Editing Appendix/Ramblings/T6CoordinatesPt3 (section)

====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!