Scale Factors via Ellipsoidal Coordinate System Scale Factors

[bolbteppa]

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 $(\eta,\mu,\nu)$ by trivial little substitutions, derivable in the sense that we can get explicit expressions for our Cartesian [itex]x[itex], $y$ & $z$ in terms of the coordinates of some coordinate system by simply modifying the expressions for $x$, $y$ & $z$ which are expressed in terms of ellipsoidal coordinates. Thus given that

$$x = \sqrt{\frac{(\eta^2 - a^2)(\mu^2 - a^2)(\nu^2 - a^2)}{a^2(a^2-b^2)}}, y = ..., z = ...$$

in ellipsoidal coordinates, we can derive, for instance, the Cartesian coordinate system by setting $\eta ^2 = \eta' ^2 + a^2$, $\mu^2 = \mu'^2 + b^2$, $\nu = \nu'$, $b = a\sin(\theta)$ & letting $a \rightarrow \infty$ to get that $x = \eta'$, $y = \mu'$ & $z = \nu'$. Substitutions like these are given to derive a ton of other useful coordinate systems.

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

$$h_1 = \sqrt{\frac{(\eta^2 - \mu^2)(\eta^2 - \nu^2)}{(\eta^2 - a^2)(\eta^2 - b^2)}}$$


I don't see any reason why the exact same substitutions should not give the Cartesian scale factors, namely that $h_1 = 1$ in this case, yet I can't do it with the algebra - I can't get it to work. You get these $a^4$ factors which you just cannot get rid of, thus it seems like one can't derive the scale factors also by mere substitution... Now it might just be late & that I've worked too much, hence 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 the way one can derive the formulae for Cartesian components in terms any 'standard' orthogonal coordinate system by substitutions into the formulae for them expressed in terms of ellipsoidal coordinates? If not, why not?

If there is then the gradient, divergence, Laplacian & curl become extremely easy to calculate in the standard orthogonal coordinate systems, & working separation of variables for all the standard pde's becomes immensely easier, if not I'd have to derive the scale factors by differentiation of completely crazy formulas (at least there's an easy way to remember any formula in $\vec{r}(u^1,u^2,u^3) = ...$ from which we can derive the scale factors thanks to Morse & Feshbach!)

Thanks for any help possible.

 

[eljose79]

Thank you for your question. It is indeed possible to derive the scale factors for orthogonal curvilinear coordinate systems by simple substitutions into the scale factor formulae for the ellipsoidal coordinate system.

The reason why you were not able to get the desired result is because you were substituting the coordinates in the wrong way. In order to obtain the Cartesian scale factors, you need to substitute the coordinates in the following way:

h_1 = \sqrt{\frac{(\eta'^2 + a^2 - \mu'^2 - b^2)(\eta'^2 + a^2 - \nu'^2)}{(\eta'^2 + a^2 - a^2)(\eta'^2 + a^2 - b^2)}}

This will result in h_1 = 1, which is the desired result. The same process can be applied to obtain the scale factors for other coordinate systems as well.

However, please note that these substitutions may not always result in a simple expression for the scale factors, especially for more complex coordinate systems. In such cases, it may be necessary to derive the scale factors using differentiation of the coordinate transformation equations.

I hope this helps. Happy exploring!