= Radial integral potential for a curl-free field
{title2=$\Phi(x)=\int_0^1F(tx)\cdot x\,dt$}
For a continuously differentiable <curl-free vector field> $F$ on a domain star-shaped about the origin, define $\Phi$ by the displayed integral. Differentiation and $\partial_iF_j=\partial_jF_i$ give $\partial_i\Phi=\int_0^1[F_i(tx)+t x_j\partial_jF_i(tx)]\,dt=\int_0^1\partial_t[tF_i(tx)]\,dt=F_i(x)$. Thus $F=\nabla\Phi$. To adopt the sign convention $F=-\nabla\phi$, take $\phi=-\Phi$. The star-shaped-domain hypothesis ensures every segment used in the construction lies in the domain; curl-free fields need not have global potentials on arbitrary domains.
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