Swami Ramanand Teerth Marathwada University, Nanded.

M.A. /M. Sc Mathematics I and II Year Syllabus (Campus School) CGPA

Semester-I

Paper Periods Per week Max. Marks

Theory Paper-I- Algebra-I(Groups & Rings) 04 50

Theory Paper-II-Real Analysis-I 04 50

Theory Paper-III -Complex Analysis 04 50

Theory Paper-IV- Numerical Analysis (With practical

on Computer

04 50

Any one of the following of V (A) to V (D). 04 50

Theory-Paper- V (A) - Multivariate Calculus

Theory-Paper- V (B) - Advanced Discrete Mathematics-I

Theory-Paper- V (C) - Differential Geometry of Manifolds- I

Theory-Paper- V (D) - Dynamics and Continuum Mechanics-I

Theory-Paper- VI- -Tutorial-I (Compulsory for all) 125 Marks

Semester-II

Paper Periods Per week Max. Marks

Theory Paper-VII- Linear Algebra 04 50

Theory Paper-VIII- Real Analysis-II 04 50

Theory Paper-IX- Topology 04 50

Theory Paper-X- Elementary Number Theory 04 50

Any one of the following of XI (A) to XI (D). 04 50

Theory-Paper-XI (A) - Differential Equations

Theory-Paper-XI (B) -Advanced Discrete Mathematics-II

Theory-Paper-XI (C) -Differential Geometry of Manifolds- II

Theory-Paper-XI (D) - Dynamics and Continuum Mechanics-II

Theory-Paper-XII - Group Project of 125 marks (Compulsory for all)

Semester-III

Paper Periods Per week Max. Marks

Theory Paper-XIII - Algebra II

(Field Theory and Galois Theory )

04 50

Theory Paper-XIV--Functional Analysis 04 50

Three papers to be chosen from XV to XXII which will

be taught in the department.

04 50

Theory-Paper- XV- Graph Theory

Theory-Paper- XVI- Operations Research

Theory-Paper- XVII- Advanced Number Theory

Theory-Paper- XVIII- Integral Equations

Theory-Paper- XIX -Programming in C (with ANSI features)

(Theory and Practical)

Theory-Paper- XX - Wave Propagation

Theory-Paper- XXI-Theory of Linear operators

Theory-Paper- XXII-Commutative Algebra

Theory-Paper- XXIII-Tutorial-II (Compulsory for all) 125 Marks

Semester-IV

Paper Periods Per week Max. Marks

Theory Paper-XXIV- Boundary Value Problems 04 50

Theory Paper-XXV- Mechanics 04 50

Any three Papers to be Chosen from the following three

groups which will be taught in the department.

Theory Paper-XXVI To XXXVIII

04 50

A) Computational Mathematics

B) Applied Mathematics

C) Pure Mathematics

Theory-Paper-XXXIX-Project Work (Compulsory for all) 125 Marks.

Course Structure

1) Each semester will have five Theory papers and assessment for each theory paper will be

of 125 Marks (50 External Exam+ 50 Internal Exam(02 tests each of 15 Marks+10

Marks for assignment+ 10 Marks for Class Performance)and 25 Marks for seminar Or

Project work OR practical).= 625

2) Total marks for I sem+ II sem+III sem + IV sem = 2500.

3) Total degree is of 2500 Marks, converted in the form of 100 credits CGPA system.

One credit is of 25 marks.

NOTE: (1) Minimum 40% Marks are required for passing in each of the above head i.e. separate passing for

External Exam, Internal Exam. (2) Project or Practical will be evaluated by one external examiner (out of

University examiner) and one internal examiner.

M.A. /M. Sc. Syllabus (Mathematics)

SEMESTER-I

Distribution of marks for Semester I:

Name of Paper External

Exam(Theory

conducted

by University)

Internal Exam

(Tests+Assignments

+ Class Performance)

(By Dept.)

