Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / B / 6 / i / Solution

The nonvanishing top-degree form ${\omega }^m$ is a volume form. Declare an ordered tangent basis positive when its value under ${\omega }^m$ is positive. Smooth nonvanishing ensures these choices agree on overlapping oriented charts and vary continuously. Hence a symplectic manifold is orientable; this is its symplectic orientation.