# RTMNU MSc 3 Sem Mathematics Syllabus 2020

RTMNU MSc Sem 3 New Syllabus 2020 | RTMNU M.Sc Revised Sem 3 Syllabus 2020 | Nagpur University M.Sc Syllabus Part 2| RTMNU Second 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 Second Year New Semester Online Detail syllabus given below.

### Complex Analysis

Unit 1
Impossibility of ordering Complex numbers. Extended complex plane and stereographic projection. Elementary properties and examples of analytic Functions: Power series, analytic functions.

Unit 2
Analytic functions as mappings, Mobius transformations. Power series representation of analytic functions, zeros of an analytic function, index of a closed curve.

Unit 3
Cauchy’s theorem and integral formula, the homotopic version of cauchy’s theorem and simple connectivity, counting zeros; the open mapping theorem, Goursat’s theorem, Classification of singularities, residues, the argument principle.

Unit 4
The maximum principle. Schwarz’s lemma. convex functions and Hadamards three circles theorem. Phragmen-Lindelof theorem.

Text Book
Functions of one complex variable: John B. Conway, Second edition, Springer international Student Edition.

Reference Book
Complex Analysis, L.V. Ahlfors. Mc-Graw Hill, 1966

___________________________________________________________________________

### Functional Analysis

Unit 1
Normed spaces, Banach spaces, Further properties of normed spaces. Finite dimensional normed spaces and subspaces. Compactness and finite dimension. Bounded and continuous linear operators.

Unit 2
Linear functionals. Normed spaces of operators. Dual spaces. Inner product space. Hilbert space. Further properties of inner product spaces. Orthogonal complements and direct sums. Orthonormal sets and sequences. Total orthonormal sets and sequences.

Unit 3
Representation of functionals on Hilbert spaces. Hilbert adjoint operators, self adjoint, unitary and normal operators. Hahn-Banach Theorem, Hahn-Banach Theorem for complex vector spaces and normed spaces. Reflexive spaces.

Unit 4
Category theorem, Uniform boundedness theorem, strong and weak convergence, Convergence of sequences of operators and functionals. Open mapping theorem, Closed linear operators and closed graph theorem.

Text Book
Introductory Functional Analysis with Applications by E. Kreyszig, John Wiley and Sons.

Reference Books
1. Introduction to Functional Analysis by A.E. Taylor and D.C. Lay, John Wiley and Sons.
2. Introduction to Topology and Modern Analysis: G.F. Simmons, Mc Graw Hill

__________________________________________________________________________

M. Sc Mathematics
Semester-III
Paper – XIII (Code 3T3)

Mathematical Methods

Unit 1
Fourier integral theorem. Fourier transform. Fourier cosine and sine transform. The convolution integral. Multiple Fourier transform. Solution of partial differential equation by means of Fourier transform.

Unit 2
Calculations of the Laplace transform of some elementary functions. Laplace transform of derivatives. The convolution of two functions. Inverse formula for the Laplace transform. Solutions of ordinary differential equations by Laplace transform.

Unit 3
Finite Fourier transform. Finite Sturm-Liouville transforms. Generalized finite Fourier transform.

Unit 4
Finite Hankel transform. Finite Legendre transform. Finite Mellin transform.

Text Book
The use of integral transforms: I N. Sneddon, Tata Mc Graw Hill Publishing Company Ltd.

References Books
Modern Mathematics For Engineers: Edwin F Beckenbach, Second series, Mc Graw Hill Book Company.

__________________________________________________________________________

### Fluid Dynamics-I

Unit 1
Real fluids and ideal fluids. Velocity of a fluid at a point. Stream lines and path lines. Steady and unsteady flows. Velocity potential. Velocity vector. Local and particle rate of change. Equation of continuity. Acceleration of a fluid. Condition at a rigid boundary. General analysis of fluid motion. Euler’s equation of motion. Bernoulli’s equation. Worked examples. Discussion of the case of steady motion under conservative body forces. Some further aspects of vortex motion.

Unit 2
Sources, sinks and doublets. Images in a rigid infinite plane. Images in solid spheres. Axisymmetric flows. Stokes’ stream function. The complex potential for twodimensional irrotational, incompressible flow. Complex velocity potential for standard two dimensional flow. Uniform stream. Line source and line sink. Line doublets. Line vortices. Two dimensional image systems. The Milne-Thomson circle theorem. Circle Theorem. Some applications of circle theorem. Extension of circle theorem. The theorem of Blasius.

Unit 3
The equations of state of a substance, the first law of thermodynamics, internal energy of a gas, functions of state, entropy, Maxwell’s thermodynamic relation, Isothermal Adiabatic and Isentropic processes. Compressibility effects in real fluids, the elements of wave motion. One dimensional wave equation, wave equation in two and three dimensions, spherical waves, progressive and stationary waves.

Unit 4
The speed of sound in a gas, equation of motion of a gas. Sonic, subsonic, supersonic flows; isentropic gas flow. Reservoir discharge through a channel of varying section, investigation of maximum mass flow through a nozzle, shock waves, formation of shock waves, elementary analysis of normal shock waves.

Text Book
F. Chorlton, Text book of Fluid Dynamics, CBS Publishers, Delhi 1985.

