Geography Reference
In-Depth Information
d
Λ
d
Φ
=
d
λ
,
d
λ
d
φ
=
d
Λ
.
d
φ
d
Φ
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
M
∂u
μ
∂U
N
∂u
ν
∂U
N
=
g
μν
δ
M
δ
N
,
=
G
MN
δ
μ
δ
ν
,
C
μν
=
G
MN
=J
r
G
l
J
r
=
A
1
cos
2
φ
,
=J
l
G
r
J
l
=
r
2
cos
2
Φ
0
r
2
,
C
r
=
0
1
−E
2
sin
2
0
C
l
=
{
c
MN
}
{
C
μν
}
A
1
(1
−E
2
)
2
(1
0
0
−E
2
sin
2
φ
)
3
(1.25)
A
1
cos
2
φ
A
1
(1
E
2
)
2
−
d
s
2
=
r
2
cos
2
Φ
d
Λ
2
+
r
2
d
Φ
2
,
d
S
2
=
E
2
sin
2
φ
d
λ
2
+
E
2
sin
2
φ
)
3
d
φ
2
.
1
−
(1
−
By means of the
left Cauchy-Green tensor
, we have succeeded to represent the right metric or
the metric of the right manifold
2
2
M
r
in the coordinates of the left manifold
M
l
.Orwemaysay
∈
∗
T
λ,φ
M
2
∈
∗
T
Λ,Φ
M
2
that we have
pulled back
(d
λ,
d
φ
)
l
, namely from the right
cotangent space to the left cotangent space. By means of the
right Cauchy-Green tensor
,wehave
been able to represent the left metric or the metric of the left manifold
r
to (d
Λ,
d
Φ
)
2
M
l
in the coordinates
r
.Orwemaysaythatwehave
pushed forward
(d
Λ,
d
Φ
)
∗
T
Λ,Φ
2
of the right manifold
M
∈
M
l
to
∈
∗
T
λ,φ
r
, namely from the left cotangent space to the right cotangent space.
(d
λ,
d
φ
)
M
End of Example.
There exists an intriguing representation of the matrix of deformation gradients J as well as of
the matrix of Cauchy-Green deformation C, namely the
polar decomposition
. It is a generalization
to matrices of the familiar polar representation of a complex number
z
=
r
exp i
φ,
(
r ≥
0) and is
defined in Corollary
1.3
.
Corollary 1.3 (Polar decomposition).
n×n
. Then there exists a unique orthonormal matrix R
SO(
n
) (called
rotation matrix
)
and a unique symmetric positive-definite matrix S (called
stretch
) such that (
1.26
) holds and the
expressions (
1.27
) are a polar decomposition of the matrix of Cauchy-Green deformation.
Let J
∈
R
∈
J=RS
,
R
∗
R=I
n
,
S=S
∗
,
(1.26)
C
l
=J
l
G
r
J
l
=S
l
R
∗
G
r
RS
l
S
r
R
∗
G
l
RS
r
=J
r
G
l
J
r
=C
r
.
versus
(1.27)
End of Corollary.
Question: “How can we compute the polar decomposition
of the
Jacobi matrix
?” Answer: “An elegant way is the
sin-
gular value decomposition
defined in Corollary
1.4
.”
Search WWH ::
Custom Search