1 00:00:01,000 --> 00:00:03,000 2 00:00:01,000 --> 00:00:03,000 The following content is provided under a Creative 3 00:00:03,000 --> 00:00:05,000 Commons license. Your support will help MIT 4 00:00:05,000 --> 00:00:08,000 OpenCourseWare continue to offer high quality educational 5 00:00:08,000 --> 00:00:13,000 resources for free. To make a donation or to view 6 00:00:13,000 --> 00:00:18,000 additional materials from hundreds of MIT courses, 7 00:00:18,000 --> 00:00:23,000 visit MIT OpenCourseWare at ocw.mit.edu. 8 00:00:23,000 --> 00:00:33,000 Today I am going to tell you about flux of a vector field for 9 00:00:33,000 --> 00:00:35,000 a curve. In case you have seen flux in 10 00:00:35,000 --> 00:00:37,000 physics, probably you have seen flux in 11 00:00:37,000 --> 00:00:39,000 space, and we are going to come to 12 00:00:39,000 --> 00:00:41,000 that in a couple of weeks, but for now we are still doing 13 00:00:41,000 --> 00:00:44,000 everything in the plane. So bear with me if you have 14 00:00:44,000 --> 00:00:47,000 seen a more complicated version of flux. 15 00:00:47,000 --> 00:00:50,000 We are going to do the easy one first. 16 00:00:50,000 --> 00:00:59,000 What is flux? Well, flux is actually another 17 00:00:59,000 --> 00:01:10,000 kind of line integral. Let's say that I have a plane 18 00:01:10,000 --> 00:01:18,000 curve and a vector field in the plane. 19 00:01:18,000 --> 00:01:27,000 Then the flux of F across a curve C is, by definition, 20 00:01:27,000 --> 00:01:35,000 a line integral, but I will use notation F dot n 21 00:01:35,000 --> 00:01:38,000 ds. I have to explain to you what 22 00:01:38,000 --> 00:01:43,000 it means, but let me first box that because that is the 23 00:01:43,000 --> 00:01:50,000 important formula to remember. That is the definition. 24 00:01:50,000 --> 00:01:55,000 What does that mean? First, mostly I have to tell 25 00:01:55,000 --> 00:02:01,000 you what this little n is. The notation suggests it is a 26 00:02:01,000 --> 00:02:05,000 normal vector, so what does that mean? 27 00:02:05,000 --> 00:02:16,000 I have a curve in the plane and I have a vector field. 28 00:02:16,000 --> 00:02:24,000 Let's see. The vector field will be yellow 29 00:02:24,000 --> 00:02:28,000 today. And I will want to integrate 30 00:02:28,000 --> 00:02:33,000 along the curve the dot product of F with the normal vector to 31 00:02:33,000 --> 00:02:37,000 the curve, a unit normal vector to the curve. 32 00:02:37,000 --> 00:02:44,000 That means a vector that is at every point of the curve 33 00:02:44,000 --> 00:02:51,000 perpendicular to the curve and has length one. 34 00:02:51,000 --> 00:03:06,000 N everywhere will be the unit normal vector to the curve C 35 00:03:06,000 --> 00:03:17,000 pointing 90 degrees clockwise from T. 36 00:03:17,000 --> 00:03:19,000 What does that mean? That means I have two normal 37 00:03:19,000 --> 00:03:22,000 vectors, one that is pointing this way, one that is pointing 38 00:03:22,000 --> 00:03:24,000 that way. I have to choose a convention. 39 00:03:24,000 --> 00:03:28,000 And the convention is that the normal vector that I take goes 40 00:03:28,000 --> 00:03:32,000 to the right of the curve as I am traveling along the curve. 41 00:03:32,000 --> 00:03:36,000 You mentioned that you were walking along this curve, 42 00:03:36,000 --> 00:03:40,000 then you look to your right, that is that direction. 43 00:03:40,000 --> 00:03:44,000 What we will do is just, at every point along the curve, 44 00:03:44,000 --> 00:03:48,000 the dot product between the vector field and the normal 45 00:03:48,000 --> 00:03:51,000 vector. And we will sum that along the 46 00:03:51,000 --> 00:03:58,000 various pieces of the curve. What this notation means is 47 00:03:58,000 --> 00:04:09,000 that if we actually break C into small pieces of length delta(s) 48 00:04:09,000 --> 00:04:17,000 then the flux will be the limit, as the pieces become smaller 49 00:04:17,000 --> 00:04:25,000 and smaller, of the sum of F dot n delta S. 50 00:04:25,000 --> 00:04:29,000 I take each small piece of my curve, I do the dot product 51 00:04:29,000 --> 00:04:32,000 between F and n and I multiply by the length of a piece. 52 00:04:32,000 --> 00:04:35,000 And then I add these together. That is what the line integral 53 00:04:35,000 --> 00:04:38,000 means. Of course that is, 54 00:04:38,000 --> 00:04:41,000 again, not how I will compute it. 55 00:04:41,000 --> 00:04:49,000 Just to compare this with work, conceptually it is similar to 56 00:04:49,000 --> 00:05:00,000 the line integral we did for work except the line integral 57 00:05:00,000 --> 00:05:10,000 for work -- Work is the line integral of F dot dr, 58 00:05:10,000 --> 00:05:15,000 which is also the line integral of F dot T ds. 59 00:05:15,000 --> 00:05:20,000 That is how we reformulated it. That means we take our curve 60 00:05:20,000 --> 00:05:26,000 and we figure out at each point how big the tangent component -- 61 00:05:26,000 --> 00:05:33,000 I guess I should probably take the same vector field as before. 62 00:05:33,000 --> 00:05:37,000 Let's see. My field was pointing more like 63 00:05:37,000 --> 00:05:40,000 that way. What I do at any point is 64 00:05:40,000 --> 00:05:45,000 project F to the tangent direction, I figure out how much 65 00:05:45,000 --> 00:05:50,000 F is going along my curve and then I sum these things 66 00:05:50,000 --> 00:06:02,000 together. I am actually summing -- -- the 67 00:06:02,000 --> 00:06:14,000 tangential component of my field F. 68 00:06:14,000 --> 00:06:17,000 Roughly-speaking the work measures, you know, 69 00:06:17,000 --> 00:06:21,000 when I move along my curve, how much I am going with or 70 00:06:21,000 --> 00:06:23,000 against F. Flux, on the other hand, 71 00:06:23,000 --> 00:06:26,000 measures, when I go along the curve, roughly how much the 72 00:06:26,000 --> 00:06:28,000 field is going to across the curve. 