Solenoidal vector field

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

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

Properties[edit]

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:

\oiint\mathbf{v} \cdot \, d\mathbf{S} = 0 ,

where d\mathbf{S} is the outward normal to each surface element.

Etymology[edit]

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.

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