Methods | Part IB, 2001

Explain what is meant by an isotropic tensor.

Show that the fourth-rank tensor

Aijkl=αδijδkl+βδikδjl+γδilδjkA_{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}

is isotropic for arbitrary scalars α,β\alpha, \beta and γ\gamma.

Assuming that the most general isotropic tensor of rank 4 has the form ()(*), or otherwise, evaluate

Bijkl=r<axixj2xkxl(1r)dVB_{i j k l}=\int_{r<a} x_{i} x_{j} \frac{\partial^{2}}{\partial x_{k} \partial x_{l}}\left(\frac{1}{r}\right) d V

where x\mathbf{x} is the position vector and r=xr=|\mathbf{x}|.

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