Graphs, Level Curves, and Level Surfaces

Author:Ku, Yin Bon (Albert)

Multivariable functions In this chapter, we mainly study multivariable functions i.e. real-valued functions of more than one variable. Suppose

is a real-valued function of two variables. We can express this function as follows:

where

is any point in the domain

, a subset of

containing all the possible inputs of the function. Then a unique real number

is assigned by the function for each

. The range of

is the set of all possible outputs of the function i.e. the set of all possible values of

. Example: Suppose

. It domain is

and its range is

. Similarly, we can also consider a function of three variables

, or in general, a function of

variables

. As we know, we can visualize a function of one variable

as a graph in

i.e. it is a set of all points of the form

. Similarly, we can visualize a function of two variable

as a graph in

i.e. it is a set of all points of the form

. In the applet below, you can draw the graph of

and then find the point

on the graph corresponding to the point

. You can try the following functions:

Level curves Consider the graph of

. The contour curve is the intersection of the graph and the horizontal plane

for some real number

. The level curve is the projection of the contour curve onto xy-plane. It can be represented by the equation

. In the applet below, the level curve

is shown in the left panel and the corresponding contour curve is shown in the right panel. You can view the level curves for the following functions:

Moreover, you can use the applet to draw multiple level curves (e.g.

for

) to form a contour map of the graph using the following method: First, enter the command "k=slider(0,5,1)". Then right-click on the level curve and check the box "Show trace". Drag the slider to generate the level curves for

Level surfaces For functions of three variables

, it is not easy to visualize its graph since it is a three-dimensional object in four-dimensional space. However, we can visualize the intersection between the graph and

as it is a two-dimensional object i.e. a surface. It is called the level surface of

. Its equation is

. Example: Let

. For

, its level surface has the equation

i.e. a sphere centered at the origin with radius

. In the applet below, you can visualize the level surface

for various functions. (Note: GeoGebra can only handle the plotting of level surface for quadratic polynomials in three variables.)

Graphs, Level Curves, and Level Surfaces

New Resources

Discover Resources

Discover Topics