73 00:06:28,000 --> 00:06:31,000 Counting positively what goes to the right, 74 00:06:31,000 --> 00:06:34,000 negatively what goes to the left. 75 00:06:34,000 --> 00:06:45,000 Flux is integral F dot n ds, and that one corresponds to 76 00:06:45,000 --> 00:06:54,000 summing the normal component of a vector field. 77 00:06:54,000 --> 00:06:57,000 But apart from that conceptually it is the same kind 78 00:06:57,000 --> 00:06:59,000 of thing. Just the physical 79 00:06:59,000 --> 00:07:01,000 interpretations will be very different, 80 00:07:01,000 --> 00:07:10,000 but for a mathematician these are two line integrals that you 81 00:07:10,000 --> 00:07:17,000 set up and compute in pretty much the same way. 82 00:07:17,000 --> 00:07:21,000 Let's see. I should probably tell you what 83 00:07:21,000 --> 00:07:22,000 it means. Why do we make this definition? 84 00:07:22,000 --> 00:07:27,000 What does it correspond to? Well, the interpretation for 85 00:07:27,000 --> 00:07:31,000 work made a lot of sense when F was representing a force. 86 00:07:31,000 --> 00:07:36,000 The line integral was actually the work done by the force. 87 00:07:36,000 --> 00:07:40,000 The interpretation for flux makes more sense if you think of 88 00:07:40,000 --> 00:07:50,000 F as a velocity field. What is the interpretation? 89 00:07:50,000 --> 00:07:55,000 Let's say that for F is a velocity field. 90 00:07:55,000 --> 00:08:00,000 That means I am thinking of some fluid that is moving, 91 00:08:00,000 --> 00:08:04,000 maybe water or something, and it is moving at a certain 92 00:08:04,000 --> 00:08:08,000 speed. And my vector field represents 93 00:08:08,000 --> 00:08:14,000 how things are moving at every point of the plane. 94 00:08:14,000 --> 00:08:31,000 I claim that flux measures how much fluid passes through -- -- 95 00:08:31,000 --> 00:08:41,000 the curve C per unit time. If you imagine that maybe you 96 00:08:41,000 --> 00:08:45,000 have a river and you are somehow building a damn here, 97 00:08:45,000 --> 00:08:48,000 a damn with holes in it so that the water still passes through, 98 00:08:48,000 --> 00:08:53,000 then this measures how much water passes through your 99 00:08:53,000 --> 00:08:57,000 membrane per unit time. Let's try to figure out why 100 00:08:57,000 --> 00:09:00,000 this is true. Why does this make sense? 101 00:09:00,000 --> 00:09:07,000 Let's look at what happens on a small portion of our curve C. 102 00:09:07,000 --> 00:09:21,000 I am zooming in on my curve C. I guess I need to zoom further. 103 00:09:21,000 --> 00:09:26,000 That is a little piece of my curve, of length delta S, 104 00:09:26,000 --> 00:09:30,000 and there is a fluid flow. On my picture things are 105 00:09:30,000 --> 00:09:33,000 flowing to the right. Here I am drawing a constant 106 00:09:33,000 --> 00:09:38,000 vector field because if you zoom in enough then your vectors will 107 00:09:38,000 --> 00:09:40,000 pretty much be the same everywhere. 108 00:09:40,000 --> 00:09:44,000 If you enlarge the picture enough then things will be 109 00:09:44,000 --> 00:09:48,000 pretty much a uniform flow. Now, how much stuff goes 110 00:09:48,000 --> 00:09:51,000 through this little piece of curve per unit time? 111 00:09:51,000 --> 00:09:57,000 Well, what happens over time is the fluid is moving while my 112 00:09:57,000 --> 00:10:04,000 curve is staying the same place so it corresponds to something 113 00:10:04,000 --> 00:10:09,000 like this. I claim that what goes through 114 00:10:09,000 --> 00:10:16,000 C in unit time is actually going to be a parallelogram. 115 00:10:16,000 --> 00:10:21,000 Here is a better picture. I claim that what will be going 116 00:10:21,000 --> 00:10:24,000 through C is this shaded parallelogram to the left of C. 117 00:10:24,000 --> 00:10:32,000 Let's see. If I move for unit time it 118 00:10:32,000 --> 00:10:35,000 works. That is the stuff that goes 119 00:10:35,000 --> 00:10:38,000 through my curve, for a small portion of curve in 120 00:10:38,000 --> 00:10:40,000 unit time. And, of course, 121 00:10:40,000 --> 00:10:43,000 I would need to add all of these together to get the entire 122 00:10:43,000 --> 00:10:47,000 curve. Let's try to understand how big 123 00:10:47,000 --> 00:10:50,000 this parallelogram is. To know how big this 124 00:10:50,000 --> 00:10:53,000 parallelogram is I would like to use base times height or 125 00:10:53,000 --> 00:10:56,000 something like that. And maybe I want to actually 126 00:10:56,000 --> 00:10:59,000 flip my picture so that the base and the height make more sense 127 00:10:59,000 --> 00:11:04,000 to me. Let me actually turn it this 128 00:11:04,000 --> 00:11:10,000 way. And, in case you have trouble 129 00:11:10,000 --> 00:11:21,000 reading the rotated picture, let me redo it on the board. 130 00:11:21,000 --> 00:11:31,000 What passes through a portion of C in unit time is the 131 00:11:31,000 --> 00:11:40,000 contents of a parallelogram whose base is on C. 132 00:11:40,000 --> 00:11:49,000 So it has length delta s. That is a piece of C. 133 00:11:49,000 --> 00:12:06,000 And the other side is going to be given by my velocity vector 134 00:12:06,000 --> 00:12:11,000 F. And to find the height of this 135 00:12:11,000 --> 00:12:17,000 thing, I need to know what actually the normal component of 136 00:12:17,000 --> 00:12:24,000 this vector is. If I call n the unit normal 137 00:12:24,000 --> 00:12:35,000 vector to the curve then the area is base times height. 138 00:12:35,000 --> 00:12:42,000 The base is delta S and the height is the normal component 139 00:12:42,000 --> 00:12:48,000 of F, so it is F dot n. And so you see that when you 140 00:12:48,000 --> 00:12:54,000 sum these things together you get, what I said, 141 00:12:54,000 --> 00:12:56,000 flux. Now, if you are worried about 142 00:12:56,000 --> 00:13:00,000 the fact that actually -- If your unit time is too long then 143 00:13:00,000 --> 00:13:03,000 of course things might start changing as it flows. 