## Frequent Links

# Solenoidal vector field

In vector calculus a **solenoidal vector field** (also known as an **incompressible vector field** or a **divergence free vector field** ) is a vector field **v** with divergence zero at all points in the field:

- <math> \nabla \cdot \mathbf{v} = 0.\, </math>

## 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:

- <math>\mathbf{v} = \nabla \times \mathbf{A}</math>

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

- <math>\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.</math>

The converse also holds: for any solenoidal **v** there exists a vector potential **A** such that <math>\mathbf{v} = \nabla \times \mathbf{A}.</math> (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:

where <math>d\mathbf{S}</math> 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**is solenoidal (see Maxwell's equations); - the velocity field of an incompressible fluid flow is solenoidal;
- the vorticity field is solenoidal
- the electric field
**E**in neutral regions (<math>\rho_e = 0</math>); - the current density
**J**where the charge density is unvarying, <math>\frac{\partial \rho_e}{\partial t} = 0</math>.

## See also

## References

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