Divergence Theorem

The theorem states that if \(S\) is a closed surface enclosing a region \(D\), oriented with \(\hat{n}\) pointing outwards, and a vector field \(\vec{F}\) defined and differentiable everywhere in \(D\), then the following is true:

\begin{align} {\subset\!\supset} \llap{\iint}_S \vec{F} \cdot \text{ d}\vec{S} = \iiint_D \text{div }\vec{F} \text{ d}V \end{align}

where \(\text{div}(P\hat{i} + Q\hat{j} + R\hat{k}) = P_x+Q_y+R_z\). We can also use del notation to represent divergence as \(\nabla \cdot \vec{F}\).

The divergence of a vector field \(\vec{F}\) can be seen as the "source rate" or the amount of flux generated per unit volume. Therefore, if we think of \(\vec{F}\) as the velocity of some incompressible fluid flow, then summing all of the divergences of \(\vec{F}\) in the region \(D\) bounded by \(S\) is the same as the flux out of that region \(D\) — in other words, the amount of fluid leaving \(D\) per unit time.

Therefore, the divergence itself measures the amount of sources or sinks per unit volume at a given place.