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 --> 00:00:27.000 So, last week we learned how to do triple integrals in 00:00:27.000 --> 00:00:31.000 rectangular and cylindrical coordinates. 00:00:31.000 --> 00:00:41.000 And, now we have to learn about spherical coordinates, 00:00:41.000 --> 00:00:49.000 which you will see are a lot of fun. 00:00:49.000 --> 00:00:53.000 So, what's the idea of spherical coordinates? 00:00:53.000 --> 00:00:58.000 Well, you're going to represent a point in space using the 00:00:58.000 --> 00:01:02.000 distance to the origin and two angles. 00:01:02.000 --> 00:01:06.000 So, in a way, you can think of these as a 00:01:06.000 --> 00:01:09.000 space analog of polar coordinates because you just use 00:01:09.000 --> 00:01:12.000 distance to the origin, and then you have to use angles 00:01:12.000 --> 00:01:15.000 to determine in which direction you're going. 00:01:15.000 --> 00:01:22.000 So, somehow they are more polar than cylindrical coordinates. 00:01:22.000 --> 00:01:25.000 So, how do we do that? So, let's say that you have a 00:01:25.000 --> 00:01:29.000 point in space at coordinates x, y, z. 00:01:29.000 --> 00:01:32.000 Then, instead of using x, y, z, you will use, 00:01:32.000 --> 00:01:37.000 well, one thing you'll use is the distance from the origin. 00:01:37.000 --> 00:01:41.000 OK, and that is denoted by the Greek letter which looks like a 00:01:41.000 --> 00:01:43.000 curly p, but actually it's the Greek R. 00:01:43.000 --> 00:01:57.000 So -- That's the distance from the origin. 00:01:57.000 --> 00:02:02.000 And so, that can take values anywhere between zero and 00:02:02.000 --> 00:02:06.000 infinity. Then, we have to use two other 00:02:06.000 --> 00:02:09.000 angles. And, so for that, 00:02:09.000 --> 00:02:16.000 let me actually draw the vertical half plane that 00:02:16.000 --> 00:02:22.000 contains our point starting from the z axis. 00:02:22.000 --> 00:02:25.000 OK, so then we have two new angles. 00:02:25.000 --> 00:02:27.000 Well, one of them is not really new. 00:02:27.000 --> 00:02:30.000 One is new. That's phi is the angle 00:02:30.000 --> 00:02:34.000 downwards from the z axis. And the other one, 00:02:34.000 --> 00:02:39.000 theta, is the angle counterclockwise from the x 00:02:39.000 --> 00:02:51.000 axis. OK, so phi, let me do it better. 00:02:51.000 --> 00:02:54.000 So, there's two ways to draw the letter phi, 00:02:54.000 --> 00:02:56.000 by the way. And, I recommend this one 00:02:56.000 --> 00:02:58.000 because it doesn't look like a rho. 00:02:58.000 --> 00:03:07.000 So, that's easier. That's the angle that you have 00:03:07.000 --> 00:03:13.000 to go down from the positive z axis. 00:03:13.000 --> 00:03:17.000 And, so that angle varies from zero 00:03:17.000 --> 00:03:23.000 when you're on the z axis, increase to pi over two when 00:03:23.000 --> 00:03:28.000 you are on the xy plane all the way to pi or 180� when you are 00:03:28.000 --> 00:03:38.000 on the negative z axis. It doesn't go beyond that. 00:03:38.000 --> 00:03:44.000 OK, so -- Phi is always between zero and pi. 00:03:44.000 --> 00:03:46.000 And, finally, the last one, 00:03:46.000 --> 00:03:51.000 theta, is just going to be the same as before. 00:03:51.000 --> 00:03:55.000 So, it's the angle after you project to the xy plane. 00:03:55.000 --> 00:04:00.000 That's the angle counterclockwise from the x 00:04:00.000 --> 00:04:02.000 axis. OK, so that's a little bit 00:04:02.000 --> 00:04:05.000 overwhelming not just because of the new letters, 00:04:05.000 --> 00:04:07.000 but also because there is a lot of angles in there. 00:04:07.000 --> 00:04:11.000 So, let me just try to, you know, suggest two things 00:04:11.000 --> 00:04:13.000 that might help you a little bit. 00:04:13.000 --> 00:04:17.000 So, one is, these are called spherical coordinates because if 00:04:17.000 --> 00:04:21.000 you fix the value of rho, then you are moving on a sphere 00:04:21.000 --> 00:04:27.000 centered at the origin. OK, so let's look at what 00:04:27.000 --> 00:04:34.000 happens on a sphere centered at the origin, so, 00:04:34.000 --> 00:04:41.000 with equation rho equals a. Well, then phi measures how far 00:04:41.000 --> 00:04:44.000 south you are going, measures the distance from the 00:04:44.000 --> 00:04:46.000 North Pole. So, 00:04:46.000 --> 00:04:48.000 if you've learned about latitude and longitude in 00:04:48.000 --> 00:04:51.000 geography, well, phi and theta you can 00:04:51.000 --> 00:04:54.000 think of as latitude and longitude except with slightly 00:04:54.000 --> 00:04:59.000 different conventions. OK, so, phi is more or less the 00:04:59.000 --> 00:05:05.000 same thing as latitude in the sense that it measures how far 00:05:05.000 --> 00:05:09.000 north or south you are. The only difference is in 00:05:09.000 --> 00:05:12.000 geography, latitude is zero on the equator 00:05:12.000 --> 00:05:16.000 and becomes something north, something south, 00:05:16.000 --> 00:05:18.000 depending on how far you go from the equator. 00:05:18.000 --> 00:05:21.000 Here, you measure a latitude starting from the North Pole 00:05:21.000 --> 00:05:24.000 which is zero, increasing all the way to the 00:05:24.000 --> 00:05:29.000 South Pole, which is at pi. And, theta or you can think of 00:05:29.000 --> 00:05:33.000 as longitude, which measures how far you are 00:05:33.000 --> 00:05:36.000 east or west. So, the Greenwich Meridian 00:05:36.000 --> 00:05:39.000 would be here, now, the one on the x axis. 00:05:39.000 --> 00:05:46.000 That's the one you use as the origin for longitude, 00:05:46.000 --> 00:05:48.000 OK? Now, if you don't like 00:05:48.000 --> 00:05:51.000 geography, here's another way to think about it. 00:05:51.000 --> 00:05:56.000 So -- Let's start again from cylindrical coordinates, 00:05:56.000 --> 00:06:02.000 which hopefully you're kind of comfortable with now. 00:06:02.000 --> 00:06:06.000 OK, so you know about cylindrical coordinates where we 00:06:06.000 --> 00:06:09.000 have the z coordinates stay z, and the xy plane we do R and 00:06:09.000 --> 00:06:13.000 theta polar coordinates. And now, let's think about what 00:06:13.000 --> 00:06:18.000 happens when you look at just one of these vertical planes 00:06:18.000 --> 00:06:21.000 containing the z axis. So, you have the z axis, 00:06:21.000 --> 00:06:25.000 and then you have the direction away from the z axis, 00:06:25.000 --> 00:06:29.000 which I will call r, just because that's what r 00:06:29.000 --> 00:06:32.000 measures. Of course, r goes all around 00:06:32.000 --> 00:06:35.000 the z axis, but I'm just doing a slice through one of these 00:06:35.000 --> 00:06:39.000 vertical half planes, fixing the value of theta. 