User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits - Revision history

Tohline: /* Coming Up to Speed */

2017-10-01T19:14:40Z

Coming Up to Speed

← Older revision		Revision as of 19:14, 1 October 2017
Line 2,926:		Line 2,926:
	I am returning to this problem after letting it lay dormant for more than 14 months. These notes are being recorded as I try to decipher the fortran code that I have previously developed. Here is a screenshot from philip.hpc.lsu.edu that will help me archive the dates various fortran routines were last edited.		I am returning to this problem after letting it lay dormant for more than 14 months. These notes are being recorded as I try to decipher the fortran code that I have previously developed. Here is a screenshot from philip.hpc.lsu.edu that will help me archive the dates various fortran routines were last edited.
	<div align="center">		<div align="center">
	[[File:PhilipScreenshot01.png\|400px\|Screenshot of philip.hpc.lsu.edu]][[File:PlayWithPieces.png\|350px\|Early playing with ZoneII pieces]]		[[File:PhilipScreenshot01.png\|400px\|Screenshot of philip.hpc.lsu.edu]][[File:WarpedSurfaceZ0.png\|400px\|Added Z0 = 0 seam to warped surface]]
			<!--[[File:PlayWithPieces.png\|350px\|Early playing with ZoneII pieces]]-->
	</div>		</div>
	Let's start by reviewing the pair of routines (''mainI_03.for'',''volumeI_03.for'') which, presumably, integrate over the region labeled, [[#Zone_I:\|Zone I]], above.		Let's start by reviewing the pair of routines (''mainI_03.for'',''volumeI_03.for'') which, presumably, integrate over the region labeled, [[#Zone_I:\|Zone I]], above.

Tohline: /* Integration Limits */

2017-09-30T19:08:42Z

Integration Limits

← Older revision		Revision as of 19:08, 30 September 2017
Line 1,623:		Line 1,623:
	</table>		</table>
	Excellent! This matches the earlier supposition.		Excellent! This matches the earlier supposition.

			Also, given that <math>~C = 1</math> and <math>~A = B^2</math>, the pair of values, <math>~(\varpi_i/a)_\pm</math>, is degenerate and given by the expression,

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

			<tr>
			<td align="right">
			<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\kappa}{2a^2 \mathrm{B}}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2a^2} \biggl[ (\varpi_t^2 - r_t^2) -a^2\biggr] \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^{-1} \, .
			</math>
			</td>
			</tr>
			</table>

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

			<tr>
			<td align="right">
			<math>~\xi_2\biggr\|_+</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\xi_1\biggl\{1 - \biggl[\biggl( \frac{\varpi_t}{a} \biggr)\frac{ (\xi_1^2-1)^{1 / 2}}{\xi_1} - 1 \biggr]
			2a^2 \biggl[ (\varpi_t^2 - r_t^2) -a^2\biggr]^{-1} \biggr\}
			</math>
			</td>
			</tr>
			</table>


	===Zone III===		===Zone III===

Tohline: /* Special Case of Z0 = 0 */

2017-09-29T23:32:53Z

Special Case of Z0 = 0

← Older revision		Revision as of 23:32, 29 September 2017
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	<td align="right">		<td align="right">
	<math>		<math>
	~\xi_1\|_\mathrm{~~max~~}		~\xi_1\|_\mathrm{min}
	</math>		</math>
	</td>		</td>
Line 1,347:		Line 1,347:
	<td align="right">		<td align="right">
	<math>		<math>
	~\xi_1\|_\mathrm{~~min~~}		~\xi_1\|_\mathrm{max}
	</math>		</math>
	</td>		</td>
Line 1,506:		Line 1,506:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \frac{1}{2}		<math>~ \frac{a}{2}
	\biggl\{\frac{a^2}{(\varpi_t \pm ~~r_t~~ )} + (\~~varpi_t~~ \pm ~~r_t~~) \~~biggr~~\} \, .		\biggl[\frac{a}{(\varpi_t \pm r_t )} + \frac{(\varpi_t \pm r_t)}{a} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \biggl( \frac{\varpi_t - \beta_\pm}{a}\biggr)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl(\frac{\Upsilon_\pm^2 + a^2}{2a\Upsilon_\pm} \biggr) \, ,
			</math>
			</td>
			</tr>
			</table>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Upsilon_\pm</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\varpi_t \pm r_t \, .</math>
			</td>
			</tr>
			</table>
			</div>
			This also means that,

