(iii) On face BFGC the unit normal to the surface is the vector n =ˆe 2 and yhas the
value 1 everywhere so that
∫∫
R
F·dS=
∫ 1
0
∫ 1
0
F·n dS =
∫ 1
0
∫ 1
0
(x−z)dxdz =
∫ 1
0
1
2 dz −
∫ 1
0
z dz = 0
(iv) On face AE0D the unit normal to the surface is the vector n =−ˆe 2 and yhas
the value 0 everywhere so that
∫∫
R
F·dS=
∫ 1
0
∫ 1
0
F·n dS =
∫ 1
0
∫ 1
0
z dzdx =
∫ 1
0
z dz =
1
2
(v) On face ABFE the unit normal to the surface is the vector n =ˆe 3 and zhas the
value 1 everywhere so that
∫∫
R
F·dS=
∫ 1
0
∫ 1
0
F·n dS =
∫ 1
0
∫ 1
0
(x+y)dxdy =
∫ 1
0
1
2
dy +
∫ 1
0
y dy = 1
(vi) On face DCG0 the unit normal to the surface is the vector n =−ˆe 3 and zhas
the value 0 everywhere so that
∫∫
R
F·dS=
∫ 1
0
∫ 1
0
F·n dS =
∫ 1
0
∫ 1
0
−(x+y)dxdy =− 1
A summation of the surface integrals over each face gives
∫∫
R
F·dS=^3
2
−^1
2
+ 0 +^1
2
+ 1 −1 =^3
2
.
Example 7-30. Evaluate the surface integral
∫∫
R
f(x, y, z )dS, where Sis the
surface of the plane
G(x, y, z ) = 2x+ 2y+z−1 = 0
which lies in the first octant and f=f(x, y, z)is the scalar field given by f=xyz.
Solution The given surface is sketched in figure 7-22. The unit normal at any point
on the surface is
eˆn=
grad G
|grad G|=
2
3 ˆe^1 +
2
3 ˆe^2 +
1
3 ˆe^3.