144 00:13:03,000 --> 00:13:07,000 You have to take the time unit and the length unit that are 145 00:13:07,000 --> 00:13:11,000 sufficiently small so that really this approximation where 146 00:13:11,000 --> 00:13:15,000 C is a straight line and where flow is at constant speed are 147 00:13:15,000 --> 00:13:17,000 valid. You want to take maybe a 148 00:13:17,000 --> 00:13:20,000 segment here that is a few micrometers. 149 00:13:20,000 --> 00:13:24,000 And the time unit might be a few nanoseconds or whatever, 150 00:13:24,000 --> 00:13:28,000 and then it is a good approximation. 151 00:13:28,000 --> 00:13:31,000 What I mean by per unit time is, well, actually, 152 00:13:31,000 --> 00:13:35,000 that works, but you want to think of a really, 153 00:13:35,000 --> 00:13:39,000 really small time. And then the amount of matter 154 00:13:39,000 --> 00:13:44,000 that passes in that really, really small time is the flux 155 00:13:44,000 --> 00:13:48,000 times the amount of time. Let's be a tiny bit more 156 00:13:48,000 --> 00:13:50,000 careful. And what I am saying is the 157 00:13:50,000 --> 00:13:53,000 amount of stuff that passes through C depends actually on 158 00:13:53,000 --> 00:13:56,000 whether n is going this way or the opposite way. 159 00:13:56,000 --> 00:14:00,000 Actually, what is implicit in this 160 00:14:00,000 --> 00:14:05,000 explanation is that I am counting positively all the 161 00:14:05,000 --> 00:14:11,000 stuff that flows across C in the direction of n and negatively 162 00:14:11,000 --> 00:14:15,000 what flows in the opposite direction. 163 00:14:15,000 --> 00:14:32,000 What flows to the right of C, well, across C from left to 164 00:14:32,000 --> 00:14:47,000 right is counted positively. While what flows right to left 165 00:14:47,000 --> 00:14:53,000 is counted negatively. So, in fact, 166 00:14:53,000 --> 00:15:00,000 it is the net flow through C per unit time. 167 00:15:00,000 --> 00:15:06,000 Any questions about the definition or the interpretation 168 00:15:06,000 --> 00:15:16,000 or things like that? Yes? 169 00:15:16,000 --> 00:15:19,000 Well, you can have both not in the same small segment. 170 00:15:19,000 --> 00:15:24,000 But it could be that, well, imagine that my vector 171 00:15:24,000 --> 00:15:28,000 field accidentally goes in the opposite direction then this 172 00:15:28,000 --> 00:15:32,000 part of the curve, while things are flowing to the 173 00:15:32,000 --> 00:15:35,000 left, contributes negatively to flux. 174 00:15:35,000 --> 00:15:39,000 And here maybe the field is tangent so the normal component 175 00:15:39,000 --> 00:15:42,000 becomes zero. And then it becomes positive 176 00:15:42,000 --> 00:15:47,000 and this part of the curve contributes positively. 177 00:15:47,000 --> 00:15:51,000 For example, if you imagine that you have a 178 00:15:51,000 --> 00:15:54,000 round tank in which the fluid is rotating and you put your dam 179 00:15:54,000 --> 00:15:57,000 just on a diameter across then things are going one way on one 180 00:15:57,000 --> 00:15:59,000 side, the other way on the other 181 00:15:59,000 --> 00:16:04,000 side, and actually it just evens out. 182 00:16:04,000 --> 00:16:06,000 We don't have complete information. 183 00:16:06,000 --> 00:16:13,000 It is just the total net flux. OK. 184 00:16:13,000 --> 00:16:19,000 If there are no other questions then I guess we will need to 185 00:16:19,000 --> 00:16:25,000 figure out how to compute this guy and how to actually do this 186 00:16:25,000 --> 00:16:33,000 line integral. Well, let's start with a couple 187 00:16:33,000 --> 00:16:43,000 of easy examples. Let's say that C is a circle of 188 00:16:43,000 --> 00:16:55,000 radius (a) centered at the origin going counterclockwise. 189 00:16:55,000 --> 00:17:02,000 And let's say that our vector field is xi yj. 190 00:17:02,000 --> 00:17:09,000 What does that look like? Remember, xi plus yj is a 191 00:17:09,000 --> 00:17:15,000 vector field that is pointing radially away from the origin. 192 00:17:15,000 --> 00:17:19,000 Because at every point it is equal to the vector from the 193 00:17:19,000 --> 00:17:25,000 origin to that point. Now, if we have a circle and 194 00:17:25,000 --> 00:17:30,000 let's say we are going counterclockwise. 195 00:17:30,000 --> 00:17:32,000 Actually, I have a nicer picture. 196 00:17:32,000 --> 00:17:48,000 Let me do it here. That is my curve and my vector 197 00:17:48,000 --> 00:17:55,000 field. And the normal vector, see, 198 00:17:55,000 --> 00:17:57,000 when you go counterclockwise in a closed curve, 199 00:17:57,000 --> 00:18:01,000 this convention that a normal vector points to the right of 200 00:18:01,000 --> 00:18:04,000 curve makes it point out. The usual convention, 201 00:18:04,000 --> 00:18:08,000 when you take flux for a closed curve, is that you are counting 202 00:18:08,000 --> 00:18:11,000 the flux going out of the region enclosed by the curve. 203 00:18:11,000 --> 00:18:13,000 And, of course, if you went clockwise it would 204 00:18:13,000 --> 00:18:18,000 be the other way around. You choose to do it the way you 205 00:18:18,000 --> 00:18:27,000 want, but the most common one is to count flux going out of the 206 00:18:27,000 --> 00:18:31,000 region. Let's see what happens. 207 00:18:31,000 --> 00:18:35,000 Well, if I am anywhere on my circle, see, the normal vector 208 00:18:35,000 --> 00:18:38,000 is sticking straight out of the circle. 209 00:18:38,000 --> 00:18:43,000 That is a property of the circle that the radial direction 210 00:18:43,000 --> 00:18:49,000 is perpendicular to the circle. Actually, let me complete this 211 00:18:49,000 --> 00:18:52,000 picture. If I take a point on the 212 00:18:52,000 --> 00:18:59,000 circle, I have my normal vector that is pointing straight out so 213 00:18:59,000 --> 00:19:05,000 it is parallel to F. Along C we know that F is 214 00:19:05,000 --> 00:19:10,000 parallel to n, so F dot n will be equal to the 215 00:19:10,000 --> 00:19:16,000 magnitude of F times, well, the magnitude of n, 216 00:19:16,000 --> 00:19:20,000 but that is one. Let me put it anywhere, 217 00:19:20,000 --> 00:19:23,000 but that is the unit normal vector. 