Solenoidal vector field

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} \equiv 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} \equiv \nabla \times \mathbf {A}

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

\nabla \cdot \mathbf {v} \equiv \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:

$\oiint$

\mathbf {v} \cdot \,d\mathbf {S} \equiv 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

References

Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0-486-66110-5

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Solenoidal vector field

Properties

Etymology

Examples

See also

References