Solenoidal vector field

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:

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

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:

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

where ${\displaystyle d\mathbf {S} }$ 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:

$${\displaystyle \mathbf {v} =\nabla \times \mathbf {A} }$$

$${\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.}$$

${\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}$

Etymology

Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.

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 (${\displaystyle \rho _{e}=0}$);
• 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

See also

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

Notes

References

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