Exact Differential

A differential of the form

(1)

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

(2)

so and must be of the form

(3)

But

(4)

(5)

so

(6)

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

(7)

Then the notation , sometimes referred to as constrained variable notation, means "the partial derivative of with respect to with held constant." Extending this notation a bit leads to the identity

(8)

where it is understood that on the left-hand side is treated as a variable that can itself be held constant.