Vector Calculus | Part IA, 2008

Find the effect of a rotation by π/2\pi / 2 about the zz-axis on the tensor

(S11S12S13S21S22S23S31S32S33)\left(\begin{array}{lll} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{array}\right)

Hence show that the most general isotropic tensor of rank 2 is λδij\lambda \delta_{i j}, where λ\lambda is an arbitrary scalar.

Prove that there is no non-zero isotropic vector, and write down without proof the most general isotropic tensor of rank 3 .

Deduce that if TijklT_{i j k l} is an isotropic tensor then the following results hold, for some scalars μ\mu and ν\nu : (i) ϵijkTijkl=0\epsilon_{i j k} T_{i j k l}=0; (ii) δijTijkl=μδkl\delta_{i j} T_{i j k l}=\mu \delta_{k l}; (iii) ϵijmTijkl=νϵklm\epsilon_{i j m} T_{i j k l}=\nu \epsilon_{k l m}.

Verify these three results in the case Tijkl=αδijδkl+βδikδjl+γδilδjkT_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}, expressing μ\mu and ν\nu in terms of α,β\alpha, \beta and γ\gamma.

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