Editing Appendix/Ramblings/EllipticCylinderCoordinates

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=Elliptic Cylinder Coordinates=

==Background==
Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosines|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, here we detail the properties of [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates Elliptic Cylinder Coordinates].  First, we present this coordinate system in the manner described by [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>]; second, we provide an alternate presentation, obtained from Wikipedia; then, third, we investigate whether or not a related coordinate system based on ''concentric'' (rather than ''confocal'') elliptic surfaces can be satisfactorily described.

It is useful to keep in mind various properties of a set of ''[https://en.wikipedia.org/wiki/Confocal_conic_sections#Confocal_ellipses confocal ellipses]'' in which the location of the pair of foci is fixed at, <math>~(x, y) = (\pm~ c, 0)</math>, and the semi-major axis, <math>~a</math>, is the parameter.  The relevant prescriptive relation is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2}</math>&nbsp; &nbsp; &nbsp; for, &nbsp; <math>~a > c\, .</math>
  </td>
</tr>
</table>
The semi-minor axis length, <math>~b</math>, and the eccentricity, <math>~e</math>, of the ellipse are, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a^2 - c^2)^{1 / 2} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e\equiv \biggl[1 - \frac{b^2}{a^2} \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c}{a} \, .</math>
  </td>
</tr>
</table>
The length, <math>~\ell_1</math>, of the chord that connects one focus to a point, <math>~P(x,y)</math>, on the ellipse is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ell_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a + \biggl(\frac{c}{a}\biggr)x \, ;</math>
  </td>
</tr>
</table>
and the length, <math>~\ell_2</math>, of the chord that connects the second focus to that same point on the ellipse is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ell_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a - \biggl(\frac{c}{a}\biggr)x \, .</math>
  </td>
</tr>
</table>
It is easy to see that, for any point on the ellipse, the sum of these two lengths is, <math>~2a</math>.  It is worth noting as well that the associated <math>~y</math> coordinate of the relevant point can be obtained from the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \ell_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y^2 + (c+x)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ (ay)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a \ell_1)^2 - (ac+ ax)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a^2 + cx )^2 - (ac+ ax)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a^4 + 2a^2 cx + c^2x^2) - (a^2c^2 + 2a^2 cx + a^2x^2)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a^4  + c^2x^2) - (a^2c^2 + a^2x^2)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a^2-x^2)(a^2 - c^2) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm~\frac{1}{a}\biggl[ (a^2-x^2)(a^2 - c^2) \biggr]^{1 / 2}  \, .</math>
  </td>
</tr>
</table>

==MF53==

===Definition===
From [[User:Tohline/Appendix/References#MF53|MF53]]'s ''Table of Separable Coordinates in Three Dimensions'' (see their Chapter 5, beginning on p. 655), we find the following description of '''Elliptic Cylinder Coordinates''' (p. 657).

<table border="1" cellpadding="10" align="center" width="80%">
<tr><td align="center">
'''Elliptic Cylindrical Coordinates'''<br />([[User:Tohline/Appendix/References#MF53|MF53]] Primary Definition)</td></tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 \xi_2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_3 </math>
  </td>
</tr>
</table>
</td></tr>
<tr><td align="center">
'''Alternate Definition'''
</td></tr>
<tr><td align="left">
Making the substitutions, <math>~\xi_3 \rightarrow z</math>, <math>~\xi_2 \rightarrow \cos\nu</math>, and <math>~\xi_1 \rightarrow d\cosh\mu</math>, we equally well obtain:
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d\cosh\mu \cdot \cos\nu </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d \sinh\mu \cdot \sin\nu </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z </math>
  </td>
</tr>
</table>

</td></tr></table>


Notice that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x^2}{d^2 \cosh^2\mu} + \frac{y^2}{d^2 \sinh^2\mu}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos^2\nu + \sin^2\nu = 1 \, .</math>
  </td>
</tr>
</table>
Hence, as is pointed out in a related [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates#Basic_definition Wikipedia discussion], "&hellip; this shows that curves of constant <math>~\mu</math> form ellipses."  For a given choice of <math>~\mu</math> &#8212; say, <math>~\mu_0</math> &#8212; let's see how the shape of the resulting ellipse relates to the standard ellipses described in our [[#Background|background discussion]], above.  The semi-major axis of the selected ellipse must be,
<div align="center">
<math>a = d\cosh\mu_0 \, .</math>
</div>

And its eccentricity must be obtainable from the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2 - c^2 = a^2(1 - e^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d^2 \sinh^2\mu_0</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2 \tanh^2\mu_0 = a^2 \biggl(1 - \frac{1}{\cosh^2\mu_0} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ e^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\cosh^2\mu_0} \, .</math>
  </td>
</tr>
</table>
We note, as well, that the x-coordinate location of the focus of the selected ellipse is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c^2 = a^2 e^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d^2\, .</math>
  </td>
</tr>
</table>
This emphasizes a key property of the MF53 Elliptic Cylindrical Coordinate system, viz., the family of ellipses that result from selecting various values of <math>~\mu_0</math> is a family of ''confocal'' ellipses.

===Scale Factors===

====Primary====

Appreciating that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \xi_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 / 2}\xi_1(1-\xi_2^2) \, ,
</math> &nbsp; &nbsp; &nbsp; and that,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \xi_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 / 2}\xi_2(\xi_1^2 - d^2) \, ,
</math>
  </td>
</tr>
</table>

we find that the respective [[User:Tohline/Appendix/Ramblings/DirectionCosines#Scale_Factors|scale factors]] are given by the expressions,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ h_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_2^2 +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 }\xi_1^2 (1-\xi_2^2)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (\xi_1^2 - d^2)^{- 1 } [  (\xi_1^2 - d^2)\xi_2^2 +\xi_1^2 (1-\xi_2^2) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{  \xi_1^2 - d^2 \xi_2^2 }{\xi_1^2 - d^2} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ h_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\partial x}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_2} \biggr)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1^2 + \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 }\xi_2^2(\xi_1^2 - d^2)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 - \xi_2^2)^{- 1 } [\xi_1^2(1 - \xi_2^2)  + \xi_2^2(\xi_1^2 - d^2) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{ \xi_1^2  - d^2 \xi_2^2  }{1 - \xi_2^2} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ h_3^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\partial x}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_3} \biggr)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, . </math>
  </td>
</tr>
</table>
These match the scale-factor expressions found in [[User:Tohline/Appendix/References#MF53|MF53]].

