Editing 2DStructure/ToroidalCoordinates (section)

==Total Mass==
How do I know whether or not I have made mistakes in these derivations?  Aside from the published work of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)], how do I know what the correct answer for the gravitational potential of a uniform-density torus is?  

A degree of assurance can be drawn by carrying out a similar pair of nested integrations to determine the total mass (or volume) of the torus that is defined by our chosen test-mass distribution.  As can be found in many mathematics handbooks, or [https://en.wikipedia.org/wiki/Torus online], the volume of our defined torus should be,
<div align="center">
<math>~V_\mathrm{torus} = 2\pi^2 \varpi_t r_t^2 \, .</math>
</div>

===Using Cylindrical Coordinates===
====Total Volume====
Let's start by integrating the three-dimensional volume element, <math>~dV_{3D}</math>, over the azimuthal angle to obtain an expression for the two-dimensional differential volume element that is written in terms of the meridional-plane differential area, <math>~d\sigma</math>, as used in our definition of [[#Expression_for_the_Axisymmetric_Potential|<math>~q_0</math>, above]].  Using the notation of MF53 (but employing the opposite sign convention from them), in cylindrical coordinates we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dV_{3D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- [h_1 d\xi_1 ] [h_2 d\xi_2] [h_3 d\xi_3] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~d\xi_1  \biggl[\frac{\xi_1}{\sqrt{1-\xi_2^2}} d\xi_2 \biggr] d\xi_3 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~d\varpi  \biggl[\frac{\varpi d(\cos\varphi) }{\sqrt{1-\cos^2\varphi}} \biggr] dz </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>+~[\varpi d\varphi ] d\sigma \, , </math>
  </td>
</tr>
</table>
</div>
where, [[#Expression_for_the_Axisymmetric_Potential|as before]] in cylindrical coordinates, <math>~d\sigma = d\varpi dz</math>.  From this we obtain,

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

<tr>
  <td align="right">
<math>~dV_{2D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi d\sigma \int_0^{2\pi} d\varphi  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \varpi d\sigma \, .</math>
  </td>
</tr>
</table>
</div>

Now in order to finish the volume integration, we need the limits of integration in the meridional plane.  These can be obtained from the [[#Chosen_Test_Mass_Distribution|above algebraic description]] of the (pink) test-mass torus as an off-center circle.  Specifically, we have,

<div align="center">
<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>~2\pi \int\limits_{(\varpi_t - r_t)}^{(\varpi_t + r_t)} \varpi d\varpi  \int\limits_{-\sqrt{r_t^2 - (\varpi_t - \varpi)^2}}^{\sqrt{r_t^2 - (\varpi_t - \varpi)^2}} dz </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int\limits_{(\varpi_t - r_t)}^{(\varpi_t + r_t)} \sqrt{r_t^2 - (\varpi_t - \varpi)^2} \varpi d\varpi  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\pi}{3} \biggl\{
3r_t^2 \varpi_t \tan^{-1}\biggl[ \frac{\varpi - \varpi_t}{ \sqrt{r_t^2 - (\varpi_t - \varpi)^2 }} \biggr] 
- [r_t^2 - (\varpi_t - \varpi)^2 ]^{1/2}~(2r_t^2 + \varpi_t^2 + \varpi_t \varpi - 2\varpi^2)
\biggr\}_{(\varpi_t - r_t)}^{(\varpi_t + r_t)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{2\pi}{3} \biggl\{3r_t^2 \varpi_t \tan^{-1}\biggl[ \frac{r_t}{0 } \biggr] - 0 \biggr\}
-~\frac{2\pi}{3} \biggl\{3r_t^2 \varpi_t \tan^{-1}\biggl[ \frac{-r_t}{0} \biggr] - 0 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{2\pi}{3} \biggl[ 3r_t^2 \varpi_t \biggl(\frac{\pi}{2}\biggr) \biggr] +~\frac{2\pi}{3} \biggl[ 3r_t^2 \varpi_t \biggl(\frac{\pi}{2}\biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~2\pi^2 r_t^2 \varpi_t \, ,
</math>
  </td>
</tr>
</table>
</div>
where we have carried out the second integration using the [http://www.wolframalpha.com/calculators/integral-calculator/ WolframAlpha online integral calculator]:

[[File:WolframAlphaTorusVolume.png|350px|center|WolframAlpha integration result]]

This is the answer for the volume of a torus that we expected.  Good!

====Volume with Cropped Top====

[[File:CropTopB.png|300px|right|Diagram of "Cropped Top" Torus]]
During my development of a computer program to integrate over the volume of a circular torus while using toroidal coordinates, I decided that it would be useful to determine the volume of a torus that has a "cropped top" as illustrated in the accompanying diagram, shown here on the right.  Let's evaluate this "cropped-top" volume using cylindrical coordinates so that we will know what the correct answer is when developing an integration scheme using toroidal coordinates.  Specifically, let's determine the volume of the "green" portion for a specified value of <math>~h < r_t</math>.

