Suppose that we are given a metric $$ds^2=-\left(1-\frac{r_s}{r}\right)c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d \theta^2+r^2\sin^2(\theta)d \phi^2.$$ Given the Catalogs of Spacetimes pdf we are given some mathematica code to calculate Christoffel symbols of the second kind $\Gamma_{\alpha\nu}^{\lambda}$, the Ricci and Riemann curvature tensors and the scalar curvature: $R_{\mu\nu\rho\sigma}$ and $R_{\mu\nu}$ and $R$. Given all these computations, what I do not have to calculate is the Christoffel symbols of the first kind $\Gamma_{\lambda\alpha\nu}$ or the (1,3)-Riemann Curvature Tensor $R_{\mu\nu\alpha}^{\lambda}$. The code for the compitations I am fiven is given by: 
and 
.
Is there any code I can use without having to refer to packages like xtensor or gr to calculate the christoffel symbols of the first kind in a non-coordinate basis? P.S the answer given in this post about the Christoffel symbols of the first kind was unhelpful for my situation.
