Page 419 - Advanced engineering mathematics

12.7 Lifting Green’s Theorem to R 3 399

is normal to the sphere and oriented outward. This vector has magnitude 2/z, so a unit outer
normal is
z x y 1
n = i + j + k = (xi + yj + zk).
2 z z 2
We need
1 2 2 2
F · n = (x + y + z ).
2
Then
1

2
2
2
flux = (x + y + z )dσ
2

1 x 2 y 2
2
2
2
= (x + y + z ) 1 + + dA
2 D z 2 z 2

1 2 2 2 x + y + z 2
2
2

= (x + y + z ) dA
2 D z 2
1 1
2 3/2
2
2
= (x + y + z ) dA
2 D 4 − x − y 2
2

1
= 4 dA.
2
D 4 − x − y 2
2
2
2
Here we used the fact that x + y + z = 4on . Converting to polar coordinates, we have
√
2π 3 1
flux = 4 √ rdr dθ
0 0 4 −r 2
√
2 1/2 3
= 8π[−(4 −r ) ] 0 = 8π.
SECTION 12.6 PROBLEMS
2
2
2
2
In each of Problems 1 through 6, find the mass and center x + y = 1and x + y = 9, with δ(x, y, z) = xy/

2
of mass of the shell . 1 + 4x + 4y .
2
2
2
1. is a triangle with vertices (1,0,0),(0,3,0) and 5. is the paraboloid z = 6 − x − y for z ≥ 0, with
2 2
(0,0,2), with δ(x, y, z) = xz + 1. δ(x, y, z) = 1 + 4x + 4y .
2. is the part of the sphere x + y + z = 9 above the 6. is the part of the sphere of radius 1 about the origin,
2
2
2
plane z = 1, and the density function is constant. lying in the first octant. The density is constant.
2 2 7. Find the flux of F = xi + yj − zk across the part of the
3. is the cone z = x + y for x + y ≤9, δ = constant
2
2
= K. plane x + 2y + z = 8 in the first octant.
2
4. is the part of the paraboloid z = 16 − x − y 2 8. Find the flux of F = xzi − yk across the part of the
2
2
2
in the first octant and between the cylinders sphere x + y + z = 4 above the plane z = 1.
12.7 Lifting Green’s Theorem to R 3
The fundamental results of vector integral calculus are the theorems of Gauss and Stokes. In this
section we will show how both can be viewed as natural generalizations of Green’s theorem from
two to three dimensions.
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October 14, 2010 14:53 THM/NEIL Page-399 27410_12_ch12_p367-424