A Constructive Proof of Helmholtz’s Theorem

It is a known result that any vector field ${\boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $\Omega\subset \mathbb{R}^3$ can be written on $\Omega$ as the sum of a gradient and a curl. Should $\Omega$ be unbounded, no conditions are required on the behaviour of ${\boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.