Let $$\omega=xz\,dx+yz\,dy-z^2\,dz$$ be a $1$-form in $\Bbb{R}^3$ and let $H$ be the hyperboloid $x^2+y^2-z^2=1$. I want to show that, if $M\subset H$ is a surface that is diffeomorphic to the sphere $S^1$, so $\int_M \omega=0.$
My idea is to find a $0$-form $\eta$ such that, in $M$, $d\eta=\omega$. My first try was
$$\eta=z\frac{x^2}{2}+z\frac{y^2}{2}-\frac{z^3}{3}\implies d\eta=xz\,dx+yz\,dy+\left(\frac{x^2+y^2-2z^2}{2}\right)\, dz $$
My hope was to use $x^2+y^2-z^2=1$ in the equation of $d\eta$ to cancel some terms and get $d\eta=\omega$, but I didn't get it.
What can I do?
Since
\begin{align} \omega &= z x\, dx + zy\, dy - z^2\, dz\\ &= z( x\, dx + y\, dy - z\, dz ) \\ &=\frac{z}{2} d (x^2 +y^2 -z^2), \end{align}
$\omega$ restricts to the zero 1-form on $H$. Then of course $\int _M \omega = 0$.