Reference Books
1. G.K. Batchelor, An Introduction to fluid Mechanics, Foundation Books, New Delhi 1994.
2. M.D. Raisinghania, fluid Mechanics, S. Chand and Company, Delhi.

_________________________________________________________________________

### General Relativity

Unit 1
Tensor Algebra, Riemannian geometry, Curvature Tensor: Covariant Curvature tensor, Ricci tensor, Einstein Tensor, The Bianchi identity.

Unit 2
The principle of covariance, The principle of equivalence, Geodesic principle, Newton’s equations of motion as an approximation of geodesic equations, Poisson’s equations as an approximation to Einstein field equations.

Unit 3
Gravitational field equations in free space, Exterior Schwarzchild’s solution and its isotropic form, Birkhoff’s theorem, Schwarzchild singularity, planetary orbit, Advance of Perihelion of a planet, Bending of light rays in the gravitational filed, Gravitational Red shift in the spectral lines.

Unit 4
Gravitational field equations for non empty space, Linearization of the field equations, The Weyl’s solution of linearized Field equations, Interior Schwarzchild’s solution.

Text Book
Introduction to General Relativity: Ronald Adler, Maurice Bezin and Manamen Schiffer, McGraw-Hill Kogakusha Ltd.

References Books
1. Introduction to theory of relativity, Rosser W.G.V., ELBS(1972).
2. Lecture on General Relativity, Sonu Nilu Publication (2004) by T M Karade, G S Khadekar and Maya S Bendre
3. Relativity Special, General and Cosmology, Rindler W., Pub. Oxford University Press (2003).
4. The Classical Theory of Fields By Landau I.D. and Lifshitz E.M., Pub. Pergamon Press (1978).

________________________________________________________________________

### Algebraic Topology- I

Unit 1
The Elements of Homotopy theory: Introduction. Homotopic mappings. Essential and inessential mappings. Homotopically equivalent spaces. Fundamental group. Knots and related embedding problems. Higher homotopy groups. Covering spaces.

Unit 2
Polytopes and triangulated spaces: En as a vector space over E1 .Barycentric coordinates. Geometrical complexes and polytopes. Barycentric subdivision. Simplicial mappings and simplicial approximation theorem.

Unit 3
Abstract simplicial complexes. Embedding theorem for polytopes. Simplicial homology theory: Introduction. Oriented complexes. Incidence numbers. Chains, cycles and groups.

Unit 4
Decomposition theorem for abelian groups. Betti numbers and torsion coefficients. Zero dimensional homology groups. Universal coefficients. Euler Poincare formula. Universal coefficients.

Text Book
Topology : J.G. Hocking and G.S. Young : Addison Wesley, 1961

Reference Books
1. Topology : J.R.Munkres, Prentice Hall, Second Edition, 2000
2. Basic Concepts of Algebraic Topology : Fred H.Croom , Springer Verlag 1978.

_________________________________________________________________________

### Non-linear Programming-I

Unit 1
The non-linear programming problem and its fundamental ingredients. Linear inequalities and the theorem of the alternative. The optimality criteria of linear programming. Tucker’s lemma and existence theorems.

Unit 2
Theorems of the alternative Convex sets – Separation theorems. Convex and concave functions – basic properties and some fundamental theorems for convex functions. Generalised Gordan theorem. Bohnenblust – Karlin – Shapley theorem. Saddle point optimality criteria without differentiability – The minimization and the local minimization problems and some basic results.

Unit 3
Sufficient optimality theorem. Fritz John Saddle point necessary optimality theorem. Slater’s and Karlin’s constraint qualifications and their equivalence. The strict constraint qualification. Kuhn – Tucker saddle point optimality theorems. Differentiable concave and convex functions – Some basic properties. Twice differentiable convex and concave functions. Theorems in cases of strict convexity and concavity of functions.

Unit 4
Optimality criteria with differentiability- Optimality theorems, Fritz John stationary point necessary optimality theorem. The Arrow – Hurwicz – Uzawa constraint qualification. Kuhn – Tucker stationary – point necessary optimality theorem.

Text Book
O.L. Mangasarian, Non- linear programming. Mc Graw Hill, New York.

Reference Book
Mokhtar S. Bazaraa and C.M.Shetty, Non- linear programming, Theory and
Algorithms, Wiley, New York.

__________________________________________________________________________

### Operator Theory

Unit 1
Basic concepts about spectrum. Spactral properties of bounded linear operators. Further properties of resolvent and spectrum. Use of complex analysis in spectral theory.

Unit 2
Banach Algebras. Further properties of Banach Algebras. Compact linear operators on normed spaces. Further properties of Compact linear operators. Spectral properties of compact linear operators.

Unit 3
Further spectral properties of Compact linear operators. Operator equations involving compact linear operators. Further theorems of Fredholm type. Fredholm alternative.

Unit 4
Spectral properties of bounded self adjoint linear operators. Further Spectral properties of bounded self adjoint linear operators. Positive operators. Square roots of a positive operator. Projection operator. Further properties of projections. Spectral family. Statement of spectral representation theorem.

Text Book
Introductory Functional Analysis with Applications by E. Kreyszig, John Wiley and Sons

Reference Book
Introduction to Functional Analysis by A.E.Taylor and D.C.Lay, John Wiley and Sons