WEBVTT 00:00:00.035 --> 00:00:03.159 Let me tell you what Stokes' theorem actually says. 00:00:03.262 --> 00:00:04.503 It is one sentence. 00:00:04.586 --> 00:00:07.791 Not a page of conditions, not a machine you turn a handle on. 00:00:07.988 --> 00:00:10.865 One sentence, which then gets written down three times, 00:00:10.948 --> 00:00:15.291 in three different numbers of dimensions, and picks up a different name each time. 00:00:16.595 --> 00:00:19.693 Before the sentence, put its three regions side by side. 00:00:19.776 --> 00:00:24.546 A chunk of the number line, a path through the plane, and a two dimensional patch. 00:00:24.629 --> 00:00:26.289 These regions have different dimensions, 00:00:26.289 --> 00:00:29.018 but each has an edge one dimension smaller. 00:00:29.815 --> 00:00:31.659 Start with the chunk of line. 00:00:31.742 --> 00:00:34.773 Its edge is two dots, one at each end. 00:00:34.969 --> 00:00:39.219 Those two endpoints are the complete boundary of the interval from a to b. 00:00:40.016 --> 00:00:42.525 The path has the same kind of boundary. 00:00:42.721 --> 00:00:45.518 Its edge is the point P where the journey starts, 00:00:45.601 --> 00:00:48.202 together with the point Q where it stops. 00:00:48.999 --> 00:00:50.590 The patch is different. 00:00:50.694 --> 00:00:52.759 Its edge is not a pair of points. 00:00:52.842 --> 00:00:56.067 It is the closed curve running all the way around the outside, 00:00:56.150 --> 00:00:57.927 walked in one direction. 00:00:58.724 --> 00:01:00.373 Now look at the pattern. 00:01:00.570 --> 00:01:03.215 The edge of a piece of line is a couple of points. 00:01:03.298 --> 00:01:05.828 The edge of a path is its two endpoints. 00:01:05.911 --> 00:01:08.164 The edge of a blob is a curve. 00:01:08.360 --> 00:01:10.878 In every case the edge is one dimension smaller, 00:01:10.961 --> 00:01:13.852 and here is the sentence all three theorems share: 00:01:14.049 --> 00:01:16.810 add up how fast something changes inside a region, 00:01:16.893 --> 00:01:19.483 and the answer depends only on its edge. 00:01:20.996 --> 00:01:23.551 And here is that sentence written in symbols. 00:01:23.747 --> 00:01:25.348 This is Stokes' theorem. 00:01:25.431 --> 00:01:27.475 The whole of it, on one line. 00:01:28.272 --> 00:01:30.896 You are not supposed to be able to read that yet. 00:01:31.000 --> 00:01:33.067 I only want you to have seen it once. 00:01:33.264 --> 00:01:36.655 Omega is the region, whatever it happens to be. 00:01:36.759 --> 00:01:40.205 Del Omega is its edge. And the line says: 00:01:40.288 --> 00:01:43.818 what you get by integrating inside is decided on the edge. 00:01:44.522 --> 00:01:47.968 And you have almost certainly met that line three times already, 00:01:48.051 --> 00:01:49.805 under three different names. 00:01:50.257 --> 00:01:52.684 The fundamental theorem of calculus. 00:01:52.881 --> 00:01:55.680 The fundamental theorem for line integrals. 00:01:55.876 --> 00:01:57.549 And Green's theorem. 00:01:58.461 --> 00:02:00.517 So let's take them one at a time. 00:02:00.714 --> 00:02:02.362 One dimension, then a curve 00:02:02.362 --> 00:02:06.984 through the plane, then a flat patch, and watch the same line come back every time. 00:02:08.127 --> 00:02:12.227 Start in one dimension, where you have known this theorem the longest. 00:02:12.539 --> 00:02:15.210 Here is the question it actually answers. 00:02:15.407 --> 00:02:18.473 You know how fast something is changing at every moment: 00:02:18.576 --> 00:02:22.444 you're driving, and you can watch the speedometer the whole way. 00:02:22.756 --> 00:02:24.417 How far did you go? 00:02:24.901 --> 00:02:28.929 The fundamental theorem of calculus says this. If you integrate the derivative 00:02:28.929 --> 00:02:35.327 of f from a to b, you don't get anything complicated. You get f at b, minus f at a. 00:02:35.868 --> 00:02:37.912 Read that out loud in words. 00:02:38.364 --> 00:02:42.533 Add up all the little changes over the whole interval, and you get the net change 00:02:42.533 --> 00:02:43.889 from one end to the other. 00:02:43.972 --> 00:02:46.342 The speedometer tells you the odometer. 