Department of
MATHEMATICS






Syllabus for
Master of Science (Mathematics)
Academic Year  (2020)

 
1 Semester - 2020 - Batch
Paper Code
Paper
Hours Per
Week
Credits
Marks
MTH131 REAL ANALYSIS 4 4 100
MTH132 ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS 4 4 100
MTH133 ADVANCED ALGEBRA 4 4 100
MTH134 FLUID MECHANICS 4 4 100
MTH135 ELEMENTARY GRAPH THEORY 4 4 100
MTH151 INTRODUCTION TO FOSS TOOLS 3 3 50
2 Semester - 2020 - Batch
Paper Code
Paper
Hours Per
Week
Credits
Marks
MTH231 GENERAL TOPOLOGY 4 4 100
MTH232 COMPLEX ANALYSIS 4 4 100
MTH233 LINEAR ALGEBRA 4 4 100
MTH234 ADVANCED FLUID MECHANICS 4 4 100
MTH235 ALGORITHMIC GRAPH THEORY 4 4 100
MTH251 PYTHON PROGRAMMING FOR MATHEMATICS 3 3 50
3 Semester - 2019 - Batch
Paper Code
Paper
Hours Per
Week
Credits
Marks
MTH331 MEASURE THEORY AND LEBESGUE INTEGRATION 4 4 100
MTH332 NUMERICAL ANALYSIS 4 4 100
MTH333 CLASSICAL MECHANICS 4 4 100
MTH334 LINEAR ALGEBRA 4 4 100
MTH335 ADVANCED GRAPH THEORY 4 4 100
MTH351 NUMERICAL METHODS USING PYTHON 3 3 50
MTH381 INTERNSHIP 0 2 0
4 Semester - 2019 - Batch
Paper Code
Paper
Hours Per
Week
Credits
Marks
MTH431 DIFFERENTIAL GEOMETRY 4 4 100
MTH432 COMPUTATIONAL FLUID DYNAMICS 4 4 100
MTH433 FUNCTIONAL ANALYSIS 4 4 100
MTH4401 CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS 4 4 100
MTH4402 MAGNETOHYDRODYNAMICS 4 4 100
MTH4403 WAVELET THEORY 4 4 100
MTH4404 MATHEMATICAL MODELLING 4 4 100
MTH4405 ATMOSPHERIC SCIENCE 4 4 100
MTH4406 ADVANCED LINEAR PROGRAMMING 4 4 100
MTH4407 DESIGN AND ANALYSIS OF ALGORITHMS 4 4 100
MTH4408 INTRODUCTION OF THEORY OF MATROIDS 4 4 100
MTH4409 TOPOLOGICAL GRAPH THEORY 4 4 100
MTH4410 ALGEBRAIC GRAPH THEORY 4 4 100
MTH451 NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON 3 3 50
MTH481 PROJECT 2 2 100
        

  

Assesment Pattern

Assessment Pattern

 

SEMESTER

COURSE CODE

COURSE TITLE

CIA (Max Marks)

Attendance (Max Marks)

ESE (Max Marks)

1

 

(Batch:2019-2021 - First Year)

     

MTH131

Real Analysis

45

5

50

MTH132

Ordinary and Partial Differential Equations

45

5

50

MTH133

Advanced Algebra

45

5

50

MTH134

Fluid Mechanics

45

5

50

MTH135

Elementary Graph Theory

45

5

50

MTH111 Research Methodology G --- ---
         

MTH151

 Introduction to Latex and FOSS tools

 50  ---  ---

2

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

MTH211

 Mathematics Lab using Python

Machine Learning

 50

G

 --

--

--

--

3

 

(Batch:2019-2021 - Second Year)

     

MTH331

Measure Theory and Lebesgue Integration

45

5

50

MTH332

Numerical Analysis

45

5

50

MTH333

Classical Mechanics

45

5

50

MTH334

Linear Algebra

45

5

50

MTH335

Advanced Graph Theory

45

5

50

MTH351

 Numerical Methods using Python

     

MTH371

 Internship in PG Course

     

4

MTH431

 Differential Geometry

45

5

50

MTH432

 Computational Fluid Dynamics

45

5

50

MTH433

 Functional Analysis

45

5

50

MTH451

 Numerical Methods for Boundary Value Problem using Python

     

MTH481

 Project

     

MTH44X

 Elective 1

45

5

50

MTH44Y

 Elective 2

45

5

50

 

 List of Elective Courses

     

