The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Function f of Two Variables

By the graph of a function f of two variables we mean the graph of the equation z = f(x, y). Recall that a real function of two variables is a set of ordered triples (x, y, z) such that for each (x, y) there is at most one z with z = f(x,y). Geometrically this means that the graph of a function intersects each vertical line through (x, y) in at most one point (x, y, z). The value of z is the height of the surface above (x, y). Figure 11.1.3 shows part of a surface z = f(x, y).


Figure 11.1.3

Whenever one quantity depends on two others we have a function of two variables. The height of a surface above (x, y) is one example. A few other examples are: the density of a plane object at (x, y), the area of a rectangle of length x and width y, the size of a wheat crop in a season with rainfall r and average temperature t, the number of items which can be sold if the price is p and the advertising budget is a, and the force of the sun's gravity on an object of mass m at distance d.

A rough sketch of the graph can be very helpful in understanding a function of two variables or an equation in three variables. In this section we do two things. First we describe a class of surfaces whose equations are simple and easily recognized, the quadric surfaces. After that we shall give a general method for sketching the graph of an equation. Graph paper with lines in the x, y, and z directions is available in many bookstores.

The graph of a second degree equation in x, y, and z is called a quadric surface. These surfaces correspond to the conic sections in the plane. There are several types of quadric surfaces. We shall present each of them in its simplest form.

Last Update: 2006-11-17