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 left things with this statement of the 00:00:29.000 --> 00:00:33.000 divergence theorem. So, the divergence theorem 00:00:33.000 --> 00:00:36.000 gives us a way to compute the flux of a vector field for a 00:00:36.000 --> 00:00:41.000 closed surface. OK, it says if I have a closed 00:00:41.000 --> 00:00:47.000 surface, s, bounding some region, D, 00:00:47.000 --> 00:00:54.000 and I have a vector field defined in space, 00:00:54.000 --> 00:00:59.000 so that I can try to compute the flux of my vector field 00:00:59.000 --> 00:01:04.000 through my surface. Double integral of F.dS or 00:01:04.000 --> 00:01:08.000 F.ndS if you want, and to set this up, 00:01:08.000 --> 00:01:11.000 of course, I need to use the geometry of the surface 00:01:11.000 --> 00:01:13.000 depending on what the surface is. 00:01:13.000 --> 00:01:17.000 We've seen various formulas for how to set up the double 00:01:17.000 --> 00:01:20.000 integral. But, we've also seen that if 00:01:20.000 --> 00:01:24.000 it's a closed surface, and if a vector field is 00:01:24.000 --> 00:01:29.000 defined everywhere inside, then we can actually reduce 00:01:29.000 --> 00:01:34.000 that to a calculation of the triple integral of the 00:01:34.000 --> 00:01:38.000 divergence of F inside, OK? 00:01:38.000 --> 00:01:40.000 So, concretely, if I use 00:01:40.000 --> 00:01:43.000 coordinates, let's say that the coordinates 00:01:43.000 --> 00:01:48.000 of my vector field are, sorry, the components are P, 00:01:48.000 --> 00:01:52.000 Q, and R dot ndS, then that will become the 00:01:52.000 --> 00:02:00.000 triple integral of, well, so, divergence is P sub x 00:02:00.000 --> 00:02:07.000 plus Q sub y plus R sub z. OK, so by the way, 00:02:07.000 --> 00:02:10.000 how to remember this formula for divergence, 00:02:10.000 --> 00:02:14.000 and other formulas for other things as well. 00:02:14.000 --> 00:02:22.000 Let me just tell you quickly about the del notation. 00:02:22.000 --> 00:02:27.000 So, this guy usually pronounced as 00:02:27.000 --> 00:02:29.000 del, rather than as pointy triangle 00:02:29.000 --> 00:02:32.000 going downwards or something like that, 00:02:32.000 --> 00:02:37.000 it's a symbolic notation for an operator. 00:02:37.000 --> 00:02:42.000 So, you're probably going to complain about putting these 00:02:42.000 --> 00:02:46.000 guys into a vector. But, let's think of partial 00:02:46.000 --> 00:02:48.000 with respect to x, with respect to y, 00:02:48.000 --> 00:02:51.000 and with respect to z as the components of some formal 00:02:51.000 --> 00:02:53.000 vector. Of course, it's not a real 00:02:53.000 --> 00:02:55.000 vector. These are not like anything. 00:02:55.000 --> 00:03:02.000 These are just symbols. But, so see for example, 00:03:02.000 --> 00:03:06.000 the gradient of function, well, if you multiply this 00:03:06.000 --> 00:03:09.000 vector by scalar, which is a function, 00:03:09.000 --> 00:03:14.000 then you will get partial, partial x of f, 00:03:14.000 --> 00:03:20.000 partial, partial y of f, partial, partial z, f, 00:03:20.000 --> 00:03:25.000 well, that's the gradient. That seems to work. 00:03:25.000 --> 00:03:29.000 So now, the interesting thing about divergence is I can think 00:03:29.000 --> 00:03:33.000 of divergence as del dot a vector field. 00:03:33.000 --> 00:03:42.000 See, if I do the dot product between this guy and my vector 00:03:42.000 --> 00:03:46.000 field P, Q, R, well, it looks like I will 00:03:46.000 --> 00:03:51.000 indeed get partial, partial x of P plus partial Q 00:03:51.000 --> 00:03:56.000 partial y plus partial R partial z. 00:03:56.000 --> 00:04:06.000 That's the divergence. and of course, similarly, 00:04:06.000 --> 00:04:08.000 when we have two variables only, x and y, 00:04:08.000 --> 00:04:11.000 we could have thought of the same notation, 00:04:11.000 --> 00:04:13.000 just with a two component vector, 00:04:13.000 --> 00:04:16.000 partial, partial x, partial, partial y. 00:04:16.000 --> 00:04:20.000 So, now, this is like of slightly limited usefulness so 00:04:20.000 --> 00:04:22.000 far. It's going to become very handy 00:04:22.000 --> 00:04:25.000 pretty soon because we are going to see curl. 00:04:25.000 --> 00:04:28.000 And, the formula for curl in the plane was kind of 00:04:28.000 --> 00:04:31.000 complicated. But, if you thought about it in 00:04:31.000 --> 00:04:34.000 terms of this, it was actually the determinant 00:04:34.000 --> 00:04:36.000 of del and f. And now, in space, 00:04:36.000 --> 00:04:39.000 we are actually going to do del cross f. 00:04:39.000 --> 00:04:40.000 But, I'm getting ahead of things. 00:04:40.000 --> 00:04:44.000 So, let's not do anything with that. 00:04:44.000 --> 00:04:52.000 Curl will be for next week. Just getting you used to the 00:04:52.000 --> 00:04:54.000 notation, especially since you might be using it in physics 00:04:54.000 --> 00:04:59.000 already. So, it might be worth doing. 00:04:59.000 --> 00:05:03.000 OK, so the other thing I wanted to say is, what does this 00:05:03.000 --> 00:05:06.000 theorem say physically? How should I think of this 00:05:06.000 --> 00:05:09.000 statement? So, I think I said that very 00:05:09.000 --> 00:05:13.000 quickly at the end of last time, but not very carefully. 00:05:13.000 --> 00:05:22.000 So, what's the physical interpretation of a divergence 00:05:22.000 --> 00:05:26.000 field? So, 00:05:26.000 --> 00:05:30.000 I want to claim that the divergence of a vector field 00:05:30.000 --> 00:05:35.000 corresponds to what I'm going to call the source rate, 00:05:35.000 --> 00:05:52.000 which is somehow the amount of flux generated per unit volume. 00:05:52.000 --> 00:05:56.000 So, to understand what that means, let's think of what's 00:05:56.000 --> 00:06:00.000 called an incompressible fluid. OK, so an incompressible fluid 00:06:00.000 --> 00:06:02.000 is something like water, for example, 00:06:02.000 --> 00:06:06.000 where a fixed mass of it always occupies the same amount of 00:06:06.000 --> 00:06:09.000 volume. So, guesses are compressible. 00:06:09.000 --> 00:06:13.000 Liquids are incompressible, basically. 00:06:13.000 --> 00:06:24.000 So, if you have an incompressible fluid flow -- -- 00:06:24.000 --> 00:06:34.000 well, so, again, what that means is really, 00:06:34.000 --> 00:06:44.000 given mass occupies always a fixed volume. 