Calculation on a differential manifold

Question

I do not know how to calculate: Let $\alpha$ be a differential 1-form on $\mathbb{R}^{2}$ and let $V$ and $W$ be vector fields on $\mathbb{R}^{2}$. Compute $d \alpha(V, W)$ at the point $(0,1),$ where $$ V=x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}, W=y \frac{\partial}{\partial y}, $$ and $$ \alpha(V)=2 x-y^{2}, \alpha(W)=2 y. $$

Answer 1

From $$ \alpha(V)=2 x-y^{2}, \alpha(W)=2 y, V=x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}, W=y \frac{\partial}{\partial y}, $$ we have $\alpha=ydx+2dy$. \begin{equation*} \begin{aligned} d \alpha(V, W)&=V\alpha(W)-W\alpha(V)-\alpha([V,W])=V\alpha(W)-W\alpha(V)-\alpha(VW-WV)\\ &=2Vy-W(2 x-y^{2})-\alpha(x \frac{\partial}{\partial y}+y \frac{\partial}{\partial x})\\ &=2x+2y^2-(2x+y^2)=y^2. \end{aligned} \end{equation*}