Do Carmo first defines a regular surface $S$ to be orientable if there is a set of charts $(\phi_\alpha,U_\alpha)$, where $\bigcup_{\alpha}U_{\alpha}=S$, such that the transition map is orientation-preserving, i.e. det$(J(\phi_{\beta}^{-1}\circ\phi_{\alpha}))>0$ for any $\alpha,\beta.$
After that, he proceeds by prove the equivalence between this definition and the fact that $S$ has a unit normal vector field. He first introduced how we can compute a unit normal as follows:
The wedge sign simply means cross product. In formula (2), he attempted to show us that the normal vector will be "flipped" if the Jacobian determinant is negative, so the unit normal defined by the two parametrizations will not be consistent. However, what confuses me is that following that in the proof of the equivalence as I mentioned, there is something which I feel is contradictory. The following is the part of the proof proving $S$ has a unit normal vector field implies the definition of orientability.
In the last line, he said that we can INTERCHANGE $u$ and $v$ in the parametrization. If we can do so, then in formula (2), we can just INTERCHANGE $u$ and $v$ so that the unit normal becomes consistent! So I am so confused about this. Can anyone help me clarify this? Thank you.
The unit normal field $N$ is already assumed to be given. What he's arguing is that you can choose connected coordinate neighborhoods so that $\mathbf x_u\wedge\mathbf x_v$ is a positive multiple of $N$ at each point. (The discussion of interchanging the variables $u,v$ is to arrange this.) Once you've done this, (2) tells you that you'll have a covering of the surface by charts for which the jacobian determinant is positive on overlaps.