Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 37A / c / Solution

Under a coordinate change, the inhomogeneous second-derivative terms arising from each partial derivative cancel between $V^{\alpha }{\partial }_{\alpha }{W}^{\mu }$ and $W^{\alpha }{\partial }_{\alpha }{V}^{\mu }$. The remainder transforms with one contravariant Jacobian, so the commutator is a vector field.