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(Under Section 3 of the UGC Act, 1956)
B.Sc., Mathematics (Honours)
Syllabus
Banga lore , Kar na ta ka , Ind ia , 2012
ii
Contents
1 COURSE OBJECTIVE AND METHODOLOGY ............................................................................................. 1
1.1 COURSE OBJECTIVE ............................................................................................................ 1
1.2 METHODOLOGY .................................................................................................................. 2
2 PAPER OBJECTIVES ....................................................................................................................................... 3
2.1 B.Sc., MATHEMATICS (Honours): ........................................................................................ 3
2.1.1 MAT 131: INTRODUCTORY ALGEBRA: ...................................................................... 3
2.1.2 MAT 231: CALCULUS: ................................................................................................... 3
2.1.3 MAT 331: DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS: ......................... 3
2.1.4 MAT 431: ANALYTICAL GEOMETRY AND VECTOR CALCULUS: .......................... 3
2.1.5 MAT 531: ALGEBRA: ..................................................................................................... 3
2.1.6 MAT 532: INTEGRAL TRANSFORMS AND LINEAR PROGRAMMING: ................... 4
2.1.7 MAT 533: INTRODUCTION TO TOPOLOGY AND DIFFERENTIAL GEOMETRY: ... 4
2.1.8 MAT 534: PROGRAMMING IN C: ................................................................................. 4
2.1.9 MAT 535: GRAPH THEORY: .......................................................................................... 4
2.1.10 MAT 536: NUMBER THEORY: .................................................................................... 4
2.1.11 MAT 631: REAL AND COMPLEX ANALYSIS: ........................................................... 5
2.1.12 MAT 632: NUMERICAL METHODS: ........................................................................... 5
2.1.13 MAT 633: ORDINARY DIFFERENTIAL EQUATIONS AND MATHEMATICAL
MODELLING: .......................................................................................................................... 5
2.1.14 MAT 634: INTRODUCTION TO MATHEMATICAL PACKAGES: ............................. 5
2.1.15 MAT 635: OPERATIONS RESEARCH: ........................................................................ 6
2.1.16 MAT 636: MECHANICS: ............................................................................................... 6
2.2 CERTIFICATE COURSES: .................................................................................................... 6
2.2.1 MAT 101: FOUNDATION OF MATHEMATICS: ........................................................... 6
2.2.2 MAT 201: INTRODUCTION TO MATHEMATICA: ....................................................... 6
2.2.3 MAT 301: QUANTITATIVE TECHNIQUES FOR MANAGERS: ................................... 7
2.2.4 MAT 401: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS: ...... 7
3 COURSE STRUCTURE .................................................................................................................................... 8
3.1 B.Sc MATHEMATICS: ........................................................................................................... 8
3.2 CERTIFICATE COURSES ..................................................................................................... 9
iii
4 SYLLABI FOR REGULAR PAPERS .............................................................................................................. 10
4.1 MAT 131: INTRODUCTORY ALGEBRA ........................................................................... 10
4.2 MAT 231: CALCULUS ........................................................................................................ 12
4.3 MAT 331: DIFFERENTIAL EQUATIONS AND APPLICATIONS ..................................... 14
4.4 MAT 431: ANALYTICAL GEOMETRY AND VECTOR CALCULUS ............................... 16
4.5 MAT 531: ALGEBRA .......................................................................................................... 18
4.6 MAT 532: INTEGRAL TRANSFORMS AND LINEAR PROGRAMMING ......................... 20
4.7 MAT 533: INTRODUCTION TO TOPOLOGY AND DIFFERENTIAL GEOMETRY ......... 22
4.8 MAT 534: PROGRAMMING IN C ....................................................................................... 24
4.9 MAT 535: GRAPH THEORY ............................................................................................... 26
4.10 MAT 536: NUMBER THEORY .......................................................................................... 28
4.11 MAT 631: REAL AND COMPLEX ANALYSIS ................................................................ 30
4.12 MAT 632: NUMERICAL METHODS ................................................................................ 32
4.13 MAT 633: ORDINARY DIFFERENTIAL EQUATIONS AND MATHEMATICAL
MODELING ............................................................................................................................... 34
4.14 MAT 634: INTRODUCTION TO MATHEMATICAL PACKAGES ................................... 36
4.15 MAT 635: OPERATIONS RESEARCH .............................................................................. 38
4.16 MAT 636: MECHANICS .................................................................................................... 40
5 SYLLABI FOR CERTIFICATE COURSES .................................................................................................... 42
5.1 MAT 101: FOUNDATIONS OF MATHEMATICS............................................................... 42
5.2 MAT 201: INTRODUCTION TO MATHEMATICA ............................................................ 43
5.3 MAT 301: QUANTITATIVE TECHNIQUES FOR MANAGERS ........................................ 44
5.4 MAT 401: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS ........... 45
1
COURSE OBJECTIVE AND METHODOLOGY
1.1 COURSE OBJECTIVE
An educational institution that does not lead to research and specialization will
remain on the way side of the higher education missing the golden opportunities
for pushing farther the frontiers of knowledge and opening up new vistas of
scientific endeavor to the young. Keeping the above in mind it has been
proposed to continue with 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 various concepts 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 endeavors. 8. To pursue higher studies in top notch institutions.
The course curriculum is intensive and extensive and includes 16 core
papers covering all major topics in mathematics. Students who opt for
Mathematics Honours programme will have to follow three major syllabi of
PCM or PME or CMS or CME combinations in the first four semesters. There
will be one mathematics paper in each of the first four semesters and six papers
in each of the fifth and sixth semesters. Each of the sixteen papers will carry
100 marks. In each paper the minimum marks for pass will be 40. Courses like
“Foundations of Mathematics”, “Introduction to Mathematica”, “Quantitative
2
methods for managers” and “Quantitative aptitude for competitive
examinations” are offered as certificate courses.
After completing three years of honours degree course, students can opt
for higher studies; get into the institutions like ISRO, NAL etc or IT oriented
service sections.
1.2 METHODOLOGY
In 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
communications technology.
3
2
PAPER OBJECTIVES
2.1 B.Sc., MATHEMATICS (Honours):
2.1.1 MAT 131: INTRODUCTORY ALGEBRA:
This paper emphasizes general techniques of problem solving and explores the
creation of mathematical patters. It aims at introducing a course that initiates the
students into the world of Discrete Mathematics. It includes the topic like
Mathematical Logic, Set Theory, Relations, Functions, Mathematical Induction,
Recursive relations and Matrices.
2.1.2 MAT 231: CALCULUS:
This paper aims at enabling the students to know various concepts and
principles of differential and integral calculus. Sound Knowledge of calculus is
essential for the students of mathematics for the better perceptions of the
subject and its development.
2.1.3 MAT 331: DIFFERENTIAL EQUATIONS AND I TS APPLICATIONS:
This paper enables the students to know the beauty of an important branch of
mathematics, viz. Differential Equations and its applications in Physics.
2.1.4 MAT 431: ANALYTICAL GEOMETRY AND VECT OR CALCULUS:
Three Dimensional Geometry is one of the fundamental areas of Mathematics.
The course is designed to lay a strong foundation of Geometry and Vector
Calculus.
2.1.5 MAT 531: ALGEBRA:
This paper aims at developing the ability to write the mathematical proofs. It
helps the students to understand and appreciate the beauty of the abstract nature
4
of mathematics and also to develop a solid foundation of theoretical
mathematics.
2.1.6 MAT 532: INTEGRAL TRANSFORMS AND LINE AR PROGRAMMING:
This paper aims at providing a solid foundation upon the fundamental theories
and transformations of Fourier Transforms and Laplace Transforms. It also
covers the introduction to linear programming techniques.
2.1.7 MAT 533: INTRODUCTION TO TOPOLOGY AND DIFFERENTIAL GEOMETRY:
Differential geometry is the study of geometrical objects using Calculus. This
paper aims at increasing the students understanding in differential geometry due
to its connection and significance to other areas of Mathematics. It helps
students in understanding some of the fundamental concepts in differential
geometry through curves and surfaces in R3
2.1.8 MAT 534: PROGRAMMING IN C:
The objective of this paper is to help students learn various aspects of C
programming and make them skilled in C programming. Also helps to evaluate
critically the C programs written by others.
2.1.9 MAT 535: GRAPH THEORY:
This paper aims at imparting in the minds of the students the fundamental
concepts in graph theory. Also helps in appreciating the application of the
subject in a variety of fields, to learn, to understand and to construct
mathematical proofs involving graphs.
