Download Mathematics syllabus 2013 - Christ University

Table of Contents

1 COURSE OBJECTIVE AND METHODOLOGY .........................................................................................................1COURSE OBJECTIVE ............................................................................................................................................1METHODOLOGY .................................................................................................................................................1

2 B.Sc., MATHEMATICS – PAPER OBJECTIVES ......................................................................................................2MAT 131: INTRODUCTORY ALGEGRA.................................................................................................................2MAT 132: INTRODUCTION TO MATRIX THEORY AND APPLICATIONS................................................................2MAT 231: DIFFERENTIAL CALCULUS...................................................................................................................2MAT 232: INTEGRAL CALCULUS .........................................................................................................................2MAT 331: ORDINARY DIFFERENTIAL EQUATIONS AND APPLICATIONS .............................................................2MAT 332: PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS .................................................................2MAT 431: ANALYTICAL GEOMETRY AND SCILAB................................................................................................3MAT 432: VECTOR CALCULUS ...........................................................................................................................3MAT 531: ALGEBRA............................................................................................................................................3MAT 532: FOURIER SERIES AND INTEGRAL TRANSFORMS ................................................................................3MAT 533: NUMERICAL ANALYSIS ......................................................................................................................3MAT 631: REAL AND COMPLEX ANALYSIS .........................................................................................................3MAT 632: LINEAR ALGEBRA ...............................................................................................................................3MAT 633: NUMBER THEORY ..............................................................................................................................3

3 CERTIFICATE COURSES – PAPER OBJECTIVES .....................................................................................................4MAT 101: FOUNDATION OF MATHEMATICS: ....................................................................................................4MAT 201: INTRODUCTION TO MATHEMATICA:.................................................................................................4MAT 301: QUANTITATIVE TECHNIQUES FOR MANAGERS: ................................................................................4MAT 401: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS: .......................................................4

4 COURSE STRUCTURE ...........................................................................................................................................5B.Sc MATHEMATICS: ..........................................................................................................................................5CERTIFICATE COURSES .......................................................................................................................................5

5 SYLLABI FOR REGULAR PAPERS ..........................................................................................................................6MAT 131: INTRODUCTORY ALGEGRA.................................................................................................................6MAT 132: INTRODUCTION TO MATRIX THEORY AND APPLICATIONS................................................................8MAT 231: DIFFERENTIAL CALCULUS...................................................................................................................9MAT 232: INTEGRAL CALCULUS .......................................................................................................................11MAT 331: ORDINARY DIFFERENTIAL EQUATIONS AND APPLICATIONS ...........................................................12MAT 332: PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS ...............................................................14MAT 431: ANALYTICAL GEOMETRY AND SCILAB..............................................................................................15MAT 432: VECTOR CALCULUS .........................................................................................................................17MAT 531: ALGEBRA..........................................................................................................................................18MAT 532: FOURIER SERIES AND INTEGRAL TRANSFORMS ..............................................................................20MAT 533: NUMERICAL ANALYSIS ....................................................................................................................22MAT 631: REAL AND COMPLEX ANALYSIS .......................................................................................................24MAT 632: LINEAR ALGEBRA .............................................................................................................................26MAT 633: NUMBER THEORY ............................................................................................................................28

6 SYLLABI FOR CERTIFICATE COURSES.................................................................................................................30MAT 101: FOUNDATIONS OF MATHEMATICS..................................................................................................30MAT 201: INTRODUCTION TO MATHEMATICAL PACKAGES ............................................................................31MAT 301: QUANTITATIVE TECHNIQUES FOR MANAGERS ...............................................................................32MAT 401: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS ......................................................33

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B.Sc Mathematics - SyllabusDEPARTMENT OF MATHEMATICS

1. Course Objective and Methodology

COURSE OBJECTIVE

An educational institution that does not lead to research and specialization willremain on the way side of the higher education missing the golden opportunities forpushing farther the frontiers of knowledge and opening up new vistas of scientificendeavor to the young. Keeping the above in mind it has been proposed to continuewith triple main system with mathematics as one of the core subjects. The B.Sc.course aims at to fulfill the following broad objectives:

1. Developing a respectable intellectual level seeking to expose the variousconcepts in mathematics.

2. To enhance the students reasoning, analytical and problem solving skills.3. To cultivate a mathematicians habit of thought and reasoning.4. To enlighten the student that the mathematical ideas are relevant for oneself

no matter what his/her interests are.5. To cultivate a research culture in young minds.6. Development of students’ competence by evolving a learner-centered

curriculum.7. To encourage the students to uphold scientific integrity and objectivity in

professional endeavours.8. To pursue higher studies in top notch institutions.

The course curriculum is comprehensive and includes 14 major papers coveringall major topics in mathematics. There will be two papers in each of the first foursemesters and three papers in each of the fifth and sixth semesters. MAT131,MAT231, MAT331, MAT431, MAT531, MAT532, MAT533, MAT631, MAT632,MAT633 will carry 100 marks each and MAT132, MAT232, MAT332 and MAT432 willcarry 50 marks each. In each paper the minimum marks for pass will be 40 percent.Courses like “Foundations of Mathematics”, “Introduction to Mathematica”,“Quantitative methods for managers” and “Quantitative aptitude for competitiveexaminations” are offered as certificate courses.

After completing these three years degree course, students can opt for higherstudies, get into institutions like ISRO, NAL etc or IT oriented service sections.

METHODOLOGYIn order to realize the objectives, a methodology based on the combination of

the following will be adopted: Case studies, Debates, Project work, Team teaching,Reflective diary writing, Seminars, Field visits, Information and communicationstechnology.

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2. B.Sc., Mathematics - Paper Objectives

MAT 131: INTRODUCTORY ALGEBRA

This paper emphasizes general techniques of problem solving and explores thecreation of mathematical patterns. It aims at introducing a course that initiatesthe students into the world of Discrete Mathematics. It includes the topic likeMathematical Logic, Set Theory, Relations, Functions, Mathematical Inductionand Recursive relations.

MAT 132: INTRODUCTION TO MATRIX THEORY AND APPLICATIONS

This paper provides a rigorous introduction to the fundamentals of matrixtheory and enables the students in applying matrices as functions, in Discretedynamical systems, in calculating powers of Graph vertices and incryptography.

MAT 231: DIFFERENTIAL CALCULUS

This paper aims at enabling the students to know various concepts andprinciples of differential calculus. Sound Knowledge of differential calculus isessential for the students of mathematics for the better perceptions of thesubject and its development.

MAT 232: INTEGRAL CALCULUS

This paper aims at enabling the students to know various principles, problemsolving skills in integral calculus and enables the students in applying it infinding length of arcs, surface areas and volumes of solids of revolution,improper integrals.

MAT 331: ORDINARY DIFFERENTIAL EQUATIONS AND APPLICATIONS

This paper enables the students to become familiar with the beauty of animportant branch of mathematics, viz., ordinary differential equations and itsvaried applications including those classical problems in Physics.

MAT 332: PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

This paper aims at imparting the fundamentals of partial differential equationsand the related problem solving skills. Three classical PDE’s, viz., heat, waveand Poisson equations are studied in the context of applications.

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MAT 431: ANALYTICAL GEOMETRY AND SCILAB

Three dimensional geometry is one of the fundamental areas of Mathematics.The course is designed to lay a strong foundation of three-dimensionalgeometry. The open source mathematical package Scilab is introduced tosensitize the students to the programming skills and visual treatment.

MAT 432: VECTOR CALCULUS

This paper aims to enlighten students with the fundamental concepts of vectoranalysis such as gradient, divergence, curl, and the evaluation of line, surfaceand volume integrals. The three classical theorems, viz., Green’s theorem,Gauss divergence theorem and the Stoke’s theorem are also covered.

