Solenoidal vector field
From Wikipedia, the free encyclopedia
In vector calculus a solenoidal vector field is a vector field v with divergence zero:
This condition is satisfied whenever v has a vector potential, because if
then
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.)
Gauss's theorem, gives the equivalent integral definition of a solonoidal field; namely that for any closed surface S, the net total flux out through the surface must be zero:
,
where 
is the outward normal to each surface element.
[edit] Examples
- one of Maxwell's equations states that the magnetic flux density B is solenoidal;
- the velocity field of an incompressible fluid flow is solenoidal.
