This article is about the general concept in the mathematical theory of vector fields. For the vector potential in electromagnetism, see Magnetic vector potential. For the vector potential in fluid mechanics, see Stream function.
If a vector field admits a vector potential , then from the equality
(divergence of the curl is zero) one obtains
which implies that must be a solenoidal vector field.
Let
be a solenoidal vector field which is twice continuously differentiable. Assume that decreases at least as fast as for . Define
where denotes curl with respect to variable . Then is a vector potential for . That is,
The integral domain can be restricted to any simply connected region . That is, also is a vector potential of , where
A generalization of this theorem is the Helmholtz decomposition theorem, 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 is a vector potential for , then so is
where 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.