Page 259 - Schaum's Outline of Theory and Problems of Advanced Calculus

250 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

so that
ðð ð ðð
@A 3
A 3 k n dS
@z
ð1Þ dV ¼
V S
Similarly, by projecting S on the other coordinate planes,
ððð ðð
@A 1
A 1 i n dS
@x
ð2Þ dV ¼
V S
ððð ðð
@A 2
A 2 j n dS
ð3Þ dV ¼
@y
V S
Adding (1), (2), and (3),
ððð ðð
@A 1 @A 2 @A 3
ðA 1 i þ A 2 j þ A 3 kÞ n dS
@x þ @y þ @z dV ¼
V S
ðð ð ðð
or r A dV ¼ A n dS
V S
The theorem can be extended to surfaces which are such that lines parallel to the coordinate axes meet
them in more than two points. To establish this extension, subdivide the region bounded by S into
subregions whose surfaces do satisfy this condition. The procedure is analogous to that used in Green’s
theorem for the plane.
2
2
10.23. Verify the divergence theorem for A ¼ð2x zÞi þ x yj xz k taken over the region bounded by
x ¼ 0; x ¼ 1; y ¼ 0; y ¼ 1; z ¼ 0; z ¼ 1.
ðð
We first evaluate A n dS where S is the surface of the cube in Fig. 10-19.
S
Face DEFG: n ¼ i; x ¼ 1. Then
1 1
ðð ð ð
2
fð2 zÞi þ j z kg i dy dz
A n dS ¼
0 0
DEFG
1 1
ð ð
ð2 zÞ dy dz ¼ 3=2
¼
0 0
Face ABCO: n ¼ i; x ¼ 0. Then
1 1
ðð ð ð
ð ziÞ ð iÞ dy dz
A n dS ¼
0 0
ABCO
1
ð ð 1
zdy dz ¼ 1=2
¼
0 0
Fig. 10-19
Face ABEF: n ¼ j; y ¼ 1. Then
1 1 1 1
ðð ð ð ð ð
2 2 2
A n dS ¼ fð2x zÞi þ x j xz kg j dx dz ¼ x dx dz ¼ 1=3
0 0 0 0
ABEF
Face OGDC: n ¼ j; y ¼ 0. Then
1 1
ðð ð ð
2
fð2x zÞi xz kg ð jÞ dx dz ¼ 0
A n dS ¼
0 0
OGDC