VIKRAMA SIMHAPURI UNIVERSITY :: KAKUTURU VENKATACHALAM MANDALAM, SPSR NELLORE DISTRICT. DEPARTMENT OF MATHEMATICS COURSE STRUCTURE (with effect from 2017-18) Name of the Course : M.Sc., Applied Mathematics Title of the Paper Work load/ Per week Sessional University Max. Marks SEMESTER – I AM 101 – Methods of Applied Mathematics 5 40 80 120 AM 102 – Real Analysis 5 40 80 120 AM 103 – Ordinary Differential Equations 5 40 80 120 AM 104 – Numerical Methods 5 40 80 120 AM 105 – Complex Analysis 5 40 80 120 AM 106 – Human Values and Professional Ethics – I SEMESTER – II AM 201 – Discrete Mathematics 5 40 80 120 AM 202 – Measure and Integration 5 40 80 120 AM 203 – Partial Differential Equations 5 40 80 120 AM 204 – Topology 5 40 80 120 AM 205 – Advanced Complex Analysis 5 40 80 120 AM 206 – Foundation course in Computer Applications SEMESTER – III AM 301 – Continuum Mechanics 5 40 80 120 AM 302 – Functional Analysis 5 40 80 120 AM 303 – Differential Geometry 5 40 80 120 OPTIONALS: AM 304(A) – Number Theory 5 40 80 120 AM 304(B) – Cryptography 5 40 80 120 AM 305(A) – Classical Mechanics 5 40 80 120 AM 305(B) – Mathematicl Modeling 5 40 80 120 AM 306 - Human Values and Professional Ethics – II SEMESTER – IV AM 401 – Fluid Dynamics 5 40 80 120 AM 402 – Operations Research 5 40 80 120 AM 403 – Graph Theory 5 40 80 120 OPTIONALS: AM 404(A) – Fuzzy sets and fuzzy logic 5 40 80 120 AM 404(B) – Approximation Theory 5 40 80 120 AM 405(A) – Mathematical Statistics 5 40 80 120 AM 405(B) – Bio-Mechanics 5 40 80 120 AM 406 – Advanced course in Computer Applications
AM 105 : COMPLEX ANALYSIS (Common with paper MA 105 of Branch I(A) Mathematics) UNIT –I : Differentiation: Analytic Functions : Derivative Rules for Differentiating Complex Functions- The Cauchy-Riemann Equations –Analytic Functions-Geometrical Interpretation of Arg f 1(z) and f 1(z) - Conformal Mapping –The Mapping w = –Conformal Mapping of the Extended Plane. UNIT – II: Mobius Transformations: The Group Property of Mobius Transformations – The Circle –Preserving Property of Mobius Transformations-Fixed points of a Mobius Transformation-Invariance of Cross Ratio-Mapping of a circle onto a Circle – Symmetry Transformations. UNIT – III: Complex Integrals: Cauchy Integral Theorem: Rectifiable Curves-Complex Integrals-The Case of Smooth Curves-Cauchy’s Integral Theorem-The Key Lemma proof of Cauchy’s Integral Theorem-Application to the Evaluation of Definite Integrals Cauchy’s Integral Theorem for a system of Contours. Cauchy’s Integral Formula – Morera’s Theorem – Cauchy’s Inequalities. UNIT-IV: Power Series: The Cauchy-Hadamard Theorem – Taylor Series. The Uniqueness Theorem for Power series-Expansion of an Analytic Function in a power series – Liouville's Theorem. The Uniqueness Theorem for Analytic functions-A Points and Zeros-Weierstrass’ Double Series Theorem-Substitution of One Power Series into Another- Division of Power series. Scope and Standard as in Chapters 3,5,7,8 and 10 of “ Introductory Complex Analysis” by Richard A. Silverman Dover Publications, Inc. (1972). New York References : 1 Complex Variables - . Schaum outline series, 2/E by Speigel 2. An Introductions to Complex Analysis, by C.L. Siegel : North Holland, 1989
