Equivariant map

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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 vector field A such that

$𝐯=\mathrm{\nabla }\times 𝐀.$

If a vector field v admits a vector potential A, then from the equality

$\mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$

(divergence of the curl is zero) one obtains

$\mathrm{\nabla }\cdot 𝐯=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0,$

which implies that v must be a solenoidal vector field.

Theorem

Let

$𝐯:{ℝ}^3\to {ℝ}^3$

be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define

$𝐀(𝐱)=\frac{1}{4\pi }\mathrm{\nabla }\times {\int }_{{ℝ}^3}\frac{𝐯(𝐲)}{\Vert 𝐱-𝐲\Vert }d𝐲.$

Then, A is a vector potential for v, that is,

$\mathrm{\nabla }\times 𝐀=𝐯.$

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.

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is

$𝐀+\mathrm{\nabla }m$

where m 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 analysis
• Magnetic potential
• Solenoid

References

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