# Potential

*potential function*

A characteristic of a vector field.

A scalar potential is a scalar function $v(M)$ such that $\mathbf{a}(M) = \mathrm{grad}\,v(M)$ at every point of the domain of definition of the field $\mathbf{a}$ (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 $\mathbf{A}(M)$ such that $\mathbf{a}(M) = \mathrm{curl}\,\mathbf{A}(M)$ (cf. Curl) at every point of the domain of definition of the field $\mathbf{a}$. If such a function exists, the vector field $\mathbf{A}(M)$ 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, $\mathbf{a} = \mathrm{grad}\,v + \mathrm{curl}\,A$.

**How to Cite This Entry:**

Potential.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Potential&oldid=42618