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/
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