Editing SSC/PerspectiveReconciliation (section)

==Step 4:==
Finally, Rosseland sets out to swap the order of the time- and space-derivatives in the term,
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<math>~\frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] \, .</math>
</div>
But, whereas the ''partial'' time derivative commutes with the ''partial'' radial derivative, the total time derivative does not commute with the spatial derivative.  Generically, however, for any scalar variable, <math>~q</math>, we can swap between total and partial time derivatives via the operator relation,
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<math>~\frac{dq}{dt} = \frac{\partial q}{\partial t} + v_r \frac{\partial q}{\partial r} \, .</math>
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Hence, if <math>~q</math> is replaced by <math>~P</math>, we can write,
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<math>~ v_r \biggl(\frac{\partial P}{\partial r}\biggr)  = \frac{dP}{dt}  - \frac{\partial P}{\partial t}\, ;</math>
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while, if <math>~q</math> is replaced by <math>~\partial P/\partial r</math>, we have,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial t}\biggl( \frac{\partial P}{\partial r} \biggr) + v_r \frac{\partial }{\partial r} \biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ v_r \biggl( \frac{\partial P}{\partial r} \biggr)\biggr] 
- \biggl( \frac{\partial P}{\partial r} \biggr)\frac{\partial v_r}{\partial r} \, ,</math>
  </td>
</tr>
</table>
</div>
where, in making this last step, we have swapped the order of the ''partial'' derivatives in the first term on the right-hand side.  Combining these two relations and incorporating the form of Euler's equation that was highlighted in step #3 gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  - \frac{\partial P}{\partial t} \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r} \, .</math>
  </td>
</tr>
</table>
</div>
Inserting this into the equation found at the end of step #3, then inserting the expression for <math>~dP/dt</math> derived in step #1 gives, 

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \gamma_g P \nabla\cdot \vec{v}\biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
This matches Rosseland's equation (2.18) with the nonadiabatic terms set to zero.




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