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

[Tevian Dray]

Most traditional texts discuss scalar line integrals (e.g. arclength or
total charge) before vector line integrals (e.g. work); the same is true
for surface integrals.  At the other extreme, the Calculus Consortium
book (McCallum, Hughes Hallett, Gleason et al; MHG) doesn't ever refer
to scalar integrals at all -- surface integrals to find area, for
instance, are not discussed.

I would like to encourage discussion of these alternatives.  There is
of course the question of whether to cover scalar integrals.  But as
someone who covers them anyway, the more interesting question to me is
which type to introduce first in the classroom.

Here's what I do for line integrals:

* I introduce scalar line integrals first, starting with the amount of
  chocolate on a pretzel.  I postulate a dipping procedure such that the
  density of chocolate on the pretzel is proportional to depth -- the
  deeper parts of the pretzel stay in the melted chocolate longer, and
  hence emerge with more chocolate.  I first dip a straight pretzel
  vertically, then at an angle, then finally a semicircular pretzel, all
  of the same length.

* I then discuss work geometrically, emphasizing the need to determine
  the amount of force *along* the path.  I quickly move on to other
  examples, especially the circulation of the magnetic field around a
  wire.

I find the use of pretzels dipped in chocolate to be a good way to
motivate thinking of integration as "chopping and adding", as well as a
good way to make the transition from single integrals to line integrals.
I should, however, emphasize that my students have seen dr-vector before
we start line integrals, and that the chocolate-dipping lecture is set
up to lead students to calculate ds as the magnitude of dr-vector.
Thus, even though I do scalar line integrals first, I still make
dr-vector fundamental; I only ever mention the traditional relationship
"dr-vector=T dot ds" as an afterthought, if at all.

Before starting on surface integrals, I first spend a day "reviewing"
the properties of the cross product (which I do *not* cover earlier),
emphasizing its use in constructing directed area.  I then spend most of
a class period discussing curves and surfaces.  First, I ask students to
help me make a list of representations of curves -- functions, their
graphs, equations, descriptions in words, smooth 1-d collections of
points, parameterizations, etc.  Then I do the same for surfaces.  Then
I draw (or have the students draw) a vector field and a curve, and ask
what questions one can reasonable ask (work/circulation).  Then I do the
same for surfaces -- and am naturally led to the notion of flux, and the
relevance of directed area.

Now I'm finally ready to discuss surface integrals:

* First I'll talk about a constant water flow through simple geometric
  shapes, usually in several different orientations.

* I quickly move on to situations where the vector field isn't constant.
  I especially like Example 2 in Section 19.1 of MHG, which discusses
  the flux of the magnetic field due to a wire through a square whose
  base is on the x-axis.  After doing the example as stated, I rotate
  the square so that its base is on the line y=x.  Many students will
  see quickly that the answer must be the same; some will also see that
  the computation in cylindrical coordinates is formally identical to
  the previous computation.  But I then go on to insist on doing the
  computation in rectangular coordinates, which provides a natural
  introduction to the relationship dA = dr1 x dr2 (all vectors).

* I then return to dipping chocolate, this time on an ice cream cone
  (one of the labs).  This problem can be solved geometrically, but
  leads naturally to the use of dr-vector and dA = dr1 x dr2.

So here I'm firmly on the side of covering flux before scalar surface
integrals.  Again, I emphasize that (scalar) dA is the magnitude of
dA-vector, and that the latter is fundamental, rather than constructing
dA-vector as (scalar) dA times the unit normal vector and having to
provide a separate construction of (scalar) dA.

Comments?  What do you do?

Tevian