Vector potential

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.

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

$\mathbf{v} = \nabla \times \mathbf{A}.$

If a vector field v admits a vector potential A, then from the equality

$\nabla \cdot (\nabla \times \mathbf{A}) = 0$

(divergence of the curl is zero) one obtains

$\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,$

which implies that v must be a solenoidal vector field.

Theorem

Let

$\mathbf{v} : \mathbb R^3 \to \mathbb R^3$

be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define

$\mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}.$

Then, A is a vector potential for v, that is,

$\nabla \times \mathbf{A} =\mathbf{v}.$

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

$\mathbf{A} + \nabla m$

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.