An example of a 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:

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:

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

The converse also holds: for any solenoidal v there exists a vector potential A such that (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:

\oiint ,

where 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.

Examples

  • The magnetic field B (see Maxwell's equations)
  • The velocity field of an incompressible fluid flow
  • The vorticity field
  • The electric field E in neutral regions ();
  • The current density J where the charge density is unvarying, .
  • The magnetic vector potential A in Coulomb gauge

This article uses material from the Wikipedia article Solenoidal vector field, which is released under the Creative Commons Attribution-Share-Alike License 3.0.