Divergence Theorem

Closed Surface

A surface $\sigma$ is a closed surface iff it is the boundary of a solid region in $\mathbb{R}^3$.

For a closed surface, the positive orientation is the outward orientation.

Statement

Let $\sigma$ be a closed, piecewise smooth, orientable surface that encloses a solid region $R$ in $\mathbb{R}^3$. Suppose $\boldsymbol{F}$ is a vector field whose component functions have continuous first-order partial derivatives on an open set containing $R$ and its boundary $\sigma$. If $\hat{n}$ is the outward unit normal field for $\sigma$,

Divergence as Flux Density

Let $G$ be a small solid region centered at $P_0$ with boundary surface $\sigma(G)$ oriented outward, and let $\boldsymbol{F}$ be a vector field whose component functions have continuous first-order partial derivatives on an open set containing $G$. Denote the volume of $G$ by $\text{Vol}(G)$ and the flux of $\boldsymbol{F}$ across $\sigma(G)$ by $\Phi(G)$.

As $G$ shrinks to $P_0$ in all directions,

If $\boldsymbol{F}$ is the velocity field of a fluid in steady-state flow, $\text{div}\,\boldsymbol{F}$ is the limiting outward flux per unit volume at a point.