# Proof of kinematic relationship between updated and total Lagrangian 2nd Piola-Kirchhoff stresses

On page 587 of Finite Element Procedures by Bathe the author gives the following kinematic transformations

$${}^t\tau_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^t_oS_{rs} \; {}^t_ox_{j,s}$$

and

$${}^{t + \Delta t}_{\;\;\;\;t} S_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^{t + \Delta t}_{\;\;\;o}S_{rs} \; {}^t_ox_{j,s}$$

I can derive the first statement but not the second. Specifically, I do not understand how the LHS of the second equation is the 2nd Piola-Kirchhoff stress $${}^{t + \Delta t}_{\;\;\;\;t} S_{ij}$$ and not of the Cauchy stress term $${}^{t + \Delta t}_{\;\;\;\;t} \tau_{ij}$$.