Solenoidal vector field

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

\[ \nabla \cdot \mathbf{v} = 0.\, \]

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:

\[\mathbf{v} = \nabla \times \mathbf{A}\]

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 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:

x45px\(\mathbf{v} \cdot \, d\mathbf{s} = 0 \),

where \(d\mathbf{s}\) 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 flux density B is solenoidal (see Maxwell's equations);
• the velocity field of an incompressible fluid flow is solenoidal;
• the vorticity field is solenoidal
• the electric flux density D in regions where there is no charge (ρ_e = 0);
• the current density J where the charge density is unvarying, \(\frac{\partial \rho_e}{\partial t} = 0\).

See also

• Longitudinal and transverse vector fields

References

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