WEBVTT 00:00:00.035 --> 00:00:04.180 Vector calculus ends with the divergence theorem and Stokes' theorem. 00:00:04.377 --> 00:00:06.920 They make the same startling promise: 00:00:07.117 --> 00:00:09.427 what a field does throughout a region can be recovered 00:00:09.427 --> 00:00:11.355 from what it does on the boundary. 00:00:12.752 --> 00:00:16.965 The divergence theorem turns total divergence inside a solid into flux 00:00:17.048 --> 00:00:19.023 through its closed surface. 00:00:19.219 --> 00:00:24.212 Stokes' theorem turns total curl through a surface into circulation around its edge. 00:00:24.525 --> 00:00:28.206 Both replace an interior measurement with a boundary measurement. 00:00:29.603 --> 00:00:32.157 Here is a vector field in the plane. 00:00:32.354 --> 00:00:34.302 At every point there's an arrow, 00:00:34.385 --> 00:00:37.962 and the honest way to read it is as the velocity of a fluid. 00:00:38.159 --> 00:00:42.499 Put a speck of dust down somewhere, and the arrow tells you which way it drifts, 00:00:42.582 --> 00:00:44.034 and how fast. 00:00:44.738 --> 00:00:49.104 Now draw a little closed loop around the origin and ask one simple question. 00:00:49.208 --> 00:00:52.088 Is more fluid leaving this loop than entering it? 00:00:52.284 --> 00:00:56.708 Here, obviously yes. Every single arrow crosses it on the way out. 00:00:56.905 --> 00:00:59.216 Something inside is making fluid. 00:01:00.013 --> 00:01:02.196 Compare that with this second field. 00:01:02.393 --> 00:01:05.319 Same plane, but now the fluid is turning. 00:01:05.423 --> 00:01:09.731 It goes round and round the origin, and it moves faster the further out you look. 00:01:10.783 --> 00:01:12.325 Draw the same loop on it. 00:01:12.408 --> 00:01:14.487 And now nothing crosses. 00:01:14.684 --> 00:01:17.727 The arrows run along the loop, never through it. 00:01:17.923 --> 00:01:21.477 There is plenty of motion here, and no outflow at all. 00:01:22.273 --> 00:01:26.649 That contrast is what divergence measures. Not how fast the field is, 00:01:26.732 --> 00:01:30.715 not which way it points, but how much of it is flowing out of a point. 00:01:30.818 --> 00:01:33.431 Positive divergence means a source. 00:01:33.628 --> 00:01:35.486 Negative means a sink. 00:01:35.590 --> 00:01:38.644 Zero means whatever comes in goes out again. 00:01:39.441 --> 00:01:43.180 Hold on to that picture, because both of today's theorems are built out of it. 00:01:43.377 --> 00:01:45.653 Now let's turn it into a formula. 00:01:46.708 --> 00:01:49.229 So let's turn that into a formula. 00:01:49.332 --> 00:01:50.598 If our field is F, 00:01:50.598 --> 00:01:56.253 then its divergence is written div F, or nabla dot F, and here it is. 00:01:56.449 --> 00:02:01.245 The components P, Q and R are functions of x, y and z. 00:02:02.041 --> 00:02:05.897 Take the first component and differentiate it with respect to x. 00:02:06.093 --> 00:02:09.647 Take the second, and differentiate with respect to y. 00:02:09.843 --> 00:02:12.224 Take the third with respect to z. 00:02:12.537 --> 00:02:14.616 Then add the three numbers up. 00:02:15.413 --> 00:02:19.834 That is genuinely all it is: one derivative from each component, 00:02:19.917 --> 00:02:22.147 each matched to its own variable. 00:02:22.251 --> 00:02:24.875 And notice what kind of object comes out. 00:02:25.072 --> 00:02:28.602 F is a vector field, an arrow at every point. 00:02:28.706 --> 00:02:31.632 Div F is a number at every point. 00:02:32.936 --> 00:02:34.469 Let's compute one. 00:02:34.666 --> 00:02:37.787 Our spreading field was F equals x, y. 00:02:37.870 --> 00:02:41.480 The first component is x, so its x derivative is one. 00:02:41.563 --> 00:02:45.032 The second is y, so its y derivative is one. 00:02:45.115 --> 00:02:46.184 Add them. 00:02:47.399 --> 00:02:51.440 Two, and positive, which is exactly what the picture said. 00:02:51.636 --> 00:02:54.049 Fluid is being created at every point, 00:02:54.132 --> 00:02:57.593 so any loop you draw has more coming out of it than going in. 00:02:57.790 --> 00:02:59.102 Here the divergence happens 00:02:59.102 --> 00:03:02.887 to be constant, but in general it is a function of position. 00:03:04.284 --> 00:03:05.817 Now the swirl. 