Vector potential

In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a "scalar potential", which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a "vector potential" is a vector field A such that 
Consequence.
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.
Theorem.
Let 
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.
Nonuniqueness.
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.