Given tensor: $$g = \begin{pmatrix} g_{00} & g_{01}\\ g_{10} & g_{11} \end{pmatrix}$$
I need to find, for example, this: $\Gamma^1_{10}$ What does this mean and how to do it? What is the difference between $\Gamma^0_{10}$ and $\Gamma^1_{10}$? I understood it in this way: $\Gamma^1_{10} = g_{10}$. Is this correct?
Succinctly $$\Gamma^1{}_{10}= \frac{1}{2}g^{10}[g_{00,1}+g_{10,0}-g_{10,0}]+ \frac{1}{2}g^{11}[g_{10,1}+g_{01,0}-g_{10,1}]$$ where $g_{00,1}=\frac{g_{00}}{\partial x^1}$ for example. The $g^{ij}$ are the entries of $g^{-1}$.
This can be simplified to $$\Gamma^1{}_{10}= \frac{1}{2}g^{10}g_{00,1}+ \frac{1}{2}g^{11}g_{01,0}.$$