The radial integration will be evaluated between the limits: (lower) <math>~(\varpi_t - b) </math> and (upper) <math>~(\varpi_t + b) </math>, where,
<div align="center">
<math>~b = \sqrt{r_t^2-h^2} \, .</math>
</div>
And the limits on the vertical integration will be:  (lower) <math>~h</math> and, as before when integrating over the entire volume, (upper) <math>~\sqrt{r_t^2 - (\varpi_t - \varpi)^2} \, .</math>  With these limits, the volume integration gives,

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

<tr>
  <td align="right">
<math>~V_\mathrm{green}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \int\limits_{(\varpi_t - b)}^{(\varpi_t + b)} \varpi d\varpi  \int\limits_{h}^{\sqrt{r_t^2 - (\varpi_t - \varpi)^2}} dz </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \int\limits_{(\varpi_t - b)}^{(\varpi_t + b)} \varpi d\varpi \biggl[\sqrt{r_t^2 - (\varpi_t - \varpi)^2} - h \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2\pi h \int\limits_{(\varpi_t - b)}^{(\varpi_t + b)} \varpi d\varpi  
+ 2\pi \int\limits_{(\varpi_t - b)}^{(\varpi_t + b)} \varpi \sqrt{X} d\varpi \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~X</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~a_0 + a_1\varpi + a_2\varpi^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-( \varpi_t^2 - r_t^2) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2\varpi_t \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-1 \, .</math>
  </td>
</tr>
</table>
</div>
Carrying out this pair of integrations gives,

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

<tr>
  <td align="right">
<math>~V_\mathrm{green}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\pi h \biggl[ (\varpi_t + b)^2 -  (\varpi_t - b)^2  \biggr]  
+ 2\pi \biggl[\frac{X\sqrt{X}}{3a_2} - \frac{a_1(2a_2\varpi + a_1)}{8a_2^2} \sqrt{X} \biggr]_{(\varpi_t - b)}^{(\varpi_t + b)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{2\pi a_1 ( 4a_0 a_2 - a_1^2)}{(4a_2)^2} \int\limits_{(\varpi_t - b)}^{(\varpi_t + b)} \frac{d\varpi}{\sqrt{X}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\pi h \biggl[\varpi_t^2 + 2b\varpi_t + b^2 - (\varpi_t^2 - 2b\varpi_t + b^2)  \biggr]
+ \frac{\pi}{12} \biggl\{-8[ r_t^2 - (\varpi_t - \varpi)^2]^{3/2} + 6\varpi_t(2\varpi - 2\varpi_t) [r_t^2 - (\varpi_t - \varpi)^2 ]^{1/2} \biggr\}_{(\varpi_t - b)}^{(\varpi_t + b)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{2\pi a_1 ( 4a_0 a_2 - a_1^2)}{(4a_2)^2} 
\biggl\{ - \frac{1}{\sqrt{-a_2}} ~\sin^{-1}\biggl[ \frac{2a_2 \varpi + a_1}{\sqrt{a_1^2 - 4a_0 a_2}} \biggr] \biggr\}_{(\varpi_t - b)}^{(\varpi_t + b)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-4\pi h b\varpi_t  
+ \frac{\pi}{3} \biggl[3\varpi_t b (r_t^2 - b^2 )^{1/2} -2( r_t^2 - b^2)^{3/2} \biggr]
+ \frac{\pi}{3} \biggl[3\varpi_t b (r_t^2 - b^2 )^{1/2} +2( r_t^2 - b^2)^{3/2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \pi \varpi_t ( a_0  + \varpi_t^2)
\biggl\{ \sin^{-1}\biggl[ \frac{\varpi_t - \varpi }{\sqrt{\varpi_t^2 + a_0}} \biggr] \biggr\}_{(\varpi_t - b)}^{(\varpi_t + b)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-4\pi h b\varpi_t  + 2\pi\varpi_t b (r_t^2 - b^2 )^{1/2} 
- \pi \varpi_t  r_t^2  \biggl\{ \sin^{-1}\biggl[ \frac{-b}{r_t} \biggr] - \sin^{-1}\biggl[ \frac{b }{r_t} \biggr] \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-4\pi h \varpi_t (r_t^2 -h^2 )^{1/2} + 2\pi\varpi_t h (r_t^2 -h^2 )^{1/2} 
+2 \pi \varpi_t  r_t^2  \sin^{-1}\biggl[1 - \biggl(\frac{h }{r_t}\biggr)^2\biggr]^{1/2}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \pi \varpi_t  r_t^2  \biggl\{ \sin^{-1}\biggl[1 - \biggl(\frac{h }{r_t}\biggr)^2\biggr]^{1/2}   
- \biggl(\frac{h}{r_t}\biggr) \biggl[ 1 - \biggl(\frac{h}{r_t}\biggr)^2 \biggr]^{1/2} \biggr\}\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the ''fractional'' volume is,

<div align="center" id="GreenAnalytic">
<table border="1" cellpadding="8"><tr><th align="center" colspan="2">
Analytic Expression for Green Volume</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\frac{V_\mathrm{green}}{V_\mathrm{torus}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\pi}  \biggl\{ \sin^{-1}\biggl[1 - \biggl(\frac{h }{r_t}\biggr)^2\biggr]^{1/2}   
- \biggl(\frac{h}{r_t}\biggr) \biggl[ 1 - \biggl(\frac{h}{r_t}\biggr)^2 \biggr]^{1/2} \biggr\}\, .
</math>
  </td>
</tr>
</table>

