I am trying to calculate the entries of the Riemann curvature tensor $R^m\_{\phantom{m}ijk}$ for the metric $g\_{ij}$. 


The Riemann-Christoffel tensor is given as
\begin{align}
R^m\_{\phantom{m}ijk} = \frac{\partial}{\partial x^j}{m \brace i\,\,k} - \frac{\partial}{\partial x^k}{m \brace i\,\,j} + 
 {n \brace i\,\,k}{m \brace n\,\,j} - {n \brace i\,\,j}{m \brace n\,\,k}
\end{align}
where the Christoffel symbol of second kind are given as
\begin{align}
{m\brace b\,\,c} = {m\brace c\,\,b} = g^{mb} \left[ac,b\right],
\end{align}
and the Christoffel symbol of first kind 
\begin{align}
[ac, b] &= [ca,b] =
\frac{1}{2} \left[\frac{\partial g\_{ab}}{\partial x^c} + \frac{\partial g\_{bc}}{\partial x^a} -
 \frac{\partial g\_{ac}}{\partial x^b}\right].
\end{align}


I have to basically calculate all these symbols and insert them in the Riemann curvature tensor. Is there any way I can easily keep track of the elements of this fourth-order tensor ? (Visual tricks, or simply using simplifications to end up with trivial results.) The metric $g\_{ij}$ is diagonal. So obviously there will be many trivial terms.



Calculating the Riemann Christoffel tensor for a diagonal metric
I managed to find a post on Physics StackExchange :


<https://physics.stackexchange.com/questions/179393/any-tips-on-evaluating-riemann-tensor>


which details what I feared. The process it self is too tedious. The best solution is to use a symbolic package which can do all these calculations, like for example the python package SymPy. 


Another answer (from the same thread) is based on some thing called "Cartan formalism" (it is also the answer which has been up-voted the most).


