Department of


Syllabus for

1 Semester  2020  Batch  
Paper Code 
Paper 
Hours Per Week 
Credits 
Marks 
MTH131  REAL ANALYSIS  4  4  100 
MTH132  ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS  4  4  100 
MTH133  ADVANCED ALGEBRA  4  4  100 
MTH134  FLUID MECHANICS  4  4  100 
MTH135  ELEMENTARY GRAPH THEORY  4  4  100 
MTH151  INTRODUCTION TO FOSS TOOLS  3  3  50 
2 Semester  2020  Batch  
Paper Code 
Paper 
Hours Per Week 
Credits 
Marks 
MTH231  GENERAL TOPOLOGY  4  4  100 
MTH232  COMPLEX ANALYSIS  4  4  100 
MTH233  LINEAR ALGEBRA  4  4  100 
MTH234  ADVANCED FLUID MECHANICS  4  4  100 
MTH235  ALGORITHMIC GRAPH THEORY  4  4  100 
MTH251  PYTHON PROGRAMMING FOR MATHEMATICS  3  3  50 
3 Semester  2019  Batch  
Paper Code 
Paper 
Hours Per Week 
Credits 
Marks 
MTH331  MEASURE THEORY AND LEBESGUE INTEGRATION  4  4  100 
MTH332  NUMERICAL ANALYSIS  4  4  100 
MTH333  CLASSICAL MECHANICS  4  4  100 
MTH334  LINEAR ALGEBRA  4  4  100 
MTH335  ADVANCED GRAPH THEORY  4  4  100 
MTH351  NUMERICAL METHODS USING PYTHON  3  3  50 
MTH381  INTERNSHIP  0  2  0 
4 Semester  2019  Batch  
Paper Code 
Paper 
Hours Per Week 
Credits 
Marks 
MTH431  DIFFERENTIAL GEOMETRY  4  4  100 
MTH432  COMPUTATIONAL FLUID DYNAMICS  4  4  100 
MTH433  FUNCTIONAL ANALYSIS  4  4  100 
MTH4401  CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS  4  4  100 
MTH4402  MAGNETOHYDRODYNAMICS  4  4  100 
MTH4403  WAVELET THEORY  4  4  100 
MTH4404  MATHEMATICAL MODELLING  4  4  100 
MTH4405  ATMOSPHERIC SCIENCE  4  4  100 
MTH4406  ADVANCED LINEAR PROGRAMMING  4  4  100 
MTH4407  DESIGN AND ANALYSIS OF ALGORITHMS  4  4  100 
MTH4408  INTRODUCTION OF THEORY OF MATROIDS  4  4  100 
MTH4409  TOPOLOGICAL GRAPH THEORY  4  4  100 
MTH4410  ALGEBRAIC GRAPH THEORY  4  4  100 
MTH451  NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON  3  3  50 
MTH481  PROJECT  2  2  100 
 
Assesment Pattern  
Assessment Pattern
 
Examination And Assesments  
EXAMINATION AND ASSESSMENTS (Theory)
EXAMINATION AND ASSESSMENTS (Practicals) The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
 
Department Overview:  
Department of Mathematics, CHRIST (Deemed to be University) is one of the oldest departments of the University. It offers programmes in Mathematics at the under graduate level, post graduate level as well as M.Phil and Ph.D. The department aims to
* enhance the logical, reasoning, analytical and problem solving skills of students.
* cultivate a research culture in young minds.
* foster aesthetic appreciation for mathematical thinking.
* encourage students for pursuing higher studies in mathematics.
* motivate students to uphold scientific integrity and objectivity in professional endeavors.  
Mission Statement:  
Vision
Excellence and Service
Mission
To organize, connect, create and communicate mathematical ideas effectively, through 4D?s; Dedication, Discipline, Direction and Determination  
Introduction to Program:  
The M.Sc. course in Mathematics aims at developing mathematical ability in students with acute and abstract reasoning. The course will enable students to cultivate a mathematician?s habit of thought and reasoning and will enlighten students with mathematical ideas relevant for oneself and for the course itself.
COURSE DESIGN: Masters in Mathematics is a two year programme spreading over four semesters. In the first two semesters focus is on the basic courses in mathematics such as Algebra, Topology, Analysis, Discrete Mathematics and Number Theory/Cryptography along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and skillbased courses including Measure Theory and Lebesgue Integration, Functional Analysis, Computational Fluid Dynamics, Advanced Graph Theory. Classical Mechanics, Linear Algebra and Numerical Analysis. Important feature of the curriculum is that one course on the topic Fluid Mechanics and Graph Theory is offered in each semester with a project on these topics in the fourth semester, which will help the students to pursue the higher studies in these topics. To gain proficiency in software skills, Mathematics Lab papers are introduced in each semester. Special importance is given to the skill enhancement courses Teaching Technology and Research Methodology in Mathematics and service learning, Introduction to Free and OpenSource Software (FOSS) T  
Program Objective:  
Programme Objective:
POBJ1. To provide learners with an indepth knowledge, abilities and insight in different topics in Mathematics.
POBJ2. To learn to apply mathematics to practical problems and help in problem solving.
POBJ3. To encourage collaborative learning through projects and research activities so that they can pursue research programmes.
POBJ4. To provide a platform for the learner to engage in various academic activities independently or in a group.
POBJ5. To make the learner familiar with FOSS tools such as Python, MAXIMA, Scilab and with the tool Mathematica.
POBJ6. To make the learner familiar with the Teaching Technology and Research Methodology in Mathematics.
Programme Outcomes:
PO1. To be able to explain mathematical principle to formulate, model and hence find solutions to practical problems.
PO2. To be able explain the advantages, limitations, importance of mathematics and its techniques to solve real life problems.
PO3. To acquire the skills which are necessary to do research/higher studies in the areas of the learner?s choice.
PO4. To be capable in formulating and analysis of mathematical models of practical problems.
PO5. To acquire skills to use mathematical FOSS tools efficiently in practical problems.
PO6. The learner will be able to become a good teacher or researcher in mathematics so that he/she can communicate the subject with others in an efficient way.  
MTH131  REAL ANALYSIS (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 
Max Marks:100 
Credits:4 
Course Objectives/Course Description 

Course Description: This course will help students to understand the concepts of functions of single and several variables. This course includes such concepts as RiemannStieltjes integral, sequences and series of functions, Special Functions, and the Implicit Function Theorem.
Course objectives: This course will help the learner to COBJ1. Develop in a rigorous and selfcontained manner the elements of real variable functions COBJ2. Integrate functions of a real variable in the sense of Riemann – Stieltjes COBJ3. Classify sequences and series of functions which are pointwise convergent and uniform Convergent COBJ4. Demonstrate the ability to manipulate and use of special functions COBJ5. Use and operate functions of several variables. 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. determine the RiemannStieltjes integrability of a bounded function. CO2. recognize the difference between pointwise and uniform convergence of sequence/series of functions. CO3. illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability. CO4. analyze and interpret the special functions such as exponential, logarithmic, trigonometric and Gamma functions. CO5. gain in depth knowledge on functions of several variables and the use of Implicit Function Theorem. 
UNIT 1 
Teaching Hours:15 

