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).

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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, $\mathbf{a}=\mathrm{grad}v+\mathrm{curl}A$.