1 00:00:01 --> 00:00:03 The following content is provided under a Creative 2 00:00:03 --> 00:00:05 Commons license. Your support will help MIT 3 00:00:05 --> 00:00:08 OpenCourseWare continue to offer high quality educational 4 00:00:08 --> 00:00:13 resources for free. To make a donation or to view 5 00:00:13 --> 00:00:18 additional materials from hundreds of MIT courses, 6 00:00:18 --> 00:00:23 visit MIT OpenCourseWare at ocw.mit.edu. 7 00:00:25 --> 00:00:30 We are going to continue to look at stuff in space. 8 00:00:30 --> 00:00:34 We have been working with triple integrals and seeing how 9 00:00:34 --> 00:00:37 to set them up in all sorts of coordinate systems. 10 00:00:37 --> 00:00:49 And the next topic we will be looking at are vector fields in 11 00:00:49 --> 00:00:52 space. And so, in particular, 12 00:00:52 --> 00:00:55 we will be learning about flux and work. 13 00:00:55 --> 00:00:58 So, just for a change, we will be starting with flux 14 00:00:58 --> 00:01:00 first. And we will do work, 15 00:01:00 --> 00:01:04 actually, after Thanksgiving. Just to remind you, 16 00:01:04 --> 00:01:08 a vector field in space is just the same thing as in the plane. 17 00:01:08 --> 00:01:19 At every point you have a vector, and the components of 18 00:01:19 --> 00:01:26 this vector depend on the coordinates x, 19 00:01:26 --> 00:01:34 y and z. Let's say the components might 20 00:01:34 --> 00:01:42 be P, Q, R, or your favorite three letters, 21 00:01:42 --> 00:01:53 where each of these things is a function of coordinates x, 22 00:01:53 --> 00:01:56 y, z. You have seen that in the plane 23 00:01:56 --> 00:01:58 it is already pretty hard to draw a vector field. 24 00:01:58 --> 00:02:01 Usually, in space, we won't really try too hard. 25 00:02:01 --> 00:02:04 But it is still useful to try to have a general idea for what 26 00:02:04 --> 00:02:07 the vectors in there are doing, whether they are all going in 27 00:02:07 --> 00:02:09 the same direction, whether they may be all 28 00:02:09 --> 00:02:12 vertical or horizontal, pointing away from the origin, 29 00:02:12 --> 00:02:15 towards it, things like that. 30 00:02:15 --> 00:02:19 But, generally-speaking, we won't really bother with 31 00:02:19 --> 00:02:24 trying to draw a picture because that is going to be quite hard. 32 00:02:24 --> 00:02:30 Just to give you examples, well, the same kinds of 33 00:02:30 --> 00:02:37 examples as the plane, you can think of force fields. 34 00:02:37 --> 00:02:48 For example, the gravitational attraction -- 35 00:02:48 --> 00:03:01 -- of a solid mass, let's call this mass big M, 36 00:03:01 --> 00:03:13 at the origin on a mass M at point x, y, z. 37 00:03:13 --> 00:03:20 That would be given by a vector field that points toward the 38 00:03:20 --> 00:03:27 origin and whose magnitude is inversely proportional to the 39 00:03:27 --> 00:03:31 square of a distance from the origin. 40 00:03:31 --> 00:03:39 Such a field would be directed towards the origin and its 41 00:03:39 --> 00:03:47 magnitude would be of the order of a constant over pho squared 42 00:03:47 --> 00:03:53 where pho is the distance from the origin. 43 00:03:53 --> 00:03:57 The picture, if I really wanted to draw a 44 00:03:57 --> 00:04:02 picture, would be everywhere it is a field that points towards 45 00:04:02 --> 00:04:09 the origin. And if I am further away then 46 00:04:09 --> 00:04:13 it gets smaller. And, of course, 47 00:04:13 --> 00:04:16 I am not going to try to draw all these vectors in there. 48 00:04:16 --> 00:04:27 If I wanted to give a formula for that -- A formula for that 49 00:04:27 --> 00:04:36 might be something of a form minus c times x, 50 00:04:36 --> 00:04:41 y, z over pho cubed. Let's see. 51 00:04:41 --> 00:04:44 Well, the direction of this vector, this vector is 52 00:04:44 --> 00:04:46 proportional to negative x, y, z. 53 00:04:46 --> 00:04:47 54 00:04:47 --> 00:04:49 is the vector that goes from the origin to your point. 55 00:04:49 --> 00:04:52 The negative goes towards the origin. 56 00:04:52 --> 00:04:56 Then the magnitude of this guy, well, the magnitude of x, 57 00:04:56 --> 00:05:00 y, z is just the distance from the origin rho. 58 00:05:00 --> 00:05:05 So the magnitude of this thing is one over rho cubed times some 59 00:05:05 --> 00:05:10 constant factor. That would be an example of a 60 00:05:10 --> 00:05:14 vector field that comes up in physics. 61 00:05:14 --> 00:05:20 Well, other examples would be electric fields. 62 00:05:20 --> 00:05:23 Actually, if you look at the electric field generated by a 63 00:05:23 --> 00:05:27 charged particle at the origin, it is given by exactly the same 64 00:05:27 --> 00:05:36 kind of formula, and there are magnetic fields 65 00:05:36 --> 00:05:42 and so on. Another example comes from 66 00:05:42 --> 00:05:46 velocity fields. If you have a fluid flow, 67 00:05:46 --> 00:05:49 for example, if you want to study wind 68 00:05:49 --> 00:05:52 patterns in the atmosphere. Well, wind, most of the time, 69 00:05:52 --> 00:05:55 is kind of horizontal, but maybe it depends on the 70 00:05:55 --> 00:05:58 altitude. At high altitude you have jet 71 00:05:58 --> 00:06:03 streams, and the wind velocity is not the same at all 72 00:06:03 --> 00:06:08 altitudes. And, just to give you more 73 00:06:08 --> 00:06:15 examples, in math we have seen that the gradient of a function 74 00:06:15 --> 00:06:21 of three variables gives you a vector field. 75 00:06:21 --> 00:06:25 If you have a function u of x, y, z then its gradient field 76 00:06:25 --> 00:06:30 has just components, u sub x, u sub y and u sub z. 77 00:06:30 --> 00:06:32 And, of course, the cases are not mutually 78 00:06:32 --> 00:06:34 exclusive. For example, 79 00:06:34 --> 00:06:38 the electric field or gravitational field is given by 80 00:06:38 --> 00:06:42 the gradient of the gravitational or electric 81 00:06:42 --> 00:06:46 potential. So, these are not like 82 00:06:46 --> 00:06:51 different cases. There is overlap. 