I'm being honest in that your explanation does not actually explain how to do an inverse transformation. It just says do it. I've fudged something to get the answer I need now but not sure I understand how I got it to to be honest.
so I get ##\tilde{g}_{\mu\nu} (x + \xi) = (\delta^{\rho}{ }_{\mu} - \partial{_{\mu}}\xi^{\rho})(\delta^{\sigma}{ }_{\nu} - \partial{_{\nu}}\xi^{\sigma})g_{\rho\sigma}(x)##?
yes I see how you get the forward one, I did the forward one already and you get the kronecka deltas and minus partial derivatives, still not sure how to do LHS
Still not sure would this be ##g_{\mu\nu}(x) = \frac{\partial{y_\rho}}{\partial{x_\mu}}\frac{\partial{y_\sigma}}{\partial{x_\nu}}\tilde{g}_{\rho\sigma} (y)## ?
is ##\tilde{g}_{\mu \nu}## different from ##g_{\mu \nu}## in terms of it is the same function but expressed in different co-ordinates or are they completely different functions ?
yeah I think lots of Uk universities are more tight with uploading accessible content, yeah I wasn't sure if I could have help on the screenshots - I know other books go more abstract as you've said. how does $$\tilde{g}_{uv}$$ differ from $${g}_{\rho\sigma}$$