= Covariance of the Jaumann derivative
Use the derivative-index-first <velocity gradient> $G_{ij}=\partial_i u_j$ and $\omega=G-G^T$. For an <objective time derivative> the <tensor> transformation $A^*=QAQ^T$ under $x^*=Q(t)x+c(t)$ must imply $\mathcal D A^*/\mathcal Dt=Q(\mathcal D A/\mathcal Dt)Q^T$. If $R=\dot Q Q^T$, then $G^*=QGQ^T-R$ and
$$
D_t A^*=Q(D_t A)Q^T+RA^*-A^*R,\qquad
\omega^*=Q\omega Q^T-2R.
$$
Consequently the extra terms cancel in $D_t A+\tfrac12(\omega A-A\omega)$, proving covariance. With the component-index-first gradient $L_{ij}=\partial_j u_i$, the same physical rate is $D_t A-WA+AW$, with <spin tensor> $W=(L-L^T)/2=-\omega/2$. Changing the gradient convention without changing the commutator sign destroys objectivity.
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