</td>
<td align="center">
[[File:CropTopB.png|150px|right|Diagram of "Cropped Top" Torus]]
</td></tr>
</table>
</div>


REALITY CHECK:  This should give a zero (green) volume if <math>~h = r_t</math>; and the fractional volume should be one-half if <math>~h = 0</math>.  In the former case, our expression gives,

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

<tr>
  <td align="right">
<math>~\frac{V_\mathrm{green}}{V_\mathrm{torus}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\pi} \sin^{-1} (0) = 0 \, ,  
</math>
  </td>
</tr>
</table>
</div>
as expected.  And in the latter case we have,

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

<tr>
  <td align="right">
<math>~\frac{V_\mathrm{green}}{V_\mathrm{torus}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\pi}  \sin^{-1} (1) =  \frac{1}{2}  \, ,  
</math>
  </td>
</tr>
</table>
</div>
which means that <math>~V_\mathrm{green}</math> is indeed half of the total torus volume, [[2DStructure/ToroidalCoordinates#Total_Volume|as derived earlier]].

===Using Toroidal Coordinates with Special Alignment===

If, instead, we use a toroidal coordinate system, we have (see p. 666 of MF53),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dV_{3D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- [h_1 d\xi_1 ] [h_2 d\xi_2] [h_3 d\xi_3] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\biggl[ \frac{a d\xi_1}{ (\xi_1-\xi_2)\sqrt{\xi_1^2-1} } \biggr]~ 
\biggl[\frac{a d\xi_2}{ (\xi_1-\xi_2)\sqrt{1-\xi_2^2} } \biggr] ~
\biggl[ \biggl( \frac{\xi_1^2 - 1}{1 - \xi_3^2} \biggr)^{1/2} \frac{a d\xi_3}{(\xi_1-\xi_2)} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- d\sigma~
\biggl[ \frac{a(\xi_1^2 - 1)^{1/2}}{(\xi_1-\xi_2)} \frac{d(\cos\varphi)}{\sqrt{1 - \cos^2\varphi}} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>+~[\varpi d\varphi ] d\sigma \, , </math>
  </td>
</tr>
</table>
</div>
where, as employed above, a differential area element in the meridional plane of a toroidal-coordinate system is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2\biggl[ \frac{d\xi_1}{ (\xi_1-\xi_2)\sqrt{\xi_1^2-1} } \biggr]~ 
\biggl[\frac{d\xi_2}{ (\xi_1-\xi_2)\sqrt{1-\xi_2^2} } \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
As in the case of the cylindrical coordinate system, because the torus is axisymmetric, integration over the azimuthal angular coordinate in a toroidal coordinate system gives,

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

<tr>
  <td align="right">
<math>~dV_{2D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi d\sigma \int_0^{2\pi} d\varphi  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \varpi d\sigma \, .</math>
  </td>
</tr>
</table>
</div>

<table border="1" align="center" width="90%" cellpadding="8">
<tr><th align="center">[https://en.wikipedia.org/wiki/Toroidal_coordinates#Scale_factors Cross-check:  Wikipedia's Differential Volume Element]</th></tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dV_{3D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[h_1 d\tau ] [h_2 d\theta] [h_3 d\varphi] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^3 \biggl[\frac{ d\tau}{\cosh\tau - \cos\theta} \biggr]  \biggl[\frac{ d\theta}{\cosh\tau - \cos\theta} \biggr] 
\biggl[\frac{\sinh\tau ~d\varphi }{\cosh\tau - \cos\theta} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d\sigma \varpi d\varphi \, , </math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{ a}{\cosh\tau - \cos\theta} \biggr]^2 d\tau d\theta \, , 
</math>
  </td>
</tr>
</table>
and, [[#Properties|as above]], 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\tau }{\cosh\tau - \cos\theta}  \, .
</math>
  </td>
</tr>
</table>
Hence, in agreement with the expression derived using the notation of MF53,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dV_{2D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi d\sigma \int_0^{2\pi} d\varphi  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \varpi d\sigma \, .</math>
  </td>
</tr>
</table>

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

Now, if we choose a toroidal coordinate system whose origin is located exactly as defined in our [[#Special_Case|special case, above]], the ''radial''-coordinate integration limits should be,
<div align="center">
<math>~\xi_1|_\mathrm{min} = \frac{\varpi_t}{r_t} </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~~\xi_1|_\mathrm{max} = \infty  \, ;</math>
</div>
and the integral over the "angular" toroidal coordinate will have the simple limits,
<div align="center">
<math>~\xi_2|_- = -1  </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~\xi_2|_+ = +1 \, .</math>
</div>
[<font color="red">COMMENT:</font> Actually, these limits will only capture integration over either the upper hemisphere (<math>~Z</math> positive) or the lower hemisphere (<math>~Z</math> negative).  So I will probably need to double the volume expression that results from these limits.]


