Editing Appendix/Ramblings/HybridSchemeOld (section)

===Eulerian Representation===
We can shift any of the PDEs from a Lagrangian to an Eulerian representation &#8212; and thereby use them to follow the time-rate of change of physical variables at a point in space that is fixed with respect to the chosen frame of reference &#8212; by using the following transformation to replace each total time derivative with a partial time derivative:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt}
</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>
\frac{\partial \psi}{\partial t} + \vec{u} \cdot \nabla\psi \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the "new" generic hyperbolic PDE derived above can be rewritten as,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial\psi}{\partial t} + \vec{u} \cdot \nabla\psi + \psi \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, ,
</math>
  </td>
</tr>
</table>
</div>
or, more succinctly,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial\psi}{\partial t} + \nabla\cdot (\psi \vec{u} )
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>
This equation also is broadly recognized as a conservation statement because, when <math>~S = 0</math>, the variable <math>~\psi</math> will represent the volume density of a conserved quantity. 

We should emphasize that the inertial-frame version of this Eulerian conservation equation can be retrieved straightforwardly by setting <math>~\Omega_0 = 0</math>, which is equivalent to setting <math>~\vec{u} = \vec{v}</math>.  It is,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial\psi}{\partial t} + \nabla\cdot (\psi \vec{v})
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>
The physics of the flow that is being described by this PDE is identical to the physics that is described by the preceding PDE.  But an important distinction must be made regarding how the two equations are ''interpreted.''  The "inertial frame" version of the equation (explicitly containing <math>~\vec{v}</math>) provides the time-rate of change of <math>~\psi</math> at a fixed point in ''inertial'' space, while the "new" version (explicitly containing <math>~\vec{u}</math>) provides the time-rate of change of <math>~\psi</math> at a fixed point in our "new" ''rotating'' coordinate frame.