Gradient volume-to-boundary identity (${\int }_V\nabla \Omega dV={\int }_{\partial V}\Omega \mathbf{n}dS$)

For a continuously differentiable scalar $\Omega$ on a bounded region $V$ with outward boundary normal, ${\int }_V\nabla \Omega dV={\int }_{\partial V}\Omega \mathbf{n}dS$. Apply the divergence theorem to $\mathbf{c}\Omega$ for every constant vector $\mathbf{c}$; equality of all scalar projections gives the vector identity. The theorem applies to piecewise smooth boundaries, with singular points removed by a limiting argument when necessary.