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 . 410. Let E be the solid bounded by the xy -plane and paraboloid z = 4 − x 2 − y 2 so that S is the surface of the paraboloid piece together with the disk in the xy -plane that farms its bottom. If F ( x , y , z ) = ( x z sin ( y z ) + x 3 ) i + cos ( y z ) j + ( 3 z y 2 − e x 2 + y 2 ) k , find ∬ s F ⋅ d S using the divergence theorem.
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 . 410. Let E be the solid bounded by the xy -plane and paraboloid z = 4 − x 2 − y 2 so that S is the surface of the paraboloid piece together with the disk in the xy -plane that farms its bottom. If F ( x , y , z ) = ( x z sin ( y z ) + x 3 ) i + cos ( y z ) j + ( 3 z y 2 − e x 2 + y 2 ) k , find ∬ s F ⋅ d S using the divergence theorem.
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.
410. Let E be the solid bounded by the xy-plane and paraboloid
z
=
4
−
x
2
−
y
2
so that S is the surface of the paraboloid piece together with the disk in the xy-plane that farms its bottom. If
F
(
x
,
y
,
z
)
=
(
x
z
sin
(
y
z
)
+
x
3
)
i
+
cos
(
y
z
)
j
+
(
3
z
y
2
−
e
x
2
+
y
2
)
k
, find
∬
s
F
⋅
d
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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