The Chemistry Maths Book, Second Edition

9 Functions of several variables

9.1 Concepts

When the equation of state of the ideal gas is written in the form

it is implied that the volume Vof the gas is determined by the values of the pressure

p, the temperature T, and the amount of substance n; that is, Vis a function of the

threevariables p, T,and n. Functions of more than one variable occur widely in

the physical sciences; examples are the thermodynamic functions of state, as in the

above example, and all those physical properties of a system whose values depend

on position. For example, mass density and potential energy were discussed in

Chapter 5 as functions of one variable only, the position along a line. More generally,

functions of positionare functions of the three coordinates of a point in ordinary

three-dimensional space.

Let the variable zbe a function of the two variables xand y. For example, the equation

z 1 = 1 x

2

1 − 12 xy 1 − 13 y

2

gives zas a particular function of xand y. The expression on the right of the equation

defines the function

f(x, y) 1 = 1 x

2

1 − 12 xy 1 − 13 y

2

(9.1)

whose value for a given pair of values of xand yis to be assigned to the variable z. The

variables xand yare called independent variablesif no relation exists between them

such that the value of one depends on the value of the other.

EXAMPLE 9.1The values of the function (9.1) when(x, y) 1 = 1 (2, 1),(x, y) 1 = 1 (1, 0),

and(x, y) 1 = 1 (0, 1)are

f(2, 1) 1 = 12

2

1 − 121 × 121 × 111 − 131 × 11

2

1 = 1 − 3

f(1, 0) 1 = 11

2

1 − 121 × 111 × 101 − 131 × 10

2

1 = 11

f(0, 1) 1 = 10

2

1 − 121 × 101 × 111 − 131 × 11

2

1 = 1 − 3

0 Exercises 1, 2

VfpTn

nRT

p

=,,=()