Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.23 EXERCISES

26.23 A fourth-order tensorTijklhas the properties

Tjikl=−Tijkl,Tijlk=−Tijkl.

Prove that for any such tensor there exists a second-order tensorKmnsuch that

Tijkl=ijmklnKmn

and give an explicit expression forKmn. Consider two (separate) special cases, as

follows.

(a) Given thatTijklis isotropic andTijji= 1, show thatTijklis uniquely deter-

mined and express it in terms of Kronecker deltas.

(b) If nowTijklhas the additional property

Tklij=−Tijkl,

show thatTijklhas only three linearly independent components and find an

expression forTijklin terms of the vector

Vi=−^14 jklTijkl.

26.24 Working in cylindrical polar coordinatesρ, φ, z, parameterise the straight line
(geodesic) joining (1, 0 ,0) to (1,π/ 2 ,1) in terms ofs, the distance along the line.
Show by substitution that the geodesic equations, derived at the end of section
26.22, are satisfied.
26.25 In a general coordinate systemui,i=1, 2 ,3, in three-dimensional Euclidean
space, a volume element is given by

dV=|e 1 du^1 ·(e 2 du^2 ×e 3 du^3 )|.

Show that an alternative form for this expression, written in terms of the deter-

minantgof the metric tensor, is given by

dV=

√

gdu^1 du^2 du^3.

Show that, under a general coordinate transformation to a new coordinate
systemu′i, the volume elementdVremains unchanged, i.e. show that it is a scalar
quantity.
26.26 By writing down the expression for the square of the infinitesimal arc length (ds)^2
in spherical polar coordinates, find the componentsgijofthemetrictensorinthis
coordinate system. Hence, using (26.97), find the expression for the divergence
of a vector fieldvin spherical polars. Calculate the Christoffel symbols (of the
second kind) Γijkin this coordinate system.
26.27 Find an expression for the second covariant derivativevi;jk≡(vi;j);kof a vector
vi(see (26.88)). By interchanging the order of differentiation and then subtracting
the two expressions, we define the componentsRlijkof theRiemann tensoras

vi;jk−vi;kj≡Rlijkvl.

Show that in a general coordinate systemuithese components are given by

Rlijk=

∂Γlik

∂uj

−

∂Γlij

∂uk

+ΓmikΓlmj−ΓmijΓlmk.

By first considering Cartesian coordinates, show that all the componentsRlijk≡ 0

foranycoordinate system in three-dimensional Euclidean space.

In such a space, therefore, we may change the order of the covariant derivatives

without changing the resulting expression.