Corrected tzpos and simplified notation.

examples/theory/multipoles.tex

@@ -127,42 +127,37 @@ In the B-H approximation, the potential at position $\p{i}$ due to a set of
 particles $A$ is given by
 
 \begin{equation}
- \phi(\p{i}) = -\sum_{\bf n} M_{A, \bf n} D_{\bf n}(\p{i} - \muu_A)
+ \phi(\p{i}) = -\sum_{\bf n} M_{A, \bf n} D_{\bf n}(\muu_A - \p{i})
 \end{equation}
 
 Keeping only the terms up to second order (i.e. letting the sum run over 
-all vectors with $|{\bf n}|\leq2$) and writing $\rr = \p{i} - \muu_A$, we get:
+all vectors with $|{\bf n}|\leq2$) and writing $\rr_{i,A} = \muu_A - \p{i}$, we 
+get:
 
 \begin{eqnarray}
- \phi(\p{i}) &=& -M_{{\rm tot},A} \frac{G}{|\rr|} -\frac{1}{2}\sum_\alpha 
-I_{A,\alpha} \left(\frac{3Gr_\alpha^2}{|\rr|^5} - 
-\frac{G}{|\rr|^3}\right) \\ 
-& &- \frac{1}{2}\sum_{\alpha,\beta} \delta_{\alpha\beta}I_{A,\alpha\beta} 
-\frac{3Gr_\alpha r_\beta}{|\rr|^5} \\
-             &=& -M_{{\rm tot},A} \frac{G}{|\rr|} +
-\frac{1}{2}\left( I_{A,xx} + I_{A,yy} +I_{A,zz}\right)\frac{G}{|\rr|^3}  \\
+ \phi(\p{i}) &=& -M_{{\rm tot},A} \frac{G}{|\rr_{i,A}|} -\frac{1}{2}\sum_\alpha 
+I_{A,\alpha} \left(\frac{3Gr_\alpha^2}{|\rr_{i,A}|^5} - 
+\frac{G}{|\rr_{i,A}|^3}\right) \\ 
+& &+ \frac{1}{2}\sum_{\alpha,\beta} (1-\delta_{\alpha\beta})I_{A,\alpha\beta} 
+\frac{3Gr_\alpha r_\beta}{|\rr_{i,A}|^5} \\
+             &=& -M_{{\rm tot},A} \frac{G}{|\rr_{i,A}|} +
+\frac{1}{2}\left( I_{A,xx} + I_{A,yy} +I_{A,zz}\right)\frac{G}{|\rr_{i,A}|^3}  
+\\
  & & - \frac{1}{2}\sum_{\alpha,\beta} I_{A,\alpha\beta}  \frac{3Gr_\alpha 
-r_\beta}{|\rr|^5} \\
-&=& -M_{{\rm tot},A} \frac{G}{|\rr|} + \frac{G}{2} \frac{{\rm 
-tr}(\underline{I_A})}{|\rr|^3} - \frac{3G}{2}\frac{\rr^T \cdot 
-\underline{I_A} \cdot \rr}{|\rr|^5}
+r_\beta}{|\rr_{i,A}|^5} \\
+&=& -G \left[ \frac{M_{{\rm tot},A}}{|\rr_{i,A}|} - \frac{1}{2} \frac{{\rm 
+tr}(\underline{I_A})}{|\rr_{i,A}|^3} + \frac{3}{2}\frac{\rr_{i,A}^T \cdot 
+\underline{I_A} \cdot \rr_{i,A}}{|\rr_{i,A}|^5}\right]
 \end{eqnarray}
 
-The accelerations $\acc{i} = -\nabla_{\rr}\phi(\p{i})$ are then given by:
+The accelerations $\acc{i} = -\nabla_{\rr_{i,A}}\phi(\p{i})$ are then given by:
 
 \begin{eqnarray}
- a_{i,x} &=& \left[M_{{\rm tot},A} \frac{G}{|\rr|^3} - \frac{3G}{2} 
-\frac{{\rm tr}(\underline{I_A})}{|\rr|^5} - \frac{3G\underline{I_A}}{|\rr|^5} + 
-\frac{15G}{2}\frac{(\rr^T \cdot 
-\underline{I_A} \cdot \rr)}{|\rr|^7}\right] r_x \\
- a_{i,y} &=& \left[M_{{\rm tot},A} \frac{G}{|\rr|^3} - \frac{3G}{2} 
-\frac{{\rm tr}(\underline{I_A})}{|\rr|^5} - \frac{3G\underline{I_A}}{|\rr|^5} + 
-\frac{15G}{2}\frac{(\rr^T \cdot 
-\underline{I_A} \cdot \rr)}{|\rr|^7}\right] r_y\\
- a_{i,z} &=& \left[M_{{\rm tot},A} \frac{G}{|\rr|^3} - \frac{3G}{2} 
-\frac{{\rm tr}(\underline{I_A})}{|\rr|^5} - \frac{3G\underline{I_A}}{|\rr|^5} + 
-\frac{15G}{2}\frac{(\rr^T \cdot 
-\underline{I_A} \cdot \rr)}{|\rr|^7}\right] r_z
+\acc{i} = G\left[\frac{M_{{\rm tot},A} \rr_{i,A}}{|\rr_{i,A}|} 
+-\frac{3}{2}\frac{{\rm tr}(\underline{I_A})\rr_{i,A}}{|\rr_{i,A}|^5}
+-\frac{3\underline{I_A}\cdot\rr_{i,A}}{|\rr_{i,A}|^5}
++\frac{15}{2}\frac{(\rr_{i,A}^T \cdot \underline{I_A} \cdot 
+\rr_{i,A})\rr_{i,A}}{|\rr_{i,A}|^7}\right]
 \end{eqnarray}