83 00:06:51 --> 00:06:54 Anyway, hopefully, you are kind of convinced that 84 00:06:54 --> 00:06:57 you should learn about vector fields. 85 00:06:57 --> 00:06:59 What are we going to do with them? 86 00:06:59 --> 00:07:16 Well, let's start with flux. Remember not so long ago we 87 00:07:16 --> 00:07:24 looked at flux of a two-dimensional field of a 88 00:07:24 --> 00:07:32 curve. We had a curve in the plane and 89 00:07:32 --> 00:07:41 we had a vector field. And we looked at the component 90 00:07:41 --> 00:07:47 of a vector field in the direction that was normal to the 91 00:07:47 --> 00:07:51 curve. We formed the flux integral 92 00:07:51 --> 00:07:54 that was a line integral F dot n ds. 93 00:07:54 --> 00:07:59 And that measured how much the vector field was going across 94 00:07:59 --> 00:08:01 the curve. If you were thinking of a 95 00:08:01 --> 00:08:05 velocity field, that would measure how much 96 00:08:05 --> 00:08:08 fluid is passing through the curve in unit time. 97 00:08:08 --> 00:08:10 Now let's say that we were in space. 98 00:08:10 --> 00:08:14 Well, we cannot really think of flux as a line integral. 99 00:08:14 --> 00:08:18 Because, if you have a curve in space and say that you have wind 100 00:08:18 --> 00:08:22 or something like that, you cannot really ask how much 101 00:08:22 --> 00:08:24 air is flowing through the curve. 102 00:08:24 --> 00:08:28 See, to have a flow through something you need a surface. 103 00:08:28 --> 00:08:32 If you have a net maybe then you can ask how much stuff is 104 00:08:32 --> 00:08:37 passing through that surface. There is going to be a big 105 00:08:37 --> 00:08:44 difference here. In the three-dimensional space, 106 00:08:44 --> 00:08:51 flux will be measured through a surface. 107 00:08:51 --> 00:08:54 And so it will be a surface integral, not a line integral 108 00:08:54 --> 00:08:59 anymore. That means we will be 109 00:08:59 --> 00:09:10 integrating, we will be summing over all the pieces of a surface 110 00:09:10 --> 00:09:13 in space. Because a surface is a 111 00:09:13 --> 00:09:15 two-dimensional object, that will end up being a double 112 00:09:15 --> 00:09:17 integral. But, of course, 113 00:09:17 --> 00:09:19 we will have to set it up properly because the surface 114 00:09:19 --> 00:09:22 that is in space, and we will probably have x, 115 00:09:22 --> 00:09:24 y and z to deal with at the same time, 116 00:09:24 --> 00:09:28 and we will have to somehow get rid of one variable so that we 117 00:09:28 --> 00:09:31 can set up and evaluate a double integral. 118 00:09:31 --> 00:09:35 So conceptually it is very similar to line integrals. 119 00:09:35 --> 00:09:39 In the line integral in the plane, you had two variables 120 00:09:39 --> 00:09:42 that you reduced to one by figuring out what the curve was. 121 00:09:42 --> 00:09:51 Here you have three variables that you will reduce to two by 122 00:09:51 --> 00:09:56 figuring out what the surface is. 123 00:09:56 --> 00:10:00 Let me give you a definition of flux in 3D. 124 00:10:00 --> 00:10:15 Let's say that we have a vector field and s, a surface in space. 125 00:10:15 --> 00:10:17 Let me draw some kind of a picture. 126 00:10:17 --> 00:10:21 I have my surface and I have my vector field F. 127 00:10:21 --> 00:10:25 Well, at every point it changes with a point. 128 00:10:25 --> 00:10:28 Well, I want to figure out how much my vector field is going 129 00:10:28 --> 00:10:34 across that surface. That means I want to figure out 130 00:10:34 --> 00:10:40 the normal component of my vector field, 131 00:10:40 --> 00:10:45 so I will use, as in the plane case, 132 00:10:45 --> 00:10:53 the unit normal vector to s. I take my point on the surface 133 00:10:53 --> 00:10:59 and build a unit vector that is standing on it perpendicularly. 134 00:10:59 --> 00:11:05 Now, we have to decide which way it is standing. 135 00:11:05 --> 00:11:09 We can build our normal vector to go this way or to go the 136 00:11:09 --> 00:11:12 other way around. There are two choices. 137 00:11:12 --> 00:11:16 Basically, whenever you want to set up a flux integral you have 138 00:11:16 --> 00:11:19 to choose one side of the surface. 139 00:11:19 --> 00:11:23 And you will count positively what flows toward that side and 140 00:11:23 --> 00:11:26 negatively what flows towards the other side. 141 00:11:26 --> 00:11:40 There are two choices for n. We need to choose a side of the 142 00:11:40 --> 00:11:45 surface. In the case of curves, 143 00:11:45 --> 00:11:50 we made that choice by deciding that because we were going along 144 00:11:50 --> 00:11:54 some direction on the curve we could choose one side by saying 145 00:11:54 --> 00:11:57 let's rotate clockwise from the tangent vector. 146 00:11:57 --> 00:12:00 And, in a way, what we were doing was really 147 00:12:00 --> 00:12:04 it was a recipe to choose for us one of the two sides. 148 00:12:04 --> 00:12:09 Here we don't have a notion of orienting the surface other than 149 00:12:09 --> 00:12:14 by precisely choosing one of the two possible normal vectors. 150 00:12:14 --> 00:12:15 So, in fact, this is called choosing an 151 00:12:15 --> 00:12:18 orientation of a surface. When you are saying you are 152 00:12:18 --> 00:12:22 orienting the surface that really means you are deciding 153 00:12:22 --> 00:12:31 which side is which. Let's call that orientation. 154 00:12:31 --> 00:12:35 Now, there is no set convention that will work forever. 