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Recognitions:
 Quote by Mentz114 I get a different result for the Ricci scalar, viz $$\frac{\partial rf}{re^{f}}$$ where $\partial r$ is differentiation wrt r.
The Ricci scalar is $R = \frac{2 f^\prime}{r} \, \exp(-2f)$ in the notation I used above. (The Ricci tensor, with components evaluated wrt the coframe field I gave, is diagonal with both diagonal components equal to the Riemann curvature component I computed.)
$$K = \frac{R_{r \phi r \phi}}{\exp(2f) \, r^2} = \frac{r \, f^\prime}{ r^2 \, \exp(2f)} = \frac{f^\prime}{r} \, \exp(-2f) = R_{1212}$$
Exercise: write an orthogonal chart for the general Riemannian two-manifold in the form $ds^2 = A^2 \, du^2 + B^2 \, dv^2$, where A,B can be functions of u,v (although this isn't neccessary; without loss of generality we could impose further restrictions), and adopt the coframe field
$\sigma^1 = A \, du, \; \sigma^2 = B \, dv$. Using the method of curvature two-forms, show that
$$-R_{1212} = \frac{ \left( \frac{A_u}{B} \right)_u + \left( \frac{B_v}{A} \right)_v }{AB}$$