CHRIST (Deemed to University), BangaloreDEPARTMENT OF MATHEMATICSSchool of Sciences 

Syllabus for

1 Semester  2021  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH111  RESEARCH METHODOLOGY  Skill Enhancement Courses  2  2  0 
MTH131  REAL ANALYSIS  Core Courses  4  4  100 
MTH132  ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS  Core Courses  4  4  100 
MTH133  ADVANCED ALGEBRA  Core Courses  4  4  100 
MTH134  INTRODUCTORY FLUID MECHANICS  Core Courses  4  4  100 
MTH135  ELEMENTARY GRAPH THEORY  Core Courses  4  4  100 
MTH151  PYTHON PROGRAMMING FOR MATHEMATICS  Core Courses  3  3  50 
2 Semester  2021  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH211  MACHINE LEARNING  Skill Enhancement Courses  2  2  0 
MTH231  GENERAL TOPOLOGY  Core Courses  4  4  100 
MTH232  COMPLEX ANALYSIS  Core Courses  4  4  100 
MTH233  LINEAR ALGEBRA  Core Courses  4  4  100 
MTH234  ADVANCED FLUID MECHANICS  Core Courses  4  4  100 
MTH235  ALGORITHMIC GRAPH THEORY  Core Courses  4  4  100 
MTH251  COMPUTATIONAL MATHEMATICS USING PYTHON  Core Courses  3  3  50 
3 Semester  2020  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH311  TEACHING TECHNOLOGY AND SERVICE LEARNING  Skill Enhancement Courses  2  2  0 
MTH331  MEASURE THEORY AND LEBESGUE INTEGRATION  Core Courses  4  4  100 
MTH332  NUMERICAL ANALYSIS  Core Courses  4  4  100 
MTH333  DIFFERENTIAL GEOMETRY  Core Courses  4  4  100 
MTH341A  BOUNDARY LAYER THEORY  Discipline Specific Elective Courses  4  4  100 
MTH341B  ADVANCED GRAPH THEORY  Discipline Specific Elective Courses  4  4  100 
MTH341C  PRINCIPLES OF DATA SCIENCE  Discipline Specific Elective Courses  4  4  100 
MTH342A  MAGNETOHYDRODYNAMICS  Discipline Specific Elective Courses  4  4  100 
MTH342B  THEORY OF DOMINATION IN GRAPHS  Discipline Specific Elective Courses  4  4  100 
MTH342C  NEURAL NETWORKS AND DEEP LEARNING  Discipline Specific Elective Courses  4  4  100 
MTH351  NUMERICAL METHODS USING PYTHON  Core Courses  3  3  50 
MTH381  INTERNSHIP  Core Courses  2  2  0 
4 Semester  2020  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH431  CLASSICAL MECHANICS  Core Courses  4  4  100 
MTH432  FUNCTIONAL ANALYSIS  Core Courses  4  4  100 
MTH433  ADVANCED LINEAR PROGRAMMING  Core Courses  4  4  100 
MTH441A  COMPUTATIONAL FLUID MECHANICS  Discipline Specific Elective Courses  4  4  100 
MTH441B  ATMOSPHERIC SCIENCE  Discipline Specific Elective Courses  4  4  100 
MTH441C  WAVELET THEORY  Discipline Specific Elective Courses  4  4  100 
MTH441D  MATHEMATICAL MODELLING  Discipline Specific Elective Courses  4  4  100 
MTH442A  ALGEBRAIC GRAPH THEORY  Discipline Specific Elective Courses  4  4  100 
MTH442B  TOPOLOGICAL GRAPH THEORY  Discipline Specific Elective Courses  4  4  100 
MTH442C  INTRODUCTION TO THE THEORY OF MATROIDS  Discipline Specific Elective Courses  4  4  100 
MTH442D  ALGORITHMS FOR NETWORKS AND NUMBER THEORY  Discipline Specific Elective Courses  4  4  100 
MTH443A  REGRESSION ANALYSIS  Discipline Specific Elective Courses  4  4  100 
MTH443B  DESIGN AND ANALYSIS OF ALGORITHMS  Discipline Specific Elective Courses  4  4  100 
MTH451A  NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON  Discipline Specific Elective Courses  3  3  50 
MTH451B  NETWORK SCIENCE WITH PYTHON AND NETWORKX  Discipline Specific Elective Courses  3  3  50 
MTH451C  PROGRAMMING FOR DATA SCIENCE IN R  Discipline Specific Elective Courses  3  3  50 
MTH481  PROJECT  Core Courses  4  4  100 
 