====Alternatively====

Alternatively, the [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates#Scale_factors Wikipedia discussion] gives,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_\mu = h_\nu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d\sqrt{ \sinh^2\mu + \sin^2\nu}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{d^2(\sinh^2\mu + \sin^2\nu)} \biggl[ \frac{\partial^2 \Phi}{\partial \mu^2} + \frac{\partial^2 \Phi}{\partial \nu^2} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} \, .
</math>
  </td>
</tr>
</table>

===Inverting Coordinate Mapping===

Inverting the original coordinate mappings, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\xi_1^2 - a^2)\biggl[ 1 - \biggl(\frac{x}{\xi_1}\biggr)^2 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\xi_1^2 - a^2) ( \xi_1^2 - x^2 ) - \xi_1^2 y^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\xi_1^2 - a^2) \xi_1^2  - (\xi_1^2 - a^2) x^2  - \xi_1^2 y^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi_1^4  - \xi_1^2 (a^2 + x^2  + y^2) + a^2 x^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \xi_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{
(a^2 + x^2 + y^2) \pm \biggl[ (a^2 + x^2 + y^2)^2 - 4a^2 x^2 \biggr]^{1 / 2}
\biggr\}
</math>
  </td>
</tr>
</table>
Only the ''superior'' &#8212; that is, only the ''positive'' &#8212; sign will ensure positive values of <math>~\xi_1^2</math>, so in summary we have,
<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">'''Coordinate Transformation'''</td></tr>
<tr>
  <td align="right">
<math>~\xi_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\sqrt{2}}\biggl\{
\biggl[ (a^2 + x^2 + y^2)^2 - 4a^2 x^2\biggr]^{1 / 2} + (a^2 + x^2 + y^2) 
\biggr\}^{1 / 2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{\xi_1} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

===Alternative Wikipedia Definition===

This same MF53 coordinate system &#8212; with different variable notation &#8212; is referred to in a [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates#Alternative_definition Wikipedia discussion] as an "alternative and geometrically intuitive set of elliptic coordinates."  The relevant mapping is, <math>~(d\sigma, \tau, z)_\mathrm{Wikipedia} = (\xi_1, \xi_2, \xi_3)_\mathrm{MF53}</math>.  The identified mapping to Cartesian coordinates is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(d\sigma)\tau </math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 \xi_2 \, ;</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d \biggl[ (\sigma^2 - 1 )(1 - \tau^2)  \biggr]^{1 / 2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_3 \, .</math>
  </td>
</tr>
</table>

According to the Wikipedia discussion, the three scale factors are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_\sigma^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
d^2\biggl[\frac{\sigma^2 - \tau^2}{\sigma^2 - 1}  \biggr] \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~h_\tau^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
d^2\biggl[\frac{\sigma^2 - \tau^2}{1 - \tau^2}  \biggr] \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~h_z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 \, .
</math>
  </td>
</tr>
</table>

Interestingly, the Wikipedia discussion also includes the following expression for the Laplacian in this elliptic cylindrical coordinate system:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{d^2(\sigma^2 - \tau^2)}
\biggl[ \sqrt{\sigma^2 - 1} \frac{\partial}{\partial\sigma}\biggl( \sqrt{\sigma^2 - 1} \frac{\partial\Phi}{\partial\sigma} \biggr)
+
\sqrt{1 - \tau^2 } \frac{\partial}{\partial\tau}\biggl( \sqrt{1 - \tau^2} \frac{\partial\Phi}{\partial\tau} \biggr) \biggr] + \frac{\partial^2\Phi}{\partial z^2} \, .
</math>
  </td>
</tr>
</table>

=T5 Coordinates=

==Introduction==

As has been made clear in our above review of the Elliptic Cylinder Coordinate system <math>~(\xi_1, \xi_2, \xi_3) = (d\cosh\mu, \cos\nu, z)</math>, individual curves within a family of ''confocal'' ellipses are identified by one's choice of the "radial" coordinate parameter, <math>~\mu</math>, or, alternatively, <math>~\xi_1</math>.  Specifically, while the two foci of every ellipse are positioned along the x-axis at the same points &#8212; namely, <math>~(x, y) = (\pm~d, 0)</math> &#8212; the length of the semi-major axis is given by, <math>~a = \xi_1 = d\cosh\mu</math>.

In a [[User:Tohline/Appendix/Ramblings/T3Integrals|separate chapter]] we have introduced a different orthogonal curvilinear coordinate system that we refer to as, "[[User:Tohline/Appendix/Ramblings/T3Integrals|T3 Coordinates]]."  In this coordinate system, <math>~(\lambda_1, \lambda_2, \lambda_3)</math>, individual surfaces within a family of ''concentric'' spheroids are identified by one's choice of a different "radial" coordinate parameter, <math>~\lambda_1</math>.  Here we will adopt essentially this same set of orthogonal coordinates, using <math>~\lambda_1</math> and <math>~\lambda_2</math> to describe a family of ''concentric'' ellipses that is independent of the vertical-coordinate.  We will refer to it as the &hellip;

<table border="1" cellpadding="10" align="center" width="80%">
<tr><td align="center">
'''T5 Coordinate System'''</td></tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\lambda_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x \cosh \zeta </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2 y^2)^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x (\sinh\zeta)^{1/(1-q^2)} </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x^{q^2}}{qy} \biggr)^{1/(q^2-1)}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z </math>
  </td>
</tr>
</table>

where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\zeta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\sinh^{-1}\biggl( \frac{qy}{x} \biggr) </math>
  </td>
</tr>
</table>
and, <math>~0 < q < \infty</math> is the (fixed) parameter used to specify the eccentricity, <math>~e = [(q^2-1)^{1 / 2}/q]</math>, of every <math>~\lambda_1 = </math> constant curve within the family of ''concentric'' ellipses.

