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

Consequence

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.

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 ${\displaystyle 1/\|\mathbf {x} \|}$ 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

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

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

See also

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

References

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