Editing PGE/FirstLawOfThermodynamics (section)

===Initial Recognition===
Multiplying the ''Adiabatic Form of the First Law of Thermodynamics'' through by {{ Template:Math/VAR_Density01 }} and rearranging terms, we find that,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d(\rho\epsilon)}{dt} - \epsilon \frac{d\rho}{dt} -  \frac{P}{\rho} \frac{d\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
\frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon) \frac{1}{\rho}\frac{d\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
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<math>
\frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon)\frac{d\ln\rho}{dt} \, ,
</math>
  </td>
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</table>
is an equally valid statement of the conservation of specific entropy in an adiabatic flow.  In combination, first, with
 
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<span id="IdealGasB"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation

{{ Template:Math/EQ_EOSideal02 }}
</div>

and, second, with the
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<span id="Continuity"><font color="#770000">'''Lagrangian Form'''</font></span><br />
of the Continuity Equation

{{ Template:Math/EQ_Continuity01 }}
</div>

we may furthermore rewrite this expression as,

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  <td align="right">
<math>\frac{d(\rho\epsilon)}{dt}</math>
  </td>
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<math>=</math>
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<math>
\gamma_g (\rho\epsilon)\frac{d\ln\rho}{dt} 
</math>
  </td>
</tr>

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<math>\Rightarrow ~~~ \frac{1}{\gamma_g} \frac{d\ln(\rho\epsilon)}{dt}</math>
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<math>=</math>
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<math>
\frac{d\ln\rho}{dt} 
</math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow ~~~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \nabla\cdot\vec{v} \, . 
</math>
  </td>
</tr>
</table>

This relation has the classic form of a conservation law.  It certifies that, within the context of adiabatic flows, the ''entropy tracer,''

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<math>\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, ,</math>
</div>

is the volume density of a conserved quantity.  In this case, that conserved quantity is the entropy of each fluid element.