The RiemannStieltjes Integration


Definition and Existence of RiemannStieltjes Integral, Linearity Properties of RiemannStieltjes Integral, The RiemannStieltjes Integral as the Limit of Sums, Integration and Differentiation, Integration of Vectorvalued Functions, Rectifiable Curves.  
UNIT 2 
Teaching Hours:15 

Sequences and Series of Functions


Pointwise and uniform convergence, Uniform Convergence: Continuity, Integration and Differentiation, Equicontinuous Families of Functions, The StoneWeierstrass Theorem  
UNIT 3 
Teaching Hours:15 

Some Special Functions


Power Series, The Exponential and Logarithmic Functions, The Trigonometric Functions, The Algebraic Completeness of the Complex Field, Fourier Series, The Gamma Function  
UNIT 4 
Teaching Hours:15 

Functions of Several Variables


Linear Transformations, Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem, The Rank Theorem, Determinants, Derivatives of Higher Order, Differentiation of Integrals  
Text Books And Reference Books: W. Rudin, Principles of Mathematical Analysis, 3rd ed., New Delhi: McGrawHill (India), 2016.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH132  ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description : This helps students understand the beauty of the important branch of mathematics, namely, differential equations. This course includes a study of second order linear differential equations, adjoint and selfadjoint equations, Eigen values and Eigen vectors of the equations, power series method for solving differential equations, second order partial differential equations like wave equation, heat equation, Laplace equations and their solutions by Eigen function method. Course objectives : This course will help the learner to COBJ1. Solve adjoint differential equations, hypergeometric differential equation and power series. COBJ2. Solve partial differential equation of the type heat equation, wave equation and Laplace equations. COBJ3. Also solving initial boundary value problems. 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Understand concept of Linear differential equation, Fundamental set Wronskian. CO2. Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Lagrange's Identity, Green’s formula, Eigen value and Eigen functions. CO3. Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial. CO4. Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod. CO5. Demonstrate on the canonical form of second order PDE. CO6. Demonstrate initial value boundary problem for homogeneous and nonhomogeneous PDE. CO7. Demonstrate on boundary value problem by Dirichlet and Neumann problem. 
UNIT 1 
Teaching Hours:20 

Linear Differential Equations


Linear differential equations, fundamental sets of solutions, Wronskian, Liouville’s theorem, adjoint and selfadjoint equations, Lagrange identity, Green’s formula, zeros of solutions, comparison and separation theorems. Legendre, Bessel's, Chebeshev's, Eigenvalues and Eigenfunctions, related examples.  
UNIT 2 
Teaching Hours:10 

Power series solutions


Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, hypergeometric differential equation and its polynomial solutions, standard properties.  
UNIT 3 
Teaching Hours:15 

Partial Differential Equations


Formation of PDE, solutions of first and second order PDE, mathematical models representing stretched string, vibrating membrane, heat conduction in solids and the gravitational potentials, secondorder equations in two independent variables, canonical forms and general solution  
UNIT 4 
Teaching Hours:15 

Solutions of PDE


The Cauchy problem for homogeneous wave equation, D’Alembert’s solution, domain of influence and domain of dependence, the Cauchy problem for nonhomogeneous wave equation, the method of separation of variables for the onedimensional wave equation and heat equation. Boundary value problems, Dirichlet and Neumann problems in Cartesian coordinates, solution by the method of separation of variables. Solution by the method of eigenfunctions  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH133  ADVANCED ALGEBRA (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory. Course objectives: This course will help the learner to COBJ1. enhance the knowledge on advanced level algebra COBJ2. understand the proof techniques for the theorems on advanced group theory, Rings and Galois Theory 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Demonstrate knowledge of conjugates, the Class Equation and Sylow theorems CO2. Demonstrate knowledge of polynomial rings and associated properties CO3. Derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals CO4. Demonstrate the characteristic of a field and the prime subfield; CO5. Demonstrate Factorization and ideal theory in the polynomial ring; the structure of a primitive polynomials; Field extensions and characterization of finite normal extensions as splitting fields; The structure and construction of finite fields; Radical field extensions;Galois group and Galois theory 
Unit1 
Teaching Hours:15 

Advanced Group Theory


Automorphisms, Cayley’s theorem, Cauchy’s theorem, permutation groups, symmetric groups, alternating groups, simple groups, conjugate elements and class equations of finite groups, Sylow theorems, direct products, finite abelian groups, solvable groups.  
Unit2 
Teaching Hours:15 

Rings


Euclidean Ring, Polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings.  
Unit3 
Teaching Hours:15 

Fields


Extension fields, roots of polynomials, construction with straightedge and compass, more about roots.  
Unit4 
Teaching Hours:15 

Galois theory


The elements of Galois theory, solvability by radicals, Galois group over the rationals, finite fields  
Text Books And Reference Books: I. N. Herstein, Topics in algebra, Second Edition, John Wiley and Sons, 2007.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH134  FLUID MECHANICS (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course aims at introducing the fundamentals of fluid mechanics. This course aims at imparting the knowledge on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of nonNewtonian fluids.
Course objectives: This course will help the learner to COBJ1. understand the basic concept of tensors and their representative COBJ2. physics and mathematics behind the basics of fluid mechanics COBJ3. familiar with two or three dimensional incompressible flows COBJ4. classifications of nonNewtonian fluids COBJ5. familiar with standard two or three dimensional viscous flows 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem. CO2. able to understand the basics laws of Fluid mechanics and their physical interpretations. CO3. able to understand two or three dimension flows incompressible flows. CO4. able to understand the viscous flows, their mathematical modelling and physical interpretations. 
Unit1 
Teaching Hours:15 

Cartesian tensors and continuum hypothesis


Cartesian tensors: Cartesian tensors, basic properties, transpose, symmetric and skew symmetric tensors, gradient, divergence and curl in tensor calculus, integral theorems. Continuum hypothesis: deformation gradient, strain tensors, infinitesimal strain, compatibility relations, principal strains, material and local time derivatives, transport formulas, stream lines, path lines.  
Unit2 
Teaching Hours:20 

Stress, Strain and basic physical laws


Stress and Strain: stress components and stress tensor, normal and shear stresses, principal stresses, transformation of the rate of strain and stress, relation between stress and rate of strain. Fundamental basic physical laws: The equation of continuity, conservation of mass, equation of motion (NavierStokes equations), conservation of momentum, the energy equation, conservation of energy.  
Unit3 
Teaching Hours:15 

