:34:32.000 --> 00:34:39.000 actually yz^3 plus an integration constant. 00:34:39.000 --> 00:34:42.000 That constant, again, does not depend on y, 00:34:42.000 --> 00:34:46.000 but it can still depend on z because we still have not said 00:34:46.000 --> 00:34:49.000 anything about partial with respect to z. 00:34:49.000 --> 00:34:54.000 In fact, that constant I will write as a function h of z. 00:34:54.000 --> 00:35:00.000 If I have this function of z and I take its partial with 00:35:00.000 --> 00:35:04.000 respect to y, I will still get z^3 no matter 00:35:04.000 --> 00:35:06.000 what h was. Now, how do I find h? 00:35:06.000 --> 00:35:09.000 Well, obviously, I have to look at f sub z. 00:35:41.000 --> 00:35:47.000 F sub z. We know from the given vector 00:35:47.000 --> 00:35:53.000 field that we want it to be 3yz^2 minus 4z^3. 00:35:53.000 --> 00:35:56.000 In case you are wondering where that came from, 00:35:56.000 --> 00:36:01.000 that was R. But that is also obtained by 00:36:01.000 --> 00:36:09.000 differentiating with respect to z what we had so far. 00:36:09.000 --> 00:36:15.000 Sorry. What did we have so far? 00:36:15.000 --> 00:36:20.000 Well, we had f equals x^2y plus g. 00:36:20.000 --> 00:36:30.000 And we said g is actually yz^3 plus h of z. 00:36:30.000 --> 00:36:37.000 That is what we have so far. If we take the derivative of 00:36:37.000 --> 00:36:44.000 that with respect to z, we will get zero plus 3yz^2 00:36:44.000 --> 00:36:51.000 plus h prime of z, or dh dz as you want. 00:36:51.000 --> 00:36:59.000 Now, if we compare these two, we will get the derivative of 00:36:59.000 --> 00:37:03.000 h. It will tell us that h prime is 00:37:03.000 --> 00:37:08.000 negative for z3. That means that h is negative 00:37:08.000 --> 00:37:13.000 z^4 plus a constant. And this it is at last an 00:37:13.000 --> 00:37:17.000 actual constant. Because it does not depend on z 00:37:17.000 --> 00:37:21.000 and there is nothing else for it to depend on. 00:37:21.000 --> 00:37:33.000 Now we plug this into what we had before, and that will give 00:37:33.000 --> 00:37:44.000 us our function f. We get that f=x^2y yz^3 - z^4 00:37:44.000 --> 00:37:49.000 plus constant. If you just wanted to find one 00:37:49.000 --> 00:37:51.000 potential, you can just forget the constant. 00:37:51.000 --> 00:37:55.000 This guy was a potential. If you want all the potentials 00:37:55.000 --> 00:37:58.000 they differ by this constant. OK. 00:37:58.000 --> 00:38:01.000 Just to recap the method what did we do? 00:38:01.000 --> 00:38:04.000 We started with -- And, of course, you can do it in 00:38:04.000 --> 00:38:07.000 whichever order you prefer, but you have to still follow 00:38:07.000 --> 00:38:11.000 the systematic method. You start with f sub x and you 00:38:11.000 --> 00:38:13.000 integrate that with respect to x. 00:38:13.000 --> 00:38:19.000 That gives you f up to a function of y and z only. 00:38:19.000 --> 00:38:25.000 Now you compare f sub y as given to you by the vector field 00:38:25.000 --> 00:38:32.000 with the formula you get from this expression for f. 00:38:32.000 --> 00:38:36.000 And, of course, this one will involve g sub y. 00:38:36.000 --> 00:38:41.000 Out of this, you will get the value of g sub 00:38:41.000 --> 00:38:44.000 y. When you have g sub y that 00:38:44.000 --> 00:38:48.000 gives you g up to a function of z only. 00:38:48.000 --> 00:38:52.000 And so now you have f up to a function of z only. 00:38:52.000 --> 00:38:57.000 And what you will do is look at the derivative with respect to 00:38:57.000 --> 00:38:59.000 z, the one you want coming from 00:38:59.000 --> 00:39:02.000 the vector field and the one you have coming from this formula 00:39:02.000 --> 00:39:05.000 for f, match them and that will tell 00:39:05.000 --> 00:39:09.000 you h prime. You will get h and then you 00:39:09.000 --> 00:39:20.000 will get f. Any questions? 00:39:20.000 --> 00:39:25.000 Who still prefers this method? OK, still most of you. 00:39:25.000 --> 00:39:29.000 Who is thinking that maybe the other method was not so bad 00:39:29.000 --> 00:39:33.000 after all? OK. That is still a minority. 00:39:33.000 --> 00:39:36.000 You can choose whichever one you prefer. 00:39:36.000 --> 00:39:42.000 I would encourage you to get some practice by trying both on 00:39:42.000 --> 00:39:47.000 least a couple of examples just to make sure that you know how 00:39:47.000 --> 00:39:53.000 to do them both and then stick to whichever one you prefer. 