00:02:47.556 --> 00:02:48.994 Let's make that concrete, 00:02:49.077 --> 00:02:52.978 because a number you can check beats a formula you can only nod at. 00:02:53.082 --> 00:02:57.007 Take f of x equal to x squared over four. 00:02:57.111 --> 00:03:01.152 Its derivative is x over two, which is this blue line. 00:03:01.348 --> 00:03:05.494 Fix the left end at one, and let the right endpoint b start at two. 00:03:06.291 --> 00:03:10.123 The left hand side is the shaded area from one to b. 00:03:10.319 --> 00:03:12.724 Now slide b out to four. 00:03:12.827 --> 00:03:16.798 The region grows with it, and the running total up in the corner counts all the way 00:03:16.798 --> 00:03:18.621 to three point seven five. 00:03:19.406 --> 00:03:21.160 Now check the other side. 00:03:21.345 --> 00:03:23.517 f of four is four. 00:03:23.946 --> 00:03:25.920 f of one is a quarter. 00:03:26.117 --> 00:03:29.670 Four minus a quarter is three point seven five. 00:03:29.867 --> 00:03:31.841 Same number, obviously. 00:03:32.893 --> 00:03:35.120 But here is the thing I really want you to notice. 00:03:35.203 --> 00:03:38.420 The right hand side never looks inside the interval at all. 00:03:39.217 --> 00:03:41.249 It only asks two questions. 00:03:41.562 --> 00:03:45.534 What is f at the right end, and what is f at the left end? 00:03:45.846 --> 00:03:47.937 Everything in the middle cancels. 00:03:48.316 --> 00:03:50.847 So here is the piece of language you need. 00:03:51.160 --> 00:03:56.060 The interval from a to b is the region. Its boundary is just the two endpoints. 00:03:56.257 --> 00:03:59.497 That word, boundary, is the whole lecture. 00:04:00.293 --> 00:04:02.869 The boundary is not the number b minus a. 00:04:02.952 --> 00:04:05.168 It is a formal sum of two points: 00:04:05.251 --> 00:04:09.533 plus b at the right hand end, together with negative a at the left hand end. 00:04:09.616 --> 00:04:14.853 Those signs are exactly why the answer is f of b minus f of a. 00:04:15.464 --> 00:04:19.830 That sign is orientation, the direction in which you travel. 00:04:20.143 --> 00:04:24.274 Watch the traveller move from the negative endpoint to the positive endpoint. 00:04:24.357 --> 00:04:26.097 Keep that direction in mind. 00:04:26.180 --> 00:04:28.700 It comes straight back in Green's theorem. 00:04:29.766 --> 00:04:31.602 Now go up a dimension. 00:04:31.798 --> 00:04:34.075 Same sentence, new region. 00:04:34.271 --> 00:04:37.499 This time the region is not a chunk of the number line. 00:04:37.696 --> 00:04:42.515 It is a route, drawn through the plane, from a point P to a point Q. 00:04:43.312 --> 00:04:47.457 And instead of a function of one variable, there are now two. 00:04:47.654 --> 00:04:53.796 Picture it as a hill: f of x, y is the height of the ground above the point x, y. 00:04:53.993 --> 00:04:57.256 These grey rings are its contour lines. 00:04:57.960 --> 00:05:02.198 A contour is the set of places at one particular height. 00:05:02.511 --> 00:05:07.258 So put a walker on the ring where f is nine, and send it the whole way round. 00:05:07.341 --> 00:05:12.067 It climbs nothing at all, because every point of that ring is at the same height. 00:05:12.771 --> 00:05:17.090 Here is theorem one again, so you can see what is coming. Integrate 00:05:17.090 --> 00:05:21.514 the derivative across the region, and read the answer off the two ends. 00:05:22.217 --> 00:05:27.463 And here is theorem two. Integrate the gradient of f along the curve C, 00:05:27.546 --> 00:05:31.448 and what you get is f at Q minus f at P. 00:05:31.552 --> 00:05:33.698 Look at those two lines together. 00:05:33.781 --> 00:05:36.011 It is the same sentence twice. 00:05:36.807 --> 00:05:41.626 Read the left side as, add up every little bit of climbing as you walk. 00:05:41.939 --> 00:05:43.400 Read the right side as, 00:05:43.483 --> 00:05:46.793 the height where you finished minus the height where you started. 00:05:46.990 --> 00:05:50.450 It is the odometer story again, on a hill. 00:05:51.247 --> 00:05:54.115 And that hands you something genuinely useful. 00:05:54.312 --> 00:05:56.182 The route does not matter. 00:05:56.378 --> 00:05:58.478 Take hold of it here, in the middle, 00:05:58.561 --> 00:06:02.718 and bend it wherever you like, keeping P and Q where they are. 00:06:03.515 --> 00:06:05.638 Up over the shoulder of the hill. 00:06:05.721 --> 00:06:07.893 Or down round the bottom instead. 