Problem 8.17. (a) Let M and N be oriented manifolds, and let ω and η be an n-form and an m-form with compact support, on M and N , respectively. We will orient M ×N by agreeing that v1, . . . , vn, w1, . . . , wm is positively oriented in (M×N)(p,q) ∼= Mp⊕Nq if v1, . . . , vn and w1, . . . , wm are positively oriented in Mp and Nq, respectively. If πi : M ×N →M or N is projection onto the ith f...