I have silly question about the evaluation of the Ricci scalar curvature at singularities.
Suppose $\theta=(x,m) \in \mathbb{R}^{2}$ a classic two-dimensional manifold $\mathcal{M}$ with diagonal metric,
$d\ell^{2}=g_{11}dx^{2}+g_{22}dm^{2}$,
where
$g_{11}=8x^{2}\quad and \quad g_{22}=2$.
The metric imposes a scalar curvature R written as follows,
$\mbox{R}=\frac{-1}{2\sqrt{g}}\left[\frac{\partial}{\partial x}\left(\frac{1}{\sqrt{g}}\frac{\partial g_{22}}{\partial x}\right)+\frac{\partial}{\partial m}\left(\frac{1}{\sqrt{g}}\frac{\partial g_{11}}{\partial m}\right)\right]$.
Suppose we want to evaluate R at $x(m)=\pm \sqrt{m}$. Should I first calculate $g_{11}$ and $g_{22}$ at $x(m)=\pm \sqrt{m}$, and then substitute the given results into R? or should I calculate R, and then evaluate the final expression at $x(m)=\pm \sqrt{m}$?