Editing ThreeDimensionalConfigurations/ChallengesPt6 (section)

==Tipped Frame==

===For Arbitrary Tip Angles===

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

<tr>
  <td align="right">
<math>x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>x' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>y</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
y' \cos\theta - z'\sin\theta 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(z - z_0)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
z' \cos\theta + y'\sin\theta 
\, ,</math>
  </td>
</tr>
</table>
the constraint becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P'(x', y', z')</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{y'\cos\theta - z'\sin\theta}{b}\biggr]^2 
+ \biggl[\frac{z_0 + z'\cos\theta + y'\sin\theta}{c}\biggr]^2
+\biggl(\frac{x'}{a}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>

<span id="gradP">Hence,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla P'(x', y', z')</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl(\frac{2x'}{a^2}\biggr)
+ \boldsymbol{\hat\jmath'} \biggl\{ \biggl[\frac{2( y'\cos\theta - z'\sin\theta )\cos\theta}{b^2}\biggr] 
+ \biggl[\frac{2( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta}{c^2}\biggr] \biggr\}

+ \mathbf{\hat{k}'}\biggl\{- \biggl[\frac{2( y'\cos\theta - z'\sin\theta) \sin\theta}{b^2}\biggr] 
+ \biggl[\frac{2 ( z_0 + z'\cos\theta + y'\sin\theta) \cos\theta}{c^2}\biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>

And, in the ''tipped'' frame we find,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[\boldsymbol{\hat\jmath'} \cos\theta  - \mathbf{\hat{k}'} \sin\theta \biggr] \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 x'
\biggr\}
+
\biggl[\boldsymbol{\hat\jmath'} \sin\theta  + \mathbf{\hat{k}'} \cos\theta \biggr] \biggl\{ 
- 
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x'
\biggr\} \, .
</math>
  </td>
</tr>
</table>

As a result, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathbf{u'}_\mathrm{EFE} \cdot \nabla P'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2x'}{a^2}\biggr)\biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\biggl\{
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 \cos\theta ~ x' - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2\sin\theta ~x' 
\biggr\}
\biggl\{ \biggl[\frac{2( y'\cos\theta - z'\sin\theta )\cos\theta}{b^2}\biggr] 
+ \biggl[\frac{2( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta}{c^2}\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- \biggl\{\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 \sin\theta~ x'
+ \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cos\theta~ x'
\biggr\}
\biggl\{- \biggl[\frac{2( y'\cos\theta - z'\sin\theta) \sin\theta}{b^2}\biggr] 
+ \biggl[\frac{2 ( z_0 + z'\cos\theta + y'\sin\theta) \cos\theta}{c^2}\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[\frac{1}{2x'}  \biggr] \mathbf{u'}_\mathrm{EFE} \cdot \nabla P'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[\frac{1}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{1}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3    
\biggl\{ \biggl[\frac{( y'\cos\theta - z'\sin\theta )\cos^2\theta}{b^2}\biggr] 
+ \biggl[\frac{( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta \cos\theta}{c^2}\biggr] \biggr\}
-
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggl\{ \biggl[\frac{( y'\cos\theta - z'\sin\theta )\sin\theta\cos\theta}{b^2}\biggr] 
+ \biggl[\frac{( z_0 + z'\cos\theta + y'\sin\theta) \sin^2\theta}{c^2}\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
\biggl\{\biggl[\frac{( y'\cos\theta - z'\sin\theta) \sin^2\theta}{b^2}\biggr] 
- \biggl[\frac{ ( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta \cos\theta}{c^2}\biggr] \biggr\}

+ \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2
\biggl\{\biggl[\frac{( y'\cos\theta - z'\sin\theta) \sin\theta \cos\theta}{b^2}\biggr] 
- \biggl[\frac{ ( z_0 + z'\cos\theta + y'\sin\theta) \cos^2\theta}{c^2}\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[\frac{1}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{1}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3    
\biggl[\frac{( y'\cos\theta - z'\sin\theta )}{b^2}\biggr] 
-
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggl[\frac{( z_0 + z'\cos\theta + y'\sin\theta)}{c^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{( y'\cos\theta - z'\sin\theta )}{a^2 + b^2}\biggr] \zeta_3    
- \biggl[\frac{1}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{1}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
-
\biggl[ \frac{( z_0 + z'\cos\theta + y'\sin\theta)}{a^2 + c^2} \biggr] \zeta_2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0 \, .
</math>
  </td>
</tr>
</table>

Q. E. D.

===For Specific Tip Angle===

So &hellip; we have demonstrated that the velocity vectors are everywhere orthogonal to the normal to the ellipsoid for <b>all</b> values of the "tip" angle, <math>\theta</math>.  So why have we been unable to demonstrate the same result in the case where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\beta \Omega_2}{\gamma \Omega_3} = - \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, .
</math>
  </td>
</tr>
</table>
Remember that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1 + \tan^2\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sin^2\theta + \cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \cos^2\theta = \frac{1}{1 + \tan^2\theta}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>
\sin^2\theta = \frac{\tan^2\theta}{1 + \tan^2\theta} \, .
</math>
  </td>
</tr>
</table>

Rearranging terms in the expression for the "tipped plane" Riemann flow velocity, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \cos\theta \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y' - z'\tan\theta) 
+ \cos\theta \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (y'\tan\theta + z')
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\boldsymbol{\hat\jmath'} \biggl\{   
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
\tan\theta 
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\} x' \cos\theta
-
\mathbf{\hat{k}'} \biggl\{
\tan\theta  
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
+
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\} x' \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \cos\theta \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y' - z'\tan\theta) 
+ \cos\theta \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (y'\tan\theta + z')
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\boldsymbol{\hat\jmath'} \biggl\{   
1 
-
\tan\theta 
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggl[ \frac{a^2 + b^2}{b^2}\biggr] \frac{1}{\zeta_3}
\biggr\} x' \cos\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3
-
\mathbf{\hat{k}'} \biggl\{
\tan\theta  
+
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[ \frac{a^2 + b^2}{b^2}\biggr] \frac{1}{\zeta_3} 
\biggr\} x' \cos\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
</math>
  </td>
</tr>
</table>


<span id="SpecificTipAngle">Then, for this specific tip angle</span>, the Riemann flow velocity is,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\boldsymbol{\hat\imath'} \biggl\{ 
\zeta_3 (z'\tan\theta - y' ) \cos\theta 
+ \cos\theta \biggl[ \frac{a^2+b^2}{a^2 + c^2}\biggr] \zeta_2 (y'\tan\theta + z')
\biggr\}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\boldsymbol{\hat\jmath'} \biggl\{   
1 + \tan^2\theta 
\biggr\} x' \cos\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3
-
\mathbf{\hat{k}'} \biggl\{
0\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\boldsymbol{\hat\imath'} \biggl\{ 
\zeta_3 (z'\tan\theta - y' )\cos\theta 
- \cos\theta \biggl[ \frac{b^2 \zeta_3 \tan\theta}{c^2 \zeta_2}\biggr] \zeta_2 (y'\tan\theta + z')
\biggr\}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
+
\boldsymbol{\hat\jmath'} \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3~x' }{\cos\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\boldsymbol{\hat\imath'} \biggl\{ 
c^2 (z'\tan\theta - y' )\cos\theta 
- b^2 \sin\theta (y'\tan\theta + z')
\biggr\}\frac{\zeta_3 }{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
+
\boldsymbol{\hat\jmath'} \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3~x' }{\cos\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\boldsymbol{\hat\imath'} \biggl\{ 
z' (c^2 - b^2 )\tan\theta  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
+
\boldsymbol{\hat\jmath'} \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3~x' }{\cos\theta}
</math>
  </td>
</tr>
</table>

As we have [[#gradP|already stated]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla P'(x', y', z')</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl(\frac{2x'}{a^2}\biggr)
+ \boldsymbol{\hat\jmath'} \biggl\{ \biggl[\frac{2( y'\cos\theta - z'\sin\theta )\cos\theta}{b^2}\biggr] 
+ \biggl[\frac{2( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta}{c^2}\biggr] \biggr\}

+ \mathbf{\hat{k}'}\biggl\{- \biggl[\frac{2( y'\cos\theta - z'\sin\theta) \sin\theta}{b^2}\biggr] 
+ \biggl[\frac{2 ( z_0 + z'\cos\theta + y'\sin\theta) \cos\theta}{c^2}\biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>

<span id="Orthogonal">Hence,</span>

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

<tr>
  <td align="right">
<math>\biggl( \frac{1}{2 x'}\biggr)\boldsymbol{u'}_\mathrm{EFE} \cdot \nabla P'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{1}{a^2}\biggr)\biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\biggl(\frac{1}{a^2}\biggr)\biggl\{ 
z' (c^2 - b^2 )\tan\theta  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3 }{\cos\theta}
\biggl\{ \biggl[\frac{( y'\cos\theta - z'\sin\theta )\cos\theta}{b^2}\biggr] 
+ \biggl[\frac{( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta}{c^2}\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[ \frac{\zeta_3 }{c^2(a^2 + b^2)}\biggr] b^2 z_0 \tan\theta
+
\biggl\{ 
z' (c^2 - b^2 )\tan\theta  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\cos\theta \biggl[\frac{\zeta_3}{c^2(a^2+b^2)}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\biggl[ \frac{\zeta_3 }{c^2 (a^2 + b^2)}\biggr] \frac{1}{\cos\theta}
\biggl\{ \biggl[c^2( y'\cos\theta - z'\sin\theta )\cos\theta\biggr] 
+ \biggl[b^2 ( z_0 + z'\cos\theta + y'\sin\theta) \sin\theta\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{c^2(a^2 + b^2)}{\zeta_3 }\biggr] \biggl( \frac{1}{2 x'}\biggr)\boldsymbol{u'}_\mathrm{EFE} \cdot \nabla P'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  b^2 z_0 \tan\theta
+
\biggl\{ 
z' (c^2 - b^2 )\tan\theta  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\cos\theta  
+
\biggl[c^2( y'\cos\theta - z'\sin\theta )\biggr] 
+ \biggl[b^2 ( z_0 + z'\cos\theta + y'\sin\theta) \tan\theta\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
z' (c^2 - b^2 )\tan\theta  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\cos\theta  
+
\biggl[c^2( y'\cos\theta - z'\sin\theta )\biggr] 
+ \biggl[b^2 ( z'\cos\theta + y'\sin\theta) \tan\theta\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
y' \cos\theta \biggl[- c^2 - b^2\tan^2\theta + c^2 + b^2\tan^2\theta  \biggr]
+
z' \biggl[(c^2 - b^2) \sin\theta - c^2\sin\theta + b^2\sin\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>