Multivariate Calculus - Bruce E. Shapiro

LECTURE 8. FUNCTIONS OF TWO VARIABLES 55
described by the equation
z = xye −(x2 +y 2 )
It can be represented by a surface something like the one illustrated in figure 8.2.
In addition to the surface plant, another useful tool is the contour plot, illustrated
in the fight-hand illustration of figure 8.2. If we take a surface plot and slice it
along planes parallel to the xy plane we will obtain a series of sections that can be
represented by curves in the xy plane. Each curve, called a countour, represents a
particular slice, or height, above the xy plane. Countour plots are commonly used
by geographers, for example, to draw topographic (e.g., figure 8.3) or weather maps
( see figure 8.4), among other things. Each curve in a contour plot is called a level
curve.
Figure 8.3: Example of a contour plot used to illustrate altitude on a topographic
map. Contours are drawn for every 20 feet of altitude; small numbers next to
contours annotate the contours at 100 foot intervals. [Taken from USGS Digital
Raster file 044072d6.tif for the Stowe, VT, USA quadrangle, as published at
http://en.wikipedia.org/ article ”Topographic Maps.” This image is in the public
domain.]
Example 8.6 Find the level curves of the function
z = 2 − x − y 2
for z = 1, 0, −1.
Math 250, Fall 2006 Revised December 6, 2006.