Divergence Theorem

Prerequisites

• Divergence

Divergence theorem (Fundamental Theorems for Divergences)

Let $\vec{F}$ be a vector field. Then the fundamental theorem for divergences / divergence theorem states that:

$${\oint }_S\vec{F}\cdot d\vec{a}={\int }_V(\nabla \cdot \vec{F})d\tau$$

Where $S$ is a closed surface that encloses a volume $V$. It is implied that $\vec{a}$ is a vector that is normal and outwards from the surface $S$, and that $d\tau$ is a scalar representing the differentiable volume at some point inside the volume.