Editing Appendix/Ramblings/T6CoordinatesPt2 (section)

===Speculation5===

====Spherical Coordinates====

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

<tr>
  <td align="right">
<math>~r\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z \, ,</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\tan\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{y}{x} \, .</math>
  </td>
</tr>
</table>

====Use &lambda;<sub>1</sub> Instead of r ====

Here, as above, we define,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x^2 + q^2 y^2 + p^2 z^2 </math>
  </td>
</tr>
</table>

Using this expression to eliminate "x" (in favor of &lambda;<sub>1</sub>) in each of the three spherical-coordinate definitions, we obtain,

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

<tr>
  <td align="right">
<math>~r^2 \equiv x^2 + y^2 + z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1^2 - y^2(q^2-1) - z^2(p^2-1) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tan^2\theta \equiv \frac{x^2 + y^2}{z^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{z^2}\biggl[
\lambda_1^2 -y^2(q^2-1) -p^2z^2
\biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\tan^2\varphi} \equiv \frac{x^2}{y^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda_1^2 - p^2z^2}{y^2} - q^2 \, .
</math>
  </td>
</tr>
</table>

After a bit of additional algebraic manipulation, we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{z^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (1+\tan^2\varphi)}{\mathcal{D}^2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{y^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{ \mathcal{D}^2 \tan^2\varphi -  p^2 \tan^2\varphi (1+\tan^2\varphi)}{(1+q^2\tan^2\varphi) \mathcal{D}^2}  \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{x^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - q^2\biggl(\frac{y^2}{\lambda_1^2} \biggr) - p^2\biggl(\frac{z^2}{\lambda_1^2}\biggr)
\, ,
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\mathcal{D}^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
(1 + q^2\tan^2\varphi)(p^2 + \tan^2\theta) - p^2(q^2-1)\tan^2\varphi
\biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
As a check, let's set <math>~q^2 = p^2 = 1</math>, which should reduce to the normal spherical coordinate system.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1^2</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
r^2 \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{D}^2</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
(1 + \tan^2\varphi)(1 + \tan^2\theta)
\biggr] \, .
</math>
  </td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{z^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
\frac{1}{1+\tan^2\theta} = \cos^2\theta = \frac{z^2}{r^2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{y^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(1 + \tan^2\varphi)(1 + \tan^2\theta) \tan^2\varphi -  \tan^2\varphi (1+\tan^2\varphi)}{(1+\tan^2\varphi) (1 + \tan^2\varphi)(1 + \tan^2\theta)}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tan^2\varphi}{(1 + \tan^2\varphi)}
\biggl[\frac{\tan^2\theta }{ (1 + \tan^2\theta)}  \biggr]
= \sin^2\theta \sin^2\varphi
= \frac{y^2}{r^2} \,;
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
1 - \sin^2\theta \sin^2\varphi - \cos^2\theta
=
- \sin^2\theta \sin^2\varphi + \sin^2\theta
=
\sin^2\theta \cos^2\varphi = \frac{x^2}{r^2}
\, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

====Relationship To T3 Coordinates====

If we set, <math>~q = 1</math>, but continue to assume that <math>~p > 1</math>, we expect to see a representation that resembles our previously discussed, [[User:Tohline/Appendix/Ramblings/T3Integrals#Integrals_of_Motion_in_T3_Coordinates|T3 Coordinates]].  Note, for example, that the new "radial" coordinate is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1^2</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
(\varpi^2 + p^2z^2) \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{D}^2</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
(1 + \tan^2\varphi)(p^2 + \tan^2\theta) 
\biggr]  \, .
</math>
  </td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{p^2z^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{ p^2}{(p^2 + \tan^2\theta)} = \frac{ 1}{(1 + p^{-2} \tan^2\theta)}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\varpi^2}{\lambda_1^2} = \frac{x^2}{\lambda_1^2} + \frac{y^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~1 -  p^2 \biggl( \frac{z^2}{\lambda_1^2}\biggr)
=
\biggl[1 -  \frac{ 1}{(1 + p^{-2}\tan^2\theta)} \biggr] \, .</math>
  </td>
</tr>
</table>

