[BUG] scalar or vector integral first?

[Tevian Dray]

> One should keep in mind that integrals of scalar-dr and scalar-dA do
> not depend on an orientation, whereas integrals of vector-dr and
> vector-dA change sign.

Yes -- and I would argue that this is most naturally seen by regarding
the scalar line/surface elements as the magnitude of the corresponding
vector line/surface element -- magnitudes do not depend on orientation.

My sense is that students who learn scalar line/surface elements first
often have trouble getting the sign of vector integrals right, whereas
those who first learn to compute vector line/surface elements have
relatively little trouble -- and are also comfortable using absolute
values to get the sign of scalar integrals right.

> ... because scalar-dA can be defined
> for a non-orientable surface, scalar integrals feel more "general."

You can always pick an orientation locally, so there's no problem
defining dA-vector on suitably small regions even on a non-orientable
surface.  It is then an interesting observation that, for some surfaces,
only the magnitude dA can be defined globally.  Perhaps a treatment
along these lines could make the point that scalar integrals are more
general (in this sense) without having to cover them first.

Tevian