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

Tohline: /* New Insight */

2021-03-27T18:08:23Z

New Insight

← Older revision		Revision as of 18:08, 27 March 2021
Line 2,525:		Line 2,525:
	</table>		</table>

	This seems to be a promising method of attack because — in all three cases, i = 1,3 — the derivative of <math>~\lambda_2</math> with respect to <math>~x_i</math> does not depend on <math>~x_i</math>. Perhaps this simplification will help us identify the function that defines <math>~\lambda_2</math>. This proposed prescription for <math>~h_2(x, y, z)</math> and some of its implications are reflected in the following "New Insight" table.		This seems to be a promising method of attack because — in all three cases, i = 1,3 — the derivative of <math>~\lambda_2</math> with respect to <math>~x_i</math> does not depend on <math>~x_i</math>. Perhaps this simplification will help us identify the function that defines <math>~\lambda_2</math>. This proposed prescription for <math>~h_2(x, y, z)</math> and some of its implications are reflected in the following "New Insight" table. (Keep in mind that, although the expressions for <math>~\gamma_{21}, \gamma_{22}, ~\mathrm{and}~ \gamma_{23}</math> remain correct, the tabulated expression is a ''guess'' for <math>~h_2</math> and, hence, the tabulated expressions for all three <math>~\partial \lambda_2/\partial x_i</math> are pure speculation.)
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr>		<tr>

Tohline: /* New Insight */

2021-03-27T03:50:31Z

New Insight

← Older revision		Revision as of 03:50, 27 March 2021
Line 2,473:		Line 2,473:
	==New Insight==		==New Insight==

	Following the development of our [[#Better_Organized\|above, ''Better Organized'']] discussion, we reverted to several hours of pen & paper derivations, primarily investigating whether it will help us to rewrite various expressions using the MF53 [[User:Tohline/Appendix/Mathematics/ScaleFactors#DirectionCosineRelations\|Direction-Cosine Relations]]. We discovered that if we set,		Following the development of our [[#Better_Organized\|above, ''Better Organized'']] discussion, we reverted to several hours of pen & paper derivations, primarily investigating whether it will help us to rewrite various expressions using the [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] [[User:Tohline/Appendix/Mathematics/ScaleFactors#DirectionCosineRelations\|Direction-Cosine Relations]]. We discovered that if we set,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* New Insight */

2021-03-27T03:48:10Z

New Insight

← Older revision		Revision as of 03:48, 27 March 2021
Line 2,473:		Line 2,473:
	==New Insight==		==New Insight==

	Following the development of our [[#Better_Organized\|above, ''Better Organized'']] discussion, we reverted to several hours of pen & paper derivations, primarily investigating whether it will help us to rewrite various expressions using the MF53 [[User:Tohline/Appendix/Mathematics/ScaleFactors#DirectionCosineRelations\|Direction-Cosine Relations]]. We discovered that if we set		Following the development of our [[#Better_Organized\|above, ''Better Organized'']] discussion, we reverted to several hours of pen & paper derivations, primarily investigating whether it will help us to rewrite various expressions using the MF53 [[User:Tohline/Appendix/Mathematics/ScaleFactors#DirectionCosineRelations\|Direction-Cosine Relations]]. We discovered that if we set,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* New Insight */

2021-03-27T03:47:08Z

New Insight

← Older revision		Revision as of 03:47, 27 March 2021
Line 2,525:		Line 2,525:
	</table>		</table>

