In calculating the mean curvature for a surface of the form $(f(z)\cos(\theta), f(z)\sin(\theta), z)$ I need to use some checks to ensure I haven't made any mistakes along the way. The one I would normally do is a spherical cap, but I dont really know how to parameterise it in this situation. I think that $\sqrt{a^2 - z^2}$ would generate a hemisphere of radius a but geometry never was my strong point.
As a note, changing coordinate systems in the main calculation is not really possible due to the main problem having axisymmetry making cylindrical coordinates the natural choice