00:06:39.000 --> 00:06:43.000 Then, r of course is a polar coordinate seen from the point 00:06:43.000 --> 00:06:46.000 of view of the xy plane. But here, it looks more like 00:06:46.000 --> 00:06:48.000 you have rectangular coordinates again. 00:06:48.000 --> 00:06:51.000 So the idea of spherical coordinate is you're going to 00:06:51.000 --> 00:06:54.000 polar coordinates again in the rz plane. 00:06:54.000 --> 00:07:01.000 OK, so if I have a point here, then rho will be the distance 00:07:01.000 --> 00:07:06.000 from the origin. And phi will be the angle, 00:07:06.000 --> 00:07:10.000 except it's measured from the positive z axis, 00:07:10.000 --> 00:07:16.000 not from the horizontal axis. But, the idea in here, 00:07:16.000 --> 00:07:18.000 see, let me put that between quotes 00:07:18.000 --> 00:07:21.000 because I'm not sure how correct that is, 00:07:21.000 --> 00:07:28.000 but in a way, you can think of this as polar 00:07:28.000 --> 00:07:34.000 coordinates in the rz plane. So, in particular, 00:07:34.000 --> 00:07:38.000 that's the key to understanding how to switch between spherical 00:07:38.000 --> 00:07:41.000 coordinates and cylindrical coordinates, 00:07:41.000 --> 00:07:44.000 and then all the way to x, y, z if you want, 00:07:44.000 --> 00:07:48.000 right, because this picture here tells 00:07:48.000 --> 00:07:53.000 us how to express z and r in terms of rho and phi. 00:07:53.000 --> 00:08:03.000 So, let's see how that works. If I project here or here, 00:08:03.000 --> 00:08:12.000 so, this line is z. But, it's also rho times cosine 00:08:12.000 --> 00:08:19.000 phi. So, I get z equals rho cos phi. 00:08:19.000 --> 00:08:21.000 And, if I look at r, it's the same thing, 00:08:21.000 --> 00:08:31.000 but on the other side. So, r will be rho sine phi. 00:08:31.000 --> 00:08:34.000 OK, so you can use this to switch back and forth between 00:08:34.000 --> 00:08:37.000 spherical and cylindrical. And of course, 00:08:37.000 --> 00:08:43.000 if you remember what x and y were in terms of r and theta, 00:08:43.000 --> 00:08:49.000 you can also keep doing this to figure out, oops. 00:08:49.000 --> 00:08:57.000 So, x is r cos theta. That becomes rho sine phi cos 00:08:57.000 --> 00:09:01.000 theta. Y is r sine theta. 00:09:01.000 --> 00:09:06.000 So, that becomes rho sine phi sine theta. 00:09:06.000 --> 00:09:15.000 And z is rho cos phi. But, basically you don't really 00:09:15.000 --> 00:09:19.000 need to remember these formulas as long as you remember how to 00:09:19.000 --> 00:09:22.000 express r in terms of rho sine phi, 00:09:22.000 --> 00:09:29.000 and x equals r cos theta. So, now, of course, 00:09:29.000 --> 00:09:31.000 we're going to use spherical coordinates in situations where 00:09:31.000 --> 00:09:33.000 we have a lot of symmetry, and in particular, 00:09:33.000 --> 00:09:35.000 where the z axis plays a special role. 00:09:35.000 --> 00:09:38.000 Actually, that's the same with cylindrical coordinates. 00:09:38.000 --> 00:09:40.000 Cylindrical and secure coordinates are set up so that 00:09:40.000 --> 00:09:44.000 the z axis plays a special role. So, that means whenever you 00:09:44.000 --> 00:09:47.000 have a geometric problem, and you are not told how to 00:09:47.000 --> 00:09:51.000 choose your coordinates, it's probably wiser to try to 00:09:51.000 --> 00:09:57.000 center things on the z axis. That's where these coordinates 00:09:57.000 --> 00:10:01.000 are the best adapted. And, 00:10:01.000 --> 00:10:03.000 in case you ever need to switch backwards, 00:10:03.000 --> 00:10:07.000 I just want to point out, so, rho is the square root of r 00:10:07.000 --> 00:10:11.000 squared plus z squared, which means it's the square 00:10:11.000 --> 00:10:15.000 root of x squared plus y squared plus z squared. 00:10:15.000 --> 00:10:20.000 OK, so that's basically all the formulas about spherical 00:10:20.000 --> 00:10:28.000 coordinates. OK, any questions about that? 00:10:28.000 --> 00:10:31.000 OK, let's see, who had seen spherical 00:10:31.000 --> 00:10:34.000 coordinates before just to see? OK, that's not very many. 00:10:34.000 --> 00:10:36.000 So, I'm sure for, one of you saw it twice. 00:10:36.000 --> 00:10:42.000 That's great. Sorry, oops, 00:10:42.000 --> 00:10:49.000 OK, so let's just look quickly at equations of some of the 00:10:49.000 --> 00:10:54.000 things. So, as I've said, 00:10:54.000 --> 00:11:02.000 if I set rho equals a, that will be just a sphere of 00:11:02.000 --> 00:11:11.000 radius a centered at the origin. More interesting things: 00:11:11.000 --> 00:11:14.000 let's say I give you phi equals pi over four. 00:11:14.000 --> 00:11:18.000 What do you think that looks like? 00:11:18.000 --> 00:11:29.000 Actually, let's take a quick poll on things. 00:11:29.000 --> 00:11:31.000 OK, yeah, everyone seems to be saying it's a cone, 00:11:31.000 --> 00:11:33.000 and that's indeed the correct answer. 00:11:33.000 --> 00:11:41.000 So, how do we see that? Well, remember, 00:11:41.000 --> 00:11:44.000 phi is the angle downward from the z axis. 00:11:44.000 --> 00:11:49.000 So, let's say that I'm going to look first at what happens if 00:11:49.000 --> 00:11:53.000 I'm in the right half of a plane of a blackboard, 00:11:53.000 --> 00:11:56.000 so, in the yz plane. Then, phi is the angle downward 00:11:56.000 --> 00:11:58.000 from here. So, if I want to get pi over 00:11:58.000 --> 00:12:01.000 four, that's 45�. That means I'm going to go 00:12:01.000 --> 00:12:03.000 diagonally like this. Of course, if I'm in the left 00:12:03.000 --> 00:12:06.000 half of a plane of a blackboard, it's going to be the same. 00:12:06.000 --> 00:12:10.000 I also take pi over four. And, I get the other half. 00:12:10.000 --> 00:12:13.000 And, because the equation does not involve theta, 00:12:13.000 --> 00:12:17.000 it's all the same if I rotate my vertical plane around the z 00:12:17.000 --> 00:12:21.000 axis. So, I get the same picture in 00:12:21.000 --> 00:12:27.000 any of these vertical half planes, actually. 00:12:27.000 --> 00:12:32.000 OK, now, so this is phi equals pi over four. 00:12:32.000 --> 00:12:35.000 And, just in case, to point out to you what's 00:12:35.000 --> 00:12:39.000 going on, when phi equals pi over four, cosine and sine are 00:12:39.000 --> 00:12:42.000 equal to each other. They are both one over root two. 00:12:42.000 --> 00:12:46.000 So, you can find, again, the equation of this 00:12:46.000 --> 00:12:51.000 thing in cylindrical coordinates, which I'll remind 00:12:51.000 --> 00:12:54.000 you was z equals r. OK, in general, 00:12:54.000 --> 00:12:58.000 phi equals some given number, or z equals some number times 00:12:58.000 --> 00:13:01.000 r. That will be a cone centered on 00:13:01.000 --> 00:13:04.000 the z axis. OK, a special case: 00:13:04.000 --> 00:13:07.000 what if I say phi equals pi over two? 00:13:07.000 --> 00:13:09.000 Yeah, it's just going to be the xy plane. 