WEBVTT 00:00:01.000 --> 00:00:03.000 The following content is provided under a Creative 00:00:03.000 --> 00:00:05.000 Commons license. Your support will help MIT 00:00:05.000 --> 00:00:08.000 OpenCourseWare continue to offer high quality educational 00:00:08.000 --> 00:00:13.000 resources for free. To make a donation or to view 00:00:13.000 --> 00:00:18.000 additional materials from hundreds of MIT courses, 00:00:18.000 --> 00:00:23.000 visit MIT OpenCourseWare at ocw.mit.edu. 00:00:25.000 --> 00:00:30.000 We are going to continue to look at stuff in space. 00:00:30.000 --> 00:00:34.000 We have been working with triple integrals and seeing how 00:00:34.000 --> 00:00:37.000 to set them up in all sorts of coordinate systems. 00:00:37.000 --> 00:00:49.000 And the next topic we will be looking at are vector fields in 00:00:49.000 --> 00:00:52.000 space. And so, in particular, 00:00:52.000 --> 00:00:55.000 we will be learning about flux and work. 00:00:55.000 --> 00:00:58.000 So, just for a change, we will be starting with flux 00:00:58.000 --> 00:01:00.000 first. And we will do work, 00:01:00.000 --> 00:01:04.000 actually, after Thanksgiving. Just to remind you, 00:01:04.000 --> 00:01:08.000 a vector field in space is just the same thing as in the plane. 00:01:08.000 --> 00:01:19.000 At every point you have a vector, and the components of 00:01:19.000 --> 00:01:26.000 this vector depend on the coordinates x, 00:01:26.000 --> 00:01:34.000 y and z. Let's say the components might 00:01:34.000 --> 00:01:42.000 be P, Q, R, or your favorite three letters, 00:01:42.000 --> 00:01:53.000 where each of these things is a function of coordinates x, 00:01:53.000 --> 00:01:56.000 y, z. You have seen that in the plane 00:01:56.000 --> 00:01:58.000 it is already pretty hard to draw a vector field. 00:01:58.000 --> 00:02:01.000 Usually, in space, we won't really try too hard. 00:02:01.000 --> 00:02:04.000 But it is still useful to try to have a general idea for what 00:02:04.000 --> 00:02:07.000 the vectors in there are doing, whether they are all going in 00:02:07.000 --> 00:02:09.000 the same direction, whether they may be all 00:02:09.000 --> 00:02:12.000 vertical or horizontal, pointing away from the origin, 00:02:12.000 --> 00:02:15.000 towards it, things like that. 00:02:15.000 --> 00:02:19.000 But, generally-speaking, we won't really bother with 00:02:19.000 --> 00:02:24.000 trying to draw a picture because that is going to be quite hard. 00:02:24.000 --> 00:02:30.000 Just to give you examples, well, the same kinds of 00:02:30.000 --> 00:02:37.000 examples as the plane, you can think of force fields. 00:02:37.000 --> 00:02:48.000 For example, the gravitational attraction -- 00:02:48.000 --> 00:03:01.000 -- of a solid mass, let's call this mass big M, 00:03:01.000 --> 00:03:13.000 at the origin on a mass M at point x, y, z. 00:03:13.000 --> 00:03:20.000 That would be given by a vector field that points toward the 00:03:20.000 --> 00:03:27.000 origin and whose magnitude is inversely proportional to the 00:03:27.000 --> 00:03:31.000 square of a distance from the origin. 00:03:31.000 --> 00:03:39.000 Such a field would be directed towards the origin and its 00:03:39.000 --> 00:03:47.000 magnitude would be of the order of a constant over pho squared 00:03:47.000 --> 00:03:53.000 where pho is the distance from the origin. 00:03:53.000 --> 00:03:57.000 The picture, if I really wanted to draw a 00:03:57.000 --> 00:04:02.000 picture, would be everywhere it is a field that points towards 00:04:02.000 --> 00:04:09.000 the origin. And if I am further away then 00:04:09.000 --> 00:04:13.000 it gets smaller. And, of course, 00:04:13.000 --> 00:04:16.000 I am not going to try to draw all these vectors in there. 00:04:16.000 --> 00:04:27.000 If I wanted to give a formula for that -- A formula for that 00:04:27.000 --> 00:04:36.000 might be something of a form minus c times x, 00:04:36.000 --> 00:04:41.000 y, z over pho cubed. Let's see. 00:04:41.000 --> 00:04:44.000 Well, the direction of this vector, this vector is 00:04:44.000 --> 00:04:46.000 proportional to negative x, y, z. 00:04:47.000 --> 00:04:49.000 is the vector that goes from the origin to your point. 00:04:49.000 --> 00:04:52.000 The negative goes towards the origin. 00:04:52.000 --> 00:04:56.000 Then the magnitude of this guy, well, the magnitude of x, 00:04:56.000 --> 00:05:00.000 y, z is just the distance from the origin rho. 00:05:00.000 --> 00:05:05.000 So the magnitude of this thing is one over rho cubed times some 00:05:05.000 --> 00:05:10.000 constant factor. That would be an example of a 00:05:10.000 --> 00:05:14.000 vector field that comes up in physics. 00:05:14.000 --> 00:05:20.000 Well, other examples would be electric fields. 00:05:20.000 --> 00:05:23.000 Actually, if you look at the electric field generated by a 00:05:23.000 --> 00:05:27.000 charged particle at the origin, it is given by exactly the same 00:05:27.000 --> 00:05:36.000 kind of formula, and there are magnetic fields 00:05:36.000 --> 00:05:42.000 and so on. Another example comes from 00:05:42.000 --> 00:05:46.000 velocity fields. If you have a fluid flow, 00:05:46.000 --> 00:05:49.000 for example, if you want to study wind 00:05:49.000 --> 00:05:52.000 patterns in the atmosphere. Well, wind, most of the time, 00:05:52.000 --> 00:05:55.000 is kind of horizontal, but maybe it depends on the 00:05:55.000 --> 00:05:58.000 altitude. At high altitude you have jet 00:05:58.000 --> 00:06:03.000 streams, and the wind velocity is not the same at all 00:06:03.000 --> 00:06:08.000 altitudes. And, just to give you more 00:06:08.000 --> 00:06:15.000 examples, in math we have seen that the gradient of a function 00:06:15.000 --> 00:06:21.000 of three variables gives you a vector field. 00:06:21.000 --> 00:06:25.000 If you have a function u of x, y, z then its gradient field 00:06:25.000 --> 00:06:30.000 has just components, u sub x, u sub y and u sub z. 00:06:30.000 --> 00:06:32.000 And, of course, the cases are not mutually 00:06:32.000 --> 00:06:34.000 exclusive. For example, 00:06:34.000 --> 00:06:38.000 the electric field or gravitational field is given by 00:06:38.000 --> 00:06:42.000 the gradient of the gravitational or electric 00:06:42.000 --> 00:06:46.000 potential. So, these are not like 00:06:46.000 --> 00:06:51.000 different cases. There is overlap. 