MTH441

 Calculus of Variations and Integral Equations

MTH442

 Magnetohydrodynamics

MTH443

 Wavelet Theory

MTH444

 Mathematical Modelling

MTH445

 Atmospheric Science

MTH446

 Advanced Linear Programming

MTH447

Design and Analysis of Algorithms

MTH448

 Combinatorial Mathematics

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

Department Overview:
Department of Mathematics, CHRIST (Deemed to be University) is one of the oldest departments of the University. It offers programmes in Mathematics at the under graduate level, post graduate level as well as M.Phil and Ph.D. The department aims to * enhance the logical, reasoning, analytical and problem solving skills of students. * cultivate a research culture in young minds. * foster aesthetic appreciation for mathematical thinking. * encourage students for pursuing higher studies in mathematics. * motivate students to uphold scientific integrity and objectivity in professional endeavors.
Mission Statement:
Vision Excellence and Service Mission To organize, connect, create and communicate mathematical ideas effectively, through 4D?s; Dedication, Discipline, Direction and Determination
Introduction to Program:
The M.Sc. course in Mathematics aims at developing mathematical ability in students with acute and abstract reasoning. The course will enable students to cultivate a mathematician?s habit of thought and reasoning and will enlighten students with mathematical ideas relevant for oneself and for the course itself. COURSE DESIGN: Masters in Mathematics is a two year programme spreading over four semesters. In the first two semesters focus is on the basic courses in mathematics such as Algebra, Topology, Analysis, Discrete Mathematics and Number Theory/Cryptography along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and 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) T
Program Objective:
Programme Objective: POBJ1. To provide learners with an in-depth knowledge, abilities and insight in different topics in Mathematics. POBJ2. To learn to apply mathematics to practical problems and help in problem solving. POBJ3. To encourage collaborative learning through projects and research activities so that they can pursue research programmes. POBJ4. To provide a platform for the learner to engage in various academic activities independently or in a group. POBJ5. To make the learner familiar with FOSS tools such as Python, MAXIMA, Scilab and with the tool Mathematica. POBJ6. To make the learner familiar with the Teaching Technology and Research Methodology in Mathematics. Programme Outcomes: PO1. To be able to explain mathematical principle to formulate, model and hence find solutions to practical problems. PO2. To be able explain the advantages, limitations, importance of mathematics and its techniques to solve real life problems. PO3. To acquire the skills which are necessary to do research/higher studies in the areas of the learner?s choice. PO4. To be capable in formulating and analysis of mathematical models of practical problems. PO5. To acquire skills to use mathematical FOSS tools efficiently in practical problems. PO6. The learner will be able to become a good teacher or researcher in mathematics so that he/she can communicate the subject with others in an efficient way.

MTH131 - REAL ANALYSIS (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course will help students to understand the concepts of functions of single and several variables. This course includes such concepts as 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.

Learning Outcome

On successful completion of the course, the students should be able to

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.

Essential Reading / Recommended Reading
  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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description : This helps students understand the beauty of the important branch of mathematics, namely, differential equations. This course includes a study of second order linear differential equations, adjoint and 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.

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Understand concept of Linear differential equation, Fundamental set Wronskian.

CO2. Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Lagrange's Identity, Green’s formula, Eigen value and Eigen functions.

CO3. Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial.

CO4. Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod.

CO5. Demonstrate on the canonical form of second order PDE.

CO6. Demonstrate initial value boundary problem for homogeneous and 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. Christian 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.
Essential Reading / Recommended Reading
  1. K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics andEngineering, Cambridge, 2005.
  2. Edwards Penney, Differential Equations and Boundary Value Problems, Pearson Education, 2005.
  3. J. David Logan, Partial Differential Equations, 2nd ed., New York: Springer, 2002.
  4. Alan 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. Tyn 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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory.

Course objectives​: This course will help the learner to

COBJ1. enhance the knowledge on advanced level algebra

COBJ2. understand the proof techniques for the theorems on advanced group theory, Rings and Galois Theory

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Demonstrate knowledge of conjugates, the Class Equation and Sylow  theorems

CO2. Demonstrate knowledge of polynomial rings and associated properties

CO3. Derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals

CO4. Demonstrate the characteristic of a field and the prime subfield;

CO5. Demonstrate Factorization and ideal theory in the polynomial ring; the structure of a primitive polynomials; Field extensions and characterization of finite normal extensions as splitting fields; The structure and construction of finite fields; Radical field extensions;Galois group and Galois theory

Unit-1
Teaching Hours:15
Advanced Group Theory
 

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

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.

Essential Reading / Recommended Reading
  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 - FLUID MECHANICS (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course aims at introducing the fundamentals of fluid mechanics.  This course aims at imparting the knowledge on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of 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

Learning Outcome

On successful completion of the course, the students should be able to

CO1. confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem.

CO2. able to understand the basics laws of Fluid mechanics and their physical interpretations.

CO3. able to understand two or three dimension flows incompressible flows.

CO4. able to understand the viscous flows, their mathematical modelling and physical interpretations.

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.
Essential Reading / Recommended Reading
  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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course is an introductory course to the basic concepts of Graph Theory. This includes definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths.