00:13:09.000 --> 00:13:13.000 OK, that's the flattest of all cones. 00:13:13.000 --> 00:13:20.000 OK, so phi equals pi over two is going to be just the xy 00:13:20.000 --> 00:13:22.000 plane. And, in general, 00:13:22.000 --> 00:13:24.000 if phi is less than pi over two, then you are in the upper 00:13:24.000 --> 00:13:28.000 half space. If phi is more than pi over 00:13:28.000 --> 00:13:32.000 two, you'll be in the lower half space. 00:13:32.000 --> 00:13:36.000 OK, so that's pretty much all we need to know at this point. 00:13:36.000 --> 00:13:45.000 So, what's next? Well, remember we were trying 00:13:45.000 --> 00:13:52.000 to do triple integrals. So now we're going to triple 00:13:52.000 --> 00:13:59.000 integrals in spherical coordinates. 00:13:59.000 --> 00:14:01.000 And, for that, we first need to understand 00:14:01.000 --> 00:14:06.000 what the volume element is. What will be dV? 00:14:06.000 --> 00:14:12.000 OK, so dV will be something, d rho, d phi, 00:14:12.000 --> 00:14:18.000 d theta, or in any order that you want. 00:14:18.000 --> 00:14:23.000 But, this one is usually the most convenient. 00:14:23.000 --> 00:14:27.000 So, to find out what it is, well, we should look at how we 00:14:27.000 --> 00:14:29.000 are going to be slicing things now. 00:14:29.000 --> 00:14:32.000 OK, so if you integrate d rho, d phi, d theta, 00:14:32.000 --> 00:14:37.000 it means that you are actually slicing your solid into little 00:14:37.000 --> 00:14:40.000 pieces that live, somehow, 00:14:40.000 --> 00:14:45.000 if you set an interval of rows, OK, 00:14:45.000 --> 00:14:48.000 sorry, maybe I should, so, if you first integrate over 00:14:48.000 --> 00:14:51.000 rho, it means that you will actually 00:14:51.000 --> 00:14:57.000 choose first the direction from the origin even by phi and 00:14:57.000 --> 00:15:00.000 theta. And, in that direction, 00:15:00.000 --> 00:15:04.000 you will try to figure out, how far does your region 00:15:04.000 --> 00:15:07.000 extend? And, of course, 00:15:07.000 --> 00:15:11.000 how far that goes might depend on phi and theta. 00:15:11.000 --> 00:15:16.000 Then, you will vary phi. So, you have to know, 00:15:16.000 --> 00:15:21.000 for a given value of theta, how far down does your solid 00:15:21.000 --> 00:15:22.000 extend? And, finally, 00:15:22.000 --> 00:15:25.000 the value of theta will correspond to, 00:15:25.000 --> 00:15:28.000 in which directions around the z axis do we go? 00:15:28.000 --> 00:15:31.000 So, we're going to see that in examples. 00:15:31.000 --> 00:15:34.000 But before we can do that, we need to get the volume 00:15:34.000 --> 00:15:36.000 element. So, what I would like to 00:15:36.000 --> 00:15:40.000 suggest is that we need to figure out, 00:15:40.000 --> 00:15:46.000 what is the volume of a small piece of solid which corresponds 00:15:46.000 --> 00:15:49.000 to a certain change, delta rho, 00:15:49.000 --> 00:15:52.000 delta phi, and delta theta? 00:15:52.000 --> 00:15:56.000 So, delta rho means that you have two concentric spheres, 00:15:56.000 --> 00:16:01.000 and you are looking at a very thin shell in between them. 00:16:01.000 --> 00:16:05.000 And then, you would be looking at a piece of that spherical 00:16:05.000 --> 00:16:08.000 shell corresponding to small values of phi and theta. 00:16:08.000 --> 00:16:14.000 So, because I am stretching the limits of my ability to draw on 00:16:14.000 --> 00:16:18.000 the board, here's a picture. I'm going to try to reproduce 00:16:18.000 --> 00:16:21.000 on the board, but so let's start by looking 00:16:21.000 --> 00:16:24.000 just at what happens on the sphere of radius a, 00:16:24.000 --> 00:16:28.000 and let's try to figure out the surface area elements on the 00:16:28.000 --> 00:16:30.000 sphere in terms of phi and theta. 00:16:30.000 --> 00:16:39.000 And then, we'll add the rho direction. 00:16:39.000 --> 00:16:49.000 OK, so -- So, let me say, let's start by 00:16:49.000 --> 00:17:02.000 understanding surface area on a sphere of radius a. 00:17:02.000 --> 00:17:12.000 So, that means we'll be looking at a little piece of the sphere 00:17:12.000 --> 00:17:21.000 corresponding to angles delta phi and in that direction here 00:17:21.000 --> 00:17:26.000 delta theta. OK, so when you draw a map of 00:17:26.000 --> 00:17:29.000 the world on a globe, that's exactly what the grid 00:17:29.000 --> 00:17:33.000 lines form for you. So, what's the area of this guy? 00:17:33.000 --> 00:17:35.000 Well, of course, all the sides are curvy. 00:17:35.000 --> 00:17:37.000 They are all on the sphere. None of them are straight. 00:17:37.000 --> 00:17:41.000 But still, if it's small enough and it looks like a rectangle, 00:17:41.000 --> 00:17:46.000 so let's just try to figure out, what are the sides of your 00:17:46.000 --> 00:17:49.000 rectangle? OK, so, let's see, 00:17:49.000 --> 00:17:55.000 well, I think I need to draw a bigger picture of this guy. 00:17:55.000 --> 00:17:59.000 OK, so this guy, so that's a piece of what's 00:17:59.000 --> 00:18:05.000 called a parallel in geography. That's a circle that goes 00:18:05.000 --> 00:18:07.000 east-west. So now, 00:18:07.000 --> 00:18:10.000 this parallel as a circle of radius, 00:18:10.000 --> 00:18:14.000 well, the radius is less than a because if your vertical is to 00:18:14.000 --> 00:18:17.000 the North Pole, it will be actually much 00:18:17.000 --> 00:18:19.000 smaller. So, that's why when you say 00:18:19.000 --> 00:18:22.000 you're going around the world it depends on whether you do it at 00:18:22.000 --> 00:18:28.000 the equator or the North Pole. It's much easier at the North 00:18:28.000 --> 00:18:33.000 Pole. So, anyway, this is a piece of 00:18:33.000 --> 00:18:40.000 a circle of radius, well, the radius is what I 00:18:40.000 --> 00:18:49.000 would call r because that's the distance from the z axis. 00:18:49.000 --> 00:18:51.000 OK, that's actually pretty hard to see now. 00:18:51.000 --> 00:18:58.000 So if you can see it better on this one, then so this guy here, 00:18:58.000 --> 00:19:03.000 this length is r. And, r is just rho, 00:19:03.000 --> 00:19:07.000 well, what was a times sine phi. 00:19:07.000 --> 00:19:09.000 Remember, we have this angle phi in here. 00:19:09.000 --> 00:19:14.000 I should use some color. It's getting very cluttered. 