= Cartesian tensor
{c}
{title2=$T'_{i_1\cdots i_n}=R_{i_1j_1}\cdots R_{i_nj_n}T_{j_1\cdots j_n}$}
A Cartesian tensor has components in each <orthonormal basis> of a <Euclidean space>, related by the displayed rule under an orthogonal component change $R$. Each index contributes one transformation matrix. The Euclidean inner product identifies upper and lower indices in orthonormal coordinates, so the distinction between covariant and contravariant components does not change this rule. An <isotropic tensor> has unchanged components under all proper <rotations>; requiring invariance under reflections as well excludes the ordinary odd-parity cases. A <Cartesian second-rank tensor> is the matrix case, with transformation $T'=RTR^T$.
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