Source of the problem, 3b here.
Problem Question
Electricity density in cylinder coordinates is $\bar{J}=e^{-r^2}\bar{e}_z$. Current creates magnetic field of the form $\bar{H}=H(r)\bar{e}_{\phi}$ where $H(0)=0$. Define $H(r)$ from the maxwell equations
$\nabla\times\bar{H}=\frac{1}{r}\begin{vmatrix}\bar{e}_r & r\bar{e}_\phi & \bar{e}_z \\ \partial_r & \partial_\phi & \partial_z \\ H_r & r H_\phi & H_z \\ \end{vmatrix} = \bar{J}.$
So
$\begin{align} \nabla\times\bar{H} &= \frac{1}{r} \bar{e}_r \begin{vmatrix} \partial_\phi & \partial_z \\ r H_\phi & H_z \\ \end{vmatrix} - r\bar{e}_\phi \begin{vmatrix} \partial_r & \partial_z \\ H_r & H_z \\ \end{vmatrix} + \bar{e}_z \begin{vmatrix} \partial_r & \partial_\phi \\ H_r & r H_\phi \\ \end{vmatrix} \\ &= \frac{1}{r} \left( \bar{e}_r (\partial_\phi H_z-\partial_z H_\phi) - r\bar{e}_\phi (\partial_r H_z-\partial_z H_r) + \bar{e}_z (\partial_r rH_\phi-\partial_\phi H_r) \right). \end{align}$
I messed the calculations here up when I tried to go back to Cartesian coordinates because it is otherwise hard for me to see the math. So I tried to think things with them
$\begin{cases} x=r\cos(\phi) \\ y=r\sin(\phi) \\ z=h \\ \end{cases}$
but it took me many pages of erroneous calculations and I could not finish on time. Now my friend suggested the below.
My friend's approach which I could not understand yet or his purpose, something to do with independence
$\begin{align} H &= H(r)e_\phi\\ &=0+H_\phi e_\phi+0 \end{align}$
where
$\begin{cases} H_\phi = H(r) <---\text{ independent of R}\\ H_r = 0 \\ H_2 =0 \\ \end{cases}.$
Could someone explain what I am doing wrong in going back to the Cartesian? I know it is not wrong but it is extremely slow way of doing things. I am not sure whether I was meant to remember the page 817 here or what is really essential to solve this problem?