= Solution
For a constant orthogonal Cartesian coordinate change $x'_i=R_{ij}x_j$, a <Cartesian second-rank tensor> has components satisfying
$$
\boxed{T'_{ij}(x')=R_{ik}R_{jl}T_{kl}(x).}
$$
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 $U'_i=R_{ik}U_k$.
Define $U_i=\partial T_{ij}/\partial x_j$. 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
$$
\begin{aligned}
U'_i&=\frac{\partial T'_{ij}}{\partial x'_j}
=R_{jm}R_{ik}R_{jl}\frac{\partial T_{kl}}{\partial x_m}\\
&=R_{ik}\delta_{ml}\frac{\partial T_{kl}}{\partial x_m}
=R_{ik}\frac{\partial T_{kl}}{\partial x_l}=R_{ik}U_k.
\end{aligned}
$$
The identity $R_{jm}R_{jl}=\delta_{ml}$ follows from orthogonality. Hence \b[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.
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