User:Tohline/SR/PoissonOrigin - Revision history

Tohline: /* Step 3 */

2017-07-06T01:38:16Z

Step 3

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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
	</div>		</div>
	which serves as one of the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] in our examination of the '''~~structure~~, ~~stability~~, ~~and dynamics~~ of ~~self~~-~~gravitating fluids~~'''.		which serves as one of the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] in our examination of the '''Structure, Stability, & Dynamics of Self-Gravitating Fluids'''.

	=See Also=		=See Also=

Tohline: /* Origin of the Poisson Equation */

2017-07-06T01:37:19Z

Origin of the Poisson Equation

← Older revision		Revision as of 01:37, 6 July 2017
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	where, {{ User:Tohline/Math/C_GravitationalConstant }} is the universal gravitational constant.		where, {{ User:Tohline/Math/C_GravitationalConstant }} is the universal gravitational constant.

	~~Now, in~~ the astrophysics literature, it is customary to adopt the following definition of the,		==Step 1==
			In the astrophysics literature, it is customary to adopt the following definition of the,
	<div align="center" id="GravitationalPotential">		<div align="center" id="GravitationalPotential">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

			==Step 2==
			Next, we realize that the divergence of the gravitational acceleration takes the form,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla_x \cdot \vec{a}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\nabla_x \cdot \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{\|\vec{x}^{~'} - \vec{x}\|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x'
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\int G\rho(\vec{x}^{~'}) \biggl\{ \nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{\|\vec{x}^{~'} - \vec{x}\|^3}\biggr] \biggr\} d^3 x' \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-6)
			</td>
			</tr>
			</table>
			</div>
			Examining the expression inside the curly braces, we find that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{\|\vec{x}^{~'} - \vec{x}\|^3}\biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{3}{\|\vec{x}^{~'} - \vec{x}\|^3}
			+ 3 \biggl[ \frac{ (\vec{x}^{~'} - \vec{x}) \cdot (\vec{x}^{~'} - \vec{x}) }{\|\vec{x}^{~'} - \vec{x}\|^5}\biggr]
			</math>
			</td>
			</tr>
			</table>
			</div>
			(Note:   Ostensibly, this last expression is the same as equation 2-7 of [<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], but apparently there is a typesetting error in the BT87 publication. As printed, the denominator of the first term on the right-hand side is <math>~\|\vec{x}^{~'} - \vec{x}\|^1</math>, whereas it should be <math>~\|\vec{x}^{~'} - \vec{x}\|^3</math> as written here.) <font color="#007700">When <math>~(\vec{x}^{~'} - \vec{x}) \ne 0</math>, we may cancel the factor <math>~\|\vec{x}^{~'} - \vec{x}\|^2</math> from top and bottom of the last term in this equation to conclude that</font>,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{\|\vec{x}^{~'} - \vec{x}\|^3}\biggr] = 0</math>
			</td>
			<td align="center">
			when,
			</td>
			<td align="left">
			<math>~
			(\vec{x}^{~'} \ne \vec{x}) \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-8)
			</td>
			</tr>
			</table>
			</div>
			<font color="#007700">Therefore, any contribution to the integral must come from the point <math>~\vec{x}^{~'} = \vec{x}</math>, and we may restrict the volume of integration to a small sphere … centered on this point. Since</font>, for a sufficiently small sphere, <font color="#007700">the density will be almost constant through this volume, we can take <math>~\rho(\vec{x}~{'}) = \rho(\vec{x})</math> out of the integral.</font> Via the divergence theorem (for details, see appendix 1.B — specifically, equation 1B-42 — of [<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>]), the remaining volume integral may be converted into a surface integral over the small volume centered on the point <math>~\vec{x}^{~'} = \vec{x}</math> and, in turn, this surface integral may be written in terms of an integral over the solid angle, <math>~d^2\Omega</math>, to give:
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla_x \cdot \vec{a}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-G\rho(\vec{x}) \int d^2\Omega
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-4\pi G\rho(\vec{x}) \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 32, Eq. (2-9b)
			</td>
			</tr>
			</table>
			</div>

			==Step 3==
			Finally, combining the results of ''Step 1'' and ''Step 2'' gives the desired,
			<div align="center">
			<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

			{{User:Tohline/Math/EQ_Poisson01}}
			</div>
			which serves as one of the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] in our examination of the '''structure, stability, and dynamics of self-gravitating fluids'''.

