[BUG] maps for vector calculus

[Martin Jackson]

At 5:10 PM -0700 4/25/04, Tevian Dray wrote:
>  >>>>> Martin Jackson writes:
>
>     MJ> I've developed the attached maps" to give some visual help
>     MJ> with this.  The first version includes some special cases
>     MJ> relevant only to planar fields ("normal line integrals",
>     MJ> a planar divergence theorem, and Green's theorem).  The
>     MJ> second includes only Green's theorem as a special planar case.
>     MJ> The second version is obviously a lot cleaner.  I suspect
>     MJ> Tevian will object to the first version.  I include it because
>     MJ> the reality of doing vector calculus in a third-semester
>     MJ> multivariate calculus course often means "bailing out" at
>     MJ> the level of planar fields if there is not time to do surface
>     MJ> integrals justice.
>
>Beautiful!  Yes, I prefer the second.  It's not so much that I object to the
>first, but that I'm not sure how helpful it is.  First and foremost, the
>"planar divergence theorem" is a special case of Green's Theorem, and hence
>should be connected to Stokes' Theorem rather than the (3-d) Divergence Thm.
>This really does make it the odd one out in the list of theorems, and I'm not
>at all sure what to do with it -- so my first choice would be to omit it.  I
>also don't like the label "normal line integral"; normal is ambiguous.
>
>One possible way around both of these problems would be to explicitly mention
>work and flux, instead of, or possibly in addition to, the boxes labeled "line
>integral" and "surface integral".  The point is that flux goes with a
>divergence theorem in both 2 and 3 (and in fact any number of) dimensions.
>However, this also isn't ideal, since not all vector line integrals represent
>work, but perhaps that doesn't matter in this context.
>
>Tevian
>
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I agree that the usual line of reasoning places the "planar 
divergence theorem" (PDT for short in the following) as a special 
case of Green's theorem.  That is, one first proves Green's theorem 
and then uses Green's theorem to give an easy and clean proof of the 
PDT.  I don't find this proof using Green's theorem to be very 
illuminating.  An alternate approach would be to give a direct proof 
of the PDT and then use the PDT to give a proof of Green's Theorem. 
I think you could go either way so perhaps the neutral thing to say 
is that Green's Theorem and the PDT are equivalent.

In the past, I have chosen a direct proof of the PDT as a path to a 
proof of the (3D) Divergence Theorem.  I really just outline an 
argument for the PDT so someone demanding rigor would find this 
lacking.  The outline is fairly standard:  first, argue that 
divergence div F is equal to flux density (flux per unit area in the 
planar case); second, argue that
	 d(flux)=(flux density)(dA)=(div F)(dA)
is additive for adjacent rectangles; and third, argue that total flux 
is thus the sum of (div F)(dA) on the one hand and the "normal" line 
integral on the other hand.  Having done this, it seems more natural 
to view the PDT as a special case of the (3D) Divergence Theorem.

This semester, I have gone straight to the theorems in three 
dimensions.  In class today, I stated Stokes' and the Divergence 
Theorem after having tried to build some intuition within my students 
for "fluid flow" interpretations of line integral, surface integral, 
divergence, and curl.  Very roughly, I used

line integral=measure of how much fluid flow helps in moving a bead 
along a rigid wire immersed in the flow (with circulation as the 
special case of a closed loop);

surface integral=measure of how much fluid flows across a rigid net 
immersed in the flow (with some designation of positive and negative 
sides of the surface);

divergence=rate of change in area/volume of a collection of points 
moving with the flow; and

curl=rate of rotation for a paddlewheel moving with the flow.

All of these are standard, although the "bead-on-a-wire" view is 
recent to me.  I like this because it eliminates the need to think 
about why the fluid isn't pushing the particle off the designated 
path.  (I also had fun having students recall that child's toy which 
consists of three or four brightly colored rigid wires bent into 
interesting shapes and attached to a base with beads on each to push 
around.  My story is that we jumped into a river with this toy and 
are playing with it in the current.)  I don't emphasize the concept 
of work for building intuition because most of my students have not 
had university physics and I think they work to be a fairly abstract 
concept.  On the other hand, the fluid flow interpretations do 
introduce time into what is really a static picture.

In any case, my plan is to outline arguments for Stokes' Theorem and 
the Divergence Theorem in the next class meeting.  Right now, I think 
I will do divergence=flux density in space, i.e., with flux density 
as flux per unit volume.  The argument is not much more involved than 
the planar argument since there is only one additional pair of faces 
for which to account.  My concern is with whether my students are 
really ready to think about vector fields in space.  I'm not sure how 
much they can "see" of vector fields in space since almost every 
picture of a vector field we have drawn is planar.  This has been one 
of the primary reasons I have done the planar cases first in the past.

Getting back to Tevian's comments about the "vector calculus map,"  I 
had considered adding a region called "Interpretations" with flux, 
flux density, circulation and circulation density as specific things 
within this region.  There would be connections:  flux to surface 
integral, flux density to divergence, circulation to line integral, 
and circulation density to curl.   I decided against this because of 
the overhead in complexity.  If I find time, I'll try a version with 
this. It would be very nice to have a tool allowing one to switch 
features in the map on and off.  There could be a switch to turn this 
"interpretations" region on (visible) or off (invisible).

In case anyone is still reading at this point, I am looking for help 
with a small and specific thing.  In the "map" there is a connection 
between curl and conservative vector field.  Since posting the map I 
revised this connection to make it a double-headed arrow with a 
dashed line to indicate a different type of relation than the solid 
line arrows.  In the key, the arrows with non-filled heads indicate a 
relation of "special case."  I'm trying to think of how to describe 
the type of relation that exists between curl and conservative vector 
fields.  It's different than the standard arrows which indicate a 
hierarchical dependency (e.g., the arrows leading to Stokes' Theorem 
tell us what we need to know before we can state Stokes'.)  I would 
appreciate any suggestions for describing this other type of relation.

OK, enough rambling for now.  I find myself writing a lot here 
because I'm processing a lot of different ideas as my course comes to 
a close.  It's helpful to imagine an audience of others who have 
thought about these issues as I compose my thoughts.  Please 
challenge pieces or the entirety of what I'm putting out.

Martin
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