= Uniqueness of the Neumann problem
On a connected bounded domain, two harmonic potentials with the same normal <derivative> differ by a constant. Their difference $u$ has $\nabla^2u=0$ and $\partial_nu=0$. <Green's first identity> gives $\int|\nabla u|^2dV=\int_{\partial\Omega}u\partial_nu\,dA=0$, so the <gradient> vanishes. In a multiply connected magnetic core, also fix the <circulation> around each independent hole: a curl-free <magnetic field intensity> with nonzero <circulation> need not be the <gradient> of a single-valued scalar potential. After those <circulations> have been matched, apply the scalar-potential uniqueness argument to the difference field.
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