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The following question come to my mind when I am viewing this Wiki page:

"Laplace's equation is separable in 13 orthogonal coordinate systems, and the Helmholtz equation is separable in 11 orthogonal coordinate systems."

Is it possible to explain why this is the case?

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A partial answer; separability of coordinates requires certain relations among the scale factors of the coordinate system. For the Helmholtz equation there is the Robertson condition which when solved yields the 11 coordinate systems.

Chapter 5 of Morse and Feshbach "Methods of Theoretical Physics", Volume 1 covers this in detail.

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