Relation between Jacobian and poloidal angle $\theta$

In $(\psi , \theta , \phi )$ coordinates, a line element is written

$$d\mathbf{l}= \frac{\partial \mathbf{r}}{\partial \psi} d \psi + \... ...}}{\partial \theta} d \theta + \frac{\partial \mathbf{r}}{\partial \phi} d \phi$$ (185)

The line element that lies on a magnetic surface (i.e., $d \psi = 0$) and on a poloidal plane (i.e., $d \phi = 0$) is then written

$$d\ensuremath{\boldsymbol{\ell}}_p = \frac{\partial \mathbf{r}}{\partial \theta} d \theta$$

$$= \mathcal{J} \nabla \phi \times \nabla \psi d \theta .$$ (186)

We use the convention that $d \ell_p$ and $d \theta$ take the same sign, i.e.,

$$d \ell_p = \vert \mathcal{J} \nabla \phi \times \nabla \psi \vert d \theta .$$ (187)

Using the fact that $\nabla \psi$ and $\nabla \phi$ is orthogonal and $\nabla \phi = \hat{\ensuremath{\boldsymbol{\phi}}} / R$, the above equation is written as

$$d \theta = \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d l_p$$ (188)

Given $\vert\mathcal{J} \nabla \psi \vert$, Eq. (188) can be used to determine the $\theta$ coordinate of points on a magnetic surface.