Geoscience Reference
In-Depth Information
to
x
1
,x
2
,x
3
:
x
1
=
Φ
1
(
x, y, z
)
,
2
=
Φ
2
(
x, y, z
)
,
x
3
=
K
1
[
Φ
1
(
x, y, z
)
,Φ
2
(
x, y, z
)]
+
K
2
[
Φ
1
(
x, y, z
)
,Φ
2
(
x, y, z
)
2
]
Ψ
(
x, y, z
)
.
(6.31)
With the summation convention, the element length is
d
s
2
=
g
ik
dx
i
d
x
k
.
Components of the metric tensor
g
ik
expressed in terms of Lam´ecoe-
cients
{
h
1
,h
2
,h
3
}
and transformation coecients (6.30)
K
1
,K
2
are given in
Appendix 6.A.
The vectors of the local basis
e
1
=
x
1
,
e
2
=
x
2
,
e
3
=
x
3
∇
∇
∇
are orthogonal to the coordinate surfaces and the vectors of the local basis
e
i
=
g
ik
e
k
(
i
=1
,
2
,
3) are tangential to the coordinate lines. These two basis
are biorthogonal, as it follows immediately from their definitions, that is
e
i
·
e
k
=
δ
i
.
For an arbitrary vector written in the first basis
F
=
F
1
e
1
+
F
2
e
2
+
F
3
e
3
,
where the covariant components by virtue of the biorthogonal condition are
determined by the equality
F
k
=
F
e
k
. The vector
F
can be written as a
sum of two vectors
F
⊥
and
F
ν
:
F
⊥
normal to the field-line,
F
⊥
=
F
1
e
1
+
F
2
e
2
and the vector
F
ν
=
F
3
e
3
,
orthogonal to the coordinate surface
x
3
=const.
A vector
F
expanded in the second basis is
·
F
=
F
1
e
1
+
F
2
e
2
+
F
3
e
3
,
where the contravariant components
F
k
=
F
e
k
. A vector
F
can be presented
as a sum of the vector
F
parallel to the field-line,
F
=
F
3
e
3
,
and the vector
F
τ
=
F
1
e
1
+
F
2
e
2
tangential to the coordinate surface
x
3
=const.
Equations for the boundary surfaces (6.28) and (6.29) in new coordinates
have the form
·
d
2
,
x
3
=
1
=const
.
±
(6.32)
The region
Ω
is a parallelepiped II in the coordinates
{
x
i
}
(see Fig. 6.2b).
Dipole Coordinates
In the important special case of a dipole magnetic field, special notation is used
for the orthogonal coordinates
y
1
=
ν
,
y
2
=
ϕ
,
y
3
=
µ
. The potential
Ψ
of the
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