[BUG] Which comes first: vector or scalar integrals?

[Tevian Dray]

> My current approach to these topics differs from what Tevian describes 

Perhaps not as much as you think!

> In introducing scalar line and surface integrals, I use the theme of
> total of a quantity=sum of (quantity density)(measure of region).
> I start this theme much earlier in the course.

So integrals are fundamentally about "chopping and adding"!

> Despite the differences in our approaches, I think Tevian and I agree 
> on what not to do (Tevian, correct me if I'm wrong on your view).  

I completely agree: Don't emphasize area/volume interpretations; don't
emphasize the tangent/normal vectors as part of the "function" being
integrated (but rather as part of dr-vector or dA-vector).

> Tevian and I differ most significantly in our approach to surface 
> integrals.  

>> In computing surface integrals, I encourage students to come up with
>> vector dA and then get its magnitude.

I now realize that you've done for surface integrals what I do for line
integrals: do the scalar case first *BUT* describe the surface using
vectors.  I encourage students to use dr-vector to do scalar line
integrals (pretzels dipped in chocolate); you encourage them to use
dA-vector to do scalar surface integrals.

The more I think about this, the less I feel we disagree.  The issue in
my mind is not whether scalar or vector integrals come first, but
whether one uses vectors (dr and dA) for all line and surface integrals,
including scalar integrals.  Put another way, line and surface elements
are fundamentally vector objects, not scalar -- the latter are derived
from the former, as in ds=|dr-vector|, dA=|dA-vector|, and not the other
way around, namely dr-vector=T ds, dA-vector = n dA.

Tevian