One, Two and Three Dimensional Invisid Incompressible Flow


Bernoulli equation, applications of Bernoulli equation, circulation theorems, circulation concept, Kelvin’s theorem, constancy of circulation, Laplace equations, stream functions in two and three dimensional motion. Two dimensional flow: Rectilinear flow, source and sink, the theorem of Blasius.  
Unit4 
Teaching Hours:10 

Two Dimensional Flows of Viscous Fluid


Flow between parallel flat plates, Couette flow, plane Poiseuille flow, the HagenPoiseuille flow, flow between two concentric rotating cylinders  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern Examination and Assessments
 
MTH135  ELEMENTARY GRAPH THEORY (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course is an introductory course to the basic concepts of Graph Theory. This includes definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths. Course objectives:This course will help the learner to COBJ1. know the history and development of graph theory COBJ2. understand all the elementary concepts and proof techniques in Graph Theory 

Learning Outcome 

Course outcomes: The successful completion of this course the students will be able to: CO1. write precise and accurate mathematical definitions of basics concepts in graph theory CO3. provide appropriate examples and counterexamples to illustrate the basic concepts CO3. demonstrate and apply various proof techniques in proving theorems in graph theory CO4. showcase mastery in using graph drawing tools 
Unit1 
Teaching Hours:15 

Introduction to Graphs


Definition and introductory concepts, Graphs as Models, Matrices and Isomorphism, Decomposition and Special Graphs, Connection in Graphs, Bipartite Graphs, Eulerian Circuits.  
Unit2 
Teaching Hours:15 

Vertex Degrees and Directed Graphs


Counting and Bijections, Extremal Problems, Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and Tournaments.  
Unit3 
Teaching Hours:15 

Trees and Distance


Properties of Trees, Distance in Trees and Graphs, Enumeration of Trees, Spanning Trees in Graphs, Decomposition and Graceful Labellings, Minimum Spanning Tree, Shortest Paths.  
Unit4 
Teaching Hours:15 

Connectivity and Paths


Connectivity, Edge  Connectivity, Blocks, 2  connected Graphs, Connectivity in Digraphs, k  connected and kedgeconnected Graphs, Maximum Network Flow, Integral Flows.  
Text Books And Reference Books: D.B. West, Introduction to Graph Theory, New Delhi: PrenticeHall of India, 2011.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH151  INTRODUCTION TO FOSS TOOLS (2020 Batch)  
Total Teaching Hours for Semester:45 
No of Lecture Hours/Week:3 

Max Marks:50 
Credits:3 

Course Objectives/Course Description 

Course Description: This course aims at introducing the mathematical software packages “WxMaxima” and “Scilab”, for learning basic operations on matrix manipulation, plotting graphs etc.,. These software packages will also help students to solve problems / applied problems on Mathematics. Course objectives: This course will help the learner to: COBJ1. use the FOSS tool WxMaxima effectively. COBJ2. use the FOSS tool Scilab effectively 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Use the basic commands in WxMaxima including the 2D and 3D plots CO2. Have a strong command on the inbuilt commands required for the learning and analyzing mathematics CO3. Solve problems / applied problems on mathematics by using Scilab. 
Unit1 
Teaching Hours:15 

Introduction to WxMaxima


Introduction to WxMaxima Interface  Maxima expressions, numbers, operators, constants and reserved words  input and output in WxMaxima  2D and 3D plots in WxMaxima  symbolic computations in WxMaxima  Solving Ordinary differential equations in WxMaxima.  
Unit2 
Teaching Hours:15 

Introduction to Scilab


Introduction to Scilab and commands connected with Matrices  Computations with Matrices  2D Plots: plot, plot2d, plot2d2, plot2d3, Histplot, Matplot, Grayplot, 3D Plots: plot3d,, plot3d1, contour, hist3d Script Files and Function Files  
Unit3 
Teaching Hours:15 

Solving problems using Scilab / WxMaxima


Solving systems of equation and explain consistence  Find the values of some standard trigonometric functions in radians as well as in degree  Create polynomials of different degrees and find its real roots  Display Fibonacci series  Display nonFibonacci series  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
 
MTH231  GENERAL TOPOLOGY (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc.,. Course objectives: This course will help the learner to: COBJ1. provide precise definitions and appropriate examples and counter examples of fundamental concepts in general topology COBJ2. acquire knowledge about generalization of the concept of continuity and related properties COBJ3. appreciate the beauty of deep mathematical results such as Uryzohn’s lemma and understand and apply various proof techniques 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. define topological spaces, give examples and counterexamples on concepts like open sets, basis and subspaces CO2. establish equivalent definitions of continuity and apply the same in proving theorems CO3. understand the concepts of metrizability, connectedness, compactness and learn the related theorems 
Unit1 
Teaching Hours:15 

Topological Spaces


Elements of topological spaces, basis for a topology, the order topology, the product topology on X x Y, the subspace topology, Closed sets and limit points.  
Unit2 
Teaching Hours:15 

Continuous Functions


Continuous functions, the product topology, metric topology.  
Unit3 
Teaching Hours:15 

Connectedness and Compactness


Connected spaces, connected subspaces of the Real Line, components and local connectedness, compact spaces, Compact Subspaces of the Real Line, limit point compactness, local compactness.  
Unit4 
Teaching Hours:15 

Countability and Separation Axioms


The countability axioms, the separation axioms, normal spaces, the Urysohn lemma, the Urysohn metrization theorem, Tietze extension theorem.  
Text Books And Reference Books: J.R. Munkres,Topology, Second Edition, Prentice Hall of India, 2007.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH232  COMPLEX ANALYSIS (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course will help students learn about the essentials of complex analysis. This course includes important concepts such as power series, analytic functions, linear transformations, Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem, argument principle, Schwarz lemma , Rouche’s theorem and Hadamard’s 3circles theorem. Course objectives: This course will help the learner to COBJ1. enhance the understanding the advanced concepts in Complex Analysis COBJ2. Acquire problem solving skills in Complex Analysis. 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. apply the concept and consequences of analyticity and the CauchyRiemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra CO2. compute complex contour integrals in several ways: directly using parameterization, using the CauchyGoursat theorem Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula CO3. represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem CO4. use conformal mappings and know about meromorphic functions 
Unit1 
Teaching Hours:18 

Power Series


Power series, radius and circle of convergence, power series and analytic functions, Line and contour integration, Cauchy’s theorem, Cauchy integral formula, Cauchy integral formula for derivatives, Cauchy integral formula for multiply connected domains, Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus principles..  
Unit2 
Teaching Hours:15 

