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

Syllabus for

1 Semester  2022  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH111  RESEARCH METHODOLOGY  Skill Enhancement Course  2  2  0 
MTH131  REAL ANALYSIS  Core Courses  4  4  100 
MTH132  ABSTRACT ALGEBRA  Core Courses  4  4  100 
MTH133  ORDINARY DIFFERENTIAL EQUATIONS  Core Courses  4  4  100 
MTH134  LINEAR ALGEBRA  Core Courses  4  4  100 
MTH135  DISCRETE MATHEMATICS  Core Courses  4  4  100 
MTH151  PYTHON PROGRAMMING FOR MATHEMATICS  Core Courses  3  3  50 
2 Semester  2022  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH211  MACHINE LEARNING    2  2  0 
MTH231  GENERAL TOPOLOGY    4  4  100 
MTH232  COMPLEX ANALYSIS    4  4  100 
MTH233  PARTIAL DIFFERENTIAL EQUATIONS    4  4  100 
MTH234  GRAPH THEORY    4  4  100 
MTH235  INTRODUCTORY FLUID MECHANICS    4  4  100 
MTH251  COMPUTATIONAL MATHEMATICS USING PYTHON    3  3  50 
3 Semester  2021  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH311  TEACHING TECHNOLOGY AND SERVICE LEARNING  Skill Enhancement Course  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  4  4  100 
MTH341B  ADVANCED GRAPH THEORY  Discipline Specific Elective  4  4  100 
MTH341C  PRINCIPLES OF DATA SCIENCE  Discipline Specific Elective  4  4  100 
MTH342A  MAGNETOHYDRODYNAMICS  Discipline Specific Elective  4  4  100 
MTH342B  THEORY OF DOMINATION IN GRAPHS  Discipline Specific Elective  4  4  100 
MTH342C  NEURAL NETWORKS AND DEEP LEARNING  Discipline Specific Elective  4  4  100 
MTH351  NUMERICAL METHODS USING PYTHON  Core Courses  3  3  50 
MTH381  INTERNSHIP  Core Courses  2  2  0 
4 Semester  2021  Batch  
Course Code 
Course 
Type 
Hours Per Week 
Credits 
Marks 
MTH431  CLASSICAL MECHANICS    4  4  100 
MTH432  FUNCTIONAL ANALYSIS    4  4  100 
MTH433  ADVANCED LINEAR PROGRAMMING    4  4  100 
MTH441A  COMPUTATIONAL FLUID DYNAMICS    4  4  100 
MTH441B  ATMOSPHERIC SCIENCE    4  4  100 
MTH441C  WAVELET THEORY    4  4  100 
MTH441D  MATHEMATICAL MODELLING    4  4  100 
MTH442A  ALGEBRAIC GRAPH THEORY    4  4  100 
MTH442B  STRUCTURAL GRAPH THEORY    4  4  100 
MTH442C  APPLIED GRAPH THEORY    4  4  100 
MTH442D  ALGORITHMS FOR NETWORKS AND NUMBER THEORY    4  4  100 
MTH443A  REGRESSION ANALYSIS    4  4  100 
MTH443B  DESIGN AND ANALYSIS OF ALGORITHMS    4  4  100 
MTH451A  NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON    3  3  50 
MTH451B  NETWORK SCIENCE WITH PYTHON AND NETWORKX    3  3  50 
MTH451C  PROGRAMMING FOR DATA SCIENCE IN R    3  3  50 
MTH481  PROJECT    4  4  100 
 
Introduction to Program:  
The MSc 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 and Graph Theory 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, Numerical Analysis and courses on Data Science . Important feature of the curriculum is that students can specialize in any one of areas (i) Fluid Mechanics, (ii) Graph Theory (and (iii) Data Science with a project on these topics in the fourth semester, which will help the students to pursue research in these topics or grab the opportunities in the industry. To gain proficiency in software skills, four Mathematics Lab papers are introduced one in each semester. viz. Python Programming for Mathematics, Computational Mathematics using Python, Numerical Methods using Python and Numerical Methods for Boundary Value Problem using Python / Network Science with Python and NetworkX / Programming for Data Science in R respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning and Teaching Technology and Service learning.  
Programme Outcome/Programme Learning Goals/Programme Learning Outcome: PO1: Engage in continuous reflective learning in the context of technology and scientific advancementPO2: Identify the need and scope of the Interdisciplinary research PO3: Enhance research culture and uphold the scientific integrity and objectivity PO4: Understand the professional, ethical and social responsibilities PO5: Understand the importance and the judicious use of technology for the sustainability of the environment PO6: Enhance disciplinary competency, employability and leadership skills Programme Specific Outcome: PSO1: Attain mastery over pure and applied branches of Mathematics and its applications in multidisciplinary fieldsPSO2: Demonstrate problem solving, analytical and logical skills to provide solutions for the scientific requirements PSO3: Develop critical thinking with scientific temper PSO4: Communicate the subject effectively and express proficiency in oral and written communications to appreciate innovations in research PSO5: Understand the importance and judicious use of mathematical software's for the sustainable growth of mankind PSO6: Enhance the research culture in three areas viz. Graph theory, Fluid Mechanics and Data Science and uphold the research integrity and objectivity  
Assesment 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 (2022 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 in Mathematics. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project and also to the preparation of a research article. 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 (2022 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. Explore the properties of special functions. COBJ5. Understand and apply the 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 indepth 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, 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.  
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  ABSTRACT ALGEBRA (2022 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 of 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 factorisation and ideal theory in the polynomial ring; the structure of 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
 
