Let $M$ be a manifold. To specify locally a covariant derivative operator $\nabla$ it suffices to fix a local chart $\{x_i\}$ and provide the functions $\Gamma_{ij}^k$ such that
$$ \nabla_{\partial_{x_i}}\partial_{x_j}=\Gamma_{ij}^k \partial_{x_k} $$which are called the Christoffel symbols (of second kind) when the covariant derivative is coming from the Levi-Civita connection of a Riemannian metric! They correspond to the vector bundle connection#Connection form.
The formula to compute the Christoffel symbols from the Riemannian metric can be deduced from the defining conditions 1. and 2. of the Levi-Civita connection (see, for example, wikipedia or this video giving rise to
$$ \Gamma_{ij}^k = \frac{1}{2} g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}) $$In 2 dimensions:
For $k=1$:
1) $\Gamma_{11}^1 =\sum_{l=1}^2 \frac{1}{2} g^{1l}(\partial_1 g_{1l} + \partial_1 g_{1l} - \partial_l g_{11}) = \frac{1}{2} g^{11}\partial_1 g_{11} + \frac{1}{2} g^{12}(2 \partial_1 g_{21} - \partial_2 g_{11})$
2) $\Gamma_{12}^1 =\sum_{l=1}^2 \frac{1}{2} g^{1l}(\partial_1 g_{2l} + \partial_2 g_{1l} - \partial_l g_{12}) = \frac{1}{2} g^{11}\partial_2 g_{11} + \frac{1}{2} g^{12}\partial_1 g_{22}$
3) $\Gamma_{21}^1 =\sum_{l=1}^2 \frac{1}{2} g^{1l}(\partial_2 g_{1l} + \partial_1 g_{2l} - \partial_l g_{21}) = \frac{1}{2} g^{11}\partial_2 g_{11} + \frac{1}{2} g^{12} \partial_1 g_{22}$
4) $\Gamma_{22}^1 =\sum_{l=1}^2 \frac{1}{2} g^{1l}(\partial_2 g_{2l} + \partial_2 g_{2l} - \partial_l g_{22}) = \frac{1}{2} g^{11}(2\partial_2 g_{21} - \partial_1 g_{22}) + \frac{1}{2} g^{12} \partial_2 g_{22}$
For $k=2$:
1) $\Gamma_{11}^2 = \frac{1}{2} g^{21}\partial_1 g_{11} + \frac{1}{2} g^{22}(2 \partial_1 g_{21} - \partial_2 g_{11})$
2) $\Gamma_{12}^2 = \frac{1}{2} g^{21}\partial_2 g_{11} + \frac{1}{2} g^{22}\partial_1 g_{22}$
3) $\Gamma_{21}^2 =\frac{1}{2} g^{21}\partial_2 g_{11} + \frac{1}{2} g^{22} \partial_1 g_{22}$
4) $\Gamma_{22}^2 = \frac{1}{2} g^{21}(2\partial_2 g_{21} - \partial_1 g_{22}) + \frac{1}{2} g^{22} \partial_2 g_{22}$
Since the metric tensor $g_{ij}$ is symmetric (i.e., $g_{ij} = g_{ji}$), it follows that the Christoffel symbols are symmetric in their lower indices, i.e., $\Gamma_{ij}^k = \Gamma_{ji}^k$.
I think that for other local frame $\{e_i\}$ of $TM$, the functions $\Gamma_{ij}^k$ such that
$$ \nabla_{e_{i}}e_j=\Gamma_{ij}^k e_k, $$playing the same role of Christoffel symbols.
This idea works also for a vector bundle connection on a vector bundle $E$, not only the particular case of $TM$. This case is explained here: it is called the connection form, not to be confused with the connection 1-form of a connection on a general bundle, although there is a relation explained here.
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Author of the notes: Antonio J. Pan-Collantes
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