Orientability of a vector bundle total space ($\det TE\cong {\pi }^{\ast }(\det E\otimes \det TM)$)

The total space of a real vector bundle $E\to M$ has the exact sequence $0\to {\pi }^{\ast }E\to TE\to {\pi }^{\ast }TM\to 0$. Taking determinant line bundles gives the displayed identity. If $M$ is an orientable smooth manifold, the total space is orientable exactly when $\det E$ is trivial. Sufficiency follows by pullback; necessity follows by restricting the total-space determinant identity to the zero section. For the cotangent bundle, base and fibre coordinate determinants cancel, giving an orientation even when the base is not orientable.