User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2 - Revision history

Tohline: /* See Also */

2021-05-11T21:51:28Z

Tohline: /* Off-Center Ellipse */

2021-05-03T21:34:46Z

Off-Center Ellipse

← Older revision		Revision as of 21:34, 3 May 2021
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	<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.		<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.
	<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">		<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
	<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''</td></tr>		<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''<br />(compare with [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4#Result2\|Result 2]])</td></tr>
	<tr><td align="left">		<tr><td align="left">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Off-Center Ellipse */

2021-05-03T21:31:13Z

Off-Center Ellipse

← Older revision		Revision as of 21:31, 3 May 2021
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	</tr>		</tr>
	</table>		</table>
	which, as indicated, allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.		<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.
	<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">		<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
	<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''</td></tr>		<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''</td></tr>

Tohline: /* Off-Center Ellipse */

2021-05-03T21:22:05Z

Off-Center Ellipse

← Older revision		Revision as of 21:22, 3 May 2021
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	</tr>		</tr>
	</table>		</table>
	which, as indicated, allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>. Dividing through by the leading coefficient,		which, as indicated, allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.
			<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
			<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''</td></tr>
			<tr><td align="left">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{y_0}{z_0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\frac{b^2\sin\theta}{c^2\cos^2\theta + b^2\sin^2\theta}
			=
			-\frac{\sin\theta}{c^2\kappa^2}
			</math>
			</td>
			</tr>
			</table>

			</td></tr>
			</table>

			Dividing through by the leading coefficient,
	<div align="center">		<div align="center">
	<math>~\kappa^2 \equiv \frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2} \, ,</math>		<math>~\kappa^2 \equiv \frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2} \, ,</math>

Tohline: /* See Also */

2021-04-30T16:22:04Z

Tohline: /* Off-Center Ellipse */

2021-04-28T17:27:29Z

Off-Center Ellipse

← Older revision		Revision as of 17:27, 28 April 2021
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	which identifies the appropriate expression for <math>~x^2_\mathrm{max}</math>. As viewed from the "tipped plane" (primed) coordinate frame, then, the equation for the orbit of each Lagrangian fluid element is that of an …		<span id="OffCenter">which identifies the appropriate</span> expression for <math>~x^2_\mathrm{max}</math>. As viewed from the "tipped plane" (primed) coordinate frame, then, the equation for the orbit of each Lagrangian fluid element is that of an …

	<table border="1" align="center" cellpadding="8" width="90%"><tr><td align="left">		<table border="1" align="center" cellpadding="8" width="90%"><tr><td align="left">

Tohline: /* See Also */

2021-04-25T21:54:38Z

Tohline: /* Try Tipped Plane Again */

2021-04-23T19:44:11Z

Try Tipped Plane Again

← Older revision		Revision as of 19:44, 23 April 2021
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	b^2 + a^2 \cos^2\theta		b^2 + a^2 \cos^2\theta
	\biggr\}\frac{\zeta_3}{a^2 + b^2} \cdot \tan\theta		\biggr\}\frac{\zeta_3}{a^2 + b^2} \cdot \tan\theta
			+ \boldsymbol{\hat{k}} ~ \biggl\{
			b^2
			+ a^2 \cos^2\theta
			\biggr\}\frac{\zeta_3}{a^2 + b^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\boldsymbol{\hat\jmath} ~ \biggl[ \frac{a^2 \zeta_2}{a^2 + c^2} \biggr]
			+
			\boldsymbol{\hat\jmath}~ \biggl\{
			b^2 + a^2 \cos^2\theta
			\biggr\}\frac{\zeta_2}{a^2 + c^2} \cdot \frac{c^2}{b^2}
	+ \boldsymbol{\hat{k}} ~ \biggl\{		+ \boldsymbol{\hat{k}} ~ \biggl\{
	b^2		b^2
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	</tr>		</tr>
	</table>		</table>


	----		----

Tohline: /* Try Tipped Plane Again */

2021-04-23T17:38:53Z

Try Tipped Plane Again

← Older revision		Revision as of 17:38, 23 April 2021
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	----		----



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

			<tr>
			<td align="right">
			<math>~\tan\theta</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr] \frac{c^2}{b^2}
			=
			- \frac{\beta \Omega_2}{\gamma \Omega_3} \, ,
			</math>
			</td>
			</tr>
			</table>

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

Tohline: /* Try Tipped Plane Again */

2021-04-23T17:34:39Z

Try Tipped Plane Again

← Older revision		Revision as of 17:34, 23 April 2021
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	<td align="left">		<td align="left">
	<math>~		<math>~
	(\boldsymbol{\hat\jmath} ~~\cos^2\theta )~~~ \biggl[ \frac{a^2 \zeta_2}{a^2 + c^2} \biggr]		\boldsymbol{\hat\jmath} ~ \biggl[ \frac{a^2 \zeta_2}{a^2 + c^2} \biggr]
	~~+ (~~-\boldsymbol{\hat\jmath} ~~\sin\theta )~~~ \biggl\{		-\boldsymbol{\hat\jmath}~ \biggl\{
	~~\frac{\zeta_3}{\cos\theta} \biggl[ \frac{~~b^2~~}{a^2 + b^2} \biggr]~~		b^2 + a^2 \cos^2\theta
	+ ~~\biggl[ \frac{~~a^2 \~~zeta_3}{a~~^2 ~~+ b^2 } \biggr] \cos~~\theta		\biggr\}\frac{\zeta_3}{a^2 + b^2} \cdot \tan\theta
	- \~~biggl[~~ \frac{~~a^2~~ \~~zeta_2~~}{a^2 + c^2} \~~biggr]~~ \~~sin~~\theta
	~~\biggr\}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	+ \boldsymbol{\hat{k}} ~ \biggl\{		+ \boldsymbol{\hat{k}} ~ \biggl\{
	b^2		b^2

← Older revision		Revision as of 21:51, 11 May 2021
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	* [[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 16:22, 30 April 2021
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	* [[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]]

← Older revision		Revision as of 21:54, 25 April 2021
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	=See Also=		=See Also=
	* [[User:Tohline/ThreeDimensionalConfigurations/Challenges\|Construction Challenges (Pt. 1)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]		* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
			* [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
			* [[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]].



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