Helmholtz decomposition

The Helmholtz theorem states that any vector $𝐅$ may be decomposed into an irrotational (curl-free) and a solenoidal (divergence-free) part under certain conditions (given below). More precisely, it may be written in the form:

$$𝐅=-\nabla \phi +\nabla \times 𝐀$$(1)

where $\phi$ is a scalar potential and $𝐀$ is a vector potential. By the definitions of scalar and vector potentials it follows that the first term on the right-hand side is irrotational and the second is solenoidal. The general conditions for this to be true are:

1. The divergence of $𝐅$ must vanish at infinity.

2. The curl of $𝐅$ must also vanish at infinity.