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) Tools: (GNU Octave) and Statistics.  
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:

MTH111  RESEARCH METHODOLOGY (2021 Batch)  
Total Teaching Hours for Semester:30 
No of Lecture Hours/Week:2 
Max Marks:0 
Credits:2 
Course Objectives/Course Description 

Course Description: This course is intended to assist students in acquiring necessary skills on the use of research methodology. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project.
Course objectives: This course will help the learner to COBJ 1: Know the general research methods COBJ 2: Get hands on experience in methods of research that can be employed for research in Mathematics 

Course Outcome 

CO1: Foster a clear understanding about research design that enables students in analyzing and evaluating the published research. CO2: Obtain necessary skills in understanding the mathematics research articles. CO3: Acquire skills in preparing scientific documents using MS Word, Origin, LaTeX and Tikz Library. 
Unit1 
Teaching Hours:10 
Research Methodology


Introduction to research and research methodology, Scientific methods, Choice of research problem, Literature survey and statement of research problem, Reporting of results, Roles and responsibilities of research student and guide.  
Unit2 
Teaching Hours:10 
Mathematical research methodology


Introducing mathematics Journals, Reading a Journal article, Mathematics writing skills. Standard Notations and Symbols, Using Symbols and Words, Organizing a paper, Defining variables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detail, Tools for checking Grammar and Plagiarism  
Unit3 
Teaching Hours:10 
Type Setting research articles


Package for Mathematics Typing, MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, Producing various types of documents using TeX.  
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: < marks to be converted to credits >  
MTH131  REAL ANALYSIS (2021 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. 

Course Outcome 

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 (2021 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. 

Course Outcome 

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 (2021 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 

Course Outcome 

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  INTRODUCTORY FLUID MECHANICS (2021 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 

Course Outcome 

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 (2021 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 

Course Outcome 

CO1: Write precise and accurate mathematical definitions of basics concepts in graph theory. CO2: Provide appropriate examples and counterexamples to illustrate the basic concepts. CO3: Demonstrate various proof techniques in proving theorems. CO4: Use algorithms to investigate Graph theoretic parameters. 
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:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH151  PYTHON PROGRAMMING FOR MATHEMATICS (2021 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. 

Course Outcome 

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 

Basic of Python


Installation, IDE, Variables, Builtin functions, Input and Output, Modules and packages, Data types and data structures, use of mathematical operators and mathematical functions, Programming Structures (Conditional structure, The for loop, The while loop, nested statements)  
Unit2 
Teaching Hours:15 

Symbolic and Numeric Computations


Use of Sympy package, Symbols, Calculus, Differential Equations, Series expressions, Linear and Nonlinear equations, List, Tuples and Arrays.  
Unit3 
Teaching Hours:15 

Data Visualization


Standard plots (2D, 3D), Scatter plots, Slope fields, Vector fields, Contour plots, stream lines, Manipulating and data visualizing data with Pandas, Mini Project.  
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:
 
MTH211  MACHINE LEARNING (2021 Batch)  
Total Teaching Hours for Semester:30 
No of Lecture Hours/Week:2 

Max Marks:0 
Credits:2 

Course Objectives/Course Description 

Course Description: This course aims at introducing the elementary notions on Machining learning and focuses on some simple application of machinelearning, algorithms on supervised machine learning and unsupervised learning. Course Objective: This course will help the learner to: COBJ1. Be proficient on the idea of machine learning COBJ2. Implement Supervised Machine Learning Algorithms COBJ3. Handle computational skills related to unsupervised learning and Preprocessing 

