Mathematical Methods for Physics and Engineering - Matematica.NET

26.23 EXERCISES26.23 A fourth-order tensor T ijkl has the propertiesT jikl = −T ijkl , T ijlk = −T ijkl .Prove that for any such tensor there exists a second-order tensor K mn such thatT ijkl = ɛ ijm ɛ kln K mnand give an explicit expression for K mn . Consider two (separate) special cases, asfollows.(a) Given that T ijkl is isotropic and T ijji = 1, show that T ijkl is uniquely determinedand express it in terms of Kronecker deltas.(b) If now T ijkl has the additional propertyT klij = −T ijkl ,show that T ijkl has only three linearly independent components and find anexpression for T ijkl in terms of the vectorV i = − 1 ɛ 4 jklT ijkl .26.24 Working in cylindrical polar coordinates ρ, φ, z, parameterise the straight line(geodesic) joining (1, 0, 0) to (1,π/2, 1) in terms of s, the distance along the line.Show by substitution that the geodesic equations, derived at the end of section26.22, are satisfied.26.25 In a general coordinate system u i , i =1, 2, 3, in three-dimensional Euclideanspace, a volume element is given bydV = |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 determinantg of the metric tensor, is given bydV = √ gdu 1 du 2 du 3 .Show that, under a general coordinate transformation to a new coordinatesystem u ′i , the volume element dV remains unchanged, i.e. show that it is a scalarquantity.26.26 By writing down the expression for the square of the infinitesimal arc length (ds) 2in spherical polar coordinates, find the components g ij ofthemetrictensorinthiscoordinate system. Hence, using (26.97), find the expression for the divergenceof a vector field v in spherical polars. Calculate the Christoffel symbols (of thesecond kind) Γ i jkin this coordinate system.26.27 Find an expression for the second covariant derivative v i; jk ≡ (v i; j ) ; k of a vectorv i (see (26.88)). By interchanging the order of differentiation and then subtractingthe two expressions, we define the components R l ijkof the Riemann tensor asv i; jk − v i; kj ≡ R l ijkv l .Show that in a general coordinate system u i these components are given byR l ijk = ∂Γl ik∂u j− ∂Γl ij∂u k+Γ m ikΓ l mj − Γ m ijΓ l mk.By first considering Cartesian coordinates, show that all the components R l ijk ≡ 0for any coordinate system in three-dimensional Euclidean space.In such a space, therefore, we may change the order of the covariant derivativeswithout changing the resulting expression.981