</td></tr></table>


Checking these expressions, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2 \equiv x (\sinh\zeta)^{1/(1-q^2)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x \biggl( \frac{qy}{x} \biggr)^{1/(1-q^2)} = x \biggl( \frac{x}{qy} \biggr)^{1/(q^2-1)} 
= \biggl( \frac{x^{q^2}}{qy} \biggr)^{1/(q^2-1)} \, .</math>
  </td>
</tr>
</table>

And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1 \equiv x\cosh\zeta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x\biggl[ 1 + \sinh^2\zeta\biggr]^{1 / 2} = x\biggl[ 1 + \biggl( \frac{qy}{x} \biggr)^2\biggr]^{1 / 2}
= ( x^2 + q^2 y^2 )^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Comparing this last expression with the [[#Background|above background description of ellipses]], we see that <math>~\lambda_1 = </math> constant &#8212; for example, <math>~\lambda_0</math> &#8212; is synonymous with an ellipse having &hellip;
<ul>
<li>A semi-major axis of length, <math>~a = \lambda_0</math>;</li>
<li>An eccentricity, <math>~e \equiv (1 - b^2/a^2)^{1 / 2} = [(q^2-1)/q^2]^{1 / 2}</math>;</li>
<li>A pair of foci whose coordinate locations along the major axis are, <math>~(x, y) = (\pm~c, 0)</math>, where, <math>~c = ae</math>.</li>
</ul>

==Invert Coordinate Mapping==

Solving for <math>~x(\lambda_1, \lambda_2)</math>, we find &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~( x^2 + q^2 y^2 )^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~y^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{q^2}\biggl[ \lambda_1^2 - x^2 \biggr] \, .</math>
  </td>
</tr>
</table>

And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{x^{q^2}}{qy} \biggr)^{1/(q^2-1)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~  y^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{q^2} \biggl[ x^{2q^2} \lambda_2^{2(1-q^2)}\biggr] \, .</math>
  </td>
</tr>
</table>

Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x^{2q^2} \lambda_2^{2(1-q^2)} + x^2 - \lambda_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .  </math>
  </td>
</tr>
</table>
Alternatively, solving for <math>~y(\lambda_1, \lambda_2)</math>, we find &hellip;

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~( x^2 + q^2 y^2 )^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~x^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1^2 - q^2 y^2 \, .</math>
  </td>
</tr>
</table>
And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{x^{q^2}}{qy} \biggr)^{1/(q^2-1)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(qy)^{1/q^2}~\lambda_2^{(q^2-1)/q^2} \, .</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(qy)^{2/q^2}~\lambda_2^{2(q^2-1)/q^2} -\lambda_1^2 + q^2 y^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>

<span id="InvertedRelations">&nbsp;</span>
<table border="1" align="center" width="80%" cellpadding="10">
<tr><td align="center">'''Summary of Inverted Relations'''</td></tr>
<tr><td align="left">

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2^2\biggl( \frac{x}{\lambda_2}\biggr)^{2q^2} + x^2 - \lambda_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2^2 \biggl( \frac{qy}{\lambda_2} \biggr)^{2/q^2} + q^2 y^2 - \lambda_1^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</td></tr>
<tr><td align="left">
'''Example:''' &nbsp; &nbsp; <math>~q^2 = 2</math>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^{4} \lambda_2^{-2} + x^2 - \lambda_1^2</math>
  </td>
  <td align="right"><math>~\leftarrow</math>&nbsp; &nbsp;'''Quadratic Eq.''' in x<sup>2</sup></td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~x^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda_2^2}{2} 
\biggl\{ \biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2}  \biggr]^{1 / 2} - 1 \biggr\} 
=
\frac{\lambda_2^2}{2} (\Lambda - 1) 
\, ;
</math>
  </td>
  <td align="right">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2y^2 + (2^{1 / 2}\lambda_2)~y -\lambda_1^2 </math>
  </td>
  <td align="right"><math>~\leftarrow</math>&nbsp; &nbsp;'''Quadratic Eq.''' in y</td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~y </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\biggl\{
-2^{1 / 2} \lambda_2 ~\pm ~ \biggl[2\lambda_2^2 + 8\lambda_1^2  \biggr]^{1 / 2}
\biggr\}
</math>
  </td>
  <td align="right">&nbsp;</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda_2}{2^{3 / 2}}
\biggl\{ \biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} - 1 \biggr\} 
=
\frac{\lambda_2}{2^{3 / 2}}(\Lambda - 1)
\, .
</math>
  </td>
  <td align="right">&nbsp;</td>
</tr>
</table>

where,
<div align="center">
<math>~\Lambda \equiv \biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} \, .</math>
</div>

Note &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{y}{x^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda_2}{2^{3 / 2}} \cdot \frac{2}{\lambda_2^2} = \frac{1}{\sqrt{2} \lambda_2} \, ;</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{4y^2}{x^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} - 1
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{1}{\ell^2} \equiv (x^2 +  4y^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2\biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} 
= x^2 \Lambda
=\frac{\lambda_2^2}{2} \Lambda(\Lambda - 1)
</math>&nbsp; &nbsp; &nbsp; or,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\ell^2} \equiv (x^2 +  4y^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{2} \lambda_2 y \biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} 
= \sqrt{2} \lambda_2 y \Lambda 
= \frac{\lambda_2^2}{2} \cdot \Lambda (\Lambda - 1)
\, .
</math>
  </td>
</tr>
</table>
Note as well that, <math>~\ell^{-2} = 2\lambda_1^2 \Lambda/(\Lambda + 1) \, .</math>
</td></tr>

