Understanding Engineering Mathematics

(x,y)

z

x

x = f(x,y) y

Figure 16.1Surface defined byz=f(x,y).

x

y

z

ax+by

+c =^0

z

= ax

+c

z =

by+

c

Figure 16.2The linear functionz=ax+by+c.

For example, thelinear function:

z=ax+by+c

represents aplanein 3-dimensional space, as shown in Figure 16.2 (only the portion in
the first octant is sketched).
In the theory of single variable we measure the rate of increase or slope of a curve by
the slope of the tangent to the curve at a given point. In the case of functions with two
variables, where we have to deal with asurface,wemeasuretherate of increaseorslope
of a surfaceby theorientationorslopeof atangent planeto the surface. Thus at any
point on the surface we can define auniqueplane, the tangent plane, which touches the
surface at just that one point,locally. The orientation of this plane clearly gives a measure