Solenoid Vector Potential

Please show that the Magnetic vector potential of a solenoid is consistent with the required values of it’s divergence, curl and laplacian in all regions.

$A_ϕ=μnI/2 s*ϕ ̂ $ inside the and $A_ϕ=μnI/2s R^2*ϕ ̂ $ outside

Cylindrical coordinates $(s,ϕ,z)$

$∇A=frac{1}{s}*frac{∂A_s}{∂s}*s ̂+frac{1}{s}*frac{∂A_ϕ}{∂ϕ} ϕ ̂+frac{∂A_z}{∂z}$

$∇×A=begin{vmatrix}s ̂/s&ϕ ̂&z ̂/s\∂/∂s&∂/∂ϕ&∂/∂z\0&sA_ϕ&0)end{vmatrix}=$

$=-frac{s ̂}{s}*frac{(∂(sA_ϕ))}{∂z}+frac{z ̂}{s} frac{(∂(sA_ϕ))}{∂s}$

and $∇^2 A=frac{1}{s}*frac{∂}{∂s} (s*frac{∂A}{∂s})$

For solenoid one has

$∇A=0$ ,$∇×A=B$ and $∆A=μ_0*J$

Inside

$∇A=frac{1}{s}*frac{(∂A_ϕ)}{∂ϕ} ϕ ̂=0$

$∇×A=frac{z ̂}{s} frac{(∂(sA_ϕ))}{∂s}=frac{z ̂}{s}*frac{∂}{∂s} (μnI/2 s^2 )=μnI=B$

$∇^2 A=frac{1}{s}*frac{∂}{∂s} (μnI/2 s)=μnI/2s=μNI/2sL=(μI_{tot})/Area=μJ$

Outside

$∇A=frac{1}{s}*frac{(∂A_ϕ)}{∂ϕ} ϕ ̂=0$

$∇×A=frac{z ̂}{s} frac{(∂(sA_ϕ))}{∂s}=frac{z ̂}{s}*frac{∂}{∂s} (μnI/2 R^2 )=0$

$∇^2 A=frac{1}{s}*frac{∂}{∂s} (μnI/2 R^2 )=0$ since $J_{outside}=0$