Course objectives:This course will help the learner to

COBJ1. know the history and development of graph theory

COBJ2. understand all the elementary concepts and proof techniques  in Graph Theory

Learning Outcome

Course outcomes: The successful completion of this course the students will be able to:

CO1. write precise and accurate mathematical definitions of basics concepts in graph theory

CO3. provide appropriate examples and counterexamples to illustrate the basic concepts

CO3. demonstrate and apply various proof techniques in proving theorems in graph theory

CO4. showcase mastery in using graph drawing tools

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:

D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

Essential Reading / Recommended Reading
  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

MTH151 - INTRODUCTION TO FOSS TOOLS (2020 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:50
Credits:3

Course Objectives/Course Description

 

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

Course objectives​: This course will help the learner to:

COBJ1. use the FOSS tool WxMaxima effectively.

COBJ2. use the FOSS tool Scilab effectively

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Use the basic commands in WxMaxima including the 2D and 3D plots

CO2. Have a strong command on the inbuilt commands required for the learning and analyzing mathematics

CO3. Solve problems / applied problems on mathematics by using Scilab.

Unit-1
Teaching Hours:15
Introduction to WxMaxima
 

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

Unit-2
Teaching Hours:15
Introduction to Scilab
 

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

Unit-3
Teaching Hours:15
Solving problems using Scilab / WxMaxima
 

Solving systems of equation and explain consistence - Find the values of some standard trigonometric functions in radians as well as in degree - Create polynomials of different degrees and find its real roots  - Display Fibonacci series - Display non-Fibonacci series

Text Books And Reference Books:
  1. C. Gomez, C. Bunks, J. P. Chancelier, F. Delebecque, M. Goursat, R. Nikouhah and S. Steer, Engineering and scientific computing with scilab. Birkhauser, 2013.
  2. Z. Hannan, wxMaxima for calculus I. Zachary Hannan, 2015.
  3. Z. Hannan, wxMaxima for calculus II. Zachary Hannan, 2015.

 

Essential Reading / Recommended Reading
  1. E.L.Woollett, Maxima by Example, Online resource(https://web.csulb.edu/~woollett/), 2009.
  2. Sourceforge,  Introduction to Graphs,  Online resource (http://maxima.sourceforge.net/docs/manual/de/maxima_50.html#SEC290).
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



MTH231 - GENERAL TOPOLOGY (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc.,.

Course objectives​: This course will help the learner to:

COBJ1. provide precise definitions and  appropriate examples and counter examples of  fundamental concepts in general topology

COBJ2. acquire knowledge about generalization of the concept of continuity and related properties

COBJ3. appreciate the beauty of deep mathematical results such as  Uryzohn’s lemma and understand and apply various proof techniques

Learning Outcome

On successful completion of the course, the students should be able to

CO1. define topological spaces, give examples and counterexamples on concepts like open sets, basis and subspaces

CO2. establish equivalent definitions of continuity and apply the same in proving theorems

CO3. understand the concepts of metrizability, connectedness, compactness and learn the related theorems

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.

Essential Reading / Recommended Reading
  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. Crump, 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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course will help students learn about the essentials of complex analysis. This course includes important concepts such as power series, analytic functions, linear transformations, Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem, argument principle, Schwarz lemma , Rouche’s theorem and Hadamard’s 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.

Learning Outcome

On successful completion of the course, the students should be able to

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.
Essential Reading / Recommended Reading
  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. David 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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms.

Course Objective: This course will help learner to

COBJ1. gain proficiency on the theories of Linear Algebra

COBJ2. enhance problem solving skills in Linear Algebra

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Have thorough understanding of the Linear transformations

CO2.Demonstrate the elementary canonical forms, rational and Jordan forms.

CO3. Apply the inner product space

CO4. Express familiarity in using bilinear forms

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.

Essential Reading / Recommended Reading
  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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems.  This course also includes the analysis Prandtlboundry layer, porous media and Non-Newtonian fluid.

Course objectives​: This course will help the learner to

COBJ1. understand the different modes of heat transfer and their applications.

COBJ2. understand the importance of doing the non-dimensionalization of basic equations.

COBJ3. understand the boundary layer flows.

COBJ4. familiarity with porous medium and non-Newtonian fluids

Learning Outcome

On successful completion of the course, the students should be able to

CO1. understand the basic laws of heat transfer and understand the fundamentals of convective heat transfer process.

CO2. solve Rayleigh - Benard problem and their physical phenomenon.

CO3. solve and understand different boundary layer problems

CO4. give an introduction to the basic equations with porous medium and solution methods for mathematical modeling of viscous fluids and elastic matter

UNIT 1
Teaching Hours:15
Dimensional Analysis and Similarity
 

Introduction to heat transfer, different modes of heat transfer- conduction, convection and radiation, steady and unsteady heat transfer, free and forced convection. 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
Prandtl Boundry Layer
 

Boundary layer concept, the boundary layer equations in two-dimensional flow, the boundary layer along a flat plate, the Blasius solution. Stagnation point flow. Falkner-Skan family of equations.