Project or Practical

(External Exam )

Algebra –I

(Groups & Rings)

50 Marks 50 Marks 25 Marks

Real Analysis- I 50 Marks 50 Marks 25 Marks

Complex Analysis 50 Marks 50 Marks 25 Marks

Numerical Analysis 50 Marks 50 Marks 25 Marks

Optional Paper V (A To

D Any One )

50 Marks 50 Marks 25 Marks

Total Marks for sem-I 625

PAPER- I: ALGEBRA-I (Groups & Rings) MTU 101

Max. Periods: 60

UNIT 1: (Prerequisites: Introduction to Groups, Definition and Examples, Elementary properties

of Groups, Finite Groups and Subgroups, Subgroup Tests, Examples of Subgroups).

Cyclic Groups, Properties of Cyclic Groups, Classification of Subgroups of Cyclic

Groups, Permutation Groups, Definition and Notation, Cycle Notation, Properties of

Permutations, A Check-Digit Scheme Based on D4.

UNIT 2: Isomorphisms, Definition and Examples, Cayley's Theorem, Properties of

Isomorphisms, Automorphisms, Cosets and Lagrange's Theorem, An Application of

Cosets to Permutaion Groups, Normal Subgroups, Factor Groups, Application of

Factor Groups, Internal Direct Product, Group Homomorphisms and their properties,

The First Isomorphism Theorem, The Fundamental Theorem, Isomorphism Classes

of Abelian Groups, Proof of Fundamental Theorem, The Class Equation, The Sylow

Theorem.

UNIT 3: Introduction to Rings, Examples and Properties, Integral Domains, Fields,

Characteristic of a Ring, Ideals and Factor Rings, Ring Homomorphisms, Polynomial

Rings, Factorization of Polynomials, Divisibility in Integral Domains, Unique

Factorization Domains, Euclidean Domains.

Text Book:

J. A. Gallian, Contemporary Abstract Algebra, Fourth edition, Narosa Publishing House.

Scope:

Reference Books:

1. D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd Ed., John Wiley, 2002.

2. M. Artin, Algebra, Prentice-Hall of India Pvt. Ltd.

3. I. N. Herstein, Topics in Algebra, Macmillan, Indian Edition.

4. J. B. Fraleigh, Abstract Algebra, 5th Edition.

5. I. S. Luthar, I. B. S. Passi, Algebra, Vol. 1, Groups, Narosa Publishing House.

6. P. B. Bhattacharyya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (2e),

Cambridge Univ. Press, Indian Edition, 1997.

PAPER- II Real Analysis-I. (MTU102)

(Maximum Number of Periods: 60)

Unit 1 (Pre-requisites)

The real numbers, Limits and continuity, Equivalence and cardinality, The Cantor set,

Monotone functions, Metric spaces, Normed vector spaces, More Inequalities, Limit in

metric spaces, Open sets, Closed sets, The relative metric.

Unit 2:

Continuous functions, Homeomorphism, The space of continuous functions, connected

sets, Completeness, Totally Bounded sets, Complete Metric spaces, Fixed points,

completions.

Unit 3:

Compactness, Compact metric space, Uniform continuity, Equivalent Metrics,

Discontinuous functions, The Baire category theorem.

Unit 4:

Sequences of functions, Historical Background, Point wise and uniform convergence,

Interchanging limits, the space of bounded functions, The space of continuous functions, the

Weierstrass theorem, Trigonometric polynomials, Infinitely differentiable functions.

Unit 5 :

Equicontinuity, Continuity and category, The Stone Weierstrass theorem.

Text Book: - N.L. Carothers, “Real Analysis”, Cambridge university press.

Scope : - Chapters 1 to 12.

Reference Books:

1. Sudhir R. Ghorpade and Balmohan V. Limaye, “A Course in Calculus and Real

Analysis”, Springer Publications.

2. T. M. Apostol, “Mathematical Analysis”, Narosa Publishing House.

3. G. F. Simmons, “Introduction to Topology and Modern Analysis”, Mc Graw Hill.

4. S. Kumaresan, “Topology of Metric Spaces”, Narosa Publishing House.

5. W. Rudin, “Principles of Mathematical Analysis”, Mc Graw Hill.

PAPER- III Complex Analysis. (MTU103)

(Maximum Number of Periods: 60)

Unit 1: Functions, Limit and Continuity, one-to-one and onto functions, Concepts of limit and

continuity, Sequences and series of functions, Analytic functions and power series,

Differentiability and Cauchy-Riemann Equations, Harmonic functions, Power series as an

analytic Function, Exponential and Trigonometric functions, Logarithmic functions, Inverse

functions.

