Tensors for Physics

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