[BUG] scalar or vector integral first?

[Smith, Alexander J.]

We have been discussing the pros and cons of introducing scalar-dr integrals before vector-dr, and the parallel idea for dA.

I tend to agree that introducing the scalar version first makes practical sense as far as learning goes, even if the scalar variants end up getting dwarfed by the orientation sensitive vector integrals because of the importance of Stokes' and Gauss' Theorems.

But think a minute about where  students are coming from. They cut their baby teeth on the Riemann integral with dx, which is orientation sensitive! (If you reverse the orientation of the curve [a,b] the integral changes sign.)

 So after all is said and done, we must admit that we have already inculcated students to the vector version of dr.

If we want to be pure and honor our conviction that the scalar version should precede the vector version, then maybe we should replace the Riemann integral in Calc II with the Lebesgue integral: Riemann dx changes sign while Lebesgue measure d_mu does not under an orientation switch.

I admit that I am suggesting this tongue-in-cheek, and  this is not ever going to happen  in American universities. But math majors in European universities do study the Lebesgue integral, and not the Riemann integral.