Singularities


Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles  
Unit3 
Teaching Hours:15 

Mappings


Rational functions, behavior of functions in the neighborhood of an essential singularity, Cauchy’s residue theorem, contour integration problems, Mobius transformation, conformal mappings.  
Unit4 
Teaching Hours:12 

Meromorphic functions


Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3circles theorem.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH233  LINEAR ALGEBRA (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms. Course Objective: This course will help learner to COBJ1. gain proficiency on the theories of Linear Algebra COBJ2. enhance problem solving skills in Linear Algebra 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Have thorough understanding of the Linear transformations CO2.Demonstrate the elementary canonical forms, rational and Jordan forms. CO3. Apply the inner product space CO4. Express familiarity in using bilinear forms 
Unit1 
Teaching Hours:15 

Linear Transformations and Determinants


Vector Spaces: Recapitulation, Linear Transformations: Algebra of Linear Transformations  Isomorphism – Representation of Transformation by Matrices – Linear Functionals – The transpose of a Linear Transformation, Determinants: Commutative Rings – Determinant Functions – Permutation and the Uniqueness of Determinants – Additional Properties of Determinants  
Unit2 
Teaching Hours:20 

Elementary Canonical Forms, Rational and Jordan Forms


Elementary Canonical Forms : Characteristic Values – Annihilating Polynomials – Invariant Subspaces – Simultaneous Triangulation and Diagonalization – Direct sum Decomposition – Invariant Dual Sums  The Primary Decomposition Theorem. The Rational and Jordan Forms: Cyclic subspaces and Annihilators – Cyclic Decompositions and the Rational Form – The Jordan Form – Computation of Invariant Factors – SemiSimple Operators.  
Unit3 
Teaching Hours:15 

Inner Product Spaces


Inner Products – Inner Product Spaces – Linear Functionals and Adjoints – Unitary Operators – Normal Operators – Forms on Inner Product Spaces – Positive Forms – Spectral Theory – Properties of Normal Operators.  
Unit4 
Teaching Hours:10 

Bilinear Forms


Bilinear Forms – Symmetric Bilinear Forms – SkewSymmetric Bilinear Forms – Groups Preserving Bilinear Forms.  
Text Books And Reference Books: K. Hoffman and R. Kunze, Linear Algebra, 2nd ed. New Delhi, India: PHI Learning Private Limited, 2011.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH234  ADVANCED FLUID MECHANICS (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and nonlinear problems. This course also includes the analysis Prandtlboundry layer, porous media and NonNewtonian fluid. Course objectives: This course will help the learner to COBJ1. understand the different modes of heat transfer and their applications. COBJ2. understand the importance of doing the nondimensionalization of basic equations. COBJ3. understand the boundary layer flows. COBJ4. familiarity with porous medium and nonNewtonian fluids 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. understand the basic laws of heat transfer and understand the fundamentals of convective heat transfer process. CO2. solve Rayleigh  Benard problem and their physical phenomenon. CO3. solve and understand different boundary layer problems CO4. give an introduction to the basic equations with porous medium and solution methods for mathematical modeling of viscous fluids and elastic matter 
UNIT 1 
Teaching Hours:15 

Dimensional Analysis and Similarity


Introduction to heat transfer, different modes of heat transfer conduction, convection and radiation, steady and unsteady heat transfer, free and forced convection. Nondimensional parameters determined from differential equations – Buckingham’s Pi Theorem – Nondimensionalization of the Basic Equations  Nondimensional parameters and dynamic similarity. .  
UNIT 2 
Teaching Hours:20 

Heat Transfer and Thermal Instability


Shear Instability: Stability of flow between parallel shear flows  Squire’s theorem for viscous and inviscid theory – Rayleigh stability equation – Derivation of OrrSommerfeld equation assuming that the basic flow is strictly parallel. Basic concepts of stability theory – Linear and Nonlinear theories – Rayleigh Benard Problem – Analysis into normal modes – Principle of Exchange of stabilities – first variation principle – Different boundary conditions on velocity and temperature.  
UNIT 3 
Teaching Hours:10 

Prandtl Boundry Layer


Boundary layer concept, the boundary layer equations in twodimensional flow, the boundary layer along a flat plate, the Blasius solution. Stagnation point flow. FalknerSkan family of equations.  
UNIT 4 
Teaching Hours:15 

Porous Media and Non  Newtonian Fluids


Introduction to porous medium, porosity, Darcy’s Law, Extension of Darcy Law – accelerations and inertial effects, Brinkman’s equation, effects of porosity variations, Bidisperse porous media. Constitutive equations of Maxwell, Oldroyd, Ostwald , Ostwald de waele, Reiner – Rivlin and Micropolar fluid. Weissenberg effect and Tom’s effect.Equation of continuity, Conservation of momentum for nonNewtonian fluids..  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH235  ALGORITHMIC GRAPH THEORY (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description: This course helps the students to understand the colouring of graphs, Planar graphs, edges and cycles. Course objectives: This course will help the learner to COBJ1. Construct examples and proofs pertaining to the basic theorems COBJ2. Apply the theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory COBJ3. Write graph theoretic ideas in a coherent and technically accurate manner. 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. understand the basic concepts and fundamental results in matching, domination, coloring and planarity. CO2. reason from definitions to construct mathematical proofs CO3. obtain a solid overview of the questions addressed by graph theory and will be exposed to emerging areas of research 
Unit1 
Teaching Hours:15 

Colouring of Graphs


Definition and Examples of Graph Colouring, Upper Bounds, Brooks’ Theorem, Graph with Large Chromatic Number, Extremal Problems and Turan’s Theorem, ColourCritical Graphs, Counting Proper Colourings.  
Unit2 
Teaching Hours:15 

Matchings and Factors


Maximum Matchings, Hall’s Matching Condition, MinMax Theorem, Independent Sets and Covers, Maximum Bipartite Matching, Weighted Bipartite Matching, Tutte’s 1factor Theorem, Domination.  
Unit3 
Teaching Hours:15 

Planar Graphs


Drawings in the Plane, Dual Graphs, Euler’s Formula, Kuratowski’s Theorem, Convex Embeddings, Coloring of Planar Graphs, Thickness and Crossing Number  
Unit4 
Teaching Hours:15 

Edges and Cycles Edge


Colourings, Characterisation of Line Graphs, Necessary Conditions of Hamiltonian Cycles, Sufficient Conditions of Hamiltonian Cycles, Cycles in Directed Graphs, Tait’s Theorem, Grinberg’s Theorem, Flows and Cycle Covers  
Text Books And Reference Books: D.B. West, Introduction to Graph Theory, New Delhi: PrenticeHall of India, 2011.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH251  PYTHON PROGRAMMING FOR MATHEMATICS (2020 Batch)  
Total Teaching Hours for Semester:45 
No of Lecture Hours/Week:3 

Max Marks:50 
Credits:3 

Course Objectives/Course Description 

Course description: This course aims at introducing the programming language Python and its uses in solving problems on discrete mathematics and differential equations. Course objectives: This course will help the learner to COBJ1.Acquire skill in usage of suitable functions/packages of Python to solve mathematical problems. COBJ2.Gain proficiency in using Python to solve problems on Differential equations. COBJ3. The built in functions required to deal withcreating and visualizing Graphs, Digraphs, MultiGraph. 

