An expression of scalar curvature

Question

In the book "Hamilton's Ricci flow" by B. Chow, P. Lu and L. Ni, at the page- 99, Exercise 2.8, the statement says:
If $(M,h)$ is a Riemannian surface and $g=uh$ for some function $u$ on $M$, then
$$R_g=u^{-1}(R_h-\Delta_h\log u),$$
where $R_h$ and $R_g$ are scalar curvature of the metric $h$ and $g$ respectively. I want to know how this expression is derived?

Answer 1

One place to find a proof is in my Introduction to Riemannian Manifolds (2nd ed.), Theorem 7.30. (The notation is a little different from yours, but it should be easy to translate between the two notations.)