00:06:51.000 --> 00:06:54.000 Anyway, hopefully, you are kind of convinced that 00:06:54.000 --> 00:06:57.000 you should learn about vector fields. 00:06:57.000 --> 00:06:59.000 What are we going to do with them? 00:06:59.000 --> 00:07:16.000 Well, let's start with flux. Remember not so long ago we 00:07:16.000 --> 00:07:24.000 looked at flux of a two-dimensional field of a 00:07:24.000 --> 00:07:32.000 curve. We had a curve in the plane and 00:07:32.000 --> 00:07:41.000 we had a vector field. And we looked at the component 00:07:41.000 --> 00:07:47.000 of a vector field in the direction that was normal to the 00:07:47.000 --> 00:07:51.000 curve. We formed the flux integral 00:07:51.000 --> 00:07:54.000 that was a line integral F dot n ds. 00:07:54.000 --> 00:07:59.000 And that measured how much the vector field was going across 00:07:59.000 --> 00:08:01.000 the curve. If you were thinking of a 00:08:01.000 --> 00:08:05.000 velocity field, that would measure how much 00:08:05.000 --> 00:08:08.000 fluid is passing through the curve in unit time. 00:08:08.000 --> 00:08:10.000 Now let's say that we were in space. 00:08:10.000 --> 00:08:14.000 Well, we cannot really think of flux as a line integral. 00:08:14.000 --> 00:08:18.000 Because, if you have a curve in space and say that you have wind 00:08:18.000 --> 00:08:22.000 or something like that, you cannot really ask how much 00:08:22.000 --> 00:08:24.000 air is flowing through the curve. 00:08:24.000 --> 00:08:28.000 See, to have a flow through something you need a surface. 00:08:28.000 --> 00:08:32.000 If you have a net maybe then you can ask how much stuff is 00:08:32.000 --> 00:08:37.000 passing through that surface. There is going to be a big 00:08:37.000 --> 00:08:44.000 difference here. In the three-dimensional space, 00:08:44.000 --> 00:08:51.000 flux will be measured through a surface. 00:08:51.000 --> 00:08:54.000 And so it will be a surface integral, not a line integral 00:08:54.000 --> 00:08:59.000 anymore. That means we will be 00:08:59.000 --> 00:09:10.000 integrating, we will be summing over all the pieces of a surface 00:09:10.000 --> 00:09:13.000 in space. Because a surface is a 00:09:13.000 --> 00:09:15.000 two-dimensional object, that will end up being a double 00:09:15.000 --> 00:09:17.000 integral. But, of course, 00:09:17.000 --> 00:09:19.000 we will have to set it up properly because the surface 00:09:19.000 --> 00:09:22.000 that is in space, and we will probably have x, 00:09:22.000 --> 00:09:24.000 y and z to deal with at the same time, 00:09:24.000 --> 00:09:28.000 and we will have to somehow get rid of one variable so that we 00:09:28.000 --> 00:09:31.000 can set up and evaluate a double integral. 00:09:31.000 --> 00:09:35.000 So conceptually it is very similar to line integrals. 00:09:35.000 --> 00:09:39.000 In the line integral in the plane, you had two variables 00:09:39.000 --> 00:09:42.000 that you reduced to one by figuring out what the curve was. 00:09:42.000 --> 00:09:51.000 Here you have three variables that you will reduce to two by 00:09:51.000 --> 00:09:56.000 figuring out what the surface is. 00:09:56.000 --> 00:10:00.000 Let me give you a definition of flux in 3D. 00:10:00.000 --> 00:10:15.000 Let's say that we have a vector field and s, a surface in space. 00:10:15.000 --> 00:10:17.000 Let me draw some kind of a picture. 00:10:17.000 --> 00:10:21.000 I have my surface and I have my vector field F. 00:10:21.000 --> 00:10:25.000 Well, at every point it changes with a point. 00:10:25.000 --> 00:10:28.000 Well, I want to figure out how much my vector field is going 00:10:28.000 --> 00:10:34.000 across that surface. That means I want to figure out 00:10:34.000 --> 00:10:40.000 the normal component of my vector field, 00:10:40.000 --> 00:10:45.000 so I will use, as in the plane case, 00:10:45.000 --> 00:10:53.000 the unit normal vector to s. I take my point on the surface 00:10:53.000 --> 00:10:59.000 and build a unit vector that is standing on it perpendicularly. 00:10:59.000 --> 00:11:05.000 Now, we have to decide which way it is standing. 00:11:05.000 --> 00:11:09.000 We can build our normal vector to go this way or to go the 00:11:09.000 --> 00:11:12.000 other way around. There are two choices. 00:11:12.000 --> 00:11:16.000 Basically, whenever you want to set up a flux integral you have 00:11:16.000 --> 00:11:19.000 to choose one side of the surface. 00:11:19.000 --> 00:11:23.000 And you will count positively what flows toward that side and 00:11:23.000 --> 00:11:26.000 negatively what flows towards the other side. 00:11:26.000 --> 00:11:40.000 There are two choices for n. We need to choose a side of the 00:11:40.000 --> 00:11:45.000 surface. In the case of curves, 00:11:45.000 --> 00:11:50.000 we made that choice by deciding that because we were going along 00:11:50.000 --> 00:11:54.000 some direction on the curve we could choose one side by saying 00:11:54.000 --> 00:11:57.000 let's rotate clockwise from the tangent vector. 00:11:57.000 --> 00:12:00.000 And, in a way, what we were doing was really 00:12:00.000 --> 00:12:04.000 it was a recipe to choose for us one of the two sides. 00:12:04.000 --> 00:12:09.000 Here we don't have a notion of orienting the surface other than 00:12:09.000 --> 00:12:14.000 by precisely choosing one of the two possible normal vectors. 00:12:14.000 --> 00:12:15.000 So, in fact, this is called choosing an 00:12:15.000 --> 00:12:18.000 orientation of a surface. When you are saying you are 00:12:18.000 --> 00:12:22.000 orienting the surface that really means you are deciding 00:12:22.000 --> 00:12:31.000 which side is which. Let's call that orientation. 00:12:31.000 --> 00:12:35.000 Now, there is no set convention that will work forever. 