Re: function of state vs exact differentials

From: Alfred Einstead (whopkins_at_csd.uwm.edu)
Date: 06/30/04 

Date: 30 Jun 2004 11:30:52 -0700

"Anja" <anja@no.spam.com> wrote:
> if X is a function of state then is dX an exact differential?
> Is it actually a biimplication?

Correction on my previous post:
For 2-forms W = A dy^dz + B dz^dx + C dx^dy
W is exact, W = dX, with X = H dx + J dy + K dz, if and only if
       W = d(H dx + J dy + K dz)
         = (dH/dx dx + dH/dy dy + dH/dz dz)^dx + (similar terms for J,K)
         = dH/dy dy^dx + dH/dz dz^dx + dJ/dx dx^dy + dJ/dz dz^dy
         + dK/dx dx^dz + dK/dy dy^dz
or
       W = (dJ/dx - dH/dy) dx^dy
         + (dH/dz - dK/dx) dz^dx
         + (dK/dy - dJ/dz) dy^dz
since dx^dx = dy^dy = dz^dz = 0 and dy^dx = -dx^dy, dx^dz = -dz^dx
and dz^dy = -dy^dz; or
           A = dK/dy - dJ/dz, B = dH/dz - dK/dx, C = dJ/dx - dH/dy.

The relevant theorem for 2-forms is then
          dA/dx + dB/dy + dC/dz
          if and only if
          A = dK/dy - dJ/dz, B = dH/dz - dK/dx, C = dJ/dx - dH/dy;
with the theorem applicable over regions S that have the property
mentioned in the previous article.