Editing Appendix/Ramblings/T6CoordinatesPt3 (section)

==Best Thus Far==

===Part A===

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T6 Coordinates'''</td>
</tr>
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\lambda_n</math></td>
  <td align="center"><math>~h_n</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial x}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial y}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial z}</math></td>
  <td align="center"><math>~\gamma_{n1}</math></td>
  <td align="center"><math>~\gamma_{n2}</math></td>
  <td align="center"><math>~\gamma_{n3}</math></td>
</tr>

<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math></td>
  <td align="center"><math>~\lambda_1 \ell_{3D}</math></td>
  <td align="center"><math>~\frac{x}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{q^2 y}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{p^2 z}{\lambda_1}</math></td>
  <td align="center"><math>~(x) \ell_{3D}</math></td>
  <td align="center"><math>~(q^2 y)\ell_{3D}</math></td>
  <td align="center"><math>~(p^2z) \ell_{3D}</math></td>
</tr>

<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~\frac{x y^{1/q^2}}{ z^{2/p^2}}</math></td>
  <td align="center"><math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y p^2 z}{ \mathcal{D}}\biggr] </math></td>
  <td align="center"><math>~\frac{\lambda_2}{x}</math></td>
  <td align="center"><math>~\frac{\lambda_2}{q^2 y}</math></td>
  <td align="center"><math>~-\frac{2\lambda_2}{p^2 z}</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{x}\biggr)</math></td>
  <td align="center"><math>~ \frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{q^2y}\biggr)</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(-\frac{2}{p^2z}\biggr)</math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~-\frac{\ell_{3D}}{\mathcal{D}}\biggl[ x(2 q^4y^2 + p^4z^2 ) \biggl]</math></td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ q^2 y(p^4z^2 + 2x^2 )  \biggl]</math></td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ p^2z( x^2 - q^4y^2 ) \biggl]</math></td>
</tr>

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>
  </td>
</tr>

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

<span id="ABderivatives">&nbsp;</span>
<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A \equiv \ell_{3D}^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 ) \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~B \equiv \mathcal{D}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\partial A}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial A}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2q^4 y \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial A}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2p^4 z\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \frac{\partial B}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2x( 4q^4y^2 + p^4 z^2 ) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial B}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2q^4 y (p^4 z^2 + 4x^2) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial B}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2p^4 z(q^4 y^2 + x^2)\, .</math>
  </td>
</tr>
</table>

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

Try &hellip;

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

<tr>
  <td align="right">
<math>~\lambda_3</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{B}{A}\biggr)^{m/2} = (D \ell_{3D})^m
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{AB}{m\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial x_i}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{ 
A \cdot \frac{\partial B}{\partial x_i} - B \cdot \frac{\partial A}{\partial x_i}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln x_i}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{x_i}{2}\biggl\{ 
A \cdot \frac{\partial B}{\partial x_i} - B \cdot \frac{\partial A}{\partial x_i}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

In this case we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x}{2}\biggl\{~~\biggr\}_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2(2q^4 y^2 + p^4z^2)^2 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{y}{2}\biggl\{~~\biggr\}_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
q^4y^2(2x^2 + p^4 z^2)^2 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{z}{2}\biggl\{~~\biggr\}_z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^4 z^2(x^2 - q^4y^2)^2 \, .
</math>
  </td>
</tr>
</table>

The scale factor is, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_3^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{i=1}^3 \biggl( \frac{\partial\lambda_3}{\partial x_i}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{i=1}^3 \biggl\{ 
\biggl[ \frac{m\lambda_3}{2AB} \biggr] \biggl[ 
A \cdot \frac{\partial B}{\partial x_i} - B \cdot \frac{\partial A}{\partial x_i}
\biggr]
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{m\lambda_3}{AB} \biggr]^2 
\biggl\{ 
\biggl[ 
x(2q^4 y^2 + p^4z^2)^2 
\biggr]^2
+
\biggl[ 
q^4y(2x^2 + p^4 z^2)^2
\biggr]^2
+
\biggl[ 
p^4 z(x^2 - q^4y^2)^2
\biggr]^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~h_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{AB}{m\lambda_3} \biggr] 
\biggl\{ 
\biggl[ 
x(2q^4 y^2 + p^4z^2)^2 
\biggr]^2
+
\biggl[ 
q^4y(2x^2 + p^4 z^2)^2
\biggr]^2
+
\biggl[ 
p^4 z(x^2 - q^4y^2)^2
\biggr]^2
\biggr\}^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>

