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.\,

A common way of expressing this property is to say that the field has no sources or sinks.^{[note 1]}

Properties

The divergence theorem gives an 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} =0

,

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

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 subject to certain technical conditions on v, see Helmholtz decomposition.)

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

Notes

^ This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.

References

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

[1] This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.

[note 1]