UNIT 4
Teaching Hours:15
Porous Media and Non - Newtonian Fluids
 

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

Text Books And Reference Books:
  1. Drazin and Reid, Hydrodynamic instability, Cambridge University Press, 2006.
  2. S. Chardrasekhar, Hydrodynamic and hydrodmagnetic stability, Oxford University Press, 2007 (RePrint).
Essential Reading / Recommended Reading
  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 (2020 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description: This course helps the students to understand the colouring of graphs, Planar graphs, edges and cycles.

Course objectives: This course will help the learner to

COBJ1. Construct examples and proofs pertaining to the basic theorems

COBJ2. Apply the theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory

COBJ3. Write graph theoretic ideas in a coherent and technically accurate manner.

Learning Outcome

On successful completion of the course, the students should be able to

CO1. understand the basic concepts and fundamental results in matching, domination, coloring and planarity.

CO2. reason from definitions to construct mathematical proofs

CO3. obtain a solid overview of the questions addressed by graph theory and will be exposed  to emerging areas of research

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.

Essential Reading / Recommended Reading
  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 - PYTHON PROGRAMMING FOR MATHEMATICS (2020 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:50
Credits:3

Course Objectives/Course Description

 

Course description: This course aims at introducing the programming language Python and its uses in solving problems on discrete mathematics and differential equations.

Course objectives: This course will help the learner to

COBJ1.Acquire skill in usage of suitable functions/packages of Python to solve mathematical problems.

COBJ2.Gain proficiency in using Python to solve problems on Differential equations. 

COBJ3. The built in functions required to deal withcreating and visualizing Graphs, Digraphs, MultiGraph.

Learning Outcome

By the end of the course the learner will be able to:

CO1. Acquire proficiency in using different functions of Python to compute solutions of basic mathematical problems. 

CO2. Demonstrate the use of Python to solve differential equations along with visualize the solutions.

CO3. Be familiar with the built-in functions to deal with  Graphs and Digraphs.

Unit-1
Teaching Hours:15
Introduction to Python Programming
 

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

Unit-2
Teaching Hours:15
Differential Equations using Python
 

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

Unit-3
Teaching Hours:15
Discrete Mathematics using Python
 

Creating and visualizing Graphs, Digraphs, MultiGraphs and MultiDiGraph - Python methods  for reporting nodes, edges and neighbours of the given graph / digraph - Python methods for counting nodes, edges and neighbours of the given graph / digraph.

Text Books And Reference Books:
  1. I. N. SNEDDON, Elements of PDE’s , McGraw Hill Book company Inc. 2009.
  2. L DEBNATH , Nonlinear PDE’s for Scientists and Engineers, Birkhauser, Boston, 2008.
  3. C. L. Liu, Elements of Discrete Mathematics, Tata McGraw-Hill, 2000.
  4. P. Farrell, Math adventures with Python: an illustrated guide to exploring math with code. No Starch Press, 2019
Essential Reading / Recommended Reading
  1. J.P. Tremblay and R.P. Manohar : Discrete Mathematical Structures with applications to computer science, McGraw Hill, 1997.
  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.
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

 

MTH331 - MEASURE THEORY AND LEBESGUE INTEGRATION (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

 

Course description: The Coursecovers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue.

Course objectives: This course will help the learner to 

COBJ1. enhance the understanding of the advanced notions from Mathematical Analysis

 

COBJ2. know more about the Measure theory and Lebesgue Integration

Learning Outcome

On successful completion of the course, the students should be able to

CO1. understand the fundamental concepts of Mathematical Analysis.

CO2. tate some of the classical theorems in of Advanced Real Analysis.

CO3. be familiar with measurable sets and functions.

CO4. integrate a measurable function

CO5. understand the properties of Lp Spaces

 

Unit-1
Teaching Hours:20
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:10
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.

Essential Reading / Recommended Reading
  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 (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description: This course deals with the theory and application of various advanced methods of numerical approximation. These methods or techniques help us to approximate the solutions of problems that arise in science and engineering. The emphasis of the course will be the thorough study of numerical algorithms to understand the guaranteed accuracy that various methods provide, the efficiency and scalability for large scale systems and issues of stability.

Course objectives: This course will help the learner

 

COBJ1. to develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use.
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.

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Derive numerical methods for approximating the solution of problems of algebraic and transcendental equations, ordinary differential equations and boundary value problems.