00:06:08.089 --> 00:06:11.826 The climb you total up is the same number every single time, 00:06:11.909 --> 00:06:15.462 because the right hand side never mentions the route at all. 00:06:16.514 --> 00:06:19.615 Because the formula only ever looks at the endpoints. 00:06:19.928 --> 00:06:21.786 There is that word again. 00:06:22.099 --> 00:06:27.046 The boundary of a curve is its two ends, and the answer depends on nothing else. 00:06:27.842 --> 00:06:31.489 Which hands you a corollary that falls straight out. 00:06:31.685 --> 00:06:35.657 Suppose the route is a closed loop, like this circle Gamma. 00:06:35.853 --> 00:06:37.815 You finish exactly where you started, 00:06:37.815 --> 00:06:41.311 so the start point and the end point are the same point. 00:06:41.922 --> 00:06:44.326 So the endpoint term is f of P 00:06:44.430 --> 00:06:46.346 minus f of P. 00:06:46.659 --> 00:06:49.922 The two copies cancel, and the result is zero. 00:06:50.533 --> 00:06:54.736 Around any closed loop, a gradient field integrates to zero. 00:06:54.933 --> 00:06:59.575 You can see it coming without computing anything: closed curve, gradient field, 00:06:59.658 --> 00:07:00.960 nothing to do. 00:07:01.756 --> 00:07:05.449 Same theorem. Region, boundary, done. 00:07:05.762 --> 00:07:08.421 Now the one that looks hardest and really isn't. 00:07:09.496 --> 00:07:11.412 So, the flat case. 00:07:11.609 --> 00:07:15.243 Green's theorem. This is the one that looks frightening the first time you meet 00:07:15.243 --> 00:07:17.937 it, and it is the same sentence again. 00:07:18.734 --> 00:07:23.736 Our region is now genuinely two dimensional: a flat patch R in the plane, 00:07:23.819 --> 00:07:25.805 carrying a field of arrows. 00:07:26.002 --> 00:07:29.880 The rectangle keeps the boundary easy to read, but the theorem does not depend 00:07:29.880 --> 00:07:31.111 on that shape. 00:07:31.908 --> 00:07:36.041 Its boundary is the closed curve that runs all the way around the outside. 00:07:36.238 --> 00:07:37.792 We write it as del R. 00:07:37.875 --> 00:07:42.578 Orientation matters, exactly as the plus and minus mattered on the interval. 00:07:43.003 --> 00:07:46.557 Travel counterclockwise, so the region stays on your left. 00:07:46.753 --> 00:07:50.434 Starting here, one complete trip comes back to the point it left. 00:07:50.631 --> 00:07:53.499 That choice of direction is the same choice as the plus sign 00:07:53.499 --> 00:07:55.601 on the interval's right hand end. 00:07:56.305 --> 00:07:57.673 Here is the statement. 00:07:57.756 --> 00:08:01.356 The double integral over every point of the region lives on the left. 00:08:01.552 --> 00:08:04.769 One line integral around its boundary lives on the right. 00:08:04.966 --> 00:08:07.776 Inside on the left, edge on the right. 00:08:08.572 --> 00:08:12.741 Read the left side as the total swirl at all points inside R. 00:08:12.938 --> 00:08:13.867 Read the right side 00:08:13.867 --> 00:08:18.453 as the total push along one counterclockwise trip around del R. 00:08:18.766 --> 00:08:23.689 Every arrow inside contributes to the first count; the rim supplies the second. 00:08:24.486 --> 00:08:27.549 Green's theorem says those two numbers are equal. 00:08:27.632 --> 00:08:32.567 Total swirl inside equals total push around the rim. That is the whole theorem. 00:08:33.364 --> 00:08:34.941 Now for the reason. 00:08:35.024 --> 00:08:37.135 Chop R into little tiles. 00:08:37.218 --> 00:08:40.644 Each tile contributes a walk around its own boundary. 00:08:41.440 --> 00:08:42.751 Choose this tile. 00:08:42.834 --> 00:08:45.516 Its contribution includes this little edge. 00:08:45.620 --> 00:08:49.824 Adding the tiles means reading that edge once from each tile that touches it. 00:08:50.736 --> 00:08:53.393 One tile walks the shared edge upward. 00:08:53.476 --> 00:08:56.391 Its neighbour walks the very same edge downward. 00:08:56.588 --> 00:09:01.801 The geometric edge is one object, but its two induced directions are opposites. 00:09:02.714 --> 00:09:04.944 Those two contributions cancel. 00:09:05.558 --> 00:09:11.016 The same happens at every shared edge, so the entire interior grid deletes itself. 00:09:12.068 --> 00:09:14.529 Only edges with no neighbour remain: 00:09:14.726 --> 00:09:16.352 the outside of R. 00:09:16.456 --> 00:09:20.660 Their induced walk is the surviving boundary, with the region on its left. 00:09:20.856 --> 00:09:22.598 That is Green's theorem. 