# Syllabus for Master of Science (Mathematics) Academic Year  (2021)

 1 Semester - 2021 - Batch Course Code Course Type Hours Per Week Credits Marks MTH111 RESEARCH METHODOLOGY Add On Course 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 Course 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 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 - 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 4 4 100 MTH441B ATMOSPHERIC SCIENCE Discipline Specific Elective 4 4 100 MTH441C WAVELET THEORY Discipline Specific Elective 4 4 100 MTH441D MATHEMATICAL MODELLING Discipline Specific Elective 4 4 100 MTH442A ALGEBRAIC GRAPH THEORY Discipline Specific Elective 4 4 100 MTH442B TOPOLOGICAL GRAPH THEORY Discipline Specific Elective 4 4 100 MTH442C INTRODUCTION TO THE THEORY OF MATROIDS Discipline Specific Elective 4 4 100 MTH442D ALGORITHMS FOR NETWORKS AND NUMBER THEORY Discipline Specific Elective 4 4 100 MTH443A REGRESSION ANALYSIS Discipline Specific Elective 4 4 100 MTH443B DESIGN AND ANALYSIS OF ALGORITHMS Discipline Specific Elective 4 4 100 MTH451A NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON Discipline Specific Elective 3 3 50 MTH451B NETWORK SCIENCE WITH PYTHON AND NETWORKX Discipline Specific Elective 3 3 50 MTH451C PROGRAMMING FOR DATA SCIENCE IN R Discipline Specific Elective 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 skill-based 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 Open-Source Software (FOSS) Tools: (GNU Octave) and Statistics.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO1: Engage in continuous reflective learning in the context of technology and scientific advancement.

PO2: 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 fields

PSO2: 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

Assessment Pattern

 Course Code Title CIA (Max Marks) Attendance (Max Marks) ESE (Max Marks) MTH131 Real Analysis 45 5 50 MTH132 Ordinary and Partial Differential Equations 45 5 50 MTH133 Advanced Algebra 45 5 50 MTH134 Introductory Fluid Mechanics 45 5 50 MTH135 Elementary Graph Theory 45 5 50 MTH151 Python Programming for Mathematics 50 -- -- MTH111 Research Methodology G -- -- MTH231 General Topology 45 5 50 MTH232 Complex Analysis 45 5 50 MTH233 Linear Algebra 45 5 50 MTH234 Advanced Fluid Mechanics 45 5 50 MTH235 Algorithmic Graph Theory 45 5 50 MTH251 Computational Mathematics using Python 50 -- -- MTH211 Machine Learning G -- -- MTH331 Measure Theory and Lebesgue Integration 45 5 50 MTH332 Numerical Analysis 45 5 50 MTH333 Differential Geometry 45 5 50 MTH341A Boundary Layer Theory 45 5 50 MTH341B Advanced Graph Theory 45 5 50 MTH341C Principles of Data Science 45 5 50 MTH342A Magnetohydrodynamics 45 5 50 MTH342B Theory of Domination in Graphs 45 5 50 MTH342C Neural Networks and Deep Learning 45 5 50 MTH351 Numerical Methods using Python 50 -- -- MTH381 Internship G -- -- MTH311 Teaching Technology and Service learning G -- -- MTH431 Classical Mechanics 60 100 4 MTH432 Functional Analysis 60 100 4 MTH433 Advanced Linear Programming 45 5 50 MTH441A Computational Fluid Mechanics 45 5 50 MTH441B Atmospheric Science 45 5 50 MTH441C Wavelet Theory 45 5 50 MTH441D Mathematical Modelling 45 5 50 MTH442A Algebraic Graph theory 45 5 50 MTH442B Topological Graph Theory 45 5 50 MTH442C Introduction to the Theory of Matroids 45 5 50 MTH442D Algorithms for Networks and Number Theory 45 5 50 MTH443A Regression Analysis 45 5 50 MTH443B Design and Analysis of Algorithms 45 5 50 MTH451A Numerical Methods for Boundary Value Problem using Python 50 -- -- MTH451B Network Science with Python and NetworkX 50 -- -- MTH451C Programming for Data Science in R 50 -- -- MTH481 Project 100 -- --
Examination And Assesments

EXAMINATION AND ASSESSMENTS (Theory)

 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Lab Assignments 10 CIA II Familiarity with the commands and execution of them in solving problems. Analytical and Problem Solving skills Lab Work Problem Solving 10 CIA III Conceptual clarity and analytical skills in solving Problems using Mathematical Package / Programming Lab Exam based on the Lab exercises 25 Attendance Regularity and Punctuality Lab attendance 05                                 =100%:5     97 – <100% :4     94 – < 97%  :3     90 – <94%  :2     85 – <90%  :1               <85% :0 Total 50
 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.
Unit-1
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.

Unit-2
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

Unit-3
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:

.

