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 $$\mathbf{v} : \R^3 \to \R^3 $$ 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 $$\mathbf{p}\in \R^3$$. 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.