= Flow definition of the Lie derivative of a tensor field
{title2=$\mathcal L_XT=\left.\frac d{dt}\right|_0\phi_t^*T$}
For the <local flow> $\phi_t$ of a smooth <vector field> $X$, pull a <tensor field> $T$ back to the fixed base point before differentiating. Pullback uses $d\phi_t^{-1}$ on vector factors and the dual of $d\phi_t$ on covector factors. The <chain rule> gives $\mathcal L_Xf=Xf$. In coordinates, $d\phi_t^{-1}Y(\phi_t(x))=Y+t(dY\,X-dX\,Y)+O(t^2)$, so $\mathcal L_XY=[X,Y]$. Differentiation of pullback preserves <tensor contractions> and obeys the <Leibniz rule> for <tensor products>; hence this is the <tensor derivation> defining the <Lie derivative of a tensor field>.
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