91Ó°ÊÓ

Two isotropic tensor${\mathrm{Ï\mu }}_{ijk}{\mathrm{Ï\mu }}_{lmn}$ and${\mathrm{Ï\mu }}_{ijk}{\mathrm{Ï\mu }}_{lmn}{\mathrm{δ}}_{jn}$

Isotropic tensor is defined as a tensor with components that remain unaltered when the coordinate system is rotated arbitrarily.

Consider two isometric tensors A and B of rank n and mrespectively.

Find the direct product.

${(A⊗B)}_{i_1…i_n{j}_1…j_m}=A_{i_1…i_n}{B}_{j_1…j_m}$

Consider the following system:

$\begin{array}{c}(A⊗B{)}_{i_1^{\text{'}}…i_n^{\text{'}}{j}_1^{\text{'}}…j_m^{\text{'}}}=A_{i_1^{\text{'}}…i_n^{\text{'}}}{B}_{j_1^{\text{'}}…j_m^{\text{'}}}\\ =A_{i_1…i_n}{B}_{j_1…j_m}\\ =(A⊗B{)}_{i_1…i_n{j}_1…j_m}\end{array}$

Hence $A⊗B$is an isomorphic tensor.

Assume that T is an arbitrary n-rank isotropic tensor. By induction, show that contraction of T maintains the isotropic quality, then contraction of the direct product of an arbitrary number of isotropic tensors will similarly preserve it.

Consider the contracted tensor Tin p-th and q-th index by$C_q^{p}T$.

Write its components below:

${\left(C_q^{p}T\right)}_{i_1…i_{p-1}{i}_{p+1}…i_{q-1}{i}_{q+1}…i_n}=T_{i_1…i_{p-1}\mathrm{Î\pm }{i}_{p+1}…i_{q-1}\mathrm{Î\pm }{i}_{q+1}…i_n}$

Therefore, consider the following expression and expand.

$\begin{array}{c}{\left(C_q^{p}T\right)}_{i_1^{\text{'}}…i_{p-1}{i}_{p+1}^{\text{'}}…i_{q-1}^{\text{'}}{i}_{q+1}^{\text{'}}…i_n^{\text{'}}}=T_{i_1^{\text{'}}…i_{p-1}^{\text{'}}\mathrm{Î\pm }{i}_{p+1}^{\text{'}}…i_{q-1}^{\text{'}}\mathrm{Î\pm }{i}_{q+1}^{\text{'}}…i_n^{\text{'}}}\\ =T_{i_1…i_{p-1}\mathrm{Î\pm }{i}_{p+1}…i_{q-1}\mathrm{Î\pm }{i}_{q+1}…i_n}\\ ={\left(C_q^{p}T\right)}_{i_1…i_{p-1}{i}_{p+1}…i_{q-1}{i}_{q+1}…i_n}\end{array}$

Hence $C_q^{p}T$is an isomorphic tensor.

Rank of${\mathrm{Ï\mu }}_{ijk}{\mathrm{Ï\mu }}_{lmn}$is .$3+3=6$

Rank of ${\mathrm{Ï\mu }}_{ijk}{\mathrm{Ï\mu }}_{lmn}{\mathrm{δ}}_{jn}$is$3+3+2−2×2=4$