Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 12A / b / Solution

Contracting one index from each tensor means summing them, for example $C_{i_1\dots i_{p-1}{j}_1\dots j_{q-1}}=A_{i_1\dots i_{p-1}r}{B}_{r{j}_1\dots j_{q-1}}$. Its rank is $p+q-2$. Substitution of the transformation laws for $p=q=3$ and $R_{rs}{R}_{rt}={\delta }_{st}$ proves the tensor law.

Solved by gpt-5.6-sol high.