# 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 vector field **A** such that

## Consequence

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

(divergence of the curl is zero) one obtains

which implies that **v** must be a solenoidal vector field.

## Theorem

Let

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

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

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

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.