<tr><td align="left">
'''Example:''' &nbsp; &nbsp; <math>~q^2 = \frac{3}{2}</math>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_2^2\biggl( \frac{x}{\lambda_2}\biggr)^{3} + x^2 - \lambda_1^2</math>
  </td>
  <td align="left"> &nbsp; &nbsp; &nbsp; &nbsp;<math>~\leftarrow</math>&nbsp; &nbsp;'''[[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|Cubic Eq.]]''' in x</td>
</tr>

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_2^2 \biggl( \frac{3 y^2}{2\lambda_2^2} \biggr)^{2/3} + \frac{3}{2} y^2 - \lambda_1^2 </math>
  </td>
  <td align="left"> &nbsp; &nbsp; &nbsp; &nbsp;<math>~\leftarrow</math>&nbsp; &nbsp;'''[[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|Cubic Eq.]]''' in y<sup>2/3</sup></td>
</tr>
</table>
</td></tr>

<tr><td align="left">
'''Example:''' &nbsp; &nbsp; <math>~q^2 = 3</math>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_2^2\biggl( \frac{x}{\lambda_2}\biggr)^{6} + x^2 - \lambda_1^2</math>
  </td>
  <td align="left"> &nbsp; &nbsp; &nbsp; &nbsp;<math>~\leftarrow</math>&nbsp; &nbsp;'''[[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|Cubic Eq.]]''' in x<sup>2</sup></td>
</tr>

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_2^2 \biggl( \frac{3y^2}{\lambda_2^2} \biggr)^{1/3} + 3 y^2 - \lambda_1^2 </math>
  </td>
  <td align="left"> &nbsp; &nbsp; &nbsp; &nbsp;<math>~\leftarrow</math>&nbsp; &nbsp;'''[[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|Cubic Eq.]]''' in y<sup>2/3</sup></td>
</tr>
</table>
</td></tr>

<tr><td align="left">
'''Example:''' &nbsp; &nbsp; <math>~q^2 = 4</math>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_2^2\biggl( \frac{x}{\lambda_2}\biggr)^{8} + x^2 - \lambda_1^2</math>
  </td>
  <td align="left"> &nbsp; &nbsp; &nbsp; &nbsp;<math>~\leftarrow</math>&nbsp; &nbsp;'''[[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Roots_of_Quartic_Equation|Quartic Eq.]]''' in x<sup>2</sup></td>
</tr>

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_2^2 \biggl( \frac{2y}{\lambda_2} \biggr)^{1 / 2} + 4 y^2 - \lambda_1^2 </math>
  </td>
  <td align="left"> &nbsp; &nbsp; &nbsp; &nbsp;<math>~\leftarrow</math>&nbsp; &nbsp;'''[[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Roots_of_Quartic_Equation|Quartic Eq.]]''' in y<sup>1/2</sup></td>
</tr>
</table>

</td></tr></table>
&nbsp;

==Relevant Partial Derivatives==

Before moving forward, we need to evaluate a number of relevant partial derivatives.

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_1}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial x} \biggl[x^2 + q^2y^2\biggr]^{1 / 2} = \frac{1}{2}\biggl[x^2 + q^2y^2\biggr]^{- 1 / 2} 2x
= \frac{x}{\lambda_1} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_1}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial y} \biggl[x^2 + q^2y^2\biggr]^{1 / 2} = \frac{1}{2}\biggl[x^2 + q^2y^2\biggr]^{- 1 / 2} 2q^2 y
= \frac{q^2 y}{\lambda_1} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial x} \biggl[x^{q^2/(q^2-1)} (q y)^{-1/(q^2-1)}\biggr] 
= \biggl[\frac{q^2}{q^2-1}\biggr] \frac{\lambda_2}{x} 
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial y} \biggl[x^{q^2/(q^2-1)} (q y)^{-1/(q^2-1)}\biggr]
= - \biggl[ \frac{1}{q^2-1} \biggr]\frac{\lambda_2}{y}  
\, .
</math>
  </td>
</tr>
</table>

<span id="ComplementaryDerivatives">We may also need the set of complementary partial derivatives</span>.  Even though we are unable to explicitly invert the coordinate mappings, once we have in hand expressions for the three scale factors (see immediately below), we can determine expressions for the set of complementary partial derivatives via the [[User:Tohline/Appendix/Ramblings/DirectionCosines#Basic_Definitions_and_Relations|generic relation]],

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial x_i}{\partial\lambda_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h_n^2 \cdot \frac{\partial \lambda_n}{\partial x_i} \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="10">
<tr><td align="left">
'''Example:''' &nbsp; &nbsp; <math>~q^2 = 2</math>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda_2}{2^{3 / 2}}\biggl\{
\Lambda - 1
\biggr\} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~x^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda_2^2}{2} 
\biggl\{
\Lambda - 1
\biggr\} \, ,
</math> &nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left" colspan="1">
<math>~
\biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2}  \biggr]^{1 / 2}
\, .</math>
  </td>
</tr>
</table>
Noting that,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial \Lambda}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\Lambda \lambda_1}\biggl[ \frac{4\lambda_1^2}{\lambda_2^2} \biggr]</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \Lambda}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\Lambda \lambda_2}\biggl[ \frac{4\lambda_1^2}{\lambda_2^2} \biggr] \, ,</math>
  </td>
</tr>
</table>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{\lambda_2}{2^{3 / 2}} \cdot \frac{\partial \Lambda}{\partial \lambda_1}
=
\frac{\sqrt{2} \lambda_1}{\lambda_2} \biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2}  \biggr]^{- 1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^{3 / 2}} \biggl[\Lambda - 1\biggr] + \frac{\lambda_2}{2^{3 / 2}} \cdot \frac{\partial \Lambda}{\partial \lambda_2}
=
\frac{(\Lambda - 1) }{2^{3 / 2}} - \frac{\sqrt{2}\lambda_1^2}{\Lambda \lambda_2^2} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\Lambda (\Lambda - 1) - 4 \lambda_1^2/\lambda_2^2 }{2^{3 / 2}\Lambda }  
=
\frac{\Lambda^2 - \Lambda - 4 \lambda_1^2/\lambda_2^2 }{2^{3 / 2}\Lambda }  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (1 - \Lambda) }{2^{3 / 2}\Lambda }  \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>


