Scale Factors via Ellipsoidal Coordinate System Scale Factors

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In Morse & Feshbach (P512 - 514) they show how 10 different orthogonal coordinate systems (mentioned on this page) are derivable from the confocal ellipsoidal coordinate system [latex](x,y,z)[/latex] by trivial little substitutions, derivable in the sense that we can get explicit expressions for our Cartesian [latex]x'[/latex], [latex]y'[/latex] & [latex]z'[/latex] in terms of the coordinates of some coordinate system by simply modifying the expressions for [latex]x'[/latex], [latex]y'[/latex] & [latex]z'[/latex] which are expressed in terms of ellipsoidal coordinates (thus allowing us to find the position vector, line element, area, volume etc... for those orthogonal systems easily enough). Thus given that

[latex]x' = \sqrt{\frac{(x^2 - a^2)(y^2 - a^2)(z^2 - a^2)}{a^2(a^2-b^2)}}, y' = ..., z' = ...[/latex]

in ellipsoidal coordinates, we can derive, for instance, the Cartesian coordinate system by setting [latex] x^2 = x^2 + a^2 [/latex], [latex]y^2 = y^2 + b^2[/latex], [latex]z = z[/latex], [latex]b = a\sin(\theta)[/latex] & letting [latex]a \rightarrow \infty[/latex] to get that [latex]x' = x[/latex], [latex]y' = y[/latex] & [latex]z' = z[/latex]. Substitutions like these are given to derive a ton of other useful coordinate systems - really beats re-deriving everything from the geometry of the situation, saves time & really cuts down on memorization (wish I'd come across it earlier!!!).

I don't see why one shouldn't be able to use the exact same substitutions on the scale factors. Thus given

[latex]h_1 = \sqrt{\frac{(x^2 - y^2)(x^2 - z^2)}{(x^2 - a^2)(x^2 - b^2)}}[/latex]


I don't see any reason why the exact same substitutions should not give the scale factors for any coordinate system, namely that [latex]h_1 = 1[/latex] in our Cartesian case, yet I can't do it with the algebra - I just cannot get it to work. You get these [latex]a^4[/latex] factors which you can't get rid of, or end up with extra [latex]y^2/x^2[/latex] terms, or end up dividing by zero or some nonsense, thus it seems like one can't derive the scale factors also by mere substitution... But again, it seems as though it should just work with no problem

Thus my question is: Is it possible to get the scale factors for orthogonal curvilinear coordinate systems by simple substitutions into the scale factor formulae for the ellipsoidal coordinate system, analogous to how one can express Cartesian components in terms any of the 'standard' orthogonal coordinate systems by little substitutions of variables in the ellipsoidal coordinate formulae? If not, why not?

If there is then the gradient, divergence, Laplacian, curl, Stackel matrices & all that stuff becomes extremely easy to calculate in any of the standard orthogonal coordinate systems, & working separation of variables for all the standard pde's becomes immensely easier & quicker. If not I'd have to derive the scale factors by differentiation of completely crazy formulas which isn't that bad, but I'm sure we can do better :cool: (& at least there's an easy unifying way to remember any formula in [latex]\vec{r}(u^1,u^2,u^3) = ...[/latex] from which we can derive the scale factors thanks to Morse & Feshbach!).

I would give nearly anything for something as simple as this to work, so thanks for any help possible.