1. E. B. Wilson, An introduction to scientific research, Reprint, Courier Corporation, 2012.
2. R. Ahuja, Research Methods, Rawat Publications, 2001.
3. G. L. Jain, Research Methodology, Mangal Deep Publications, 2003.
4. B. C. Nakra and K. K. Chaudhry, Instrumentation, measurement and analysis, TMH Education, 2003.
5. L. Radhakrishnan, Write Mathematics Right: Principles of Professional Presentation, Exemplified with Humor and Thrills, Alpha Science International, Limited, 2013.
6. G. Polya, How to solve it: a new aspect of mathematical method. Princeton, N.J.: Princeton University Press, 1957
7. R. Hamming, You and your research, available at https://www.cs.virginia.edu/~robins/YouAndYourResearch.html
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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Assignments 10 CIA II Analytical and Problem Solving skills Problem Solving (or) Assessment on software skills (if any) 10 CIA III Conceptual clarity and analytical skills in solving Problems (using Mathematical Package / Programming, if any) Problem Solving (or) Assessment on software skills (if any) 25 Attendance Regularity and Punctuality Attendance 05                                  =100%:5      97 – <100% :4      94 – < 97%  :3      90 – <94%  :2      85 – <90%  :1                <85% :0 Total 50

< 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 Riemann-Stieltjes 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 self-contained 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 Riemann-Stieltjes 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 Riemann-Stieltjes Integration

Definition and Existence of Riemann-Stieltjes Integral, Linearity Properties of Riemann-Stieltjes Integral, The Riemann-Stieltjes Integral as the Limit of Sums, Integration and Differentiation, Integration of Vector-valued 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 Stone-Weierstrass 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: McGraw-Hill (India), 2016.

1. T.M. Apostol, Mathematical Analysis, New Delhi: Narosa, 2004.
2. E.D. Bloch, The Real Numbers and Real Analysis, New York: Springer, 2011.
3. J.M. Howie, Real Analysis, London: Springer, 2005.
4. J. Lewin, Mathematical Analysis, Cambridge: Cambridge University Press, 2003.
5. F. Morgan, Real Analysis, New York: American Mathematical Society, 2005.
6. S. Ponnusamy, Foundations of Mathematical Analysis, illustrated ed., Birkhauser, 2012.
7. S.C. Malik and S. Arora, Mathematics Analysis, 4th ed., New Age International, 2012.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 self-adjoint 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 non-homogeneous 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 self-adjoint 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, second-order 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 non-homogeneous wave equation, the method of separation of variables for the one-dimensional 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:
1. C. Constanda, Solution Techniques for Elementary Partial Differential Equations, New York: Chapman & Hall, 2010.
2. S.J.Farlow, An Introduction to Differential Equations and their Applications, reprint, Dover Publications Inc., 2012.
1. K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics andEngineering, Cambridge, 2005.
2. E. Penney, Differential Equations and Boundary Value Problems, Pearson Education, 2005.
3. J. D. Logan, Partial Differential Equations, 2nd ed., New York: Springer, 2002.
4. A. Jeffrey, Applied Partial Differential Equations: An Introduction, California: Academic Press, 2003.
5. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, 2nd ed., New York: Springer, 2004.
6. L.C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.
7. K. Sankara Rao, Introduction to Partial Differential Equations, 2nd ed., New Delhi: Prentice-Hall of India, 2006.
8. R.C. McOwen, Partial Differential Equations: Methods and Applications, 2nd ed., New York: Pearson Education, 2003.
9. E. A. Coddington, Introduction to ordinary differential equations, Reprint: McGraw Hill, 2006.
10. G. F. Simmons, Differential equations with applications and historical notes, Tata McGraw Hill, 2003. (Unit I and II).
11. T. Myint-U and L. Debnath, Linear Partial Differential Equations, Boston: Birkhauser, 2007.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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.

Unit-1
Teaching Hours:15

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.

Unit-2
Teaching Hours:15
Rings

Euclidean Ring, Polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings.

Unit-3
Teaching Hours:15
Fields

Extension fields, roots of polynomials, construction with straightedge and compass, more about roots.

Unit-4
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.

1. S. Lang, Algebra, 3rd revised ed., Springer, 2002.
2. S. Warner, Modern Algebra, Reprint, Courier Corporation, 2012.
3. G. Birkhoff and S.M. Lane, A Survey of ModernAlgebra, 3rd ed., A K Peters/CRC Press, 2008.
4. J. R. Durbin, Modern algebra: An introduction, 6th ed., Wiley, 2008.
5. N. Jacobson, Basic algebra – I, 2nd ed., Dover Publications, 2009.
6. J. B. Fraleigh, A first course in abstract algebra, 7th ed., Addison-Wesley Longman, 2002.
7. D.M. Dummit and R.M.Foote, Abstract Algebra, 3rd  ed., John Wiley and Sons, 2003.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 non-Newtonian 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 non-Newtonian 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.

Unit-1
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.

