Divergence, and the Two Great Theorems

Vector calculus ends with the divergence theorem and Stokes' theorem. They make the same startling promise: what a field does throughout a region can be recovered from what it does on the boundary. The divergence theorem turns total divergence inside a solid into flux through its closed surface. Stokes' theorem turns total curl through a surface into circulation around its edge. Both replace an interior measurement with a boundary measurement. Here is a vector field in the plane. At every point there's an arrow, and the honest way to read it is as the velocity of a fluid. Put a speck of dust down somewhere, and the arrow tells you which way it drifts, and how fast. Now draw a little closed loop around the origin and ask one simple question. Is more fluid leaving this loop than entering it? Here, obviously yes. Every single arrow crosses it on the way out. Something inside is making fluid. Compare that with this second field. Same plane, but now the fluid is turning. It goes round and round the origin, and it moves faster the further out you look. Draw the same loop on it. And now nothing crosses. The arrows run along the loop, never through it. There is plenty of motion here, and no outflow at all. That contrast is what divergence measures. Not how fast the field is, not which way it points, but how much of it is flowing out of a point. Positive divergence means a source. Negative means a sink. Zero means whatever comes in goes out again. Hold on to that picture, because both of today's theorems are built out of it. Now let's turn it into a formula.

So let's turn that into a formula. If our field is F, then its divergence is written div F, or nabla dot F, and here it is. The components P, Q and R are functions of x, y and z. Take the first component and differentiate it with respect to x. Take the second, and differentiate with respect to y. Take the third with respect to z. Then add the three numbers up. That is genuinely all it is: one derivative from each component, each matched to its own variable. And notice what kind of object comes out. F is a vector field, an arrow at every point. Div F is a number at every point. Let's compute one. Our spreading field was F equals x, y. The first component is x, so its x derivative is one. The second is y, so its y derivative is one. Add them. Two, and positive, which is exactly what the picture said. Fluid is being created at every point, so any loop you draw has more coming out of it than going in. Here the divergence happens to be constant, but in general it is a function of position. Now the swirl. F equals minus y, x. The first component is minus y, and we differentiate it with respect to x. There is no x in it, so that derivative is nothing. The second component is x, differentiated with respect to y, and that is nothing too. So the divergence is zero, everywhere. And again the picture agrees. The fluid is moving, quite fast in places, but none of it ever leaves. Motion and divergence are two different questions, and this field is the cleanest reminder of that I know.

Now the first theorem. Take a solid region in space, call it E, and a vector field F living on it. The skin of that region, the closed surface wrapping it up, is written del E. There are two completely different things you could measure here. First, go inside. Compute the divergence at every point of E and add it all up over the volume. That is the total rate at which fluid is being produced in there. Second, forget the inside completely. Stand on the surface, and at each point take the part of F pointing straight out, F dot n hat. Add that up over the whole skin. That is the flux, the net rate at which fluid crosses the boundary. The divergence theorem says those two numbers are equal. Always. Total production inside equals total flow across the boundary, which, once you say it out loud, is really just conservation of stuff. So why is it true? Keep the solid in view and take one flat slice through it. The slice stands for a layer of tiny cells filling the volume. We can understand the whole solid by inspecting two neighbours in that layer. Here are those two neighbours enlarged. For each cell, the flux across its boundary is its divergence contribution. When all the cells are added, the question is what happens along every shared wall. The cell on the left counts that shared wall with its outward normal pointing into the cell on the right. That direction sets the sign of its flux contribution. The cell on the right counts the same wall with its outward normal pointing back into the cell on the left. The two directions are opposite, so the two flux contributions have opposite signs. Equal and opposite. Every interior wall cancels against its partner, in pairs, all the way through the solid. Remove those interior boundaries and what survives is only the outside skin. That is the entire proof, in spirit. The divergence theorem is a bookkeeping identity. Interior faces cancel; boundary faces don't.

The second theorem needs a different measurement. Divergence asked how much of a field flows out of a point. Curl asks something else: how much the field turns around a point. Drop a tiny paddle wheel into the fluid, pinned at its centre so that it can only spin. If the flow turns it, the field has curl there. Curl is a vector: it points along the axis the wheel spins about, and its length says how fast. In the plane, that axis is always straight up out of the page, so only one number survives. For our swirling field, minus y and x, it comes out as two k hat, a constant amount of spin at every single point. Now Stokes' theorem. Take a surface in space, and this time not a closed one. A piece of one, like a butterfly net. Then del S is the curve running round its rim. Choose a normal direction for the surface, and walk that rim the matching way: the way that keeps the surface on your left when your head points along n hat. Pairing those two directions up is the only fiddly part of the whole theorem. And Stokes says this. The total curl passing through the surface equals the circulation of F once around its rim. The left-hand side is a surface integral: at each point of S, how much curl is aimed through it. The right-hand side is a line integral: F dotted with the direction you are travelling, added up all the way round the closed loop. And the reason is the same trick as before. Chop the surface into little tiles, and walk round the edge of every single tile, in the direction the normal tells you to. Every interior edge belongs to two tiles, and those two tiles walk it in opposite directions. So it cancels, exactly as the shared walls did a moment ago. All the little circulations collapse, and what is left is the walk round the outer rim. Meanwhile each tiny loop's circulation is its own curl times its own area. Add them all up and you get the surface integral on one side and the boundary loop on the other. Same bookkeeping, one dimension down.

Let's put the two theorems one above the other, because the resemblance is the whole point. First the divergence theorem: a triple integral of div F over a solid region, equal to the flux of F out through the closed surface around it. And then Stokes: a surface integral of curl F over a piece of surface, equal to the circulation of F once round the curve that bounds it. Now look at the shape of them. On the left of each one, a derivative of F, integrated over a region. On the right of each one, F itself, integrated over the boundary of that region. Div and curl are different derivatives, and a solid and a surface are different regions. And you have met that sentence before. The fundamental theorem of calculus: integrate f prime across an interval, and the answer is f at the two endpoints. The boundary of an interval is just its two ends. Same sentence, every time. Integrate a derivative over a region, and the answer is already written on the edge. That is what these theorems buy you: a hard integral over something fat, traded for an easier one over its skin. So. Divergence measures outflow, curl measures spin, and each of them has a theorem saying that the inside tells you nothing the boundary has not already said.