I calculated the curl of the vector field $\vec j = \delta(z) f(\rho) \vec e_\phi$ in cylindrical coordinates (ρ,φ,z). For the $\vec e_ρ$ unti vector I got: $$-\delta'(z)f(\rho)$$
The main question is when does this expression equal 0 if $f(\rho)\neq0$ and real?
It's trivial that $\delta'(z\neq0)=0$ because it's true for δ(z), thus for its derivative too.
Which leads to the question of how does δ'(z) behave around 0? I expect to show that somehow it will be 0 (which I dont think is true) in order to be consistent with the result of counting only in polar coordinates, ignoring z dependency thus not getting an $\vec e_ρ$ component for the curl. I dont think its true because if i try to calculate the limits, I expect to have undefined values of $\infty$ from the left side and $-\infty$ from the right side.
I am not familiar with distribution theory, but I found that they define δ' by using integration by parts on it. In my case that yields: $$\int_a^b \delta'(z)f(\rho)dz=\left[ \delta(z)f(\rho)\right]_a^b-\int_a^b\delta(z){df(\rho)\over dz}dz$$
For any $a,b \neq 0$ the first term is 0. I don't think its possible to give an interpretation other than 'undefined' for a or b =0. My previous analysis of the limits also implies this behavior of δ'(z), there I would also get 0 if I integrate it including 0 and undefined values if 0 is one of the borders.
The second term is always 0 because ${df(\rho)\over dz}=0$
However, I have read that in this integration I should use a bump test function which my currently used f(ρ) doesn't satisfy. Then what function I should use to see how δ' behaves around 0? What did I miss or was I doing everything alright?
To summarize, my main problem is that if I would just ignore z dependency in polar coordinates and calculate the curl in the z=0 plane, I would not get an $\vec e_ρ$ component for the curl. If I calculate with δ(z) in cylindrical I get that vector component that behaves undefined at z=0. I don't know how to make the results of the two approaches consistent, I dont think its a valid reasoning that I should ignore z dependency just because it behaves undefined at the origin.