9.7 Exact differentials 273
is exact when there exists a functionz 1 = 1 z(x, y)such that
(9.44)
The differential can then be equated to the total differential of z:
(9.45)
The general condition that a differential in two variables be exact is that the functions
Fand Gsatisfy
(9.46)
By (9.44), each of these partial derivatives is equal to the mixed second derivative of z,
∂
2z 2 ∂x∂y. The condition (9.46) is sometimes called the Euler reciprocity relation, and
is used in thermodynamics to derive a number of relations, called Maxwell relations,
amongst thermodynamic properties (see Example 9.22).
6The significance of exactness
is discussed in the following section.
EXAMPLE 9.21Test of exactness
(i)F dx 1 + 1 G dy 1 = 1 (x
21 − 1 y
2)dx 1 + 12 xydy.
We have and , and the differential is not exact.
(ii)F dx 1 + 1 G dy 1 = 1 (2ax 1 + 1 by)dx 1 + 1 (bx 1 + 12 cy)dy.
We have and the differential is exact.
It is readily verified that it is the total differential ofax
21 + 1 bxy 1 + 1 cy
2.
0 Exercises 53–55
xy
F
y
b
G
x
∂
∂
==
∂
∂
,
y
G
x
y
∂
∂
=+ 2
x
F
y
y
∂
∂
=− 2
xy
F
y
G
x
∂
∂
=
∂
∂
Fdx Gdy dz
z
x
dx
z
y
dy
yx
+==
∂
∂
∂
∂
F
z
x
G
z
y
yx
=
∂
∂
=
∂
∂
and
6The equality of the mixed second partial derivatives was used by Clairaut in 1739 to test a differential for
exactness (and also by Euler at about the same time). Alexis Claude Clairaut (1713–1765) was one of a family of
20, only one of whom survived the father. He read a paper on geometry to the Académie des Sciences at the age of
13, and was a member at 18 (a younger brother, known as ‘le cadet Clairaut’, published a book on the calculus in
1731 at the age of 15, and died of smallpox a year later). His Recherches sur les courbes à double courbure(Research
on curves of double curvature) in 1731 marked the beginning of the development of a cartesian geometry of three
dimensions.