Editing Appendix/Mathematics/ScaleFactors (section)

=Examples=

Once expressions for the nine separate direction cosines are known for a system of orthogonal coordinates, then the following hold:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\hat{g}}_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k} \gamma_{n3} \, ;</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat\imath</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 \gamma_{11}
+ \mathbf{\hat{g}}_2 \gamma_{21}
+ \mathbf{\hat{g}}_3 \gamma_{31}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat\jmath</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 \gamma_{12}
+ \mathbf{\hat{g}}_2 \gamma_{22}
+ \mathbf{\hat{g}}_3 \gamma_{32}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{k}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 \gamma_{13}
+ \mathbf{\hat{g}}_2 \gamma_{23}
+ \mathbf{\hat{g}}_3 \gamma_{33}
\, .
</math>
  </td>
</tr>
</table>

Hence, the position vector is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}} = \hat\imath x + \hat\jmath y + \hat{k}z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{g}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
+
\mathbf{\hat{g}}_3 (\gamma_{31} x + \gamma_{32} y + \gamma_{33} z) 
</math>
  </td>
</tr>
</table>

==Cylindrical Coordinates==

This is drawn principally from Example #1 (starting on p. 148) of [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Kelly].
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\varpi \equiv \biggl[(x^1)^2 + (x^2)^2 \bigg]^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\tan^{-1}\biggl[\frac{x^2}{x^1} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
x^3 \, .
</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for Cylindrical Coordinates'''</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>~\varpi \equiv (x^2 + y^2 )^{1 / 2} </math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~\frac{x}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{y}{\lambda_1}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{x}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{y}{\lambda_1}</math></td>
  <td align="center"><math>~0</math></td>
</tr>

<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~\varphi \equiv \tan^{-1}\biggl[\frac{y}{x}\biggr]</math></td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~- \frac{y}{\varpi^2}</math></td>
  <td align="center"><math>~\frac{x}{\varpi^2}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~- \frac{y}{\varpi}</math></td>
  <td align="center"><math>~\frac{x}{\varpi}</math></td>
  <td align="center"><math>~0</math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~z</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
</tr>
</table>


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

<tr>
  <td align="right">
<math>~\mathbf{ \hat{g}}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \gamma_{11} + \hat\jmath \gamma_{12} + \hat{k} \gamma_{13} 
=
\hat\imath \biggl( \frac{x}{\varpi} \biggr) + \hat\jmath \biggl( \frac{y}{\varpi} \biggr)
=
\hat\imath \cos\varphi + \hat\jmath \sin\varphi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathbf{ \hat{g}}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \gamma_{21} + \hat\jmath \gamma_{22} + \hat{k} \gamma_{23} 
=
- \hat\imath \biggl( \frac{y}{\varpi} \biggr) + \hat\jmath \biggl( \frac{x}{\varpi} \biggr)
=
\hat\imath \sin\varphi + \hat\jmath \cos\varphi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathbf{ \hat{g}}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \gamma_{31} + \hat\jmath \gamma_{32} + \hat{k} \gamma_{33} 
=
\hat{k} \, .
</math>
  </td>
</tr>
</table>

And the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{g}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
+
\mathbf{\hat{g}}_3 (\gamma_{31} x + \gamma_{32} y + \gamma_{33} z) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 \biggl[\frac{x^2}{\varpi}  + \frac{y^2}{\varpi} \biggr]
+
\mathbf{\hat{g}}_2 \biggl[-\frac{xy}{\varpi}  + \frac{xy}{\varpi} \biggr]
+
\mathbf{\hat{g}}_3 z 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 \varpi
+
\mathbf{\hat{g}}_3 z \, .
</math>
  </td>
</tr>
</table>

The line element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ds^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{i=1}^3 h_i^2 d\lambda_i^2
=
d\lambda_1^2 + \varpi^2 d\lambda_2^2 + d\lambda_3^2 \, .
</math>
  </td>
</tr>
</table>

