BSc Semester 5 Mathematics Paper II (C) Differential Geometry & Tensor Analysis Exam 2023 PDF

## CODE-4397 **B.Sc. (SEMESTER - V)** **EXAMINATION - 2023** **MATHEMATICS** **PAPER - II (C)** **DIFFERENTIAL GEOMETRY & TENSOR ANALYSIS** **Time:** 02 Hours **Maximum Marks:** 75 **Note:** This question paper has three sections A, B and C. Follow instructions in each section. Section A, B and...

## CODE-4397 **B.Sc. (SEMESTER - V)** **EXAMINATION - 2023** **MATHEMATICS** **PAPER - II (C)** **DIFFERENTIAL GEOMETRY & TENSOR ANALYSIS** **Time:** 02 Hours **Maximum Marks:** 75 **Note:** This question paper has three sections A, B and C. Follow instructions in each section. Section A, B and C are of 9, 36 and 30 marks respectively. **Section A - (खण्ड-अ)** **Note:** This section have five sub questions attempt any three questions. Each sub question carries three marks. **Q.No.1** (a) Define Kronecker delta. (b) Define Christoffel symbols of first and second kind. (c) Find Osculating plane at t on the helix $x = a cos t, y = a sin t, Z = c t$ (d) Give the statement of curvature and torsion. (e) Define normal curvature. **Section-B - (खण्ड-ब)** **Note:** This section has seven sub questions. Attempt any four questions. Each sub questions carries nine (9) marks. **Q.No.2** (a) State and prove that Euler's Theorem. (b) To prove that $ \int \frac{1 }{\log x \sqrt{ x } } dx = 2 log (log x) + c $ (c) Prove that first fundamental from (i.e. metric) is a positive definite quadratic form in du, dv. (d) State and Prove Weingarten equation. (e) Show that a symmetric tensor A or A" has $ \frac{n(n+1)}{2} $ independent components in V. (f) Prove that $ div A" = V, A" = A', \frac{1}{\sqrt{g}} \frac{d}{dx^i} ( A^{ik} \sqrt{g} ) $ **What form does the above equation assume if A^{ik} is skew symmetric.** (g) State and prove Bonnet's Theorem. **Section-C - (खण्ड-स)** **Note:** This section has four sub questions. Attempt any two questions. Each sub questions carries 15 (Fifteen) marks. **Q.No.3** (a) State and prove serret Frenet formula. (b) State and prove that quotient laws. (c) For the curve $x = 3t, y =3t², Z = 2t$ Show that $ p = - \sigma = \frac{ 3 (1 + 2t²)² }{ 2} $ **Where p is radius of curvature and σ is the radius of torsion.** (d) To show that metric tensors are covariant constant w.r to Christoffel symbols.