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## 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, $\mathbf {v} (x,y)=(y,-x)$ $\nabla \cdot \mathbf {v} =0.$ 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:

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

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.[dubious ]

## Examples

• The magnetic field B (see Gauss's law for magnetism)
• 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, ${\textstyle {\frac {\partial \rho _{e}}{\partial t}}=0}$ .
• The magnetic vector potential A in Coulomb gauge