User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT12Coordinates - Revision history

Tohline: /* Partial Derivatives */

2021-03-16T20:07:28Z

Partial Derivatives

← Older revision		Revision as of 20:07, 16 March 2021
Line 1,392:		Line 1,392:
	<math>~		<math>~
	(m-n)p^4[ a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2 ]		(m-n)p^4[ a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2 ]
	- n (aq^2y~~^2 - bx~~^2)^2		- n (bx^2 - aq^2y^2 )^2
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(m-n)p^4\mathfrak{B}
			- n (bx^2 - aq^2y^2 )^2
			</math>
			</td>
			</tr>

	~~Now, if we set n~~ = m, ~~this expression reduces to~~,		<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			m p^4\mathfrak{B} -n [p^4\mathfrak{B}+ (bx^2 - aq^2y^2 )^2 ] \, .
			</math>
			</td>
			</tr>
			</table>
			Reflecting back on the first line of the "example" derivation, we recognize that,
			<table border="0" cellpadding="5" align="center">

	<~~table border~~="0" ~~cellpadding~~="5" align="~~center~~">		<tr>
			<td align="right">
			<math>~p^4\mathfrak{B}+ (bx^2 - aq^2y^2 )^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\mathfrak{A} (a^2 y^2 + b^2x^2)
			</math>
			</td>
			</tr>

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \~~frac{1}{2z}\biggl[ \frac{~~2~~\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial z}~~</math>		<math>~\Rightarrow ~~~(bx^2 - aq^2y^2 )^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,412:		Line 1,452:
	<td align="left">		<td align="left">
	<math>~		<math>~
	(~~xy)~~^2~~[mc~~^2p^~~4 - na~~^2 ~~- nb^2q^4]~~		\mathfrak{A} (a^2 y^2 + b^2x^2)
	- ~~n a^2 q^4y^4 - nb^2x~~^4		-p^4\mathfrak{B}
	</math>		</math>
	</td>		</td>
Line 1,419:		Line 1,459:
	</table>		</table>


	~~<!-- Not promising at all because there is no way to get rid of terms <math>~\mathcal{O}(y^4)</math> and <math>~\mathcal{O}(x^4)</math>. -->~~
	</td></tr></table>		</td></tr></table>

Tohline: /* Partial Derivatives */

2021-03-16T19:30:27Z

Partial Derivatives

← Older revision		Revision as of 19:30, 16 March 2021
Line 1,325:		Line 1,325:
	+ (xy)^2[mc^2p^4 - na^2 - nb^2q^4]		+ (xy)^2[mc^2p^4 - na^2 - nb^2q^4]
	- n a^2 q^4y^4 - nb^2x^4		- n a^2 q^4y^4 - nb^2x^4
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(m-n)p^4[ a^2(yz)^2 + b^2(xz)^2 ]
			+ (xy)^2[mc^2p^4 - n(a^2 + b^2q^4)]
			- n [ (aq^2y^2 \pm bx^2)^2 \mp 2abq^2(xy)^2] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			Choosing the ''inferior'' sign then enforcing the above-derived <font color="red">One-Two Perpendicular Constraint</font> by setting, <math>~(a + bq^2) = -cp^2</math>, gives,

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

			<tr>
			<td align="right">
			<math>~ \frac{1}{2z}\biggl[ \frac{2\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial z}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(m-n)p^4[ a^2(yz)^2 + b^2(xz)^2 ]
			+ (xy)^2[mc^2p^4 - n(a^2 + b^2q^4 + 2abq^2)]
			- n (aq^2y^2 - bx^2)^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(m-n)p^4[ a^2(yz)^2 + b^2(xz)^2 ]
			+ (xy)^2[mc^2p^4 - n(-c p^2 )^2]
			- n (aq^2y^2 - bx^2)^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(m-n)p^4[ a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2 ]
			- n (aq^2y^2 - bx^2)^2
			</math>
			</td>
			</tr>
			</table>


	Now, if we set n = m, this expression reduces to,		Now, if we set n = m, this expression reduces to,

Tohline: /* Partial Derivatives */

2021-03-15T19:28:50Z

Partial Derivatives

← Older revision		Revision as of 19:28, 15 March 2021
Line 1,330:		Line 1,330:
	</table>		</table>

	Not promising at all because there is no way to get rid of terms <math>~\mathcal{O}(y^4)</math> and <math>~\mathcal{O}(x^4)</math>.		Now, if we set n = m, this expression reduces to,

