A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant
(1)
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in which are functions of alone is called a Stäckel determinant. A coordinate system is separable if it obeys the Robertson condition, namely that the scale factors in the Laplacian
(2)
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can be rewritten in terms of functions defined by
(3)
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such that can be written
(4)
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When this is true, the separated equations are of the form
(5)
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The s obey the minor equations
(6)
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(7)
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(8)
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which are equivalent to
(9)
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(10)
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(11)
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(Morse and Feshbach 1953, p. 509). This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655-666) give not only the Stäckel determinants for common coordinate systems, but also the elements of the determinant (although it is not clear how these are derived).