User:Tohline/AxisymmetricConfigurations/PGE - Revision history

Tohline: /* Cylindrical Coordinate Base */

2019-08-08T22:15:02Z

Cylindrical Coordinate Base

← Older revision		Revision as of 22:15, 8 August 2019
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	</tr>		</tr>
	</table>		</table>
	From this last expression — the so-called ''convective operator'' — we conclude as well that, for axisymmetric systems,		<span id="CYLconvectiveOperator">From this last expression — the so-called ''convective operator'' — we conclude as well that, for axisymmetric systems,</span>

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

Tohline: /* Spherical Coordinate Base */

2019-08-07T16:39:45Z

Spherical Coordinate Base

← Older revision		Revision as of 16:39, 7 August 2019
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	<td align="left">		<td align="left">
	<math>		<math>
	\hat{e}_r \biggl[ F_r \frac{\partial B_r}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{F_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} - \frac{(F_\theta B_\theta + F_\varphi B_\varphi)}{r}\biggr]		\hat{e}_r \biggl[ F_r \frac{\partial B_r}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_r}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} } - \frac{(F_\theta B_\theta + F_\varphi B_\varphi)}{r}\biggr]
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	+ \hat{e}_\theta \biggl[ F_r \frac{\partial B_\theta}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\theta}{\partial \theta } + \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi}		+ \hat{e}_\theta \biggl[ F_r \frac{\partial B_\theta}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\theta}{\partial \theta } + \cancel{ \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi} }
	+ \frac{F_\theta B_r}{r} - \frac{F_\varphi B_\varphi \cot\theta}{r} \biggr]		+ \frac{F_\theta B_r}{r} - \frac{F_\varphi B_\varphi \cot\theta}{r} \biggr]
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	+ \hat{e}_\varphi \biggl[ F_r \frac{\partial B_\varphi}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi}		+ \hat{e}_\varphi \biggl[ F_r \frac{\partial B_\varphi}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi} }
	+ \frac{F_\varphi B_r}{r} + \frac{F_\varphi B_\theta \cot\theta}{r} \biggr] \, .		+ \frac{F_\varphi B_r}{r} + \frac{F_\varphi B_\theta \cot\theta}{r} \biggr] \, .
	</math>		</math>
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	From this last expression — the so-called ''convective operator'' — we conclude as well that, for axisymmetric systems,		From this last expression — the so-called ''convective operator'' — we conclude as well that, for axisymmetric systems,

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

			<tr>
			<td align="right">
			<math>
			(\vec{v} \cdot \nabla )\vec{v}
			</math>
			</td>
			<td align="center">
			=
			</td>
			<td align="left">
			<math>
			\hat{e}_r \biggl[ v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta} - \frac{(v_\theta^2 + v_\varphi^2 )}{r}\biggr]
			+ \hat{e}_\theta \biggl[ v_r \frac{\partial v_\theta}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_\theta}{\partial \theta }
			+ \frac{v_\theta v_r}{r} - \frac{v_\varphi^2 \cot\theta}{r} \biggr]
			+ \hat{e}_\varphi \biggl[ v_r \frac{\partial v_\varphi}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_\varphi}{\partial \theta}
			+ \frac{v_\varphi v_r}{r} + \frac{v_\varphi v_\theta \cot\theta}{r} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

	<li>Express all vector time-derivatives in spherical coordinates:		<li>Express all vector time-derivatives in spherical coordinates:

Tohline: /* Spherical Coordinate Base */

2019-08-07T16:32:55Z

Spherical Coordinate Base

← Older revision		Revision as of 16:32, 7 August 2019
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	</tr>		</tr>
	</table>		</table>

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

			<tr>
			<td align="right">
			<math>
			(\vec{F} \cdot \nabla )\vec{B}
			</math>
			</td>
			<td align="center">
			=
			</td>
			<td align="left">
			<math>
			\hat{e}_r \biggl[ F_r \frac{\partial B_r}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{F_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} - \frac{(F_\theta B_\theta + F_\varphi B_\varphi)}{r}\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>
			+ \hat{e}_\theta \biggl[ F_r \frac{\partial B_\theta}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\theta}{\partial \theta } + \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi}
			+ \frac{F_\theta B_r}{r} - \frac{F_\varphi B_\varphi \cot\theta}{r} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>
			+ \hat{e}_\varphi \biggl[ F_r \frac{\partial B_\varphi}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi}
			+ \frac{F_\varphi B_r}{r} + \frac{F_\varphi B_\theta \cot\theta}{r} \biggr] \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-55)
			</td>
			</tr>
			</table>
			From this last expression — the so-called ''convective operator'' — we conclude as well that, for axisymmetric systems,


	<li>Express all vector time-derivatives in spherical coordinates:		<li>Express all vector time-derivatives in spherical coordinates:

