تانسور

${\displaystyle {\hat{\mathbf{e}}}_i=\sum _{j=1}^n{\mathbf{e}}_j{R}_i^j={\mathbf{e}}_j{R}_i^j.}$

${\displaystyle {\hat{v}}^i={\left(R^{-1}\right)}_j^i{v}^j,}$

${\displaystyle {\hat{w}}_i=w_j{R}_i^j}$

${\displaystyle \mathbf{v}={\hat{v}}^i{\hat{\mathbf{e}}}_i=\left({\left(R^{-1}\right)}_j^i{v}^j\right)\left({\mathbf{e}}_k{R}_i^k\right)=\left({\left(R^{-1}\right)}_j^i{R}_i^k\right)v^j{\mathbf{e}}_k={\delta }_j^k{v}^j{\mathbf{e}}_k=v^k{\mathbf{e}}_k=v^i{\mathbf{e}}_i}$

${\displaystyle {\left(\widehat{Tv}\right)}^{i^{\prime }}={\hat{T}}_{j^{\prime }}^{i^{\prime }}{\hat{v}}^{j^{\prime }}=\left[{\left(R^{-1}\right)}_i^{i^{\prime }}{T}_j^i{R}_{j^{\prime }}^j\right]\left[{\left(R^{-1}\right)}_j^{j^{\prime }}{v}^j\right]={\left(R^{-1}\right)}_i^{i^{\prime }}(Tv{)}^i.}$

${\displaystyle {\hat{T}}_{j_1^{\prime },\dots ,j_q^{\prime }}^{i_1^{\prime },\dots ,i_p^{\prime }}={\left(R^{-1}\right)}_{i_1}^{i_1^{\prime }}\cdots {\left(R^{-1}\right)}_{i_p}^{i_p^{\prime }}}$

${\displaystyle \mathbf{f}R=\left({\mathbf{f}}_i{R}_1^i,\dots ,{\mathbf{f}}_i{R}_n^i\right)}$

${\displaystyle T(FR)=\rho \left(R^{-1}\right)T(F).}$