> [!NOTE] Theorem (Divergence) > Let $\underline{F}:\mathbb{R}^{3}\to \mathbb{R}^{3}$ be differentiable [[Vector Field on Subset of Real n-Space|vector field]]. Let $V\subset \mathbb{R}^{3}$ be a finite volume bounded by a closed [[Surface|surface]] $S.$ Then $ \int \int_{S} \underline{F} \cdot \underline{\hat{n}} \, dS =\int \int \int_{V} \nabla \cdot \underline{F} \, dV $where $\underline{\hat{n}}$ is the outward-pointing normal to the surface $S$; $\nabla \cdot F$ denotes [[Divergence of Vector Field on Real 3-Space|divergence]] of $\underline{F}.$ **Proof**: ... # Applications See [[Stokes' Theorem]] & [[Green's Theorem]].