262 Chapter 9Functions of several variables
One of the principal uses of the total differential in the physical sciences is in the
formulation of the laws of thermodynamics (see Examples 9.22 and 9.27). In the
following sections we use (9.16) and its limiting form (9.17) to derive a number of
differential and integral properties of functions.
9.6 Some differential properties
The total derivative
In the function of two variables,z 1 = 1 f(x, y), let xand ybe functions of a third
variable t:
x 1 = 1 x(t), y 1 = 1 y(t) (9.19)
Thenz 1 = 1 f(x(t), y(t))is essentially a function of the single variable t, and there exists
an ordinary derivativedz 2 dt. This derivative is called the total derivativeof zwith
respect to t, and can be obtained directly by substituting the functionsx(t)andy(t)
for the variables inf(x,y)and differentiating the resulting function of t. It can also be
obtained indirectly, by dividing the expression (9.16) by ∆tand taking the limit
∆t 1 → 10. Thus, division of
by ∆tgives
(9.20)
and, in the limit∆t 1 → 10 ,
(9.21)
Alternatively, the same result is obtained by dividing the total differential (9.17) by
(infinitesimal) dt.
Equation (9.21) is a generalization of the chain rule (see Section 4.6). For a function
of nvariables,u 1 = 1 f(x
1,x
2,x
3,1=,x
n), in which the variables are all functions of t,
(9.22)
=
∂
∂
=∑
iniiu
x
dx
dt
1du
dt
u
x
dx
dt
u
x
dx
dt
=
∂
∂
∂
∂
1122+
∂
∂
u
x
dx
dt
nndz
dt
z
x
dx
dt
z
y
dy
dt
yx
=
∂
∂
∂
∂
∆
∆
≈
∂
∂
∆
∆
∂
∂
∆
∆
z
t
z
x
x
t
z
y
y
t
yx
∆≈
∂
∂
∆+
∂
∂
z ∆
z
x
x
z
y
y
yx