a. A function F is said to have spherical svmmetiy if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as F(x. y. z) = f(p). where ρ = x 2 + y 2 + z 2 . Show that ∭ B F ( x , y , z ) d V = 2 π ∫ a b ρ 2 f ( ρ ) d ρ , where B is the region between the upper concentric hemispheres of radii a and b centered at the origin, with 0 < a < b and F a spherical function defined on B. b. Use the previous result to show that ∭ B ( x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 d V = 21 π where B = { ( x , y , z ) | 1 ≤ x 2 + y 2 + z 2 ≤ 2 , z ≥ 0 }
a. A function F is said to have spherical svmmetiy if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as F(x. y. z) = f(p). where ρ = x 2 + y 2 + z 2 . Show that ∭ B F ( x , y , z ) d V = 2 π ∫ a b ρ 2 f ( ρ ) d ρ , where B is the region between the upper concentric hemispheres of radii a and b centered at the origin, with 0 < a < b and F a spherical function defined on B. b. Use the previous result to show that ∭ B ( x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 d V = 21 π where B = { ( x , y , z ) | 1 ≤ x 2 + y 2 + z 2 ≤ 2 , z ≥ 0 }
a. A function F is said to have spherical svmmetiy if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as F(x. y. z) = f(p). where
ρ
=
x
2
+
y
2
+
z
2
.
Show that
∭
B
F
(
x
,
y
,
z
)
d
V
=
2
π
∫
a
b
ρ
2
f
(
ρ
)
d
ρ
,
where B is the region between the upper concentric hemispheres of radii a and b centered at the origin, with 0 < a < b and F a spherical function defined on B.
b. Use the previous result to show that
∭
B
(
x
2
+
y
2
+
z
2
)
x
2
+
y
2
+
z
2
d
V
=
21
π
where
B
=
{
(
x
,
y
,
z
)
|
1
≤
x
2
+
y
2
+
z
2
≤
2
,
z
≥
0
}
Finite Mathematics & Its Applications (12th Edition)
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