Editing Appendix/Ramblings/T3Integrals (section)

==Definition==
By defining the dimensionless angle,
<div align="center">
<math>
\Zeta \equiv \sinh^{-1}\biggl( \frac{qz}{\varpi} \biggr) ,
</math>
</div>
the two key "T3" coordinates will be written as,

<table align="center" border="0" cellpadding="2">
<tr>
  <td align="right">
<math>
\lambda_1
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\varpi \cosh\Zeta = ( \varpi^2 + q^2z^2 )^{1/2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>
\lambda_2
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\varpi [\sinh\Zeta ]^{1/(1-q^2)} = \biggl[\frac{\varpi^{q^2}}{qz}\biggr]^{1/(q^2-1)}</math>
  </td>
</tr>
</table>

===Relevant Partial Derivatives===
Here are some relevant partial derivatives:  

<table align="center" border="1" cellpadding="5">
<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial x}
</math>
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial y}
</math>
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial z}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\lambda_1</math>
  </td>
  <td align="center">
<math>
\frac{x}{\lambda_1}
</math>
  </td>
  <td align="center">
<math>
\frac{y}{\lambda_1}
</math>
  </td>
  <td align="center">
<math>
\frac{q^2 z}{\lambda_1}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\lambda_2</math>
  </td>
  <td align="center">
<math>
\frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) x
</math><br />
<math>
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi^2} \biggr)
</math><br />
<math>
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi} \biggr)
</math>
  </td>
  <td align="center">
<math>
\frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) y
</math><br />
<math>
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)}  \biggl( \frac{y}{\varpi^2} \biggr)
</math><br />
<math>
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)}  \biggl( \frac{y}{\varpi} \biggr)
</math>
  </td>
  <td align="center">
<math>
- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\zeta} \biggr]^{q^2/(q^2-1)} \frac{q}{\varpi^{q^2}}
</math><br />
<math>
=- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \frac{1}{z}
</math>
<br />
<math>
=- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz^{q^2}} \biggr]^{1/(q^2-1)}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\lambda_3</math>
  </td>
  <td align="center">
<math>
- \frac{y}{\varpi^{2}}
</math>
  </td>
  <td align="center">
<math>
+ \frac{x}{\varpi^{2}}
</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
</tr>
</table>

Alternatively, partials can be taken with respect to the cylindrical coordinates, <math>\varpi</math>, <math>z</math> and <math>\phi</math>.  (Incidentally, I have reversed the traditional order of the <math>\phi</math> and <math>z</math> coordinates in an attempt to parallelize structure between cylindrical and T3 coordinates since <math>\lambda_3 \equiv \phi</math>.)

<table align="center" border="1" cellpadding="5">
<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial \varpi}
</math>
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial z}
</math>
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial \phi}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>{\lambda_1}</math>
  </td>
  <td align="center">
<math>
\frac{\varpi}{\lambda_1}
</math>
  </td>
  <td align="center">
<math>
\frac{q^2 z}{\lambda_1}
</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\lambda_2</math>
  </td>
  <td align="center">
<math>
\frac{q^2}{q^2-1} \left( \frac{\varpi}{qz} \right)^{1/(q^2-1)}
</math><br />
  </td>
  <td align="center">
<math>
-\frac{1}{q^2-1} \left( \frac{\varpi^{q^2}}{qz^{q^2}} \right)^{1/(q^2-1)}
</math><br />
  </td>
  <td align="center">
<math>
0
</math><br />
  </td>
</tr>

<tr>
  <td align="center">
<math>\lambda_3</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
  <td align="center">
<math>
1
</math>
  </td>
</tr>
</table>

Furthermore, the inverted partials are

<table align="center" border="1" cellpadding="5">
<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial \lambda_1}
</math>
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial \lambda_2}
</math>
  </td>
  <td align="center">
<math>
\frac{\partial}{\partial \lambda_3}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>{\varpi}</math>
  </td>
  <td align="center">
<math>
\varpi \ell^2 \lambda_1
</math>
  </td>
  <td align="center">
<math>
(q^2-1) q^2 \varpi z^2 \ell^2 / \lambda_2
</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>z</math>
  </td>
  <td align="center">
<math>
q^2 z \ell^2 \lambda_1
</math><br />
  </td>
  <td align="center">
<math>
- (q^2-1) \varpi^2 z \ell^2 / \lambda_2
</math><br />
  </td>
  <td align="center">
<math>
0
</math><br />
  </td>
</tr>

<tr>
  <td align="center">
<math>\phi</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
  <td align="center">
<math>
0
</math>
  </td>
  <td align="center">
<math>
1
</math>
  </td>
</tr>
</table>

===Scale Factors===
The scale factors are,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>h_1^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{\partial\lambda_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\lambda_1^2 \ell^2 
</math>
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>h_2^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{\partial\lambda_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(q^2-1)^2 \biggl(\frac{\varpi z \ell}{\lambda_2} \biggr)^2 
</math>
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>h_3^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{\partial\lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\varpi^2 
</math>
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="left" colspan="7">
where, &nbsp; &nbsp; &nbsp; &nbsp;<math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>.
  </td>
</tr>
</table>

===Direction Cosines===
The following table contains expressions for the [[User:Tohline/Appendix/Ramblings/DirectionCosines#Basic_Definitions_and_Relations|nine direction cosines]].