218 00:19:23,000 --> 00:19:27,000 Now, what is the magnitude of this vector field if I am at a 219 00:19:27,000 --> 00:19:29,000 point x, y? Well, it is square root of x 220 00:19:29,000 --> 00:19:32,000 squared plus y squared, which is the same as the 221 00:19:32,000 --> 00:19:36,000 distance from the origin. So if this distance, 222 00:19:36,000 --> 00:19:46,000 if this radius is a then the magnitude of F will just be a. 223 00:19:46,000 --> 00:19:51,000 In fact, F dot n is constant, always equal to a. 224 00:19:51,000 --> 00:19:57,000 So the line integral will be pretty easy because all I have 225 00:19:57,000 --> 00:20:04,000 to do is the integral of F dot n ds becomes the integral of a ds. 226 00:20:04,000 --> 00:20:07,000 (a) is a constant so I can take it out. 227 00:20:07,000 --> 00:20:16,000 And integral ds is just a length of C which is 2pi a, 228 00:20:16,000 --> 00:20:24,000 so I will get 2pi a squared. And that is positive, 229 00:20:24,000 --> 00:20:28,000 as we expected, because stuff is flowing out of 230 00:20:28,000 --> 00:20:36,000 the circle. Any questions about that? 231 00:20:36,000 --> 00:20:41,000 No. OK. 232 00:20:41,000 --> 00:20:45,000 Just out of curiosity, let's say that we had taken our 233 00:20:45,000 --> 00:20:52,000 other favorite vector field. Let's say that we had the same 234 00:20:52,000 --> 00:20:57,000 C, but now the vector field . 235 00:20:57,000 --> 00:21:05,000 Remember, that one goes counterclockwise around the 236 00:21:05,000 --> 00:21:09,000 origin. If you remember what we did 237 00:21:09,000 --> 00:21:12,000 several times, well, along the circle that 238 00:21:12,000 --> 00:21:16,000 vector field now is tangent to the circle. 239 00:21:16,000 --> 00:21:19,000 If it is tangent to the circle it doesn't have any normal 240 00:21:19,000 --> 00:21:22,000 component. The normal component is zero. 241 00:21:22,000 --> 00:21:25,000 Things are not flowing into the circle or out of it. 242 00:21:25,000 --> 00:21:30,000 They are just flowing along the circle around and around so the 243 00:21:30,000 --> 00:21:38,000 flux will be zero. F now is tangent to C. 244 00:21:38,000 --> 00:21:51,000 F dot n is zero and, therefore, the flux will be 245 00:21:51,000 --> 00:21:55,000 zero. These are examples where you 246 00:21:55,000 --> 00:21:57,000 can compute things geometrically. 247 00:21:57,000 --> 00:22:00,000 And I would say, generally speaking, 248 00:22:00,000 --> 00:22:03,000 with flux, well, if it is a very complicated 249 00:22:03,000 --> 00:22:06,000 field then you cannot. But, if a field is fairly 250 00:22:06,000 --> 00:22:08,000 simple, you should be able to get some 251 00:22:08,000 --> 00:22:11,000 general feeling for whether your answer should be positive, 252 00:22:11,000 --> 00:22:15,000 negative or zero just by thinking about which way is my 253 00:22:15,000 --> 00:22:20,000 flow going. Is it going across the curve 254 00:22:20,000 --> 00:22:32,000 one way or the other way? Still no questions about these 255 00:22:32,000 --> 00:22:36,000 examples? The next thing we need to know 256 00:22:36,000 --> 00:22:40,000 is how we will actually compute these things because here, 257 00:22:40,000 --> 00:22:43,000 yeah, it works pretty well, but what if you don't have a 258 00:22:43,000 --> 00:22:47,000 simple geometric interpretation. What if I give you a really 259 00:22:47,000 --> 00:22:50,000 complicated curve and then you have trouble finding the normal 260 00:22:50,000 --> 00:22:53,000 vector? It is going to be annoying to 261 00:22:53,000 --> 00:22:56,000 set up things this way. Actually, there is a better way 262 00:22:56,000 --> 00:22:59,000 to do it in coordinates. Just as we do work, 263 00:22:59,000 --> 00:23:04,000 when we compute this line integral, usually we don't do it 264 00:23:04,000 --> 00:23:08,000 geometrically like this. Most of the time we just 265 00:23:08,000 --> 00:23:12,000 integrate M dx plus N dy in coordinates. 266 00:23:12,000 --> 00:23:16,000 That is a similar way to do it because it is, 267 00:23:16,000 --> 00:23:20,000 again, a line integral so it should work the same way. 268 00:23:20,000 --> 00:23:21,000 Let's try to figure that out. 269 00:23:21,000 --> 00:24:05,000 270 00:24:05,000 --> 00:24:22,000 How do we do the calculation in coordinates, or I should say 271 00:24:22,000 --> 00:24:29,000 using components? That is the general method of 272 00:24:29,000 --> 00:24:33,000 calculation when we don't have something geometric to do. 273 00:24:33,000 --> 00:24:41,000 Remember, when we were doing things for work we said this 274 00:24:41,000 --> 00:24:49,000 vector dr, or if you prefer T ds, we said just becomes 275 00:24:49,000 --> 00:24:56,000 symbolically dx and dy. When you do the line integral 276 00:24:56,000 --> 00:25:01,000 of F dot dr you get line integral of n dx plus n dy. 277 00:25:01,000 --> 00:25:07,000 Now let's think for a second about how we would express n ds. 278 00:25:07,000 --> 00:25:11,000 Well, what is n ds compared to T ds? 279 00:25:11,000 --> 00:25:15,000 Well, M is just T rotated by 90 degrees, so n ds is T ds rotated 280 00:25:15,000 --> 00:25:19,000 by 90 degrees. That might sound a little bit 281 00:25:19,000 --> 00:25:23,000 outrageous because these are really symbolic notations but it 282 00:25:23,000 --> 00:25:25,000 works. I am not going to spend too 283 00:25:25,000 --> 00:25:28,000 much time trying to convince you carefully. 