A beautiful flux integral Consider the potential function ϕ ( x, y, z ) = G ( p ), where G is any twice differentiable function and ρ = x 2 + y 2 + z 2 ; therefore, G depends only on the distance from the origin. a. Show that the gradient vector field associated with ϕ is F = ∇ φ = G ′ ( ρ ) r ρ , where r = 〈 x , y , z 〉 and ρ = | r |. b. Let S be the sphere of radius a centered at the origin and let D be the region enclosed by S. Show that the flux of F across S is ∬ s F ⋅ n d S = 4 π a 2 G ′ ( a ) . c. Show that ∇ ⋅ F = ∇ ⋅ ∇ φ = 2 G ′ ( ρ ) ρ + G ″ ( ρ ) . d. Use part (c) to show that the flux across S (as given in part (b)) is also obtained by the volume integral ∭ D ∇ ⋅ F d V . (Hint: use spherical coordinates and integrate by parts.)
A beautiful flux integral Consider the potential function ϕ ( x, y, z ) = G ( p ), where G is any twice differentiable function and ρ = x 2 + y 2 + z 2 ; therefore, G depends only on the distance from the origin. a. Show that the gradient vector field associated with ϕ is F = ∇ φ = G ′ ( ρ ) r ρ , where r = 〈 x , y , z 〉 and ρ = | r |. b. Let S be the sphere of radius a centered at the origin and let D be the region enclosed by S. Show that the flux of F across S is ∬ s F ⋅ n d S = 4 π a 2 G ′ ( a ) . c. Show that ∇ ⋅ F = ∇ ⋅ ∇ φ = 2 G ′ ( ρ ) ρ + G ″ ( ρ ) . d. Use part (c) to show that the flux across S (as given in part (b)) is also obtained by the volume integral ∭ D ∇ ⋅ F d V . (Hint: use spherical coordinates and integrate by parts.)
Solution Summary: The author explains the gradient vector field associated with phi .
A beautiful flux integral Consider the potential function ϕ(x, y, z) = G(p), where G is any twice differentiable function and
ρ
=
x
2
+
y
2
+
z
2
; therefore, G depends only on the distance from the origin.
a. Show that the gradient vector field associated with ϕ is
F
=
∇
φ
=
G
′
(
ρ
)
r
ρ
, where r = 〈x, y, z〉 and ρ = |r|.
b. Let S be the sphere of radius a centered at the origin and let D be the region enclosed by S. Show that the flux of F across S is
∬
s
F
⋅
n
d
S
=
4
π
a
2
G
′
(
a
)
.
c. Show that
∇
⋅
F
=
∇
⋅
∇
φ
=
2
G
′
(
ρ
)
ρ
+
G
″
(
ρ
)
.
d. Use part (c) to show that the flux across S (as given in part (b)) is also obtained by the volume integral
∭
D
∇
⋅
F
d
V
. (Hint: use spherical coordinates and integrate by parts.)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Propyl alcohol with a density of d = 0.2 g/cm2 flows over the closed curve r(t) = (sin t)i - (cos t)j, 0<= t<= 2pai, according to the vector field F = dv, where v = (x - y)i + x2j is a velocity field measured in centimeters per second. Find the circulation of F around the curve r(t).
Calculate the vector field flow g⟶ (x,y,z) = (y) î - (x) j + (x + y) k, counterclockwise, along the curve intersecting the surfaces z = x2 + y2 and z = 1. Calculate in two ways:
a) Through direct calculation of the line integral
b) Through Stokes' theorem
(a)Show that F is a conservative vector field.
f(x,y,z) = < yzcos(xz)-y^(2), sin(xz) - 2xy, 3z^2 + xycos(xz) >
(b)Find a potential function for F.
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