For the following exercises, use Stokes’ theorem to evaluate ∬ s ( c u r l F ⋅ N ) d S for the vector fields and surface. 364. Let S be paraboloid z = a ( 1 − x 2 − y 2 ) , for z ≥ 0 , where a > 0 is a real number. Let F = 〈 x − y , y + z , z − x 〉 . For what value(s) of a (if any) does ∬ s ( ∇ × F ) ⋅ n d S have its maximum value?
For the following exercises, use Stokes’ theorem to evaluate ∬ s ( c u r l F ⋅ N ) d S for the vector fields and surface. 364. Let S be paraboloid z = a ( 1 − x 2 − y 2 ) , for z ≥ 0 , where a > 0 is a real number. Let F = 〈 x − y , y + z , z − x 〉 . For what value(s) of a (if any) does ∬ s ( ∇ × F ) ⋅ n d S have its maximum value?
For the following exercises, use Stokes’ theorem to evaluate
∬
s
(
c
u
r
l
F
⋅
N
)
d
S
for the vector fields and surface.
364. Let S be paraboloid
z
=
a
(
1
−
x
2
−
y
2
)
, for
z
≥
0
, where
a
>
0
is a real number. Let
F
=
〈
x
−
y
,
y
+
z
,
z
−
x
〉
. For what value(s) of a (if any) does
∬
s
(
∇
×
F
)
⋅
n
d
S
have its maximum value?
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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