Unit 2: Complex Integration, Curves in the complex plane, Properties of complex line integrals,

Cauchy-Goursat theorem, Simple connectivity, Cauchy Integral Formula, Morera’s Theorem,

Existence of Harmonic conjugate, Zeros of an analytic function, Laurent series.

Unit 3: Conformal Mappings, Principle of conformal mapping, Basic properties of Möbius map,

Fixed points and Möbius maps, Triples to Triples under Möbius map (cross ratio and its

invariance properties).

Unit 4: Maximum principle, Schwarz’s Lemma and Liouville’s Theorem, Maximum modulus

principle, Hadamard’s three circles/ Lines theorem, Schwarz Lemma and its consequences, ,

Doubly periodic entire functions, Fundamental theorem of Algebra, Zero’s of certain

polynomials.

Unit 5: Classifications of singularities, Isolated and non-isolated singularities, Removable

singularities, Poles, isolated singularities at infinity, Meromorphic functions, Essential

singularities and Picard’s theorem, Residue at a finite point, Residue at infinity, Residue

theorem, Number of zeros and poles, Rouche’s theorem.

Text Book: - S.Ponnusamy, “Foundation of Complex Analysis”, Narosa Publication,

Second Edition.

Scope: Chapters 1 to 8.

Reference Books:

1. John B. Convey, “Functions of one Complex Variable”, Narosa Publishing House.

2. L. V. Ahlfors, “Complex Analysis”, Mc Graw Hill.

3. Ruel V. Churchil, J.W. Brown, “Complex Variables and Applications”, Mc Graw Hill.

4. H.Silverman, “Functions of Complex Variables”.

5. T.W.Gamelin, Complex Analysis, Springer Publications.

PAPER- IV - Numerical Analysis (MTU104) Max.Periods: 60

Unit 1:

Transcendental and Polynomial equations: Introduction, Bisection method,

Iteration methods based on first degree equations and second degree equations, Rate of

convergence, Polynomial Equations, Model problems.

Unit 2:

System of Linear algebraic equations and Eigen value problems: Introduction,

direct methods, Iteration methods, Eigen value and Eigen vectors, Model problems.

Unit 3:

Interpolations and approximations: Introduction, Lagrange’s, Newtonian

Interpolation, finite difference operators, Interpolating polynomials using finite

differences, Approximations, Least Square approximations.

Unit 4:

Ordinary Differential Equation, Initial Value Problems: Difference Equations,

Numerical methods, Stability Analysis of single step Method, Multistep Method,

Stability Analysis of multistep Method, Initial Value Problem Method , Finite difference

Method , Finite Element Methods .

Note: - One credit of 25 marks reserved for practicals on Computer software.

Text Book : 1. M.K. Jain, SRK Iyengar, R.K. Jain, “Numerical methods for Scientific

and Engineering computations.” New Age International Limited Pub.

: Chap2: Art. 2.1 to 2.5, 2.8, 2.9 Chap3: Art 3.1, 3.2 3.4 3.5 3.6

Chap4 Art 4.1 to 4.4, 4.8, 4.9 Chap 6: Art 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.8.

Chap 7: Art 7.1, 7.2, 7.3, 7.4.

Reference

Books

: 1. S.S. Sastry, “Introductory methods of Numerical Analysis” Prentice-

Hall of India Private Ltd. (Second Edition) 1997.

2. E.V. Krishnamurthi & Sen. “Numerical Algorithm,”

Affiliate East. West press. Private Limited 1986.

PAPER- V(A) Multivariate Calculus. (MTU105)

(Maximum Number of Periods: 60)

Unit 1 :

Introduction: - Level sets and Tangent spaces, Lagrange’s multipliers, Maxima &

Minima on open sets.

Unit 2 :

Line integral, Frenet-Serret equations, Double Integration parameterized surfaces in R3. ,

Surface Area, Surface Integral.

