Other expressions for magnetic field

The expression of the magnetic field in Eq. (250) can be rewritten in terms of the flux function $\Psi _p$ and $\Psi_t$ discussed in Sec. 6.3. Equation (250) is

$$\mathbf{B}= \nabla \Psi \times \nabla \zeta + q \nabla \theta \times \nabla \Psi,$$ (328)

which, by using Eq. (222), i.e., $\nabla \Psi = \nabla \Psi_p / (2 \pi)$, is rewritten as

$$\mathbf{B}= \frac{1}{2 \pi} (\nabla \zeta \times \nabla \Psi_p + q \nabla \Psi_p \times \nabla \theta),$$ (329)

which, by using Eq. (229), i.e., $q = d \Psi_t / d \Psi_p$, is further written as

$$\mathbf{B}= \frac{1}{2 \pi} (\nabla \zeta \times \nabla \Psi_p + \nabla \Psi_t \times \nabla \theta) .$$ (330)