Bsc maths - Lecture notes 1-6 PDF

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PERIYAR UNIVERSITY PERIYAR PALKALAI NAGAR SALEM – 636 011

DEGREE OF BACHELOR OF SCIENCE CHOICE BASED CREDIT SYSTEM

SYLLABUS FOR B.Sc. MATHEMATICS

FOR THE STUDENTS ADMITTED FROM THE ACADEMIC YEAR 2012 – 2013 ONWARDS

1

1. OBJECTIVES OF THE COURSE Mathematics is a key to success in the field of science and engineering. Today, the students need a thorough knowledge of fundamental basic principles, methods, results and a clear perception of the power of mathematical ideas and tools to use them effectively in modeling, interpreting and solving the real world problems. Mathematics plays an important role in the context of globalization of Indian economy, modern technology, and computer science and information technology. This syllabus is aimed at preparing the students to hope with the latest developments and compete with students from other universities and put them on the right track. 2. ELIGIBILITY FOR ADMISSION A Pass in the Higher Secondary Examination of TamilNadu Higher Secondary Board or some other Board accepted by the Syndicate as equivalent thereto with Mathematics (other than Business mathematics) as one of the subjects. 3. DURATION OF THE COURSE The course of study shall be based on semester pattern with internal assessment under Choice Based Credit System. The course shall consist of six semesters and a total period of three years with 140 credits. The course of study will comprise of the following subjects according to the syllabus and is given in the scheme of Examinations and books prescribed from time to time. 4. EXAMINATIONS The theory of examination shall be of three hours duration for each paper at the end of each semester. The candidate failing in any subject(s) will be permitted to appear for each failed subject(s) in the subsequent examinations. The practical examinations for UG course shall be conducted at the end of the even semesters only. 5. SCHEME OF EXAMINATIONS The Scheme of examinations for different semesters shall be as follows: B. Sc .Mathematics – Course Structure under Choice Based Credit System. (Applicable to the candidates admitted from the year 2012 – 2013 onwards)

2

SEM

I

PART

I II III

IV II

III

I II III

IV I II III

COURSE CODE 12UFTA01 12UFEN01 12UMA01

12UES01 12UVE01 12UMAS01 12UFTA02 12UFEN02 12UMA02 12UMAE01

12UES01 12UFTA03 12UFEN03 12UMA03 12UMA04

IV

V

IV I II III

12UNE01 12UFTA04 12UFEN04 12UMA05

IV

12UMAS02 12UMAS03 12UNE02 12UMA06

III

12UMA07 12UMA08 12 UMA09 12UMAE02 IV VI

III

12UMAS04 12UMAS05 12UMA10 12UMA11

COURSE TITILE Tamil Course –I English Course –I Core Course I-Algebra &Trigonometry Allied I –Course I –Theory Allied I –Practical Environmental Studies Value Education Skill Based Elective Course I Tamil Course II English Course II Core Course II-Calculus Elective course I-From Group Allied I - Course II-Theory Allied I -Practical Environmental studies Tamil course III English Course III Core Course III –Differential Equations & Laplace Transforms Core Course IV –Statics Allied II-Course I-Theory Allied II-Practical Non Major Elective Course I Tamil Course IV English Course IV Core Course V-Dynamics Allied II Course II –Theory Allied II –Practical Skill Based Elective Course II Skill Based Elective Course III Non Major Elective Course II Core Course VIAlgebraic structures I Core Course VII-Real Analysis I Core Course VIII – Discrete Mathematics Core Course IX – Numerical Analysis Elective Course II – From Group B Skill Based Elective Course IV Skill based Elective Course V Core Course X – Algebraic Structure II Core course XI – Real Analysis II

HOURS / WEEK

CREDI T

EXA HOU RS

INT

MARKS EXT TOTAL

Lect.

Tut.

Pra.

