Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the solution using Rabcd=gae(∂cΓedb-∂dΓecb+ΓecfΓfdb-ΓedfΓfcb) With e and f as summation indices I have assumed that where the e and f occur in the same Christoffel symbol a summation of all Christoffel symbols, of all combinations of the two variables, should be summed. For R0101 I have the equation to be R0101=g00(∂0(Γ011+Γ111)-∂1(Γ001+Γ101)+(Γ000+Γ001+Γ100+Γ101)(Γ011+Γ111)-(Γ010+Γ110+Γ011+Γ111)(Γ001+Γ101) Yielding the solution R0101=(1/2)eλ(∂2λ/∂t2) - (1/4)eλ(∂λ/∂t)(∂v/∂t)-(1/4)eλ(∂λ/dt)2-(1/2)evv''-(1/4)ev(v')^2-(1/4)evv'λ'-(1/4)ev(λ'(∂v/∂t)-(∂λ/∂t)v') where ' represents ∂/∂r. The only term that is not in the solution given in the textbook is -(1/4)ev(λ'(∂v/∂t)-(∂λ/∂t)v') . The relevant Christoffel symbols for R0101 are Γ000=1/2 (∂v/dt) , Γ001=1/2 v' Γ011=(1/2)e(λ-v)(∂λ/∂t) , Γ100e(v-λ)v' Γ101=(1/2)(∂λ/∂t) , Γ111=(1/2)λ' I feel like I'm missing something rather simple as I have yet to come across a thread or example where the covariant Riemann Tensor has been calculated and the workings have been displayed. I may of course be using the wrong search terms for finding such a thing.