So, integration over the remaining two (meridional-plane) coordinates gives,
<div align="center">
<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>~2\pi \int_\Sigma \varpi d\sigma </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\int\limits_{-1}^{1} \frac{(\xi_1^2 - 1)^{1/2}}{(\xi_1-\xi_2)} \biggl[ \frac{1}{ (\xi_1-\xi_2)\sqrt{\xi_1^2-1} } \biggr]~ 
\biggl[\frac{1}{ (\xi_1-\xi_2)\sqrt{1-\xi_2^2} } \biggr] d\xi_2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\int\limits_{-1}^{1}  \biggl[\frac{d\xi_2}{ (\xi_1-\xi_2)^3\sqrt{1-\xi_2^2} } \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
If, following MF53, we make the substitution, 
<div align="center">
<math>~\xi_2 ~\rightarrow \cos\zeta</math>
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; &nbsp; 
<math>~d\xi_2 ~\rightarrow -\sin\zeta d\zeta \, ,</math>
</div>
we can write,
<div align="center">
<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>~2\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\int\limits_{0}^{\pi}  \biggl[\frac{d\zeta}{ (\xi_1-\cos\zeta)^3 } \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\biggl\{
\frac{\sin\zeta [ 4\xi_1^2 - 3\xi_1 \cos\zeta - 1]}{(\xi_1^2-1)^2 (\xi_1 - \cos\zeta)^2}
- \frac{2(2\xi_1^2 + 1)\tanh^{-1}\biggl[\frac{(\xi_1 + 1)\tan(\zeta/2)}{\sqrt{1-\xi_1^2}}  \biggr]}{(1-\xi_1^2)^{5/2}}
\biggr\}_{0}^{\pi} \, ,
</math>
  </td>
</tr>

<!-- COMMENT OUT MANY LINES ...
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\biggl\{
\frac{\sin\zeta [ 4\xi_1^2 - 3\xi_1 \cos\zeta - 1]}{(\xi_1^2-1)^2 (\xi_1 - \cos\zeta)^2}
\biggr\}_{0}^{\pi/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1 \biggl[ \frac{ 4\xi_1^2 - 1}{(\xi_1^2-1)^2 \xi_1^2 } \biggr]
=
2\pi a^3 \int\limits_{\cosh^{-1}(\varpi_t/r_t)}^\infty dx \biggl[ \frac{ 4\cosh^2x - 1}{\sinh^3 x \cosh^2 x } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\pi a^3 \biggl\{
-3\sinh^{-2}(x/2) - 3 \cosh^{-2}(x/2) + \frac{8}{\cosh(x)} - 4\ln[\sinh(x/2)] + 4\ln[\cosh(x/2)]
\biggr\}_{\cosh^{-1}(\varpi_t/r_t)}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\pi a^3 \biggl\{
-3\biggl[\frac{1}{\sinh^{2}(x/2)} + \frac{1}{\cosh^{2}(x/2)}\biggr] + \frac{8}{\cosh(x)} + \ln[\coth^4(x/2)]
\biggr\}_{\cosh^{-1}(\varpi_t/r_t)}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\pi a^3 \biggl\{
-3\biggl[\frac{2}{(\cosh x - 1)} + \frac{2}{(\cosh x + 1)}\biggr] + \frac{8}{\cosh(x)} + \ln\biggl[\frac{\cosh x + 1}{\cosh x - 1} \biggr]^2
\biggr\}_{\cosh^{-1}(\varpi_t/r_t)}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\pi a^3 \biggl\{
-3\biggl[\frac{1}{(\xi_1 - 1)} + \frac{1}{(\xi_1 + 1)}\biggr] + \frac{4}{\xi_1} + \ln\biggl[\frac{1 + 1/\xi_1}{1 - 1/\xi_1} \biggr]
\biggr\}_{\varpi_t/r_t}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\pi a^3 \biggl\{
-3\biggl[\frac{2\xi_1}{(\xi_1^2 - 1)} \biggr] + \frac{4}{\xi_1} + \ln\biggl[\frac{1 + 1/\xi_1}{1 - 1/\xi_1} \biggr]
\biggr\}_{\varpi_t/r_t}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \biggl\{
\frac{-(2+\xi_1^2)}{\xi_1(\xi_1^2 - 1)}  + \ln\biggl[\frac{1 + 1/\xi_1}{1 - 1/\xi_1} \biggr]^{1/2}
\biggr\}_{\varpi_t/r_t}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \biggl\{
\frac{2+(\varpi_t/r_t)^2}{\varpi_t/r_t[(\varpi_t/r_t)^2 - 1]}  - \ln\biggl[\frac{\varpi_t + r_t}{\varpi_t - r_t} \biggr]^{1/2}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi \biggl\{ r_t \biggl[\biggl( \frac{\varpi_t}{r_t} \biggr)^2 - 1  \biggr]^{1/2} \biggr\}^3 \biggl\{
\frac{r_t(2r_t^2+\varpi_t^2)}{\varpi_t(\varpi_t^2 - r_t^2)}  - \ln\biggl[\frac{\varpi_t + r_t}{\varpi_t - r_t} \biggr]^{1/2}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi (\varpi_t^2 - r_t^2)^{3/2} \biggl\{
\frac{r_t(2r_t^2+\varpi_t^2)}{\varpi_t(\varpi_t^2 - r_t^2)}  - \ln\biggl[\frac{\varpi_t + r_t}{\varpi_t - r_t} \biggr]^{1/2}
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

At first glance, this does not appear to give the correct expression for the volume of a torus.