Tohline: /* Cylindrical Coordinate Base */

2019-08-07T16:15:17Z

Cylindrical Coordinate Base

← Older revision		Revision as of 16:15, 7 August 2019
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	<td align="left">		<td align="left">
	<math>		<math>
	\hat{e}_\varpi \biggl[ F_\varpi \frac{\partial B}{\partial\varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_\varpi}{\partial\varphi} + F_z \frac{\partial B_\varpi}{\partial z} - \frac{F_\varphi B_\varphi}{\varpi} \biggr]		\hat{e}_\varpi \biggl[ F_\varpi \frac{\partial B_\varpi}{\partial\varpi} + \cancel{\frac{F_\varphi}{\varpi} \frac{\partial B_\varpi}{\partial\varphi}} + F_z \frac{\partial B_\varpi}{\partial z} - \frac{F_\varphi B_\varphi}{\varpi} \biggr]
	+ \hat{e}_\varphi \biggl[ F_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}{\varpi} \biggr]		+ \hat{e}_\varphi \biggl[ F_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \cancel{ \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi}} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}{\varpi} \biggr]
	+ \hat{e}_z \biggl[ F_\varpi \frac{\partial B_z}{\partial\varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_z}{\partial \varphi} + F_z \frac{\partial B_z}{\partial z} \biggr] \, .		+ \hat{e}_z \biggl[ F_\varpi \frac{\partial B_z}{\partial\varpi} +\cancel{ \frac{F_\varphi}{\varpi} \frac{\partial B_z}{\partial \varphi}} + F_z \frac{\partial B_z}{\partial z} \biggr] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-54)
			</td>
			</tr>
			</table>
			From this last expression — the so-called ''convective operator'' — we conclude as well that, for axisymmetric systems,

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

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	<math>		<math>
	+ \~~hat~~{e}_\~~varphi~~ \~~biggl[ F_~~\~~varpi \frac{\partial B_\varphi}{\partial \varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}~~{~~\varpi~~} ~~\biggr]~~		(\vec{v} \cdot \nabla )\vec{v}
	</math>		</math>
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	</td>		</td>
	<td align="center">		<td align="center">
	~~ ~~		=
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	+ \hat{e}_z \biggl[ F_\varpi \frac{\partial ~~B_z~~}{\partial\varpi} + \frac{F_\varphi}{\varpi} \frac{\partial ~~B_z~~}{\partial \~~varphi~~} + ~~F_z~~ \frac{\partial ~~B_z~~}{\partial z} \biggr] \, .		\hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi} \biggr]
			+ \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi} + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi} \biggr]
			+ \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi} + v_z \frac{\partial v_z}{\partial z} \biggr] \, .
	</math>		</math>
	</td>		</td>
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-54)~~
	~~</td>~~
	</tr>		</tr>
	</table>		</table>

	~~From this last expression — the so-called ''convective operator'' — we conclude as well that,~~

Tohline: /* Cylindrical Coordinate Base */

2019-08-07T16:04:32Z

Cylindrical Coordinate Base

← Older revision		Revision as of 16:04, 7 August 2019
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	</tr>		</tr>
	</table>		</table>

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

			<tr>
			<td align="right">
			<math>
			(\vec{F} \cdot \nabla )\vec{B}
			</math>
			</td>
			<td align="center">
			=
			</td>
			<td align="left">
			<math>
			\hat{e}_\varpi \biggl[ F_\varpi \frac{\partial B}{\partial\varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_\varpi}{\partial\varphi} + F_z \frac{\partial B_\varpi}{\partial z} - \frac{F_\varphi B_\varphi}{\varpi} \biggr]
			+ \hat{e}_\varphi \biggl[ F_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}{\varpi} \biggr]
			+ \hat{e}_z \biggl[ F_\varpi \frac{\partial B_z}{\partial\varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_z}{\partial \varphi} + F_z \frac{\partial B_z}{\partial z} \biggr] \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>
			+ \hat{e}_\varphi \biggl[ F_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}{\varpi} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>
			+ \hat{e}_z \biggl[ F_\varpi \frac{\partial B_z}{\partial\varpi} + \frac{F_\varphi}{\varpi} \frac{\partial B_z}{\partial \varphi} + F_z \frac{\partial B_z}{\partial z} \biggr] \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-54)
			</td>
			</tr>
			</table>

			From this last expression — the so-called ''convective operator'' — we conclude as well that,


	<li>Express all vector time-derivatives in cylindrical coordinates:		<li>Express all vector time-derivatives in cylindrical coordinates:

Tohline: /* Eulerian Formulation (SPH.) */

2019-08-06T03:49:41Z

Eulerian Formulation (SPH.)

← Older revision		Revision as of 03:49, 6 August 2019
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	<td align="right">		<td align="right">
	<math>		<math>
	r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{~~\theta~~} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta		r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{\theta}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta
	</math>		</math>
	</td>		</td>

Tohline: /* Axisymmetric Configurations (Part I) */

2019-08-04T00:17:26Z

Axisymmetric Configurations (Part I)

← Older revision		Revision as of 00:17, 4 August 2019
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	<!-- __FORCETOC__ will force the creation of a Table of Contents -->		<!-- __FORCETOC__ will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Axisymmetric Configurations (~~Part I~~)=		=Axisymmetric Configurations (Governing Equations)=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	If the self-gravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our [[User:Tohline/PGE\|principal governing equations]] can be simplified to a coupled set of two-dimensional PDEs.		If the self-gravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our [[User:Tohline/PGE\|principal governing equations]] can be simplified to a coupled set of two-dimensional PDEs.

