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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 left-hand side f(x,y)=f is treated as a variable that can itself be held constant.


See also

Inexact Differential, Partial Derivative, Pfaffian Form

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References

Thomas, G. B., Jr. and Finney, R. L. Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, 1996.

Referenced on Wolfram|Alpha

Exact Differential

Cite this as:

Weisstein, Eric W. "Exact Differential." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExactDifferential.html

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