Paper 3, Section II, A

Vector Calculus | Part IA, 2014

(a) Show that any rank 2 tensor tijt_{i j} can be written uniquely as a sum of two rank 2 tensors sijs_{i j} and aija_{i j} where sijs_{i j} is symmetric and aija_{i j} is antisymmetric.

(b) Assume that the rank 2 tensor tijt_{i j} is invariant under any rotation about the zz-axis, as well as under a rotation of angle π\pi about any axis in the (x,y)(x, y)-plane through the origin.

(i) Show that there exist α,βR\alpha, \beta \in \mathbb{R} such that tijt_{i j} can be written as

tij=αδij+βδi3δj3.t_{i j}=\alpha \delta_{i j}+\beta \delta_{i 3} \delta_{j 3} .

(ii) Is there some proper subgroup of the rotations specified above for which the result ()(*) still holds if the invariance of tijt_{i j} is restricted to this subgroup? If so, specify the smallest such subgroup.

(c) The array of numbers dijkd_{i j k} is such that dijksijd_{i j k} s_{i j} is a vector for any symmetric matrix sijs_{i j}.

(i) By writing dijkd_{i j k} as a sum of dijksd_{i j k}^{s} and dijkad_{i j k}^{a} with dijks=djiksd_{i j k}^{s}=d_{j i k}^{s} and dijka=djikad_{i j k}^{a}=-d_{j i k}^{a}, show that dijksd_{i j k}^{s} is a rank 3 tensor. [You may assume without proof the Quotient Theorem for tensors.]

(ii) Does dijkad_{i j k}^{a} necessarily have to be a tensor? Justify your answer.

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