Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 3 / 12C / b

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- Solution b

Solution

In three dimensions, the most general isotropic tensors of ranks zero through three are

T = α, T_{i} = 0, T_{ij} = α δ_{ij}, T_{ijk} = α ε_{ijk} .

The scalar and zero vector are unchanged by rotations. For every

R \in SO (3)

,

R_{ia} R_{jb} δ_{ab} = δ_{ij}

and

R_{ia} R_{jb} R_{k c} ε_{ab c} = (det R) ε_{ijk} = ε_{ijk}

, proving isotropy.

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