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In-Depth Information
f
g
2
(
V
)+
h
2
(
V
)
Λ
2
=
f
cos
n−
1
V
(
n
+1)
2
cos
2
Vsin
2
V
+
α
2
[
cos
2
V
(
n
+
a
)
− n
]
2
=
(23.110)
Box 23.25 (Conformal mapping of the church tower of onion shape
M
:=
Z
2
onto the circular
2
cylinder
C
g
(
v
)
).
f
=
g
2(
V
)+
h
2(
V
)
g
(
V
)
g
(
V
)
Λ
1
(V) =
Λ
2
(V)
∀
V
↔
(23.111)
f
(
V
)=
g
V
g
2(
V
)+
h
2(
V
)
g
(
V
)
dV
+
c
(23.112)
f
(
V
)=0
→
c
= 0
(23.113)
(n + 1)
2
cos
2
V
∗
sin
2
V
∗
+
a
2
[cos
2
V
∗
(n + 1)
(
n
+1)
(
n
+1)
/
2
V
n]
2
−
n
n/
2
d
V
∗
(23.114)
f
(
V
)=
cos
2
V
∗
sin
2
V
∗
V
For the
case
(
β
)ofan
equiareal mapping
the “
canonical postulate
”
Λ
1
Λ
2
=1
,
the product
identity of the eigenvalues (left principal stretches), the
first order differential equation
(
23.115
)
is generates which is solved by direct integration (
23.116
). The integration constant
c
is
again
fixed by the
boundary condition
(
23.117
), namely
y
(
V
)=
f
(
V
)=0for
ˆ
V=
arcos
n/
(
n
+1).
The final
mapping equation y
=
f
(
V
)
,
namely (
23.118
) is left as an integral to be numerically
treated.
Finally based upon
case
(
γ
), the identity of the second eigenvalues (left principal stretches)
Λ
2
=
1
,
the “
canonical postulate
” generates the
first order differential equation
(
23.119
)ofBox
23.27
.
Solved by direct integration (
23.120
), the integration constant
c
is
again
fixed by the
boundary
condition
(
23.121
). The final
mapping equation
(
23.122
) of an equidistant mapping of the
church
tower of onion shape
onto a developed circular cylinder is presented in an integral form, suited
for numerical integration.
Box 23.26 (Equiareal mapping of the church tower of the onion shape
M
:=
Z
onto the
circular cylinder
C
2
g
(
V
)
).
g
(
V
)
g
2
(
V
)+
h
2
(
V
)
Λ
1
(V)
Λ
2
(V) = 1
∀
V
↔ f
=
g
(
V
)
(23.115)
g
(
V
)
g
2
(
V
)+
h
2
(
V
)
dV
+
c
1
g
V
f
(
V
)=
(23.116)
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