A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant
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(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
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(2)
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can be rewritten in terms of functions 
defined by
![1/(h_1h_2h_3)partial/(partialu_i)((h_1h_2h_3)/(h_i^2)partial/(partialu_i))
=(g(u_(i+1),u_(i+2)))/(h_1h_2h_3)partial/(partialu_i)[f_i(u_i)partial/(partialu_i)]
=1/(h_i^2f_i)partial/(partialu_i)(f_ipartial/(partialu_i))](/images/equations/StaeckelDeterminant/NumberedEquation3.svg) |
(3)
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such that 
can be written
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(4)
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When this is true, the separated equations are of the form
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(5)
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The 
s
obey the minor equations
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(6)
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 |  |  |
(7)
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 |  |  |
(8)
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which are equivalent to
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(9)
|
 |
(10)
|
 |
(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).
