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

Consequence

If a vector field v admits a vector potential A, 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 v 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 decreases at least as fast as $1/\, \mathbf\,$ for $\, \mathbf\, \to \infty$.
Define
$\mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf.$

Then, A is a vector potential for , that is,　$\nabla \times \mathbf =\mathbf.$
Here, $\nabla_y \times$ is curl for variable y.
Substituting curl '' for the current density j of the retarded potential, you will get this formula. In other words, v corresponds to the H-field.

You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v;
$\mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf.$

A generalization of this theorem is the Helmholtz decomposition 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 Biot-Savart's law, the following $\boldsymbol(\textbf)$ is also qualify as a vector potential for v.
:$\boldsymbol(\textbf) =\int_\Omega \frac d^3 \boldsymbol$

Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law.

Let $\textbf\in \mathbb$ and let the Ω be a star domain centered on the ''p'' then,
translating Poincaré's lemma for differential forms into vector fields world, the followng $\boldsymbol(\boldsymbol)$ is also a vector potential for the $\boldsymbol$

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

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If is a vector potential for , 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
* Solenoid
* 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