# M.Sc in Mathematics

Bharathiar University
In Coimbatore

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## Important information

- Master
- Coimbatore

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CoimbatoreBharathiar University, Coimbatore, 641046., Tamil Nadu, India See map |

## Frequent Asked Questions

**· Requirements**

A candidate who has passed the Degree Examination in B.Sc. (Mathematics) or B. Sc. (Mathematics with Computer Applications) of this University or an examination of some other University accepted by the syndicate as equivalent thereto shall be eligible for admission to the Master Degree of this University.

## Course programme

UNIT-I:

Group Theory: Direct products- Group Action on a Set: Isotropy Subgroups- Orbits-

Counting Theorems- p-Groups- The Sylow Theorems

UNIT-II:

Applications of the Sylow Theory: Applications to p-Groups and the Class Equation-

Further Applications

Ring Theory: Ring of Polynomials: Polynomials in an Inderterminate- The

Equation Homomorphisms- Factorization of Polynomials over a Field

UNIT-III:

Field Theory: Extension Fields-Algebraic and Transcendental Elements-Irreducible

polynomial over F-Simple Extension- Algebraic Extensions: Finite Extensions- Structure

of Finite Fields

UNIT-IV:

Automorphisms of Fields- Conjugation Isomorphisms- Automorphisms and Fixed

Fields- The Frobenius Automorphism- Splitting Fields.

UNIT-V:

Separable Extensions- Galois Theory: Normal Extensions- The Main Theorem-

Illustrations of Galois Theory: Symmetric Functions

Text book: "A First Course in Abstract Algebra" by J.B.Fraleigh, Fifth Edition, Addition

Wesly Longman, Inc, Reading Massachusetts, 1999.

UNIT-I: Chapter 2, Section: 2.4 (Direct Product only)

Chapter 3, Sections: 3.6, 3.7

UNIT-II: Chapter 4, Section: 4.3, Chapter 5, Sections: 5.5, 5.6

UNIT-III: Chapter 8, Sections: 8.1, 8.3 (Finite Extensions Only), 8.5

UNIT-IV: Chapter 9, Sections: 9.1, 9.3

UNIT-V: Chapter 9, Sections: 9.4, 9.6, 9.7 (Symmetric Functions only)

MATAC02 : REAL ANALYSIS

RIEMANN STILTJES INTEGRAL:

Unit-I.

Definition and Existence of the Integral - properties of the integral - Integration and

differentiation - Integration of vector valued function - rectifiable curves

Unit-II.

Uniform convergence and continuity - uniform convergence and integration - uniform

convergence and differentiation - equicontinuous families of functions - The Stone

Weirstrass theorem

FUNCTIONS OF SEVERAL VARIABLES

Unit-III.

Linear transformation - contraction principle - Inverse function theorem - Implicit

function theorem - determinants - derivatives of higher order - differentiation of

integrals

LEBESGUE MEASURE:

Unit-IV.

Outer measure - Measurable sets and Lebesgue measure - Measurable functions -

Littlewood's Theorem

Unit-V.

The Lebesgue integral of bounded functions over a set of finite measure - integral of a

non - negative function - General Lebesgue Integral - convergence in measure

Text Book:

For Unit-I to III relevant chapters from : Principles of Mathematical Analysis by W.

Rudin, McGraw Hill, New York, 1976

For Unit-IV and V relevant chapters from : Real Analysis by H.L. Roydon, Third Edition,

Macmillan, New York, 1988.MATAC03: ORDINARY DIFFERENTIAL EQUATIONS

Unit-I.

Linear Equations with constant coefficients - Second order Homogeneous equations -

Initial value problems - Linear dependence and independence Wronskian and a formula

for Wronskian - Non Homogeneous equation of order two.

Unit-II.

Homogeneous and Non - Homogeneous Equations of order n - Initial value problems -

annihilator Method to solve a non - homogeneous equation - Algebra of constant

coefficients operators.

Unit-III.

Linear Equations with variable coefficients - Initial value problems - Existence and

Uniqueness Theorems - Solutions to a non - homogeneous equation - Wronskian and

Linear dependence - reduction of the order of a homogeneous equation - Homogeneous

equation with analytic coefficients - The Legendre equation.

Unit-IV.

Linear Equation with regular singular points - Euler Equation - Second order equations

with regular singular points - Exceptional cases - Bessel equation.

Unit-V.

Existence and Uniqueness of solutions to first order equations - Equation with variables

separated - Exact Equations - Method of successive approximations - The Lipschitz

condition - convergence of the successive approximations and the existence theorem.

MATAC04: COMPLEX ANALYSIS

Unit-I:

Introduction to the concept of analytic function: Limits and continuity - Analytic

functions - Polynomials - Rational functions - Conformality: Arcs and closed curves -

Analytic functions in regions - Conformal Mapping - Length and Area - Linear

Transformations: The Linear group - The Cross ratio - Elementary Riemann Surfaces.

Unit-II:

Complex Integration: Line Integrals Rectifiable Arcs - Line Integrals as Functions of

Arcs - Cauchy's theorem for a rectangle - Cauchy's theorem in a disk, Cauchy's Integral

formula: The Index of a point with respect to a closed curve - The Integral formula -

Higher derivatives Removable singularities, Taylor's Theorem - Zeros and Poles - The

Local Mapping - The Maximum principle - chains and cycles.

Unit-III:

The Calculus of Residues The Residue theorem - The Argument principle - Evaluation

of definite integrals. Harmonic functions: The Definitions and basic Properties - Mean

value property - Poisson's Formula.

Unit-IV:

Series and Product Developments: Weierstrass Theorem - The Taylor Series - The

Laurent Series - Partial fractions and Factorization: Partial Fractions - Infinite Products -

Canonical Products.