00:12:35.000 --> 00:12:39.000 But the usually traditional settings would be to take your 00:12:39.000 --> 00:12:43.000 normal vector pointing maybe out of the solid region because then 00:12:43.000 --> 00:12:48.000 you will be looking at flux that is coming out of that region of 00:12:48.000 --> 00:12:51.000 space. Or, if you have a surface that 00:12:51.000 --> 00:12:55.000 is not like closed or anything but maybe you will want the flux 00:12:55.000 --> 00:12:59.000 going up through the region. Or, there are various 00:12:59.000 --> 00:13:02.000 conventions. Concretely, on problem sets it 00:13:02.000 --> 00:13:05.000 will either say which choice you have to make or you get to 00:13:05.000 --> 00:13:07.000 choose which one you want to make. 00:13:07.000 --> 00:13:10.000 And, of course, if you choose the other one 00:13:10.000 --> 00:13:12.000 then the sign becomes the opposite. 00:13:12.000 --> 00:13:17.000 Now, once we have made a choice then we can define the flux 00:13:17.000 --> 00:13:21.000 integral. It will just be the double 00:13:21.000 --> 00:13:26.000 integral over a surface of F dot n dS. 00:13:26.000 --> 00:13:33.000 Now I am using a big dS. That stands for the surface 00:13:33.000 --> 00:13:39.000 area element on this surface. I am using dS rather than dA 00:13:39.000 --> 00:13:43.000 because I still want to think of dA as maybe the area in one of 00:13:43.000 --> 00:13:47.000 the coordinate planes like the one we had in double integrals. 00:13:47.000 --> 00:13:51.000 You will see later where this comes in. 00:13:51.000 --> 00:13:54.000 But conceptually it is very similar. 00:13:54.000 --> 00:13:59.000 Concretely what this means is I cut my surface into little 00:13:59.000 --> 00:14:03.000 pieces. Each of them has area delta S. 00:14:03.000 --> 00:14:07.000 And, for each piece, I take my vector field, 00:14:07.000 --> 00:14:13.000 I take my normal vector, I dot them and I multiply by 00:14:13.000 --> 00:14:17.000 this surface area and sum all these things together. 00:14:17.000 --> 00:14:23.000 That is what a double integral means. 00:14:23.000 --> 00:14:25.000 In particular, an easy case where you know you 00:14:25.000 --> 00:14:28.000 can get away without computing anything is, of course, 00:14:28.000 --> 00:14:32.000 if your vector field is tangent to the surface because then you 00:14:32.000 --> 00:14:36.000 know that there is no flux. Flux is going to be zero 00:14:36.000 --> 00:14:38.000 because nothing passes through the surface. 00:14:38.000 --> 00:14:42.000 Otherwise, we have to figure out how to compute these things. 00:14:42.000 --> 00:14:50.000 That is what we are going to learn now. 00:14:50.000 --> 00:14:51.000 Well, maybe I should box this formula. 00:14:51.000 --> 00:14:57.000 I have noticed that some of you seem to like it when I box the 00:14:57.000 --> 00:15:03.000 important formulas. (APPLAUSE) By the way, 00:15:03.000 --> 00:15:12.000 a piece of notation before I move on, sometimes you will also 00:15:12.000 --> 00:15:18.000 see the notation vector dS. What is vector dS? 00:15:18.000 --> 00:15:24.000 Vector dS is this guy n dS put together. 00:15:24.000 --> 00:15:30.000 Vector dS is a vector which points perpendicular to the 00:15:30.000 --> 00:15:35.000 surface and whose length corresponds to the surface 00:15:35.000 --> 00:15:37.000 element. And the reason for having this 00:15:37.000 --> 00:15:41.000 shortcut notation, well, it is not only laziness 00:15:41.000 --> 00:15:44.000 like saving one n, but it is because this guy is 00:15:44.000 --> 00:15:49.000 very often easier to compute than it is to set up n and dS 00:15:49.000 --> 00:15:52.000 separately. Actually, if you remember in 00:15:52.000 --> 00:15:56.000 the plane, we have seen that vector n little ds can be 00:15:56.000 --> 00:15:59.000 written directly as dy, - dx. 00:15:59.000 --> 00:16:03.000 That was easier than finding n and ds separately. 00:16:03.000 --> 00:16:15.000 And here the same is going to be true in many cases. 00:16:15.000 --> 00:16:23.000 Well, any questions before we do examples? 00:16:23.000 --> 00:16:24.000 No. OK. 00:16:24.000 --> 00:16:38.000 Let's do examples. The first example for today is 00:16:38.000 --> 00:16:51.000 we are going to look at the flux of vector field xi yj xk through 00:16:51.000 --> 00:17:02.000 the sphere of radius a -- -- centered at the origin. 00:17:02.000 --> 00:17:18.000 What does the picture look like? We have a sphere of radius a. 00:17:18.000 --> 00:17:22.000 I have my vector field. Well, , see, 00:17:22.000 --> 00:17:25.000 that is a vector field that is equal to the vector from the 00:17:25.000 --> 00:17:35.000 origin to the point where I am, so it is pointing radially away 00:17:35.000 --> 00:17:42.000 from the origin. My vector field is really 00:17:42.000 --> 00:17:49.000 sticking out everywhere away from the origin. 00:17:49.000 --> 00:17:56.000 Now I have to find the normal vector to the sphere if I want 00:17:56.000 --> 00:18:04.000 to set up double integral over the sphere of F dot vector ds, 00:18:04.000 --> 00:18:09.000 or if you want F dot n dS. What does the normal vector to 00:18:09.000 --> 00:18:12.000 the sphere look like? Well, it depends, 00:18:12.000 --> 00:18:14.000 of course, whether I choose it pointing out or in. 00:18:14.000 --> 00:18:18.000 Let's say I am choosing it pointing out then it will be 00:18:18.000 --> 00:18:20.000 sticking straight out of a sphere as well. 00:18:20.000 --> 00:18:27.000 Hopefully, you can see that if I take a normal vector to the 00:18:27.000 --> 00:18:34.000 sphere it is actually pointing radially out away from the 00:18:34.000 --> 00:18:38.000 origin. In fact, our vector field and 00:18:38.000 --> 00:18:41.000 our normal vector are parallel to each other. 00:18:41.000 --> 00:18:45.000 Let's think a bit more about what a normal vector looks like. 00:18:45.000 --> 00:18:47.000 I said it is sticking straight out. 00:18:47.000 --> 00:18:49.000 It is proportional to this vector field. 00:18:49.000 --> 00:18:51.000 Maybe I should start by writing 00:18:52.000 --> 00:18:56.000 because that is the vector that goes from the origin to my point 00:18:56.000 --> 00:18:59.000 so it points radially away from the origin. 00:18:59.000 --> 00:19:00.000 Now there is a small problem with that. 00:19:00.000 --> 00:19:04.000 It is not a unit vector. So what is its length? 00:19:04.000 --> 00:19:08.000 Well, its length is square root of x^2 y^2 z^2. 