A <local flow> of $v$ is a smooth family $\Phi^t$ defined near $\{0\}\times X$ such that $\Phi^0=\operatorname{id}$, $\partial_t\Phi^t(x)=v(\Phi^t(x))$, and $\Phi^{s+t}=\Phi^s\circ\Phi^t$ whenever defined. The <Lie derivative of a differential form> is
$$
\mathcal L_v\alpha=\left.\frac d{dt}\right|_{t=0}(\Phi^t)^*\alpha.
$$
<Cartan's magic formula> is
$$
\boxed{\mathcal L_v\alpha=d(\iota_v\alpha)+\iota_v(d\alpha).}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-115/3/a/solution