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When we define a differential map on $x \in M$ as an aplication between differential manifolds $f : M \to N$ such that $d(\psi \circ f \circ \phi^{-1})_{\phi(x)} : \mathbb{R}^m \to \mathbb{R}^n$ is differential in usual sense.

My question is if we can remove $\phi(x)$ in the definition of $d(\psi \circ f \circ \phi^{-1})_{\phi(x)}$ because you can pass directly to $d(\psi \circ f)(x)$, yeah? Or is it simple notation and you ever must write it?

Thanks for advance!

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The lower $\phi(x)$ is a subscript, and it is not part of a composition against the "neighbor" $\phi^{-1}$ that may cancel. It is a location marker, telling you where you are taking the real derivative.

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