Editing PGE/PoissonOrigin (section)

==Step 2==
Next, we realize that the divergence of the gravitational acceleration takes the form,
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<math>\nabla_x \cdot \vec{a}(\vec{x})</math>
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<math>=</math>
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<math>
\nabla_x \cdot \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x'
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&nbsp;
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<math>=</math>
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<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>
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-6)
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Examining the expression inside the curly braces, we find that,
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<math>\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] </math>
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<math>=</math>
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<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]
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Note: &nbsp; 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''.  
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<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>,
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<math>\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] = 0</math>
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&nbsp; &nbsp; &nbsp; when, &nbsp; &nbsp; &nbsp; 
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<math>
(\vec{x}^{~'} \ne \vec{x}) \, .
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-8)
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<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 &hellip; 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 &#8212; specifically, equation 1B-42 &#8212; of [<b>[[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:
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<math>\nabla_x \cdot \vec{a}(\vec{x})</math>
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<math>=</math>
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<math>
-G\rho(\vec{x}) \int d^2\Omega
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&nbsp;
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<math>=</math>
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<math>
-4\pi G\rho(\vec{x}) \, .
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 32, Eq. (2-9b)
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