Unit-V:

The Riemann Mapping Theorem - Statement and Proff - Boundary Behaviour - Use of

the reflection principle - Analytic arcs - Conformal mapping of Polygons: The

Behaviour at an angle - The Schwarz - Christoffel Formula - Mapping on a rectangle.

MATAC05: PARTIAL DIFFERENTIAL EQUATIONS

Unit I:

Nonlinear Partial Differential Equations of the first order - Cauchy's method of

characteristics - Compatible systems of first order equations - Charpit's method- Special

types of First order equations - Jacobi's method.

Unit II:

Partial Differential Equations of Second order - The origin of Second-order Equations -

Linear Partial Differential Equations with constant coefficients - Equations with variable

coefficients - Characteristics curves of second -order equations- Characteristics of

equations in three variables.

Unit III:

The Solution of Linear Hyperbolic Equations - Separation of variables - The Method of

Integral Transforms - Nonlinear Equations of the second order.

Unit IV:

Laplace's Equation - The occurrence of Laplace's Equation in Physics- Elementary

solution of Laplace's Equation - Families of Equipotential surfaces Boundary value

problems - Separation of variables- Problems with axial symmetry.

Unit V:

The wave equation - The occurrence of wave equation in Physics - Elementary solutions

of the one-dimensional wave equation - Vibrating Membranes: Applications of the

calculus of variations - Three dimensional problems.

The Diffusion Equations: Elementary solutions of the Diffusion Equation - Separation of

variables- The use of Integral transforms.

MATAC06: MECHANICS

Unit-I:

INDRODUCTORY CONCEPTS: Mechanical system - Generalized Coordinates -

Constraints - Virtual Work - Energy and Momentum.

Unit-II:

LAGRANGE'S EQUATIONS: Derivations of Lagrange's Equations: Derivations of

Lagrange's Equations - Examples - Integrals of Motion.

Unit-III:

HAMITON'S EQUATIONS: Hamilton's Principle - Hamilton's Equations.

Unit-IV:

HAMILTON - JACOBI THEORY: Hamilton's Principle function - Hamilton - Jacobi

Equation - Separability.

Unit-V:

CANONICAL TRANSFORMATIONS: Differential forms and Generating Functions -

Lagrange and Poisson Brackets.

MATAC07- TOPOLOGY AND FUNCTIONAL ANALYSIS

TOPOLOGY:

Unit-I:

Spaces and Maps.

Unit-II:

Separability Axioms and Compactness.

Unit-III:

Connectedness - Pathwise connectedness - Imbedding and Extension theorems.

FUNCTIONAL ANALYSIS

Unit-IV:

Banach spaces - Definition and Examples - continuous linear transformations - Hahn

Banach theorem - Natural Imbedding - open mapping theorem - conjugate of an

operator.

Unit-V:

Hilbert spaces - Definition and simple properties - Orthogonal Complements -

Orthonormal basis - conjugate space.

MATAC08: FLUID DYNAMICS

Unit-I:

Introductory Notions- Velocity- Stream Lines and Path Lines- Stream Tubes and

Filaments- Fluid Body- Density- Pressure. Differentiation following the Fluid- Equation of

continuity- Boundary conditions- Kinematical and physical- Rate of change of linear

momentum- Equation of motion of an inviscid fluid.

Unit-II:

Euler's momentum Theorem- Conservative forces- Bernoulli's theorem in steady motionenergy

equation for inviscid fluid- circulation- Kelvin's theorem- vortex motion-

Helmholtz equation.

Unit-III:

Two Dimensional Motion- Two Dimensional Functions- Complex Potential Basic

singularities- source- sink- Vortex- doublet- Circle theorem. Flow past a Circular cylinder

with circulation- Blasius Theorem- Lift force. (Magnus effect)

Unit-IV:

Viscous flows- Navier - stokes equations- some exact solutions of Navier Stokes

equations- Flow between parallel flat plates- Couette flow- Plane Poiseuille flow- Steady

flow in pipes: Flow through a pipe- The Hagen Poiseuille flow.

Unit-V:

Laminar Boundary Layer in incompressible flow: Boundary Layer concept- Boundary

Layer equations- Boundary Layer along a flat plate- The Blasius solution- Shearing stress

and boundary layer thickness- Displacement thickness, momentum thickness- Momentum

integral theorem for the boundary layer- The Von Karman Integral relation, The Von

Karman Integral relation by momentum law.

MATAC09: MATHEMATICAL METHODS

Unit-I: FOURIER TRANSFORMS: Fourier Transforms - Defn. Inversion theorem -

Fourier cosine transforms - Fourier sine transforms - Fourier transforms of derivatives -

Fourier transforms of some simple functions - Fourier transforms of rational functions -

The convolution integral - convolution theorem - Parseval's relation for Fourier

transforms - solution of PDE by Fourier transform.

Laplace's Equation in Half plane

Laplace's Equation in an infinite strip

The Linear diffusion equation on a semi-infinite line

The two-dimensional diffusion equation.

Unit-II: HANKEL TRANSFORMS: Definition - Elementary properties of Hankel

Transforms - Hankel Transforms of Derivatives of functions - Hankel Transforms of

some elementary functions - The Parseval relation for Hankel transforms - Relation

between Fourier and Hankel transforms - Application to PDE.

Axisymmetric Dirichlet problem for a half - space.

Axisymmetric Dirichlet problem for a thick plate

Unit-III: INTEGRAL EQUATIONS: Types of Integral equations - Equation with

separable kernel - Fredholm Alternative Approximate method - Volterra integral

equations - Classical Fredholm theory - Fredholm's First, Second, Third theorems.

