[math-fun] (no subject)

I'm reading a book on general relativity, and the author makes a statement giving a proof that I don't find convincing. Suppose we wish to construct a scalar R (i.e. R(x') = R(x) under coordinate transformations) from the metric tensor and two powers of differentiation.  That is, R must consist of terms of the form (∂∂g) and (∂g)(∂g).  The claim is that R must be the scalar curvature, except for a multiplicative constant.  Can someone point me to a proof? The importance of this result is that the Einstein-Hilbert action for the dynamics of spacetime is then uniquely determined.  Well, unique in the absence of the cosmological constant.   --  Gene