Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 309 / 3 / c

Use Cartan's magic formula. The exterior derivative and interior product of a differential form giveThus $d({\iota }_X\alpha )=-2xdx-2ydy+2zdz$, and the first two components cancel:The rotation in the $xy$-plane leaves $ydx-xdy$ invariant, while the dilation in $z$ doubles the infinitesimal weight of $zdz$, providing a geometric check of this Lie derivative.

Integrating the two vector fields gives their complete flow maps:The first is a simultaneous rotation about the z-axis through angle $t$ and a dilation by $e^t$ in the $z$-direction. The second is an isotropic dilation about the origin. Both satisfy the group law for all real $t$ and have inverse given by time $-t$. In particular ${\varphi }_t$ is not a pure rotation unless restricted to the plane $z=0$.

The Jacobian determinant of ${\varphi }_t$ is $e^t$, while every coordinate and every coordinate differential is multiplied by $e^t$ under ${\psi }_t$. Hence the pullback of a differential form operation givesTo compare directly with Cartan's magic formula, note $d\mu =0$ and computeThe first two differentiated terms vanish because their coordinate differentials repeat; the last contributes $dz\wedge dx\wedge dy=\mu$. AlsoTheir sum is $2\alpha$, as required. Equivalently the divergence of $X$ relative to the standard volume form is one.