In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero:

\nabla \cdot \mathbf = 0.\,

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:

\mathbf = \nabla \times \mathbf

automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

\nabla \cdot \mathbf = \nabla \cdot (\nabla \times \mathbf) = 0.

The converse also holds: for any solenoidal v there exists a vector potential A such that \mathbf = \nabla \times \mathbf. (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 \cdot \, d\mathbf = 0 ,

where d\mathbf is the outward normal to each surface element.

Etymology

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.

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.