Learning Outcome 

By the end of the course the learner will be able to: CO1. Acquire proficiency in using different functions of Python to compute solutions of basic mathematical problems. CO2. Demonstrate the use of Python to solve differential equations along with visualize the solutions. CO3. Be familiar with the builtin functions to deal with Graphs and Digraphs. 
Unit1 
Teaching Hours:15 

Introduction to Python Programming


Python commands: Comments, Number and other data types, Expressions, Operators, Variables and assignments, Decisions, Loops, Lists, Strings  plotting using “matplotlib”  Basic operations , Simplification, Calculus, Solvers and Matrices using Sympy.  
Unit2 
Teaching Hours:15 

Differential Equations using Python


Solving ODE’s using Python  Libraries for Differential equations in Python, PDE’s using sympy user functions pde_seperate(), pde_seperate_add(). pde_seperate_mul(), pdsolve(), classify_pde(), checkpdesol(), pde_1st_linear_constant_coeff_homogeneous, pde_1st_linear_constant_coeff, pde_1st_linear_variable_coeff.  
Unit3 
Teaching Hours:15 

Discrete Mathematics using Python


Creating and visualizing Graphs, Digraphs, MultiGraphs and MultiDiGraph  Python methods for reporting nodes, edges and neighbours of the given graph / digraph  Python methods for counting nodes, edges and neighbours of the given graph / digraph.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
 
MTH331  MEASURE THEORY AND LEBESGUE INTEGRATION (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description: The Coursecovers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue. Course objectives: This course will help the learner to COBJ1. enhance the understanding of the advanced notions from Mathematical Analysis
COBJ2. know more about the Measure theory and Lebesgue Integration 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. understand the fundamental concepts of Mathematical Analysis. CO2. tate some of the classical theorems in of Advanced Real Analysis. CO3. be familiar with measurable sets and functions. CO4. integrate a measurable function CO5. understand the properties of Lp Spaces

Unit1 
Teaching Hours:20 

Lebesgue Measure


Lebesgue Outer Measure, The sAlgebra of Lebesgue Measurable Sets, Outer and Inner Approximation of Lebesgue Measurable Sets, Countable Additivity, Continuity and the BorelCantelli Lemma, Nonmeasurable Sets, The Cantor Set and the CantonLebesgue Function, Sums, Products and Compositions of Lebesgue Measurable Functions, Sequential Pointwise Limits and Simple Approximation, Littlewood’s three principles, Egoroff’s Theorem and Lusin’s Theorem.  
Unit2 
Teaching Hours:15 

The Lebesgue Integration


The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure, The Lebesgue Integral of a Measurable Nonnegative Function; The General Lebesgue Integral; Countable Additivity and Continuity of Integration, Uniform Integrability, Uniform Integrability and Tightness, Convergence in measure, Characterizations of Riemann and Lebesgue Integrability.  
Unit3 
Teaching Hours:15 

Differentiation and Lebesgue Integration


Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives.  
Unit4 
Teaching Hours:10 

The Lp Spaces


Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The L^{p} spaces, Approximation and Separability, The Riesz Representation for the Dual of L^{p}, Weak Sequential Convergence in L^{p}, Weak Sequential Compactness, The Minimization of Convex Functionals.  
Text Books And Reference Books: H.L. Royden and P.M. Fitzpatrick, “Real Analysis,” 4th ed. New Jersey: Pearson Education Inc., 2013.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH332  NUMERICAL ANALYSIS (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description: This course deals with the theory and application of various advanced methods of numerical approximation. These methods or techniques help us to approximate the solutions of problems that arise in science and engineering. The emphasis of the course will be the thorough study of numerical algorithms to understand the guaranteed accuracy that various methods provide, the efficiency and scalability for large scale systems and issues of stability. Course objectives: This course will help the learner
COBJ1. to develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use. 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Derive numerical methods for approximating the solution of problems of algebraic and transcendental equations, ordinary differential equations and boundary value problems. CO2. Implement a variety of numerical algorithms appropriately in various situations CO3. interpret, analyse and evaluate results from numerical computations 
Unit1 
Teaching Hours:20 

Solution of algebraic and transcendental equations


Fixed point iterative method, convergence criterion, Aitken’s process, Sturm sequence method to identify the number of real roots, NewtonRaphson methods (includes the convergence criterion for simple roots), Bairstow’s method, Graeffe’s root squaring method, BirgeVieta method, Muller’s method. Solution of Linear System of Algebraic Equations: LUdecomposition methods (Crout’s, Choleky and Delittle methods), consistency and illconditioned system of equations, Tridiagonal system of equations, Thomas algorithm.  
Unit2 
Teaching Hours:15 

Interpolation and Numerical Integration


Lagrange, Hermite, Cubicspline’s (Natural, Not a Knot and Clamped)  with uniqueness and error term, for polynomial interpolation. Chebychev and Rational function approximation. Gaussian quadrature, GaussLegendre, GaussChebychev formulas.  
Unit3 
Teaching Hours:15 

Numerical solution of ordinary differential equations


Initial value problems, RungeKutta methods of second and fourth order, multistep method, AdamsMoulton method, stability (convergence and truncation error for the above methods), boundary value problems, second order finite difference method.  
Unit4 
Teaching Hours:10 

Boundary Value Problems


Numerical solutions of second order boundary value problems (BVP) of first, second and third types by shooting method, RayleighRitz Method, Gelarkin Method.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH333  CLASSICAL MECHANICS (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description: Classical Mechanics is the study of mechanics using Mathematical methods. This course deals with some of the key ideas of classical mechanics. The concepts covered in the course include generalized coordinates, Lagrange’s equations, Hamilton’s equations and Hamilton  Jacobi theory. Course objectives: This course will help the learner to
COBJ1. derive necessary equations of motions based on the chosen configuration space. 