00:19:08.000 --> 00:19:13.000 But, if I am on the sphere, then that length is just equal 00:19:13.000 --> 00:19:16.000 to a because distance from the origin is a. 00:19:16.000 --> 00:19:23.000 In fact, I get my normal vector by scaling this guy down by a 00:19:23.000 --> 00:19:27.000 factor of a. And let me write it down just 00:19:27.000 --> 00:19:34.000 in case you are still unsure. This is unit because square 00:19:34.000 --> 00:19:43.000 root of x^2 y^2 z^2 is equal to a on the sphere. 00:19:43.000 --> 00:19:48.000 OK. Any questions about this? 00:19:48.000 --> 00:19:52.000 No. It looks OK? I see a lot of blank faces. 00:19:52.000 --> 00:19:58.000 That physics test must have been hard. 00:19:58.000 --> 00:20:03.000 Yes? I could have put a rho but I 00:20:03.000 --> 00:20:06.000 want to emphasize the fact that here it is going to be a 00:20:06.000 --> 00:20:09.000 constant. I mean rho has this connotation 00:20:09.000 --> 00:20:13.000 of being a variable that I will need to then maybe integrate 00:20:13.000 --> 00:20:17.000 over or do something with. Yes, it would be correct to put 00:20:17.000 --> 00:20:20.000 rho but I then later will want to replace it by its actual 00:20:20.000 --> 00:20:24.000 value which is a number. And the number is a. 00:20:24.000 --> 00:20:28.000 It is not going to actually change from point to point. 00:20:28.000 --> 00:20:30.000 For example, if this was the unit sphere 00:20:30.000 --> 00:20:32.000 then I would just put x, y, z. 00:20:32.000 --> 00:20:41.000 I wouldn't divide by anything. Now let's figure out F dot n. 00:20:41.000 --> 00:20:47.000 Let's do things one at a time. Well, F and n are parallel to 00:20:47.000 --> 00:20:53.000 each other. F dot n, the normal component 00:20:53.000 --> 00:21:00.000 of F, is actually equal to the length of F. 00:21:00.000 --> 00:21:05.000 Well, times the length of n if you want, but that is going to 00:21:05.000 --> 00:21:09.000 be a one since F and n are parallel to each other. 00:21:09.000 --> 00:21:12.000 And what is the magnitude of F if I am on the sphere? 00:21:12.000 --> 00:21:18.000 Well, the magnitude of F in general is square root of x^2 00:21:18.000 --> 00:21:23.000 y^2 z^2 on the sphere that is going be a. 00:21:23.000 --> 00:21:25.000 The other way to do it, if you don't want to think 00:21:25.000 --> 00:21:28.000 geometrically like that, is to just to do the dot 00:21:28.000 --> 00:21:31.000 product x, y, z doted with x over a, 00:21:31.000 --> 00:21:35.000 y over a, z over a. You will be x^2 y^2 z^2 divided 00:21:35.000 --> 00:21:40.000 by a. That will simplify to a because 00:21:40.000 --> 00:21:45.000 we are on the sphere. See, we are already using here 00:21:45.000 --> 00:21:47.000 the relation between x, y and z. 00:21:47.000 --> 00:21:49.000 We are not letting x, y and z be completely 00:21:49.000 --> 00:21:51.000 arbitrary. But the slogan is everything 00:21:51.000 --> 00:21:54.000 happens on the surface where we are doing the integral. 00:21:54.000 --> 00:21:56.000 We are not looking at anything inside or outside. 00:21:56.000 --> 00:21:58.000 We are just on the surface. 00:22:34.000 --> 00:22:43.000 Now what do I do with that? Well, I have turned my integral 00:22:43.000 --> 00:22:50.000 into the double integral of a dS. 00:22:50.000 --> 00:22:53.000 And a is just a constant, so I am very lucky here. 00:22:53.000 --> 00:22:59.000 I can just say this will be a times the double integral of dS. 00:22:59.000 --> 00:23:02.000 And, of course, some day I will have to learn 00:23:02.000 --> 00:23:05.000 how to tackle that beast, but for now I don't actually 00:23:05.000 --> 00:23:08.000 need to because the double integral of dS just means I am 00:23:08.000 --> 00:23:11.000 summing the area of each little piece of the sphere. 00:23:11.000 --> 00:23:16.000 I am just going to get the total area of the sphere which I 00:23:16.000 --> 00:23:23.000 know to be 4pi a2. This guy here is going to be 00:23:23.000 --> 00:23:29.000 the area of S. I know that to be 4pi a^2. 00:23:29.000 --> 00:23:37.000 So I will get 4pi a^3. That one was relatively 00:23:37.000 --> 00:23:44.000 painless. That was too easy. 00:23:44.000 --> 00:23:49.000 Let's do a second example with the same sphere. 00:23:49.000 --> 00:23:56.000 But now my vector field is going to be just z times k. 00:23:56.000 --> 00:23:58.000 Well, let me give it a different name. 00:23:58.000 --> 00:24:04.000 Let me call it H instead of f or something like that just so 00:24:04.000 --> 00:24:08.000 that it is not called F anymore. Well, the initial part of the 00:24:08.000 --> 00:24:11.000 setup is still the same. The normal vector is still the 00:24:11.000 --> 00:24:13.000 same. What changes is, 00:24:13.000 --> 00:24:16.000 of course, my vector field is no longer sticking straight out 00:24:16.000 --> 00:24:18.000 so I cannot use this easy geometric argument. 00:24:18.000 --> 00:24:22.000 It looks like I will have to compute F dot n and then figure 00:24:22.000 --> 00:24:24.000 out how to integrate that with dS. 00:24:24.000 --> 00:24:36.000 Let's do that. We still have that n is 00:24:44.000 y, z>/a. That tells us that H dot n will 00:24:44.000 --> 00:24:49.000 be dot 00:24:57.000 z> / a. It looks like I will be left 00:24:57.000 --> 00:25:10.000 with z^2 over a. H dot n is z^2 over a. 00:25:10.000 --> 00:25:18.000 The double integral for flux now becomes double integral on 00:25:18.000 --> 00:25:25.000 the sphere of z^2 over a dS. Well, we can take out one over 00:25:25.000 --> 00:25:29.000 a, that is fine, but it looks like we will have 00:25:29.000 --> 00:25:33.000 to integrate z^2 on the surface of the sphere. 00:25:33.000 --> 00:25:37.000 How do we do that? Well, we have to figure out 00:25:37.000 --> 00:25:41.000 what is dS in terms of our favorite set of two variables 00:25:41.000 --> 00:25:45.000 that we will use to integrate. Now, what is the best way to 00:25:45.000 --> 00:25:47.000 figure out where you are on the sphere? 00:25:47.000 --> 00:25:51.000 Well, you could try to use maybe theta and z. 00:25:51.000 --> 00:25:55.000 If you know how high you are and where you are around, 00:25:55.000 --> 00:25:58.000 in principle you know where you are on the sphere. 00:25:58.000 --> 00:26:02.000 But since spherical coordinates we have actually learned about 00:26:02.000 --> 00:26:06.000 something much more interesting, namely spherical coordinates. 