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1
1
g
11
b
1
,
b
1
=
b
1
b
1
=
b
1
·
1
b
2
·
1
g
22
b
2
,
b
2
=
b
2
b
2
=
(1.105)
1
1
g
33
b
3
.
b
3
=
b
3
b
3
=
b
3
·
Taking the scalar product of each of these equations with its corresponding recip-
rocal unitary base vector, and using the orthogonality relation (1.7), we recover the
required squared magnitude ratios (1.104).
In an orthogonal co-ordinate system, with the abbreviations
h
1
=
√
g
11
,
h
2
=
√
g
22
,
h
3
=
√
g
33
,
(1.106)
the infinitesimal displacements, (1.52) and (1.54), along the co-ordinate directions
become
h
1
du
1
h
2
du
2
h
3
du
3
ds
1
=
,
ds
2
=
,
ds
3
=
,
(1.107)
while the surface elements, (1.58), (1.59) and (1.60), are
h
2
h
3
du
2
du
3
h
3
h
1
du
3
du
1
h
1
h
2
du
1
du
2
da
1
=
,
da
2
=
,
da
3
=
.
(1.108)
The volume element (1.68) is
=
√
g
du
1
du
2
du
3
h
1
h
2
h
3
du
1
du
2
du
3
d
v
=
.
(1.109)
The o
-diagonal elements of the determinant (1.42) defining g vanish, so it is
simply g
11
g
22
g
33
, giving
ff
h
1
h
2
h
3
.
g
=
(1.110)
The physical components of an arbitrary vector,
V
, in an orthogonal system,
are expressed by its projections on the three unit vectors, as in (1.20), except that
now the unit vectors are orthogonal, giving
e
i
i
·
e
j
=
δ
j
. From (1.19) and (1.106),
the unitary base system of vectors is
b
i
=
h
i
e
i
, while from (1.105), the reciprocal
base system vectors are
b
i
=
e
i
/
h
i
. From (1.21) and (1.106), the contravariant com-
i
ponents of
V
are v
V
i
/
h
i
, while from (1.16), the covariant components of
V
are
v
i
=
h
i
V
i
. In these last four expressions, no summation is implied by the repeated
subscript.
By substitution in the formula (1.70) for the cross product of two vectors,
V
and
W
, it is found that
=
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