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

			<tr>
			<td align="right">
			<math>~1 - \biggl( \frac{a}{\varpi_t - \beta_\pm}\biggr)^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 1 -
			\biggl(\frac{2a\Upsilon_\pm}{\Upsilon_\pm^2 + a^2} \biggr)^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{(\Upsilon_\pm^2 + a^2)^2 - 4a^2\Upsilon_\pm^2}{(\Upsilon_\pm^2 + a^2)^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{\Upsilon_\pm^2 - a^2}{\Upsilon_\pm^2 + a^2} \biggr]^2 \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			</div>
			That is,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl[ \xi_1\|_\mathrm{max}\biggr]^{-2} = \biggl[ \frac{\Upsilon_+^2 - a^2}{\Upsilon_+^2 + a^2} \biggr]^2</math>
			</td>
			<td align="center">
			<math>~\Rightarrow</math>
			</td>
			<td align="left">
			<math>~\xi_1\|_\mathrm{max} =
			\frac{\Upsilon_+^2 + a^2}{\Upsilon_+^2 - a^2} \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\biggl[ \xi_1\|_\mathrm{min} \biggr]^{-2} = \biggl[ \frac{\Upsilon_-^2 - a^2}{\Upsilon_-^2 + a^2} \biggr]^2 </math>
			</td>
			<td align="center">
			<math>~\Rightarrow</math>
			</td>
			<td align="left">
			<math>~\xi_1\|_\mathrm{min} =
			\frac{\Upsilon_-^2 + a^2}{\Upsilon_-^2 - a^2} \, .
			</math>
			</td>
			</tr>
			</table>
			Excellent! This matches the earlier supposition.

	===Zone III===		===Zone III===

Tohline: /* Integration Limits */

2017-09-29T21:18:20Z

Integration Limits

Show changes

Tohline: /* Coming Up to Speed */

2017-09-27T00:33:16Z

Coming Up to Speed

← Older revision		Revision as of 00:33, 27 September 2017
Line 2,476:		Line 2,476:
	I am returning to this problem after letting it lay dormant for more than 14 months. These notes are being recorded as I try to decipher the fortran code that I have previously developed. Here is a screenshot from philip.hpc.lsu.edu that will help me archive the dates various fortran routines were last edited.		I am returning to this problem after letting it lay dormant for more than 14 months. These notes are being recorded as I try to decipher the fortran code that I have previously developed. Here is a screenshot from philip.hpc.lsu.edu that will help me archive the dates various fortran routines were last edited.
	<div align="center">		<div align="center">
	[[File:PhilipScreenshot01.png\|~~500px~~\|Screenshot of philip.hpc.lsu.edu]]		[[File:PhilipScreenshot01.png\|400px\|Screenshot of philip.hpc.lsu.edu]][[File:PlayWithPieces.png\|350px\|Early playing with ZoneII pieces]]
	</div>		</div>
	Let's start by reviewing the pair of routines (''mainI_03.for'',''volumeI_03.for'') which, presumably, integrate over the region labeled, [[#Zone_I:\|Zone I]], above.		Let's start by reviewing the pair of routines (''mainI_03.for'',''volumeI_03.for'') which, presumably, integrate over the region labeled, [[#Zone_I:\|Zone I]], above.