	This seems to be a promising method of attack because the derivative of <math>~\lambda_2</math> with respect to <math>~x_i</math> does not depend on <math>~x_i</math>. Perhaps this simplification will help us identify the function that defines <math>~\lambda_2</math>. This proposed prescription for <math>~h_2(x, y, z)</math> and some of its implications are reflected in the following "New Insight" table.		This seems to be a promising method of attack because — in all three cases, i = 1,3 — the derivative of <math>~\lambda_2</math> with respect to <math>~x_i</math> does not depend on <math>~x_i</math>. Perhaps this simplification will help us identify the function that defines <math>~\lambda_2</math>. This proposed prescription for <math>~h_2(x, y, z)</math> and some of its implications are reflected in the following "New Insight" table.
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr>		<tr>
Line 2,557:		Line 2,557:
	<td align="center"><math>~2</math></td>		<td align="center"><math>~2</math></td>
	<td align="center">---</td>		<td align="center">---</td>
	<td align="center">~~---~~</td>		<td align="center"><math>~\ell_q \ell_{3D} (xq^2y)</math></td>
	<td align="center">~~---~~</td>		<td align="center"><math>~\frac{p^2z}{q^2y}</math></td>
	<td align="center">~~---~~</td>		<td align="center"><math>~\frac{p^2z}{x}</math></td>
	<td align="center">---</td>		<td align="center"><math>~-\biggl[ \frac{x}{q^2y} + \frac{q^2y}{x} \biggr]</math></td>
	<td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{p^2z}{q^2y} \biggr]</math></td>		<td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{p^2z}{q^2y} \biggr]</math></td>
	<td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{ p^2z}{x} \biggr] </math></td>		<td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{ p^2z}{x} \biggr] </math></td>

Tohline: /* New Insight */

2021-03-27T03:40:58Z

New Insight

← Older revision		Revision as of 03:40, 27 March 2021
Line 2,473:		Line 2,473:
	==New Insight==		==New Insight==

	~~<span id="Table1NewInsight">This modified third-coordinate expression means that~~ the ~~last row~~ of ~~the~~ above ~~table changes~~, ~~as follows~~.<~~/span~~>		Following the development of our [[#Better_Organized\|above, ''Better Organized'']] discussion, we reverted to several hours of pen & paper derivations, primarily investigating whether it will help us to rewrite various expressions using the MF53 [[User:Tohline/Appendix/Mathematics/ScaleFactors#DirectionCosineRelations\|Direction-Cosine Relations]]. We discovered that if we set
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~h_2^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~[(xq^2y)\ell_q \ell_{3D}]^2 \, ,</math>
			</td>
			</tr>
			</table>
			then,

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

			<tr>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial x} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{p^2 z}{q^2y} \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial y} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{p^2z}{x} \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial z} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl[ \frac{x}{q^2y} + \frac{q^2y}{x} \biggr] \, .</math>
			</td>
			</tr>
			</table>

			This seems to be a promising method of attack because the derivative of <math>~\lambda_2</math> with respect to <math>~x_i</math> does not depend on <math>~x_i</math>. Perhaps this simplification will help us identify the function that defines <math>~\lambda_2</math>. This proposed prescription for <math>~h_2(x, y, z)</math> and some of its implications are reflected in the following "New Insight" table.
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr>		<tr>

Tohline: /* Insight? */

2021-03-27T03:09:28Z

Insight?

← Older revision		Revision as of 03:09, 27 March 2021
Line 2,403:		Line 2,403:
	</table>		</table>

	===~~Insight?~~===		===Struggling===

	I have noticed that, in this last set of expressions, there are recurring terms of the form, <math>~(qy\lambda_3)</math> and <math>~(y^{2/q^2})</math>. So, while keeping the same definition of the ccordinate, <math>~\lambda_1</math>, let's replace <math>~\lambda_2</math> and <math>~\lambda_3</math> with a pair of coordinates defined as follows:		I have noticed that, in this last set of expressions, there are recurring terms of the form, <math>~(qy\lambda_3)</math> and <math>~(y^{2/q^2})</math>. So, while keeping the same definition of the ccordinate, <math>~\lambda_1</math>, let's replace <math>~\lambda_2</math> and <math>~\lambda_3</math> with a pair of coordinates defined as follows:
Line 2,469:		Line 2,469:
	</tr>		</tr>
	</table>		</table>
	Is this a set of orthogonal coordinates? Well …		Is this a set of orthogonal coordinates? Well …       <font color="red">No!</font>

