Flux surface

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

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

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

$\overrightarrow{B}\cdot \overrightarrow{\mathrm{\nabla }}f=0$

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.