Upper-convected rate as a stress push-forward ($\dot{\mathbf{\tau }}-L\mathbf{\tau }-\mathbf{\tau }{L}^T=F\dot{S}{F}^T$)

Differentiate the Kirchhoff stress tensor relation $\mathbf{\tau }=FS{F}^T$, using $\dot{F}=LF$ for the velocity gradient. Subtracting the two deformation terms gives the displayed upper-convected derivative. This removes the transport generated solely by deformation of a fixed material stress tensor.