In terms of Cartesian basis vector, this is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ds^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \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 \biggl]^2
+
h_2^2 \biggl[ \biggl( \frac{\partial \lambda_2}{\partial x}\biggr)dx + \biggl( \frac{\partial \lambda_2}{\partial y}\biggr)dy + \biggl( \frac{\partial \lambda_2}{\partial z}\biggr)dz \biggl]^2
+
h_3^2 \biggl[ \biggl( \frac{\partial \lambda_3}{\partial x}\biggr)dx + \biggl( \frac{\partial \lambda_3}{\partial y}\biggr)dy + \biggl( \frac{\partial \lambda_3}{\partial z}\biggr)dz \biggl]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \gamma_{11} dx + \gamma_{12} dy + \gamma_{13} dz \biggl]^2
+
\biggl[ \gamma_{21} dx + \gamma_{22} dy + \gamma_{23} dz \biggl]^2
+
\biggl[ \gamma_{31} dx + \gamma_{32} dy + \gamma_{33} dz \biggl]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl(\frac{x}{\varpi}\biggr) dx + \biggl(\frac{y}{\varpi}\biggr) dy \biggl]^2
+
\biggl[- \biggl(\frac{y}{\varpi}\biggr) dx + \biggl(\frac{x}{\varpi}\biggr) dy \biggl]^2
+
dz^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{x}{\varpi}\biggr)^2 dx^2 + \biggl(\frac{y}{\varpi}\biggr)^2 dy^2 + 2 \biggl(\frac{x}{\varpi}\biggr)\biggl(\frac{y}{\varpi}\biggr)dxdy
~-~ 2\biggl(\frac{y}{\varpi}\biggr)\biggl(\frac{x}{\varpi}\biggr)  dx dy + \biggl(\frac{y}{\varpi}\biggr)^2 dx^2 + \biggl(\frac{x}{\varpi}\biggr)^2 dy^2
+
dz^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl(\frac{x}{\varpi}\biggr)^2+ \biggl(\frac{y}{\varpi}\biggr)^2 \biggr] dx^2 + \biggl[ \biggl(\frac{y}{\varpi}\biggr)^2  + \biggl(\frac{x}{\varpi}\biggr)^2 \biggr] dy^2
+
dz^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
dx^2 + dy^2 + dz^2 \, .
</math>
&nbsp; &nbsp; &nbsp; &nbsp; <font color="red"><b>Yes!</b></font>
  </td>
</tr>
</table>

And, when written in terms of Cartesian coordinates, the "cylindrical" differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dV = h_1 h_2 h_3~d\lambda_1 d\lambda_2 d\lambda_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \gamma_{11} dx + \gamma_{12} dy + \gamma_{13} dz \biggl]
\biggl[ \gamma_{21} dx + \gamma_{22} dy + \gamma_{23} dz \biggl]
\biggl[ \gamma_{31} dx + \gamma_{32} dy + \gamma_{33} dz \biggl]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{x}{\varpi} \biggr) dx + \biggl( \frac{y}{\varpi} \biggr)  dy \biggl]
\biggl[ \biggl( -\frac{y}{\varpi} \biggr) dx + \biggl( \frac{x}{\varpi} \biggr) dy \biggl]
\biggl[ dz \biggl]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~dz \biggl\{
\biggl(\frac{x}{\varpi} \biggr) dx
\biggl[ \biggl( -\frac{y}{\varpi} \biggr) dx + \biggl( \frac{x}{\varpi} \biggr) dy \biggl]
+
\biggl( \frac{y}{\varpi} \biggr)  dy 
\biggl[ \biggl( -\frac{y}{\varpi} \biggr) dx + \biggl( \frac{x}{\varpi} \biggr) dy \biggl]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~dz \biggl\{
\biggl(\frac{-xy}{\varpi^2} \biggr) dx^2
+
\biggl(\frac{x^2}{\varpi^2} \biggr) dx dy
-
\biggl( \frac{y^2}{\varpi^2} \biggr)  dxdy 
+
\biggl( \frac{xy}{\varpi^2} \biggr)  dy^2 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{dz}{\varpi^2} \biggl\{
xy (dy^2 - dx^2)
+
(x^2 - y^2) dx dy
\biggr\} \, .
</math>
  </td>
</tr>
</table>

==T10 Coordinates==

===Position Vector===

Pulling from our accompanying [[Appendix/Ramblings/ConcentricEllipsoidalCoordinates#PartBCoordinatesT10|Table of Direction Cosine Components for T10 Coordinates]], the position vector is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{g}}_2 (\gamma_{41} x + \gamma_{42} y + \gamma_{43} z)
+
\mathbf{\hat{g}}_3 (\gamma_{51} x + \gamma_{52} y + \gamma_{53} z) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{g}}_1 \biggl[ 
\lambda_1^2 \ell_{3D}
\biggr]
+
\mathbf{\hat{g}}_2 \biggl[
1 + \frac{1}{q^2}  - \frac{2}{p^2}
\biggr]\frac{xq^2y p^2z}{\mathcal{D}}
+
\mathbf{\hat{g}}_3 \biggl[
-x^2(2q^4y^2 + p^4z^2) + q^2y^2(p^4z^2 + 2x^2) + p^2z^2(x^2-q^4y^2)
\biggr] \frac{\ell_{3D}}{\mathcal{D}}
</math>
  </td>
</tr>
</table>