	==New Insight==		==New Insight==

Tohline: /* New Insight */

2021-03-27T03:08:14Z

New Insight

← Older revision		Revision as of 03:08, 27 March 2021
Line 2,472:		Line 2,472:

	==New Insight==		==New Insight==

	<span id="Table1NewInsight">This modified third-coordinate expression means that the last row of the above table changes, as follows.</span>		<span id="Table1NewInsight">This modified third-coordinate expression means that the last row of the above table changes, as follows.</span>

Line 2,509:		Line 2,510:
	<td align="center">---</td>		<td align="center">---</td>
	<td align="center">---</td>		<td align="center">---</td>
	<td align="center"><math>~[\ell_q \ell_{3D} (xq^y)~~] (xp~~^2z)</math></td>		<td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{p^2z}{q^2y} \biggr]</math></td>
	<td align="center"><math>~[ \ell_q \ell_{3D} (xq^y)~~] (q^2 y~~ p^2z) </math></td>		<td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{ p^2z}{x} \biggr] </math></td>
	<td align="center"><math>~- ~~(x^2 + q^4y^2) [~~ \ell_q \ell_{3D} (xq^y)]</math></td>		<td align="center"><math>~- \ell_q \ell_{3D} (xq^2y) \biggl[ \frac{x}{q^2y} + \frac{q^2y}{x} \biggr]</math></td>
	</tr>		</tr>

Tohline: /* Daring Attack */

2021-03-27T03:02:25Z

Daring Attack

← Older revision		Revision as of 03:02, 27 March 2021
Line 2,470:		Line 2,470:
	</table>		</table>
	Is this a set of orthogonal coordinates? Well …		Is this a set of orthogonal coordinates? Well …

			==New Insight==
			<span id="Table1NewInsight">This modified third-coordinate expression means that the last row of the above table changes, as follows.</span>

			<table border="1" cellpadding="8" align="center">
			<tr>
			<td align="center" colspan="9">'''New Insight'''</td>
			</tr>
			<tr>
			<td align="center"><math>~n</math></td>
			<td align="center"><math>~\lambda_n</math></td>
			<td align="center"><math>~h_n</math></td>
			<td align="center"><math>~\frac{\partial \lambda_n}{\partial x}</math></td>
			<td align="center"><math>~\frac{\partial \lambda_n}{\partial y}</math></td>
			<td align="center"><math>~\frac{\partial \lambda_n}{\partial z}</math></td>
			<td align="center"><math>~\gamma_{n1}</math></td>
			<td align="center"><math>~\gamma_{n2}</math></td>
			<td align="center"><math>~\gamma_{n3}</math></td>
			</tr>

			<tr>
			<td align="center"><math>~1</math></td>
			<td align="center"><math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math></td>
			<td align="center"><math>~\lambda_1 \ell_{3D}</math></td>
			<td align="center"><math>~\frac{x}{\lambda_1}</math></td>
			<td align="center"><math>~\frac{q^2 y}{\lambda_1}</math></td>
			<td align="center"><math>~\frac{p^2 z}{\lambda_1}</math></td>
			<td align="center"><math>~(x) \ell_{3D}</math></td>
			<td align="center"><math>~(q^2 y)\ell_{3D}</math></td>
			<td align="center"><math>~(p^2z) \ell_{3D}</math></td>
			</tr>

			<tr>
			<td align="center"><math>~2</math></td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center"><math>~[\ell_q \ell_{3D} (xq^y)] (xp^2z)</math></td>
			<td align="center"><math>~[ \ell_q \ell_{3D} (xq^y)] (q^2 y p^2z) </math></td>
			<td align="center"><math>~- (x^2 + q^4y^2) [ \ell_q \ell_{3D} (xq^y)]</math></td>
			</tr>

