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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 ) 2 ( x, y, z )]
+ K 2 [ Φ 1 ( x, y, z ) 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 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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