From 96d82b792525249ebf8c0ed4028cadc981c94397 Mon Sep 17 00:00:00 2001
From: Matthieu Schaller <matthieu.schaller@durham.ac.uk>
Date: Tue, 21 Apr 2015 22:38:49 +0100
Subject: [PATCH] Added expressions for the derivatives of the potential.
---
examples/theory/multipoles.tex | 31 ++++++++++++++++++++++++++++++-
1 file changed, 30 insertions(+), 1 deletion(-)
diff --git a/examples/theory/multipoles.tex b/examples/theory/multipoles.tex
index e986929..8e938e1 100644
--- a/examples/theory/multipoles.tex
+++ b/examples/theory/multipoles.tex
@@ -10,7 +10,7 @@
\newcommand{\acc}{\mathbf{a}}
\newcommand{\muu}{\boldsymbol{\mu}}
-\title{FMM and B-H equations up to quadrupole terms}
+\title{B-H and FMM equations up to quadrupole terms}
\author{Matthieu Schaller}
\begin{document}
@@ -86,4 +86,33 @@ M_B\mu_{B,\alpha}\mu_{B,\beta} \right) - \frac{1}{M_A}
\section{Derivatives of the potential}
+The gravitational potential is given by $\phi(\rr) = G/|\rr|$. Its derivatives
+are given by the following expressions. The first LHS term uses Dehnen's
+notation.\\
+
+0-th order:
+\begin{equation}
+ D_{(0,0,0)}(\rr) = \phi(\rr) = \frac{G}{|\rr|}
+\end{equation}
+
+1-st order:
+\begin{eqnarray}
+D_{(1,0,0)}(\rr) &=& \partial_x \phi(\rr)~=\frac{G}{|\rr|^3}r_x\\
+D_{(0,1,0)}(\rr) &=& \partial_y \phi(\rr)~=\frac{G}{|\rr|^3}r_y\\
+D_{(0,0,1)}(\rr) &=& \partial_z \phi(\rr)~=\frac{G}{|\rr|^3}r_z
+\end{eqnarray}
+
+2-nd order:
+\begin{eqnarray}
+D_{(2,0,0)} &=& \partial_{xx} \phi(\rr)~= \frac{3Gr_x^2}{|\rr|^5} -
+\frac{G}{|\rr|^3}\\
+D_{(0,2,0)} &=& \partial_{yy} \phi(\rr)~= \frac{3Gr_y^2}{|\rr|^5} -
+\frac{G}{|\rr|^3}\\
+D_{(0,0,2)} &=& \partial_{zz} \phi(\rr)~= \frac{3Gr_z^2}{|\rr|^5} -
+\frac{G}{|\rr|^3}\\
+D_{(1,1,0)} &=& \partial_{xy} \phi(\rr)~= \frac{3Gr_xr_y}{|\rr|^5} \\
+D_{(0,1,1)} &=& \partial_{yz} \phi(\rr)~= \frac{3Gr_yr_z}{|\rr|^5}\\
+D_{(1,0,1)} &=& \partial_{xz} \phi(\rr)~= \frac{3Gr_xr_z}{|\rr|^5}
+\end{eqnarray}
+
\end{document}
--
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