Solenoidal vector field

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: {{block indent|em=1.6|text={{oiint | integrand=\;\; \mathbf{v} \cdot \, d\mathbf{S} = 0 ,}}}} where 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: \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 subject to certain technical conditions on v, see Helmholtz decomposition.) ==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 (\rho_e = 0); • The current density J where the charge density is unvarying, \frac{\partial \rho_e}{\partial t} = 0. • The magnetic vector potential A in Coulomb gauge ==See also==