Everything Is Stokes' Theorem

Let me tell you what Stokes' theorem actually says. It is one sentence. Not a page of conditions, not a machine you turn a handle on. One sentence, which then gets written down three times, in three different numbers of dimensions, and picks up a different name each time. Before the sentence, put its three regions side by side. A chunk of the number line, a path through the plane, and a two dimensional patch. These regions have different dimensions, but each has an edge one dimension smaller. Start with the chunk of line. Its edge is two dots, one at each end. Those two endpoints are the complete boundary of the interval from a to b. The path has the same kind of boundary. Its edge is the point P where the journey starts, together with the point Q where it stops. The patch is different. Its edge is not a pair of points. It is the closed curve running all the way around the outside, walked in one direction. Now look at the pattern. The edge of a piece of line is a couple of points. The edge of a path is its two endpoints. The edge of a blob is a curve. In every case the edge is one dimension smaller, and here is the sentence all three theorems share: add up how fast something changes inside a region, and the answer depends only on its edge. And here is that sentence written in symbols. This is Stokes' theorem. The whole of it, on one line. You are not supposed to be able to read that yet. I only want you to have seen it once. Omega is the region, whatever it happens to be. Del Omega is its edge. And the line says: what you get by integrating inside is decided on the edge. And you have almost certainly met that line three times already, under three different names. The fundamental theorem of calculus. The fundamental theorem for line integrals. And Green's theorem. So let's take them one at a time. One dimension, then a curve through the plane, then a flat patch, and watch the same line come back every time.

Start in one dimension, where you have known this theorem the longest. Here is the question it actually answers. You know how fast something is changing at every moment: you're driving, and you can watch the speedometer the whole way. How far did you go? The fundamental theorem of calculus says this. If you integrate the derivative of f from a to b, you don't get anything complicated. You get f at b, minus f at a. Read that out loud in words. Add up all the little changes over the whole interval, and you get the net change from one end to the other. The speedometer tells you the odometer. Let's make that concrete, because a number you can check beats a formula you can only nod at. Take f of x equal to x squared over four. Its derivative is x over two, which is this blue line. Fix the left end at one, and let the right endpoint b start at two. The left hand side is the shaded area from one to b. Now slide b out to four. The region grows with it, and the running total up in the corner counts all the way to three point seven five. Now check the other side. f of four is four. f of one is a quarter. Four minus a quarter is three point seven five. Same number, obviously. But here is the thing I really want you to notice. The right hand side never looks inside the interval at all. It only asks two questions. What is f at the right end, and what is f at the left end? Everything in the middle cancels. So here is the piece of language you need. The interval from a to b is the region. Its boundary is just the two endpoints. That word, boundary, is the whole lecture. The boundary is not the number b minus a. It is a formal sum of two points: plus b at the right hand end, together with negative a at the left hand end. Those signs are exactly why the answer is f of b minus f of a. That sign is orientation, the direction in which you travel. Watch the traveller move from the negative endpoint to the positive endpoint. Keep that direction in mind. It comes straight back in Green's theorem.

Now go up a dimension. Same sentence, new region. This time the region is not a chunk of the number line. It is a route, drawn through the plane, from a point P to a point Q. And instead of a function of one variable, there are now two. Picture it as a hill: f of x, y is the height of the ground above the point x, y. These grey rings are its contour lines. A contour is the set of places at one particular height. So put a walker on the ring where f is nine, and send it the whole way round. It climbs nothing at all, because every point of that ring is at the same height. Here is theorem one again, so you can see what is coming. Integrate the derivative across the region, and read the answer off the two ends. And here is theorem two. Integrate the gradient of f along the curve C, and what you get is f at Q minus f at P. Look at those two lines together. It is the same sentence twice. Read the left side as, add up every little bit of climbing as you walk. Read the right side as, the height where you finished minus the height where you started. It is the odometer story again, on a hill. And that hands you something genuinely useful. The route does not matter. Take hold of it here, in the middle, and bend it wherever you like, keeping P and Q where they are. Up over the shoulder of the hill. Or down round the bottom instead. The climb you total up is the same number every single time, because the right hand side never mentions the route at all. Because the formula only ever looks at the endpoints. There is that word again. The boundary of a curve is its two ends, and the answer depends on nothing else. Which hands you a corollary that falls straight out. Suppose the route is a closed loop, like this circle Gamma. You finish exactly where you started, so the start point and the end point are the same point. So the endpoint term is f of P minus f of P. The two copies cancel, and the result is zero. Around any closed loop, a gradient field integrates to zero. You can see it coming without computing anything: closed curve, gradient field, nothing to do. Same theorem. Region, boundary, done. Now the one that looks hardest and really isn't.

