**M.Sc. MATHEMATICS **

**SEMESTER - III **

**CORE VIII - PARTIAL DIFFERENTIAL EQUATIONS**

** UNIT I:**

Second order Partial Differential Equations: Origin of second order partial differential equations – Linear differential equations with constant coefficients – Method of solving partial (linear ) differential equation – Classification of second order partial differential equations – Canonical forms – Adjoint operators – Riemann method. (Chapter 2 : Sections 2.1 to 2.5) .

**UNIT II:**

Elliptic Differential Equations: Elliptic differential equations – Occurrence of Laplace and Poisson equations – Boundary value problems – Separation of variables method – Laplace equation in cylindrical – Spherical co-ordinates, Dirichlet and Neumann problems for circle – Sphere.(Chapter 3 : Sections 3.1 to 3.9).

**UNIT III : ** Parabolic Differential Equations: Parabolic differential equations – Occurrence of the diffusion equation – Boundary condition – Separation of variable method – Diffusion equation in cylindrical – Spherical co-ordinates. (Chapter 4: Sections 4.1 to 4.5).

** UNIT IV: ** Hyperbolic Differential Equations: Hyperbolic differential equations – Occurrence of wave equation – One dimensional wave equation – Reduction to canonical form – D’Alembertz solution – Separation of variable method – Periodic solutions – Cylindrical – Spherical co-ordinates – Duhamel principle for wave equations.(Chapter 5 : Sections 5.1 to 5.6 and 5.9).

** UNIT V:** Integral Transform: Laplace transforms – Solution of partial differential equation – Diffusion equation – Wave equation – Fourier transform – Application to partial differential equation – Diffusion equation – Wave equation – Laplace equation. (Chapter 6 : Sections 6.2 to 6.4).

** M.Sc. MATHEMATICS**

** SEMESTER - III**

** CORE IX - TOPOLOGY **

**UNIT I :**

Topological spaces: Topological spaces - Basis for a topology – The Order Topology - The Product Topology on XxY – The Subspace Topology – Closed sets and Limit points. (Chapter 2: sections 12 to 17).

**UNIT II :**

Continuous functions: Continuous Functions– The Product Topology – The Metric Topology. (Chapter 2: Sections 18 to 21).

**UNIT III :**

Connectedness: Connected Spaces – Connected Subspaces of the Real line – Components and Local Connectedness. (Chapter 3: Sections 23 to 25)

** UNIT IV :**

Compactness: Compact spaces – Compact Subspace of the real line –Limit Point Compactness – Local Compactness. (Chapter 3: Sections 26 to 29).

** UNIT V :**

Countability and Separation axioms: The Countability Axioms – The Separation Axioms – Normal Spaces – The Urysohn Lemma – The Urysohn Metrization Theorem – The Tietze extension theorem. (Chapter 4: Sections 30 to 35).

** M.Sc. MATHEMATICS **

**SEMESTER - III **

**CORE X - MEASURE THEORY AND INTEGRATION**

** UNIT I :**

Lebesgue Measure: Lebesgue Measure – Introduction – Outer measure – Measurable sets and Lebesgue measure – Measurable functions – Little Woods’ Three Principles. (Chapter 3: Sections 1 to 3, 5 and 6).

**UNIT II :**

Lebesgue integral : Lebesgue integral – The Riemann integral – Lebesgue integral of bounded functions over a set of finite measure – The integral of a nonnegative function – The general Lebesgue integral. (Chapter 4: Sections 1 to 4).

** UNIT III :**

Differentiation and Integration : Differentiation and Integration – Differentiation of monotone functions – Functions of bounded variation – Differentiation of an integral – Absolute continuity. (Chapter 5: Sections 1 to 4).

** UNIT IV** :

General Measure and Integration : General Measure and Integration – Measure spaces – Measurable functions – Integration – Signed Measure – The Radon – Nikodym theorem. (Chapter 11: Sections 1 to 3, 5 and 6) .

