Chatting with an engineer, he suggested me to have a look to a certain book in order to understand what fluid mechanics is about (I know nothing about the subject). But this question is not about fluid mechanics in general, it's a bit more specific (and much, much more basic).
Starting to read the aforementioned book, I couldn't help stopping by the first sentence because I felt the need to 'translate' things into mathematiquese in order to understand. What is a fluid motion? Well, just like a rigid motion on $M=\mathbb{R}^3$ is a smooth map $\varphi:\mathbb{R}\to\mathrm{Euc}^{+}(3)$, we can take a fluid motion to be a smooth map $\varphi:\mathbb{R}\times M\to M$ such that the induced map is a diffeomorphism at every time, $\varphi:\mathbb{R}\to\mathrm{Diff}(M)$, $(t,x)\mapsto\varphi_t(x)$. (We also assume $\varphi_0=\mathrm{id}_M$). $\varphi$ is not assumed to be a flow in general, i.e. need not verify $\varphi_t\circ\varphi_s=\varphi_{t+s}$ and $\varphi_t^{-1}=\varphi_{-t}$. Then, the notion of velocity field $v(t,x)$ of the fluid was mentioned, which is assumed to be, at a time $t$ and point $x\in M=\mathbb{R}^3$, the velocity of a material particle passing through position $x$ at time $t$. The immediate thought was that $v(t,x)$ must just be the velocity $d\varphi_t(x)/dt$ of the curve $t\mapsto \varphi_t(x)$. But it does not live in $T_xM$. So, let's take it back to $x$, i.e. define $v(t,x)$ as $(\varphi_t^{-1})_{*}(d\varphi_t(x)/dt)$, so that now it lies in $T_xM$. This is still 'physically' wrong, as one can see by considering a waterfall with a horizontal part (with almost constant velocity) and a vertical part in which water accelerates. So, I came up with the following definition:
$v(t,x):=\frac{d}{ds}|_{s=t}(\varphi_s(\varphi_t^{-1}(x)))=\dot{\varphi}_t(\varphi_t^{-1}(x))$ .
First question:
Is my definition the one usually (implicitely or explicitely) taken in fluid dynamics?
If we assume the motion is affine, i.e. $\varphi_t(x)=A(t)\cdot x + \beta(t)$ with $A(t)\in\mathrm{GL}(3)$, we get:
$v(t,x)=\dot{A}\cdot A^{-1}\cdot x - \dot{A}\cdot A^{-1}\cdot\beta+\dot{\beta}$,
in which the linear term reminds me of the Maurer-Cartan form $\omega_{MC}=g^{-1}\cdot\mathrm{d}g$ on a matrix Lie group $G$.
Second question:
Does $v(t,x)$ actually have anything to do with a Maurer-Cartan form?
