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 V be the sphere 0⩽r⩽a. Explain briefly why
Ti1…in=∫Vxi1…xin dV
is an isotropic tensor for any n. Hence show that
∫Vxixj dV=αδij,∫Vxixjxk dV=0 and ∫Vxixjxkxl dV=β(δijδkl+δikδjl+δilδjk)
for some scalars α and β, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of
∫Vx×(Ω×x)dV
where Ω is a constant vector.
[You may assume that the most general isotropic tensor of rank 4 is
λδijδkl+μδikδjl+νδilδjk
where λ,μ and ν are scalars.]