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 $\mathbf$, a ''vector potential'' is a $C^2$ vector field $\mathbf$ such that
$\mathbf = \nabla \times \mathbf.$

Consequence

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

Theorem

Let
$\mathbf : \R^3 \to \R^3$
be a solenoidal vector field which is twice continuously differentiable. Assume that $\mathbf(\mathbf)$ decreases at least as fast as $1/\, \mathbf\,$ for $\, \mathbf\, \to \infty$. Define
$\mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf$
where $\nabla_y \times$ denotes curl with respect to variable $\mathbf$. Then $\mathbf$ is a vector potential for $\mathbf$. That is,
$\nabla \times \mathbf =\mathbf.$

The integral domain can be restricted to any simply connected region $\mathbf$. That is, $\mathbf$ also is a vector potential of $\mathbf$, where
$\mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf.$

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, $\mathbf(\mathbf)$ also qualifies as a vector potential for $\mathbf$, where

:$\mathbf(\mathbf) =\int_\Omega \frac d^3 \mathbf$.

Substituting $\mathbf$ (current density) for $\mathbf$ and $\mathbf$ ( H-field) for $\mathbf$, yields the Biot-Savart law.

Let $\mathbf$ be a star domain centered at the point $\mathbf$, where $\mathbf\in \R^3$. Applying Poincaré's lemma for differential forms to vector fields, then $\mathbf(\mathbf)$ also is a vector potential for $\mathbf$, where

$\mathbf(\mathbf) =\int_0^1 s ((\mathbf-\mathbf)\times ( \mathbf( s \mathbf + (1-s) \mathbf ))\ ds$

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If $\mathbf$ is a vector potential for $\mathbf$, then so is
$\mathbf + \nabla f,$
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.

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.

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Concepts in physics
Potentials
Vector calculus
Vector physical quantities