Paper 3, Section II, C

Vector Calculus | Part IA, 2011

Write down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic.

Let VV be the sphere 0ra0 \leqslant r \leqslant a. Explain briefly why

Ti1in=Vxi1xin dVT_{i_{1} \ldots i_{n}}=\int_{V} x_{i_{1}} \ldots x_{i_{n}} \mathrm{~d} V

is an isotropic tensor for any nn. Hence show that

Vxixj dV=αδij,Vxixjxk dV=0 and Vxixjxkxl dV=β(δijδkl+δikδjl+δilδjk)\int_{V} x_{i} x_{j} \mathrm{~d} V=\alpha \delta_{i j}, \quad \int_{V} x_{i} x_{j} x_{k} \mathrm{~d} V=0 \quad \text { and } \int_{V} x_{i} x_{j} x_{k} x_{l} \mathrm{~d} V=\beta\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right)

for some scalars α\alpha and β\beta, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of

Vx×(Ω×x)dV\int_{V} \mathbf{x} \times(\boldsymbol{\Omega} \times \mathbf{x}) \mathrm{d} V

where Ω\boldsymbol{\Omega} is a constant vector.

[You may assume that the most general isotropic tensor of rank 4 is

λδijδkl+μδikδjl+νδilδjk\lambda \delta_{i j} \delta_{k l}+\mu \delta_{i k} \delta_{j l}+\nu \delta_{i l} \delta_{j k}

where λ,μ\lambda, \mu and ν\nu are scalars.]

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