Editing Appendix/Ramblings/T3Integrals (section)

==Vector Derivatives==

For orthogonal coordinate systems, the time-rate-of-change of the three unit vectors are given by the expressions,
<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d}{dt}\hat{e}_1 
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\hat{e}_2 A + \hat{e}_3 B
</math>
  </td>
</tr>
<tr>
  <td align="right">  
<math>
\frac{d}{dt}\hat{e}_2 
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
- \hat{e}_1 A + \hat{e}_3 C
</math>
  </td>
</tr>
<tr>
  <td align="right">  
<math>
\frac{d}{dt}\hat{e}_3 
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
- \hat{e}_1 B - \hat{e}_2 C
</math>
  </td>
</tr> 
</table>

where,
<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
A
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
\frac{\dot{\lambda}_2}{h_1} \frac{\partial h_2}{\partial \lambda_1} -
\frac{\dot{\lambda}_1}{h_2} \frac{\partial h_1}{\partial \lambda_2}
</math>
  </td>
</tr>
<tr>
  <td align="right">  
<math>
B
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
\frac{\dot{\lambda}_3}{h_1} \frac{\partial h_3}{\partial \lambda_1} -
\frac{\dot{\lambda}_1}{h_3} \frac{\partial h_1}{\partial \lambda_3}
</math>
  </td>
</tr>
<tr>
  <td align="right">  
<math>
C
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
\frac{\dot{\lambda}_3}{h_2} \frac{\partial h_3}{\partial \lambda_2} -
\frac{\dot{\lambda}_2}{h_3} \frac{\partial h_2}{\partial \lambda_3}
</math>
  </td>
</tr> 
</table>

Another way of expressing this involves Christoffel symbols and is most easily written using index notation.
  <div align="center">
<math>
\frac{d}{dt} \hat{e}_c = \frac{h_a}{h_c} \ \Gamma^a_{bc} \dot{\lambda}_b \ \hat{e}_a \ \ (a \ne c)
</math>
  </div>

Here, we have been admittedly slopping with the placement and notation of indices in order to best accommodate the notation we have been using up to here.  The <math>b</math> index is summed over all the coordinates.  The <math>a</math> index is summed over all coordinates EXCEPT the <math>c</math> coordinate.  The <math>c</math> index is NOT summed over because it is a free index, meaning that it can equal any of the coordinates depending on which unit vector you want to differentiate.  

Writing this out for each of the individual unit vectors, and striking through terms that are automatically zero when dealing with an orthogonal coordinate system, produces

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d}{dt} \hat{e}_1
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\overbrace{\frac{h_2}{h_1} \left( \Gamma^2_{11} \dot{\lambda}_1 + \Gamma^2_{21} \dot{\lambda}_2 + \cancel{\Gamma^2_{31} \dot{\lambda}_3} \right)}^{A} \hat{e}_2 + \overbrace{\frac{h_3}{h_1} \left( \Gamma^3_{11} \dot{\lambda}_1 + \cancel{\Gamma^3_{21} \dot{\lambda}_2} + \Gamma^3_{31} \dot{\lambda}_3 \right)}^{B} \hat{e}_3
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{d}{dt} \hat{e}_2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\overbrace{\frac{h_1}{h_2} \left( \Gamma^1_{12} \dot{\lambda}_1 + \Gamma^1_{22} \dot{\lambda}_2 + \cancel{\Gamma^1_{32} \dot{\lambda}_3} \right)}^{-A} \hat{e}_1 + \overbrace{\frac{h_3}{h_2} \left( \cancel{\Gamma^3_{12} \dot{\lambda}_1} + \Gamma^3_{22} \dot{\lambda}_2 + \Gamma^3_{32} \dot{\lambda}_3 \right)}^{C} \hat{e}_3
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{d}{dt} \hat{e}_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\overbrace{\frac{h_1}{h_3} \left( \Gamma^1_{13} \dot{\lambda}_1 + \cancel{\Gamma^1_{23} \dot{\lambda}_2} + \Gamma^1_{33} \dot{\lambda}_3 \right)}^{-B} \hat{e}_1 + \overbrace{\frac{h_2}{h_3} \left( \cancel{\Gamma^2_{13} \dot{\lambda}_1} + \Gamma^2_{23} \dot{\lambda}_2 + \Gamma^2_{33} \dot{\lambda}_3 \right)}^{-C} \hat{e}_2
</math>
  </td>
</tr>
</table>

where, quite generally, the 27 Christoffel symbols are,

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\Gamma^1_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial_1 h_1}{h_1}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^1_{12} = \Gamma^1_{21}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial_2 h_1}{h_1}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^1_{13} = \Gamma^1_{31}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^1_{22}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{h_2}{h_1} \frac{\partial_1 h_2}{h_1}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\cancel{\Gamma^1_{23}} = \cancel{\Gamma^1_{32}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^1_{33}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{h_3}{h_1} \frac{\partial_1 h_3}{h_1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Gamma^2_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{h_1}{h_2} \frac{\partial_2 h_1}{h_2}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^2_{12} = \Gamma^2_{21}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial_1 h_2}{h_2}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\cancel{\Gamma^2_{13}} = \cancel{\Gamma^2_{31}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^2_{22}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial_2 h_2}{h_2}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^2_{23} = \Gamma^2_{32}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^2_{33}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{h_3}{h_2} \frac{\partial_2 h_3}{h_2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Gamma^3_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\cancel{\Gamma^3_{12}} = \cancel{\Gamma^3_{21}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^3_{13} = \Gamma^3_{31}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial_1 h_3}{h_3}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^3_{22}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^3_{23} = \Gamma^3_{32}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial_2 h_3}{h_3}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\Gamma^3_{33}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>
</table>
Notice that it is the Christoffel symbols labeled with all three of the coordinate indices that are automatically zero for orthogonal coordinate systems.

For additional details surrounding the Christoffel symbols (the significant role they play in the field equations, and how they can be calculated) visit this page [[User:Jaycall/KillingVectorApproach|coming soon]].