Tohline: /* Eulerian Formulation (SPH.) */

2019-08-03T18:36:01Z

Eulerian Formulation (SPH.)

← Older revision		Revision as of 18:36, 3 August 2019
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	<td align="right">		<td align="right">
	<math>		<math>
	\frac{d\dot{r}}{dt} - r {\dot\~~theta~~}~~^2 -~~ \biggl[ \frac{~~j^2~~}{~~r^3~~ \~~sin^3~~\theta} \biggr]\~~sin~~\theta		\biggl\{ \frac{\partial \dot{r}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{r}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\}
			- r {\dot\theta}^2
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] \~~, ,~~		- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr]
	</math>		</math>
	</td>		</td>
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	<td align="right">		<td align="right">
	<math>		<math>
	\frac{~~d(r~~\dot\theta)}{dt} + \dot{r} \dot\theta - \biggl[ \frac{~~j^2~~}{~~r^3~~ \~~sin^3~~\theta} \biggr] \~~cos~~\theta		r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{\theta} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	- \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] \~~, .~~		- \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] + \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
	</math>		</math>
	</td>		</td>

Tohline: /* Eulerian Formulation (SPH.) */

2019-08-03T18:25:18Z

Eulerian Formulation (SPH.)

← Older revision		Revision as of 18:25, 3 August 2019
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	</div>		</div>

	Making this substitution throughout the set of governing relations gives:		Making this substitution throughout the set of governing relations gives:

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d\rho}{dt} + ~~\rho~~ \biggl[ \frac{1}{r^2} \frac{\partial (r^2 \dot{r})}{\partial r}		<math>~
	+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \dot\theta r \sin\theta \biggr)		\frac{\partial \rho}{\partial t}
			+ \biggl[ \frac{1}{r^2} \frac{\partial (\rho r^2 \dot{r})}{\partial r}
			+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \rho \dot\theta r \sin\theta \biggr)
	\biggr]</math>		\biggr]</math>
	</td>		</td>
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	<span id="PGE:Euler">		<span id="PGE:Euler">The Two Relevant Components of the<br />
	<font color="#770000">'''Euler Equation'''</font>		<font color="#770000">'''Euler Equation'''</font>
	</span><br />		</span><br />
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	<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />		<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

	{{~~User:Tohline~~/~~Math~~/~~EQ_FirstLaw02}}~~		<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			\biggl\{ \frac{\partial \epsilon}{\partial t} + \biggl[ \dot{r} \frac{\partial \epsilon}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \epsilon}{\partial\theta} \biggr] \biggr\}
			+ P\biggl\{ \frac{\partial }{\partial t} \biggl( \frac{1}{\rho}\biggr)
			+ \biggl[ \dot{r} \frac{\partial }{\partial r} \biggl( \frac{1}{\rho}\biggr) \biggr]
			+ \biggl[ \dot\theta \frac{\partial }{\partial\theta} \biggl( \frac{1}{\rho}\biggr) \biggr] \biggr\}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0</math>
			</td>
			</tr>
			</table>

Tohline: /* Eulerian Formulation (SPH.) */

2019-08-03T18:08:59Z

Eulerian Formulation (SPH.)

← Older revision		Revision as of 18:08, 3 August 2019
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right"><math>~{\hat{e}}_r</math>:    </td>
			<td align="right">
			<math>
			\frac{d\dot{r}}{dt} - r {\dot\theta}^2 - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr]\sin\theta
			</math>
			</td>
			<td align="center">
			=
			</td>
			<td align="left">
			<math>
			- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] \, ,
			</math>
			</td>
			</tr>

	<tr>		<tr>
			<td align="right"><math>~{\hat{e}}_\theta</math>:    </td>
	<td align="right">		<td align="right">
	<math>~		<math>
	~~{\hat{e}}_r \biggl[ \frac{d\dot{r}}{dt} - r {\dot\theta}^2 - r {\dot\varphi}^2 \sin^2\theta \biggr]~~		\frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
	~~+ {\hat{e}}_\theta \biggl[~~ \frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta - ~~r { \dot\varphi }^2 \sin\theta \cos\theta \biggr]~~
	~~+ {\hat{e}}_\varphi~~ \biggl[ \frac{~~d(r \sin\theta \dot\varphi)}{dt~~} ~~+ \dot~~{r~~} \dot\varphi~~ \sin\theta ~~+ r~~ \~~dot\theta \dot\varphi~~ \cos\theta ~~\biggr]~~
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	~~<math>~~~=~~</math>~~		=
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~- {\hat{e}}_r \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr]~~		<math>
	- ~~{\hat{e}}_\theta~~ \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr]		- \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>


	<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />		<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />