Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 37D / a / Solution

In coordinates $x^{\mu }$, write the type-$(1,1)$ tensor asFor a covector $\eta ={\eta }_{\mu }d{x}^{\mu }$ and vector $V=V^{\nu }{\partial }_{\nu }$,Since this must equal $\eta (V)={\eta }_{\nu }{V}^{\nu }$ for every $\eta$ and $V$, the components areThus $T$ is the identity tensor.

The tensor component transformation law givesThe components are therefore the same Kronecker delta in every coordinate system.