Covariant and contravariant representation of equilibrium magnetic field

The axisymmetric equilibrium magnetic field is given by Eq. (54), i.e.,

$$\mathbf{B}= \nabla \Psi \times \nabla \phi + g (\Psi) \nabla \phi .$$ (170)

In $(\psi , \theta , \phi )$ coordinate system, the above expression can be written as

$$\mathbf{B}= - \Psi_{\psi} \nabla \phi \times \nabla \psi - \Psi_{\theta} \nabla \phi \times \nabla \theta + g (\Psi) \nabla \phi,$$ (171)

where the subscripts denote the partial derivatives with the corresponding subscripts. Note that Eq. (171) is a mixed representation, which involves both covariant and contravariant basis vectors. Equation (171) can be converted to contravariant form by using the metric tensor, giving

$$\mathbf{B}= - \Psi_{\psi} \nabla \phi \times \nabla \psi - \Psi_{... ... \theta + g (\Psi) \frac{1}{R^2} \mathcal{J} \nabla \psi \times \nabla \theta .$$ (172)

Similarly, Eq. (171) can also be transformed to covariant form, giving

$$\mathbf{B}= \left( \Psi_{\psi} \frac{\mathcal{J}}{R^2} \nabla \ps... ... \nabla \theta \cdot \nabla \psi \right) \nabla \theta + g (\Psi) \nabla \phi .$$ (173)

For the convenience of notation, define

$$h^{\alpha \beta} = \frac{\mathcal{J}}{R^2} \nabla \alpha \cdot \nabla \beta,$$ (174)

then Eq. (173) is written as

$$\mathbf{B}= (\Psi_{\psi} h^{\psi \theta} + \Psi_{\theta} h^{\thet... ...i \psi} - \Psi_{\theta} h^{\psi \theta}) \nabla \theta + g (\Psi) \nabla \phi .$$ (175)