284 00:25:28,000 --> 00:25:33,000 But if you go back to where we wrote this and how we tried to 285 00:25:33,000 --> 00:25:36,000 justify this and you work your way through it, 286 00:25:36,000 --> 00:25:42,000 you will see that n ds can be analyzed the same way. 287 00:25:42,000 --> 00:25:51,000 N is T rotated 90 degrees clockwise. 288 00:25:51,000 --> 00:25:57,000 That tells us that n ds is -- How do we rotate a vector by 90 289 00:25:57,000 --> 00:26:00,000 degrees? Well, we swept the two 290 00:26:00,000 --> 00:26:05,000 components and we put a minus sign. 291 00:26:05,000 --> 00:26:07,000 You have dy and dx. And you have to be careful 292 00:26:07,000 --> 00:26:11,000 where to put the minus sign. Well, if you are doing it 293 00:26:11,000 --> 00:26:13,000 clockwise, it is in front of dx. 294 00:26:13,000 --> 00:26:26,000 295 00:26:26,000 --> 00:26:29,000 Well, actually, let me just convince you 296 00:26:29,000 --> 00:26:32,000 quickly. Let's say we have a small piece 297 00:26:32,000 --> 00:26:36,000 of C. If we do T delta S, 298 00:26:36,000 --> 00:26:44,000 that is also vector delta r. That is going to be just the 299 00:26:44,000 --> 00:26:48,000 vector that goes along the curve given by this. 300 00:26:48,000 --> 00:26:54,000 Its components will be indeed the change in x, 301 00:26:54,000 --> 00:27:00,000 delta x, and the change in y, delta y. 302 00:27:00,000 --> 00:27:07,000 And now, if I want to get n delta S, well, 303 00:27:07,000 --> 00:27:15,000 I claim now that it is perfectly valid and rigorous to 304 00:27:15,000 --> 00:27:24,000 just rotate that by 90 degrees. If I want to rotate this by 90 305 00:27:24,000 --> 00:27:31,000 degrees clockwise then the x component will become the same 306 00:27:31,000 --> 00:27:36,000 as the old y component. And the y component will be 307 00:27:36,000 --> 00:27:40,000 minus delta x. Then you take the limit when 308 00:27:40,000 --> 00:27:44,000 the segment becomes shorter and shorter, and that is how you can 309 00:27:44,000 --> 00:27:47,000 justify this. That is the key to computing 310 00:27:47,000 --> 00:27:50,000 things in practice. It means, actually, 311 00:27:50,000 --> 00:27:55,000 you already know how to compute line integrals for flux. 312 00:27:55,000 --> 00:28:05,000 Let me just write it explicitly. Let's say that our vector field 313 00:28:05,000 --> 00:28:08,000 has two components. And let me just confuse you a 314 00:28:08,000 --> 00:28:12,000 little bit and not call them M and N for this time just to 315 00:28:12,000 --> 00:28:16,000 stress the fact that we are doing a different line integral. 316 00:28:16,000 --> 00:28:22,000 Let me call them P and Q for now. 317 00:28:22,000 --> 00:28:31,000 Then the line integral of F dot n ds will be the line integral 318 00:28:31,000 --> 00:28:39,000 of dot product . 319 00:28:39,000 --> 00:28:46,000 That will be the integral of - Q dx P dy. 320 00:28:46,000 --> 00:28:50,000 Well, I am running out of space here. 321 00:28:50,000 --> 00:29:01,000 It is integral along C of negative Q dx plus P dy. 322 00:29:01,000 --> 00:29:04,000 And from that point onwards you just do it the usual way. 323 00:29:04,000 --> 00:29:10,000 Remember, here you have two variables x and y but you are 324 00:29:10,000 --> 00:29:14,000 integrating along a curve. If you are integrating along a 325 00:29:14,000 --> 00:29:18,000 curve x and y are related. They depend on each other or 326 00:29:18,000 --> 00:29:21,000 maybe on some other parameter like T or theta or whatever. 327 00:29:21,000 --> 00:29:28,000 You express everything in terms of a single variable and then 328 00:29:28,000 --> 00:29:36,000 you do a usual single integral. Any questions about that? 329 00:29:36,000 --> 00:29:39,000 I see a lot of confused faces so maybe I shouldn't have called 330 00:29:39,000 --> 00:29:41,000 my component P and Q. 331 00:29:41,000 --> 00:30:04,000 332 00:30:04,000 --> 00:30:14,000 If you prefer, if you are really sentimentally 333 00:30:14,000 --> 00:30:27,000 attached to M and N then this new line integral becomes the 334 00:30:27,000 --> 00:30:35,000 integral of - N dx M dy. If a problem tells you compute 335 00:30:35,000 --> 00:30:37,000 flux instead of saying compute work, 336 00:30:37,000 --> 00:30:41,000 the only thing you change is instead of doing M dx plus N dy 337 00:30:41,000 --> 00:30:45,000 you do minus N dx plus M dy. And I am sorry to say that I 338 00:30:45,000 --> 00:30:49,000 don't have any good way of helping you remember which one 339 00:30:49,000 --> 00:30:52,000 of the two gets the minus sign, so you just have to remember 340 00:30:52,000 --> 00:30:58,000 this formula by heart. That is the only way I know. 341 00:30:58,000 --> 00:31:04,000 Well, you can try to go through this argument again, 342 00:31:04,000 --> 00:31:10,000 but it is really best if you just remember that formula. 343 00:31:10,000 --> 00:31:15,000 I am not going to do an example because we already know how to 344 00:31:15,000 --> 00:31:19,000 do line integrals. Hopefully you will get to see 345 00:31:19,000 --> 00:31:23,000 one at least in recitation on Monday. 346 00:31:23,000 --> 00:31:29,000 That is all pretty good. Let me tell you now what if I 347 00:31:29,000 --> 00:31:35,000 have to compute flux along a closed curve and I don't want to 348 00:31:35,000 --> 00:31:39,000 compute it? Well, remember in the case of 349 00:31:39,000 --> 00:31:43,000 work we had Green's theorem. We saw yesterday Green's 350 00:31:43,000 --> 00:31:45,000 theorem. Let's us replace a line 351 00:31:45,000 --> 00:31:48,000 integral along a closed curve by a double integral. 352 00:31:48,000 --> 00:31:51,000 Well, here it is the same. We have a line integral along a 353 00:31:51,000 --> 00:31:53,000 curve. If it is a closed curve, 354 00:31:53,000 --> 00:31:57,000 we should be able to replace it by a double integral. 