For the following exercises, use Stokes’ theorem to evaluate ∬ s ( c u r l F ⋅ N ) d S for the vector fields and surface. 342. [T] Use a CAS and Stokes’ theorem to evaluate ∬ s c u r l F ⋅ d S , where F ( x , y , z ) = z 2 i − 3 x y j + x 3 y 3 k and S is the top part of z = 5 − x 2 − y 2 above plane z = 1 , and S is oriented upward.
For the following exercises, use Stokes’ theorem to evaluate ∬ s ( c u r l F ⋅ N ) d S for the vector fields and surface. 342. [T] Use a CAS and Stokes’ theorem to evaluate ∬ s c u r l F ⋅ d S , where F ( x , y , z ) = z 2 i − 3 x y j + x 3 y 3 k and S is the top part of z = 5 − x 2 − y 2 above plane z = 1 , and S is oriented upward.
For the following exercises, use Stokes’ theorem to evaluate
∬
s
(
c
u
r
l
F
⋅
N
)
d
S
for the vector fields and surface.
342. [T] Use a CAS and Stokes’ theorem to evaluate
∬
s
c
u
r
l
F
⋅
d
S
, where
F
(
x
,
y
,
z
)
=
z
2
i
−
3
x
y
j
+
x
3
y
3
k
and S is the top part of
z
=
5
−
x
2
−
y
2
above plane
z
=
1
, and S is oriented upward.
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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