Vector potential

Theorem

Let $${\displaystyle \mathbf{v}:{\mathbb{R}}^3\to {\mathbb{R}}^3}$$ be a solenoidal vector field which is twice continuously differentiable. Assume that ${\displaystyle \mathbf{v}(\mathbf{x})}$ decreases at least as fast as ${\displaystyle 1/\Vert \mathbf{x}\Vert }$ for ${\displaystyle \Vert \mathbf{x}\Vert \to \mathrm{\infty }}$. Define $${\displaystyle \mathbf{A}(\mathbf{x})=\frac{1}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{{\mathrm{\nabla }}_{\mathbf{s}}\times \mathbf{v}(\mathbf{s})}{\Vert \mathbf{x}-\mathbf{s}\Vert }{d}^3\mathbf{s}}$$ where ${\displaystyle {\mathrm{\nabla }}_{\mathbf{s}}\times }$ denotes curl with respect to variable ${\displaystyle \mathbf{s}}$. Then ${\displaystyle \mathbf{A}}$ is a vector potential for ${\displaystyle \mathbf{v}}$. That is, $${\displaystyle \mathrm{\nabla }\times \mathbf{A}=\mathbf{v}.}$$

The integral domain can be restricted to any simply connected region ${\displaystyle \mathrm{\Omega }}$. That is, ${\displaystyle {\mathbf{A}}^{\prime }}$ also is a vector potential of ${\displaystyle \mathbf{v}}$, where $${\displaystyle {\mathbf{A}}^{\prime }(\mathbf{x})=\frac{1}{4\pi }{\int }_{\mathrm{\Omega }}\frac{{\mathrm{\nabla }}_{\mathbf{s}}\times \mathbf{v}(\mathbf{s})}{\Vert \mathbf{x}-\mathbf{s}\Vert }{d}^3\mathbf{s}.}$$

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, ${\displaystyle {\mathbf{A}}^{″}(\mathbf{x})}$ also qualifies as a vector potential for ${\displaystyle \mathbf{v}}$, where

$${\displaystyle {\mathbf{A}}^{″}(\mathbf{x})={\int }_{\mathrm{\Omega }}\frac{\mathbf{v}(\mathbf{s})\times (\mathbf{x}-\mathbf{s})}{4\pi {\left|\mathbf{x}-\mathbf{s}\right|}^3}{d}^3\mathbf{s}}$$

Substituting ${\displaystyle \mathbf{j}}$ (current density) for ${\displaystyle \mathbf{v}}$ and ${\displaystyle \mathbf{H}}$ (H-field) for ${\displaystyle \mathbf{A}}$, yields the Biot–Savart law.

Let ${\displaystyle \mathrm{\Omega }}$ be a star domain centered at the point ${\displaystyle \mathbf{p}}$, where ${\displaystyle \mathbf{p}\in {\mathbb{R}}^3}$. Applying Poincaré's lemma for differential forms to vector fields, then ${\displaystyle {\mathbf{A}}^{‴}(\mathbf{x})}$ also is a vector potential for ${\displaystyle \mathbf{v}}$, where

$${\displaystyle {\mathbf{A}}^{‴}(\mathbf{x})={\int }_0^{1}s\left[(\mathbf{x}-\mathbf{p})\times \mathbf{v}(s\mathbf{x}+(1-s)\mathbf{p})\right]ds}$$

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If ${\displaystyle \mathbf{A}}$ is a vector potential for ${\displaystyle \mathbf{v}}$, then so is $${\displaystyle \mathbf{A}+\mathrm{\nabla }f,}$$ where ${\displaystyle f}$ 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.