			<tr>
			<td align="center"><math>~3</math></td>
			<td align="center"><math>~\frac{y^{1/q^2}}{x} </math></td>
			<td align="center"><math>~\frac{xq^2 y \ell_q}{\lambda_3}</math></td>
			<td align="center"><math>~-\frac{\lambda_3}{x}</math></td>
			<td align="center"><math>~+\frac{\lambda_3}{q^2y}</math></td>
			<td align="center"><math>~0</math></td>
			<td align="center"><math>~-q^2 y \ell_q</math></td>
			<td align="center"><math>~x\ell_q</math></td>
			<td align="center"><math>~0</math></td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Speculation */

2021-03-26T22:38:23Z

Speculation

← Older revision		Revision as of 22:38, 26 March 2021
Line 2,175:		Line 2,175:
	\frac{ \ell_q \ell_{3D} }{h_2} \, .		\frac{ \ell_q \ell_{3D} }{h_2} \, .
	</math>		</math>
			</td>
			</tr>
			</table>

			This key, leading coefficient function is unity — and, hence, is independent of position — if, as in our [[#First\|''First'' speculation]] above, we ''guess'' that <math>~h_2^2 = (\ell_q \ell_{3D})^2</math>. If, as in our [[#Second\|''Second'' speculation]] above, we ''guess'' that <math>~h_2^2 = (\ell_q \ell_{3D})^{-2}</math>, we find that, <math>~A = (\ell_q \ell_{3D})^2</math>. Our above [[#Third\|''Third'' speculation]] is replicated if we ''guess'' that <math>~h_2^2 = (\lambda_2 \ell_q \ell_{3D})^2</math>; we immediately see that, in this ''Third'' case,

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

			<tr>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial x} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{xp^2 z}{\lambda_2} \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial y} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{q^2y p^2z}{\lambda_2} \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial z} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\frac{(x^2 + q^4y^2)}{\lambda_2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Better Organized */

2021-03-26T22:18:58Z

Better Organized

← Older revision		Revision as of 22:18, 26 March 2021
Line 2,042:		Line 2,042:

	====Better Organized====		====Better Organized====

			From our above [[#Table1DaringAttack\|Daring Attack Table]], we appreciate that the three direction cosines associated with the (as yet unknown) second curvilinear coordinate are,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\gamma_{21}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\ell_q \ell_{3D} (xp^2z) \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\gamma_{22}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\ell_q \ell_{3D} (q^2 y p^2z) \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\gamma_{23}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\ell_q \ell_{3D} (x^2 + q^4 y^2) \, .</math>
			</td>
			</tr>
			</table>

			It is easy to see that the desired ''orthogonality'' relationship,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\sum_{i=1}^3 (\gamma_{2i})^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 \, ,</math>
			</td>
			</tr>
			</table>
			is satisfied because,
			<table border="0" cellpadding="5" align="center">

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

			Now, as we attempt to determine the functional form of the second curvilinear coordinate, <math>~\lambda_2(x, y, z)</math>, a seemingly useful intermediate step is to determine the functional form of each of the three partial derivatives of this key coordinate function, namely, <math>~\partial \lambda_2/\partial x_i</math>, for i = 1, 3. Here, we will accomplish this intermediate step by ''guessing'' the functional form of the second scale factor, <math>~h_2(x, y, z)</math>, then applying the relation,

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

			<tr>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial x_i}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\gamma_{2i}}{h_2} \, .</math>
			</td>
			</tr>
			</table>
			Notice that, without violating the above-state ''orthogonality'' relationship, we can adopt virtually any functional form for <math>~h_2(x, y, z)</math> and deduce that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial x}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~A(x, y, z) (xp^2z) \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial y}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~A(x, y, z) (q^2 y p^2z) \, ,</math>
			</td>
			<td align="center">     </td>
			<td align="right">
			<math>~\frac{\partial \lambda_2}{\partial z}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-A(x, y, z) (x^2 + q^4 y^2) \, ,</math>
			</td>
			</tr>
			</table>
			as long as,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(x, y, z)</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{ \ell_q \ell_{3D} }{h_2} \, .
			</math>
			</td>
			</tr>
			</table>

	==Study the Functional Forms==		==Study the Functional Forms==