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Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW UNIVERSITY, CUTTACK, ODISHA 753003 Scheme of UG Course Mathematics Under Choice Based Credit System (CBCS) Courses for Honours Students

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Page 1: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

Mathematics (Hons.)

Syllabus

Choice Based Credit System (CBCS)

w.e.f. ACADEMIC SESSION 2016-17

SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW UNIVERSITY, CUTTACK, ODISHA

753003

Scheme of UG Course Mathematics Under Choice Based Credit System (CBCS)

Courses for Honours Students

Page 2: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

Scheme of UG Courses in Mathematics Under Choice Based Credit System (CBCS)

A: Course for Honors Students

Sem Course Course Code

Title Remarks

I Core C101 Calculus Compulsory C102 Algebra

Generic Elective G101 _____ Students from Arts stream to choose course from School of Social Sciences/Statistics Students from Science stream to choose course from Physics/Chemistry/Statistics /Geology/Economics

Ability Enhancement

AE101 Environmental Science Compulsory

II Core C203 Real Analysis Compulsory

C204 Differential Equations Generic Elective G202 _____ Students from Arts stream to choose course

from School of Social Sciences/Statistics Students from Science stream to choose course from Physics/Chemistry/ Statistics/Geology/Economics

Ability Enhancement

AE202 Communicative English Compulsory

Skill Enhancement

SE201 MIL (Odia/Hindi/Alt. English)

Compulsory; the students will have to choose one of the courses

III Core C305 Theory of Real Functions Compulsory C306 Group Theory-I C307 PDE and Systems of ODE

Generic Elective G303 _____ Students from Arts stream to choose course from School of Social Sciences/Statistics Students from Science stream to choose course from Physics/Chemistry/Statistics /Geology/Economics

Skill Enhancement

SE302 Computer Science Compulsory

IV

Core C408 Numerical Methods Compulsory C409 Riemann Integration and

Series of Functions

C410 Ring Theory and Linear Algebra I

Generic Elective G404 _____ Students from Arts stream to choose course from School of Social Sciences/Statistics

Page 3: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

Students from Science stream to choose course from Physics/Chemistry/Statistics/ Geology/Economics

Skill Enhancement

SE403 Programming in C Compulsory

V Core C511 Multivariate Calculus Compulsory C512 Group Theory-II

Discipline Specific Elective

D501 A. Number Theory Students will have to choose any one of the two courses

B. Mechanics

D502 A. Probability and Statistics

Students will have to choose any one of the two courses

B. Differential Geometry

Skill Enhancement

SE504 Logic and Sets Compulsory

VI Core C613 Metric Spaces and Complex Analysis

Compulsory

C614 Ring Theory and Linear Algebra-II

Discipline Specific Elective

D603 A. Linear Programming Students will have to choose any one of the two courses B. Theory of Equations

D604 Project Compulsory

Page 4: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

B: Generic Electives For students opting Mathematics as Generic Elective.

Semester Code Course Title Remarks I G101 Calculus and

Differential Equations For the students who will choose Mathematics as

Generic Elective II G202 Group Theory

For the students who will choose Mathematics as Generic Elective

III G303 Numerical Methods

For the students who will choose Mathematics as Generic Elective

IV G404 Linear Programming

For the students who will choose Mathematics as Generic Elective

Mark Distribution of the mentioned papers is as follows:

COURSE CREDIT MARKS Core Course 14 x 6 = 84 14 x 100 =1400

Discipline Specific Elective Course

4 x 6 = 24 4 x 100 = 400

Generic Elective Course 4 x 6 = 24 4 x 100 = 400 Ability Enhancement

Course 2 x 2 = 4 2 x 50 = 100

Skill Enhancement Course

4 x 2 = 8 4 x 50 = 200

TOTAL 144 2500

Page 5: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER I

CORE COURSE C101 – Calculus (6 Credits)

Full Mark 100 (Theory-70, Practical-30)

