Divergence Theorem
Table of Contents
1. Divergence Theorem in 2D Space
Let \( \partial D \) be the boundary of a closed and bounded region \( D \) in \( \mathbb{R}^{2} \). Suppose \( \partial D \) is the union of finitely many simple closed piecewise \( C^{1} \) curves.
Let \( \mathbf{F}:X\mapsto\mathbb{R}^{2} \) be a vector field of class \( C^{1} \) where \( X\subset \mathbb{R}^{2} \) contains \( D \). If \( \mathbf{n} \) is the outward unit normal vector to \( D \), then
\[ \oint_{\partial D} \mathbf{F}\cdot \mathbf{n}\, ds = \iint_{D} \nabla\cdot\mathbf{F}\, dA \]
2. Divergence Theorem in 3D Space, aka Gauss’s Theorem
Let \( D \) be a bounded solid region in \( \mathbb{R}^{3} \). Suppose the boundary \( \partial D \) of \( D \) is the union of finitely many piecewise smooth, closed orientable surfaces which are oriented by normals pointing away from \( D \). Let \( \mathbf{F}:X\mapsto\mathbb{R}^{3} \) be a vector field of class \( C^{1} \), where \( X\subset\mathbb{R}^{3} \) contains \( D \). Then
\[ \oiint_{\partial D} \mathbf{F}\cdot d\mathbf{S} = \iiint_{D} \nabla\cdot \mathbf{F}\, dV \]
Proof.