2.1.10 MAT 536: NUMBER THEORY:
The objective of the paper is to help the students to learn long and interesting
history of Number Theory and its problems that have attracted many of the
greatest mathematicians. Consequently the study of number theory is an
5
excellent introduction to the development and the achievement of Mathematics.
With its discrete precise nature is an ideal topic which helps the students to
perform Numerical Experiments and Calculations; also to explore its wide
spread applications in Cryptography.
2.1.11 MAT 631: REAL AND COMPLEX ANALYSIS:
This paper enables the students to understand the basic techniques and theories
of real and complex analysis, two traditionally separated subjects.
2.1.12 MAT 632: NUMERICAL METHODS:
This paper will help the students to have an in depth knowledge of various
advanced numerical methods and some interpolation techniques.
2.1.13 MAT 633: ORDINARY DIFFERENTIAL EQUATIONS AND MATHEMATICAL
MODELLING:
Mathematical modeling is a mathematical tool for solving real world problems.
In this paper, students study a problem-solving process and learn how to
identify a problem, construct or select appropriate models, find solutions and
implement the model. Emphasis lies on model construction in order to promote
student creativity and demonstrate the link between theoretical mathematics and
real world applications.
2.1.14 MAT 634: INTRODUCTION TO MATHEMATICAL PAC KAGES:
This paper gives a comprehensive introduction to the basic commands and
operations in MATLAB. It aims at introducing the students to write the script
files and function files and effective use them in implementing the built in
features of 2-D ploting and 3-D ploting. Also aims at introducing the
programming with applications.
6
2.1.15 MAT 635: OPERATIONS RESEARCH:
As Operations Research (OR) applications continue to grow and flourish in a
number of decision making fields, a number of applications of linear
programming are covered in this paper. This skill-based paper aims at imparting
theoretical knowledge of optimization techniques which are widely used in the
industry to optimize available resources.
2.1.16 MAT 636: MECHANICS:
This paper helps the students to develop an ability to apply fundamental
knowledge of mathematics and science. It aims at analyzing and interpretation
of mechanical laws and theorems. It also helps the students to identify,
formulate, and solve problems involving mechanics and to appreciate the
application of mechanics in various fields.
2.2 CERTIFICATE COURSES:
2.2.1 MAT 101: FOUNDATION OF MATHEMATICS:
This course is designed as a foundation course in Mathematics for those who
have not been exposed to any Mathematics course earlier. This enables the
students to improve their analytical, reasoning and problem solving skills.
Topics included are Set Theory, Theory of Equations, Matrices, Determinants,
Differential Calculus and Integral Calculus.
2.2.2 MAT 201: INTRODUCTION TO MATHEMATIC A:
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.
7
2.2.3 MAT 301: QUANTITATIVE TECHNIQUES FO R MANAGERS:
This skill based paper aims at imparting theoretical knowledge of optimization
techniques. These techniques are widely used in the industry to optimize available
resources. This will help the student to apply the mathematical techniques to real life
situations.
2.2.4 MAT 401: QUANTITATIVE APTITUDE FOR COMPETITIVE EXAMINATIONS:
The quantitative aptitude occupies a very important place in any business
school entrance examination. This skill based paper aims at imparting the
aptitude knowledge required for competitive examination and provides a well-
knitted path to success. This knowledge acquisition will help the students to
overcome the hurdles of competitive examinations like CAT, MAT, XAT,
JMET, GMAT, SWAT, etc.,
8
3
COURSE STRUCTURE
3.1 B.Sc MATHEMATICS:
Semester Paper Code Title of the Paper Hrs /
Week Marks Credit
I MAT 131 INTRODUCTORY ALGEBRA 6 100 4
II MAT 231 CALCULUS 6 100 4
III MAT 331 DIFFERENTIAL EQUATIONS AND ITS
APPLICATIONS 6 100 4
IV MAT 431 ANALYTICAL GEOMETRY AND VECTOR
CALCULUS 6 100 4
V
MAT 531 ALGEBRA 5 100 4
MAT 532 INTEGRAL TRANSFORMS AND LINEAR
PROGRAMMING 5 100 4
MAT 533 INTRODUCTION TO TOPOLOGY AND
DIFFERENTIAL GEOMETRY 5 100 4
MAT 534 PROGRAMMIN IN C 5 100 4
MAT 535 GRAPH THEORY 5 100 4
MAT 536 NUMBER THEORY 5 100 4
VI
MAT 631 REAL AND COMPLEX ANALYSIS 5 100 4
MAT 632 NUMERICAL METHODS 5 100 4
MAT 633 ORDINARY DIFFERENTIAL EQUATIONS
AND MATHEMATICAL MODELLING 5 100 4
MAT 634 INTRODUCTION TO MATHEMATICAL
PACKAGES 5 100 4
MAT 635 OPERATIONS RESEARCH 5 100 4
MAT 636 MECHANICS 5 100 4
9
3.2 CERTIFICATE COURSES
Semester Subject Code Paper Name Hrs /
Week Credit
I MAT 101 FOUNDATIONS OF MATHEMATICS 4 2
II MAT 201 INTRODUCTION TO MATHEMATICA 4 2
III MAT 301 QUANTITATIVE TECHNIQUES FOR
MANAGERS. 4 2
IV MAT 401 QUANTITATIVE APTITUDE FOR
COMPETITIVE EXAMINATIONS. 4 2
10
4 SYLLABI FOR REGULAR PAPERS
4.1 MAT 131: INTRODUCTORY ALGEBRA
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – I Semester
MAT 131: INTRODUCTORY ALGEBRA
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 and Contrapostive of Propositions – open sentences – Quantifiers – Logical implications involving quantifiers – Rules of inferences.
UNIT II: SETS, RELATIONS AND FUNCTIONS: (25 Hrs)
Sets :– 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 of sets. Relations : - Properties of special binary relations - equivalence relations - ordering relations - Hasse Diagram. Functions : - Types of functions – composition of functions – invertible functions – inverse of compositions ( Standard theorems and related problems ).
UNIT III : MATHEMATICAL INDUCTION AND RECURCIVE REL ATION (15 Hrs)
Mathematical induction – induction principle - related examples – Recursive Definitions – Fibonacci Sequence
– Lucas sequence – Eulerian numbers – Ackermann’s numbers – Other recursive definitions – Union and
intersection of n sets – conjunction and disjunction of n-proposition – well formed formulae.
UNIT IV : MATRIX THEORY (20 Hrs)
Recapitulation of fundamentals of matrix algebra – Symmetric and skew-symmetric – Hermitian and skew 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 vectors and properties – Cayley Hamilton theorem and its use in finding inverse and powers of a matrix.
Total Hours: (75+15)
11
TEXT BOOKS:
1. Doris L C Chen K T Lueng, Elementary Set Theory – Part I, Reprint.: Hong Kong University Press, 2009. (Chapter 2: A, B, C, D, E, F, G, H, I, J; Chapter 3: A, B )
2. D.S.Chandrasekharaiah, Discrete Mathematical Structures, 2nd ed.: PRISM Book Pvt. Ltd., 2005 ( 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. B S Vatssa, Theory of Matrices. New Delhi: New Age International Publishers., 2005. (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 )
4. Kenneth H Rosen, Discrete Mathematics and its Applications, 5th ed.: WCB / McGraw – Hill., 1999 . ( 10.1, 10.2, 10.3, 10.4 ) Books for Reference:
1. Tremblay & Manohar, Discrete Mathematical Structures with Application to Computer Science, 5th ed.: Tata McGraw Hill Book Company, 2000.
2. Shanti Narayan, Text book of Matrices, 5th ed. New Delhi: S Chand and Co., 1968.
Suggested Web links:
1. http://www.cs.columbia.edu/~zeph/3203s04/lectures.html 2. http://home.scarlet.be/math/matr.htm 3. http://www.cut-the-knot.org/induction.shtml 4. http://www.themathpage.com/ 5. http://www.abstractmath.org/ 6. http://mathworld.wolfram.com/DiscreteMathematics.html
FORMAT OF QUESTION PAPER
MAT 131: INTRODUCTORY ALGEBRA
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 2
10 1 10 Unit II 3
Unit III 2
Unit IV 3
B
Unit I 2
9 2 18 Unit II 3
Unit III 2
Unit IV 3
C Unit I and II 5
8 6 48 Unit III and IV 5
D Unit I, II, III and IV 4 3 8 24
Total 100
12
4.2 MAT 231: CALCULUS
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – II Semester
MAT 231: CALCULUS
UNIT I: SUCCESSIVE AND PARTIAL DIFFERENTIATIONS (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 and composite functions – Jacobians.