Course Outcome 

CO1: Demonstrate some simple applications of Machine learning. CO2: Use supervised machine learning algorithms on knearest neighbor, linear model, decisions trees. CO3: Showcase the skill using the unsupervised learning and preprocessing. 
Unit1 
Teaching Hours:10 
Introduction to Machine Learning


Introduction  Simple Machine Learning Applications: Classifying Iris Species: Meet the data, Training and Testing Data, Pair Plot of Iris dataset  knearest neighbours model, Evaluating model.  
Unit2 
Teaching Hours:10 
Supervised Learning


Classification and Regression  Generalization, Overfitting and Underfitting  Relation of Model Complexity to Dataset Size, Supervised Machine Learning Algorithms: kNearest Neighbour algorithm: kNeighbors classification, kneighbors regression, Strengths, Weakness and parameters of kNN algorithm, Linear Models: Linear models for regression, Linear models for classification, Linear models for multiclass classification, Strengths, Weakness and parameters of linear models, Decision Trees: Building decision trees, controlling complexity of decision trees, Analyzing decision trees, Strengths, Weakness and parameters of decision trees.  
Unit3 
Teaching Hours:10 
Unsupervised Learning and Preprocessing


Types of unsupervised learning, Challenges in unsupervised learning, Preprocessing and scaling: Different kinds of preprocessing, Applying Data transformations, Scaling training and test data, Principal component analysis, Nonnegative matrix factorization.  
Text Books And Reference Books: A. C. Müller and S. Guido, Introduction to machine learning with Python, O’Reilly, 2017.  
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: < marks to be converted to credits >  
MTH231  GENERAL TOPOLOGY (2021 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 

Course Outcome 

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 (2021 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. 

Course Outcome 

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 (2021 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 

Course Outcome 

CO1: Have thorough understanding of the Linear transformations. CO2: Demonstrate the elementary canonical forms, rational and Jordan forms. CO3: Familiarity with inner product spaces. 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 (2021 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 mathematical modelling of nanoliquids Course objectives: This course will help the learner to COBJ 1: Understand the different modes of heat transfer and their applications. COBJ 2: Understand the importance of doing the nondimensionalization of basic equations. COBJ 3: Understand the boundary layer flows. COBJ 4: Familiarity with porous medium and nonNewtonian fluids 

Course Outcome 

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 

Mathematical Modelling of Nanoliquids for Thermal Applications


Buongiorno Model (Two phase model): Nanoparticle/Fluid Slip : Inertia, Brownian Diffusion, Thermophoresis, Diffusiophoresis, Magnus Effect, Fluid Drainage, Gravity, Relative importance of the Nanoparticle Transport Mechanisms. Conservation Equation for two phase Nanoliquids: The Continuity equation, The Momentum equation and The energy equation.  
UNIT 4 
Teaching Hours:15 

Porous Media and NonNewtonian 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 (2021 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 shall 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. 

Course Outcome 

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  COMPUTATIONAL MATHEMATICS USING PYTHON (2021 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 Python programming using the libraries of Python programming language for Mathematical modelling, Linear Algebra and Fluid Mechanics. Course objectives: This course will help the learner to COBJ1: Acquire skill in using suitable libraries of Python to solve problems on Mathematical modelling. COBJ2: Gain proficiency in using Python to solve problems on Linear Algebra. COBJ3: Build userdefined functions to deal with the problem on Fluid Mechanics. 

Course Outcome 

CO1: Demonstrate the use of Python libraries for handling problems on Mathematical Modelling. CO2: Compute the problems on Linear Algebra using Python libraries. CO3: Handle the Python libraries for solving problems on Fluid dynamics. 
Unit1 
Teaching Hours:20 

Mathematical Modelling using Python: (Case Studies)


Linear and nonLinear Model – Growth and decay, Halflife, Newton’s law of cooling / warming, Mixtures, Computational Models with Quadratic Growth, A PredatorPrey Model, LotkaVolterra predatorprey model, competition models, Concentration of a Nutrient, Spring/Mass Systems: Free Undamped Motion, Free Damped Motion, Driven Motion, Nonlinear Springs, Simple Pendulum, Projectile, Double Pendulum  
Unit2 
Teaching Hours:15 

