Ok, I have managed to show (i) by direct computation; by just plugging in the partial derivatives and it pops out. I've tried to do (ii) but it's not equal to zero............HELP!!
Curl in orthogonal curvilinear 3D coordinates is:
Where are Lamé coefficients and is Levi-Civita symbol.
In spherical coordinates , , .
Since in our case has only r component can have only and components.
We can see that
because doesn't depend on and
The same argument for component: