Let $M$ be the torus $\Bbb{T}^2 = \Bbb{S}^1 \times \Bbb{S}^1 \subset \Bbb{R}^4$ by $\{(w,x,y,z) \in \Bbb{R}^4: w^2+x^2=y^2+z^2=1\}$. Let $\omega$ be the $2-$form
$$\omega = xyz\, dw \wedge dy.$$
Compute $\int_M \omega$.
Attempt:
The torus can be parameterized as a function $F:\Bbb{R}^2 \to \Bbb{R}^4$ defined via
$$(\phi,\theta) \mapsto(\cos \phi, \sin \phi, \cos \theta, \sin \theta).$$
So its enough to compute
$$\int_{[0,2\pi)^2}F^* \omega.$$
A quick computation shows $F^* \omega$ is
$$F^* \omega = \sin^2 \phi \sin^2 \theta \cos \theta \, d \phi \wedge d \theta.$$
So we have
\begin{align} \int_M F^* \omega&=\int_0^{2\pi}\int_0^{2\pi} \sin^2 \phi \sin^2 \theta \cos \theta d \phi d \theta\\ &=0. \end{align}
Is my solution correct? Also, this is exercise $16-2$ out of Lee's intro to smooth manifolds.