in the spatial configuration associated with an arbitrary node $I$ is denoted $\b{\Phi}^I_t$. By choosing the arbitrary field in the material view as the spatial placement $\b{x}$ one obtains $\b{F}$, while choosing the arbitrary field in the spatial view as the material placement $\b{X}$ one obtains $\b{f}$.

The volume Jacobian determinant $J$ and its inverse $j$ are determined from the ratio of the Jacobian determinant of the map from the isoparametric configuration to the reference configuration $J_{\square,0}$ to the inverse of the Jacobian determinant of the map from the isoparametric configuration (reference cell) to the spatial configuration $J_{\square}$.

An identical approach to the volume is followed on the surface. The functionality provided by the *codimension routines* in [deal.II]{} allows the surface gradient of a field to be evaluated in a straightforward manner. The surface deformation gradient and its inverse are thus computed without needing to invert $\h{\b{F}}$. This is critical as $\h{\b{F}}$ is rank deficient. The surface Jacobian determinant $\h{J}$ and its inverse $\h{j}$ are determined from the ratio of the Jacobian determinant of the map from the isoparametric configuration to the reference configuration $\h{J}_{\square,0}$ to the inverse of the Jacobian determinant of the map from the isoparametric configuration to 