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. However, often this statement is misunderstood and leads to the conclusion that the field lines of a solenoidal field are either closed loops or end at infinity. Actually, these properties are sufficient, but not strictly required, to satisfy the equation above.

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.

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

See also

• Longitudinal and transverse vector fields
• Stream function
• Conservative vector field

References

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