I'm studying general relativity and am running into an ambiguity with the covariant derivative. The covariant derivative acting on a scalar is, in a co-ordinate basis, simply $$\nabla_X f = X^a \nabla_a f = X^a \partial_a f $$ whilst when acting on vectors or tensors, the covariant derivative includes extra terms involving the connection coefficients $\Gamma$. My problem is that I'm not sure what happens when the covariant derivative acts on the components of a vector. So the $X^a$ in the above expression are each scalar functions, and so when we write $\nabla_Y X^a$, do we assume the covariant derivative acts on $X^a$ as scalars, or do we include connection coefficient terms?
To clarify, consider $$ \nabla_X Y = X^a \nabla_a(Y^b e_b) = X^a (\nabla_a Y^b) e_b + X^a Y^b (\nabla_a e_b) = X^a (\partial_a Y^b) e_b + X^a Y^b \Gamma^c{}_{ba} e_c $$ As you can see, when the covariant derivative acts on the $Y^b$, it acts as a simple partial derivative, since the $Y^b$ are here considered functions premultiplying the vectors $e_b$.
However, in this equation from this page: $$(\nabla_X \nabla_Y Z)^a = X^c \nabla_c Y^b \nabla_b Z^a = X^c Y^b \nabla_c \nabla_b Z^a + (X^c \nabla_c Y^b) \nabla_b Z^a = (\nabla^2_{X,Y} Z)^a + (\nabla_{\nabla_X Y} Z)^a$$ When the covariant derivative acts on the $Y^b$ term it treats it like a vector.
What's going on here? Thanks.
I think that the confusion in such a matter is generated by a slight misuse of notation that is usually made by physicists (such as myself).
The correct action of the covariant derivative, say, on a vector $A=A^a e_a$, for a coordinate basis $e_a=\partial_a$, is of course $$ \nabla_bA\equiv \nabla_{b}\left(A^a e_a\right)=\left(\nabla_bA^a\right)e_a+A^a\nabla_be_a =\left(\partial_bA^a+\Gamma^a_{bc}A^c\right)e_a, $$ where $$ \nabla_bA^a=\partial_bA^a $$ and the definition of connection $$ \nabla_ae_b=\Gamma^c_{ab}e_c $$ have been used.
However, since the $a$-th component of the covariant derivative $\nabla_b$ of a vector of components $A^a$ is $\partial_bA^a+\Gamma^a_{bc}A^c$, physics textbooks tend to use the improper notation $$ \nabla_bA^a=\partial_bA^a+\Gamma^a_{bc}A^c $$ in the calculations, with the convention that this is in fact a shorthand for the above computation. In practice, this writing leaves the basis vectors ''implicit''.