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sritejatheboss
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Mon Aug 31, 2009 2:05 am
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Post subject: Exact Differential 
A differential of the form
df=P(x,y)dx+Q(x,y)dy
(1)
is exact (also called a total differential) if intdf is path-independent. This will be true if
df=(partialf)/(partialx)dx+(partialf)/(partialy)dy,
(2)
so P and Q must be of the form
P(x,y)=(partialf)/(partialx) Q(x,y)=(partialf)/(partialy).
(3)
But
(partialP)/(partialy)=(partial^2f)/(partialypartialx)
(4)
(partialQ)/(partialx)=(partial^2f)/(partialxpartialy),
(5)
so
(partialP)/(partialy)=(partialQ)/(partialx).
(6)
There is a special notation encountered especially often in statistical thermodynamics. Consider an exact differential
df=((partialf)/(partialx))_ydx+((partialf)/(partialy))_xdy.
(7)
Then the notation (partialf/partialx)_y, sometimes referred to as constrained variable notation, means "the partial derivative of f with respect to x with y held constant." Extending this notation a bit leads to the identity
((partialy)/(partialx))_f=-(((partialf)/(partialx))_y)/(((partialf)/(partialy))_x),
(8)
where it is understood that on the last-hand side f(x,y)=f is treated as a variable that can itself be help constant.
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TommyKenneth
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Thu May 13, 2010 4:27 pm
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Post subject: Re: Exact Differential
Great tips to share.. Very helpful.. 
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