We also see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi^2}{p^2z^2}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~
(1 + p^{-2}\tan^2\theta)\biggl[1 -  \frac{ 1}{(1 + p^{-2}\tan^2\theta)} \biggr]
=
p^{-2}\tan^2\theta \, .
</math>
  </td>
</tr>
</table>

====Again Consider Full 3D Ellipsoid====
Let's try to replace everywhere, <math>~[\varpi/(pz)]^2 = p^{-2}\tan^2\theta</math> with <math>~\lambda_2</math>.  This gives,

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

<tr>
  <td align="right">
<math>~\frac{\mathcal{D}^2}{p^2}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
(1 + q^2\tan^2\varphi)\lambda_2 - (q^2-1)\tan^2\varphi
\biggr] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\frac{p^2 z^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (1+\tan^2\varphi)}{[(1 + q^2\tan^2\varphi)\lambda_2 - (q^2-1)\tan^2\varphi]}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (1+\tan^2\varphi)/\tan^2\varphi}{[q^2 \lambda_2 (1 + q^2\tan^2\varphi)/(q^2\tan^2\varphi) - (q^2-1)]} 
= \frac{1/\sin^2\varphi}{[q^2\lambda_2 Q^2 - (q^2-1) ]}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{q^2y^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ q^2 \tan^2\varphi }{(1+q^2\tan^2\varphi) }  
-
\frac{ q^2 \tan^2\varphi (1+\tan^2\varphi)}{(1+q^2\tan^2\varphi) [(1 + q^2\tan^2\varphi)\lambda_2 - (q^2-1)\tan^2\varphi] }
</math>
  </td>
</tr>

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

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

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

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

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

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

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

<tr>
  <td align="right">
<math>~\frac{x^2 + q^2y^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1
- \biggl[1 + \frac{ q^2 \tan^2\varphi }{(1+q^2\tan^2\varphi) }\biggr] \biggl\{
\frac{ (1+\tan^2\varphi)}{ [(1 + q^2\tan^2\varphi)\lambda_2 - (q^2-1)\tan^2\varphi] }
\biggr\} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{q^2 y^2 Q^2}{\lambda_1^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{[~~]\sin^2\varphi} \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x^2}{\lambda_1^2} + \frac{1}{Q^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{[~~]\sin^2\varphi} \, .</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Q^4 \biggl( \frac{q^2 y^2}{\lambda_1^2} \biggr) - Q^2 \biggl( \frac{x^2}{\lambda_1^2} \biggr) - 1</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Q^4  - Q^2 \biggl( \frac{x^2}{q^2 y^2} \biggr) - \biggl( \frac{\lambda_1^2}{q^2 y^2} \biggr) \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1 + q^2\tan^2\varphi}{q^2\tan^2\varphi} \, .</math>
  </td>
</tr>
</table>
Solving the quadratic equation, we have,
<table border="0" cellpadding="5" align="center">

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

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

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<div align="center">'''Tentative Summary'''</div>
<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^2 + q^2y^2 + p^2 z^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>~
\frac{(x^2 + y^2)^{1 / 2}}{pz} 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3 = Q^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x^2}{2q^2 y^2} \biggr) \biggl\{ 1  \pm \biggl[ 1 + 4\biggl( \frac{\lambda_1^2 q^2 y^2}{x^4} \biggr) \biggr]^{1 / 2}  \biggr\} \, .</math>
  </td>
</tr>
</table>

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

====Partial Derivatives &amp; Scale Factors====
=====First Coordinate=====

<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{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{q^2 y}{\lambda_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_1}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2 z}{\lambda_1} \, .</math>
  </td>
</tr>
</table>

<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 \lambda_1}{\partial x} \biggr)^2
+ \biggl( \frac{\partial \lambda_1}{\partial y} \biggr)^2
+ \biggl( \frac{\partial \lambda_1}{\partial z} \biggr)^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{x}{\lambda_1} \biggr)^2 
+ \biggl( \frac{q^2 y}{\lambda_1} \biggr)^2 
+ \biggl( \frac{p^2 z}{\lambda_1} \biggr)^2 
\, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ h_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
\lambda_1 \ell_{3D} \, ,
</math>
  </td>
</tr>
</table>
where,
<div align="center"><math>\ell_{3D} \equiv (x^2 + q^4y^2 + p^4z^2)^{-1 / 2} \, .</math></div>
As a result, the associated unit vector 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} h_1 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)
+ \hat{\jmath} h_1 \biggl( \frac{\partial \lambda_1}{\partial y} \biggr)
+ \hat{k} h_1 \biggl( \frac{\partial \lambda_1}{\partial z} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{\imath} x \ell_{3D}
+ \hat{\jmath} q^2 y\ell_{3D}
+ \hat{k} p^2 z \ell_{3D}
\, .
</math>
  </td>
</tr>
</table>

