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# 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 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 also holds: for any solenoidal v there exists a vector potential A such that $\mathbf{v} = \nabla \times \mathbf{A}.$ (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: $\iint_S \mathbf{v} \cdot \, d\mathbf{s} = 0$,

where $d\mathbf{s}$ is the outward normal to each surface element.

## Examples

• the magnetic field B is solenoidal (see Maxwell's equations);
• the velocity field of an incompressible fluid flow is solenoidal;
• the electric field in regions where ρe = 0;
• the current density, J, if əρe/ət = 0.