Editing 2DStructure/ToroidalCoordinates (section)

====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!