I was given the $\mathbb{R^2}$ metric in polar coordinates, as follows:
$$ ds^2=dr^2+r^2d\theta^2. $$
In this context we denote $e_1=\partial_r=(\cos(\theta), \sin(\theta))$, $e_2=\partial_{\theta}=(-r\sin(\theta), r\cos(\theta))$. 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: $$ \tilde{\Gamma}^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\rho}\left(\partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\mu\rho}-\partial_{\rho}g_{\mu\nu}\right). $$
I thought about applying the covariant derivative to the basis vectors: $\nabla_{\mu}e_{\rho}=\partial_{\mu}e_{\rho}-{\Gamma}^{\lambda}_{\mu\rho}e_{\lambda}$, but I was not able to show the left part should vanish. What should I do to show the equivalence between those definitions?