Let's compare by drawing from the expressions for <math>~\ell^2</math>, above, and for <math>~h_n^2</math> derived below.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ h_1^2 \cdot \frac{\partial \lambda_1}{\partial y} \biggr]_{q^2=2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \lambda_1^2 \ell^2 \biggl( \frac{q^2 y}{\lambda_1} \biggr) \biggr]_{q^2=2}
=
\biggl[ 2\lambda_1 \ell^2 y \biggr]_{q^2=2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\lambda_1 \biggl\{
\frac{1}{\sqrt{2}\lambda_2 \Lambda}
\biggr\} 
=
\frac{\sqrt{2}\lambda_1}{\lambda_2\Lambda} \, .
</math>
  </td>
</tr>
</table>
<font color="red">'''Yes!'''</font>  This, indeed matches the just-derived expression for <math>~\partial y/\partial \lambda_1</math>.  And we also have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ h_2^2 \cdot \frac{\partial \lambda_2}{\partial y} \biggr]_{q^2=2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{  - \biggl[ \frac{1}{q^2-1} \biggr]\frac{\lambda_2}{y} \biggl[ \frac{(q^2-1)xy \ell}{\lambda_2}\biggr]^2\biggr\}_{q^2=2}
=
- \biggl[ \frac{(q^2-1)x^2 y \ell^2}{\lambda_2}\biggr]_{q^2=2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(1 - \Lambda )}{2^{3 / 2} \Lambda}  \, .
</math>
  </td>
</tr>
</table>

<font color="red">'''Yes, again!'''</font>

==Scale Factors, Direction Cosines &amp; Unit Vectors==

From our [[User:Tohline/Appendix/Ramblings/DirectionCosines#Usage|accompanying generic discussion of direction cosines]], we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial \lambda_1}{\partial y } \biggr)^2 + \biggl( \frac{\partial \lambda_1}{\partial z} \biggr)^2\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2 + \biggl( \frac{q^2 y}{\lambda_1}  \biggr)^2 \biggr]^{-1}
=
\lambda_1^2 \biggl[ x^2 + q^4 y^2 \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda_1^2 \ell^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~h_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl( \frac{\partial \lambda_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial \lambda_2}{\partial y } \biggr)^2 + \biggl( \frac{\partial \lambda_2}{\partial z} \biggr)^2\biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{  \biggl[\frac{q^2}{q^2-1}\biggr]^2 \biggl(\frac{\lambda_2}{x}\biggr)^2 +  \biggl[ \frac{1}{q^2-1} \biggr]^2 \biggl(\frac{\lambda_2}{y} \biggr)^2 \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(q^2-1)xy \ell}{\lambda_2}\biggr]^2\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~h_3^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \biggl( \frac{\partial \lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial \lambda_3}{\partial y } \biggr)^2 + \biggl( \frac{\partial \lambda_3}{\partial z} \biggr)^2\biggr\}^{-1} = 1 \, ;</math>
  </td>
</tr>
</table>
where,
<div align="center">
<math>~\ell \equiv (x^2 + q^4 y^2)^{- 1 / 2} \, .</math>
</div>


<table border="1" cellpadding="8" align="center" width="60%">
<tr>
  <td align="center" colspan="4">
'''Direction Cosines for T5 Coordinates'''
<br />
<math>~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center" width="10%"><math>~n</math></td>
  <td align="center" colspan="3"><math>~i = x, y, z</math>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="center">&nbsp;<br />
<math>~x\ell</math><br />&nbsp;</td>
  <td align="center"><math>~q^2 y \ell</math></td>
  <td align="center"><math>~0</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center">
&nbsp;<br />
<math>~q^2 y \ell </math>
<br /></td>
  <td align="center">
<math>~ - x\ell
</math></td>
  <td align="center"><math>~0</math></td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">&nbsp;<br /><math>~0</math><br />&nbsp;</td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
</tr>
</table>

The unit vectors are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k}\gamma_{n3} \, ,
</math>
  </td>
</tr>
</table>
that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (x\ell) + \hat\jmath (q^2 y \ell)  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (q^2 y\ell) - \hat\jmath (x \ell)  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{k}  \, .
</math>
  </td>
</tr>
</table>
Notice that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_1 \cdot \hat{e_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
q^2 xy\ell^2 - q^2 xy\ell^2 = 0 \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_1 \cdot \hat{e_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2\ell^2 + q^4 y^2 \ell^2 
= \ell^2 (x^2 + q^4 y^2) = 1 \, .
</math>
  </td>
</tr>
</table>
These are both desired orthogonality conditions.  Alternatively,
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\hat\imath
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\hat{e}_1 \gamma_{11} + \hat{e}_2 \gamma_{21} + \hat{e}_3 \gamma_{31} 
=
\hat{e}_1 (x\ell) + \hat{e}_2 (q^2 y\ell)  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\hat\jmath
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\hat{e}_1 \gamma_{12} + \hat{e}_2 \gamma_{22} + \hat{e}_3 \gamma_{32} 
=
\hat{e}_1 (q^2y\ell) - \hat{e}_2 (x\ell) \,  ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\hat{k}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\hat{e}_1 \gamma_{13} + \hat{e}_2 \gamma_{23} + \hat{e}_3 \gamma_{33} = \hat{e}_3 \, .
</math>
  </td>
</tr>
</table>