00:06:44.000 --> 00:06:51.000 Then, well, let's say that we have such a fluid with velocity 00:06:51.000 --> 00:06:57.000 given by our vector field. OK, so we're thinking of F as 00:06:57.000 --> 00:07:03.000 the velocity and maybe something containing water, 00:07:03.000 --> 00:07:08.000 a pipe, or something. So, what does the divergence 00:07:08.000 --> 00:07:14.000 theorem say? It says that if I take a region 00:07:14.000 --> 00:07:18.000 in space, let's call it D, 00:07:18.000 --> 00:07:23.000 sorry, D is the inside, and S is the surface around it, 00:07:23.000 --> 00:07:27.000 well, so if I sum the divergence in D, 00:07:27.000 --> 00:07:35.000 well, I'm going to get the flux going out through this surface, 00:07:35.000 --> 00:07:37.000 S. I should have mentioned it 00:07:37.000 --> 00:07:39.000 earlier. The convention in the 00:07:39.000 --> 00:07:43.000 divergence theorem is that we orient the surface with a normal 00:07:43.000 --> 00:07:47.000 vector pointing always outwards. OK, so now, we know what flux 00:07:47.000 --> 00:07:49.000 means. Remember, we've been 00:07:49.000 --> 00:07:53.000 describing, flux means how much fluid is passing through this 00:07:53.000 --> 00:08:00.000 surface. So, that's the amount of fluid 00:08:00.000 --> 00:08:11.000 that's leaving the region, D, per unit time. 00:08:11.000 --> 00:08:13.000 And, of course, when I'm saying that, 00:08:13.000 --> 00:08:16.000 it means I'm counting everything that's going out of D 00:08:16.000 --> 00:08:18.000 minus everything that's coming into D. 00:08:18.000 --> 00:08:22.000 That's what the flux measures. So, now, if there is stuff 00:08:22.000 --> 00:08:26.000 coming into D or going out of D, well, it must come from 00:08:26.000 --> 00:08:28.000 somewhere. So, one possibility would be 00:08:28.000 --> 00:08:32.000 that your fluid is actually being compressed or expanded. 00:08:32.000 --> 00:08:34.000 But, I've said, no, I'm looking at something 00:08:34.000 --> 00:08:37.000 like water that you cannot squish into smaller volume. 00:08:37.000 --> 00:08:40.000 So, in that case, the only explanation is that 00:08:40.000 --> 00:08:44.000 there is something it here that actually is sucking up water or 00:08:44.000 --> 00:08:47.000 producing more water. And so, integrating the 00:08:47.000 --> 00:08:52.000 divergence gives you the total amount of sources minus the 00:08:52.000 --> 00:08:56.000 amount of syncs that are inside this region. 00:08:56.000 --> 00:09:01.000 So, the divergence itself measures basically the amount of 00:09:01.000 --> 00:09:06.000 sources or syncs per unit volume in a given place. 00:09:06.000 --> 00:09:07.000 And now, if you think about it that way, 00:09:07.000 --> 00:09:12.000 well, it's basically the divergence 00:09:12.000 --> 00:09:17.000 theorem is just stating something completely obvious 00:09:17.000 --> 00:09:23.000 about all the matter that is leaving this region must come 00:09:23.000 --> 00:09:28.000 from somewhere. So, that's basically how we 00:09:28.000 --> 00:09:30.000 think about it. Now, of course, 00:09:30.000 --> 00:09:33.000 if you're doing 8.02, then you might actually have 00:09:33.000 --> 00:09:35.000 seen the divergence theorem already being used for things 00:09:35.000 --> 00:09:39.000 that are more like force fields, say, electric fields and so on. 00:09:39.000 --> 00:09:42.000 Well, I'll try to say a few things about that during the 00:09:42.000 --> 00:09:45.000 last week of classes. But, then this kind of 00:09:45.000 --> 00:09:48.000 interpretation doesn't quite work. 00:09:48.000 --> 00:09:51.000 OK, any questions, generally speaking, 00:09:51.000 --> 00:09:56.000 before we move on to the proof and other applications? 00:09:56.000 --> 00:10:05.000 Yes? Oh, not the gradient. 00:10:05.000 --> 00:10:09.000 So, yeah, the divergence of F measures the amount of sources 00:10:09.000 --> 00:10:11.000 or syncs in there. Well, what makes it happen? 00:10:11.000 --> 00:10:13.000 If you want, in a way, it's this theorem. 00:10:13.000 --> 00:10:16.000 Or, in another way, if you think about it, 00:10:16.000 --> 00:10:20.000 try to look at your favorite vector fields and compute their 00:10:20.000 --> 00:10:23.000 divergence. And, if you take a vector field 00:10:23.000 --> 00:10:25.000 where maybe everything is rotating, 00:10:25.000 --> 00:10:29.000 a flow that's just rotating about some axis, 00:10:29.000 --> 00:10:31.000 then you'll find that its divergence is zero. 00:10:31.000 --> 00:10:37.000 If you, sorry? No, divergence is not equal to 00:10:37.000 --> 00:10:39.000 the gradient. Sorry, there's a dot here that 00:10:39.000 --> 00:10:42.000 maybe is not very big, but it's very important. 00:10:42.000 --> 00:10:44.000 OK, so you take the divergence of a vector field. 00:10:44.000 --> 00:10:46.000 Well, you take the gradient of a function. 00:10:46.000 --> 00:10:49.000 So, if the gradient of a function is a vector, 00:10:49.000 --> 00:10:52.000 the divergence of a vector field is a function. 00:10:52.000 --> 00:10:56.000 So, somehow these guys go back and forth between. 00:10:56.000 --> 00:10:59.000 So, I should have said, with new notations comes new 00:10:59.000 --> 00:11:04.000 responsibility. I mean, 00:11:04.000 --> 00:11:07.000 now that we have this nice, nifty notation that will let us 00:11:07.000 --> 00:11:10.000 do gradient divergence and later curl in a unified way, 00:11:10.000 --> 00:11:12.000 if you choose this notation you have to be really, 00:11:12.000 --> 00:11:17.000 really careful what you put after it because otherwise it's 00:11:17.000 --> 00:11:21.000 easy to get completely confused. OK, so divergence and gradients 00:11:21.000 --> 00:11:24.000 are completely different things. The only thing they have in 00:11:24.000 --> 00:11:26.000 common is that both are what's called a first order 00:11:26.000 --> 00:11:29.000 differential operator. That means it involves the 00:11:29.000 --> 00:11:33.000 first partial derivatives of whatever you put into it. 00:11:33.000 --> 00:11:35.000 But, one of them goes from functions to vectors. 00:11:35.000 --> 00:11:38.000 That's gradient. The other one goes from vectors 00:11:38.000 --> 00:11:41.000 to functions. That's divergence. 00:11:41.000 --> 00:11:43.000 And, curl later will go from vectors to vectors. 00:11:43.000 --> 00:11:57.000 But, that will be later. Let's see, more questions? 00:11:57.000 --> 00:12:03.000 No? OK, so let's see, 00:12:03.000 --> 00:12:12.000 so how are we going to actually prove this theorem? 