** UNIT V :**

Measure and Outer Measure : Measure and outer measure – outer measure and measurability – The Extension theorem – Product measures. (Chapter 12: Sections 1, 2 and 4).

**M.Sc. MATHEMATICS**

** SEMESTER - III**

** CORE XI - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS **

**UNIT I** :

Variational Problems with Fixed Boundaries: The concept of variation and its properties – Euler’s equation- Variational problems for Functionals – Functionals dependent on higher order derivatives – Functions of several independent variables – Some applications to problems of Mechanics. (Chapter 1: Sections 1.1 to 1.7 of [1]).

** UNIT II** :

Variational Problems with Moving Boundaries: Movable boundary for a functional dependent on two functions – one-side variations - Reflection and Refraction of extermals - Diffraction of light rays. (Chapter 2: Sections 2.1 to 2.5 of [1]).

**UNIT III : **

Integral Equation: Introduction – Types of Kernels – Eigen Values and Eigen functions – Connection with differential equation – Solution of an integral equation – Initial value problems – Boundary value problems. (Chapter 1: Section 1.1 to 1.3 and 1.5 to 1.8 of [2]).

**UNIT IV : **

Solution of Fredholm Integral Equation: Second kind with separable kernel – Orthogonality and reality eigen function – Fredholm Integral equation with separable kernel – Solution of Fredholm integral equation by successive substitution – Successive approximation – Volterra Integral equation – Solution by successive substitution. (Chapter 2: Sections 2.1 to 2.3 and Chapter 4 Sections 4.1 to 4.5 of [2]).

** UNIT V :**

Hilbert – Schmidt Theory: Complex Hilbert space – Orthogonal system of functions- Gram Schmit orthogonlization process – Hilbert – Schmit theorems – Solutions of Fredholm integral equation of first kind. (Chapter 3: Section 3.1 to 3.4 and 3.8 to 3.9 of [2]).

**M.Sc. MATHEMATICS**

** SEMESTER - III **

**CORE VIII - PARTIAL DIFFERENTIAL EQUATIONS **

**UNIT I:**

Second order Partial Differential Equations: Origin of second order partial differential equations – Linear differential equations with constant coefficients – Method of solving partial (linear ) differential equation – Classification of second order partial differential equations – Canonical forms – Adjoint operators – Riemann method. (Chapter 2 : Sections 2.1 to 2.5) .

** UNIT II :**

Elliptic Differential Equations: Elliptic differential equations – Occurrence of Laplace and Poisson equations – Boundary value problems – Separation of variables method – Laplace equation in cylindrical – Spherical co-ordinates, Dirichlet and Neumann problems for circle – Sphere.(Chapter 3 : Sections 3.1 to 3.9).

**UNIT III :**

Parabolic Differential Equations: Parabolic differential equations – Occurrence of the diffusion equation – Boundary condition – Separation of variable method – Diffusion equation in cylindrical – Spherical co-ordinates. (Chapter 4: Sections 4.1 to 4.5).

**UNIT IV:**

Hyperbolic Differential Equations: Hyperbolic differential equations – Occurrence of wave equation – One dimensional wave equation – Reduction to canonical form – D’Alembertz solution – Separation of variable method – Periodic solutions – Cylindrical – Spherical co-ordinates – Duhamel principle for wave equations.(Chapter 5 : Sections 5.1 to 5.6 and 5.9).

** UNIT V :**

Integral Transform: Laplace transforms – Solution of partial differential equation – Diffusion equation – Wave equation – Fourier transform – Application to partial differential equation – Diffusion equation – Wave equation – Laplace equation. (Chapter 6 : Sections 6.2 to 6.4).

** M.Sc. MATHEMATICS **

**SEMESTER - III**

** CORE IX - TOPOLOGY **

**UNIT I :**

Topological spaces: Topological spaces - Basis for a topology – The Order Topology - The Product Topology on XxY – The Subspace Topology – Closed sets and Limit points. (Chapter 2: sections 12 to 17).