155 00:12:35 --> 00:12:39 But the usually traditional settings would be to take your 156 00:12:39 --> 00:12:43 normal vector pointing maybe out of the solid region because then 157 00:12:43 --> 00:12:48 you will be looking at flux that is coming out of that region of 158 00:12:48 --> 00:12:51 space. Or, if you have a surface that 159 00:12:51 --> 00:12:55 is not like closed or anything but maybe you will want the flux 160 00:12:55 --> 00:12:59 going up through the region. Or, there are various 161 00:12:59 --> 00:13:02 conventions. Concretely, on problem sets it 162 00:13:02 --> 00:13:05 will either say which choice you have to make or you get to 163 00:13:05 --> 00:13:07 choose which one you want to make. 164 00:13:07 --> 00:13:10 And, of course, if you choose the other one 165 00:13:10 --> 00:13:12 then the sign becomes the opposite. 166 00:13:12 --> 00:13:17 Now, once we have made a choice then we can define the flux 167 00:13:17 --> 00:13:21 integral. It will just be the double 168 00:13:21 --> 00:13:26 integral over a surface of F dot n dS. 169 00:13:26 --> 00:13:33 Now I am using a big dS. That stands for the surface 170 00:13:33 --> 00:13:39 area element on this surface. I am using dS rather than dA 171 00:13:39 --> 00:13:43 because I still want to think of dA as maybe the area in one of 172 00:13:43 --> 00:13:47 the coordinate planes like the one we had in double integrals. 173 00:13:47 --> 00:13:51 You will see later where this comes in. 174 00:13:51 --> 00:13:54 But conceptually it is very similar. 175 00:13:54 --> 00:13:59 Concretely what this means is I cut my surface into little 176 00:13:59 --> 00:14:03 pieces. Each of them has area delta S. 177 00:14:03 --> 00:14:07 And, for each piece, I take my vector field, 178 00:14:07 --> 00:14:13 I take my normal vector, I dot them and I multiply by 179 00:14:13 --> 00:14:17 this surface area and sum all these things together. 180 00:14:17 --> 00:14:23 That is what a double integral means. 181 00:14:23 --> 00:14:25 In particular, an easy case where you know you 182 00:14:25 --> 00:14:28 can get away without computing anything is, of course, 183 00:14:28 --> 00:14:32 if your vector field is tangent to the surface because then you 184 00:14:32 --> 00:14:36 know that there is no flux. Flux is going to be zero 185 00:14:36 --> 00:14:38 because nothing passes through the surface. 186 00:14:38 --> 00:14:42 Otherwise, we have to figure out how to compute these things. 187 00:14:42 --> 00:14:50 That is what we are going to learn now. 188 00:14:50 --> 00:14:51 Well, maybe I should box this formula. 189 00:14:51 --> 00:14:57 I have noticed that some of you seem to like it when I box the 190 00:14:57 --> 00:15:03 important formulas. (APPLAUSE) By the way, 191 00:15:03 --> 00:15:12 a piece of notation before I move on, sometimes you will also 192 00:15:12 --> 00:15:18 see the notation vector dS. What is vector dS? 193 00:15:18 --> 00:15:24 Vector dS is this guy n dS put together. 194 00:15:24 --> 00:15:30 Vector dS is a vector which points perpendicular to the 195 00:15:30 --> 00:15:35 surface and whose length corresponds to the surface 196 00:15:35 --> 00:15:37 element. And the reason for having this 197 00:15:37 --> 00:15:41 shortcut notation, well, it is not only laziness 198 00:15:41 --> 00:15:44 like saving one n, but it is because this guy is 199 00:15:44 --> 00:15:49 very often easier to compute than it is to set up n and dS 200 00:15:49 --> 00:15:52 separately. Actually, if you remember in 201 00:15:52 --> 00:15:56 the plane, we have seen that vector n little ds can be 202 00:15:56 --> 00:15:59 written directly as dy, - dx. 203 00:15:59 --> 00:16:03 That was easier than finding n and ds separately. 204 00:16:03 --> 00:16:15 And here the same is going to be true in many cases. 205 00:16:15 --> 00:16:23 Well, any questions before we do examples? 206 00:16:23 --> 00:16:24 No. OK. 207 00:16:24 --> 00:16:38 Let's do examples. The first example for today is 208 00:16:38 --> 00:16:51 we are going to look at the flux of vector field xi yj xk through 209 00:16:51 --> 00:17:02 the sphere of radius a -- -- centered at the origin. 210 00:17:02 --> 00:17:18 What does the picture look like? We have a sphere of radius a. 211 00:17:18 --> 00:17:22 I have my vector field. Well, , see, 212 00:17:22 --> 00:17:25 that is a vector field that is equal to the vector from the 213 00:17:25 --> 00:17:35 origin to the point where I am, so it is pointing radially away 214 00:17:35 --> 00:17:42 from the origin. My vector field is really 215 00:17:42 --> 00:17:49 sticking out everywhere away from the origin. 216 00:17:49 --> 00:17:56 Now I have to find the normal vector to the sphere if I want 217 00:17:56 --> 00:18:04 to set up double integral over the sphere of F dot vector ds, 218 00:18:04 --> 00:18:09 or if you want F dot n dS. What does the normal vector to 219 00:18:09 --> 00:18:12 the sphere look like? Well, it depends, 220 00:18:12 --> 00:18:14 of course, whether I choose it pointing out or in. 221 00:18:14 --> 00:18:18 Let's say I am choosing it pointing out then it will be 222 00:18:18 --> 00:18:20 sticking straight out of a sphere as well. 223 00:18:20 --> 00:18:27 Hopefully, you can see that if I take a normal vector to the 224 00:18:27 --> 00:18:34 sphere it is actually pointing radially out away from the 225 00:18:34 --> 00:18:38 origin. In fact, our vector field and 226 00:18:38 --> 00:18:41 our normal vector are parallel to each other. 227 00:18:41 --> 00:18:45 Let's think a bit more about what a normal vector looks like. 228 00:18:45 --> 00:18:47 I said it is sticking straight out. 229 00:18:47 --> 00:18:49 It is proportional to this vector field. 230 00:18:49 --> 00:18:51 Maybe I should start by writing 231 00:18:51 --> 00:18:52 232 00:18:52 --> 00:18:56 because that is the vector that goes from the origin to my point 233 00:18:56 --> 00:18:59 so it points radially away from the origin. 234 00:18:59 --> 00:19:00 Now there is a small problem with that. 235 00:19:00 --> 00:19:04 It is not a unit vector. So what is its length? 236 00:19:04 --> 00:19:08 Well, its length is square root of x^2 y^2 z^2. 