coordinates of a vector field

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 $U$.

• You have a local basis for the tangent vectors given at a point $x$ by the $\partial_i$'s, of course these tangent vectors are only defined on a neighborhood $U$ of $x$. If $X$ is global vector field on $M$ you can restrictit it to $U$ and you get the fact that for every $u$ in $U$, $X(u)=\sum f_i(u)\partial_i$ with the $f_i$ smooth. The left hand side should be labelled $X_{|U}$ but we never do that, especially since we could do that at every point $x\in X$ (of course we would get different $\partial_i$'s). – A.Rod May 18 '17 at 12:22
• Yes, the equality only holds on $U$. – mathematician May 18 '17 at 12:22
• Correct: When writes a decomposition of a vector field w.r.t. a local frame, the decomposition, by definition, only applies on the domain $U$ of that frame. Strictly speaking one should write $X\vert_U = \cdots$. – Travis May 18 '17 at 12:23
• Ok. So when we write (to find Christofel symbols) $\nabla_XY = \nabla_{\sum_iX^i\partial_{x^i}}\sum_iX^i\partial_{x^i}$. It should actually be $\nabla_{X\vert_U}Y\vert_U$ (which is ok since we know that a connexion is local – tomak May 18 '17 at 12:45