Do I substitute [itex]A_\mu + \partial_\mu \lambda[/itex] everywhere [itex]A_\mu[/itex] appears, then expand out? Do I substitute a contravariant form of the substitution for [itex]A^\mu[/itex] as well? (If so, do I use a metric to convert it first?) I’m new to Ricci calculus; an explanation as to the meaning of raised and lowered indices here would be greatly appreciated. Everything I've found online so far has been (unnecessarily, I feel) complicated.