Page 258 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 249

is an example of a one-sided surface. This is sometimes called a nonorientable surface. A two-sided surface
is orientable.

THE DIVERGENCE THEOREM
10.22. Prove the divergence theorem. (See Fig. 10-18.)

Fig. 10-18

Let S be a closed surface which is such that any line parallel to the coordinate axes cuts S in at most two
points. Assume the equations of the lower and upper portions, S 1 and S 2 ,tobe z ¼ f 1 ðx; yÞ and z ¼ f 2 ðx; yÞ,
respectively. Denote the projection of the surface on the xy plane by r. Consider

ððð ðð ð ðð ð

f 2 ðx;yÞ
@A 3 @A 3 @A 3
dz dy dx
@z dV ¼ @z dz dy dx ¼ z¼f 1 ðx;yÞ @z
V V r
2
ðð f ðð
½A 3 ðx; y; f 2 Þ A 3 ðx; y; f 1 ފ dy dx
¼ A 3 ðx; y; zÞ dy dx ¼

z¼f 1
r r
For the upper portion S 2 , dy dx ¼ cos
2 dS 2 ¼ k n 2 dS 2 since the normal n 2 to S 2 makes an acute angle

2 with k.
For the lower portion S 1 , dy dx ¼ cos
1 dS 1 ¼ k n 1 dS 1 since the normal n 1 to S 1 makes an obtuse
angle
1 with k.
ðð ðð
Then A 3 ðx; y; f 2 Þ dy dx ¼ A 3 k n 2 dS 2
r S 2
ðð ðð
A 3 k n 1 dS 1
A 3 ðx; y; f 1 Þ dy dx ¼
r S 1
and
ðð ðð ðð ðð
A 3 k n 1 dS 1
A 3 ðx; y; f 2 Þ dy dx A 3 ðx; y; f 1 Þ dy dx ¼ A 3 k n 2 dS 2 þ
r r S 2 S 1
ðð
A 3 k n dS
¼
S