Tensor Analysis: Solving Isotropic Second-Rank Tensors in 3-D Space

[s8wardwe]

Homework Statement

I am having trouble solving this problem. From an analysis of the behavior of a general second-rank tensor under 90 degree and 180 degree rotations about the coordinate axes, show that an isotropic second-rank tensor in 3-D space must be a multiplier of delta ij.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Well, what does "isotropic" mean here? And how does a tensor change under rotations?
For this problem you don't really need to look at general rotations: rotation by 90 degrees changes x to y and y to -x. Rotation by 180 degrees changes x to -x and y to -y.