provided $A-B$ is small enough. Thus, when linear approximation is allowable, first definition is the good approximation of the second. For instance $$ln(g^{-1}C_{3}(\phi ))=ln(I+g^{-1}(C_{3}(\phi )-g))\approx
g^{-1}(C_{3}(\phi )-g),$$ provided $C_{3}(\phi )-g$ is small.

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Strain Rate Tensor
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One can also define the material elastic strain rate tensor as follows $${\dot{\hat E}}^{el}={\mathcal L}_{u_{\phi}}{\hat E}^{el},$$ as well as inelastic strain rate tensor $${\dot{\hat E}}^{in}={\mathcal L}_{u_{\phi}}{\hat E}^{in}.$$ In the case where $G=G_{0}$ and $\phi ^{0}=T$, elastic strain rate tensor defined in (4.17) has, the same spatial components as the conventional strain rate tensor ([@MH]).

Denote by $\dot{G}$ the the Lie derivative ${\dot{G}}={\mathcal
L}_{u_{\phi }}G$ of the metric tensor $G$ with respect to the flow vector $\vec{u}_{\phi }$. Then the calculation of the Lie derivative in (4.17) results in the following relation $${\dot{\hat E}}^{el}={\mathcal
L}_{u_{\phi}}{\hat E}^{el}=-G^{-1}{\dot{G}}{\hat E}^{el}
+\frac{1}{2}G^{-1}({\dot{C_{4}(\phi )}}-{K}),$$ where $K={\mathcal L}_{u_{\phi }}\gamma$ is the [**extrinsic curvature tensor**]{} of the slices $B_{\phi ,t}$.

Here we are using material coordinates and tensors only. In order to obtain the corresponding “laboratory” quantities (seen by an external observer), one defines the laboratory (Euler) Elastic Strain Tensor $$\epsilon
^{el\ i}_{j}=\phi ^{i}_{,A}\phi ^{-1\ B}_{j}E^{el\ A}_{B}$$ and recalculate all the other quantities accordingly.

Figure 5 presents the above decomposition of 