MTH133  ORDINARY DIFFERENTIAL EQUATIONS (2022 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, existence and uniqueness of solutions, Eigenvalues and Eigenvectors of the equations, power series method for solving differential equations. Nonlinear autonomous system of equations. Course Objectives: This course will help the learner to COBJ 1: Solve adjoint differential equations and understand the zeros of solutions COBJ 2:Understand the existence and uniqueness of solutions of differential equations and to solve the StrumLiouville problems. COBJ 3:Solve the differential equations by power series method and also hypergeometric equations. COBJ 4:Understand and solve the nonlinear autonomous system of equations. 

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, Eigenvalue and Eigenfunctions. CO3: Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial. CO4: Understand the basic concepts existence and uniqueness of solutions. CO5: Understand basic concept of solving the linear and nonlinear autonomous systems of equations. CO6: Understand the concept of critical point and stability of the system. 
UNIT 1 
Teaching Hours:15 

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.  
UNIT 2 
Teaching Hours:15 

Existence and Uniqueness of solutions


Fundamental existence and uniqueness theorem, Dependence of solutions on initial conditions, existence and uniqueness theorem for higher order and system of differential equations, Eigenvalue Problems, StrumLiouville problems, Orthogonality of eigenfunctions.  
UNIT 3 
Teaching Hours:15 

Power series solutions


Ordinary and singular points of the differential equations, Classification of singular points, Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, Hermite, Laguerre, Chebyshev and Hypergeometric differential equation and its polynomial solutions, standard properties.  
UNIT 4 
Teaching Hours:15 

Linear and nonlinear Autonomous differential equations


Linear system of homogeneous and nonhomogeneous equations, Nonlinear autonomous system of equations, Phase plane, Critical points, Stability, Liapunov direct method, limit cycle and periodic solutions, Bifurcation of plane autonomous systems.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH134  LINEAR ALGEBRA (2022 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 Objectives: This course will help the learner to COBJ 1: Have thorough understanding of Linear transformations and its properties. COBJ 2: Understand and apply the elementary canonical forms, rational and Jordan forms in real life problems. COBJ 3: Gain knowledge on Inner product space and the orthogonalisation process. COBJ 4: Explore different types of bilinear forms and their properties. 

Course Outcome 

CO1: Gain indepth knowledge on Linear transformations. CO2: Demonstrate the elementary canonical forms, rational and Jordan forms. CO3: Apply the inner product space in orthogonality. CO4: Gain familiarity in using bilinear forms. 
Unit1 
Teaching Hours:15 

Linear Transformations and Determinants


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
 
MTH135  DISCRETE MATHEMATICS (2022 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 fundamental concepts and tools in discrete mathematics with emphasis on their applications to mathematical writing, enumeration, recurrence relations and analysis of algorithms. Course Objectives: The course will help the learner to COBJ 1: develop mathematical foundations to understand and create mathematical arguments. COBJ 2: demonstrate logical reasoning to solve a variety of practical problems. COBJ 3: implement enumeration techniques in a variety of reallife problems. COBJ 4: develop efficient algorithms and determine their efficiency. COBJ 5: communicate the basic and advanced concepts of the subject precisely and effectively.


Course Outcome 

CO1: Perform correct mathematical arguments. CO2: comprehend the fundamental and advanced concepts of relations, functions and discrete structures. CO3: demonstrate enumeration skills in various reallife problems. CO4: formulate and solve problems using generating functions and recurrence relations. CO5: apply mathematical logic to write mathematical proofs and solve problems. 
Unit1 
Teaching Hours:15 

Mathematical Logic


Sets: Cardinality and countability, recursively defined sets, relations, equivalence relations and equivalence classes, partial and total ordering, representation of relations, closure of relations, functions, bijection, inverse functions. Logic: Propositions, logical equivalences, normal forms, rules of inference, predicates, quantifiers, nested quantifiers, arguments, formal proof methods and strategies.  
Unit2 
Teaching Hours:15 

Enumeration Relations and Functions


Fundamental principles, pigeonhole principle, permutations – with and without repetitions, combinations with and without repetitions, binomial theorem, binomial coefficients, the principle of inclusion and exclusion, derangements, arrangements with forbidden positions, rook polynomial.  
Unit3 
Teaching Hours:15 

Recurrence Relations


Ordinary and exponential generating functions, recurrence relations, firstorder linear recurrence relations, higherorder linear homogeneous recurrence relations, nonhomogeneous recurrence relations, solving recurrence relations using generating functions.  
Unit4 
Teaching Hours:15 