MAT 531: ALGEBRA

This paper aims at developing the ability to write the mathematical proofs. Ithelps the students to understand and appreciate the beauty of the abstractnature of mathematics and also to develop a solid foundation of theoreticalmathematics.

MAT 532: FOURIER SERIES AND INTEGRAL TRANSFORMS

This paper aims at providing a solid foundation upon the fundamental theoriesand transformations of Fourier Series, Fourier Transforms and LaplaceTransforms.

MAT 533: NUMERICAL ANALYSIS

This paper will help the students to have an in depth knowledge of variousnumerical methods required in Scientific and Technological Applications.

MAT 631: REAL AND COMPLEX ANALYSIS:

This paper enables the students to understand the basic techniques andtheories of real and complex analysis, two traditionally separated subjects.

MAT 632: LINEAR ALGEBRA

This paper enables the students to understand the basic concepts of vectorspaces, linear transformations and inner product space.

MAT 633: NUMBER THEORY

This paper is concerned with the basics of analytical number theory. Topicssuch as divisibility, congruence’s, quadratic residues and functions of numbertheory are covered in this paper. Some of the applications of the said conceptsare also included.

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3. Certificate Courses - Paper Objectives

MAT 101: FOUNDATION OF MATHEMATICS:

This course is designed as a foundation course in Mathematics for those who havenot been exposed to any Mathematics course earlier. This enables the students toimprove their analytical, reasoning and problem solving skills. Topics included are SetTheory, Theory of Equations, Matrices, Determinants, Differential Calculus andIntegral Calculus.

MAT 201: INTRODUCTION TO MATHEMATICA:

This paper can be used by students in Mathematics as an introduction to the

fundamental ideas of MATHEMATICA PACKAGE and as a foundation for the

development of more advanced concepts in MATHEMATICA. Study of this paper

promotes the development of Basic Programming skills in MATHEMATICA.

MAT 301: QUANTITATIVE TECHNIQUES FOR MANAGERS:

This skill based paper aims at imparting theoretical knowledge of optimizationtechniques. These techniques are widely used in the industry to optimize availableresources. This will help the student to apply the mathematical techniques to reallife situations.

MAT 401: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS:

The quantitative aptitude occupies a very important place in any business schoolentrance examination. This skill based paper aims at imparting the aptitude knowledgerequired for competitive examination and provides a well-knitted path to success. Thisknowledge acquisition will help the students to overcome the hurdles of competitiveexaminations like CAT, MAT, XAT, JMET, GMAT, SWAT, etc.,

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4. Course Structure

B.Sc Mathematics

Semester PaperCode

Paper Title Hrs/week TotalHrs.

Marks Credit

1MAT131 INTRODUCTORY ALGEBRA 4 60 100 3

MAT132INTRODUCTION TO MATRIX THEORYAND APPLICATIONS

2 30 50 1

2MAT231 DIFFERENTIAL CALCULUS 4 60 100 3MAT232 INTEGRAL CALCULUS 2 30 50 1

3MAT331

ORDINARY DIFFERENTIAL EQUATIONSAND APPLICATIONS

4 60 100 3

MAT332PARTIAL DIFFERENTIAL EQUATIONSAND APPLICATIONS

2 30 50 1

4MAT431 ANALYTICAL GEOMETRY AND SCILAB 4 60 100 3MAT432 VECTOR CALCULUS 2 30 50 1

5

MAT531 ALGEBRA 3 45 100 2

MAT532FOURIER SERIES AND INTEGRALTRANSFORM

3 45 100 2

MAT533 NUMERICAL ANALYSIS 3 45 100 2

6MAT631 REAL AND COMPLEX ANALYSIS 3 45 100 2MAT632 LINEAR ALGEBRA 3 45 100 2MAT633 NUMBER THEORY 3 45 100 2

Total 630 1200 28

CERTIFICATE COURSES

Semester PaperCode Paper Name Hrs /

WeekTotalHrs. Credit

1 MAT 101 FOUNDATIONS OF MATHEMATICS 4 45 22 MAT 201 INTRODUCTION TO MATHEMATICA 4 45 23 MAT 301 QUANTITATIVE TECHNIQUES FOR MANAGERS. 4 45 2

4 MAT 401QUANTITATIVE APTITUDE FOR COMPETITIVEEXAMINATIONS.

4 45 2

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5. Syllabi for Regular Papers

I Semester

PAPER NAME: INTRODUCTORY ALGEBRA

PAPER CODE: MAT131

UNIT I: MATHEMATICAL LOGIC: (15 Hrs)

Propositions and Truth values – Connectives, their truth values – Tautology and contradiction –Logical equivalence – Standard Theorems – Problems on Negation – Converse, Inverse andContrapostive of Propositions – open sentences – Quantifiers – Logical implications involvingquantifiers – Rules of inferences.

UNIT II: SETS AND RELATIONS: (15 Hrs)

Sets - finite sets – sets of real and complex numbers – topology of real line - Axiom of extension –sub sets and empty set – Venn diagrams – Unordered pairs and Singletons – Intersections – Unions –complements – power sets – Union and intersection of subsets – Ordered pairs – Cartesian product ofsets. Relations : - Properties of special binary relations - equivalence relations - ordering relations -Hasse Diagram.

UNIT III: FUNCTIONS: (15 Hrs)

Types of functions – graphical representation of functions – composition of functions – invertiblefunctions – inverse of compositions (Standard theorems and related problems).

UNIT IV: MATHEMATICAL INDUCTION AND RECURRENCE RELATION (15 Hrs)

Mathematical induction – induction principle - related examples – Recursive Definitions – FibonacciSequence – Lucas sequence – Eulerian numbers – Ackermann’s numbers – Other recursive definitions– Union and intersection of n sets – conjunction and disjunction of n-proposition – well formedformulae.

Total Hours: 60Total Marks: 100

TEXT BOOKS:

1. K. T. Lueng and D. L. Chen, Elementary Set Theory – Part I, Reprint.: Hong Kong UniversityPress, 2009.(Chapter 2: A, B, C, D, E, F, G, H, I, J; Chapter 3: A, B )

2. D. S. Chandrasekharaiah, Discrete Mathematical Structures, 4th ed., India: PRISM Book Pvt. Ltd.,2012 ( Sections : 2.1, 2.1.1, 2.1.2, 2.2, 2.2.1, 2.2.3, 2.2.4, 2.3, 2.4 , 2.4.1, 3.1, 3.2, 4.1, 4.2, 4.2.1,4.4, 4.5, 4.6, 5.1, 5.2, 5.3, 5.4 )

3. K. H. Rosen, Discrete Mathematics and its Applications, 5th ed. USA: WCB / McGraw – Hill.,1999 . ( 10.1, 10.2, 10.3, 10.4 )

4. J. Stewart, L. Redlin and S. Watson, Precalculus-Mathematics for Calculus, 6th ed., USA:Brooks/cole, Cenage Learning, 2012. (Problems from sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7)

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BOOKS FOR REFERENCE:

1. J. P. Tremblay and R. Manohar, Discrete Mathematical Structures with Application to ComputerScience, Reprint, India: Tata McGraw Hill Education, 2008.

2. J.A. Dossey, A.D.Otto, L. E. Spence and C.V. Eynden, Discrete Mathematics, 5th ed., USA:Pearson Education, 2006.

3. L. J. Goldstein, D. I. Schneider and M. J. Siegel, Finite Mathematics and its Applications, 10th

ed., USA: Pearson, 2008.4. E. Gossett, Discrete Mathematics with proof, 2nd ed., USA: John Wiley and Sons, 2009.5. E. D. Goodaire and M. M. Parmenter, Discrete Mathematics with Graph Theory, 3rd ed., USA:

Pearson, 2005.

