Lower-convected stress pull-back ($\dot{(F^T\tau F)}=F^T(\dot{\tau }+L^T\tau +\tau L)F$)

For $L=\dot{F}{F}^{-1}$, the product rule proves the displayed lower-convected derivative identity. Under a superposed rotation $Q(t)$, ${\tau }^{\ast }=Q\tau Q^T$ and $L^{\ast }=QL{Q}^T+\dot{Q}{Q}^T$. The two rotation terms in ${\dot{\tau }}^{\ast }$ cancel those in $(L^{\ast }{)}^T{\tau }^{\ast }+{\tau }^{\ast }{L}^{\ast }$, proving that the lower-convected rate is an objective time derivative.