00:12:12.000 --> 00:12:15.000 Well, if you remember how we prove Green's theorem a while 00:12:15.000 --> 00:12:18.000 ago, the answer is we're going to do it exactly the same way. 00:12:18.000 --> 00:12:22.000 So, if you don't remember, then I'm going to explain. 00:12:22.000 --> 00:12:24.000 OK, so the first thing we need to do is actually a 00:12:24.000 --> 00:12:28.000 simplification. So, instead of proving the 00:12:28.000 --> 00:12:33.000 divergence theorem, namely, the equality up there, 00:12:33.000 --> 00:12:38.000 I'm going to actually prove something easier. 00:12:38.000 --> 00:12:44.000 I'm going to prove that the flux of a vector field that has 00:12:44.000 --> 00:12:52.000 only a z component is actually equal to the triple integral of, 00:12:52.000 --> 00:12:58.000 well, the divergence of this is just R sub z dV. 00:12:58.000 --> 00:13:00.000 OK, now, how do I go back to the general case? 00:13:00.000 --> 00:13:03.000 Well, I will just prove the same thing for a vector field 00:13:03.000 --> 00:13:07.000 that has only an x component or only a y component. 00:13:07.000 --> 00:13:10.000 And then, I will add these things together. 00:13:10.000 --> 00:13:12.000 So, if you think carefully about what happens when you 00:13:12.000 --> 00:13:15.000 evaluate this, you will have some formula for 00:13:15.000 --> 00:13:16.000 ndS, and when you do the dot 00:13:16.000 --> 00:13:18.000 product, you'll end up with the sum, 00:13:18.000 --> 00:13:21.000 P times something plus Q times something plus R times 00:13:21.000 --> 00:13:22.000 something. And basically, 00:13:22.000 --> 00:13:26.000 we are just dealing with the last term, R times something, 00:13:26.000 --> 00:13:28.000 and showing that it's equal to what it should be. 00:13:28.000 --> 00:13:30.000 And then, we the three such terms together. 00:13:30.000 --> 00:13:44.000 We'll get the general case. OK, so then we get the general 00:13:44.000 --> 00:14:01.000 case by summing one such identity for each component. 00:14:01.000 --> 00:14:08.000 I should say three such identities, one for each 00:14:08.000 --> 00:14:13.000 component, whatever. Now, let's make a second 00:14:13.000 --> 00:14:17.000 simplification because I'm still not feeling confident I can 00:14:17.000 --> 00:14:19.000 prove this right away for any surface. 00:14:19.000 --> 00:14:23.000 I'm going to do it first or what's called a vertically 00:14:23.000 --> 00:14:26.000 simple region. OK, so vertically simple means 00:14:26.000 --> 00:14:30.000 it will be something which I can setup an integral over the z 00:14:30.000 --> 00:14:36.000 variable first easily. So, it's something that has a 00:14:36.000 --> 00:14:44.000 bottom face, and a top face, and then some vertical sides. 00:14:44.000 --> 00:14:53.000 OK, so let's say first what happens if the given region, 00:14:53.000 --> 00:15:02.000 D, is vertically simple. So, vertically simple means it 00:15:02.000 --> 00:15:09.000 looks like this. It has top. 00:15:09.000 --> 00:15:16.000 It has a bottom. And, it has some vertical sides. 00:15:16.000 --> 00:15:20.000 So, if you want, if I look at it from above, 00:15:20.000 --> 00:15:25.000 it projects to some region in the xy plane. 00:15:25.000 --> 00:15:30.000 Let's call that R. And, it lives between the top 00:15:30.000 --> 00:15:34.000 face and the bottom face. Let's say the top face is z 00:15:34.000 --> 00:15:37.000 equals z2 of (x, y). 00:15:37.000 --> 00:15:42.000 Let's say the bottom face is z equals z1(x, y). 00:15:42.000 --> 00:15:44.000 OK, and I don't need to know actual formulas. 00:15:44.000 --> 00:15:47.000 I'm just going to work with these and prove things 00:15:47.000 --> 00:15:50.000 independently of what the formulas will be for these 00:15:50.000 --> 00:15:52.000 functions. OK, so anyway, 00:15:52.000 --> 00:15:56.000 a vertically simple region is something that lives above a 00:15:56.000 --> 00:15:59.000 part of the xy plane, and is between two graphs of 00:15:59.000 --> 00:16:03.000 two functions. So, let's see what we can do in 00:16:03.000 --> 00:16:10.000 that case. So, the right-hand side of this 00:16:10.000 --> 00:16:20.000 equality, so that's the triple integral, let's start computing 00:16:20.000 --> 00:16:23.000 it. OK, so of course we will not be 00:16:23.000 --> 00:16:26.000 able to get a number out of it because we don't know, 00:16:26.000 --> 00:16:28.000 actually, formulas for anything. 00:16:28.000 --> 00:16:32.000 But at least we can start simplifying because the way this 00:16:32.000 --> 00:16:36.000 region looks like, I should say this is D, 00:16:36.000 --> 00:16:40.000 tells me that I can start setting up the triple integral 00:16:40.000 --> 00:16:45.000 at least in the order where I integrate first over z. 00:16:45.000 --> 00:16:53.000 OK, so I can actually do it as a triple integral with Rz dz 00:16:53.000 --> 00:16:57.000 dxdy or dydx, doesn't matter. 00:16:57.000 --> 00:17:01.000 So, what are the bounds on z? See, this is actually good 00:17:01.000 --> 00:17:04.000 practice to remember how we set up triple integrals. 00:17:04.000 --> 00:17:06.000 So, remember, when we did it first over z, 00:17:06.000 --> 00:17:09.000 we start by fixing a point, x and y, 00:17:09.000 --> 00:17:12.000 and for that value of x and y, we look at a small vertical 00:17:12.000 --> 00:17:16.000 slice and see from where to where we have to go. 00:17:16.000 --> 00:17:21.000 Well, we start at z equals whatever the value is at the 00:17:21.000 --> 00:17:28.000 bottom, so, z1 of x and y. And, we go up to the top face, 00:17:28.000 --> 00:17:32.000 z2 of x and y. Now, for x and y, 00:17:32.000 --> 00:17:37.000 I'm not going to actually set up bounds because I've already 00:17:37.000 --> 00:17:41.000 called R the quantity that I'm integrating. 00:17:41.000 --> 00:17:45.000 So let me change this to, let's say, U or something like 00:17:45.000 --> 00:17:47.000 that. If you already have an R, 00:17:47.000 --> 00:17:49.000 I mean, there's not much risk for confusion, 00:17:49.000 --> 00:17:53.000 but still. OK, so we're going to call U 00:17:53.000 --> 00:17:59.000 the shadow of my region instead. So, now I want to integrate 00:17:59.000 --> 00:18:01.000 over all values of x and y that are in the shadow of my region. 