TITLE: Why is the tangent bundle orientable?
QUESTION [29 upvotes]: Let $M$ be a smooth manifold. How do I show that the tangent bundle $TM$ of $M$ is orientable?

REPLY [6 votes]: Since the cotangent bundle $T^*M$ has a natural symplectic form $\omega,$ and, as for any symplectic form, $\omega^n$, for $2n=dimT^*M$ is a volume form, the cotangent bundle is orientable. Since the $TM$ is diffeomorphic to $T^*M$ via, say, any choice of a metric on $M$, the tangent bundle is orientable.