[BUG] scalar or vector integral first?

[Smith, Alexander J.]

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. This reminds me of Gaussian curvature and
mean curvature for a surface. If you switch the normal vector, the
Gaussian curvature is invariant but the mean curvature switches
sign. Because of this analogy and because scalar-dA can be defined
for a non-orientable surface, scalar integrals feel more "general."

In our course, we do introduce scalar-dr before vector-dr, and
scalar-dA before vector-dA. But I find it necessary to give plenty
of time to Stokes' Theorem and Divergence Theorem, and so we end
hardly ever using scalar-dr or scalar-dA after they are introduced.
Many times at the end of the semester I wonder if there was any
point to even covering the scalar variants. At the end of the course
they look like a speck.

Let me change subject a little. Take a Mobius strip. It's edge is a
single loop. Make the edge be a wire, and let a current flow around
the loop. It seems to me like this would correspond to a changing
magnetic flux through the surface, except we cannot calculate or
define the flux because the surface is not orientable. But you would
think there must be some physical thing corresponding to a magnetic
flux through the surface that could be measured. Any thoughts?
Maybe it is simply 0, corresponding to some sort of double cover
of the strip.