[BUG] partial derivative product rule

[Matthias Kawski]

I am not the physicist here -- but recall from Marsden's book (?)
(or simialr) some plausible examples from theromdynamics (a little
more sophisticated models than Boyle's law) where it might be hard
experimentally to measure one desired derivative (forgot which was
difficult to hold constant), but this nice relationship allowed
one to measure two other partials (supposedly easy to hold the
corresponding quantities fixed), and then obtain the desired one
via simple arithmetic (-- but the minus sign is critical!).

Tevian should fill us in about the physics details -- don't down-
play the physics.... I would not be suprised if there are similar
applications from the life sciences aned economics -- anytime
there is a conservation law, the same reasoning applies!

Regarding notation -- this is what the scientists supposedly want,
differentiate w.r.t. variables/quantities, as opposed to taking
derivatives like f --> f'.

Re geometry, instead of drawing any complicated surface
F(x,y,z)=const, it is perfectly sound to consider linear
(or affine F).
Draw a little three-sided pyramid w/ corners
(x0,y0,z0), (x0+dx,y0,z0), (x0,y0+dy,z0),(x0,y0,z0+dz).
May just as well take (x0,y0,z0)=(0,0,0).....

Now comes the key -- of course dx, dy, dz can each have either
sign -- but depending on the linear function L only some combi-
nations may occur. So rather than just plotting one pyramid, plot
all 8 of them, giving the usual octahedron (basically the faces
correspond to parts of the surfaces of F(x,y,z)= eps_x * x + eps_y
*y + eps_z * z = eps_0 with each eps either 1 or -1.  Holding any
of the variables constant corresponds to the inter- section of
this plane (face) w/ one of the coord planes. Each of the
resulting lines has either + or - slope (recall that the sign of
dy/dx is the same as the sign of dx/dy).
Thus, we can look at which of the 12 edges have positive and which
have negative slopes in the corresponding coordinate planes. Not
unexpected there will be 6+ and 6-, but they are NOT perfectly
symmetric: The - edges form 2 loops of 3 edges, while the + edges
form a single loop of 6 edges.
As a consequence, each face (each corresponding to a different
conse5rvation law F=const) is bounded by either 3 - edges or
1 - edge and 2 + edges.
The product of all 3 is necessarily - , i.e. the reason is
purely combinatorial.
Below is some naive maple code to plot in color these edges --
rotate to convince yourself.

> restart;
> with(plots):
> x:=[-1,0,0];X:=[1,0,0];y:=[0,-1,0];Y:=[0,1,0];z:=[0,0,-1];Z:=[0,0,1];
> display([spacecurve([X,Y],color=red,thickness=4)
>         ,spacecurve([X,Z],color=red,thickness=4)
>         ,spacecurve([Y,Z],color=red,thickness=4)
>         ,spacecurve([x,z],color=red,thickness=4)
>         ,spacecurve([x,y],color=red,thickness=4)
>         ,spacecurve([y,z],color=red,thickness=4)
>         ,spacecurve([y,X],color=blue,thickness=4)
>         ,spacecurve([z,X],color=blue,thickness=4)
>         ,spacecurve([y,Z],color=blue,thickness=4)
>         ,spacecurve([x,Y],color=blue,thickness=4)
>         ,spacecurve([x,Z],color=blue,thickness=4)
>         ,spacecurve([z,Y],color=blue,thickness=4)
>    ],axes=normal,tickmarks=[3,3,3],
> orientation=[14,78]);





Matthias
**********************************************************
Matthias Kawski                http://math.asu.edu/~kawski
Dept. of Mathematics and Statistics         kawski at asu.edu
Arizona State University            office: (480) 965 3376
Tempe, Arizona 85287-1804           home:   (480) 893 0107
**********************************************************