Geology Reference
In-Depth Information
Multiplying (1.28) through byg
kj
and summing over
j
, the new covariant compon-
ents of
V
are found to be given by
∂
u
k
∂
u
m
∂
u
l
∂
u
j
∂
u
j
v
k
=
i
∂
u
i
g
lm
v
,
(1.33)
where
∂
u
j
∂
u
j
∂
u
m
∂
u
m
∂
u
i
=
δ
m
∂
u
i
=
i
.
(1.34)
Hence, (1.33) reduces to
v
k
=
∂
u
l
=
∂
u
l
i
g
lm
v
i
∂
u
k
δ
∂
u
k
v
l
.
(1.35)
Thus, the transformation law for the covariant components of the vector
V
is
v
j
=
∂
u
i
∂
u
j
v
i
.
(1.36)
1.1.4 Tensors
In the analysis of the state of stress in a solid, it was realised that physical quant-
ities more complicated than vectors were required for the description of the state
of stress. Considering an imaginary surface within the stressed solid, with orient-
ation described by its outward normal vector, the force on the surface is di
erent
for each orientation of the surface. Thus, the stress associates a force vector with
each spatial direction. This has led to the definition of a
second-order tensor
as
a physical quantity that associates a vector (
a first-order tensor
) with each spatial
direction, and the generalisation that a tensor of order
n
associates a tensor of order
n
ff
1 with each spatial direction.
The transformation laws, for contravariant vectors (1.28) and for covariant vec-
tors (1.36), are easily generalised to those for tensors of second and higher order.
In fact, we have already met a
second-order, twice covariant tensor
, the metrical
coe
−
cient g
ij
, whose transformation law is given by (1.32) as
g
kl
=
∂
u
i
∂
u
k
∂
u
j
∂
u
l
g
ij
.
(1.37)
Replacing the index
j
by
k
in both of the relations (1.12), and
i
by
j
in the second,
gives
du
i
ik
du
k
,
du
k
=
g
kj
du
j
=
g
.
(1.38)
Then,
du
i
ik
du
k
ik
g
kj
du
j
=
g
=
g
.
(1.39)
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