===Part B (25 February 2021)===

Now, from [[#Eureka|above]], we know that,
<table border="0" cellpadding="5" align="center">

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

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="4">
'''Example:'''  <br /><math>~(q^2, p^2) = (2, 5)</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>~(x, y, z) = (0.7, \sqrt{0.23}, 0.1)~~\Rightarrow~~ \lambda_1 = 1.0</math>
  </td>
</tr>

<tr>
  <td align="center"><math>~\biggl[ x(2 q^4y^2 + p^4z^2 ) \biggl]^2</math></td>
  <td align="center"><math>~\biggl[ q^2 y(p^4z^2 + 2x^2 )  \biggl]^2</math></td>
  <td align="center"><math>~\biggl[ p^2z( x^2 - q^4y^2 )  \biggl]^2</math></td>
  <td align="center"><math>~\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2}</math></td>
</tr>

<tr>
  <td align="center">2.14037</td>
  <td align="center">1.39187</td>
  <td align="center">0.04623</td>
  <td align="center">3.57847</td>
</tr>
</table>

As an aside, note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~AB</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln x}
+
\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln y}
+
\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln z}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \ln \lambda_3}{\partial \ln x}
+
\frac{\partial \ln \lambda_3}{\partial \ln y}
+
\frac{\partial \ln \lambda_3}{\partial \ln z} \, .
</math>
  </td>
</tr>
</table>

We realize that this ratio of lengths may also be written in the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
6x^2 q^4y^2 p^4 z^2
+
x^4(4q^4y^2 + p^4z^2)
+
q^8y^4(4x^2 + p^4z^2)
+
p^8z^4(x^2 + q^4y^2) \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="5">
'''Same Example, but Different Expression:'''  <br /><math>~(q^2, p^2) = (2, 5)</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>~(x, y, z) = (0.7, \sqrt{0.23}, 0.1)~~\Rightarrow~~ \lambda_1 = 1.0</math>
  </td>
</tr>

<tr>
  <td align="center"><math>~6x^2 q^4y^2 p^4 z^2</math></td>
  <td align="center"><math>~x^4(4q^4y^2 + p^4z^2)</math></td>
  <td align="center"><math>~q^8y^4(4x^2 + p^4z^2)</math></td>
  <td align="center"><math>~p^8z^4(x^2 + q^4y^2)</math></td>
  <td align="center"><math>~\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2}</math></td>
</tr>

<tr>
  <td align="center">0.67620</td>
  <td align="center">0.94359</td>
  <td align="center">1.87054</td>
  <td align="center">0.08813</td>
  <td align="center">3.57847</td>
</tr>
</table>


Let's try &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x(2 q^4y^2 + p^4z^2 ) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 y(p^4z^2 + 2x^2 ) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p^2z( x^2 - q^4y^2 ) \, .</math>
  </td>
</tr>
</table>
This means that the relevant scale factor is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_5^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -x(2 q^4y^2 + p^4z^2 ) \biggr]^2
+
\biggl[ q^2 y(p^4z^2 + 2x^2 ) \biggr]^2
+
\biggl[ p^2z( x^2 - q^4y^2 ) \biggr]^2
=
\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~h_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, ,
</math>
  </td>
</tr>
</table>
and the three associated direction cosines are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{51} = h_5 \biggl( \frac{\partial \lambda_5}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x(2 q^4y^2 + p^4z^2 )\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{52} = h_5 \biggl( \frac{\partial \lambda_5}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 y(p^4z^2 + 2x^2 )\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{53} = h_5 \biggl( \frac{\partial \lambda_5}{\partial z} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p^2z( x^2 - q^4y^2 )\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, .</math>
  </td>
</tr>
</table>
<span id="PartBCoordinatesT10">These direction cosines</span> exactly match what is required in order to ensure that the coordinate, <math>~\lambda_5</math>, is everywhere orthogonal to both <math>~\lambda_1</math> and <math>~\lambda_4</math>.  <font color="red">'''GREAT!'''</font>  The resulting summary table is, therefore:

