[BUG] maps for vector calculus

[Tevian Dray]

>>>>> 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