00:19:14.000 --> 00:19:19.000 So, we have this phi, and so r is going to be rho 00:19:19.000 --> 00:19:21.000 sine phi. That rho is a. 00:19:21.000 --> 00:19:29.000 So, let me just put a sine phi. OK, and the corresponding angle 00:19:29.000 --> 00:19:32.000 is going to be measured by theta. 00:19:32.000 --> 00:19:48.000 So, the length of this is going to be a sine phi delta theta. 00:19:48.000 --> 00:19:54.000 That's for this side. Now, what about that side, 00:19:54.000 --> 00:19:56.000 the north-south side? Well, if you're moving 00:19:56.000 --> 00:19:58.000 north-south, it's not like east-west. 00:19:58.000 --> 00:20:01.000 You always have to go all the way from the North Pole to the 00:20:01.000 --> 00:20:04.000 South Pole. So, that's actually a great 00:20:04.000 --> 00:20:08.000 circle meridian of length, well, I mean, 00:20:08.000 --> 00:20:13.000 well, the radius is the radius of the sphere. 00:20:13.000 --> 00:20:22.000 Total length is 2pi a. So, this is a piece of a circle 00:20:22.000 --> 00:20:27.000 of radius a. And so, now, 00:20:27.000 --> 00:20:34.000 the length of this one is going to be a delta phi. 00:20:34.000 --> 00:20:41.000 OK, so, just to recap, this is a sine phi delta theta. 00:20:41.000 --> 00:20:46.000 And, this guy here is a delta phi. 00:20:46.000 --> 00:20:59.000 So, you can't read it because it's -- And so, 00:20:59.000 --> 00:21:02.000 that tells us if I take a small piece of the sphere, 00:21:02.000 --> 00:21:06.000 then its surface area, delta s, 00:21:06.000 --> 00:21:15.000 is going to be approximately a sine phi delta theta times a 00:21:15.000 --> 00:21:22.000 delta phi, which I'm going to rewrite as a 00:21:22.000 --> 00:21:27.000 squared sine phi delta phi delta theta. 00:21:27.000 --> 00:21:31.000 So, what that means is, say that I want to integrate 00:21:31.000 --> 00:21:34.000 something just on the surface of a sphere. 00:21:34.000 --> 00:21:37.000 Well, I would use phi and theta as my coordinates. 00:21:37.000 --> 00:21:46.000 And then, to know how big a piece of a sphere is, 00:21:46.000 --> 00:21:55.000 I would just take a squared sine phi d phi d theta. 00:21:55.000 --> 00:21:59.000 OK, so that's the surface element in a sphere. 00:21:59.000 --> 00:22:03.000 And now, what about going back into the third dimension, 00:22:03.000 --> 00:22:05.000 so, adding some depth to these things? 00:22:05.000 --> 00:22:10.000 Well, I'm not going to try to draw a picture because you've 00:22:10.000 --> 00:22:17.000 seen that's slightly tricky. Well, let me try anyway just 00:22:17.000 --> 00:22:24.000 you can have fun with my completely unreadable diagrams. 00:22:24.000 --> 00:22:28.000 So anyway, if you look at, now, something that's a bit 00:22:28.000 --> 00:22:33.000 like that piece of sphere, but with some thickness to it. 00:22:33.000 --> 00:22:38.000 The thickness will be delta rho, and so the volume will be 00:22:38.000 --> 00:22:44.000 roughly the area of the thing on the sphere times the thickness. 00:22:44.000 --> 00:22:48.000 So, I claim that we will get basically the volume element 00:22:48.000 --> 00:22:51.000 just by multiplying things by d rho. 00:22:51.000 --> 00:23:10.000 So, let's see that. So now, if I have a sphere of 00:23:10.000 --> 00:23:19.000 radius rho, and another one that's slightly bigger of radius 00:23:19.000 --> 00:23:27.000 rho plus delta rho, and then I have a little box in 00:23:27.000 --> 00:23:29.000 here. Then, 00:23:29.000 --> 00:23:34.000 I know that the volume of this thing will be essentially, 00:23:34.000 --> 00:23:38.000 well, its thickness, the thickness is going to be 00:23:38.000 --> 00:23:42.000 delta rho times the area of its base, 00:23:42.000 --> 00:23:44.000 although it doesn't really matter, 00:23:44.000 --> 00:23:48.000 which is what we've called delta s. 00:23:48.000 --> 00:23:55.000 OK, so we will get, sorry, a becomes rho now. 00:23:55.000 --> 00:23:57.000 Square sine phi, delta rho, 00:23:57.000 --> 00:24:00.000 delta phi, delta theta, 00:24:00.000 --> 00:24:04.000 and so out of that we get the volume element and spherical 00:24:04.000 --> 00:24:08.000 coordinates, which is rho squared sine phi d 00:24:08.000 --> 00:24:09.000 rho, d phi, 00:24:09.000 --> 00:24:14.000 d theta. And, that's a formula that you 00:24:14.000 --> 00:24:17.000 should remember. OK, so whenever we integrate a 00:24:17.000 --> 00:24:20.000 function, and we decide to switch to 00:24:20.000 --> 00:24:25.000 spherical coordinates, then dx dy dz or r dr d theta 00:24:25.000 --> 00:24:33.000 dz will become rho squared sine phi d rho d phi d theta. 00:24:33.000 --> 00:24:40.000 OK, any questions on that? No? 00:24:40.000 --> 00:24:58.000 OK, so let's -- Let's see how that works. 00:24:58.000 --> 00:25:04.000 So, as an example, remember at the end of the last 00:25:04.000 --> 00:25:11.000 lecture, I tried to set up an example where we were looking at 00:25:11.000 --> 00:25:16.000 a sphere sliced by a slanted plane. 00:25:16.000 --> 00:25:20.000 And now, we're going to try to find the volume of that 00:25:20.000 --> 00:25:23.000 spherical cap again, but using spherical coordinates 00:25:23.000 --> 00:25:26.000 instead. So, I'm going to just be 00:25:26.000 --> 00:25:29.000 smarter than last time. So, last time, 00:25:29.000 --> 00:25:33.000 we had set up these things with a slanted plane that was cutting 00:25:33.000 --> 00:25:35.000 things diagonally. And, 00:25:35.000 --> 00:25:37.000 if I just want to find the volume of this cap, 00:25:37.000 --> 00:25:41.000 then maybe it makes more sense to rotate things so that my 00:25:41.000 --> 00:25:45.000 plane is actually horizontal, and things are going to be 00:25:45.000 --> 00:25:49.000 centered on the z axis. So, in case you see that it's 00:25:49.000 --> 00:25:52.000 the same, then that's great. If not, then it doesn't really 00:25:52.000 --> 00:25:55.000 matter. You can just think of this as a 00:25:55.000 --> 00:26:01.000 new example. So, I'm going to try to find 00:26:01.000 --> 00:26:10.000 the volume of a portion of the unit sphere -- -- that lies 00:26:10.000 --> 00:26:20.000 above the horizontal plane, z equals one over root two. 00:26:20.000 --> 00:26:22.000 OK, one over root two was the distance from the origin to our 00:26:22.000 --> 00:26:24.000 slanted plane. So, after you rotate, 00:26:24.000 --> 00:26:28.000 that say you get this value. Anyway, it's not very important. 