Learning Outcome 

On successful completion of the course, the students should be able to:
CO1. Interpret mechanics through the configuration space. 
Unit1 
Teaching Hours:12 

Introductory concepts


The mechanical system  Generalised Coordinates  constraints  virtual work  Energy and momentum.  
Unit2 
Teaching Hours:20 

Lagrange's equation


Derivation and examples  Integrals of the Motion  Small oscillations. Special Applications of Lagrange’s Equations: Rayleigh’s dissipation function  impulsive motion  velocity dependent potentials.  
Unit3 
Teaching Hours:13 

Hamilton's equations


Hamilton's principle  Hamilton’s equations  Other variational principles  phase space.  
Unit4 
Teaching Hours:15 

Hamilton  Jacobi Theory


Hamilton's Principal Function – The Hamilton  Jacobi equation  Separability.  
Text Books And Reference Books: Donald T. Greenwood, Classical Dynamics, Reprint, USA: Dover Publications, 2012.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH334  LINEAR ALGEBRA (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms. Course Objective: This course will help learner to COBJ1. gain proficiency on the theories of Linear Algebra COBJ2. enhance problem solving skills in Linear Algebra 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Have thorough understanding of the Linear transformations CO2. Demonstrate the elementary canonical forms, rational and Jordan forms. CO3. Apply the inner product space CO4. Express familiarity in using bilinear forms 
Unit1 
Teaching Hours:15 

Linear Transformations and Determinants


Vector Spaces: Recapitulation, Linear Transformations: Algebra of Linear Transformations  Isomorphism – Representation of Transformation by Matrices – Linear Functionals – The transpose of a Linear Transformation, Determinants: Commutative Rings – Determinant Functions – Permutation and the Uniqueness of Determinants – Additional Properties of Determinants  
Unit2 
Teaching Hours:20 

Elementary Canonical Forms, Rational and Jordan Forms


Elementary Canonical Forms : Characteristic Values – Annihilating Polynomials – Invariant Subspaces – Simultaneous Triangulation and Diagonalization – Direct sum Decomposition – Invariant Dual Sums  The Primary Decomposition Theorem. The Rational and Jordan Forms: Cyclic subspaces and Annihilators – Cyclic Decompositions and the Rational Form – The Jordan Form – Computation of Invariant Factors – SemiSimple Operators.  
Unit3 
Teaching Hours:15 

Inner Product Spaces


Inner Products – Inner Product Spaces – Linear Functionals and Adjoints – Unitary Operators – Normal Operators – Forms on Inner Product Spaces – Positive Forms – Spectral Theory – Properties of Normal Operators.  
Unit4 
Teaching Hours:10 

Bilinear Forms


Bilinear Forms – Symmetric Bilinear Forms – SkewSymmetric Bilinear Forms – Groups Preserving Bilinear Forms  
Text Books And Reference Books: K. Hoffman and R. Kunze, Linear Algebra, 2nd ed. New Delhi, India: PHI Learning Private Limited, 2011.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH335  ADVANCED GRAPH THEORY (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description:Domination of Graphs, perfect graphs, chromatic graph theory and Eigenvalues of Graphs are dealt with in the detail in this course. Course objectives: This course will help the learner to COBJ1. understand the advanced topics in Graph Theory COBJ2. enhance the understanding of techniques of writing proofs for advanced topics in Graph Theory 

Learning Outcome 

By the end of the course the learner will be able to: CO1. have thorough understanding of the concepts in domination and perfect graphs CO2. familiarity in implementing the acquired knowledge appropriately CO3. mastery in employing proof techniques 
Unit1 
Teaching Hours:15 

Domination in Graphs


Domination in Graphs, Bounds in terms of Order, Bounds in terms of Order, Degree and Packing, Bounds in terms of Order and Size, Bounds in terms of Degree, Diameter and Girth, Bounds in terms of Independence and Covering.  
Unit2 
Teaching Hours:15 

Chromatic Graph Theory


TColourings, L(2,1)colourings, Radio Colourings, Hamiltonian Colourings, Domination and Colourings.  
Unit3 
Teaching Hours:15 

Perfect Graphs


The Perfect Graph Theorem, Chordal Graphs Revisited, Other Classes of Perfect Graphs, Imperfect Graphs, The Strong Perfect Graph Conjecture  
Unit4 
Teaching Hours:15 

Eigenvalues of Graphs


The Characteristic Polynomial, Eigenvalues and Graph Parameters, Eigenvalues of Regular Graphs, Eigenvalues and Expanders, Strongly Regular Graphs  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH351  NUMERICAL METHODS USING PYTHON (2019 Batch)  
Total Teaching Hours for Semester:45 
No of Lecture Hours/Week:3 

Max Marks:50 
Credits:3 

Course Objectives/Course Description 

Course description: In this course programming Numerical Methods in Python will be focused. How to program the numerical methods step by step to create the most basic lines of code that run on the computer efficiently and output the solution at the required degree of accuracy. Course objectives: This course will help the learner to COBJ1. Program the numerical methods to create simple and efficient Python codes that output the numerical solutions at the required degree of accuracy. COBJ2. Use the plotting functions of matplotlib to visualize the results graphically. COBJ3. Acquire skill in usage of suitable functions/packages of Python to solve initial value problems numerically. 

Learning Outcome 

By the end of the course the learner will be able to: CO1. Acquire proficiency in using different functions of Python to compute solutions of system of equations. CO2. Demonstrate the use of Python to solve initial value problem numerically along with graphical visualization of the solutions . CO3. Be familiar with the builtin functions to deal with numerical methods. 
Unit1 
Teaching Hours:15 

Introduction to Python and Roots of HighDegree Equations


Introduction and Simple Iterations Method, Finite Differences Method  
Unit2 
Teaching Hours:15 

Systems of Linear Equations


Introduction & Gauss Elimination Method: Algorithm, Gauss Elimination Method, Jacobi's Method, GaussSeidel's Method, Linear System Solution in NumPy and SciPy & Summary  
Unit3 
Teaching Hours:15 

Numerical differentiation, Integration and Ordinary Differential Equations


Introduction & Euler's Method, Second Order RungeKutta's Method, Fourth Order RungeKutta's Method, Fourth Order RungeKutta's Method: Plot Numerical and Exact Solutions.  
Text Books And Reference Books: J. Kiusalaas, Numerical methods in engineering with Python 3. Cambridge University Press, 2013.  
Essential Reading / Recommended Reading Hans Fangohr, Introduction to Python for Computational Science and Engineering (A beginner’s guide), University of Southampton, 2015.  
Evaluation Pattern The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
 
MTH381  INTERNSHIP (2019 Batch)  
Total Teaching Hours for Semester:45 
No of Lecture Hours/Week:0 

Max Marks:0 
Credits:2 

Course Objectives/Course Description 

The objective of this course is to provide the students an opportunity to gain work experience in the relevant institution, connected to their subject of study. The experienced gained in the workplace will give the students a competetive edge in their career. 

