Helmholtz's theorem revisited

Helmholtz’s theorem for three‐vectors is applicable to electrostatics and magnetostatics, but it must be generalized to antisymmetric second‐rank tensors to be applicable to electromagnetism. A proof of Helmholtz’s theorem for three‐vectors, which is shorter than the usual one, is given. Helmholtz’s theorem is shown to be a special case of the Hodge decomposition theorem in the theory of differential forms.