Given a smooth manifold $M$ and a smooth vector field $V$ on $M$, can we extend any integral curve $\gamma$ to a maximal integral curve $\gamma'$?
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What did you try? – Arctic Char Oct 18 '20 at 12:11
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I guess if I take the union of all integral curves $\gamma_\alpha$ satisfying $\gamma_\alpha(0)=m\in M$ and $|\gamma'(0)|=1$, then it gives a maximal integral curve? I don't know whether my argument works because the definition of maximal integral curve Lee gives is an integral curve that cannot be extended to an integral curve on any larger open interval, which I don't understand. – kid111 Oct 18 '20 at 12:47