Let us assume I have sampled a vector field of "velocity" vectors : $\bf x \to {\bf v}({\bf x})$
How can I calculate a vector-field of orts-vectors where I end up after some time $t$?
Let us call say (${\bf x},t) \to {\bf s}_t({\bf x})$
Simple example (unless I am mistaken) :
$${\bf v}({\bf x}) = \begin{bmatrix}0&-1\\1&0\end{bmatrix}{\bf x} = \begin{bmatrix}-x_2\\x_1\end{bmatrix}$$
Should give $${\bf s}_t({\bf x}) = \begin{bmatrix}\cos(t)&\sin(t)\\-\sin(t)&\cos(t)\end{bmatrix}{\bf x}$$
If we "follow" the velocity field of a revolution, we will end up on a circle. Compare the in physics circular motion with constant speed.
My intuition tells me that this should be conceptually similar to integration in some sense. I also have some memory of a professor once telling me there is a connection to abstract algebra and Lie groups and Lie algebras. There was a corresponding "infinitesimal" rotation which by exponentiation was related to this integral. I am sorry if this all sounds very fuzzy.
What I mostly am after is some reference to literature of where to read more on this, although an explanation how to calculate the example is also welcome.
You are correct that this is reminiscent of integration — you are trying to solve a differential equation. Namely, if you have a relationship $\vec{v}=\vec{F}(\vec{x})$, then, since the definition of velocity is $\vec{v}=\frac{d\vec{x}}{dt}$, one obtains the relation $\frac{d\vec{x}}{dt}=\vec{F}(\vec{x})$.
This ODE does not have a general solution, but the specific example you cite, $$\frac{d\vec{x}}{dt}=\begin{bmatrix} 0&-1\\1&0 \end{bmatrix}\vec{x}\mathrm{,}$$ does. In particular, let $A$ be any matrix. Then, for the same reason $x=Ce^{at}$ solves $\frac{dx}{dt}=ax$, the function $$\vec{x}=M\left(\sum_{j=0}^{\infty}{\frac{(At)^j}{j!}}\right)\vec{x}_0$$ (where $M$ is an arbitrary constant matrix, and $\vec{x}_0$ an arbitrary constant vector) also satisfies $\frac{d\vec{x}}{dt}=A\vec{x}$.