This problem is one of three verification exercises designed to fix some fundamental ideas

about stress and change of basis system. Please show all your algebraic steps clearly in your

verification.

We had considered the transformation of components of a vector upon a rotation of axes

about the $e_3$ axis as shown in Fig. $1.$ For a vector such as $u,$ shown in the figure, we found the

transformation rule to be

$u_i^{\text{'}}={}_{ij}{u}_j,$ with

$e_i^{\text{'}}={}_{ij}{e}_j$;

this is discussed further below in the next problem. We recall that in eq$.2b$ the rotated axes

Figure $1$: Rotated base system about $e_3$ axis.

are simply expressed via their components on the original $e_i,i=1,2,3$ axes. In class we argued

that the transformation rule for a second rank tensor, like stress, the rule would be

${}_{ij}^{\text{'}}={}_{ir}{}_{js}{}_{rs}$

since there were two base vectors to transform in $={}_{rs}{e}_r{e}_s.$ Your task here is to use our

same procedures for transforming components of vectors to verify eq$.3.$