Tensors for Physics

136 8 Integration of Fields

For a sphere with radiusa/2, the tensor has the same form, just withΘ= 101 Ma^2.
Obviously, on the level of second rank tensors, cubic symmetry cannot be distin-
guished from spherical symmetry. This is different for tensors of rank four, as they
occur, e.g. in elasticity.
The traceΘμμ= 3 Θ ̄, whereΘ ̄is the mean moment of inertia, is given by

3 Θ ̄= 2

∫

V

ρ(r)r^2 d^3 r. (8.69)

In the case of uniaxial symmetry, 3Θ ̄is equal toΘ‖+ 2 Θ⊥,see(8.68). In some
applications, it is preferable to computeΘ‖and 3Θ ̄and to inferΘ⊥from( 3 Θ ̄−
Θ‖)/2.

8.4 Exercise: Moment of Inertia Tensor of a Half-Sphere
Compute the moment of inertia tensor of a half-sphere with radiusaand constant
mass densityρ 0. The orientation is specified by the unit vectoru, pointing from the
center of the sphere to the center of mass of the half-sphere.

Hint: make use of the symmetry. First calculate the moments of inertia with respect
to the center of the sphere, then use the Steiner law (8.66) to find the moments of
inertia with respect to the center of mass of the half-sphere, see also Sect.8.3.2.

8.3.4 Generalized Gauss Theorem.

When the integrand of a volume integral is the spatial derivative of a function, the
integral over the volumeVcan be transformed into a surface integral over the bound-
ing surfaceA=∂V. Thus the “three dimensional” integration is reduced to a “two
dimensional” one. Here, a generalized version of the Gauss theorem is stated first
and the standard Gauss theorem is obtained as a special case.
Letf=f(r)be a differentiable function,Va volume with a well defined surface
∂VinR^3 .Thegeneralized Gauss theoremreads

Vμ≡

∫

V

∇μfd^3 r=

∮

∂V

fnμd^2 s, (8.70)

wherenis the outer normal ofV, cf. Fig.8.16at its bounding surface∂Vand d^2 sis
the scalar surface element.
Whenfoccurring in (8.70) is the component of a tensor of rankthe integrals
on both sides of the equation are tensors of rank+1. In particular, forfbeing the
componentvνof a vector fieldv, the quantities occurring on both sides of (8.70)are
second rank tensors:

Vμν≡

∫

V

∇μvνd^3 r=

∮

∂V

vνnμd^2 s. (8.71)