Unit-2
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 (Navier-Stokes equations), conservation of momentum, the energy equation, conservation of energy.

Unit-3
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.

Unit-4
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:
1. S. W. Yuan, Foundations of  fluid mechanics, Prentice Hall of India, 2001.
2. M. D. Raisinghania, Fluid Dynamics, S. Chand and Company Ltd., 2010.
1. D. S. Chandrasekharaiah and L. Debnath, Continuum mechanics, Academic Press, 2014 (Reprint).
2. P. K. Kundu, Ira M. Cohen and David R. Dowling, Fluid Mechanics, Fifth Edition , 2010.
3. G.K. Batchelor, An introduction to fluid mechanics, Cambridge University Press, 2000.
4. F. Chorlton, Text book of fluid dynamics, New Delhi: CBS Publishers & Distributors, 2004.
5. J. F. Wendt, J.D. Anderson, G. Degrez and E. Dick, Computational fluid dynamics: An introduction, Springer-Verlag, 1996.
6. F. M White, Fluid Mechanics, Tata Mcgraw Hill. 2010.
Evaluation Pattern

Examination and Assessments

 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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.

Unit-1
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.

Unit-2
Teaching Hours:15
Vertex Degrees and Directed Graphs

Counting and Bijections, Extremal Problems, Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and Tournaments.

Unit-3
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.

Unit-4
Teaching Hours:15
Connectivity and Paths

Connectivity, Edge - Connectivity, Blocks, 2 - connected Graphs, Connectivity in Digraphs, k - connected and k-edge-connected Graphs, Maximum Network Flow, Integral Flows.

Text Books And Reference Books:
1. D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.
2. G. Chartrand and P. Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill, 2006
1. B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005.
2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.
3. N. Deo, Graph Theory with applications to engineering and computer science, Delhi: Prentice Hall of India, 1979.
4. G. Chatrand and L. Lesniak, Graphs and Digraphs, 4 ed., Boca Raton: CRC Press, 2004.
5. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008
6. J. Clark and D. A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.
7. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, New Delhi, Springer, 2008
8. R. Diestel, Graph Theory, New Delhi: Springer, 2006.
9. V. K. Balakrishnan Graph Theory, Schaum’s outlines, New Delhi:Tata Mcgrawhill, 2004.
10. S.A. Choudum, A first Course in Graph Theory, MacMillan Publishers India Ltd, 2013.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 built-in functions to deal with Graphs and Digraphs.

Unit-1
Teaching Hours:15
Basic of Python

Installation, IDE, Variables, Built-in 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)

Unit-2
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.

Unit-3
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:
1. S. Linge and H. P. Langtangen, Programming for computations- Python - A gentle Introduction to Numerical Simulations with Python 3.6, Springer Open, Second Edm. 2020.
2. H. Fangohr, Introduction to Python for Computational Science and Engineering (A beginner’s guide), University of Southampton, 2015. (https://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf)
3. H. P. Langtangen, A Primer on Scientific Programming with Python, 2nd ed., Springer, 2016.
1. W. A Strauss, Partial Differential Equations - An Introduction, John Wiley and Sons, 2007.
2. K. H. Rosen and K. Krithivasan, Discrete mathematics and its applications. McGraw-Hill, 2013.
3. S. Rao, Partial Differential Equations, Prentice Hall of India, 2009.
4. B. E. Shapiro, Scientific Computation: Python Hacking for Math Junkies, Sherwood Forest Books, 2015..
5. C. Hill, Learning Scientific Programming with Python, Cambridge University Press, 2016.
6. J. Kiusalaas, Numerical methods in engineering with Python 3, Cambridge University Press, 2013.
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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Lab Assignments 10 CIA II Familiarity with the commands and execution of them in solving problems. Analytical and Problem Solving skills Lab Work Problem Solving 10 CIA III Conceptual clarity and analytical skills in solving Problems using Mathematical Package / Programming Lab Exam based on the Lab exercises 25 Attendance Regularity and Punctuality Lab attendance 05                                  =100%:5      97 – <100% :4      94 – < 97%  :3      90 – <94%  :2      85 – <90%  :1                <85% :0 Total 50

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 machine-learning, 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 k-nearest neighbor, linear model, decisions trees.

CO3: Showcase the skill using the unsupervised learning and preprocessing.

Unit-1
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 - k-nearest neighbours model, Evaluating model.

Unit-2
Teaching Hours:10
Supervised Learning

Classification and Regression - Generalization, Overfitting and Underfitting - Relation of Model Complexity to Dataset Size, Supervised Machine Learning Algorithms: k-Nearest Neighbour algorithm: k-Neighbors classification, k-neighbors regression, Strengths, Weakness and parameters of k-NN 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.

Unit-3
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, Non-negative matrix factorization.

