Hi, I have some trouble with Theorem 2.2.1 in Wald's GR book p.15.(adsbygoogle = window.adsbygoogle || []).push({});

He derived formula 2.2.5 by using 2.2.4. Here, the ##f## in ##v(f)## is a map ##f:M\rightarrow\mathbb R##, but 2.2.4 is the expression for ##f## only in the domain ##O\subseteq M## and we don't know the expression for ##f## outside ##O##. So how can 2.2.5 be valid. ##v## is a map ##v:\mathcal F\rightarrow\mathbb R##, but ##x^{\mu}\circ\psi## is a map:##O\rightarrow\mathbb R## which doesn't belong to##\mathcal F##, so ##v(x^{\mu}\circ\psi)## in 2.2.5 & 2.2.7 doesn't make sense.

Is there anything assumed in advance by the author that make these wrong expressions in the formulas become meaningful?

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# Wald Theorem 2.2.1

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