Guillemin and Pollack give quite confusing (at least for me) definition of the preimage orientation (see below).
I don't understand the part starting from the last display. Namely:
- How exactly does the last display follow?
- Why does $T_xS$ contain $\ker df_x$ and how does it follow from this that $df_x$ restricted to $N_x(S;X)$ is injective (I believe that's what is claimed)?
- At which point does this prove use the fact that every vector from $N_x(S; X)$ is orthogonal to every vector from $T_x S$?