2450_Level Curves and Surfaces

Functions of Two Variables

Recall that a function is like a machine: given an input

the function returns a value

. A function of two variables operates similarly: given an input

the function returns a value

When working with a function of two variables, it may be written as

. We refer to

and

as the independent variables and

as the dependent variable. The set of ordered pairs

that produce an output is the domain of the function and the outputs

that correspond to them make up the range.

Given the function

, its level curves are the two-dimensional curves we get by setting

(where

is a constant). The equations of the level curves are

In the applet below, the left shows the level curves on the

plane while the right shows the surface

. To explore the level curves for different functions, choose an example and change the value of

, using the slider, to see different level curves.