Unit-IV: Application of Integral equation to ordinary differential equation - initial value

problems - Boundary value problems - singular integral equations - Abel Integral

equation

Unit-V: CALCULUS OF VARIATIONS: Variation and its properties - Euler's equation

- Functionals of the integral forms Functional dependent on higher order derivatives -

functionals dependent on the functions of several independent variables - variational

problems in parametric form.

MATAC10 - NON LINEAR DIFFERENTIAL EQUATIONS

Unit-I:

First order systems in two variables and linearization: The general phase plane-some

population models - Linear approximation at equilibrium points - Linear systems in

matrix form.

Unit-II:

Averaging Methods: An energy balance method for limit cycles - Amplitude and

frequency estimates - slowly varying amplitudes - nearly periodic solutions - periodic

solutions: harmony balance - Equivalent linear equation by harmonic balance - Accuracy

of a period estimate.

Unit-III:

Perturbation Methods: Outline of the direct method - Forced Oscillations far from

resonance - Forced Oscillations near resonance with Weak excitation - Amplitude

equation for undamped pendulum - Amplitude Perturbation for the pendulum equation -

Lindstedt's Method - Forced oscillation of a self - excited equation - The Perturbation

Method and Fourier series.

Unit-IV:

Linear Systems: Time Varying Systems - Constant coefficient System - Periodic

Coefficients - Floquet Theory - Wronskian.

Unit-V:

Stability: Poincare stability - solutions, paths and norms - Liapunov stability Stability of

linear systems - Comparison theorem for the zero solutions of nearly - linear systems.MATAC11: CONTROL THEORY

Unit-I:

OBSERVABILITY: Linear Systems - Observability Grammian - Constant coefficient

systems - Reconstruction kernel - Nonlinear Systems

Unit-II:

CONTROLLABILITY: Linear systems - Controllability Grammian - Adjoint systems -

Constant coefficient systems - steering function - Nonlinear systems

Unit-III:

STABILITY: Stability - Uniform Stability - Asymptotic Stability of Linear Systems -

Linear time varying systems - Perturbed linear systems - Nonlinear systems

Unit-IV:

STABILIZABILITY: Stabilization via linear feedback control - Bass method -

Controllable subspace - Stabilization with restricted feedback

Unit-V:

OPTIMAL CONTROL: Linear time varying systems with quadratic performance criteria

- Matrix Riccati equation - Linear time invariant systems - Nonlinear Systems

MATAE01: NUMERICAL METHODS

Unit-I:

SOLUTION OF NONLINEAR EQUATIONS: Newton's method - Convergence of

Newton's method - Bairstow"s Method for quadratic factors NUMERICAL

DIFFERENTIATION AND INTEGRATION: Derivatives from Differences tables -

Higher order derivatives - Divided difference, Central-Difference formulas - Composite

formula of Trapezoidal rule - Romberg integration - Simpson's rules.

Unit-II:

SOLUTION OF SYSTEM OF EQUATIONS: The Elimination method - Gauss and

Gauss Jordan methods - LU Decomposition method - Matrix inversion by Gauss-Jordan

method - Methods of Iteration - Jacobi and Gauss Seidal Iteration - Relaxation method -

Systems of Nonlinear equations.

Unit-III:

SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS: Taylor series method -

Euler and Modified Euler methods - Rungekutta methods - Multistep methods - Milne's

method - Adams Moulton method.

Unit-IV:

BOUNDARY VALUE PROBLEMS AND CHARACTERISTIC VALUE PROBLEMS:

The shooting method - solution through a set of equations - Derivative boundary

conditions - Characteristic value problems - Eigen values of a matrix by Iteration - The

power method.

Unit-V:

NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS: (Solutions

of Elliptic, Parabolic and Hyperbolic partial differential equations) Representation as a

difference equation - Laplace's equation on a rectangular region - Iterative methods for

Laplace equation - The Poisson equation - Derivative boundary conditions - Solving the

equation for time-dependent heat flow (i) The Explicit method (ii) The Crank Nicolson

method - solving the wave equation by Finite Differences.

MATAE02: COMPUTER PROGRAMMING AND LAB I

THEORY(50 Marks)

Unit-I:

Fortran 77 - Representation of Integer and Real constants - Variable names - Arithmetic

operators and modes for Expression - Integer Expressions. Real Expressions - Hierarchy

of Operations in Expressions - Arithmetic statement - Defining Variables - Mixed Mode

Expressions - Special Functions - Input Output statements.

Unit-II:

Format description for Read statement - Format description for print statement - multi

record formats - Hollerith field declaration - specifications in a Format - Generalized

input / output statements - Logical constants, variables and Logic Expressions.

Unit-III:

Control statements - Relational Operator - Logical IF statement - GO TO Statements -

Nested Logical IF statement - Arithmetic IF statement - computed GOTO statement.

Unit-IV:

The DO Statement - Rules to be followed in Utilizing Do Loops - RRPEAT WHILE

structure - Subscripted Variables - Subscript Expressions - Dimension statement - DO

loops with subscripts.

Unit-V:

Functions and subroutines - Function subprograms - Subroutines - Common declaration

- Implicit declaration - Equivalence declaration.

MATAE03 - COMPUTER PROGRAMMING AND LAB II

THEORY(50 Marks)

Unit-I:

Overview of C - Constants. Variables and Data Types - Character set - C tokens -

Keywords & Identifiers - constants - variables - Data types - Declaration of variables -

Assigning values to variables - Defining symbolic constants.

Unit-II:

Arithmetic of operators - Relational operators - Logical operators - Assignment

operator - Increment and decrement operators - conditional operator - Bitwise operators

- special operators - Arithmetic Expressions - Evaluation of Expressions - Precedence

of arithmetic operators - Type conversions in Expressions - Operator Precedence and

Associativity - Mathematical Functions.

