User:Tohline/ThreeDimensionalConfigurations/Challenges - Revision history

Tohline: /* See Also */

2021-05-11T21:52:03Z

See Also

← Older revision		Revision as of 21:52, 11 May 2021
Line 6,466:		Line 6,466:
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
			* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

Tohline: /* See Also */

2021-04-30T16:23:01Z

See Also

← Older revision		Revision as of 16:23, 30 April 2021
Line 6,465:		Line 6,465:
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
			* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

Tohline: /* See Also */

2021-04-25T21:53:17Z

See Also

← Older revision		Revision as of 21:53, 25 April 2021
Line 6,462:		Line 6,462:
	=See Also=		=See Also=

			* [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]

Tohline: /* See Also */

2021-04-25T21:51:57Z

See Also

← Older revision		Revision as of 21:51, 25 April 2021
Line 6,462:		Line 6,462:
	=See Also=		=See Also=

	*[[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
			* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
			* Related discussions of models viewed from a rotating reference frame:
			** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
			** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[User:Tohline/Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* See Also */

2021-04-05T21:43:52Z

See Also

← Older revision		Revision as of 21:43, 5 April 2021
Line 6,461:		Line 6,461:

	=See Also=		=See Also=

			*[[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Trial#5 */

2020-10-23T14:57:21Z

Trial#5

← Older revision		Revision as of 14:57, 23 October 2020
Line 5,156:		Line 5,156:
	\frac{2\Omega_f}{\rho_c} \cdot \frac{q}{q-1}		\frac{2\Omega_f}{\rho_c} \cdot \frac{q}{q-1}
	\biggl[ \boldsymbol{\hat\jmath} \biggl( \frac{\partial\sigma^{q-1}}{\partial y} \biggr)		\biggl[ \boldsymbol{\hat\jmath} \biggl( \frac{\partial\sigma^{q-1}}{\partial y} \biggr)
	+ \boldsymbol{\hat\imath} ~~\frac{q}{(q-1)}~~ \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)\biggr]		+ \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)\biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* All Together */

2020-10-22T21:58:14Z

All Together

← Older revision		Revision as of 21:58, 22 October 2020
Line 6,417:		Line 6,417:
	\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-4)/q} \, .		\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-4)/q} \, .
	</math>		</math>
			</td>
			</tr>
			</table>

			Note that, for the [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#Invert_Coordinate_Mapping\|specific ''example'' case of]] <math>~\kappa^2 = 2</math>,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl[ \frac{1}{\ell^2} \biggr]_{\kappa^2=2} = (x^2 + 4y^2)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			x^2 \Lambda
			=
			\frac{\Lambda \lambda_2^2}{2} \biggl[ \Lambda - 1 \biggr]
			=
			2\lambda_1^2 \biggl[ \frac{\Lambda}{\Lambda + 1} \biggr]
			\, .
			</math>
			</td>
			</tr>
			</table>
			where,

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

			<tr>
			<td align="right">
			<math>~\Lambda</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl[ 1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* All Together */

2020-10-18T22:53:24Z

All Together

← Older revision		Revision as of 22:53, 18 October 2020
Line 6,414:		Line 6,414:
	\biggl[ \frac{2q(1 + \kappa^2)\Psi_0}{\rho_c^2 a^2} \biggr] \biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}		\biggl[ \frac{2q(1 + \kappa^2)\Psi_0}{\rho_c^2 a^2} \biggr] \biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
	-		-
	\biggl[ \frac{4 q(q-2) \Psi_0}{\rho_c^2 a^4} \biggr] (x^2 + \kappa^2 y^2)		\biggl[ \frac{4 q(q-2) \Psi_0}{\rho_c^2 a^4} \biggr] (x^2 + \kappa^4 y^2)
	\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-4)/q} \, .		\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-4)/q} \, .
	</math>		</math>

Tohline: /* Trial #7 */

2020-10-18T22:22:30Z

Trial #7

← Older revision		Revision as of 22:22, 18 October 2020
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	-		-
	\biggl[ \frac{2(q-1)(1 + \kappa^2)}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q} \, .		\biggl[ \frac{2(q-1)(1 + \kappa^2)}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q} \, .
			</math>
			</td>
			</tr>
			</table>

			====All Together====

			Putting this all together, we obtain,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~ g(\Psi) </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl(\frac{2\Omega_f}{\rho_c}\biggr)\biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
			-
			\biggl[ \frac{q\Psi_0}{\rho_c^2 (q-1) } \biggr] \biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
			\biggl\{
			\frac{4(q-1)(q-2)}{a^4\ell^2}
			\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
			-
			\biggl[ \frac{2(q-1)(1 + \kappa^2)}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl(\frac{2\Omega_f}{\rho_c}\biggr)\biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
			+
			\biggl[ \frac{2q(1 + \kappa^2)\Psi_0}{\rho_c^2 a^2} \biggr] \biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
			-
			\biggl[ \frac{4 q(q-2) \Psi_0}{\rho_c^2 a^4} \biggr] (x^2 + \kappa^2 y^2)
			\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-4)/q} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Trial #7 */

2020-10-18T21:54:23Z

Trial #7

← Older revision		Revision as of 21:54, 18 October 2020
Line 6,352:		Line 6,352:
	- \biggl[ \frac{2(q-1)\lambda_1}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}		- \biggl[ \frac{2(q-1)\lambda_1}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
	\biggr\}		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{4(q-1)(q-2)}{a^4\ell^2}
			\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
			-
			\biggl[ \frac{2(q-1)(1 + \kappa^2)}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q} \, .
	</math>		</math>
	</td>		</td>

← Older revision		Revision as of 21:52, 11 May 2021
Line 6,466:		Line 6,466:
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
			* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

← Older revision		Revision as of 16:23, 30 April 2021
Line 6,465:		Line 6,465:
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
			* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]