Advanced Mathematical Physics I — DU B.Sc. (Hons) Physics Previous Year Question Papers

Roll Number:...................... (DSE Paper) (a) Show that the contraction of two cartesian tensors lowered the rank by any even numbers or 2. (b) IF Aguy = Ayav then show that symmetry properties of a tensor are invariant. (c) Prove the identity Eilm€jim = 28ij. (d) Find expression for gradient of a tensor. (e) Prove that divergence of curl of a tensor is zero. (f) What are the generators of a cyclic group G= (a) of order 10. (g) List all the elements of A4, where A4 denotes the set of all even permutations in S4. Q.2. (a) Show that in 3-dimensional space alternating tensor has 27 components. Further shows that it is an anti-symmetric tensor of order 3. (10) (b) What is skew symmetric tensor? Determine how many numbers of different nonzero components in skew symmetric tensor. If we assume Wij is skew-symmetric tensor of order two, with indices i and j running from 1 to 50. Calculate the number of independent components. (8) Q.3. (a) What are equal tensors? If w = aijA;Aj. Show that we can always write ∞ = bjiAjAj , where, bj is symmetric tensor? (9) Q.4. By using Cartesian tensor prove that Q.5. (a) Obtain an expression for moment of inertia tensor. What is rank of this tensor and prove that it is symmetric tensor. (9) (b) If there is a mapping from G(of part 4(a)) to C*(group of non-zero complex numbers with respect to multiplication) given by fiG → C" such that f_b 2= a tib, then prove that this mapping is an isomorphism. (9) Q.6.(a) From a group table of a multiplicative group Gi generated from two square matrices A and a , each of order n, subject to the relations Where, E is the identity matrix of order n. (b) Consider a set G2= { I,-1,I,-i }, prove that G2 form a group w.r.t. multiplication and check ifGi and G2 can be isomoxphic to each other or not. (6) (c ) Consider a cyclic group of order 4, G3= {e,a,a2,a3}. If G3 has 4 classes, find its irreducible representations.