= Gradient volume-to-boundary identity
{title2=$\int_V\nabla\Omega\,dV=\int_{\partial V}\Omega\mathbf n\,dS$}
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.
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