00:26:06.000 --> 00:26:09.000 It looks like longitude / latitude is the way to go when 00:26:09.000 --> 00:26:12.000 trying to figure out where you are on a sphere. 00:26:12.000 --> 00:26:19.000 We are going to use phi and theta. 00:26:19.000 --> 00:26:24.000 And, of course, we have to figure out how to 00:26:24.000 --> 00:26:28.000 express dS in terms of d phi and d theta. 00:26:28.000 --> 00:26:32.000 Well, if you were paying really, really close attention 00:26:32.000 --> 00:26:36.000 last time, you will notice that we have actually already done 00:26:36.000 --> 00:26:41.000 that. Last time we saw that if I have 00:26:41.000 --> 00:26:48.000 a sphere of radius a and I take a little piece of it that 00:26:48.000 --> 00:26:56.000 corresponds to small changes in phi and theta then we said that 00:26:56.000 --> 00:27:01.000 -- Well, we argued that this side here, 00:27:01.000 --> 00:27:08.000 the one that is going east-west was a piece of the circle that 00:27:08.000 --> 00:27:14.000 has a radius a sin phi because that is r, 00:27:14.000 --> 00:27:19.000 so that side is a sin phi delta theta. 00:27:19.000 --> 00:27:22.000 And the side that goes north-south is a piece of the 00:27:22.000 --> 00:27:26.000 circle of radius a corresponding to angle delta phi, 00:27:26.000 --> 00:27:32.000 so it is a delta phi. And so, just to get to the 00:27:32.000 --> 00:27:40.000 answer, we got dS equals a^2 sin phi d phi d theta. 00:27:40.000 --> 00:27:45.000 When we set up a surface integral on the surface of a 00:27:45.000 --> 00:27:48.000 sphere, most likely we will be using 00:27:48.000 --> 00:27:52.000 phi and theta as our two variables of integration and dS 00:27:52.000 --> 00:27:55.000 will become this. Now, it is OK to think of them 00:27:55.000 --> 00:27:58.000 as spherical coordinates, but I would like to encourage 00:27:58.000 --> 00:28:01.000 you not to think of them as spherical coordinates. 00:28:01.000 --> 00:28:05.000 Spherical coordinates are a way of describing points in space in 00:28:05.000 --> 00:28:09.000 terms of three variables. Here it is more like we are 00:28:09.000 --> 00:28:12.000 parameterizing the sphere. We are finding a parametric 00:28:12.000 --> 00:28:15.000 equation for the sphere using two variables phi and theta 00:28:15.000 --> 00:28:18.000 which happen to be part of the spherical coordinate system. 00:28:18.000 --> 00:28:22.000 But, see, there is no rho involved in here. 00:28:22.000 --> 00:28:26.000 I am not using any rho ever, and I am not going to in this 00:28:26.000 --> 00:28:28.000 calculation. I have two variable phi and 00:28:28.000 --> 00:28:37.000 theta. That is it. 00:28:37.000 --> 00:28:40.000 It is basically in the same way as when you parameterize a line 00:28:40.000 --> 00:28:45.000 integral in the circle, we use theta as the parameter 00:28:45.000 --> 00:28:50.000 variable and never think about r. 00:28:50.000 --> 00:28:52.000 That being said, well, we are going to use phi 00:28:52.000 --> 00:28:54.000 and theta. We know what dS is. 00:28:54.000 --> 00:28:58.000 We still need to figure out what z is. 00:28:58.000 --> 00:29:01.000 There we want to think a tiny bit about spherical coordinates 00:29:01.000 --> 00:29:08.000 again. And we will know that z is just 00:29:08.000 --> 00:29:15.000 a cos phi. In case you don't quite see it, 00:29:15.000 --> 00:29:25.000 let me draw a diagram. Phi is the angle down from the 00:29:25.000 --> 00:29:31.000 positive z axes, this distance is a, 00:29:31.000 --> 00:29:38.000 so this distance here is a cos phi. 00:29:38.000 --> 00:29:44.000 Now I have everything I need to compute my double integral. 00:29:44.000 --> 00:29:49.000 z^2 over a dS will become a double integral. 00:29:49.000 --> 00:30:00.000 z^2 becomes a^2 cos^2 phi over a times, ds becomes, 00:30:00.000 --> 00:30:07.000 a^2 sin phi d phi d theta. Now I need to set up bounds. 00:30:07.000 --> 00:30:12.000 Well, what are the bounds? Phi goes all the way from zero 00:30:12.000 --> 00:30:19.000 to pi because we go all the way from the north pole to the south 00:30:19.000 --> 00:30:23.000 pole, and theta goes from zero to 2pi. 00:30:23.000 --> 00:30:27.000 And, of course, I can get rid of some a's in 00:30:27.000 --> 00:30:34.000 there and take them out. Let's look at what number we 00:30:34.000 --> 00:30:37.000 get. First of all, 00:30:37.000 --> 00:30:43.000 we can take out all those a's and get a^3. 00:30:43.000 --> 00:30:50.000 Second, in the inner integral, we are integrating cos^2 phi 00:30:50.000 --> 00:30:54.000 sin phi d phi. I claim that integrates to cos3 00:30:54.000 --> 00:30:57.000 up to some factor, and that factor should be 00:30:57.000 --> 00:31:02.000 negative one-third. If you look at cos3 phi and you 00:31:02.000 --> 00:31:07.000 take its derivative, you will get that guy with a 00:31:07.000 --> 00:31:12.000 negative three in front between zero and pi. 00:31:12.000 --> 00:31:16.000 And, while integrating over theta, we will just multiply 00:31:16.000 --> 00:31:24.000 things by 2pi. Let me add the 2pi in front. 00:31:24.000 --> 00:31:27.000 Now, if I evaluate this guy between zero and pi, 00:31:27.000 --> 00:31:32.000 well, at pi cos^3 is negative one, at zero it is one, 00:31:32.000 --> 00:31:35.000 I will get two-thirds out of this. 00:31:35.000 --> 00:31:39.000 I end up with four-thirds pi a^3. 00:31:39.000 --> 00:31:46.000 Sorry I didn't write very much because I am trying to save 00:31:46.000 --> 00:31:52.000 blackboard space. Yes? 00:31:52.000 --> 00:31:55.000 That is a very natural question. That looks a lot like somebody 00:31:55.000 --> 00:31:58.000 we know, like the volume of a sphere. 00:31:58.000 --> 00:32:03.000 And ultimately it will be. Wait until next class when we 00:32:03.000 --> 00:32:07.000 talk about the divergence theorem. 00:32:07.000 --> 00:32:11.000 I mean the question was is this related to the volume of a 00:32:11.000 --> 00:32:14.000 sphere, and ultimately it is, but for now it is just some 00:32:14.000 --> 00:32:23.000 coincidence. Yes? 00:32:23.000 --> 00:32:26.000 The question is there is a way to do it M dx plus N dy plus 00:32:26.000 --> 00:32:28.000 stuff like that? The answer is unfortunately no 00:32:28.000 --> 00:32:30.000 because it is not a line integral. 