Jacobian for equal-arc-length poloidal angle

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

$$\mathcal{J}= \frac{R}{\vert \nabla \psi \vert} .$$ (194)

Then $\overline{\theta}_{i, j}$ in Eq. (191) is written

$$\overline{\theta}_{i, j} = \frac{2 \pi}{\oint d l_p} \int_0^{\mathbf{x}_{i, j}} d l_p .$$ (195)

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

$$\mathcal{J}_{\ensuremath{\operatorname{new}}} = \pm \frac{\oint d... ...c{\oint d l_p}{2 \pi} \frac{R}{\vert \nabla \Psi \vert} \frac{d \Psi}{d \psi} .$$ (196)

Equation (195) indicates a set of poloidal points with equal arc intervals corresponds to a set of uniform $\theta_i$ points. Therefore this choice of the Jacobian is called the equal-arc-length Jacobian. Note that Eq. (195) does not involve the radial coordinate $\psi$. Therefore the values of $\theta$ of points on any magnetic surface can be determined before the radial coordinate is chosen. By using Eq. (195) (to calculate $\theta$) and the definition of $\psi$ (to calculate $\psi$), we obtain a magnetic surface coordinate system $(\psi, \overline{\theta}, \phi)$.