I've found two substantially different notions of a *material derivative* of a shape-dependent function.

The first one, which seems more naturally to me, is as follows: Let $d\in\mathbb N$, $D\subseteq\mathbb R^d$ be open, $\mathcal A\subseteq 2^D$, $E_\Omega\subseteq\mathbb R^\Omega$ be a $\mathbb R$-Banach space for $\Omega\in\mathcal A$, $E:=\bigcup_{\Omega\in\mathcal A}E_\Omega$ and $F:\mathcal A\to E$ with $$F(\Omega)\in E_\Omega\;\;\;\text{for all }\Omega\in\mathcal A.$$

Now fix $\Omega\in\mathcal A$ and a family of $C^1$-diffeomorphisms $T_t:D\to D$ for $t\in[0,\tau)$, $\tau>0$. Then $F$ is "differentiable" with respect to the (velocity induced by) $(T_t)_{t\in[0,\:\tau)}$ if $$F(\Omega_t)\circ\left.T_t\right|_{\Omega}\in E_\Omega\;\;\;\text{for all }t\in[0,\tau)\tag1$$ and $$[0,\tau)\to E_\Omega\;,\;\;\;t\mapsto F(\Omega_t)\circ\left.T_t\right|_{\Omega}\tag2$$ is Fréchet differentiable at $0$. In that case, the material derivative is defined to be the Fréchet derivative of $(2)$ at $0$.

Now the second definition is that $F$ is differentiable if $$\lim_{t\to0+}\frac{F(\Omega_t)\circ\left.T_t\right|_{\Omega}-F(\Omega)}t\tag3$$ exists in $L^1(\Omega)$.

Since I'm new to this topic, I wonder whether one of these definitions is more common than the other and what are the advantages/disadvantages of them when we are working with them.