Editing Appendix/Ramblings/DirectionCosines (section)

===Orthogonality===
How can we check to make sure that the coordinate <math>\xi_1</math> is everywhere orthogonal to the coordinate <math>\xi_2</math>?  Well, for an orthogonal system, the unit vectors should be everywhere perpendicular to one another, that is, the dot product of two (different) unit vectors should be zero at all coordinate positions.  Drawing on the above unit-vector transformation expressions, this means that, for <math>m \ne n</math>,
<div align="center">
<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 
</math><br />

<math>
\Rightarrow ~~~~~ \sum_{s=1}^3 \gamma_{ms}\gamma_{ns} = 0 .
</math>
</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>.

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">
<math>
\frac{\partial\varpi}{\partial x} = \frac{\partial}{\partial x} (x^2 + y^2)^{1/2} = \frac{x}{\varpi} ,
</math>
</div>
and,
<div align="center">
<math>
\frac{\partial\varpi}{\partial y} = \frac{\partial}{\partial x} (x^2 + y^2)^{1/2} = \frac{y}{\varpi} ,
</math>
</div>
Hence,
<div align="center">
<math>
\frac{\partial\xi_i}{\partial x} = \frac{\partial\xi_i}{\partial \varpi}\frac{\partial\varpi}{\partial x} = \biggl(\frac{x}{\varpi}\biggr) \frac{\partial\xi_i}{\partial \varpi} ,
</math>
</div>
and,
<div align="center">
<math>
\frac{\partial\xi_i}{\partial y} = \frac{\partial\xi_i}{\partial \varpi}\frac{\partial\varpi}{\partial y} = \biggl(\frac{y}{\varpi}\biggr) \frac{\partial\xi_i}{\partial \varpi} .
</math>
</div>
Therefore also,
<div align="center">
<math>
\biggl( \frac{\partial\xi_i}{\partial x}  \biggr)^2 + \biggl( \frac{\partial\xi_i}{\partial y }  \biggr)^2 = \biggl( \frac{\partial\xi_i}{\partial\varpi} \biggr)^2
</math><br />

<math>
\Rightarrow ~~~~~ h_i^2 =  \biggl[ \biggl(\frac{\partial\xi_i}{\partial \varpi} \biggr)^2 + \biggl(\frac{\partial\xi_i}{\partial z} \biggr)^2 \biggr]^{-1} .
</math>

</div>

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">
<math>~\sum_s \gamma_{1s}\gamma_{2s} = 0 .</math>
</div>
This means that if <math>~\xi_1</math> is orthogonal to <math>~\xi_2</math>,
<div align="center">
<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 y} \cdot h_2 \frac{\partial\xi_2}{\partial y} + 
h_1 \frac{\partial\xi_1}{\partial z} \cdot h_2 \frac{\partial\xi_2}{\partial z}= 0
</math><br /><br />

<math>
\Rightarrow ~~~~~ 
h_1 h_2\biggl[ \biggl( \frac{x^2}{\varpi^2} \biggr) \frac{\partial\xi_1}{\partial \varpi} \cdot  \frac{\partial\xi_2}{\partial \varpi} + 
\biggl( \frac{y^2}{\varpi^2} \biggr) \frac{\partial\xi_1}{\partial \varpi} \cdot \frac{\partial\xi_2}{\partial \varpi} + 
\frac{\partial\xi_1}{\partial z} \cdot \frac{\partial\xi_2}{\partial z} \biggr] = 0 .

</math>
</div>

Hence,
<span id="DC.02"><table align="right" border="1" cellpadding="10" width="10%">
<tr><th><font color="darkblue">DC.02</font></th></tr>
</table></span>
<table align="center" border="1" cellpadding="10">
<tr>
  <th align="center">
<font color="blue">
An Example Orthogonality Condition
</font>
  </th>
</tr>
<tr>
  <td align="center">
<math>
\frac{\partial\xi_1}{\partial \varpi} \cdot  \frac{\partial\xi_2}{\partial \varpi} = - 
\frac{\partial\xi_1}{\partial z} \cdot \frac{\partial\xi_2}{\partial z} .
</math>
  </td>
</tr>
</table>