CO2. Implement a variety of numerical algorithms appropriately in various situations

CO3. interpret, analyse and evaluate results from numerical computations

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, Choleky and Delittle methods), consistency and ill-conditioned system of equations, Tri-diagonal 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, Gelarkin 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
Essential Reading / Recommended Reading
  1. R.L. Burden and J. Douglas 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.
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 - CLASSICAL MECHANICS (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description: Classical Mechanics is the study of mechanics using Mathematical methods. This course deals with some of the key ideas of classical mechanics. The concepts covered in the course include generalized coordinates, Lagrange’s equations, Hamilton’s equations and Hamilton - Jacobi theory.

Course objectives: This course will help the learner to

 

COBJ1. derive necessary equations of motions based on the chosen configuration space.
COBJ2. gain sufficient skills in using the derived equations in solving the applied problems in Classical Mechanics.

Learning Outcome

On successful completion of the course, the students should be able to:

 

CO1. Interpret mechanics through the configuration space.
CO2. solve problems on mechanics by using hamilton’s principle
CO3. Illustrate the use of Hamilton –Jacobi theory in finding equations of motions

Unit-1
Teaching Hours:12
Introductory concepts
 

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

Unit-2
Teaching Hours:20
Lagrange's equation
 

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

Unit-3
Teaching Hours:13
Hamilton's equations
 

Hamilton's principle - Hamilton’s equations - Other variational principles - phase space.

Unit-4
Teaching Hours:15
Hamilton - Jacobi Theory
 

Hamilton's Principal Function – The Hamilton - Jacobi equation - Separability.

Text Books And Reference Books:

Donald T. Greenwood, Classical Dynamics, Reprint, USA: Dover Publications, 2012.

Essential Reading / Recommended Reading
  1. H. Goldstein, Classical Mechanics, Second edition, 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.
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

MTH334 - LINEAR ALGEBRA (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms.

Course Objective: This course will help learner to

COBJ1. gain proficiency on the theories of Linear Algebra

COBJ2. enhance problem solving skills in Linear Algebra

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Have thorough understanding of the Linear transformations

CO2. Demonstrate the elementary canonical forms, rational and Jordan forms.

CO3. Apply the inner product space

CO4. Express familiarity in using bilinear forms

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.

Essential Reading / Recommended Reading
  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

MTH335 - ADVANCED GRAPH THEORY (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description:Domination of Graphs, perfect graphs, chromatic graph theory and Eigenvalues of Graphs are dealt with in the detail in this course.

Course objectives: This course will help the learner to

COBJ1. understand the advanced topics in Graph Theory

COBJ2. enhance the understanding of techniques of writing proofs for advanced topics in Graph Theory

Learning Outcome

By the end of the course the learner will be able to:

CO1. have thorough understanding of the concepts in domination and perfect graphs

CO2. familiarity in implementing the acquired knowledge appropriately

CO3. mastery in employing proof techniques

Unit-1
Teaching Hours:15
Domination in Graphs
 

Domination in Graphs, Bounds in terms of Order, Bounds in terms of Order, Degree and Packing, Bounds in terms of Order and Size, Bounds in terms of Degree, Diameter and Girth, Bounds in terms of Independence and Covering.

Unit-2
Teaching Hours:15
Chromatic Graph Theory
 

T-Colourings,  L(2,1)-colourings,  Radio Colourings,  Hamiltonian Colourings, Domination and Colourings.

Unit-3
Teaching Hours:15
Perfect Graphs
 

The Perfect Graph Theorem, Chordal Graphs Revisited, 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. T.W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs. Reprint, CRC Press, 2000.

    2. D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011. 

    3. J. Bang-Jensen and G. Gutin, Digraphs. London: Springer, 2009.

    4. G. Chartrand and P. Zhang, Chromatic Graph Theory. New York: CRC Press, 2009.

Essential Reading / Recommended Reading
  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. J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008
  5. J. Clark and D.A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.
  6. R. Balakrishnan and K Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.
  7. R. Diestel, Graph Theory, New Delhi: Springer, 2006.
  8. 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

MTH351 - NUMERICAL METHODS USING PYTHON (2019 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:50
Credits:3

Course Objectives/Course Description

 

Course description: In this course programming Numerical Methods in Python will be focused. How to program the numerical methods step by step to create the most basic lines of code that run on the computer efficiently and output the solution at the required degree of accuracy.

Course objectives: This course will help the learner to

COBJ1. Program the numerical methods to create simple and efficient Python codes that output the numerical solutions at the required degree of accuracy.

COBJ2. Use the plotting functions of matplotlib to visualize the results graphically.

COBJ3. Acquire skill in usage of suitable functions/packages of Python to solve initial value problems numerically.

Learning Outcome

By the end of the course the learner will be able to:

CO1. Acquire proficiency in using different functions of Python to compute solutions of system of equations.

CO2. Demonstrate the use of Python to solve initial value problem numerically along with graphical visualization of the solutions .

CO3. Be familiar with the 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.

Essential Reading / Recommended Reading

Hans Fangohr, Introduction to Python for Computational Science and Engineering (A beginner’s guide), University of Southampton, 2015.