237 00:19:08 --> 00:19:13 But, if I am on the sphere, then that length is just equal 238 00:19:13 --> 00:19:16 to a because distance from the origin is a. 239 00:19:16 --> 00:19:23 In fact, I get my normal vector by scaling this guy down by a 240 00:19:23 --> 00:19:27 factor of a. And let me write it down just 241 00:19:27 --> 00:19:34 in case you are still unsure. This is unit because square 242 00:19:34 --> 00:19:43 root of x^2 y^2 z^2 is equal to a on the sphere. 243 00:19:43 --> 00:19:48 OK. Any questions about this? 244 00:19:48 --> 00:19:52 No. It looks OK? I see a lot of blank faces. 245 00:19:52 --> 00:19:58 That physics test must have been hard. 246 00:19:58 --> 00:20:03 Yes? I could have put a rho but I 247 00:20:03 --> 00:20:06 want to emphasize the fact that here it is going to be a 248 00:20:06 --> 00:20:09 constant. I mean rho has this connotation 249 00:20:09 --> 00:20:13 of being a variable that I will need to then maybe integrate 250 00:20:13 --> 00:20:17 over or do something with. Yes, it would be correct to put 251 00:20:17 --> 00:20:20 rho but I then later will want to replace it by its actual 252 00:20:20 --> 00:20:24 value which is a number. And the number is a. 253 00:20:24 --> 00:20:28 It is not going to actually change from point to point. 254 00:20:28 --> 00:20:30 For example, if this was the unit sphere 255 00:20:30 --> 00:20:32 then I would just put x, y, z. 256 00:20:32 --> 00:20:41 I wouldn't divide by anything. Now let's figure out F dot n. 257 00:20:41 --> 00:20:47 Let's do things one at a time. Well, F and n are parallel to 258 00:20:47 --> 00:20:53 each other. F dot n, the normal component 259 00:20:53 --> 00:21:00 of F, is actually equal to the length of F. 260 00:21:00 --> 00:21:05 Well, times the length of n if you want, but that is going to 261 00:21:05 --> 00:21:09 be a one since F and n are parallel to each other. 262 00:21:09 --> 00:21:12 And what is the magnitude of F if I am on the sphere? 263 00:21:12 --> 00:21:18 Well, the magnitude of F in general is square root of x^2 264 00:21:18 --> 00:21:23 y^2 z^2 on the sphere that is going be a. 265 00:21:23 --> 00:21:25 The other way to do it, if you don't want to think 266 00:21:25 --> 00:21:28 geometrically like that, is to just to do the dot 267 00:21:28 --> 00:21:31 product x, y, z doted with x over a, 268 00:21:31 --> 00:21:35 y over a, z over a. You will be x^2 y^2 z^2 divided 269 00:21:35 --> 00:21:40 by a. That will simplify to a because 270 00:21:40 --> 00:21:45 we are on the sphere. See, we are already using here 271 00:21:45 --> 00:21:47 the relation between x, y and z. 272 00:21:47 --> 00:21:49 We are not letting x, y and z be completely 273 00:21:49 --> 00:21:51 arbitrary. But the slogan is everything 274 00:21:51 --> 00:21:54 happens on the surface where we are doing the integral. 275 00:21:54 --> 00:21:56 We are not looking at anything inside or outside. 276 00:21:56 --> 00:21:58 We are just on the surface. 277 00:21:58 --> 00:22:34 278 00:22:34 --> 00:22:43 Now what do I do with that? Well, I have turned my integral 279 00:22:43 --> 00:22:50 into the double integral of a dS. 280 00:22:50 --> 00:22:53 And a is just a constant, so I am very lucky here. 281 00:22:53 --> 00:22:59 I can just say this will be a times the double integral of dS. 282 00:22:59 --> 00:23:02 And, of course, some day I will have to learn 283 00:23:02 --> 00:23:05 how to tackle that beast, but for now I don't actually 284 00:23:05 --> 00:23:08 need to because the double integral of dS just means I am 285 00:23:08 --> 00:23:11 summing the area of each little piece of the sphere. 286 00:23:11 --> 00:23:16 I am just going to get the total area of the sphere which I 287 00:23:16 --> 00:23:23 know to be 4pi a2. This guy here is going to be 288 00:23:23 --> 00:23:29 the area of S. I know that to be 4pi a^2. 289 00:23:29 --> 00:23:37 So I will get 4pi a^3. That one was relatively 290 00:23:37 --> 00:23:44 painless. That was too easy. 291 00:23:44 --> 00:23:49 Let's do a second example with the same sphere. 292 00:23:49 --> 00:23:56 But now my vector field is going to be just z times k. 293 00:23:56 --> 00:23:58 Well, let me give it a different name. 294 00:23:58 --> 00:24:04 Let me call it H instead of f or something like that just so 295 00:24:04 --> 00:24:08 that it is not called F anymore. Well, the initial part of the 296 00:24:08 --> 00:24:11 setup is still the same. The normal vector is still the 297 00:24:11 --> 00:24:13 same. What changes is, 298 00:24:13 --> 00:24:16 of course, my vector field is no longer sticking straight out 299 00:24:16 --> 00:24:18 so I cannot use this easy geometric argument. 300 00:24:18 --> 00:24:22 It looks like I will have to compute F dot n and then figure 301 00:24:22 --> 00:24:24 out how to integrate that with dS. 302 00:24:24 --> 00:24:36 Let's do that. We still have that n is 00:24:44 y, z>/a. That tells us that H dot n will 304 00:24:44 --> 00:24:49 be dot 00:24:57 z> / a. It looks like I will be left 306 00:24:57 --> 00:25:10 with z^2 over a. H dot n is z^2 over a. 307 00:25:10 --> 00:25:18 The double integral for flux now becomes double integral on 308 00:25:18 --> 00:25:25 the sphere of z^2 over a dS. Well, we can take out one over 309 00:25:25 --> 00:25:29 a, that is fine, but it looks like we will have 310 00:25:29 --> 00:25:33 to integrate z^2 on the surface of the sphere. 311 00:25:33 --> 00:25:37 How do we do that? Well, we have to figure out 312 00:25:37 --> 00:25:41 what is dS in terms of our favorite set of two variables 313 00:25:41 --> 00:25:45 that we will use to integrate. Now, what is the best way to 314 00:25:45 --> 00:25:47 figure out where you are on the sphere? 315 00:25:47 --> 00:25:51 Well, you could try to use maybe theta and z. 316 00:25:51 --> 00:25:55 If you know how high you are and where you are around, 317 00:25:55 --> 00:25:58 in principle you know where you are on the sphere. 318 00:25:58 --> 00:26:02 But since spherical coordinates we have actually learned about 319 00:26:02 --> 00:26:06 something much more interesting, namely spherical coordinates. 