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

			<tr>
			<td align="right">
			<math>~ \frac{1}{2z}\biggl[ \frac{2\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial z}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(xy)^2[mc^2p^4 - na^2 - nb^2q^4]
			- n a^2 q^4y^4 - nb^2x^4
			</math>
			</td>
			</tr>
			</table>


			<!-- Not promising at all because there is no way to get rid of terms <math>~\mathcal{O}(y^4)</math> and <math>~\mathcal{O}(x^4)</math>. -->
	</td></tr></table>		</td></tr></table>

Tohline: /* Partial Derivatives */

2021-03-12T20:22:00Z

Partial Derivatives

← Older revision		Revision as of 20:22, 12 March 2021
Line 1,330:		Line 1,330:
	</table>		</table>

			Not promising at all because there is no way to get rid of terms <math>~\mathcal{O}(y^4)</math> and <math>~\mathcal{O}(x^4)</math>.
	</td></tr></table>		</td></tr></table>

Tohline: /* Partial Derivatives */

2021-03-12T19:43:12Z

Partial Derivatives

← Older revision		Revision as of 19:43, 12 March 2021
Line 1,308:		Line 1,308:
	-		-
	n [(x^2 + q^4y^2 + p^4z^2)a^2 y^2 + (x^2 + q^4y^2 + p^4z^2)b^2x^2]		n [(x^2 + q^4y^2 + p^4z^2)a^2 y^2 + (x^2 + q^4y^2 + p^4z^2)b^2x^2]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(m-n)a^2p^4(yz)^2
			+ (m-n)b^2p^4(xz)^2
			+ (xy)^2[mc^2p^4 - na^2 - nb^2q^4]
			- n a^2 q^4y^4 - nb^2x^4
	</math>		</math>
	</td>		</td>

Tohline: /* Partial Derivatives */

2021-03-12T18:29:23Z

Partial Derivatives

← Older revision		Revision as of 18:29, 12 March 2021
Line 1,260:		Line 1,260:
	</tr>		</tr>
	</table>		</table>
	Then we have,		Then we have, for example,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,266:		Line 1,266:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \biggl[ \frac{2\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial z}</math>		<math>~ \frac{1}{2z}\biggl[ \frac{2\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial z}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,273:		Line 1,273:
	<td align="left">		<td align="left">
	<math>~		<math>~
	m\mathfrak{B} \biggl[ 2p^4z \biggr]		m \mathfrak{B} \biggl[ p^4 \biggr]
	-		-
	n \mathfrak{A}\biggl[ 2z(a^2 y^2 + b^2x^2)\biggr]		n \mathfrak{A}\biggl[ (a^2 y^2 + b^2x^2)\biggr]
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~

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

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~\mathfrak{A} \equiv \ell_{3D}^{-2}</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,291:		Line 1,288:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(x^2 + q^4y^2 + p^4z^2) ~~\, ,~~</math>		<math>~
			mp^4 [a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2]
			-
			n (x^2 + q^4y^2 + p^4z^2)(a^2 y^2 + b^2x^2)
			</math>
	</td>		</td>
	</tr>		</tr>
Line 1,297:		Line 1,298:
	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~\mathfrak{B} \equiv [\mathfrak{L}(abc)]^2</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,304:		Line 1,305:
	<td align="left">		<td align="left">
	<math>~		<math>~
	[a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2] ~~\, .~~		mp^4 [a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2]
			-
			n [(x^2 + q^4y^2 + p^4z^2)a^2 y^2 + (x^2 + q^4y^2 + p^4z^2)b^2x^2]
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>


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

Tohline: /* Guess Third Coordinate Expression */

2021-03-12T18:02:37Z

Guess Third Coordinate Expression

← Older revision		Revision as of 18:02, 12 March 2021
Line 1,280:		Line 1,280:
	</tr>		</tr>
	</table>		</table>

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

			<tr>
			<td align="right">
			<math>~\mathfrak{A} \equiv \ell_{3D}^{-2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(x^2 + q^4y^2 + p^4z^2) \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\mathfrak{B} \equiv [\mathfrak{L}(abc)]^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			[a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2] \, .
			</math>
			</td>
			</tr>
			</table>


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

Tohline: /* Guess Third Coordinate Expression */

2021-03-12T17:59:51Z

Guess Third Coordinate Expression

← Older revision		Revision as of 17:59, 12 March 2021
Line 1,223:		Line 1,223:
	</tr>		</tr>
	</table>		</table>


