MSc Sem I & II- Syllabus.pptx
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This document outlines the proposed syllabus for a Master of Science in Mathematics program to be implemented in the 2020-21 academic year. It includes 10 core courses to be taken over two semesters. The courses cover topics in group theory, ring theory, real analysis, topology, linear algebra, field theory, measure theory, advanced topology, classical mechanics, and students can choose one core elective course from options like numerical analysis, differential equations, calculus of variations, number theory, fuzzy mathematics, and others. For each course, the document lists the course code, credit hours, and a brief description of the topics to be covered in 4 units. Required textbooks and reference materials are also provided for each course.
MSc Sem I & II- Syllabus.pptx
- 1. MAHATMA GANDHI ARTS, SCIENCE AND LATE N.P. COMMERCE COLLEGE ARMORI Proposed Syllabus For M.Sc. Mathematics Semester-I and Semester-II Under Choice Based Credit System (C.B.C.S.) With effect from AcademicYear: 2020-21 (Considered and approved by B.O.S. in Mathematics) 1
- 2. M.Sc. Mathematics Semester wise Syllabus M.Sc. Semester- I CORE COURSES 2 PSCMTH01: PSCMTH02: PSCMTH03: PSCMTH04: Group Theory & Ring Theory RealAnalysis Topology LinearAlgebra PSCMTH05: CORE ELECTIVE COURSE (Opt any one of the following) (a) NumericalAnalysis (b) Ordinary Differential Equations (c) Calculus of Variations (d) Number Theory (e) Fuzzy Mathematics- I M.Sc. Semester- II CORE COURSES PSCMTH06: PSCMTH07: PSCMTH08: PSCMTH09: Field Theory Lebesgue Measure Theory Advanced Topics in Topology Classical Mechanics PSCMTH10: CORE ELECTIVE COURSE (Opt any one of the following) (a) Differential Geometry (b) Coding Theory (c) Cryptography (d) SCILAB Programming (e) Fuzzy Mathematics- II
- 3. 3 SEMESTER-I Core Course Code: – PSCMTH01 Credit - 05 Group Theory & Ring Theory UNIT-I Permutation Groups, Normal subgroups and quotient groups, Isomorphism Theorems,Automorphisms, Conjugacy and G- Sets. UNIT-II Normal Series, Solvable groups, Nilpotent groups, Cyclic decomposition,Alternating groupsAn,, Simplicity ofAn. UNIT-III Direct products, Sylow theorems, Groups of order p2 and pq. UNIT-IV Ideals, Homomorphisms, Sum and direct sum of ideals, Maximal and prime ideals, Nilpotent and Nil ideals, Zorn’s Lemma. Text Book: Basic Abstract Algebra (Second Edition): P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press. Reference Books: 1. Topics inAlgebra (Second Edition): I. N. Herstein, John Wiley. 2. AbstractAlgebra: David Summit and Richard M. Foote, John Wiley.
- 4. 4 Core Course Code: - PSCMTH02 Credit - 05 RealAnalysis UNIT-I Uniform convergence, Uniform convergence and continuity, Uniform convergence and integration, Uniform convergence and differentiation, Equicontinuous families of functions, The Stone-Weierstrass theorem. UNIT-II Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem, UNIT-III Topological Manifolds, Compatible Charts, Smooth Manifolds, Examples of Smooth Manifolds. UNIT-IV Smooth functions on a Manifold, Smooth Maps between Manifolds, Diffeomorphisms, Smoothness in terms of Components, Examples on Smooth maps, Partial derivatives, The inverse function theorem. Text Books: 1. Principles of Mathematical Analysis (Third Edition): Walter Rudin, McGRAW – HILLBook Company. 2. An Introduction to Manifolds (Second Edition): Loring W. Tu, Springer. Reference Books: 1.Methods of Real Analysis: R.R. Goldberg, John Wiley. 2.Calculus of Several Variables: C Goffman, Harper and Row.
- 5. 5 Core Course Code:- PSCMTH03 Credit- 05 Topology UNIT-I Equipotent sets, Cardinal Numbers, Order Types, Ordinal Numbers. UNIT-II Open sets and limit points, Closed sets and closure, Operators and Neighbourhoods, Bases and Relative topologies. UNIT-III Connected sets and components, Compact and countably compact spaces, Continuous functions, Homeomorphisms. UNIT-IV T0- and T1- spaces, T2- spaces and sequences, Axioms of countability, Separability.,Regular and Normal spaces. Text Book: Foundations of General Topology: W. J. Pervin,Academic press, 1964. Reference Books: 1.Topology (Second edition): J. R. Munkres, Prentice Hall of India, 2002. 2. Introduction to Topology and Modern Analysis: G. F. Simmons, Mc Graw Hill 1963. 3. General Topology: J.L. Kelley, Van Nostrand, 1995. 4. Introduction to general Topology: K.D.Joshi, Wiley Eastern Ltd., 1983.