SUGGESTED WEB LINKS:

1. http://www.cs.columbia.edu/~zeph/3203s04/lectures.html2. http://home.scarlet.be/math/matr.htm3. http://www.cut-the-knot.org/induction.shtml4. http://www.themathpage.com/5. http://www.abstractmath.org/6. http://mathworld.wolfram.com/DiscreteMathematics.html

FORMAT OF QUESTION PAPERMAT 131: INTRODUCTORY ALGEBRA

Part Unit and No. of subdivisions to be setin the unit

No. ofsubdivisions to

be answered

Marks foreach

subdivision

Max. marksfor the part

A

Unit I 2

10 1 10Unit II 3Unit III 2Unit IV 3

B

Unit I 2


CUnit I and II 5

8 6 48Unit III and IV 5

D Unit I, II, III and IV 4 3 8 24Total 100

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I Semester

PAPER NAME: INTRODUCTION TO MATRIX THEORY AND APPLICATIONS

PAPER CODE: MAT 132

UNIT I: BASIC CONCEPTS, DEFINITIONS AND PROPERTIES (20 Hrs)Recapitulation of fundamentals of matrix algebra – Symmetric and skew-symmetric – Hermitian andskew Hermitian matrices – Idempotent, Nilpotent, Orthogonal, Unitary matrices and their properties –Rank of a matrix – Normal form – Finding the inverse of a matrix by elementary transformation –System of linear equations and consistency - Characteristic equations – Eigen values, Eigen vectorsand properties – Cayley Hamilton theorem and its use in finding inverse and powers of a matrix.

UNIT II: APPLICATIONS OF MATRICES (10 Hrs)Matrix application as a function : Scaling Operator, Shearing Operator, Concatenation of Scalingand Shearing Operator, Rotation Operator, Reflection Operator – Discrete Dynamical Systems:Markov Chain, Structured population model, Calculating power of Graph Vertices : DominanceDirected Graphs, Adjacency Matrix, Vertex Power, Matrix application to cryptography: Forminguncoded row matrices 1x2 and 1x3, writing cryptogram for a message, decoding a message,


TEXT BOOKS:

1. B. S. Vatssa, Theory of Matrices, 2nd ed., New Delhi: New Age International Publishers.,2007.(Sections : 1.5, 3.5, 2.6, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 6.1, 6.2, 6.3, 6.4, 8.1, 8.2, 8.3,8.4, 8.5, 8.6 )

2. T. S. Shores, Applied Linear Algebra and Matrix Analysis, 1st ed.,USA: Springer, 2007.(Sections :2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7)

3. M. J. Thobias, Matrices in Engineering Problems, Reprint, USA: Morgan and ClaypoolPublishers., 2011. (Section : 2.7)

4. Larson, College Algebra, 8th ed., USA: Brooks/Cole Cengage Learning, 2011. (Section: 7.5)


1. S. Narayan and P.K. Mittal, Text book of Matrices, 10th ed. New Delhi: S Chand and Co. Ltd,2004.

2. R. Bronson, Schaum’s Outline of Matrix Operations, 1st ed., USA: McGraw Hill Professional,1998.

FORMAT OF QUESTION PAPERMAT 132: INTRODUCTION TO MATRIX THEORY AND APPLICATIONS

PartUnit and No. of subdivisions to be set in

the unit

No. ofsubdivisions tobe answered

Marks for eachsubdivision

Max. marks forthe part

A Unit I 6 6 1 6B Unit I 8 6 2 12

CUnit I 4

4 6 24Unit II 2

D Unit II 2 1 8 8Total 50

9


II Semester

PAPER NAME: DIFFERENTIAL CALCULUS

PAPER CODE: MAT 231

UNIT I: LIMITS, CONTINUITY AND DIFFERENTIABILITY: (10 Hrs)

Definition of the limit of a function ( − ) form – Continuity – Types of discontinuities – Propertiesof continuous functions on a closed interval – Differentiability – Differentiability implies continuity –Converse not true.

UNIT II: MEAN VALUE THEOREMS: (15 Hrs)

Rolle’s theorem – Lagrange’s and Cauchy’s First Mean Value Theorems – Taylor’s theorem(Lagrange’s form) – Maclaurin’s theorem and expansions – Evaluation of limits by L’Hospital’s rule.

UNIT III: SUCCESSIVE AND PARTIAL DIFFERENTIATION: (15 Hrs)

Successive differentiation – nth derivatives of functions – Leibnitz theorem and its applications –Partial differentiation – First and higher order derivatives – Differentiation of homogeneous functions– Euler’s theorem – Total derivative and differential – Differentiation of implicit functions andcomposite functions – Jacobians.

UNIT IV: DERIVATIVES OF ARCS: (20 Hrs)

Polar coordinates – Angle between the radius vector and the tangent – Angle of intersection of curves(polar form) – Polar subtangent and polar subnormal – Perpendicular from pole on the tangent – Pedalequations – Derivative of an arc in Cartesian, parameter and polar forms – Equation of a conic inpolar form – Convexity, concavity and curvature of plane curves – Formula for radius of curvature inCartesian, parametric, polar and pedal forms – Centre of curvature – Evolutes and involutes –Envelopes.


TEXT BOOKS:

1. S. Narayan and P.K.Mittal, Differential Calculus, Reprint. New Delhi: S.Chand & Company Ltd.,2011 (Sections : 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 4.1, 5.1, 5.2, 5.3, 5.4, 5.5, 6.1, 6.2, 7.3, 7.5,7.6, 7.7, 7.8, 7.9, 7.10, 7.11, 7.12, 7.13,, 8.1, 8.2, 8.5, 10.1, 10.2, 10.3, 10.4, 10.5, 11.2, 11.3, 11.4,11.5, 11.6, 11.7, 11.8, 12.1, 12.2, 12.3, 12.4, 13.2, 14.1, 14.2, 14.3, 14.4, 14.5, 14.7, 15.1, 15.2,15.3, 16.1, 16.2, 16.3, 16.4, 18.1, 18.4, 18.8, 18.11)

2. J. Stewart, Single Variable Essential Calculus: Early Transcendentals, 2nd ed.: Belmont, USA:Brooks/Cole Cengage Learning., 2013. (Problems from sections: 1.3, 1.4, 1.5, 1.6)


1. G. B. Thomas and R. L. Finney, Calculus and Analytical geometry, 10th ed., USA: Addison – Wesley,2000.

2. S. Narayanan & T. K. M. Pillay, Calculus, Reprint, India: S. Viswanathan Pvt. Ltd., 2009. (vol. I& II.)

3. J. Edwards, An elementary treatise on the differential calculus: with applications and numerousexample, Reprint, Charleston, USA: BiblioBazaar, 2010.

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4. G. K. Ranganath, Text book of B.Sc., Mathematics, Revised ed. New Delhi, India: S Chand andCo., 2013.

5. F. Ayres and E. Mendelson, Schaum's Outline of Calculus, 6th ed. USA: Mc. Graw Hill., 2013.6. N. P. Bali, Differential Calculus, New ed. New Delhi, India: Laxmi Publications (P) Ltd.., 2012.


1. http://ocw.mit.edu/courses/mathematics/2. http://planetmath.org/encyclopedia/TopicsOnCalculus.html3. http://ocw.mit.edu/OcwWeb/Mathematics/18-01Fall-2005/CourseHome/index.htm4. http://mathworld.wolfram.com/Calculus.html

Recommended Assignment Topics:

Asymptotes – Singular points – Cusp, node and conjugate points. Tracing of standard Cartesian andpolar curves

FORMAT OF QUESTION PAPERMAT 231: DIFFERENTIAL CALCULUS



be answered

Marks foreach

subdivision


A

Unit I 2


B

Unit I 2


CUnit I and II 5



11


II Semester

PAPER NAME: INTEGRAL CALCULUS

PAPER CODE: MAT 232

UNIT I: ELEMENTS OF INTEGRAL CALCULUS (20 Hrs)Recapitulation of methods of integration and Definite Integral: Fundamental theorem of calculus –properties of integration – integration of standard elementary functions:– polynomial, exponential,logarithmic, rational, trigonometric, inverse trigonometric, integration of powers and products of sinesand cosines – method of substitution – integration by parts – completing squares method – integrationby partial fractions – Reduction Formulae – Leibnitz rule for differentiation under integral sign.

