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

Tohline: /* Other Coordinate Pair in the Tangent Plane */

2021-05-27T22:39:50Z

Other Coordinate Pair in the Tangent Plane

← Older revision		Revision as of 22:39, 27 May 2021
Line 288:		Line 288:
	<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">		<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
	<div align="center">		<div align="center">
	'''Tangent Plane'''		'''Tangent Plane'''<br />
			(See, for example, [http://math.furman.edu/~dcs/courses/math21/ Dan Sloughter's] ([https://www.furman.edu Furman University]) 2001 Calculus III class lecture notes — specifically [http://math.furman.edu/~dcs/courses/math21/lectures/l-15.pdf Lecture 15])
	</div>		</div>

			----

	The two-dimensional plane that is tangent to the <math>~\lambda_1</math> = constant surface ''at this point'' is given by the expression,		The two-dimensional plane that is tangent to the <math>~\lambda_1</math> = constant surface ''at this point'' is given by the expression,

Tohline: /* Part B (25 February 2021) */

2021-03-03T00:12:53Z

Part B (25 February 2021)

← Older revision		Revision as of 00:12, 3 March 2021
Line 6,296:		Line 6,296:
	</tr>		</tr>
	</table>		</table>
	These direction cosines exactly match what is required in order to ensure that the coordinate, <math>~\lambda_5</math>, is everywhere orthogonal to both <math>~\lambda_1</math> and <math>~\lambda_4</math>. <font color="red">'''GREAT!'''</font> The resulting summary table is, therefore:		<span id="PartBCoordinatesT10">These direction cosines</span> exactly match what is required in order to ensure that the coordinate, <math>~\lambda_5</math>, is everywhere orthogonal to both <math>~\lambda_1</math> and <math>~\lambda_4</math>. <font color="red">'''GREAT!'''</font> The resulting summary table is, therefore:

	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">

Tohline: /* Understanding the Volume Element */

2021-03-01T22:46:15Z

Understanding the Volume Element

← Older revision		Revision as of 22:46, 1 March 2021
Line 6,657:		Line 6,657:
	</table>		</table>

	First, let's make sure that we understand how to relate the components of the Cartesian line element with the components of our T10 coordinates. MF53 claim that the following relation gives the various expressions for the scale factors; we will go ahead and incorporate the expectation that, since our coordinate system is orthogonal, the off-diagonal elements are zero.		First, let's make sure that we understand how to relate the components of the Cartesian line element with the components of our T10 coordinates.

			====Line Element====

			MF53 claim that the following relation gives the various expressions for the scale factors; we will go ahead and incorporate the expectation that, since our coordinate system is orthogonal, the off-diagonal elements are zero.
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 7,361:		Line 7,365:
	</table>		</table>
	… Not sure this is headed anywhere useful!		… Not sure this is headed anywhere useful!

			====Volume Element====

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

			<tr>
			<td align="right">
			<math>~(h_1 h_4 h_5) d\lambda_1 d\lambda_4 d\lambda_5</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(h_1 h_4 h_5)
			\biggl[
			\biggl( \frac{\partial \lambda_1}{\partial x}\biggr) dx
			+ \biggl( \frac{\partial \lambda_1}{\partial y}\biggr) dy
			+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr) dz
			\biggr]
			\biggl[
			\biggl( \frac{\partial \lambda_4}{\partial x}\biggr) dx
			+ \biggl( \frac{\partial \lambda_4}{\partial y}\biggr) dy
			+ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr) dz
			\biggr]
			\biggl[
			\biggl( \frac{\partial \lambda_5}{\partial x}\biggr) dx
			+ \biggl( \frac{\partial \lambda_5}{\partial y}\biggr) dy
			+ \biggl( \frac{\partial \lambda_5}{\partial z}\biggr) dz
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[
			\biggl( \gamma_{11}\biggr) dx
			+ \biggl( \gamma_{12}\biggr) dy
			+ \biggl( \gamma_{13}\biggr) dz
			\biggr]
			\biggl[
			\biggl( \gamma_{41}\biggr) dx
			+ \biggl( \gamma_{42}\biggr) dy
			+ \biggl( \gamma_{43}\biggr) dz
			\biggr]
			\biggl[
			\biggl( \gamma_{51} \biggr) dx
			+ \biggl( \gamma_{52} \biggr) dy
			+ \biggl( \gamma_{53} \biggr) dz
			\biggr]
			</math>
			</td>
			</tr>
			</table>

	=COLLADA=		=COLLADA=

Tohline: /* Understanding the Volume Element */

2021-03-01T16:18:52Z

Understanding the Volume Element

← Older revision		Revision as of 16:18, 1 March 2021
Line 6,795:		Line 6,795:
	h_4^2\biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2		h_4^2\biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2
	+		+
	h_5^2 \biggl( \frac{\partial \lambda_5}{\partial }\biggr)^2		h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
	\biggr] dz^2 \, .		\biggr] dz^2 \, .
	</math>		</math>
Line 7,360:		Line 7,360:
	</tr>		</tr>
	</table>		</table>
			… Not sure this is headed anywhere useful!