Total

4 4 4

2 2 2

-

6 6 6

3 3 5

3 3 3

25 25 25

75 75 75

100 100 100

5 1 2 2 4 4 5 5 5 1 4 4 4

2 2 2 2 -

2 2 -

5 2 1 2 2 6 6 5 5 5 2 1 6 6 4

4 2 2 3 3 4 5 3 3 2 3 3 4

3 * * 3 3 3 3 3 3 3 3 3 3 3 3

25 25 25 25 25 25 25 25 40 25 25 25 25

75 75 75 75 75 75 75 75 60 75 75 75 75

100 100 100 100 100 100 100 100 100 100 100 100 100

5 5 2 4 4 5 4 2 2 2 6

2 2 -

2 3 -

5 5 2 2 6 6 5 4 3 2 2 2 6

4 3 ** 2 3 3 4 3 3 2 2 2 5

3 3 3 3 3 3 3 3 3 3 3 3

25 25 25 25 25 25 25 40 25 25 25 25

75 75 75 75 75 75 75 60 75 75 75 75

100 100 100 100 100 100 100 100 100 100 100 100

5 5

-

-

5 5

5 5

3 3

25 25

75 75

100 100

5

-

-

5

5

3

25

75

100

5

-

-

5

5

3

25

75

100

2 2 5

-

-

2 2 6

2 2 5

3 3 3

25 25 25

75 75 75

100 100 100

6

-

-

6

5

3

25

75

100

3

12UMA12 12UMA13 12UMAE03 IV V

12UMAS06

Core Course XII – Complex Analysis Core Course XIII-Graph Theory Elective Course III – From Group Skill Based Elective Course VI – Practical Extension Activates

6

-

-

6

5

3

25

75

100

5 5

-

-

5 5

5 5

3 3

25 25

75 75

100 100

2

-

-

2

2

3

40

60

100

-

-

-

-

1

-

-

-

* - Examination at the end of Second Semester. ** - Examination at the end of Fourth Semester ALLIED SUBJECTS FOR B.Sc. MATHEMATICS PHYSICS / CHEMISTRY / STATISTICS/ELECTRONICS/ACCOUNTANCY Any two of the above subjects can be chosen as Allied subjects. Subject Allied Physics – I Allied Physics – II Allied Physics – Practical Allied Chemistry – I Allied Chemistry –II Allied Chemistry – Practical Allied Statistics – I Allied Statistics – II Allied Statistics – Practical Allied Electronics – I Allied Electronics - II Allied Electronics – Practical Allied Accountancy – I Allied Accountancy – II Allied Accountancy – Practical

Code 12 UPHA01 12 UPHA02 12 UPHAP01 12 UCHA01 12 UCHA02 12 UCHAP01 12 USTA01 12 USTA02 12 USTAP01

ALLIED MATHEMATICS FOR B.Sc. STATISTICS, PHYSICS, COMPUTER SCIENCE, ELECTRONICS, BIOINFORMATICS & BCA MAJOR STUDENTS ALLIED MATHEMATICS – GROUP – I 1. Paper I– Algebra, Calculus and Fourier series 2. Paper II – Differential Equation and Laplace Transforms 3. Paper III – Allied Mathematics – Praticals

4

ALLIED MATHEMATICS GROUP – II 1. Paper I – Discrete Mathematics 2. Paper II – Numerical Method 3. Paper III – Graph Theory ELECTIVE SUBJECTS: Subject

Subject code

From Group A : Vector Analysis Financial Mathematics From Group B : Linear Programming Number Theory Combinatorics From Group C: Operations Research Astronomy Probability Theory

U12MAE01 U12MAE02 U12MAE03 U12MAE04 U12MAE05 U12MAE06 U12MAE07 U12MAE08

SKILL BASED ELECTIVE COURSES: Aptitude Examination - I Aptitude Examination – II Aptitude Examination -III Aptitude Examination – IV Programming in C C – Programming Practical

U12MAS01 U12MAS02 U12MAS03 U12MAS04 U12MAS05 U12MASP06

NON - MAJOR ELECTIVE COURSES: Non-Major Elective Course - I 1. Competitive Examination – Paper – I 2. Matrix Algebra 3. Linear Programming Non-Major Elective Course - II 1. Competitive Examination – Paper – II 2. Numerical Methods 3. Operations Research

5

6. UNIFORMITY IN THE NUMBER OF UNITS IN EACH PAPER: Each theory paper shall consist of five units. The Question paper shall consist of questions uniformly distributed among all the units. For theory paper without practicals, Max marks is 75. 7. A. QUESTION PAPER PATTERN FOR ALL UG COURSES WITHOUT PRACTICAL: Time: Three Hours

Maximum Marks: 75 Part A: (10 x 2 = 20) Answer ALL Questions (Two Questions from Each Unit) Part B: (5 x 5 = 25)

Answer ALL Questions (One Question From Each Unit with internal choice) Part C: ( 3 x 10 = 30) Answer Any Three Questions out of Five Questions (One Question from Each Unit) B. SKILL BASED ELECTIVE COURSE – C PROGRAMMING – PRATICAL QUESTION PATTERN EXTERNAL MARK: 60 INTERNAL MARK: 40 RECORD WORK – 15 Part – A: (2X15 =30) Answer any two out of Four Questions Part – B: (1X5=15) Answer any one out of two questions Practical - 45 Mark Allotment: 60 – External Record - 15 40 – Internal 6

