Let $F: \Bbb{R}^2 \to \Bbb{R}^3$ be the smooth map given via
$$(\theta,\phi) \mapsto ((2+\cos \phi)\cos\theta,(2+\cos\phi)\sin\theta,\sin\phi).$$
Let $M$ be $F(\Bbb{R}^2)$ the torus.
Let $\omega$ be the $2-$form
$$\omega=x^2 \, dy \wedge dz.$$
Compute $F^* \omega$ and use this to compute $\int_M \omega$.
Attempt:
As the torus is orientation preserving, we can use
$$\int_M \omega = \int_0^{2\pi}\int_0^{2\pi}F^*\omega.$$
So I computed $F^* \omega$ as follows:
\begin{align} F^*\omega&=F^*(x^2 \, dy \wedge dz)\\ &=(2+\cos\phi)^2\cos^2\theta \, d((2+\cos\phi)\sin\theta) \wedge d(\sin\phi)\\ &=(2+\cos\phi)^2\cos^2\theta(-\sin\phi \sin \theta\, d \phi+\cos\theta(2+\cos\phi)\,d\theta) \wedge \cos\phi\, d\phi\\ &=(2+\cos\phi)^3\cos^3\theta\cos\phi\, d \theta \wedge d \phi \end{align}
And I computed
$$\int_0^{2\pi}\int_0^{2\pi}(2+\cos\phi)^3\cos^3\theta\cos\phi d \theta d \phi$$
to be $0$. Sorry I have been asking lots of questions since yesterday I have a big diff geo final today and I wanna make sure I am computing the integrals of these forms correctly.