diff --git a/examples/theory/multipoles.tex b/examples/theory/multipoles.tex index e986929ab5341972a8f7d6e00b74e2d2b7952d68..8e938e12c6e7588877a2040681496e6249f1003b 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}