UNIT II: APPLICATIONS OF INTEGRAL CALCULUS (10 Hrs)Application of Integral Calculus : Length of arcs – Surface areas and Volumes of solids of revolutionsfor standard curves in Cartesian and Polar forms, Improper Integrals – beta and gamma functions –properties – relation between beta and gamma functions


TEXT BOOKS:

1. S. Narayan, Integral Calculus, 10th revised ed. New Delhi: S. Chand and Company Ltd., 2005.(Sections : 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.10, 7.1, 7.2, 7.3, 8.1, 8.2, 8.3, 9.1, 9.2, 9.3, 9.4, 9.5 )

2. J. Stewart, Single Variable Essential Calculus: Early Transcendentals, 2nd ed.: Belmont, USA:BROOKS/COLE Cengage Learning, 2013. (Problems from sections: 5.1. 5.2, 5.3, 5.4, 5.5, 6.1,6.2, 6.3, 6.6, 7.1, 7.2, 7.3)


1. G. B. Thomas and R. L. Finney, Calculus and Analytical geometry, 10th ed., USA: Addison –Wesley, 2000.

2. S. Narayanan & T. K. M. Pillay, Calculus, Reprint, India: S. Viswanathan Pvt. Ltd., 2009. ( vol. I& II.)

3. G. K. Ranganath, Text book of B.Sc., Mathematics, Revised ed., New Delhi, India: S Chand andCo., 2013.

4. F. Ayres and E. Mendelson, Schaum's Outline of Calculus, 6th ed. USA: Mc. Graw Hill., 2013.5. N. P. Bali, Integral Calculus, 11th ed. New Delhi, India: Laxmi Publications (P) Ltd.., 2011.6. D. Bhardwaj, Integral Calculus made easy, 1st ed. NewDelhi: Laxmi Publications (P) Ltd., 2006.7. M. Spivak, Calculus, 3rd ed., Cambridge University Press, 2006.8. T.M. Apostol, Calculus vol-1, 2nd ed., Wiley India Pvt. Ltd., 2011.9. T.M. Apostol, Calculus vol-2, 2nd ed., Wiley India Pvt. Ltd., 2007.

SUGGESTED WEB LINKS:1. http://ocw.mit.edu/courses/mathematics/2. http://planetmath.org/encyclopedia/TopicsOnCalculus.html3. http://ocw.mit.edu/OcwWeb/Mathematics/18-01Fall-2005/CourseHome/index.htm4. http://mathworld.wolfram.com/Calculus.html

12


FORMAT OF QUESTION PAPERMAT 232: INTEGRAL CALCULUS



be answered

Marks foreach

subdivision


A Unit I 6 6 1 6B Unit I 8 6 2 12

CUnit I 4

4 6 24Unit II 2


13


III Semester

PAPER NAME: ORDINARY DIFFERENTIAL EQUATIONS AND APPLICATIONS

PAPER CODE: MAT 331

UNIT I : FIRST ORDER ODE’s (15 Hrs)

Solution of ordinary differential equations of first order and first degree – Variable separable andreducible to variable separable forms – Homogeneous and reducible to homogeneous forms – Linearequations and Bernoulli equations – Exact equations, equations reducible to exact form with standardintegrating factors – Clairaut’s equation – singular solution for Clairaut’s equation. Orthogonaltrajectories.

UNIT II : MODELLING WITH ODE’s (15 Hrs)

Heating and cooling problems, fluid mixing problems, falling objects, the hanging chain, pursuitcurves, population growth and decay problems, radioactive decay and carbon dating, economics andfinance.

UNIT III : SECOND ORDER AND HIGHER ORDER ODE’s (15 Hrs)

Second and higher order ordinary linear differential equations with constant coefficients –Cauchy-Euler differential equations – Simultaneous differential equations (two variables) with constantcoefficients. Second order linear differential equations with variable coefficients by the followingmethods: (i) when a part of complementary functions is given, (ii) reducing to normal form, (iii)variation of parameters and (iv) method of undetermined co-efficient (v) by finding the first integral(exact equation).

UNIT IV : PARTICLE DYNAMICS (15 Hrs)

Simple Harmonic motion, projectiles – horizontal plane - trajectory – velocity of projection – angleof projection – Range - time of flight – greatest height - projectiles on inclined plane. Central orbitand Central forces – differential equation of a path – pedal equation of a differential equation –velocity at any point of a central orbit – areal velocity – Kepler’s laws of planetary motion.


TEXT BOOKS:

1. Frank Ayres, Differential Equations, Reprint, New York: Schaums Outline Series, 1989.(Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 & 21)

2. Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th ed.,New York: John Wiley & Sons, 2012. (Chapters 2 & 3)

3. S. Narayanan, Dynamics, New Delhi: S. Chand and Co., 1986. (Chapter 8 )4. N. P. Bali, Dynamics, Golden Series, New Delhi: Lakshmi Publications, 2005. (Chapters 5, 6 &7)


1. S. Narayanan and T. K. Manicavachogam Pillay , Differential Equations,New Delhi: S.V. Publishers, 1981.

2. P. Duraipandian , Mechanics, 4th ed., New Delhi: S. Chand and Co., 1995.3. G. K. Ranganath, Text book of B.Sc Mathematics, Reprint, New Delhi: S. Chand and Co., 2006.4. G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd ed., New York:

McGraw Hill, 2006.

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1. http://ocw.mit.edu/courses/mathematics/2. http://www.analyzemath.com/3. http://tutorial.math.lamar.edu/classes/de/de.aspx4. http://www.sosmath.com/diffeq/diffeq.html5. http://www.analyzemath.com/calculus/Differential_Equations/applications.html

FORMAT OF QUESTION PAPERMAT 331: ORDINARY DIFFERENTIAL EQUATIONS AND APPLICATIONS

Part Unit and No. of subdivisions to beset in the unit

No. ofsubdivisions

to beanswered

Marks foreach

subdivision


A

Unit I 3


B

Unit I 3


CUnit I and II 5



15


III Semester

PAPER NAME: PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

PAPER CODE: MAT 332

UNIT I : PARTIAL DIFFERENTIAL EQUATIONS (20 Hrs)

Total and Simultaneous differential equations. Partial differential equations of first order –Lagrange’s solution – Charpit’s general method of solution. Partial differential equations of 2nd order– Classification of linear partial differential equation of 2nd order – Homogeneous and non-homogeneous equations with constant coefficients – Partial differential equations reducible toequations with constant coefficients.

UNIT II: MODELLING WITH PDE’s (10 Hrs)

Initial and boundary value problems. One-dimensional heat equation, one-dimensional waveequation and two dimensional Laplace equation. Method of separation of variables for the solution ofheat, wave and Laplace equations.


TEXT BOOKS:

1. Frank Ayres, Differential Equations, Reprint, New York: Schaums Outline Series, 1989.(Chapters 22, 23, 28, 29, 30, 31, 32 & 33)

2. Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists andEngineers, 4th ed., Boston: Birkhauser, 2009. (Chapters 1, 2 & 3)


1. I. N. Sneddon, Elements of Partial Differential Equations, Reprint, New York: Dover, 2006.2. G. K. Ranganath, Text book of B.Sc, Mathematics, Reprint, New Delhi: S. Chand and Co., 2006.3. G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd ed.,

New York: McGraw Hill, 2006.4. K. Sankar Rao, Introduction to Partial Differential Equations, New Delhi: Prentice Hall, 1997.5. M. D. Raisingania, Ordinary and Partial Differential Equations, New Delhi: S. Chand and Co.,

1993.


