Divergence operator in general coordinates $(\psi , \theta , \zeta )$

To calculate the divergence of a vector, the vector should be in the contravariant representation since we can make use of the fact that

$$\nabla \cdot (\nabla \alpha \times \nabla \beta) = 0,$$ (133)

for any scalar quantities $\alpha$ and $\beta$. Therefore we write vector $\mathbf{A}$ as

$$\mathbf{A}= A^1 \mathcal{J} \nabla \theta \times \nabla \zeta + A... ...la \zeta \times \nabla \psi + A^3 \mathcal{J} \nabla \psi \times \nabla \theta,$$ (134)

where $A^1 =\mathbf{A} \cdot \nabla \psi$, etc.. Then the divergence of $\mathbf{A}$ is readily calculated as

$$\nabla \cdot \mathbf{A} = \nabla (A^1 \mathcal{J}) \cdot (\nabla \theta \times \nabla \zeta... ...nabla \psi) + \nabla (A^3 \mathcal{J}) \cdot (\nabla \psi \times \nabla \theta)$$ (135)

$$= \frac{1}{\mathcal{J}} \left( \frac{\partial A^1 \mathcal{J}}{\par... ...J}}{\partial \theta} + \frac{\partial A^3 \mathcal{J}}{\partial \zeta} \right),$$ (136)

where the second equality is obtained by using Eqs. (129), (130), and (131).