00:18:01.000 --> 00:18:04.000 That means it's a double integral over this region, 00:18:04.000 --> 00:18:06.000 U, which I haven't described to you. 00:18:06.000 --> 00:18:09.000 So, I can't actually set up bounds for x and y. 00:18:09.000 --> 00:18:12.000 But, I'm going to just leave it like this. 00:18:12.000 --> 00:18:16.000 OK, now you see, 00:18:16.000 --> 00:18:19.000 if you look at how you would start evaluating this, 00:18:19.000 --> 00:18:22.000 well, the inner integral certainly is not scary because 00:18:22.000 --> 00:18:25.000 you're integrating the derivative of R with respect to 00:18:25.000 --> 00:18:27.000 z, integrating that with respect 00:18:27.000 --> 00:18:33.000 to z. So, you should get R back. 00:18:33.000 --> 00:18:39.000 OK, so triple integral over D of Rz dV becomes, 00:18:39.000 --> 00:18:42.000 well, we'll have a double integral over U of, 00:18:42.000 --> 00:18:49.000 so, the inner integral becomes R at the point on the top. 00:18:49.000 --> 00:18:53.000 So, that means, remember, R is a function of x, 00:18:53.000 --> 00:18:56.000 y, and z. And, in fact, 00:18:56.000 --> 00:19:03.000 I will plug into it the value of z at the top, 00:19:03.000 --> 00:19:13.000 so, z of xy minus the value of R at the point on the bottom, 00:19:13.000 --> 00:19:16.000 x, y, z1 of x, y. 00:19:16.000 --> 00:19:26.000 OK, any questions about this? No? 00:19:26.000 --> 00:19:29.000 Is it looking vaguely believable? 00:19:29.000 --> 00:19:32.000 Yeah? OK. So, now, let's compute the 00:19:32.000 --> 00:19:34.000 other side because here we are stuck. 00:19:34.000 --> 00:19:36.000 We won't be able to do anything else. 00:19:36.000 --> 00:19:39.000 So, let's look at the flux integral. 00:19:39.000 --> 00:19:43.000 OK, we have to look at the flux of this vector field through the 00:19:43.000 --> 00:19:46.000 entire surface, S, which is the whole boundary 00:19:46.000 --> 00:19:51.000 of D. So, that consists of a lot of 00:19:51.000 --> 00:19:56.000 pieces, namely the top, bottom, and the sides. 00:19:56.000 --> 00:20:04.000 OK, so the other side -- So, let me just remind you, 00:20:04.000 --> 00:20:12.000 S is bottom plus top plus side of this vector field, 00:20:12.000 --> 00:20:19.000 dot ndS equals, OK, so what do we have? 00:20:19.000 --> 00:20:21.000 So first, we have to look at the bottom. 00:20:21.000 --> 00:20:23.000 No, let's start with the top actually. 00:20:23.000 --> 00:20:35.000 Sorry. OK, so let's start with the top. 00:20:35.000 --> 00:20:43.000 So, just remind you, let's do all of them. 00:20:43.000 --> 00:20:50.000 So, let's look at the top first. So, we need to set up the flux 00:20:50.000 --> 00:20:52.000 integral for a vector field dot ndS. 00:20:52.000 --> 00:20:56.000 We need to know what ndS is. Well, fortunately for us, 00:20:56.000 --> 00:20:59.000 we know that the top face is going to be the graph of some 00:20:59.000 --> 00:21:03.000 function of x and y. So, we've seen a formula for 00:21:03.000 --> 00:21:06.000 ndS in this kind of situation, OK? 00:21:06.000 --> 00:21:11.000 We have seen that ndS, sorry, so, just to remind you 00:21:11.000 --> 00:21:16.000 this is the graph of a function z equals z2 of x, 00:21:16.000 --> 00:21:21.000 y. So, we've seen ndS for that is 00:21:21.000 --> 00:21:30.000 negative partial derivative of this function with respect to x, 00:21:30.000 --> 00:21:35.000 negative partial z2 with respect to y, 00:21:35.000 --> 00:21:38.000 one, dxdy. OK, and, well, 00:21:38.000 --> 00:21:44.000 we can't compute these guys, but it's not a big deal because 00:21:44.000 --> 00:21:47.000 if we do the dot product with 00:21:48.000 --> 00:21:51.000 dot ndS, that will give us, 00:21:51.000 --> 00:21:53.000 well, if you dot this with zero, zero, R, 00:21:53.000 --> 00:22:03.000 these terms go away. You just have R dxdy. 00:22:03.000 --> 00:22:11.000 So, that means that the double integral for flux through the 00:22:11.000 --> 00:22:19.000 top of R vector field dot ndS becomes double integral of the 00:22:19.000 --> 00:22:23.000 top of R dxdy. Now, how do we evaluate that, 00:22:23.000 --> 00:22:28.000 actually? Well, so R is a function of x, 00:22:28.000 --> 00:22:29.000 y, z. But we said, 00:22:29.000 --> 00:22:32.000 we have only two variables that we're going to use. 00:22:32.000 --> 00:22:35.000 We're going to use x and y. We're going to get rid of z. 00:22:35.000 --> 00:22:38.000 How do we get rid of z? Well, if we are on the top 00:22:38.000 --> 00:22:41.000 surface, z is given by this formula, z2 of x, 00:22:41.000 --> 00:22:45.000 y. So, I plug z equals z2 of x, 00:22:45.000 --> 00:22:50.000 y into the formula for R, whatever it may be. 00:22:50.000 --> 00:22:54.000 Then, I integrate dxdy. And, what's the range for x and 00:22:54.000 --> 00:22:57.000 y? Well, my surface sits exactly 00:22:57.000 --> 00:23:01.000 above this region U in the xy plane. 00:23:01.000 --> 00:23:08.000 So, it's double integral over U, OK? 00:23:08.000 --> 00:23:17.000 Any questions about how I set up this flux integral? 00:23:17.000 --> 00:23:21.000 No? OK, let me close the door, 00:23:21.000 --> 00:23:26.000 actually. OK, so we've got one of the two 00:23:26.000 --> 00:23:31.000 terms that we had over there. Let's try to get the others. 00:23:44.000 --> 00:23:49.000 [LAUGHTER] No comment. OK, so, we need to look, 00:23:49.000 --> 00:23:56.000 also, at the other parts of our surface for the flux integral. 00:23:56.000 --> 00:24:00.000 So, the bottom, well, it will work pretty much 00:24:00.000 --> 00:24:03.000 the same way, right, because it's the graph 00:24:03.000 --> 00:24:06.000 of a function, z equals z1 of x, 00:24:06.000 --> 00:24:10.000 y. So, we should be able to get 00:24:10.000 --> 00:24:17.000 ndS using the same method, negative partial with respect 00:24:17.000 --> 00:24:23.000 to x, negative partial with respect to y, 00:24:23.000 --> 00:24:26.000 one dxdy. Now, there's a small catch. 00:24:26.000 --> 00:24:30.000 OK, we have to think of it carefully about orientations. 00:24:30.000 --> 00:24:34.000 So, remember, when we set up the 00:24:34.000 --> 00:24:38.000 divergence theorem, we need the normal vectors to 00:24:38.000 --> 00:24:42.000 point out of our region, which means that on the top 00:24:42.000 --> 00:24:46.000 surface, we want n pointing up. 00:24:46.000 --> 00:24:50.000 But, on the bottom face, we want n pointing down. 00:24:50.000 --> 00:24:52.000 So, in fact, we shouldn't use this formula 00:24:52.000 --> 00:24:55.000 here because that one corresponds to the other 00:24:55.000 --> 00:24:58.000 orientation. Well, we could use it and then 00:24:58.000 --> 00:25:02.000 subtract, but I was just going to say that ndS is actually the 00:25:02.000 --> 00:25:06.000 opposite of this. So, I'm going to switch all my 00:25:06.000 --> 00:25:09.000 signs. OK, that's the other side of 00:25:09.000 --> 00:25:13.000 the formula when you orient your graph with n pointing downwards. 