320 00:26:06 --> 00:26:09 It looks like longitude / latitude is the way to go when 321 00:26:09 --> 00:26:12 trying to figure out where you are on a sphere. 322 00:26:12 --> 00:26:19 We are going to use phi and theta. 323 00:26:19 --> 00:26:24 And, of course, we have to figure out how to 324 00:26:24 --> 00:26:28 express dS in terms of d phi and d theta. 325 00:26:28 --> 00:26:32 Well, if you were paying really, really close attention 326 00:26:32 --> 00:26:36 last time, you will notice that we have actually already done 327 00:26:36 --> 00:26:41 that. Last time we saw that if I have 328 00:26:41 --> 00:26:48 a sphere of radius a and I take a little piece of it that 329 00:26:48 --> 00:26:56 corresponds to small changes in phi and theta then we said that 330 00:26:56 --> 00:27:01 -- Well, we argued that this side here, 331 00:27:01 --> 00:27:08 the one that is going east-west was a piece of the circle that 332 00:27:08 --> 00:27:14 has a radius a sin phi because that is r, 333 00:27:14 --> 00:27:19 so that side is a sin phi delta theta. 334 00:27:19 --> 00:27:22 And the side that goes north-south is a piece of the 335 00:27:22 --> 00:27:26 circle of radius a corresponding to angle delta phi, 336 00:27:26 --> 00:27:32 so it is a delta phi. And so, just to get to the 337 00:27:32 --> 00:27:40 answer, we got dS equals a^2 sin phi d phi d theta. 338 00:27:40 --> 00:27:45 When we set up a surface integral on the surface of a 339 00:27:45 --> 00:27:48 sphere, most likely we will be using 340 00:27:48 --> 00:27:52 phi and theta as our two variables of integration and dS 341 00:27:52 --> 00:27:55 will become this. Now, it is OK to think of them 342 00:27:55 --> 00:27:58 as spherical coordinates, but I would like to encourage 343 00:27:58 --> 00:28:01 you not to think of them as spherical coordinates. 344 00:28:01 --> 00:28:05 Spherical coordinates are a way of describing points in space in 345 00:28:05 --> 00:28:09 terms of three variables. Here it is more like we are 346 00:28:09 --> 00:28:12 parameterizing the sphere. We are finding a parametric 347 00:28:12 --> 00:28:15 equation for the sphere using two variables phi and theta 348 00:28:15 --> 00:28:18 which happen to be part of the spherical coordinate system. 349 00:28:18 --> 00:28:22 But, see, there is no rho involved in here. 350 00:28:22 --> 00:28:26 I am not using any rho ever, and I am not going to in this 351 00:28:26 --> 00:28:28 calculation. I have two variable phi and 352 00:28:28 --> 00:28:37 theta. That is it. 353 00:28:37 --> 00:28:40 It is basically in the same way as when you parameterize a line 354 00:28:40 --> 00:28:45 integral in the circle, we use theta as the parameter 355 00:28:45 --> 00:28:50 variable and never think about r. 356 00:28:50 --> 00:28:52 That being said, well, we are going to use phi 357 00:28:52 --> 00:28:54 and theta. We know what dS is. 358 00:28:54 --> 00:28:58 We still need to figure out what z is. 359 00:28:58 --> 00:29:01 There we want to think a tiny bit about spherical coordinates 360 00:29:01 --> 00:29:08 again. And we will know that z is just 361 00:29:08 --> 00:29:15 a cos phi. In case you don't quite see it, 362 00:29:15 --> 00:29:25 let me draw a diagram. Phi is the angle down from the 363 00:29:25 --> 00:29:31 positive z axes, this distance is a, 364 00:29:31 --> 00:29:38 so this distance here is a cos phi. 365 00:29:38 --> 00:29:44 Now I have everything I need to compute my double integral. 366 00:29:44 --> 00:29:49 z^2 over a dS will become a double integral. 367 00:29:49 --> 00:30:00 z^2 becomes a^2 cos^2 phi over a times, ds becomes, 368 00:30:00 --> 00:30:07 a^2 sin phi d phi d theta. Now I need to set up bounds. 369 00:30:07 --> 00:30:12 Well, what are the bounds? Phi goes all the way from zero 370 00:30:12 --> 00:30:19 to pi because we go all the way from the north pole to the south 371 00:30:19 --> 00:30:23 pole, and theta goes from zero to 2pi. 372 00:30:23 --> 00:30:27 And, of course, I can get rid of some a's in 373 00:30:27 --> 00:30:34 there and take them out. Let's look at what number we 374 00:30:34 --> 00:30:37 get. First of all, 375 00:30:37 --> 00:30:43 we can take out all those a's and get a^3. 376 00:30:43 --> 00:30:50 Second, in the inner integral, we are integrating cos^2 phi 377 00:30:50 --> 00:30:54 sin phi d phi. I claim that integrates to cos3 378 00:30:54 --> 00:30:57 up to some factor, and that factor should be 379 00:30:57 --> 00:31:02 negative one-third. If you look at cos3 phi and you 380 00:31:02 --> 00:31:07 take its derivative, you will get that guy with a 381 00:31:07 --> 00:31:12 negative three in front between zero and pi. 382 00:31:12 --> 00:31:16 And, while integrating over theta, we will just multiply 383 00:31:16 --> 00:31:24 things by 2pi. Let me add the 2pi in front. 384 00:31:24 --> 00:31:27 Now, if I evaluate this guy between zero and pi, 385 00:31:27 --> 00:31:32 well, at pi cos^3 is negative one, at zero it is one, 386 00:31:32 --> 00:31:35 I will get two-thirds out of this. 387 00:31:35 --> 00:31:39 I end up with four-thirds pi a^3. 388 00:31:39 --> 00:31:46 Sorry I didn't write very much because I am trying to save 389 00:31:46 --> 00:31:52 blackboard space. Yes? 390 00:31:52 --> 00:31:55 That is a very natural question. That looks a lot like somebody 391 00:31:55 --> 00:31:58 we know, like the volume of a sphere. 392 00:31:58 --> 00:32:03 And ultimately it will be. Wait until next class when we 393 00:32:03 --> 00:32:07 talk about the divergence theorem. 394 00:32:07 --> 00:32:11 I mean the question was is this related to the volume of a 395 00:32:11 --> 00:32:14 sphere, and ultimately it is, but for now it is just some 396 00:32:14 --> 00:32:23 coincidence. Yes? 397 00:32:23 --> 00:32:26 The question is there is a way to do it M dx plus N dy plus 398 00:32:26 --> 00:32:28 stuff like that? The answer is unfortunately no 399 00:32:28 --> 00:32:30 because it is not a line integral. 