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Let $M$ be a Hausdorff manifold. I'm trying to prove that a vector field $Y:M\to TM$ is smooth if and only if the derivation induced by $Y$ for all globablly defined smooth functions is smooth. That is, $Yf:M\to \mathbb{R}$ is smooth for all $f\in C_M^\infty$. In Lee, there is a lemma saying $Y$ is smooth iff for every open set $U\subset M$ and every $f\in C_U^\infty$, the function $Yf:U\to \mathbb{R}$ is smooth. One direction of the proof involves taking coordinate functions of a chart of $U$. I'm not sure how to do this for the global case.

D. Huang
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    Sounds as if you're looking at the first edition of my book. In the second edition, there's a proof of this fact (Prop. 8.14). The basic idea is that if $f$ smooth on $M$ $\implies$ $Yf$ smooth on $M$, you can use bump functions to apply this locally to a smooth function defined only on an open subset. – Jack Lee Nov 04 '13 at 15:51
  • Thank you very much! I figured out how to solve it using bump functions. – D. Huang Nov 05 '13 at 03:46

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