Theory (Mid Semester 20, End Semester 50)

Unit-I Hyperbolic functions, higher order derivatives, Leibniz rule and its applications to problems of 푒 푠푖푛푥,푒 푐표푠푥, (ax+ b) sinx, (ax + b) cosx, concavity and inflection points, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves

Unit-II L’Hospital’s rule, applications in business, economics and life sciences.Reduction formulae, derivations and illustrations of reduction formulae of the type

sinnxdx , cosnxdx , tannxdx , secnxdx , (log x) dx , 푠푖푛 푥cos xdx

Unit-III volumes by slicing, disks and washers methods, volumes by cylindrical shells, parametric equations, parameterizing a curve, arc length, arc length of parametric curves, area of surface of revolution.

Unit-IV Techniques of sketching conics, reflection properties of conics, rotation of axes and second degree equations, classification into conics using the discriminant, polar equations of conics.

Unit-V

Triple product, introduction to vector functions, operations with vector-valued functions, limits and continuity of vector functions, differentiation and integration of vector functions, tangent and normal components of acceleration, modeling ballistics and planetary motion, Kepler’s second law. Practical-30 (End Semester evaluation)

List of Practicals (using any software) (i) Plotting of graphs of function

푒 , 푙표푔(푎푥 + 푏),1

(푎푥 + 푏) , 푠푖푛(푎푥 + 푏), 푐표푠(푎푥 + 푏), |푎푥 + 푏|

and to illustrate the effect of 푎and 푏 on the graph. (ii) Plotting the graphs of polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them.

Page 6: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

(iii) Sketching parametric curves (Eg. Trochoid, cycloid, epicycloids, hypocycloid, SkekchingFolim of Descartes, lemniscate of Bernoulli, Astroid). (iv) Obtaining surface of revolution of curves. (v) Tracing of conics in cartesian coordinates/ polar coordinates. (vi) Sketching ellipsoid, hyperboloid of one and two sheets, elliptic cone, elliptic, paraboloid, hyperbolic paraboloid using cartesian coordinates. (vii) Matrix operation (addition, multiplication, inverse, transpose). Books Recommended

1. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) P. Ltd.(Pearson Education), Delhi, 2007: Chapters:4(4.3,4.4,4.5 and 4.7), 9(9.4), 10(10.1-10.4).

2. H. Anton, I. Bivens and S. Davis, Calculus, 7th Ed., JohnWiley and Sons (Asia) P. Ltd., Singapore, 2002: Chapters: 6, (6.2-6.5),7(7.8), 8(8.2-8.3, Pages:532-538), 11(11.1),13(13.5)

Books for Reference

1. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005. 2. R. Courant and F. John, Introduction to Calculus and Analysis (Volumes I & II), Springer- Verlag, New York, Inc., 1989.

Page 7: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER I

CORE COURSE C102 – Algebra(6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Polar representation of complex numbers, nthroots of unity, De Moivre’s theorem for rational indices and its applications.

Unit-II Equivalence relations, Functions, Composition of functions, Invertible functions, One to one correspondence and cardinality of a set, Well-ordering property of positive integers.

Unit-III Division algorithm, Divisibility and Euclidean algorithm, Congruence relation between integers, Principles of Mathematical Induction, statement of Fundamental Theorem of Arithmetic.

Unit-IV

Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation Ax=b, solution sets of linear systems, applications of linear systems, linear independence.