00:03:06.014 --> 00:03:08.508 F equals minus y, x. 00:03:08.591 --> 00:03:13.584 The first component is minus y, and we differentiate it with respect to x. 00:03:13.781 --> 00:03:18.795 There is no x in it, so that derivative is nothing. The second component is x, 00:03:18.878 --> 00:03:22.501 differentiated with respect to y, and that is nothing too. 00:03:23.042 --> 00:03:25.966 So the divergence is zero, everywhere. 00:03:26.049 --> 00:03:27.777 And again the picture agrees. 00:03:27.860 --> 00:03:32.238 The fluid is moving, quite fast in places, but none of it ever leaves. 00:03:32.434 --> 00:03:36.173 Motion and divergence are two different questions, and this field is the cleanest 00:03:36.173 --> 00:03:37.834 reminder of that I know. 00:03:38.909 --> 00:03:40.162 Now the first theorem. 00:03:40.245 --> 00:03:45.061 Take a solid region in space, call it E, and a vector field F living on it. 00:03:45.144 --> 00:03:49.766 The skin of that region, the closed surface wrapping it up, is written del E. 00:03:50.376 --> 00:03:54.960 There are two completely different things you could measure here. First, go inside. 00:03:55.043 --> 00:03:59.639 Compute the divergence at every point of E and add it all up over the volume. 00:03:59.722 --> 00:04:03.252 That is the total rate at which fluid is being produced in there. 00:04:04.049 --> 00:04:06.927 Second, forget the inside completely. 00:04:07.010 --> 00:04:11.072 Stand on the surface, and at each point take the part of F pointing straight out, 00:04:11.155 --> 00:04:12.746 F dot n hat. 00:04:12.942 --> 00:04:15.323 Add that up over the whole skin. 00:04:15.520 --> 00:04:20.269 That is the flux, the net rate at which fluid crosses the boundary. 00:04:21.066 --> 00:04:24.747 The divergence theorem says those two numbers are equal. 00:04:24.944 --> 00:04:26.047 Always. 00:04:26.244 --> 00:04:29.655 Total production inside equals total flow across the boundary, 00:04:29.738 --> 00:04:33.965 which, once you say it out loud, is really just conservation of stuff. 00:04:35.269 --> 00:04:36.895 So why is it true? 00:04:37.092 --> 00:04:40.898 Keep the solid in view and take one flat slice through it. 00:04:40.981 --> 00:04:44.488 The slice stands for a layer of tiny cells filling the volume. 00:04:44.685 --> 00:04:48.587 We can understand the whole solid by inspecting two neighbours in that layer. 00:04:49.383 --> 00:04:51.683 Here are those two neighbours enlarged. 00:04:51.880 --> 00:04:56.605 For each cell, the flux across its boundary is its divergence contribution. 00:04:56.709 --> 00:05:01.365 When all the cells are added, the question is what happens along every shared wall. 00:05:01.906 --> 00:05:05.796 The cell on the left counts that shared wall with its outward normal pointing 00:05:05.796 --> 00:05:10.719 into the cell on the right. That direction sets the sign of its flux contribution. 00:05:11.423 --> 00:05:15.173 The cell on the right counts the same wall with its outward normal pointing back 00:05:15.173 --> 00:05:16.785 into the cell on the left. 00:05:16.868 --> 00:05:18.480 The two directions are opposite, 00:05:18.563 --> 00:05:21.827 so the two flux contributions have opposite signs. 00:05:22.368 --> 00:05:27.474 Equal and opposite. Every interior wall cancels against its partner, in pairs, 00:05:27.557 --> 00:05:29.358 all the way through the solid. 00:05:29.461 --> 00:05:34.513 Remove those interior boundaries and what survives is only the outside skin. 00:05:35.565 --> 00:05:38.270 That is the entire proof, in spirit. 00:05:38.722 --> 00:05:41.788 The divergence theorem is a bookkeeping identity. 00:05:41.892 --> 00:05:45.375 Interior faces cancel; boundary faces don't. 00:05:46.428 --> 00:05:49.030 The second theorem needs a different measurement. 00:05:49.133 --> 00:05:52.444 Divergence asked how much of a field flows out of a point. 00:05:52.756 --> 00:05:54.766 Curl asks something else: 00:05:54.870 --> 00:05:57.424 how much the field turns around a point. 00:05:58.220 --> 00:06:00.459 Drop a tiny paddle wheel into the fluid, 00:06:00.542 --> 00:06:03.714 pinned at its centre so that it can only spin. 00:06:03.910 --> 00:06:06.928 If the flow turns it, the field has curl there. 00:06:07.242 --> 00:06:11.164 Curl is a vector: it points along the axis the wheel spins about, 00:06:11.248 --> 00:06:13.628 and its length says how fast. 00:06:14.540 --> 00:06:15.432 In the plane, 00:06:15.516 --> 00:06:20.266 that axis is always straight up out of the page, so only one number survives. 