Surfaces

In this module we study functions of two variables

z = f(x, y)

These functions are important because there are many situations in which one quantity, z, depends on two other quantities, x and y. One of the most common uses of such functions is to describe height or elevation in hilly terrain. In this situation x and y represent the location of a point measured at sea level -- x is its location along an east-west axis and y is its location along a north-south axis. Then z represents the elevation of the terrain above sea level. Because we have so much familarity with this situation it helps us study these functions more generally.

The variables x and y are called independent variables and the variable z is called the dependent variable because it depends on the variables x and y. Functions of two variables, like functions of one variable, describe a situation in which the dependent variable is uniquely and completely determined by the independent variable(s). For example, we cannot use a function of two variables to describe the terrain shown below because of the arch -- at each point (x, y) where the arch is located there are three elevations -- the ground, the underside of the arch, and the top of the arch.

Missing photograph

The photographer who took the picture of the arch above took it near sunset so that the lighting would accentuate the colors of the arch and its three dimensionality. In the graph below we used three light sources illuminating the surface from different directions to help us visualize its three-dimensionality.

The graph shown below shows the surface, or terrain, described by the function

z = exp(-(x2 + y 2))

Missing graph

The tools we have been studying give us many different ways of looking at two dimensional objects like surfaces. We can look at these objects from various different perspectives, using various different kinds of lighting and shading. We can even do something more artificial like the graph below. This graph is colored using a completely different technique. The color of each point on the surface is determined by its height.

Missing graph

Contour Maps:

Next we look at contour maps. You have probably seen many contour maps although you may not have used this term before. Hikers often use contour maps to help plan a trip in the mountains although they usually call them topographic maps or, more familiarly "tope sheets." Contour maps are often used to describe the weather -- for example, temperatures or barometric pressure. We begin our study of contour maps by looking at the function

z = 2 x + 3 y

shown in the graph below.

Missing Graph

The following remarks may help understand this function and the surface it describes.

The graph below shows a contour map for this function. The horizontal axis is the x-axis and the vertical axis is the y-axis. Each of the diagonal lines represents one value of z.

  • The red line represents z = 0 -- that is, 

    
  0 = 2x + 3y
-3y = 2x

         2
  y = - --- x
         3

  • The green line represents z = 5 -- that is, 

    
  5 = 2x + 3y
-3y = 2x - 5

         2       5
  y = - --- x + ---
         3       3

  • The blue line represents z = -5 -- that is, 

    
 -5 = 2x + 3y
-3y = 2x + 5

         2       5
  y = - --- x - ---
         3       3

  • The other diagonal lines represent z = -9, -8, ... 9.

Missing graph

There is a lot of information to be read from a contour map like this one. Notice, for example, that if you start at the origin and walk one unit north -- that is, along the y-axis from the origin to the point (0, 1) -- then you cross three bands, climbing from z = 0 to z = 3. If, however, you walk one unit east from the origin along the x-axis to the point (1, 0) then you cross only two bands, climbing from z = 0 to z = 2. Notice that the closeness of the contour lines as you walk along represents the steepness of the surface.

The two graphs below examine the function

z = x2 + y2

The graph on the left shows a three-dimensional view of this surface colored by height. The graph on the right is a contour plot enhanced by color, with the same colors as the first graph. Notice that as you move further from the origin, the contour lines get closer together because the surface is getting steeper.

Missing graph Missing

Click here to see a map from the University of Wisconsin Space Science and Engineering Center that uses color to represent ocean surface temperatures.

Use the tools in your CAS window to study each of the surfaces below. Use both three-dimensional plots and contour plots. Then describe each surface in words.

Copyright c 1997 by Frank Wattenberg, Department of Mathematics, Montana State University, Bozeman, MT 59717