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T10 Coordinates'''</td>
</tr>
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\lambda_n</math></td>
  <td align="center"><math>~h_n</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial x}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial y}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial z}</math></td>
  <td align="center"><math>~\gamma_{n1}</math></td>
  <td align="center"><math>~\gamma_{n2}</math></td>
  <td align="center"><math>~\gamma_{n3}</math></td>
</tr>

<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math></td>
  <td align="center"><math>~\lambda_1 \ell_{3D}</math></td>
  <td align="center"><math>~\frac{x}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{q^2 y}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{p^2 z}{\lambda_1}</math></td>
  <td align="center"><math>~(x) \ell_{3D}</math></td>
  <td align="center"><math>~(q^2 y)\ell_{3D}</math></td>
  <td align="center"><math>~(p^2z) \ell_{3D}</math></td>
</tr>

<tr>
  <td align="center"><math>~4</math></td>
  <td align="center"><math>~\frac{x y^{1/q^2}}{ z^{2/p^2}}</math></td>
  <td align="center"><math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y p^2 z}{ \mathcal{D}}\biggr] </math></td>
  <td align="center"><math>~\frac{\lambda_2}{x}</math></td>
  <td align="center"><math>~\frac{\lambda_2}{q^2 y}</math></td>
  <td align="center"><math>~-\frac{2\lambda_2}{p^2 z}</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{x}\biggr)</math></td>
  <td align="center"><math>~ \frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{q^2y}\biggr)</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(-\frac{2}{p^2z}\biggr)</math></td>
</tr>

<tr>
  <td align="center"><math>~5</math></td>
  <td align="center">---</td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}</math></td>
  <td align="center"><math>~-x(2 q^4y^2 + p^4z^2)</math></td>
  <td align="center"><math>~q^2 y(p^4z^2 + 2x^2 )</math></td>
  <td align="center"><math>~p^2z( x^2 - q^4y^2 )</math></td>
  <td align="center"><math>~-\frac{\ell_{3D}}{\mathcal{D}}\biggl[ x(2 q^4y^2 + p^4z^2 ) \biggl]</math></td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ q^2 y(p^4z^2 + 2x^2 )  \biggl]</math></td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ p^2z( x^2 - q^4y^2 ) \biggl]</math></td>
</tr>

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>
  </td>
</tr>

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


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

<tr>
  <td align="right">
<math>~\lambda_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^{-2q^4} \cdot y^{2q^2}
+
y^{q^2p^4} \cdot z^{-q^4p^2}
+
x^{-p^4} \cdot z^{p^2} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{x^{2q^4 + p^2} z^{q^4p^2} }\biggl\{
\mathfrak{F_5}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2q^4}{x}\biggl( \frac{ y^{2q^2} }{ x^{2q^4} } \biggr)
- \frac{p^4}{x}\biggl( \frac{ z^{p^2} }{ x^{p^2} } \biggr)
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~x^{2q^4 + p^2 + 1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{ z^{q^4p^2} }\biggl\{
\frac{\mathfrak{F_5}}{\lambda_5}
\biggr\} 
\, ,
</math>
  </td>
</tr>
</table>
we can also write,

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{ z^{q^4p^2} }{x}\biggl\{
\frac{\lambda_5}{\mathfrak{F_5}}
\biggr\} 

\biggl[ 2q^4y^{2q^2} x^{p^2} + p^4 x^{2q^4}z^{p^2} \biggr] 
</math>
  </td>
</tr>
</table>
Similarly,

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2q^2}{y} \biggl( \frac{ y^{2q^2} }{ x^{2q^4} } \biggr)
+
\frac{q^2p^4}{y} \biggl( \frac{ y^{q^2p^4} }{ z^{q^4p^2} } \biggr)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x^{p^2}}{ y } \biggl\{ \frac{\lambda_5}{\mathfrak{F_5}} \biggr\}
\biggl[ 2q^2 \biggl( y^{2q^2} z^{q^4p^2} \biggr)
+
q^2p^4 \biggl( y^{q^2p^4} x^{2q^4} \biggr)\biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{q^4p^2}{z} \biggl( \frac{ y^{q^2p^4} }{ z^{q^4p^2} } \biggr)
+
\frac{p^4}{z} \biggl( \frac{ z^{p^4} }{ x^{p^2} } \biggr)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x^{2q^4} }{z} \biggl\{ \frac{\lambda_5}{\mathfrak{F_5}} \biggr\} \biggl[
p^4 \biggl( z^{p^4 + q^4p^2 }  \biggr)
- 
q^4p^2 \biggl( x^{p^2} y^{q^2p^4}  \biggr)\biggr]
</math>
  </td>
</tr>
</table>

===Understanding the Volume Element===

Let's see if the expression for the volume element makes sense; that is, does
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(h_1 h_4 h_5) d\lambda_1 d\lambda_4 d\lambda_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~dx dy dz \, ?</math>
  </td>
</tr>
</table>

First, let's make sure that we understand how to relate the components of the Cartesian line element with the components of our T10 coordinates. 

