= Vector gradient form of the divergence theorem
{title2=$\int_V\nabla f\,dV=\int_{\partial V}f\,n\,dS$}
For a continuously differentiable scalar-valued function $f$ on a bounded sufficiently regular region, apply the <divergence theorem> to $fk$, where $k$ is any constant vector. Since $\nabla\cdot(fk)=k\cdot\nabla f$, the dot products of the two displayed vector integrals with every $k$ agree. The vector integrals are therefore equal. This is the scalar-gradient analogue of the usual flux identity.
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