Learning Outcome 

Course Outcomes On successful completion of the course, the students should be able to
COBJ1. Expose to the field of their professional interest 
Unit1 
Teaching Hours:45 
Internship in PG Mathematics course


M.Sc. Mathematics students have to undertake a mandatory internship of not less than 45 working days at any of the following: reputed research centers, recognized educational institutions, summer research fellowships, programmes like M.T.T.S or any other approved by the P.G. coordinator and H.O.D. In the present scenario of COVID 19 pandemic, the students unable to do internship in an organization, have to complete one MOOC in Mathematics that suits the academic interest of the student in consultation with the assigned internship supervisors and a dissertation based on a detailed review of two research articles. The duration of the course has to be at least 30 hours and should be completed on or before 20 June 2020. However, if a student chooses to go ahead with the internship, then they should complete at least 25 working days in the organization on or before 31 May 2020, in which case submission of the dissertation is not necessary. The internship is to be undertaken at the end of second semester (during first year vacation). The report submission and the presentation on the report will be held during the third semester and the credits will appear in the mark sheet of the third semester. The students will have to give an internship proposal with the following details: Organization where the student proposes to do the internship, reasons for the choice, nature of internship, period on internship, relevant permission letters, if available, name of the mentor in the organization, email, telephone and mobile numbers of the person in the organization with whom Christ University could communicate matters related to internship. Typed proposals will have to be given at least one month before the end of the second semester. The coordinator of the programme in consultation with the HOD will assign faculty members from the department as guides at least two weeks before the end of second semester. The students will have to be in touch with the guides during the internship period either through personal meetings, over the phone or through email. At the place of internship, students are advised to be in constant touch with their mentors. At the end of the required period of internship, the candidates will submit a report in a specified format adhering to department guidelines. The report should be submitted within the first 10 days of the reopening of the University for the third semester. The students doing the MOOCs are expected to prepare course notes on their own using all the resources accessible and this is to be given as the first part of the internship report. Within 20 days from the day of reopening, the department must hold a presentation by the students. During the presentation, the supervisor or a nominee of the supervisor should be present and be one of the evaluators. Students should preferably be encouraged to make a presentation of their report. A minimum of 10 minutes should be given for each of the presenters. The maximum limit is left to the discretion of the evaluation committee.
Students will get 2 credits on successful completion of internship.  
Text Books And Reference Books: .  
Essential Reading / Recommended Reading .  
Evaluation Pattern .  
MTH431  DIFFERENTIAL GEOMETRY (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 
Max Marks:100 
Credits:4 
Course Objectives/Course Description 

Course description: Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in threedimensional Euclidean space and will be acquainted with the ways of generalising these concepts to higher dimensions.
Course objectives: This course will help the learner to CO BJ1. write proofs for the theorems on Curves and Surfaces in R3. COBJ2. implement the properties of curves and surfaces in solving problems described in terms of tangent vectors / vector fields / forms etc., 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. express sound knowledge on the basic concepts in geometry of curves and surfaces in Euclidean space, especially E3. CO2. demonstrate mastery in solving typical problems associated with the theory. CO3. extend the knowledge in generalizing the concepts learned to higher dimensions. 
UNIT 1 
Teaching Hours:15 

Calculus on Euclidean Geometry


Euclidean Space  Tangent Vectors  Directional derivatives  Curves in E^{3}  1Forms  Differential Forms  Mappings.  
UNIT 2 
Teaching Hours:15 

Frame Fields and Euclidean Geometry


Dot product  Curves  vector field  The Frenet Formulas  Arbitrary speed curves  cylindrical helix  Covariant Derivatives  Frame fields  Connection Forms  The Structural equations.  
UNIT 3 
Teaching Hours:15 

Euclidean Geometry and Calculus on Surfaces


Isometries of E^{3}  The derivative map of an Isometry  Surfaces in E^{3}  patch computations  Differential functions and Tangent vectors  Differential forms on a surface  Mappings of Surfaces.  
UNIT 4 
Teaching Hours:15 

Shape Operators


The Shape operator of M in E^{3}  Normal Curvature  Gaussian Curvature  Computational Techniques  Special curves in a surface  Surfaces of revolution.  
Text Books And Reference Books: B.O’Neill, Elementary Differential geometry, 2^{nd} revised ed., New York: Academic Press, 2006.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH432  COMPUTATIONAL FLUID DYNAMICS (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course helps the students to learn the solutions of partial differential equations using finite difference and finite element methods. This course also helps them to know how solve the Burger’s equations using finite difference equation, quasilinearization of nonlinear equations. Course objectives: This course will help the students to COBJ1. be familiar with solving PDE using finite difference method and finite element method COBJ2. Understand the nonlinear equation Burger’s equation using finite difference method COBJ3.Understand the compressible fluid flow using ACM, PCM and SIMPLE methods COBJ4.Solve differential equations using finite element method usingdifferent shape functions 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Solve both linear and nonlinear PDE using finite difference methods CO2. Understand both physics and mathematical properties of governing NavierStokes equations and define proper boundary conditions for solution CO3. Understanding of physics of compressible and incompressible fluid flows CO4. Write the programming in MATLAB to solve PDE using finite difference method 
Unit1 
Teaching Hours:15 

Numerical solution of elliptic partial differential equations


Review of classification of partial differential equations, classification of boundary conditions, numerical analysis, basic governing equations of fluid mechanics. Difference methods for elliptic partial differential equations, difference schemes for Laplace and Poisson’s equations, iterative methods of solution by Jacobi and GaussSiedel, solution techniques for rectangular and quadrilateral regions.  
Unit2 
Teaching Hours:15 

Numerical solution of parabolic and hyperbolic partial differential equations


Difference methods for parabolic equations in onedimension, methods of Schmidt, Laasonen, CrankNicolson and DufortFrankel, stability and convergence analysis for Schmidt and CrankNicolson methods, ADI method for twodimensional parabolic equation, explicit finite difference schemes for hyperbolic equations, wave equation in one dimension.  
Unit3 
Teaching Hours:15 

Finite Difference Methods for nonlinear equations


Finite difference method to nonlinear equations, coordinate transformation for arbitrary geometry, Central schemes with combined spacetime discretizationLaxFriedrichs, LaxWendroff, MacCormack methods, Artificial compressibility method, pressure correction method – Lubrication model, convection dominated flows – Euler equation – Quasilinearization of Euler equation, Compatibility relations, nonlinear Burger equation.  
Unit4 
Teaching Hours:15 

