Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

26.28 A curver(t) is parameterised by a scalar variablet. Show that the length of the
curve between two points,AandB,isgivenby

L=

∫B

A

√

gij

dui

dt

duj

dt

dt.

Using the calculus of variations (see chapter 22), show that the curver(t)that

minimisesLsatisfies the equation

d^2 ui

dt^2

+Γijk

duj

dt

duk

dt

=

̈s

̇s

dui

dt

,

wheresis the arc length along the curve, ̇s=ds/dtand ̈s=d^2 s/dt^2. Hence, show
that if the parametertis of the formt=as+b,whereaandbare constants,
then we recover the equation for a geodesic (26.101).
[ A parameter which, liket, is the sum of a linear transformation ofsand a
translation is called anaffineparameter. ]
26.29 We may define Christoffel symbols of the first kind by

Γijk=gilΓljk.

Show that these are given by

Γkij=

1

2

(

∂gik

∂uj

+

∂gjk

∂ui

−

∂gij

∂uk

)

.

By permuting indices, verify that

∂gij

∂uk

=Γijk+Γjik.

Using the fact that Γljk=Γlkj, show that

gij;k≡ 0 ,

i.e. that the covariant derivative of the metric tensor is identically zero in all

coordinate systems.

26.24 Hints and answers

26.1 (a)u′ 1 =x 1 cos(φ−θ)−x 2 sin(φ−θ), etc.;
(b)u′ 11 =s^2 x^21 − 2 scx 1 x 2 +c^2 x^22 =c^2 x^22 +csx 1 x 2 +scx 1 x 2 +s^2 x^21.
26.3 (a) (1/

√

2)(

√

2 , 0 ,0; 0, 1 ,−1; 0, 1 ,1). (b) (1/

√

2)(1, 0 ,−1; 0,

√

2 ,0; 1, 0 ,1).

r=(2

√

2 , −1+

√

2 , − 1 −

√

2)T.

26.5 Twice contract the array with the outer product of (x, y, z) with itself, thus
obtaining the expression−(x^2 +y^2 +z^2 )^2 , which is an invariant and therefore a
scalar.
26.7 WriteAj(∂Ai/∂xj)as∂(AiAj)/∂xj−Ai(∂Aj/∂xj).
26.9 (i) Write out the expression for|AT|, contract both sides of the equation withlmn
and pick out the expression for|A|ontheRHS.Notethatlmnlmnis a numerical
scalar.
(iii) Each non-zero term on the RHS contains any particular row index once and
only once. The same can be said for the Levi–Civita symbol on the LHS. Thus
interchanging two rows is equivalent to interchanging two of the subscripts of
lmn, and thereby reversing its sign. Consequently, the magnitude of|A|remains
the same but its sign is changed.
(v) If, say,Api=λApj, for some particular pair of valuesiandjand allpthen,