Unit 3 :

Stoke’s Theorem, Triple integral, The Divergence theorem.

Unit 4 :

Geometry of surfaces in R3, Gaussian curvatures, Geodesic curvature.

Tex Book: Sean Dineen, “Multivariate Calculus and Geometry”, Springer Verlag.

Scope: Chapters 11 to 18.

Reference Books:

1. Sudhir R. Ghorpade and Balmohan V. Limaye, “A course in Multivariate Calculus and

Analysis”, Springer Verlag.

2. T. M. Apostol, “Calculus”, Vol. 2, Second Edition, John Wiley and Sons, Inc.

3. J. A. Thorpe, “Elementary Topics in Differential Geometry”, Springer Verlag.

4. Devinatz, “Advanced Calculus”.

5. B. Oneill, Elementary Differential Geometry.

6. J. E. Marsden, A. J. Tromba, A. Weinstein, Basic Multivariable Calculus,

Springer Internationsl Edition, Springer Verlag.

Paper V (B)-Advanced Discrete Mathematics –I (MTU105)

(Maximum Number of Periods: 60)

Unit I: Formal Logic:

Statements Symbolic Representation and Tautologies. Quantifiers, Predicates and

validity, Prepositional Logic.

Unit II: Semi groups and Monoids:

Definitions and example of Semigroups and Monoids (including those pertaining to

concatenation operations) Homomorphism of semi groups and monoids. Congruence relation

and quotient semigroups, Subsemigroup and submonoids, Direct products ,Basic

Homomorphism Theorem.

Unit III: Lattices:

Lattices as partially ordered sets. their properties. Lattices as Algebraic systems.

Sublattices. Direct products and Homomorphisms Some special lattices e.g. complete.

Complemented and Distributive Lattices.

Unit IV: Boolean Algebras:

Boolean Algebras as Lattices, Various Boolean Identities, The switching Algebra.

Example, subalgebras,Direct Products and Homomorphisms, Joint-irreducible elements.

Atoms and Minterms, Boolean forms and their equivalence, Minterm Boolean forms. Sum

of Products, Canonical forms,Minimization of Boolean functions,Applications of Boolean

Algebra to Switching Theory (using AND, OR and NOT gates.) The Karnaugh Map

method.

Reference Books:

1. Discrete Mathematical Structures with applications to Computer Science, By J. P. Trembley

and Manohar, McGraw-Hill Book Co. 1997.

2. Finite Mathematics (International edition 1983), By Seymour Lipschutz, McGraw-Hill Book

Co, New York.

3. Discrete Mathematics - A Unified Approach, By S. Wiitala, McGraw-Hill Book Co. New

York.

4. Mathematical Structures for Computer Science, (3rd

edition) By J. L. Gersting.

Paper V(C) -Differential Geometry of Manifolds -1 (MTU105)

(Maximum Number of Periods: 60)

Unit I: Differentiable Manifolds:

Definition and examples of differentiable manifolds. Tangent spaces, Jocobian map. One

parameter group of transformations Lie -derivatives. Immersions and imbedding. Distributions.

Exterior algebra. Exterior Derivative.

Unit II: Lie Groups and Lie Algebras:

Topological groups. Lie groups and Lie algebras. Product of two Liegroups. One

parameter subgroups and exponential maps.

Examples of Liegroups. Homomorphism and Isomorphism. Lie transformation groups. General

linear groups.

Reference Books:

1. A course in tensors with applications to Riemannian geometry, By R. S. Mishra,

Potishala (Pvt) Ltd. 1965.

2. Structures on a differentiable manifold and their applications, By R. S. Mishra,

Chandrama Prakashan, Allahabad, 1984.

3. An Introduction to Modern Differential Geometry, By B. B. Sinha, Kalyani Publishers,

NewDelhi, 1982.

4.. Structure of Manifolds, By K. Yono and M. Kon, World Scientific Publishing Co. Pvt.

Ltd. 1984.

Paper V (D): Dynamics and continuum Mechanics-I (MTU105)

(Max No. of Periods - 60)

Unit I: Vector Methods:

Vector moment about a point and scalar moment about an axis, Vector and scalar

couples, Centroids, Vector calculus.

