Relation between the partial derivatives in $(\psi , \theta , \phi )$ and $(\psi , \theta , \zeta )$ coordinates

Noting the simple fact that

$$\frac{d}{d x} = \frac{d}{d (x + c)},$$ (251)

where $c$ is a constant, we conclude that

$$\left( \frac{\partial f}{\partial \zeta} \right)_{\psi, \theta} = \left( \frac{\partial f}{\partial \phi} \right)_{\psi, \theta},$$ (252)

(since $\phi = \zeta + q (\psi) \delta (\psi, \theta)$, where the part $q (\psi) \delta (\psi, \theta)$ acts as a constant when we hold $\psi$ and $\theta$ constant), i.e., the symmetry property with respect to the new toroidal angle $\zeta$ is identical with the one with respect to the old toroidal angle $\phi$. On the other hand, generally we have

$$\left( \frac{\partial f}{\partial \psi} \right)_{\theta, \zeta} \neq \left( \frac{\partial f}{\partial \psi} \right)_{\theta, \phi}$$ (253)

and

$$\left( \frac{\partial f}{\partial \theta} \right)_{\psi, \zeta} \neq \left( \frac{\partial f}{\partial \theta} \right)_{\psi, \phi} .$$ (254)

In the special case that $f$ is axisymmetric (i.e., $f$ is independent of $\phi$ in $(\psi , \theta , \phi )$ coordinates), then two sides of Eqs. (253) and (254) are equal to each other. Note that the partial derivatives $\partial / \partial \psi$ and $\partial / \partial \theta$ in Sec. 7.1 and 7.2 are taken in $(\psi , \theta , \phi )$ coordinates. Because the quantities involved in Sec. 7.1 and 7.2 are axisymmetric, these partial derivatives are equal to their counterparts in $(\psi , \theta , \zeta )$ coordinates.