Unit-III:

Managing Input and Output operators - Reading a character - Writing a character -

formatted input - formatted output - Decision making IF statement - IF - ELSE -

statement - Nesting of IF ELSE statements - The Switch statement - The GO TO

statement.

Unit-IV:

The WHILE statement DO statement - FOR statement -Jumps in Loops - Onedimensional

Array - Two dimensional Arrays - Initializing two dimensional arrays -

Multidimensional arrays.

Unit-V:

Need for User defined functions - A multi-function program - the form of C Functions -

Return Values and their Types calling a function - Category of functions - Arguments

but no return values - Arguments with return values In file management in C - Defining

and with return values - In file management in C - Defining and opening a file - closing

a file- Input / Output operations on files.

COMPUTER PROGRAMMING AND LAB II

PAPER - I

LATEX

Unit I:

Text formatting, TEX and its offspring, What's different in LATEX 2є, Distinguishing

LATEX 2є , Basics of a LATEX file.

Unit II:

Commands and Environments-Command names and arguments, Environments,

Declarations, Lengths, Special Characters, Fragile Commands, Exercises.

Unit III:

Document Layout and Organization - Document class, Page style, Parts of the document,

Table of contents, Fine - Tuning text, Word division.

Displayed Text - Changing font, Centering and indenting, Lists, Generalized lists,

Theorem-like declarations, Tabulator stops, Boxes.

Unit IV:

Tables, Printing literal text, Footnotes and marginal notes. Drawing pictures with

LATEX.

Unit V:

Mathematical Formulas - Mathematical environments, Main elements of math mode,

Mathematical symbols, Additional elements, Fine-tuning mathematics.

PAPER - II

MATLAB

Unit - I

Introduction - Basics of MATLAB, Input - Output, File trypes - Platform dependence -

General commands.

Unit - II

Interactive Computation: Matrices and Vectors - Matrix and Array operations - Creating

and Using Inline functions - Using Built-in Functions and On-line Help - Saving and

loading data - Plotting simple graphs.

Unit - III

Programming in MATLAB: Scripts and Functions - Script files - Functions files-

Language specific features - Advanced Data objects.

Unit - IV

Applications - Linear Algebra - Curve fitting and Interpolation - Data analysis and

Statistics - Numerical Integration - Ordinary differential equations - Nonlinear

Algebraic Equations.

Unit - V

Graphics: Basic 2-D Plots - Using subplot to Layout multiple graphs - 3 - D Plots -

Handle Graphics - Saving and printing Graphs - Errors.

PAPER - III

MATHEMATICA

Unit - I: Introduction to Mathematica

Running Mathematica - Numerical Calculations - Building Up calculations - Using the

Mathematica system - Algebraic calculations - Symbolic Mathematics - Numerical

Mathematics.

Unit - II

Functions and Programs - Lists - Graphics - Input and Output in Notebooks - The

structure of Graphics.

Unit - III: Advanced Mathematics in Mathematica

Mathematical Functions - Algebraic Manipulation - Manipulating Equations - Calculus.

Unit - IV

Series, Limits and Residues - Linear Algebra - Constructing matrices - Getting pieces of

matrices - Scalars, Vectors and Matrices - Operations on scalars, vectors and matrices -

Multiplying Vectors and matrices - Matrix inversion - Basic matrix operations - Solving

linear systems - Eigen values and Eigen vectors.

Unit - V

Numerical operations on data - Curve fitting - Approximate functions and Interpolation

- Fourier Transforms.

Numerical operations on functions - Numerical Integration - Numerical evaluation of

sums and products - Numerical Solution of Polynomial equations - Numerical root

finding - Numerical solution of Differential equations -

PAPER IV

PRACTICALS

Implementing the Algorithms of any one of the software in Papers I to III above.

SUPPORTIVE: APPLIED MATHEMATICS - I

UNIT I: ORDINARY DIFFERENTIAL EQUATIONS

Second and higher order linear ODE - Homogeneous linear equations with constant and

variable coefficients - Nonhomogeneous equations - Solutions by variation of

parameters.

UNIT II: FUNCTIONS OF SEVERAL VARIABLES

Partial derivatives - Total differential - Taylor's expansions - Maxima and Minima of

functions - Differentiation under integral sign.

UNIT III: PARTIAL DIFFERENTIAL EQUATIONS

Formation of PDE by elimination of arbitrary constants and functions - Solutions -

General and singular solution- Lagrange's Linear equation - Linear PDE of second and

higher order with constant coefficients.

UNIT IV: FOURIER SERIES

Dirichlet's conditions - General Fourier series - Half range Sine and Cosine series -

Parseval's identity - Harmonic Analysis.

UNIT V: BOUNDARY VALUE PROBLEMS

Classifications of PDE - Solutions by separation of variables - One dimensional heat and wave equation.

SUPPORTIVE: APPLIED MATHEMATICS - II

UNIT I: LAPLACE TRANSFORM

Transform of elementary functions - Transforms of derivatives and integrals - Initial and

final value theorems - Inverse Laplace transform - Convolution theorem - Solutions of

linear ODE with constant coefficients.

UNIT II: FOURIER TRANSFORMS

Fourier integral theorem - Fourier transform pairs- Fourier Sine and Cosine transforms -

Properties - Transforms of simple functions - Convolution theorem - Parseval's identity.

UNIT III: MULTIPLE INTEGRALS

Double integration - Cartesian and polar co-ordinates - Change of order of integration -

Area as a double integral - Triple integration - Volume as a triple integral.

UNIT IV: VECTOR CALCULUS

Gradient, Divergence and Curl - Directional derivative - Irrotational and solenoidal

vector fields - Vector integration - Green's theorem, Gauss divergence theorem and

Stoke's theorem.

