Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
College Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search