For a function dependent on two variables x and y, a change in f is brought about by changes in x and y. For an infinitesimal change df we have
df = ( f/x ) y dx + (f/y) x dy
(21.15)
The quantity df is called an exact differential if the second partial derivative, taken in two different orders are identical i.e., if eq.(21.17) is satisfied. Here (f /x ) y is the partial derivatives of f with respect to x when y is held constant. Let
df = Mdx + Ndy
(21.16)
where M = (f/ x) y and N = ( f/ y) x . Now, f is an exact differential if and only if
( M/ y) x = (N/ x) y
(21.17)
These relations can be easily verified for typical functions such as f (x, y) = x 3 e y. The thermodynamic functions U, H and S are exact differentials, and so are P and V.
Applying eq.(21.17) to dU = TdS-PdV, we have
( T/ V) S = -( P/ S) v
(21.18)
Similarly H = U + PV; dH = TdS – PdV +VdP+ PdV and