Text Books And Reference Books:

A. C. Müller and S. Guido, Introduction to machine learning with Python,  O’Reilly, 2017.

1. D. Julian, Designing machine learning systems with python. Packt Publishing Limited, 2016.
2. D. Cielen, M. A. D. B., and A. Mohamed, Introducing data science: big data, machine learning, and more, using Python tools. Manning., 2016.
3. M. Bowles, Machine Learning in Python: essential techniques for predictive analysis. John Wiley &amp; Sons, 2019.
4. R. Garreta and G. Moncecchi, Learning scikit-learn: machine learning in python. Packard publishing limited, 2013.
5. S. Raschka and V. Mirjalili, Python machine learning machine learning and deep learning with Python, scikit-learn, and TensorFlow. Packt Publishing, 2018.
6. L. P. Coelho and W. Richert, Building machine learning systems with Python. Packt Publishing, 2015
7. G. James, D. Witten, T. Hastie, and R. Tibshirani, An introduction to statistical learning: with applications in R. Springer, 2017.
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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Assignments 10 CIA II Analytical and Problem Solving skills Problem Solving (or) Assessment on software skills (if any) 10 CIA III Conceptual clarity and analytical skills in solving Problems (using Mathematical Package / Programming, if any) Problem Solving (or) Assessment on software skills (if any) 25 Attendance Regularity and Punctuality Attendance 05                                  =100%:5      97 – <100% :4      94 – < 97%  :3      90 – <94%  :2      85 – <90%  :1                <85% :0 Total 50

< 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.

Unit-1
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.

Unit-2
Teaching Hours:15
Continuous Functions

Continuous functions, the product topology, metric topology.

Unit-3
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.

Unit-4
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.

1. G.F.Simmons, Introduction to topology and modern analysis, Tata McGraw Hill Education, 2004.
2. J. Dugundji, Topology, Prentice Hall of India, 2000.
3. S. Willard, General topology, Courier-Corporation, 2012.
4. C. W. Baker, Introduction to topology, Krieger Publishing Company, 2000.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 3-circles 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 Cauchy-Riemann 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 Cauchy-Goursat 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.

Unit-1
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..

Unit-2
Teaching Hours:15
Singularities

Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles

Unit-3
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.

Unit-4
Teaching Hours:12
Meromorphic functions

Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem.

Text Books And Reference Books:
1. M.J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, 2003.
2. J.B.Conway, Functions of One Complex Variable, 2nd ed., New York: Springer, 2000.
1. J.H. Mathews and R.W. Howell, Complex Analysis for Mathematics and Engineering, 6th ed., London: Jones and Bartlett Learning, 2011.
2. J.W. Brown and R.V. Churchill, Complex Variables and Applications, 7th ed., New York: McGraw-Hill, 2003.
3. L.S. Hahn and B. Epstein, Classical Complex Analysis, London: Jones and Bartlett Learning, 2011.
4. D. Wunsch, Complex Variables with Applications, 3rd ed., New York: Pearson Education, 2009.
5. D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., Boston: Jones and Bartlett Learning, 2010.
6. E.M. Stein and Rami Sharchi, Complex Analysis, New Jersey: Princeton University Press, 2003.
7. T.W.Gamblin, Complex Analysis, 1st ed., Springer, 2001.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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.

Unit-1
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.

Unit-2
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 – Semi-Simple Operators.

Unit-3
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.

Unit-4
Teaching Hours:10
Bilinear Forms

Bilinear Forms – Symmetric Bilinear Forms – Skew-Symmetric 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.

1. S. Lang, Introduction to Linear Algebra, Undergraduate Texts in Mathematics, 2nd ed. New York: Springer, 1997.
2. P. D. Lax, Linear Algebra and its Applications, 2nd ed., John Wiley and Sons, 2013.
3. S. Roman, Advanced Linear Algebra, 3rd ed., Springer Science and Business Media, 2013.
4. G. Strang, Linear Algebra and its Applications, 15th Re-print edition, Cengage Learning, 2014.
5. S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 4th ed., Prentice Hall, 2003.
Evaluation Pattern

 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 non-linear problems. This course also includes the mathematical modelling of nano-liquids

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 non-dimensionalization of basic equations.

COBJ 3: Understand the boundary layer flows.

COBJ 4: Familiarity with porous medium and non-Newtonian 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. Non-dimensional parameters determined from differential equations – Buckingham’s Pi Theorem – Non-dimensionalization of the Basic Equations - Non-dimensional 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 Orr-Sommerfeld equation assuming that the basic flow is strictly parallel. Basic concepts of stability theory – Linear and Non-linear 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 Nano-liquids 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 Non-Newtonian Fluids

Introduction to porous medium, porosity, Darcy’s Law, Extension of Darcy Law – accelerations and inertial effects, Brinkman’s equation, effects of porosity variations, Bidisperse porous media. Constitutive equations of Maxwell, Oldroyd, Ostwald, Ostwald de waele, Reiner – Rivlin and Micropolar fluid. Weissenberg effect and Tom’s effect. Equation of continuity, Conservation of momentum for non-Newtonian fluids.

