Editing 2DStructure/ToroidalGreenFunction (section)

==Basic Elements of a Toroidal Coordinate System==

Given the meridional-plane coordinate location of a toroidal-coordinate system's axisymmetric ''anchor ring'', <math>~(\varpi,z) = (a,Z_0)</math>, the relationship between toroidal coordinates <math>~(\eta,\theta,\psi) </math>and Cartesian coordinates <math>~(x, y, z)</math> is,
<div align="center">
<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>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
</tr>

<tr>
  <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>
This set of coordinate relations appears as equations 2.1 - 2.3 in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].  This set of relations may also be found, for example, on p. 1301 within eq. (10.3.75) of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>]; in [http://dlmf.nist.gov/14.19 &sect;14.19 of NIST's ''Digital Library of Mathematical Functions'']; or even within [https://en.wikipedia.org/wiki/Toroidal_coordinates#Definition Wikipedia].  (In most cases the implicit assumption is that <math>~Z_0 = 0</math>.)  It is clear, of course, that the cylindrical radial coordinate is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = (x^2 + y^2)^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>


Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
<div align="center">
<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>
</tr>

<tr>
  <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>

<tr>
  <td align="right">
<math>~\tan\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{y}{x} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<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>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.

<table border="1" cellpadding="10" align="center" width="85%">
<tr>
  <td align="center">
<!-- [[File:Hicks1881TitlePage.png|500px|Title Page of Hicks (1881)]] -->
[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
''"On Toroidal Functions"''<br />
Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652
</td>
</tr>
<tr><td align="left">
[http://rstl.royalsocietypublishing.org/content/172/609.full.pdf+html W. M. Hicks (1881)] presents one of the first &#8212; if not ''the'' first &#8212; discussions of toroidal coordinates and associated toroidal functions.  Equation (4) on p. 614 of his article provides the following definition of the pair of meridional-plane coordinates <math>~(u,v)</math>, written in terms of the traditional cylindrical-coordinate pair <math>~(\rho,z)</math> and the specified ''anchor ring'' radius, <math>~a</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln \biggl[ \frac{z^2 + (\rho+a)^2}{z^2 + (\rho - a)^2} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tan^{-1} \biggl[ \frac{2az}{\rho^2 + z^2 - a^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
His equation (5) presents the reverse mapping, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a\sinh u}{\cosh u - \cos v} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a\sin v}{\cosh u - \cos v} \, .
</math>
  </td>
</tr>
</table>

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

<span id="DiffVolumeElement">&nbsp;</span>
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 16 August 2017:  In equation (2.17) of his &sect;IIB &#8212; when Wong (1973) introduces the differential volume element &#8212; the variable used to represent the azimuthal coordinate angle switches from &psi; to &#x003A6;.  We will stick with the &psi; notation, here.]]According to p. 1301, eq. (10.3.75) of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>] &#8212; or, for example, as found in [https://en.wikipedia.org/wiki/Toroidal_coordinates#Scale_factors Wikipedia] &#8212; in toroidal coordinates the differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h_\eta h_\theta h_\psi d\eta d\theta 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>