Evaluation Pattern

The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:

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 (2019 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:0
Max Marks:0
Credits:2

Course Objectives/Course Description

 

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

Learning Outcome

Course Outcomes

On successful completion of the course, the students should be able to

 

COBJ1. Expose to the field of their professional interest
COBJ2. Explore an opportunity to get practical experience in the field of their interest
COBJ3. Strengthen the research culture

 

Unit-1
Teaching Hours:45
Internship in PG Mathematics course
 

M.Sc. Mathematics students have to undertake a mandatory internship of not less than 45 working days at any of the following: reputed research centers, recognized educational institutions, summer research fellowships, programmes like M.T.T.S or any other approved by the P.G. coordinator and H.O.D.

In the present scenario of COVID 19 pandemic, the students unable to do internship in an organization, have to complete one MOOC  in Mathematics that suits the academic interest of the student in consultation with the assigned internship supervisors and a dissertation based on a detailed review of two research articles. The duration of the course has to be at least 30 hours and should be completed on or before 20 June 2020. However, if a student chooses to go ahead with the internship, then they should complete at least 25 working days in the organization on or before 31 May 2020, in which case submission of the dissertation is not necessary.

The internship is to be undertaken at the end of second semester (during first year vacation). The report submission and the presentation on the report will be held during the third semester and the credits will appear in the mark sheet of the third semester. 

The students will have to give an internship proposal with the following details: Organization where the student proposes to do the internship, reasons for the choice, nature of internship, period on internship, relevant permission letters, if available, name of the mentor in the organization, email, telephone and mobile numbers of the person in the organization with whom Christ University could communicate matters related to internship. Typed proposals will have to be given at least one month before the end of the second semester.

The coordinator of the programme in consultation with the HOD will assign faculty members from the department as guides at least two weeks before the end of second semester. The students will have to be in touch with the guides during the internship period either through personal meetings, over the phone or through email.

At the place of internship, students are advised to be in constant touch with their mentors. 

At the end of the required period of internship, the candidates will submit a report in a specified format adhering to department guidelines. The report should be submitted within the first 10 days of the reopening of the University for the third semester.  

The students doing the MOOCs are expected to prepare course notes on their own using all the resources accessible and this is to be given as the first part of the internship report.

Within 20 days from the day of reopening, the department must hold a presentation by the students. During the presentation, the supervisor or a nominee of the supervisor should be present and be one of the evaluators. Students should preferably be encouraged to make a presentation of their report. A minimum of 10 minutes should be given for each of the presenters. The maximum limit is left to the discretion of the evaluation committee. 

 

Students will get 2 credits on successful completion of internship.

Text Books And Reference Books:

.

Essential Reading / Recommended Reading

.

Evaluation Pattern

.

MTH431 - DIFFERENTIAL GEOMETRY (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description: Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in three-dimensional Euclidean space and will be acquainted with the ways of generalising these concepts to higher dimensions.

 

Course objectives: This course will help the learner to

CO BJ1. write proofs for the theorems on Curves and Surfaces in R3.

COBJ2. implement the properties of curves and surfaces in solving problems described in terms of tangent vectors / vector fields / forms etc.,

Learning Outcome

On successful completion of the course, the students should be able to

CO1. express  sound knowledge on the basic concepts in geometry of curves and surfaces in Euclidean space, especially E3.

CO2. demonstrate  mastery in solving typical problems associated with the theory.

CO3. extend the knowledge in generalizing the concepts learned to higher dimensions.

UNIT 1
Teaching Hours:15
Calculus on Euclidean Geometry
 

Euclidean Space - Tangent Vectors  - Directional derivatives - Curves in 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
Euclidean Geometry and Calculus on Surfaces
 

Isometries of E3 - The derivative map of an Isometry - Surfaces in E3 - patch computations - Differential functions and Tangent vectors - Differential forms on a surface - Mappings of Surfaces.

UNIT 4
Teaching Hours:15
Shape Operators
 

The Shape operator of M in E3 - Normal Curvature - Gaussian Curvature - Computational Techniques - Special curves in a surface - Surfaces of revolution.

Text Books And Reference Books:

B.O’Neill, Elementary Differential geometry, 2nd revised ed., New York: Academic Press, 2006.

Essential Reading / Recommended Reading
  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.
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 - COMPUTATIONAL FLUID DYNAMICS (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course helps the students to learn the solutions of partial differential equations using finite difference and finite element methods. This course also helps them to know how solve the Burger’s equations using finite difference equation, quasi-linearization of non-linear equations.

