User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates - Revision history

Tohline: /* Scale Factors, Direction Cosines & Unit Vectors */

2020-10-27T17:58:16Z

Scale Factors, Direction Cosines & Unit Vectors

← Older revision		Revision as of 17:58, 27 October 2020
Line 1,768:		Line 1,768:
	<td align="center"><math>~1</math></td>		<td align="center"><math>~1</math></td>
	<td align="center"> <br />		<td align="center"> <br />
	<math>~x\ell</math><br />		<math>~x\ell</math><br /> </td>
	<td align="center"><math>~q^2 y \ell</math>		<td align="center"><math>~q^2 y \ell</math></td>
	<td align="center"><math>~0</math>		<td align="center"><math>~0</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	<br />		<br />
	<math>~q^2 y \ell </math>		<math>~q^2 y \ell </math>
	<br />~~ ~~		<br /></td>
	<td align="center">		<td align="center">
	<math>~ - x\ell		<math>~ - x\ell
	</math>		</math></td>
	<td align="center"><math>~0</math>		<td align="center"><math>~0</math></td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~3</math></td>		<td align="center"><math>~3</math></td>
	<td align="center"> <br /><math>~0</math><br />		<td align="center"> <br /><math>~0</math><br /> </td>
	<td align="center"><math>~0</math>		<td align="center"><math>~0</math></td>
	<td align="center"><math>~1</math>		<td align="center"><math>~1</math></td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Trials */

2020-10-25T21:05:21Z

Trials

← Older revision		Revision as of 21:05, 25 October 2020
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	</tr>		</tr>
	</table>		</table>


			<font color="green">'''(25 October 2020) I give up … for now.'''</font>

	=See Also=		=See Also=

Tohline: /* Trials */

2020-10-24T20:26:56Z

Trials

← Older revision		Revision as of 20:26, 24 October 2020
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	</tr>		</tr>
	</table>		</table>
	~~Now, the LHS may be rewritten as~~,
			If <math>~\alpha = 4</math>,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~LHS~~ <math>~~~= 4~~\~~pi G~~\~~rho_c~~ \~~biggl[1 - \frac~~{\~~lambda_1^~~2}{a^2} \biggr] </math>		<math>~
			\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2} \biggl[ \beta \Lambda +1 \biggr]
			</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4\pi G\rho_c - \frac{~~4\pi G\rho_c~~\lambda_1^2}{a^2} </math>		<math>~
			4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] -8A\lambda_1^2
			- 32A\lambda_1^{2} \Lambda
			</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	&~~nbsp~~;		<math>~\Rightarrow ~~~
			8B\beta \lambda_2^{\beta-2} \biggl[ \beta \Lambda +1 \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl\{ 4\pi G\rho_c
			- 32A\lambda_1^{2} \Lambda
			- \lambda_1^2 \biggl[\frac{4\pi G \rho_c}{a^2} + 8A \biggr] \biggr\}
			(\Lambda^2 - 2\Lambda + 1) \, .
			</math>
			</td>
			</tr>
			</table>
			If, then, <math>~8Aa^2 = -4\pi G\rho_c</math>,

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

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			8B\beta \lambda_2^{\beta-2} \biggl[ \beta \Lambda +1 \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			4\pi G\rho_c\biggl\{ 1
			+ \biggl[ \frac{4\lambda_1^{2}}{a^2} \biggr] \Lambda
			\biggr\}
			(\Lambda^2 - 2\Lambda + 1) \, .
			</math>
			</td>
			</tr>
			</table>
			But, we also know that, <math>\lambda_1^2 = \lambda_2^2(\Lambda^2-1)/4</math>, so …

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

			<tr>
			<td align="right">
			<math>~
			8a^2 B\beta \lambda_2^{\beta-2} \biggl[ \beta \Lambda +1 \biggr]
			</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4\pi G\rho_c - \~~frac{4~~\~~pi G~~\~~rho_c~~\~~lambda_1~~^2~~}{a^~~2} </math>		<math>~
			4\pi G\rho_c \biggl\{ a^2
			+ \lambda_2^2 (\Lambda^2-1)\Lambda
			\biggr\}
			(\Lambda^2 - 2\Lambda + 1) \, .
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Trials */

2020-10-24T18:01:52Z

Trials

← Older revision		Revision as of 18:01, 24 October 2020
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	<td align="right">		<td align="right">
	LHS <math>~= 4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>		LHS <math>~= 4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~4\pi G\rho_c - \frac{4\pi G\rho_c\lambda_1^2}{a^2} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Trials */

2020-10-24T18:00:47Z

Trials

← Older revision		Revision as of 18:00, 24 October 2020
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	+		+
	\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2} \biggl\{ \beta \Lambda +1		\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2} \biggl\{ \beta \Lambda +1
	\biggr\}		\biggr\} \, .
	</math>		</math>
			</td>
			</tr>
			</table>
			Now, the LHS may be rewritten as,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			LHS <math>~= 4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~4\pi G\rho_c - \frac{4\pi G\rho_c\lambda_1^2}{a^2} </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Example (q2 = 2) Poisson Equation */