00:25:13.000 --> 00:25:18.000 Now, if I do things the same way as before, 00:25:18.000 --> 00:25:24.000 I will get that <0,0, R> dot ndS will be negative 00:25:24.000 --> 00:25:27.000 R dxdy. And so, 00:25:27.000 --> 00:25:34.000 when I do the double integral over the bottom, 00:25:34.000 --> 00:25:39.000 I'm going to get the double integral over the bottom of 00:25:39.000 --> 00:25:42.000 negative R dxdy, which, if I try to evaluate 00:25:42.000 --> 00:25:46.000 that, well, I actually will have to 00:25:46.000 --> 00:25:48.000 integrate. Sorry, first I'll have to 00:25:48.000 --> 00:25:53.000 substitute the value of z. The value of z that I will want 00:25:53.000 --> 00:25:57.000 to plug into R will be given by, now, z1 of x, 00:25:57.000 --> 00:26:00.000 y. And, the bounds of integration 00:26:00.000 --> 00:26:04.000 will be given, again, by the shadow of our 00:26:04.000 --> 00:26:07.000 surface, which is, again, this guy, 00:26:07.000 --> 00:26:09.000 U. OK, so we seem to be all set, 00:26:09.000 --> 00:26:12.000 except we are still missing one part of our surface, 00:26:12.000 --> 00:26:14.000 S, because we also need to look at the sides. 00:26:14.000 --> 00:26:20.000 Well, what about the sides? Well, our vector field, 00:26:20.000 --> 00:26:23.000 , is actually vertical. 00:26:23.000 --> 00:26:29.000 It's parallel to the z axis. OK, so my vector field does 00:26:29.000 --> 00:26:35.000 something like this everywhere. And, why that makes it very 00:26:35.000 --> 00:26:38.000 interesting on the top and bottom, that means that on the 00:26:38.000 --> 00:26:40.000 sides, really not much is going on. 00:26:40.000 --> 00:26:45.000 No matter is passing through the vertical sides. 00:26:45.000 --> 00:26:57.000 So, the side -- The sides are vertical. 00:26:57.000 --> 00:27:05.000 So, <0,0, R> is tangent to the side, 00:27:05.000 --> 00:27:14.000 and therefore, the flux through the sides is 00:27:14.000 --> 00:27:23.000 just going to be zero. OK, no calculation needed. 00:27:23.000 --> 00:27:26.000 That's because, of course, that's the reason why a 00:27:26.000 --> 00:27:31.000 simplified first things so that my vector field would only have 00:27:31.000 --> 00:27:35.000 a z component, well, not just that but that's 00:27:35.000 --> 00:27:39.000 the main place where it becomes very useful. 00:27:39.000 --> 00:27:42.000 So, now, if I compare my double integral and, 00:27:42.000 --> 00:27:45.000 sorry, my triple integral and my flux integral, 00:27:45.000 --> 00:27:47.000 I get that they are, indeed, the same. 00:28:03.000 --> 00:28:05.000 Well, that's the statement of the theorem we are trying to 00:28:05.000 --> 00:28:17.000 prove. I shouldn't erase it, OK? 00:28:17.000 --> 00:28:22.000 [LAUGHTER] So, just to recap, 00:28:22.000 --> 00:28:32.000 we've got a formula for the triple integral of R sub z dV. 00:28:32.000 --> 00:28:36.000 It's up there at the very top. And, we got formulas for the 00:28:36.000 --> 00:28:39.000 flux through the top and the bottom, and the sides. 00:28:39.000 --> 00:28:41.000 And, when you add them together, 00:28:41.000 --> 00:28:47.000 you get indeed the same formula, 00:28:47.000 --> 00:29:03.000 top plus bottom -- -- plus sides of, 00:29:03.000 --> 00:29:08.000 OK, and so we have, actually, completed the proof for this 00:29:08.000 --> 00:29:11.000 part. Now, well, that's only for a 00:29:11.000 --> 00:29:14.000 vertically simple region, OK? 00:29:14.000 --> 00:29:24.000 So, if D is not vertically simple, what do we do? 00:29:24.000 --> 00:29:39.000 Well, we cut it into vertically simple pieces. 00:29:39.000 --> 00:29:44.000 OK so, concretely, I wanted to integrate over a 00:29:44.000 --> 00:29:48.000 solid doughnut. Then, that's not vertically 00:29:48.000 --> 00:29:52.000 simple because here in the middle, I have not only does top 00:29:52.000 --> 00:29:56.000 in this bottom, but I have this middle face. 00:29:56.000 --> 00:29:59.000 So, the way I would cut my doughnut would be probably I 00:29:59.000 --> 00:30:03.000 would slice it not in the way that you'd usually slice the 00:30:03.000 --> 00:30:06.000 doughnut or a bagel, but at it's probably more 00:30:06.000 --> 00:30:09.000 spectacular if you think that it's a bagel. 00:30:09.000 --> 00:30:15.000 Then, a mathematician's way of slicing it is like this into 00:30:15.000 --> 00:30:17.000 five pieces, OK? And, that's not very convenient 00:30:17.000 --> 00:30:20.000 for eating, but that's convenient for 00:30:20.000 --> 00:30:24.000 integrating over it because now each of these pieces has a 00:30:24.000 --> 00:30:26.000 well-defined top and bottom face, 00:30:26.000 --> 00:30:32.000 and of course you've introduced some vertical sides for two 00:30:32.000 --> 00:30:35.000 reasons. One is that we've said the flux 00:30:35.000 --> 00:30:40.000 through them is zero anyway. So, who cares? 00:30:40.000 --> 00:30:43.000 Why bother? But, also, if you sum the flux 00:30:43.000 --> 00:30:47.000 through the surface of each little piece, 00:30:47.000 --> 00:30:50.000 well, you will see that you will be integrating twice over 00:30:50.000 --> 00:30:52.000 each of these vertical cuts. Once, when you do this piece, 00:30:52.000 --> 00:30:56.000 you will be taking the flux through this red guy with normal 00:30:56.000 --> 00:31:00.000 vector pointing to the right, and once, when you take this 00:31:00.000 --> 00:31:03.000 middle little piece, you will be taking the flux 00:31:03.000 --> 00:31:07.000 through that cut surface again, but now with normal vector 00:31:07.000 --> 00:31:09.000 pointing the other way around. So, in fact, 00:31:09.000 --> 00:31:12.000 you'll be summing the flux through these guys twice with 00:31:12.000 --> 00:31:15.000 opposite orientations. They cancel out. 00:31:15.000 --> 00:31:18.000 Well, and again, because of what you are doing 00:31:18.000 --> 00:31:20.000 actually, the integral was just zero anyway. 