===Line Element===

====T10 Example====
In terms of Cartesian basis vector, the line element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ds^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \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 \biggl]^2
+
h_2^2 \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 \biggl]^2
+
h_3^2 \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 \biggl]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \gamma_{11} dx + \gamma_{12} dy + \gamma_{13} dz \biggl]^2
+
\biggl[ \gamma_{41} dx + \gamma_{42} dy + \gamma_{43} dz \biggl]^2
+
\biggl[ \gamma_{51} dx + \gamma_{52} dy + \gamma_{53} dz \biggl]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ (x) dx + (q^2 y) dy + (p^2 z) dz \biggl]^2 \ell_{3D}^2
+
\biggl[ (q^2y p^2z) dx +( x p^2z) dy - ( 2xq^2y) dz \biggl]^2 \frac{1}{\mathcal{D}^2}
+
\biggl[ -x(2q^4y^2 + p^4z^2) dx + q^2y(p^4z^2 + 2x^2) dy + p^2z(x^2 - q^4y^2) dz \biggl]^2 \frac{\ell_{3D}^2}{\mathcal{D}^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{\mathcal{D}}{\ell_{3D}}\biggr)^2 ds^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ (x) dx + (q^2 y) dy + (p^2 z) dz \biggl]^2 \mathcal{D}^2
+
\biggl[ (q^2y p^2z) dx +( x p^2z) dy - ( 2xq^2y) dz \biggl]^2 (\ell_{3D})^{-2}
+
\biggl[ -x(2q^4y^2 + p^4z^2) dx + q^2y(p^4z^2 + 2x^2) dy + p^2z(x^2 - q^4y^2) dz \biggl]^2 \, .
</math>
  </td>
</tr>
</table>
Notice that the coefficient that corresponds to each term is given by the following expressions:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[dy^2]:</math>
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
q^4y^2 \mathcal{D}^2
+ x^2p^4z^2 \ell_{3D}^{-2}
+ q^4y^2(p^4z^2 + 2x^2)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
q^4y^2 \biggl[ q^4y^2p^4z^2 + x^2p^4z^2 + 4x^2q^4y^2 \biggr]
+ x^2p^4z^2 \biggl[ x^2 + q^4y^2 + p^4z^2 \biggr]
+ q^4y^2(p^8z^4 + 4x^2p^4z^2 + 4x^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
q^8y^4(4x^2 +  p^4z^2)
+
6x^2 q^4y^2 p^4z^2 
+ p^8z^4( x^2 + q^4y^2)
+ x^4( p^4z^2 + 4q^4y^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mathcal{D}}{\ell_{3D}}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~[dx^2]:</math>
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
x^2 \mathcal{D}^2 + q^4y^2 p^4z^2 \ell_{3D}^{-2} + x^2(2q^4y^2 + p^4z^2)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2 (q^4y^2p^4z^2 + x^2p^4z^2 + 4x^2q^4y^2) 
+ 
q^4y^2 p^4z^2 (x^2 + q^4 y^2 + p^4z^2)
+ 
x^2(4q^8y^4 + 4q^4y^2p^4z^2 + p^8z^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
6x^2 q^4y^2p^4z^2 
+ x^4(4q^4y^2 + p^4z^2)
+q^8y^4(4x^2 + p^4z^2)
+ p^8z^2(x^2 + q^4y^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mathcal{D}}{\ell_{3D}}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~[dx\cdot dy]:</math>
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
2xq^2y \mathcal{D}^2 +2q^2yp^2z \cdot xp^2z (\ell_{3D})^{-2} - 2x(2q^4y^2 + p^4z^2)q^2y(p^4z^2 + 2x^2)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2xq^2y[q^4y^2p^4z^2 + x^2p^4z^2 + 4x^2q^4y^2 + p^4z^2(x^2 + q^4 y^2 + p^4 z^2) - 2q^4y^2 (p^4z^2 + 2x^2) - p^4z^2(p^4z^2 + 2x^2)]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2xq^2y[ ~0~]  = 0\, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~ds^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \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 \biggl]^2
+
h_2^2 \biggl[ \biggl( \frac{\partial \lambda_2}{\partial x}\biggr)dx + \biggl( \frac{\partial \lambda_2}{\partial y}\biggr)dy + \biggl( \frac{\partial \lambda_2}{\partial z}\biggr)dz \biggl]^2
+
h_3^2 \biggl[ \biggl( \frac{\partial \lambda_3}{\partial x}\biggr)dx + \biggl( \frac{\partial \lambda_3}{\partial y}\biggr)dy + \biggl( \frac{\partial \lambda_3}{\partial z}\biggr)dz \biggl]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \gamma_{11} dx + \gamma_{12} dy + \gamma_{13} dz \biggl]^2
+
\biggl[ \gamma_{21} dx + \gamma_{22} dy + \gamma_{23} dz \biggl]^2
+
\biggl[ \gamma_{31} dx + \gamma_{32} dy + \gamma_{33} dz \biggl]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
dx^2 \biggl[ \gamma_{11}^2  + \gamma_{21}^2  + \gamma_{31}^2 \biggr]
+
dy^2 \biggl[ \gamma_{12}^2 + \gamma_{22}^2 + \gamma_{32}^2  \biggr]
+
dz^2 \biggl[  \gamma_{13}^2  + \gamma_{23}^2 dz^2  + \gamma_{33}^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
2dxdy \biggl[ \gamma_{11}\gamma_{12} + \gamma_{21}\gamma_{22} + \gamma_{31}\gamma_{32} \biggr]
+
2dx dz \biggl[ \gamma_{11}\gamma_{13} + \gamma_{21}\gamma_{23} + \gamma_{31}\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">
&nbsp;
  </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]
+
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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~dx^2 + dy^2 + dz^2 \, .
</math>
  </td>
</tr>
</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>], &sect;1.3, p. 23, Eq. (1.3.1b)
  </td>
</tr>
</table>
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">&nbsp; &nbsp;</span>
<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

