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26.23 A fourth-order tensor To has the properties Til = -Tijl, Tijl = -Tiki- Prove that for any such tensor there exists a second-order tensor K, such that Till = Eijmekin K man and give an explicit expression for K. Consider two (separate) special cases, as follows. (a) Given that Tow is isotropic and Tix = 1, show that Tow is uniquely deter- mined and express it in terms of Kronecker deltas. (b) If now Tijd has the additional property show that The has only three linearly independent components and find an expression for The in terms of the vector VI = -Tej Tips. 26.24 Working in cylindrical polar coordinates p. o, z, parameterise the straight line geodesic) joining (1, 0, 0) to (1, m/2, 1) in terms of s, 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 system i', i = 1,2,3, in three-dimensional Euclidean space, a volume element is given by dV = le, du' . (ez du' x ex du )|. Show that an alternative form for this expression, written in terms of the deter- minant g of the metric tensor, is given by dV = g du' du du'. Show that, under a general coordinate transformation to a new coordinate system u", the volume element dl' remains unchanged, i.e. show that it is a scalar quantity. 26.26 By writing down the expression for the square of the infinitesimal are length (ds)- in spherical polar coordinates, find the components go of the metric tensor in this coordinate system. Hence, using (26.97), find the expression for the divergence of a vector field v in spherical polars. Calculate the Christoffel symbols (of the second kind) I' in this coordinate system. 26.27 Find an expression for the second covariant derivative via = (v;; ;);k of a vector (see (26.88)). By interchanging the order of differentiation and then subtracting the two expressions, we define the components Rig of the Riemann tensor as Show that in a general coordinate system u' these components are given by Rijk = durk By first considering Cartesian coordinates, show that all the components R', = 0 for any coordinate 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