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subject to the integrability conditions
2 x
∂U∂V =
2 x
∂V ∂U ,
2 y
∂U∂V =
2 y
∂V ∂U .
(2.27)
End of Lemma.
2
l
r =
2 μν
2 , exterior calculus).
Box 2.1 (Conformal diffeomorphism
M
M
{ R
}
=
E
Diffeomorphism :
d x
d y
=J l d U
d U
d V
=J r d x
or
d V
d y
J l =J 1
(2.28)
r
J r =J l .
Right Cauchy-Green matrix for a conformal diffeomorphism:
C r =J r G l J r = λ 2 I 2
(2.29)
C r =J l G l J l = λ 2 I 2 .
The rows of the left Jacobi matrix are G l orthogonal :
2
x M d U M , 1 := D U x = x U , 2 := D V x = x V .
d x = x U d U + x d V d V =
(2.30)
M =1
Hodge star operator:
2
e LM det [G l ] G MN y N d U L ,
d y :=
(2.31)
L,M,N =1
subject to
y 1 := D Uy = y U , 2 := D Vy = y V .
Permutation symbol:
+1 for an even permutation of the indices L, M
∈{
1 , 2
}
e LM =
1 for an odd permutation of the indices L, M
∈{
1 , 2
}
.
(2.32)
0 th rw e
Korn-Lichtenstein equations in exterior calculus:
2
2
e LM det [G l ] G MN y N d U L =d y
x M d U M =
d x =
M =1
L,M,N =1
(2.33)
∂U L = e LM det [G l ] G MN ∂y
∂x
∂U N , d x =d y .
 
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