Linear Algebra using Python


Matrix construct, eye, zeros matrices, Addition, Subtraction, Multiplication of matrices, powers and invers of a matrix. Accessing Rows and Columns, Deleting and Inserting Rows and Columns, Determinant, reduced row echelon form, nullspace, column space, Rank, Solving systems of linear equations (Gauss Elimination Method, Gauss Jordan Method, LU decomposition Method), Eigenvalues, Eigenvectors, and Diagonalization, Linear combinations, Linearly independence and dependence, basis and dimension, Linear Transformation, Orthogonal set, orthogonal projection of a vector, Orthonormal, GramSchmidt Process.  
Unit3 
Teaching Hours:10 

Fluid Mechanics using Python


Stream Lines, Path lines, Vortex lines and their plots, Calculating Rayleigh number for RayleighBenard convection with external constraints magnetic field, rotation, non uniform temperature gradients, solution of Lorenz equations – Nusselt number.  
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:
 
MTH311  TEACHING TECHNOLOGY AND SERVICE LEARNING (2020 Batch)  
Total Teaching Hours for Semester:30 
No of Lecture Hours/Week:2 

Max Marks:0 
Credits:2 

Course Objectives/Course Description 

Course Description: This course is intended to assist the students in acquiring necessary skills on the use of modern technology in teaching, they are exposed to the principles, procedures and techniques of planning and implementing teaching techniques. Through service learning they will apply the knowledge in realworld situations and benefit the community. Course objectives: This course will help the learner to COBJ 1: Understand the pedagogy of teaching. COBJ 2: Able to use various ICT tools for effective teaching. COBJ 3: Apply the knowledge in realworld situation. COBJ 4: Enhances academic comprehension through experiential learning. 

Course Outcome 

CO1: Gain necessary skills on the use of modern technology in teaching. CO2: Understand the components and techniques of effective teaching. CO3: Obtain necessary skills in understanding the mathematics teaching. CO4: Strengthen personal character and sense of social responsibility through service learning module. CO5: Contribute to the community by addressing and meeting community need. 
Unit1 
Teaching Hours:10 
Teaching Technology


Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids (AudioVisual), Teaching aids (projected and nonprojected), Communication skills, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching.  
Unit2 
Teaching Hours:5 
Service Learning


Concept of difference between social service and service learning, Case study of best practices, understanding contemporary societal issues, Intervention in the community, Assessing need and demand of the chosen community.  
Unit3 
Teaching Hours:15 
Community Service


A minimum of fifteen (15) hours documented service is required during the semester. A student must keep a log of the volunteered time and write the activities of the day and the services performed. A student must write a reflective journal containing an analysis of the learning objectives.  
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: < marks to be converted to credits >  
MTH331  MEASURE THEORY AND LEBESGUE INTEGRATION (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: 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 

Course Outcome 

CO1: Understand the fundamental concepts of Mathematical Analysis. CO2: State 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:15 

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:15 

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 (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 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. COBJ2. To become familiar with the methods which will help to obtain solution of algebraic and transcendental equations, linear system of equations, finite differences, interpolation numerical integration and differentiation, numerical solution of differential equations and boundary value problems. 

Course Outcome 

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, Cholesky and Doolittle's 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, Galerkin Method.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH333  DIFFERENTIAL GEOMETRY (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:: Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. This course helps learners to acquire active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in Euclidean space.
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. 