Unit II: Kinematics of Particles and Rigid Bodies:

Velocity and acceleration of a Particle along a curve, Motion in plane – radial and

transverse components, Relative velocity and acceleration, Vector angular velocity, General

motion of rigid body, Moving axes.

Unit III: Newtons Laws of Motion:

Mass, Momentum, Force, Newton’s laws of motion, Work, Energy and Power,

Conservative forces- potential energy, Impulsive forces.

Unit IV: Motion of a system of particles:

Linear momentum of system of particles, Angular momentum and rate of change of

angular momentum. Use of centroids, Moving origins, Impulsive force.

Unit V: Introduction to Rigid Body Dynamics:

Moments and products of Inertia, The theorem of parallel and perpendicular axes,

Angular Momentum, Principal axes, Kinetic Energy of a rigid body, Momental Ellipsoid,

Coplanar distribution, General motion of a rigid body,

Unit VI: Two Dimensional Rigid Body Dynamics:

Problems illustrating the laws of motion, Problems illustrating the law of conservation of

energy, Problems illustrating the impulsive motion.

Text Book:

F.Chorlton: A text book of Dynamics (E.L.B.S.) (2nd

Edition)

Reference Books:

1. J.L. Synge and Griffith: Classical Mechanics.

2. Atkin R.H. : Classical Dynamics

SYLLABUS FOR CAMPUS SCHOOL

SEMESTER ––II

Distribution of marks Semester II :

Name of Paper External

Exam(Theory

conducted

by University)

Internal Exam

(Tests+Assignments

+ Class Performance)

(By Dept.)

Project or Practical

(External Exam )

Linear Algebra 50 Marks 50 Marks 25 Marks

Real Analysis-II 50 Marks 50 Marks 25 Marks

Topology 50 Marks 50 Marks 25 Marks

Elementary Number

Theory 50 Marks 50 Marks 25 Marks

Optional Paper XI A To D 50 Marks 50 Marks 25 Marks

Total Marks for one semester 625

Paper-VII : Linear Algebra (MTU 201)

(Maximum Number of Periods: 60)

Unit 1: (Pre-requisites):

Basic theory of fields, Field extension, Examples, matrices, determinants, polynomials.

Unit 2: 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 3: 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 4: Elementary Matrix Operations and system of linear equations (Revision):

Elementary Matrix Operations and elementary matrices, the rank of a matrix, System of linear

equations-Theoretical Aspects, System of linear equations-Computational Aspects.

Unit 5: Diagonalization:

Eigen values and eigen vectors, Diagonalizabity, Invariant Subspaces and the

Cayley-Hamilton Theorem.

Unit 6: 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.

Unit 7: Canonical Forms:

Jordan Canonical form I, Jordan Canonical form II, The minimal polynomial, Rational

Canonical form.

Text Book: S.H.Friedberg, A.J.Insel, L.E.Spence: Linear Algebra, Prentice-Hall

International, Inc., 3rd

Edition.

Scope: Ch 1: Art.1.1 to 1.7, Ch 2:Art. 2.1 to 2.7, Ch 3:Art 3.1 to 3.4, 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. Vivek Sahai, Vikas Bist: Linear Algebra, Narosa Publishing House, 2nd

Edition.

2. I. N. Herstein, “Topics in Algebra”, Macmillan, Indian Edition.

3. S.Lang:Introduction to Linear algebra,Springer International Edition,2nd Edition.

4. K.Hoffman,R.Kunze: Linear Algebra. Prentice Hall of India.

5. J.H.Kwak, S.Hong: Linear Algebra, Birkhäuser Verlag, 2nd Edition.

6. Harvey E.Rose:Linear Algebra.A pure Mathematical Approach, Birkhäuser Verlag.

Paper-VIII: REAL ANALYSIS –II (MTU 202)

(Maximum Number of Periods: 60)

Unit 1: Functions of Bounded Variations & the Riemann- Stieltjes Integrals

Functions of Bounded variations, Helly’s first Theorem, Weights and measures,

The Riemann- Stieltjes Intergrals, The Space of Integrable Functions, Integration of

Bounded Variations , The Riemann Integral , The Riesz Representation Theorem,

Other properties .

