ABSTRACT
Consider a function of two variables x and y defined by z= f (x, y)=3x2−2y. If (x, y)=(0,0), then f (0,0)=0 and if (x , y)=(2,1), then f (2,1)=10. Each pair of numbers, (x, y), may be represented by a point P in the (x, y) plane of a rectangularCartesian co-ordinate system, as shown in Figure 62.1. The corresponding value of z= f (x, y) may be represented by a line PP ′ drawn parallel to the z-axis. Thus, if, for example, z=3x2−2y, as above, and P is the co-ordinate (2, 3) then the length of PP ′ is 3(2)2−2(3)=6. Figure 62.2 shows that when a large number of (x, y) co-ordinates are taken for a function f (x, y), and then f (x, y) calculated for each, a large number of lines such as PP ′ can
be constructed, and in the limit when all points in the (x, y) plane are considered, a surface is seen to result as shown in Figure 62.2. Thus the function z= f (x, y) represents a surface and not a curve.
