I have a question on a direct computation. How would one compute the following $$ (dx \odot dy ) \left(\frac{1+uv+xy}{1+xy} \frac{\partial}{\partial v}, \frac{\partial}{\partial y} \right) $$ and$$ (dy \odot dv ) \left(\frac{1+uv+xy}{1+xy} \frac{\partial}{\partial v}, \frac{\partial}{\partial y} \right) $$
I know the general formula is $(\omega \odot \eta )(a,b)= \frac{1}{2}(\omega(a)\eta(b) + \omega(b)\eta(a))$, but I am not sure how to do it with the certain vector fields above.
Hint: A differential one-form on a smooth manifold $M$ is a $C^{\infty}(M)$-linear map $TM \to C^{\infty}(M)$. So, if $\alpha \in \Omega^1(M)$, $f \in C^{\infty}(M)$, and $V$ is a smooth vector field on $M$,
$$\alpha(fV) = f\alpha(V).$$
Furthermore, in coordinates $(x^1, \dots, x^n)$, the vector fields $\dfrac{\partial}{\partial x^i}$ form a basis, and the one-forms $dx^j$ are the dual basis, so
$$dx^j\left(\frac{\partial}{\partial x^i}\right) = \delta_i^j =\begin{cases} 1 & i = j\\ 0 & i \neq j.\end{cases}$$