00:32:30.000 --> 00:32:35.000 It is a surface integral, so we need to have to variables 00:32:35.000 --> 00:32:38.000 in there. In a way you would end up with 00:32:38.000 --> 00:32:41.000 things like some dx dy maybe and so on. 00:32:41.000 --> 00:32:45.000 I mean it is not practical to do it directly that way because 00:32:45.000 --> 00:32:49.000 you would have then to compute Jacobians to switch from dx dy 00:32:49.000 --> 00:32:52.000 to something else. We are going to see various 00:32:52.000 --> 00:32:54.000 ways of computing it. Unfortunately, 00:32:54.000 --> 00:32:57.000 it is not quite as simple as with line integrals. 00:32:57.000 --> 00:32:59.000 But it is not much harder. It is the same spirit. 00:32:59.000 --> 00:33:04.000 We just use two variables and set up everything in terms of 00:33:04.000 --> 00:33:12.000 these two variables. Any other questions? 00:33:12.000 --> 00:33:14.000 No. OK. 00:33:51.000 --> 00:33:54.000 By the way, just some food for thought. 00:33:54.000 --> 00:34:01.000 Never mind. Conclusion of looking at these 00:34:01.000 --> 00:34:05.000 two examples is that sometimes we can use geometric. 00:34:05.000 --> 00:34:07.000 The first example, we didn't actually have to 00:34:07.000 --> 00:34:11.000 compute an integral. But most of the time we need to 00:34:11.000 --> 00:34:14.000 learn how to set up double integrals. 00:34:14.000 --> 00:34:26.000 Use geometry or you need to set up for double integral of a 00:34:26.000 --> 00:34:30.000 surface. And so we are going to learn 00:34:30.000 --> 00:34:33.000 how to do that in general. As I said, we need to have two 00:34:33.000 --> 00:34:37.000 parameters on the surface and express everything in terms of 00:34:37.000 --> 00:34:43.000 these. Let's look at various examples. 00:34:43.000 --> 00:34:46.000 We are going to see various situations where we can do 00:34:46.000 --> 00:34:49.000 things. Well, let's start with an easy 00:34:49.000 --> 00:34:53.000 one. Let's call that number zero. 00:34:53.000 --> 00:35:02.000 Say that my surface S is a horizontal plane, 00:35:02.000 --> 00:35:07.000 say z equals a. When I say a horizontal plane, 00:35:07.000 --> 00:35:09.000 it doesn't have to be the entire horizontal plane. 00:35:09.000 --> 00:35:14.000 It could be a small piece of it. It could even be, 00:35:14.000 --> 00:35:16.000 to trick you, maybe an ellipse in there or a 00:35:16.000 --> 00:35:19.000 triangle in there or something like that. 00:35:19.000 --> 00:35:23.000 What you have to recognize is my surface is a piece of just a 00:35:23.000 --> 00:35:27.000 flat plane, so I shouldn't worry too much about what part of a 00:35:27.000 --> 00:35:30.000 plane it is. Well, it will become important 00:35:30.000 --> 00:35:32.000 when I set up bounds for integration. 00:35:32.000 --> 00:35:36.000 But, when it comes to looking for the normal vector, 00:35:36.000 --> 00:35:40.000 be rest assured that the normal vector to a horizontal plane is 00:35:40.000 --> 00:35:44.000 just vertical. It is going to be either k or 00:35:44.000 --> 00:35:49.000 negative k depending on whether I have chosen to orient it 00:35:49.000 --> 00:35:54.000 pointing up or down. And which one I choose might 00:35:54.000 --> 00:35:57.000 depend on what I am going to try to do. 00:35:57.000 --> 00:36:02.000 The normal vector is just sticking straight up or straight 00:36:02.000 --> 00:36:05.000 down. Now, what about dS? 00:36:05.000 --> 00:36:11.000 Well, it is just going to be the area element in a horizontal 00:36:11.000 --> 00:36:14.000 plane. It just looks like it should be 00:36:14.000 --> 00:36:16.000 dx dy. I mean if I am moving on a 00:36:16.000 --> 00:36:18.000 horizontal plane, to know where I am, 00:36:18.000 --> 00:36:26.000 I should know x and y. So dS will be dx dy. 00:36:26.000 --> 00:36:31.000 If I play the game that way, I have my vector field F. 00:36:31.000 --> 00:36:34.000 I do F dot n. That just gives me the z 00:36:34.000 --> 00:36:37.000 component which might involve x, y and z. 00:36:37.000 --> 00:36:40.000 x and y I am very happy with. They will stay as my variables. 00:36:40.000 --> 00:36:43.000 Whenever I see z, well, I want to get rid of it. 00:36:43.000 --> 00:36:46.000 That is easy because z is just equal to a. 00:36:46.000 --> 00:36:50.000 I just plug that value and I am left with only x and y, 00:36:50.000 --> 00:36:53.000 and I am integrating that dx dy. 00:36:53.000 --> 00:36:58.000 It is actually ending up being just a usual double integral in 00:36:58.000 --> 00:37:00.000 x, y coordinates. And, of course, 00:37:00.000 --> 00:37:02.000 once it is set up anything is fair game. 00:37:02.000 --> 00:37:05.000 I might want to switch to polar coordinates or something like 00:37:05.000 --> 00:37:09.000 that. Or, I can set it up dx dy or dy 00:37:09.000 --> 00:37:12.000 dx. All the usual stuff applies. 00:37:12.000 --> 00:37:17.000 But, for the initial setup, we are just going to use these 00:37:17.000 --> 00:37:21.000 and express everything in terms of x and y. 00:37:21.000 --> 00:37:27.000 A small variation on that. Let's say that we take vertical 00:37:27.000 --> 00:37:35.000 planes that are parallel to maybe the blackboard plane, 00:37:35.000 --> 00:37:42.000 so parallel to the yz plane. That might be something like x 00:37:42.000 --> 00:37:47.000 equals some constant. Well, what would I do then? 00:37:47.000 --> 00:37:52.000 It could be pretty much the same. 00:37:52.000 --> 00:37:55.000 The normal vector for this guy would be sticking straight out 00:37:55.000 --> 00:37:59.000 towards me or away from me. Let's say I am having it come 00:37:59.000 --> 00:38:03.000 to the front. The normal vector would be 00:38:03.000 --> 00:38:07.000 plus/minus i. And the variables that I would 00:38:07.000 --> 00:38:11.000 be using, to find out my position on this guy, 00:38:11.000 --> 00:38:14.000 would be y and z. In terms of those, 00:38:14.000 --> 00:38:19.000 the surface element is just dy dz. 00:38:19.000 --> 00:38:25.000 Similarly for planes parallel to the xz plane. 