2020-10-24T17:26:36Z

Example (q2 = 2) Poisson Equation

← Older revision		Revision as of 17:26, 24 October 2020
Line 2,273:		Line 2,273:

	==Example (q<sup>2</sup> = 2) Poisson Equation==		==Example (q<sup>2</sup> = 2) Poisson Equation==

			===Setup===

	Let's see if we can solve the,		Let's see if we can solve the,
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	</tr>		</tr>
	</table>		</table>

			===Trials===

	Try,		Try,
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\lambda_1^2 \lambda_2^2 \~~, ,~~</math>		<math>~A \lambda_1^\alpha + B\lambda_2^\beta </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ \frac{\partial \Phi}{\partial \lambda_1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~A\alpha \lambda_1^{\alpha-1} \, ,</math>
			</td>
			<td align="center">      and,       </td>
			<td align="right">
			<math>~ \frac{\partial \Phi}{\partial \lambda_2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~B\beta \lambda_2^{\beta - 1} \, .</math>
			</td>
			</tr>
			</table>
			In this case we find,

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

			<tr>
			<td align="right">
			<math>~4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2}{\lambda_1} \biggl\{ \Lambda \lambda_1 \frac{\partial}{\partial \lambda_1}\biggl[ \frac{\partial \Phi}{\partial \lambda_1}\biggr]
			+
			(\Lambda + 1) \biggl[ \frac{\partial \Phi}{\partial \lambda_1} \biggr] \biggr\}
			+
			\frac{8}{\lambda_2 (\Lambda-1)^2}\biggl\{
			\Lambda \lambda_2 \frac{\partial}{\partial \lambda_2}\biggl[ \frac{\partial \Phi}{\partial \lambda_2}\biggr]
			+
			(\Lambda+1) \biggl[\frac{\partial \Phi}{\partial \lambda_2} \biggr]
			\biggr\}
			</math>
	</td>		</td>
	<~~td align="center"~~>~~      or,      ~~ <~~/td~~>		</tr>

			<tr>
	<td align="right">		<td align="right">
	<math>~\~~Phi~~</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2A}{\lambda_1} \biggl\{ \Lambda \lambda_1 \frac{\partial}{\partial \lambda_1}\biggl[ \alpha \lambda_1^{\alpha-1} \biggr]
			+
			(\Lambda + 1) \biggl[ \alpha \lambda_1^{\alpha-1} \biggr] \biggr\}
			+
			\frac{8B}{\lambda_2 (\Lambda-1)^2}\biggl\{
			\Lambda \lambda_2 \frac{\partial}{\partial \lambda_2}\biggl[ \beta \lambda_2^{\beta - 1} \biggr]
			+
			(\Lambda+1) \biggl[ \beta \lambda_2^{\beta - 1} \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2A\alpha }{\lambda_1} \biggl\{ \Lambda (\alpha-1) \lambda_1^{\alpha-1}
			+
			(\Lambda + 1) \lambda_1^{\alpha-1} \biggr\}
			+
			\frac{8B\beta }{\lambda_2 (\Lambda-1)^2}\biggl\{
			\Lambda(\beta-1) \lambda_2^{\beta - 1}
			+
			(\Lambda+1) \lambda_2^{\beta - 1}
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2A\alpha \lambda_1^{\alpha-2} \biggl\{ \Lambda (\alpha-1)
			+
			(\Lambda + 1) \biggr\}
			+
			\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2}\biggl\{
			\Lambda(\beta-1)
			+
			(\Lambda+1)
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\~~frac~~{1}{\~~lambda_1~~ \lambda_2} \~~, .~~</math>		<math>~
			2A\alpha \lambda_1^{\alpha-2}
			\biggl\{ \alpha \Lambda + 1 \biggr\}
			+
			\frac{8B\beta \lambda_2^{\beta-2}}{ (\Lambda-1)^2} \biggl\{ \beta \Lambda +1
			\biggr\}
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Example (q2 = 2) Poisson Equation */

2020-10-24T05:02:05Z

Example (q2 = 2) Poisson Equation

← Older revision		Revision as of 05:02, 24 October 2020
Line 2,577:		Line 2,577:
	\biggr\} \, .		\biggr\} \, .
	</math>		</math>
			</td>
			</tr>
			</table>

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

			<tr>
			<td align="right">
			<math>~\Phi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\lambda_1^2 \lambda_2^2 \, ,</math>
			</td>
			<td align="center">      or,       </td>
			<td align="right">
			<math>~\Phi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{\lambda_1 \lambda_2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Example (q2 = 2) Poisson Equation */

