 Lie derivative of a differential form

Let $X$ be a smooth vector field with local flow ${\varphi }_t$. The Lie derivative of a differential form isPulling 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: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 ${\varphi }_t(x)=e^{t}x$. Hence ${\varphi }_t^{\ast }dx=e^{t}dx$ and ${\mathcal{L}}_{X}dx=dx$. 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.