Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 6 / a

For a Cartesian change of components

x_{i}^{'} = R_{ij} x_{j}

, a rank-

n

tensor transforms according to

T_{i_{1} \dots i_{n}}^{'} = R_{i_{1} j_{1}} \dots R_{i_{n} j_{n}} T_{j_{1} \dots j_{n}},

(1)

with repeated indices summed. An isotropic tensor has unchanged components under every proper rotation matrix.

For rank two write

T^{'} = RT R^{T}

. Invariance under quarter-turns implies invariance under their squares, the coordinate half-turns

R_{x} (π) = diag (1, - 1, - 1), R_{y} (π) = diag (- 1, 1, - 1) .

(2)

Each off-diagonal entry changes sign under at least one of these, so all off-diagonal entries vanish. A quarter-turn about the third axis then interchanges the first two diagonal entries, making them equal; a quarter-turn about the first axis equates the other two. Therefore