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

To take the curl of a vector, it should be in the covariant representation since we can make use of the fact that $\nabla \times \nabla \alpha = 0$. Thus the curl of $\mathbf{A}$ is written as

$$\nabla \times \mathbf{A} = \nabla \times (A_1 \nabla \psi + A_2 \nabla \theta + A_3 \nabla \zeta)$$

$$= \nabla A_1 \times \nabla \psi + \nabla A_2 \times \nabla \theta + \nabla A_3 \times \nabla \zeta$$

$$= \left( \frac{\partial A_1}{\partial \theta} \nabla \theta + \frac... ... \frac{\partial A_3}{\partial \theta} \nabla \theta \right) \times \nabla \zeta$$

$$= \frac{1}{\mathcal{J}} \left( \frac{\partial A_2}{\partial \psi} -... ...al A_2}{\partial \zeta} \right) \mathcal{J} \nabla \theta \times \nabla \zeta .$$ (139)

Note that taking the curl of a vector in the covariant form leaves the vector in the contravariant form.