			----


			<table border="1" cellpadding="10" align="center" width="80%"><tr><td align="left">
			Alternatively, suppose
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\lambda_3</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\mathfrak{A}^{m / 2} \mathfrak{B}^{-n/ 2} </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ \biggl[ \frac{2\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial x_i}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			m \mathfrak{B}\cdot \frac{\partial \mathfrak{A}}{\partial x_i}
			-
			n \mathfrak{A}\cdot \frac{\partial \mathfrak{B}}{\partial x_i} \, .
			</math>
			</td>
			</tr>
			</table>
			Then we have,

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

			<tr>
			<td align="right">
			<math>~ \biggl[ \frac{2\mathfrak{A}\mathfrak{B}}{\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial z}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			m\mathfrak{B} \biggl[ 2p^4z \biggr]
			-
			n \mathfrak{A}\biggl[ 2z(a^2 y^2 + b^2x^2)\biggr]
			</math>
			</td>
			</tr>
			</table>
			</td></tr></table>


			----


	Similarly, we find,		Similarly, we find,

Tohline: /* Dot With 1st Unit Vector */

2021-03-12T16:45:12Z

Dot With 1st Unit Vector

← Older revision		Revision as of 16:45, 12 March 2021
Line 1,888:		Line 1,888:
	+ q^2y^2(a^2p^4z^4 - 2acx^2p^2z^2 + c^2x^4)		+ q^2y^2(a^2p^4z^4 - 2acx^2p^2z^2 + c^2x^4)
	+ p^2z^2 (a^2q^4y^4 - 2aq^2y^2 bx^2 + b^2x^4)		+ p^2z^2 (a^2q^4y^4 - 2aq^2y^2 bx^2 + b^2x^4)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ -2x^2 y^2 z^2[cq^2 bp^2 + aq^2cp^2 + aq^2 bp^2 ]
			+ x^2 (c^2q^4y^4 + b^2p^4z^4)
			+ q^2y^2(a^2p^4z^4 + c^2x^4)
			+ p^2z^2 (a^2q^4y^4 + b^2x^4) \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			It does not appear as though the RHS of this expression can be zero for all values of the Cartesian coordinates, (x, y, z). Hence <math>~[\hat{e}_3]_\mathrm{guess}</math> is not orthogonal to <math>~\hat{e}_1</math>.

	=See Also=		=See Also=

Tohline: /* Dot With 1st Unit Vector */

2021-03-12T16:33:09Z

Dot With 1st Unit Vector

← Older revision		Revision as of 16:33, 12 March 2021
Line 1,842:		Line 1,842:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~-\hat{e}_1 \cdot [\hat{e}_3]_\mathrm{guess}</math>		<math>~-\hat{e}_1 \cdot [\hat{e}_3]_\mathrm{guess} \biggl[ \frac{\mathfrak{C}}{\ell_{3D}} \biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,848:		Line 1,848:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ ~~\frac{\ell_{3D}}{\mathfrak{C}}~~ \biggl\{		<math>~ \biggl\{
	\hat\imath (x) + \hat\jmath (q^2y ) + \hat{k} (p^2 z)		\hat\imath (x) + \hat\jmath (q^2y ) + \hat{k} (p^2 z)
	\biggl\} \cdot		\biggl\} \cdot
Line 1,868:		Line 1,868:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ ~~\frac{\ell_{3D}}{\mathfrak{C}} \biggl\{~~		<math>~
	~~\biggl[~~ x^2 (cq^2y^2 - bp^2z^2)^2 ~~\biggr]~~		x^2 (cq^2y^2 - bp^2z^2)^2
	+ ~~\biggl[~~ q^2y^2(ap^2z^2 - cx^2)^2 ~~\biggr]~~		+ q^2y^2(ap^2z^2 - cx^2)^2
	+ ~~\biggl[~~ p^2z^2 (aq^2y^2 - bx^2)^2 ~~\biggr]~~		+ p^2z^2 (aq^2y^2 - bx^2)^2
	~~\biggr\}~~		</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			x^2 (c^2q^4y^4 - 2cq^2y^2 bp^2z^2 + b^2p^4z^4)
			+ q^2y^2(a^2p^4z^4 - 2acx^2p^2z^2 + c^2x^4)
			+ p^2z^2 (a^2q^4y^4 - 2aq^2y^2 bx^2 + b^2x^4)
	</math>		</math>
	</td>		</td>