Solenoidal field

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tubular field

A vector field in 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 such that . Examples of solenoidal fields are field of velocities of an incompressible liquid and the magnetic field within an infinite solenoid.


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 be a vector field (on ) with . 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.


[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. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098