= Lie derivative of a differential form
{c}
{title2=$\mathcal L_X$}
= Lie derivative
{c}
{synonym}
Let $X$ be a smooth <vector field> with <local flow> $\phi_t$. The <Lie derivative of a differential form> is
$$
\mathcal L_X\alpha=\left.\frac{d}{dt}\right|_{t=0}\phi_t^*\alpha.
$$
Pulling back before taking the <derivative> compares forms in the same <cotangent space>. The <Lie derivative of a differential form> preserves degree and obeys the <Leibniz rule> for the <wedge product of differential forms>.
<Cartan's magic formula> expresses it using two intrinsic operations:
$$
\mathcal L_X\alpha=d(\iota_X\alpha)+\iota_X(d\alpha),
$$
where $d$ is the <exterior derivative> and $\iota_X$ is the <interior product>. In particular, it commutes with the <exterior derivative>.
On the real line, $X=x\partial_x$ has <local flow> $\phi_t(x)=e^t x$. Hence $\phi_t^*dx=e^t dx$ and $\mathcal L_Xdx=dx$. The <vector field> vanishes at zero, but its differential still stretches the <one-form> there. For a <volume form> $\omega$, the equation $\mathcal L_X\omega=0$ defines a <volume-preserving vector field>.
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