Text Books And Reference Books:
1. P. G. Drazin and W. H. Reid, Hydrodynamic instability, Cambridge University Press, 2006.
2. S. Chardrasekhar,Hydrodynamic and hydrodmagnetic stability, Oxford University Press, 2007 (RePrint).
1. P. K. Kundu, Ira M. Cohen and David R Dowling, Fluid Mechanics, 5th ed., Academic Press, 2011.
2. F. M White, Fluid Mechanics, Tata Mcgraw Hill. 2011.
3. D. A. Nield and Adrian Bejan, Convection in Porous Media”, Third edition, Springer, 2006.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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.

Unit-1
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, Colour-Critical Graphs, Counting Proper Colourings.

Unit-2
Teaching Hours:15
Matchings and Factors

Maximum Matchings, Hall’s Matching Condition, Min-Max Theorem, Independent Sets and Covers, Maximum Bipartite Matching, Weighted Bipartite Matching, Tutte’s 1-factor Theorem, Domination.

Unit-3
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

Unit-4
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: Prentice-Hall of India, 2011.

1. B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005.
2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.
3. G. Chartrand and P. Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill, 2006.
4. G. Chatrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, Boca Raton: CRC Press, 2004
5. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.
6. J. Clark and D. A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.
7. R. Balakrishnan and K. Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.
8. R. Diestel, Graph Theory, New Delhi: Springer, 2006.
9. V. K. Balakrishnan, Graph Theory, Schaum’s outlines, New Delhi: Tata Mcgrahill, 2004.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 user-defined 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.

Unit-1
Teaching Hours:20
Mathematical Modelling using Python: (Case Studies)

Linear and non-Linear Model – Growth and decay, Half-life, Newton’s law of cooling / warming, Mixtures, Computational Models with Quadratic Growth, A Predator-Prey Model, Lotka-Volterra predator-prey 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

Unit-2
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, Gram-Schmidt Process.

Unit-3
Teaching Hours:10
Fluid Mechanics using Python

Stream Lines, Path lines, Vortex lines and their plots, Calculating Rayleigh number for Rayleigh-Benard convection with external constraints magnetic field, rotation, non -uniform temperature gradients, solution of Lorenz equations – Nusselt number.

Text Books And Reference Books:
1. D. G. Zill, First Course in Differential Equations with Modelling Applications. (Unit 1)
2. H.S. Son, Linear Algebra Coding with Python. (Unit 2)
3. Fluid Mechanics Book and Research Papers. (Unit 3)
1. H. Fangour, Introduction to Python for computational science and Engineering, 2015.
2. H. P. Langtangen, A premier on scientific programming with Python, 3rd ed., 2012.
3. S. Linge and H. P. Langtangen, Programming for computations - Python, 2nd ed., 2018.
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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Lab Assignments 10 CIA II Familiarity with the commands and execution of them in solving problems. Analytical and Problem Solving skills Lab Work Problem Solving 10 CIA III Conceptual clarity and analytical skills in solving Problems using Mathematical Package / Programming Lab Exam based on the Lab exercises 25 Attendance Regularity and Punctuality Lab attendance 05                                  =100%:5      97 – <100% :4      94 – < 97%  :3      90 – <94%  :2      85 – <90%  :1                <85% :0 Total 50

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 real-world 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 real-world 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.

Unit-1
Teaching Hours:10
Teaching Technology

Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids (Audio-Visual), Teaching aids (projected and non-projected), Communication skills, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching.

Unit-2
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.

Unit-3
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:
1. R. Varma, Modern trends in teaching technology, Anmol publications Pvt. Ltd., New Delhi 2003.
2. U. Rao, Educational teaching, Himalaya Publishing house, New Delhi 2001.
3. C. B. Kaye, The Complete Guide to Service Learning: Proven, Practical Ways to Engage Students in Civic Responsibility, Academic Curriculum, & Social Action, 2009.
1. J. Mohanthy, Educational teaching, Deep & Deep Publications Pvt. Ltd., New Delhi 2001.
2. K. J. Sree and D. B. Rao, Methods of teaching sciences, Discovery publishing house, 2010.
3. D. Butin, Service-Learning in Theory and Practice-The Future of Community Engagement in Higher Education, Palgrave Macmillan US., 2010.
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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Assignments 10 CIA II Analytical and Problem Solving skills Problem Solving (or) Assessment on software skills (if any) 10 CIA III Conceptual clarity and analytical skills in solving Problems (using Mathematical Package / Programming, if any) Problem Solving (or) Assessment on software skills (if any) 25 Attendance Regularity and Punctuality Attendance 05                                  =100%:5      97 – <100% :4      94 – < 97%  :3      90 – <94%  :2      85 – <90%  :1                <85% :0 Total 50

< 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

Unit-1
Teaching Hours:15
Lebesgue Measure

Lebesgue Outer Measure, The s-Algebra of Lebesgue Measurable Sets, Outer and Inner Approximation of Lebesgue Measurable Sets, Countable Additivity, Continuity and the Borel-Cantelli Lemma, Nonmeasurable Sets, The Cantor Set and the Canton-Lebesgue 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.

