F-3 Deformation analysis of vertically/horizontally averaged equiareal cylindric mappings 581
Proof (of (F.49)).
For the proof of (F.49), we depart from the horizontal mixed equiareal cylindric mapping (F.43)–(F.46),
compute the J acobi matrix of partial derivatives of {x(Λ, Φ), y(Φ)} and constitute the coordinates of
the left Cauchy–Green deformation tensor c AB as well as the matrix C l G
−1
l
implementing (F.14)and
finally the trace tr
C l G
−1
l
.
x(Λ, Φ) =
AΛ
α + β
α +
β cos Φ
1 − E 2 sin
2 Φ
,
y(Φ) = A(1 − E
2 )(α + β)
Φ
0
cos Φ
∗ (1 − E
2 sin
2 Φ
∗ )
−3/2
α
1 − E 2 sin
2 Φ ∗ + β cos Φ ∗
dΦ
∗ ,
(F.56)
x Λ =
αA + βL
α + β
, y Λ = 0 , x Φ = −
β sin ΦM Λ
α + β
, y Φ =
(α + β)LM
αA + βL
,
(F.57)
c 11 = x
2
Λ + y
2
Λ =
(αA + βL)
2
(α + β) 2 , c 12 = x Λ x Φ + y Λ y Φ = −
β sin ΦM (αA + βL)Λ
(α + β) 2
,
c 22 = x
2
Φ + y
2
Φ = M
2
β
2 sin
2 ΦΛ
2
(α + β) 2 +
(α + β)
2 L
2
(αA + βL) 2
,
(F.58)
det [c AB ]= det [ G AB ]=
A
4 (1 − E
2 )
2 cos
2 Φ
(1 − E 2 sin
2 Φ) 4 ,
tr
C l G
−1
l
=
c 11
G 11
+
c 22
G 22
=
(αA + βL)
2
(α + β) 2 L 2 +
β
2 Λ
2 sin
2 Φ(αA + βL)
2 + (α + β)
4 L
2
(α + β) 2 (αA + βL) 2
.
(F.59)
End of Proof.
Box F.3 (Left Cauchy–Green deformation tensor (vertical mixed equiareal cylindric mapping)).
The coordinates of the left Cauchy–Green deformation tensor
for the vertical mixed equiareal cylindric mapping:
c 11 =
A
2 (α + β)
2 L
2
(αL + βA) 2 , c 12 = −
A
3 (α + β)
2 β sin ΦLM Λ
(αL + βA) 3
,
c 22 = M
2
»
A
6 (α + β)
4 β
2 sin
2 ΦΛ
2 + (αL + βA)
6
A 2 (α + β) 2 (αL + βA) 4
–
.
(F.60)
Box F.4 (Left Cauchy–Green deformation tensor (horizontal mixed equiareal cylindric mapping)).
The coordinates of the left Cauchy–Green deformation tensor
for the horizontal mixed equiareal cylindric mapping:
c 11 =
(αA + βL)
2
(α + β) 2 , c 12 = −
β sin ΦM (αA + βL)Λ
(α + β) 2
,
c 22 = M
2
»
β
2 Λ
2 sin
2 Φ(αA + βL)
2 + (α + β)
4 L
2
(α + β) 2 (αA + βL) 2
–
.
(F.61)