# Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:

An example of a solenoidal vector field, ${\displaystyle \mathbf {v} (x,y)=(y,-x)}$
${\displaystyle \nabla \cdot \mathbf {v} =0.\,}$

A common way of expressing this property is to say that the field has no sources or sinks. The field lines of a solenoidal field are either closed loops or end at infinity.

## 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:

${\displaystyle \mathbf {v} =\nabla \times \mathbf {A} }$

automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

${\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.}$

The converse also holds: for any solenoidal v there exists a vector potential A such that ${\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}$  (Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition.)

## Etymology

Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained as if in a pipe, so with a fixed volume.