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F
22
Λ
2
G
11
,
1
G
11
=1
c
12
−
Λ
2
G
12
c
11
−Λ
2
Λ
2
G
12
c
12
−
G
12
−
−
G
11
G
12
G
22
c
11
−
1
=1
F
22
G
12
Λ
2
c
12
−
G
12
Λ
2
G
12
Λ
2
G
12
G
11
c
12
−
G
12
c
12
−
−
⇐⇒
−
Λ
2
G
11
,G
22
−
c
11
−Λ
2
G
11
c
11
−
c
11
−
Λ
2
G
11
1
F
22
G
22
−
(
c
11
− Λ
2
G
11
)
2
=1
Λ
2
G
12
Λ
2
G
12
)
2
2
G
12
c
12
−
c
11
− Λ
2
G
11
+
G
11
(
c
12
−
⇐⇒
Λ
2
G
11
c
11
−
=
⇒
F
22
=
±
(
c
11
−
Λ
2
G
11
)
2
G
22
−
2(
c
11
−
Λ
2
G
11
)(
c
12
−
Λ
2
G
12
)
G
12
+(
c
12
−
Λ
2
G
12
)
2
G
11
(1.69)
=
F
22
=
F
12
F
22
c
12
−Λ
2
G
12
c
11
−Λ
2
G
11
1
−
⇐⇒
1
=
±
(
c
11
−
Λ
2
G
12
)
2
G
11
×
Λ
2
G
11
)
2
G
22
−
2(
c
11
−
Λ
2
G
11
)(
c
12
−
Λ
2
G
12
)
G
12
+(
c
12
−
−
q. e. d.
Λ
2
G
12
)
(
c
12
−
×
Λ
2
G
11
c
11
−
End of Proof.
S
1
, left Tissot ellipse
E
Λ
1
,Λ
2
, the tangent vectors are
∂/∂U
Fig. 1.7.
Left Cauchy-Green tensor, left Tissot circle
and
∂/∂V
The canonical forms of the metric, namely d
S
2
and d
s
2
, have been interpreted as the following
pairs:
1
1
λ
1
,λ
2
left Tissot circle
S
right Tissot ellipse
E
versus
and
versus
(1.70)
left Tissot ellipse
E
Λ
1
,Λ
2
,
right Tissot circle
S
1
.
Figure
1.7
illustrates the pair
{
left Cauchy-Green deformation tensor, left metric tensor
}
by
1
and the left Tissot ellipse
1
Λ
1
,Λ
2
2
means of the left Tissot circle
S
E
on the left tangent space
T
M
l
.
In contrast, by means of Fig.
1.8
, we aim at illustrating the pair
{
right Cauchy-Green deformation
1
λ
1
,λ
2
tensor, right metric tensor
}
by means of the right Tissot ellipse
E
and the right Tissot circle
1
on the right tangent space
T
2
S
M
r
. The left eigenvectors span canonically the left tangent space
2
2
T
M
l
, while the right eigenvectors span the right tangent space
T
M
r
,namely
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