I have a few question, I hope you can help me on some of them. 1.Show that if A (subset of R^n) is a submanifold with dimension, n, with boundary then dA (the boundary of A) is orientable. 2. Show that a torus in R^3 is orientable. 3.Show that a mobius band isn't orientable. 4. Let M,N be two connected oriented manifolds. Let f:M->N be a diffeomorphism. Show that [tex]df_x:T_x M\rightarrow T_f(x) N[/tex] either preserves or reverses orientation for all x in M simultaneously. Here is what I thought of: 1)I think that the standard orientation on R^n is induced to the boundary of A. 4) I need to prove that the determinant of df_x is always positive or negative, now from the definition of orientation on M and N, we have two diffeomorphism [tex]\psi , \phi[/tex] such that for every x in M there's a neighbourhood U, such that: psi is a local diffeomorphism of U onto an open set V of R^N, and for every z in U [tex]d\psi_z : T_z M\rightarrow R^m[/tex] keeps the orientation, the same for N. Now if f can be broken into two diffeomorphisms one from [tex]T_x M \rightarrow R^m[/tex] the other from [tex] T_f(x) N \rightarrow R^m[/tex], then the determinant of df_x would be equal the product of two determinants which both of them have a plus sign cause they keep the orientation. 2. a torus is [tex]S^1 x S^1[/tex] where S^1 is a circle, intuitively I understand why it's orintebale but how to prove it rigourosly? I mean I think I need to show that if I induce the standrad orientation of R^3 onto the torus, it keeps orientation, not sure. 3. the same for 2, just inducing the standard orientation and to show the determinat changes signs from some point.