Notice that,
<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 + q^4 y^2 + p^4 z^2) \ell_{3D}^2 = 1 \, .
</math>
  </td>
</tr>
</table>


=====Second Coordinate (1<sup>st</sup> Try)=====
<table border="0" cellpadding="5" align="center">

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

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

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

<tr>
  <td align="right">
<math>~h_2^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\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
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{p^2 z^2} 
+ \frac{(x^2 + y^2)}{p^2 z^4} 
=
\frac{(x^2 + y^2 + z^2)}{p^2 z^4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~h_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p z^2}{r }
</math>
  </td>
</tr>
</table>
As a result, the associated unit vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{\imath} h_2 \biggl( \frac{\partial \lambda_2}{\partial x} \biggr)
+ \hat{\jmath} h_2 \biggl( \frac{\partial \lambda_2}{\partial y} \biggr)
+ \hat{k} h_2 \biggl( \frac{\partial \lambda_2}{\partial z} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{\imath}  \biggl[ \frac{xz}{r(x^2 + y^2)^{1 / 2}} \biggr]
+ \hat{\jmath}  \biggl[ \frac{yz}{r(x^2 + y^2)^{1 / 2}} \biggr]
- \hat{k} \biggl[ \frac{(x^2 + y^2)^{1 / 2}}{r} \biggr] \, .
</math>
  </td>
</table>

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

<tr>
  <td align="right">
<math>~\hat{e}_2 \cdot \hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x^2 z^2}{r^2(x^2 + y^2)} \biggr]
+ \biggl[ \frac{y^2 z^2}{r^2(x^2 + y^2)} \biggr]
+ \biggl[ \frac{(x^2 + y^2)}{r^2} \biggr] 
= 1 \, .
</math>
  </td>
</tr>
</table>

Let's check to see if this "second" unit vector is orthogonal to the "first."
<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>~
x\ell_{3D} \biggl[ \frac{xz}{r(x^2 + y^2)^{1 / 2}} \biggr]
+ q^2 y\ell_{3D} \biggl[ \frac{yz}{r(x^2 + y^2)^{1 / 2}} \biggr]
- p^2 z \ell_{3D} \biggl[ \frac{(x^2 + y^2)^{1 / 2}}{r} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ell_{3D} \biggl\{ \biggl[ \frac{x^2z}{r(x^2 + y^2)^{1 / 2}} \biggr]
+ \biggl[ \frac{q^2 y^2 z}{r(x^2 + y^2)^{1 / 2}} \biggr]
-  \biggl[ \frac{p^2 z(x^2 + y^2)^{1 / 2}}{r} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{z\ell_{3D}}{r (x^2 + y^2)^{1 / 2}} \biggl\{ \biggl[ x^2\biggr]
+ \biggl[ q^2 y^2 \biggr]
-  \biggl[ p^2 (x^2 + y^2) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\ne</math>
  </td>
  <td align="left">
<math>~
0 \ .
</math>
  </td>
</tr>
</table>