AM 203 : PARTIAL DIFFERENTIAL EQUATIONS (Common with paper MA 203 of Branch I(A) Mathematics) UNIT –I : Ordinary Differential Equations in more than two variables : Methods of solutions of dx/P =dy/Q = dz/R-Orthogonal trajectories of a system of curves on asuface-Pfaffian differential forms and equations in Three variables . (Sections 3,4,5 and of Chapter 1) UNIT – II: Partial Differential Equations of the First order: Partial Differential equations-Origins of first order partial differential equations- Cauchy’s problems for first order equations-Linear equations of first order-Integral surfaces passing through a given curve –Surfaces orthogonal to a given system of surfaces-Charpit’s method.(Sections 1,2,3, 4,5,6 and 10 of Chapter 2) UNIT – III: Partial Differential Equations of the Second order: The Origin of second order equations –Linear partial differential with constant coefficients-Equations with variable coefficients.(Sections 1,4 and 5 of Chapter 3) UNIT-IV: Laplace’s Equations : Elementary solution of Laplace’s equation-Families of equipotential surfaces-Boundary value problems – Separation of variables.( Sections 2,3,4 and 5 of Chapter 4) Scope and Standard as in “ Elements of Partial Differential Equations” by IAN Sneddon Chapter 1: Section 1 to 6, Chapter 2: Sections 1,2,4,5,6,10 Chapter 3: Sections 1,4,5, chapter 4: Sections 2,3,4,5, Chapter 5: Sec2, Chapter 6: Section 3 and 4. Reference : 1. Ordinary and Partial Differential Equations by M.D. Raisinghania
AM 205 : ADVANCED COMPLEX ANALYSIS (Common with paper MA 205 of Branch I(A) Mathematics) UNIT –I : Laurent Series-Singular Points: Laurent Series-Laurent’s Theorem-Poles and Essential Singular points-Behavior at an Essential Singular point. Picard’s at infinity. UNIT – II: The Residue Theorem and its Applications: The Residue Theorem-Residues at infinity-Jordan’s Lemma-Evaluation of Definite Integrals – The Argument Theorems of Rouche and Hurwitz-Local Behavior of Analytic Mappings-The Maximum Modulus principle and Schwarz’s Lemma. UNIT – III: Harmonic Functions: Laplace’s Equations-Conjugate Harmonic Functions-Poisson’s integral. Schwarz’s Formula-The Dirichlet problem. Conformal Mapping: General Principles of Conformal Mapping –Mapping of the Upper Half-Plane onto a Rectangle –The Schwarz-Christoffel Transformation. UNIT-IV: Infinite product and Partial Fraction Expansions: Preliminary Results- Infinite Products-Weierstrass’ Theorem –Mittage – Leffer’s Theorem – The gamma Functions –Cauchy’s Theorem on Partial Fraction Expansions. Scope and Standard as in “ Introductory Complex Analysis” by Richard A. Silverman, Dover Publications, Inc. New York (1972) Chapter 11 to 15. References : 1. Fundamentals of Complex Analysis- Edward B. Saff, Arthur David Snider, Pearson Education 2. Foundations of Complex Analysis by S. Ponnusamy- Narosa Publications
AM 302 : FUNCTIONAL ANALYSIS (Common with paper MA 302 of Branch I(A) Mathematics) UNIT –I : The definitions and some examples –continuous –linear transformations-the Theorem. UNIT – II: Natural imbedding of N in N**-Open mapping theorem –Conjugate of an Operator. UNIT – III: Definition and Simple Properties –Orthogonal Complements- Orthonormal sets – Conjugate spaces-Adjoint of an Operator. UNIT-IV: Self adjoint operators –Normal and Unitary Operators-Projection –Spectral theorem. Scope and Standard as in Sections 46 to 51 of Chapter 9, section 52 to 59 of chapter 10, section 62 of chapter 11 of “ Introduction to Topological and Modern analysis by G.F. Simmons Mc Graw Hill Book Company. References : 1. “ Foundations of Functional Analysis” by S. Ponnusamy-Narosa Publications 2. “ Text book of Functional Analysis – A Problem oriented Approach” by V.K. Krishnan-Prentice Halls of India Publishers 3. “ Functional Analysis” by B.V. Limaye New age International Publishers