00:26:28.000 --> 00:26:31.000 You can just treat that as a good example if you want. 00:26:31.000 --> 00:26:36.000 OK, so we can compute this in actually pretty much any 00:26:36.000 --> 00:26:39.000 coordinate system. And also, of course, 00:26:39.000 --> 00:26:42.000 we can set up not only the volume, but we can try to find 00:26:42.000 --> 00:26:44.000 the moment of inertia about the central axis, 00:26:44.000 --> 00:26:47.000 or all sorts of things. But, we are just doing the 00:26:47.000 --> 00:26:49.000 volume for simplicity. So, actually, 00:26:49.000 --> 00:26:52.000 this would go pretty well in cylindrical coordinates. 00:26:52.000 --> 00:26:55.000 But let's do it in spherical coordinates because that's the 00:26:55.000 --> 00:26:57.000 topic of today. A good exercise: 00:26:57.000 --> 00:27:01.000 do it in cylindrical and see if you get the same thing. 00:27:01.000 --> 00:27:08.000 So, how do we do that? Well, we have to figure out how 00:27:08.000 --> 00:27:14.000 to set up our triple integral in spherical coordinates. 00:27:14.000 --> 00:27:18.000 So, remember we'll be integrating one dV. 00:27:18.000 --> 00:27:28.000 So, dV will become rho squared sign phi d rho d phi d theta. 00:27:28.000 --> 00:27:32.000 And, now as we start, we're already facing some 00:27:32.000 --> 00:27:35.000 serious problem. We want to set up the bounds 00:27:35.000 --> 00:27:37.000 for rho for a given, phi and theta. 00:27:37.000 --> 00:27:39.000 So, that means we choose latitude/longitude. 00:27:39.000 --> 00:27:42.000 We choose which direction we want to aim for, 00:27:42.000 --> 00:27:45.000 you know, which point of the sphere we want to aim at. 00:27:45.000 --> 00:27:50.000 And, we are going to shoot a ray from the origin towards this 00:27:50.000 --> 00:27:55.000 point, and we want to know what portion of the ray is in our 00:27:55.000 --> 00:28:03.000 solid. So -- We are going to choose a 00:28:03.000 --> 00:28:11.000 value of phi and theta. And, we are going to try to 00:28:11.000 --> 00:28:16.000 figure out what part of our ray is inside this side. 00:28:16.000 --> 00:28:20.000 So, what should be clear is at which point we leave the solid, 00:28:20.000 --> 00:28:23.000 right? What's the value of rho here? 00:28:23.000 --> 00:28:25.000 It's just one. The sphere is rho equals one. 00:28:25.000 --> 00:28:29.000 That's pretty good. The question is, 00:28:29.000 --> 00:28:33.000 where do we enter the region? So, we enter the region when we 00:28:33.000 --> 00:28:38.000 go through this plane. And, the plane is z equals one 00:28:38.000 --> 00:28:41.000 over root two. So, what does that tell us 00:28:41.000 --> 00:28:44.000 about rho? Well, it tells us, 00:28:44.000 --> 00:28:50.000 so remember, z is rho cosine phi. 00:28:50.000 --> 00:28:55.000 So, the plane is z equals one over root two. 00:28:55.000 --> 00:29:00.000 That means rho cosine phi is one over root two. 00:29:00.000 --> 00:29:05.000 That means rho equals one over root two cosine phi or, 00:29:05.000 --> 00:29:11.000 as some of you know it, one over root two times second 00:29:11.000 --> 00:29:17.000 phi. OK, so if we want to set up the 00:29:17.000 --> 00:29:27.000 bounds, then we'll start with one over root two second phi all 00:29:27.000 --> 00:29:32.000 the way to one. Now, what's next? 00:29:32.000 --> 00:29:35.000 Well, so we've done, I think that's basically the 00:29:35.000 --> 00:29:38.000 hardest part of the job. Next, we have to figure out, 00:29:38.000 --> 00:29:41.000 what's the range for phi? So, the range for phi, 00:29:41.000 --> 00:29:44.000 well, we have to figure out how far to the north and to the 00:29:44.000 --> 00:29:48.000 south our region goes. Well, the lower bound for phi 00:29:48.000 --> 00:29:51.000 is pretty easy, right, because we go all the 00:29:51.000 --> 00:29:56.000 way to the North Pole direction. So, phi starts at zero. 00:29:56.000 --> 00:29:59.000 The question is, where does it stop? 00:29:59.000 --> 00:30:02.000 To find out where it stops, we have to figure out, 00:30:02.000 --> 00:30:06.000 what is the value of phi when we hit the edge of the region? 00:30:06.000 --> 00:30:10.000 OK, so maybe you see it. Maybe you don't. 00:30:10.000 --> 00:30:15.000 One way to do it geometrically is to just, it's always great to 00:30:15.000 --> 00:30:19.000 draw a slice of your region. So, if you slice these things 00:30:19.000 --> 00:30:22.000 by a vertical plane, or actually even better, 00:30:22.000 --> 00:30:25.000 a vertical half plane, something to delete one half of 00:30:25.000 --> 00:30:28.000 the picture. So, I'm going to draw these r 00:30:28.000 --> 00:30:33.000 and z directions as before. So, my sphere is here. 00:30:33.000 --> 00:30:38.000 My plane is here at one over root two. 00:30:38.000 --> 00:30:43.000 And, my solid is here. So now, the question is what is 00:30:43.000 --> 00:30:49.000 the value of phi when I'm going to stop hitting the region? 00:30:49.000 --> 00:30:54.000 And, if you try to figure out first what is this direction 00:30:54.000 --> 00:30:57.000 here, that's also one over root two. 00:30:57.000 --> 00:31:03.000 And so, this is actually 45�, also known as pi over four. 00:31:03.000 --> 00:31:09.000 The other way to think about it is at this point, 00:31:09.000 --> 00:31:16.000 well, rho is equal to one because you are on the sphere. 00:31:16.000 --> 00:31:22.000 But, you are also on the plane. So, rho cos phi is one over 00:31:22.000 --> 00:31:26.000 root two. So, if you plug rho equals one 00:31:26.000 --> 00:31:31.000 into here, you get cos phi equals one over root two which 00:31:31.000 --> 00:31:34.000 gives you phi equals pi over four. 00:31:34.000 --> 00:31:37.000 That's the other way to do it. You can do it either by 00:31:37.000 --> 00:31:39.000 calculation or by looking at the picture. 00:31:39.000 --> 00:31:43.000 OK, so either way, we've decided that phi goes 00:31:43.000 --> 00:31:48.000 from zero to pi over four. So, this is pi over four. 00:31:48.000 --> 00:31:54.000 Finally, what about theta? Well, because we go all around 00:31:54.000 --> 00:32:00.000 the z axis we are going to go just zero to 2pi. 00:32:00.000 --> 00:32:06.000 OK, any questions about these bounds? 00:32:06.000 --> 00:32:10.000 OK, so note how the equation of this horizontal plane in 00:32:10.000 --> 00:32:13.000 spherical coordinates has become a little bit weird. 00:32:13.000 --> 00:32:16.000 But, if you remember how we do 00:32:16.000 --> 00:32:19.000 things, say that you have a line in 00:32:19.000 --> 00:32:21.000 polar coordinates, and that line does not pass 00:32:21.000 --> 00:32:23.000 through the origin, then you also end up with 00:32:23.000 --> 00:32:26.000 something like that. You get something like r equals 00:32:26.000 --> 00:32:31.000 a second theta or a cos second theta for horizontal or vertical 00:32:31.000 --> 00:32:33.000 lines. And so, it's not surprising you 00:32:33.000 --> 00:32:38.000 should get this. That's in line with the idea 00:32:38.000 --> 00:32:44.000 that we are just doing again, polar coordinates in the rz 00:32:44.000 --> 00:32:46.000 directions. So of course, 00:32:46.000 --> 00:32:48.000 in general, things can be very messy. 