Finite Element Methods


Introduction to finite element methods, oneand twodimensional bases functions – Lagrange and Hermite polynomials elements, triangular and rectangular elements, Finite element method for onedimensional problem and twodimensional problems: model equations, discretization, interpolation functions, evaluation of element matrices and vectors and their assemblage.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH433  FUNCTIONAL ANALYSIS (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This abstract course imparts an indepth analysis of Banach spaces, Hilbert spaces, conjugate spaces, etc. This course also includes a few important applications of functional analysis to other branches of both pure and applied mathematics. Course Objective. This course will help learner to COBJ1: know the notions behind Functional Analysis COBJ2. enhance the problem solving ability in Functional Analysis 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Explain the fundamental concepts of functional analysis. CO2. Understand the approximation of continuous functions. CO3. Understand concepts of Hilbert and Banach spaces with l2 and lp spaces serving as examples. CO4. Understand the definitions of linear functional and prove the HahnBanach theorem, open mapping theorem, uniform boundedness theorem, etc. CO5. Define linear operators, self adjoint, isometric and unitary operators on Hilbert spaces. 
Unit1 
Teaching Hours:15 

Banach spaces


Normed linear spaces, Banach spaces, continuous linear transformations, isometric isomorphisms, functionals and the HahnBanach theorem, the natural embedding of a normed linear space in its second dual.  
Unit2 
Teaching Hours:12 

Mapping theorems


The open mapping theorem and the closed graph theorem, the uniform boundedness theorem, the conjugate of an operator.  
Unit3 
Teaching Hours:15 

Inner products


Inner products, Hilbert spaces, Schwarz inequality, parallelogram law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormal sets.
 
Unit4 
Teaching Hours:18 

Conjugate space


The conjugate space, the adjoint of an operator, selfadjoint, normal and unitary operators, projections, finite dimensional spectral theory.
 
Text Books And Reference Books: G.F. Simmons, Introduction to topology and modern Analysis, Reprint, Tata McGrawHill, 2004.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH4401  CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course description: This course introduces fundamental concepts and some standard results of calculus of variations and the integral equations. It plays an important role for solving various engineering sciences problems. Therefore, it has tremendous applications in diverse fields in engineering sciences. Course objectives: This course will help the learners to study extrema of functional, the Brachistochrone problem, Euler’s equation, variational derivative and invariance of Euler’s equations. It also contains Fredholm and Volterra integral equations and their solutions using various methods such as Neumann series, resolvent kernels, Green’s function approach and transform methods 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Derive some classical differential equations by using principles of calculus of variations. CO2. Knowledge of Variational Problems, EulerLagrange Condition, Second Variation,Generalizations of the Variational Problem. CO3. find maximum or minimum of a functional using calculus of variations Technique, solve Volterra integral equations and Fredholm integral equations CO4. Reduce the differential equations to integral equations. 
Unit1 
Teaching Hours:18 

Euler equations and variational notations


Maxima and minima, method of Lagrange multipliers, the simplest case, Euler equation,extremals, stationary function, geodesics, Brachistochrone problem, natural boundary conditions and transition conditions, variational notation, the more general case.  
Unit2 
Teaching Hours:16 

Advanced variational problems


Galerkian Technique, the RayleighRitz method.  
Unit3 
Teaching Hours:12 

Linear integral equations


Definitions, integral equation, Fredholm and Volterra equations, kernel of the integral equation, integral equations of different kinds, relation between differential and integral equations, symmetric kernels, the Green’s function.  
Unit4 
Teaching Hours:14 

Methods for solutions of linear integral equations


Fredholm equations with separable kernels, homogeneous integral equations, characteristic values and characteristic functions of integral equations, HilbertSchmidt theory, iterative methods for solving integral equations of the second kind, the Neumann series.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH4402  MAGNETOHYDRODYNAMICS (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course provides the fundamentals of Magnetohydrodynamics, which include theory of Maxwell’s equations, basic equations, exact solutions and applications of classical MHD. Course objectives: This course will help the students to COBJ1.Understand mathematical form of Gauss’s Law, Faraday’s Law and Ampere’s Law and corresponding boundary conditions COBJ2. Derive the basic governing equations and boundary conditions of MHD flows. COBJ3. Finding the exact solutions of MHD governing equations. COBJ4. Understand the Alfven waves and derive their corresponding equations. 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. Derive the MHD governing equations using Faraday’s law and Ampere’s law. CO2. Solve the Fluid Mechanics problems with magnetic field. CO3.Understand the properties of force free magnetic field. CO4. Understand the application of Alfven waves, heating of solar corona, earth’s magnetic field. 
Unit1 
Teaching Hours:12 

Electrodynamics


Outline of electromagnetic units and electrostatics, derivation of Gauss law, Faraday’s law, Ampere’s law and solenoidal property, dielectric material, conservation of charges, electromagnetic boundary conditions.  
Unit2 
Teaching Hours:13 

Basic Equations


Outline of basic equations of MHD, magnetic induction equation, Lorentz force, MHD approximations, nondimensional numbers, velocity, temperature and magnetic field boundary conditions.  
Unit3 
Teaching Hours:20 

Exact Solutions


Hartmann flow, generalized Hartmann flow, velocity distribution, expression for induced current and magnetic field, temperature discribution, Hartmann couette flow, magnetostaticforce free magnetic field, abnormality parameter, Chandrashekar theorem, application of magnetostaticBennett pinch.  
Unit4 
Teaching Hours:15 

Applications


Classical MHD and Alfven waves, Alfven theorem, Frozeninphenomena, Application of Alfven waves, heating of solar corana, earth’s magnetic field, Alfven wave equation in an incompressible conducting fluid in the presence of an vertical magnetic field, solution of Alfven wave equation, Alfven wave equation in a compressible conducting nonviscous fluid, Helmholtz vorticity equation, Kelvin’s circulation theorem, Bernoulli’s equation.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading D. J. Griffiths, Introduction to electrodynamics, 4^{th} ed., Prentice Hall of India, 2012.
 
Evaluation Pattern
 
MTH4403  WAVELET THEORY (2019 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Course Description: This course aims at studying the fundamentals of wavelet theory. This includes the concept on the continuous and discrete wavelet transform and wavelet packets like construction and measure of wavelet sets and construction of wavelet spaces. Course objectives: This course will help the students to COBJ1. understand Fourier series, Fourier transform and Wavelet transformation and their interdependence. COBJ2. construct the wavelet transforms. COBJ3. learn the applications of Wavelet transforms 

Learning Outcome 

On successful completion of the course, the students should be able to CO1. construct the Euler’s formula of complex exponential function and convolutions CO2. understand the discrete wavelet theory in Haar transforms CO3. understand applications of wavelet transform 
Unit1 
Teaching Hours:15 
Introduction


Limitations of Fourier Series and Transforms, need of wavelet theory, Complex numbers and basic operation, the space L2(R), inner products, bases and projections, Euler’s formula and complex exponential function, Fourier series, Fourier transforms, Convolutions and BSplines, the wavelet, requirements for wavelet.  
Unit2 
Teaching Hours:15 
The Continuous wavelet transform

