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- Jan 26, 2012

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**Problem**: Let $M$ be a manifold. Let $\alpha$ be a $k$-form on $M$ and let $X$ and $Y$ be vector fields on $M$.

(a) Prove that $L_X(\iota_Y\alpha)=\iota_Y(L_X\alpha) + \iota_{(L_X Y)}\alpha$.

(b) Prove that $L_{[X,Y]}\alpha=0$ whenever $L_X\alpha=0$ and $L_Y\alpha=0$.

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Here, $\iota$ is your inclusion map and $[X,Y]$ is your standard Lie bracket.

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