UNIT II : DERIVATIVES OF ARCS (25 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 – Pedal equations –
Derivative of an arc in Cartesian, parameter and polar forms – Equation of a conic in polar form – Convexity,
concavity and curvature of plane curves – Formula for radius of curvature in Cartesian, parametric, polar and pedal
forms – Centre of curvature – Evolutes and involutes – Envelopes – Asymptotes – Singular points – Cusp, node and
conjugate points. Tracing of standard Cartesian and polar curves.
UNIT III : LIMITS, CONTINUITY AND MEAN VALUE THEORE MS (15 Hrs)
Definition of the limit of a function (ε - δ) form – Continuity – Types of discontinuities – Properties of
continuous functions on a closed interval – Differentiability – Differentiability implies continuity – Converse not
true – 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 IV: INTEGRAL CALCULUS (20 Hrs)
Integral Calculus: Recapitulation of methods of integration and Definite Integral – Reduction
Formulae – Application of Integral Calculus – Length of arcs – Surface areas and Volumes of solids of
revolutions for standard curves in Cartesian and Polar forms, Improper Integrals – beta and gamma functions –
properties – relation between beta and gamma functions.
Total Hours: (75+15)
13
TEXT BOOKS:
1. Shanthi 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. Shanthi Narayan, Integral Calculus, Reprint. New Delhi: S. Chand and Company Ltd., 2004. (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 )
Books for Reference:
1. G B Thomas and R L Finney, Calculus and Analytical geometry, 10th ed.: Addison – Wesley, 2000. 2. S. Narayanan & T. K. Manicavachogam Pillay, Calculus.: S. Viswanathan Pvt. Ltd., 1996, vol. I & II. 3. S.Narayanan and T.K.Manicavachogam Pillay, Calculus ( I & II). Chennai, India: S. Viswanathan Pvt. Ltd., 1996. 4. Joseph Edwards, An elementary treatise on the differential calculus: with applications and numerous example,
Reprint. Charleston, USA: BiblioBazaar, 2010. 5. G K Ranganath, Text book of B.Sc., Mathematics, Revised ed. New Delhi, India: S Chand and Co., 2011. 6. Frank Ayres and Elliott Mendelson, Schaum's Outline of Calculus, 5th ed. USA: Mc. Graw Hill., 2008. 7. N. P. Bali, Differential Calculus, New ed. New Delhi, India: Laxmi Publications (P) Ltd.., 2010.
Suggested Web links:
1. http://ocw.mit.edu/courses/mathematics/ 2. http://planetmath.org/encyclopedia/TopicsOnCalculus.html 3. http://ocw.mit.edu/OcwWeb/Mathematics/18-01Fall-2005/CourseHome/index.htm 4. http://mathworld.wolfram.com/Calculus.html
FORMAT OF QUESTION PAPER MAT 231: CALCULUS
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 2
10 1 10 Unit II 3
Unit III 2
Unit IV 3
B
Unit I 2
9 2 18 Unit II 3
Unit III 2
Unit IV 3
C Unit I and II 5
8 6 48 Unit III and IV 5
D Unit I, II, III and IV 4 3 8 24
Total 100
14
4.3 MAT 331: DIFFERENTIAL EQUATIONS AND A PPLICATIONS
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – III Semester
MAT 331: DIFFERENTIAL EQUATIONS AND ITS APPLICATION S
UNIT I : FIRST ORDER DIFFERENTIAL EQUATIONS (15 Hrs)
Solution of ordinary differential equations of first order and first degree – Variable separable and reducible to variable separable forms – Homogeneous and reducible to homogeneous forms – Linear equations and Bernoulli equations – Exact equations, equations reducible to exact form with standard integrating factors – Clairaut’s equation – singular solution for Clairaut’s equation. Orthogonal trajectories.
UNIT II : HIGHER ORDER DIFFERENTIAL EQUATIONS (25 Hrs)
Second and higher order ordinary linear differential equations with constant coefficients –Cauchy-Euler differential equations – Simultaneous differential equations (two variables) with constant coefficients. Second order linear differential equations with variable coefficients by the following methods: (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) - Total and Simultaneous differential equation.
UNIT III : PARTIAL DIFFERENTIAL EQUATIONS (20 Hrs)
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 to equations with constant coefficients.
UNIT IV : APPLICATIONS OF DIFFERENTIAL EQUATIONS (15 Hrs)
Particle dynamics: Simple Harmonic motion – Projectiles: – horizontal plane - trajectory – velocity of projection – angle of projection – Range - time of flight – greatest height - projectiles on inclined plane. Central orbit and 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.
Total Hours: (75+15)
15
TEXT BOOKS:
1. Frank Ayres, Schaum's outline of theory and problems of Differential Equations, 1st ed. USA: McGraw-Hill, 1972. (Chapters: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 31, 32, 33)
2. S Narayanan, Dynamics, 16th ed. New Delhi: S Chand and Company Ltd., 1986. (Chapter: 8 ) 3. N.P. Bali, Dynamics (Golden Series), Latest ed. New Delhi, India: Lakshmi Publications (p) Limited, 2004.
(Chapters:5,6,7)
Books for Reference:
1. I N Sneddon, Elements of Partial Differential Equations, 3rd ed.: Mc. Graw Hill., 1980. 2. S Narayanan & T K Manicavachogam Pillay, Differential Equations.: S V Publishers Private Ltd., 1981. 3. P Duraipandian, Mechanics, 4th ed. New Delhi: S Chand and Company Ltd., 1995. 4. G K Ranganath, Text book of B.Sc, Mathematics, Revised ed. New Delhi, India: S Chand and Company Ltd., 2011. 5. George F Simmons, Differential equation with Applications and historical notes, 2nd ed.: McGraw-Hill Publishing
Company, Oct 1991.
Suggested Web links:
1. http://ocw.mit.edu/courses/mathematics/ 2. http://www.analyzemath.com/ 3. http://tutorial.math.lamar.edu/classes/de/de.aspx 4. http://www.sosmath.com/diffeq/diffeq.html 5. http://www.analyzemath.com/calculus/Differential_Equations/applications.html
FORMAT OF QUESTION PAPER
MAT 331: DIFFERENTIAL EQUATIONS AND APPLICATIONS
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 2
10 1 10 Unit II 3
Unit III 3
Unit IV 2
B
Unit I 2
9 2 18 Unit II 3
Unit III 3
Unit IV 2
C Unit I and II 5
8 6 48 Unit III and IV 5
D Unit I, II, III and IV 4 3 8 24
Total 100
16
4.4 MAT 431: ANALYTICAL GEOMETRY AND VECTOR CALCU LUS
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – IV Semester
MAT 431: Analytical Geometry and Vector Calculus
UNIT I: ANALYTICAL GEOMETRY(3D) - (Lines and Plane s) (25 Hrs)
Direction cosines of a line – Direction ratios of the join of two points - Projection on a line – Angle between the lines – Area of a triangle and volume of a tetrahedron with given vertices - Equation of line in different forms – Perpendicular from a point onto a line - Equation of a plane in different forms – Perpendicular from a point onto a plane - Angle between two planes – Line of intersection of two planes - Plane co-axial with given planes – Planes bisecting the angle between two planes – Angle between a line and a Plane – Co-Planarity of two lines – Shortest distance between two lines.
UNIT II : ANALYTICAL GEOMETRY(3D) - (Spheres, Cyli nders & Cone) (15 Hrs)
Equation of the sphere in its general form – Determination of the centre and radius of a sphere with the 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.
UNIT III : VECTOR DIFFERENTIAL CALCULUS, LINE AND M ULTIPLE INTEGRALS (25 Hrs)
Vector Differentiation – Gradient – Divergence – Curl and Laplacian Operators – Vector identities - Line integral and basic properties – examples on evaluation of the integrals – Definition of a double integral – its conversion to iterated integrals – evaluation of double integral by change of order of integration and by change of variables – surface areas as double integrals – Definition of a triple integral and evaluation – change of variables – volume as a triple integral – Line, surface and volume integrals of vector functions
UNIT IV: INTEGRAL THEOREMS (10 Hrs)
Green’s theorem in the plane (statement and proof) – Direct consequences of the theorem – The Divergence theorem (statement only) – Direct consequences of the theorem – The Stoke’s theorem (statement only) – Direct consequences of the theorem.
