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.
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Solenoidal field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solenoidal_field&oldid=19139