MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 186

x
sin I ,

x 1 = x = x 1 sin I,

− −

x 2 = y , y = x 2 ,

x 3
sin I .

x 3 = x cos I z sin I, z = x 1 cos I − − − (7.3)

The contravariant basis

x n = ∂x n

∂x x + ∂x n

y + ∂x n
∂z

l n = ∇ z

∂y

of the oblique coordinate system is

1
sin I x ,

l 1 = x l 2 = y = y , l 3 = z =cos I x − sin I z . − cot I z = −

The covariant basis

∂ r
∂x n

l n =

is

1
sin I z .

sin I x − cos I z , l 2 = y = y , l 1 = x = − l 3 =cot I x + z = −

These two bases are mutually orthonormal:
l i

l k = δ k .

·

Here δ k =1if i = k and δ k

=0if i = k . The metric tensor of the oblique

coordinate system is

⎛ ⎞

1 0 cot I

⎝ ⎠

g ik =( l i · l k )= 0 1 0

0si − 2 I cot I

and

⎛ ⎞

sin − 2 I

0 − cot I

g ik = l i l k =

⎝ ⎠ .

· 0 1 0

− cot I 0 1

Determinant of the metric tensor det( g ik ) = 1, so that

l 1 = l 2 l 3 , l 2 = l 3 l 1 , l 3 = l 1 l 2 . × × ×

l 1 , l 2 , l 3 Then, for operator ∇ in { } basis, we get

∂
∂x 1 + l 2

∂
∂x 2 + l 3

∂
∂x 3 .

= l 1

∇

Hence the contravariant components of a = ∇ × c are