it is permanently attached to the substrate (i.e. the boundary of the volume). Therefore $\c{S}_t = \p\c{B}_t$ and $\h{\b{x}}=\b{x}|_{\p{\c{B}_t}}$. This assumption implies that the motion of the surface $\h{\b{\varphi}}$ is the restriction of the volume motion $\b{\varphi}$ to the surface, i.e. $\h{\b{\varphi}} = \b{\varphi}|_{\p{\c{B}_0}}$.

Material line elements on the surface in the material and spatial configurations are denoted $\d\h{\b{X}} \in T \c{S}_0$ and $\d\h{\b{x}} \in T \c{S}_t$, respectively, and are related as $\d\h{\b{x}} = \h{\b{F}} \cdot \d\h{\b{X}}$, where $\h{\b{F}}:=\p \h{\b{x}} / \p \h{\b{X}}  = {\h{\b{g}}}_\alpha \dyad {\h{\b{G}}}{}^\alpha$ denotes the rank-deficient, and thus non-invertible, surface deformation gradient. The co- and contravariant surface basis vectors in the material and spatial configurations are denoted $\h{\b{G}}_\alpha$ and $\h{\b{g}}^\alpha$, respectively. Various key concepts from differential geometry are given in Summary \[summary:surface-1\]. The inverse deformation gradient $\h{\b{f}}$ is defined by $\h{\b{f}}:=\p \h{\b{X}} / \p \h{\b{x}}  = {\h{\b{G}}}_\alpha \dyad {\h{\b{g}}}{}^\alpha$ and is related to the surface deformation gradient as follows: $$\begin{aligned}
    \h{\b{f}} \cdot \h{\b{F}} = \h{\b{I}} =: \b{I} - \b{N}\otimes\b{N}
    && \text{and} &&
    \h{\b{F}} \cdot \h{\b{f}} = \h{\b{i}} =: \b{i} - \b{n}\otimes\b{n} \, .\end{aligned}$$ The (surface) determinant of the surface deformation gradient and its inverse are denoted $\h{J} = \h{\det} \h{\b{F}} > 0$ and $\h{j} = \h{\det} 