The Divergence Theorem

The divergence theorem simply says you can count what leaves a room in two ways. Stand at the doors and windows and measure everything flowing out, or go inside and add up every source of stuff in the room. Either way you get the same total. On the left, $\iint_S\vec F\cdot d\vec S$ is the outward flux through the closed surface $S$ (the walls). On the right, $\nabla\cdot\vec F$ is the source strength at each point inside, and $\iiint_E$ adds it up over the solid $E$ (the room). It's also called Gauss's theorem. It looks like a big formula, but it's bookkeeping: nothing appears or disappears without a source.

Why it's true. Chop the solid into tiny boxes. Divergence was defined as outflow per unit volume, so the flux out of each box is about (divergence) × (volume). Now add up all the boxes. Think about a face shared by two neighbors: whatever flows out of one box through that face flows into the other, so it counts once as positive and once as negative and cancels. Every interior face cancels like that, leaving only the faces on the outer surface. So the total of (divergence × volume) inside equals the flux out through the boundary. It's the same cancellation trick as Green's and Stokes' theorems, with boxes instead of squares.

Use it either way:

• A hard flux integral over a closed surface becomes a triple integral, often easy.
• If $\nabla\cdot\vec F = 0$ inside, the net flux is zero: whatever flows in flows out.
• For a surface that isn't closed, close it off with a simple lid, use the theorem on the whole thing, and subtract the flux through the lid. The challenge problem does exactly this.

A common mistake. The theorem only applies to a closed surface, one that completely encloses a solid, with the outward normal. It's tempting to use it on an open surface like a hemisphere by itself. That would give the wrong answer, because some of the flow could leave through the missing base without being counted.

It's the last of the great theorems: the Fundamental Theorem of Calculus, the Fundamental Theorem for line integrals, Green, Stokes and Gauss all say "the integral of a derivative over a region equals the integral of the function over its boundary."

Gauss's law for gravity and electricity, conservation of mass in fluid dynamics, and the heat equation itself are all derived from the divergence theorem.