- #1
Euge
Gold Member
MHB
POTW Director
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Here is this week's POTW:
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Let $M$ be a smooth manifold, $X, Y$ smooth vector fields on $M$, and $\phi_t$ the flow of $X$. The Lie derivative of $Y$ along $X$, $\mathcal{L}_XY$, is given by
$$\mathcal{L}_XY:= \frac{d}{dt}\bigg|_{t=0} \phi_{-t*}Y.$$
Show that $\mathcal{L}_XY$ is equal to the Lie bracket field $[X,Y]$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
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Let $M$ be a smooth manifold, $X, Y$ smooth vector fields on $M$, and $\phi_t$ the flow of $X$. The Lie derivative of $Y$ along $X$, $\mathcal{L}_XY$, is given by
$$\mathcal{L}_XY:= \frac{d}{dt}\bigg|_{t=0} \phi_{-t*}Y.$$
Show that $\mathcal{L}_XY$ is equal to the Lie bracket field $[X,Y]$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
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