I'm studying fluid dynamics and I'm really struggling with notations and passages in some proofs that simply vanish.
In this question I need some guide-lines to show how to link the defomation map of a fluid and the velocity gradient: Let $B_0$ be a body that after a certain amount of time $t$ takes a different shape $B_t$. Supposing the change of shape is smooth enought into time, I used to call $M_t$ the deformation map which takes $B_0 \to B_t$.
Now, a part the clarifications about eulerian and lagrangian descripition of motion that is almost clear to me, I struggling to prove a theorem that states: $$L_v = \dot{M}_t \cdot M_t^{-1}$$ where $L_v$ is used ad a notation of the gradient of velocity (which one: the eulero's or lagrange's one?) and the dot operator is the convective/material derivative.
Can anyone help me to dissolve my doubts and give me a sketch of proof (in the case the theorem I stated is true)?
$M_t=\frac{\partial\mathbf{x}(t)}{\partial\mathbf{x}(0)}$, so $$ \begin{split} \dot{M}_t M_t^{-1} = \left[\frac{D}{\mathrm{d}t}\frac{\partial\mathbf{x}(t)}{\partial\mathbf{x}(0)}\right]\left[\frac{\partial\mathbf{x}(t)}{\partial\mathbf{x}(0)}\right]^{-1}\\ =\left[\frac{\partial\mathbf{v}(t)}{\partial\mathbf{x}(0)}\right]\cdot\left[\frac{\partial\mathbf{x}(0)}{\partial\mathbf{x}(t)}\right]=\frac{\partial\mathbf{v}(t)}{\partial\mathbf{x}(t)}=L_\mathbf{v} \end{split} $$ This is an Eulerian quantity because we are taking derivative with respect to the current $\mathbf{x}(t)$ not a fixed reference $\mathbf{x}(0)$.