Unit-V

Introduction to linear transformations, matrix of a linear transformation, inverse of a matrix, characterizations of invertible matrices. Subspaces of Rn, dimension of subspaces of Rn and rank of a matrix, Eigen values, Eigen Vectors and Characteristic Equation of a matrix. Books Recommended: 1. TituAndreescu and DorinAndrica, Complex Numbers from A to Z, Birkhauser, 2006: Chapter:2 2. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rd Ed., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2005: Chapters:2(2.4), 3,4(4.1-4.1.6, 4.2-4.2.11, 4.4(4.1-4.4.8),4.3-4.3.9, 5(5.1-5.1.4). 3. David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education Asia, Indian Reprint,2007: Chapters:1(1.1-1.9), 2(2.1-2.3, 2.8, 2.9), 5(5.1,5.2)

Page 8: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS Semester – I

Generic Elective Course G101 - Calculus and Differential Equations (6 Credit)

Full marks: 100 (Mid Term-20+End Term-80)

UNIT-I

Concavity and inflection points, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves

UNIT-II

Functions of several variables, limit and continuity of functions of two variables, Partial differentiation, total differentiability and differentiability, sufficient condition for differentiability. Chain rule for one and two independent parameters, directional derivatives, the gradient, maximal and normal property of the gradient, tangent planes

UNIT-III Differential equations and mathematical models. General, particular, explicit, implicit and singular solutions of a differential equation. Exact differential equations and integrating factors, separable equations and equations reducible to this form, linear equation and Bernoulli equations, special integrating factors and transformations.

UNIT-IV

General solution of homogeneous equation of second order, principle of super position for homogeneous equation, Wronskian: its properties and applications, Linear homogeneous and non-homogeneous

UNIT-V Equations of higher order with constant coefficients, Euler’s equation, method of undetermined coefficients, method of variation of parameters. Books Recommended

1. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) P. Ltd.(Pearson Education), Delhi, 2007

2. S. L. Ross, Differential Equations, 3rd Edition, John Wiley & Sons, India, 2004. 3. H. Anton, I. Bivens and S. Davis, Calculus, 7th Ed., JohnWiley and Sons (Asia) P. Ltd.,

Singapore, 2002 Books for Reference:

1. J.Sinha Roy and S. Padhy, A Course of Ordinary and Partial Di erential Equations, Kalyani Publishers, New Delhi.

Page 9: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER I

Ability Enhancement Compulsory Course AE101- Environmental Science (2 Credits)

Full Mark: 50 (End semester evaluation) Unit I: Ecosystems

Ecosystem- Structure and function of ecosystem (Abiotic and Biotic factors); Energy flow in an ecosystem.

Environmental Problems: global warming and Climate change, ozone layer depletion. Deforestation, acid rain; impacts of environmental disturbances.

Unit II: Natural Resources and Biodiversity Energy resources: Renewable and non-renewable energy sources. Biodiversity patterns and

global biodiversity hot spots, India as a mega-biodiversity nation; Endangered and endemic species of India, Threats to biodiversity, Conservation of biodiversity: In-situ and Ex-situ conservation of biodiversity.

Unit III: Environmental Pollution, Impact and Management

Environmental pollution: Air, water, soil and noise pollution, Nuclear hazards and human health risks, Solid waste management: Control measures of urban and industrial waste.

Environment Protection Act and International agreements: Montreal and Kyoto protocols and Convention on Biological Diversity (CBD).

Suggested Readings:

1. Carson, R. 2002. Silent Spring. Houghton Mifflin Harcourt. 2. Gadgil, M., & Guha, R. 1993. This Fissured Land: An Ecological History of India. Univ. of

California Press. 3. Gleeson, B. and Low, N. (eds.) 1999. Global Ethics and Environment, London, Routledge. 4. Gleick, P. H. 1993. Water in Crisis. Pacific Institute for Studies in Dev., Environment &

Security. Stockholm Env. Institute, Oxford Univ. Press. 5. Groom, Martha J., Gary K. Meffe, and Carl Ronald Carroll. Principles of Conservation

Biology. Sunderland: Sinauer Associates, 2006. 6. Grumbine, R. Edward, and Pandit, M.K. 2013. Threats from India’s Himalaya

dams. Science, 339: 36-37. 7. McCully, P. 1996. Rivers no more: the environmental effects of dams (pp. 29-64). Zed Books. 8. McNeill, John R. 2000. Something New Under the Sun: An Environmental History of the

Twentieth Century. 9. Odum, E.P., Odum, H.T. & Andrews, J. 1971. Fundamentals of Ecology. Philadelphia:

Saunders. 10. Pepper, I.L., Gerba, C.P. & Brusseau, M.L. 2011. Environmental and Pollution Science.

