My text defines differentiability of [itex]f:M\rightarrow \mathbb{R}[/itex] at a point p on a manifold M as the differentiability of [itex]f\circ \phi^{-1}:\phi(V) \rightarrow \mathbb{R}[/itex](adsbygoogle = window.adsbygoogle || []).push({}); on the whole of phi(V)for any chart (U,[itex]\phi [/itex]) containing p, where V is an open neighbourhood of p contained in U.

Is this customary? Why not simply ask that [itex]f\circ \phi^{-1}:\phi(U) \rightarrow \mathbb{R}[/itex] be differentiable at [itex]\phi(p)[/itex]??

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# Definition of differentiability on a manifold

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