I want to use integral to prove the cartan's magic formula, i.e., it's enough to prove that for all small disk $D$ of dim=k=deg $\alpha$ in a manifold $M$, we have $$ \int_D L_X\alpha = \int_D (d(i_X\alpha)+i_X d\alpha. $$ To prove that, we consider a flow of X $\phi: D\times [0,\epsilon]\to M$, then $$ \int_D L_X\alpha= \int_D \frac{d}{dt}|_{t=0}\phi^*_t\alpha=\lim_{\epsilon\to0}(1/\epsilon)\int_D(\phi^*_{\epsilon}\alpha-\alpha). $$ What can we do to continue?
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Hint: Try to use Stokes's Theorem on $D\times [0,\epsilon]$ to compute that derivative by the definition. – Ted Shifrin Feb 07 '15 at 04:19