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 \mathbf{v}, a vector potential is a C^2 vector field \mathbf{A} such that

Consequence

If a vector field \mathbf{v} admits a vector potential \mathbf{A}, then from the equality (divergence of the curl is zero) one obtains which implies that \mathbf{v} must be a solenoidal vector field.

Theorem

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 \mathbf{y}. Then \mathbf{A} is a vector potential for \mathbf{v}. That is, The integral domain can be restricted to any simply connected region. That is, \mathbf{A'} also is a vector potential of \mathbf{v}, 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. By analogy with the Biot-Savart law, also qualifies as a vector potential for \mathbf{v}, where Substituting \mathbf{j} (current density) for \mathbf{v} and \mathbf{H} (H-field) for \mathbf{A}, yields the Biot-Savart law. Let be a star domain centered at the point \mathbf{p}, where. Applying Poincaré's lemma for differential forms to vector fields, then also is a vector potential for \mathbf{v}, where

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If \mathbf{A} is a vector potential for \mathbf{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.