Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 37D / a / i / Solution

WriteBy the chain rule, ${\tilde{V}}^{\alpha }=J^{\alpha }{}_{\mu }{V}^{\mu }$, ${\tilde{W}}^{\alpha }=J^{\alpha }{}_{\mu }{W}^{\mu }$, and ${\tilde{V}}^{\beta }{\tilde{\partial }}_{\beta }=V^{\nu }{\partial }_{\nu }$. ThereforeThe last term vanishes because its first factor is antisymmetric in $\mu ,\nu$ and its second is symmetric. Thus the coordinate invariance of the Lie bracket is