Unit-2
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.

Unit-3
Teaching Hours:15
Differentiation and Lebesgue Integration

Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives.

Unit-4
Teaching Hours:15
The Lp Spaces

Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The Lp spaces, Approximation and Separability, The Riesz Representation for the Dual of Lp, Weak Sequential Convergence in Lp, 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.

1. P.  R. Halmos, Measure Theory, Springer, 2014,
2. M. E. Munroe, Introduction to measure and integration, Addison Wesley, 1959.
3. G. de Barra, Measure theory and integration, New Age, 1981.
4. P. K. Jain and V.P. Gupta, Lebesgue measure and integration, New Age, 1986.
5. F. Morgan, Geometric measure theory – A beginner’s guide, Academic Press, 1988.
6. F. Burk, Lebesgue measure and integration: An introduction, Wiley, 1997.
7. D. H. Fremlin, Measure theory, Torres Fremlin, 2000.
8. M. M. Rao, Measure theory and integration, 2nd ed., Marcel Dekker, 2004.
Evaluation Pattern

 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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.
COBJ3. Understand accuracy, consistency, stability and convergence of numerical methods.

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

Unit-1
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, Newton-Raphson methods (includes the convergence criterion for simple roots), Bairstow’s method, Graeffe’s root squaring method, Birge-Vieta method, Muller’s method. Solution of Linear System of Algebraic Equations: LU-decomposition methods (Crout’s, Cholesky and Doolittle's methods), consistency and ill-conditioned system of equations, Tridiagonal system of equations, Thomas algorithm.

Unit-2
Teaching Hours:15
Interpolation and Numerical Integration

Lagrange, Hermite, Cubic-spline’s (Natural, Not a Knot and Clamped) - with uniqueness and error term, for polynomial interpolation. Chebychev and Rational function approximation. Gaussian quadrature, Gauss-Legendre, Gauss-Chebychev formulas.

Unit-3
Teaching Hours:15
Numerical solution of ordinary differential equations

Initial value problems, Runge-Kutta methods of second and fourth order, multistep method, Adams-Moulton method, stability (convergence and truncation error for the above methods), boundary value problems, second order finite difference method.

Unit-4
Teaching Hours:10
Boundary Value Problems

Numerical solutions of second order boundary value problems (BVP) of first, second and third types by shooting method, Rayleigh-Ritz Method, Galerkin Method.

Text Books And Reference Books:
1. M. K. Jain, S. R. K. Iyengar and R. K. Jain, Numerical Methods for Scientific and Engineering Computation, 5th ed., New Delhi: New Age International, 2007.
2. S. S. Sastry, Introductory Methods of Numerical Analysis, 4th ed., New Delhi: Prentice-Hall of India, 2006.
1. R. L. Burden and J. D. Faires, Numerical Analysis, 9th ed., Boston: Cengage Learning, 2011.
2. S. C. Chopra and P. C. Raymond, Numerical Methods for Engineers, New Delhi: Tata McGraw-Hill, 2010.
3. C. F. Gerald and P. O. Wheatley, Applied Numerical Methods, 7th ed., New York: Pearson Education, 2009.
4. J. Milton, T. Ohira, Mathematics as a Laboratory Tool: Dynamics, Delays and Noise, Springer-Verlag New York, 2014.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 – 1-Forms – 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 nets-Dupin indicatrix.

Text Books And Reference Books:
1. B. O’Neill, Elementary Differential geometry, 2nd revised ed., New York: Academic Press, 2006. (Unit 1 and Unit 2)
2. L. P. Eiserhart, An introduction to Differential Geometry with use of the Tensor Calculus (Unit 3 and Unit 4)
1. J. A. Thorpe, Elementary topics in differential geometry, 2nd ed., Springer, 2004.
2. A. Pressley, Elementary differential geometry, 2nd ed., Springer, 2010.
3. Mittal and Agarwal, Differential geometry, 36th ed., Meerut: Krishna Prakashan Media (P) Ltd., 2010.
4. K. S. Amur, D. J. Shetty and C. S. Bagewadi, An introduction to differential geometry, Oxford, U.K: Alpha Science International, 2010.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 objectives: This course will help the learner to
COBJ 1:Understands the boundary layer flows and its applications.
COBJ 2:Understands the findings of exact and approximation solution of two-dimensional boundary layer equations.
COBJ 3:Understands the axial symmetry boundary layer flow.
COBJ 4:Understands the three dimensional boundary layer flow.
COBJ 5:Understands the unsteady boundary layer flow and its 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.

Unit-1
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.

Unit-2
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, Prandtl-Mises 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.

Unit-3
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.