Course Outcome 

CO1: Express sound knowledge on the basic concepts in geometry of curves and surfaces in Euclidean space. CO2: Demonstrate mastery in solving typical problems associated with the theory. 
UNIT 1 
Teaching Hours:15 

Calculus on Euclidean Geometry


Euclidean Space – Tangent Vectors – Directional derivatives – Curves in E3 – 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 

Intrinsic geometry of Surface


First fundamental quadratic form of a surface  Angle of two intersecting curves in a surface  element of area  Family of curves in a surface  principle directions  isometric surfaces  The Riemannian curvature tensor, the Gaussian curvature of a surface.  
UNIT 4 
Teaching Hours:15 

Surfaces in Space


Second fundamental form of a surface  equation of Gauss and equations of Codazzi  Normal curvature of surface  lines of curvature of a surface  Isometric conjugate netsDupin indicatrix.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH341A  BOUNDARY LAYER 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 basic concepts of boundary layer theory and gains the knowledge of the flow of air and other fluids with small viscosity and its engineering applications.


Course Outcome 

CO1: Able to apply the flow of small viscosity fluids with mathematical analysis CO2: Able to solve many applications problems with the aid of boundary layer theory. CO3: Able to understand the application of laminar and turbulent boundary layer with mathematical and physical justification. 
Unit1 
Teaching Hours:15 

Laminar Boundary Layer:


Boundary layer concept, Boundary layer equation for two dimensional incompressible flow, Separation of boundary layer, skin – friction, Boundary layer along a plate – The Blasius solution, Boundary layer of higher order, Similar solutions.  
Unit2 
Teaching Hours:20 

Solution of steady state two dimensional boundary layer equations:


Exact solution: Flow past wedge, Flow in a convergent channel, Flow past cylinder, Flow on a flat plate at zero incidence, Gortler series method, Plane free jet, PrandtlMises transformation and its applications of plane free jet. Approximate solution: Von Karman’s integral equation – Momentum integral equation – Energy integral equation, Applications of Von Karman’s integral equation – absence of pressure gradient – with pressure gradient – Van Karman Pohlhausen method.  
Unit3 
Teaching Hours:15 

Axially symmetry and three dimensional boundary layer:


Axially symmetrical boundary layer on a body of revolution, Mangler’s transformation, Three dimensional boundary layer – boundary layer flow on yawed cylinder and on other bodies.  
Unit4 
Teaching Hours:10 

Unsteady boundary layer:


Unsteady boundary layer equation, Method of successive approximations, Boundary layer after impulsive start of motion, Periodic boundary layer flow. Free convection from heated vertical plate, Thermal –Energy integral equation – approximate solution.  
Text Books And Reference Books: H. Schilichting, Boundary Layer Theory, Mc GrawHill Book Company, 2002.  
Essential Reading / Recommended Reading .  
Evaluation Pattern
 
MTH341B  ADVANCED 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:Theory of intersection 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. 

Course Outcome 

CO1: have thorough understanding of the concepts in chordal graphs and perfect graphs. CO2: familiarity in implementing the acquired knowledge appropriately. CO3: mastery in employing proof techniques 
Unit1 
Teaching Hours:15 

Chromatic Graph Theory


TColourings, Lcolourings, Radio Colourings, Hamiltonian Colourings, Domination and Colourings.  
Unit2 
Teaching Hours:15 

Intersection Graph Theory


Intersection Graphs, clique graphs, line graphs, hypergraphs, interval graphs, chordal graphs, weakly and strongly chordal graphs  
Unit3 
Teaching Hours:15 

Perfect Graphs


Vertex multiplication, Perfect graphs, The Perfect Graph Theorem, 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
 
MTH341C  PRINCIPLES OF DATA SCIENCE (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

Data Science is an interdisciplinary, problemsolving oriented subject that learns to apply scientific techniques to practical problems. This course provides strong foundation for data science and application area related to information technology and understand the underlying core concepts and emerging technologies in data science 

Course Outcome 

CO1: The managerial understanding of the tools and techniques used in Data Science process CO2: Analyze data analysis techniques for applications handling large data CO3: Apply techniques used in Data Science and Machine Learning algorithms to make data driven, real time, day to day organizational decisions CO4: Present the inference using various Visualization tools CO5: Learn to think through the ethics surrounding privacy, data sharing and algorithmic decisionmaking 
UNIT 1 
Teaching Hours:12 