00:39:53.000 --> 00:39:59.000 Any questions on that? No. I guess I already asked. 00:39:59.000 --> 00:40:07.000 Still no questions? OK. 00:40:07.000 --> 00:40:13.000 The next logical thing is going to be curl. 00:40:13.000 --> 00:40:17.000 And the theorem that is going to replace Green's theorem for 00:40:17.000 --> 00:40:21.000 work in this setting is going to be called Stokes' theorem. 00:40:21.000 --> 00:40:34.000 Let me start by telling you about curl in 3D. 00:40:34.000 --> 00:40:39.000 Here is the statement. The curl is just going to 00:40:39.000 --> 00:40:43.000 measure how much your vector field fails to be conservative. 00:40:43.000 --> 00:40:47.000 And, if you want to think about it in terms of motions, 00:40:47.000 --> 00:40:51.000 that also will measure the rotation part of the motion. 00:40:51.000 --> 00:40:55.000 Well, let me first give a definition. 00:40:55.000 --> 00:41:00.000 Let's say that my vector field has components P, 00:41:00.000 --> 00:41:06.000 Q and R. Then we define the curl of F to 00:41:06.000 --> 00:41:16.000 be R sub y minus Q sub z times i plus P sub z minus R sub x times 00:41:16.000 --> 00:41:21.000 j plus Q sub x minus P sub y times k. 00:41:21.000 --> 00:41:25.000 And of course nobody can remember this formula, 00:41:25.000 --> 00:41:28.000 so what is the structure of this formula? 00:41:28.000 --> 00:41:35.000 Well, you see, each of these guys is one of 00:41:35.000 --> 00:41:43.000 the things that have to be zero for our field to be 00:41:43.000 --> 00:41:52.000 conservative. If F is defined in a simply 00:41:52.000 --> 00:42:05.000 connected region then we have that F is conservative and is 00:42:05.000 --> 00:42:15.000 equivalent to if and only if curl F is zero. 00:42:15.000 --> 00:42:19.000 Now, an important difference between curl here and curl in 00:42:19.000 --> 00:42:23.000 the plane is that now the curl of a vector field is again a 00:42:23.000 --> 00:42:27.000 vector field. These expressions are functions 00:42:27.000 --> 00:42:31.000 of x, y, z and together you form a vector out of them. 00:42:31.000 --> 00:42:36.000 The curl of a vector field in space is actually a vector 00:42:36.000 --> 00:42:43.000 field, not a scalar function. I have delayed the inevitable. 00:42:43.000 --> 00:42:47.000 I have to really tell you how to remember this evil formula. 00:42:47.000 --> 00:42:55.000 The secret is that, in fact, you can think of this 00:42:55.000 --> 00:43:01.000 as del cross f. Maybe you have seen that in 00:43:01.000 --> 00:43:04.000 physics. This is really where this del 00:43:04.000 --> 00:43:08.000 notation becomes extremely useful, because that is 00:43:08.000 --> 00:43:13.000 basically the only way to remember the formula for curl. 00:43:30.000 --> 00:43:34.000 Remember we introduced the dell operator. 00:43:34.000 --> 00:43:42.000 That was this symbolic vector operator in which the components 00:43:42.000 --> 00:43:47.000 are the partial derivative operators. 00:43:47.000 --> 00:43:59.000 We have seen that if you apply this to a scalar function then 00:43:59.000 --> 00:44:08.000 that will give you the gradient. And we have seen that if you do 00:44:08.000 --> 00:44:13.000 the dot product between dell and a vector field, 00:44:13.000 --> 00:44:19.000 maybe I should give it components P, 00:44:19.000 --> 00:44:24.000 Q and R, you will get partial P over 00:44:24.000 --> 00:44:30.000 partial x plus partial Q over partial y plus partial R over 00:44:30.000 --> 00:44:36.000 partial z, which is the divergence. 00:44:36.000 --> 00:44:47.000 And so now what is new is that if I try to do dell cross F, 00:44:47.000 --> 00:44:53.000 well, what is dell cross F? I have to set up a 00:44:53.000 --> 00:44:58.000 cross-product between this strange thing that is not really 00:44:58.000 --> 00:45:02.000 a vector. I mean, I cannot really think 00:45:02.000 --> 00:45:05.000 of partial over partial x as a number. 00:45:05.000 --> 00:45:10.000 And my vector field

