Show that if $M$ and $N$ are oriented corresponding to forms $\omega \in \Omega^m(M)$ and $\eta \in \Omega^n(N)$ then $M \times N$ has an orientation corresponding to the form $\pi_M^*\omega \wedge \pi_N^*\eta$, where $\pi_M$ and $\pi_N$ denote projection onto $M$ and $N$, respectively. This is called the product orientation.
So far I have a, what I think is basic, start at a proof.
To show $M \times N$ has an orientation corresponding to the form $\pi_{M}^{*} \omega \wedge \pi_{N}^{*} \eta$, we need to show that it is non-vanishing. We already know that $\omega$ and $\eta$ are non-vanishing forms if they are restricted to $M$ and $N$ respectivley. Thus, if we pull back along the projections $\pi_M$ and $\pi_N$ the form $\pi_{M}^{*} \omega \wedge \pi_{N}^{*} \eta$ will be non-vanishing since if have coordinates for $M$ and $N$ they will remain independent in $M \times N$ from the definition of $\times$.
I'm not sure how to prove that the basis for $\pi_M^* \omega$ and $\pi_N^* \eta$ will be independent. I've heard words like transverse thrown around occasionally which I think is related. Would that help here?