Formally, given a vector field v, a vector potential is a vector field A such that
If a vector field v admits a vector potential A, then from the equality
which implies that v must be a solenoidal vector field.
Then, A is a vector potential for v, that is,
The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is
where m 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.
See also 
- Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.