Covariant Derivative of Smooth Vector Field along Smooth Vector Field in Smooth Local Frame

Theorem
Let $M$ be a smooth manifold with or without boundary.

Let $TM$ be the tangent bundle of $M$.

Let $\nabla$ be a connection in $TM$.

Let $U \subseteq M$ be an open subset.

Suppose $\tuple {E_i}$ is a smooth local frame over $U$.

Let $\set {\Gamma^k_{ij}}$ be the connection coefficients of $\nabla$ $\tuple {E_i}$.

Let $\map {\mathfrak{X}} U$ be the space of smooth vector fields on $U$.

Let $X, Y \in \map {\mathfrak{X}} U$ be smooth vector fields.

Suppose $X$ and $Y$ read as $X = X^i E_i$ and $Y = Y^i E_i$ $\tuple {E_i}$.

Then:


 * $\ds \nabla_X Y = \paren {\map X {Y^k} + X^i Y^j \Gamma^k_{ij}} E_k$