Volume and Surface Flux
The Divergence Theorem (Gauss's Theorem) relates the flux through a closed surface \(S\) to the divergence within the volume \(V\) it encloses:
\[\iint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) dV\]
It means the total amount of "stuff" flowing out of a balloon is equal to the total amount being generated inside.
Worked Examples
Example 1: Electric Charge
The divergence of the electric field is proportional to the charge density (\(\nabla \cdot \vec{E} = \rho/\epsilon_0\)). Integrating this over a volume and applying the Divergence Theorem proves that the flux through a surface is exactly proportional to the total enclosed charge. This is the first of Maxwell's Equations.
The Bridge to Quantum Mechanics
The Divergence Theorem is the ultimate proof for the Conservation of Probability. By integrating the continuity equation \(\nabla \cdot \vec{J} + \partial P / \partial t = 0\) over all space and applying the Divergence Theorem, we show that as long as the probability flux \(\vec{J}\) goes to zero at infinity, the total probability of finding the particle remains exactly 1 for all time. This "Global Conservation" is what makes the probabilistic interpretation of Quantum Mechanics consistent.