Total Hours: (75+15)
17
TEXT BOOKS:
1. S.P.Mahajan & Ajay Aggarwal, Comprehensive Solid Geometry , 1st ed.: Anmol Publications , 2000. (Sections : 1.8, 1.15, 3.2, 3.3,3.7, 3.8, 3.9, 3.10, 3.12, 3.13, 4.2, 4.3, 5.3, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 6.10 )
2. B Spain, Vector Analysis, 2nd ed. Calcutta: Radha Publishing Co., 1988. ( Chapter 3 : Sections 17, 18, 19, 20, 21, Chapter 5 : Sections 32, 33, 34, 35, 36, Chapter 6 : Sections 37, 38, 39, Chapter 7 : 41, 42, 43, Chapter 8 : Sections 44, 45, 46, 47, 48, 49, Chapter 9 : Sections 50, 51, 53, 54, 55, Chapter 10 : Sections 56, 57, 58, 59, 60, Chapter 11: section 66 )
Books for Reference: 1. Shanthi Narayan, Analytical Solid Geometry. New Delhi: S. Chand and Co. Pvt. Ltd., 2004. 2. S Narayanan & Manicavachogam Pillay, Vector Algebra and Analysis, 4th ed.: S V Publishers, 1986. 3. Raisinghania Md, Saxena Hc, and Dass Hk, Simplified course in Vector Calculus, 1st ed. New Delhi, India: S.Chand
and Company Ltd., 2002. 4. G K Ranganath, Text book of B.Sc. Mathematics, Revised ed. NewDelhi, India: S Chand and Co., 2011.
Suggested Web links :
1. http://ocw.mit.edu/courses/mathematics/ 2. http://www.univie.ac.at/future.media/moe/galerie.html 3. http://mathworld.wolfram.com/AnalyticGeometry.html 4. http://www.math.gatech.edu/~harrell/calc/
FORMAT OF QUESTION PAPER
MAT 431: ANALYTICAL GEOMETRY AND VECTOR CALCULUS
Part Unit and No. of subdivisions to be set in the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 2
10 1 10 Unit II 3
Unit III 3
Unit IV 2
B
Unit I 2
9 2 18 Unit II 3
Unit III 3
Unit IV 2
C Unit I and II 5
8 6 48 Unit III and IV 5
D Unit I, II, III and IV 4 3 8 24
Total 100
18
4.5 MAT 531: ALGEBRA
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – V Semester
MAT 531: ALGEBRA
UNIT I : GROUPS (25 Hrs)
Groups – Subgroups – Cyclic groups – Cosets – Lagrange’s theorem – Normal Subgroup – Quotient Group – Homomorphism of groups – Fundamental Theorem of Homomorphism – Isomorphism - Cauchy’s theorem for abelian groups – Permutation groups.
UNIT II : RINGS, INTEGRAL DOMAINS AND FIELDS (15 Hrs)
Rings – Properties – Subrings – Ideals, Principal, Prime and Maximal ideal in a commutative ring – Homomorphism and Isomorphism of Rings –Integral domains – Fields – Properties following the definition.
UNIT III : LINEAR ALGEBRA (20 Hrs)
Vector space: Definition – Examples – Properties – Subspaces – Span of a set – Linear dependence and independence – Dimension and Basis
Linear transformation: Definition and examples – Range and Kernel of a linear map – Rank and Nullity – inverse of a linear transformation – consequence of Rank-nullity theorem.
Inner product space: Definition – examples – orthonormal sets – Schwarz inequality – Gram Schmidt orthogonolization process – problems.
Total Hours:(60+15)
19
TEXT BOOKS:
1. Herstein I N, Topics in Algebra, 4th ed. New Delhi, India: Vikas Publishing House Pvt. Ltd, 1991. (for Unit I and II ) .( Sections : 2.1 , 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.10, 3.1, 3.2, 3.3, 3.4, 3.5)
2. Krishnamoorty V K and Mainra V P and Arora J L, An Introduction to Linear Algebra, Reprint. New Delhi, India: Affiliated East West Press Pvt. Ltd., 2003. (for Unit III) . ( Sections: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 4.1, 4.2, 4.3, 4.4, 4.5, 7.2.1, 7.2.2, 7.2.3, 7.2.4, 7.2.5, 7.2.6, 7.2.7, 7.2.8, 7.2.9, 7.2.10, 7.2.11, 7.2.12)
Books for Reference:
1. John B Fraleigh, A First course in Abstract Algebra, 3rd ed.: Narosa Publishing House., 1990. 2. Vashista, A First Course in Modern Algebra, 11th ed.: Krishna Prakasan Mandir, 1980. 3. R. Balakrishan and N.Ramabadran, A Textbook of Modern Algebra, 1st ed. New Delhi, India: Vikas
publishing house pvt. Ltd., 1991. 4. G K Ranganath, Text book of B.Sc. Mathematics, Revised ed. NewDelhi, India: S Chand and Co., 2011. 5. Michael Artin, Algebra, 2nd ed. New Delhi, India: PHI Learning Pvt. Ltd., 2011.
Suggested Web links :
1. http://ocw.mit.edu/courses/mathematics/ 2. http://www.extension.harvard.edu/openlearning/math222/ 3. http://mathworld.wolfram.com/Algebra.html 4. http://www.math.niu.edu/~beachy/aaol/ 5. http://planetmath.org/encyclopedia/Inverse.html
FORMAT OF QUESTION PAPER
MAT 531: ALGEBRA
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 4
9 2 18 Unit II 3
Unit III 3
C Unit I 4
8 6 48 Unit II and III 6
D Unit I, II and III 4 3 8 24
Total 100
20
4.6 MAT 532: INTEGRAL TRANSFORMS AND LINEAR PROGR AMMING
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – V Semester
MAT 532 : INTEGRAL TRANSFORMS AND LINEAR PROGRAMMIN G
UNIT I : FOURIER SERIES AND FOURIER TRANSFORMS (25 Hrs)
Introduction to integral transforms - Fourier Series of functions with period 2π and period 2L – half range cosine and sine series – Finite Fourier cosine and sine transforms – transform of some common functions.
The Fourier Integral – Complex Fourier Transforms – Basic Properties - - Transform of the derivative and t he derivative of the transform – Convolution theorem – Parseval’s Identity – Fourier sine and cosine transforms – transforms for first and second order derivatives.
UNIT II : LAPLACE TRANSFORM (15 Hrs)
Laplace Transform of standard functions – Laplace transform of periodic functions – Inverse
Laplace transform– Solution of ordinary differential equation with constant coefficient using Laplace
transform.
UNIT III : LINEAR PROGRAMMING (20 Hrs)
Introduction - General form of Linear Programming Problem – Graphical Method of solution – Simplex Method for Maximization of L.P.P. standard form – Minimization problem in standard form – Big M method – Two phase method.
Introduction to Transportation Problem – Initial Basic Feasible solution – Moving towards Optimality – Degeneracy in Transportation Problems – Unbalanced Transportation Problem – Mathematical formulation of Assignment Problems – Hungarian method for solving Assignment problem – Unbalanced Assignment problem – Travelling salesman problem – Formulation of Travelling salesman problem as an assignment problem and solution procedure.
Total Hours:(60+15)
21
TEXT BOOKS:
1. Erwin Kreyszig, Advanced Engineering Mathematics, 8th ed. New Delhi, India: Wiley India Pvt. Ltd., 2010. (Sections : 5.1, 5.2, 5.3, 5.4,5.5,5.6,5.7, 5.8, 5.9, 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, 10.8, 10.9,10.10,10.11).
2. S Dharani Venkata Krishnan, Operations Research - Principles and Problems, 3rd ed.: Keerthi Publishing House house(p) ltd., 1992.(Sections : 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7.1, 2.7.2 2.7.3, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7).
Books for Reference:
1. Sneddon I N, Fourier Transform, 1st ed. New York, USA: Dover Publications., 1995. 2. Raisinghania M.D., Laplace and Fourier Transforms. New Delhi, India: S. Chand and Co. Ltd. , 1995. 3. G K Ranganath, Text book of B.Sc. Mathematics, Revised ed. New Delhi, India: S Chand and Company Ltd.,
2011. 4. G Hadley, Linear Programming, Reprint. New Delhi, India: Narosa Publishing House, 2002. 5. Hamdy A Taha, Operations Research – an introduction, 6th ed. New Delhi, India: Prentice Hall of India.,
1996. 6. V K Kapoor, Operations Research, Reprint. New Delhi, India: Sultan Chand & Sons., 1994.
