# Curl of a vector field with two different systems of coordinates

Let

$$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$

be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$, so in the $(x,y,z)$ system of coordinates.

Question 1: If we computed its curl in a new system of coordinates, that is $(x' = x, y' = y, z' = -z)$, how would we do it?

$$\nabla \times \mathbf{H} = \mathrm{det} \begin{vmatrix} \mathbf{u}_{x'} & \mathbf{u}_{y'} & \mathbf{u}_{z'}\\ H_x? & H_y? & H_z?\\ \displaystyle \frac{\partial}{\partial x'} & \frac{\partial}{\partial y'} & \frac{\partial}{\partial z'} \end{vmatrix}$$

In other words, which quantities should we put in the second line? $H_x$, $H_y$ and $H_z$ are along $\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$ and not $\mathbf{u}_{x'}$, $\mathbf{u}_{y'}$ and $\mathbf{u}_{z'}$.

Question 2: And if we would like to keep $H_x$, $H_y$ and $H_z$ in the second line, which would be their meaning in this computation?

These questions are strictly relative (but not equal) to my previous one.

1) Forget you ever had $x,y,z$ coordinate system and plug $H_{i'}$ into determinant.
2) Compute curl in $x,y,z$ coordinates and see how it looks in $x',y',z'$.