Contravariant form of magnetic field in $(\psi , \theta , \zeta )$ coordinates

Recall that the contravariant form of the magnetic field in $(\psi , \theta , \phi )$ coordinates is given by Eq. (180), i.e.,

$$\mathbf{B}= - \Psi' (\nabla \phi \times \nabla \psi + \hat{q} \nabla \psi \times \nabla \theta) .$$ (248)

Next, let us derive the corresponding form in $(\psi , \theta , \zeta )$ coordinates. Using the definition of $\zeta$, equation (248) is written as

$$\mathbf{B} = - \Psi' \nabla (\zeta + q \delta) \times \nabla \psi - \Psi' \hat{q} \nabla \psi \times \nabla \theta$$

$$= - \Psi' \nabla \zeta \times \nabla \psi - \Psi' \nabla (q \delta) \times \nabla \psi - \Psi' \hat{q} \nabla \psi \times \nabla \theta$$

$$= - \Psi' \nabla \zeta \times \nabla \psi - \Psi' q \frac{\partial ... ...abla \theta \times \nabla \psi - \Psi' \hat{q} \nabla \psi \times \nabla \theta$$ (249)

Using Eq. (240), the above equation is simplified as

$$\mathbf{B} = - \Psi' (\nabla \zeta \times \nabla \psi + q \nabla \psi \times \nabla \theta) .$$ (250)

Equation (250) is the contravariant form of the magnetic field in $(\psi , \theta , \zeta )$ coordinates.