Solenoidal vector field

In vector calculus a solenoidal vector field is a vector field v with divergence zero:

$\nabla \cdot \mathbf{v} = 0.\,$

This condition is clearly satisfied whenever v has a vector potential, because if

$\mathbf{v} = \nabla \times \mathbf{A}$

then

$\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.$

The converse holds: for any solenoidal v there exists a vector potential A such that $\mathbf{v} = \nabla \times \mathbf{A}$ . (Strictly, this holds only subject to certain technical conditions on v.)

Examples:

one of Maxwell's equations states that the magnetic field B is solenoidal;
the velocity field of an incompressible fluid flow is solenoidal.

Solenoidal vector field

See also: Solenoidal vector field, Divergence, Incompressible fluid flow, Magnetic field, Maxwell's equations, Vector calculus, Vector field, Vector potential, Velocity field