Geology Reference
In-Depth Information
In the transformation from co-ordinates (
x
1
,
x
2
,
x
3
) to a new system of right-
handed Cartesian co-ordinates (
x
1
,
x
2
,
x
3
), the partial derivatives
∂
u
j
∂
u
i
and
∂
u
i
∂
u
j
(1.148)
both become cos(
x
j
,
x
i
)
c
ji
, the direction cosine between
x
j
and
x
i
,
which is equal to the direction cosine between
x
i
and
x
j
. Thus, in Cartesian co-
ordinate systems, the transformation laws (1.28) and (1.36) for the components
(
V
1
,
V
2
,
V
3
) of an arbitrary vector
V
become
V
j
=
cos(
x
i
,
x
j
)
=
=
c
ji
V
i
.
(1.149)
In contrast to the partial derivatives (1.148) in curvilinear co-ordinates, the direc-
tion cosines
c
ji
are not functions of position. The transformation of co-ordinates
follows the same law as that for vectors,
x
j
=
c
jk
x
k
.
(1.150)
The inverse transformation from the new co-ordinates back to the original co-
ordinates follows
x
k
=
c
lk
x
l
.
(1.151)
Then,
x
j
=
c
jk
c
lk
x
l
.
(1.152)
The co-ordinates
x
j
and
x
l
are independent if
j
l
,andif
j
=
l
they are identical.
Hence,
c
jk
c
lk
=
δ
jl
,
(1.153)
analogous to (1.34), where δ
jl
is the Kronecker delta, originally defined following
(1.7) and shown to follow the transformation law (1.49) for a second-order mixed
tensor. In Cartesian co-ordinates, it is a simple second-order tensor as our notation
implies.
If
a
,
b
,
c
are three arbitrary vectors, the volume of the parallelepiped they
bound in a right-handed co-ordinate system is the triple scalar product (1.112).
In a Cartesian co-ordinate system, the
i
th component of the cross product
b
×
c
,
from (1.81), becomes
(
b
×
c
)
i
=
ξ
ijk
b
j
c
k
,
(1.154)
using the permutation symbol defined by (1.80). Then the triple scalar product
is given by the triple contraction ξ
ijk
a
i
b
j
c
k
. Since this leads to the scalar volume
of the bounding parallelepiped, the permutation symbol in right-handed Cartesian
co-ordinates is a third-order tensor, called the
alternating tensor
.
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