Fake Plot

{n,k}={2,3};
r0=BesselJZero[n,k];
f[r_,θ_]= BesselJ[n,r r0] Sin[n θ] 
Plot3D[ f[r,θ], {r, 0, 1},{θ,0, 2π}]

ContourPlot[ f[r,θ], {r, 0, 1},{θ,0, 2π}]

ParametricPlot3D[ {r Cos[θ], r Sin[θ],f[r,θ]}, 
{r, 0, 1},{θ,0, π}]

Question A

All about Bessel Functions.

D[ BesselJ[n, r (Sqrt[λ])], r]

n=2;
Plot[
 BesselJ[-1+n,μ]-BesselJ[1+n, μ],
{μ, 0, 23}]

FindRoot[
 BesselJ[-1+n,μ]-BesselJ[1+n, μ]==0, 
{μ, 16}
]

Question B

How do I separate the pieces when there is a left over “r” in the equation. Appropriate for Q2.

Spherical Coordinates

FullSimplify[
Laplacian[ u[ρ,θ,ϕ],{ρ,θ,ϕ},"Spherical"]
]