Solenoidal vector field


In vector calculus a solenoidal 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.

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

Etymology

Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές meaning pipe-shaped, from σωλην or pipe.

Examples