00:09:23.395 --> 00:09:27.390 The equality now reads from the region to its surviving boundary. 00:09:27.702 --> 00:09:29.130 The derivative is integrated 00:09:29.130 --> 00:09:34.831 over R on the left; the original field is integrated over del R on the right. 00:09:35.628 --> 00:09:41.074 If a line integral around a closed curve is horrible, read the equality backward. 00:09:41.177 --> 00:09:45.439 Swap the loop integral to the left and the double integral to the right. 00:09:45.635 --> 00:09:47.679 Often that reversal is the whole 00:09:47.679 --> 00:09:51.162 difference between an integral you can do and one you cannot. 00:09:52.237 --> 00:09:56.406 Put the three theorems side by side with the three regions we started from. 00:09:56.602 --> 00:10:00.643 For each theorem, name its region and then its complete boundary. 00:10:01.556 --> 00:10:06.584 For the fundamental theorem of calculus, the region is the interval from a to b. 00:10:06.781 --> 00:10:10.033 Its boundary is the two endpoints a and b. 00:10:10.736 --> 00:10:15.193 For line integrals, the region is the curve C. Its boundary is the point P 00:10:15.276 --> 00:10:18.818 where the curve starts and the point Q where it ends. 00:10:19.614 --> 00:10:22.527 For Green's theorem, the region is the patch R. 00:10:22.610 --> 00:10:27.684 Its boundary is the closed loop del R, walked with the region on the left. 00:10:28.597 --> 00:10:31.036 Now read the two right hand columns. 00:10:31.139 --> 00:10:33.808 This one names the region being integrated over. 00:10:33.891 --> 00:10:37.909 This one names its boundary, always one dimension smaller. 00:10:38.013 --> 00:10:40.254 Every row is saying the same thing. 00:10:41.050 --> 00:10:44.604 And that shared structure is the line I showed you at the start. 00:10:44.800 --> 00:10:49.166 Here it is again, and this time it should read as ordinary English. 00:10:49.963 --> 00:10:53.842 Omega names the region, whatever dimension it has. 00:10:54.038 --> 00:10:58.706 Del Omega names its boundary, always one dimension smaller. 00:10:59.619 --> 00:11:04.193 The symbol d means take the appropriate derivative: an ordinary derivative 00:11:04.193 --> 00:11:09.209 on an interval, a gradient along a curve, or curl over a patch. 00:11:09.313 --> 00:11:10.323 It is the operation 00:11:10.323 --> 00:11:14.005 that turns the original quantity into the inside quantity. 00:11:14.708 --> 00:11:18.030 The left side integrates that derivative over the region. 00:11:18.133 --> 00:11:21.861 The right side integrates the original quantity over the boundary. 00:11:22.476 --> 00:11:26.807 Read at three different dimensions, this one line is all three theorems. 00:11:27.832 --> 00:11:30.582 And notice what that line does not say. 00:11:30.665 --> 00:11:33.545 It never says how many dimensions the region has. 00:11:33.741 --> 00:11:38.908 So take one more: a surface in space, a sheet of it hanging in the air. 00:11:39.821 --> 00:11:40.738 Let me turn it round, 00:11:40.738 --> 00:11:45.952 because a curved sheet and a flat one look exactly alike until something moves. 00:11:46.566 --> 00:11:51.432 Now you can see it is bowed upward, like a cloth held up by its four corners. 00:11:52.344 --> 00:11:56.837 Its boundary is the closed curve running round the rim, walked so that the surface 00:11:56.837 --> 00:11:58.359 stays on your left. 00:11:58.556 --> 00:12:02.109 A region, and an edge one dimension smaller. 00:12:02.422 --> 00:12:05.604 The same two things as every other picture today. 00:12:06.401 --> 00:12:10.708 Read that same line over a surface like this one, and what you get is the theorem 00:12:10.708 --> 00:12:13.089 that actually carries Stokes' name. 00:12:13.192 --> 00:12:17.012 Read it one dimension higher again and it is the divergence theorem. 00:12:17.012 --> 00:12:19.985 Nobody had to invent a new idea for either of them. 00:12:20.781 --> 00:12:21.906 So there it is. 00:12:21.989 --> 00:12:24.103 That line is Stokes' theorem. 00:12:24.555 --> 00:12:29.837 Omega is whatever region you have, del Omega is its edge, and adding 00:12:29.837 --> 00:12:34.168 up the change inside always leaves you something that was settled on that edge.