Suggested Web links:
1. http://ocw.mit.edu/courses/mathematics/ 2. http://www.fourier-series.com/ 3. http://mathworld.wolfram.com/ 4. http://www.princeton.edu/~rvdb 5. http://www.zweigmedia.com/RealWorld/Summary4.html 6. http://people.brunel.ac.uk/~mastjjb/jeb/or/contents.html 7. http://people.brunel.ac.uk/~mastjjb/jeb/or/lpmore.html
FORMAT OF QUESTION PAPER MAT 532: INTEGRAL TRANSFORMS AND LINEAR PROGRAMMING
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 4
9 2 18 Unit II 3
Unit III 3
C Unit I 4
8 6 48 Unit II and III 6
D Unit I, II and III 4 3 8 24
Total 100
22
4.7 MAT 533: INTRODUCTION TO TOPOLOGY AND DIFFERENTIAL GEOMETRY
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – V Semester
MAT 533: INTRODUCTION TO TOPOLOGY AND DIFFERENTIAL GEOMETRY
Unit I: INTRODUCTION TO TOPOLOGY (20 Hrs)
Metric Spaces – Open sets and closed sets – convergence and completeness – spaces of
continuous functions – Topological space – Definition and examples – Open base and Open
sub bases – The function algebras C(X,R), C(X,C).
Unit II: THE THEORY OF CURVE (20 Hrs)
Regular representations – regular curves – Orthogonal projections – Implicit representations of
a curve – Regular curves of class Cm – Definition of arc length – Arc length as a parameter –
unit tangent vector - Tangent line and normal plane – curvature – principal normal unit vector
– Principal normal line and osculating plane – binormal – moving trihedron – torsion –
spherical indicatrices – Frenet Equations – The fundamental existence and uniqueness theorem.
Unit III: CONCEPT OF A SURFACE (20 Hrs)
Regular parametric representation – coordinate patches – Definition of a simple surface –
Tangent plane and Normal line – Topological properties of simple surfaces – First fundamental
form – arc length and surface area – second fundamental form – normal curvature – principal
curvatures and directions – Gaussian curvature and mean curvature.
Total Hours: (60+15)
23
Text Book:
1. G F Simmons, Introduction To Topology and Modern Analysis, 1st ed. New Delhi, India: Tata Mc Graw-Hill Publishing Company Limited., 2004.
2. Martin Lipschultz, Schaum's Outline of Differential Geometry, Reprint. New Delhi, India: Tata McGraw-Hill Publishing Company Limited, 2005.
Reference Books:
1. Munkers, Topology , 2nd ed.: Pearson Education Pvt. Limited. , 2006. 2. Erwin Kreysizig, Differential Geometry, New ed. NewYork: Dover publications inc. , 1991. 3. T J Willmore, An introduction to Differential Geometry, 21st Indian Impression: Oxford University
Press, 2006..
Suggested Web links: 1. http://mathworld.wolfram.com/Topology.html 2. http://140.177.205.65/infocenter/Courseware/259/ 3. http://www.trillia.com/online-math/geometry.html 4. http://www.trillia.com/online-math/geometry 5. http://www.wisdom.weizmann.ac.il/~yakov/Geometry/ 6. http://people.math.gatech.edu/~ghomi/LectureNotes/index.html
FORMAT OF QUESTION PAPER MAT 533: INTRODUCTION TO TOPOLOGY AND DIFFERENTIAL GEOMETRY
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
24
4.8 MAT 534: PROGRAMMING IN C
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours)– V Semester
MAT 534: PROGRAMMING IN C
UNIT I: BASICS AND CONTROL STRUCTURES (20 HRS)
Basics: Algorithm and Flow chart. Introduction to C: Development of C, Features, constants and variables, data types, operators and expressions, library functions. I/O statements: Formatted and Unformatted I / O, Scanf(), Print(), getchar() and putchar() functions. Control Structures: Conditional and Unconditional, if, for, while and do… while, switch, break and continue, goto statement. Lab Programs on the topics.
UNIT II: ARRAYS, FUNCTIONS AND POINTERS (20 HRS)
Arrays: one and Multi dimensional arrays, Strings and String functions. Linear and binary search. Functions: Definition, different types, advantages, calling a function, passing parameters, call by reference and call by value, local and global variables, recursive functions. Pointers: Declaration, operation on pointers, relationship. Lab Programs on the topics.
UNIT III: STRUCTURE, UNION AND FILES (20 HRS)
Structure and Unions: Defining a structure, classification, union, user-defined data types, pointers to a structure, structure as an argument to a function. Lab Programs on the topics. Files: Sequential files, file pointers random files, processing a data file, unformatted data file, file error handling, implementation of copy and merge commands.
LAB PROGRAMS:
1. LARGEST OF THREE NUMBERS. 2. SUM AND AVERAGE OF N NUMBERS. 3. FACTORIAL OF A NUMBER. 4. PRIME NUMBER. 5. REVERSE OF A NUMBER. 6. FIBONACCI SERIES. 7. FAHRENHEIT-TO- CELSIUS CONVERSION TABLE 8. ADDITION AND MULTIPLICATION OF TWO MATRICES. 9. READING, WRITING AND REVERSING AN INTEGER ARRAY. 10. LINEAR SEARCH 11. BINARY SEARCH. 12. PALINDROME 13. LENGTH OF A STRING. 14. STRING COPY, COMPARE, CONCATENATE. 15. PROGRAMS BASED ON FUNCTIONS 16. FUNCTIONS-CALL BY VALUE , CALL BY REFERENCE. 17. RECURSIVE FUNCTIONS. 18. STRUCTURE AND UNION- STUDENT PROFILE, LIBRARY DETAILS 19. FILE OPERATIONS 20. TO FIND THE SOLUTION OF AN EQUATION F(X)=0 USING BISECTION METHOD.
Total Hours: (60+15)
25
TEXT BOOKS:
1. Yashavant Kanetkar, LET US C, 8th ed.: Infinity Science Press., 2008. 2. Byron S Gottfried, Schaum's Outline of Programming with C, 2nd ed.: TATA MCGRAW-HILL. , 1996.
REFERENCE BOOKS:
1. E.Balagurusamy, Programming in ANSI C, 5th ed.: Tata Mc-Graw Hill., 2010. 2. P.J.Deitel and H.M.Deitel, C How to Program, 2nd ed.: Prentice Hall - gale, 1994.
Suggested Web Links:
1. http://www.cs.cf.ac.uk/Dave/C/CE.html
2. http://www.cprogramming.com/
3. http://www.howstuffworks.com/c.htm
FORMAT OF QUESTION PAPER
MAT 534: PROGRAMMING IN C
Part Unit and No. of subdivisions to be set in the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
26
4.9 MAT 535: GRAPH THEORY
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – V Semester
MAT 535: GRAPH THEORY
UNIT I : GRAPHS, PATHS, CYCLES AND TREES (20 Hrs)
Introduction -Definition-Examples-Famous problems in graph theory- Paths Cycles-Connectivity-Eulerian graphs-Hamiltonian graphs-Trees-Elementary properties of trees-counting trees- Shortest path algorithms- Algorithms on trees -Application of trees.
UNIT II: PLANARITY OF GRAPHS AND COLORING (20 Hrs)
Planar graphs-Euler's formula-Graphs on other surfaces-Dual graphs-Infinite graphs- Coloring graphs- Coloring vertices-Brook's theorem-coloring maps-coloring edges-chromatic polynomials-Algorithm on planarity testing.
UNIT III : DIGRAPHS, MATCHING AND MENGER’S THEOREM (20 Hrs)
Digraphs- Eulerian digraphs and tournaments-Markov chains- Matching-Hall's 'marriage' theorem-Transversal theory-Application of Hall's theorem-Menger's theorem-Network flows.
Total Hours: (60+15)
27
TEXT BOOK:
Dr.Robin.J.Wilson, Introduction to Graph Theory, 4th ed. New Delhi, India: Pearson, 2003.
