# Solenoidal field

*tubular field*

A vector field in $ \mathbf R ^ {3} $ having neither sources nor sinks, i.e. its divergence vanishes at all its points. The flow of a solenoidal field through any closed piecewise-smooth oriented boundary of any domain is equal to zero. Solenoidal fields are characterized by their so-called vector potential, that is, a vector field $ A $ such that $ \mathbf a = \mathop{\rm curl} A $. Examples of solenoidal fields are field of velocities of an incompressible liquid and the magnetic field within an infinite solenoid.

#### Comments

A solenoid is a long spiral coil of wire, usually cylindrical, through which a current can be passed to produce a magnetic field. More abstractly, let $ \mathbf a $ be a vector field (on $ \mathbf R ^ {3} $) with $ \mathop{\rm div} ( \mathbf a ) = 0 $. Consider a surface consisting of a cylinder along the vector lines together with surfaces normal to the lines at both ends. Such a tube is called a solenoid.

#### References

[a1] | E.A. Hylleras, "Mathematical and theoretical physics" , 1 , Wiley (Interscience) (1970) pp. 70ff |

[a2] | B.G. Levich, "Theoretical physics" , 1. Theory of the electromagnetic field , North-Holland (1970) pp. 6; 364; 366 |

[a3] | G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1974) pp. 75; 167 |

[a4] | K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 272 |

**How to Cite This Entry:**

Solenoidal field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Solenoidal_field&oldid=48746