Geology Reference
In-Depth Information
i
and v
j
in (1.14) by these expressions gives
Replacing v
=
b
i
V
b
i
ij
b
i
b
j
V
=
g
v
j
b
i
=
·
v
j
b
i
·
(1.17)
and
=
b
j
V
b
j
i
b
j
i
b
j
V
=
g
ji
v
=
b
j
·
b
i
v
·
.
(1.18)
i
are called the
contravariant components
of the vector
V
while the components v
j
are called the
covariant components
of
V
. By conven-
tion, contravariant components are indicated by a superscripted index and covariant
components are indicated by a subscripted index.
Physical components of
V
can be defined by resolving the vector along a system
of vectors of
unit
length. These may be defined as parallel to the unitary base
system, with each reduced to unit length, by
The components v
b
1
√
b
1
·
1
√
g
11
b
1
,
1
√
g
22
b
2
,
1
√
g
33
b
3
,
e
1
=
b
1
=
e
2
=
e
3
=
(1.19)
thus,
V
=
V
1
e
1
+
V
2
e
2
+
V
3
e
3
,
(1.20)
with physical components
=
√
g
11
v
=
√
g
22
v
=
√
g
33
v
1
2
3
V
1
,
V
2
,
V
3
.
(1.21)
Hence, the physical components,
V
i
, are of the same dimensions as the vector
V
itself.
We can now give dimensions to the vector space. The di
erential vector
d
r
represents a displacement from the point
P
with co-ordinates (
u
1
ff
,
u
2
,
u
3
)tothe
point with co-ordinates (
u
1
du
1
,
u
2
du
2
,
u
3
du
3
). Denoting the magnitude of
+
+
+
this displacement by
ds
,
ds
2
b
j
du
i
du
j
b
i
b
j
du
i
du
j
,
=
d
r
·
d
r
=
b
i
·
=
·
(1.22)
or
ds
2
=
g
ij
du
i
du
j
ij
du
i
du
j
.
=
g
(1.23)
ij
appear in bilinear forms expressing the square of the
incremental displacement in terms of the increments in the co-ordinates
u
i
,orin
terms of the increments in the reciprocal co-ordinates
u
i
. They are called
metrical
coe
The coe
cients g
ij
and g
cients
.
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