# Quadrupole moment for a right triangle

The authors define a quadrupole moment for a right triangle in

Lazić, Predrag, Hrvoje Štefančić, and Hrvoje Abraham. “The Robin Hood Method – A Novel Numerical Method for Electrostatic Problems Based on a Non-Local Charge Transfer.” Journal of Computational Physics 213, no. 1 (March 2006): 117–40. https://doi.org/10.1016/j.jcp.2005.08.006.

PDF on ArXiv

page 122, equation 9 and 10. I am struggeling a bit on their notation, as it seems that some indices are not defined or they are using something like Einstein notation.

The equations are:

$$Q = \frac{2}{a_jb_j}\begin{pmatrix} \frac{a_j^3 b_j}{18} - \frac{a_j b_j^3}{36} & - \frac{a_j^2 b_j^2}{24} & 0 \\ -\frac{a_j^2 b_j^2}{24} & - \frac{a_j^3 b_j}{36} + \frac{a_j b_j^3}{18} & 0 \\ 0 & 0 & - \frac{a_j^3 b_j}{36} - \frac{a_j b_j^3}{36} \end{pmatrix}$$

and

$$I_{ij} = k\left[\frac{1}{|\vec{x}_i-\vec{x}_j|}+\frac{1}{6} Q_{nm} \frac{(\vec{x}_i - \vec{x}_j)_m (\vec{x}_i - \vec{x}_j)_n}{|\vec{x}_i - \vec{x}_j|^5}\right]$$

In the second equation I do not get the indices $$n$$ and $$m$$. Should there be a summation over $$n, m$$ and $$Q_{nm}$$ refers to a matrix element (i.e. $$\sum_{n,m} Q_{nm}\dots$$) or are there $$Q_{nm}$$ matrices defined for all combinations of $$n, m$$ (triangle points or edges)? and is $$(\vec{x}_i - \vec{x}_j)_m (\vec{x}_i - \vec{x}_j)_n$$ a scalar product or are the $$n$$, $$m$$ indexing vector elements as well?

For Einstein notation, there should be upper/lower indices to indicate what will be multiplied with each other, but here are only lower indices used.

Note that the authors used $$n$$, $$m$$ before, but it seems they are not related. Before they used them as indices of points of minimal/maximal potential.

• For work in Euclidean (flat) spaces it's extremely common for Einstein summations to be only over repeated dummy subscripts, e.g. $\mathbf{a}\cdot\mathbf{b} = a_n b_n$, but I admit that doesn't really help with the abuse of notation in this case. mathworld.wolfram.com/EinsteinSummation.html – origimbo May 2 at 14:43