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## Main Question or Discussion Point

##\int d^4 x \sqrt {g} ... ##

if I am given an action like this , were the ##\sqrt{\pm g} ## , sign depending on the signature , is to keep the integral factor invariant, when finding an eom via variation of calculus, often one needs to integrate by parts. When you integrate by parts, with this integration factor, is it now, always, the covariant derivative, replacing the partial derivative as in normal calculus when integrating by parts ? ..

if I am given an action like this , were the ##\sqrt{\pm g} ## , sign depending on the signature , is to keep the integral factor invariant, when finding an eom via variation of calculus, often one needs to integrate by parts. When you integrate by parts, with this integration factor, is it now, always, the covariant derivative, replacing the partial derivative as in normal calculus when integrating by parts ? ..