UNIT-V: NUMERICAL SOLUTIONS OF ODEs

Solution by Taylor's series Method - Euler's Method - Modified Euler Method, Runge-

Kutta Method - Solving simultaneous equations.

Group Theory: Direct products- Group Action on a Set: Isotropy Subgroups- Orbits-

Counting Theorems- p-Groups- The Sylow Theorems

UNIT-II:

Applications of the Sylow Theory: Applications to p-Groups and the Class Equation-

Further Applications

Ring Theory: Ring of Polynomials: Polynomials in an Inderterminate- The

Equation Homomorphisms- Factorization of Polynomials over a Field

UNIT-III:

Field Theory: Extension Fields-Algebraic and Transcendental Elements-Irreducible

polynomial over F-Simple Extension- Algebraic Extensions: Finite Extensions- Structure

of Finite Fields

UNIT-IV:

Automorphisms of Fields- Conjugation Isomorphisms- Automorphisms and Fixed

Fields- The Frobenius Automorphism- Splitting Fields.

UNIT-V:

Separable Extensions- Galois Theory: Normal Extensions- The Main Theorem-

Illustrations of Galois Theory: Symmetric Functions

Text book: "A First Course in Abstract Algebra" by J.B.Fraleigh, Fifth Edition, Addition

Wesly Longman, Inc, Reading Massachusetts, 1999.

UNIT-I: Chapter 2, Section: 2.4 (Direct Product only)

Chapter 3, Sections: 3.6, 3.7

UNIT-II: Chapter 4, Section: 4.3, Chapter 5, Sections: 5.5, 5.6

UNIT-III: Chapter 8, Sections: 8.1, 8.3 (Finite Extensions Only), 8.5

UNIT-IV: Chapter 9, Sections: 9.1, 9.3

UNIT-V: Chapter 9, Sections: 9.4, 9.6, 9.7 (Symmetric Functions only)

MATAC02 : REAL ANALYSIS

RIEMANN STILTJES INTEGRAL:

Unit-I.

Definition and Existence of the Integral - properties of the integral - Integration and

differentiation - Integration of vector valued function - rectifiable curves

Unit-II.

Uniform convergence and continuity - uniform convergence and integration - uniform

convergence and differentiation - equicontinuous families of functions - The Stone

Weirstrass theorem

FUNCTIONS OF SEVERAL VARIABLES

Unit-III.

Linear transformation - contraction principle - Inverse function theorem - Implicit

function theorem - determinants - derivatives of higher order - differentiation of

integrals

LEBESGUE MEASURE:

Unit-IV.

Outer measure - Measurable sets and Lebesgue measure - Measurable functions -

Littlewood's Theorem

Unit-V.

The Lebesgue integral of bounded functions over a set of finite measure - integral of a

non - negative function - General Lebesgue Integral - convergence in measure

Text Book:

For Unit-I to III relevant chapters from : Principles of Mathematical Analysis by W.

Rudin, McGraw Hill, New York, 1976

For Unit-IV and V relevant chapters from : Real Analysis by H.L. Roydon, Third Edition,

Macmillan, New York, 1988.MATAC03: ORDINARY DIFFERENTIAL EQUATIONS

Unit-I.

Linear Equations with constant coefficients - Second order Homogeneous equations -

Initial value problems - Linear dependence and independence Wronskian and a formula

for Wronskian - Non Homogeneous equation of order two.

Unit-II.

Homogeneous and Non - Homogeneous Equations of order n - Initial value problems -

annihilator Method to solve a non - homogeneous equation - Algebra of constant

coefficients operators.

Unit-III.

Linear Equations with variable coefficients - Initial value problems - Existence and

Uniqueness Theorems - Solutions to a non - homogeneous equation - Wronskian and

Linear dependence - reduction of the order of a homogeneous equation - Homogeneous

equation with analytic coefficients - The Legendre equation.

Unit-IV.

Linear Equation with regular singular points - Euler Equation - Second order equations

with regular singular points - Exceptional cases - Bessel equation.

Unit-V.

Existence and Uniqueness of solutions to first order equations - Equation with variables

separated - Exact Equations - Method of successive approximations - The Lipschitz

condition - convergence of the successive approximations and the existence theorem.

MATAC04: COMPLEX ANALYSIS

Unit-I:

Introduction to the concept of analytic function: Limits and continuity - Analytic

functions - Polynomials - Rational functions - Conformality: Arcs and closed curves -

Analytic functions in regions - Conformal Mapping - Length and Area - Linear

Transformations: The Linear group - The Cross ratio - Elementary Riemann Surfaces.

Unit-II:

Complex Integration: Line Integrals Rectifiable Arcs - Line Integrals as Functions of

Arcs - Cauchy's theorem for a rectangle - Cauchy's theorem in a disk, Cauchy's Integral

formula: The Index of a point with respect to a closed curve - The Integral formula -

Higher derivatives Removable singularities, Taylor's Theorem - Zeros and Poles - The

Local Mapping - The Maximum principle - chains and cycles.

Unit-III:

The Calculus of Residues The Residue theorem - The Argument principle - Evaluation

of definite integrals. Harmonic functions: The Definitions and basic Properties - Mean

value property - Poisson's Formula.

Unit-IV:

Series and Product Developments: Weierstrass Theorem - The Taylor Series - The

Laurent Series - Partial fractions and Factorization: Partial Fractions - Infinite Products -

Canonical Products.

Unit-V:

The Riemann Mapping Theorem - Statement and Proff - Boundary Behaviour - Use of

the reflection principle - Analytic arcs - Conformal mapping of Polygons: The

Behaviour at an angle - The Schwarz - Christoffel Formula - Mapping on a rectangle.