1. http://ocw.mit.edu/courses/mathematics/2. http://www.analyzemath.com/3. http://tutorial.math.lamar.edu/classes/de/de.aspx4. http://www.sosmath.com/diffeq/diffeq.html5. http://www.analyzemath.com/calculus/Differential_Equations/applications.html

FORMAT OF QUESTION PAPERMAT 332: PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

Part Unit and No. of subdivisions tobe set in the unit


be answered

Marks foreach

subdivision


A Unit I 6 6 1 6B Unit I 8 6 2 12

CUnit I 4

4 6 24Unit II 2


16


IV Semester

PAPER NAME: ANALYTICAL GEOMETRY AND SCILAB

PAPER CODE: MAT 431

UNIT I: SCILAB (20 Hrs)

Theory: Introduction to Scilab, Arguments and entries in Scilab, Operators, Priorities inmathematical operations, Script files, in-built functions, user defined functions, conditional statements(If then, If then else and Nested if), Looping structures (For, While). Practical: Two and threedimensional plots, Parametric plots, Polar plots, Matrix Operations, Matrix inversions, Solvingsystem of equations. Evaluation of definite integrals, Generating prime numbers, Illustration ofRolle’s and Mean value theorems.

UNIT II: ANALYTICAL GEOMETRY (3D) - (Key concepts) (10 Hrs)

Direction cosines of a line – Direction ratios of the join of two points - Projection on a line – Anglebetween the lines – Area of a triangle and volume of a tetrahedron with given vertices.

UNIT III : ANALYTICAL GEOMETRY (3D) - (Lines and Planes) (15 Hrs)

Equation of line in different forms – Perpendicular from a point onto a line - Equation of a plane indifferent forms – Perpendicular from a point onto a plane - Angle between two planes – Line ofintersection of two planes - Plane co-axial with given planes – Planes bisecting the angle betweentwo planes – Angle between a line and a Plane – Co-Planarity of two lines – Shortest distancebetween two lines.

UNIT IV : ANALYTICAL GEOMETRY (3D) - (Spheres, Cylinders & Cone) (15 Hrs)

Equation of the sphere in its general form – Determination of the centre and radius of a sphere withthe given ends of a diameter – section of sphere by a plane – tangent plane - orthogonal spheres -Equations of Right circular cones and right circular cylinders – Problems.


TEXT BOOKS:

1. S.P. Mahajan and Ajay Aggarwal, Comprehensive Solid Geometry, New Delhi: AnmolPublications, 2000. (Chapters 1, 3, 4, 5, 6)2. Vinu V. Das and D.L. Shah, Programming in Scilab, New Delhi: New Age Publishers, 2008.

(Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

BOOKS FOR REFERENCE:1. Shanthi Narayan, Analytical Solid Geometry, New Delhi: S. Chand and Co., 2004.2. G. K. Ranganath, Text book of B.Sc. Mathematics, Reprint, New Delhi: S. Chand and Co., 2006.3. S. L. Campbell, J. P. Chancelier and R. Nikoukhah, Modelling and Simulation in Scilab/Sciocos,

2nd ed., New York: Springer, 2010.4. Claude Gomez, Engineering and Scientific Computing with Scilab, Boston: Birkhauser, 1999.


1. http://ocw.mit.edu/courses/mathematics2. http://www.univie.ac.at/future.media/moe/galerie.html3. http://mathworld.wolfram.com/AnalyticGeometry.html4. http://www.math.gatech.edu/~harrell/calc5. http://www-irma.u-strasbg.fr/~sonnen/SCILAB_HELP

17


FORMAT OF QUESTION PAPERMAT 431: ANALYTICAL GEOMETRY AND SCILAB

Part Unit and No. of subdivisions tobe set in the unit


be answered

Marks foreach

subdivision


A

Unit I 2


B

Unit I 2


CUnit I and II 5


D Unit I, II, III andIV 4 3 8 24

Total 100

18


IV Semester

PAPER NAME: VECTOR CALCULUS

PAPER CODE: MAT 432

UNIT I : VECTOR DIFFERENTIAL CALCULUS, LINEAND MULTIPLE INTEGRALS (20 Hrs)

Vector Differentiation – Gradient – Divergence – Curl and Laplacian Operators – Vectoridentities - Line integral and basic properties – examples on evaluation of the integrals – Definition ofa double integral – its conversion to iterated integrals – evaluation of double integral by change oforder of integration and by change of variables – surface areas as double integrals – Definition of atriple integral and evaluation – change of variables – volume as a triple integral – Line, surface andvolume integrals of vector functions.

UNIT II: INTEGRAL THEOREMS (10 Hrs)

Green’s theorem in the plane (statement and proof) – Direct consequences of the theorem – TheDivergence theorem (statement and interpretation) – Direct consequences of the theorem – TheStoke’s theorem (statement and interpretation) – Direct consequences of the theorem.


TEXT BOOKS:

1. B. Spain, Vector Analysis, 2nd ed., Calcutta: Radha Publishers, 1988.(Chapters 3, 5, 6, 7, 8, 9, 10 & 11)


1. S. Narayanan and & M. Pillay, Vector Algebra and Analysis, 4th ed., New Delhi: S.V. Publishers,1986.

2. M. D. Raisinghania, H. K. Dass and H. C. Saxena, Simplified Course in Vector Calculus,New Delhi: S. Chand & Co., 2002.

3. G. K. Ranganath, Text book of B.Sc. Mathematics, Reprint, New Delhi: S. Chand and Co., 2006.


1. http://ocw.mit.edu/courses/mathematics2. http://www.univie.ac.at/future.media/moe/galerie.html3. http://mathworld.wolfram.com/AnalyticGeometry.html4. http://www.math.gatech.edu/~harrell/calc

FORMAT OF QUESTION PAPERMAT 432: VECTOR CALCULUS

Part Unit and No. of subdivisions to beset in the unit


be answered

Marks foreach

subdivision


A Unit I 6 6 1 6B Unit I 8 6 2 12

CUnit I 4

4 6 24Unit II 2


19


V Semester

PAPER NAME: INTRODUCTION TO ABSTRACT ALGEBRA

PAPER CODE: MAT 531

UNIT I : GROUPS (20 Hrs)

Groups, Subgroups, Cyclic groups, Cosets, Lagrange’s theorem, Normal Subgroup, QuotientGroup, Homomorphism of groups, Fundamental Theorem of Homomorphism, Isomorphism,Cauchy’s theorem for abelian group, Permutation groups.

UNIT II : RINGS, INTEGRAL DOMAINS AND FIELDS (15 Hrs)

Rings, Properties, Integral domains, Fields, Subrings, Quotient Rings, Ideals, Principal, Prime andMaximal ideal in a commutative ring , Homomorphism and Isomorphism of Rings, Fields.

UNIT III : APPLICATIONS OF GROUPS THEORY (10 Hrs)

Dihedral Group, Group of symmetries, Symmetries of an equilateral triangle, Symmetries of asquare, simple applications of rings.


TEXT BOOKS:

1. I N Herstein , Topics in Algebra, Second Edition. Wiley India (P) Ltd.New Delhi, India: VikasPublishing House Pvt. Ltd, 2006.

2. Dr D S Chandrasekariah, Discrete Mathematical Structures 2e(vtu)(iii Sem) Cse & Ise Branches,Fifth Edition., Prism Publications, 2005.


1. J B Fraleigh, A First course in Abstract Algebra, Seventh Edition, Dorling Kindersely (India)Pvt. Ltd., 2008.

2. G K Ranganath, Text book of B.Sc. Mathematics, Revised Edition. NewDelhi, India: S Chand andCo., 2011.

3. M Artin, Algebra, Second Edition. New Delhi, India: PHI Learning Pvt. Ltd., 2011.3. V K Krishnamoorty and V P Mainra and J L Arora, An Introduction to Linear Algebra, Reprint.