**UNIT II : **

Continuous functions: Continuous Functions– The Product Topology – The Metric Topology. (Chapter 2: Sections 18 to 21).

** UNIT III : **

Connectedness: Connected Spaces – Connected Subspaces of the Real line – Components and Local Connectedness. (Chapter 3: Sections 23 to 25)

**UNIT IV :**

Compactness: Compact spaces – Compact Subspace of the real line –Limit Point Compactness – Local Compactness. (Chapter 3: Sections 26 to 29).

**UNIT V : **

Countability and Separation axioms: The Countability Axioms – The Separation Axioms – Normal Spaces – The Urysohn Lemma – The Urysohn Metrization Theorem – The Tietze extension theorem. (Chapter 4: Sections 30 to 35).

**M.Sc. MATHMATICS**

** SEMESTER - III**

** CORE X - MEASURE THEORY AND INTEGRATION**

** UNIT I : **

Lebesgue Measure: Lebesgue Measure – Introduction – Outer measure – Measurable sets and Lebesgue measure – Measurable functions – Little Woods’ Three Principles. (Chapter 3: Sections 1 to 3, 5 and 6).

** UNIT II :**

Lebesgue integral : Lebesgue integral – The Riemann integral – Lebesgue integral of bounded functions over a set of finite measure – The integral of a nonnegative function – The general Lebesgue integral. (Chapter 4: Sections 1 to 4).

** UNIT III :**

Differentiation and Integration : Differentiation and Integration – Differentiation of monotone functions – Functions of bounded variation – Differentiation of an integral – Absolute continuity. (Chapter 5: Sections 1 to 4).

** UNIT IV : **

General Measure and Integration : General Measure and Integration – Measure spaces – Measurable functions – Integration – Signed Measure – The Radon – Nikodym theorem. (Chapter 11: Sections 1 to 3, 5 and 6) .

**UNIT V : **

Measure and Outer Measure : Measure and outer measure – outer measure and measurability – The Extension theorem – Product measures. (Chapter 12: Sections 1, 2 and 4).

**M.Sc. MATHEMATICS**

** SEMESTER - III **

**CORE XI - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS**

** UNIT I : **

Variational Problems with Fixed Boundaries: The concept of variation and its properties – Euler’s equation- Variational problems for Functionals – Functionals dependent on higher order derivatives – Functions of several independent variables – Some applications to problems of Mechanics. (Chapter 1: Sections 1.1 to 1.7 of [1]).

**UNIT II : **

Variational Problems with Moving Boundaries: Movable boundary for a functional dependent on two functions – one-side variations - Reflection and Refraction of extermals - Diffraction of light rays. (Chapter 2: Sections 2.1 to 2.5 of [1]).

** UNIT III : **

Integral Equation: Introduction – Types of Kernels – Eigen Values and Eigen functions – Connection with differential equation – Solution of an integral equation – Initial value problems – Boundary value problems. (Chapter 1: Section 1.1 to 1.3 and 1.5 to 1.8 of [2]).

** UNIT IV :**

Solution of Fredholm Integral Equation: Second kind with separable kernel – Orthogonality and reality eigen function – Fredholm Integral equation with separable kernel – Solution of Fredholm integral equation by successive substitution – Successive approximation – Volterra Integral equation – Solution by successive substitution. (Chapter 2: Sections 2.1 to 2.3 and Chapter 4 Sections 4.1 to 4.5 of [2]).

**UNIT V :**

Hilbert – Schmidt Theory: Complex Hilbert space – Orthogonal system of functions- Gram Schmit orthogonlization process – Hilbert – Schmit theorems – Solutions of Fredholm integral equation of first kind. (Chapter 3: Section 3.1 to 3.4 and 3.8 to 3.9 of [2]).

