Exact Differential

A differential of the form

is exact (also called a total differential) if intdf is path-independent. This will be true if

so P and Q must be of the form

But

so

There is a special notation encountered especially often in statistical thermodynamics. Consider an exact differential

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

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

Explore with Wolfram|Alpha

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 Resource. https://mathworld.wolfram.com/ExactDifferential.html

Subject classifications