= Cotangent bundle orientation
{title2=$(d\lambda)^n\ne0$}
Every <cotangent bundle> $T^*M$ 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 $d\lambda=\sum_i da_i\wedge dx^i$. Its $n$th wedge power is the nowhere-zero form $n!\,da_1\wedge dx^1\wedge\cdots\wedge da_n\wedge dx^n$, which defines an <orientation> of the $2n$-dimensional total space. Using $-d\lambda$ gives another standard sign convention; either is a <symplectic form>.
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