### The idea behind the divergence theorem

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 $\dlvf$ represents the flow of a fluid, then the divergence of $\dlvf$ represents the expansion or compression of the fluid. The divergence theorem says that the total expansion of the fluid inside some three-dimensional region $\dlv$ equals the total flux of the fluid out of the boundary of $\dlv$. In math terms, this means the triple integral of $\div \dlvf$ over the region $\dlv$ is equal to the flux integral (or surface integral) of $\dlvf$ over the surface $\partial \dlv$ that is the boundary of $\dlv$ (with outward pointing normal): \begin{align*} \iiint_\dlv \div \dlvf\, dV = \sint{\partial \dlv}{\dlvf}. \end{align*}

I hope that this makes sense intuitively from the above two examples.
In the first example, the gas expanding meant $\div \dlvf >0$
everywhere in $\dlv$, the inside of the container. Therefore, the net
flux out of $\dlv$, $\sint{\partial \dlv}{\dlvf}$, must
also be greater than zero, i.e., the gas must leak out through the
container walls $\partial \dlv$. In the second example, by pumping
air into the tire $\dlv$, I insisted that the net flux out of the
tire, $\sint{\partial \dlv}{\dlvf}$, must be negative
(since there was a net flux \textbf{into} the tire, and we are
assuming an **outward** pointing normal). By the divergence
theorem, the total expansion inside $\dlv$,
$\displaystyle\iiint_\dlv \div \dlvf\, dV$, must be negative,
meaning the air was compressing.

Notice that the divergence theorem equates a surface integral with a triple integral over the volume inside the surface. In this way, it is analogous to Green's theorem, which equates a line integral with a double integral over the region inside the curve. Remember that Green's theorem applies only for closed curves. For the same reason, the divergence theorem applies to the surface integral \begin{align*} \dsint \end{align*} only if the surface $\dls$ is a closed surface. Just like a closed curve, a closed surface has no boundary. A closed surface has to enclose some region, like the surface that represents a container or a tire. In other words, the surface has to be a boundary of some $\dlv$ (i.e., $\dls= \partial \dlv$), as described above. You cannot use the divergence theorem to calculate a surface integral over $\dls$ if $\dls$ is an open surface, like part of a cone or a paraboloid. If you want to use the divergence theorem to calculate the ice cream flowing out of a cone, you have to include a top to your cone to make your surface a closed surface.

Here are some divergence theorem examples.

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