Academic Press.

Page 10: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER II

CORE COURSE C203 – Real Analysis (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Review of Algebraic and Order Properties of R, 훿-neighborhood of a point in 푅, Idea of countable sets, uncountable sets and uncountability of R. Bounded above sets, Bounded below sets, Bounded Sets, Unbounded sets, Suprema and Infima.

Unit-II The Completeness Property of R, The Archimedean Property, Density of Rational (and Irrational) numbers in R, Intervals.

Unit-III Interior point, Limit point, boundary pointsof a set, Isolated points, open sets, closed sets, closure, derived set, cantor set, compact sets, Heine Borel Theorem, Bolzano-Weierstrass theorem for sets.

Unit-IV Sequences, Bounded sequence, Convergent sequence, Limit of a sequence. Limit Theorems, Monotone Sequences, Monotone Convergence Theorem. Subsequences, Divergence Criteria, Monotone Subsequence Theorem (statement only), Bolzano Weierstrass Theorem for Sequences. Cauchy sequence, Cauchy’s Convergence Criterion.

Unit-V

Infinite series, convergence and divergence of infinite series, Cauchy Criterion, Tests for convergence: Comparison test, Limit Comparison test, Ratio Test, Cauchy’s nthroot test, Integral test, Alternating series, Leibniz test, Absolute and Conditional convergence. Books Recommended

1. R.G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002. (Chapter 1(1.3), 2,3, 4(4.1.1,4.1.2,4.1.3), 9(9.1,9.2,9.3), 11(11.1-11.2,11.3),Appendix B)

Books for Reference 1. G. Das and S. Pattanayak, Fundamentals of Mathematics Analysis, TMH Publishing

Co. 2. Gerald G. Bilodeau , Paul R. Thie, G.E. Keough, An Introduction to Analysis, 2nd

Ed., Jones & Bartlett, 2010. 3. Brian S. Thomson, Andrew. M. Bruckner and Judith B. Bruckner, Elementary Real

Analysis, Prentice Hall, 2001. 4. S.K. Berberian, A First Course in Real Analysis, Springer Verlag, New York, 1994.

Page 11: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER II

CORE COURSE C204 – Differential Equations (6 Credits) Full Mark 100 (Theory-70, Practical-30)

Theory (Mid Semester 20, End Semester 50)

Unit-I Differential equations and mathematical models. General, particular, explicit, implicit and singular solutions of a differential equation. Exact differential equations and integrating factors, separable equations and equations reducible to this form, linear equation and Bernoulli equations, special integrating factors and transformations.

Unit-II Applications of first order equations: Problems in mechanics(Falling body problem, Frictional forces), rate problems( Rate of growth and decay, population growth, mixture problem).

Unit-III

General solution of homogeneous equation of second order, principle of super position for homogeneous equation, Wronskian: its properties and applications, Linear homogeneous and non-homogeneous

Unit-IV Equations of higher order with constant coefficients, Euler’s equation, method of undetermined coefficients, method of variation of parameters.

Unit-V Series Solutions of linear differential equations: Power series solutions about an ordinary point, solutions about singular points; method of Frobenius, Bessel’s equation and Bessel functions. Practical-30 (End Semester evaluation)

List of Practicals (using any software) 1. Plotting of second order solution family of differential equation. 2. Plotting of third order solution family of differential equation. 3. Growth model (exponential case only). 4. Decay model (exponential case only). 5. Economic problems. 6. Mixture problems. 7. Vibration problems. 8. Oxygen debt model. 9. Falling body problems. 10. Frictional forces problems.