00:31:20.000 --> 00:31:25.000 So, it didn't matter. But, even if it hadn't 00:31:25.000 --> 00:31:30.000 simplified, that would do it for us. 00:31:30.000 --> 00:31:32.000 OK, so that's how we do it with the general region. 00:31:32.000 --> 00:31:34.000 And then, as I said at the beginning, 00:31:34.000 --> 00:31:37.000 when we can do it for a vector field that has only a z 00:31:37.000 --> 00:31:39.000 component, well, we can also do it for a 00:31:39.000 --> 00:31:42.000 vector field that has only an x or only a y component. 00:31:42.000 --> 00:31:45.000 And then, we sum together and we get the general case. 00:31:45.000 --> 00:31:52.000 So, that's the end of the proof. OK, so you see, 00:31:52.000 --> 00:31:55.000 the idea is really the same as for Green's theorem. 00:31:55.000 --> 00:32:00.000 Yes? Oh, there's only four pieces, 00:32:00.000 --> 00:32:05.000 thank you. Yes, there's three kinds of 00:32:05.000 --> 00:32:13.000 mathematicians: those who know how to count, 00:32:13.000 --> 00:32:30.000 and those who don't. Well, OK. 00:32:30.000 --> 00:32:34.000 So, OK, now I hope that you can see already the interest of this 00:32:34.000 --> 00:32:38.000 theorem for the divergence theorem for computing flux 00:32:38.000 --> 00:32:42.000 integrals just for the sake of computing flux integrals like 00:32:42.000 --> 00:32:46.000 might happen on the problem set or on the next test. 00:32:46.000 --> 00:32:49.000 But let me tell you also why it's important physically to 00:32:49.000 --> 00:32:54.000 understand equations that had been observed empirically well 00:32:54.000 --> 00:32:57.000 before they were actually understood in terms of how 00:32:57.000 --> 00:33:03.000 things go. So, let's look at something 00:33:03.000 --> 00:33:10.000 called the diffusion equation. So, let me explain to you what 00:33:10.000 --> 00:33:13.000 it does. So, the diffusion equation is 00:33:13.000 --> 00:33:16.000 something that governs, well, what's called diffusion. 00:33:16.000 --> 00:33:19.000 Diffusion is when you have a fluid in which you are 00:33:19.000 --> 00:33:24.000 introducing some substance, and you want to figure out how 00:33:24.000 --> 00:33:27.000 that thing is going to spread out, 00:33:27.000 --> 00:33:30.000 the technical term is diffuse, into the ambient fluid. 00:33:30.000 --> 00:33:36.000 So, for example, that governs the motion of, 00:33:36.000 --> 00:33:43.000 say, smoke in the air, or if you put dye in the 00:33:43.000 --> 00:33:49.000 solution or things like that. That will tell you, 00:33:49.000 --> 00:33:53.000 say that you drop some ink into a glass of water. 00:33:53.000 --> 00:33:57.000 Well, you can imagine that obviously it will get diluted 00:33:57.000 --> 00:33:59.000 into there. And, that equation will tell 00:33:59.000 --> 00:34:04.000 you how exactly over time this thing is going to spread out and 00:34:04.000 --> 00:34:09.000 start filling the entire glass. So, what's the equation? 00:34:09.000 --> 00:34:12.000 Well, we need, first, to know what the unknown 00:34:12.000 --> 00:34:13.000 will be. So, it's a partial differential 00:34:13.000 --> 00:34:16.000 equation, OK? So the unknown is a function, 00:34:16.000 --> 00:34:20.000 and the equation will relate the partial derivatives of that 00:34:20.000 --> 00:34:26.000 function to each other. So, u, the unknown, 00:34:26.000 --> 00:34:36.000 will be the concentration at a given point. 00:34:36.000 --> 00:34:38.000 And, of course, that depends on the point where 00:34:38.000 --> 00:34:40.000 you are. So, that depends on x, 00:34:40.000 --> 00:34:42.000 y, z, the location where you are. 00:34:42.000 --> 00:34:45.000 But, since the goal is also to understand how things spread 00:34:45.000 --> 00:34:47.000 over time, it should also depend on time. 00:34:47.000 --> 00:34:51.000 Otherwise, we're not going to get very far. 00:34:51.000 --> 00:34:53.000 And, in fact, what the equation will give us 00:34:53.000 --> 00:34:55.000 is the derivative of u with respect to t. 00:34:55.000 --> 00:34:59.000 It will tell us how the concentration at a given point 00:34:59.000 --> 00:35:03.000 varies over time in terms of how the concentration varied in 00:35:03.000 --> 00:35:06.000 space. So, it will relate partial u 00:35:06.000 --> 00:35:10.000 partial t to partial derivatives with respect to x, 00:35:10.000 --> 00:35:11.000 y, and z. 00:35:42.000 --> 00:35:43.000 [APPLAUSE] OK, [LAUGHTER] 00:35:43.000 --> 00:35:48.000 so what's the equation? The equation is partial u 00:35:48.000 --> 00:35:55.000 partial t equals some constant. Let me call it constant k times 00:35:55.000 --> 00:36:01.000 something I will call del squared u, which is also called 00:36:01.000 --> 00:36:05.000 the Laplacian of u, and what is that? 00:36:05.000 --> 00:36:09.000 Well, that means, 00:36:09.000 --> 00:36:14.000 OK, so just to scare you, del squared is this, 00:36:14.000 --> 00:36:20.000 which means it's the divergence of gradient u that we've used 00:36:20.000 --> 00:36:25.000 this notation for gradient. OK, so if you actually expand 00:36:25.000 --> 00:36:29.000 it in terms of variables, that becomes partial u over 00:36:29.000 --> 00:36:35.000 partial x squared plus partial squared u over partial y squared 00:36:35.000 --> 00:36:40.000 plus partial squared u over partial z squared. 00:36:40.000 --> 00:36:48.000 OK, so the equation is this equals that. 00:36:48.000 --> 00:36:51.000 OK, so that's a weird looking equation. 00:36:51.000 --> 00:36:54.000 And, I'm going to have to explain to you, 00:36:54.000 --> 00:36:57.000 where does it come from? OK, but before I do that, 00:36:57.000 --> 00:37:02.000 well, let me point out actually that the equation is not just 00:37:02.000 --> 00:37:10.000 the diffusion equation. It's also known as the heat 00:37:10.000 --> 00:37:15.000 equation. And, that's because exactly the 00:37:15.000 --> 00:37:21.000 same equation governs how temperature changes over time 00:37:21.000 --> 00:37:25.000 when you have, again, so, sorry I should have 00:37:25.000 --> 00:37:28.000 been actually more careful. I should have said this is in 00:37:28.000 --> 00:37:31.000 air that's not moving, OK? 00:37:31.000 --> 00:37:32.000 OK, and same thing with the solution. 00:37:32.000 --> 00:37:35.000 If you drop some ink into your glass of water, 00:37:35.000 --> 00:37:38.000 well, if you start stirring, obviously it will mix much 00:37:38.000 --> 00:37:40.000 faster than if you don't do anything. 