====Line Element====

MF53 claim that the following relation gives the various expressions for the scale factors; we will go ahead and incorporate the expectation that, since our coordinate system is orthogonal, the off-diagonal elements are zero.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ds^2 = dx^2 + dy^2 + dz^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{i=1,4,5} h_i^2 d\lambda_i^2 \, .
</math>
  </td>
</tr>
</table>
Let's see.  The first term on the RHS is,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \biggl[  
\biggl( \frac{\partial \lambda_1}{\partial x}\biggr)^2 dx^2
+
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 dy^2
+
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 dz^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\cancel{2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)} dx~dy
+
\cancel{2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)} dx~dz
+
\cancel{2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)} dy~dz
\biggr] \, ;
</math>
  </td>
</tr>
</table>
the other two terms assume easily deduced, similar forms.  When put together and after regrouping terms, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\sum_{i=1,4,5} h_i^2 d\lambda_i^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)^2 
+
h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2
+
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2
\biggr] dx^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[  
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
+
h_4^2\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2
+
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2
\biggr] dy^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[  
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
+
h_4^2\biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2
+
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggr] dz^2 \, .
</math>
  </td>
</tr>
</table>

Given that this summation should also equal the square of the Cartesian line element, <math>~(dx^2 + dy^2 + dz^2)</math>, we conclude that the three terms enclosed inside each of the pair of brackets must sum to unity.  Specifically, from the coefficient of <math>~dx^2</math>, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_1^2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>
Using this relation to replace <math>~h_1^2</math> in each of the other two bracketed expressions, we find for the coefficients of <math>~dy^2</math> and <math>~dz^2</math>, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ 
1 - h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 - h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 
\biggr]
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 
\biggr]\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[ 
1 - h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 - h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 
\biggr]
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggr]\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \, .
</math>
  </td>
</tr>
</table>

We can use the first of these two expressions to solve for <math>~h_4^2</math> in terms of <math>~h_5^2</math>, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
- h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
- h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
h_4^2
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr] 
</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  
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
</math>
  </td>
</tr>
</table>
Analogously, the second of these two expressions gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
h_4^2 \biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
 </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 z}\biggr)^2 
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
</math>
  </td>
</tr>
</table>
Eliminating <math>~h_4</math> between the two gives the desired overall expression for <math>~h_5</math>, namely,
<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>~\biggl[
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
\biggr]
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-
\biggl[
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
\biggr]
\biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_5^2 \biggl\{
\biggl[ \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
-
\biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \biggr]
\biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-
\biggl[ \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
-
\biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \biggr]
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\biggl[ \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 \biggr]
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[ \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  \biggr]
\biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{h_5^2}{h_1^4 h_4^2 h_5^2}\biggl\{
\biggl[ \gamma_{51}^2 \gamma_{12}^2  
-
\gamma_{52}^2 \gamma_{11}^2 \biggr]
\biggl[ \gamma_{43}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{13}^2 
\biggr]
-
\biggl[ \gamma_{51}^2 \gamma_{13}^2
-
\gamma_{53}^2 \gamma_{11}^2 
\biggr]
\biggl[
\gamma_{42}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{12}^2 
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{h_4^2 h_1^2}\biggl\{
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
\biggl[ \gamma_{43}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{13}^2 
\biggr]
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggl[ \gamma_{43}^2\gamma_{11}^2
-~ \gamma_{41}^2 \gamma_{13}^2 
\biggr]
- \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
\biggl[
\gamma_{42}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{12}^2
\biggr]
+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggl[
\gamma_{42}^2 \gamma_{11}^2
- \gamma_{41}^2\gamma_{12}^2 
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{h_5^2}{h_1^4 h_4^2 h_5^2}\biggl\{
\biggl[ \gamma_{51}^2 \gamma_{12}^2  
-
\gamma_{52}^2 \gamma_{11}^2 \biggr]

