= Solution
The <Lie derivative of a function> is differentiation along the flow, while the <Lie derivative of a vector field> compares a vector with its pullback by that flow. Their index-free expressions are
$$
\boxed{\mathcal L_X f=X(f)=df(X),\qquad \mathcal L_XY=[X,Y].}
$$
Here $X(f)$ is the <directional derivative> of the <smooth function>, and $[X,Y]$ is the <Lie bracket of vector fields>, characterized by $[X,Y](f)=X(Y(f))-Y(X(f))$. To see the second expression directly, if $\phi_t$ is the <local flow> of $X$, differentiating $(d\phi_t)^{-1}Y\circ\phi_t$ at $t=0$ gives $dY(X)-dX(Y)=[X,Y]$. This uses the <flow definition of the Lie derivative of a tensor field>.
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