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 Incoming links: Divergence of a Cartesian second-rank tensor

For a constant orthogonal Cartesian coordinate change

x_{i}^{'} = R_{ij} x_{j}

T_{ij}^{'} (x^{'}) = R_{ik} R_{j l} T_{k l} (x) .

Both indices transform by the coordinate-change matrix; summation over repeated indices is understood by the Einstein summation convention. A vector would have the one-index rule

U_{i}^{'} = R_{ik} U_{k}

U_{i} = \partial T_{ij} / \partial x_{j}

. Since

x_{m} = R_{jm} x_{j}^{'}

, the chain rule gives

\partial / \partial x_{j}^{'} = R_{jm} \partial / \partial x_{m}

. The rotation coefficients are constant, so

U_{i}^{'} = \frac{\partial T _{ij}^{'}}{\partial x _{j}^{'}} = R_{jm} R_{ik} R_{j l} \frac{\partial T _{k l}}{\partial x _{m}} = R_{ik} δ_{m l} \frac{\partial T _{k l}}{\partial x _{m}} = R_{ik} \frac{\partial T _{k l}}{\partial x _{l}} = R_{ik} U_{k} .

The identity

R_{jm} R_{j l} = δ_{m l}

follows from orthogonality. Hence the contracted derivative transforms as a vector. This proves the divergence of a Cartesian second-rank tensor rule for global Cartesian rotations. In curvilinear coordinates the corresponding construction uses a covariant derivative; ordinary partial derivatives alone would not have the stated tensorial transformation law.