On Christoffel symbol and vector fields Take the defining equation of Christoffel symbols: $$\nabla_{\frac{\partial}{\partial x^\mu}}\frac{\partial}{\partial x^\nu}=\Gamma^{\sigma}_{\mu\nu}\frac{\partial}{\partial x^{\sigma}}$$ Both sides of the above definition are vector fields, in fact, the right side being a linear combination of coordinate vector fields with the coefficients of the combination being precisely the Christoffel symbols $\Gamma^{\sigma}_{\mu\nu}$ that do not transform as tensor. The above fact motivates my following question: If one has a vector field $X$ written out in a chart $x^\mu$ as $X=X^{\mu}\frac{\partial}{\partial x^\mu}$, is it the case that the smooth functions $X^\mu$ on the manifold should always transform as a vector? The right side of the definition above for Christoffel symbols suggest that this claim is not true.
 A: If you have some $\{f^\mu\}\subset C^\infty(U)$ where $U$ is some coordinate domain, then $$\tag{$1$}X=f^\mu \partial_\mu$$ is indeed a vector field in $U$. If with respect to some other coordinate system, we have
$$X=g^{\mu'}\partial_{\mu'},$$
then we will of course have $$\tag{$2$}g^{\mu'}=\frac{\partial x^{\mu'}}{\partial x^\mu} f^\mu.$$
But if $\{f^\mu\}$ has a transformation law other than (2), it does not have to be that 
$$g^{\mu'}=f^{\mu'}.$$
So basically, if you define a vector field by (1) and then transform, you have to forget whatever auxiliary transformation law the multiplet $\{f^\mu\}$ has. 
A: A geometric object can be defined by its local expression in a particular coordinate system combined with rules governing coordinate change.
So if you treated the lower indices as mere labels, for each choice $n,k$ of $\nu, \kappa$ you could take the right-hand side as the definition of a local vector field
$$
Y_{nk} = \Gamma^{\mu}_{nk}\frac{\partial}{\partial x^{\mu}}
$$
However, these fields would need to transform under coordinate change according to
$$
{\tilde Y}^\mu_{nk} = \frac{\partial {\tilde x}^\mu}{\partial x^{\alpha}} \Gamma^{\alpha}_{nk}
$$
which in general no longer agrees with the Christoffel symbols expressed in the new coordinates, which are given by
$$
\tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over \partial \tilde x^\nu}{\partial x^\gamma \over \partial \tilde x^\kappa} + {\partial ^2 x^\alpha \over \partial \tilde x^\nu \partial \tilde x^\kappa} \right ]
$$
intstead.
Because the rules governing coordinate change are different, you're dealing with different geometric objects, whose coordinate expression may just coincidentally agree in a particular coordinate system.
