Editing VE/RiemannEllipsoids (section)

=Appendices: &nbsp;Various Integrals Over Ellipsoid Volume=
Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, .
</math>
  </td>
</tr>
</table>

==Appendix A: &nbsp;Volume==

Here we seek to find the volume of the ellipsoid via the ''Cartesian'' integral expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~V</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint  dx ~dy ~dz \, .
</math>
  </td>
</tr>
</table>

===Preliminaries===
First, we will integrate over <math>~x</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\ell</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ;
</math>
  </td>
</tr>
</table>

second, we will integrate over <math>~z</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_\ell</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ;
</math>
  </td>
</tr>
</table>
third, we will integrate over <math>~y</math> and set the limits of integration as <math>~\pm b</math>.  

===Carry Out the Integration===
Following thestrategy that has just been outlined, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~V</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iint  dy ~dz \int_{-x_\ell}^{+x_\ell} dx
=
\iint  dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell}
= 
2\int dy \int x_\ell ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
=
\frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2a}{c} \int \frac{dy}{2}  \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{|z_\ell |} \biggr) \biggr]_{-z_\ell}^{+z_\ell}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy
=
\frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr)  dy
=
\pi a c  \biggl[ y - \frac{y^3}{3b^2}  \biggr]_{-b}^{+b}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi}{3} \cdot a b c\, .
</math>
  </td>
</tr>
</table>

==Appendix B: &nbsp;Coriolis Component u<sub>1</sub>x<sub>2</sub>==
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\iiint  [u_1 y] ~dx ~dy ~dz</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz
+
 \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx 
+
 \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz  
+
 \biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz  
+
 \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} ~dz 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz  
~+~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int_{-z_\ell}^{+z_\ell} z~\biggl[ z_\ell^2 - z^2 \biggr]^{1 / 2} ~dz 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z \sqrt{z_\ell^2 - z^2} + z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}  
~-~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \cdot \frac{1}{3} \biggl\{ \biggl[ z_\ell^2 - z^2 \biggr]^{3 / 2} \biggr\}_{-z_\ell}^{+z_\ell}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}  
=
- \pi a~c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int_{-b}^b y^2 \biggl[1 - \frac{y^2}{b^2}  \biggr] dy  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \pi ac\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{y^3}{3} - \frac{y^5}{5b^2}  \biggr]_{-b}^{+b}  
=
- 2\pi a b^3 c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{2}{15}  \biggr]  
=
- \frac{4\pi abc}{3} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{b^2}{5}  \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3   \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, p. 130, Eq. (9a)</font> ]</td></tr>
</table>

==Appendix C: &nbsp;Coriolis Component u<sub>1</sub>x<sub>3</sub>==

Here we will additionally make use of the integration limits,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_\ell^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~b^2 \biggl(1 - \frac{z^2}{c^2}\biggr) \, .</math>
  </td>
</tr>
</table>
Integration over the relevant Coriolis component gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\iiint  [u_1 z] ~dx ~dy ~dz</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} z ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3\iiint \cancelto{0}{y  z ~dx ~dy ~dz}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint  z^2 ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2a\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \biggl\{ \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int_{-y_\ell}^{+y_\ell} \biggl[ y_\ell^2 - y^2 \biggr]^{1 / 2} dy
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \cdot \frac{1}{2}\biggl\{ y \sqrt{y_\ell^2 - y^2} + y_\ell^2 \sin^{-1}\biggr( \frac{y}{|y_\ell |} \biggr)\biggr\}_{-y_\ell}^{+y_\ell}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2  \biggl\{ \frac{\pi}{2} y_\ell^2 \biggr\} dz
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2  \biggl\{ 1 - \frac{z^2}{c^2} \biggr\} dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{z^3}{3} - \frac{z^5}{5c^2} \biggr\}_{-c}^{+c}
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{1}{3} - \frac{1}{5} \biggr\}2c^3
=
\frac{4 \pi a b c}{3}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{c^2}{5} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+
~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, p. 130, Eq. (9b)</font> ]</td></tr>
</table>


==Appendix D: &nbsp; The Other Four Coriolis Components ==

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

<tr>
  <td align="right">
<math>~\iiint  [u_2 x] ~dx ~dy ~dz</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z} + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x \biggr\} x ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\iiint  [u_2 z] ~dx ~dy ~dz</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z + \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x} \biggr\} z ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\iiint  [u_3 x] ~dx ~dy ~dz</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x + \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y} \biggr\} x ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\iiint  [u_3 y] ~dx ~dy ~dz</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x} + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y \biggr\} y ~dx ~dy ~dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1  \, .
</math>
  </td>
</tr>
</table>

==Appendix E: &nbsp; Kinetic Energy Components ==

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

<tr>
  <td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V  u_1 u_1 d^3x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  [u_1^2] ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  \biggl\{
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2 
- 2\cancelto{0}{\biggl[ \frac{a^2}{a^2 + b^2}\biggr]  
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3} yz 
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2 
\biggr\} ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \iiint  y^2  ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2\iiint z^2 ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{22}}{\rho} \biggr]
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{33}}{\rho} \biggr] \, .</math>
  </td>
</tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{22} = \int_V  u_2 u_2 d^3x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  [u_2^2] ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x  \biggr\}^2 ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2  \biggl[ \frac{I_{33}}{\rho} \biggr]
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{11}}{\rho} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{33} = \int_V  u_3 u_3 d^3x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  [u_2^2] ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}^2 ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2  \biggl[ \frac{I_{11}}{\rho} \biggr]
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{22}}{\rho} \biggr] \, .</math>
  </td>
</tr>
</table>

===Off-Diagonal Elements===
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{23} = \int_V  u_2 u_3 d^3x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  [u_2 u_3] ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\} 
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} 
~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z \biggr\} 
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} 
~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ \iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\} 
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} 
~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3  \biggr\} 
x^2~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- ~\frac{I_{11}}{\rho}
\biggl[ \frac{b^2}{b^2+a^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3  
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, p. 130, Eq. (8)</font> ]</td></tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{12} = \int_V  u_1 u_2 d^3x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  [u_1 u_2] ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} 
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\} 
~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~  
\biggl[ \frac{a^2}{a^2+c^2}\biggr]  
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2  
\iiint z^2~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~  \frac{I_{33}}{\rho}
\biggl[ \frac{a^2}{a^2+c^2}\biggr]  
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2  
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{31} = \int_V  u_3 u_1 d^3x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  [u_3 u_1] ~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint  
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} 
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} 
~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -~ 
\biggl[ \frac{c^2}{c^2+b^2}\biggr]  
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3  
\iiint y^2~dx ~dy ~dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -~ \frac{I_{22}}{\rho}
\biggl[ \frac{c^2}{c^2+b^2}\biggr]  
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3  \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\mathfrak{T}_{32}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{T}_{23} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mathfrak{T}_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{T}_{12} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and,&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mathfrak{T}_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{T}_{31} \, .</math>
  </td>
</tr>
</table>