New Delhi India: Affiliated East West Press Pvt. Ltd., 2003.


1. http://ocw.mit.edu/courses/mathematics/2. http://www.extension.harvard.edu/openlearning/math222/3. http://mathworld.wolfram.com/Algebra.html4. http://www.math.niu.edu/~beachy/aaol/5. http://planetmath.org/encyclopedia/Inverse.html

20


FORMAT OF QUESTION PAPERMAT 531: ALGEBRA

Section Unit and No. of subdivisions to beset in the Unit

No. ofsubdivisions

to beanswered

Marks foreach

subdivision

MaximumMarks forthe Part

AUnit I 4

10 1 10Unit II 3Unit III 3

BUnit I 4


CUnit I 4


DUnit I 2


TOTAL 100

21


V Semester

PAPER NAME: FOURIER SERIES AND INTEGRAL TRANSFORMS

PAPER CODE: MAT 532

UNIT I : FOURIER SERIES AND FOURIER INTEGRALS (15 Hrs)

Fourier Series of functions with period 2π and period 2L, half range cosine and sine series.

UNIT II : FOURIER TRANSFORM (20 Hrs)

Finite Fourier cosine and sine transforms, transform of some common functions, the FourierIntegral, Complex Fourier Transforms-Basic Properties, Transform of the derivative, Convolutiontheorem, Parseval’s Identity – Fourier sine and cosine transforms, Solution of differential equationsusing Fourier transforms.

UNIT III : LAPLACE TRANSFORM (10 Hrs)Laplace Transform of standard functions, Laplace transform of periodic functions, Inverse

Laplace transform, Solution of ordinary differential equation with constant coefficient using Laplacetransform, Solution of Ordinary differential equations with constant and variable coefficients,Solution of simultaneous Ordinary differential equations , Solution of Partial differential equations.


TEXT BOOKS:

1. E Kreyszig, Advanced Engineering Mathematics, Eighth Edition New Delhi, India: Wiley IndiaPvt. Ltd., 2010.

2. Dr. B. S. Grewal, Higher Engineering Mathematics, Thirty ninth Edition, Khanna Publishers, July2005.

Books for Reference:

1. K Sankara Rao, Introduction to Partial Differential Equations, PHI Learning Pvt. Ltd., NewDelhi: 2012.

2. N H Shah, Ordinary and Partial Differential Equations Theory and Applications, PHI LearningPvt. Ltd., New Delhi: 2010.

3. M. D. Raisinghania, Ordinary and Partial Differential Equation, Chand (S.) & Co.Ltd., India:March 17, 2005..

4. G K Ranganath, Text book of B.Sc. Mathematics, Revised Edition New Delhi, India: S Chand andCompany Ltd., 2011.

5. G F. Simmons and S G. Krantz, Differential Equation, Theory, Technique and Practice, TataMcGraw – Hill, 2006.

Suggested Web links:

1. http://ocw.mit.edu/courses/mathematics/2. http://math.fullerton.edu/mathews/c2003/ComplexUndergradMod.html3. http://www.fourier-series.com/4. http://mathworld.wolfram.com/5. http://www.princeton.edu/~rvdb6. http://www.zweigmedia.com/RealWorld/Summary4.html7. http://people.brunel.ac.uk/~mastjjb/jeb/or/contents.html8. http://people.brunel.ac.uk/~mastjjb/jeb/or/lpmore.html

22


FORMAT OF QUESTION PAPERMAT 532: FOURIER SERIES AND INTEGRAL TRANSFORMS


No. ofsubdivisions

to beanswered

Marks foreach

subdivision


AUnit I 3


BUnit I 3


CUnit I 3


DUnit I 1


TOTAL 100

23


V Semester

PAPER NAME: NUMERICAL ANALYSIS

PAPER CODE: MAT 533

UNIT I : NUMERICAL SOLUTION OF ALGEBRAIC ANDTRANSCEDENTAL EQUATIONS (15 Hrs)

Errors and their analysis , Floating point representation of numbers, Solution of Algebraic andTranscendental Equations : Bisection method, Iteration method, the method of False Position,Newton Raphson method, Solution of linear systems -Gaussian Elimination method, Iterativemethods- Gauss Seidel method, Gauss Jacobi method, Method of factorization.

UNIT II : INTERPOLATION, NUMERICAL DIFFERENTIATIONAND NUMERICAL INTEGRATION (20 Hrs)

Finite differences: Forward difference, Backward difference and Shift Operators, Separationof symbols, Newton’s Formulae for interpolation, Lagranges interpolation formulae, Hermite, Cubic-Spline interpolation formulas, Bivariate interpolation and least square approximation, Numericaldifferentiation, Numerical integration : Trapezoidal rule, Simpson’s one-third rule and Simpson’sthree-eighth rule.

UNIT III : NUMERICAL SOLUTION OF ORDINARYDIFFERENTIAL EQUATIONS (10 Hrs)

Numerical solution of ordinary differential equations - Taylor’s series method, Picard’s method ,Euler’s method, Modified Euler’s method, Runge Kutta methods - Second order (with proof) andfourth order (without proof).


TEXT BOOKS:

1. S S Sastry, Introductory methods of Numerical Analysis, Fourth Edition New Delhi, India:Prentice Hall of India, 2006.

2. M K Jain, S R K Iyengar, and R K Jain, Numerical Methods for Scientific and EngineeringComputation, Fifth Edition, New Delhi, India: New Age International, 2007.


1. E Kreyszig, Advanced Engineering Mathematics, Eighth Edition, New Delhi, India: Wiley IndiaPvt. Ltd., 2010.

2. Dr. B. S. Grewal, Higher Engineering Mathematics, Thirty ninth Edition, Khanna Publishers,July 2005.

3. F Scheid, Schaum's Outline of Numerical Analysis, Revised Edtion: Mc.Graw Hill., 2006.4. R L Burden, J.Douglas Faires, Numerical Analysis, Seventh Edition, Thomson Brooks/Cole,

2005.5. G K Ranganath, Text book of B.Sc. Mathematics, Revised Edition New Delhi, India: S Chand

and Company Ltd., 2011.6. P Kandasamy, K Thilagavathy, and K Gunavathy, Engineering Mathematics Volume III, S.

Chand & Company Ltd., New Delhi, 2003.


1. http://www.amtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/index.htm

2. http://math.fullerton.edu/mathews/numerical.html

3. http://www.onesmartclick.com/engineering/numerical-methods.html

24


FORMAT OF QUESTION PAPERMAT 533: NUMERICAL ANALYSIS


No. ofsubdivisions

to beanswered

Marks foreach

subdivision


AUnit I 3


BUnit I 3


CUnit I 3


DUnit I 1


TOTAL 100

25


VI Semester

PAPER NAME: REAL AND COMPLEX ANALYSIS

PAPER CODE: MAT 631

UNIT I : SETS AND SEQUENCES IN R (10 Hrs)

Limit of a set, Open sets, Closed sets, closure of a set, countable and uncountable sets,topology of real line. Sequences: Definition of Sequences, limit of a sequence, algebra of limits of asequence, convergent, divergent and oscillatory sequences, problems thereon. Bounded sequences,Monotonic sequences and their properties, Cauchy sequence.

UNIT II: INFINITE SERIES (15 Hrs)

Definition of convergence, divergence and oscillation of series, properties of convergent series,Cauchy’s criterion (Statement only), Geometric series. Tests of convergence of series, p series,comparison tests, D’Alembert’s test, Raabe’s test, Cauchy’s root test. Absolute and conditionalconvergence, Leibnitz test for alternating series.

