Tensors for Physics

132 8 Integration of Fields

For (8.54) one findsMV=mNV,for(8.55) the result is

MV=

∑NV

i= 1

mi,

NVis the number of particles located within the volumeV.
In applications, where the atomistic structure of matter is not relevant, e.g. in
hydrodynamics, the mass densityρ=ρ(r)is treated as a continuous field function.
For the dynamic phenomena discussed in Sects.8.4.1and8.4.2, the density should
also be a differentiable function. Differentiability plays no role for the global prop-
erties, which are obtained via volume integrals. The position of the center of mass
and the moment of inertia tensor are of this type.
The position vectorRof thecenter of massof a substance characterized by the
mass densityρ, and confined within the volumeV, is determined by

MRμ=

∫

V

rμρ(r)d^3 r, (8.58)

whereM=

∫

Vρ(r)d

(^3) ris the total mass.
An example, instructive for the computation of volume integrals, is a homoge-
neous density, confined by a spherical cap. Its cross section is shown in Fig.8.14.
The cap has uniaxial symmetry, characterized by the unit vectoruwhich is parallel
to the vector pointing from the geometric center to the North pole of the cap. In the
figure, the geometric center is put at the origin of the coordinate system and the
direction of thez-axis is chosen parallel tou. The inner and outer radii are denoted
bya 1 anda 2 , respectively. As conventional, the angleθis counted from thez-axis,
its maximum isθmax. Withρ=ρ 0 =const. within the cap andρ=0 outside, the
Fig. 8.14Cross section of a
spherical cap