AM 303 : DIFFERENTIAL GEOMETRY (Common with paper MA 304 of Branch I(A) Mathematics) UNIT –I : The Theory Space Curves: Introductory remarks about space curves –Definitions – Arc length-Tangent, normal, and binormal –Curvature and torsion of a curve given as the intersection of two surfaces –Contact between curves and surfaces-Tangent surface, involutes and evolutes.( Sections 1 to 7 of Chapter 1). UNIT – II: The Metric: Local Intrinsic Properties of a Surface: Definitions of a Surface- Curves on a surface-Surfaces of revolution –Helicoids- Metric-Direction of Curves –Isometric correspondence –Intrinsic properties. ( Sections 1 to 9 of Chapter 11). UNIT – III: Geodesics-Canonical Geodesic Equations-Normal Property of geodesics –Existence theorems-Geodesic parallels-Geodesic curvature-Gauss-Bonnet theorems –Gaussian curvature surfaces of constant curvature –conformal mapping-Geodesic mapping (Sections 10 to 20 of Chapter 11). UNIT-IV: The second Fundamental Form: Local non – intrinsic properties of a surface: The second fundamental form-principal curvatures –Lines of curvature -Developables associated with space curves-Developables associated with curves on surfaces – Minimal surfaces-Ruled surfaces-The fundamental equations of surface theory – Parallel surfaces. ( Sections 1 to 10 Chapter III). Scope and Standard as in Sections and chapters as specified above of the book “ An Introduction to Differential Geometry" of T.J Willmore, Oxford University Press, Thirteenth Impression, 1997. References : 1. A first course in Differential Geometry- D. Soma sundaram – Narosa Publications Families
AM 305 (A) CLASSICAL MECHANICS (IE) (Common with paper MA 305 of Branch I(A) Mathematics) UNIT –I : D’Alembert’s Principle and Lagrange’s Equations: Some Definitions-Classification of Dynamical System-Some Examples of Constraints Virtual Displacement-Principle of Virtual Work –Generalised Force in Holonomic System-Mathematical Expression for the principle of Virtual work-D Alembert’s principle-Lagrange’s Equations for a Holonomic system-Velocity-dependent potential –Lagrange’s Equations of Motion for conservative , Non-holonomic system-physcial Significance of 1 –Harmonic Oscillator. UNIT – II: Variational Principle and Lagrange’s Equations: Variational Principle-Calculus of Variations-Hamilton Principle-Derivation of Hamilton’s Principle from Lagrange’s Equations-Derivation of Lagrange’s Equations from Hamilton’s Principle –Extension of Hamilton’s Principle –Hamilton’s Principle for Non-conservative, Non-holonomic System –Generlised Force in Dynamic system-Hamilton Principle for holonomic System -Lagrange’s Equations for Non –conservative –Holonomic System –Cyclic or ignorable Coordinates –Conservation Theorem-Conservation of Linear Momentum in Lagrangian Formulation-Conservation of Angular Momentum in Lagrangian Formulation –Conservation of Angular Momentum –Conservation of Energy in Lagrangian Formulation. UNIT – III: Hamilton’s Equations of Motion: Derivation of Hamilton’s Equations of Motion (using Lagrange’s Equations)-Routh’s Procedure-Equations of Motion-Derivation of Hamilton’s Equations from Hamilton’s Principle –Principle of Least between Hamilton’s Principle and Principle of Least Action.