MATAC05: PARTIAL DIFFERENTIAL EQUATIONS

Unit I:

Nonlinear Partial Differential Equations of the first order - Cauchy's method of

characteristics - Compatible systems of first order equations - Charpit's method- Special

types of First order equations - Jacobi's method.

Unit II:

Partial Differential Equations of Second order - The origin of Second-order Equations -

Linear Partial Differential Equations with constant coefficients - Equations with variable

coefficients - Characteristics curves of second -order equations- Characteristics of

equations in three variables.

Unit III:

The Solution of Linear Hyperbolic Equations - Separation of variables - The Method of

Integral Transforms - Nonlinear Equations of the second order.

Unit IV:

Laplace's Equation - The occurrence of Laplace's Equation in Physics- Elementary

solution of Laplace's Equation - Families of Equipotential surfaces Boundary value

problems - Separation of variables- Problems with axial symmetry.

Unit V:

The wave equation - The occurrence of wave equation in Physics - Elementary solutions

of the one-dimensional wave equation - Vibrating Membranes: Applications of the

calculus of variations - Three dimensional problems.

The Diffusion Equations: Elementary solutions of the Diffusion Equation - Separation of

variables- The use of Integral transforms.

MATAC06: MECHANICS

Unit-I:

INDRODUCTORY CONCEPTS: Mechanical system - Generalized Coordinates -

Constraints - Virtual Work - Energy and Momentum.

Unit-II:

LAGRANGE'S EQUATIONS: Derivations of Lagrange's Equations: Derivations of

Lagrange's Equations - Examples - Integrals of Motion.

Unit-III:

HAMITON'S EQUATIONS: Hamilton's Principle - Hamilton's Equations.

Unit-IV:

HAMILTON - JACOBI THEORY: Hamilton's Principle function - Hamilton - Jacobi

Equation - Separability.

Unit-V:

CANONICAL TRANSFORMATIONS: Differential forms and Generating Functions -

Lagrange and Poisson Brackets.

MATAC07- TOPOLOGY AND FUNCTIONAL ANALYSIS

TOPOLOGY:

Unit-I:

Spaces and Maps.

Unit-II:

Separability Axioms and Compactness.

Unit-III:

Connectedness - Pathwise connectedness - Imbedding and Extension theorems.

FUNCTIONAL ANALYSIS

Unit-IV:

Banach spaces - Definition and Examples - continuous linear transformations - Hahn

Banach theorem - Natural Imbedding - open mapping theorem - conjugate of an

operator.

Unit-V:

Hilbert spaces - Definition and simple properties - Orthogonal Complements -

Orthonormal basis - conjugate space.

MATAC08: FLUID DYNAMICS

Unit-I:

Introductory Notions- Velocity- Stream Lines and Path Lines- Stream Tubes and

Filaments- Fluid Body- Density- Pressure. Differentiation following the Fluid- Equation of

continuity- Boundary conditions- Kinematical and physical- Rate of change of linear

momentum- Equation of motion of an inviscid fluid.

Unit-II:

Euler's momentum Theorem- Conservative forces- Bernoulli's theorem in steady motionenergy

equation for inviscid fluid- circulation- Kelvin's theorem- vortex motion-

Helmholtz equation.

Unit-III:

Two Dimensional Motion- Two Dimensional Functions- Complex Potential Basic

singularities- source- sink- Vortex- doublet- Circle theorem. Flow past a Circular cylinder

with circulation- Blasius Theorem- Lift force. (Magnus effect)

Unit-IV:

Viscous flows- Navier - stokes equations- some exact solutions of Navier Stokes

equations- Flow between parallel flat plates- Couette flow- Plane Poiseuille flow- Steady

flow in pipes: Flow through a pipe- The Hagen Poiseuille flow.

Unit-V:

Laminar Boundary Layer in incompressible flow: Boundary Layer concept- Boundary

Layer equations- Boundary Layer along a flat plate- The Blasius solution- Shearing stress

and boundary layer thickness- Displacement thickness, momentum thickness- Momentum

integral theorem for the boundary layer- The Von Karman Integral relation, The Von

Karman Integral relation by momentum law.

MATAC09: MATHEMATICAL METHODS

Unit-I: FOURIER TRANSFORMS: Fourier Transforms - Defn. Inversion theorem -

Fourier cosine transforms - Fourier sine transforms - Fourier transforms of derivatives -

Fourier transforms of some simple functions - Fourier transforms of rational functions -

The convolution integral - convolution theorem - Parseval's relation for Fourier

transforms - solution of PDE by Fourier transform.

Laplace's Equation in Half plane

Laplace's Equation in an infinite strip

The Linear diffusion equation on a semi-infinite line

The two-dimensional diffusion equation.

Unit-II: HANKEL TRANSFORMS: Definition - Elementary properties of Hankel

Transforms - Hankel Transforms of Derivatives of functions - Hankel Transforms of

some elementary functions - The Parseval relation for Hankel transforms - Relation

between Fourier and Hankel transforms - Application to PDE.

Axisymmetric Dirichlet problem for a half - space.

Axisymmetric Dirichlet problem for a thick plate

Unit-III: INTEGRAL EQUATIONS: Types of Integral equations - Equation with

separable kernel - Fredholm Alternative Approximate method - Volterra integral

equations - Classical Fredholm theory - Fredholm's First, Second, Third theorems.

Unit-IV: Application of Integral equation to ordinary differential equation - initial value

problems - Boundary value problems - singular integral equations - Abel Integral

equation

Unit-V: CALCULUS OF VARIATIONS: Variation and its properties - Euler's equation

- Functionals of the integral forms Functional dependent on higher order derivatives -

functionals dependent on the functions of several independent variables - variational

problems in parametric form.