	=COLLADA=		=COLLADA=

Tohline: /* Understanding the Volume Element */

2021-02-28T21:27:32Z

Understanding the Volume Element

← Older revision		Revision as of 21:27, 28 February 2021
Line 7,328:		Line 7,328:
	<math>~		<math>~
	\frac{1}{h_1^4 h_4^2 }\biggl\{		\frac{1}{h_1^4 h_4^2 }\biggl\{
			\biggl[
			(- \gamma_{51} \gamma_{12}
	-		-
	\~~biggl[~~ \gamma_{51} \gamma_{12}		\gamma_{52} \gamma_{11} )
			\gamma_{43}
			+
			\gamma_{12}^2

			-\gamma_{11}^{2}
	\biggr]		\biggr]
	~~\gamma_{43}~~
	(\gamma_{43} \gamma_{11}		(\gamma_{43} \gamma_{11}
	+ \gamma_{41} \gamma_{13} )		+ \gamma_{41} \gamma_{13} )
	\gamma_{52}		\gamma_{52}
	-
	~~\biggl[~~
	~~\gamma_{52} \gamma_{11} \biggr]~~
	~~\gamma_{43}~~
	~~(\gamma_{43} \gamma_{11}~~
	~~+ \gamma_{41} \gamma_{13} )~~
	~~\gamma_{52}~~

	+		+
	\biggl[ \gamma_{51} \gamma_{13}		\biggl[
			(\gamma_{51} \gamma_{13}
	+		+
	\gamma_{53} \gamma_{11}		\gamma_{53} \gamma_{11} )
	~~\biggr]~~
	\gamma_{42}		\gamma_{42}
	~~\biggl[~~
	~~\gamma_{42} \gamma_{11}~~
	~~+ \gamma_{41} \gamma_{12}~~
	~~\biggr] \gamma_{53}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	+
	~~\gamma_{12}^2~~
	~~(\gamma_{43}\gamma_{11}~~
	~~+ \gamma_{41} \gamma_{13} )\gamma_{52}~~
	~~-\gamma_{11}^{2}~~
	~~\biggl[ \gamma_{43} \gamma_{11}~~
	~~+ \gamma_{41} \gamma_{13}~~
	~~\biggr]\gamma_{52}~~


	- \gamma_{11}^{2}		- \gamma_{11}^{2}
	~~\biggl[~~
	~~\gamma_{42} \gamma_{11}~~
	~~+ \gamma_{41} \gamma_{12}~~
	~~\biggr]\gamma_{53}~~


	+ \gamma_{13}^2		+ \gamma_{13}^2
	\~~biggl[~~		\biggr]
	\gamma_{42} \gamma_{11}		(\gamma_{42} \gamma_{11}
	+ \gamma_{41}\gamma_{12}		+ \gamma_{41}\gamma_{12} )
	~~\biggr]~~\gamma_{53}		\gamma_{53}
	\biggr\}		\biggr\}
	</math>		</math>

Tohline: /* Understanding the Volume Element */

2021-02-28T20:52:51Z

Understanding the Volume Element

← Older revision		Revision as of 20:52, 28 February 2021
Line 7,330:		Line 7,330:
	-		-
	\biggl[ \gamma_{51} \gamma_{12}		\biggl[ \gamma_{51} \gamma_{12}
	+		\biggr]
			\gamma_{43}
			(\gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13} )
			\gamma_{52}
			-
			\biggl[
	\gamma_{52} \gamma_{11} \biggr]		\gamma_{52} \gamma_{11} \biggr]
	\gamma_{43}		\gamma_{43}
	~~\biggl[~~ \gamma_{43} \gamma_{11}		(\gamma_{43} \gamma_{11}
	+ \gamma_{41} \gamma_{13}		+ \gamma_{41} \gamma_{13} )
	~~\biggr]~~
	\gamma_{52}		\gamma_{52}