00:32:48.000 --> 00:32:51.000 But, generally speaking, the kinds of regions that we 00:32:51.000 --> 00:32:55.000 will be setting up things for are no more complicated or no 00:32:55.000 --> 00:32:59.000 less complicated than what we would do in the plane in polar 00:32:59.000 --> 00:33:00.000 coordinates. OK, so there's, 00:33:00.000 --> 00:33:03.000 you know, a small list of things that you should know how 00:33:03.000 --> 00:33:07.000 to set up. But, you won't have some 00:33:07.000 --> 00:33:18.000 really, really strange thing. Yes? 00:33:18.000 --> 00:33:20.000 D rho? Oh, you mean the bounds for rho? 00:33:20.000 --> 00:33:23.000 Yes. So, in the inner integral, 00:33:23.000 --> 00:33:26.000 we are going to fix values of phi and theta. 00:33:26.000 --> 00:33:29.000 So, that means we fix in advance the direction in which 00:33:29.000 --> 00:33:31.000 we are going to shoot a ray from the origin. 00:33:31.000 --> 00:33:35.000 So now, as we shoot this ray, we are going to hit our region 00:33:35.000 --> 00:33:37.000 somewhere. And, we are going to exit, 00:33:37.000 --> 00:33:40.000 again, somewhere else. OK, so first of all we have to 00:33:40.000 --> 00:33:43.000 figure out where we enter, where we leave. 00:33:43.000 --> 00:33:46.000 Well, we enter when the ray hits the flat face, 00:33:46.000 --> 00:33:50.000 when we hit the plane. And, we would leave when we hit 00:33:50.000 --> 00:33:52.000 the sphere. So, the lower bound will be 00:33:52.000 --> 00:33:56.000 given by the plane. The upper bound will be given 00:33:56.000 --> 00:33:58.000 by the sphere. So now, you have to get 00:33:58.000 --> 00:34:01.000 spherical coordinate equations for both the plane and the 00:34:01.000 --> 00:34:02.000 sphere. For the sphere, that's easy. 00:34:02.000 --> 00:34:05.000 That's rho equals one. For the plane, 00:34:05.000 --> 00:34:08.000 you start with z equals one over root two. 00:34:08.000 --> 00:34:11.000 And, you switch it into spherical coordinates. 00:34:11.000 --> 00:34:14.000 And then, you solve for rho. And, that's how you get these 00:34:14.000 --> 00:34:19.000 bounds. Is that OK? 00:34:19.000 --> 00:34:26.000 All right, so that's the setup part. 00:34:26.000 --> 00:34:29.000 And, of course, the evaluation part goes as 00:34:29.000 --> 00:34:30.000 usual. 00:34:42.000 --> 00:34:46.000 And, since I'm running short of time, I'm not going to actually 00:34:46.000 --> 00:34:52.000 do the evaluation. I'm going to let you figure out 00:34:52.000 --> 00:34:58.000 how it goes. Let me just say in case you 00:34:58.000 --> 00:35:07.000 want to check your answers, so, at the end you get 2pi over 00:35:07.000 --> 00:35:13.000 three minus 5pi over six root two. 00:35:13.000 --> 00:35:17.000 Yes, it looks quite complicated. That's basically because you 00:35:17.000 --> 00:35:20.000 get one over, well, you get a second square 00:35:20.000 --> 00:35:23.000 when you integrate C. When you integrate rho squared, 00:35:23.000 --> 00:35:24.000 you will get rho cubed over three. 00:35:24.000 --> 00:35:27.000 But that rho cubed will give you a second cube for the lower 00:35:27.000 --> 00:35:29.000 bound. And, when you integrate sine 00:35:29.000 --> 00:35:31.000 phi second cubed phi, you do a substitution. 00:35:31.000 --> 00:35:37.000 You see that integrates to one over second squared with a 00:35:37.000 --> 00:35:42.000 factor in front. So, in the second square, 00:35:42.000 --> 00:35:49.000 when you plug in, no, that's not quite all of it. 00:35:49.000 --> 00:35:51.000 Yeah, well, the second square is one thing, 00:35:51.000 --> 00:35:53.000 and also the other bound you get sine phi which integrates to 00:35:53.000 --> 00:35:56.000 cosine phi. So, anyways, 00:35:56.000 --> 00:36:04.000 you get lots of things. OK, enough about it. 00:36:04.000 --> 00:36:07.000 So, next, I have to tell you about applications. 00:36:07.000 --> 00:36:13.000 And, of course, well, there's the same 00:36:13.000 --> 00:36:14.000 applications that we've seen that last time, 00:36:14.000 --> 00:36:16.000 finding volumes, finding masses, 00:36:16.000 --> 00:36:19.000 finding average values of functions. 00:36:19.000 --> 00:36:22.000 In particular, now, we could say to find the 00:36:22.000 --> 00:36:26.000 average distance of a point in this solid to the origin. 00:36:26.000 --> 00:36:28.000 Well, spherical coordinates become 00:36:28.000 --> 00:36:32.000 appealing because the function you are averaging is just rho 00:36:32.000 --> 00:36:35.000 while in other coordinate systems it's a more complicated 00:36:35.000 --> 00:36:37.000 function. So, if you are asked to find 00:36:37.000 --> 00:36:41.000 the average distance from the origin, spherical coordinates 00:36:41.000 --> 00:36:43.000 can be interesting. Also, 00:36:43.000 --> 00:36:47.000 well, there's moments of inertia, 00:36:47.000 --> 00:36:50.000 preferably the one about the z axis because if you have to 00:36:50.000 --> 00:36:52.000 integrate something that involves x or y, 00:36:52.000 --> 00:36:55.000 then your integrand will contain that awful rho sine phi 00:36:55.000 --> 00:36:57.000 sine theta or rho sine phi cosine theta, 00:36:57.000 --> 00:37:00.000 and then it won't be much fun to evaluate. 00:37:00.000 --> 00:37:05.000 So, that anyway, there's the usual ones. 00:37:05.000 --> 00:37:08.000 And then there's a new one. So, in physics, 00:37:08.000 --> 00:37:16.000 you've probably seen things about gravitational attraction. 00:37:16.000 --> 00:37:19.000 If not, well, it's what causes apples to fall 00:37:19.000 --> 00:37:22.000 and other things like that as well. 00:37:22.000 --> 00:37:26.000 So, anyway, physics tells you that if you have two masses, 00:37:26.000 --> 00:37:30.000 then they attract each other with a force that's directed 00:37:30.000 --> 00:37:33.000 towards each other. And in intensity, 00:37:33.000 --> 00:37:37.000 it's proportional to the two masses, and inversely 00:37:37.000 --> 00:37:41.000 proportional to the square of the distance between them. 00:37:41.000 --> 00:37:45.000 So, if you have a given solid with 00:37:45.000 --> 00:37:50.000 a certain mass distribution, and you want to know how it 00:37:50.000 --> 00:37:53.000 attracts something else that you will put nearby, 00:37:53.000 --> 00:37:58.000 then you actually have to, the first approximation will be 00:37:58.000 --> 00:37:59.000 to say, well, let's just put a point 00:37:59.000 --> 00:38:02.000 mass at its center of mass. But, if you're solid is 00:38:02.000 --> 00:38:04.000 actually not homogenous, or has a weird shape, 00:38:04.000 --> 00:38:07.000 then that's not actually the exact answer. 