Deο¬ne (r,π,Ο) coordinates by
![]() | (574) |
![]() | (575) |
where (R,Ο,Z) are the cylindrical coordinates and R0 is a constant. The above transformation is shown graphically in Fig. 40.
The Jacobian of (r,π,Ο) coordinates can be calculated using the deο¬nition. Using x = R cosΟ, y = R sinΟ, and z = Z, the Jacobian (with respect to the Cartesian coordinates (x,y,z)) is written as
Next, we transform the GS equation from (R,Z) coordinates to (r,π) coordinates. Using the relations R = R0 + r cosπ and Z = r sinπ, we have
![]() | (577) |
![]() | (578) |
![]() | (579) |
![]() | (580) |
The GS equation in (R,Z) coordinates is given by
![]() | (581) |
The term βΞ¨ββZ is written as
Using Eq. (582), β2Ξ¨ββZ2 is written as
![]() | (584) |
sinπ = ![]() |
cosπ![]() ![]() |
![]() | (585) |
cosπ = ![]() |
![]() | (586) |
Summing the the right-hand-side of Eq. (583) and the expression on line (587) yields
![]() | (589) |
Using these, the GS equation is written as
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
which can be arranged in the form
![]() | (590) |
which agrees with Eq. (3.6.2) in Wesssonβs book[27], where f is deο¬ned by f = RBΟβΞΌ0, which is diο¬erent from g β‘ RBΟ by a 1βΞΌ0 factor.