Lower-convected derivative

The lower-convected derivative of a covariant tensor is $\stackrel{△}{\mathbf{C}}=D\mathbf{C}/Dt+(\nabla \mathbf{u}{)}^T\mathbf{C}+\mathbf{C}\nabla \mathbf{u}$.

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.