Tensor Analysis and Application in Physics — DU B.Sc. (Hons) Physics Previous Year Question Papers

B.Sc. Hons Physics NEP: UGCF-2022 (a) If Ag# is a mixed tensor of rank 4, prove that A8# is a mixed tensor of rank 2. (b) Given a vector A+ = (2, 6,-3). Find the matrix elements of the second order skew-symmetric tensor associated with it. (c) Show that gradient of a scalar is a covariant tensor of order one. (d) Write down equations for two intersecting lines passing through a common point and having direction cosines Z{ and m!. Hence determine an expression for angle between the lines. (e) Prove that if g = gfj is a covariant tensor of order two, and U = UZ and V = V'£ are contravariant vectors, then the double inner product guv = gijU!Vj is lan invariant. ® Show that there is no distinction between contravariant and covariant vectors when we restrict to rectangular cartesian transformation of coordinates. (g) Any second order tensor can be written as a sum of symmetric Sij and an dntisymmetric tensor A[j. Find the corresponding symmetric and antisymmetric tensors associated with (a) Prove the following vector identities using cartesian tensors. (12) (b) Define isotropic tensors and show that the only isotropic tensor of rank one is the null vector. (a) A covariant tensor has components (xy, 2y - z', XZ) in rectangular coordinates. Find its covariant components in cylindrical coordinates. (b) Using the covariant components in cylindrical coordinates obtained in part (a), find the contravariant components of the tensor in cylindrical coordinates. (8) (a) In a Cartesian coordinate system, components of a tensor Tij are given by - b -a where a, b and c are constants. Find the components of this tensor in a new cartesian coordínate which is obtained by rotating the coordinate system about the z-axis by a fixed angle B. (10) Hence find tabcEabs (8) (a) Given that the stress tensor Pij and strain tensor ekz are both symmetric tensors. Prove that rank four elastic tensor Gijkı must be symmetric in its first pair of indices (Gijk = Cjüet), its last pair of indices (Gijk = Gijic), and also symmetric under the exchange of these pairs (Gijkz = Cklj). Assume the existence of an elastic potential energy density, (b) A body undergoes a deformation described by the displacement vector i = (kxy? , 2KX1X2, 0) Find the strain tensor eij. (8) (a) Check whether the Christoffel symbols of the second kind transform like a tensor or not. (b) Prove that the divergence of a contravariant vector of rank one, AP is 1 д div AP = Alp = where g is the determinant of the metric tensor gij and A, corresponds to the covariant derivative of AP with respect to x9. Use For = 2xr Im((9).