Solenoidal vector field

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

$\nabla \cdot {\mathbf {v} }=0.\,$ Properties

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:

${\mathbf {v} }=\nabla \times {\mathbf {A} }$ automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

$\nabla \cdot {\mathbf {v} }=\nabla \cdot (\nabla \times {\mathbf {A} })=0.$ The converse also holds: for any solenoidal v there exists a vector potential A such that ${\mathbf {v} }=\nabla \times {\mathbf {A} }.$ (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)

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

Template:Oiint

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

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.