. 00:45:10.000 --> 00:45:14.000 See, that is really a completely perverted use of a 00:45:14.000 --> 00:45:18.000 determinant notation. Initially, determinants were 00:45:18.000 --> 00:45:21.000 just supposed to be you had a three by three table of numbers 00:45:21.000 --> 00:45:23.000 and you computed a number out of them. 00:45:23.000 --> 00:45:28.000 These guys are functions so they count as numbers, 00:45:28.000 --> 00:45:33.000 but these are vectors and these are partial derivatives. 00:45:33.000 --> 00:45:37.000 It doesn't really make much sense, except this notation. 00:45:37.000 --> 00:45:43.000 If you try to enter this into a calculator or computer, 00:45:43.000 --> 00:45:48.000 it will just yell back at you saying are you crazy. 00:45:48.000 --> 00:45:50.000 [LAUGHTER] We just use that as a notation 00:45:50.000 --> 00:45:53.000 to remember what is in there. Let's try and see how that 00:45:53.000 --> 00:45:55.000 works. The component of i in this 00:45:55.000 --> 00:45:58.000 cross-product, remember that is this smaller 00:45:58.000 --> 00:46:01.000 determinant, that smaller determinant is 00:46:01.000 --> 00:46:07.000 partial over partial y of R minus partial over partial z of 00:46:07.000 --> 00:46:10.000 Q, the coefficient of i. 00:46:10.000 --> 00:46:14.000 And that seems to be what I had over there. 00:46:14.000 --> 00:46:18.000 If not then I made a mistake. Minus the next determinant 00:46:18.000 --> 00:46:20.000 times z. Remember there is always a 00:46:20.000 --> 00:46:23.000 minus sign in front of a j component when you do a 00:46:23.000 --> 00:46:27.000 cross-product. The other one is partial over 00:46:27.000 --> 00:46:33.000 partial x R minus partial over partial z of P plus the 00:46:33.000 --> 00:46:39.000 component of z which is going to be partial over partial x Q 00:46:39.000 --> 00:46:46.000 minus partial over partial y P. And that is indeed going to be 00:46:46.000 --> 00:46:48.000 the curl of F. In practice, 00:46:48.000 --> 00:46:51.000 if you have to compute the curl of a vector field, 00:46:51.000 --> 00:46:53.000 you know, don't try to remember this formula. 00:46:53.000 --> 00:46:59.000 Just set up this cross-product with whatever formulas you have 00:46:59.000 --> 00:47:04.000 for the components of a field and then compute it. 00:47:04.000 --> 00:47:15.000 Don't bother to try to remember the general formula, 00:47:15.000 --> 00:47:25.000 just remember this. What is the geometric 00:47:25.000 --> 00:47:35.000 interpretation of curl, just to finish? 00:47:35.000 --> 00:47:48.000 In a way, I will say just curl measures the rotation component 00:47:48.000 --> 00:47:55.000 in a velocity field. An exercise that you can do, 00:47:55.000 --> 00:47:59.000 which is actually pretty easy to check, is say that we have a 00:47:59.000 --> 00:48:04.000 fluid that is just rotating about the x-axis uniformly. 00:48:04.000 --> 00:48:10.000 Your fluid is just rotating like that about the z-axis. 00:48:10.000 --> 00:48:15.000 If I take a rotation about the z-axis. 00:48:15.000 --> 00:48:28.000 That is given by a velocity field with components at angular 00:48:28.000 --> 00:48:35.000 velocity omega. That will be negative omega 00:48:35.000 --> 00:48:41.000 times y, then omega x and zero. And the curl of that you can 00:48:41.000 --> 00:48:45.000 compute, and you will find two omega times k. 00:48:45.000 --> 00:48:48.000 Concretely, this curl gives you the angular 00:48:48.000 --> 00:48:51.000 velocity of the rotation, well, with a factor two but 00:48:51.000 --> 00:48:53.000 that doesn't matter, and the axis of rotation, 00:48:53.000 --> 00:48:56.000 the direction of the axis of rotation. 00:48:56.000 --> 00:48:59.000 It tells you it is rotating about a vertical axis. 00:48:59.000 --> 00:49:01.000 And, in general, if you have a complicated 00:49:01.000 --> 00:49:03.000 motion some of it might be, you know, there is a 00:49:03.000 --> 00:49:05.000 translation. And then within that 00:49:05.000 --> 00:49:08.000 translation there is maybe expansion and rotation and 00:49:08.000 --> 00:49:11.000 sharing and everything. And the curl will compute how 00:49:11.000 --> 00:49:14.000 much rotation is taking place. It will tell you, 00:49:14.000 --> 00:49:16.000 say that you have a very small solid, 00:49:16.000 --> 00:49:19.000 I don't know like a ping pong ball in your flow, 00:49:19.000 --> 00:49:21.000 and it is just going with the flow, 00:49:21.000 --> 00:49:25.000 it tells you how it is going to start rotating. 00:49:25.000 --> 00:49:32.000 That is what curl measures. On Thursday we will see Stokes' 00:49:32.000 --> 00:49:37.000 theorem, which will be the last ingredient before the next exam. 00:49:37.000 --> 00:49:40.000 And then on Friday we will review stuff.