# Separation of variables is possible only in 11 coordinate systems?

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• Trying2Learn
In summary, eleven coordinate systems are possible for separation of variables. These systems are Cartesian, Circular-cylinder, Elliptic-cylinder, Parabolic-cylinder, Prolate spheroidal, Oblate spheroidal, Parabolic, Conical, Ellipsoidal, and Paraboloidal.
Trying2Learn
TL;DR Summary
Is such separation possible only in 11 coordinate systems?
I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems.

I list them below:

1.Cartesian coordinates
2.Circular-cylinder coordinates
3.Elliptic-cylinder coordinates
4.Parabolic-cylinder coordinates
5.Spherical coordinates
6.Prolate spheroidal coordinates
7.Oblate spheroidal coordinates
8.Parabolic coordinates
9.Conical coordinates
10.Ellipsoidal coordinates
11.Paraboloidal coordinatesIf this is true, why eleven?
Or is my memory foggy, and this is not true?

jedishrfu said:
Separation of variables I think needs an orthogonal coordinate system to work.

https://en.wikipedia.org/wiki/Separation_of_variables#Curvilinear_coordinates

In the applicability section they mention this requirement briefly with a reference to the MathWorld website.

https://mathworld.wolfram.com/OrthogonalCoordinateSystem.html

This article mentions the 11 coordinate systems of degree two or less in 3D.
OK, but why 11?

There were 12 Apostles, so why not 12 (please forgive my facetiousness, but I am trying to the bottom of this that has bothered me for 30 years)

Why 11 or 8 or 14 or some other number?

Or are there moments in math, where we do not ask such questions?

I am neither sufficiently bright nor sufficiently schooled to directly answer your question. But is it not exactly similar to "why are there only 5 space filling regular polyhedra" in 3D. Seems an easier question to ponder.......

Trying2Learn
hutchphd said:
I am neither sufficiently bright nor sufficiently schooled to directly answer your question. But is it not exactly similar to "why are there only 5 space filling regular polyhedra" in 3D. Seems an easier question to ponder.......
Damn, I forgot about that one. Now my day is ruined.

berkeman
Trying2Learn said:
There were 12 Apostles
But Judas Iscariot killed himself

Trying2Learn
malawi_glenn said:
But Judas Iscariot killed himself
Oh, now that's just great: throw in a singularity and ruin the week

i'm sure there are a dozen more with computer graphic coordinates and several different color vector coordinate systems, bipolar 2 stings and a 3D angle, toroidal-elliptic -parabolic, hyperbolic coordinates . etc. but take my wife, please, she is not coordinated ;)

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