Tensors for Physics

10.1 Descending Multipoles 165

∇μ∇νr−^1 =Xμν(r)−

4 π

3

δμνδ(r), (10.7)

whereXμν(r)is given by (10.5).
To verify (10.7), integrate it over a sphere with a finite radiusR, centered atr=0.
More specifically, use the Gauss theorem for

∫

∇μ(∇νr−^1 )d^3 rto obtain

∫

∇μ∇νr−^1 d^3 r=R^2

∫

rˆμ

(

∇νr−^1

)

r=R

d^2 rˆ=−

∫

rˆμrˆνd^2 rˆ=−( 4 π/ 3 )δμν.

Notice that d^3 r=r^2 drd^2 rˆ, that the integral of the symmetric traceless tensorXμν

over the sphere vanishes since

∫

rˆμrˆνd^2 rˆ =0, and that theδ-function has the
property

∫

δ(r)d^3 r=1.

The trace of (10.7) yields

Δr−^1 =− 4 πδ(r). (10.8)

The Poisson equation of electrostatics or the electric potentialφ∼r−^1 in vacuum,
caused by a point chargeqlocated atr=0, viz.ε 0 Δφ=− 4 πqδ(r), is mathemat-
ically equivalent to (10.8). The Poisson equation, in turn, is a consequence of the
Gauss law of electrodynamics, cf. Sect.7.5.

10.1.4 General Properties of Multipole Potentials

In general, the-th multipole potential can be written as

Xμ 1 μ 2 ···μ=( 2 − 1 )!!r−(^2 +^1 )rμ 1 rμ 2 ···rμ, (10.9)

or equivalently

Xμ 1 μ 2 ···μ(r)=r−(+^1 )Yμ 1 μ 2 ···μ(rˆ), (10.10)

with the tensors

Yμ 1 μ 2 ···μ(rˆ)=( 2 − 1 )!!̂rμ 1 ̂rμ 2 ···̂rμ. (10.11)

The irreducible tensorsY...depend on the direction ofronly. With the unit vector
ˆrexpressed in terms of the polar angles, the components of the-th rank Cartesian
tensor are isomorphic to the spherical harmonicsYm.
Clearly, cf. (10.10), the-th rank multipole potential is proportional to

r−(+^1 ).