Editing Appendix/Ramblings/Radiation/RadHydro (section)

====First Law====

By combining the continuity equation with the

<div id="PGE:FirstLaw" align="center">
<font color="#770000">'''First Law of Thermodynamics'''</font>

{{ Math/EQ_FirstLaw01 }}
</div>

we can write,
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<tr>
  <td align="right">
<math>\rho T\frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
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  <td align="left">
<math>
\rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, .
</math>
  </td>
</tr>
</table>
Given that the specific internal energy <math>(\epsilon)</math> and the internal energy density <math>(e)</math> are related via the expression, <math>\epsilon = e/\rho</math>, we appreciate that the first of the above-identified ''energy-conservation-based'' dynamical equations is simply a restatement of the 1<sup>st</sup> Law of Thermodynamics in the context of a physical system whose fluid elements gain or lose entropy as a result of the (radiation-transport-related) source and sink terms,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho T \frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
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  <td align="left">
<math>c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>
  </td>
</tr>
</table>