Editing ThreeDimensionalConfigurations/ChallengesPt2 (section)

===Try Tipped Plane Again===

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

<tr>
  <td align="right">
<math>~\frac{\dot{x}'}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (y' - y_0) \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \, ,
</math>
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~\frac{\dot{y}'}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \, .
</math>
  </td>
</tr>
</table>

<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{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2} 
=
- \frac{\beta \Omega_2}{\gamma \Omega_3}  \, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
</math>
  </td>
</tr>
</table>
Also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2 (1+\tan^2\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{b^2 c^2}  (c^2 + b^2\tan^2\theta)  \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\dot\varphi}^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~y_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{(z_0 + z'\cos\theta) b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta} \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="10" width="60%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{\jmath}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat{\jmath}'}\cos\theta  - \boldsymbol{\hat{k}'}\sin\theta \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat{\jmath}'}\sin\theta  + \boldsymbol{\hat{k}'}\cos\theta \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\boldsymbol{\Omega} = \boldsymbol{\hat\jmath} \Omega_2 + \boldsymbol{\hat{k}} \Omega_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2 ( \boldsymbol{\hat{\jmath}'}\cos\theta  - \boldsymbol{\hat{k}'}\sin\theta )
+
\Omega_3 ( \boldsymbol{\hat{\jmath}'}\sin\theta  + \boldsymbol{\hat{k}'}\cos\theta )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath'} (\Omega_2 \cos\theta + \Omega_3\sin\theta) + \boldsymbol{\hat{k}'} (- \Omega_2\sin\theta + \Omega_3\cos\theta) \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

In the inertial reference frame,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{u'}^{(0)} = \bold{u'} + \boldsymbol{\Omega \times}\bold{x'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( \boldsymbol{\hat\imath'} \dot{x}'
+ \boldsymbol{\hat\jmath'} \dot{y}'
+ \boldsymbol{\hat{k}'} \cancelto{0}{\dot{z}' })
+
[  \boldsymbol{\hat\jmath'} (\Omega_2 \cos\theta + \Omega_3\sin\theta) + \boldsymbol{\hat{k}'} (- \Omega_2\sin\theta + \Omega_3\cos\theta) ]
\boldsymbol\times (\boldsymbol{\hat\imath'} x' + \boldsymbol{\hat\jmath'}y' + \boldsymbol{\hat{k'}}z')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \dot{x}'
+ \boldsymbol{\hat\jmath'} \dot{y}'
+
[  -\boldsymbol{\hat{k'}} (\Omega_2 \cos\theta + \Omega_3\sin\theta) x']
+
[  \boldsymbol{\hat\imath'} (\Omega_2 \cos\theta + \Omega_3\sin\theta) z']
+
[  \boldsymbol{\hat\jmath'} (- \Omega_2\sin\theta + \Omega_3\cos\theta)x' ]
+
[ - \boldsymbol{\hat\imath'} (- \Omega_2\sin\theta + \Omega_3\cos\theta) y']
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'}  \biggl[ \dot{x}' + (\Omega_2\sin\theta - \Omega_3\cos\theta) y'  + (\Omega_2 \cos\theta + \Omega_3\sin\theta) z' \biggr]
+ \boldsymbol{\hat\jmath'} \biggl[ \dot{y}' +  (- \Omega_2\sin\theta + \Omega_3\cos\theta)x'  \biggr]
~-~\boldsymbol{\hat{k'}} \biggl[ (\Omega_2 \cos\theta + \Omega_3\sin\theta) x' \biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<div align="center">'''Inertial-Frame Vorticity in Primed Frame'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'}^{(0)} \equiv \boldsymbol{\nabla \times}\bold{u}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ \frac{\partial u_z' }{\partial y'} - \frac{\partial u_y'}{\partial z'} \biggr]^{(0)}
+ \boldsymbol{\hat\jmath'} \biggl[ \frac{\partial u_x'}{\partial z'} - \frac{\partial u_z'}{\partial x'} \biggr]^{(0)}
+ \bold{\hat{k}'} \biggl[ \frac{\partial u_y'}{\partial x'} - \frac{\partial u_x'}{\partial y'} \biggr]^{(0)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ 0 \biggr]
+ \boldsymbol{\hat\jmath'} \biggl[\frac{\partial \dot{x}'}{\partial z'} +  (\Omega_2 \cos\theta + \Omega_3\sin\theta)  + (\Omega_2 \cos\theta + \Omega_3\sin\theta) \biggr]
+ \bold{\hat{k}'} \biggl[\frac{\partial \dot{y}'}{\partial x'} + (- \Omega_2\sin\theta + \Omega_3\cos\theta) - \frac{\partial \dot{x}'}{\partial y'} - (\Omega_2\sin\theta - \Omega_3\cos\theta) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\boldsymbol{\Omega} +
\boldsymbol{\hat\jmath'} \biggl[ \frac{\partial \dot{x}'}{\partial z'} \biggr]
+ \bold{\hat{k}'} \biggl[\frac{\partial \dot{y}'}{\partial x'} - \frac{\partial \dot{x}'}{\partial y'} \biggr] 
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial \dot{x}'}{\partial z'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial z'} \biggl[
(y_0 - y' ) \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\biggr]
=
\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\frac{\partial y_0}{\partial z'} 
=
-\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'}^{(0)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\boldsymbol{\Omega} -
\boldsymbol{\hat\jmath'} \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) + \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \biggr]\dot\varphi \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

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

<tr>
  <td align="right">
<math>~{\dot\varphi} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \, ,
</math>
  </td>
</tr>
</table>

and rearranging terms, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'}^{(0)} - 2\boldsymbol{\Omega}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr) \biggl\{
-\boldsymbol{\hat\jmath'} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)^2
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ 1+ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{
-\boldsymbol{\hat\jmath'} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)^2
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ 1+ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr]
\biggr\}
</math>
  </td>
</tr>

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

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

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

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

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

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

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

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

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

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


----


<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{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2} 
=
- \frac{\beta \Omega_2}{\gamma \Omega_3}  \, ,
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~-\frac{\zeta_3}{c^2} \biggl[ \frac{a^2b^2}{a^2 + b^2} \biggr] \sin\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2 \zeta_2}{a^2 + c^2} \biggr]  \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{1}{\cos\theta} \biggl[ \frac{\zeta_3b^2}{a^2 + b^2} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl[ \frac{c^2 \zeta_2}{a^2 + c^2} \biggr]  \frac{1}{\sin\theta}
</math>
  </td>
</tr>
</table>