Tensors for Physics

8.3 Volume Integrals, Gauss 131

As a more specific example, a half-sphere with radiusR, located above thex–y-
plane, is chosen. When, furthermore, the functionfdoes not depend on the angleφ,
the volume integral reduces to

V = 2 π

∫R

0

r^2 dr

∫ 1

0

dζ f(r(r, ζ )). (8.53)

The minus sign occurring in (8.52) is taken care of by an exchange of the integration
limits,θ=0 andπ/2 correspond toζ=1 andζ=0.
Forf=1thevolumeV=( 2 / 3 )πR^3 of the half-sphere is obtained.

8.3.2 Application: Mass Density, Center of Mass

The macroscopic description of matter, be it a gas, a liquid, or a solid, is based on
the mass densityρ=ρ(r). Its microscopic interpretation, for a substance composed
ofNparticles with the massm, is provided by

ρ(r)d^3 r=mdN(r), (8.54)

where dN(r)is the number of particles found within a small volume element d^3 r,
located at the positionr. Alternatively, and even more general, the mass density
of a substance composed of particles with massesmi, located at positionsri, with
i= 1 , 2 ,...,N, is given by

ρ(r)=

∑N

i= 1

miδ(r−ri). (8.55)

Hereδ(r)is the three dimensional delta-distribution functionδ(r), with the property

∫

δ(r−s)f(r)d^3 r=f(s), (8.56)

which applies when the functionfis single valued at the positions. The integrals of
both expressions forρ, over a volumeV, yield the massMVof the substance within
this volume,

MV=

∫

V

ρ(r)d^3 r. (8.57)