END COMMENTED-OUT    -->
</table>
</div>
where the expression obtained after integrating over <math>~\zeta</math> was obtained from WolframAlpha's online integrator.  

[<font color="red">COMMENT:</font> This result is problematic because it was derived without enforcing the condition, <math>~\xi_1^2 > 1</math>.  Notice, in particular, that the last term includes a couple of square-roots of expressions that will naturally be negative.]

Carrying out this same integration (specifying wider integration limits based on the toroidal-coordinate specification [https://en.wikipedia.org/wiki/Toroidal_coordinates#Definition described in Wikipedia]) via multiple steps using the integral tables published by [http://www.mathtable.com/gr/ Gradshteyn &amp; Ryzhik], I have obtained,
<div align="center">
<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>~2\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\int\limits_{-\pi}^{\pi}  \biggl[\frac{d\zeta}{ (\xi_1-\cos\zeta)^3 } \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\biggl\{
\frac{\sin\zeta [ 4\xi_1^2 - 3\xi_1 \cos\zeta - 1]}{(\xi_1^2-1)^2 (\xi_1 - \cos\zeta)^2}
+ \biggl[ \frac{2(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \tan^{-1}\biggl[\tan\biggl(\frac{\zeta}{2}\biggr) \cdot \frac{(\xi_1 + 1)^{1/2}}{(\xi_1 - 1)^{1/2}}  \biggr]
\biggr\}_{-\pi}^{\pi} \, .
</math>
  </td>
</tr>

</table>
</div>

This expression makes more sense; at least the arguments of the square-roots are all positive.  Now, evaluating the limits:  (1) The first term inside the curly braces goes to zero at both limits; and (2) the argument of the arctangent is <math>~\pm \infty.</math>  Hence, the result of taking the arctangent is <math>~+\tfrac{\pi}{2}</math>, at the upper limit, and is <math>~-\tfrac{\pi}{2}</math> at the lower limit.  Hence, we have,
<div align="center">
<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>~\pi a^3 \int\limits_{\varpi_t/r_t}^\infty d\xi_1
\biggl\{\biggl[ \frac{2(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \biggl(\frac{\pi}{2} + \frac{\pi}{2}\biggr)  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 a^3 \int\limits_{\varpi_t/r_t}^\infty 
\biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] d\xi_1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 a^3 \biggl[ - \frac{\xi_1}{(\xi_1^2 - 1)^{3/2}} \biggr]_{\varpi_t/r_t}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 a^3 \biggl[ \frac{\varpi_t r_t^2}{(\varpi_t^2 - r_t)^{3/2}} \biggr]\, .
</math>
  </td>
</tr>
</table>
</div>

Now, in the special case we are considering here,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_t \biggl[\biggl( \frac{\varpi_t}{r_t} \biggr)^2 - 1  \biggr]^{1/2} 
= [\varpi_t^2 - r_t^2]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<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>~2\pi^2 \varpi_t r_t^2 \, ,
</math>
  </td>
</tr>
</table>
</div>
which is the answer we were expecting for the volume of the (pink) torus.

<!--  COMMENT OUT
There are some differences between the last term obtained via the printed table of integrals versus via WolframAlpha's online integrator.  Eventually, I would like to reconcile these differences.  In the meantime, the integration limits appear to be a problem because (good news, perhaps) the first term involving <math>~\sin\zeta</math> goes to zero no matter how you slice it; but (bad news), the tangent function in the last term goes to <math>~\pm \infty</math>.  So I'm not sure, yet, how to assess this result.


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}\pi a^3 \biggl\{
\frac{2(\xi_1^2 + 2)}{\xi_1(1-\xi_1^2)} + \ln\biggl[ \frac{1+\xi_1}{1-\xi_1} \biggr]
\biggr\}_{\varpi_t/r_t}^\infty \, .
</math>
  </td>
</tr>
At first glance, this doesn't work because at both limiting values, the argument of the logarithm is negative.  Maybe I need to change to cosh function before integrating.
-->
<!-- NO LONGER INTERESTED IN THE FOLLOWING...
Let's try switching the order of the integrations.
<div align="center">
<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>~2\pi a^3 
\int\limits_{-\pi}^{\pi} d\zeta \int\limits_{\varpi_t/r_t}^\infty \biggl[\frac{d\xi_1}{ (\xi_1-\cos\zeta)^3 } \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \pi a^3 \int\limits_{-\pi}^{\pi} d\zeta 
\biggl\{\frac{1}{(\xi_1-\cos\zeta)^2}
\biggr\}_{\varpi_t/r_t}^\infty
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{-\pi}^{\pi}  
\biggl[\frac{d\zeta}{(\varpi_t/r_t- \cos\zeta)^2}
\biggr]
</math>
  </td>
</tr>