Unit 2: Fourier series

Preliminaries, Dirichlet’s Formula, Fejer’s Theorem, complex Fourier series.

Unit 3: Lebesgue Measure and Measurable functions

The problem of measure, Lebesgue outer measure, Riemann Integrability

Measurable sets, a non measurable set. Measurable functions, Extended real-valued

functions, sequence of Measurable functions, Approximation of Measurable functions

Unit 4 : The Lebesgue Integral:

Simple functions, Non-negative functions, Lebesgue’s Dominated Convergence

Theorem, Approximation of integrable function

Convergence in measure, The Lp spaces, Approximation of Lp functions.

Unit 5 : Differentiation:

Lebesgue’s Differentiation Theorem, Absolute continuity.

Text Book: - N.L. Carothers, “Real Analysis”, Cambridge university press.

Scope: - Chapters 13 to 20.

Reference Books:

i) P.K. Jain and V.P Gupta : Lebesgue measure and Integrtion

New age international (P) ltd publishing, New Delhi (Reprint 2000.)

ii) P.R. Halmos: Measure theory, Van Nostranel Princeton 1950

iii) Inder K.Rana : An Introduction to measure and Integration.

Norosa publishing House ,Delhi : 1997.

iv) G. de. Barra; Measure theory and Integration,

Willey Eastern Limited (First Willey Eastern Reprint 1987)

Paper-IX: TOPOLOGY (MTU 203)

(Maximum Number of Periods: 60)

Unit 1 :( Prerequisites):

Topological Spaces, Basis for Topology,

Unit 2: The Order Topology, The product Topology, The Subspace Topology , Closed Sets

and Limit Points ,Continuous functions , The Metric Topology.

Unit 3: Connectedness

Connected Spaces, Connected Subspace on Real Line,

Unit 4: Compactness

Compact Spaces, Compact Subspace on the Real Line, Limit Point Compactness, Local

Compactness.

Unit 5: Countability and Separation Axioms

The Countailibity Axioms, the Separation Axioms, Normal Spaces, the Urysohn’s Lemma,

The Urysohan Metrization Theorem, the Tietze Extension Theorem, the Tychonoff Theorem.

. Text Book:

James R. Munkres: Topology, A first course, Prentice Hall of India. Pvt. Ltd. New Delhi-2000.

Scope:-

Chapter 2: Articles 12 to 20

Chapter 3 : Articles 23, 24, 26, 27,28, 29

Chapter 4: Articles 30 to 35.

Chapter 5: Articles 37

Reference Books:

i] J. Dugundji Allya and Bacon. Topology, (1966) reprinted: Prentice Hall of India.

ii] W. J. Pervin: Foundations of general topology, academic press Inc. N.Y. Hi] S. T.Hu:

Elements of general topology. Holden day Inc. 1965.

iii] Stephen Willard, "General Topology", Addison-Wesley Publishing Company, 1970

iv] Sheldon W. Davis, Topology (The Walter Rudin Student Series in Advanced

Mathematics), TATA McGraw-Hill.2006.

Paper-X: Elementary Number Theory (MTU 204)

Maximum Number of Periods: 60

Unit 1 : Divisibility Theory in the Integers

Division Algorithm, the Greatest common Divisor, The Euclidean Algorithm, The

Diophantine Equations ax+by = c, Fundamental Theorem of Arithmetic,

Unit 2: Theory of Congruences

Basic Properties of Congruences, Binary and Decimal Representations of Integers, Linear

congruence and the Chinese Remainder Theorem.

Unit 3: Fermat Theorem

Fermat Little theorem and Pseudo primes, Wilson’s Theorem, The Farmat –Kraitchik

Factorization Method , The Equation x2+y

2= z

2 , Fermat’s last Theorem.

Unit 4 :Euler’s Generalization of Fermat’s Theorem

Sum and Number of divisors, The Mobius Inversion Formula, The greatest Integer

function, Euler’s Phi- Function, Euler’s theorem, Properties of Phi function.