Books for Reference:
1. G.Chartrand and P.Zhang, Introduction to Graph Theory, 1st ed. New Delhi: Tata McGraw-Hill, 2006.
2. G.Chartrand and L.Lesniak, Graphs and Digraps, 4th ed. Boca Raton, USA: CRC Press, 2004. 3. F.Harary, Graph Theory, 1st ed.: Narosa Publishing House, 2001. 4. S.Arumugam and S. Ramachandran, Invitation to Graph Theory, 1st ed. Chennai, India: Scitech,
2006. 5. Narsingh Deo, Graph Theory with Applications to Engineering and Computer Scinece, 2nd ed.
New Delhi, India: PHI Learning, 2009.
Suggested Web links:
1. http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html 2. http://www.personal.kent.edu/~rmuhamma/GraphTheory/graphTheory.htm 3. http://www.utm.edu/departments/math/graph/ 4. http://diestel-graph-theory.com/ 5. http://www.ti.inf.ethz.ch/ew/courses/GT05/ 6. http://users.utu.fi/harju/graphtheory/graphtheory.p df 7. http://www.mathcove.net/petersen/ 8. http://www.maths.nottingham.ac.uk/personal/pmznd/g13gra/sylgra.html 9. http://www3.interscience.wiley.com/journal/35334/home 10. http://people.math.gatech.edu/~thomas/FC/fourcolor.html
FORMAT OF QUESTION PAPER MAT 535: GRAPH THEORY
Part Unit and No. of subdivisions to be set
in the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
28
4.10 MAT 536: NUMBER THEORY
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – V Semester
MAT 536: NUMBER THEORY
UNIT – I : DIVISIBILITY THEORY IN INTEGERS (20 Hrs)
The division algorithm – The greatest common divisor – The Euclid algorithm – Basic properties of
congruence – special divisibility tests – Linear congruencies - The Diophantine Equations ax + by =
c.
UNIT – II : FERMATS THEOREM (20 Hrs)
The Little Fermat’s theorem and pseudoprimes – Wilson Theroem – The Fermat-Kraitchik
Factorization Method - The function τ and σ - The Mobius inversion formula – The greatest integer
function .
UNIT – III : EULER’S THEOREM AND SUM OF SQUARES (20 Hrs)
Euler’s Phi-function – Euler’s theorem – Some properties of the Phi-function – Euler’s criterion –
The Legendre’s symbol and its properties – Quadratic reciprocity – The Diophantine Equations
x2 + y2 = z2, x4 + y4 = z4.
Total Hours: (60+15)
29
Text Book:
David M. Burton, Elementary Number Theory, 6th ed. New Delhi, India: Tata McGraw-Hill, 2008.
Reference Books:
1. J.Niven and H.Zuckerman, An Introduction to the Theory of Numbers, 5th ed. New York, USA: John Wiley & Sons Inc., 1991.
2. Kenneth Rosen, Elementary Number Theory and its applications, 2nd ed.: Reading Mass Addision – Wesley., 1987.
3. Gareth.A.Jones and Josephine.M.Jones, Elementary Number Theory, 2nd ed.: Springer, 2009.
Suggested Web links:
1. http://www.numbertheory.org/
2. http://www.mathpages.com/home/inumber.htm
3. http://mathworld.wolfram.com/NumberTheory.html
4. http://www.math.niu.edu/~rusin/known-math/index/11AXX.html
FORMAT OF QUESTION PAPER MAT 536: NUMBER THEORY
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
30
4.11 MAT 631: REAL AND COMPLEX ANALYSIS
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – VI Semester
MAT 631: REAL AND COMPLEX ANALYSIS
UNIT I : OPEN SETS, CLOSED SETS, COUNTABLE SETS AND SEQUENCES (10 Hrs) Limit of a set, Open sets, Closed sets, closure of a set, countable and uncountable sets -Sequences: Definition of Sequences – limit of a sequence – algebra of limits of a sequence – convergent, divergent and oscillatory sequences - problems thereon. Bounded sequences – every convergent sequence is bounded – converse is not true – Monotonic sequences and their properties – problems using the properties – Cauchy sequence – Results related to Cauchy’s sequences.
UNIT II: INFINITE SERIES (25 Hrs)
Infinite Series: Definition of convergence, divergence and oscillation of series – properties of convergent series – a series of positive terms either convergences or diverges – 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 conditional convergence, Leibnitz test for alternating series. – Summation of Binomial, Exponential and Logarithmic series.
UNIT III : COMPLEX ANALYSIS (25 Hrs)
Continuity and Differentiability of complex functions – Analytic Functions – Cauchy-Riemann equations – Harmonic functions – Definite integrals – Contours – Line integrals – The Cauchy’s Integral theorem and its direct consequences – Cauchy’s Integral formula for the functions and derivatives – Morera’s theorem – Applications to the Evaluation of Simple line integrals – Cauchy’s inequality – Liouville’s Theorem – Fundamental theorem of Algebra – power series – Taylors series – Laurent’s series – circle and radius of convergence – sum functions - Transformations – Definition of Conformal Mapping – Bilinear Transformation – Cross ratio – Properties – Inverse points – Bilinear Transformation transforms Circles into circles or lines – Problems there on – Discussion of various Transformations.
Total Hours: (60+15)
31
TEXT BOOKS:
1. S.C.Malik and Savita Arora, Mathematical Analysis, 2nd ed. New Delhi, India: New Age international (P) Ltd., 1992 (Chapter2, Chapter 3 : Sections 1, 2, 3, 4, 5, 6, 7, 9 ; Chapter 4 : Sections 1.1, 1.2, 1.3, 1.4, 2, 2.1, 2.2, 2.3, 3, 3.1, 3.2, 4, 5, 6, 10, 10.1, 10.2)
2. R V Churchil & J W Brown, Complex Variables and Applications, 5th ed.: Mc. Graw Hill Companies., 1989. (Sections:14,15,16,17,18,19,20,21,22,23,24,25,36,37,38,39,40,41,47,48,49,51, 52, 53,,54, 55, 56, 57,58, 83, 84, 85, 86, 87, 88, 89, 90, 94)
Books for Reference:
1. Richard R Goldberg, Methods of Real Analysis, Indian ed. New Delhi, India: Oxford and IBH Publishing Co., 1970.
2. L V Ahlfors, Complex Analysis, 3rd ed.: Mc Graw Hill. , 1979. 3. G K Ranganath, Text book of B.Sc. Mathematics, Revised ed. New Delhi, India: S Chand and Company
Ltd., 2011.
Suggested Web links:
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
FORMAT OF QUESTION PAPER
MAT 631: REAL AND COMPLEX ANALYSIS
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 2
10 1 10 Unit II 4
Unit III 4
B
Unit I 2
9 2 18 Unit II 4
Unit III 4
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
32
4.12 MAT 632: NUMERICAL METHODS
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – VI Semester
MAT 632: NUMERICAL METHODS
UNIT I. NUMERICAL SOLUTION OF ALGEBRAIC AND SYSTEM OF EQUATIONS (20 Hrs)
Errors and their analysis – Floating point representation of numbers – Solution of Algebraic and
Transcendental Equations : Bisection method, Iteration method, the method of False Position, Newton
Raphson method.
Solution of linear systems – Matrix inversion method – Gaussian Elimination method – Modification
of the Gauss method to compute the inverse – Method of factorization – Iterative methods.
UNIT II. FINITE DIFFERENCES AND INTERPOLATION (20 Hrs)
Finite differences: Forward difference, Backward difference and Shift Operators – Separation of
symbols – Newton’s Formulae for interpolation – Lagranges interpolation formulae - Hermite, Cubic-Spline
interpolation formulas, Bivariate interpolation and least square approximation.
UNIT III. NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS (20 Hrs)
Numerical differentiation – Numerical integration: Trapezoidal rule, Simpson’s one-third rule and
Simpson’s three-eighth rule.
Numerical solution of ordinary differential equations – Taylor’s series – Picard’s method – Euler’s
method – Modified Euler’s method – Runge Kutta methods - second order (with proof) and fourth order
(without proof).
Total Hours: (60+15)
33
TEXT BOOKS:
1. S S Sastry, Introductory methods of Numerical Analysis, 3rd ed. New Delhi, India: Prentice Hall of India, 1999. (Sections : 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2.1, 2.2, 2.3, 2.4, 2.5, 5.3.1, 5.3.2, 5.3.3, 5.4, 3.3.1, 3.3.2, 3.3.3, 3.9.1, 3.9.3, 3.10, 4.2, 4.4.1).
