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In Nakahara's book, this example is given:

Example 5.11: If X is applied to the coordinate functions ##\phi : M \to R^n## [that come with the manifold] along a curve ##c: R \to M## such that ##\phi (c(t))=x^ \mu (t)##, we have $$X (x ^\mu) = (\frac {dx^\mu}{dt}) (\frac{\partial x ^ \mu}{\partial x^ \nu}) = \frac {dx^\mu (t)}{dt}$$

The thing I don't understand is why Nakahara uses the letter ##x## in both the basis of ##T_pM## and for the coordinate functions (by him saying that ## \phi (c(t))=x^ \mu (t) ## ). Is it implied that he

*chose*the basis of the tangent space at p as the basis of coordinate chart? The simplification ##\delta ^\mu _\nu = \frac{\partial x ^ \mu}{\partial x^ \nu} ## is only possible because of this. And doesn't him saying that ##\phi (c(t))=x^ \mu (t) ## defines ##x## to be $$x = \phi \circ c$$ but it is not clear why $$\frac{\partial}{\partial x^\nu} (\phi \circ c) ^\mu= \delta ^\mu _\nu$$.

My point is that ##x## already has a meaning because it is used to write the basis of the tangent space at p. How can we stretch its meaning and say it has anything to do with the coordinate charts or a curve?