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:
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.
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:
where 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:
automatically results in the identity (as can be shown, for example, using Cartesian coordinates):
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.
- Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0-486-66110-5