A similar series of summation expressions &#8212; with the order of indices flipped &#8212; 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>], &sect;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">

<tr>
  <td align="right">
<math>~\mathbf{\hat{g}}_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k} \gamma_{n3} \, .</math>
  </td>
</tr>
</table>
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 details how each one of the <math>~\gamma</math>'s is related to various algebraic combinations of the others.

<!-- Table Detailing Orthogonality Conditions -->

<table  border="1" align="center" cellpadding="10">
<tr>
  <td align="left">
<table border="0" cellpadding="5" align="left">

<tr>
  <td align="right">
<math>~\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="center">
<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>~\gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<table border="0" cellpadding="5" align="right">

<tr>
  <td align="right">
<math>~\gamma_{31}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{32}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{33}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21} \, .</math>
  </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>], &sect;1.3, p. 23, Eq. (1.3.2)
  </td>
</tr>
</table>

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

===Volume Element===

<!-- DIRECTION COSINE RELATIONS -->

<table  border="1" align="center" cellpadding="10">
<tr>
  <td align="left">
<table border="0" cellpadding="5" align="left">

<tr>
  <td align="right">
<math>~\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{44}\gamma_{55} - \gamma_{45}\gamma_{54} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{14}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{45}\gamma_{51} - \gamma_{41}\gamma_{55} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{15}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{41}\gamma_{54} - \gamma_{44}\gamma_{51} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{41}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{54}\gamma_{15} - \gamma_{55}\gamma_{14} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{44}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{55}\gamma_{11} - \gamma_{51}\gamma_{15} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{45}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{51}\gamma_{14} - \gamma_{54}\gamma_{11} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<table border="0" cellpadding="5" align="right">

<tr>
  <td align="right">
<math>~\gamma_{51}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{14}\gamma_{45} - \gamma_{15}\gamma_{44} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{54}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{15}\gamma_{41} - \gamma_{11}\gamma_{45} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{55}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{11}\gamma_{44} - \gamma_{14}\gamma_{41} \, .</math>
  </td>
</tr>
</table>
</td></tr></table>