00:37:40.000 --> 00:37:43.000 OK, so that's the case where we don't actually, 00:37:43.000 --> 00:37:47.000 the fluid is not moving. And, the heat equation now does 00:37:47.000 --> 00:37:51.000 the same, but for temperature in a fluid that's at rest, 00:37:51.000 --> 00:37:55.000 that's not moving. It tells you how the heat goes 00:37:55.000 --> 00:37:58.000 from the warmest parts to the coldest parts, 00:37:58.000 --> 00:38:03.000 and eventually temperatures should somehow even out. 00:38:03.000 --> 00:38:08.000 So, in the heat equation, that would just be, 00:38:08.000 --> 00:38:15.000 this u would just measure the temperature for temperature of 00:38:15.000 --> 00:38:19.000 your fluid at a given point. Actually, it doesn't have to be 00:38:19.000 --> 00:38:23.000 a fluid. It could be a solid for that 00:38:23.000 --> 00:38:26.000 heat equation. So, for example, 00:38:26.000 --> 00:38:31.000 say that you have a big box made of metal or something, 00:38:31.000 --> 00:38:34.000 and you do some heating at one side. 00:38:34.000 --> 00:38:38.000 You want to know how quickly is the other side going to get hot? 00:38:38.000 --> 00:38:40.000 Well, you can use the equation to figure out, 00:38:40.000 --> 00:38:44.000 you know, say that you have a metal bar, and you keep one side 00:38:44.000 --> 00:38:46.000 at 400� because it's in your oven. 00:38:46.000 --> 00:38:52.000 How quickly will the other side get warm? 00:38:52.000 --> 00:38:57.000 OK, so it's the same equation for both phenomena even though 00:38:57.000 --> 00:39:00.000 they are, of course, different phenomena. 00:39:00.000 --> 00:39:02.000 Well, the physical reason why they're the same is actually 00:39:02.000 --> 00:39:05.000 that phenomena are different, but not all that much. 00:39:05.000 --> 00:39:07.000 They involve, actually, how the atoms and 00:39:07.000 --> 00:39:11.000 molecules are actually moving, and hitting each other inside 00:39:11.000 --> 00:39:14.000 this medium. OK, so anyway, 00:39:14.000 --> 00:39:17.000 what's the explanation for this? 00:39:17.000 --> 00:39:20.000 So, to understand the explanation, and given what 00:39:20.000 --> 00:39:22.000 we've been doing, we should have a vector field 00:39:22.000 --> 00:39:26.000 in there. So, I'm going to think of the 00:39:26.000 --> 00:39:30.000 flow of, well, let's imagine that we are doing 00:39:30.000 --> 00:39:35.000 motion of smoke in air. So, that's the flow of the 00:39:35.000 --> 00:39:39.000 smoke: that means at every point, it's a vector whose 00:39:39.000 --> 00:39:43.000 direction tells me in which direction the smoke is actually 00:39:43.000 --> 00:39:47.000 moving. And, its magnitude tells me how 00:39:47.000 --> 00:39:52.000 fast it's moving, and also what amount of smoke 00:39:52.000 --> 00:39:56.000 is moving. So, there's two things to 00:39:56.000 --> 00:40:01.000 understand. One is how the disparities in 00:40:01.000 --> 00:40:06.000 the concentration between different points causes the flow 00:40:06.000 --> 00:40:10.000 to be there, how smoke will flow from the 00:40:10.000 --> 00:40:14.000 regions where there's more smoke to the regions where there's 00:40:14.000 --> 00:40:17.000 less smoke. And, that's actually physics. 00:40:17.000 --> 00:40:24.000 But, in a way, it's also common sense. 00:40:24.000 --> 00:40:40.000 So, physics and common sense tell us that the smoke will flow 00:40:40.000 --> 00:40:56.000 from high concentration towards low concentration regions. 00:40:56.000 --> 00:41:01.000 OK, so if we think of this function, U, 00:41:01.000 --> 00:41:04.000 that measures concentration, that means we are always going 00:41:04.000 --> 00:41:07.000 to go in the direction where the concentration decreases the 00:41:07.000 --> 00:41:09.000 fastest. Well, what's that? 00:41:09.000 --> 00:41:25.000 That's negative the gradient. So, F is directed along minus 00:41:25.000 --> 00:41:32.000 gradient u. Now, how big is F going to be? 00:41:32.000 --> 00:41:35.000 Well, they are, you have to come up with some 00:41:35.000 --> 00:41:39.000 intuition for how the two are related. 00:41:39.000 --> 00:41:42.000 And, the easiest relation I can think of is that they might be 00:41:42.000 --> 00:41:44.000 just proportional. So, the steeper the differences 00:41:44.000 --> 00:41:47.000 in concentration, the faster the flow will be, 00:41:47.000 --> 00:41:50.000 or the more there will be flow. And, if you try to think about 00:41:50.000 --> 00:41:53.000 it as molecules moving in random directions, you will see it 00:41:53.000 --> 00:41:56.000 makes actually complete sense. Anyway, it should be part of 00:41:56.000 --> 00:42:00.000 your physics class, not part of what I'm telling 00:42:00.000 --> 00:42:04.000 you. So, I'm just going to accept 00:42:04.000 --> 00:42:12.000 that the flow is just proportional to the gradient of 00:42:12.000 --> 00:42:13.000 u. So, if you want, 00:42:13.000 --> 00:42:16.000 the differences between concentrations of different 00:42:16.000 --> 00:42:18.000 points are very small, then the flow will be very 00:42:18.000 --> 00:42:22.000 gentle. And, if on the other hand you 00:42:22.000 --> 00:42:26.000 have huge disparities, then things will try to even 00:42:26.000 --> 00:42:31.000 out faster. OK, so that's the first part. 00:42:31.000 --> 00:42:35.000 Now, we need to understand the second part, which is once we 00:42:35.000 --> 00:42:38.000 know how flow is going, how does that affect the 00:42:38.000 --> 00:42:40.000 concentration? If smoke is going that way, 00:42:40.000 --> 00:42:43.000 then it means the concentration here should be decreasing. 00:42:43.000 --> 00:42:45.000 And there, it should be increasing. 00:42:45.000 --> 00:42:58.000 So, that's the relation between F and partial u partial t. 00:42:58.000 --> 00:43:07.000 At that part is actually math, namely, the divergence theorem. 00:43:07.000 --> 00:43:19.000 So, let's try to understand that part more carefully. 00:43:19.000 --> 00:43:25.000 So, let's take a small piece of a small region in space, 00:43:25.000 --> 00:43:28.000 D, bounded by a surface, S. 00:43:28.000 --> 00:43:33.000 So, I want to figure out what's going on in here. 00:43:33.000 --> 00:43:42.000 So, let's look at the flux out of D through S. 00:43:42.000 --> 00:43:49.000 Well, we said that this flux would be given by double 00:43:49.000 --> 00:43:58.000 integral on S of F dot n dS. So, this flux measures how much 00:43:58.000 --> 00:44:05.000 smoke is passing through S per unit time. 00:44:05.000 --> 00:44:14.000 That's the amount of smoke through S per unit time. 00:44:14.000 --> 00:44:19.000 But now, how can I compute that differently? 