==Spatial Operators==

<table border="1" align="center" cellpadding="10" width="80%">
<tr>
  <td align="center">'''Summary Reminder'''</td>
</tr>
<tr><td align="left">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~h_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda_1^2 \ell^2 \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~h_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(q^2-1)xy \ell}{\lambda_2}\biggr]^2\, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
   <td align="right">
<math>~h_3^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, ;</math>
  </td>
</tr>
</table>

</td></tr></table>

In T5 Coordinates, a couple of relevant operators are:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla F</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_1 \biggl[\frac{1}{h_1} \frac{\partial F}{\partial \lambda_1} \biggr]
+
\hat{e}_2 \biggl[\frac{1}{h_2} \frac{\partial F}{\partial \lambda_2} \biggr]
+
\hat{e}_3 \biggl[\frac{1}{h_3} \frac{\partial F}{\partial \lambda_3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_1 \biggl( \frac{1}{\lambda_1 \ell} \biggr) \frac{\partial F}{\partial \lambda_1} 
+
\hat{e}_2 \biggl[\frac{\lambda_2}{(q^2-1)xy\ell} \biggr] \frac{\partial F}{\partial \lambda_2} 
+
\hat{e}_3 \frac{\partial F}{\partial \lambda_3} \, .
</math>
  </td>
</tr>
</table>


<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2 F</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{h_1 h_2 h_3} \biggl[
\frac{\partial}{\partial \lambda_1} \biggl( \frac{h_2 h_3}{h_1} \cdot \frac{\partial F}{\partial \lambda_1}\biggr)
+
\frac{\partial}{\partial \lambda_2} \biggl( \frac{h_3 h_1}{h_2} \cdot \frac{\partial F}{\partial \lambda_2}\biggr)
+
\frac{\partial}{\partial \lambda_3} \biggl( \frac{h_1 h_2}{h_3} \cdot \frac{\partial F}{\partial \lambda_3}\biggr)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_2}{\lambda_1 (q^2-1)xy\ell^2} \biggl\{
\frac{\partial}{\partial \lambda_1} \biggl[ \frac{(q^2-1)xy }{\lambda_1 \lambda_2 } \cdot \frac{\partial F}{\partial \lambda_1}\biggr]
+
\frac{\partial}{\partial \lambda_2} \biggl[ \frac{\lambda_1 \lambda_2}{(q^2-1) xy } \cdot \frac{\partial F}{\partial \lambda_2}\biggr]
+
\frac{\partial}{\partial \lambda_3} \biggl[ \frac{\lambda_1  (q^2-1) xy \ell^2 }{\lambda_2} \cdot \frac{\partial F}{\partial \lambda_3}\biggr]
\biggr\}
</math>
  </td>
</tr>
</table>

And if <math>~F</math> is a function only of <math>~\lambda_1</math>, then,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2 F</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_2}{\lambda_1 (q^2-1)xy\ell^2} \biggl\{
\frac{\partial}{\partial \lambda_1} \biggl[ \frac{(q^2-1)xy }{\lambda_1 \lambda_2 } \cdot \frac{\partial F}{\partial \lambda_1}\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_2}{\lambda_1 (q^2-1)xy\ell^2} \biggl\{
\biggl[ \frac{(q^2-1)xy }{\lambda_1 \lambda_2 } \biggr] \biggl[ \frac{\partial^2 F}{\partial \lambda_1^2}\biggr]
+
\frac{\partial F}{\partial \lambda_1} \cdot \frac{\partial}{\partial \lambda_1} \biggl[ \frac{(q^2-1)xy }{\lambda_1 \lambda_2 } \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\lambda_1^2  \ell^2} \biggr] 
\biggl[ \frac{\partial^2 F}{\partial \lambda_1^2}\biggr]
+
\biggl[ \frac{1}{\lambda_1 xy\ell^2} \biggr]
\frac{\partial F}{\partial \lambda_1} \cdot \frac{\partial}{\partial \lambda_1} \biggl[ \frac{xy }{\lambda_1  } \biggr] \, .
</math>
  </td>
</tr>
</table>
In order to complete this evaluation, we need a couple of "complementary partial derivatives."  Referencing the [[#ComplementaryDerivatives|relation provided above]], we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial \lambda_1} \biggl[ \frac{xy }{\lambda_1  } \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
xy \biggl[ \frac{\partial}{\partial \lambda_1} \biggl(\lambda_1^{-1}\biggr)\biggr]
+
\frac{y}{\lambda_1} \biggl[ \frac{\partial x}{\partial \lambda_1} \biggr]
+
\frac{x}{\lambda_1} \biggl[ \frac{\partial y}{\partial \lambda_1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{xy}{\lambda_1^2} 
+
\frac{y}{\lambda_1} \biggl[h_1^2 \frac{\partial \lambda_1}{\partial x} \biggr]
+
\frac{x}{\lambda_1} \biggl[h_1^2 \frac{\partial \lambda_1}{\partial y} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{xy}{\lambda_1^2} 
+
\frac{h_1^2}{\lambda_1} \biggl[\frac{xy}{\lambda_1}  
+
\frac{q^2 x y}{\lambda_1}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{xy}{\lambda_1^2} \biggl[\lambda_1^2 \ell^2 (1 + q^2)  - 1 \biggr] 
\, .
</math>
  </td>
</tr>
</table>
<span id="Laplacian">Hence,</span>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2 F</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\lambda_1^2  \ell^2} \biggr] 
\biggl[ \frac{\partial^2 F}{\partial \lambda_1^2}\biggr]
+
\frac{xy}{\lambda_1^2} \biggl[\lambda_1^2 \ell^2 (1 + q^2)  - 1 \biggr]
\biggl[ \frac{1}{\lambda_1 xy\ell^2} \biggr]
\frac{\partial F}{\partial \lambda_1}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\lambda_1^2  \ell^2} \biggr] 
\biggl[ \frac{\partial^2 F}{\partial \lambda_1^2}\biggr]
+
\biggl[\lambda_1^2 \ell^2 (1 + q^2)  - 1 \biggr]
\biggl[ \frac{1}{\lambda_1^3 \ell^2} \biggr]
\frac{\partial F}{\partial \lambda_1}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\lambda_1^2  \ell^2} \biggr] 
\biggl[ \frac{\partial^2 F}{\partial \lambda_1^2}\biggr]
-
\biggl[ \frac{1}{\lambda_1^3 \ell^2} \biggr]
\frac{\partial F}{\partial \lambda_1}  
+
\biggl[ \frac{(1 + q^2)}{\lambda_1 } \biggr]
\frac{\partial F}{\partial \lambda_1}  \, .
</math>
  </td>
</tr>
</table>

