I have a small bit of confusion with the definition my text is providing me with for a tangent vector.
Given a manifold $M$, it is first stated that to define a tangent vector, a curve $c:(a,b) \rightarrow M$, $a < 0 < b$, and a smooth function $f : M \rightarrow \mathbb{R}$ are needed. With $c$ and $f$, the directional derivative of a function $f(c(t))$ along $c(t)$ at $t = 0$ is $$\frac{\text{d}f(c(t))}{\text{d}t} \bigg|_{t = 0}.$$ Given a chart $(U,\phi)$ such that $c(0) = p \in U$, the directional derivative in terms of local coordinates becomes $$\left(\frac{\partial f\phi^{-1}(x)}{\partial x^\mu}\right) \left(\frac{\text{d}x^\mu(c(t))}{\text{d}t}\right)\Bigg|_{t=0}.$$
The text then states that $\text{d}f(c(t))/\text{d}t|_{t=0}$ is obtained by applying $X$ to $f$ where $$X = X^\mu \left(\frac{\partial}{\partial x^\mu}\right), \quad X^\mu = \frac{\text{d}x^\mu(c(t))}{\text{d}t}\Bigg|_{t=0}.$$ After which it defines $X$ as the tangent vector to $M$ at $p = c(0).$
The bit I would like clarity on is $\mu$. I know that if $M$ is an $m$ dimensional manifold, then the local coordinate of $p$ is $\phi(p) = x = (x^1,\dots,x^m)$ so $1 \leq \mu \leq m$. Is $X$ really $$X = \left(X^1 \left(\frac{\partial}{\partial x^1}\right),\dots,X^m \left(\frac{\partial}{\partial x^m}\right)\right),$$ and $X$ as defined is a shorthand way of writing that? The only conflict I really have with this idea though is that $f$ maps to $\mathbb{R}$ and so $$\frac{\text{d}f(c(t))}{\text{d}t} \bigg|_{t = 0}$$ should only be mapping to a single value. So is a single $\mu$ chosen? Would anyone be able to give me some clarity on this.
Here the Einstein summation rule is with respect to the index $\mu$ used: $$ X = X^{\mu} \frac{\partial}{\partial x^{\mu}} = X^1 \frac{\partial}{\partial x^1} + \dots + X^n \frac{\partial}{\partial x^n} $$ or, in other words, $$ X = \left( X^1,\dots, X^n \right) $$ in the coordinate system $(U,\phi)$. The $\frac{\partial}{\partial x^{\mu}}$ are the coordinate vectors at point $p$, where $\mu = 1,\dots,n$.