Solenoidal vector field An example of a solenoidal vector field, $\mathbf {v} (x,y)=(y,-x)$ In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:

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

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational 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:

$\mathbf {v} =\nabla \times \mathbf {A}$ automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

$\nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.$ 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, the net total flux through the surface must be zero: $\ \ \ \mathbf {v} \cdot \,d\mathbf {S} =0$ ,

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

Etymology

Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained as if in a pipe, so with a fixed volume.

Examples

• The magnetic field B (see Maxwell's equations)
• The velocity field of an incompressible fluid flow
• The vorticity field
• The electric field E in neutral regions ($\rho _{e}=0$ );
• The current density J where the charge density is unvarying, ${\frac {\partial \rho _{e}}{\partial t}}=0$ .
• The magnetic vector potential A in Coulomb gauge