From f90c531937a64e4b92b53353ec437efd2f28d95b Mon Sep 17 00:00:00 2001
From: Matthieu Schaller <matthieu.schaller@durham.ac.uk>
Date: Thu, 27 Nov 2014 18:32:36 +0000
Subject: [PATCH] Correction of a typo in the multipole expansion file
---
examples/theory/multipoles.tex | 10 ++++++----
1 file changed, 6 insertions(+), 4 deletions(-)
diff --git a/examples/theory/multipoles.tex b/examples/theory/multipoles.tex
index e9b6b6c..0947a5d 100644
--- a/examples/theory/multipoles.tex
+++ b/examples/theory/multipoles.tex
@@ -83,17 +83,19 @@ Keeping only the $xx$ term of the Hessian matrix in the Taylor expansion and int
\sum_im_ip_{i,x}^2$, we get for the accelerations:
\begin{equation}
- a_u(\rr) = Mf_u(\rr-\muu) + \frac{1}{2}(\sigma_{xx}^2 - M\mu_x)\nabla^2f_u(\dd)_{vv},
+ a_u(\rr) = Mf_u(\rr-\muu) + \frac{1}{2}(\sigma_{xx}^2 - M\mu_x^2)\nabla^2f_u(\dd)_{vv},
\end{equation}
with both $v=u$ or $v\neq u$. Expanding this coordinate by coordinate, we get:
\begin{eqnarray}
- a_x(\rr) &=& M\frac{G}{|\dd|^3} d_x + \frac{1}{2}\left(\sigma_{xx}^2 - M\mu_x\right)\left(\frac{15Gd_x^3}{|\dd|^7} -
+ a_x(\rr) &=& M\frac{G}{|\dd|^3} d_x + \frac{1}{2}\left(\sigma_{xx}^2 - M\mu_x^2\right)\left(\frac{15Gd_x^3}{|\dd|^7} -
\frac{9Gd_x}{|\dd|^5}\right)\\
- a_y(\rr) &=& M\frac{G}{|\dd|^3} d_y + \frac{1}{2}\left(\sigma_{xx}^2 - M\mu_x\right)\left(\frac{15Gd_xd_y^2}{|\dd|^7}-
+ a_y(\rr) &=& M\frac{G}{|\dd|^3} d_y + \frac{1}{2}\left(\sigma_{xx}^2 -
+M\mu_x^2\right)\left(\frac{15Gd_xd_y^2}{|\dd|^7}-
\frac{3Gd_x}{|\dd|^5}\right) \\
- a_z(\rr) &=& M\frac{G}{|\dd|^3} d_z + \frac{1}{2}\left(\sigma_{xx}^2 - M\mu_x\right)\left(\frac{15Gd_xd_z^2}{|\dd|^7}-
+ a_z(\rr) &=& M\frac{G}{|\dd|^3} d_z + \frac{1}{2}\left(\sigma_{xx}^2 -
+M\mu_x^2\right)\left(\frac{15Gd_xd_z^2}{|\dd|^7}-
\frac{3Gd_x}{|\dd|^5}\right)
\end{eqnarray}
--
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