C. ALLIED – MATHEMATICS PRATICAL (3x15 =45) Answer any Three out of Five Questions Practical - 45 Mark Allotment: 60 – External Record - 15 8. PASSING MINIMUM: The Candidates shall be declared to have passed the examination if the candidates secure not less than 30 marks in the University examination in each theory paper without practical. 9. CLASSIFICATION OF SUCCESSFUL CANDIDATES: Candidates who secure not less than 60% of the aggregate marks in the whole examination shall be declared to have passed the examination in the First class .All other successful candidates shall be declared to have passed in the second class. Candidates who obtain 75% of the marks in the aggregate shall be deemed to have passed the examination in First Class with Distinction provided they pass all the examinations prescribed for the course at the first appearance. Candidates who pass all the examinations prescribed for the course in the first attempt and within a period of three academic years from the year of admission to the course only eligible for University Ranking. 10. COMMENCEMENT OF THIS REGULATION: The CBCS regulations shall take effect from the academic year 2012-2013 ie, for the students who are admitted to the first year of the course during the academic year 2012-2013 and thereafter. 11. TRANSITARY PROVISION: Candidates who were admitted to the UG course of study prior to 2012-2013 shall be permitted to appear for the examinations under those regulations for a period of three years ie, up to and inclusive of the examinations of April/May 2018.Thereafter they shall be permitted to appear for the examination only under the regulations then in force. 12. NOTE: 1. The Non Major Elective Course Papers Syllabus will be given at the end of this book. 2. This Paper should be handling and valued by Mathematics Department. 3. For University Practical Examination both Internal and External Examiners should be appointed from Mathematics Department. 7

FIRST SEMESTER Core Paper I – Algebra and Trigonometry Paper code: 12UMA01

Max Marks: 75

Unit I Characteristic equation - Characteristic roots and Characteristic vectors – properties – problems - Cayley – Hamilton theorem (statement only) and its problems – Diagonalisation of Matrices – problems. Unit II Polynomial equations – Imaginary and Irrational roots – relation between roots and coefficients of equations – Symmetric functions of roots in terms of coefficients of third degree equation - problems. Unit III Sum of the powers of the roots of an equation – Newton’s Theorem on the sum of the powers of the roots – Transformation of equations – Roots with sign changed – Roots multiplied by a given number – Reciprocal equations – problems. Unit IV To increase or decrease the roots of a given equation by a given quantity. Removal of terms - Square of the roots – Transformations in general – Descarte’s rule of signs – problems. Unit V Expansions of sin nθ,Cos nθ and Tan nθ – Expansions of sinnθ , cosnθ -Expansions of sinθ , cosθ and tanθ in terms of θ – Hyperbolic and inverse hyperbolic functions and their properties – Logarithm of a complex number – General principal values – problems.

8

Text Books:S.No

Title of the Book

Author

Publishing Company

1.

Algebra-Volume I

2.

Trigonometry

3.

Algebra,calculus and Trigonometry

T.K.Manickava sagam Pillai and S. Narayanan. T.K.Manickava sagam Pillai and S. Narayanan Dr.P.R.Vittal.

Vijay Nicole Imprints Pvt, Ltd,#c-7,Nelson Manickam Road,Chennai-600029 Vijay Nicole Imprints Pvt, Ltd,#c-7,Nelson Manickam Road, Chennai-600029 Margham publications,24,Rameswa ram Road, T.Nager, Chennai-600017.

Author

Publishing Company

Year of Publication 2004

2004

2000

Reference Books:-

S.No

Title of the Book

1.

Trigonometry.

N.P.Bali.

2.

Algebra.

Burnside and Pantern.

9

Krishna Prakasan mandir,9, Shivaji Road,Meerut(UP)250001 Macmillan publishers,U.K.

Year of Publication 1994

1976

FIRST SEMESTER Skill Based Elective Paper I – Aptitude Examination - I

Paper Code – 12UMAS01

Max Marks: 75

Unit I Numbers, H.C.F. and L.C.M. of numbers , Decimal Fractions. Unit II Simplification , Square roots and Cube Roots , Average. Unit III Problems on numbers , problems on Ages. Unit IV Surds and Indices , Percentage , Profit and Loss. Unit V Ratio and Proportion , Partnership.

Text Books:S.No

Tiltle of the Book

1.

Quantitative Aptitude for competitive Examination Quantitative Aptitude and Reasoning

2.

Author R.S.Aggarwal.

Praveen

10

Publishing Company S.Chand and company Ltd,152,Anna salai,Chennai. PHI P.Ltd.

