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:

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:

- ,

where 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:

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

The converse also holds: for any solenoidal **v** there exists a vector potential **A** such that (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 (); - The current density
**J**where the charge density is unvarying, . - The magnetic vector potential
**A**in Coulomb gauge

## See also

## 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