Difference between revisions of "User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits"

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(→‎Preamble: Introduce schematic illustrations of Zone I and Zone II)
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   <th align="center" colspan="2"><font size="+1">Schematic Zones</font></th>
   <th align="center" colspan="3"><font size="+1">Schematic Zones</font></th>
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   <th align="center" colspan="1"><font size="+1">Zone I</font><p></p>
   <th align="center" colspan="1"><font size="+1">Zone I</font><p></p>
<math>~Z_0 > r_t</math>&nbsp; for any <math>~a</math></th>
<math>~Z_0 > r_t</math>&nbsp; for any <math>~a</math></th>
  <th align="center" colspan="1"><font size="+1">Zone II</font><p></p>
<math>~r_t > Z_0 > 0</math>&nbsp; and &nbsp; <math>~a < (\varpi_t-r_t)</math></th>
   <th align="center" colspan="1"><font size="+1">Zone II</font><p></p>
   <th align="center" colspan="1"><font size="+1">Zone II</font><p></p>
<math>~r_t > Z_0 > 0</math>&nbsp; and &nbsp; <math>~a < (\varpi_t-r_t)</math></th>
<math>~r_t > Z_0 > 0</math>&nbsp; and &nbsp; <math>~a < (\varpi_t-r_t)</math></th>
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[[File:Apollonian_myway7B.png|300px|Apollonian Circles]]
[[File:Apollonian_myway7B.png|300px|Apollonian Circles]]
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[[File:Apollonian_myway8B.png|300px|Apollonian Circles]]
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Revision as of 00:31, 10 November 2015

Toroidal-Coordinate Integration Limits

In support of our accompanying discussion of the gravitational potential of a uniform-density circular torus, here we explain in detail what limits of integration must be specified in order to accurately determine the volume — and, hence also the total mass — of such a torus using toroidal coordinates.


Whitworth's (1981) Isothermal Free-Energy Surface
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Preamble

Apollonian Circles (schematic)

(see also Wikipedia's Apollonian Circles)

Apollonian Circles

Apollonian Circles

Quantitative Illustration of Employed Toroidal Coordinate System

Diagram of Torus and Toroidal Coordinates

Diagram of Torus and xi_2-constant Toroidal Coordinate curve


Schematic Zones
Zone I

<math>~Z_0 > r_t</math>  for any <math>~a</math>
Zone II

<math>~r_t > Z_0 > 0</math>  and   <math>~a < (\varpi_t-r_t)</math>
Zone II

<math>~r_t > Z_0 > 0</math>  and   <math>~a < (\varpi_t-r_t)</math>

Apollonian Circles

Apollonian Circles

Apollonian Circles

See Also

 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation