\h{\b{f}} > 0$, respectively, where $\h{j} = 1 / \h{J}$.

![The material and spatial configurations of a continuum body and its boundary and the associated deformation maps and deformation gradients.[]{data-label="fig:motion"}](./Figs/motion){width="\textwidth"}

![image](./Figs/summary-surface-1){width="100.00000%"}

$$\begin{array}{c}
    \d\b{r} = \d\b{r} (\h{\b{\xi}}) = \d\b{r} (\h{\xi}{}^1,\h{\xi}{}^2)\\[12pt]
    \h{\b{g}}_\alpha = \f{\p\b{r}}{\p\h{\xi}^\alpha} \qquad , \qquad \h{\b{g}}^\alpha = \f{\p\h{\xi}^\alpha}{\p\b{r}} \qquad \text{with} \qquad \alpha \in \{ 1,2 \} \\[15pt]
    \h{\b{g}}{}^3 = \h{\b{g}}_1 \times \h{\b{g}}_2 \quad , \quad \h{\b{g}}_3 = [\h{g}{}^{33}]\inv \, \h{\b{g}}{}^3 \quad , \quad \b{n} = \sqrt{\h{g}{}_{33}} \, \h{\b{g}}^3 = \sqrt{ \h{g}{\,}^{33}} \, \h{\b{g}_3} \\[15pt]
    \h{\grad}\hsth = \f{\p\hsth}{\p\h{\xi}{}^\alpha} \dyad \h{\b{g}}_\alpha \qquad , \qquad \h{\div}\hsth = \f{\p\hsth}{\p\h{\xi}{}^\alpha} \cdot \h{\b{g}}_\alpha = \h{\grad}\hsth : \h{\b{i}} \\[15pt]
    \h{\det}\hsth = \f{\big|[\hsth\cdot\h{\b{g}}_1]\times[\hsth\cdot\h{\b{g}}_2]\big|}{| \h{\b{g}}_1 \times \h{\b{g}}_2|} 
 \end{array}$$

$$\begin{array}{c}
    \h{\b{g}}_\alpha = \h{g}_{\alpha\beta} \, \h{\b{g}}^\beta \, , \quad \h{g}_{\alpha\beta} = \h{\b{g}}_\alpha \cdot \h{\b{g}}_\beta \, , \qquad \h{\b{g}}{}^\alpha = \h{g}\,{}^{\alpha\beta} \, \h{\b{g}}_\beta \, , \quad \h{g}\,{}^{\alpha\beta} = \h{\b{g}}^\alpha \cdot \h{\b{g}}^\beta \, , \qquad [\h{g}_{\alpha\beta}]=[\h{g}\,{}^{\alpha\beta}]\inv \, , \qquad \h{g} = |[\h{g}_{\alpha\beta}]|  \\[10pt]
    \h{\v{g}}: \text{surface permutation tensor} \, , \, \h{\v{g}} = \h{g}_{\alpha\beta} \, \h{\b{g}}{}^\alpha\dyad\h{\b{g}}{}^\beta\dyad\b{g}^3 = \h{g}\,{}^{\alpha\beta} \, \h{\b{g}}_\alpha\dyad\h{\b{g}}_\beta\dyad\b{g}_3   \, , \qquad \h{g}_{\alpha\beta} = \h{g}\,{}^{\alpha\beta} \ \h{g} = \sqrt{\, \h{g}}\; \h{e}_{\alpha\beta} \\[12pt]
    \h{e}_{\alpha\beta} = \lc \begin{array}{cl}
                           1 & \text{if $\alpha\beta$ is 12}\\
                           -1 & \text{if $\alpha\beta$ is 21}\\
                           0 & \text{otherwise}
                          \end{array} \right. 