Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 37D (General Relativity)

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

For every smooth function $f$, equality of mixed partial derivatives givesConsequently the commuting coordinate basis property is

Write $V={\sum }_a{f}_a{Z}^{(a)}$ and $W={\sum }_b{g}_b{Z}^{(b)}$. Relation (2) reduces $[V,W]$ to brackets of individual summands. Applying relations (3), (1), and (3), in that order, givesThe middle term vanishes because the $Z^{(a)}$ commute, and every remaining term lies in their smooth span. ThereforeThis is the involutivity condition in the Frobenius theorem.

For plane polar coordinates, $x=r\cos \theta$ and $y=r\sin \theta$. The chain rule givesTheir Euclidean norms are $\Vert {\partial }_r\Vert =1$ and $\Vert {\partial }_{\theta }\Vert =r$ for $r>0$.

The unit fields are $\hat{r}={\partial }_r$ and $\hat{\theta }=r^{-1}{\partial }_{\theta }$. Relation (3) and $[{\partial }_r,{\partial }_{\theta }]=0$ giveA coordinate basis must commute, soThis is the noncommuting orthonormal polar frame.