In the beginning of the chapter 11 of Bott&Tu's book Differential Forms in Algebraic Topology, we want to find a global closed form of a sphere bundle $E\rightarrow M$ which restricts to a generator of the cohomology on each fiber. In their book, suppose {$U_\alpha$} is a good cover of $M$, they found a global form of $E$, which restricts to the $d$-cohomology class of $E|_{U_\alpha}$ on all $U_\alpha$, and they claimed that this global closed form restricrs to a generator of the cohomology on each fiber. I wonder why, wish someone could help.
PS: This problem is equivalent to a generator of the cohomology of a sphere bundle over a manifold which is diffeomorphic to $\mathbb R ^n$ restricts to a generator of the cohomology on each fiber.