Page 192 - Schaum's Outline of Theory and Problems of Advanced Calculus

Applications of Partial

Derivatives

APPLICATIONS TO GEOMETRY
The theoretical study of curves and surfaces began
more than two thousand years ago when Greek phi-
losopher-mathematicians explored the properties of
conic sections, helixes, spirals, and surfaces of revolu-
tion generated from them. While applications were
not on their minds, many practical consequences
evolved. These included representation of the ellipti-
cal paths of planets about the sun, employment of the
focal properties of paraboloids, and use of the special
properties of helixes to construct the double helical
model of DNA.
The analytic tool for studying functions of more
than one variable is the partial derivative. Surfaces are
a geometric starting point, since they are represented
by functions of two independent variables. Vector
forms of many of these these concepts were introduced
in the previous chapter. In this one, corresponding
coordinate equations are exhibited. Fig. 8-1

1. Tangent Plane to a Surface. Let Fðx; y; zÞ¼ 0be the equation of a surface S such as shown in
Fig. 8-1. We shall assume that F, and all other functions in this chapter, is continuously differentiable
unless otherwise indicated. Suppose we wish to find the equation of a tangent plane to S at the point
Pðx 0 ; y 0 ; z 0 Þ. A vector normal to S at this point is N 0 ¼rFj , the subscript P indicating that the
P
gradient is to be evaluated at the point Pðx 0 ; y 0 ; z 0 Þ.
If r 0 and r are the vectors drawn respectively from O to Pðx 0 ; y 0 ; z 0 Þ and Qðx; y; zÞ on the plane, the
equation of the plane is
ðr r 0 Þ N 0 ¼ðr r 0 Þ rFj P ¼ 0 ð1Þ
since r r 0 is perpendicular to N 0 .
183
Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.