Editing Appendix/Ramblings/ConcentricEllipsoidalCoordinates (section)

===Other Coordinate Pair in the Tangent Plane===

Let's focus on a particular point on the <math>\lambda_1</math> = constant surface, <math>(x_0, y_0, z_0)</math>, that necessarily satisfies the function, <math>F(x_0, y_0, z_0) = 0</math>.  We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely,
<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>
\hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, ,
</math>
  </td>
</tr>
</table>
where, for this specific point on the surface,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\ell_{3D}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<div align="center">
'''Tangent Plane'''<br />
[See, for example, [http://math.furman.edu/~dcs/courses/math21/ Dan Sloughter's] ([https://www.furman.edu Furman University]) 2001 Calculus III  class lecture notes &#8212; specifically [http://math.furman.edu/~dcs/courses/math21/lectures/l-15.pdf Lecture 15]]
</div>

----

The two-dimensional plane that is tangent to the <math>\lambda_1</math> = constant surface ''at this point'' is given by the expression,
<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 - x_0) \biggl[ \frac{\partial \lambda_1}{\partial x} \biggr]_0 
+ (y - y_0) \biggl[\frac{\partial \lambda_1}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[\frac{\partial \lambda_1}{\partial z} \biggr]_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(x - x_0) \biggl[ \frac{\partial F}{\partial x} \biggr]_0  
+ (y - y_0) \biggl[\frac{\partial F}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[ \frac{\partial F}{\partial z} \biggr]_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(x - x_0) \biggl( \frac{x}{\lambda_1}\biggr)_0 + (y - y_0)\biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
x \biggl( \frac{x}{\lambda_1}\biggr)_0 + y \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_0 \biggl( \frac{x}{\lambda_1}\biggr)_0 + y_0 \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z_0 \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
x x_0  + q^2 y y_0  + p^2 z z_0 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_0^2  +  q^2 y_0^2 + p^2 z_0^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
x x_0  + q^2 y y_0  + p^2 z z_0 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\lambda_1^2)_0 \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

Fix the value of <math>\lambda_1</math>.  This means that the relevant ellipsoidal surface is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\lambda_1^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>x^2 + q^2y^2 + p^2z^2 \, .</math>
  </td>
</tr>
</table>
If <math>z = 0</math>, the semi-major axis of the relevant x-y ellipse is <math>\lambda_1</math>, and the square of the semi-minor axis is <math>\lambda_1^2/q^2</math>.  At any other value, <math>z = z_0 < c</math>, the square of the semi-major axis of the relevant x-y ellipse is, <math>~(\lambda_1^2 - p^2z_0^2)</math> and the square of the corresponding semi-minor axis is, <math>(\lambda_1^2 - p^2z_0^2)/q^2</math>.  Now, for any chosen <math>x_0^2 \le (\lambda_1^2 - p^2z_0^2)</math>, the y-coordinate of the point on the <math>~\lambda_1</math> surface is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>y_0^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{q^2}\biggl[ \lambda_1^2 - p^2 z_0 -x_0^2 \biggr] \, .</math>
  </td>
</tr>
</table>
The slope of the line that lies in the <math>z = z_0</math> plane and that is tangent to the ellipsoidal surface at <math>(x_0, y_0)</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>m \equiv \frac{dy}{dx}\biggr|_{z_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{x_0}{q^2y_0}</math>
  </td>
</tr>
</table>