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