00:06:20.880 --> 00:06:24.082 For our swirling field, minus y and x, 00:06:24.166 --> 00:06:29.390 it comes out as two k hat, a constant amount of spin at every single point. 00:06:31.206 --> 00:06:33.098 Now Stokes' theorem. 00:06:33.294 --> 00:06:36.938 Take a surface in space, and this time not a closed one. 00:06:37.022 --> 00:06:39.542 A piece of one, like a butterfly net. 00:06:39.645 --> 00:06:42.712 Then del S is the curve running round its rim. 00:06:43.252 --> 00:06:47.709 Choose a normal direction for the surface, and walk that rim the matching way: 00:06:47.792 --> 00:06:51.590 the way that keeps the surface on your left when your head points along n hat. 00:06:51.692 --> 00:06:55.466 Pairing those two directions up is the only fiddly part of the whole theorem. 00:06:55.950 --> 00:06:58.178 And Stokes says this. The total 00:06:58.178 --> 00:07:02.906 curl passing through the surface equals the circulation of F once around its rim. 00:07:03.702 --> 00:07:06.335 The left-hand side is a surface integral: 00:07:06.418 --> 00:07:09.307 at each point of S, how much curl is aimed through it. 00:07:09.390 --> 00:07:13.350 The right-hand side is a line integral: F dotted with the direction 00:07:13.350 --> 00:07:17.078 you are travelling, added up all the way round the closed loop. 00:07:18.102 --> 00:07:20.634 And the reason is the same trick as before. 00:07:20.739 --> 00:07:25.113 Chop the surface into little tiles, and walk round the edge of every single tile, 00:07:25.196 --> 00:07:27.694 in the direction the normal tells you to. 00:07:28.490 --> 00:07:31.100 Every interior edge belongs to two tiles, 00:07:31.184 --> 00:07:34.354 and those two tiles walk it in opposite directions. 00:07:34.667 --> 00:07:38.592 So it cancels, exactly as the shared walls did a moment ago. 00:07:38.696 --> 00:07:40.829 All the little circulations collapse, 00:07:40.912 --> 00:07:43.862 and what is left is the walk round the outer rim. 00:07:44.914 --> 00:07:49.942 Meanwhile each tiny loop's circulation is its own curl times its own area. 00:07:50.047 --> 00:07:50.848 Add them all up 00:07:50.848 --> 00:07:55.214 and you get the surface integral on one side and the boundary loop on the other. 00:07:55.410 --> 00:07:58.047 Same bookkeeping, one dimension down. 00:07:59.133 --> 00:08:01.407 Let's put the two theorems one above the other, 00:08:01.490 --> 00:08:03.859 because the resemblance is the whole point. 00:08:04.474 --> 00:08:09.685 First the divergence theorem: a triple integral of div F over a solid region, 00:08:09.768 --> 00:08:13.635 equal to the flux of F out through the closed surface around it. 00:08:14.431 --> 00:08:20.351 And then Stokes: a surface integral of curl F over a piece of surface, 00:08:20.434 --> 00:08:24.417 equal to the circulation of F once round the curve that bounds it. 00:08:25.329 --> 00:08:27.176 Now look at the shape of them. 00:08:27.791 --> 00:08:32.668 On the left of each one, a derivative of F, integrated over a region. 00:08:32.865 --> 00:08:34.361 On the right of each one, 00:08:34.444 --> 00:08:38.554 F itself, integrated over the boundary of that region. 00:08:38.751 --> 00:08:41.245 Div and curl are different derivatives, 00:08:41.328 --> 00:08:44.812 and a solid and a surface are different regions. 00:08:45.516 --> 00:08:47.827 And you have met that sentence before. 00:08:48.024 --> 00:08:50.660 The fundamental theorem of calculus: 00:08:50.857 --> 00:08:56.581 integrate f prime across an interval, and the answer is f at the two endpoints. 00:08:57.033 --> 00:09:00.587 The boundary of an interval is just its two ends. 00:09:01.197 --> 00:09:03.427 Same sentence, every time. 00:09:03.624 --> 00:09:09.047 Integrate a derivative over a region, and the answer is already written on the edge. 00:09:09.243 --> 00:09:11.250 That is what these theorems buy you: 00:09:11.333 --> 00:09:16.617 a hard integral over something fat, traded for an easier one over its skin. 00:09:17.413 --> 00:09:18.377 So. 00:09:18.690 --> 00:09:24.505 Divergence measures outflow, curl measures spin, and each of them has a theorem 00:09:24.588 --> 00:09:29.558 saying that the inside tells you nothing the boundary has not already said.