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;

$\nabla \cdot \mathbf{v} = 0.\,$

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 S, the net total flux through the surface must be zero:

$\iint_S \mathbf{v} \cdot \, d\mathbf{s} = 0$,

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) and meaning pipe-shaped. This contains σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained like 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 D in regions where ρ_e = 0;
• the current density, J, if $\frac{\partial \rho_e}{\partial t} = 0$.

See also

• Longitudinal and transverse vector fields

References

• Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0486661105