# Flux and trajectories through vector field

I have here a very simple vector field $F(u, v) = 2\pi \binom{a}{b}$ where a and b are fix and $a \neq 0$. I have a parametrization of the surface of the Torus T with

$\Phi(u, v) =$ $$\begin{matrix} (R + r\cos(v))\cdot\cos(u) \\ (R + r\cos(v))\cdot\sin(v) \\ r\cdot\sin(v) \end{matrix}$$

Im now looking for the vector field $\hat F$ on T. According to my notes its:

$\hat F(\Phi(u, v)) = D\Phi_{(u,v)}(F(u,v))$

This step is pretty straight forward, i've just calculated $D\phi$ and used $2\pi\cdot a$ and $2\pi\cdot b$ as the arguments which gives a nice 2x3 matrix.

For the next step I need to calculate the flux and the trajectories of $\hat F$.

Again according to my notes its:

$\hat\phi(t,\Phi(u,v)) = \Phi(\phi(t,u,v))$ where $\phi(t,u,v)$ is the flux of the vector field F.

Calculating the flux for a vector field of form $F(X) = A \cdot X$ where A is an nxn matrix we did quite often, but how do I calculate the flux for the vector field $F(u, v) = 2\pi \binom{a}{b}$. Im in general not really sure what this flux exactly is or how we calculate it, there was just like an algorithm for 2x2 matrices for vector fields like the one above. How do I have to start? And from the flux how do i get the trajectories? Im a bit lost here... Any help is greatly appreciated!

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