MATAC10 - NON LINEAR DIFFERENTIAL EQUATIONS

Unit-I:

First order systems in two variables and linearization: The general phase plane-some

population models - Linear approximation at equilibrium points - Linear systems in

matrix form.

Unit-II:

Averaging Methods: An energy balance method for limit cycles - Amplitude and

frequency estimates - slowly varying amplitudes - nearly periodic solutions - periodic

solutions: harmony balance - Equivalent linear equation by harmonic balance - Accuracy

of a period estimate.

Unit-III:

Perturbation Methods: Outline of the direct method - Forced Oscillations far from

resonance - Forced Oscillations near resonance with Weak excitation - Amplitude

equation for undamped pendulum - Amplitude Perturbation for the pendulum equation -

Lindstedt's Method - Forced oscillation of a self - excited equation - The Perturbation

Method and Fourier series.

Unit-IV:

Linear Systems: Time Varying Systems - Constant coefficient System - Periodic

Coefficients - Floquet Theory - Wronskian.

Unit-V:

Stability: Poincare stability - solutions, paths and norms - Liapunov stability Stability of

linear systems - Comparison theorem for the zero solutions of nearly - linear systems.MATAC11: CONTROL THEORY

Unit-I:

OBSERVABILITY: Linear Systems - Observability Grammian - Constant coefficient

systems - Reconstruction kernel - Nonlinear Systems

Unit-II:

CONTROLLABILITY: Linear systems - Controllability Grammian - Adjoint systems -

Constant coefficient systems - steering function - Nonlinear systems

Unit-III:

STABILITY: Stability - Uniform Stability - Asymptotic Stability of Linear Systems -

Linear time varying systems - Perturbed linear systems - Nonlinear systems

Unit-IV:

STABILIZABILITY: Stabilization via linear feedback control - Bass method -

Controllable subspace - Stabilization with restricted feedback

Unit-V:

OPTIMAL CONTROL: Linear time varying systems with quadratic performance criteria

- Matrix Riccati equation - Linear time invariant systems - Nonlinear Systems

MATAE01: NUMERICAL METHODS

Unit-I:

SOLUTION OF NONLINEAR EQUATIONS: Newton's method - Convergence of

Newton's method - Bairstow"s Method for quadratic factors NUMERICAL

DIFFERENTIATION AND INTEGRATION: Derivatives from Differences tables -

Higher order derivatives - Divided difference, Central-Difference formulas - Composite

formula of Trapezoidal rule - Romberg integration - Simpson's rules.

Unit-II:

SOLUTION OF SYSTEM OF EQUATIONS: The Elimination method - Gauss and

Gauss Jordan methods - LU Decomposition method - Matrix inversion by Gauss-Jordan

method - Methods of Iteration - Jacobi and Gauss Seidal Iteration - Relaxation method -

Systems of Nonlinear equations.

Unit-III:

SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS: Taylor series method -

Euler and Modified Euler methods - Rungekutta methods - Multistep methods - Milne's

method - Adams Moulton method.

Unit-IV:

BOUNDARY VALUE PROBLEMS AND CHARACTERISTIC VALUE PROBLEMS:

The shooting method - solution through a set of equations - Derivative boundary

conditions - Characteristic value problems - Eigen values of a matrix by Iteration - The

power method.

Unit-V:

NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS: (Solutions

of Elliptic, Parabolic and Hyperbolic partial differential equations) Representation as a

difference equation - Laplace's equation on a rectangular region - Iterative methods for

Laplace equation - The Poisson equation - Derivative boundary conditions - Solving the

equation for time-dependent heat flow (i) The Explicit method (ii) The Crank Nicolson

method - solving the wave equation by Finite Differences.

MATAE02: COMPUTER PROGRAMMING AND LAB I

THEORY(50 Marks)

Unit-I:

Fortran 77 - Representation of Integer and Real constants - Variable names - Arithmetic

operators and modes for Expression - Integer Expressions. Real Expressions - Hierarchy

of Operations in Expressions - Arithmetic statement - Defining Variables - Mixed Mode

Expressions - Special Functions - Input Output statements.

Unit-II:

Format description for Read statement - Format description for print statement - multi

record formats - Hollerith field declaration - specifications in a Format - Generalized

input / output statements - Logical constants, variables and Logic Expressions.

Unit-III:

Control statements - Relational Operator - Logical IF statement - GO TO Statements -

Nested Logical IF statement - Arithmetic IF statement - computed GOTO statement.

Unit-IV:

The DO Statement - Rules to be followed in Utilizing Do Loops - RRPEAT WHILE

structure - Subscripted Variables - Subscript Expressions - Dimension statement - DO

loops with subscripts.

Unit-V:

Functions and subroutines - Function subprograms - Subroutines - Common declaration

- Implicit declaration - Equivalence declaration.

MATAE03 - COMPUTER PROGRAMMING AND LAB II

THEORY(50 Marks)

Unit-I:

Overview of C - Constants. Variables and Data Types - Character set - C tokens -

Keywords & Identifiers - constants - variables - Data types - Declaration of variables -

Assigning values to variables - Defining symbolic constants.

Unit-II:

Arithmetic of operators - Relational operators - Logical operators - Assignment

operator - Increment and decrement operators - conditional operator - Bitwise operators

- special operators - Arithmetic Expressions - Evaluation of Expressions - Precedence

of arithmetic operators - Type conversions in Expressions - Operator Precedence and

Associativity - Mathematical Functions.

Unit-III:

Managing Input and Output operators - Reading a character - Writing a character -

formatted input - formatted output - Decision making IF statement - IF - ELSE -

statement - Nesting of IF ELSE statements - The Switch statement - The GO TO

statement.