2. Francis Scheid, Schaum's Outline of Numerical Analysis, Revised ed.: Mc.Graw Hill., 2006. Books for Reference:
1. M K Jain, S R K Iyengar, and R K Jain, Numerical Methods for Scientific and Engineering Computation, 4th ed. New Delhi, India: New Age International, 2003.
2. G K Ranganath, Text book of B.Sc. Mathematics, Revised ed. New Delhi, India: S Chand and Company Ltd., 2011.
Suggested Web links:
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
FORMAT OF QUESTION PAPER MAT 632: NUMERICAL METHODS
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
34
4.13 MAT 633: ORDINARY DIFFERENTIAL EQUATIONS AND MATHEM ATICAL MODELING
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – VI Semester
MAT 633: ORDINARY DIFFERENTIAL EQUATIONS AND MATHEM ATICAL
MODELLING
Unit 1: ORDINARY DIFFERENTIAL EQUATIONS (15 Hours)
Power Series solutions: Series solutions of first order equations, Second order equations – Ordinary
points, Regular singular points and Method of Frobenius.
Unit 2: SPECIAL FUNCTIONS (15 Hours)
Special functions: Legender Differential equations – Legender polynomials, Legender Series and
Orthogonolity property. Bessel Differential equations Gamma function, General solution of Bessel
equation, Identities and Orthogonality property.
Unit3: MATHEMATICAL MODELING (30 Hours)
Mathematical Modeling; Need, Techniques, Classifications and Simple Illustrations through
Geometry, Algebra and Calculus. Limitations of Mathematical Modelling. Modeling through
Ordinary Differential equations: First order – Linear and non linear Growth and Decay Models,
Compartment Models, Mathematical Modelling in Dynamics, Mathematical modeling of Geometrical
problems. Population dynamics, Epidemics, Mathematical modeling in Economics, Models in
medicines, Arm race,. Battles. Second order – Planetary motion, Circular motion, Satellites motion.
Mathematical modeling through second order linear differential equations.
Total Hours: (60+15)
35
Text Books:
1. George F. Simmions, Differential Equations with Applications and Historical Notes, 2nd ed.: Tata Mcghaw Hill., 2003. (Unit 1 and 2)
2. J. N. Kapur, Mathematical Modelling, 1st ed.: New age international Publishers, 1998. (Unit 3 and 4)
Reference Books:
1. Giordano, Weir, and Fox, A First Course in Mathematical Modeling, 3rd ed.: Thomson Brooks/Cole, 2003.
Suggested Web Links:
1. http://ocw.mit.edu/OcwWeb/Mathematics/18-306Fall-2009/MATLABResources/index.htm
2. http://www.sosmath.com/diffeq/modeling/modeling.html
3. http://www.encyclopedia.com/doc/1G1-200987672.html
4. http://serc.carleton.edu/sencer/ode_real_world/index.html
5. http://www.worldscibooks.com/mathematics/7573.html
FORMAT OF QUESTION PAPER MAT 633: ORDINARY DIFFERENTIAL EQUATIONS AND MATHEM ATICAL MODELLING
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 3
10 1 10 Unit II 3
Unit III 4
B
Unit I 3
9 2 18 Unit II 3
Unit III 4
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
36
4.14 MAT 634: INTRODUCTION TO MATHEMATICAL PACKAGES
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – VI Semester
MAT 634: INTRODUCTION TO MATHEMATICAL PACKAGES
Unit I: INTRODUCTION (20 Hrs)
Starting with MATLAB, Creating Arrays – Mathematical Operations with Arrays: Addition and
Subtraction – Array Multiplication – Array division – Element by element operations – Using Arrays
in Matlab built in math functions – Built in functions for analyzing arrays – Script Files : Creating
and Saving script files – global variables – input / output commands - functions and functions files.
Unit II: PROGRAMMING AND GRAPHICS (20 Hrs)
Relational and Logical Operators – Conditional Statements – The Switch-Case Statement – Loops –
Nested Loops and Nested Conditional Statements – The break and continue commands – Two
dimensional Plots : The plot command – the fplot command – Plotting multiple graphs in the same
plot – Formatting plot – Plot with Logarithmic axes – Plots with special graphics – polar plots –
plotting multiple plots on the same page – Three dimensional plots: Line plots – Mesh and Surface
plots – Plots with special graphics – The view command.
Unit III: APPLICATIONS IN MATHEMATICS (20 Hrs)
Bracketing Methods for Locating a Root : Bisections Method of Bolzano, Regula Falsi Method –
Solution of Linear Systems AX=B: PA=LU Factorization with pivoting, Jacobi Iteration Method,
Gauss seidel Iteration method - Trigonometric polynomials – Numerical Differentiation:
Differentiation Using Limits, Differentiation Using Extrapolation – Numerical Integration: Composite
Trapezoidal Rule , Composite Simpsons Rule – Numerical Optimization: Golden Search for
minimum, Fibonacci Search for a Minimum.
Total Hours: (60+15)
37
Text Books:
1. Amos GilaT, MATLAB-An introduction with Applications. New Delhi, India: Wiley India, 2009. (Chapter 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 9.1, 9.2, 9.3, 9.4)
2. John H Mathews and Kurtis D Fink, Numerical Methods Using MATLAB, 4th ed.: Pearson Prentice Hall, 2008. (Chapter 2.2, 3.5, 3.6, 5.4, 6.1, 7.2, 8.1, 8.2)
Reference Books:
1. Desmond.J.Higham and Nicholas.J.Higham, MATLAB Guide, 2nd ed.: SIAM, 2005. 2. Brain.R.Hunt and Ronald.L.Lipsman, A Guide to MATLAB for Beginners and Experienced Users,
Reprint ed. India: Replika Press Private limited, 2006.
Suggested Web Link:
1. http://www.mathworks.com/access/helpdesk/help/techdoc/ 2. http://www.nd.edu/~dtl/cheg258/notes/doc/ReferenceTOC.html 3. http://www.artefact.tk/software/matlab/m2html/doc/
FORMAT OF THE QUESTION PAPER
MAT 634: INTRODUCTION TO MATHEMATICAL PACKAGES
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
38
4.15 MAT 635: OPERATIONS RESEARCH
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – VI Semester
MAT 635: OPERATIONS RESEARCH
Unit I: Introduction to linear and non-linear progr amming (20 Hrs) Introduction to simplex algorithm –Special cases in the Simplex Method –Definition of the Dual Problem – Primal Dual relationships – Dual simplex methods – Unconstrained problems , Necessary and sufficient conditions – Constrained Problems, equality constraints and inequality constraints (Karush-Kuhn-Tucker conditions). Unit II Queuing Theory and game theory (30 Hrs) Elements of a queuing Model – Pure Birth Model – Pure Death Model – Specialized Poisson Queues – Steady state Models: (M/M/1):(GD/∞ / ∞ ) – (M/M/1):(FCFS/∞ / ∞ ) - (M/M/1):(GD/N/ ∞ ) – (M/M/c):(GD/∞ / ∞ ) – (M/M/ ∞ ):(GD/∞ / ∞ ). Game Theory: Optimal solution of two person zero – sum games – Solution of Mixed strategy Games (both graphical and Linear programming solution) – Goal Programming: - Formulation – Tax Planning Problem – Goal programming algorithms – The weights method – preemptive method.
Unit III Network Models (10 Hrs) Linear programming formulation of the shortest-route Problem. Maximal Flow model:- Enumeration of cuts – Maximal Flow Algorithm – Linear Programming Formulation of Maximal Flow Model. CPM and PERT:- Network Representation – Critical path computations – Construction of the Time Schedule – Linear Programming formulation of CPM – PERT calculations.
Total Hours: (60+15)
39
Text Book:
A.H. Taha, Operations research, 8th ed.: Pearson Education, 2007.