</table>
</div>

===Another Simple Toroidal-System Alignment===
Okay, because the struggle with our previous volume integration had to do with the limits on integration, let's try moving the origin of the toroidal-coordinate system to a position that lies outside of the (pink) torus &#8212; let's choose, <math>~0 < a < \varpi_t</math>.  For simplicity, however, let's keep the origin in the equatorial plane, that is, set <math>~Z_0 = 0</math>.  Given that <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math>, from [[#Associated_Analytic_Expressions|our above derivations]] of the "radial-coordinate" limits of integration, we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2  - (\varpi_t^2 - r_t^2)  = a^2 - \tfrac{1}{2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
= -\frac{1}{2}\biggl(a^2 - \frac{1}{2} \biggr)\biggl[ \frac{\varpi_t \mp r_t }{(\varpi_t^2 - r_t^2)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{2} - a^2\biggr)\biggl[\frac{3}{4} \mp \frac{1}{4}  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Let's try, <math>~a = \tfrac{1}{2}</math>, in which case,
<div align="center">
<math>\beta_+ = \frac{1}{8}</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\beta_- = \frac{1}{4}\, ,</math>
</div>
and,
<div align="center">
<math>\xi_1|_\mathrm{max} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
= \biggl[1-\biggl( \frac{4}{5} \biggr)^2  \biggr]^{-1/2}
</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\xi_1|_\mathrm{min} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\, ,</math>
</div>

FINISHED COMMENTING-OUT LINES THAT ARE NO LONGER OF INTEREST -->

===Move General Case===
<font color="red"><b>NOTE:</b></font>  A complete prescription of the toroidal-coordinate integration limits that are appropriate for a determination of the volume or the gravitational potential of a circular torus can be found in [[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits|an accompanying discussion]].

In the more general case, the expression for the volume integral should be the same; all we should have to do is incorporate the more general integration ''limits'' as specified in our above evaluation of the gravitational potential.  Hence, in the more general case we should have,
<div align="center">
<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>~\pi a^3 \int\limits_{\xi_1|_\mathrm{min}}^{\xi_1|_\mathrm{max}}  d\xi_1
\biggl\{
\frac{\sin\zeta [ 4\xi_1^2 - 3\xi_1 \cos\zeta - 1]}{(\xi_1^2-1)^2 (\xi_1 - \cos\zeta)^2}
+ \biggl[ \frac{2(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \tan^{-1}\biggl[\tan\biggl(\frac{\zeta}{2}\biggr) \cdot \frac{(\xi_1 + 1)^{1/2}}{(\xi_1 - 1)^{1/2}}  \biggr]
\biggr\}_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} \, .
</math>
  </td>
</tr>

</table>
</div>

Next, referencing various trigonometric relations, we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\biggl(\frac{\zeta}{2}\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{1-\cos\zeta}{1+\cos\zeta} \biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{1-\xi_2}{1+\xi_2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, in our expression for the torus volume, the argument of the arctangent may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\biggl(\frac{\zeta}{2}\biggr) \cdot \frac{(\xi_1 + 1)^{1/2}}{(\xi_1 - 1)^{1/2}}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{1-\xi_2}{1+\xi_2} \biggr]^{1/2} \cdot \frac{(\xi_1 + 1)^{1/2}}{(\xi_1 - 1)^{1/2}}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{(1-\xi_2)}{(1+\xi_2)} \cdot \frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{\xi_1+1-\xi_1\xi_2 - \xi_2}{\xi_1 - 1 +\xi_1\xi_2 - \xi_2}  \biggr]^{1/2}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{(\xi_1- \xi_2)-(\xi_1\xi_2-1) }{(\xi_1- \xi_2) +(\xi_1\xi_2- 1 ) }  \biggr]^{1/2}  </math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan\biggl(\frac{\alpha}{2}\biggr) \, , </math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\cos\alpha \equiv \frac{(\xi_1\xi_2- 1 )}{(\xi_1- \xi_2)} </math>
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; 
<math>~\alpha \equiv \cos^{-1}\biggl[ \frac{(\xi_1\xi_2- 1 )}{(\xi_1- \xi_2)} \biggr]
= \cos^{-1}\biggl[ \frac{(\xi_1\cos\zeta - 1 )}{(\xi_1- \cos\zeta)} \biggr] \, .</math>
</div>
Hence, the volume integral may be written as,
<div align="center" id="GeneralVolumeIntegration">
<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>~\pi a^3 \int\limits_{\xi_1|_\mathrm{min}}^{\xi_1|_\mathrm{max}}  d\xi_1
\biggl\{
\frac{\sin\zeta [ 4\xi_1^2 - 3\xi_1 \cos\zeta - 1]}{(\xi_1^2-1)^2 (\xi_1 - \cos\zeta)^2}
+ \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[ \frac{(\xi_1\cos\zeta - 1 )}{(\xi_1- \cos\zeta)} \biggr] 
\biggr\}_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{\xi_1|_\mathrm{min}}^{\xi_1|_\mathrm{max}}  d\xi_1
\biggl\{
\frac{(1-\xi_2^2)^{1/2} [ 4\xi_1^2 - 3\xi_1 \xi_2 - 1]}{(\xi_1^2-1)^2 (\xi_1 - \xi_2)^2}
+ \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[ \frac{(\xi_1\xi_2 - 1 )}{(\xi_1- \xi_2)} \biggr] 
\biggr\}_{\xi_2|_-}^{\xi_2|_+} \, .
</math>
  </td>
</tr>

