Page 198 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 189

Gðx 1 ; x 2 ; ... ; x n Þ F þ 1 1 þ 2 2 þ þ k k ð23Þ
subject to the (necessary) conditions
@G @G @G
¼ 0; ¼ 0; ... ; 0 ð24Þ
@x 1 @x 2 @x n
where 1 ; 2 ; ... ; k , which are independent of x 1 ; x 2 ; ... ; x n , are the Lagrange multipliers.

APPLICATIONS TO ERRORS
The theory of differentials can be applied to obtain errors in a function of x; y; z, etc., when the
errors in x; y; z, etc., are known. See Problem 8.28.

Solved Problems

TANGENT PLANE AND NORMAL LINE TO A SURFACE
2 2
8.1. Find equations for the (a) tangent plane and (b) normal line to the surface x yz þ 3y ¼
2
2xz 8z at the point ð1; 2; 1Þ.
2
2
2
(a) The equation of the surface is F ¼ x yz þ 3y 2xz þ 8z ¼ 0. A normal to the surface at ð1; 2; 1Þ is
2 2 2
N 0 ¼rFj ð1;2; 1Þ ¼ð2xyz 2z Þi þðx z þ 6yÞj þðx y 4xz þ 8Þkj ð1;2; 1Þ
¼ 6i þ 11j þ 14k
Referring to Fig. 8-1, Page 183:
The vector from O to any point ðx; y; zÞ on the tangent plane is r ¼ xi þ yj þ zk.
The vector from O to the point ð1; 2; 1Þ on the tangent plane is r 0 ¼ i þ 2j k.
The vector r r 0 ¼ðx 1Þi þð y 2Þj þðz þ 1Þk lies in the tangent plane and is thus perpen-
dicular to N 0 .
Then the required equation is
ðr r 0 Þ N 0 ¼ 0 i:e:; fðx 1Þi þð y 2Þj þðz þ 1Þkg f 6i þ 11j þ 14kg¼ 0
6ðx 1Þþ 11ð y 2Þþ 14ðz þ 1Þ¼ 0 or 6x 11y 14z þ 2 ¼ 0
(b)Let r ¼ xi þ yj þ zk be the vector from O to any point ðx; y; zÞ of the normal N 0 . The vector from O to
the point ð1; 2; 1Þ on the normal is r 0 ¼ i þ 2j k. The vector r r 0 ¼ðx 1Þi þð y 2Þj þðz þ 1Þk
is collinear with N 0 . Then

i j
k

ðr r 0 Þ N 0 ¼ 0 i:e:; x 1 y 2 z þ 1 ¼ 0

6 11 14
which is equivalent to the equations
11ðx 1Þ¼ 6ð y 2Þ; 14ð y 2Þ¼ 11ðz þ 1Þ; 14ðx 1Þ¼ 6ðz þ 1Þ
These can be written as
x 1 y 2 z þ 1
6 ¼ 11 ¼ 14
often called the standard form for the equations of a line. By setting each of these ratios equal to the
parameter t,we have
x ¼ 1 6t; y ¼ 2 þ 11t; z ¼ 14t 1
called the parametric equations for the line.