I was trying to understand the properties of conformal transformations in $d$ dimensions, and I am confused about something very basic. The idea is discussed in many books but not very clearly. For example page 22-23 of "Lectures on CFT". I have a few questions about the discussion there:
1) From considering infinitesimal coordinate transformation, I can obtain Eqn-2.5 $\partial_\mu \epsilon_\nu + \partial_\nu \epsilon_\mu = \frac{2}{d}(\partial.\epsilon) \eta_{\mu\nu}$
Few lines below, the author applies $\partial_\nu$ to both sides to get Eqn-2.6.
$\partial_\mu (\partial.\epsilon) + \partial^2 \epsilon_\mu = \frac{2}{d} \partial_\mu (\partial . \epsilon)$
Here somehow the derivative $\partial_{\nu}$ does not seem to act on $\eta_{\mu\nu}$. Why is this so ? Why do I not have an additional term that looks like
$(\partial . \epsilon) \partial^\nu \eta_{\mu\nu}~? $
I suppose the argument is that $\eta_{\mu\nu}(x)$ before the transformation is the constant flat metric. Is this correct?
2) This is a subjective/general question. What is the general motivation / structure for the sequence of arguments here?
3) A more minor question about [page 22 of "Lectures on CFT"]: Shouldn't the transformation above Eqn 2.1 be
$g'_{\rho\sigma}(x') = \frac{\partial x^\mu}{\partial x'^\rho} \frac{\partial x^\nu}{\partial x'^\sigma} g_{\mu\nu}(x)$
instead of what is given in the notes ? This will be consistent with this post.