	=See Also=		=See Also=

Tohline at 00:43, 6 July 2017

2017-07-06T00:43:18Z

← Older revision		Revision as of 00:43, 6 July 2017
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	</table>		</table>
	</div>		</div>
	where, {{ User:Tohline/Math/C_GravitationalConstant }} is the ~~Newtonian~~ gravitational constant.		where, {{ User:Tohline/Math/C_GravitationalConstant }} is the universal gravitational constant.

	Now, in the astrophysics literature, it is customary to adopt the following definition of the,		Now, in the astrophysics literature, it is customary to adopt the following definition of the,
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	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-5)		[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-5)
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	~~=Drawn from Other Wiki Pages=~~
	It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr>~~
	~~<td align="right">~~
	~~<math>~ - \Phi = \mathfrak{B} </math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{\|\vec{x} - \vec{x}^{~'}\|} d^3x^' \, .</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] into this H_Book.

	… which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>. The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] for details) the,
	~~<div align="center">~~
	~~<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />~~
	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />~~
	~~[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)~~
	</td>		</td>
	</tr>		</tr>

Tohline: /* Origin of the Poisson Equation */

2017-07-05T23:42:32Z

Origin of the Poisson Equation

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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \Phi</math>		<math>~ \Phi(\vec{x})</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ -G \int~~\limits_V~~ \frac{\rho(\vec{x}^{~'})}{\|\vec{x}^{~'} - \vec{x}\|} d^3x^' \, .</math>		<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{\|\vec{x}^{~'} - \vec{x}\|} d^3x^' \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>
	</div>		</div>
	<font color="#007700">we find that we may write</font> the gravitational acceleration as,		and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the ''primed'' <math>~(\vec{x}~{'})</math>, rather than the unprimed <math>~(\vec{x})</math>, coordinate system, <font color="#007700">we find that we may write</font> the gravitational acceleration as,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="right">		<td align="right">
	<math>~\vec{a}(\vec{x})</math>		<math>~\vec{a}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\int G\rho(\vec{x}^{~'}) \nabla_x \biggl[ \frac{1}{\|\vec{x}^{~'} - \vec{x}\|} \biggr]d^3 x'
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \nabla_x \biggl\{ G \int \biggl[ \frac{\rho(\vec{x}^{~'}) }{\|\vec{x}^{~'} - \vec{x}\|} \biggr]d^3 x'\biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

Tohline: /* Origin of the Poisson Equation */

2017-07-05T22:59:01Z

Origin of the Poisson Equation

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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)		[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
			[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
			[<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>

	[Note ~~that, as~~ we have detailed in a [[User:Tohline/VE#Setting_the_Stage\|separate discussion]], throughout [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] Chandrasekhar adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.]		(Note:   As we have detailed in a [[User:Tohline/VE#Setting_the_Stage\|separate discussion]], throughout [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] Chandrasekhar adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.) Recognizing that the gradient of the function, <math>~\|\vec{x}^{~'} - \vec{x}\|^{-1}</math>, with respect to <math>~\vec{x}</math> is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla_x \biggl[ \frac{1}{\|\vec{x}^{~'} - \vec{x}\|} \biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\vec{x}^{~'} - \vec{x}}{\|\vec{x}^{~'} - \vec{x}\|^3} \, ,
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-4)
			</td>
			</tr>
			</table>
			</div>
			<font color="#007700">we find that we may write</font> the gravitational acceleration as,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{a}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\nabla_x \Phi \, .</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-5)
			</td>
			</tr>
			</table>
			</div>

	=Drawn from Other Wiki Pages=		=Drawn from Other Wiki Pages=

Tohline at 22:38, 5 July 2017

2017-07-05T22:38:46Z

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	__FORCETOC__		__FORCETOC__
	=Origin of the Poisson Equation=		=Origin of the Poisson Equation=
	~~{{LSU_HBook_header}}~~

	~~The~~		In deriving the,
	<div align="center">		<div align="center">
	<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />		<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />
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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
	</div>		</div>
	~~is derived straightforwardly from Isaac Newton's inverse-square law of gravitation. In presenting this derivation~~ we follow closely the presentation found in §2.1 of [<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>]		we will follow closely the presentation found in §2.1 of [<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>].


