Question

Neumann boundary condition for Laplace's equation in 2D axisymmetric coordinates?

Answer 1

Neither condition you wrote can be described as "constant gradient of density condition". The former says that at infinity the two non-angular components of the gradient are in fact constant, but there is the further constraint that these components are equal. The latter condition does not regard the gradient at infinity but only the normal component of it referred to a larger and larger spherical surface of equation $l=(r^2+z^2)^{1/2}$

Answer 2

Axis coordinates makes the edges in very Perpendicular shape for the help of the reader.