Year of Publication 2001

SECOND SEMESTER Core Paper II - Calculus Paper code: 12UMA02

Max Marks: 75

Unit I Curvature - Radius of curvature , Circle of curvature and Center of curvature in Cartesian co-ordinates and Polar co-ordinates - Evolutes and Envelopes – definition Method of finding envelopes - Problems in all sections. Unit II Asymptotes:- Definition - Methods of finding asymptotes of plane algebraic curves – special cases – problems. Slope of the tangent in polar co-ordinates - Angle of intersection of two curves - Pedal equation of a curve – Problems. Unit III Integration - Bernoulli’s formula - Reduction formula for

∫0Л/2 sin

n

x dx , ∫0Л/2 a

cosn x dx , ∫0Л/4 tann x dx , ∫secn x dx , ∫cosecn x dx , ∫cosm x sinn x dx , ∫cotn x dx ,

xn eax 0

-x

n

m

n

dx, ∫e x dx , x (logx) dx - Problems for all the above cases. Unit IV Beta and Gamma functions – Definition – properties – problems - relation between Beta and Gamma functions - Applications to evaluate the definite integrals. Unit V Fourier series - Definition – Fourier coefficients – Fourier series of periodic functions of period 2Л - Even and Odd functions – Half Range series – problems.

11

Text Books:S.No

Tiltle of the Book

Author

Publishing Company Vijay Nicole Imprints Pvt Ltd,#C-7,Nelson Chambers,115,Nelson Manickam Road,Chennai-600029 Vijay Nicole Imprints Pvt Ltd,#C-7,Nelson Chambers,115,Nelson Manickam Road,Chennai-600029 Margham publications , 24,Rameswaram road, T.Nagar,Chennai 17.

1.

Calculus Volume. I

T.K.Manichava sagam Pillai and S.Narayanan

2.

Calculus Volume. II

T.K.Manichava sagam Pillai and S.Narayanan

3.

Calculus

Dr.P.R.Vittal.

Year of Publication 2004

2004

2000

Reference Books:S.No

Tiltle of the Book

Author

1.

Calculus.

N.P.Bali.

2.

Calculus

D.Sudha.

Publishing Company Krishna prakasan Mandhir,9,Shivaji Road,Meerut.(UP) Emerald Publishers,135,Anna Salai,Chennai-600002

12

Year of Publication 1994

1988

FIRST YEAR SECOND SEMESTER Elective Paper I - Vector Analysis Paper Code – 12UMAE01

Max Marks :75

Unit I Vector differentiation: Limit of a vector function – continuity and derivative of vector function - Geometrical and Physical significance of vector differentiation - Partial derivative of vector function – gradient and directional derivative of scalar point functions – Equations of tangent plane and normal line to a level surface.

Unit II Vector point function: Divergence and curl of a vector point function – solenoidal and irrational functions – physical interpretation of divergence and curl of a vector point function. Unit III Vector identities – Laplacian operator. Unit IV Integration of vector functions – Line , surface and volume intergrals. Unit V Guass - Divergence Theorem – Green’sTheorem – Stoke’s Theorem (Statements only). Verification of theorems and simple problems using the theorems.

13

Text Books :S.No

Tiltle of the Book

1.

Vector Analysis

2.

Vector Analysis

3.

Vector Analysis

Author

Publishing Company

P.Duraipandian and S.Viswanathan and others co, 38, McnicalsRoad, Chetpet,Chennai 31. Dr.P.R.Vittal Margham publications, 24, Rameswaram Road, T.nagar, Chennai– 17. T.K. Vijay Nicole Imprints Manickavasagam Pvt Ltd, # c-7 Nelson and others. Chambers, 115, Nelson Manickam Road, Chennai – 29.

Year of Publication 1984

1997

2004

Reference Books :-

S.No

Tiltle of the Book

Author

Publishing Company

1.

Vector Calculus

K.Viswanathan & S. Selvaraj

2.

Vector Calculus

J.N. Sharma & A.R. Vasishtha

3.

Vector Algebra

M.D. Raisinghania and others.

Emerald Publishers, 135,Anna Salai Chennai-2. Krishna Prakasan Mandhir,9,Shivaji Road, Meerut(UP). S. Chand & Co,Ltd., Ram Nagar New Delhi 110055.

14

Year of Publication 1984

1999

SECOND SEMESTER Elective Paper II – Financial Mathematics Paper code: 12UMAE02

Max Marks :75

Unit I Probability – Probabilities and Events – Conditional probability – Random Variables and Expected Values – Covariance and correlation – Continuous Random variables –Normal Random Variables – Properties of Normal Random Variables – The central limit Theorem – Simple Problems. Unit II Geometric Brownian Motion – G.B.M. as a limit of simple models – Brownian Motion – Simple problems - Interest rates – Present value analysis – Rate of return – Continution of varying interest rates – An example of option pricing – other examples of pricing via arbitrage. Unit III The Arbitage theorem – The multi period Binomial model – proof of the Arbitrage theorem - Black Scholes formula – properties of the Black Scholes option cost – Derivation of Black Scholes formula – simple problems. Unit IV Additional results on options – Call options on Di...