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I have the following question: Let $M$ be a smooth manifold and let $p \in M$. Furthermore let $X$ and $Y$ be two vector fields in a neighbourhood $U$ of $p$ and consider their flows $\varphi^{X}(t,x)$ and $\varphi^{Y}(t,x)$. Define now $s(t):= \varphi^{X}_{\sqrt{t}} \circ \varphi^{Y}_{\sqrt{t}} \circ \varphi^{X}_{-\sqrt{t}} \circ \varphi^{Y}_{-\sqrt{t}}$. How can one show that $\dot{s}(0) = [X,Y]|_{0}$ ? Or is this statement true? If yes, why? Thanks in advance.


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This is more or less a few applications of the chain rule. – Qiaochu Yuan Jun 26 '12 at 5:38
really? how? I already tried and came up with nothing. – eric Jun 26 '12 at 5:47
the derivative of $\sqrt{t}$ in zero is not defined. – eric Jun 26 '12 at 5:53
It doesn't have to be. For example, neither $1 + \sqrt{t}$ nor $1 - \sqrt{t}$ have well-defined derivatives at zero but their product $1 - t$ does. – Qiaochu Yuan Jun 26 '12 at 6:22
yes I see. So ow am I applying the chain rule here ? – eric Jun 26 '12 at 6:27

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