355 00:31:57,000 --> 00:32:09,000 There is a version of Green's theorem for flux. 356 00:32:09,000 --> 00:32:13,000 And you will see it is not scarier than the other one. 357 00:32:13,000 --> 00:32:18,000 It is perhaps less scarier or perhaps just as scary or just 358 00:32:18,000 --> 00:32:22,000 not as scary, depending on how you feel about 359 00:32:22,000 --> 00:32:26,000 it, but it works pretty much the same way. 360 00:32:26,000 --> 00:32:30,000 What does Green's theorem for flux say? 361 00:32:30,000 --> 00:32:39,000 It says if C is a curve that encloses a region R 362 00:32:39,000 --> 00:32:51,000 counterclockwise and if I have a vector field that is defined 363 00:32:51,000 --> 00:32:56,000 everywhere, not just on C but also inside, 364 00:32:56,000 --> 00:33:11,000 so also on R. Well, maybe I should give names 365 00:33:11,000 --> 00:33:14,000 to the components. If you will forgive me for a 366 00:33:14,000 --> 00:33:16,000 second, I will still use P and Q for now. 367 00:33:16,000 --> 00:33:23,000 You will see why. It is defined and 368 00:33:23,000 --> 00:33:30,000 differentiable in R. Then I can actually -- -- 369 00:33:30,000 --> 00:33:40,000 replace the line integral for flux by a double integral over R 370 00:33:40,000 --> 00:33:47,000 of some function. And that function is called the 371 00:33:47,000 --> 00:33:58,000 divergence of F dA. This is the divergence of F. 372 00:33:58,000 --> 00:34:08,000 And I have to define for you what this guy is. 373 00:34:08,000 --> 00:34:15,000 The divergence of a vector field with components P and Q is 374 00:34:15,000 --> 00:34:20,000 just P sub x Q sub y. This one is actually easier to 375 00:34:20,000 --> 00:34:23,000 remember than curl because you just take the x component, 376 00:34:23,000 --> 00:34:26,000 take its partial with respect to x, 377 00:34:26,000 --> 00:34:29,000 take the y component, take its partial with respect 378 00:34:29,000 --> 00:34:31,000 to y and add them together. No signs. 379 00:34:31,000 --> 00:34:36,000 No switching things around. This one is pretty 380 00:34:36,000 --> 00:34:43,000 straightforward. The picture again is if I have 381 00:34:43,000 --> 00:34:50,000 my curve C going counterclockwise around a region 382 00:34:50,000 --> 00:34:58,000 R and I want to find the flux of some vector field F that is 383 00:34:58,000 --> 00:35:03,000 everywhere in here. Maybe some parts of C will 384 00:35:03,000 --> 00:35:06,000 contribute positively and some parts will contribute 385 00:35:06,000 --> 00:35:10,000 negatively. Just to reiterate what I said, 386 00:35:10,000 --> 00:35:14,000 positively here means, because we are going 387 00:35:14,000 --> 00:35:19,000 counterclockwise, the normal vector points out of 388 00:35:19,000 --> 00:35:31,000 the region. This guy here is the flux out 389 00:35:31,000 --> 00:35:39,000 of R through C. That is the formula. 390 00:35:39,000 --> 00:35:45,000 Any questions about what the statement says or how to use it 391 00:35:45,000 --> 00:35:48,000 concretely? No. 392 00:35:48,000 --> 00:35:51,000 OK. It is pretty similar to Green's 393 00:35:51,000 --> 00:35:58,000 theorem for work. Actually, I should say -- This 394 00:35:58,000 --> 00:36:07,000 is called Green's theorem in normal form also. 395 00:36:07,000 --> 00:36:17,000 Not that the other one is abnormal, but just that the old 396 00:36:17,000 --> 00:36:23,000 one for work was, you could say, 397 00:36:23,000 --> 00:36:28,000 in tangential form. That just means, 398 00:36:28,000 --> 00:36:32,000 well, Green's theorem, as seen yesterday was for the 399 00:36:32,000 --> 00:36:36,000 line integral F dot T ds, integrating the tangent 400 00:36:36,000 --> 00:36:39,000 component of F. The one today is for 401 00:36:39,000 --> 00:36:43,000 integrating the normal component of F. 402 00:36:43,000 --> 00:36:47,000 OK. Let's prove this. Good news. 403 00:36:47,000 --> 00:36:50,000 It is much easier to prove than the one we did yesterday because 404 00:36:50,000 --> 00:36:53,000 we are just going to show that it is the same thing just using 405 00:36:53,000 --> 00:36:54,000 different notations. 406 00:36:54,000 --> 00:37:23,000 407 00:37:23,000 --> 00:37:29,000 How do I prove it? Well, maybe actually it would 408 00:37:29,000 --> 00:37:33,000 help if first, before proving it, 409 00:37:33,000 --> 00:37:38,000 I actually rewrite what it means in components. 410 00:37:38,000 --> 00:37:46,000 We said the line integral of F dot n ds is actually the line 411 00:37:46,000 --> 00:37:54,000 integral of - Q dx P dy. And we want to show that this 412 00:37:54,000 --> 00:38:03,000 is equal to the double integral of P sub x Q sub y dA. 413 00:38:03,000 --> 00:38:09,000 This is really one of the features of Green's theorem. 414 00:38:09,000 --> 00:38:12,000 No matter which form it is, it relates a line integral to a 415 00:38:12,000 --> 00:38:16,000 double integral. Let's just try to see if we can 416 00:38:16,000 --> 00:38:19,000 reduce it to the one we had yesterday. 417 00:38:19,000 --> 00:38:24,000 Let me forget what these things mean physically and just focus 418 00:38:24,000 --> 00:38:26,000 on the math. On the math it is a line 419 00:38:26,000 --> 00:38:29,000 integral of something dx plus something dy. 420 00:38:29,000 --> 00:38:36,000 Let's call this guy M and let's call this guy N. 421 00:38:36,000 --> 00:38:42,000 Let M equal negative Q and N equal P. 422 00:38:42,000 --> 00:38:53,000 Then this guy here becomes integral of M dx plus N dy. 423 00:38:53,000 --> 00:38:57,000 And I know from yesterday what this is equal to, 424 00:38:57,000 --> 00:39:01,000 namely using the tangential form of Green's theorem. 425 00:39:01,000 --> 00:39:05,000 Green for work. This is the double integral of 426 00:39:05,000 --> 00:39:11,000 curl of this guy. That is Nx minus My dA. 427 00:39:11,000 --> 00:39:15,000 But now let's think about what this is in terms of M and N. 428 00:39:15,000 --> 00:39:24,000 Well, we said that M is negative Q so this is negative 429 00:39:24,000 --> 00:39:29,000 My. And we said P is the same as N, 430 00:39:29,000 --> 00:39:33,000 so this is Nx. Just by renaming the 431 00:39:33,000 --> 00:39:37,000 components, I go from one form to the other one. 432 00:39:37,000 --> 00:39:38,000 So it is really the same theorem. 