400 00:32:30 --> 00:32:35 It is a surface integral, so we need to have to variables 401 00:32:35 --> 00:32:38 in there. In a way you would end up with 402 00:32:38 --> 00:32:41 things like some dx dy maybe and so on. 403 00:32:41 --> 00:32:45 I mean it is not practical to do it directly that way because 404 00:32:45 --> 00:32:49 you would have then to compute Jacobians to switch from dx dy 405 00:32:49 --> 00:32:52 to something else. We are going to see various 406 00:32:52 --> 00:32:54 ways of computing it. Unfortunately, 407 00:32:54 --> 00:32:57 it is not quite as simple as with line integrals. 408 00:32:57 --> 00:32:59 But it is not much harder. It is the same spirit. 409 00:32:59 --> 00:33:04 We just use two variables and set up everything in terms of 410 00:33:04 --> 00:33:12 these two variables. Any other questions? 411 00:33:12 --> 00:33:14 No. OK. 412 00:33:14 --> 00:33:51 413 00:33:51 --> 00:33:54 By the way, just some food for thought. 414 00:33:54 --> 00:34:01 Never mind. Conclusion of looking at these 415 00:34:01 --> 00:34:05 two examples is that sometimes we can use geometric. 416 00:34:05 --> 00:34:07 The first example, we didn't actually have to 417 00:34:07 --> 00:34:11 compute an integral. But most of the time we need to 418 00:34:11 --> 00:34:14 learn how to set up double integrals. 419 00:34:14 --> 00:34:26 Use geometry or you need to set up for double integral of a 420 00:34:26 --> 00:34:30 surface. And so we are going to learn 421 00:34:30 --> 00:34:33 how to do that in general. As I said, we need to have two 422 00:34:33 --> 00:34:37 parameters on the surface and express everything in terms of 423 00:34:37 --> 00:34:43 these. Let's look at various examples. 424 00:34:43 --> 00:34:46 We are going to see various situations where we can do 425 00:34:46 --> 00:34:49 things. Well, let's start with an easy 426 00:34:49 --> 00:34:53 one. Let's call that number zero. 427 00:34:53 --> 00:35:02 Say that my surface S is a horizontal plane, 428 00:35:02 --> 00:35:07 say z equals a. When I say a horizontal plane, 429 00:35:07 --> 00:35:09 it doesn't have to be the entire horizontal plane. 430 00:35:09 --> 00:35:14 It could be a small piece of it. It could even be, 431 00:35:14 --> 00:35:16 to trick you, maybe an ellipse in there or a 432 00:35:16 --> 00:35:19 triangle in there or something like that. 433 00:35:19 --> 00:35:23 What you have to recognize is my surface is a piece of just a 434 00:35:23 --> 00:35:27 flat plane, so I shouldn't worry too much about what part of a 435 00:35:27 --> 00:35:30 plane it is. Well, it will become important 436 00:35:30 --> 00:35:32 when I set up bounds for integration. 437 00:35:32 --> 00:35:36 But, when it comes to looking for the normal vector, 438 00:35:36 --> 00:35:40 be rest assured that the normal vector to a horizontal plane is 439 00:35:40 --> 00:35:44 just vertical. It is going to be either k or 440 00:35:44 --> 00:35:49 negative k depending on whether I have chosen to orient it 441 00:35:49 --> 00:35:54 pointing up or down. And which one I choose might 442 00:35:54 --> 00:35:57 depend on what I am going to try to do. 443 00:35:57 --> 00:36:02 The normal vector is just sticking straight up or straight 444 00:36:02 --> 00:36:05 down. Now, what about dS? 445 00:36:05 --> 00:36:11 Well, it is just going to be the area element in a horizontal 446 00:36:11 --> 00:36:14 plane. It just looks like it should be 447 00:36:14 --> 00:36:16 dx dy. I mean if I am moving on a 448 00:36:16 --> 00:36:18 horizontal plane, to know where I am, 449 00:36:18 --> 00:36:26 I should know x and y. So dS will be dx dy. 450 00:36:26 --> 00:36:31 If I play the game that way, I have my vector field F. 451 00:36:31 --> 00:36:34 I do F dot n. That just gives me the z 452 00:36:34 --> 00:36:37 component which might involve x, y and z. 453 00:36:37 --> 00:36:40 x and y I am very happy with. They will stay as my variables. 454 00:36:40 --> 00:36:43 Whenever I see z, well, I want to get rid of it. 455 00:36:43 --> 00:36:46 That is easy because z is just equal to a. 456 00:36:46 --> 00:36:50 I just plug that value and I am left with only x and y, 457 00:36:50 --> 00:36:53 and I am integrating that dx dy. 458 00:36:53 --> 00:36:58 It is actually ending up being just a usual double integral in 459 00:36:58 --> 00:37:00 x, y coordinates. And, of course, 460 00:37:00 --> 00:37:02 once it is set up anything is fair game. 461 00:37:02 --> 00:37:05 I might want to switch to polar coordinates or something like 462 00:37:05 --> 00:37:09 that. Or, I can set it up dx dy or dy 463 00:37:09 --> 00:37:12 dx. All the usual stuff applies. 464 00:37:12 --> 00:37:17 But, for the initial setup, we are just going to use these 465 00:37:17 --> 00:37:21 and express everything in terms of x and y. 466 00:37:21 --> 00:37:27 A small variation on that. Let's say that we take vertical 467 00:37:27 --> 00:37:35 planes that are parallel to maybe the blackboard plane, 468 00:37:35 --> 00:37:42 so parallel to the yz plane. That might be something like x 469 00:37:42 --> 00:37:47 equals some constant. Well, what would I do then? 470 00:37:47 --> 00:37:52 It could be pretty much the same. 471 00:37:52 --> 00:37:55 The normal vector for this guy would be sticking straight out 472 00:37:55 --> 00:37:59 towards me or away from me. Let's say I am having it come 473 00:37:59 --> 00:38:03 to the front. The normal vector would be 474 00:38:03 --> 00:38:07 plus/minus i. And the variables that I would 475 00:38:07 --> 00:38:11 be using, to find out my position on this guy, 476 00:38:11 --> 00:38:14 would be y and z. In terms of those, 477 00:38:14 --> 00:38:19 the surface element is just dy dz. 478 00:38:19 --> 00:38:25 Similarly for planes parallel to the xz plane. 