Expression of current density

Let us derive the contravariant expression for the current density. Using

$$\mu_0 \mathbf{J}= \nabla \times \mathbf{B},$$

along with magnetic field expression (177) and the curl formula (139), we obtain

$$\mu_0 \mathbf{J} = - \left[ \left( \Psi' \frac{\mathcal{J}}{R^2} \vert \nabla \psi \... ...\right] \nabla \psi \times \nabla \theta - g' \nabla \phi \times \nabla \psi,$$ (331)

which is the contravariant form of the current density vector. Next, for later use, calculate the parallel current. By using Eqs. (177) and (331), the parallel current density is written as

$$\mu_0 \mathbf{J} \cdot \mathbf{B} = - \left[ \left( \Psi' \frac{\mathcal{J}}{R^2} \vert \nabla \psi \... ... \nabla \psi \vert^2 \right) \nabla \phi \times \nabla \psi \cdot \nabla \theta$$

$$= - \left[ \left( \Psi' \frac{\mathcal{J}}{R^2} \vert \nabla \psi \... ... + g' \mathcal{J}^{- 1} \Psi' \frac{\mathcal{J}}{R^2} \vert \nabla \psi \vert^2$$

$$= - g^2 \mathcal{J}^{- 1} \left\{ \frac{1}{g} \left( \Psi' \frac{\m... ...\frac{g'}{g^2} \Psi' \frac{\mathcal{J}}{R^2} \vert \nabla \psi \vert^2 \right\}$$

$$= - g^2 \mathcal{J}^{- 1} \left[ \left( \frac{\Psi'}{g} \frac{\math... ...ac{\mathcal{J}}{R^2} \nabla \psi \cdot \nabla \theta \right)_{\theta} \right] .$$ (332)