Line 7,348:		Line 7,353:
	+ \gamma_{41} \gamma_{12}		+ \gamma_{41} \gamma_{12}
	\biggr] \gamma_{53}		\biggr] \gamma_{53}

	~~\biggr\}~~
	</math>		</math>
	</td>		</td>
Line 7,363:		Line 7,366:
	<td align="left">		<td align="left">
	<math>~		<math>~
	+ ~~\frac{1}{ h_1^4 h_4^2}\biggl\{~~		+
	\gamma_{12}^2		\gamma_{12}^2
	~~\biggl[~~ \gamma_{43}\gamma_{11}		(\gamma_{43}\gamma_{11}
	+ \gamma_{41} \gamma_{13}		+ \gamma_{41} \gamma_{13} )\gamma_{52}
	~~\biggr]~~\gamma_{52}
	-\gamma_{11}^{2}		-\gamma_{11}^{2}
	\biggl[ \gamma_{43} \gamma_{11}		\biggl[ \gamma_{43} \gamma_{11}

Tohline: /* Understanding the Volume Element */

2021-02-28T18:02:25Z

Understanding the Volume Element

← Older revision		Revision as of 18:02, 28 February 2021
Line 7,240:		Line 7,240:
	- \gamma_{41}\gamma_{12}		- \gamma_{41}\gamma_{12}
	\biggr]		\biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{h_1^4 h_4^2 }\biggl\{
			-
			\biggl[ \gamma_{51} \gamma_{12}
			+
			\gamma_{52} \gamma_{11} \biggr]
			\gamma_{43}
			\biggl[ \gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]
			\gamma_{52}

			+
			\biggl[ \gamma_{51} \gamma_{13}
			+
			\gamma_{53} \gamma_{11}
			\biggr]
			\gamma_{42}
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41} \gamma_{12}
			\biggr] \gamma_{53}

			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+ \frac{1}{ h_1^4 h_4^2}\biggl\{
			\gamma_{12}^2
			\biggl[ \gamma_{43}\gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]\gamma_{52}
			-\gamma_{11}^{2}
			\biggl[ \gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]\gamma_{52}


			- \gamma_{11}^{2}
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41} \gamma_{12}
			\biggr]\gamma_{53}


			+ \gamma_{13}^2
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41}\gamma_{12}
			\biggr]\gamma_{53}
	\biggr\}		\biggr\}
	</math>		</math>

Tohline: /* Understanding the Volume Element */

2021-02-28T17:59:55Z

Understanding the Volume Element

← Older revision		Revision as of 17:59, 28 February 2021
Line 7,254:		Line 7,254:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{~~h_5^2~~}{h_1^4 h_4~~^2 h_5~~^2}\biggl\{		\frac{1}{h_1^4 h_4^2 }\biggl\{
	~~-\biggl[ \gamma_{51}^2 \gamma_{12}^2~~
	-		-
	\gamma_{52}^2 \gamma_{11}^2 \biggr]		\biggl[ \gamma_{51} \gamma_{12}
			+
			\gamma_{52} \gamma_{11} \biggr]
			\gamma_{43}
	\biggl[ \gamma_{43} \gamma_{11}		\biggl[ \gamma_{43} \gamma_{11}
	+ \gamma_{41} \gamma_{13}		+ \gamma_{41} \gamma_{13}
Line 7,264:		Line 7,265:
	\gamma_{52}		\gamma_{52}

	-		+
	\biggl[ \gamma_{51}^2 \gamma_{13}^2		\biggl[ \gamma_{51} \gamma_{13}
	-		+
	\gamma_{53}^2 \gamma_{11}^2		\gamma_{53} \gamma_{11}
	\biggr]		\biggr]
			\gamma_{42}
	\biggl[		\biggl[
	\gamma_{42} \gamma_{11}		\gamma_{42} \gamma_{11}
Line 7,289:		Line 7,290:
	<td align="left">		<td align="left">
	<math>~		<math>~
	+ \frac{1}{h_4~~^2 h_1~~^2}\biggl\{		+ \frac{1}{ h_1^4 h_4^2}\biggl\{
	\~~biggl( \frac~~{~~\partial \lambda_1}{\partial y~~}~~\biggr)~~^2		\gamma_{12}^2
	\biggl[ \gamma_{43}\gamma_{11}		\biggl[ \gamma_{43}\gamma_{11}
	+ \gamma_{41} \gamma_{13}		+ \gamma_{41} \gamma_{13}
	\biggr]\gamma_{52}		\biggr]\gamma_{52}
	-\~~biggl( \frac~~{~~\partial \lambda_1~~}~~{\partial x} \biggr)~~^{2}		-\gamma_{11}^{2}
	\biggl[ \gamma_{43} \gamma_{11}		\biggl[ \gamma_{43} \gamma_{11}
	+ \gamma_{41} \gamma_{13}		+ \gamma_{41} \gamma_{13}
Line 7,300:		Line 7,301:


