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Surface Integrals
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Expression 17: "u" Subscript, 1 , Baseline equals 0
u
1
=
0
negative 10
−
1
0
10
1
0
17
Expression 18: "u" Subscript, 2 , Baseline equals 2 pi
u
2
=
2
π
equals
=
6.2 8 3 1 8 5 3 0 7 1 8
6
.
2
8
3
1
8
5
3
0
7
1
8
18
Expression 19: left bracket, "v" Subscript, 1 , Baseline less than "v" less than "v" Subscript, 2 , Baseline , right bracket
v
1
<
v
<
v
2
19
Expression 20: "v" Subscript, 1 , Baseline equals 0
v
1
=
0
negative 10
−
1
0
10
1
0
20
Expression 21: "v" Subscript, 2 , Baseline equals 2 pi
v
2
=
2
π
equals
=
6.2 8 3 1 8 5 3 0 7 1 8
6
.
2
8
3
1
8
5
3
0
7
1
8
21
Calculations
Calculations
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22
"S" "u" "r" "f" "a" "c" "e" "A" "r" "e" "a"
S
u
r
f
a
c
e
A
r
e
a
"S" Subscript, "u" "r" "f" "a" "c" "e" "A" "r" "e" "a" , Baseline
S
u
r
f
a
c
e
A
r
e
a
r1c1:
r1c2: 236.8 7 0 5 1
2
3
6
.
8
7
0
5
1
r1c3:
r2c1:
r2c2:
r2c3:
23
Formulas
Formulas
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24
Expression 25: Start\vec "r" undefined left parenthesis, "u" , "v" , right parenthesis equals "x" left parenthesis, "u" , "v" , right parenthesis Start\vec "i" undefined positive "y" left parenthesis, "u" , "v" , right parenthesis Start\vec "j" undefined positive "z" left parenthesis, "u" , "v" , right parenthesis Start\vec "k" undefined
→
r
u
,
v
=
x
u
,
v
→
i
+
y
u
,
v
→
j
+
z
u
,
v
→
k
25
Expression 26: StartFraction, ∂ Start\vec "r" undefined Over ∂ "u" , EndFraction equals StartFraction, ∂ "x" Over ∂ "u" , EndFraction left parenthesis, "u" , "v" , right parenthesis Start\vec "i" undefined positive StartFraction, ∂ "y" Over ∂ "u" , EndFraction left parenthesis, "u" , "v" , right parenthesis Start\vec "j" undefined positive StartFraction, ∂ "z" Over ∂ "u" , EndFraction left parenthesis, "u" , "v" , right parenthesis Start\vec "k" undefined
∂
→
r
∂
u
=
∂
x
∂
u
u
,
v
→
i
+
∂
y
∂
u
u
,
v
→
j
+
∂
z
∂
u
u
,
v
→
k
26
Expression 27: StartFraction, ∂ Start\vec "r" undefined Over ∂ "v" , EndFraction equals StartFraction, ∂ "x" Over ∂ "v" , EndFraction left parenthesis, "u" , "v" , right parenthesis Start\vec "i" undefined positive StartFraction, ∂ "y" Over ∂ "v" , EndFraction left parenthesis, "u" , "v" , right parenthesis Start\vec "j" undefined positive StartFraction, ∂ "z" Over ∂ "v" , EndFraction left parenthesis, "u" , "v" , right parenthesis Start\vec "k" undefined
∂
→
r
∂
v
=
∂
x
∂
v
u
,
v
→
i
+
∂
y
∂
v
u
,
v
→
j
+
∂
z
∂
v
u
,
v
→
k
27
Expression 28: "S" "u" "r" "f" "a" "c" "e" "A" "r" "e" "a" equals Start integral from "u" Subscript, 1 , Baseline to "u" Subscript, 2 , Baseline , end integral, Start integral from "v" Subscript, 1 , Baseline to "v" Subscript, 2 , Baseline , end integral, StartAbsoluteValue, StartFraction, ∂ Start\vec "r" undefined Over ∂ "v" , EndFraction times StartFraction, ∂ Start\vec "r" undefined Over ∂ "u" , EndFraction , EndAbsoluteValue "d" "v" "d" "u"
S
u
r
f
a
c
e
A
r
e
a
=
∫
u
2
u
1
∫
v
2
v
1
∂
→
r
∂
v
·
∂
→
r
∂
u
d
v
d
u
28
Perspective View
Perspective View
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29
Expression 30: theta Subscript, "h" "o" "r" "i" "z" "o" "n" "t" "a" "l" , Baseline equals negative 0.5
θ
h
o
r
i
z
o
n
t
a
l
=
−
0
.
5
negative pi
−
π
pi
π
30
Expression 31: theta Subscript, "v" "e" "r" "t" "i" "c" "a" "l" , Baseline equals negative 0.4 5
θ
v
e
r
t
i
c
a
l
=
−
0
.
4
5
negative pi
−
π
pi
π
31
Expression 32: theta Subscript, "c" "i" "r" "c" "u" "l" "a" "r" , Baseline equals 0
θ
c
i
r
c
u
l
a
r
=
0
negative pi
−
π
pi
π
32
Visual Representation Construction
Visual Representation Construction
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33
66
powered by
powered by
"x"
x
"y"
y
"z"
z
Perspective
"S" "u" "r" "f" "a" "c" "e" "A" "r" "e" "a" equals Start integral from "u" Subscript, 1 , Baseline to "u" Subscript, 2 , Baseline , end integral, Start integral from "v" Subscript, 1 , Baseline to "v" Subscript, 2 , Baseline , end integral, StartAbsoluteValue, StartFraction, ∂ Start\vec "r" undefined Over ∂ "v" , EndFraction times StartFraction, ∂ Start\vec "r" undefined Over ∂ "u" , EndFraction , EndAbsoluteValue "d" "v" "d" "u" equals 236.9
S
u
r
f
a
c
e
A
r
e
a
=
∫
u
2
u
1
∫
v
2
v
1
∂
→
r
∂
v
×
∂
→
r
∂
u
d
v
d
u
=
2
3
6
.
9
"x"
x
"y"
y
"a" squared
a
2
"a" Superscript, "b" , Baseline
a
b
7
7
8
8
9
9
over
÷
functions
(
(
)
)
less than
<
greater than
>
4
4
5
5
6
6
times
×
| "a" |
|
a
|
,
,
less than or equal to
≤
greater than or equal to
≥
1
1
2
2
3
3
negative
−
A B C
StartRoot, , EndRoot
pi
π
0
0
.
.
equals
=
positive
+
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