			{{LSU_HBook_header}}

	<font color="#007700">According to Isaac Newton's inverse-square law of gravitation,</font> the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho({\vec{x'}}) d^3x'</math>, as follows:		<font color="#007700">According to Isaac Newton's inverse-square law of gravitation,</font> the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho(\vec{x}^{~'}) d^3x'</math>, as follows:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\int \biggl[\frac{(\vec{x'} - \vec{x})}{\|\vec{x'} - \vec{x}\|}\biggr] G\rho(\vec{x'}) d^3 x'		\int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{\|\vec{x}^{~'} - \vec{x}\|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, ,
	</math>		</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-2)
			</td>
			</tr>
			</table>
			</div>
			where, {{ User:Tohline/Math/C_GravitationalConstant }} is the Newtonian gravitational constant.

			Now, in the astrophysics literature, it is customary to adopt the following definition of the,
			<div align="center" id="GravitationalPotential">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="center" colspan="3">
			<font color="#770000">'''Scalar Gravitational Potential'''</font>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~ \Phi</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~ -G \int\limits_V \frac{\rho(\vec{x}^{~'})}{\|\vec{x}^{~'} - \vec{x}\|} d^3x^' \, .</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>

			[Note that, as we have detailed in a [[User:Tohline/VE#Setting_the_Stage\|separate discussion]], throughout [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] Chandrasekhar adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.]

	=Drawn from Other Wiki Pages=		=Drawn from Other Wiki Pages=

Tohline: /* Origin of the Poisson Equation */

2017-07-05T21:58:56Z

Origin of the Poisson Equation

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	=Origin of the Poisson Equation=		=Origin of the Poisson Equation=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}


	[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>, §8, p. 15] <font color="#007700">A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations. The ''virial method'' … is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account. The ''virial equations'' of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations.</font> In this context, Chandrasekhar's focus is on two of the four [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] that serve as the foundation of our entire H_Book, namely, the

	The		The
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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
	</div>		</div>
			is derived straightforwardly from Isaac Newton's inverse-square law of gravitation. In presenting this derivation we follow closely the presentation found in §2.1 of [<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>]

			<font color="#007700">According to Isaac Newton's inverse-square law of gravitation,</font> the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho({\vec{x'}}) d^3x'</math>, as follows:
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{a}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\int \biggl[\frac{(\vec{x'} - \vec{x})}{\|\vec{x'} - \vec{x}\|}\biggr] G\rho(\vec{x'}) d^3 x'
			</math>
			</td>
			</tr>
			</table>
			</div>

	=Drawn from Other Wiki Pages=		=Drawn from Other Wiki Pages=

Tohline at 21:34, 5 July 2017

2017-07-05T21:34:46Z

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	__FORCETOC__		__FORCETOC__
	=Poisson Equation=		=Origin of the Poisson Equation=
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			[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>, §8, p. 15] <font color="#007700">A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations. The ''virial method'' … is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account. The ''virial equations'' of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations.</font> In this context, Chandrasekhar's focus is on two of the four [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] that serve as the foundation of our entire H_Book, namely, the

			The
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			<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

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Tohline: Created page with 'FORCETOC =Poisson Equation= {{LSU_HBook_header}} =Drawn from Other Wiki Pages= It is clear, therefore, that Chandrasekhar uses the variable $~\vec{u}$ instead of…'

2017-07-05T21:28:53Z

Created page with '__FORCETOC__ =Poisson Equation= {{LSU_HBook_header}} =Drawn from Other Wiki Pages= It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of…'

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=Poisson Equation=
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It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
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<math>~ - \Phi = \mathfrak{B} </math>
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<math>~=</math>
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<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>
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Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] into this H_Book.

… which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>. The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details) the,
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<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
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<math>~\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>
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<math>~=</math>
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<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
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[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)
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=See Also=

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User:Tohline/SR/PoissonOrigin - Revision history

Tohline: /* Step 3 */

Tohline: /* Origin of the Poisson Equation */

Tohline at 00:43, 6 July 2017

Tohline: /* Origin of the Poisson Equation */

Tohline: /* Origin of the Poisson Equation */

Tohline at 22:38, 5 July 2017

Tohline: /* Origin of the Poisson Equation */

Tohline at 21:34, 5 July 2017

Tohline: Created page with '__FORCETOC__ =Poisson Equation= {{LSU_HBook_header}} =Drawn from Other Wiki Pages= It is clear, therefore, that Chandrasekhar uses the variable ~\vec{u} instead of…'

Tohline: Created page with 'FORCETOC =Poisson Equation= {{LSU_HBook_header}} =Drawn from Other Wiki Pages= It is clear, therefore, that Chandrasekhar uses the variable $~\vec{u}$ instead of…'