Editing PGE/AStarScheme (section)

===Curvature Terms===
Converting from a Lagrangian to an Eulerian time-derivative, Equation [I.A.5] becomes,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\partial_t(\rho\boldsymbol{v}) + (\boldsymbol{v}\cdot\nabla)(\rho \boldsymbol{v}) + (\rho \boldsymbol{v})\nabla\cdot \boldsymbol{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\nabla P - \rho \nabla\Phi \, .
</math>
  </td>
</tr>
</table>
</div>
Now, if you're not working in Cartesian coordinates, care must be taken when dealing with the second term on the left-hand-side of this equation because when the gradient operator acts on a vector quantity (in this case, <math>~\rho\boldsymbol{v}</math>), various curvature terms will arise reflecting the fact that, in general, the unit vectors of your curvilinear coordinate system point in different directions as the fluid moves to different locations in space.  Quite generally, though, for the <math>~j^\mathrm{th}</math> component of the equation of motion we may isolate these curvature terms as follows:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\partial_t(\rho v_j)  + \nabla\cdot (\rho v_j \boldsymbol{v}) + (\mathrm{curvature})_j
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\nabla_j P - \rho \nabla_j \Phi \, ,
</math>
  </td>
</tr>
</table>
</div>
where, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~(\mathrm{curvature})_j
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\Sigma_{i=1,2,3} \{ [ (\rho v_i)/(h_i h_j) ] [ v_j \partial_{\xi_i} h_j - v_i \partial_{\xi_j} h_i ] \} \, .
</math>
  </td>
</tr>
</table>
</div>

So, for example, in cylindrical coordinates where <math>~h_1 = h_\varpi = 1, h_2 = h_\theta = \varpi,</math> and <math>~h_3 = h_z = 1,</math>
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~(\mathrm{curvature})_\varpi
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
[ (\rho v_\theta)/\varpi ] [ -v_\theta ] = - \rho v_\theta^2/\varpi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~(\mathrm{curvature})_\theta
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
[ (\rho v_\varpi)/\varpi ] [ v_\theta ] = \rho v_\varpi v_\theta/\varpi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~(\mathrm{curvature})_z
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0 \, ;
</math>
  </td>
</tr>
</table>
</div>
Thus, in cylindrical coordinates the three components of the equation of motion become,

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\partial_t(\rho v_\varpi)  + \nabla\cdot (\rho v_\varpi \boldsymbol{v}) 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\nabla_\varpi P - \rho \nabla_\varpi \Phi + \rho v_\theta^2/\varpi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\partial_t(\rho v_\theta)  + \nabla\cdot (\rho v_\theta \boldsymbol{v}) 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\nabla_\theta P - \rho \nabla_\theta \Phi - \rho v_\varpi v_\theta/\varpi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\partial_t(\rho v_z)  + \nabla\cdot (\rho v_z \boldsymbol{v}) 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\nabla_z P - \rho \nabla_z \Phi \, .
</math>
  </td>
</tr>
</table>
</div>