Editing Appendix/Ramblings/ConcentricEllipsoidalCoordinates (section)

===Primary (''radial-like'') Coordinate===

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,
<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^2 y^2 + p^2 z^2)^{1 / 2} \, .</math>
  </td>
</tr>
</table>
When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>.  (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.)  

A vector, <math>~\bold{\hat{n}}</math>, that is normal to the <math>~\lambda_1</math> = constant surface is given by the gradient of the function,
<table border="0" cellpadding="5" align="center">

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

In Cartesian coordinates, this means,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{\hat{n}}(x, y, z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl( \frac{\partial F}{\partial x} \biggr)
+ \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr)
+ \hat{k} \biggl( \frac{\partial F}{\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[ x(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]
+ \hat\jmath \biggl[ q^2y(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]
+ \hat{k}\biggl[ p^2 z(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \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}{\lambda_1} \biggr)
+ \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr)
+ \hat{k}\biggl(\frac{p^2 z}{\lambda_1} \biggr) \, ,
</math>
  </td>
</tr>
</table>
where it is understood that this expression is only to be evaluated at points, <math>~(x, y, z)</math>, that lie on the selected <math>~\lambda_1</math> surface &#8212; that is, at points for which the function, <math>~F(x,y,z) = 0</math>.  The length of this normal vector is given by the expression,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\lambda_1 \ell_{3D}}
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

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

It is therefore clear that the ''properly normalized'' normal unit vector that should be associated with any <math>~\lambda_1</math> = constant ellipsoidal surface is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_1 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} }
=
\hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat{k} (p^2 z \ell_{3D}) \, .
</math>
  </td>
</tr>
</table>
From our [[Appendix/Ramblings/DirectionCosines#Scale_Factors|accompanying discussion of direction cosines]], it is clear, as well, that the scale factor associated with the <math>~\lambda_1</math> coordinate is,
<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_{3D}^2 \, .</math>
  </td>
</tr>
</table>
We can also fill in the top line of our direction-cosines table, namely,

<table border="1" cellpadding="8" align="center" width="60%">
<tr>
  <td align="center" colspan="4">
'''Direction Cosines for T6 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_{3D}</math><br />&nbsp;
  <td align="center"><math>~q^2 y \ell_{3D}</math>
  <td align="center"><math>~p^2 z \ell_{3D}</math>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
</tr>
</table>