Page 244 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 235

rF @r @r @r @r

@x @y
¼ ð15Þ
jrFj @v 1 @v 2
[Now a conclusion of the theory of implicit functions is that from Fðx; y; zÞ¼ 0 (and under appro-
priate conditions) there can be produced an explicit representation z ¼ f ðx; yÞ of a portion of the surface.
This is an existence statement. The theorem does not say that this representation can be explicitly
produced.] With this fact in hand, we again let v 1 ¼ x; v 2 ¼ y; z ¼ f ðv 1 ; v 2 Þ. Then
rF ¼ F x i þ f y j þ F z k
Taking the dot product of both sides of (15) yields
1
F z
¼
@r

jrFj @r

@v 1 @v 2

The ambiguity of sign can be eliminated by taking the absolute value of both sides of the equation.
Then
2 2 2 1=2
@r
@r jrFj ½ðF x Þ þðF y Þ þðF z Þ Š

¼ ¼

@v 1 @v 2 jF z j jF z j
and the surface integral of takes the form
2 2 2 1=2
ðð
dx dy ð16Þ
½ðF x Þ þðF y Þ þðF z Þ Š
jF z j
S
The formulas (14) and (16) also can be introduced in the following nonvectorial manner.
Let S be a two-sided surface having projection r on the xy plane as in the adjoining Fig. 10-4.
Assume that an equation for S is z ¼ f ðx; yÞ, where f is single-valued and continous for all x and y in r .
Divide r into n subregions of area A p ; p ¼ 1; 2; .. . ; n, and erect a vertical column on each of these
subregions to intersect S in an area S p .

Fig. 10-4

Let ðx; y; zÞ be single-valued and continuous at all points of S. Form the sum

n
X
ð p ; p ; p Þ S p ð17Þ
p¼1