Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let and be two orientable manifolds of pure dimensions and , respectively. Suppose that is a surjective submersion. This implies that each fiber is -dimensional and that, around each point of , there is a chart on which looks like the projection from a product onto one of its factors. Fix and set . Suppose that