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

Tohline: /* See Also */

2020-10-15T15:51:56Z

Tohline at 15:26, 15 October 2020

2020-10-15T15:26:32Z

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			=See Also=

			<ul>
			<li>Elliptic Cylinder Coordinates</li>
			</ul>

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Tohline: /* Position Vector */

2020-10-14T21:14:50Z

Position Vector

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Tohline: /* Scale Factors */

2020-10-14T21:14:35Z

Scale Factors

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	<tr>		<tr>
	<td align="center" colspan="3">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, Eq. (1.3.4)</font> ]</td>		<td align="center" colspan="3">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, p. 24, Eq. (1.3.4)</font> ]</td>
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Tohline: /* Basic Definitions and Relations */

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Basic Definitions and Relations

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	<td align="center" colspan="7">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, Eq. (1.3.5)</font> ]</td>		<td align="center" colspan="7">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, p. 25, Eq. (1.3.5)</font> ]</td>
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	<td align="center" colspan="5">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, Eq. (1.3.1)</font> ]</td>		<td align="center" colspan="5">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, p. 23, Eq. (1.3.1)</font> ]</td>
	</tr>		</tr>
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Tohline: /* Orthogonality */

2020-10-14T21:12:36Z

Orthogonality

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	<math>		<math>
	\hat{e}_m \cdot \hat{e}_n = \biggl[ \hat\imath \gamma_{m1} + \hat\jmath \gamma_{m2} + \hat{k}\gamma_{m3} \biggr] \cdot \biggl[ \hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k}\gamma_{n3} \biggr] = \gamma_{m1}\gamma_{n1} + \gamma_{m2}\gamma_{n2} + \gamma_{m1}\gamma_{n2} = 0		\hat{e}_m \cdot \hat{e}_n = \biggl[ \hat\imath \gamma_{m1} + \hat\jmath \gamma_{m2} + \hat{k}\gamma_{m3} \biggr] \cdot \biggl[ \hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k}\gamma_{n3} \biggr] = \gamma_{m1}\gamma_{n1} + \gamma_{m2}\gamma_{n2} + \gamma_{m1}\gamma_{n2} = 0
	</math~~><br /~~><br />		</math><br />

	<math>		<math>

Tohline: /* Orthogonality */

2020-10-14T21:11:30Z

Orthogonality

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	<math>		<math>
	\Rightarrow ~~~~~ \sum_{s=1,3} \gamma_{ms}\gamma_{ns} = 0 .		\Rightarrow ~~~~~ \sum_{s=1}^3 \gamma_{ms}\gamma_{ns} = 0 .
	</math>		</math>
	</div>		</div>
	This is precisely the condition enforced on direction cosines in conjunction with their definition, shown above as [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.01\|Equation DC.01]]. Notice as well that, when <math>m = n</math>, Equation DC.01 is equivalent to the statement, <math>\hat{e}_m\cdot \hat{e}_m = 1</math>.		This is precisely the condition enforced on direction cosines in conjunction with their definition, shown above as [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.01\|Equation DC.01]]. Notice as well that, when <math>~m = n</math>, Equation DC.01 is equivalent to the statement, <math>~\hat{e}_m\cdot \hat{e}_m = 1</math>.

	Here we'll illustrate how orthogonality can be checked for any axisymmetric coordinate system; that is, we'll examine behavior only in the <math>(\varpi,z)</math> plane. First, note that,		Here we'll illustrate how orthogonality can be checked for any axisymmetric coordinate system; that is, we'll examine behavior only in the <math>~(\varpi,z)</math> plane. First, note that,
	<div align="center">		<div align="center">
	<math>		<math>
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	The relationship between the direction cosines when <math>m \ne n</math> gives a key orthogonality condition. Take, for example, <math>m=1</math> and <math>n=2</math>:		The relationship between the direction cosines when <math>m \ne n</math> gives a key orthogonality condition. Take, for example, <math>~m=1</math> and <math>~n=2</math>:
	<div align="center">		<div align="center">
	<math>\sum_s \gamma_{1s}\gamma_{2s} = 0 .</math>		<math>~\sum_s \gamma_{1s}\gamma_{2s} = 0 .</math>
	</div>		</div>
	This means that if <math>\xi_1</math> is orthogonal to <math>\xi_2</math>,		This means that if <math>~\xi_1</math> is orthogonal to <math>~\xi_2</math>,
	<div align="center">		<div align="center">
	<math>		<math>~
	h_1 \frac{\partial\xi_1}{\partial x} \cdot h_2 \frac{\partial\xi_2}{\partial x} +		h_1 \frac{\partial\xi_1}{\partial x} \cdot h_2 \frac{\partial\xi_2}{\partial x} +
	h_1 \frac{\partial\xi_1}{\partial y} \cdot h_2 \frac{\partial\xi_2}{\partial y} +		h_1 \frac{\partial\xi_1}{\partial y} \cdot h_2 \frac{\partial\xi_2}{\partial y} +