</table>
</div>

====Green Cropped-Top Volume====
Now, if we set <math>~Z_0 = h</math> with <math>~0 < h < r_t</math>, then the horizontal plane defined by <math>~z = Z_0</math> will cut through the circular torus, splitting it into two hemispheres &#8212; a lower, pink sub-volume and an upper, green sub-volume [[#Volume_with_Cropped_Top|as depicted in  the above diagram]].  While using toroidal coordinates to perform the volume integral, we recognize that this horizontal plane is also identified by setting the angular coordinate to <math>~\xi_2 = +1</math> [if <math>~a \leq (\varpi_t-b)</math>] or to <math>~\xi_2 = -1</math> [if <math>~a \geq (\varpi_t+b)</math>].   Then, using the former case as an example, for each value of <math>~\xi_1</math> (corresponding to a specific <math>\xi_1</math>- circle) the integral over the angular, <math>~\xi_2</math> coordinate should naturally break into two segments:  The segment falling within the green sub-volume should have integration limits, <math>~\xi_2|_+ \rightarrow 1</math>; and the segment falling within the pink sub-volume should have integration limits, <math>~1 \rightarrow \xi_2|_-</math>.  Let's see if specification of these limits allows us to derive an analytic expression for the green sub-volume that matches the expression for <math>~V_\mathrm{green}</math> [[#Volume_with_Cropped_Top|as derived above]] using cylindrical coordinates.


Notice that, for the green sub-volume, the limits on <math>~\xi_1</math> should correspond to <math>~\xi_2 = +1</math> and <math>~\varpi = (\varpi_t \pm b)</math>.  Because, in general,
<div align="center">
<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(\xi_1^2 -1)^{1/2}}{\xi_1-\xi_2} \, ,</math>
  </td>
</tr>
</table>
</div>
this means that the limits on <math>~\xi_1</math> are (valid only for <math>0 < Z_0 < r_t</math>),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi_t \pm b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a(\xi_1^2 -1)^{1/2}}{\xi_1-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \varpi_t \pm \sqrt{r_t^2 - Z_0^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a(\xi_1 +1)^{1/2}(\xi_1 -1)^{1/2}}{\xi_1-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a \biggl[\frac{\xi_1 +1}{\xi_1-1}\biggr]^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ (\xi_1 - 1)\biggl[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}\biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2(\xi_1 +1) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \xi_1\biggl\{\biggl[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}\biggr]^2-a^2\biggr\} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}\biggr]^2 + a^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \xi_1\biggr|_\pm  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}]^2-a^2} \, .</math>
  </td>
</tr>
</table>
</div>

Hence, according to our [[#GeneralVolumeIntegration|just-derived volume integral]], we have,

<div align="center" id="GreenAnalytic">
<table border="1" cellpadding="8"><tr><th align="center" colspan="2">
Toroidal-Coordinate Integral Expression for Green Volume</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~V_\mathrm{green}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{\xi_1|_-}^{\xi_1|_+}  d\xi_1
\biggl\{
\frac{(1-\xi_2^2)^{1/2} [ 4\xi_1^2 - 3\xi_1 \xi_2 - 1]}{(\xi_1^2-1)^2 (\xi_1 - \xi_2)^2}
+ \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[ \frac{(\xi_1\xi_2 - 1 )}{(\xi_1- \xi_2)} \biggr] 
\biggr\}_{1}^{\xi_2|_+} \, .
</math>
  </td>
</tr>

</table>

</td>
<td align="center">
[[File:CropTopB.png|150px|right|Diagram of "Cropped Top" Torus]]
</td></tr>
</table>
</div>

Note from J. E. Tohline:  On 4 November 2015, a fortran subroutine was used to numerically perform this 1D integration and thereby determine the volume of the green segment of a "cropped top" torus.  The name of this fortran code was &hellip;
<pre>philip.hpc.lsu.edu:/home/tohline/fortran/Toroidal/testI3.for</pre>
The following table presents results of tests run with different sets of physical parameters and different numbers of 1D integration steps (nzones); note that the analytic expression for the ''angular'' integration limit, <math>~\xi_2|_+</math>, is given in the [[#Parameters|above table of parameter expressions]].  In the following table, values of <math>~V_\mathrm{green}</math> obtained by numerical integration are compared with values obtained from the [[#GreenAnalytic|analytic expression derived above via a cylindrical-coordinate formulation]].