00:38:07.000 --> 00:38:09.000 So, in general, you would have to just take 00:38:09.000 --> 00:38:12.000 every single piece of your object and figure out how it 00:38:12.000 --> 00:38:14.000 attracts you, and then compute the sum of 00:38:14.000 --> 00:38:15.000 these. So, for example, 00:38:15.000 --> 00:38:18.000 if you want to understand why anything that you drop in this 00:38:18.000 --> 00:38:21.000 room will fall down, you have to understand that 00:38:21.000 --> 00:38:24.000 Boston is actually attracting it towards Boston. 00:38:24.000 --> 00:38:26.000 And, Somerville's attracting it towards Somerville, 00:38:26.000 --> 00:38:29.000 and lots of things like that. And, China, which is much 00:38:29.000 --> 00:38:33.000 further on the other side is going to attract towards China. 00:38:33.000 --> 00:38:35.000 But, there's a lot of stuff on the other side of the Earth. 00:38:35.000 --> 00:38:37.000 And so, overall, it's supposed to end up just 00:38:37.000 --> 00:38:41.000 going down. OK, so now, how to find this 00:38:41.000 --> 00:38:47.000 out, well, you have to just integrate over the entire Earth. 00:38:47.000 --> 00:38:52.000 OK, so let's try to see how that goes. 00:38:52.000 --> 00:38:56.000 So, the setup that's going to be easiest for us to do 00:38:56.000 --> 00:39:01.000 computations is going to be that we are going to be the test mass 00:39:01.000 --> 00:39:04.000 that's going to be falling. And, we are going to put 00:39:04.000 --> 00:39:07.000 ourselves at the origin. And, the solid that's going to 00:39:07.000 --> 00:39:10.000 attract us is going to be wherever we want in space. 00:39:10.000 --> 00:39:13.000 You'll see, putting yourself at the origin is going to be 00:39:13.000 --> 00:39:15.000 better. Well, you have to put something 00:39:15.000 --> 00:39:17.000 at the origin. And, the one that will stay a 00:39:17.000 --> 00:39:21.000 point mass, I mean, in my case not really a point, 00:39:21.000 --> 00:39:24.000 but anyway, let's say that I'm a point. 00:39:24.000 --> 00:39:27.000 And then, I have a solid attracting me. 00:39:27.000 --> 00:39:32.000 Well, so then if I take a small piece 00:39:32.000 --> 00:39:37.000 of it with the mass delta M, then that portion of the solid 00:39:37.000 --> 00:39:42.000 exerts a force on me, which is going to be directed 00:39:42.000 --> 00:39:47.000 towards it, and we'll have intensity. 00:39:47.000 --> 00:39:59.000 So, the gravitational force -- -- exerted by the mass delta M 00:39:59.000 --> 00:40:09.000 at the point of x, y, z in space on a mass at the 00:40:09.000 --> 00:40:13.000 origin. Well, we know how to express 00:40:13.000 --> 00:40:16.000 that. Physics tells us that the 00:40:16.000 --> 00:40:21.000 magnitude of this force is going to be, well, G is just a 00:40:21.000 --> 00:40:23.000 constant. It's the gravitational 00:40:23.000 --> 00:40:27.000 constant, and its value depends on which unit system you use. 00:40:27.000 --> 00:40:33.000 Usually it's pretty small, times the mass delta M, 00:40:33.000 --> 00:40:39.000 times the test mass little m, divided by the square of the 00:40:39.000 --> 00:40:43.000 distance. And, the distance from U to 00:40:43.000 --> 00:40:48.000 that thing is conveniently called rho since we've been 00:40:48.000 --> 00:40:51.000 introducing spherical coordinates. 00:40:51.000 --> 00:40:54.000 So, that's the size, that's the magnitude of the 00:40:54.000 --> 00:40:56.000 force. We also need to know the 00:40:56.000 --> 00:41:01.000 direction of the force. And, the direction is going to 00:41:01.000 --> 00:41:07.000 be towards that point. So, the direction of the force 00:41:07.000 --> 00:41:11.000 is going to be that of x, y, z. 00:41:11.000 --> 00:41:13.000 But if I want a unit vector, then I should scale this down 00:41:13.000 --> 00:41:22.000 to length one. So, let me divide this by rho 00:41:22.000 --> 00:41:32.000 to get a unit vector. So, that means that the force 00:41:32.000 --> 00:41:40.000 I'm getting from this guy is actually going to be G delta M m 00:41:40.000 --> 00:41:44.000 over rho cubed times x, y, z. 00:41:44.000 --> 00:41:50.000 I'm just multiplying the magnitude by the unit vector in 00:41:50.000 --> 00:41:54.000 the correct direction. OK, so now if I have not just 00:41:54.000 --> 00:41:56.000 that little p is delta M, but an entire solid, 00:41:56.000 --> 00:41:59.000 then I have to sum all these guys together. 00:41:59.000 --> 00:42:04.000 And, I will get the vector that gives me the total force 00:42:04.000 --> 00:42:06.000 exerted, OK? So, of course, 00:42:06.000 --> 00:42:09.000 there's actually three different calculations in one 00:42:09.000 --> 00:42:12.000 because you have to sum the x components to get the x 00:42:12.000 --> 00:42:16.000 components of a total force. Same with the y, 00:42:16.000 --> 00:42:28.000 and same with the z. So, let me first write down the 00:42:28.000 --> 00:42:36.000 actual formula. So, if you integrate over the 00:42:36.000 --> 00:42:39.000 entire solid, oh, and I have to remind you, 00:42:39.000 --> 00:42:42.000 well, what's the mass, delta M of a small piece of 00:42:42.000 --> 00:42:45.000 volume delta V? Well, it's the density times 00:42:45.000 --> 00:42:48.000 the volume. So, the mass is going to be, 00:42:48.000 --> 00:42:54.000 sorry, density is delta. There is a lot of Greek letters 00:42:54.000 --> 00:43:04.000 there, times the volume element. So, you will get that the force 00:43:04.000 --> 00:43:12.000 is the triple integral over your solid of G m x, 00:43:12.000 --> 00:43:18.000 y, z over rho cubed, delta dV. 00:43:18.000 --> 00:43:21.000 Now, two observations about that. 00:43:21.000 --> 00:43:23.000 So, the first one, well, of course, 00:43:23.000 --> 00:43:29.000 these are just constants. So, they can go out. 00:43:29.000 --> 00:43:31.000 The second observation, so here, we are integrating a 00:43:31.000 --> 00:43:33.000 vector quantity. So, what does that mean? 00:43:33.000 --> 00:43:38.000 I just mean the x component of a force is given by integrating 00:43:38.000 --> 00:43:41.000 G m x over rho cubed delta dV. The y components, 00:43:41.000 --> 00:43:43.000 same thing with y. The z components, 00:43:43.000 --> 00:43:46.000 same thing with z. OK, there's no, 00:43:46.000 --> 00:43:51.000 like, you know, just integrate component by 00:43:51.000 --> 00:43:56.000 component to get each component of the force. 