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Solenoidal |
In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero:
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field 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 S, 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) and meaning pipe-shaped. This contains σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained like in a pipe, so with a fixed volume.
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