433 00:39:38,000 --> 00:39:41,000 That's why it is also called Green's theorem. 434 00:39:41,000 --> 00:39:45,000 But the way we think about it when we use it is different, 435 00:39:45,000 --> 00:39:48,000 because one of them computes the work done by a force along a 436 00:39:48,000 --> 00:39:53,000 closed curve, the other one computes the flux 437 00:39:53,000 --> 00:39:59,000 maybe of a velocity field out of region. 438 00:39:59,000 --> 00:40:10,000 Questions? Yes? 439 00:40:10,000 --> 00:40:14,000 That is correct. If you are trying to compute a 440 00:40:14,000 --> 00:40:18,000 line integral for flux, wait, where did I put it? 441 00:40:18,000 --> 00:40:20,000 A line integral for flux just becomes this. 442 00:40:20,000 --> 00:40:25,000 And once you are here you know how to compute that kind of 443 00:40:25,000 --> 00:40:27,000 thing. The double integral side does 444 00:40:27,000 --> 00:40:29,000 not even have any kind of renaming to do. 445 00:40:29,000 --> 00:40:31,000 You know how to compute a double integral of a function. 446 00:40:31,000 --> 00:40:35,000 This is just a particular kind of function that you get out of 447 00:40:35,000 --> 00:40:38,000 a vector field, but it is like any function. 448 00:40:38,000 --> 00:40:41,000 The way you would evaluate these double integrals is just 449 00:40:41,000 --> 00:40:46,000 the usual way. Namely, you have a function of 450 00:40:46,000 --> 00:40:54,000 x and y, you have a region and you set up the bounds for the 451 00:40:54,000 --> 00:40:57,000 isolated integral. The way you would evaluate the 452 00:40:57,000 --> 00:40:59,000 double integrals is really the usual way, 453 00:40:59,000 --> 00:41:02,000 by slicing the region and setting up the bounds for 454 00:41:02,000 --> 00:41:06,000 iterated integrals in dx, dy or dydx or maybe rd, 455 00:41:06,000 --> 00:41:12,000 rd theta or whatever you want. In fact, in terms of computing 456 00:41:12,000 --> 00:41:14,000 integrals, we just have two sets of skills. 457 00:41:14,000 --> 00:41:18,000 One is setting up and evaluating double integrals. 458 00:41:18,000 --> 00:41:21,000 The other one is setting up and evaluating line integrals. 459 00:41:21,000 --> 00:41:25,000 And whether these line integrals or double integrals 460 00:41:25,000 --> 00:41:29,000 are representing work, flux, integral of a curve, 461 00:41:29,000 --> 00:41:34,000 whatever, the way that we actually 462 00:41:34,000 --> 00:41:40,000 compute them is the same. Let's do an example. 463 00:41:40,000 --> 00:41:47,000 Oh, first. Sorry. This renaming here, see, 464 00:41:47,000 --> 00:41:51,000 that is why actually I call my components P and Q because the 465 00:41:51,000 --> 00:41:54,000 argument would have gotten very messy if I had told you now I 466 00:41:54,000 --> 00:41:57,000 call M ,N and I call N minus M and so on. 467 00:41:57,000 --> 00:42:00,000 But, now that we are through with this, 468 00:42:00,000 --> 00:42:03,000 if you still like M and N better, 469 00:42:03,000 --> 00:42:15,000 you know, what this says -- The formulation of Green's theorem 470 00:42:15,000 --> 00:42:27,000 in this language is just integral of minus N dx plus M dy 471 00:42:27,000 --> 00:42:37,000 is the double integral over R of Mx plus Ny dA. 472 00:42:37,000 --> 00:42:42,000 Now let's do an example. Let's look at this picture 473 00:42:42,000 --> 00:42:51,000 again, the flux of xi plus yj out of the circle of radius A. 474 00:42:51,000 --> 00:42:53,000 We did the calculation directly using geometry, 475 00:42:53,000 --> 00:42:57,000 and it wasn't all that bad. But let's see what Green's 476 00:42:57,000 --> 00:42:58,000 theorem does for us here. 477 00:42:58,000 --> 00:43:19,000 478 00:43:19,000 --> 00:43:22,000 Example. Let's take the same example as 479 00:43:22,000 --> 00:43:28,000 last time. F equals xi yj. 480 00:43:28,000 --> 00:43:43,000 C equals circle of radius a counterclockwise. 481 00:43:43,000 --> 00:43:46,000 How do we set up Green's theorem. 482 00:43:46,000 --> 00:43:57,000 Well, let's first figure out the divergence of F. 483 00:43:57,000 --> 00:44:00,000 The divergence of this field, I take the x component, 484 00:44:00,000 --> 00:44:03,000 which is x, and I take its partial respect to x. 485 00:44:03,000 --> 00:44:08,000 And then I do the same with the y component, and I will get one 486 00:44:08,000 --> 00:44:12,000 plus one equals two. So, the divergence of this 487 00:44:12,000 --> 00:44:17,000 field is two. Now, Green's theorem tells us 488 00:44:17,000 --> 00:44:25,000 that the flux out of this region is going to be the double 489 00:44:25,000 --> 00:44:29,000 integral of 2 dA. What is R now? 490 00:44:29,000 --> 00:44:31,000 Well, R is the region enclosed by C. 491 00:44:31,000 --> 00:44:38,000 So if C is the circle, R is the disk of radius A. 492 00:44:38,000 --> 00:44:42,000 Of course, we can compute it, but we don't have to because 493 00:44:42,000 --> 00:44:46,000 double integral of 2dA is just twice the double integral of dA 494 00:44:46,000 --> 00:44:51,000 so it is twice the area of R. And we know the area of a 495 00:44:51,000 --> 00:44:54,000 circle of radius A. That is piA2. 496 00:44:54,000 --> 00:45:01,000 So, it is 2piA2. That is the same answer that we 497 00:45:01,000 --> 00:45:04,000 got directly, which is good news. 498 00:45:04,000 --> 00:45:08,000 Now we can even do better. Let's say that my circle is not 499 00:45:08,000 --> 00:45:12,000 at the origin. Let's say that it is out here. 500 00:45:12,000 --> 00:45:17,000 Well, then it becomes harder to calculate the flux directly. 