9.6 Some differential properties 267
The representative surface of the functionr 1 = 1 (x
21 + 1 y
2)
122is a vertical cone whose
contours (r 1 = 1 constant)are circles of radius rparallel to the xy-plane. The quantity
−x 2 yis the gradient at(x, y)on the circle.
(iii) The differential volume of a thermodynamic system is (see Example 9.11)
By equation (9.29),
so that
0 Exercises 42, 43
Change of constant variable
Letz 1 = 1 f(x, y)be a function of the variables xand y, and letu 1 = 1 g(x, y)be some other
function of xand y. Consider changes in the variables such that uis constant. Then,
division of the total differential dzby infinitesimal change dxat constant ugives
(9.30)
This equation shows how the partial derivative with respect to xat constant yis
related to the partial derivative with respect to xwhen some functionu(x, y)of xand
yis kept constant.
EXAMPLE 9.16Letz 1 = 1 f(x, y)andu 1 = 1 x
21 + 1 y
2. Then, by equation (9.27),
and, therefore, by equation (9.30),
uy x
z
x
z
x
x
y
z
y
∂
∂
=
∂
∂
−
∂
∂
uyx
y
x
u
x
u
y
x
y
∂
∂
=−
∂
∂
∂
∂
=−
uyxu
z
x
z
x
z
y
y
x
∂
∂
=
∂
∂
∂
∂
∂
∂
V pT
p
T
V
T
V
p
∂
∂
=−
∂
∂
∂
∂
==
α
κ
expannsivity
compressibility
−=
∂
∂
∂
∂
∂
∂
1
TV p
V
p
p
T
T
V
dV
V
T
dT
V
p
dp
pT
=
∂
∂
∂
∂
(for fixed amounttn).