Section 7.1: Functions of Several Variables

Functions of Several Variables

In general, an equation of the form:

z = f(x, y)

describes a function of two independent variables if, for each permissible ordered pair (x, y), there is one and only one value of z determined by f(x, y).

• The variables x and y are independent variables.
• The variable z is a dependent variable.
• The set of all permissible values of x and y is the domain of the function.
• The set of all corresponding values f(x, y) is the range of the function.

We can similarly define functions of three independent variables, w = f(x, y, z), or functions of four independent variables, u = f(w, x, y, z), and so on.

Three-Dimensional Coordinate Systems

Since functions of the form z = f(x, y) involve two independent variables x and y, and one dependent variable z, we need a three-dimensional coordinate system for their graphs.

• This is formed by three mutually perpendicular number lines intersecting at their origins.
• Every ordered triplet of numbers (x, y, z) can be associated with a unique point.
• Every unique point can be associated with a triplet of numbers (x, y, z).

Drag the sliders to see the point(x, y, z) plotted in a three-dimensional coordinate system.

Graphs of Functions of Two Variables

In general, the graph of any function of the form z = f(x, y) is called a surface. The graph of such a function is the set of all ordered triplets of numbers (x, y, z) that satisfy the equation.

Drag the sliders to see the graphs of the cross sections of the functions f(x, y) = x² + y² and g(x, y) = y² - x² holding either x or y constant.