126 TENsaR ANALYSIS

EXERCISES

3.2.1 If T..¡ is a tensor ofrank n, show that 8T ..;/8xj is a tensar of rank n + 1 (cartesiancoordinates).Note. In noncartesian coordinate systems the coefficients a¡jare, in general, fune-tions of the coordinates, and the simple derivative of a tensor of rank n is no!atensar except in the special case of n = O. In this case the derivative does yieldacovariant vector (tensor ofrank 1) by Eq. 3.1l.

3.2.2 If T;jk... is a tensar of rank n, show that I8T;jk.../8xj is a tensar of rank n-1(cartesian coordinates).

3.2.3 The operator

v2 - ~ 82e2 8t2

may be written as

4- 82

I 8X2'¡~1 ,

using x4-= ¡et. This is the four-dimensional Laplacian, usually called the d'Alem-bertian and denoted by 02. Show that it is a sealar operator.

::L" ->

lS seenin Section 3.2,we can easilyshow that A¡B¡re we are con cerned with a variety of inverse rela-ms as

K¡A¡=B

KijAj = B¡

(3.29a)

(3.29b)

(3.29c)

(3.29d)

(3.2ge)

r KijAjk = B¡k

K¡jklAij = Bkl

KijAk = B¡jk

In each ofthese expressions A and B are known tensors ofrank indicated bythenumber ofindices and A is arbitrary. In each case K is an unknown quantity. Wewish to establish the transformation properties of K. The quotient rule assertsthat if the equation of interest holds in all (rotated) cartesian coordinate sys-tems, K is a tensar of the indicated rank. The importance in physical theory isthat the quotient rule can establish the tensar nature of quantities. Exercise3.3.1 is a simple illustration ofthis. The quotient rule (Eq. 3.29b) shows that theinertia matrix appearing in the angular momentum equation L = 1m, Section4.6, is a tensar. And Eq. 3.29d is quoted in Section 3.6 to establish the tensarnature of the generalized Hooke's law "constant" Cijkl'

In proving the quotient rule, we consider Eq. 3.29b as a typical case. In our

i.3LoJ. j

136

3.4.3

3.4.4

3.4.5

3.4.6

TENsaR ANALYSIS

Show that

(a) ¿¡ii= 3,(b) ¿¡ij0ijk= O,(e) 0ipq0jpq= 2¿¡ij,(d) 0ijk0ijk= 6.

Show that

0ijk0pqk = ¿¡ip¿¡jq- ¿¡iq¿¡jp'

(a) Express the components of a cross-product vector C, C = A x B, in termsof 0ijkand the components of A and B.

(b) Use the antisymmetry of 0ijkto show that A. A x B = O.ANS. (a) Ci = 0ijkAjBk'

(a) Show that the inertia tensar (matrix) of Section 4.6 may be written

lij = m(xnxn¿¡ij - XiX)

fmass m at (Xl,X2,X3)'

lij = -Mi/MIj = -m0i/kXktljmXm,

1/2tilkXk' This is the contraction of two second-rank tensors~u~~ .,'-'~':;uu~m with the matrix product of Section 4.2.-=

r

3.4.7

3.4.8

3.4.9

Write V. V x A and V x V<pin 0ijknotation, so that it becomes obvious that eachexpress ion vanishes.

a aANS. V'V x A = 0ijk--Ak

aXi aXj

a a--¡p,

(V x V<p)i= 0ijkaXj aXk

Expressing cross products in terms of Levi-Civita symbols (tijk), derive the BAC-CAB rule, Eq. 1.50.Hint. The relation of Exercise 3.4.4 is helpfuL

Verify that each of the following fourth-rank tensors is isotropic, that is, it hasthe same forro independent of any rotation of the coordinate systems.(a) Aijki = ¿¡ij¿¡kio(b) Bijkl = bikbji + bilbjk'

(e) Cijki = bikbji - bi/bjk'

3.4.10 Show that the two index Levi-Civita symbol tij is a second-rank pseudotensor (intwo-dimensional space). Does this contradict the uniqueness of bij (Exercise3. 1.4)?

3.4.11 (a) Represent tij by a 2 x 2 matrix, and using the 2 x 2 rotation matrix ofSection 4.3, show that tij is invariant under orthogonal similarity trans-formations.

