Solenoidal vector field
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:
where is the outward normal to each surface element.
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.
- the magnetic flux density B is solenoidal (see Maxwell's equations);
- the velocity field of an incompressible fluid flow is solenoidal;
- the vorticity field is solenoidal
- the electric flux density D in regions where there is no charge (ρe = 0);
- the current density J where the charge density is unvarying, .