Let $\omega=xzdx+yzdy-z^2dz$ be an 1-form in $\mathbb{R}^3$. Define $H:=\big\{(x,y,z)\in \mathbb{R}^3:x^2+y^2-z^2=1\big\}$. Show that if $M\subseteq H$ is a submanifold diffeomorphic to the 1-sphere $\mathbb{S}^1$, then $\int _M\omega =0$.
I'm still learning the basic about integrations of differential forms, therefore I don't how to solve this problem.
I know that if $f:\mathbb{S}^1\to M$ is an orientation-preserving diffeomorphism, then $\int _M\omega =\int _{\mathbb{S}^1}f^*\omega $ in which $f^*\omega $ is the pullback of $\omega$ by $f$. However I don't how to use this fact to solve that problem.
Thank you for your attention.
Note that $\omega = \frac{1}{2}z (d(x^2 + y^2 - z^2))$. Then consider some closed continuously differentiable curve $\gamma : [0, 1] \to H$. Write $\gamma(t) = (x(t), y(t), z(t))$. Then we have $\int_\gamma \omega = \int_0^1 \frac{1}{2} z \frac{d(x^2(t) + y^2(t) - z^2(t))}{dt} dt$. But note that $x^2(t) + y^2(t) - z^2(t)$ is constant since the domain is $H$. So the integral is 0.