BREAKING NEWS
Solenoidal vector field

## Summary

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:

An example of a solenoidal vector field, ${\displaystyle \mathbf {v} (x,y)=(y,-x)}$
${\displaystyle \nabla \cdot \mathbf {v} =0.}$
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:

${\displaystyle \;\;\mathbf {v} \cdot \,d\mathbf {S} =0,}$

where ${\displaystyle d\mathbf {S} }$  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:

${\displaystyle \mathbf {v} =\nabla \times \mathbf {A} }$

automatically results in the identity (as can be shown, for example, using Cartesian coordinates):
${\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.}$

The converse also holds: for any solenoidal v there exists a vector potential A such that ${\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}$  (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.[dubious ]

## Notes

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.

## References

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