Course objectives​: This course will help the students to

COBJ1. be familiar with solving PDE using finite difference method and finite element method

COBJ2. Understand the non-linear equation Burger’s equation using finite difference method

COBJ3.Understand the compressible fluid flow using ACM, PCM and SIMPLE methods

COBJ4.Solve differential equations using finite element method usingdifferent shape functions

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Solve both linear and non-linear PDE using finite difference methods

CO2. Understand both physics and mathematical properties of governing Navier-Stokes equations and define proper boundary conditions for solution

CO3. Understanding of physics of compressible and incompressible fluid flows

CO4. Write the programming in MATLAB to solve PDE using finite difference method

Unit-1
Teaching Hours:15
Numerical solution of elliptic partial differential equations
 

 

Review of classification of partial differential equations, classification of boundary conditions, numerical analysis, basic governing equations of fluid mechanics. Difference methods for elliptic partial differential equations, difference schemes for Laplace and Poisson’s equations, iterative methods of solution by Jacobi and Gauss-Siedel, solution techniques for rectangular and quadrilateral regions.

Unit-2
Teaching Hours:15
Numerical solution of parabolic and hyperbolic partial differential equations
 

Difference methods for parabolic equations in one-dimension, methods of Schmidt, Laasonen, Crank-Nicolson and Dufort-Frankel, stability and convergence analysis for Schmidt and Crank-Nicolson methods, ADI method for two-dimensional parabolic equation, explicit finite difference schemes for hyperbolic equations, wave equation in one dimension.

Unit-3
Teaching Hours:15
Finite Difference Methods for non-linear equations
 

Finite difference method to nonlinear equations, coordinate transformation for arbitrary geometry, Central schemes with combined space-time discretization-Lax-Friedrichs, Lax-Wendroff, MacCormack methods, Artificial compressibility method, pressure correction method – Lubrication model, convection dominated flows – Euler equation – Quasilinearization of Euler equation, Compatibility relations, nonlinear Burger equation.

Unit-4
Teaching Hours:15
Finite Element Methods
 

Introduction to finite element methods, one-and two-dimensional bases functions – Lagrange and Hermite polynomials elements, triangular and rectangular elements, Finite element method for one-dimensional problem and two-dimensional problems: model equations, discretization, interpolation functions, evaluation of element matrices and vectors and their assemblage.

Text Books And Reference Books:
  1. T. Chung, Computational Fluid Dynamics, Cambridge University Press, 2003.
  2. J. Blazek, Computational Fluid Dynamics, Elsevier Science, 2001.
Essential Reading / Recommended Reading
  1. C. Fletcher, Computational Techniques for Fluid Dynamics 1, Springer Berlin Heidelberg, 1991.
  2. C. Fletcher, Computational Techniques for Fluid Dynamics 2, Springer Berlin Heidelberg, 1991.
  3. D. Anderson, R. Pletcher, J. Tannehill and, Computational Fluid Mechanics and Heat Transfer, McGraw Hill Book Company, 2010.
  4. K. Muralidhar and T. Sundararajan, Computational Fluid Flow and Heat Transfer, Narosa Publishing House, 2010.
  5. W. Ames, Numerical Method for Partial Differential Equation, Academic Press, 2008.
  6. T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, Springer-Verlag, 2005.
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

MTH433 - FUNCTIONAL ANALYSIS (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This abstract course imparts an in-depth analysis of Banach spaces, Hilbert spaces, conjugate spaces, etc. This course also includes a few important applications of functional analysis to other branches of both pure and applied mathematics.

Course Objective. This course will help learner to

COBJ1: know the notions behind Functional Analysis

COBJ2. enhance the problem solving ability in Functional Analysis

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Explain the fundamental concepts of functional analysis.

CO2. Understand the approximation of continuous functions.

CO3. Understand concepts of Hilbert and Banach spaces with l2 and lp spaces serving as examples.

CO4. Understand the definitions of linear functional and prove the Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem, etc.

CO5. Define linear operators, self adjoint, isometric and unitary operators on Hilbert spaces.

Unit-1
Teaching Hours:15
Banach spaces
 

Normed linear spaces, Banach spaces, continuous linear transformations, isometric isomorphisms, functionals and the Hahn-Banach theorem, the natural embedding of a normed linear space in its second dual.

Unit-2
Teaching Hours:12
Mapping theorems
 

The open mapping theorem and the closed graph theorem, the uniform boundedness theorem, the conjugate of an operator.

Unit-3
Teaching Hours:15
Inner products
 

Inner products, Hilbert spaces, Schwarz inequality, parallelogram law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormal sets. 

 

Unit-4
Teaching Hours:18
Conjugate space
 

The conjugate space, the adjoint of an operator, self-adjoint, normal and unitary operators, projections, finite dimensional spectral theory.

 

Text Books And Reference Books:

G.F. Simmons, Introduction to topology and modern Analysis, Reprint, Tata McGraw-Hill, 2004.

