Added expressions for the derivatives of the potential.

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}