Editing Solenoidal vector field

{{Short description|Vector field with zero divergence}}
[[File:Solenoidal vector field 1.svg|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 [[Helmholtz decomposition#Longitudinal and transverse fields| '''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 and sinks|sources or sinks]].<ref group="note">This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.</ref>

==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 [[Helmholtz decomposition|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 [[Vector calculus identities|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 (logic)|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 [[Material_derivative|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]]