Jacobian for straight-field-line poloidal angle

If the Jacobian $\mathcal{J}$ is chosen to be of the following form

$$\mathcal{J} (\psi, \theta) = R^2,$$

then Eq. (179) implies that $q_{\ensuremath{\operatorname{local}}}$ is a magnetic surface function, i.e., the magnetic field lines are straight on $(\psi, \theta)$ plane. The poloidal angle in Eq. (191) is written

$$\overline{\theta}_{i, j} = \frac{2 \pi}{\oint \frac{1}{R \vert \n... ... d l_p} \int_0^{\mathbf{x}_{i, j}} \frac{1}{R \vert \nabla \psi \vert} d l_p,$$

The Jacobian $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ given by Eq. (193) now takes the form

$$\mathcal{J}_{\ensuremath{\operatorname{new}}} = \pm R^2 \frac{\oint \frac{1}{R \vert \nabla \psi \vert} d l_p}{2 \pi} .$$ (200)