= Vector-valued Stokes identity
{title2=$\int_S d\mathbf S\times\nabla\phi=\oint_{\partial S}\phi\,d\mathbf x$}
For an oriented smooth surface and compatible boundary orientation, apply <Stokes theorem> to $\phi\mathbf k$ for an arbitrary constant vector $\mathbf k$. Since $\nabla\times(\phi\mathbf k)=\nabla\phi\times\mathbf k$, its surface integral is $\mathbf k\cdot\int_S d\mathbf S\times\nabla\phi$. The boundary integral is $\mathbf k\cdot\oint\phi\,d\mathbf x$. Equality for every $\mathbf k$ proves the displayed vector identity. Reversing both orientations preserves the identity.
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