Unit 5: Primitive Roots, Indices and the Quadratic Reciprocity Law

The Order of an Integer Modulo n , Primitive Roots for Primes , Composite Numbers

having primitive Roots, Theory of Indices, Euler’s Criterion , The Legendre Symbol and its

Properties , Quadratic Congruences with Composite Moduli .

Text Book : Elementary Number Theory, By David M. Burton .Tata McGRAW-

HILL,2006,

Scope : Chapter 2 to Chapter 9,

Reference

Books

: 1.A Baker, A concise Introduction to the Theory of Numbers, Cambridge

University Press 1984

2. J.P. Serre, A course in arithmetic-. GTM Vol.7, Springer Verlag 1973.

3. Tom M. Apostol. ,Introduction to Analytic number theory

Narosa Publishing house 1980.

4. I. Niven and Zuckerman, An Introduction to the Theory of Numbers,

4th

Ed Wiley, New York,1980,

5. Rosen K.H., Elementary Number Theory and its Applications Pearson

Addision Wesely, 5th

Edition.

Paper-XI A: Differential Equations. (MTU205)

Max. Periods: 60

Ordinary differential Equations:

Unit 1: Existence and uniqueness of solution to first order equations, Equation with variable

separated, Exact Equations, The Method of successive approximation, The Lipschitz

condition, convergence of successive approximations, Non local existence of solution,

Approximation to and uniqueness of solutions, Equations with complex valued functions.

Unit 2: Existence and Uniqueness of solutions to systems: Introduction, Some special equations,

Complex n-dimensional space, Systems as vector equations, Existence and Uniqueness of

solutions to systems, Existence and Uniqueness of solutions for linear systems,

Partial Differential Equations:

Unit 3: Partial differential equations of second order, Linear partial differential equations with

constant coefficients, Equation with variable coefficients, Separation of variables, The

method of integral Transform, Non-linear equation of second order.

Unit 4: Laplace Equation: Occurrence of Laplace Equation, Elementary solutions of Laplace

Equations, Separation of variables. Theory of Green’s function for Laplace equation

Unit 5: Wave Equation: Occurrence of wave equation, Elementary solution of wave equation,

General solution of wave equation, Greens function for wave equation.

Text Books:

1. E. A. Coddington, “An introduction to Ordinary differential Equation”,

Prentice-Hall of India Pvt. Ltd., New Delhi.

Scope: Chapter 5: Articles 1 to 8; Chapter 6: Articles 6, 7, 8.

2. I. N. Sneddon, “Elements of Partial Differential Equation”, Mc. Hill Book Co.

Scope: Chapter 2: Art 1, 2, 4 (Prerequisites); Chapter.3: Art 1, 2, 3, 4, 5, 9.

Chapter 4: Art. 1, 2, 3, 5, 11; Chapter 5: Art 1, 2, 6.

Reference Books:

1. G. F. Simmons, “Differential Equations with Applications and Historical Notes”,

Second Edition, Mc Graw Hill.

2. W. E. Williams, “ Partial Differential Equations”, Claredon Press Oxford.

3. G. Birkhoff and G. C. Rota, “Ordinary Differential Equations”, John Wiley and Sons.

4. E. T. Copson, “ Partial Differential Equations ”, Cambridge University Press.

Paper XI (B) - Advanced Discrete Mathematics – II (MTU205)

( Maximum Number of Periods : 60)

Graph Theory: Definition of (undirected) Graph, Paths, Circuits, Cycles and subgraphs. Induced

subgraphs. Degree of a Vertex, Connectivity. Planar Graphs and theire properties. Trees Euler's

Formula for connected planar Graphs. Complete and complete Bipartite Graphs. Kuratowski's

Theorem (statement only) and its use. Spanning Trees, Cut-sets, Fundamental cut-sets, and

Cycles, Minimal spanning Trees and Kruskal's Algorithm. Matrix Representations of graphs.

Euler's theorem on the existence of Eulerian paths and Circuits. Directed Graphs. Indegree and

outdegree of a Vertex. Weighted underected Graphs. Dijikstra's Algrotithm. Strong connvectivity

and Warshall's Algorithm. Directed Trees. Search Trees. Tree Traversals.

