# Potential

*potential function*

A characteristic of a vector field.

A scalar potential is a scalar function such that at every point of the domain of definition of the field (sometimes, for example in physics, its negative is called a potential). If such a function exists, the vector field is called a potential field.

A vector potential is a vector function such that (cf. Curl) at every point of the domain of definition of the field . If such a function exists, the vector field is called a solenoidal field.

Depending on the distribution of the mass or the charge by which the potential is generated one speaks about a potential of a point-charge, a surface potential (single-layer or double-layer), a volume potential, etc. (see Potential theory).

#### Comments

See also Double-layer potential; Logarithmic potential; Multi-field potential; Newton potential; Non-linear potential; Riesz potential.

The use of a vector potential is restricted to three-dimensional vector fields. In this case one can prove the so-called Clebsch lemma, according to which any vector field can be represented as a sum of a potential field and a solenoidal field, .

**How to Cite This Entry:**

Potential.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Potential&oldid=17644