- 6. 6 Core Course Code:- PSCMTH04 Credit – 05 LinearAlgebra UNIT-I Vector spaces: Introduction, Vector spaces, subspaces, Linear combinations and system of linear equations, linear dependence and independence, Bases and dimension, Maximal Linear Independent Subsets. UNIT-II Linear Transformations and Matrices: Linear Transformations, Null spaces, and ranges, The matrix representation of a linear transformation, Composition of linear transformations, Invertibility and Isomorphisms, The change of Coordinate matrix, Dual spaces, and Homogeneous Linear Differential equations with constant coefficients. UNIT-III Diagonalization: Eigen values and eigen vectors, Diagonalizability, Invariant Subspaces and the Cayley-Hamilton Theorem. UNIT-IV Inner Product Spaces: Inner products and Norms, The Gram-Schmidt Orthogonalization process and orthogonal complements, the adjoint of a linear operator, Normal and self-adjoint operators, Unitary and orthogonal operators and their matrices, orthogonal projections and the spectral theorem, Quadratic forms. Jordan Canonical form I, Jordan Canonical form II, The minimal polynomial, Rational Canonical form. Text Book: Linear Algebra (Fourth Edition): S. H. Friedberg, A. J. Insel and L. E. Spence, Prentice-Hall International Inc. Scope: Ch 1: Art.1.1 to 1.7, Ch 2: Art. 2.1 to 2.7, Ch 5: Art 5.1,5.2,5.4, Ch 6: Art 6.1 to 6.7, Ch 7:Art 7.1 to 7.4. Reference Books: 1. Linear Algebra (Second Edition): Vivek Sahai, Vikas Bist, Narosa Publishing House 2. Topics inAlgebra: I. N. Herstein, Macmillan, Indian Edition. 3. Introduction to LinearAlgebra (Second Edition): S. Lang, Springer International Edition. 4. LinearAlgebra: K. Hoffman, R. Kunze, Prentice Hall of India. 5. LinearAlgebra (Second Edition): J. H. Kwak, S. Hong Birkhauser Verlag. 6. LinearAlgebra:Apure MathematicalApproach: Harvey E.Rose, Birkhäuser Verlag.
- 7. 7 (b) Ordinary Differential Equations UNIT-I Linear equations of first order, The equation y′ + ay = 0 , The equation y′ + ay = b(x), The general linear equation of first order, The second order homogeneous equation, Initial value problems, Linear dependence and independence, A formula for the Wronskian, The homogeneous equation of order n, The non-homogeneous equation of order n, A special method for solving non- homogeneous equation. UNIT-II Linear equations with variable coefficients, Initial value problem for the homogeneous equation, Solution of the homogeneous equation, Wronskian and linear independence, Reduction of order of a homogeneous equation, Non- homogeneous equation, Homogeneous equations with analytic coefficients, The Legendre equation, The Euler equation, Second order equation with regular singular points, The Bessel equation. UNIT-III Equations with variables separated, Exact equation, The method of successive approximations, The Lipschitz condition, Convergence of successive approximations, Nonlocal existence of solutions, Approximations to and uniqueness of the solutions. UNIT-IV Some special equations, Complex n-dimensional space, Systems as vector equations, Existence and uniqueness of solutions to systems, Existence and uniqueness for linear systems, Equations of order n. Text Book: An Introduction to Ordinary Differential Equation: E. A. Coddington, Prentice-Hall of India Pvt. Ltd., New Delhi. Scope: Chapter 1 to 6. Reference Books: 1. Differential Equations with Applications and Historical Notes (Second edition): G. F. Simmons, McGraw Hill Book Co. 2. Ordinary Differential Equations: G. Birkhoff and G. C. Rota, John Wiley and Sons. 3. Ordinary Differential Equations - A First Course: D. Somasundaram,, Narosa Publishing House.
