66 5 Symmetric Second Rank Tensors
with some positive numberX. In many applications, all principal values are posi-
tive,S(i)>0. Then (5.35) represents an ellipsoid in “x-space” with the semi-axes
X/
√
S(i). In the uniaxial case it is an ellipsoid of revolution. It reduces to a sphere
with radiusX, when all principal values are equal.
In many applications in physics, bilinear forms are encountered, where the prin-
cipal values of the second rank tensor cannot be negative. An example is the kinetic
energyTof a solid body rotating with angular velocityw:
T=
1
2
wμΘμνwν≥ 0. (5.36)This inequality has to hold true for any direction of the angular velocity, in particular
forwparallel to any one of the principal axes. All principal values must not be
negative. Thus the equation (5.36), withT=const.>0, describes an ellipsoid in
w-space.
5.4.2 Linear Mapping
A second geometric interpretation of a symmetric tensor is based on thelinear
mapping
yμ=Sμνxν (5.37)
fromx-space intoy-space.
Firstly, notice that the vectoryas related toxvia (5.37), in general, is not parallel
tox, unlessxis parallel to one of the principal directions. Thus the cross product
x×yis not zero, provided that the symmetric traceless part of the tensorSis not
isotropic. More specifically, one has
(x×y)μ=εμνλxνyλ=εμνλxνSλκxκ=εμνλxνSλκ xκ. (5.38)In the uniaxial case, cf. (5.8), with the symmetry axis parallel to the unit vectore,
this relation reduces to
(x×y)μ=εμνλxνyλ=(S‖−S⊥)εμνλxνeλeνxκ=(S‖−S⊥)x·e(x×e)μ.(5.39)Clearly, this expression vanishes when the principal valuesS‖andS⊥are equal to
each other. Furthermore, (5.39) gives zero both forxparallel and perpendicular toe,
and it assumes extremal values, whenxencloses an angle of 45◦with the symmetry
axis.
Secondly, when the end point of the vectorxscans a unit sphere, the end point
of the vectorywill be on an ellipsoid provided that all principal values are positive.
In this case the semi-axes are equal toS(i). Again biaxial and uniaxial ellipsoids are
generated by biaxial and uniaxial tensors, and the ellipsoid degenerates to a sphere