 Cotangent bundle orientation

Every cotangent bundle $T^{\ast }M$ is an orientable smooth manifold even when its base is not. The intrinsic canonical one-form on a cotangent bundle ${\lambda }_{(p,\alpha )}(W)=\alpha (d\pi W)$ has $d\lambda ={\sum }_{i}d{a}_i\wedge d{x}^i$. Its $n$th wedge power is the nowhere-zero form $n!d{a}_1\wedge d{x}^1\wedge \cdots \wedge d{a}_n\wedge d{x}^n$, which defines an orientation of the $2n$-dimensional total space. Using $-d\lambda$ gives another standard sign convention; either is a symplectic form.