An <orientable smooth manifold> is a smooth $n$-manifold that admits a smoothly varying orientation of its tangent spaces. Equivalently, it has an atlas whose coordinate-transition <Jacobian determinants> are positive, or a nowhere-vanishing smooth top-degree <differential form>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 1 / a / Solution

 Solution (source code)