# 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[edit]

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

**v**must be a solenoidal vector field.

## Theorem[edit]

Let

**v**(

**x**) decreases at least as fast as for . Define

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

**y**. Substituting

**curl[v]**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**;

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 is also qualify as a vector potential for **v**.

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

Let 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 is also a vector potential for the

## Nonuniqueness[edit]

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

This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.

## See also[edit]

- Fundamental theorem of vector calculus
- Magnetic vector potential
- Solenoid
- Closed and Exact Differential Forms

## References[edit]

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