Page 423 - Advanced engineering mathematics

12.8 The Divergence Theorem of Gauss 403

EXAMPLE 12.24

2
Let be the piecewise smooth surface consisting of the surface 1 of the cone z = x + y 2
2
2
2
2
for x + y ≤ 1, together with the flat cap 2 consisting of the disk x + y ≤ 1 in the plane
z = 1. is shown in Figure 12.21. Let F(x, y, z) = xi + yj + zk. We will calculate both sides of
equation (12.10).
The unit outer normal to 1 is
1 x y
n 1 = √ i + j − k .
2 z z
Then
1 x 2 y 2
F · n 1 = √ + − z = 0
2 z z
2
2
2
because on 1 , z = x + y . Therefore

F · n 1 dσ = 0.
1
The unit outer normal to 2 is n 2 = k,so F · n 2 = z. Since z = 1on 2 , then

F · n 2 dσ = zdσ = dσ
2 2 2
= area of 2 = π.

Therefore,

F · ndσ = F · n 1 dσ + F · n 2 dσ = π.
1 2
Now compute the triple integral. The divergence of F is
∂ ∂ ∂
∇· F = x + y + z = 3,
∂x ∂y ∂z

z
Σ 2 (0, 0, 1)

Σ 1

y

x
FIGURE 12.21 in Example 12.24.

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October 14, 2010 14:53 THM/NEIL Page-403 27410_12_ch12_p367-424