- 8. 8 SEMESTER-II Core Course Code:- PSCMTH06 Credit - 05 Field Theory UNIT-I Unique factorization domains, Principal Ideal domains, Euclidean domains, Polynomial rings over unique factorization domains. UNIT-II Irreducible polynomials and Eisenstein criterion, Adjunction of roots, Algebraic extensions,Algebraically closed fields. UNIT-III Splitting fields, Normal extensions, Multiple roots, Finite fields, Separable extensions. UNIT-IV Automorphism groups and fixed fields, Fundamental theorem of Galois theory, Fundamental theorem of algebra. Text Book: Basic Abstract Algebra (Second Edition): P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press. Reference Books: 1. Topics inAlgebra, I. N. Herstein, Second Edition, John Wiley. 2. AbstractAlgebra, David S. Dummit and Richard M. Foote, John Wiley.
- 9. 9 Core Course Code:- PSCMTH07 Credit - 05 Lebesgue Measure Theory UNIT-I Introduction, Outer measure, Measurable sets and Lebesgue measure, A non- measurable set, Measurable functions, Littlewood’s three principles. UNIT-II The Riemann integral, The Lebesgue integral of a bounded function over a set of finite measure, Integral of a non-negative function, General Lebesgue integral, Convergence in measure. UNIT-III Differentiation of monotone functions, Functions of bounded variation, Differentiation of an integral,Absolute continuity, Convex functions. UNIT-IV The LP - Spaces, The Holder and Minkowski inequalities, Convergence and Completeness, Bounded linear functionals on LP- Spaces. Text Book: RealAnalysis (Third edition): H. L. Royden, Prentice Hall India, 1988. Reference Books: 1. Measure theory and Integration: De Barra G., Wiley Eastern Limited, 1981. 2. An introduction to Measure & Integration: Inder K. Rana, Narosa Publishing House.
- 10. 10 Core Course Code:- PSCMTH08 Credit- 05 Advanced Topics in Topology UNIT-I Completely normal spaces, Completely Regular spaces, Metric spaces as Topological spaces, Topological Properties. UNIT-II Finite products, Product invariant properties, Metric products, Tichonov topology, Tichonov theorem. UNIT-III Quotient topology, Urysohn’s metrization theorem, Paracompact spaces. UNIT-IV Nets and filters. Text books: 1. Foundations of General Topology: W. J. Pervin, Academic press, 1964. (For Unit-I, II & III) 2. Introduction to General Topology: K. D. Joshi, Wiley Eastern Ltd. 1983. (For Unit -IV) Reference books: 1. Topology (Second Edition): J. R. Munkres, Prentice Hall of India, 2002. 2. Introduction to topology and modern analysis: G. F. Simmons, Mc Graw Hill 1963. 3. General Topology: J. L. Kelley, Van Nostrand, 1995.
- 11. 11 Core Course Code:- PSCMTH09 Credit - 05 Classical Mechanics UNIT-I Hamilton’s Principle, Some techniques of calculus of variations, Derivation of Lagrange equations from Hamilton’s principle, Extension of principle to nonholonomic systems, Conservation theorems and symmetry properties. UNIT-II Legendre transformations and the Hamilton equations of motion, Cyclic coordinates and conservation theorems, Routh’s procedure and oscillations about steady motion, The Hamiltonian formulation of relativistic mechanics, The Principle of least action. UNIT-III The equations of canonical transformation, Examples of canonical transformations, The symplectic approach to canonical transformations, Poisson brackets and other canonical invariants. UNIT-IV Equations of motion, Infinitesimal canonical transformations, and conservation theorems in the Poisson bracket formulation, The angular momentum Poisson bracket relations, Symmetry groups of mechanical systems, Liouville’s theorem. Text Book: Classical Mechanics (Second Edition): H. Goldstein, Narosa publishing house, New Delhi. Reference Books: 1. Lectures inAnalytic Mechanics: F. Gantmacher, MIR Publishers, Moscow, 1975. 2. Classical Mechanics: Narayan Chandra Rana and Pramod Sharad Chandra Jog, Tata Mc Graw Hill.
- 12. 12 Core Course Code:- PSCMTH10 Credit – 05 (a) Differential Geometry UNIT-I The First Fundamental Form and Local Intrinsic Properties of a Surface. UNIT-II Godesics on a Surface. UNIT-III The Second Fundamental Form and Local Non-Intrinsic Properties of a Surface. UNIT-IV The Fundamental Equations of Surface Theory. Text Book: Differential Geometry A First Course: D. Somasundaram, Eighth Reprint 2017, Narosa Publishing House. Scope: Chapter 2,3,4 &5. Reference Book: 1. Geometry of curves and surfaces: M. P. Do Carmo,Academic Press. 2. An introduction to Differential Geometry: T. J. Wilmore, Oxford University Press.