00:44:19.000 --> 00:44:23.000 Well, I can compute it just by looking at the total amount of 00:44:23.000 --> 00:44:26.000 smoke in this region, and seeing how much it changes. 00:44:26.000 --> 00:44:29.000 If I'm gaining or losing smoke, it means it must be going up 00:44:29.000 --> 00:44:32.000 there. Well, unless I have a smoker in 00:44:32.000 --> 00:44:35.000 here, but that's not part of the data. 00:44:35.000 --> 00:44:41.000 So, that should be, sorry, 00:44:41.000 --> 00:44:44.000 that's the same thing as the derivative with respect to t of 00:44:44.000 --> 00:44:47.000 the total amount of smoke in the region, 00:44:47.000 --> 00:44:50.000 which is given by the triple integral of u. 00:44:50.000 --> 00:44:52.000 If I integrate the concentration of smoke, 00:44:52.000 --> 00:44:56.000 which means the amount of smoke per unit volume over d, 00:44:56.000 --> 00:44:59.000 I will get the total amount of smoke in d, 00:44:59.000 --> 00:45:02.000 except, well, 00:45:02.000 --> 00:45:05.000 let's see. This flux is counted positively 00:45:05.000 --> 00:45:07.000 if we go out of d. So, actually, 00:45:07.000 --> 00:45:12.000 it's minus the derivative. This is the amount of smoke 00:45:12.000 --> 00:45:16.000 that we are losing per unit time. 00:45:16.000 --> 00:45:33.000 OK, so now we are almost there. Well, let me actually -- 00:45:33.000 --> 00:45:42.000 Because we know yet another way to compute this guy using the 00:45:42.000 --> 00:45:48.000 divergence theorem. Right, so this part here is 00:45:48.000 --> 00:45:53.000 just common sense and thinking about what it means. 00:45:53.000 --> 00:46:00.000 The divergence theorem tells me this is also equal to the triple 00:46:00.000 --> 00:46:06.000 integral, d, of div f dV. So, what I got is that the 00:46:06.000 --> 00:46:15.000 triple integral over d of div F dV equals this derivative. 00:46:15.000 --> 00:46:18.000 Well, let's think a bit about this derivative so, 00:46:18.000 --> 00:46:20.000 see, you are integrating function over x, 00:46:20.000 --> 00:46:22.000 y, and z. And then, you are 00:46:22.000 --> 00:46:24.000 differentiating with respect to t. 00:46:24.000 --> 00:46:28.000 I claim that you can actually switch the order in which you do 00:46:28.000 --> 00:46:30.000 things. So, when we think about it, 00:46:30.000 --> 00:46:33.000 is, here, you are taking the total amount of smoke and then 00:46:33.000 --> 00:46:37.000 see how that changes over time. So, you're taking the 00:46:37.000 --> 00:46:40.000 derivative of the sum of all the small amounts of smoke 00:46:40.000 --> 00:46:42.000 everywhere. Well, that will be the sum of 00:46:42.000 --> 00:46:47.000 the derivatives of the amounts of smoke inside each little box. 00:46:47.000 --> 00:46:55.000 So, we can actually move the derivatives into the integral. 00:46:55.000 --> 00:47:00.000 So, let's see, I said minus d dt of triple 00:47:00.000 --> 00:47:07.000 integral over d udV. And, now I'm saying this is the 00:47:07.000 --> 00:47:14.000 same as the triple integral in d of du dt dv. 00:47:14.000 --> 00:47:19.000 And the reason why this is going to work is you should 00:47:19.000 --> 00:47:24.000 think of it as d dt of a sum of u of some values. 00:47:24.000 --> 00:47:30.000 You plug in the values of your points at that given time times 00:47:30.000 --> 00:47:32.000 the small volume. You sum them, 00:47:32.000 --> 00:47:33.000 and then you take the derivative. 00:47:33.000 --> 00:47:42.000 And now, you see, the derivative of this sum is 00:47:42.000 --> 00:47:49.000 the sum of the derivatives. yi, zi, t, so, 00:47:49.000 --> 00:47:53.000 if you think about what the integral measures, 00:47:53.000 --> 00:47:58.000 that's actually pretty easy. And it's because this variable 00:47:58.000 --> 00:48:01.000 here is not the same as the variables on which we are 00:48:01.000 --> 00:48:03.000 integrating. That's why we can do it. 00:48:03.000 --> 00:48:13.000 OK, so now, if we have this for any region, d. 00:48:13.000 --> 00:48:18.000 So, we have a function of x, y, z, t, and we have another 00:48:18.000 --> 00:48:21.000 function here. And whenever we integrate them 00:48:21.000 --> 00:48:23.000 anywhere, we get the same answer. 00:48:23.000 --> 00:48:26.000 Well, that must mean they're the same. 00:48:26.000 --> 00:48:29.000 Just think of what happens if you just take d to be a tiny 00:48:29.000 --> 00:48:31.000 little box. You will get roughly the value 00:48:31.000 --> 00:48:33.000 of div f at that point times the volume of the box. 00:48:33.000 --> 00:48:36.000 Or, you will get roughly the value of du dt at that point 00:48:36.000 --> 00:48:41.000 times the value of a little box. So, the values must be the same. 00:48:41.000 --> 00:48:46.000 Well, let me actually explain that a tiny bit better. 00:48:46.000 --> 00:48:50.000 So, what I get is that one over, let me divide by the 00:48:50.000 --> 00:49:00.000 volume of D, sorry. I promise, I'm done in a minute. 00:49:00.000 --> 00:49:08.000 Is the same thing as one over volume D of negative du dt, 00:49:08.000 --> 00:49:10.000 dV. So, that means the average 00:49:10.000 --> 00:49:12.000 value, OK, maybe that's the best way 00:49:12.000 --> 00:49:17.000 of telling it, the average of div f in D is 00:49:17.000 --> 00:49:27.000 equal to the average of minus partial u partial t in D. 00:49:27.000 --> 00:49:30.000 And, that's true for any region, D, not just for some 00:49:30.000 --> 00:49:33.000 regions, but for, really, any region I can think 00:49:33.000 --> 00:49:37.000 of. So, the outcome is that 00:49:37.000 --> 00:49:43.000 actually the divergence of f is equal to minus du dt. 00:49:43.000 --> 00:49:47.000 And, that's another way to think about what divergence 00:49:47.000 --> 00:49:48.000 means. The divergence, 00:49:48.000 --> 00:49:50.000 we said, is how much stuff is actually expanding, 00:49:50.000 --> 00:49:54.000 flowing out. That's how much we're losing. 00:49:54.000 --> 00:49:58.000 And so, now, if you combine this with that, 00:49:58.000 --> 00:50:02.000 you will get that du dt is minus divergence f, 00:50:02.000 --> 00:50:08.000 which is plus K del squared u. OK, so you combine this guy 00:50:08.000 --> 00:50:10.000 with that guy, and you get the diffusion 00:50:10.000 --> 00:50:13.000 equation.