00:38:25.000 --> 00:38:33.000 You can figure that one out. These are somehow the easiest 00:38:33.000 --> 00:38:39.000 ones, because those we already know how to compute without too 00:38:39.000 --> 00:38:41.000 much trouble. What if it is a more 00:38:41.000 --> 00:38:43.000 complicated plane? We will come back to that next 00:38:43.000 --> 00:38:49.000 time. Let's explore some other 00:38:49.000 --> 00:38:56.000 situations first. Number one on the list. 00:38:56.000 --> 00:39:03.000 Let's say that I gave you a sphere of radius a centered at 00:39:03.000 --> 00:39:09.000 the origin, or maybe just half of that sphere or some portion 00:39:09.000 --> 00:39:12.000 of it. Well, we have already seen how 00:39:12.000 --> 00:39:15.000 to do things. Namely, we will be saying the 00:39:15.000 --> 00:39:18.000 normal vector is x, y, z over a, 00:39:18.000 --> 00:39:23.000 plus or minus depending on whether we want it pointing in 00:39:23.000 --> 00:39:29.000 or out. And dS will be a^2 sin phi d 00:39:29.000 --> 00:39:32.000 phi d theta. In fact, we will express 00:39:32.000 --> 00:39:35.000 everything in terms of phi and theta. 00:39:35.000 --> 00:39:37.000 If I wanted to I could tell you what the formulas are for x, 00:39:37.000 --> 00:39:40.000 y, z in terms of phi and theta. You know them. 00:39:40.000 --> 00:39:44.000 But it is actually better to wait a little bit. 00:39:44.000 --> 00:39:48.000 It is better to do F dot n, because F is also going to have 00:39:48.000 --> 00:39:49.000 a bunch of x's, y's and z's. 00:39:49.000 --> 00:39:53.000 And if there is any kind of symmetry to the problem then you 00:39:53.000 --> 00:39:57.000 might end up with things like x^2 y^2 z^2 or things that have 00:39:57.000 --> 00:40:01.000 more symmetry that are easier to express in terms of phi and 00:40:01.000 --> 00:40:05.000 theta. The advice would be first do 00:40:05.000 --> 00:40:10.000 the dot product with F, and then see what you get and 00:40:10.000 --> 00:40:17.000 then turn it into phi and theta. That is one we have seen. 00:40:17.000 --> 00:40:20.000 Let's say that I have -- It is a close cousin. 00:40:20.000 --> 00:40:30.000 Let's say I have a cylinder of radius a centered on the z-axis. 00:40:30.000 --> 00:40:37.000 What does that look like? And, again, when I say 00:40:37.000 --> 00:40:40.000 cylinder, it could be a piece of cylinder. 00:40:40.000 --> 00:40:44.000 First of all, what does the normal vector to 00:40:44.000 --> 00:40:47.000 a cylinder look like? Well, it is sticking straight 00:40:47.000 --> 00:40:50.000 out, but sticking straight out in a slightly different way from 00:40:50.000 --> 00:40:52.000 what happens with a sphere. See, the sides of a cylinder 00:40:52.000 --> 00:40:54.000 are vertical. If you imagine that you have 00:40:54.000 --> 00:40:56.000 this big cylindrical type in front of you, 00:40:56.000 --> 00:40:59.000 hopefully you can see that a normal vector is going to always 00:40:59.000 --> 00:41:02.000 be horizontal. It is sticking straight out in 00:41:02.000 --> 00:41:07.000 the horizontal directions. It doesn't have any z component. 00:41:07.000 --> 00:41:13.000 I claim the normal vector for the cylinder, 00:41:13.000 --> 00:41:21.000 if you have a point here at (x, y, z), it would be pointing 00:41:21.000 --> 00:41:27.000 straight out away from the central axis. 00:41:27.000 --> 00:41:31.000 My normal vector, well, if I am taking it two 00:41:31.000 --> 00:41:34.000 points outwards, will be going straight away 00:41:34.000 --> 00:41:38.000 from the central axis. If I look at it from above, 00:41:38.000 --> 00:41:42.000 maybe it is easier if I look at it from above, 00:41:42.000 --> 00:41:45.000 look at x, y, then my cylinder looks like a 00:41:45.000 --> 00:41:49.000 circle and the normal vector just points straight out. 00:41:49.000 --> 00:41:53.000 It is the same situation as when we had a circle in the 2D 00:41:53.000 --> 00:41:57.000 case. The normal vector for that is 00:41:57.000 --> 00:42:02.000 just going to be x, y and 0 in the z component. 00:42:02.000 --> 00:42:05.000 Well, plus/minus, depending on whether you want 00:42:05.000 --> 00:42:06.000 it sticking in or out. 00:42:41.000 --> 00:42:47.000 We said in our cylinder normal vector is plus or minus x, 00:42:47.000 --> 00:42:54.000 y, zero over a. What about the surface element? 00:42:54.000 --> 00:42:57.000 Before we ask that, maybe we should first figure 00:42:57.000 --> 00:43:00.000 out what coordinates are we going to use to locate ourselves 00:43:00.000 --> 00:43:02.000 in a cylinder. Well, yes, 00:43:02.000 --> 00:43:05.000 we probably want to use part of a cylindrical coordinate, 00:43:05.000 --> 00:43:08.000 except for, well, we don't want r because r 00:43:08.000 --> 00:43:11.000 doesn't change, it is not a variable here. 00:43:11.000 --> 00:43:15.000 Indeed, you probably want to use z to tell how high you are 00:43:15.000 --> 00:43:18.000 and theta to tell you where you are around. 00:43:18.000 --> 00:43:27.000 dS should be in terms of dz d theta. 00:43:27.000 --> 00:43:33.000 Now, what is the constant? Well, let's look at a small 00:43:33.000 --> 00:43:39.000 piece of our cylinder corresponding to a small angle 00:43:39.000 --> 00:43:44.000 delta theta and a small height delta z. 00:43:44.000 --> 00:43:47.000 Well, the height, as I said, is going to be delta 00:43:47.000 --> 00:43:50.000 z. What about the width? 00:43:50.000 --> 00:43:55.000 It is going to be a piece of a circle of radius a corresponding 00:43:55.000 --> 00:43:59.000 to the angle delta theta, so this side will be a delta 00:43:59.000 --> 00:44:05.000 theta. Delta S is a delta theta delta 00:44:05.000 --> 00:44:08.000 z. DS is just a dz d theta or d 00:44:08.000 --> 00:44:13.000 theta dz. It doesn't matter which way you 00:44:13.000 --> 00:44:16.000 do it. And so when we set up the flux 00:44:16.000 --> 00:44:21.000 integral, we will take first the dot product of f with this 00:44:21.000 --> 00:44:25.000 normal vector. Then we will stick in this dS. 00:44:25.000 --> 00:44:28.000 And then, of course, we will get rid of any x and y 00:44:28.000 --> 00:44:31.000 that are left by expressing them in terms of theta. 00:44:31.000 --> 00:44:37.000 Maybe x becomes a cos theta, y becomes a sin theta. 