2020-10-24T04:51:08Z

Example (q2 = 2) Poisson Equation

← Older revision		Revision as of 04:51, 24 October 2020
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{\lambda_1} \biggl\{ 2\Lambda \lambda_1 \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]		\frac{2}{\lambda_1} \biggl\{ \Lambda \lambda_1 \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
	+		+
	2(\Lambda + 1) \frac{\partial \Phi}{\partial \lambda_1} \biggr\}		(\Lambda + 1) \frac{\partial \Phi}{\partial \lambda_1} \biggr\}
	+		+
	\frac{8}{\lambda_2 (\Lambda-1)^2}\biggl\{		\frac{8}{\lambda_2 (\Lambda-1)^2}\biggl\{

Tohline: /* Example (q2 = 2) Poisson Equation */

2020-10-24T04:46:18Z

Example (q2 = 2) Poisson Equation

← Older revision		Revision as of 04:46, 24 October 2020
Line 2,528:		Line 2,528:
	\frac{(\Lambda-1)}{2 }\biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]		\frac{(\Lambda-1)}{2 }\biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
	+		+
	\frac{\~~partial \Phi~~}{\~~partial~~ \~~lambda_1~~} \~~cdot~~ \frac{\partial}{\partial \lambda_1} ~~\biggl[ \frac{(\Lambda-1)}{2 } \biggr]~~		\biggl[ \frac{(\Lambda^2-1)}{2 \lambda_1\Lambda} \biggr] \frac{\partial \Phi}{\partial \lambda_1}
	+		+
	\frac{2}{ (\Lambda-1) } \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]		\frac{2}{ (\Lambda-1) } \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
	+		+
	\frac{\~~partial \Phi~~}{\~~partial~~ \lambda_2} \~~cdot~~ \frac{\partial}{\partial \lambda_2} ~~\biggl[ \frac{2}{ (\Lambda-1) } \biggr]~~		\biggl[ \frac{2(\Lambda+1)}{ (\Lambda-1)\lambda_2\Lambda } \biggr] \frac{\partial \Phi}{\partial \lambda_2}
	\biggr\}		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2\Lambda \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
			+
			\biggl[ \frac{2(\Lambda + 1)}{ \lambda_1} \biggr] \frac{\partial \Phi}{\partial \lambda_1}
			+
			\frac{8 \Lambda }{ (\Lambda-1)^2 } \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
			+
			\biggl[ \frac{8(\Lambda+1)}{ (\Lambda-1)^2\lambda_2 } \biggr] \frac{\partial \Phi}{\partial \lambda_2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\lambda_1} \biggl\{ 2\Lambda \lambda_1 \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
			+
			2(\Lambda + 1) \frac{\partial \Phi}{\partial \lambda_1} \biggr\}
			+
			\frac{8}{\lambda_2 (\Lambda-1)^2}\biggl\{
			\Lambda \lambda_2 \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
			+
			(\Lambda+1) \frac{\partial \Phi}{\partial \lambda_2}
			\biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Example (q2 = 2) Poisson Equation */

2020-10-24T04:26:28Z

Example (q2 = 2) Poisson Equation

← Older revision		Revision as of 04:26, 24 October 2020
Line 2,485:		Line 2,485:
	=		=
	\frac{\Lambda + 1}{(\Lambda-1)} \biggl[ \frac{1}{\lambda_2 \Lambda} \biggr] \, .		\frac{\Lambda + 1}{(\Lambda-1)} \biggl[ \frac{1}{\lambda_2 \Lambda} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

			Hence,

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

			<tr>
			<td align="right">
			<math>~4\pi G\rho_c \biggl[1 - \frac{\lambda_1^2}{a^2} \biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{4 \Lambda}{(\Lambda-1 )} \biggl\{
			\frac{(\Lambda-1)}{2 }\biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
			+
			\frac{\partial \Phi}{\partial \lambda_1} \cdot \frac{\partial}{\partial \lambda_1} \biggl[ \frac{(\Lambda-1)}{2 } \biggr]
			+
			\frac{2}{ (\Lambda-1) } \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
			+
			\frac{\partial \Phi}{\partial \lambda_2} \cdot \frac{\partial}{\partial \lambda_2} \biggl[ \frac{2}{ (\Lambda-1) } \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{4 \Lambda}{(\Lambda-1 )} \biggl\{
			\frac{(\Lambda-1)}{2 }\biggl[ \frac{\partial^2 \Phi}{\partial \lambda_1^2}\biggr]
			+
			\frac{\partial \Phi}{\partial \lambda_1} \cdot \frac{\partial}{\partial \lambda_1} \biggl[ \frac{(\Lambda-1)}{2 } \biggr]
			+
			\frac{2}{ (\Lambda-1) } \biggl[ \frac{\partial^2 \Phi}{\partial \lambda_2^2}\biggr]
			+
			\frac{\partial \Phi}{\partial \lambda_2} \cdot \frac{\partial}{\partial \lambda_2} \biggl[ \frac{2}{ (\Lambda-1) } \biggr]
			\biggr\}
	</math>		</math>
	</td>		</td>