UNIT III : COMPLEX ANALYSIS (20 Hrs)

Continuity and Differentiability of complex functions, Analytic Functions, Cauchy-Riemannequations, Harmonic functions, Line integrals and Contour integration, The Cauchy’s Integraltheorem and its direct consequences, Cauchy’s Integral formula, Cauchy’s Integral formula forderivatives, Morera’s theorem, Cauchy’s inequality, Liouville’s Theorem, Fundamental theorem ofAlgebra.


TEXT BOOKS:1. S.C.Malik and Savita Arora, Mathematical Analysis, Second Edition, New Delhi, India: New Age

international (P) Ltd., 2005.2. W Rudin, Real and Complex analysis, Third Edition,Tata Mc-Graw Hill, 2006.3. R V Churchil & J W Brown, Complex Variables and Applications, Fifth Edition, Mc. Graw Hill

Companies., 2002.

BOOKS FOR REFERENCE:1. S Narayana and M.D. Raisinghania, Elements of Real Analysis, Revised ed., S. Chand &

Company Ltd, 2011.2. G K Ranganath, Text book of B.Sc. Mathematics, Revised Edition, New Delhi, India: S Chand and

Company Ltd., 2011.3. J H. Mathews and R W Howell ,Complex Analysis: for Mathematics and Engineering, Jones and

Bartlett Publishers, Sudbury, Massachusetts, Sixth Edition, 2012.4. W. Rudin, Real and Complex Analysis, 3rd ed., Tata McGraw-Hill Education, 2007.


1. http://www.math.unl.edu/~webnotes/contents/chapters.htm

2. http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html

3. http://web01.shu.edu/projects/reals/index.html

4. http://www.mathcs.org/analysis/reals/index.html

26


FORMAT OF QUESTION PAPERMAT 631: REAL AND COMPLEX ANALYSIS


No. ofsubdivisions

to beanswered

Marks foreach

subdivision


AUnit I 3


BUnit I 3


CUnit I 3


DUnit I 1


TOTAL 100

27


VI Semester

PAPER NAME: LINEAR ALGEBRA

PAPER CODE: MAT 632

UNIT I : VECTOR SPACE (15 Hrs)

Definition, Examples, Properties. Subspaces, Span of a set, Linear dependence andindependence, Dimension and Basis.

UNIT II : LINEAR TRANSFORMATION (15 Hrs)

Definition and examples, Range and Kernel of a linear map , Matrix of the LinearTransformation, Rank and Nullity, inverse of a linear transformation, consequence of Rank-nullitytheorem.

UNIT III : INNER PRODUCT SPACE (15 Hrs)

Definition, examples, orthonormal sets, Schwarz inequality, Gram Schmidtorthogonolization process – problems, least square approximations.


TEXT BOOKS:

1. I N Herstein , Topics in Algebra, Second Edition, Wiley India (P) Ltd.New Delhi, India: VikasPublishing House Pvt. Ltd, 2006.

2. V K Krishnamoorty and V P Mainra and J LArora, An Introduction to Linear Algebra, Reprint.New Delhi, India: Affiliated East West Press Pvt. Ltd., 2003.


1. J B Fraleigh, A First course in Abstract Algebra, Seventh Edition, Dorling Kindersely (India) Pvt.Ltd., 2008.

2. R Balakrishan and N Ramabadran, A Textbook of Modern Algebra, First Edition, New Delhi,India: Vikas publishing house pvt. Ltd., 1991.

3. G K Ranganath, Text book of B.Sc. Mathematics, Revised Edition, NewDelhi, India: S Chand andCo., 2011.

4. M Artin, Algebra, Second Edition, New Delhi, India: PHI Learning Pvt. Ltd., 2011.5. G. Strang, Algebra and its Applications, 4th ed., Thomson Brooks /Cole, 2006.


1. http://ocw.mit.edu/courses/mathematics/2. http://www.extension.harvard.edu/openlearning/math222/3. http://mathworld.wolfram.com/Algebra.html4. http://www.math.niu.edu/~beachy/aaol/5. http://planetmath.org/encyclopedia/Inverse.html

Suggested assignment topic: Coding Theory

28


FORMAT OF QUESTION PAPERMAT 632: LINEAR ALGEBRA


No. ofsubdivisions

to beanswered

Marks foreach

subdivision


A Unit I 410 1 10Unit II 3

Unit III 3B Unit I 4


C Unit I 48 6 48Unit II 3

Unit III 3D Unit I 2


TOTAL 100

NOTE : In all the Sections one Question from any unit can be increased as per the content of the unit.

29


VI Semester

PAPER NAME: NUMBER THEORY

PAPER CODE: MAT 633

UNIT I : DIVISIBILITY AND PRIMES (15 Hrs)

The Division Algorithm, The Greatest Common Divisor, The Euclidean Algorithm, TheLinear Diophantine Equation, The Fundamental Theorem of Arithmetic.

UNIT II : THE THEORY OF CONGRUENCES (15 Hrs)

Basic Properties of Congruences, Binary and Decimal Representations of Integers, LinearCongruences and Chinese Remainder Theorem, Fermat’s Little Theorem and Pseudoprimes, Wilson’sTheorem.

UNIT III : NUMBER-THEORETIC FUNCTIONS (15 Hrs)

Multiplicative Functions, The Sum and Number of Divisors, The Möbius InversionFormula, The Greatest Integer Function, Euler’s Phi-Function, Euler’s Generalization of Fermat’sTheorem, Properties of Phi-Function.


Text Book:1. D.M. Burton, Elementary Number Theory, Sixth Edition, New Delhi: Tata McGraw-Hill, 2012.

Books for Reference:

1. I. Niven, H.S. Zuckerman and H.L. Montgomery, An Introduction to The Theory of Numbers,Fifth Edition, New Delhi: John Wiley & Sons, Inc., 2012.

2. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Second Edition,New York: Springer-Verlag, 2010.

3. G. A. Jones And J. Mary Jones, Elementary Number Theory, Springer, 1998.4. J. H. Silverman, A Friendly Introduction To Number Theory, Pearson Prentice Hall, 2006.

Suggested Web link:

www.numbertheory.org.

30


FORMAT OF THE QUESTION PAPERMAT 633: NUMBER THEORY


No. ofsubdivisions

to beanswered

Marks foreach

subdivision


AUnit I 4


BUnit I 4


CUnit I 4


DUnit I 2


TOTAL 100

31


6. Syllabi for Certificate Courses

PAPER NAME: FOUNDATIONS OF MATHEMATICS

PAPER CODE : MAT 101

UNIT I : SET THEORY (10 Hrs)Set Theory – Definition – Types of Sets – Operation on sets ( Union, Intersection Complement,

Difference ) – Venn Diagram – Application problems.

UNIT II : EQUATIONS (10 Hrs)Linear Equations – solution of linear equation – Quadratic equations – solutions of Quadratic

equations – The equation x2 + 1 = 0 and introduction to complex numbers - Square roots, cube roots andfourth roots of unity.

UNIT III: MATRICES AND DETERMINANTS (10 Hrs)Matrices – Types of Matrices – Operations on Matrices – Expansion of 2nd and 3rd order

Determinants – Minors – Co-factors – Adjoint – Singular and Non-singular matrices – Inverse of a matrix –Solution of system of linear equation by matrix and determinant methods.

UNIT IV: DIFFERENTIAL AND INTEGRAL CALCULUS (15 Hrs)Limits – Differentiation – Methods of differentiation – Second order derivative – Maxima

and Minima – Applications to Revenue Function, Cost function, profit function, Elasticity of demand,Break even point. Indefinite integral – Standard results – substitution method – application to costand revenue functions.

