Solenoidal vector field

A common way of expressing this property is to say that the field has no sources or sinks, but one must be careful with this interpretation because there are vector fields without sources or sinks that still have non-zero divergence.

Properties

The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:

${\displaystyle \mathbf{v}\cdot d\mathbf{S}=0,}$

where ${\displaystyle d\mathbf{S}}$ is the outward normal to each surface element.

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: $${\displaystyle \mathbf{v}=\mathrm{\nabla }\times \mathbf{A}}$$ automatically results in the identity (as can be shown, for example, using Cartesian coordinates): $${\displaystyle \mathrm{\nabla }\cdot \mathbf{v}=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})=0.}$$ The converse also holds: for any solenoidal v there exists a vector potential A such that ${\displaystyle \mathbf{v}=\mathrm{\nabla }\times \mathbf{A}.}$ (Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition.)