Dyadic transport of material line elements ($D_{t}A=LA+A{L}^T$)

For a material line tangent $\mathbf{\ell }$, $D_t\mathbf{\ell }=L\mathbf{\ell }$ with velocity gradient $L_{ij}={\partial }_j{u}_i$. The product rule gives $D_t(\mathbf{\ell }{\mathbf{\ell }}^T)=L\mathbf{\ell }{\mathbf{\ell }}^T+\mathbf{\ell }{\mathbf{\ell }}^T{L}^T$. Consequently the upper-convected derivative vanishes for any transported sum of these outer products. Under the derivative-index-first convention $G=L^T$ the same formula is $D_{t}A=G^{T}A+AG$. The distinction prevents transposition errors in constitutive equations.