Geology Reference
In-Depth Information
On interchanging the indices
k
and
l
,aswellas
i
and
j
,wefind
∂
u
l
∂
u
i
∂
u
j
t
lk
=−
t
kl
,
∂
u
k
t
ij
=−
(1.135)
using relation (1.133). The antisymmetry of
t
ij
is preserved in the transformation to
the new co-ordinates, giving
t
kl
=−
t
lk
. Then,τ
1
t
23
,τ
2
t
31
,τ
3
t
12
. Replacing
=
=
=
1
, τ
2
, τ
3
and the antisymmetric compon-
the antisymmetric components of
t
ij
by τ
ents of
t
ij
by τ
1
, τ
2
, τ
3
,wefindthat
∂
u
2
∂
u
3
∂
u
3
∂
u
3
∂
u
2
∂
u
3
∂
u
3
−
∂
u
3
∂
u
2
∂
u
2
∂
u
2
∂
u
1
∂
u
3
−
∂
u
1
∂
u
2
∂
u
3
τ
1
1
2
=
τ
+
τ
∂
u
1
∂
u
3
∂
u
2
∂
u
2
∂
u
3
−
∂
u
2
∂
u
2
∂
u
1
3
+
τ
,
∂
u
2
∂
u
1
∂
u
3
∂
u
1
∂
u
3
∂
u
3
∂
u
1
−
∂
u
3
∂
u
3
∂
u
2
∂
u
3
∂
u
1
∂
u
1
−
∂
u
1
∂
u
3
∂
u
3
τ
2
1
2
=
τ
+
τ
(1.136)
∂
u
1
∂
u
1
∂
u
3
∂
u
2
∂
u
1
−
∂
u
2
∂
u
3
∂
u
1
3
+
τ
,
∂
u
2
∂
u
2
∂
u
3
∂
u
2
∂
u
1
∂
u
3
∂
u
2
−
∂
u
3
∂
u
1
∂
u
2
∂
u
1
∂
u
1
∂
u
2
−
∂
u
1
∂
u
1
∂
u
3
τ
3
1
2
=
τ
+
τ
∂
u
1
∂
u
2
∂
u
1
∂
u
2
∂
u
2
−
∂
u
2
∂
u
1
∂
u
1
3
+
τ
.
cients of the transformation law (1.36) for the covariant components
of a vector may be abbreviated as
The coe
a
ij
=
∂
u
j
∂
u
i
.
(1.137)
These may be regarded as the elements of a 3
×
3matrix,
⎝
⎠
∂
u
1
/∂
u
1
∂
u
2
/∂
u
1
∂
u
3
/∂
u
1
∂
u
1
/∂
u
2
∂
u
2
/∂
u
2
∂
u
3
/∂
u
2
A
=
.
(1.138)
∂
u
1
/∂
u
3
∂
u
2
/∂
u
3
∂
u
3
/∂
u
3
Let
C
be the matrix of cofactors of the determinant,
, of the matrix
A
(1.138),
with elements
c
ij
. Then, the transformation (1.136) of the quantitiesτ
|
A
|
j
to the quant-
ities τ
i
may be written
τ
i
j
=
c
ij
τ
.
(1.139)
The inverse of the matrix
A
is given by
C
A
−
1
B
T
=
|
=
,
(1.140)
|
A
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