Unit-IV:

The WHILE statement DO statement - FOR statement -Jumps in Loops - Onedimensional

Array - Two dimensional Arrays - Initializing two dimensional arrays -

Multidimensional arrays.

Unit-V:

Need for User defined functions - A multi-function program - the form of C Functions -

Return Values and their Types calling a function - Category of functions - Arguments

but no return values - Arguments with return values In file management in C - Defining

and with return values - In file management in C - Defining and opening a file - closing

a file- Input / Output operations on files.

COMPUTER PROGRAMMING AND LAB II

PAPER - I

LATEX

Unit I:

Text formatting, TEX and its offspring, What's different in LATEX 2є, Distinguishing

LATEX 2є , Basics of a LATEX file.

Unit II:

Commands and Environments-Command names and arguments, Environments,

Declarations, Lengths, Special Characters, Fragile Commands, Exercises.

Unit III:

Document Layout and Organization - Document class, Page style, Parts of the document,

Table of contents, Fine - Tuning text, Word division.

Displayed Text - Changing font, Centering and indenting, Lists, Generalized lists,

Theorem-like declarations, Tabulator stops, Boxes.

Unit IV:

Tables, Printing literal text, Footnotes and marginal notes. Drawing pictures with

LATEX.

Unit V:

Mathematical Formulas - Mathematical environments, Main elements of math mode,

Mathematical symbols, Additional elements, Fine-tuning mathematics.

PAPER - II

MATLAB

Unit - I

Introduction - Basics of MATLAB, Input - Output, File trypes - Platform dependence -

General commands.

Unit - II

Interactive Computation: Matrices and Vectors - Matrix and Array operations - Creating

and Using Inline functions - Using Built-in Functions and On-line Help - Saving and

loading data - Plotting simple graphs.

Unit - III

Programming in MATLAB: Scripts and Functions - Script files - Functions files-

Language specific features - Advanced Data objects.

Unit - IV

Applications - Linear Algebra - Curve fitting and Interpolation - Data analysis and

Statistics - Numerical Integration - Ordinary differential equations - Nonlinear

Algebraic Equations.

Unit - V

Graphics: Basic 2-D Plots - Using subplot to Layout multiple graphs - 3 - D Plots -

Handle Graphics - Saving and printing Graphs - Errors.

PAPER - III

MATHEMATICA

Unit - I: Introduction to Mathematica

Running Mathematica - Numerical Calculations - Building Up calculations - Using the

Mathematica system - Algebraic calculations - Symbolic Mathematics - Numerical

Mathematics.

Unit - II

Functions and Programs - Lists - Graphics - Input and Output in Notebooks - The

structure of Graphics.

Unit - III: Advanced Mathematics in Mathematica

Mathematical Functions - Algebraic Manipulation - Manipulating Equations - Calculus.

Unit - IV

Series, Limits and Residues - Linear Algebra - Constructing matrices - Getting pieces of

matrices - Scalars, Vectors and Matrices - Operations on scalars, vectors and matrices -

Multiplying Vectors and matrices - Matrix inversion - Basic matrix operations - Solving

linear systems - Eigen values and Eigen vectors.

Unit - V

Numerical operations on data - Curve fitting - Approximate functions and Interpolation

- Fourier Transforms.

Numerical operations on functions - Numerical Integration - Numerical evaluation of

sums and products - Numerical Solution of Polynomial equations - Numerical root

finding - Numerical solution of Differential equations -

PAPER IV

PRACTICALS

Implementing the Algorithms of any one of the software in Papers I to III above.

SUPPORTIVE: APPLIED MATHEMATICS - I

UNIT I: ORDINARY DIFFERENTIAL EQUATIONS

Second and higher order linear ODE - Homogeneous linear equations with constant and

variable coefficients - Nonhomogeneous equations - Solutions by variation of

parameters.

UNIT II: FUNCTIONS OF SEVERAL VARIABLES

Partial derivatives - Total differential - Taylor's expansions - Maxima and Minima of

functions - Differentiation under integral sign.

UNIT III: PARTIAL DIFFERENTIAL EQUATIONS

Formation of PDE by elimination of arbitrary constants and functions - Solutions -

General and singular solution- Lagrange's Linear equation - Linear PDE of second and

higher order with constant coefficients.

UNIT IV: FOURIER SERIES

Dirichlet's conditions - General Fourier series - Half range Sine and Cosine series -

Parseval's identity - Harmonic Analysis.

UNIT V: BOUNDARY VALUE PROBLEMS

Classifications of PDE - Solutions by separation of variables - One dimensional heat and wave equation.

SUPPORTIVE: APPLIED MATHEMATICS - II

UNIT I: LAPLACE TRANSFORM

Transform of elementary functions - Transforms of derivatives and integrals - Initial and

final value theorems - Inverse Laplace transform - Convolution theorem - Solutions of

linear ODE with constant coefficients.

UNIT II: FOURIER TRANSFORMS

Fourier integral theorem - Fourier transform pairs- Fourier Sine and Cosine transforms -

Properties - Transforms of simple functions - Convolution theorem - Parseval's identity.

UNIT III: MULTIPLE INTEGRALS

Double integration - Cartesian and polar co-ordinates - Change of order of integration -

Area as a double integral - Triple integration - Volume as a triple integral.

UNIT IV: VECTOR CALCULUS

Gradient, Divergence and Curl - Directional derivative - Irrotational and solenoidal

vector fields - Vector integration - Green's theorem, Gauss divergence theorem and

Stoke's theorem.

UNIT-V: NUMERICAL SOLUTIONS OF ODEs

Solution by Taylor's series Method - Euler's Method - Modified Euler Method, Runge-

Kutta Method - Solving simultaneous equations.