==Example (q<sup>2</sup> = 2) Poisson Equation==

===Setup===

Let's see if we can solve the,
<div align="center">
<font color="maroon">'''Poisson Equation'''</font>
{{ User:Tohline/Math/EQ_Poisson01 }}
</div>
obtaining an analytic expression for the gravitational potential in the case where, independent of the coordinate, <math>~z</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho = \rho_c\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c \biggl[ 1 - \biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c \biggl[ 1 - \frac{1}{a^2}\biggl(x^2 + q^2 y^2 \biggr)\biggr] \, .</math>
  </td>
</tr>
</table>
Given that the density distribution is independent of <math>~z</math>, we expect the potential to be independent of <math>~z</math> as well.  So, in terms of T5-Coordinates, the Poisson equation may be written as,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{\lambda_2}{\lambda_1 (q^2-1)xy\ell^2} \biggl\{
\frac{\partial}{\partial \lambda_1} \biggl[ \frac{(q^2-1)xy }{\lambda_1 \lambda_2 } \cdot \frac{\partial \Phi}{\partial \lambda_1}\biggr]
+
\frac{\partial}{\partial \lambda_2} \biggl[ \frac{\lambda_1 \lambda_2}{(q^2-1) xy } \cdot \frac{\partial \Phi}{\partial \lambda_2}\biggr]
\biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
  </td>
</tr>
</table>
If we specifically consider the case where <math>~q^2 = a^2/b^2 = 2</math>, this can be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_2}{\lambda_1 (q^2-1)xy\ell^2} \biggl\{
\frac{\partial}{\partial \lambda_1} \biggl[ \frac{(q^2-1)xy }{\lambda_1 \lambda_2 } \cdot \frac{\partial \Phi}{\partial \lambda_1}\biggr]
+
\frac{\partial}{\partial \lambda_2} \biggl[ \frac{\lambda_1 \lambda_2}{(q^2-1) xy } \cdot \frac{\partial \Phi}{\partial \lambda_2}\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_2}{\lambda_1 } \cdot (\Lambda-1)^{-3 / 2} \frac{2^2}{\lambda_2^2} \cdot \frac{\lambda_2^2}{2}(\Lambda-1)\Lambda \biggl\{
\frac{\partial}{\partial \lambda_1} \biggl[ \frac{(\Lambda-1)}{2 } \cdot \frac{\partial \Phi}{\partial \lambda_1}\biggr]
+
\frac{\partial}{\partial \lambda_2} \biggl[ \frac{2}{ (\Lambda-1) } \cdot \frac{\partial \Phi}{\partial \lambda_2}\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4 \Lambda}{(\Lambda-1 )} \biggl\{
\frac{\partial}{\partial \lambda_1} \biggl[ \frac{(\Lambda-1)}{2 } \cdot \frac{\partial \Phi}{\partial \lambda_1}\biggr]
+
\frac{\partial}{\partial \lambda_2} \biggl[ \frac{2}{ (\Lambda-1) } \cdot \frac{\partial \Phi}{\partial \lambda_2}\biggr]
\biggr\}
</math>
  </td>
</tr>
</table>
where we have used the following expressions [[#InvertedRelations|derived above]]:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x^2y^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda-1) \biggl[ \frac{\lambda_2^2}{2^2}(\Lambda - 1) \biggr]^2  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\ell^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_2^2}{2} (\Lambda-1)\Lambda = \frac{2\lambda_1^2 \Lambda}{(\Lambda+1)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} 
~~\Rightarrow~~ 
\frac{1}{2}(\Lambda^2 - 1)^{1 / 2} = \frac{\lambda_1}{\lambda_2}
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{xy}{\lambda_1 \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^2}\biggl[ \frac{\lambda_2}{\lambda_1}(\Lambda - 1)^{3 / 2} \biggr]  
=
\frac{1}{2}(\Lambda - 1) \, .  
</math>
  </td>
</tr>
</table>

Now,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial(\Lambda-1)}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\Lambda}\biggl( \frac{8\lambda_1}{\lambda_2^2} \biggr)
=
\frac{4}{\lambda_1 \Lambda} \biggl( \frac{\lambda_1^2}{\lambda_2^2} \biggr)
=
\frac{(\Lambda^2-1)}{\lambda_1 \Lambda} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial(\Lambda-1)^{-1}}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{(\Lambda-1)^2} \biggl[ \frac{1}{2\Lambda} \biggr] \biggl(- \frac{8\lambda_1^2}{\lambda_2^3}  \biggr)
=
\frac{4}{(\Lambda-1)^2} \biggl[ \frac{1}{\lambda_2 \Lambda} \biggr] \biggl(\frac{\lambda_1^2}{\lambda_2^2}  \biggr)
=
\frac{\Lambda + 1}{(\Lambda-1)} \biggl[ \frac{1}{\lambda_2 \Lambda} \biggr]  \, .
</math>
  </td>
</tr>
</table>

Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4 \Lambda}{(\Lambda-1 )} \biggl\{
\frac{(\Lambda-1)}{2 }\biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
+
\frac{\partial \Phi}{\partial \lambda_1} \cdot \frac{\partial}{\partial \lambda_1} \biggl[ \frac{(\Lambda-1)}{2 } \biggr]
+
\frac{2}{ (\Lambda-1) } \biggl[  \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
+
\frac{\partial \Phi}{\partial \lambda_2} \cdot \frac{\partial}{\partial \lambda_2} \biggl[ \frac{2}{ (\Lambda-1) } \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4 \Lambda}{(\Lambda-1 )} \biggl\{
\frac{(\Lambda-1)}{2 }\biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
+
\biggl[ \frac{(\Lambda^2-1)}{2 \lambda_1\Lambda} \biggr] \frac{\partial \Phi}{\partial \lambda_1} 
+
\frac{2}{ (\Lambda-1) } \biggl[  \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
+
\biggl[ \frac{2(\Lambda+1)}{ (\Lambda-1)\lambda_2\Lambda } \biggr] \frac{\partial \Phi}{\partial \lambda_2} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\Lambda \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
+
\biggl[ \frac{2(\Lambda + 1)}{ \lambda_1} \biggr] \frac{\partial \Phi}{\partial \lambda_1} 
+
\frac{8 \Lambda }{ (\Lambda-1)^2 } \biggl[  \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
+
\biggl[ \frac{8(\Lambda+1)}{ (\Lambda-1)^2\lambda_2 } \biggr] \frac{\partial \Phi}{\partial \lambda_2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{\lambda_1} \biggl\{ \Lambda \lambda_1 \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
+
(\Lambda + 1)  \frac{\partial \Phi}{\partial \lambda_1} \biggr\}
+
\frac{8}{\lambda_2 (\Lambda-1)^2}\biggl\{ 
\Lambda \lambda_2  \biggl[  \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
+
(\Lambda+1) \frac{\partial \Phi}{\partial \lambda_2} 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

===Trials===

Try,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A \lambda_1^\alpha + B\lambda_2^\beta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{\partial \Phi}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A\alpha \lambda_1^{\alpha-1}  \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~ \frac{\partial \Phi}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~B\beta  \lambda_2^{\beta - 1} \, .</math>
  </td>
</tr>
</table>
In this case we find,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{\lambda_1} \biggl\{ \Lambda \lambda_1 \frac{\partial}{\partial \lambda_1}\biggl[ \frac{\partial \Phi}{\partial \lambda_1}\biggr]
+
(\Lambda + 1)  \biggl[ \frac{\partial \Phi}{\partial \lambda_1} \biggr] \biggr\}
+
\frac{8}{\lambda_2 (\Lambda-1)^2}\biggl\{ 
\Lambda \lambda_2  \frac{\partial}{\partial \lambda_2}\biggl[  \frac{\partial \Phi}{\partial \lambda_2}\biggr]
+
(\Lambda+1) \biggl[\frac{\partial \Phi}{\partial \lambda_2} \biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2A}{\lambda_1} \biggl\{ \Lambda \lambda_1 \frac{\partial}{\partial \lambda_1}\biggl[ \alpha \lambda_1^{\alpha-1}  \biggr]
+
(\Lambda + 1)  \biggl[ \alpha \lambda_1^{\alpha-1}  \biggr] \biggr\}
+
\frac{8B}{\lambda_2 (\Lambda-1)^2}\biggl\{ 
\Lambda \lambda_2  \frac{\partial}{\partial \lambda_2}\biggl[ \beta  \lambda_2^{\beta - 1} \biggr]
+
(\Lambda+1) \biggl[ \beta  \lambda_2^{\beta - 1} \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2A\alpha }{\lambda_1} \biggl\{ \Lambda (\alpha-1)  \lambda_1^{\alpha-1} 
+
(\Lambda + 1) \lambda_1^{\alpha-1} \biggr\}
+
\frac{8B\beta }{\lambda_2 (\Lambda-1)^2}\biggl\{ 
\Lambda(\beta-1) \lambda_2^{\beta - 1}
+
(\Lambda+1) \lambda_2^{\beta - 1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2A\alpha \lambda_1^{\alpha-2} \biggl\{ \Lambda (\alpha-1)  
+
(\Lambda + 1)  \biggr\}
+
\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2}\biggl\{ 
\Lambda(\beta-1)
+
(\Lambda+1) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2A\alpha \lambda_1^{\alpha-2} 
\biggl\{ \alpha \Lambda   + 1  \biggr\}
+
\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2} \biggl\{ \beta \Lambda +1 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

If <math>~\alpha = 4</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2} \biggl[ \beta \Lambda +1 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] -8A\lambda_1^2
- 32A\lambda_1^{2} \Lambda
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
8B\beta \lambda_2^{\beta-2} \biggl[ \beta \Lambda +1 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4\pi G\rho_c 
- 32A\lambda_1^{2} \Lambda
- \lambda_1^2 \biggl[\frac{4\pi G \rho_c}{a^2} + 8A  \biggr] \biggr\}
(\Lambda^2 - 2\Lambda + 1) \, .
</math>
  </td>
</tr>
</table>
If, then, <math>~8Aa^2 = -4\pi G\rho_c</math>,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
8B\beta \lambda_2^{\beta-2} \biggl[ \beta \Lambda +1 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G\rho_c\biggl\{ 1 
+ \biggl[ \frac{4\lambda_1^{2}}{a^2} \biggr] \Lambda
\biggr\}
(\Lambda^2 - 2\Lambda + 1) \, .
</math>
  </td>
</tr>
</table>
But, we also know that, <math>\lambda_1^2 = \lambda_2^2(\Lambda^2-1)/4</math>, so &hellip;

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
8a^2 B\beta \lambda_2^{\beta-2} \biggl[ \beta \Lambda +1 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G\rho_c \biggl\{ a^2 
+  \lambda_2^2 (\Lambda^2-1)\Lambda 
\biggr\}
(\Lambda^2 - 2\Lambda + 1) \, .
</math>
  </td>
</tr>
</table>


<font color="green">'''(25 October 2020) I give up &hellip; for now.'''</font>

=See Also=

<ul>
  <li>
[[User:Tohline/Appendix/Ramblings/DirectionCosines|Direction Cosines]]
  </li>
</ul>


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