 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 $A^{\ast }=QA{Q}^T$ under $x^{\ast }=Q(t)x+c(t)$ must imply $\mathcal{D}{A}^{\ast }/\mathcal{D}t=Q(\mathcal{D}A/\mathcal{D}t)Q^T$. If $R=\dot{Q}{Q}^T$, then $G^{\ast }=QG{Q}^T-R$ andConsequently the extra terms cancel in $D_{t}A+\frac{1}{2}(\omega A-A\omega )$, 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.