The Divergence Theorem

This is now the same theorem, but for regions whose boundary are surfaces

Given a simple solid region \(E\) where \(\partial E\) denotes the boundary surface of \(E\) taken to have positive (outward) orientation, and given a vector field \(\bm{F}\) whose components have continuous partial derivatives on a open region in space containing \(E,\) \[ \iint\limits_{\partial E} \bm{F}\cdot\mathrm{d}\bm{S} = \iiint\limits_E \operatorname{div}\bm{F} \,\mathrm{d}V \,. \]

Examples

Sometimes called Gauss’s Theorem, or Ostrogradsky’s Theorem