Difference between revisions of "User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits"
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[[File:ConstantXi2.png|300px|Diagram of Torus and xi_2-constant Toroidal Coordinate curve]] | [[File:ConstantXi2.png|300px|Diagram of Torus and xi_2-constant Toroidal Coordinate curve]] | ||
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<th align="center" colspan="2"><font size="+1">Schematic Zones</font></th> | |||
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<th align="center" colspan="1"><font size="+1">Zone I</font><p></p> | |||
<math>~Z_0 > r_t</math> 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> and <math>~a < (\varpi_t-r_t)</math></th> | |||
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[[File:Apollonian_myway5B.png|300px|Apollonian Circles]] | |||
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[[File:Apollonian_myway7B.png|300px|Apollonian Circles]] | |||
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Revision as of 00:18, 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.
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Preamble
| Apollonian Circles (schematic) (see also Wikipedia's Apollonian Circles) | |
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| Quantitative Illustration of Employed Toroidal Coordinate System | |
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| Schematic Zones | |
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| 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> |
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See Also
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© 2014 - 2021 by Joel E. Tohline |