Unit-4
Teaching Hours:10

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 Graw-Hill Book Company, 2002.

.

Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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

Unit-1
Teaching Hours:15
Chromatic Graph Theory

T-Colourings, L-colourings, Radio Colourings, Hamiltonian Colourings, Domination and Colourings.

Unit-2
Teaching Hours:15
Intersection Graph Theory

Intersection Graphs, clique graphs, line graphs, hypergraphs, interval graphs, chordal graphs, weakly and strongly chordal graphs

Unit-3
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.

Unit-4
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:
1. G. Chartrand and P. Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill, 2006.
2. T. A. McKee and F.R. McMorris, Topics in Intersection Graph Theory, SIAM, 1999.
3. D. B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.
4. R. B. Bapat, Graphs and Matrices, Springer, New York, 2010
1. B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005.
2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.
3. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.
4. J. Clark and D. A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.
5. R. Balakrishnan and K. Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.
6. R. Diestel, Graph Theory, New Delhi: Springer, 2006.
7. M. Bona, A walk through combinatorics, World scientific, 2011.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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, problem-solving 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 decision-making

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.

Unit-5
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:
1. D. Cielen, A. D. B. Meysman and M. Ali, Introducing Data Science, Manning Publications Co., 1st ed., 2016
2. G. James, D. Witten, T. Hastie and R. Tibshirani, An Introduction to Statistical Learning: with Applications in R, Springer, 1st ed., 2013
3. Y. Bengio, A. Courville, Deep Learning, Ian Goodfellow, MIT Press, 1st ed., 2016
4. D. J. Patil, H. Mason, M. Loukides, O’ Reilly, Ethics and Data Science, 1st ed., 2018
1. J. Grus, Data Science from Scratch: First Principles with Python, O’Reilly, 1st ed., 2015.
2. C. O'Neil and R. Schutt, Doing Data Science, Straight Talk from the Frontline, O’Reilly, 1st ed., 2013
3. J. Leskovec, A. Rajaraman, J. D. Ullman, Mining of Massive Datasets, Cambridge University Press, 2nd ed., 2014
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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

Unit-1
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.

Unit-2
Teaching Hours:13
Basic Equations

Outline of basic equations of MHD, magnetic induction equation, Lorentz force, MHD approximations, non-dimensional numbers, velocity, temperature and magnetic field boundary conditions.

Unit-3
Teaching Hours:20
Exact Solutions

Hartmann flow, generalized Hartmann flow, velocity distribution, expression for induced current and magnetic field, temperature discribution, Hartmann couette flow, magnetostatic-force free magnetic field, abnormality parameter, Chandrashekar theorem, application of magnetostatic-Bennett pinch.

Unit-4
Teaching Hours:15
Applications

Classical MHD and Alfven waves, Alfven theorem, Frozen-in-phenomena, 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 non-viscous fluid, Helmholtz vorticity equation, Kelvin’s circulation theorem, Bernoulli’s equation.

Text Books And Reference Books:
1. P. A. Davidson, Introduction to Magnetohydrodynamics, Cambridge University Press, 2001.
2. G.W, Sutton and A, Sherman, Engineering Magnetohydrodynamics, Dover Publications Inc., 2006.

D. J. Griffiths, Introduction to electrodynamics, 4th ed., Prentice Hall of India, 2012.

Evaluation Pattern

 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 real-life 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 objectives: This course will help the learner to
COBJ 1:Understand the advanced topics of domination in graphs.
COBJ 2:Understand different types of dominations related to various real-life situations
COBJ 3:Enhance the understanding of techniques of writing proofs for advanced topics in domination theory.

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

Unit-1
Teaching Hours:15
Domination in Graphs

Dominating sets in graphs, total domination, independence domination, bipartite domination, connected domination, distance domination, Applications to real-life situations, social network theory.

Unit-2
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.

Unit-3
Teaching Hours:15
Domination, Independence & Irredundance

Hereditary and super hereditary properties, Independent ad dominating sets, irredundant sets, Domination Chain.

Unit-4
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.

1. S. T. Hedetniemi and R. C. Laskar, Topics on Domination North-Holland, 1991.
2. T. W. Haynes and S. T. Hedetniemi, M.A. Henning, Topics in Domination in Graphs, Springer, 2020
3. T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Domination in graphs: Advanced Topics, Marcel Decker, 1997.
4. M. A. Henning, A. Yeo, Total Domination in Graphs, Springer, 2013.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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.

Unit-1
Teaching Hours:12
Introduction to Artificial Neural Networks

Neural Networks-Application Scope of Neural Networks- Fundamental Concept of ANN: The Artificial Neural Network-Biological Neural Network-Comparison between Biological Neuron and Artificial Neuron-Evolution of Neural Network. Basic models of ANN-Learning Methods-Activation Functions-Importance Terminologies of ANN.

Unit-2
Teaching Hours:12
Supervised Learning Network

Shallow neural networks- Perceptron Networks-Theory-Perceptron Learning Rule-Architecture-Flowchart for training Process-Perceptron Training Algorithm for Single and Multiple Output Classes.
Back Propagation Network- Theory-Architecture-Flowchart for training process-Training Algorithm-Learning Factors for Back-Propagation Network.
Radial Basis Function Network RBFN: Theory, Architecture, Flowchart and Algorithm.

Unit-3
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

Unit-4
Teaching Hours:12
Recurrent Neural Network

Introduction- The Architecture of Recurrent Neural Network- The Challenges of Training Recurrent Networks- Echo-State Networks- Long Short-Term Memory (LSTM) - Applications of RNN

Unit-5
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:
1. S. N. Sivanandam and S. N. Deepa, Principles of Soft Computing, Wiley-India, 3rd Edition, 2018.
2. S. L. Rose, L. A. Kumar, D. K. Renuka, Deep Learning Using Python, Wiley-India, 1st ed., 2019.
1. C. C. Aggarwal, Neural Networks and Deep Learning, Springer, September 2018.
2. F. Chollet, Deep Learning with Python, Manning Publications; 1st edition, 2017.
3. J. D. Kelleher, Deep Learning (MIT Press Essential Knowledge series), The MIT Press, 2019.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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 built-in functions to deal with numerical methods.

Unit-1
Teaching Hours:15
Introduction to Python and Roots of High-Degree Equations

Introduction and Simple Iterations Method, Finite Differences Method

Unit-2
Teaching Hours:15
Systems of Linear Equations

Introduction & Gauss Elimination Method: Algorithm, Gauss Elimination Method, Jacobi's Method, Gauss-Seidel's Method, Linear System Solution in NumPy and SciPy & Summary

Unit-3
Teaching Hours:15
Numerical differentiation, Integration and Ordinary Differential Equations

Introduction & Euler's Method, Second Order Runge-Kutta's Method, Fourth Order Runge-Kutta's Method, Fourth Order Runge-Kutta'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.

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:

 Component Parameter Mode of assessment Maximum points CIA I Mastery of  the fundamentals Lab Assignments 10 CIA II Familiarity with the commands and execution of them in solving problems. Analytical and Problem Solving skills Lab Work Problem Solving 10 CIA III Conceptual clarity and analytical skills in solving Problems using Mathematical Package / Programming Lab Exam based on the Lab exercises 25 Attendance Regularity and Punctuality Lab attendance 05                                  =100%:5      97 – <100% :4      94 – < 97%  :3      90 – <94%  :2      85 – <90%  :1                <85% :0 Total 50

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.

 Unit-1 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 Viva-Voce. 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 Viva-Voce 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.
Unit-1
Teaching Hours:10
Introductory concepts

The mechanical system - Generalised Coordinates - constraints - virtual work - Energy and momentum.

Unit-2
Teaching Hours:20
Lagrange's and Hamilton's equations

Derivation of Lagrange's equations: Kinetic energy, Lagranges equations, form of equations of motion, Non-holonomic 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.

Unit-3
Teaching Hours:15
Lagrangian Mechanics on Manifolds

Introduction to differentiable manifolds, Lagrangian system on a manifold, Lagrangian system with holonomic constraints, Lagrangian non-autonomous system, Noether's theorem, equivalence of D'Alembert-Lagrange principle and the variational principle, Linearization of the Lagrangian system, small oscillations.

Unit-4
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:
1. D. T. Greenwood, Classical Dynamics, Reprint, USA: Dover Publications, 2012. (Unit 1 and Unit 2)
2. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., New York, Springer, 2013 (Unit 3 and Unit 4)

1. H. Goldstein, Classical Mechanics, 2nd ed., New Delhi: Narosa Publishing House, 2001.
2. N. C. Rana and P. S. Joag, Classical Mechanics, 29th Reprint, New Delhi: Tata McGraw- Hill, 2010.
3. J. E. Marsden, R. Abraham, Foundations of Mechanics, 2nd ed., American Mathematical Society, 2008.
4. D. D. Holm, Geometric Mechanics-Part I: Dynamics and Symmetry. World Scientific Publishing Company, 2nd ed., 2011.
5. D. D. Holm, Geometric Mechanics: Part II: Rotating, Translating and Rolling. World Scientific Publishing Company, 2008
6. D. D. Holm, T. Schma and C. Stocia, Geometry, Symmetry and Mechanics - from finite to infinite dimensions, Oxford University Press Inc., 2009.
Evaluation Pattern
 Component Mode of Assessment Parameters Points CIA I Written Assignment Reference work Mastery of the core concepts 10 CIA II Mid-semester Examination Basic, conceptual and analytical knowledge of the subject 25 CIA III Written Assignment Class Test Problem solving skills Familiarity with the proof techniques 10 Attendance Attendance Regularity and Punctuality 05 ESE Basic, conceptual and analytical knowledge of the subject 50 Total 100

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