479 00:38:25 --> 00:38:33 You can figure that one out. These are somehow the easiest 480 00:38:33 --> 00:38:39 ones, because those we already know how to compute without too 481 00:38:39 --> 00:38:41 much trouble. What if it is a more 482 00:38:41 --> 00:38:43 complicated plane? We will come back to that next 483 00:38:43 --> 00:38:49 time. Let's explore some other 484 00:38:49 --> 00:38:56 situations first. Number one on the list. 485 00:38:56 --> 00:39:03 Let's say that I gave you a sphere of radius a centered at 486 00:39:03 --> 00:39:09 the origin, or maybe just half of that sphere or some portion 487 00:39:09 --> 00:39:12 of it. Well, we have already seen how 488 00:39:12 --> 00:39:15 to do things. Namely, we will be saying the 489 00:39:15 --> 00:39:18 normal vector is x, y, z over a, 490 00:39:18 --> 00:39:23 plus or minus depending on whether we want it pointing in 491 00:39:23 --> 00:39:29 or out. And dS will be a^2 sin phi d 492 00:39:29 --> 00:39:32 phi d theta. In fact, we will express 493 00:39:32 --> 00:39:35 everything in terms of phi and theta. 494 00:39:35 --> 00:39:37 If I wanted to I could tell you what the formulas are for x, 495 00:39:37 --> 00:39:40 y, z in terms of phi and theta. You know them. 496 00:39:40 --> 00:39:44 But it is actually better to wait a little bit. 497 00:39:44 --> 00:39:48 It is better to do F dot n, because F is also going to have 498 00:39:48 --> 00:39:49 a bunch of x's, y's and z's. 499 00:39:49 --> 00:39:53 And if there is any kind of symmetry to the problem then you 500 00:39:53 --> 00:39:57 might end up with things like x^2 y^2 z^2 or things that have 501 00:39:57 --> 00:40:01 more symmetry that are easier to express in terms of phi and 502 00:40:01 --> 00:40:05 theta. The advice would be first do 503 00:40:05 --> 00:40:10 the dot product with F, and then see what you get and 504 00:40:10 --> 00:40:17 then turn it into phi and theta. That is one we have seen. 505 00:40:17 --> 00:40:20 Let's say that I have -- It is a close cousin. 506 00:40:20 --> 00:40:30 Let's say I have a cylinder of radius a centered on the z-axis. 507 00:40:30 --> 00:40:37 What does that look like? And, again, when I say 508 00:40:37 --> 00:40:40 cylinder, it could be a piece of cylinder. 509 00:40:40 --> 00:40:44 First of all, what does the normal vector to 510 00:40:44 --> 00:40:47 a cylinder look like? Well, it is sticking straight 511 00:40:47 --> 00:40:50 out, but sticking straight out in a slightly different way from 512 00:40:50 --> 00:40:52 what happens with a sphere. See, the sides of a cylinder 513 00:40:52 --> 00:40:54 are vertical. If you imagine that you have 514 00:40:54 --> 00:40:56 this big cylindrical type in front of you, 515 00:40:56 --> 00:40:59 hopefully you can see that a normal vector is going to always 516 00:40:59 --> 00:41:02 be horizontal. It is sticking straight out in 517 00:41:02 --> 00:41:07 the horizontal directions. It doesn't have any z component. 518 00:41:07 --> 00:41:13 I claim the normal vector for the cylinder, 519 00:41:13 --> 00:41:21 if you have a point here at (x, y, z), it would be pointing 520 00:41:21 --> 00:41:27 straight out away from the central axis. 521 00:41:27 --> 00:41:31 My normal vector, well, if I am taking it two 522 00:41:31 --> 00:41:34 points outwards, will be going straight away 523 00:41:34 --> 00:41:38 from the central axis. If I look at it from above, 524 00:41:38 --> 00:41:42 maybe it is easier if I look at it from above, 525 00:41:42 --> 00:41:45 look at x, y, then my cylinder looks like a 526 00:41:45 --> 00:41:49 circle and the normal vector just points straight out. 527 00:41:49 --> 00:41:53 It is the same situation as when we had a circle in the 2D 528 00:41:53 --> 00:41:57 case. The normal vector for that is 529 00:41:57 --> 00:42:02 just going to be x, y and 0 in the z component. 530 00:42:02 --> 00:42:05 Well, plus/minus, depending on whether you want 531 00:42:05 --> 00:42:06 it sticking in or out. 532 00:42:06 --> 00:42:41 533 00:42:41 --> 00:42:47 We said in our cylinder normal vector is plus or minus x, 534 00:42:47 --> 00:42:54 y, zero over a. What about the surface element? 535 00:42:54 --> 00:42:57 Before we ask that, maybe we should first figure 536 00:42:57 --> 00:43:00 out what coordinates are we going to use to locate ourselves 537 00:43:00 --> 00:43:02 in a cylinder. Well, yes, 538 00:43:02 --> 00:43:05 we probably want to use part of a cylindrical coordinate, 539 00:43:05 --> 00:43:08 except for, well, we don't want r because r 540 00:43:08 --> 00:43:11 doesn't change, it is not a variable here. 541 00:43:11 --> 00:43:15 Indeed, you probably want to use z to tell how high you are 542 00:43:15 --> 00:43:18 and theta to tell you where you are around. 543 00:43:18 --> 00:43:27 dS should be in terms of dz d theta. 544 00:43:27 --> 00:43:33 Now, what is the constant? Well, let's look at a small 545 00:43:33 --> 00:43:39 piece of our cylinder corresponding to a small angle 546 00:43:39 --> 00:43:44 delta theta and a small height delta z. 547 00:43:44 --> 00:43:47 Well, the height, as I said, is going to be delta 548 00:43:47 --> 00:43:50 z. What about the width? 549 00:43:50 --> 00:43:55 It is going to be a piece of a circle of radius a corresponding 550 00:43:55 --> 00:43:59 to the angle delta theta, so this side will be a delta 551 00:43:59 --> 00:44:05 theta. Delta S is a delta theta delta 552 00:44:05 --> 00:44:08 z. DS is just a dz d theta or d 553 00:44:08 --> 00:44:13 theta dz. It doesn't matter which way you 554 00:44:13 --> 00:44:16 do it. And so when we set up the flux 555 00:44:16 --> 00:44:21 integral, we will take first the dot product of f with this 556 00:44:21 --> 00:44:25 normal vector. Then we will stick in this dS. 557 00:44:25 --> 00:44:28 And then, of course, we will get rid of any x and y 558 00:44:28 --> 00:44:31 that are left by expressing them in terms of theta. 559 00:44:31 --> 00:44:37 Maybe x becomes a cos theta, y becomes a sin theta. 