Given the vector fields $v_n=x^{n}{\partial_x}+y^{n}{\partial_y}$.
I am not sure how can I find the push forward $\phi^t_*v$ and the pull back ${\phi^{t}}^*dx\wedge dy$ where $\phi^t$ is the local flow of $v_1$.
I think for case $k=1$, it works nicely for global flow, but I am not sure for local flow, the push forward and pull back process could be the same as in the global flow or not.