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 ${\nabla }^{2}u=0$ and ${\partial }_{n}u=0$. Green's first identity gives $\int |\nabla u{|}^{2}dV={\int }_{\partial \Omega }u{\partial }_{n}udA=0$, 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.