(2 credits ) Total Hours : 45

TEXT BOOKS:

1. D.C.Sancheti and V. K.Kapoor, Business Mathematics, 11th ed., New Delhi, India: Sultan Chandand Sons., 2012.

2. B.G.Sathtyaprasad, K.Nirmala, R.G.Saha, and C.S.Anantharaman, Business Mathematics. 1st ed.,Mumbai, India: Himalaya publishing House., 2006.


1. Shanti Narayanan and P.K. Mittal, Text book of Matrices, 10th ed.: S. Chand and Company Ltd.,2010.

2. E. Don and J. Lerner, Schaums Outlines of Basic Business Mathematics, 2nd ed.,: McGraw Hill,2000.


1. http://planetmath.org/encyclopedia/SetTheory.html2. http://plato.stanford.edu/entries/set-theory/3. http://mathworld.wolfram.com/Logarithm.html4. http://www.sosmath.com/algebra/logs/log1/log1.html5. http://www.mathagonyaunt.co.uk/STATISTICS/ESP/Perms_combs.html6. http://www.mathsisfun.com/combinatorics/combinations-permutations.html7. http://home.scarlet.be/math/matr.htm8. http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html

32


PAPER NAME: INTRODUCTION TO MATHEMATICAL PACKAGES


UNIT I : ALGEGRAIC COMPUTATION: (10 Hrs)Simplification of algebraic expression, simplification of expressions involving special

functions, built-in functions for transformations on trigonometric expressions, definite and indefinitesymbolic integration, symbolic sums and products, symbolic solution of ordinary and partialdifferential equations, symbolic linear algebra, equations solving, calculus, polynomial functions,matrix operations.UNIT II : MATHEMATICAL FUNCTIONS: (07 Hrs)

Special functions, inverse error function, gamma and beta function, hyper-geometricfunction, elliptic function, Mathieu function.

UNIT III: NUMERICAL COMPUTATION: (10 Hrs)Numerical solution of differential equations, numerical solution of initial and boundary value

problems, numerical integration, numerical differentiation, matrix manipulations and optimizationtechniques.

UNIT IV : GRAPHICS: (08 Hrs)Two and Three dimensional plots, parametric plots, contours, typesetting capabilities for

labels and text in plots, direct control of final graphics size, resolution etc.

UNIT V : PACKAGES: (10 Hrs)Algebra, linear algebra, calculus, discrete math, geometry, graphics, number theory, vector

analysis, Laplace and Fourier transforms, statistics.

Total Hours : 45(2 credits)

TEXT BOOKS:1. Stephen Wolfram, The Mathematica book.: Wolfram Research Inc. , 2003..2. Michael Trott, The Mathematica guide book for programminG, Springer, 2004.3. P.Wellin, R.Gaylord, and S.Kamin, An introduction to programming with Mathematica, 3rd ed.:Cambridge, 2005.

Suggested Web Links:1. http://www.math.montana.edu/frankw/ccp/modeling/topic.htm2. http://library.wolfram.com/

33


PAPER NAME: QUANTITATIVE TECHNIQUES FOR MANAGERS


1 : LINEAR PROGRAMMING (20 Hrs)Definitions of O.R.- Definition of Linear Programming Problem (L.P.P) - Formulation of

L.P.P. – Linear Programming in Matrix Notation – Graphical Solution of L.P.P – Simplex Method –Big M Technique – Two Phase Method - Concept of Duality – Formulation of Primal Dual Pairs –Dual Simplex Method.

2 : TRANSPORTATION AND ASSIGNMENT PROBLEMS (10 Hrs)Introduction to Transportation Problem – Initial Basic Feasible solution – Moving towards

Optimality – Degeneracy in Transportation Problems – Unbalanced Transportation Problem –Assignment Problems.

3 : GAME THEORY (15 Hrs)Games and Strategies – Introduction – Two person zero sum games – Maximin and Minimax

Principles – Games without saddle point – mixed strategies – Solution of 2 x 2 rectangular games –Graphical method – Dominance Property – Algebraic Method for m x n games


TEXTBOOK:

Kanti Swarup, P.K.Gupta, and ManMohan, Operations Research, Reprint, New Delhi, India: SultanChand & Sons, 1994.


1. G Hadley, Linear Programming, Reprint, New Delhi: Narosa Publishing House, 2002.

2. K.V.Mittal and C.Mohan, Optimization Methods in Operation Research and System Analysis,

3rd ed., New Delhi: New Age International Pvt. Ltd., 2008.

3. Hamdy A Taha, Operations Research- an introduction, 8th ed., New Delhi: Prentice Hall of India,2009.


1. http://www.zweigmedia.com/RealWorld/Summary4.html2. http://people.brunel.ac.uk/~mastjjb/jeb/or/lpmore.html3. http://www2.isye.gatech.edu/~jswann/casestudy/assign.html4. http://mathworld.wolfram.com/GameTheory.html

34


PAPER NAME: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS


1. PERCENTAGES, AVERAGES AND PROGRESSIONS: (10 Hrs)Number System,HCF and LCM: – Factors – Multiples – HCF – LCM – Product of two numbers – Difference between HCFand LCM,Fraction: Fractional part of a number – To find the fraction related to Balance amount,Square roots - Cube roots,Percentage: Fraction to Rate Percent – Rate Percent to Fraction – Rate Percent of a Number – Expressing agiven quantity as a Percentage of Another given quantity – Converting a percentage into decimal – converting adecimal into a percentage – Effect of percentage change on any quantity – Rate change and change in quantityavailable for fixed expenditure ,Average: Average of different groups – Addition or removal of items and change in average – replacement ofsome of the items,Arithmetic progression and Geometric Progression.

2 : RATIOS AND PROPORTIONS (25 Hrs)Ratio and Proportions: Properties of Ratio – Dividing a given number in the given ratio – comparison of ratios– useful results on proportion – continued proportion – relation among the quantities more than two – directproportion and indirect proportion,Profit and Loss: Gain percentage and Loss Percentage – Relation among cost price, sale price, Gain/Loss andGain% or Loss% - Discount and Marked Price,Time and work: Basic concepts – examples,Pipes and Cistern: Basic concepts – examples,Time and Distance: Definition – Average speed – distance covered is same, different – stoppage time per hourfor a train – time taken with two difference modes of transport,Boats and Streams: Introduction, Speed of Man (Boat) and stream – Important formulae,Mixture: Allegation rule – Mean Value of the Mixture – Six golden rules to solve problems on mixture –Removal and replacement by equal amount.

3 : COMMERCIAL ARITHMETICS (10 Hrs)Simple interest: Definition – Effect of change of P, R and T on simple interest – amount – amount becomes Ntimes the principal – Repayment of debt in equal installments – Rate and Time are numerically equal,Compound Interest: Basic Formula - conversion period – to find the principal/time/rate – difference betweencompound interest and simple interest – equal annual installments to pay the debt amount – growth –depreciationShares and Debentures : Basic facts – Approach to problems on stock – Approach to problems on Shares –regular problems - Debentures.


Text Book :A. Guha, Quantitative Aptitude for competitive examinations, 4th ed., New Delhi, India: Tata Mc-GrawHill, 2010.

Books for Reference:1. K. Dinesh, Quantitative Aptitude, 3rd ed., New Delhi: Pearson Education India, 2008.2. M. Muneer, How to prepare for CAT.: 3rd ed., New Delhi, India:Tata Mc Graw Hill Education,

2007.

Suggested Web links :1. http://www.ascenteducation.com2. http://www.winentrance.com/MCA-Entrance-Exam-Question-Bank-CD.html

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External Experts:

Dr. P M BALAGONDAR,Professor and Chairman,Department of Mathematics,Bangalore University, Bangalore.

Dr. KAUSHAL VERMA,Associate Professor,Department of Mathematics,Indian Institute of Science,Bangalore.

Dr. HEMALATHA,Professor and Head,Mount Carmel College,Bangalore.

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