560 00:44:37 --> 00:44:41 These various formulas, you should try to remember them 561 00:44:41 --> 00:44:45 because they are really useful, for the sphere, 562 00:44:45 --> 00:44:48 for the cylinder. And, hopefully, 563 00:44:48 --> 00:44:52 those for the planes you kind of know already intuitively. 564 00:44:52 --> 00:44:59 What about marginals or faces? Not everything in life is made 565 00:44:59 --> 00:45:08 out of cylinders and spheres. I mean it is a good try. 566 00:45:08 --> 00:45:11 Let's look at a marginal kind of surface. 567 00:45:11 --> 00:45:19 Let's say I give you a graph of a function z equals f of x, 568 00:45:19 --> 00:45:21 y. This guy has nothing to do with 569 00:45:21 --> 00:45:22 the integrand. It is not what we are 570 00:45:22 --> 00:45:24 integrating. We are just integrating a 571 00:45:24 --> 00:45:26 vector field that has nothing to do with that. 572 00:45:26 --> 00:45:31 This is how I want to describe the surface on which I will be 573 00:45:31 --> 00:45:37 integrating. My surface is given by z as a 574 00:45:37 --> 00:45:40 function of x, y. 575 00:45:40 --> 00:45:48 Well, I would need to tell you what n is and what dS is. 576 00:45:48 --> 00:45:51 That is going to be slightly annoying. 577 00:45:51 --> 00:45:54 I mean, I don't want to tell them separately because you see 578 00:45:54 --> 00:45:58 they are pretty hard. Instead, I am going to tell you 579 00:45:58 --> 00:46:01 that in this case, well, let's see. 580 00:46:01 --> 00:46:07 What variables do we want? I am going to tell you a 581 00:46:07 --> 00:46:11 formula for n dS. What variables do we want to 582 00:46:11 --> 00:46:15 express this in terms of? Well, most likely x and y 583 00:46:15 --> 00:46:19 because we know how to express z in terms of x and y. 584 00:46:19 --> 00:46:26 This is an invitation to get rid of any z that might be left 585 00:46:26 --> 00:46:30 and set everything up in terms of dx dy. 586 00:46:30 --> 00:46:31 The formula that we are going to see, 587 00:46:31 --> 00:46:36 I think we are going to see the details of why it works 588 00:46:36 --> 00:46:40 tomorrow, is that you can take negative 589 00:46:40 --> 00:46:44 partial f partial x, negative partial f partial y, 590 00:46:44 --> 00:46:48 one, dx dy. 591 00:46:48 --> 00:46:52 Plus/minus depending on which way you want it to go. 592 00:46:52 --> 00:46:57 If you really want to know what dS is, well, dS is the magnitude 593 00:46:57 --> 00:47:01 of this vector times dx dy. There will be a square root and 594 00:47:01 --> 00:47:04 some squares and some stuff. What is the normal vector? 595 00:47:04 --> 00:47:10 Well, you take this vector and you scale it down to unit 596 00:47:10 --> 00:47:14 length. Just to emphasize it, 597 00:47:14 --> 00:47:24 this guy here is not n and this guy here is not dS. 598 00:47:24 --> 00:47:27 Each of them is more complicated than that, 599 00:47:27 --> 00:47:31 but the combination somehow simplifies nicely. 600 00:47:31 --> 00:47:36 And that is good news for us. Now, concretely, 601 00:47:36 --> 00:47:40 one way you can think about it is this tells you how to reduce 602 00:47:40 --> 00:47:42 things to an integral of x and y. 603 00:47:42 --> 00:47:44 And, of course, you will have to figure out 604 00:47:44 --> 00:47:47 what are the bounds on x and y. That means you will need to 605 00:47:47 --> 00:47:51 know what does the shadow of your surface look like in the x, 606 00:47:51 --> 00:47:59 y plane. To set up bounds on whatever 607 00:47:59 --> 00:48:06 you will get dx dy, well, of course you can switch 608 00:48:06 --> 00:48:08 to dy dx or anything you would like, 609 00:48:08 --> 00:48:19 but you will need to look at the shadow of S in the x y 610 00:48:19 --> 00:48:22 plane. But only do that after you 611 00:48:22 --> 00:48:27 gotten rid of all the z. When you no longer have z then 612 00:48:27 --> 00:48:33 you can figure out what the bounds are for x and y. 613 00:48:33 --> 00:48:40 Any questions about that? Yes? 614 00:48:40 --> 00:48:42 For the cylinder. OK. 615 00:48:42 --> 00:48:44 Let me re-explain quickly how I got a normal vector for the 616 00:48:44 --> 00:48:47 cylinder. If you know what a cylinder 617 00:48:47 --> 00:48:50 looks like, you probably can see that the normal vector sticks 618 00:48:50 --> 00:48:56 straight out of it horizontally. That means the z component of n 619 00:48:56 --> 00:48:59 is going to be zero. And then the x, 620 00:48:59 --> 00:49:02 y components you get by looking at it from above. 621 00:49:02 --> 00:49:12 One last thing I want to say. What about the geometric 622 00:49:12 --> 00:49:15 interpretation and how to prove it? 623 00:49:15 --> 00:49:27 Well, if your vector field F is a velocity field then the flux 624 00:49:27 --> 00:49:37 is the amount of matter that crosses the surface that passes 625 00:49:37 --> 00:49:44 through S per unit time. And the way that you would 626 00:49:44 --> 00:49:47 prove it would be similar to the picture that I drew when we did 627 00:49:47 --> 00:49:50 it in the plane. Namely, you would consider a 628 00:49:50 --> 00:49:53 small element of a surface delta S. 629 00:49:53 --> 00:49:55 And you would try to figure out what is the stuff that flows 630 00:49:55 --> 00:49:59 through it in a second. Well, it is the stuff that 631 00:49:59 --> 00:50:05 lives in a small box whose base is that piece of surface and 632 00:50:05 --> 00:50:10 whose other side is given by the vector field. 633 00:50:10 --> 00:50:15 And then the volume of that is given by base times height, 634 00:50:15 --> 00:50:20 and the height is F dot n. It is the same argument as what 635 00:50:20 --> 00:50:22 we saw in the plane. OK. 636 00:50:22 --> 00:50:24 Next time we will see more formulas. 637 00:50:24 --> 00:50:28 We will first see how to prove this, more ways to do it, 638 00:50:28 --> 00:50:31 more examples. And then we will get to the 639 00:50:31 --> 00:50:34 divergence theorem. 640 00:50:34 --> 00:50:39