For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ∫ s F ⋅ n d S for the given choice of F and the boundary surface S . For each closed surface, assume N is the outward unit normal vector . 385. Use the divergence theorem to calculate surface integral ∬ s F ⋅ d S , where F ( x , y , z ) = ( e y 2 ) i + ( y + sin ( z 2 ) ) j + ( z − 1 ) k and S is upper hemisphere x 2 + y 2 + z 2 = 1 , z ≥ 0 , oriented upward.
For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ∫ s F ⋅ n d S for the given choice of F and the boundary surface S . For each closed surface, assume N is the outward unit normal vector . 385. Use the divergence theorem to calculate surface integral ∬ s F ⋅ d S , where F ( x , y , z ) = ( e y 2 ) i + ( y + sin ( z 2 ) ) j + ( z − 1 ) k and S is upper hemisphere x 2 + y 2 + z 2 = 1 , z ≥ 0 , oriented upward.
For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral
∫
s
F
⋅
n
d
S
for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.
385. Use the divergence theorem to calculate surface integral
∬
s
F
⋅
d
S
, where
F
(
x
,
y
,
z
)
=
(
e
y
2
)
i
+
(
y
+
sin
(
z
2
)
)
j
+
(
z
−
1
)
k
and S is upper hemisphere
x
2
+
y
2
+
z
2
=
1
,
z
≥
0
, 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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