Magnetic surface coordinate system $(\psi , \theta , \phi )$

A coordinate system $(\psi , \theta , \phi )$, where $\phi$ is the usual cylindrical toroidal angle, is called a magnetic surface coordinate system if $\Psi$ is a function of only $\psi$, i.e., $\partial \Psi / \partial \theta = 0$ (we also have $\partial \Psi / \partial \phi = 0$ since we are considering axially symmetrical case). In terms of $(\psi , \theta , \phi )$ coordinates, the contravariant form of the magnetic field, Eq. (172), is written as

$$\mathbf{B}= - \Psi' \nabla \phi \times \nabla \psi + g (\Psi) \frac{\mathcal{J}}{R^2} \nabla \psi \times \nabla \theta,$$ (176)

where $\Psi' \equiv d \Psi / d \psi$. The covariant form of the magnetic field, Eq. (173), is written as

$$\mathbf{B}= \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \cdo... ...}{R^2} \vert \nabla \psi \vert^2 \right) \nabla \theta + g (\Psi) \nabla \phi .$$ (177)