Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ]-\pi/2,\pi/2[ \rightarrow \mathbb{S}^2 \setminus \{ (0,0,1),(0,0,-1) \}$ be the function defined by $f((u,v),\theta) = (u \cos \theta, v \cos \theta, \sin \theta)$.
- $f$ is a diffeomorphism ;
- $\omega = \frac{dx_1 \wedge dx_2}{x_3}$ on $\mathbb{S}^2 \setminus \{ x_3 = 0 \}$.
Also, $f^*\omega = \sin \theta \cos^2 \theta \,du \wedge dv - v \cos \theta \,du \wedge d\theta + u \cos \theta \,dv \wedge d\theta$. (not entirely sure about that)
I want to show that $\omega$ is exact on $\mathbb{S}^2 \setminus \{ (0,0,1),(0,0,-1) \}$. To do that, I'm supposed to prove that $f^*\omega$ is exact and use the fact that $f$ is a diffeomorphism. Unfortunately, I can't find a "primitive" $\beta$ such that $d\beta = f^*\omega$.
- Does the given expression for $f^*\omega$ is correct?
- How can I prove that $f^*\omega$ is exact, without using Poincaré's lemma?
Thank you in advance! :)