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="4">
<font color="darkblue">Direction Cosines for T3 Coordinates</font><br/>
<math>\gamma_{ni} = h_n\frac{\partial\lambda_n}{\partial x_i}</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center" colspan="3">
'''<math>i</math>'''
  </td>
</tr>

<tr>
  <td align="center" valign="middle" rowspan="3">
'''<math>n</math>'''
  </td>
  <td align="center" colspan="1">
<math>\ell x</math>
  </td>
  <td align="center" colspan="1">
<math>\ell y</math>
  </td>
  <td align="center" colspan="1">
<math>q^2 \ell z</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="1">
<math>\frac{q^2 x z \ell}{\varpi}</math>
  </td>
  <td align="center" colspan="1">
<math>\frac{q^2 y z \ell}{\varpi}</math>
  </td>
  <td align="center" colspan="1">
<math>- \varpi \ell</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="1">
<math>- \frac{y}{\varpi}</math>
  </td>
  <td align="center" colspan="1">
<math>+\frac{x}{\varpi}</math>
  </td>
  <td align="center" colspan="1">
<math>0</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="4">
where:
<math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>
  </td>
</tr>
</table>

===Orthogonality Condition===
Next, let's use the example orthogonality condition [[User:Tohline/Appendix/Ramblings/DirectionCosines#Orthogonality|derived elsewhere in connection with our overview of direction cosines]].  Specifically, let's see if [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.02|Equation DC.02]] is satisfied.
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{\partial\lambda_1}{\partial \varpi} \cdot  \frac{\partial\lambda_2}{\partial \varpi} + 
\frac{\partial\lambda_1}{\partial z} \cdot \frac{\partial\lambda_2}{\partial z}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{\varpi}{\lambda_1} \biggl[ \frac{q^2}{q^2-1} \left( \frac{\varpi}{qz} \right)^{1/(q^2-1)}
 \biggr] - 
\frac{q^2z}{\lambda_1} \biggl[ \frac{1}{q^2-1} \left( \frac{\varpi^{q^2}}{qz^{q^2}} \right)^{1/(q^2-1)}
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{q^2}{(q^2-1)\lambda_1} \biggl\{ \varpi\biggl[ \left( \frac{\varpi}{qz} \right)^{1/(q^2-1)}
 \biggr] - 
z \biggl[ \left( \frac{\varpi^{q^2}}{qz^{q^2}} \right)^{1/(q^2-1)}
 \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{q^2}{(q^2-1)\lambda_1} \biggl[ q^{-1/(q^2-1)} \varpi^{1 + 1/(q^2-1)} z^{-1/(q^2-1)} - q^{-1/(q^2-1)} \varpi^{q^2/(q^2-1)} z^{1-q^2/(q^2-1)} \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>
Hence, the key orthogonality condition defined by [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.02|Equation DC.02]] is satisfied. MF53 also gives us relationships that should apply ''between'' the various direction cosines if the coordinate system is orthogonal.  Let's check a few cases to see whether <math>\gamma_{mn} = M_{mn}</math>, where "<math>M_{mn}</math> is the minor of <math>\gamma_{mn}</math> in the determinant <math>|\gamma_{mn}|</math>":
<div align="center">
<math>
M_{11} = \gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} = - (-\varpi\ell)\frac{x}{\varpi} = + \ell x .
</math> <br /><br />

<math>
M_{12} = \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} = -\varpi\ell \biggl(-\frac{y}{\varpi}\biggr) = +\ell y .
</math> <br /><br />

<math>
M_{13} = \gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31} = \biggl( \frac{q^2 x z \ell}{\varpi} \biggr) \frac{x}{\varpi} - \biggl( \frac{q^2 y z \ell}{\varpi} \biggr) \biggl( -\frac{y}{\varpi} \biggr) = q^2 \ell z.
</math> <br /><br />

<math>
M_{31} = \gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} = \ell y (-\varpi\ell) - q^2 \ell z \biggl(\frac{q^2 yz\ell}{\varpi}\biggr) = - \frac{y \ell^2}{\varpi} \biggl( \varpi^2 + q^4 z^2 \biggr) = - \frac{y}{\varpi} .
</math> <br /><br />

<math>
M_{33} = \gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21} = \ell x \biggr(\frac{q^2 y z \ell}{\varpi} \biggr) - \ell y \biggr(\frac{q^2 x z \ell}{\varpi} \biggr) = 0 .
</math>
</div>
All of these beautifully obey the relationship, <math>\gamma_{mn} = M_{mn}</math>.

===Position Vector===
The position vector is,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\vec{x}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\hat{i}x + \hat{j}y + \hat{k}z
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\hat{e}_1 (h_1 \lambda_1) + \hat{e}_2 (h_2 \lambda_2) .
</math>
  </td>
</tr>
</table>