Problem 1. Show that with our definition (basically having fibered charts so that transition maps are fiberwise linear) each fiber of a vector bundle has a natural (meaning uniquely determined) structure of a vector space.

Problem 1. Show that the space of sections of a vector bundle is naturally a vector space. Show that there is an isomorphism between the vector space of smooth functions on a manifold X, and the set of smooth sections of the trivial bundle X × ℝ over X.

Problem 1. Let M be the open Mobious band (remove the boundary circle). There is a natural projection π : M → S¹ is this a smooth vector bundle?