Editing Appendix/Mathematics/ScaleFactors

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=Scale Factors for Orthogonal Curvilinear Coordinate Systems=

Here we lean heavily on the class notes and associated references that have been provided by [https://unidirectory.auckland.ac.nz/profile/pa-kelly P. A. Kelly] in a collection titled, ''[http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/index.html Mechanics Lecture Notes: An Introduction to Solid Mechanics]'', as they appeared online in early 2021.  See especially the subsection of [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Part III] in which the properties of ''Vectors and Tensors'' are discussed.

==Getting Started==
Following [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Kelly], we will use <math>~\hat{e}_i</math> and <math>~x_i</math> when referencing, respectively, the three (''i'' = 1,3) basis vectors and coordinate "curves" of the Cartesian coordinate system; and we will  use <math>~\hat{g}_i</math> and <math>~\Theta_i</math> when referencing, respectively, the three (''i'' = 1,3) basis vectors and coordinate curves of some other, curvilinear coordinate system.

===2D Oblique Coordinate System Example===

Consider a vector, <math>~\vec{v}</math>, which in Cartesian coordinates is described by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_1 v_x + \hat{e}_2 v_y \, .
</math>
  </td>
</tr>
</table>
Referencing Figure 1.16.4 of [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Kelly's Part III], we appreciate that in a two-dimensional (2D) '''oblique''' coordinate system where <math>~\alpha</math> is the (less than 90&deg;) angle between the two basis vectors, the same vector will be represented by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{g}_1 v^1 + \hat{g}_2 v^2 \, .
</math>
  </td>
</tr>
</table>
The angle between <math>~\hat{g}_2</math> and <math>~\hat{e}_2</math> is, (&pi;/2 - &alpha;), so we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v^2\cos\biggl(\frac{\pi}{2} - \alpha \biggr) = v^2 \sin\alpha</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~v^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{v_y}{\sin\alpha} \, .</math>
  </td>
</tr>
</table>
Next, from a visual inspection of the figure, we appreciate that <math>~v_x</math> is longer than <math>~v^1</math> by the amount, <math>~v^2\cos\alpha</math>; that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v^1 + v^2\cos\alpha = v_1 + \frac{v_y}{\tan\alpha}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ v^1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_x -  \frac{v_y}{\tan\alpha} \, .</math>
  </td>
</tr>
</table>
(These are the same pair of transformation relations that appear as Eq. (1.16.3) of [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Kelly's Part III].)

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
'''<font color="red">Covarient:'''</font> &nbsp; The set of basis vectors, <math>~\hat{g}_1</math> and <math>~\hat{g}_2</math> (note the subscript indices), that are aligned with the coordinate directions, <math>~\Theta_1</math> and <math>~\Theta_2</math>, are generically referred to as '''covariant''' base vectors.

'''<font color="red">Contravarient:'''</font> &nbsp; A second set of vectors, which will be termed '''contravariant''' base vectors, <math>~\hat{g}^1</math> and <math>~\hat{g}^2</math> (denoted by superscript indices), will be aligned with a new set of coordinate directions, <math>~\Theta^1</math> and <math>~\Theta^2</math>.

This new set of base vectors is defined as follows (see Fig. 1.15.5 of [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Kelly's Part III]):  the base vector <math>~\hat{g}^1</math> is perpendicular to <math>~\hat{g}_1</math> &#8212; that is, <math>~\hat{g}^1 \cdot \hat{g}_2 = 0</math> &#8212; and the base vector <math>~\hat{g}^2</math> is perpendicular to <math>~\hat{g}_2</math> &#8212; that is, <math>~\hat{g}_1 \cdot \hat{g}^2 = 0</math>.  Further, we ensure that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{g}_1 \cdot \hat{g}^1 = 1 \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\hat{g}_2 \cdot \hat{g}^2 = 1 \, .</math>
  </td>
</tr>
</table>

(''Verbatim'' from p. 138 of [http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/ Kelly's Part III]) <font color="orange">A good trick for remembering which are the covariant and which are the contravariant is that the third letter of the word tells us whether the word is associated with subscripts or with superscripts.  In "covariant," the "v" is pointing down, so we use subscripts; for "contravariant," the "n" is (with a bit of imagination) pointing up, so we use superscripts.</font>
</td></tr></table>

Continuing with our 2D '''oblique''' coordinate system example and appreciating that Kelly has chosen to align the <math>~\hat{g}_1</math> basis vector with the <math>~\hat{e}_1</math> (Cartesian) basis vector, we see that the transformation between the two sets of '''covariant''' basis vectors is given by the relations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{g}_1 = \hat{e}_1 \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\hat{g}_2 = \hat{e}_1\cos\alpha +  \hat{e}_2\sin\alpha \, .</math>
  </td>
</tr>
</table>
These conditions lead to the following complementary set of '''contravariant''' basis vectors:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{g}^1 = \hat{e}_1 - \hat{e}_2 \biggl( \frac{1}{\tan\alpha}\biggr) \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\hat{g}^2 = \hat{e}_2 \biggl( \frac{1}{\sin\alpha} \biggr) \, .</math>
  </td>
</tr>
</table>

Note that, as defined herein, the magnitude (''i.e.,'' scalar lengths) of these contravariant basis vectors is not unity; they are, instead,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|\hat{g}^1| \equiv \biggl[ \hat{g}^1 \cdot \hat{g}^1 \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
\biggl[\hat{e}_1 - \hat{e}_2 \biggl( \frac{1}{\tan\alpha}\biggr)\biggr] \cdot \biggl[\hat{e}_1 - \hat{e}_2 \biggl( \frac{1}{\tan\alpha}\biggr)\biggr]
\biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 + \frac{1}{\tan^2\alpha}
\biggr\}^{1 / 2}
= \frac{1}{\sin\alpha} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~|\hat{g}^2| \equiv \biggl[ \hat{g}^2 \cdot \hat{g}^2 \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\sin\alpha} \, .
</math>
  </td>
</tr>
</table>

===Some Useful Relations===

=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>

=See Also=

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