Essential Reading / Recommended Reading
  1. K. Yoshida, Functional analysis, 6th ed., Springer Science and Business Media, 2013.
  2. Kreyszig, Introductory functional analysis with applications, 1st ed., John Wiley, 2007.
  3. B.V. Limaye, Functional analysis, 3rd ed., New Age International, 2014.
  4. W. Rudin, Functional analysis, 2nd ed., McGraw Hill, 2010.
  5. S. Karen, Beginning functional analysis, Reprint, Springer, 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

MTH4401 - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course description: This course introduces fundamental concepts and some standard results of calculus of variations and the integral equations. It plays an important role for solving various engineering sciences problems.  Therefore, it has tremendous applications in diverse fields in engineering sciences.

Course objectives: This course will help the learners to study extrema of functional, the Brachistochrone problem, Euler’s equation, variational derivative and invariance of Euler’s equations. It also contains Fredholm and Volterra integral equations and their solutions using various methods such as Neumann series, resolvent kernels, Green’s function approach and transform methods

Learning Outcome

On successful completion of the course, the students should be able to

CO1. Derive some classical differential equations by using principles of calculus of variations.

CO2. Knowledge of Variational Problems, Euler-Lagrange Condition, Second Variation,Generalizations of the Variational Problem.

CO3. find maximum or minimum of a functional using calculus of variations Technique, solve Volterra integral equations and Fredholm integral equations

CO4. Reduce the differential equations to integral equations.

Unit-1
Teaching Hours:18
Euler equations and variational notations
 

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

Unit-2
Teaching Hours:16
Advanced variational problems
 

Galerkian Technique, the Rayleigh-Ritz method.

Unit-3
Teaching Hours:12
Linear integral equations
 

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

Unit-4
Teaching Hours:14
Methods for solutions of linear integral equations
 

Fredholm equations with separable kernels, homogeneous integral equations, characteristic values and characteristic functions of integral equations, Hilbert-Schmidt theory, iterative methods for solving integral equations of the second kind, the Neumann series.

Text Books And Reference Books:
  1. F.B. Hildebrand, Methods of Applied Mathematics, New York: Dover, 1992.
  2. R.P. Kanwal, Linear Integral Equations: Theory and Techniques, New York: Birkhäuser,  2013.
Essential Reading / Recommended Reading
  1. B. Dacorogna, Introduction to the Calculus of Variations, London: Imperial College Press, 2004.
  2. F. Wan, Introduction to the Calculus of Variations and Its Applications, New York: Chapman/Hall, 1995.
  3. J. Jost and X. Li-Jost, Calculus of Variations, Cambridge: Cambridge University Press, 2008.
  4. C. Corduneanu, Integral Equations and Applications, Cambridge: Cambridge University Press, 2008.
  5. A. J. Jerri, Introduction to integral equations with applications. Sampling Publishing, 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

MTH4402 - MAGNETOHYDRODYNAMICS (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course provides the fundamentals of Magnetohydrodynamics, which include theory of Maxwell’s equations, basic equations, exact solutions and applications of classical MHD.

Course objectives​: This course will help the students to

COBJ1.Understand mathematical form of Gauss’s Law, Faraday’s Law and Ampere’s Law and corresponding boundary conditions

COBJ2. Derive the basic governing equations and boundary conditions of MHD flows.

COBJ3. Finding the exact solutions of MHD governing equations.

COBJ4. Understand the Alfven waves and derive their corresponding equations.

Learning Outcome

On successful completion of the course, the students should be able to

CO1.  Derive the MHD governing equations using Faraday’s law and Ampere’s law.

CO2. Solve the Fluid Mechanics problems with magnetic field.

CO3.Understand the properties of force free magnetic field.

CO4. Understand the application of Alfven waves, heating of solar corona, earth’s magnetic field.

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.

 

Essential Reading / Recommended Reading

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

MTH4403 - WAVELET THEORY (2019 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course aims at studying the fundamentals of wavelet theory. This includes the concept on the continuous and discrete wavelet transform and wavelet packets like construction and measure of wavelet sets and construction of wavelet spaces.

Course objectives​: This course will help the students to

COBJ1. understand Fourier series, Fourier transform and Wavelet transformation and their interdependence.

COBJ2. construct the wavelet transforms.

COBJ3. learn the applications of Wavelet transforms

Learning Outcome

On successful completion of the course, the students should be able to

CO1. construct the Euler’s formula of complex exponential function and convolutions

CO2. understand the discrete wavelet theory in Haar transforms

CO3. understand applications of wavelet transform

Unit-1
Teaching Hours:15
Introduction
 

Limitations of Fourier Series and Transforms, need of wavelet theory, Complex numbers and basic operation, the space L2(R), inner products, bases and projections, Euler’s formula and complex exponential function, Fourier series, Fourier transforms, Convolutions and B-Splines, the wavelet, requirements for wavelet.

Unit-2
Teaching Hours:15
The Continuous wavelet transform