Introduction to Data Science


Definition – Big Data and Data Science Hype – Why data science – Getting Past the Hype – The Current Landscape – Who is Data Scientist?  Data Science Process Overview – Defining goals – Retrieving data – Data preparation – Data exploration – Data modeling – Presentation.  
UNIT 2 
Teaching Hours:12 

Big Data


Problems when handling large data – General techniques for handling large data – Case study – Steps in big data – Distributing data storage and processing with Frameworks – Case study  
UNIT 3 
Teaching Hours:14 

Machine Learning


Machine learning – Modeling Process – Training model – Validating model – Predicting new observations –Supervised learning algorithms – Unsupervised learning algorithms. Introduction to Deep learning  
UNIT 4 
Teaching Hours:12 

Data Visualization


The Characteristic Polynomial, Eigenvalues and Graph Parameters, Eigenvalues of Regular Graphs, Eigenvalues and Expanders, Strongly Regular Graphs.  
Unit5 
Teaching Hours:10 

Ethics and Recent Trends


Data Science Ethics – Doing good data science – Owners of the data  Valuing different aspects of privacy  Getting informed consent  The Five Cs – Diversity – Inclusion – Future Trends.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH342A  MAGNETOHYDRODYNAMICS (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 provides the fundamentals of Magnetohydrodynamics, which include theory of Maxwell 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’ 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. 

Course Outcome 

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
 
MTH342B  THEORY OF DOMINATION IN GRAPHS (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 covers a large area of domination in graphs. This course discusses different types of dominations with their applications in reallife situations, the relation of domination related parameters with other graph parameters such as vertex degrees, chromatic number, independence number, packing number, matching number etc.


Course Outcome 

CO1: Have a thorough understanding on the concepts domination in graphs CO2: Apply the domination theory in various practical problems CO3: Gain mastery over the reasoning and proof writing techniques in graph theory 
Unit1 
Teaching Hours:15 

Domination in Graphs


Dominating sets in graphs, total domination, independence domination, bipartite domination, connected domination, distance domination, Applications to reallife situations, social network theory.  
Unit2 
Teaching Hours:15 

Bounds of Domination Number


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.  
Unit3 
Teaching Hours:15 

Domination, Independence & Irredundance


Hereditary and super hereditary properties, Independent ad dominating sets, irredundant sets, Domination Chain.  
Unit4 
Teaching Hours:15 

Efficiency and Redundancy in Domination


Efficient Dominating Sets, Codes and Cubes, Closed Neighbours, Computational Results, Realizability.  
Text Books And Reference Books: T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Reprint, CRC Press, 2000.  
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH342C  NEURAL NETWORKS AND DEEP LEARNING (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 

The main aim of this course is to provide fundamental knowledge of neural networks and deep learning. On successful completion of the course, students will acquire fundamental knowledge of neural networks and deep learning, such as Basics of neural networks, shallow neural networks, deep neural networks, forward & backward propagation process and build various research projects 

Course Outcome 

CO1: Understand the major technology trends in neural networks and deep learning. CO2: Build, train and apply neural networks and fully connected deep neural networks. CO3: Implement efficient (vectorized) neural networks for real time application. 
Unit1 
Teaching Hours:12 

Introduction to Artificial Neural Networks


Neural NetworksApplication Scope of Neural Networks Fundamental Concept of ANN: The Artificial Neural NetworkBiological Neural NetworkComparison between Biological Neuron and Artificial NeuronEvolution of Neural Network. Basic models of ANNLearning MethodsActivation FunctionsImportance Terminologies of ANN.  
Unit2 
Teaching Hours:12 

Supervised Learning Network


Shallow neural networks Perceptron NetworksTheoryPerceptron Learning RuleArchitectureFlowchart for training ProcessPerceptron Training Algorithm for Single and Multiple Output Classes.  
Unit3 
Teaching Hours:12 

Convolutional Neural Network


Introduction  Components of CNN Architecture  Rectified Linear Unit (ReLU) Layer Exponential Linear Unit (ELU, or SELU)  Unique Properties of CNN Architectures of CNN Applications of CNN  
Unit4 
Teaching Hours:12 

Recurrent Neural Network


Introduction The Architecture of Recurrent Neural Network The Challenges of Training Recurrent Networks EchoState Networks Long ShortTerm Memory (LSTM)  Applications of RNN  
Unit5 
Teaching Hours:12 

Auto Encoder And Restricted Boltzmann Machine


Introduction  Features of Auto encoder Types of Autoencoder Restricted Boltzmann Machine Boltzmann Machine  RBM Architecture Example  Types of RBM  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH351  NUMERICAL METHODS USING PYTHON (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: 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. 

