= Cartan's magic formula
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{title2=$\mathcal L_X=d\iota_X+\iota_Xd$}
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= Cartan formula for the Lie derivative
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The <Lie derivative of a differential form> satisfies $\mathcal L_X=d\iota_X+\iota_Xd$. Both sides agree on <functions>. On a <differential 1-form> $\eta$, evaluation at a <vector field> $Y$ gives $X(\eta(Y))-\eta([X,Y])$ on both sides, by the definition of the <exterior derivative> and the <Lie bracket of vector fields>. Both operators are degree-zero <derivations of an algebra> on the <exterior algebra>. Agreement on <functions> and coordinate <differential 1-forms> therefore proves agreement on every <differential form>. Since $d^2=0$, the formula also gives $\mathcal L_Xd=d\mathcal L_X$, so the <Lie derivative of a differential form> commutes with the <exterior derivative>.
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