Difference between revisions of "Ellipsoidal coordinates"
(→References: Jeffreys & Jeffreys (1972)) |
(→Comments: Laplace's equation, cite JJ') |
||
| Line 30: | Line 30: | ||
====Comments==== | ====Comments==== | ||
| − | + | [[Laplace equation|Laplace's equation]] expressed in ellipsoidal coordinates is separable (cf [[Separation of variables, method of]]), and leads to [[Lamé function]]s. | |
====References==== | ====References==== | ||
Revision as of 20:27, 25 October 2014
spatial elliptic coordinates
The numbers 
, 
and 
connected with Cartesian rectangular coordinates 
, 
and 
by the formulas
 |
 |
 |
where 
. The coordinate surfaces are (see Fig.): ellipses 
, one-sheet hyperbolas (
), and two-sheet hyperbolas (
), with centres at the coordinate origin.
Figure: e035420a
The system of ellipsoidal coordinates is orthogonal. To every triple of numbers 
, 
and 
correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system 
.
The Lamé coefficients are
 |
 |
 |
If one of the conditions 
in the definition of ellipsoidal coordinates is replaced by an equality, then degenerate ellipsoidal coordinate systems are obtained.
Comments
Laplace's equation expressed in ellipsoidal coordinates is separable (cf Separation of variables, method of), and leads to Lamé functions.
References
| [a1] | G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18 |
| [a2] | Harold Jeffreys, Bertha Jeffreys, Methods of Mathematical Physics, 3rd edition, Cambridge University Press (1972) Zbl 0238.00004 |
Ellipsoidal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ellipsoidal_coordinates&oldid=34026