Introductory Computability Theory: Finite state Machines and their Transition table diagrams

Equivalence of Finite State, Machines. Reduced Machines. Homomorphism. Finite automata.

Acceptors. Non-deterministic. Finite Automata and equivalence of its power to that of

deterministic Finite automata. Moore and Mealy Machines. Turing Machine and Partial

Recursive functions.

Grammars and Language: Phrase-Strucutre Grammars. Reqiriting rules. Dervation. Sentential

forms. Language generated by a Grammar. Regular, Context -Free and context sensitive

grammars and Languages. Regular sets. Regular Expressions and the pumping Lemma. Kleene

Theorem. Notions of syntax Analysis. Polish Notations. Conversion of Infix. Expressions to

polish Notations. The Reverse Polish Notation.

Reference Books:

i] Mathematical structures for computer science, (3rd

edition) By J.L. Gersting, Computer

Science Press, New York.

ii] Introduction to Automata Theory, Langauages and Computation, By J.E. Hopcraft and J.D.

Ullman, Narosa Publishing House.

iii] Elements of Discrete Mathematics, By C. L. Liu, McGraw Hill Book Co.

iv] Graph Theory with Application to Engineering and computer science, by Narsing deo

Prentice Hall of India.

Paper XI(C) -Differential Geometry of Manifolds – II (MTU205)

(Maximum Number of Periods : 60)

Bundles: Principal fibre bundle. Linear frame bundle. Associated fibre bundle. Vector

bundle. Tangent bundle. Induced bundle. Bundle Homomorphisms.

Riemannian Manifolds: Riemannian manifolds. Riemannian connection. Curvature tensors.

Sectional Curvature. Schur's theorem. Geodesics in a Riemannian manifold. Projective curvature

tensor. Con-formal curvature tensor.

Submanifolds and complex Manifolds :Submanifolds and Hypersurfaces. Normal. Gauss formulae.

Weingarten equations. Lines of Curvature. Generalized Gauss and Mainardi Codazzi equations.

Almost complex manifolds. Nijenhuis tensor. Contravariant and covariant almost analytic victor

field. F-connection.

Reference Books:

i) A course in tensors with applications to Riemannian Geometry, By R. S. Mishra,

Potishala (Pvt) Ltd. 1965.

ii) Structures on a differentiate manifold and their applications, By R. S. Misiira, Chandrama

Prakashan, Allahabad, 1984.

iii) An Introduction to Modern Differential Geometry, By B. B. Sinha, Kalyani Publishers,

NewDelhi, 1982.

iv) Structure of Manifolds, By K. Yono and M. Kon, World Scientific Publishing Co. Pvt.

Ltd. 1984.

Paper XI (D) -Dynamics and Continuum Mechanics - II(MTU205)

( Maximum Number of Periods : 60)

Tensors: Indicial Notation, Summation convention, Dumy induces, Free indices, Kroneeker

delta, Permutation symbol. Tensor-as a linear transformation, Components, Sum, Dyadic

product. Product of tensors, Identity, Transpose, Orthogonal tensors, symmetric and

antisymmetric tensors. Eigen values and eigenvectors of a tensor. The dual vector of an

antisymmetric tensor. Principal values and principal directions of real symmetric tensors. Scalar

invariants of tensor. Scalar and vector fields. Gradient, divergence and curl of these fields.

Kinematics of continuum: Description of motion of a continuum, Material and spatial

description, Material derivatives, Deformation. Principle strain, Dilatation, Rate of deformation,

Equation of conservation of mass, Compatibility conditions of infinitesimal strain components.

Stress : Stress vector, Stress tensor, Components of symmetry of stress tensor, Principle of

moment of momentum, Principal stresses, Maximum shearing stress. Equations of motion,

Principle of linear momentum.

Newtonian viscous fluids:

Fluids, Compressible and incompressible fluid, equations of hydrostatics, Newtonian Fluid

Interpretation of Incompressible Newtonian fluid.

Text Books:

1] Lai W. M. Rubin D and Kremple E, Introduction to continuum Mechanics.

Reference Book :

1] Lang R.R. Mechanics of Solids and fluids,. Prentice hall