164 10 Multipole Potentials
The same applies for a-fold spatial differentiation ofr−^1. In this spirit, Cartesian
tensors of rankare defined by
Xμ 1 μ 2 ···μ≡(− 1 )
∂
∂rμ 1 ∂rμ 2 ···∂rμ
r−^1 =(− 1 )∇μ 1 ∇μ 2 ···∇μr−^1. (10.2)
The tensorial functionsX...approach 0 forr→∞. By definition, these tensors are
symmetric. Whenever two subscripts are equal and summed over, these two spatial
derivatives are equivalent toΔand consequently 0 is obtained. Thus the spatial
differentiation (10.2) yields irreducible tensors of rank. These are thedescending
multipole potentials. Due to the definition (10.2), the-th multipole potential is
related to the−1functionby
Xμ 1 μ 2 ···μ− 1 μ=−
∂
∂rμ
Xμ 1 μ 2 ···μ− 1 =−∇μXμ 1 μ 2 ···μ− 1. (10.3)
10.1.2 Dipole, Quadrupole and Octupole Potentials.
Examples for multipole potential tensors of rank= 1 , 2 ,3arethe
dipole potential
Xμ=r−^3 rμ=r−^2 ̂rμ, (10.4)
thequadrupole potential
Xμν= 3 r−^5
(
rμrν−
1
3
r^2 δμν
)
= 3 r−^5 rμrν = 3 r−^3 ̂rμ̂rν, (10.5)
and theoctupole potential
Xμνλ= 15 r−^7 rμrνrλ = 15 r−^4 ̂rμ̂rν̂rλ. (10.6)
The reason for the namesdipole, quadrupoleandoctupolepotential is seen in
Sect.10.3.
10.1.3 Source Term for the Quadrupole Potential
The formulas presented here and in the following for the multipole potential tensors
are valid forr>0. When a source term is included atr =0, the second rank
multipole tensor obeys the relation