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The rules are simple but long, as stated in Wikipedia. I'll elaborate on them here. We start with a rank $(r,s)$ tensor $T$ in $d$ dimensions. We seek the $d^{r+s+1}$ components of $\nabla T$, a rank $(r,s+1)$ tensor. We can think of these components as collected into $(r,s)$ sets of $d^{r+s}$, one for each of the $d$ values of the free index $\gamma$ in ...
Yes. If you define $f=-\partial_\mu A^\mu$ then you can write the equation in the form $$\partial_\mu\partial^\mu\psi = f$$ This is the Klein-Gordon equation with a nonzero source ($f$) and can be solved via Green's function methods. Once you have the Klein-Gordon propagator* $G(x)$ (this is derived in any e.g. quantum field theory textbook) appropriate to ...