Is the outward pointing normal always consistent with a orientation preserving coordinate system?

Question

In Folland's Advanced Calculus, when discussing Stokes' Theorem, the surfaces are oriented by a choice of normal (either inwards or outwards). Visually, it's easy to see which direction is inwards and outwards, especially for simple shapes such as cylinders. But I have not seen a rigorous definition of

In Spivak's Calculus on Manifolds, we orient a manifold by a coordinate system that is either orientation preserving (i.e. preserves the standard orientation $e_1, \ldots, e_n$), or orientation reversing.

Must surfaces oriented by outward pointing normals (in the Folland sense) be A) orientation preserving (in the Spivak sense), or B) orientation reversing, or C) is there is no relationship at all, and depends on the manifold. Why?

Answer 1

Not all manifolds have a global orientation (ex: Möbius strip/Klein bottle). Each chart in an atlas has an orientation that you can use to get an orientation for adjacent charts, but it’s possible to go in a loop and end in a different orientation than you started.

Inside/outside orientations only work on manifolds which are embedded in an another space and are the boundary of some space. In that case inside/outside matters when you’re doing something with that inside (I.e. Generalized Stokes Theorem).

Orientation-preserving maps only make sense if you’ve already decided on a basis/orientation for the domain/codomain.