a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates ( ρ , θ , φ ) is F ( x , y , z ) = f ( ρ ) cos φ . Show that if g ( a ) = g ( b ) = 0 and ∫ a b h ( h ρ ) d ρ = 0 . then ∭ B F ( x , y , z ) d V = π 2 4 [ a h ( a ) − b h ( b ) ] where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that ∭ B z cos x 2 + y 2 + z 2 x 2 + y 2 + z 2 d V = 3 π 2 . where B is the region between the upper concentric hemispheres of radii π and 2 π centered at the origin and situated in the first octant.
a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates ( ρ , θ , φ ) is F ( x , y , z ) = f ( ρ ) cos φ . Show that if g ( a ) = g ( b ) = 0 and ∫ a b h ( h ρ ) d ρ = 0 . then ∭ B F ( x , y , z ) d V = π 2 4 [ a h ( a ) − b h ( b ) ] where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that ∭ B z cos x 2 + y 2 + z 2 x 2 + y 2 + z 2 d V = 3 π 2 . where B is the region between the upper concentric hemispheres of radii π and 2 π centered at the origin and situated in the first octant.
a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates (
ρ
,
θ
,
φ
) is
F
(
x
,
y
,
z
)
=
f
(
ρ
)
cos
φ
. Show that if
g
(
a
)
=
g
(
b
)
=
0
and
∫
a
b
h
(
h
ρ
)
d
ρ
=
0
. then
∭
B
F
(
x
,
y
,
z
)
d
V
=
π
2
4
[
a
h
(
a
)
−
b
h
(
b
)
]
where g is an antiderivative of f and h is an antiderivative of g.
b. Use the previous result to show that
∭
B
z
cos
x
2
+
y
2
+
z
2
x
2
+
y
2
+
z
2
d
V
=
3
π
2
. where B is the region between the upper concentric hemispheres of radii
π
and 2
π
centered at the origin and situated in the first octant.
Explain the key points and reasons for 12.8.2 (1) and 12.8.2 (2)
Q1:
A slider in a machine moves along a fixed straight rod. Its
distance x cm along the rod is given below for various values of the time. Find the
velocity and acceleration of the slider when t = 0.3 seconds.
t(seconds)
x(cm)
0 0.1 0.2 0.3 0.4 0.5 0.6
30.13 31.62 32.87 33.64 33.95 33.81 33.24
Q2:
Using the Runge-Kutta method of fourth order, solve for y atr = 1.2,
From
dy_2xy +et
=
dx x²+xc*
Take h=0.2.
given x = 1, y = 0
Q3:Approximate the solution of the following equation
using finite difference method.
ly -(1-y=
y = x), y(1) = 2 and y(3) = −1
On the interval (1≤x≤3).(taking h=0.5).
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