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. 381. [T] F ( x , y , z ) = x i + y j + ( z 2 − 1 ) k ; S is the surface of the solid bounded by cylinder x 2 + y 2 = 4 and planes z = 0 and z = 1 .
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. 381. [T] F ( x , y , z ) = x i + y j + ( z 2 − 1 ) k ; S is the surface of the solid bounded by cylinder x 2 + y 2 = 4 and planes z = 0 and z = 1 .
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.
381. [T]
F
(
x
,
y
,
z
)
=
x
i
+
y
j
+
(
z
2
−
1
)
k
; S is the surface of the solid bounded by cylinder
x
2
+
y
2
=
4
and planes
z
=
0
and
z
=
1
.
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.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Shading a Venn diagram with 3 sets: Unions, intersections, and...
The Venn diagram shows sets A, B, C, and the universal set U.
Shade (CUA)' n B on the Venn diagram.
U
Explanation
Check
A-
B
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3. A different 7-Eleven has a bank of slurpee fountain heads. Their available flavors are as follows: Mountain
Dew, Mountain Dew Code Red, Grape, Pepsi and Mountain Dew Livewire. You fill five different cups full
with each type of flavor. How many different ways can you arrange the cups in a line if exactly two Mountain
Dew flavors are next to each other?
3.2.1
Elementary Statistics: Picturing the World (7th Edition)
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