1-1 Cauchy–Green deformation tensor
7
Next, we are going to identify the coordinates of the left metric tensor G l and of the right metric
tensor G r , in particular, from the inner products
∂X
∂Λ
∂X
∂Λ
=
A
2
1 cos
2 Φ
1−E 2 sin 2 Φ
=: G 11 ,
∂x
∂λ
∂x
∂λ
= r
2 cos
2 φ =: g 11 ,
∂X
∂Λ
∂X
∂Φ
=
∂X
∂Φ
∂X
∂Λ
=: G 12 = 0 ,
∂x
∂λ
∂x
∂φ
=
∂x
∂φ
∂x
∂λ
=: g 12 = 0 ,
∂X
∂Φ
∂X
∂Φ
=
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 Φ) 3 =: G 22 ,
∂x
∂φ
∂x
∂φ
= r
2 =: g 22 ,
dS
2 =
A
2
1 cos
2 Φ
1−E 2 sin 2 Φ
dΛ
2 +
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 Φ) 3 dΦ
2 ,
ds
2 = r
2 cos
2 φ dλ
2 + r
2 dφ
2 .
(1.21)
Resorting to this identification, we obtain the left metric tensor, i. e. G l , and the right metric tensor,
i. e. G r , according to
G l :=
G 11 G 12
G 12 G 22
=
G MN
=
G r :=
g 11 g 12
g 12 g 22
=
g µν
=
=
A
2
1 cos
2 Φ
1−E 2 sin 2 Φ
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 Φ) 3
,
=
r
2 cos
2 φ 0
0
r
2
.
(1.22)
Finally, we implement the isoparametric mapping f = id. Applying the summation convention
over repeated indices, this is realized by
U
M
→ u
µ = f
µ (U
µ ) , u
µ = δ
µ
M U
M , u
1 = U
1 , u
2 = U
2 , λ = Λ , φ = Φ , J l = I 2 = J r ,
(1.23)
∂U
M /∂u
µ
= 1 > 0 ,
f ∗ : dU
M = δ
M
µ du
µ ,
dΛ
dΦ
=
dλ
dφ
,
∂u
µ /∂U
M
= 1 > 0 ,
f
∗ : du
µ = δ
µ
M dU
M ,
dλ
dφ
=
dΛ
dΦ
.
(1.24)
Resorting to these relations and applying again the summation convention over repeated indices, we
arrive at the left and right Cauchy–Green tensors, namely
c MN = g µν
∂u
µ
∂U M
∂u
ν
∂U N = g µν δ
µ
M δ
ν
N ,
C µν = G MN
∂U
M
∂u µ
∂U
N
∂u ν = G MN δ
M
µ δ
N
ν ,
C l =
c MN
= J
T
l G r J l =
r
2 cos
2 Φ 0
0
r
2
, C r =
C µν
= J
T
r G l J r =
A
2
1 cos
2 φ
1−E 2 sin 2 φ
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 φ) 3
,
ds
2 = r
2 cos
2 Φ dΛ
2 + r
2 dΦ
2 ,
dS
2 =
A
2
1 cos
2 φ
1−E 2 sin 2 φ
dλ
2 +
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 φ) 3 dφ
2 .
(1.25)
By means of the left Cauchy–Green tensor, we have succeeded to represent the right metric or the
metric of the right manifold M
2
r in the coordinates of the left manifold M
2
l . Or we may say that we
have pulled back (dλ, dφ) ∈
∗ T λ,φ M
2
r to (dΛ, dΦ) ∈
∗ T Λ,Φ M
2
l , namely from the right cotangent space to
the left cotangent space. By means of the right Cauchy–Green tensor, we have been able to represent
the left metric or the metric of the left manifold M
2
l in the coordinates of the right manifold M
2
r . Or
we may say that we have pushed forward (dΛ, dΦ) ∈
∗ T Λ,Φ M
2
l to (dλ, dφ) ∈
∗ T λ,φ M
2
r , namely from
the left cotangent space to the right cotangent space.
