User:Tohline/Appendix/Mathematics/ScaleFactors - Revision history

Tohline: /* More Generally */

2021-03-27T03:18:39Z

More Generally

← Older revision		Revision as of 03:18, 27 March 2021
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	where <math>~\delta_{mn}</math> is the ''[https://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta function],'' which is zero when <math>~m</math> is not equal to <math>~n</math>, unity when <math>~m=n</math>.		where <math>~\delta_{mn}</math> is the ''[https://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta function],'' which is zero when <math>~m</math> is not equal to <math>~n</math>, unity when <math>~m=n</math>.

			<span id="DirectionCosineRelations">   </span>
	<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">		<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

Tohline: /* More Generally */

2021-03-07T23:24:12Z

More Generally

← Older revision		Revision as of 23:24, 7 March 2021
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	</tr>		</tr>
	</table>		</table>
	where <math>~\delta_{mn}</math> is the ''Kronecker delta function,'' which is zero when <math>~m</math> is not equal to <math>~n</math>, unity when <math>~m=n</math>.		where <math>~\delta_{mn}</math> is the ''[https://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta function],'' which is zero when <math>~m</math> is not equal to <math>~n</math>, unity when <math>~m=n</math>.

Tohline: /* More Generally */

2021-03-07T23:22:16Z

More Generally

← Older revision		Revision as of 23:22, 7 March 2021
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	</table>		</table>
	</td></tr></table>		</td></tr></table>
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], §1.3, p. 23, Eq. (1.3.2)
			</td>
			</tr>
			</table>

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

Tohline: /* More Generally */

2021-03-07T23:18:44Z

More Generally

Show changes

Tohline: /* More Generally */

2021-03-07T22:19:54Z

More Generally

← Older revision		Revision as of 22:19, 7 March 2021
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], §1.3, p. 23, Eq. (1.3.1)		[<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], §1.3, p. 23, Eq. (1.3.1b)
	</td>		</td>
	</tr>		</tr>
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	<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">		<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
	~~Given~~ that (see [[#Examples\|above]]),
			A similar series of summation expressions — with the order of indices flipped — provides additional interrelationships between the values of the various direction cosines, namely,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\sum_{s=1}^3 \gamma_{ms}\gamma_{ns}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\delta_{mn} \, ,</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], §1.3, p. 23, Eq. (1.3.1a)
			</td>
			</tr>
			</table>

			It is particularly easy to validate each member of this set of summation expressions, given that (see [[#Examples\|above]]),
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k} \gamma_{n3} \, ;</math>		<math>~\hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k} \gamma_{n3} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	and given that the dot product of (any one of these) this unit vector with itself must be unity, we conclude that,		Each summation expression derives from the dot-product, <math>~~\mathbf{\hat{g}}_m \cdot \mathbf{\hat{g}}_n</math>, and the appreciation that (1) the dot product of any unit vector with itself (''i.e.,'' <math>~m = n</math>) gives unity, while (2) the dot product of any unit vector with either of its orthogonal partners (''i.e.,'' <math>m \ne n</math>) is zero.

			For any right-handed orthogonal coordinate system, it can also be shown that the following set of nine tabulated expressions provide one of the <math>~\gamma s</math> in terms of the others.

			and given that the dot product of any pair of the set of three unit vectors any one of these) this unit vector with itself must be unity, we conclude that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* More Generally */

2021-03-07T21:15:54Z

More Generally

← Older revision		Revision as of 21:15, 7 March 2021
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	+		+
	dz^2 \biggl[ \sum_{s=1}^3 \gamma_{s3}\gamma_{s3} \biggr]		dz^2 \biggl[ \sum_{s=1}^3 \gamma_{s3}\gamma_{s3} \biggr]
			+
			2dxdy \biggl[ \sum_{s=1}^3 \gamma_{s1}\gamma_{s2} \biggr]
			+
			2dx dz \biggl[ \sum_{s=1}^3 \gamma_{s1}\gamma_{s3} \biggr]
			+
			2dy dz \biggl[ \sum_{s=1}^3 \gamma_{s2}\gamma_{s3} \biggr]
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="center">		<td align="center">
	~~ ~~		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~dx^2 + dy^2 + dz^2 \, .
	+
	~~2dxdy \biggl[ \sum_{s=1}~~^~~3 \gamma_{s1}\gamma_{s2} \biggr]~~
	+
	~~2dx dz \biggl[ \sum_{s=1}~~^~~3 \gamma_{s1}\gamma_{s3} \biggr]~~
	+
	~~2dy~~ dz ~~\biggl[ \sum_{s=1}~~^~~3 \gamma_{s2}\gamma_{s3}~~ \~~biggr]~~
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			The last step of this derivation results from the following series of equations that interrelate the values of various direction cosines:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\sum_{s=1}^3 \gamma_{sm}\gamma_{sn}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\delta_{mn} \, ,</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], §1.3, p. 23, Eq. (1.3.1)
			</td>
			</tr>
			</table>
			where <math>~\delta_{mn}</math> is the ''Kronecker delta function,'' which is zero when <math>~m</math> is not equal to <math>~n</math>, unity when <math>~m=n</math>.

