When we calculate the Cristofel symboles, in my course it is written the following:
$(U,\varphi)$ a map and $x^1,\dots,x^n$ local coordinates then
$$ X = \sum_i X^i\frac{\partial}{\partial x^i}$$
where $X:M\rightarrow TM:x\rightarrow X_x$ is a vector field.
What does this mean? In other words, how does one give coordinates to a vector field $X$. Indeed, $X$ as a vector field is defined on the whole of $M$ however, if I understand correctly, the function $X^i$ is as follows $X^i:U\rightarrow \mathbb{R}:x\rightarrow X^i(x)$ (with $X^i(x)$ the i coordinate of $X_x$ in the basis $\frac{\partial}{\partial x^i}$ of $T_xM$) and the vector field $\frac{\partial}{\partial x^i}$ is as follows $\frac{\partial}{\partial x^i}:U\rightarrow TM: x\rightarrow \frac{\partial}{\partial x^i}(x)$ (where $\frac{\partial}{\partial x^i}(x)$ is the i basis vector of $T_xM$)
So the left hand side is defined on the whole of $M$ and the right hand side only on $U$.