Begin2.DVI

Elliptic cylindrical coordinates (ξ, η, z ) :

x= cosh ξcos η, ξ ≥ 0

y= sinh ξsin η, 0 ≤η≤ 2 π

z=z, −∞ < z < ∞

ds^2 =h^2 ξdξ^2 +h^2 ηdη^2 +h^2 zdz^2

hξ=hη=

√

sinh^2 ξ+ sin^2 η, h z= 1

gij =





h^2 ξ 0 0

0 h^2 η 0

0 0 h^2 z





(8 .81)

Elliptic coordinates (ξ, η, φ ) :

x=

√

(1 −η^2 )(ξ^2 −1) cos φ, − 1 ≤η≤ 1

y=

√

(1 −η^2 )(ξ^2 −1) sin φ, 1 ≤ξ < ∞

z=ξη, 0 ≤φ≤ 2 π

ds^2 =h^2 ξdξ^2 +h^2 ηdη^2 +h^2 φdφ^2

hξ=

√

ξ^2 −η^2

ξ^2 − 1

, h η=

√

ξ^2 −η^2

1 −η^2

, h φ=

√

(ξ^2 −1)(1 −η^2 )

gij =





h^2 ξ 0 0

0 h^2 η 0

0 0 h^2 φ





(8 .82)

Transformation of Vectors

A vector field defined by

A
=A
(x, y, z ) = A 1 (x, y, z )ˆe 1 +A 2 (x, y, z )ˆe 2 +A 3 (x, y, z )ˆe 3

represents a magnitude and direction associated to each point (x, y, z )in some region

Ror three dimensional cartesian coordinates. This vector field is to remain invariant

under a coordinate transformation. However, the form used to represent the vector

field will change. For example, under a transformation to cylindrical coordinates,

where

x=rcos θ y =rsin θ z =z, (8 .83)

the above vector can be represented in terms of the unit orthogonal vectors ˆer, ˆeθ,ˆez

in the form

A
=A
(r, θ, z) = Ar(r, θ, z )ˆer+Aθ(r, θ, z )ˆeθ+Az(r, θ, z )ˆez. (8 .84)