= Solution
An <objective time derivative> transforms as a physical <tensor> under a time-dependent rigid change of observer. If $x^*=Q(t)x+c(t)$, with $Q$ orthogonal and $A^*=QAQ^T$, objectivity requires
$$
\boxed{\left(\frac{\mathcal DA}{\mathcal Dt}\right)^*
=Q\frac{\mathcal DA}{\mathcal Dt}Q^T.}
$$
This means the <constitutive equation> makes the same physical prediction in translating and rotating frames, rather than requiring individual tensor components to be invariant.
The printed plus commutator uses the derivative-index-first <velocity gradient>, $G_{ij}=\partial_i u_j$. This is the transpose of the component-index-first convention $L_{ij}=\partial_j u_i$ often used for the <velocity gradient>. In this notation $\omega=G-G^T$, and the rate is the <Jaumann derivative>
$$
\frac{\mathcal DA}{\mathcal Dt}=D_tA+\frac12(\omega A-A\omega),\qquad
D_t=\partial_t+u\cdot\nabla.
$$
To check the <covariance of the Jaumann derivative>, let $R=\dot Q Q^T$. Direct transformation gives
$$
D_t^*A^*=Q(D_tA)Q^T+RA^*-A^*R,\qquad
\omega^*=Q\omega Q^T-2R.
$$
The observer-rotation terms cancel in the displayed corotational rate. Equivalently, in the $L$ convention the physical <spin tensor> is $W=(L-L^T)/2=-\omega/2$, and the same rate is $D_tA-WA+AW$. One must transpose the gradient and reverse the commutator together; using $L$ in the printed plus-sign expression would not be objective.
Another example is the <upper-convected derivative>, written consistently in the $L$ convention as
$$
\boxed{\overset{\triangledown}{A}=D_tA-LA-AL^T.}
$$
The extra terms in $L^*=QLQ^T+R$ cancel those in $D_t^*A^*$, so this also transforms covariantly. In contrast, the uncorrected <material derivative> $D_tA$ of a second-rank tensor is \b[not objective]: its transformation contains $RA^*-A^*R$. For instance a constant anisotropic tensor in a stationary fluid has zero material derivative in its original frame, but its components have a nonzero commutator derivative in a rotating observer's frame. The ordinary partial time derivative is likewise not an objective tensor rate.
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