# RTMNU M.Sc Mathematics 2 Sem Syllabus 2020

RTMNU M.Sc. Sem 2 New Syllabus 2020 | RTMNU M.Sc. Revised Sem 2 Syllabus 2020 | Nagpur University M.Sc. Syllabus Part 1 | RTMNU First Year M.Sc. Syllabus
RTM Nagpur University M.Sc Semester New Revised Syllabus is given below for Downloading. The students can Download the respective Syllabus from following given details. Just go through the given links & read the given syllabus carefully. Nagpur University First Year New Semester Online Detail syllabus given below.

### Algebra-II

Unit 1:
Unique factorization domains. Principal Ideal domains. Euclidean domains. Polynomial rings over unique factorization domains.

Unit 2:
Irreducible polynomials and Eisenstein criterion. Adjunction of roots. Algebraic extensions. Algebraically closed fields. Splitting fields. Normal extensions. Multiple roots.

Unit 3:
Finite fields. Separable extensions. Automorphism groups, and fixed fields. Fundamental theorem of Galois theory. Fundamental theorem of algebra.

Unit 4:
Roots of unity and Cyclotomic polynomials. Cyclic extensions. Polynomials solvable by radicals. Ruler and compass constructions.

Text Book :
Basic Abstract Algebra: Bhattacharya, Jain, Nagpaul; Second Edition, Cambridge University Press.

Reference Books :
1. Topics in Algebra, I. N. Herstein, Second Edition, John Wiley.
2. Abstract Algebra, David S.Dummit and Richard M.Foote, John Wiley.

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### Real Analysis -II

Unit 1:
Outer measure.Measurable sets and Lebesgue measure. Anon-measurable set, Measurable functions, Littlewood’s three principles.

Unit 2:
The Riemann integral. Lebesgue integral of a bounded function over a set of finite measure. Integral of a non-negative function. General Lebesgue integral. Convergence in measure. Differentiation of monotone functions. Functions of bounded variation. Differentiation of an integral.

Unit 3:
Absolute continuity.Convex functions. Lp-spaces. Holder and Minkowski inequality. Riesz Fischer theorem. Approximation in Lp. Bounded linear functionals on Lp-spaces.

Unit 4:
Compact metric spaces. Baire category theorem. Arzela Ascoli theorem. Locally compact spaces. Sigma compact spaces.

Text Book :
Real Analysis, H.L. Royden, Third edition, Prentice Hall, 1988.

Reference Books :
1. Measure theory and Integration, G. de Barra Wiley Eastern Limited, 1981.
2. An introduction to Measure & Integration, Inder K. Rana, Narosa Publishing House

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### Topology-II

Unit 1:
Urysohn’s lemma. Tietze extension theorem. Completely regular spaces.Completely normal
spaces. Compactness for metric spaces. Properties of metric spaces.

Unit 2:
Quotient topology. Nets and filters.

Unit 3:
Product topology : Finite products, product invariant properties, metric products, Tichonov
topology, Tichonov theorem.

Unit 4:
Locally finite and discrete families in topological spaces. Paracompact spaces, Urysohn’s metrization theorem.

Text books:
1. Foundations of General Topology: W.J. Pervin, Academic press, 1964.
2. Introduction to general Topology: K.D. Joshi, Wiley Eastern Ltd. 1983.

Reference books:
1. Topology: J.R.. Munkres, second edition, 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.

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### Differential Geometry

Unit 1:
Definition of surface. Curves on a surface. Surfaces of revolution. Helicoids. Metric. Direction coefficients. Families of curves. Isometric correspondence. Intrinsic properties. Geodesics. Canonical geodesic equations.

Unit 2:
Normal property of geodesics. Existence theorems. Geodesic parallels. Geodesic curvature. Gauss Bonnet theorem. Gaussian curvature. Surfaces of constant curvature. Conformal mapping. Geodesic mapping.

Unit 3:
Second fundamental form. Principal curvatures. Lines of curvature. Developable. Developable associated with space curves. Developable associated with curves on surfaces. Minimal surfaces and ruled surfaces. Fundamental equations of Surface theory. Parallel surfaces.

Unit 4: Compact surfaces whose points are umbilics. Hilbert’s lemma. Compact surfaces of constant Gaussian or mean curvature. Complete surfaces. Characterisation of complete surfaces. Hilbert’s theorem. Conjugate points on geodesics. Intrinsically defined surfaces. Triangulation. Two dimensional Riemannian manifolds. Problem of metrization. Problem of continuation.

Text Book:
An introduction to Differential Geometry: T.J. Wilmore; Oxford University Press

Reference Book:
Geometry of curves and surfaces: do Carmo, Academic Press.

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### Classical Mechanics

Unit 1:
Variational Principle and Lagrange’s equations; 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 2: 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 3:
The equations of canonical transformation. Examples of canonical transformation. The symplectic approach to canonical transformations. Poisson brackets and other canonical invariants.

Unit 4:
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: By H. Goldstein, Second Edition Narosa publishing house, New Delhi.

References:
1. Lectures in Analytic Mechanics: F. Gantmacher, MIR Publishers, Moscow, 1975.
2. Classical Mechanics: Narayan Chandra Rana and Pramod Sharad Chandra Jog, Tata Mc Graw Hill.
3. Lecture on Advanced Mechanics, Sonu Nilu Publication (2004) by T M Karade and G S Khadekar