=====Second Coordinate (2<sup>nd</sup> Try)=====

Let's try, 
<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^2 + q^2y^2 + \mathfrak{f}\cdot p^2 z^2)^{1 / 2}}{pz}  \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{\partial \lambda_2}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{pz(x^2 + q^2y^2 + \mathfrak{f}\cdot p^2 z^2)^{1 / 2} } = \frac{x}{p^2 z^2 \lambda_2}
\, ,
</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{q^2 y}{pz(x^2 + q^2y^2 + \mathfrak{f}\cdot p^2 z^2)^{1 / 2} } = \frac{q^2y}{p^2 z^2 \lambda_2}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\mathfrak{f}\cdot p^2z}{pz(x^2 + q^2y^2 + \mathfrak{f}\cdot p^2 z^2)^{1 / 2} }
-
\frac{(x^2 + q^2y^2 + \mathfrak{f}\cdot p^2 z^2)^{1 / 2}}{pz^2}  
= \frac{1}{p^2z^2 \lambda_2 } \biggl( \mathfrak{f}\cdot p^2z \biggr) - \frac{\lambda_2 }{z}
= \frac{1}{p^2z^2 \lambda_2 } \biggl( \mathfrak{f}\cdot p^2z  - p^2z \lambda_2^2 \biggr)
\, .
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right">
<math>~h_2^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\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
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \frac{x^2 + q^4 y^2}{p^4 z^4 \lambda_2^2} \biggr]
+ \biggl[ \frac{1}{z\lambda_2}\biggl( \mathfrak{f} - \lambda_2^2 \biggr) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{p^4 z^4 \lambda_2^2}
\biggl[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ h_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{p^2 z^2 \lambda_2}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \, .
</math>
  </td>
</tr>
</table>
So, the associated unit vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{\imath} h_2 \biggl( \frac{\partial \lambda_2}{\partial x} \biggr)
+ \hat{\jmath} h_2 \biggl( \frac{\partial \lambda_2}{\partial y} \biggr)
+ \hat{k} h_2 \biggl( \frac{\partial \lambda_2}{\partial z} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{\imath}  \biggl\{ \frac{x}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \biggr\}
+ \hat{\jmath}  \biggl\{ \frac{q^2 y}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \biggr\}
+ \hat{k} \biggl\{ \frac{p^2z(\mathfrak{f}-\lambda_2^2)}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \biggr\} \, .
</math>
  </td>
</table>

Checking orthogonality &hellip;
<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>~
x\ell_{3D} \biggl\{ \frac{x}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \biggr\}
+ q^2 y\ell_{3D} \biggl\{ \frac{q^2 y}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \biggr\}
+ p^2 z \ell_{3D} \biggl\{ \frac{p^2z(\mathfrak{f}-\lambda_2^2)}{[ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} } \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_{3D}}{ [ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} }
\biggl\{ x^2 + q^4y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)\biggr\} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_{3D}}{ [ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} }
\biggl\{ x^2 + q^4y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)\biggr\} \, .
</math>
  </td>
</tr>
</table>

If <math>~\mathfrak{f} = 0</math>, we have &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p^2 z (\mathfrak{f} - \lambda_2^2) </math>
  </td>
  <td align="center">
&nbsp; &nbsp; <math>~~~\rightarrow ~~~ </math>&nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
\biggl[- p^2 z \lambda_2^2\biggr]_{\mathfrak{f} = 0}
=
- p^2 z\biggl[\frac{(x^2 + q^2y^2 + \cancelto{0}{\mathfrak{f} \cdot }p^2 z^2 )^{1 / 2}}{pz}  \biggr]^2
=
- \frac{(x^2 + q^2y^2  )}{z} \, ,</math>
  </td>
</tr>
</table>
which, in turn, means &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[ x^2 + q^4 y^2 + p^4 z^2 (\cancelto{0}{\mathfrak{f}} - \lambda_2^2)^2 ]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ x^2 + q^4 y^2 + p^4 z^2 \lambda_2^4 ]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ x^2 + q^4 y^2 + p^4 z^2 \biggl[\frac{(x^2 + q^2y^2 + \cancelto{0}{\mathfrak{f}\cdot} p^2 z^2)^{1 / 2}}{pz}  \biggr]^4 \biggr\}^{1 / 2}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(x^2 + q^4 y^2)^{1 / 2}}{z} \biggl[ z^2 + (x^2 + q^2y^2 ) \biggr]^{1 / 2} \, ,
</math>
  </td>
</tr>
</table>
and,
<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>~\frac{\ell_{3D}}{ [ x^2 + q^4 y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)^2 ]^{1 / 2} }
\biggl\{ x^2 + q^4y^2 + p^4 z^2 (\mathfrak{f} - \lambda_2^2)\biggr\} \, .
</math>
  </td>
</tr>
</table>