# 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 $\mathbf {v}$, a vector potential is a $C^{2}$ vector field $\mathbf {A}$ such that $\mathbf {v} =\nabla \times \mathbf {A} .$

## Consequence

If a vector field $\mathbf {v}$ admits a vector potential $\mathbf {A}$, 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 $\mathbf {v}$ 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 $\mathbf {v} (\mathbf {x} )$ decreases at least as fast as $1/\|\mathbf {x} \|$ for $\|\mathbf {x} \|\to \infty$. Define $\mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y}$ where $\nabla _{y}\times$ denotes curl with respect to variable $\mathbf {y}$. Then $\mathbf {A}$ is a vector potential for $\mathbf {v}$. That is, $\nabla \times \mathbf {A} =\mathbf {v} .$
The integral domain can be restricted to any simply connected region $\mathbf {\Omega }$. That is, $\mathbf {A'}$ also is a vector potential of $\mathbf {v}$, where $\mathbf {A'} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\Omega }{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .$
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, $\mathbf {A''} (\mathbf {x} )$ also qualifies as a vector potential for $\mathbf {v}$, where
    $\mathbf {A''} (\mathbf {x} )=\int _{\Omega }{\frac {\mathbf {v} (\mathbf {y} )\times (\mathbf {x} -\mathbf {y} )}{4\pi |\mathbf {x} -\mathbf {y} |^{3}}}d^{3}\mathbf {y}$.
Substituting $\mathbf {j}$ (current density) for $\mathbf {v}$ and $\mathbf {H}$ (H-field) for $\mathbf {A}$, yields the Biot-Savart law.
Let $\mathbf {\Omega }$ be a star domain centered at the point $\mathbf {p}$, where $\mathbf {p} \in \mathbb {R} ^{3}$. Applying Poincaré's lemma for differential forms to vector fields, then $\mathbf {A'''} (\mathbf {x} )$ also is a vector potential for $\mathbf {v}$, where
$\mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s((\mathbf {x} -\mathbf {p} )\times (\mathbf {v} (s\mathbf {x} +(1-s)\mathbf {p} ))\ ds$

## 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 $\mathbf {A} +\nabla f,$ 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.