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I am going through Lee's smooth manifold book. I thought I understood the coordinate basis for a tangent space properly, but I am a bit confused now as Lee doesn't exactly make it clear how he constructs the coordinate basis. Basically, Lee starts out by saying that given any point $a \in \mathbb{R}^n$, there is a standard basis for its tangent space, namely the set of vectors of the form $\frac{\partial}{\partial x^i} \vert_a$ where the $x^i$ denote the standard coordinate functions, so that the basis vectors just mentioned are actually the partial derivative operators. So then given a manifold $M$ and a point $p \in M$ and a smooth chart $(U, \varphi)$ containing $p$, we can just use the differential of $\varphi^{-1}$ to pushforward the standard basis for $T_{\varphi(p)} \mathbb{R}^n$ to obtain a basis for $T_pM$. BUT, in later passages, Lee seems to be saying that we use the differential of $\varphi^{-1}$ to pushforward the vectors $\frac{\partial}{\partial x^i}\vert_{\varphi(p)}$ where the $x^i$ are the local coordinate functions of $\varphi$. But the coordinate representation of the $x^i$ generally will NOT be the standard coordinate functions on $\mathbb{R}^n$.

So when we push forward $\frac{\partial}{\partial x^i}\vert_{\varphi(p)}$, do the $x^i$ represent standard coordinates or the local coordinate functions of $\varphi$?

I hope my question is clear.

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    Your interpretation is correct. Note that the coordinate representation of $x^i$ (with respect to $\varphi$) will be by definition the standard coordinates on $\mathbb{R}^n$ (which, by abuse of notation, are also denoted by $x^i$). – 2017-02-26
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    How do I see that the coordinate representation of $x^i$ with respect to $\varphi$ are the standard coordinates on $\mathbb{R}^n$? – 2017-02-26
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    The coordinate chart is a function $\varphi \colon U \rightarrow \mathbb{R}^n$ written as $\varphi(p) = (x^1(p), \dots, x^n(p))$ where $x^i \colon U \rightarrow \mathbb{R}$ are the associated coordinate functions on $U$. Denote by $\pi^i \colon \mathbb{R}^n \rightarrow \mathbb{R}$ the standard coordinate functions on $\mathbb{R}^n$ (so that $\pi^i(p_1,\dots,p_n) = p_i$). Then by definition $\pi^i \circ \varphi = x^i$ which means that $x^i \circ \varphi^{-1} = \pi^i$. The left hand side is by definition the coordinate representation of $x^i \colon U \rightarrow \mathbb{R}$ with respect to – 2017-02-26
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    the coordinate system $\varphi$. – 2017-02-26

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