Intuitive

Let's say I have a rigid container filled with some gas. If the gas starts to expand but the container does not expand, what has to happen? Since we assume that the container does not expand (it is rigid) but that the gas is expanding, then gas has to somehow leak out of the container. (Or I suppose the container could burst, but that counts as both gas leaking out of the container and the container expanding.)

If I go to a gas station and pump air into one of my car's tires, what has to happen to the air inside the tire? (Assume the tire is rigid and does not expand as I put air inside it.) The air inside of the tire compresses.

These two examples illustrate the divergence theorem (also called Gauss's theorem). Recall that if a vector field F represents the flow of a fluid, then the divergence of F represents the expansion or compression of the fluid. The divergence theorem says that the total expansion of the fluid inside some three-dimensional region WW equals the total flux of the fluid out of the boundary of W. In math terms, this means the triple integral of divF over the region WW is equal to the flux integral (or surface integral) of F over the surface ∂Wthat is the boundary of W (with outward pointing normal):

\begin{align*} \iiint_W \text{div} F \, dV = \iint{\partial W}{F \cdot S}. \end{align*}

http://mathinsight.org/divergence_theorem_idea

Concrete

[Gauss' Divergence Theorem] follows naturally from our intuitive explanation of flux and div. We know that div is the flux density, or flux/volume. How do we find the flux given the div? It's simple: multiply by volume. This is just like calculating the mass of an object given its density and volume.

Flux = div * volume

Integration is usually required because we are considering regions of varying div over a region, just like calculating the mass of an object with varying density. To do this, we take a tiny volume, multiply by its divergence, take another tiny volume, multiply by its div, and keep adding these tiny mass elements up. Thus,

Flux = Integral(div dv)

This is the Divergence Theorem… it's pretty easy once you know how to look at flux and div. https://betterexplained.com/~kazad/resources/math/Gauss/gauss.htm

• A perfect explanation of Gauss' theorem can be found in Section 3 of Vol. 2 of Feynman's Lectures on Physics, which are available here

Abstract

Why is it interesting?

Gauss' divergence theorem allows us to rewrite integrals over a volume as integrals over a surface. This is often useful, for example, in quantum field theory. There we have to integrate over all of space. Using Gauss' theorem, we can rewrite these integrals as integrals over the surface of space. The thing is now that integrals over a surface of all of space have to vanish, because physical fields vanish at infinity. (If we integrate over all of space, the surface of space is infinitely far away and therefore is not allowed to have any influence on the physics at finite distances.)

It is also important for different problems in electrodynamics, where rewriting integrals as surface integrals often makes the problem easier.