Flux surface

A given smooth surface S with normal n is a flux surface of a smooth vector field B when

${\vec {B}}\cdot {\vec {n}}=0$

everywhere. Defining a scalar flux function (f) such that its value is constant on the surface S, this can be rewritten

${\vec {B}}\cdot {\vec {\nabla }}f=0$

In three dimensions, the only closed flux surface corresponding to a non-vanishing vector field is a topological toroid. This fact lies at the basis of the design of magnetic confinement devices.