So, the flat case. Green's theorem. This is the one that looks frightening the first time you meet it, and it is the same sentence again. Our region is now genuinely two dimensional: a flat patch R in the plane, carrying a field of arrows. The rectangle keeps the boundary easy to read, but the theorem does not depend on that shape. Its boundary is the closed curve that runs all the way around the outside. We write it as del R. Orientation matters, exactly as the plus and minus mattered on the interval. Travel counterclockwise, so the region stays on your left. Starting here, one complete trip comes back to the point it left. That choice of direction is the same choice as the plus sign on the interval's right hand end. Here is the statement. The double integral over every point of the region lives on the left. One line integral around its boundary lives on the right. Inside on the left, edge on the right. Read the left side as the total swirl at all points inside R. Read the right side as the total push along one counterclockwise trip around del R. Every arrow inside contributes to the first count; the rim supplies the second. Green's theorem says those two numbers are equal. Total swirl inside equals total push around the rim. That is the whole theorem. Now for the reason. Chop R into little tiles. Each tile contributes a walk around its own boundary. Choose this tile. Its contribution includes this little edge. Adding the tiles means reading that edge once from each tile that touches it. One tile walks the shared edge upward. Its neighbour walks the very same edge downward. The geometric edge is one object, but its two induced directions are opposites. Those two contributions cancel. The same happens at every shared edge, so the entire interior grid deletes itself. Only edges with no neighbour remain: the outside of R. Their induced walk is the surviving boundary, with the region on its left. That is Green's theorem. The equality now reads from the region to its surviving boundary. The derivative is integrated over R on the left; the original field is integrated over del R on the right. If a line integral around a closed curve is horrible, read the equality backward. Swap the loop integral to the left and the double integral to the right. Often that reversal is the whole difference between an integral you can do and one you cannot.

Put the three theorems side by side with the three regions we started from. For each theorem, name its region and then its complete boundary. For the fundamental theorem of calculus, the region is the interval from a to b. Its boundary is the two endpoints a and b. For line integrals, the region is the curve C. Its boundary is the point P where the curve starts and the point Q where it ends. For Green's theorem, the region is the patch R. Its boundary is the closed loop del R, walked with the region on the left. Now read the two right hand columns. This one names the region being integrated over. This one names its boundary, always one dimension smaller. Every row is saying the same thing. And that shared structure is the line I showed you at the start. Here it is again, and this time it should read as ordinary English. Omega names the region, whatever dimension it has. Del Omega names its boundary, always one dimension smaller. The symbol d means take the appropriate derivative: an ordinary derivative on an interval, a gradient along a curve, or curl over a patch. It is the operation that turns the original quantity into the inside quantity. The left side integrates that derivative over the region. The right side integrates the original quantity over the boundary. Read at three different dimensions, this one line is all three theorems. And notice what that line does not say. It never says how many dimensions the region has. So take one more: a surface in space, a sheet of it hanging in the air. Let me turn it round, because a curved sheet and a flat one look exactly alike until something moves. Now you can see it is bowed upward, like a cloth held up by its four corners. Its boundary is the closed curve running round the rim, walked so that the surface stays on your left. A region, and an edge one dimension smaller. The same two things as every other picture today. Read that same line over a surface like this one, and what you get is the theorem that actually carries Stokes' name. Read it one dimension higher again and it is the divergence theorem. Nobody had to invent a new idea for either of them. So there it is. That line is Stokes' theorem. Omega is whatever region you have, del Omega is its edge, and adding up the change inside always leaves you something that was settled on that edge.