[BUG] maps for vector calculus

[Tevian Dray]

> Having done this, it seems more natural 
> to view the PDT as a special case of the (3D) Divergence Theorem.

I like the analogy, but am still uncomfortable with this argument.  Green's
Theorem really is equivalent to Stokes' Theorem; I know how to argue in both
directions.  But the 2d and 3d divergence theorems are not equivalent in this
sense; I don't know how to derive either one from the other except by analogy
(or perhaps using differential forms).  Perhaps this bias reflects the
traditional algebraic viewpoint, and one should accept the geometric argument
of equivalence.  But I am still not convinced that the "special case" arrow
connecting these theorems is justified.

> ... 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.

This is excellent -- and much better than my attempts to motivate line
integrals using a handwaving discussion of the effect of wind on sailboats.

> 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.

One possibility might be to have a double-headed "equivalence" arrow, which
could be used between curl and conservative, and also between Green's Theorem
and the 2d Divergence Theorem (and also, I would argue, between Green's
Theorem and Stokes' Theorem).

Tevian