Tohline: /* Evaluation of Elliptic Integrals */

2017-09-12T00:16:38Z

Evaluation of Elliptic Integrals

← Older revision		Revision as of 00:16, 12 September 2017
Line 383:		Line 383:
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">90°<br /><math>~K(k)</math></td>		<td align="center"><table border="0" cellpadding="3"><tr><td align="center" rowspan="2">(90°)</td><td align="center"><math>~K(k)</math></td></tr><tr><td align="center"><math>~E(k)</math></td></tr></table></td>
	<td align="right">1.6200<br />1.5238</td>		<td align="right">1.6200<br />1.5238</td>
	<td align="right">1.7312<br />1.4323</td>		<td align="right">1.7312<br />1.4323</td>
Line 389:		Line 389:
	<td align="right">2.3088<br />1.1638</td>		<td align="right">2.3088<br />1.1638</td>
	<td align="right">3.1534<br />1.0401</td>		<td align="right">3.1534<br />1.0401</td>
	<td align="center">π/2<br /><math>~K(k)</math></td>		<td align="center"><table border="0" cellpadding="3"><tr><td align="center" rowspan="2">(π/2)</td><td align="center"><math>~K(k)</math></td></tr><tr><td align="center"><math>~E(k)</math></td></tr></table></td>
	<td align="right">1.62002589<br />1.52379921</td>		<td align="right">1.62002589<br />1.52379921</td>
	<td align="right">1.73124518<br />1.43229097</td>		<td align="right">1.73124518<br />1.43229097</td>

Tohline: /* Evaluation of Elliptic Integrals */

2017-09-12T00:10:59Z

Evaluation of Elliptic Integrals

← Older revision		Revision as of 00:10, 12 September 2017
Line 292:		Line 292:
	<td align="center" colspan="6">CRC's    <math>~F(k,\phi)</math> (top);  <math>~E(k,\phi)</math> (bottom)</td>		<td align="center" colspan="6">CRC's    <math>~F(k,\phi)</math> (top);  <math>~E(k,\phi)</math> (bottom)</td>
	<td align="center" colspan="1" rowspan="9" bgcolor="#DDDDDD"> </td>		<td align="center" colspan="1" rowspan="9" bgcolor="#DDDDDD"> </td>
	<td align="center" colspan="6">Numerical Recipe's    <math>~F(k,\phi)</math> (top); <math>~E(k,\phi)</math> (bottom)</td>		<td align="center" colspan="6">Numerical Recipe's    <math>~F(\phi,k)</math> (top); <math>~E(\phi,k)</math> (bottom)</td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Evaluation of Elliptic Integrals */

