Exactness Criterion for Differentials

Date: 06/14/99 at 03:58:54
From: Eric
Subject: Exact Differentials

I'm reading a text that uses the "exactness criterion" for the 
coefficients of the differentials to derive an equation. I know it 
sounds easy, but I simply cannot think of what they are talking about. 
I've checked my math books, and school's out now so I can't ask anyone 
on campus. Anyway, here are the details:

Dealing on Constant-Volume Stressed Bars

from dS = C(dT/T) + Va(do), the exactness criterion is applied to the 
coefficients of the differentials dT and do to get:

(dC/do)(@ constant T) = TV(da/dT)(@ constant o)

Thanks,
Eric Rosson

Date: 06/14/99 at 07:58:20
From: Doctor Jerry
Subject: Re: Exact Differentials

Hi Eric,

Thanks for your question. It reminds me that long ago I wrote my 
calculus instructor for help during the summer. He replied. Can I do 
less?

If you start with a function f(x,y) and form the differential df you 
get

   df = f_x*dx + f_y*dy,

where f_x and f_y are the partials of f with respect to x and y, 
respectively.

The question is this: If someone gives you an expression like

   M*dx + N*dy

when can you be sure that there is a function f for which M=f_x and 
N = f_y ? 

The "exactness criterion" I know about is that N_x must equal M_y. 
If this is satisfied, then an f exists such that df = M*dx + N*dy.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/