{{Short description|Vector field with zero divergence}} thumb|250px|An example of a solenoidal vector field, <math>\mathbf{v}(x, y) = (y, -x)</math> 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: <math display="block"> \nabla \cdot \mathbf{v} = 0. </math> A common way of expressing this property is to say that the field has no sources or sinks.
==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: {{block indent|em=1.6|text={{oiint | integrand=<math>\;\; \mathbf{v} \cdot \, d\mathbf{S} = 0 ,</math>}}}} where <math>d\mathbf{S}</math> 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: <math display="block">\mathbf{v} = \nabla \times \mathbf{A}</math> automatically results in the identity (as can be shown, for example, using Cartesian coordinates): <math display="block">\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.</math> The converse also holds: for any solenoidal '''v''' there exists a vector potential '''A''' such that <math>\mathbf{v} = \nabla \times \mathbf{A}.</math> (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 (<math>\rho_e = 0</math>); * The current density '''J''' where the charge density is unvarying, <math display="inline">\frac{\partial \rho_e}{\partial t} = 0</math>. * The magnetic vector potential '''A''' in Coulomb gauge
==See also== * Longitudinal and transverse vector fields * Stream function * Conservative vector field
==Notes==
{{Reflist|group="note"|1}}
==References== *{{citation | title=Vectors, tensors, and the basic equations of fluid mechanics | authorlink=Rutherford Aris | first=Rutherford | last=Aris | publisher=Dover | year=1989 | isbn=0-486-66110-5 |url=https://books.google.com/books?id=QcZIAwAAQBAJ&q=%22solenoidal+vector+field%22}}
Category:Vector calculus Category:Fluid dynamics