<!-- VOLUME ELEMENT -->

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

<tr>
  <td align="right">
<math>~dV = 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>~
\biggl[ \gamma_{11} dx + \gamma_{14} dy + \gamma_{15}  dz \biggl]
\biggl[ \gamma_{41} dx + \gamma_{44} dy + \gamma_{45} dz \biggl]
\biggl[ \gamma_{51} dx + \gamma_{54} dy + \gamma_{55} dz \biggl]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \gamma_{11} dx + \gamma_{14} dy + \gamma_{15}  dz \biggl]
\biggl[
dx^2 \gamma_{41}\gamma_{51}
+ dy^2 \gamma_{44}\gamma_{54}
+ dz^2 \gamma_{45}\gamma_{55}
+ dx dy ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} )
+ dx dz ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} )
+ dy dz ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
dx^3 \gamma_{11}  \gamma_{41}\gamma_{51}
+ dx dy^2 \gamma_{11} \gamma_{44}\gamma_{54}
+ dx dz^2 \gamma_{11} \gamma_{45}\gamma_{55}
+ dx^2 dy \gamma_{11}  ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} )
+ dx^2 dz \gamma_{11}  ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} )
+ dx dy dz \gamma_{11}  ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
dx^2 dy \gamma_{14}  \gamma_{41}\gamma_{51}
+ dy^3 \gamma_{14}  \gamma_{44}\gamma_{54}
+ dy dz^2 \gamma_{14}  \gamma_{45}\gamma_{55}
+ dx dy^2 \gamma_{14}  ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} )
+ dx dy dz \gamma_{14} ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} )
+ dy^2 dz \gamma_{14}  ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
dx^2 dz \gamma_{15}   \gamma_{41}\gamma_{51}
+ dy^2 dz \gamma_{15}   \gamma_{44}\gamma_{54}
+ dz^3 \gamma_{15}   \gamma_{45}\gamma_{55}
+ dx dy dz \gamma_{15}   ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} )
+ dx dz^2 \gamma_{15}   ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} )
+ dy dz^2 \gamma_{15}   ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
dx dy dz [ \gamma_{11}  ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} ) +  \gamma_{14} ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} ) + \gamma_{15}   ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} ) ]
+ dx^3 \gamma_{11}  \gamma_{41}\gamma_{51}
+ dy^3 \gamma_{14}  \gamma_{44}\gamma_{54}
+ dz^3 \gamma_{15}   \gamma_{45}\gamma_{55}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
dx^2 dy [\gamma_{14}  \gamma_{41}\gamma_{51} + \gamma_{11}  ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} ) ]
+ dy dz^2 [ \gamma_{14}  \gamma_{45}\gamma_{55} + \gamma_{15}   ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} ) ]
+ dx dy^2 [ \gamma_{14}  ( \gamma_{41}\gamma_{54} + \gamma_{44}\gamma_{51} ) + \gamma_{11} \gamma_{44}\gamma_{54} ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
dx^2 dz [ \gamma_{15}   \gamma_{41}\gamma_{51} +  \gamma_{11}  ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} ) ]
+ dy^2 dz [ \gamma_{15}   \gamma_{44}\gamma_{54} + \gamma_{14}  ( \gamma_{44}\gamma_{55} + \gamma_{45}\gamma_{54} ) ]
+ dx dz^2 [ \gamma_{15}   ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} ) + \gamma_{11} \gamma_{45}\gamma_{55} ]
</math>
  </td>
</tr>
</table>

<!--  EXAMPLES -->

For example, the coefficient of <math>~dx dz^2</math>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{15}   ( \gamma_{41}\gamma_{55} + \gamma_{45}\gamma_{51} ) + \gamma_{11} \gamma_{45}\gamma_{55}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^2z \ell_{3D} \biggl\{
\frac{q^2yp^2z \ell_{3D}}{\mathcal{D}^2}\biggl[ p^2z(x^2-q^4y^2) \biggr]
+ \frac{2xq^2y \ell_{3D}}{\mathcal{D}^2}\biggl[x(2q^4y^2 + p^4 z^2)  \biggr]
\biggr\}
-
\biggl\{
x \ell_{3D} \cdot \frac{2xq^2y \ell_{3D} }{\mathcal{D}^2} \biggl[ p^2z(x^2-q^4y^2) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^2z \biggl(\frac{ \ell_{3D}}{\mathcal{D}}\biggr)^2 \biggl\{
q^2yp^2z \biggl[ p^2z(x^2-q^4y^2) \biggr]
+ 2xq^2y \biggl[x(2q^4y^2 + p^4 z^2)  \biggr]
\biggr\}
-
\biggl\{
2x^2q^2y  \biggl[ p^2z(x^2-q^4y^2) \biggr]
\biggr\} \biggl(\frac{ \ell_{3D}}{\mathcal{D}}\biggr)^2
</math>
  </td>
</tr>
</table>


And the coefficient of <math>~dx^3</math> is:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \gamma_{11}  \gamma_{41}\gamma_{51}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \ell_{3D} \biggl( \frac{q^2y p^2z}{\mathcal{D}} \biggr) - \frac{\ell_{3D}}{\mathcal{D}} \biggl[x( 2q^4y^2 + p^4z^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
q^2y p^2z  - ( 2q^4y^2 + p^4z^2 )
\biggr] \frac{x \ell_{3D}}{\mathcal{D}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[
(q^4y^2 - 2q^2y p^2z  + p^4z^2 ) +q^4y^2 +q^2y p^2z
\biggr] \frac{x \ell_{3D}}{\mathcal{D}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[
(q^2y -  p^2 z )^2 +q^2y (q^2y + p^2z)
\biggr] \frac{x \ell_{3D}}{\mathcal{D}} \, .
</math>
  </td>
</tr>
</table>