Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 115 / 3 / a / Solution

A local flow of $v$ is a smooth family ${\Phi }^t$ defined near $\{0\}\times X$ such that ${\Phi }^0=\mathrm{id}$, ${\partial }_t{\Phi }^t(x)=v({\Phi }^t(x))$, and ${\Phi }^{s+t}={\Phi }^s\circ {\Phi }^t$ whenever defined. The Lie derivative of a differential form isCartan's magic formula is