 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/6/a/solution

For a Cartesian coordinate change

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

with

R

n

tensor obeys

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

with repeated indices summed according to Einstein notation. A tensor is isotropic if

T_{i_{1} \dots i_{n}}^{'} = T_{i_{1} \dots i_{n}}

for every proper rotation matrix

R

. Under proper rotations the Levi-Civita symbol is an isotropic third-rank tensor; if reflections are included, it is a pseudotensor. The convention using proper rotations is the one relevant to the rank-three classification here.

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