Editing VE/RiemannEllipsoids (section)

===Off-Diagonal Elements===

Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero.  Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2<sup>nd</sup>-order TVE is,
 
<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>~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x
\, .
</math>
  </td>
</tr>
</table>

For example &#8212; as is explicitly illustrated on p. 130 of EFE &#8212; for <math>~i=2</math> and <math>~j=3</math>,
 
<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>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x}
- 2\Omega_3 \int_V \rho u_1x_3 d^3x \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (3)</font> ]</td></tr>
</table>
whereas for <math>~i=3</math> and <math>~j=2</math>,
 
<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>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x
- 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x}
\, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (4)</font> ]</td></tr>
</table>

<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_V \rho u_i x_j d^3x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; if  &nbsp; &nbsp;<math>~i = j \, .</math>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (5)</font> ]</td></tr>
</table>
This has allowed us to set to zero one of the integrals in each of these last two expressions.  In what follows, we will benefit from recognizing, as well, that,
<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">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .</math>
  </td>
</tr>
</table>
</td></tr></table>

Our first off-diagonal element is, then,

<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>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \rho \int_V u_1 z d^3x 
</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+a^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_2\Omega_3 c^2 
- 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_3 \zeta_2 c^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
\Omega_2\Omega_3  
+ \biggl[ \frac{\zeta_2 a^2}{a^2 + c^2 }\biggr] \biggl[ 2\Omega_3 + \frac{\zeta_3 b^2}{b^2+a^2}\biggr]     
\biggr\} c^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
1  
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]     
\biggr\} \Omega_2\Omega_3c^2 \, .
</math>
  </td>
</tr>
</table>

The second is,
 
<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>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \rho \int_V u_1 y d^3x
</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+a^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3  a^2
- \Omega_3 \Omega_2 b^2 
- 2  \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\Omega_2 \zeta_3 b^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
\Omega_2 \Omega_3  
+ \biggl[ \frac{\zeta_3 a^2}{a^2+b^2}\biggr]  \biggl[2\Omega_2 +  \frac{\zeta_2 c^2}{c^2 + a^2}\biggr]   
\biggr\} b^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
1  
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr]  \biggl[2 +  \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]   
\biggr\} \Omega_2 \Omega_3b^2 \, .
</math>
  </td>
</tr>
</table>