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: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.} A common way of expressing this property is to say that the field has no sources or sinks.[note 1]
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:
where d S {\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: v = ∇ × A {\displaystyle \mathbf {v} =\nabla \times \mathbf {A} } automatically results in the identity (as can be shown, for example, using Cartesian coordinates): ∇ ⋅ v = ∇ ⋅ ( ∇ × A ) = 0. {\displaystyle \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 v = ∇ × A . {\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .} (Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition.)
Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.