501 00:45:17,000 --> 00:45:21,000 And it is harder even to guess exactly what will happen because 502 00:45:21,000 --> 00:45:24,000 on this side here the vector field will go into the region so 503 00:45:24,000 --> 00:45:27,000 the contribution to flux will be negative here. 504 00:45:27,000 --> 00:45:31,000 Here it will be positive because it is going out of the 505 00:45:31,000 --> 00:45:33,000 region. There are positive and negative 506 00:45:33,000 --> 00:45:35,000 terms. Well, it looks like positive 507 00:45:35,000 --> 00:45:38,000 should win because here the vector field is much larger than 508 00:45:38,000 --> 00:45:41,000 over there. But, short of computing it, 509 00:45:41,000 --> 00:45:45,000 we won't actually know what it is. 510 00:45:45,000 --> 00:45:48,000 If you want to do it by direct calculation then you have to 511 00:45:48,000 --> 00:45:51,000 parametize this circle and figure out what the line 512 00:45:51,000 --> 00:45:55,000 integral will be. But if you use Green's theorem, 513 00:45:55,000 --> 00:46:00,000 well, we never used the fact that it is the circle of radius 514 00:46:00,000 --> 00:46:03,000 A at the origin. It is true actually for any 515 00:46:03,000 --> 00:46:08,000 closed curve that the flux out of it is going to be twice the 516 00:46:08,000 --> 00:46:12,000 area of the region inside. It still will be 2piA2 even if 517 00:46:12,000 --> 00:46:16,000 my circle is anywhere else in the plane. 518 00:46:16,000 --> 00:46:18,000 If I had asked you a trick question where do you want to 519 00:46:18,000 --> 00:46:21,000 place this circle so that that the flux is the largest? 520 00:46:21,000 --> 00:46:28,000 Well, the answer is it doesn't matter. 521 00:46:28,000 --> 00:46:33,000 Now, let's just finish quickly by answering a question that 522 00:46:33,000 --> 00:46:36,000 some of you, I am sure, must have, 523 00:46:36,000 --> 00:46:40,000 which is what does divergence mean and what does it measure? 524 00:46:40,000 --> 00:46:44,000 I mean, we said for curl, curl measures how much things 525 00:46:44,000 --> 00:46:48,000 are rotating somehow. What does divergence mean? 526 00:46:48,000 --> 00:46:53,000 Well, the answer is divergence measures how much things are 527 00:46:53,000 --> 00:47:08,000 diverging. Let's be more explicit. 528 00:47:08,000 --> 00:47:20,000 Interpretation of divergence. You can think of it, 529 00:47:20,000 --> 00:47:23,000 you know, what do I want to say first? 530 00:47:23,000 --> 00:47:28,000 If you take a vector field that is a constant vector field where 531 00:47:28,000 --> 00:47:32,000 everything just translates then there is no divergence involved 532 00:47:32,000 --> 00:47:34,000 because the derivatives will be zero. 533 00:47:34,000 --> 00:47:37,000 If you take the guy that rotates things around you will 534 00:47:37,000 --> 00:47:40,000 also compute and find zero for divergence. 535 00:47:40,000 --> 00:47:43,000 This is not sensitive to translation motions where 536 00:47:43,000 --> 00:47:46,000 everything moves together or to rotation motions, 537 00:47:46,000 --> 00:47:51,000 but instead it is sensitive to explaining motions. 538 00:47:51,000 --> 00:48:04,000 A possible answer is that it measures how much the flow is 539 00:48:04,000 --> 00:48:10,000 expanding areas. If you imagine this flow that 540 00:48:10,000 --> 00:48:14,000 we have here on the picture, things are moving away from the 541 00:48:14,000 --> 00:48:16,000 origin and they fill out the plane. 542 00:48:16,000 --> 00:48:19,000 If we mention this fluid flowing out there, 543 00:48:19,000 --> 00:48:21,000 it is occupying more and more space. 544 00:48:21,000 --> 00:48:24,000 And so that is what it means to have positive divergence. 545 00:48:24,000 --> 00:48:28,000 If you took the opposite vector field that contracts everything 546 00:48:28,000 --> 00:48:31,000 to the origin that will have negative divergence. 547 00:48:31,000 --> 00:48:34,000 That is a good way to think about it if you are thinking of 548 00:48:34,000 --> 00:48:37,000 a gas maybe that can expand to fill out more volume. 549 00:48:37,000 --> 00:48:41,000 If you thinking of water, well, water doesn't really 550 00:48:41,000 --> 00:48:43,000 shrink or expand. The fact that it is taking more 551 00:48:43,000 --> 00:48:46,000 and more space actually means that there is more and more 552 00:48:46,000 --> 00:48:51,000 water. The other way to think about it 553 00:48:51,000 --> 00:48:56,000 is divergence is the source rate, 554 00:48:56,000 --> 00:49:00,000 it is the amount of fluid that is being inserted into the 555 00:49:00,000 --> 00:49:12,000 system, that is being pumped into the 556 00:49:12,000 --> 00:49:27,000 system per unit time per unit area. 557 00:49:27,000 --> 00:49:31,000 What div F equals two here means is that here you actually 558 00:49:31,000 --> 00:49:35,000 have matter being created or being pumped into the system so 559 00:49:35,000 --> 00:49:39,000 that you have more and more water filling more and more 560 00:49:39,000 --> 00:49:41,000 space as it flows. But, actually, 561 00:49:41,000 --> 00:49:43,000 divergence is not two just at the origin. 562 00:49:43,000 --> 00:49:46,000 It is two everywhere. So, in fact, 563 00:49:46,000 --> 00:49:49,000 to have this you need to have a system of pumps that actually is 564 00:49:49,000 --> 00:49:52,000 in something water absolutely everywhere uniformly. 565 00:49:52,000 --> 00:49:55,000 That is the only way to do this. I mean if you imagine that you 566 00:49:55,000 --> 00:49:57,000 just have one spring at the origin then, 567 00:49:57,000 --> 00:50:00,000 sure, water will flow out, but as you go further and 568 00:50:00,000 --> 00:50:02,000 further away it will do so more and more slowly. 569 00:50:02,000 --> 00:50:04,000 Well, here it is flowing away faster and faster. 570 00:50:04,000 --> 00:50:09,000 And that means everywhere you are still pumping more water 571 00:50:09,000 --> 00:50:11,000 into it. So, that is what divergence 572 00:50:11,000 --> 00:50:13,000 measures. 573 00:50:13,000 --> 00:50:18,000