End of Example.
7
Next, we are going to identify the coordinates of the left metric tensor G l and of the right metric
tensor G r , in particular, from the inner products
∂X
∂Λ
∂X
∂Λ
=
A
2
1 cos
2 Φ
1−E 2 sin 2 Φ
=: G 11 ,
∂x
∂λ
∂x
∂λ
= r
2 cos
2 φ =: g 11 ,
∂X
∂Λ
∂X
∂Φ
=
∂X
∂Φ
∂X
∂Λ
=: G 12 = 0 ,
∂x
∂λ
∂x
∂φ
=
∂x
∂φ
∂x
∂λ
=: g 12 = 0 ,
∂X
∂Φ
∂X
∂Φ
=
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 Φ) 3 =: G 22 ,
∂x
∂φ
∂x
∂φ
= r
2 =: g 22 ,
dS
2 =
A
2
1 cos
2 Φ
1−E 2 sin 2 Φ
dΛ
2 +
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 Φ) 3 dΦ
2 ,
ds
2 = r
2 cos
2 φ dλ
2 + r
2 dφ
2 .
(1.21)
Resorting to this identification, we obtain the left metric tensor, i. e. G l , and the right metric tensor,
i. e. G r , according to
G l :=
G 11 G 12
G 12 G 22
=
G MN
=
G r :=
g 11 g 12
g 12 g 22
=
g µν
=
=
A
2
1 cos
2 Φ
1−E 2 sin 2 Φ
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 Φ) 3
,
=
r
2 cos
2 φ 0
0
r
2
.
(1.22)
Finally, we implement the isoparametric mapping f = id. Applying the summation convention
over repeated indices, this is realized by
U
M
→ u
µ = f
µ (U
µ ) , u
µ = δ
µ
M U
M , u
1 = U
1 , u
2 = U
2 , λ = Λ , φ = Φ , J l = I 2 = J r ,
(1.23)
∂U
M /∂u
µ
= 1 > 0 ,
f ∗ : dU
M = δ
M
µ du
µ ,
dΛ
dΦ
=
dλ
dφ
,
∂u
µ /∂U
M
= 1 > 0 ,
f
∗ : du
µ = δ
µ
M dU
M ,
dλ
dφ
=
dΛ
dΦ
.
(1.24)
Resorting to these relations and applying again the summation convention over repeated indices, we
arrive at the left and right Cauchy–Green tensors, namely
c MN = g µν
∂u
µ
∂U M
∂u
ν
∂U N = g µν δ
µ
M δ
ν
N ,
C µν = G MN
∂U
M
∂u µ
∂U
N
∂u ν = G MN δ
M
µ δ
N
ν ,
C l =
c MN
= J
T
l G r J l =
r
2 cos
2 Φ 0
0
r
2
, C r =
C µν
= J
T
r G l J r =
A
2
1 cos
2 φ
1−E 2 sin 2 φ
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 φ) 3
,
ds
2 = r
2 cos
2 Φ dΛ
2 + r
2 dΦ
2 ,
dS
2 =
A
2
1 cos
2 φ
1−E 2 sin 2 φ
dλ
2 +
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 φ) 3 dφ
2 .
(1.25)
By means of the left Cauchy–Green tensor, we have succeeded to represent the right metric or the
metric of the right manifold M
2
r in the coordinates of the left manifold M
2
l . Or we may say that we
have pulled back (dλ, dφ) ∈
∗ T λ,φ M
2
r to (dΛ, dΦ) ∈
∗ T Λ,Φ M
2
l , namely from the right cotangent space to
the left cotangent space. By means of the right Cauchy–Green tensor, we have been able to represent
the left metric or the metric of the left manifold M
2
l in the coordinates of the right manifold M
2
r . Or
we may say that we have pushed forward (dΛ, dΦ) ∈
∗ T Λ,Φ M
2
l to (dλ, dφ) ∈
∗ T λ,φ M
2
r , namely from
the left cotangent space to the right cotangent space.
End of Example.