I'm looking at problem (3) in here
For (a)I'm confused how to express $\alpha$ in the coordinates (x,y). So far I know $d\pi^*: T^*M\to T^*T^*M$ is given by $d\pi^*(\eta)=\eta\circ d\pi$ and $\eta=\sum y^idx^i$ in coordinates, but I have no idea how to express $\eta\circ d\pi$ in coordinates.
For (b), $\omega=\sum dy^i\wedge dx^i$, $$\wedge^n\omega=\sum_{\sigma\in Sn}\wedge_{i\in n} dy^{\sigma(i)}\wedge dx^{\sigma(i)}$$ To show this is nonwhere-vanishing, we need to find for every (x,y) some 2n-tuple of tangent vectors $(v_i)$ that $\wedge^n\omega(v_1,...v_{2n})\neq 0$. $\wedge^n\omega(v_1,...v_{2n})=\sum_{\sigma\in Sn}(-1)^{2sign(\sigma)}det[v_1,...,v_{2n}]$, where ${v_i}$ is written as a column vector with respect to the ordered basis $(\partial y^1, \partial x^1,..., \partial y^n, \partial x^n)$. $(-1)^{2sign(\sigma)}$ is got from swapping any two $dy^i\wedge dx^i$ and $dy^j \wedge dx^j$. So as long as we choose linear independent $(v_i)$, it will work?