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

[Martin Jackson]

My current approach to these topics differs from what Tevian describes 
below.  For context, note that I am teaching a multivariable calculus 
course as the third semester of our calculus sequence.  Most of the 
students are majoring or planning to major in the natural sciences 
(chemistry, physics, biology), the social sciences (primarily 
economics), or mathematics.  A few are interested in our dual-degree 
engineering program.

My approach to line and surface integrals has changed over the years.  
I've recently settled into an approach that I think works well for 
students (a claim for which I don't have quantitative evidence to 
support).  I now do scalar line and surface integrals before vector 
versions of either.

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.  My first discussion is 
on the constant density case, being careful to distinguish length 
density, surface density, and volume density.  Following physics 
conventions, I denote these by lambda, sigma, and rho, respectively.  
In examples and problems, I use various densities such as cost, mass, 
charge, and probability.  (Think of the difficulties that arise for 
anyone who doesn't distinguish between cost density and total cost when 
talking about a problem such as minimizing the total cost of a box (of 
fixed volume) given cost densities for each of the sides.)  I often use 
charge density since it naturally allows for positive and negative 
values.

I continue developing this theme with a review of definite integrals as 
a prelude to multiple integrals.  I encourage students to downplay the 
"integral is area under the curve" interpretation.  I then use "total 
is sum of density times measure" as the default theme for double 
integrals rather than the more standard default interpretation of 
"volume under the graph".

I try to word examples and problems without reference to a coordinate 
system.  Here's an example: "Charge is arranged on a flat rectangle of 
dimensions L by W. The charge density is proportional to the square of 
the distance from one corner.  Compute the total charge."  If needed, 
I'll resolve ambiguity with a diagram (but one that does not include 
coordinate axes). I encourage students to start by writing down a 
"schematic" integral.  For this example, I want to see something like 
"Q=integral_R sigma dA" where R is the name of the relevant planar 
region.  After choosing an appropriate coordinate system, we work out 
the details to get an equivalent iterated integral in the two 
coordinate variables. Working out the details include finding a 
coordinate expression for sigma, giving a coordinate description of the 
region, and writing down a coordinate expression for dA.

After covering double and triple integrals, we move to scalar line 
integrals as a generalization of definite integrals from adding up 
along a straight thing to adding up along a curvy thing.  A typical 
problem would be "Charge is distributed on a semi-circle of radius R.  
The charge density is proportional to the distance from the diameter 
containing the ends of the semi-circle.  Compute the total charge."

Surface integrals come next as a generalization of double integrals 
from adding up over a planar region to adding up over a curvy surface.  
In computing surface integrals, I encourage students to come up with 
vector dA and then get its magnitude.

All of this happens before thinking about vector fields (other than the 
specific example of gradient vector fields from earlier in the course.) 
  After an introduction to vector fields (but before any mention of 
divergence and curl), we look at vector line integrals and vector 
surface integrals.  I emphasize a "fluid flow" interpretation in which 
we think of the vector field as giving fluid flow velocity at each 
point.  A vector line integral tells us the net help provided by the 
fluid flow in pushing a bead along a rigid wire in the shape of the 
curve.  A vector surface integral gives us flux in the way Tevian 
describes below.  I do not emphasize a work interpretation of vector 
line integral because a significant percentage of my students have not 
had a physics course.  Work is a relatively abstract concept, 
particularly when isolated from the energy and conservation of energy.  
On the other hand, the fluid flow interpretation of "net help provided 
by the flow" is not precise and introduces time into a static picture 
in a way that can be awkward or distracting.  I'm not satisfied with 
this and need to rethink my de-emphasis on a work interpretation.

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).  
Here's what not to do (at least for my audience): Define scalar line 
integral and then define vector line integral in terms of scalar line 
integral.  That is, do not define the line integral of the vector field 
F as the scalar integral of the scalar quantity F.T where T represents 
unit tangent vectors along the curve.  This approach does illustrate a 
standard mathematical style of defining new things in terms of old 
things.   This provides economy of definitions but mucks up important 
and distinct interpretations of scalar and vector line integrals.

Tevian and I differ most significantly in our approach to surface 
integrals.  One thing I like about my approach is that it allows 
students to get comfortable with the often-involved business of 
constructing vector dA and its magnitude from a description of the 
surface (geometric, implicit, parametric,...) while dealing with a 
by-then familiar theme of "quantity from density".  When it comes time 
to work with vector surface integrals, students can focus on the new 
aspects (the vectorness of dA, adding up the normal component of a 
vector field along the surface, and a flux interpretation) because they 
have some familiarity with vector dA if only in passing on the way to 
its magnitude.

Sorry for the long (and long delayed) post.

Martin



On Feb 15, 2007, at 2:37 PM, Tevian Dray wrote:

> 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
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