I am trying to show that following:
Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable.
Let $\pi_M:M\times N\to M$ and $\pi_N:M\times N\to N$ be canonical projections.
Since $M$ and $N$ are orientable, we can find non-vanishing forms $\omega \in \Omega^m(M)$ and $\eta\in \Omega^n(N)$.
I think that $\pi_M^*(\omega)\wedge \pi_N^*(\eta)$ should then be a non-vahinsghing $(m+n)$-form on $M\times N$.
I am unable to show this.
Let us write $\pi_M^*\omega$ as $\omega^*$ and $\pi_N^*\eta$ as $\eta^*$. One of the difficulties I am facing is that $\omega^*\wedge \eta^*$ is $\binom{m+n}{n}\sum_{\sigma\in S_{m+n}} {^\sigma}(\omega^*\otimes \eta^*)$.
I need to operate this thing on a basis of $T_{(p,q)}(M\times N)$, where $p\in M$ and $q\in N$. I know that $T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$ via the isomorphism $Z\mapsto (d\pi_MZ, d\pi_NZ)$.
Can somebody see what to do from here?