Let $v$ be the vector field $x_1\dfrac{\partial}{\partial x_2}-x_2\dfrac{\partial}{\partial x_1}$, and let $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be the diffeomorphism $(x_1,x_2)\rightarrow (x_2,x_1)$. What is $f_*v$, the push-forward of $v$ by $f$?
So, at a point $p=(x_1,x_2)$, the vector $v(p)$ is $(-x_2,x_1)$. By definition, $f_*v$ is the vector field $w$ such that $v$ and $w$ are $f$-related. This means that $Df_pv(p)=w(f(p))$ for all $p$. So $Df_{(x_1,x_2)}(-x_2,x_1)=w((x_2,x_1))$. How can I calculate this $w$?
[By $Df_p$ I mean the derivative of $f$ at point $p$]