\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\biggl[ \gamma_{43} \gamma_{11}
- \gamma_{41} \gamma_{13} 
\biggr]

-
\biggl[ \gamma_{51}^2 \gamma_{13}^2
-
\gamma_{53}^2 \gamma_{11}^2 
\biggr]

\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12} 
\biggr] 
\biggl[
\gamma_{42} \gamma_{11}
- \gamma_{41} \gamma_{12} 
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{h_4^2 h_1^2}\biggl\{
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}

\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\biggl[ \gamma_{43} \gamma_{11}
- \gamma_{41} \gamma_{13} 
\biggr]

-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggl[ \gamma_{43}\gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\biggl[ \gamma_{43}\gamma_{11}
-~ \gamma_{41} \gamma_{13} 
\biggr]

- \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12}
\biggr]
\biggl[
\gamma_{42} \gamma_{11}
- \gamma_{41} \gamma_{12}
\biggr]


+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41}\gamma_{12} 
\biggr]
\biggl[
\gamma_{42} \gamma_{11}
- \gamma_{41}\gamma_{12} 
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{h_1^4 h_4^2 }\biggl\{
-
\biggl[ \gamma_{51} \gamma_{12}  
+
\gamma_{52} \gamma_{11} \biggr]
\gamma_{43}
\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\gamma_{52}

+
\biggl[ \gamma_{51} \gamma_{13}
+
\gamma_{53} \gamma_{11} 
\biggr]
\gamma_{42}
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12} 
\biggr] \gamma_{53}

\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{ h_1^4 h_4^2}\biggl\{
\gamma_{12}^2 
\biggl[ \gamma_{43}\gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]\gamma_{52}
-\gamma_{11}^{2}
\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]\gamma_{52}


- \gamma_{11}^{2}
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12}
\biggr]\gamma_{53}


+ \gamma_{13}^2 
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41}\gamma_{12} 
\biggr]\gamma_{53} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{h_1^4 h_4^2 }\biggl\{
\biggl[
(- \gamma_{51} \gamma_{12}  
-
\gamma_{52} \gamma_{11} )
\gamma_{43}
+ 
\gamma_{12}^2 

-\gamma_{11}^{2}
\biggr]
(\gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} )
\gamma_{52}
+
\biggl[
(\gamma_{51} \gamma_{13}
+
\gamma_{53} \gamma_{11} )
\gamma_{42}

- \gamma_{11}^{2}

+ \gamma_{13}^2 
\biggr]
(\gamma_{42} \gamma_{11}
+ \gamma_{41}\gamma_{12} )
\gamma_{53} 
\biggr\}
</math>
  </td>
</tr>
</table>
&hellip; Not sure this is headed anywhere useful!

====Volume Element====

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

<tr>
  <td align="right">
<math>~(h_1 h_4 h_5) d\lambda_1 d\lambda_4 d\lambda_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(h_1 h_4 h_5)
\biggl[ 
\biggl( \frac{\partial \lambda_1}{\partial x}\biggr) dx
+ \biggl( \frac{\partial \lambda_1}{\partial y}\biggr) dy
+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr) dz
\biggr]
\biggl[ 
\biggl( \frac{\partial \lambda_4}{\partial x}\biggr) dx
+ \biggl( \frac{\partial \lambda_4}{\partial y}\biggr) dy
+ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr) dz
\biggr]
\biggl[ 
\biggl( \frac{\partial \lambda_5}{\partial x}\biggr) dx
+ \biggl( \frac{\partial \lambda_5}{\partial y}\biggr) dy
+ \biggl( \frac{\partial \lambda_5}{\partial z}\biggr) dz
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
\biggl( \gamma_{11}\biggr) dx
+ \biggl( \gamma_{12}\biggr) dy
+ \biggl( \gamma_{13}\biggr) dz
\biggr]
\biggl[ 
\biggl( \gamma_{41}\biggr) dx
+ \biggl( \gamma_{42}\biggr) dy
+ \biggl( \gamma_{43}\biggr) dz
\biggr]
\biggl[ 
\biggl( \gamma_{51} \biggr) dx
+ \biggl( \gamma_{52} \biggr) dy
+ \biggl( \gamma_{53} \biggr) dz
\biggr]
</math>
  </td>
</tr>
</table>