If a vector field v admits a vector potential A, then from the equality
(divergence of the curl is zero) one obtains
which implies that v must be a solenoidal vector field.
be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define
Then, A is a vector potential for v, that is,
A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.
The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is
where f is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.