Radial fields Consider the radial vector field F = r | r | p = 〈 x , y , z 〉 ( x 2 + y 2 + z 2 ) p 2 . Let S be the sphere of radius a centered at the origin. a. Use a surface integral to show that the outward flux of F across S is 4 πa 3 − p . Recall that the unit normal to the sphere is r /| r |. b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p , use the fact ( Theorem 17.10 ) that ∇ ⋅ F = 3 − p | r | p to compute the flux across S using the Divergence Theorem.
Radial fields Consider the radial vector field F = r | r | p = 〈 x , y , z 〉 ( x 2 + y 2 + z 2 ) p 2 . Let S be the sphere of radius a centered at the origin. a. Use a surface integral to show that the outward flux of F across S is 4 πa 3 − p . Recall that the unit normal to the sphere is r /| r |. b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p , use the fact ( Theorem 17.10 ) that ∇ ⋅ F = 3 − p | r | p to compute the flux across S using the Divergence Theorem.
Radial fields Consider the radial vector field
F
=
r
|
r
|
p
=
〈
x
,
y
,
z
〉
(
x
2
+
y
2
+
z
2
)
p
2
. Let S be the sphere of radius a centered at the origin.
a. Use a surface integral to show that the outward flux of F across S is 4πa3 − p. Recall that the unit normal to the sphere is r/|r|.
b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p, use the fact (Theorem 17.10) that
∇
⋅
F
=
3
−
p
|
r
|
p
to compute the flux across S using the Divergence Theorem.
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.
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