Ref. Schuller's lecture 7 on gravity and light (derivation starts at this timestamp).
I'm watching a lecture that introduces connection coefficients – specifically the part of the video showing for the first time how to calculate covariant derivative.
Relevant timestamp here, the calculation is being done for $\nabla_XY$ in the particular chart $(U,x)$. In short, it goes like this (let $X,Y$ be smooth vector fields on a smooth manifold $M$):
$$\nabla_XY=X^i\nabla_i(Y^m\partial_m)=X^i(\nabla_iY^m)\partial_m+X^iY^m(\nabla_i\partial_m)$$
Regarding the $\nabla_i\partial_m$ term, the lecturer says:
What is the result? It's a vector field because this ($\partial_m$) is a vector field. We act on a vector field to get a vector field. Whatever it is, we can expand it in the chart […] as $\partial_q$ and it has some coefficient function – let's call it $\Gamma^q$. But these coefficients coefficients also depend on which basis vectors (i.e. $i$ and $m$) we chose, so it becomes $\Gamma^q_{\ \ mi}$.
So first of all, when defining the covariant derivative towards the start of the lecture (see definition on the board here), the implication is that $\nabla$ takes in vector fields $A,B$ and gives back a vector field $\nabla_AB$. This is fine, but I want to confirm: is the reason why we can take $A=\partial_i$ and $B=\partial_m$ (which are effectively a chart map-dependent vector fields), as it has been done in the last term of the above equation, because we're confining our attention only to a particular chart $(U,x)$?
In other words, if we change the chart map (but not the neighborhood, i.e. $(U,y)$), $\nabla_i\partial_m$ will yield an entirely different vector field, and also $\nabla_AB$ will be entirely different, right?
Best Answer
The expression $\nabla_i\partial_m$ only makes sense in a coordinate chart $(U, (x^1, \dots, x^n))$. In different coordinates $(U, (y^1, \dots, y^n))$ we won't necessarily have $\partial_{x^i} = \partial_{y^i}$, so $\nabla_{\partial_{x^i}}\partial_{x^m}$ and $\nabla_{\partial_{y^i}}\partial_{y^m}$ will be different in general. Note however that $\nabla_XY$ is perfectly well-defined (when you change coordinates, the functions $X^i$ and $Y^j$ also change).