2017-09-12T00:09:34Z

Evaluation of Elliptic Integrals

← Older revision		Revision as of 00:09, 12 September 2017
Line 314:		Line 314:
	<tr>		<tr>
	<td align="center">11°</td>		<td align="center">11°</td>
	<td align="right">0.1921</td>		<td align="right">0.1921<br />0.1918</td>
	<td align="right">0.1924</td>		<td align="right">0.1924<br />0.1916</td>
	<td align="right">0.1927</td>		<td align="right">0.1927<br />0.1913</td>
	<td align="right">0.1930</td>		<td align="right">0.1930<br />0.1910</td>
	<td align="right">0.1931</td>		<td align="right">0.1931<br />0.1908</td>
	<td align="center">0.19198622</td>		<td align="center">0.19198622</td>
	<td align="right">0.192123431<br />0.191849180</td>		<td align="right">0.192123431<br />0.191849180</td>
Line 328:		Line 328:
	<tr>		<tr>
	<td align="center">23°</td>		<td align="center">23°</td>
	<td align="right">0.4027</td>		<td align="right">0.4027<br />0.4002</td>
	<td align="right">0.4049</td>		<td align="right">0.4049<br />0.3980</td>
	<td align="right">0.4078</td>		<td align="right">0.4078<br />0.3952</td>
	<td align="right">0.4105</td>		<td align="right">0.4105<br />0.3927</td>
	<td align="right">0.4123</td>		<td align="right">0.4123<br />0.3911</td>
	<td align="center">0.40142573</td>		<td align="center">0.40142573</td>
	<td align="right">0.402656868<br />0.400201278</td>		<td align="right">0.402656868<br />0.400201278</td>
Line 342:		Line 342:
	<tr>		<tr>
	<td align="center">35°</td>		<td align="center">35°</td>
	<td align="right">0.6151</td>		<td align="right">0.6151<br />0.6067</td>
	<td align="right">0.6231</td>		<td align="right">0.6231<br />0.5991</td>
	<td align="right">0.6336</td>		<td align="right">0.6336<br />0.5895</td>
	<td align="right">0.6441</td>		<td align="right">0.6441<br />0.5806</td>
	<td align="right">0.6513</td>		<td align="right">0.6513<br />0.5748</td>
	<td align="center">0.61086524</td>		<td align="center">0.61086524</td>
	<td align="right">0.615064063<br />0.606716517</td>		<td align="right">0.615064063<br />0.606716517</td>
Line 356:		Line 356:
	<tr>		<tr>
	<td align="center">56°</td>		<td align="center">56°</td>
	<td align="right">0.9930</td>		<td align="right">0.9930<br />0.9622</td>
	<td align="right">1.0250</td>		<td align="right">1.0250<br />0.9335</td>
	<td align="right">1.0725</td>		<td align="right">1.0725<br />0.8962</td>
	<td align="right">1.1285</td>		<td align="right">1.1285<br />0.8595</td>
	<td align="right">1.1743</td>		<td align="right">1.1743<br />0.8344</td>
	<td align="center">0.97738438</td>		<td align="center">0.97738438</td>
	<td align="right">0.993020463<br />0.962159483</td>		<td align="right">0.993020463<br />0.962159483</td>
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	<tr>		<tr>
	<td align="center">77°</td>		<td align="center">77°</td>
	<td align="right">1.3788</td>		<td align="right">1.3788<br />1.3104</td>
	<td align="right">1.4554</td>		<td align="right">1.4554<br />1.2457</td>
	<td align="right">1.5867</td>		<td align="right">1.5867<br />1.1580</td>
	<td align="right">1.7909</td>		<td align="right">1.7909<br />1.0643</td>
	<td align="right">2.0653</td>		<td align="right">2.0653<br />0.9917</td>
	<td align="center">1.34390352</td>		<td align="center">1.34390352</td>
	<td align="right">1.37884243<br />1.31035009</td>		<td align="right">1.37884243<br />1.31035009</td>
Line 384:		Line 384:
	<tr>		<tr>
	<td align="center">90°<br /><math>~K(k)</math></td>		<td align="center">90°<br /><math>~K(k)</math></td>
	<td align="right">1.6200</td>		<td align="right">1.6200<br />1.5238</td>
	<td align="right">1.7312</td>		<td align="right">1.7312<br />1.4323</td>
	<td align="right">1.9356</td>		<td align="right">1.9356<br />1.3055</td>
	<td align="right">2.3088</td>		<td align="right">2.3088<br />1.1638</td>
	<td align="right">3.1534</td>		<td align="right">3.1534<br />1.0401</td>
	<td align="center">π/2<br /><math>~K(k)</math></td>		<td align="center">π/2<br /><math>~K(k)</math></td>
	<td align="right">1.62002589<br />1.52379921</td>		<td align="right">1.62002589<br />1.52379921</td>

Tohline: /* Evaluation of Elliptic Integrals */

2017-09-11T23:52:04Z

Evaluation of Elliptic Integrals

← Older revision		Revision as of 23:52, 11 September 2017
Line 390:		Line 390:
	<td align="right">3.1534</td>		<td align="right">3.1534</td>
	<td align="center">π/2<br /><math>~K(k)</math></td>		<td align="center">π/2<br /><math>~K(k)</math></td>
	<td align="right">1.</td>		<td align="right">1.62002589<br />1.52379921</td>
	<td align="right">1.</td>		<td align="right">1.73124518<br />1.43229097</td>
	<td align="right">1.</td>		<td align="right">1.93558110<br />1.30553909</td>
	<td align="right">2.</td>		<td align="right">2.30878680<br />1.16382796</td>
	<td align="right">3.</td>		<td align="right">3.15338525<br />1.04011440</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Evaluation of Elliptic Integrals */

2017-09-11T23:47:40Z

Evaluation of Elliptic Integrals

← Older revision		Revision as of 23:47, 11 September 2017
Line 376:		Line 376:
	<td align="right">2.0653</td>		<td align="right">2.0653</td>
	<td align="center">1.34390352</td>		<td align="center">1.34390352</td>
	<td align="right">1.</td>		<td align="right">1.37884243<br />1.31035009</td>
	<td align="right">1.</td>		<td align="right">1.45540074<br />1.24566028</td>
	<td align="right">1.</td>		<td align="right">1.58672349<br />1.15795471</td>
	<td align="right">1.</td>		<td align="right">1.79094197<br />1.06431530</td>
	<td align="right">2.</td>		<td align="right">2.06528894<br />0.99170032</td>
	</tr>		</tr>
	<tr>		<tr>