**M.Sc. MATHeMATICS **

**SEMESTER - III **

**ELECTIVE III - PAPER I - DIFFERENTIAL GEOMETRY**

** UNIT I :**

Theory of Space Curves: Theory of space curves – Representation of space curves – Unique parametric representation of a space curve – Arc-length – Tangent and osculating plane – Principle normal and binormal – Curvature and torsion – Behaviour of a curve near one of its points – The curvature and torsion of a curve as the intersection of two surfaces. (Chapter 1 : Sections 1.1 to 1.9) .

**UNIT II : **

Theory of Space Curves (Contd.): Contact between curves and surfaces – Osculating circle and osculating sphere – Locus of centre of spherical curvature – Tangent surfaces – Involutes and Evolutes – Intrinsic equations of space curves – Fundamental Existence Theorem – Helices. (Chapter 1 : Sections 1.10 to 1.13 and 1.16 to 1.18) .

**UNIT III :**

Local Intrinsic properties of surface: Definition of a surface – Nature of points on a surface – Representation of a surface – Curves on surfaces – Tangent plane and surface normal – The general surfaces of revolution – Helicoids – Metric on a surface – Direction coefficients on a surface. (Chapter 2 : Sections 2.1 to 2.10).

**UNIT IV : **

Local Intrinsic properties of surface and geodesic on a surface: Families of curves – Orthogonal trajectories – Double family of curves – Isometric correspondence – Intrinsic properties – Geodesics and their differential equations – Canonical geodesic equations – Geodesics on surface of revolution. (Chapter 2: Sections 2.11 to 2.15 and Chapter 3: Sections 3.1 to 3.4) .

** UNIT V : **

Geodesic on a surface: Normal property of Geodesics – Differential equations of geodesics using normal property – Existence theorems – Geodesic parallels – Geodesic curvature – Gauss Bonnet Theorems – Gaussian curvature – Surface of constant curvature . (Chapter 3: Sections 3.5 to 3.8 and Sections 3.10 to 3.13) .

**M.Sc. MATHEMATICS**

** SEMESTER - III **

**ELCTIVE III - PAPER II - PROGRAMMING WITH C++**

**UNIT I : **

Software Evolution – Procedure oriented Programming – Object oriented programming paradigm – Basic concepts of object oriented programming – Benefits of oops – Object oriented Languages – Application of OOP – Beginning with C++ - what is C++ - Application of C++ - A simple C++ Program – More C++ Statements – An Example with class – Structure of C++ Program.

**UNIT II : **

Token, Expressions and control structures: Tokens – Keywords – Identifiers and Constants – Basic Data types – User defined Data types – Derived data types – Symbolic Constants in C++ - Scope resolution operator – Manipulators – Type cost operator – Expressions and their types – Special assignment expressions – Implicit Conversions – Operator Overloading – Operator precedence – Control Structure.

** UNIT III :**

Function in C++: Main Function – function prototyping – Call by reference – Return by reference – Inline functions – default arguments – Const arguments – Function overloading – Friend and Virtual functions – Math library function. Class and Objects: Specifying a class – Defining member functions – A C++ program with class – Making an outside function inline – Nesting of member functions – Private member functions – Arrays within a class – Memory allocations for objects – Static data member – Static member functions – Array of the object – Object as function arguments – Friendly functions – Returning objects – Const member functions – Pointer to members – Local classes.

** UNIT IV : **

Constructors and Destructors: Constructors – Parameterized Constructors in a Constructor – Multiple constructors in a class – Constructors with default arguments – Dynamic Initialization of objects – Copy constructors – Dynamic Constructors – Constructing Two-dimensional arrays – Const objects – Destructors. Operator overloading and type conversions: Defining operator overloading – overloading unary operators – overloading binary operators - overloading binary operators using friends – Manipulation of strings using operators – Rules for overloading operators – Type conversions.

** UNIT V : **

Files: Introduction – Class for file stream operations – opening and closing a file – detecting End-of file – More about open () File modes – File pointer and their manipulations – Sequential input and output operations. Exception Handling: Introduction – Basics of Exception Handling – Exception Handling Mechanism – Throwing Mechanism – Catching Mechanism – Rethrowing an Exception.