Geography Reference
In-Depth Information
Assuming summation over repeated indices, we specialize the deformation tensor of first order
c
KL
according to (
14.2
). In detail, we note that (
14.3
)and(
14.4
)hold.
∂u
k
∂U
K
∂u
l
∂U
l
=
δ
kl
∂u
k
∂U
K
∂u
l
∂U
l
=
c
KL
=
g
kl
∂x
k
∂U
K
∂x
l
∂U
l
,
=
(14.2)
c
11
=
∂x
∂Λ
2
+
∂y
∂Λ
2
,
c
12
=
∂x
∂Λ
∂x
∂Φ
+
∂y
∂Λ
∂y
∂Φ
,
(14.3)
c
22
=
∂x
∂Φ
2
+
∂y
∂Φ
2
,
∂x
∂Λ
=
A
1
,
∂x
∂Φ
=
∂y
∂Λ
=0
,
∂y
∂Φ
=
f
(
Φ
)
,
(14.4)
c
KL
=
A
1
.
0
f
(
Φ
)
0
At this point, let us finally review the principal stretches and let us finally give the general
structure of the coordinate lines.
Λ
1
=
c
11
/G
11
=
1
−
E
2
sin
2
Φ
cos
Φ
, Λ
2
=
c
22
/G
22
=
f
(
Φ
)(1
E
2
sin
2
Φ
)
3
/
2
A
1
(1
− E
2
)
−
,
(14.5)
x
=
A
1
Λ, y
=
f
(
Φ
)
.
(14.6)
14-2 Special Mapping Equations
Special mapping equations of cylindric mappings: normal equidistant, normal conformal, and
normal equiareal mappings.
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