Editing Appendix/Mathematics/ToroidalSynopsis01 (section)

==Basics==
Here we attempt to bring together &#8212; in as succinct a manner as possible &#8212; [[2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|our approach]] and [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong's (1973)]  approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>.  The relevant toroidal coordinate system is one based on an ''anchor ring'' of major radius, 
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<math>~a^2 \equiv R^2 - d^2 \, .</math>
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If the meridional-plane location of the ''anchor ring'' &#8212; as written in cylindrical coordinates &#8212; is, <math>~(\varpi, z) = (a,Z_0)</math>, then the preferred toroidal-coordinate system has meridional-plane coordinates, <math>~(\eta, \theta)</math>, defined such that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
  </td>
</tr>
</table>
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where,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(\varpi - a)^2 + (z-Z_0)^2 \, ,</math>
  </td>
</tr>
</table>
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and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.  Mapping the other direction, we have,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~z-Z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>
</div>
<span id="DiffVolElement">The three-dimensional differential volume element is,</span>
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="center">
<math>\varpi d\varpi ~dz ~d\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{a^3\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math>
  </td>
</tr>
</table>
</div>

Note that, if <math>~\eta_0</math> identifies the surface of the uniform-density torus, then,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{d} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{d} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\coth\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{a} \, ;</math>
  </td>
</tr>
</table>
</div>

and when the integral over the volume element is completed &#8212; that is, over all <math>~\psi</math>, over all <math>~\theta</math>, and over the "radial" interval, <math>~\eta_0 \le \eta \le \infty</math> &#8212; the resulting volume is,
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<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>~\frac{2\pi^2 \cosh\eta_0}{\sinh^3\eta_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 Rd^2 \, .</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr]</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\sinh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ e^\eta - e^{-\eta} \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\coth\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ e^\eta + e^{-\eta} \biggr]\biggl[ e^\eta - e^{-\eta} \biggr]^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{r_1}{r_2} +  \frac{r_2}{r_1} \biggr]\biggl[  \frac{r_1}{r_2} - \frac{r_2}{r_1} \biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="1">
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1 r_2}  \biggr]\biggl[  \frac{r_1^2 - r_2^2}{r_1 r_2} \biggr]^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="1">
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1^2 - r_2^2}  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
\frac{  \varpi^2 + a^2 + (z - Z_0)^2 }{ 2a\varpi } \, .
</math>
  </td>
</tr>
</table>
</div>