Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 1 / a / ii

Apply the definition to the coordinate basis, for which $[{\partial }_{\mu },{\partial }_{\nu }]=0$ and ${\nabla }_{\mu }{\partial }_{\tau }={\Gamma }^{\rho }{}_{\tau \mu }{\partial }_{\rho }$. Comparing coefficients givesequivalent to the paper's ordering after commuting scalar factors. Although the individual Christoffel symbols are not tensors, the preceding intrinsic definition proves that this complete combination is tensorial.