For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D . 421. Use the divergence theorem to evaluate ∬ s F ⋅ d S , where F ( x , y , z ) = x y i − 1 2 y 2 j + z k and S is the surface consisting of three pieces: z = 4 − 3 x 2 − 3 y 2 , 1 ≤ z ≤ 4 on the top; x 2 + y 2 = 1 , 0 ≤ z ≤ 1 on the sides; and z = 0 on the bottom.
For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D . 421. Use the divergence theorem to evaluate ∬ s F ⋅ d S , where F ( x , y , z ) = x y i − 1 2 y 2 j + z k and S is the surface consisting of three pieces: z = 4 − 3 x 2 − 3 y 2 , 1 ≤ z ≤ 4 on the top; x 2 + y 2 = 1 , 0 ≤ z ≤ 1 on the sides; and z = 0 on the bottom.
For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D.
421. Use the divergence theorem to evaluate
∬
s
F
⋅
d
S
, where
F
(
x
,
y
,
z
)
=
x
y
i
−
1
2
y
2
j
+
z
k
and S is the surface consisting of three pieces:
z
=
4
−
3
x
2
−
3
y
2
,
1
≤
z
≤
4
on the top;
x
2
+
y
2
=
1
,
0
≤
z
≤
1
on the sides; and
z
=
0
on the bottom.
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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