Analysis of Algorithms


Realvalued functions, bigO, bigOmega and bigTheta notations, orders of power functions, orders of polynomial functions, analysis of algorithm efficiency, the sequential search algorithm, exponential and logarithmic orders, binary search algorithm.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH151  PYTHON PROGRAMMING FOR MATHEMATICS (2022 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 (2022 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:7 
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:13 
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 (2022 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 counterexamples of fundamental concepts in general topology. COBJ2. Acquire knowledge about a generalisation 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 (2022 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  PARTIAL DIFFERENTIAL EQUATIONS (2022 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, partial differential equations. This course includes a study of first and second order linear and nonlinear partial differential equations, existence and uniqueness of solutions to various boundary conditions, classification of second order partial differential equations, wave equation, heat equation, Laplace equations and their solutions by Eigenfunction method and Integral Transform Method. Course Objectives: This course will help the learner to COBJ 1: Understand the occurrence of partial differential equations in physics and its applications. COBJ 2: Solve partial differential equation of the type heat equation, wave equation and Laplace equations. COBJ 3: Also solving initial boundary value problems. 

Course Outcome 

CO1: Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod. CO2: Demonstrate the canonical form of second order PDE. CO3: Demonstrate initial value boundary problem for homogeneous and nonhomogeneous PDE. CO4: Demonstrate on boundary value problem by Dirichlet and Neumann problem. 
UNIT 1 
Teaching Hours:10 

First Order Partial differential equations order


Formation of PDE, initial value problems (IVP), boundary value problems (BVP) and IBVP, solutions of first, methods of characteristics for first order PDE, linear and quasi, linear, method of characteristics for onedimensional wave equations and other hyperbolic equations.  
UNIT 2 
Teaching Hours:15 

Second order Partial Differential Equations


Origin of second order PDE, Classification of second order PDE, Initial value problems (IVP), Boundary value problems (BVP) and IBVP, Mathematical models representing stretched string, vibrating membrane, heat conduction in solids, secondorder equations in two independent variables. Cauchy’s problem for second order PDE, Canonical forms, General solutions.  
UNIT 3 
Teaching Hours:15 

Solutions of Parabolic PDE


Occurrence of heat equation in Physics, resolution of boundary value problem, elementary solutions, method of separation of variables, method of eigen function expansion, Integral transforms method, Green’s function.  
UNIT 4 
Teaching Hours:20 

Solutions of Hyperbolic and Elliptic PDE


Occurrence of wave and Laplace equations in Physics, Jury problems, elementary solutions of wave and Laplace equations, methods of separation of variables,, the theory of Green’s function for wave and Laplace equations.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH234  GRAPH THEORY (2022 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 COBJ 1: Know the history and development of Graph Theory COBJ 2: Understand all the elementary concepts and results COBJ 3: Learn proof techniques and algorithms 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


Graphs as models, degree sequences, classes of graphs, matrices, isomorphism, distances in graphs, connectivity, Eulerian and Hamiltonian graphs, Chinese postman problems, travelling salesman problem and Dijkstra’s algorithm.  
Unit2 
Teaching Hours:15 

Trees


Properties of trees, rooted trees, spanning trees, algorithms on trees Prufer’s code, Huffmans coding, searching and sorting algorithms, spanning tree algorithms.  
Unit3 
Teaching Hours:15 

Planarity


Graphical embedding, Euler’s formula, platonic bodies, homeomorphic graphs, Kuratowski’s theorem, geometric duality.  
Unit4 
Teaching Hours:15 

Graph Invariants


Vertex and edge coloring, chromatic polynomial and index, matching, decomposition, independent sets and cliques, vertex and edge covers, clique covers, digraphs and networks.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern
 
MTH235  INTRODUCTORY FLUID MECHANICS (2022 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 fundamental aspects of fluid mechanics. They will have a deep insight and general comprehension 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 representatives. COBJ2: Derive and understand the governing equations in fluid mechanics. COBJ3: Familiarize with two or threedimensional incompressible flows. COBJ4: Determine properties of inviscid and viscous fluids. COBJ5: Describe and construct analytically standard two or threedimensional viscous flows. 

Course Outcome 

CO1: Confidently calculate and derive tensor expressions using index notation, and use the divergence theorem and the transport theorem. CO2: Derive the fundamental equations of fluid mechanics and appreciate their physical interpretations. CO3: Comprehend two and three dimension flows incompressible flows. CO4: Describe twodimensional viscous flows and their classifications. 
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 threedimensional motion. Twodimensional 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 Hagen Poiseuille flow, flow between two concentric rotating cylinders.  
Text Books And Reference Books:
 
Essential Reading / Recommended Reading
 
Evaluation Pattern Examination and Assessments
 
MTH251  COMPUTATIONAL MATHEMATICS USING PYTHON (2022 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 aimsto solve mathematical models using differential equations, linear algebra and fluid mechanics using Python libraries. Course objectives: This course will help the learner to COBJ1: Acquire skill in using suitable libraries of Python to solve realworld problems giving rise to differential equations 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:45 

Proposed Topics:


 
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 (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 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 needs. 
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 (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: The Course covers 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
