PGE/PoissonOrigin - Revision history

Jet53man: /* See Also */

2022-12-18T00:35:11Z

See Also

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Jet53man: /* See Also */

2022-12-18T00:26:20Z

See Also

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 <ul>
 <li>Ulrich D. Jentschura &amp; Jonathan Sapirstein (April, 2018), arXiv:1801.10224v2 [math-ph], ''Green Function of the Poisson Equation:  <math>D = 2, 3, 4</math>''</li>
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Jet53man: /* See Also */

2022-12-17T23:36:36Z

See Also

← Older revision		Revision as of 19:36, 17 December 2022
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	=See Also=		=See Also=
	<ul>		<ul>
	<li>U. D. Jentschura & J. Sapirstein (April, 2018), arXiv:1801.10224v2 [math-ph], ''Green Function of the Poisson Equation: <math>D = 2, 3, 4</math>''</li>		<li>Ulrich D. Jentschura & Jonathan Sapirstein (April, 2018), arXiv:1801.10224v2 [math-ph], ''Green Function of the Poisson Equation: <math>D = 2, 3, 4</math>''</li>
	</ul>		</ul>


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Jet53man: /* See Also */

2022-12-17T23:34:38Z

See Also

← Older revision		Revision as of 19:34, 17 December 2022
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	=See Also=		=See Also=
			<ul>
			<li>U. D. Jentschura & J. Sapirstein (April, 2018), arXiv:1801.10224v2 [math-ph], ''Green Function of the Poisson Equation: <math>D = 2, 3, 4</math>''</li>
			</ul>


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Jet53man: /* Origin of the Poisson Equation */

2022-10-17T03:10:45Z

Origin of the Poisson Equation

← Older revision		Revision as of 23:10, 16 October 2022
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	<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />		<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

	{{~~Template:~~Math/EQ_Poisson01}}		{{ Math/EQ_Poisson01 }}
	</div>		</div>
	<br />		<br />

Jet53man: /* Step 2 */

2022-05-01T17:44:40Z

Step 2

← Older revision		Revision as of 13:44, 1 May 2022
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	(Note:   Ostensibly, this last expression is the same as equation 2-7 of [<b>[[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. In an [https://www-thphys.physics.ox.ac.uk/people/JamesBinney/web/index_files/book%201%20errors.pdf ''Errata'' to <b><font color="red">BT87</font></b>], the authors have identified this error along with its correction as the first among a list of ''innocuous errors''.) <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>,
			<table border="0" align="center" width="80%" cellpadding="8"><tr><td align="left">
			Note:   Ostensibly, this last expression is the same as equation 2-7 of [<b>[[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. In an [https://www-thphys.physics.ox.ac.uk/people/JamesBinney/web/index_files/book%201%20errors.pdf ''Errata'' to <b><font color="red">BT87</font></b>], the authors have identified this error along with its correction as the first among a list of ''innocuous errors''.
			</td></tr></table>

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

Jet53man: /* Step 2 */

2022-05-01T17:40:50Z

Step 2

← Older revision		Revision as of 13:40, 1 May 2022
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	</table>		</table>
	</div>		</div>
	(Note:   Ostensibly, this last expression is the same as equation 2-7 of [<b>[[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>,		(Note:   Ostensibly, this last expression is the same as equation 2-7 of [<b>[[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. In an [https://www-thphys.physics.ox.ac.uk/people/JamesBinney/web/index_files/book%201%20errors.pdf ''Errata'' to <b><font color="red">BT87</font></b>], the authors have identified this error along with its correction as the first among a list of ''innocuous errors''.) <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">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man at 16:31, 20 September 2021

2021-09-20T16:31:51Z

← Older revision		Revision as of 12:31, 20 September 2021
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	</div>		</div>

	(Note:   As we have detailed in a [[~~User:Tohline/~~VE#Setting_the_Stage\|separate discussion]], throughout [<b>[[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 {{ Template:Math/VAR_PositionVector01 }} is,		(Note:   As we have detailed in a [[VE#Setting_the_Stage\|separate discussion]], throughout [<b>[[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 {{ Template:Math/VAR_PositionVector01 }} is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man at 03:23, 28 July 2021

2021-07-28T03:23:38Z

← Older revision		Revision as of 23:23, 27 July 2021
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Origin of the Poisson Equation=		=Origin of the Poisson Equation=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Context\|<b>Poisson</b>]]</font>
			\|}

	In deriving the,		We will follow closely the presentation found in §2.1 of [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>] 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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	{{Template:Math/EQ_Poisson01}}		{{Template:Math/EQ_Poisson01}}
	</div>		</div>
	~~we will follow closely the presentation found in~~ &~~sect~~;~~2.1 of [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]~~</b>].		<br />


	<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, {{ Template:Math/VAR_PositionVector01 }}, 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, {{ Template:Math/VAR_PositionVector01 }}, 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:

Jet53man: /* Step 3 */

2021-07-02T19:01:33Z

Step 3

← Older revision		Revision as of 15:01, 2 July 2021
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	<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />		<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

	{{~~User~~:~~Tohline/~~Math/EQ_Poisson01}}		{{ Template: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, & Dynamics of Self-Gravitating Fluids'''.		which serves as one of the [[PGE#Principal_Governing_Equations\|principal governing equations]] in our examination of the '''Structure, Stability, & Dynamics of Self-Gravitating Fluids'''.

	=See Also=		=See Also=