# Does Jackson's result for the vector potential of current loop correct?

General form of Maxwell equation is given by $$\nabla_\mu F^{\mu\nu} = 4\pi J^\nu$$ where $$F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu$$ is the tensor of EM field. Then Maxwell equations can be rewritten as $$\nabla_\mu F^{\mu\nu} =\nabla_\mu(\nabla^\mu A^\nu-\nabla^\nu A^\mu) = 4\pi J^\nu\ , \qquad \nabla_\mu A^\mu = 0 \to Lorentz\,\,\, gauge\ ,$$ So finally we have $$\nabla^2A^\nu=\nabla_\mu\nabla^\mu A^\nu=4\pi J^\nu$$ If we choose scalar field $$\Phi$$ instead of vector field $$A^\nu$$ then we get Poisson equation $$\nabla^2\Phi = 4\pi\rho$$, $$\rho$$ is the source term and solution of Poisson equation will be in the form of Green's function. $$\Phi(r) = \frac{1}{4\pi} \int\frac{\rho(\vec r')d{\vec r'}}{|{\vec r}-{\vec r}'|}$$
Let us go back to our main equation $$\nabla^2A^\nu = 4\pi ^\nu$$. Solution of this equation will be in form of Green's function only in Cartesian coordinate $$(x,y,z)$$. Since we want to consider vector potential of current loop. It is convenience to write Maxwell equation either in spherical ($$r,\theta,\phi$$) or cylindrical ($$r,\phi,z$$) coordinates. If we follow Jackson's book solution for vector potential written as $$A_\phi = \frac{1}{4\pi}\int\frac{I(r')r'^2dr'\sin\theta'd\theta'd\phi'}{|\vec r-\vec r'|}$$ but we know that $$A_\phi\neq A^\phi$$ in spherical coordinate. That is why square of Nabla operator will be different for $$A_\phi$$ and $$A^\phi$$ in spherical coordinate. That is why I am a bit worrying about this solution. Is it correct or not? Hint: $$\nabla_\mu A^\nu = \partial_\mu A^\nu + \Gamma_{\mu\lambda}^\nu A^\lambda$$ and $$\nabla_\mu A_\nu = \partial_\mu A_\nu - \Gamma_{\mu\nu}^\lambda A_\lambda$$ where $$\Gamma_{\mu\nu}^\lambda$$ is Christoffel symbol.