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.