t:= T / N = t_{n+1} - t_n$, where $t_n = n \Delta t$. The complete state of the system is assumed known at $t_n$. The (not necessarily converged) value of a variable evaluated at an iteration $(i)$ during the $n+1$ timestep is denoted $(\bullet)_{n+1}^{(i)} \equiv (\bullet)^{(i)}$. The volume and surface residual contributions, denoted $R$ and $\h{R}$, respectively, and the directional derivatives of the residual contributions in the direction of the solution increment in the volume $\Delta \b{u} = \b{\varphi}^{(i+1)} - \b{\varphi}^{(i)}$ and on the surface $\Delta \h{\b{u}}$ are defined by $$\begin{aligned}
 R^{(i)} &:= \int_{\c{B}_0} \Grad  \delta\b{\varphi} : \b{P}^{(i)}  \, \d V 
    - \int_{\c{B}_0}  \delta \b{\varphi} \cdot  \b{b}\prs_0 \, \d V \, , \\
\h{R}^{(i)}&:=  \int_{\c{S}_0} \h{\Grad} \delta \h{\b{\varphi}} :\h{\b{P}}{}^{(i)}  \, \d A 
    - \int_{\c{S}_0^\text{\tiny N}} \delta \h{\b{\varphi}} \cdot \h{\b{b}}\prs_0 \, \d A \, ,
\intertext{and}
 \D_{\Delta \b{u}} R^{(i)} &:= \int_{\c{B}_0} \Grad  \delta\b{\varphi} : \dfrac{\p \b{P}{}^{(i)}}{\p \b{F}} : \Grad{\Delta \b{u}}  \, \d V \, , \\
\D_{\Delta \h{\b{u}}}\h{R}^{(i)} &:=\int_{\c{S}_0} \h{\Grad} \delta \h{\b{\varphi}} : \dfrac{\p \h{\b{P}}{}^{(i)}}{\p \h{\b{F}}} : \h{\Grad} \Delta \h{\b{u}}   \, \d A  \, . \end{aligned}$$ The derivatives of the Piola–Kirchhoff stress measures with respect to the deformation measures (i.e. the tangents) are given in Summary \[constiutive\_relations\]. Note, an exact 