(b) Demonstrate the pseudo nature of tij by using (~matrix.

O

),

- 1 . as the transformmg

3.4.12 Given Ak = 1tijkBij with Bij = - Bji, antisymmetric, show that

Bmn= tmnkAk'

T EXERCISES 139

Uxx = O, and so on,

Uxy = - Uyx, and so on,

then for any vector a

a.U = -U-a. (3.70)

Multiplication of a vector and an antisymmetric dyadic follows an anticom-mutation rule (See Exercise 3.5.4a).

Dyadics are rather awkward to handle in comparison with the usual tensaranalysis (once the concept oftransformation under coordinate rotation has beenabsorbed). They are quite unwieldy for representing third- or higher-ranktensors, so we shall return to tensar analysis and hm 'd.

dyadic notation.<)

So <.(

EXERCISES

3.5.1

3.5.2

3.5.3

3.5.4

3.5.5

3.5.6

3.5.7

If A and B transform as vectors, Eqs. 3.6 and 3.8, show that the dyadic ABsatisfies the tensar transformation law, Eq. 3.13.

Show that I = ii + ii + kk is a unir dyadic in the sense that for any vector V

I.V=V.

The individual dyads ii, and so on are specific examples of the projection opera-tors of quantum mechanics.

Show that Vr is equal to the unit dyadic, l.

If U is an antisymmetric dyadic and V a vector, show that(a) V.U = -U.V(b) V.U.V=O.

The two-dimensionalvectorsr = ix + jy and t = - yi + jx mayberelatedbytheten sor equation r. U = t.(a) Find the tensar U, using our earlier component description of tensors.(b) Find U and treat it as a dyadic.

In an investigation of the interaction of molecules a dyadic is formed from theunit relative distance vectors e¡2 given by

r2 - r¡

e¡2=lr2-r¡l'

For

U = I - 3e¡2e¡2

show that trace U .U = 6.lis the unit dyadic; that is I = ii + ii + kk.

Showthat Gauss's theorem holds for dyadics,that

1 dO'. D = 1 V' D dT.

!!II

""

and

EXERCISES 163

Pij = gkS[ij,k]

= !gkS{

Ogik + ogj~ - Ogij}

.2 oqJ oq' oqk

These Christoffe1symbolsand the covariant derivasection.

(3.153)

'6. ~I

~, }'

EXERCISES

3.8.1

3.8.2

3.8.3

3.8.4

3.8.4A

3.8.5

3.8.6

3.8.7

~ - ~ ~ -

Equations 3.128 and 3.129 use the scale factor hi, citíng Exercise 2.2.3. In Section2.2 we had restricted ourselves to orthogonal coordinate systems, yet Eq. 3.128holds for nonorthogonal systems. Justify the use ofEq. 3.128 for nonorthogonalsystems.

(a) Show that Eio Ej = c5j.

(b) From the result ofpart (a) show that

Fi=FoEi and F;=FoEi'

F or the special case of three-dimensional space (E1, E2, E3defining a right-handedcoordinate system, not necessarily orthogonal) show that

i - Ej X EkE--,

Ej X Ek' Eii, j, k = 1, 2, 3 and cydic permutatíons.

Note. These contravariant basis vectors, E" define the reciprocal lattice spaceof Sections 1.5 and 4.4.

Prove that the contravariant metric tensor is given by

gij = Ei o Ej.

If the covariant vectors Ei are orthogonal, show that(a) gij is diagonal,

(b) gii = l/gii (no summation),(c) IEil= l/IE.I-

Derive the covariant and contravariant metric tensors for circular cylindricalcoordinates.

Transform the right-hand side ofEq. 3.138

V¡f¡= 8¡f¡Ei8q'

into the ei basis and verify that this expression agrees with the gradient developedin Section 2.2 (for orthogonal coordinates).

Evaluate 8E;/8qj for the spherical polar coordinates, and from these resultscalcula te the spherical polar coordinate r\.Note. Exercise 2.5.1 offers a way of calculating the needed partíal derivatives.Remember

E1=rO but E2= r90 and E3= rsin8<¡1o.

..,. -