Reference Books:
1. R. Ravindran, D.T. Philips, and J.J. Solberg, Operations Research: Principles and Practice, 2nd ed.: John Wiley & Sons, 1976
2. R. Bronson, Schaum's Outline Series of Operations research, 2nd ed.: McGraw Hill., 1997.
3. F.S. Hillier and G.J. Lieberman, Introduction to operations research, 9th ed.: McGraw-Hill, 2009.
4. Chandrasekhara Rao and Shanthi Lata Mishra, Operations research.: Alpha Science International, 2005.
Web Links:
1. http://www.universalteacherpublications.com/univ/cs51contents.htm 2. http://people.brunel.ac.uk/~mastjjb/jeb/or/netex.html
FORMAT OF QUESTION PAPER
MAT 635: OPERATIONS RESEARCH
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 3
10 1 10 Unit II 5
Unit III 2
B
Unit I 3
9 2 18 Unit II 5
Unit III 2
C Unit I 3
8 6 48 Unit II and III 7
D Unit I, II and III 4 3 8 24
Total 100
40
4.16 MAT 636: MECHANICS
CHRIST UNIVERSITY DEPARTMENT OF MATHEMATICS
B.Sc Mathematics (Honours) – VI Semester
MAT 636: MECHANICS
UNIT I: STATICS (20 Hours)
Forces acting at a point-Resultant and components, Parallelogram law of forces, Magnitude and direction of the resultant forces acting at an angle α ,Components of a force in two given directions, Resolved part of a force in a given direction, Triangle law of forces, Converse, λ-µ theorem , Lami’s theorem, Converse. Moments- Geometrical interpretation, Varignon’s theorem of moments, Centre of parallel forces, Moment of force about a line. Equilibrium of rigid bodies acted on by three forces-Three forces acting on a rigid body kept in equilibrium must be coplanar, must either be concurrent or parallel, Three parallel forces in equilibrium, Three concurrent forces in equilibrium. General conditions of equilibrium of Co-planar forces acting on a rigid body-Theorems .
UNIT II: STATICS(Cont…) (20 Hours)
Friction – Types, Laws, co-efficient, Angle of friction, Centre of gravity (C.G) - C.G of a number of particles arranged in a straight line, C.G of a number of weights placed at points in a plane, C.G of a uniform thin lamina in the shape of a parallelogram, C.G of a uniform triangular lamina, C.G of three uniform rods forming a triangle or a uniform wire bent into the shape of a triangle, C.G of a compound body, Determination of C.G by integration(uniform rod, uniform tetrahedron, solid right circular cone, curved surface of a hollow right circular cone, solid hemisphere, segment of a solid sphere, hollow hemisphere, segment of a hollow sphere)Stable and unstable equilibrium- Definitions, Test for the determination of the nature of stability. UNIT III: DYNAMICS (20 Hours)
Constrained Motion in Vertical Circles - Motion on the outside and inside of a smooth vertical circle, vertical circular tube, Motion of a particle attached to the end of a string. Constrained Motion in Horizontal Circles - Motion in a horizontal circle with uniform speed, Relative rest inside a rotating sphere. Constrained Motion in a Plane-Motion of a heavy particle on a smooth curve in a vertical plane, Cycloid and related results, Motion on a smooth cycloid in a vertical plane ,Motion under constraint in various curves. Motion in Three Dimensions- Cylindrical (polar) co-ordinates, components of velocity and acceleration of a moving point in terms of cylindrical co-ordinates, Spherical polar co-ordinates, Problems.
Total Hours: (60+15)
41
TEXT BOOKS:
1. N.P.Bali, STATICS (Golden Series), 6th ed.: Laxmi Publications (P) Ltd., 2007. 2. M.D.Raisinghania, Dynamics.: S.Chand and co., 2006.
Books for Reference:
1. S.L.Loney, The elements of Statics and Dynamics. London, U.K.: Kessinger Publishing , 2008. 2. S.L.Loney, An Elementary Treatise on the Dynamics of a particle and of rigid bodies.: Kessinger
Publishing., 2007. 3. M.Ray and G.C.Sharma, ATextbook on Dynamics, 13th ed. New Delhi, India: S.Chand & Company
Ltd., 2005. 4. Russell.C.Hibbeler, Engineering Mechanics:Statics, 12th ed.: Prentice Hall, 2010. Suggested Web links:
1. http://scitation.aip.org/AMR/ 2. http://plato.stanford.edu/entries/qm/ 3. http://oli.web.cmu.edu/openlearning/forstudents/.../engineering-statics 4. http://www.aerostudents.com/files/statics/staticsFullVersion.pdf 5. http://www.physicsforums.com/forumdisplay.php?f=101
FORMAT OF QUESTION PAPER
MAT 636: MECHANICS
Part Unit and No. of subdivisions to be set in
the unit
No. of subdivisions to be answered
Marks for each subdivision
Max. marks for the part
A
Unit I 4
10 1 10 Unit II 3
Unit III 3
B
Unit I 3
9 2 18 Unit II 4
Unit III 3
C Unit I and II 6
8 6 48 Unit III 4
D Unit I, II and III 4 3 8 24
Total 100
42
5 SYLLABI FOR CERTIFICATE COURSES
5.1 MAT 101: FOUNDATIONS OF MATHEMATICS 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 and fourth 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 cost and 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 Chand and Sons., 2009.
2. B.G.Sathtyaprasad, K.Nirmala, R.G.Saha, and C.S.Anantharaman, Business Mathematics. Mumbai, India: Himalaya publishing House., 2004.
Books for Reference:
1. Shanti Narayanan and P.K. Mittal, Text book of Matrices, 10th ed.: S. Chand and Company Ltd., 2004. 2. Navaneetham, Business Mathematics, Reprint. Tanjore: Gemini Publishing House, 1997.
Suggested Web Links: 1. http://planetmath.org/encyclopedia/SetTheory.html 2. http://plato.stanford.edu/entries/set-theory/ 3. http://mathworld.wolfram.com/Logarithm.html 4. http://www.sosmath.com/algebra/logs/log1/log1.html 5. http://www.mathagonyaunt.co.uk/STATISTICS/ESP/Perms_combs.html 6. http://www.mathsisfun.com/combinatorics/combinations-permutations.html 7. http://home.scarlet.be/math/matr.htm 8. http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html
43
5.2 MAT 201: INTRODUCTION TO MATHEMATICA
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 indefinite symbolic integration,
symbolic sums and products, symbolic solution of ordinary and partial differential 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-geometric function,
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 optimization techniques.
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, Ed., 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.htm
2. http://library.wolfram.com/
44
5.3 MAT 301: 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
Total Hours : 45
(2 credits)
TEXTBOOK: Kanti Swarup, P.K.Gupta, and ManMohan, Operations Research, Reprint ed. New Delhi, India: Sultan Chand & Sons, 1994.
Books for Reference:
1. G Hadley, Linear Programming, Reprint.: Narosa Publishing House, 2002. 2. K.V.Mittal and C.Mohan, Optimization Methods in Operation Research and System Analysis, Reprint.:
New Age International Pvt. Ltd., 1996. 3. Hamdy A Taha, Operations Research- an introduction, 6th ed.: Prentice Hall of India, 1996.
Suggested Web Links:
1. http://www.zweigmedia.com/RealWorld/Summary4.html 2. http://people.brunel.ac.uk/~mastjjb/jeb/or/lpmore.html 3. http://www2.isye.gatech.edu/~jswann/casestudy/assign.html 4. http://mathworld.wolfram.com/GameTheory.html
45
5.4 MAT 401: 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 HCF and 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 a given quantity as a Percentage of Another given quantity – Converting a percentage into decimal – converting a decimal into a percentage – Effect of percentage change on any quantity – Rate change and change in quantity available for fixed expenditure ,
Average: Average of different groups – Addition or removal of items and change in average – replacement of some 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 – direct proportion and indirect proportion,
Profit and Loss: Gain percentage and Loss Percentage – Relation among cost price, sale price, Gain/Loss and Gain% 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 hour for 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 N times 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 between compound interest and simple interest – equal annual installments to pay the debt amount – growth – depreciation
Shares and Debentures : Basic facts – Approach to problems on stock – Approach to problems on Shares – regular problems - Debentures.
Total Hours : 45
(2 credits)
46
Text Book :
Abhijit Guha, Quantitative Aptitude for competitive examinations, 3rd ed.: Tata Mc-Graw Hill, 2005.
Books for Reference:
1. Dinesh Khattar, Quantitative Aptitude, 3rd ed.: Pearson Education, 2005. 2. Muhamed Muneer, How to prepare for CAT.: Tata Mc Graw Hill. , 2003.
Suggested Web links :
1. http://www.ascenteducation.com
2. http://www.winentrance.com/MCA-Entrance-Exam-Question-Bank-CD.html
External Experts:
1) Dr. Y B Maralabhavi, Professor and Chairman, Department of Mathematics, Bangalore University, Bangalore
2) Prof. Jayanthi Purandar, Professor and Head, Department of Mathematics, Jyothi Nivas College, Bangalore.
3) Dr. Reena K Abraham, Infosys Technologies Ltd, Electronics City. Hosur Road, Bangalore – 561229.