<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="11">
Comparison of "Cropped-Top" Volume Determinations
</th></tr>
<tr>
  <td align="center" rowspan="3"><math>~\varpi_t</math></td>
  <td align="center" rowspan="3"><math>~r_t</math></td>
  <td align="center" rowspan="3"><math>~a</math></td>
  <td align="center" rowspan="3"><math>~Z_0</math></td>
  <td align="center" colspan="5"><math>~V_\mathrm{green}/V_\mathrm{torus}</math></td>
</tr>
<tr>
  <td align="center" rowspan="2">Analytic</td>
  <td align="center" colspan="2">nzones = 5000</td>
  <td align="center" colspan="2">nzones = 500</td>
</tr>
<tr>
  <td align="center">1D Integration</td>
  <td align="center">Error</td>
  <td align="center">1D Integration</td>
  <td align="center">Error</td>
</tr>
<tr>
  <td align="center"><math>\tfrac{3}{4}</math></td>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="center">0.24</td>
  <td align="center">4.7727731D-03</td>
  <td align="center">4.7727731D-03</td>
  <td align="center">-1.66D-08</td>
  <td align="center">4.7727865D-03</td>
  <td align="center">-2.8D-06</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.23</td>
  <td align="center">1.3417064D-02</td>
  <td align="center">1.3417065D-02</td>
  <td align="center">-2.6D-08</td>
  <td align="center">1.3417115D-02</td>
  <td align="center">-3.8D-06</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.20</td>
  <td align="center">5.2044018D-02</td>
  <td align="center">5.2044021D-02</td>
  <td align="center">-6.3D-08</td>
  <td align="center">5.2044404D-02</td>
  <td align="center">-7.4D-06</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.15</td>
  <td align="center">1.4237849D-01</td>
  <td align="center">1.4237851D-01</td>
  <td align="center">-1.6D-07</td>
  <td align="center">1.4238095D-01</td>
  <td align="center">-1.7D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.125</td>
  <td align="center">1.9550110D-01</td>
  <td align="center">1.9550115D-01</td>
  <td align="center">-2.4D-07</td>
  <td align="center">1.9550595D-01</td>
  <td align="center">-2.5D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.10</td>
  <td align="center">2.5231578D-01</td>
  <td align="center">2.5231587D-01</td>
  <td align="center">-3.4D-07</td>
  <td align="center">2.5232459D-01</td>
  <td align="center">-3.5D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.05</td>
  <td align="center">3.7353003D-01</td>
  <td align="center">3.7353031D-01</td>
  <td align="center">-7.4D-07</td>
  <td align="center">3.7355809D-01</td>
  <td align="center">-7.5D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.01</td>
  <td align="center">4.7454199D-01</td>
  <td align="center">4.7454347D-01</td>
  <td align="center">-3.3D-06</td>
  <td align="center">4.7465374D-01</td>
  <td align="center">-2.4D-04</td>
</tr>
</table>

====Total Volume by Summing Four Zones====
Let's stick with a discussion of the situation where we set <math>~Z_0 = h</math> with <math>~0 < h < r_t</math>, and now determine the total volume by adding together four sub-volumes.  The green "cropped-top" region is the first of these sub-volume zones.  The remaining (pink) portion of the torus can be broken into three adjoining segments &#8212; left-to-right &#8212; whose two edge boundaries plus two internal interfaces are identified by the following four special values of the "radial" coordinate:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_1|_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\xi_1\biggr|_+  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t +\sqrt{r_t^2 - Z_0^2}]^2-a^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_1\biggr|_-  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_1|_\mathrm{min}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}\, .</math>
  </td>
</tr>
</table>
</div>
As presented in the following table, we have determined the values of these four boundary/interface radial coordinates for the same set of parameter values as in the previous table.  Notice that, as listed, they represent monotonically increasing values of the radial coordinate.  Between the first two boundaries and between the last two boundaries, the limits on the "angular" coordinate should be the normal, &nbsp;<math>~~~\xi_2|_-~\rightarrow ~\xi_2|_+ ~~~</math>.  But between the second and third boundaries, the integration limits on the angular coordinate should be, &nbsp;<math>~\xi_2|_- ~\rightarrow~ +1</math>.

<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="10">
Various Boundary Values for <math>~\xi_1</math>
</th></tr>
<tr>
  <td align="center" rowspan="1"><math>~\varpi_t</math></td>
  <td align="center" rowspan="1"><math>~r_t</math></td>
  <td align="center" rowspan="1"><math>~a</math></td>
  <td align="center" rowspan="1"><math>~Z_0</math></td>
  <td align="center"><math>~\xi_1|_\mathrm{max}</math></td>
  <td align="center"><math>~\xi_1|_+</math></td>
  <td align="center"><math>~\xi_1|_-</math></td>
  <td align="center"><math>~\xi_1|_\mathrm{min}</math></td>
</tr>
<tr>
  <td align="center"><math>\tfrac{3}{4}</math></td>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="center">0.24</td>
  <td align="center">1.1859</td>
  <td align="center">1.3959</td>
  <td align="center">1.6326</td>
  <td align="center">2.0312</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.23</td>
  <td align="center">1.1900</td>
  <td align="center">1.3655</td>
  <td align="center">1.7077</td>
  <td align="center">2.0630</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.20</td>
  <td align="center">1.2021</td>
  <td align="center">1.3180</td>
  <td align="center">1.8929</td>
  <td align="center">2.1603</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.15</td>
  <td align="center">1.2209</td>
  <td align="center">1.2808</td>
  <td align="center">2.1611</td>
  <td align="center">2.3213</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.125</td>
  <td align="center">1.2291</td>
  <td align="center">1.2700</td>
  <td align="center">2.2808</td>
  <td align="center">2.3963</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.10</td>
  <td align="center">1.2363</td>
  <td align="center">1.2622</td>
  <td align="center">2.3872</td>
  <td align="center">2.4637</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.05</td>
  <td align="center">1.2464</td>
  <td align="center">1.2529</td>
  <td align="center">2.5436</td>
  <td align="center">2.5638</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.01</td>
  <td align="center">1.2499</td>
  <td align="center">1.2501</td>
  <td align="center">2.5977</td>
  <td align="center">2.5985</td>
</tr>
</table>