Christoffel Symbols Different Representation?

I was given the $mathbb{R^2}$ metric in polar coordinates, as follows:

$$
ds^2=dr^2+r^2dtheta^2
$$

In this context we denote $e_1=partial_r=(cos(theta), sin(theta))$, $e_2=partial_{theta}=(-rsin(theta), rcos(theta))$. Also, here we call $z^1=r$, $z^2=theta$.

We can also define:

$$frac{partial e_{i}}{partial z^{j}}=tilde{Gamma}^{k}_{ij}e_k$$

After calculating the individual terms $tilde{Gamma}^{k}_{ij}$ I see that, in fact, they are equal to the Christoffel Symbols for the connection associated to the metric (Levi-Civita). The problem is I can’t see this as a general thing. How am I supposed to prove both definitions are equal. Meaning:

$$
tilde{Gamma}^{lambda}_{munu}=Gamma^{lambda}_{munu}=frac{1}{2}g^{lambdarho}left(partial_{mu}g_{nurho}+partial_{nu}g_{murho}-partial_{rho}g_{munu}right)
$$

I tought about applying the covariant derivative to the basis vectors: $nabla_{mu}e_{rho}=partial_{mu}e_{rho}-{Gamma}^{lambda}_{murho}e_{lambda}$ but I was not able to show the left part should vanish. What Should I do to show the equivalence between those definitions?