00:44:37.000 --> 00:44:41.000 These various formulas, you should try to remember them 00:44:41.000 --> 00:44:45.000 because they are really useful, for the sphere, 00:44:45.000 --> 00:44:48.000 for the cylinder. And, hopefully, 00:44:48.000 --> 00:44:52.000 those for the planes you kind of know already intuitively. 00:44:52.000 --> 00:44:59.000 What about marginals or faces? Not everything in life is made 00:44:59.000 --> 00:45:08.000 out of cylinders and spheres. I mean it is a good try. 00:45:08.000 --> 00:45:11.000 Let's look at a marginal kind of surface. 00:45:11.000 --> 00:45:19.000 Let's say I give you a graph of a function z equals f of x, 00:45:19.000 --> 00:45:21.000 y. This guy has nothing to do with 00:45:21.000 --> 00:45:22.000 the integrand. It is not what we are 00:45:22.000 --> 00:45:24.000 integrating. We are just integrating a 00:45:24.000 --> 00:45:26.000 vector field that has nothing to do with that. 00:45:26.000 --> 00:45:31.000 This is how I want to describe the surface on which I will be 00:45:31.000 --> 00:45:37.000 integrating. My surface is given by z as a 00:45:37.000 --> 00:45:40.000 function of x, y. 00:45:40.000 --> 00:45:48.000 Well, I would need to tell you what n is and what dS is. 00:45:48.000 --> 00:45:51.000 That is going to be slightly annoying. 00:45:51.000 --> 00:45:54.000 I mean, I don't want to tell them separately because you see 00:45:54.000 --> 00:45:58.000 they are pretty hard. Instead, I am going to tell you 00:45:58.000 --> 00:46:01.000 that in this case, well, let's see. 00:46:01.000 --> 00:46:07.000 What variables do we want? I am going to tell you a 00:46:07.000 --> 00:46:11.000 formula for n dS. What variables do we want to 00:46:11.000 --> 00:46:15.000 express this in terms of? Well, most likely x and y 00:46:15.000 --> 00:46:19.000 because we know how to express z in terms of x and y. 00:46:19.000 --> 00:46:26.000 This is an invitation to get rid of any z that might be left 00:46:26.000 --> 00:46:30.000 and set everything up in terms of dx dy. 00:46:30.000 --> 00:46:31.000 The formula that we are going to see, 00:46:31.000 --> 00:46:36.000 I think we are going to see the details of why it works 00:46:36.000 --> 00:46:40.000 tomorrow, is that you can take negative 00:46:40.000 --> 00:46:44.000 partial f partial x, negative partial f partial y, 00:46:44.000 --> 00:46:48.000 one, dx dy. 00:46:48.000 --> 00:46:52.000 Plus/minus depending on which way you want it to go. 00:46:52.000 --> 00:46:57.000 If you really want to know what dS is, well, dS is the magnitude 00:46:57.000 --> 00:47:01.000 of this vector times dx dy. There will be a square root and 00:47:01.000 --> 00:47:04.000 some squares and some stuff. What is the normal vector? 00:47:04.000 --> 00:47:10.000 Well, you take this vector and you scale it down to unit 00:47:10.000 --> 00:47:14.000 length. Just to emphasize it, 00:47:14.000 --> 00:47:24.000 this guy here is not n and this guy here is not dS. 00:47:24.000 --> 00:47:27.000 Each of them is more complicated than that, 00:47:27.000 --> 00:47:31.000 but the combination somehow simplifies nicely. 00:47:31.000 --> 00:47:36.000 And that is good news for us. Now, concretely, 00:47:36.000 --> 00:47:40.000 one way you can think about it is this tells you how to reduce 00:47:40.000 --> 00:47:42.000 things to an integral of x and y. 00:47:42.000 --> 00:47:44.000 And, of course, you will have to figure out 00:47:44.000 --> 00:47:47.000 what are the bounds on x and y. That means you will need to 00:47:47.000 --> 00:47:51.000 know what does the shadow of your surface look like in the x, 00:47:51.000 --> 00:47:59.000 y plane. To set up bounds on whatever 00:47:59.000 --> 00:48:06.000 you will get dx dy, well, of course you can switch 00:48:06.000 --> 00:48:08.000 to dy dx or anything you would like, 00:48:08.000 --> 00:48:19.000 but you will need to look at the shadow of S in the x y 00:48:19.000 --> 00:48:22.000 plane. But only do that after you 00:48:22.000 --> 00:48:27.000 gotten rid of all the z. When you no longer have z then 00:48:27.000 --> 00:48:33.000 you can figure out what the bounds are for x and y. 00:48:33.000 --> 00:48:40.000 Any questions about that? Yes? 00:48:40.000 --> 00:48:42.000 For the cylinder. OK. 00:48:42.000 --> 00:48:44.000 Let me re-explain quickly how I got a normal vector for the 00:48:44.000 --> 00:48:47.000 cylinder. If you know what a cylinder 00:48:47.000 --> 00:48:50.000 looks like, you probably can see that the normal vector sticks 00:48:50.000 --> 00:48:56.000 straight out of it horizontally. That means the z component of n 00:48:56.000 --> 00:48:59.000 is going to be zero. And then the x, 00:48:59.000 --> 00:49:02.000 y components you get by looking at it from above. 00:49:02.000 --> 00:49:12.000 One last thing I want to say. What about the geometric 00:49:12.000 --> 00:49:15.000 interpretation and how to prove it? 00:49:15.000 --> 00:49:27.000 Well, if your vector field F is a velocity field then the flux 00:49:27.000 --> 00:49:37.000 is the amount of matter that crosses the surface that passes 00:49:37.000 --> 00:49:44.000 through S per unit time. And the way that you would 00:49:44.000 --> 00:49:47.000 prove it would be similar to the picture that I drew when we did 00:49:47.000 --> 00:49:50.000 it in the plane. Namely, you would consider a 00:49:50.000 --> 00:49:53.000 small element of a surface delta S. 00:49:53.000 --> 00:49:55.000 And you would try to figure out what is the stuff that flows 00:49:55.000 --> 00:49:59.000 through it in a second. Well, it is the stuff that 00:49:59.000 --> 00:50:05.000 lives in a small box whose base is that piece of surface and 00:50:05.000 --> 00:50:10.000 whose other side is given by the vector field. 00:50:10.000 --> 00:50:15.000 And then the volume of that is given by base times height, 00:50:15.000 --> 00:50:20.000 and the height is F dot n. It is the same argument as what 00:50:20.000 --> 00:50:22.000 we saw in the plane. OK. 00:50:22.000 --> 00:50:24.000 Next time we will see more formulas. 00:50:24.000 --> 00:50:28.000 We will first see how to prove this, more ways to do it, 00:50:28.000 --> 00:50:31.000 more examples. And then we will get to the 00:50:31.000 --> 00:50:34.000 divergence theorem.