00:43:56.000 --> 00:44:01.000 So, now we could very well to this in rectangular coordinates 00:44:01.000 --> 00:44:04.000 if we want. But the annoying thing is this 00:44:04.000 --> 00:44:06.000 rho cubed. Rho cubed is going to be x 00:44:06.000 --> 00:44:10.000 squared plus y squared plus z squared to the three halves. 00:44:10.000 --> 00:44:13.000 That's not going to be a very pleasant thing to integrate. 00:44:13.000 --> 00:44:24.000 So, it's much better to set up these integrals in spherical 00:44:24.000 --> 00:44:29.000 coordinates. And, if we're going to do it in 00:44:29.000 --> 00:44:32.000 spherical coordinates, then probably we don't want to 00:44:32.000 --> 00:44:34.000 bother too much with x and y components because those would 00:44:34.000 --> 00:44:38.000 be unpleasant. It would give us rho sine phi 00:44:38.000 --> 00:44:47.000 cos theta or sine theta. So, the actual way we will set 00:44:47.000 --> 00:44:57.000 up things, set things up, is to place the solid so that 00:44:57.000 --> 00:45:04.000 the z axis is an axis of symmetry. 00:45:04.000 --> 00:45:07.000 And, of course, that only works if the solid 00:45:07.000 --> 00:45:10.000 has some axis of symmetry. Like, if you're trying to find 00:45:10.000 --> 00:45:13.000 the gravitational attraction of the Pyramid of Giza, 00:45:13.000 --> 00:45:16.000 then you won't be able to set up so that it has rotation 00:45:16.000 --> 00:45:18.000 symmetry. Well, that's a tough fact of 00:45:18.000 --> 00:45:21.000 life, and you have to actually do it in x, y, 00:45:21.000 --> 00:45:24.000 z coordinates. But, if at all possible, 00:45:24.000 --> 00:45:27.000 then you're going to place things. 00:45:27.000 --> 00:45:30.000 Well, I guess even then, you could center it on the z 00:45:30.000 --> 00:45:32.000 axis. But anyway, so you're going to 00:45:32.000 --> 00:45:37.000 mostly place things so that your solid is actually centered on 00:45:37.000 --> 00:45:41.000 the z-axis. And, what you gain by that is 00:45:41.000 --> 00:45:45.000 that by symmetry, the gravitational force will be 00:45:45.000 --> 00:45:52.000 directed along the z axis. So, you will just have to 00:45:52.000 --> 00:45:58.000 figure out the z component. So, then the force will be 00:45:58.000 --> 00:46:03.000 actually, you know in advance that it will be given by zero, 00:46:03.000 --> 00:46:11.000 zero, and some z component. And then, you just need to 00:46:11.000 --> 00:46:19.000 compute that component. And, that component will be 00:46:19.000 --> 00:46:27.000 just G times m times triple integral of z over rho cubed 00:46:27.000 --> 00:46:30.000 delta dV. OK, so that's the first 00:46:30.000 --> 00:46:35.000 simplification we can try to do. The second thing is, 00:46:35.000 --> 00:46:38.000 well, we have to choose our favorite coordinate system to do 00:46:38.000 --> 00:46:45.000 this. But, I claim that actually 00:46:45.000 --> 00:46:57.000 spherical coordinates are the best -- -- because let's see 00:46:57.000 --> 00:47:04.000 what happens. So, G times mass times triple 00:47:04.000 --> 00:47:09.000 integral, well, a z in spherical coordinates 00:47:09.000 --> 00:47:14.000 becomes rho cosine phi over rho cubed. 00:47:14.000 --> 00:47:17.000 Density, well, we can't do anything about 00:47:17.000 --> 00:47:21.000 density. And then, dV becomes rho 00:47:21.000 --> 00:47:28.000 squared sine phi d rho d phi d theta. 00:47:28.000 --> 00:47:34.000 Well, so, what happens with that? 00:47:34.000 --> 00:47:37.000 Well, you see that you have a rho, a rho squared, 00:47:37.000 --> 00:47:39.000 and a rho cubed that cancel each other. 00:47:39.000 --> 00:47:42.000 So, in fact, it simplifies quite a bit if 00:47:42.000 --> 00:47:44.000 you do it in spherical coordinates. 00:48:08.000 --> 00:48:12.000 OK, so the z component of the force, sorry, 00:48:12.000 --> 00:48:18.000 I'm putting a z here to remind you it's the z component. 00:48:18.000 --> 00:48:19.000 That is not a partial derivative, OK? 00:48:19.000 --> 00:48:27.000 Don't get things mixed up, just the z component of the 00:48:27.000 --> 00:48:35.000 force becomes Gm triple integral of delta cos phi sine phi d rho 00:48:35.000 --> 00:48:40.000 d phi d theta. And, so this thing is not dV, 00:48:40.000 --> 00:48:42.000 of course. dV is much bigger, 00:48:42.000 --> 00:48:45.000 but we've somehow canceled out most of dV with stuff that was 00:48:45.000 --> 00:48:49.000 in the integrand. And see, that's actually 00:48:49.000 --> 00:48:55.000 suddenly much less scary. OK, so just to give you an 00:48:55.000 --> 00:49:01.000 example of what you can prove it this way, you can prove Newton's 00:49:01.000 --> 00:49:06.000 theorem, which says the following thing. 00:49:06.000 --> 00:49:23.000 It says the gravitational attraction -- -- of a spherical 00:49:23.000 --> 00:49:29.000 planet, I should say with uniform 00:49:29.000 --> 00:49:32.000 density, or actually it's enough for the 00:49:32.000 --> 00:49:34.000 density to depend just on distance to the center. 00:49:34.000 --> 00:49:50.000 But we just simplify the statement is equal to that of a 00:49:50.000 --> 00:50:05.000 point mass -- -- with the same total mass at its center. 00:50:05.000 --> 00:50:11.000 OK, so what that means is that, so the way we would set it up 00:50:11.000 --> 00:50:18.000 is u would be sitting here and your planet would be over here. 00:50:18.000 --> 00:50:21.000 Or, if you're at the surface of it, then of course you just put 00:50:21.000 --> 00:50:25.000 it tangent to the xy plane here. And, you would compute that 00:50:25.000 --> 00:50:27.000 quantity. Computation is a little bit 00:50:27.000 --> 00:50:30.000 annoying if a sphere is sitting up there because, 00:50:30.000 --> 00:50:31.000 of course, you have to find bounds, 00:50:31.000 --> 00:50:33.000 and that's not going to be very pleasant. 00:50:33.000 --> 00:50:37.000 The case that we actually know how to do fairly well is if you 00:50:37.000 --> 00:50:39.000 are just at the surface of the planet. 00:50:39.000 --> 00:50:41.000 But then, what the theorem says is that 00:50:41.000 --> 00:50:44.000 the force that you're going to feel is exactly the same as if 00:50:44.000 --> 00:50:48.000 you removed all of the planet and you just put an equivalent 00:50:48.000 --> 00:50:50.000 point mass here. So, if the earth collapsed to a 00:50:50.000 --> 00:50:53.000 black hole at the center of the earth with the same mass, 00:50:53.000 --> 00:50:55.000 well, you wouldn't notice the difference immediately, 00:50:55.000 --> 00:51:00.000 or, rather, you would, but at least not in terms of 00:51:00.000 --> 00:51:04.000 your weight. OK, that's the end for today.