	- \~~biggl( \frac~~{~~\partial \lambda_1~~}~~{\partial x} \biggr)~~^{2}		- \gamma_{11}^{2}
	\biggl[		\biggl[
	\gamma_{42} \gamma_{11}		\gamma_{42} \gamma_{11}
Line 7,307:		Line 7,308:


	+ \~~biggl( \frac~~{~~\partial \lambda_1~~}~~{\partial z}\biggr)~~^2		+ \gamma_{13}^2
	\biggl[		\biggl[
	\gamma_{42} \gamma_{11}		\gamma_{42} \gamma_{11}

Tohline: /* Understanding the Volume Element */

2021-02-28T17:45:10Z

Understanding the Volume Element

← Older revision		Revision as of 17:45, 28 February 2021
Line 7,240:		Line 7,240:
	- \gamma_{41}\gamma_{12}		- \gamma_{41}\gamma_{12}
	\biggr]		\biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{h_5^2}{h_1^4 h_4^2 h_5^2}\biggl\{
			-\biggl[ \gamma_{51}^2 \gamma_{12}^2
			-
			\gamma_{52}^2 \gamma_{11}^2 \biggr]

			\biggl[ \gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]
			\gamma_{52}

			-
			\biggl[ \gamma_{51}^2 \gamma_{13}^2
			-
			\gamma_{53}^2 \gamma_{11}^2
			\biggr]

			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41} \gamma_{12}
			\biggr] \gamma_{53}

			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+ \frac{1}{h_4^2 h_1^2}\biggl\{
			\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2
			\biggl[ \gamma_{43}\gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]\gamma_{52}
			-\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
			\biggl[ \gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]\gamma_{52}


			- \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41} \gamma_{12}
			\biggr]\gamma_{53}


			+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41}\gamma_{12}
			\biggr]\gamma_{53}
	\biggr\}		\biggr\}
	</math>		</math>

Tohline: /* Understanding the Volume Element */

2021-02-28T17:31:22Z

Understanding the Volume Element

← Older revision		Revision as of 17:31, 28 February 2021
Line 7,146:		Line 7,146:
	\gamma_{42}^2 \gamma_{11}^2		\gamma_{42}^2 \gamma_{11}^2
	- \gamma_{41}^2\gamma_{12}^2		- \gamma_{41}^2\gamma_{12}^2
			\biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{h_5^2}{h_1^4 h_4^2 h_5^2}\biggl\{
			\biggl[ \gamma_{51}^2 \gamma_{12}^2
			-
			\gamma_{52}^2 \gamma_{11}^2 \biggr]

			\biggl[ \gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]
			\biggl[ \gamma_{43} \gamma_{11}
			- \gamma_{41} \gamma_{13}
			\biggr]

			-
			\biggl[ \gamma_{51}^2 \gamma_{13}^2
			-
			\gamma_{53}^2 \gamma_{11}^2
			\biggr]

			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41} \gamma_{12}
			\biggr]
			\biggl[
			\gamma_{42} \gamma_{11}
			- \gamma_{41} \gamma_{12}
			\biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+ \frac{1}{h_4^2 h_1^2}\biggl\{
			\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}

			\biggl[ \gamma_{43} \gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]
			\biggl[ \gamma_{43} \gamma_{11}
			- \gamma_{41} \gamma_{13}
			\biggr]

			-
			\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2
			\biggl[ \gamma_{43}\gamma_{11}
			+ \gamma_{41} \gamma_{13}
			\biggr]
			\biggl[ \gamma_{43}\gamma_{11}
			-~ \gamma_{41} \gamma_{13}
			\biggr]

			- \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41} \gamma_{12}
			\biggr]
			\biggl[
			\gamma_{42} \gamma_{11}
			- \gamma_{41} \gamma_{12}
			\biggr]


			+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
			\biggl[
			\gamma_{42} \gamma_{11}
			+ \gamma_{41}\gamma_{12}
			\biggr]
			\biggl[
			\gamma_{42} \gamma_{11}
			- \gamma_{41}\gamma_{12}
	\biggr]		\biggr]
	\biggr\}		\biggr\}