structural:scholar_edition, quality:corruption, structural:expansion
Uniqueness of the solution

In general, the Helmholtz decomposition is not uniquely defined.A harmonic function H (r) {\displaystyle H(\mathbf {r})} is a function that satisfies Δ H (r) = 0 {\displaystyle \Delta H(\mathbf {r})=0}.By adding H (r) {\displaystyle H(\mathbf {r})} to the scalar potential Φ (r) {\displaystyle \Phi (\mathbf {r})}, a different Helmholtz decomposition can be obtained:

G ′ (r) = ∇ (Φ (r) + H (r)) = G (r) + ∇ H (r), R ′ (r) = R (r) − ∇ H (r). {\displaystyle {\begin{aligned}\mathbf {G} '(\mathbf {r})&=\nabla (\Phi (\mathbf {r})+H(\mathbf {r}))=\mathbf {G} (\mathbf {r})+\nabla H(\mathbf {r}),\\\mathbf {R} '(\mathbf {r})&=\mathbf {R} (\mathbf {r})-\nabla H(\mathbf {r}).\end{aligned}}}

For vector fields F {\displaystyle \mathbf {F} }, decaying at infinity, it is a plausible choice that scalar and rotation potentials also decay at infinity. Because H (r) = 0 {\displaystyle H(\mathbf {r})=0} is the only harmonic function with this property, which follows from Liouville's theorem, this guarantees the uniqueness of the gradient and rotation fields.

This uniqueness does not apply to the potentials: In the three-dimensional case, the scalar and vector potential jointly have four components, whereas the vector field has only three. The vector field is invariant to gauge transformations and the choice of appropriate potentials known as gauge fixing is the subject of gauge theory. Important examples from physics are the Lorenz gauge condition and the Coulomb gauge. An alternative is to use the poloidal–toroidal decomposition.