Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.23 EXERCISES

26.10 A symmetric second-order Cartesian tensor is defined by

Tij=δij− 3 xixj.

Evaluate the following surface integrals, each taken over the surface of the unit

sphere:

(a)

∫

TijdS;(b)

∫

TikTkjdS;(c)

∫

xiTjkdS.

26.11 Given a non-zero vectorv, find the value that should be assigned toαto make

Pij=αvivj and Qij=δij−αvivj

into parallel and orthogonal projection tensors, respectively, i.e. tensors that
satisfy, respectively,Pijvj=vi,Pijuj=0andQijvj=0,Qijuj=ui,foranyvector
uthat is orthogonal tov.
Show, in particular, thatQijis unique, i.e. that if another tensorTijhas the
same properties asQijthen (Qij−Tij)wj=0foranyvectorw.
26.12 In four dimensions, define second-order antisymmetric tensors,FijandQij,and
a first-order tensor,Si, as follows:

(a) F 23 =H 1 ,Q 23 =B 1 and their cyclic permutations;

(b)Fi 4 =−Di,Qi 4 =Eifori=1, 2 ,3;

(c) S 4 =ρ,Si=Jifori=1, 2 ,3.

Then, takingx 4 astand the other symbols to have their usual meanings in

electromagnetic theory, show that the equations

∑

j∂Fij/∂xj=Siand∂Qjk/∂xi+
∂Qki/∂xj+∂Qij/∂xk= 0 reproduce Maxwell’s equations. In the latteri, j, kis any
set of three subscripts selected from 1, 2 , 3 ,4, but chosen in such a way that they
are all different.
26.13 In a certain crystal the unit cell can be taken as six identical atoms lying at the
corners of a regular octahedron. Convince yourself that these atoms can also be
considered as lying at the centres of the faces of a cube and hence that the crystal
has cubic symmetry. Use this result toprovethat the conductivity tensor for the
crystal,σij,mustbeisotropic.
26.14 Assuming that the current densityjand the electric fieldEappearing in equation
(26.44) are first-order Cartesian tensors, show explicitly that the electrical con-
ductivity tensorσijtransforms according to the law appropriate to a second-order
tensor.
The rateWat which energy is dissipated per unit volume, as a result of the
current flow, is given byE·j. Determine the limits between whichWmust lie for
a given value of|E|as the direction ofEis varied.
26.15 In a certain system of units, the electromagnetic stress tensorMijis given by

Mij=EiEj+BiBj−^12 δij(EkEk+BkBk),
where the electric and magnetic fields,EandB, are first-order tensors. Show that
Mijis a second-order tensor.
Consider a situation in which|E|=|B|, but the directions ofEandBare
not parallel. Show thatE±Bare principal axes of the stress tensor and find
the corresponding principal values. Determine the third principal axis and its
corresponding principal value.
26.16 A rigid body consists of four particles of massesm, 2 m, 3 m, 4 m, respectively
situated at the points (a, a, a), (a,−a,−a), (−a, a,−a), (−a,−a, a) and connected
together by a light framework.
(a) Find the inertia tensor at the origin and show that the principal moments of
inertia are 20ma^2 and (20± 2

√

5)ma^2.