A rule which assigns a unique number z to each point (x , y) in part or all of the (x , y )-plane is said to define a function of the two variables x and y. The

variables x and y are called independent variables and z is called the depen-

dent variable. It is usual to show this relationship by writing

zf (x, y),

where f(x , y) is the rule relating x and y to the (dependent) variable z . The region in the (x , y)-plane for which z is defined is called the domain of

definition of the function f(x , y ). Unless stated otherwise, the domain of

definition of a function f(x , y) of the two independent variables x and y is

taken to be all the points (x , y) for which f(x , y ) is defined. For example, the



is defined for all points in the (x , y)-plane, so, unless otherwise stated, its

domain of definition is the entire plane. However, if the function concerned is


where the positive square root is taken, z is only defined for all points such

that x2/y20/1, and so its largest possible domain of definition is the interior and boundary of a circle of radius 1 centred on the origin.