Geography Reference
In-Depth Information
We briefly outline the
simultaneous diagonalization
of the positive-definite, symmetric matrices
{
C
l
,
G
r
}
and
{
C
r
,
G
l
}
, respectively, which is based upon a transformation called “
Kartenwechsel
”:
left “Kartenwechsel”:
right “Kartenwechsel”:
versus
(1.48)
V
l
(
U
M
l
)
V
r
(
U
M
1
)
.
T
:
V
l
(
U
M
l
)
→
τ
:
V
r
(
U
M
r
)
→
The commutative diagram shown in Fig.
1.6
illustrates this “Kartenwechsel”. Let us pay atten-
tion to Theorem
1.1
and Corollary
1.3
, and let us present the various transformations in the
Boxes
1.2
-
1.8
.
Fig. 1.6.
Commutative diagram, canonical representation of pairs of metric tensors, “Kartenwechsel”
T
and
τ
,
canonical mapping
f
from the left chart
V
l
to the right chart
V
r
can
Box 1.2 (Left versus right Cauchy-Green deformation tensor).
Left CG:
Right CG:
∂u
μ
∂U
M
∂u
ν
f
λ
(
U
L
)
∂U
N
d
U
M
d
U
N
=d
S
2
=
G
MN
{
F
L
(
u
λ
)
d
s
2
=
g
μν
{
}
}
∂U
M
∂u
μ
∂U
N
∂u
ν
d
u
μ
d
u
ν
=
(1.49)
=
c
MN
(
U
L
)d
U
M
d
U
N
,
=
C
μν
(
u
λ
)
du
μ
du
ν
,
c
MN
(
U
L
):=
g
μν
(
U
L
)
∂u
μ
∂U
M
(
U
L
)
∂u
ν
∂U
N
(
U
L
)
.c
μν
(
u
λ
):=
G
MN
(
u
λ
)
∂U
M
∂u
μ
(
u
λ
)
∂U
N
∂u
ν
(
u
λ
)
.
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