Tohline: /* Unit Vectors */

2020-10-14T21:05:24Z

Unit Vectors

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	notice that we have liberally used the idea that, for orthogonal systems, <math>\gamma_{nm} = \gamma_{mn}</math>.		notice that we have liberally used the idea that, for orthogonal systems, <math>~\gamma_{nm} = \gamma_{mn}</math>.

	===Orthogonality===		===Orthogonality===

Tohline: /* Scale Factors */

2020-10-14T21:03:55Z

Scale Factors

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	<math>~\biggl[ \biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 \biggr] \, .</math>		<math>~\biggl[ \biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 \biggr] \, .</math>
	</td>		</td>
			</tr>
			<tr>
			<td align="center" colspan="3">[ [[User:Tohline/Appendix/References#MF53\|MF53]], <font color="#00CC00">§1.3, Eq. (1.3.4)</font> ]</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Scale Factors */

2020-10-14T21:01:37Z

Scale Factors

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	===Scale Factors===		===Scale Factors===
	The above relations can be used to define the scale factors <math>(h_1, h_2, h_3)</math>. For example,		The above relations can be used to define the scale factors <math>~(h_1, h_2, h_3)</math>. For example,
	<~~div~~ align="center">		<table border="0" cellpadding="5" align="center">
	~~<math>~~
	~~\sum_s \gamma_{1s}\gamma_{1s} = \sum_s \biggl( h_1 \frac{\partial\xi_1}{\partial x_s} \biggr)^2 = 1~~
	~~</math><br /~~>

	<math>		<tr>
	\Rightarrow ~~~~~ h_1^2 = \biggl[ \biggl(\frac{\partial\xi_1}{\partial x} \biggr)^2 + \biggl(\frac{\partial\xi_1}{\partial y} \biggr)^2 + \biggl(\frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1} ;		<td align="right">
	</~~math~~>		<math>~\sum_s \gamma_{1s}\gamma_{1s}</math>
	</~~div~~>		</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\sum_s \biggl( h_1 \frac{\partial\xi_1}{\partial x_s} \biggr)^2 = 1</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ h_1^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \biggl(\frac{\partial\xi_1}{\partial x} \biggr)^2 + \biggl(\frac{\partial\xi_1}{\partial y} \biggr)^2 + \biggl(\frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1} ;</math>
			</td>
			</tr>
			</table>
	or,		or,
	<~~div~~ align="center">		<table border="0" cellpadding="5" align="center">
	<math>
	\sum_s \gamma_{1s}\gamma_{1s} = \sum_s \biggl( \frac{1}{h_1} \frac{\partial x_s}{\partial\xi_1} \biggr)^2 = 1		<tr>
	</~~math~~><br />		<td align="right">
			<math>~\sum_s \gamma_{1s}\gamma_{1s}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\sum_s \biggl( \frac{1}{h_1} \frac{\partial x_s}{\partial\xi_1} \biggr)^2 = 1</math>
			</td>
			</tr>

	<math>		<tr>
	\Rightarrow ~~~~~ h_1^2 = \biggl[ \biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 \biggr] .		<td align="right">
	</~~math~~>		<math>~\Rightarrow ~~~ h_1^2</math>
	</~~div~~>		</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 \biggr] \, .</math>
			</td>
			</tr>
			</table>

	===Unit Vectors===		===Unit Vectors===

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	<ul>		<ul>
	<li>Elliptic Cylinder Coordinates</li>		<li>[[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates\|Elliptic Cylinder Coordinates]]</li>
	</ul>		</ul>


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