Let $M$ be a $n$-dimensional manifold. We want to deduce a basis of $T_xM$ for $x\in M$. for $x\in M$ we can find an open neighbourhood $U$ such that $x\in U$ and an homeomorphism onto an open set $\varphi(U)$ in $\Bbb{R}^n$, thus $\varphi:U\rightarrow\varphi(U)\subseteq\Bbb{R}^n$. Now let $(e_1,\ldots,e_n)$ be the standard basis of $\mathbb{R}$, then we can define a vector $\frac{\partial}{\partial x_i}:=[\varphi^{-1}(\varphi(x)+te_i)]$. Here $[\cdot]$ denotes the equivalenceclass of a curve in $M$ (with the well-known equivalence relation). We have also a differential $d$ for function $f:M\rightarrow N$ which is defined by $df([\gamma])=[f\circ\gamma]$. Therefore, since $\varphi:M\rightarrow\mathbb{R}$:
$$\frac{\partial}{\partial x_i}:=[\varphi^{-1}(\varphi(x)+te_i)]=d\varphi^{-1}[\varphi(x)+te_i]$$
Now i read that $d\varphi^{-1}[\varphi(x)+te_i]=d\varphi^{-1}(e_i)$ and here is the confusion, because $e_i$ is no curve in $\Bbb{R}^n$ and $\varphi^{-1}:\varphi(U)\rightarrow U\subseteq M$. How to interpretate this? If this is really an equality and not only abuse of notation please help me to understand this. Thank you very much.