Course Outcome 

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 H. 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 (2020 Batch)  
Total Teaching Hours for Semester:30 
No of Lecture Hours/Week:2 

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. 

Course Outcome 

CO1: Expose to the field of their professional interest. CO2: Opportunity to get practical experience in the specific field of interest for each student. CO3: Strengthen the research culture. 
Unit1 
Teaching Hours:30 
Internship in PG Mathematics course


M.Sc. Mathematics students have to undertake a mandatory internship in Mathematics for a period of not less than 45 working days. Students can chose to their internship in reputed research centers, recognized educational institutions, or participate in training or fellowship program offered by research institutes or organization subject to the approval of program coordinator and the Head of the department. 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 Head of the Department will assign faculty members from the department as supervisors 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 end of the required period of internship, the student 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. Within 20 days from the day of reopening, the department will conduct a presentation by the student followed by a VivaVoce. During the presentation, the supervisor or a nominee of the supervisor should be present and be one of the evaluators. 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 within one month of commencement of 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. The final evaluation includes a presentation by the students followed by the VivaVoce examination.  
Text Books And Reference Books: .  
Essential Reading / Recommended Reading .  
Evaluation Pattern .  
MTH431  CLASSICAL 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: Classical Mechanics is the study of mechanics using Mathematical methods. This course deals with some of the key ideas of classical mechanics like generalized coordinates, Lagranges equations and Hamilton's equations. Also, this course aims at introducing the Lagrangian Mechanics and Hamiltonian mechanics on Manifolds. Course objectives: This course will help the learner to COBJ 1: Derive necessary equations of motions based on the chosen configuration space. COBJ 2: Gain sufficient skills in using the derived equations in solving the applied problems in Classical Mechanics. COBJ 3: Deal with the Lagrangian and Hamiltonian mechanics on the manifolds. 

Course Outcome 

CO1: Interpret mechanics through the configuration space. CO2: Solve problems on mechanics by using Lagrange's and Hamilton?s principle. CO3: Demonstrate the Lagrangian and Hamiltonian Mechanics on Manifolds. 
Unit1 
Teaching Hours:10 

Introductory concepts


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

Lagrange's and Hamilton's equations


Derivation of Lagrange's equations: Kinetic energy, Lagranges equations, form of equations of motion, Nonholonomic systems, Examples: Spherical pendulum, Double pendulum, Lagrangian multiplier and constraint forces, Particle in a Whirling tube, Particle with moving support, Rheonomic constraint system, Integrals of Motion: Ignorable coordinates, Examples: the Kepler problem, Routhian function, Conservative systems, Natural systems, Liouvillie' system and examples. Hamilton's principle, Hamilton's equations.  
Unit3 
Teaching Hours:15 

Lagrangian Mechanics on Manifolds


Introduction to differentiable manifolds, Lagrangian system on a manifold, Lagrangian system with holonomic constraints, Lagrangian nonautonomous system, Noether's theorem, equivalence of D'AlembertLagrange principle and the variational principle, Linearization of the Lagrangian system, small oscillations.  
Unit4 
Teaching Hours:15 

Hamiltonian Mechanics on Manifolds


Hamiltonian vector fields, Hamiltonian Phase flows, Integral invariants, Law of conservation of energy, Lie algebra of Hamiltonian functions, Locally Hamiltonian vector fields.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH432  FUNCTIONAL ANALYSIS (2020 Batch)  
Total Teaching Hours for Semester:60 
No of Lecture Hours/Week:4 

Max Marks:100 
Credits:4 

Course Objectives/Course Description 