Tohline: /* More Generally */

2021-03-07T20:53:11Z

More Generally

← Older revision		Revision as of 20:53, 7 March 2021
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	+		+
	2dy dz \biggl[ \gamma_{12}\gamma_{13} + \gamma_{22}\gamma_{23} + \gamma_{32}\gamma_{33} \biggr]		2dy dz \biggl[ \gamma_{12}\gamma_{13} + \gamma_{22}\gamma_{23} + \gamma_{32}\gamma_{33} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			dx^2 \biggl[ \sum_{s=1}^3 \gamma_{s1}\gamma_{s1} \biggr]
			+
			dy^2 \biggl[ \sum_{s=1}^3 \gamma_{s2}\gamma_{s2} \biggr]
			+
			dz^2 \biggl[ \sum_{s=1}^3 \gamma_{s3}\gamma_{s3} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+
			2dxdy \biggl[ \sum_{s=1}^3 \gamma_{s1}\gamma_{s2} \biggr]
			+
			2dx dz \biggl[ \sum_{s=1}^3 \gamma_{s1}\gamma_{s3} \biggr]
			+
			2dy dz \biggl[ \sum_{s=1}^3 \gamma_{s2}\gamma_{s3} \biggr]
	</math>		</math>
	</td>		</td>
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	\gamma_{23}\gamma_{33} [ \gamma_{22}\gamma_{31} + \gamma_{32} \gamma_{21} ]		\gamma_{23}\gamma_{33} [ \gamma_{22}\gamma_{31} + \gamma_{32} \gamma_{21} ]
	+ \gamma_{21}\gamma_{22}( 1 - \gamma_{33}^2) + \gamma_{31}\gamma_{32}(1 -\gamma_{23}^2)		+ \gamma_{21}\gamma_{22}( 1 - \gamma_{33}^2) + \gamma_{31}\gamma_{32}(1 -\gamma_{23}^2)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\gamma_{21}\gamma_{22}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			( \gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} )( \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} )
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\gamma_{32}\gamma_{13} ( \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} ) - \gamma_{33}\gamma_{12} ( \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} )
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\gamma_{13} \gamma_{33} ( \gamma_{11}\gamma_{32} + \gamma_{12} \gamma_{31} ) - \gamma_{32} \gamma_{31}\gamma_{13}^2
			- \gamma_{33}^2 \gamma_{12}\gamma_{11}
	</math>		</math>
	</td>		</td>

Tohline: /* More Generally */

2021-03-07T19:31:17Z

More Generally

← Older revision		Revision as of 19:31, 7 March 2021
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	\gamma_{11} [ \gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} ]		\gamma_{11} [ \gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} ]
	+ \gamma_{12} [ \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} ]		+ \gamma_{12} [ \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} ]
			+ \gamma_{13} [ \gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\gamma_{11} [ \gamma_{32}\gamma_{13} ]
			+ \gamma_{12} [ - \gamma_{31}\gamma_{13} ]
	+ \gamma_{13} [ \gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} ]		+ \gamma_{13} [ \gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} ]
	</math>		</math>
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	<math>~		<math>~
	(\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} )( \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} )		(\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} )( \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} )
	+		+ \gamma_{21}\gamma_{22} + \gamma_{31}\gamma_{32}
	( \gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} )( \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} )		</math>
	+		</td>
	( \gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} )( \gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23} )		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			[ \gamma_{22}\gamma_{33} ( \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} ) - \gamma_{23}\gamma_{32} ( \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} ) ]
			+ \gamma_{21}\gamma_{22} + \gamma_{31}\gamma_{32}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\gamma_{23}\gamma_{33} [ \gamma_{22}\gamma_{31} + \gamma_{32} \gamma_{21} ]
			+ \gamma_{21}\gamma_{22}( 1 - \gamma_{33}^2) + \gamma_{31}\gamma_{32}(1 -\gamma_{23}^2)
	</math>		</math>
	</td>		</td>

Tohline: /* More Generally */

2021-03-06T18:34:37Z

More Generally

← Older revision		Revision as of 18:34, 6 March 2021
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	</table>		</table>

	~~Now, given~~ that (see [[#Examples\|above]]),
			<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
			Given that (see [[#Examples\|above]]),
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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

			Note as well that the dot product between any ''pair'' of these three unit vectors should be zero because they are, by definition, perpendicular to each other. This leads to the following three valid relations:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0 = \mathbf{\hat{g}}_1 \cdot \mathbf{\hat{g}}_2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(\hat\imath \gamma_{11} + \hat\jmath \gamma_{12} + \hat{k} \gamma_{13} )
			\cdot
			(\hat\imath \gamma_{21} + \hat\jmath \gamma_{22} + \hat{k} \gamma_{23} )
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\gamma_{11}\gamma_{21} + \gamma_{12}\gamma_{22} + \gamma_{13}\gamma_{23}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\gamma_{11} [ \gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} ]
			+ \gamma_{12} [ \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} ]
			+ \gamma_{13} [ \gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} ]
			</math>
			</td>
			</tr>
			</table>

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


	Notice that the coefficient that corresponds to each of the three remaining terms is given by the following expressions:		Notice that the coefficient that corresponds to each of the three remaining terms is given by the following expressions:

Tohline: /* More Generally */

2021-03-05T22:13:57Z

More Generally

← Older revision		Revision as of 22:13, 5 March 2021
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	<td align="left">		<td align="left">
	<math>~		<math>~
			(\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} )( \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} )
			+
			( \gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} )( \gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} )
			+
			( \gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} )( \gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23} )
	</math>		</math>
	</td>		</td>