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

If a vector field ${\displaystyle \mathbf {v} }$ admits a vector potential ${\displaystyle \mathbf {A} }$, then from the equality $${\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}$$ (divergence of the curl is zero) one obtains $${\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,}$$ which implies that ${\displaystyle \mathbf {v} }$ must be a solenoidal vector field.

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 ${\displaystyle 1/\|\mathbf {x} \|}$ for ${\displaystyle \|\mathbf {x} \|\to \infty }$. Define $${\displaystyle \mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} }$$ where ${\displaystyle \nabla _{y}\times }$ denotes curl with respect to variable ${\displaystyle \mathbf {y} }$. Then ${\displaystyle \mathbf {A} }$ is a vector potential for ${\displaystyle \mathbf {v} }$. That is, $${\displaystyle \nabla \times \mathbf {A} =\mathbf {v} .}$$

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

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

${\displaystyle \mathbf {A''} (\mathbf {x} )=\int _{\Omega }{\frac {\mathbf {v} (\mathbf {y} )\times (\mathbf {x} -\mathbf {y} )}{4\pi |\mathbf {x} -\mathbf {y} |^{3}}}d^{3}\mathbf {y} }$.

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

Let ${\displaystyle \mathbf {\Omega } }$ 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

${\displaystyle \mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s((\mathbf {x} -\mathbf {p} )\times (\mathbf {v} (s\mathbf {x} +(1-s)\mathbf {p} ))\ ds}$

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

• Fundamental theorem of vector calculus
• Magnetic vector potential
• Solenoidal vector field
• Closed and Exact Differential Forms

• Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.