Page 12: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

Books Recommended:

1. S. L. Ross, Differential Equations,3rd Edition, John Wiley & Sons, India, 2004. Chapters:1, 2, 3(3.2-3.3), 4(4.1-4.5), 5,6

Books for Reference:

1. J.Sinha Roy and S. Padhy, A Course of Ordinary and Partial Di erential Equations, Kalyani Publishers, New Delhi. Chapters: 1, 2(2.1 to 2.7), 3, 4(4.1 to 4.7), 5, 7(7.1-7.4), 9(9.1, 9.2, 9.3, 9.4, 9.5, 9.10, 9.11, 9.13)

2. Belinda Barnes and Glenn R. Fulford, Mathematical Modeling with Case Studies, A Differential Equation Approach using Maple and Matlab, 2nd Ed., Taylor and Francis group, London and New York, 2009.

3. C.H. Edwards and D.E. Penny, Differential Equations and Boundary Value problems Computing and Modeling, Pearson Education India, 2005.

4. Martha L Abell, James P Braselton, Differential Equations with MATHEMATICA, 3rd Ed., Elsevier Academic Press, 2004.

Page 13: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS Semester – II

Generic Elective Course G202 - Group Theory (6 Credit)

Full marks: 100 (Mid Term-20+ End Term-80)

Unit-I Symmetries of a square, Dihedral groups, definition and examples of groups including permutation groups and quaternion groups (illustration through matrices), elementary properties of groups.

Unit-II Subgroups and examples of subgroups, centralizer, normalizer, center of a group, product of two subgroups.

Unit-III

Properties of cyclic groups, classification of subgroups of cyclic groups.Cycle notation for permutations, properties of permutations, even and odd permutations, alternating group.

Unit-IV

Properties of cosets, Lagrange’s theorem and consequences, normal subgroups, factor groups,

Unit-V Group homomorphisms, properties of homomorphisms, Cayley’s theorem, properties of isomorphisms, First, Second and Third isomorphism theorems. Books Recommended 1. Joseph A. Gallian, Contemporary Abstract Algebra, 8th Ed., Cengage Learning India Private Limited, Delhi, 2013. (Chapter 1,2,3,4,5,6,7,9,10) Books for Reference 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995. 4. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

Page 14: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER II

Ability Enhancement Compulsory Course AE202 - Communicative English (2 Credits)

Full Mark: 50 (End semester evaluation)

This is a need based course. It aims to develop the communicative ability of the students in speaking, reading and writing skills. The course also aims to equip the students in use of grammar in meaningful contexts and express themselves in day to day life’s practical situations.

Unit I

Basics of Communication:

The process of Communication; Language as a Tool of Communication; Levels of Communication, The Flow of Communication; Communication Networks; Barriers to Communication; Technology in Communication Verbal and Non-verbal Communication: Spoken and written; Formal and Informal Style; Use of bias free English

Unit II

Principles and Practices of Writing. Grammar and Punctuation: Subject-Verb Agreement; Common Mistakes; Punctuation. Sentence Construction

Unit III Reading and Writing

Comprehension; Art of Condensation; Note making; Summarizing; Paragraph Writing; Writing a Review; Dialogue Writing

Page 15: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS)

SEMESTER II Skill Enhancement Compulsory Course

SE201- MIL(Odia/Hindi/Alternative English)(2 Credits) Full Mark: 50 (End semester evaluation)

Page 16: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

HINDI

ALTERNATIVE ENGLISH

Unit I Poetry: Shakespeare: Sonnet CXXX; William Blake: London; T S Eliot: Preludes; Thomas Hardy: The Darkling Thrush Wordsworth: It’s a Beauteous Evening

Unit II Short Story: Gabriel Garcia Marquez: Balthazar’s Marvellous Afternoon Amitav Ghosh: Ghost’s of Mrs Gandhi; Omprakash Valmiki: Jhootan

Unit III Essays: Virginia Woolf: Shakespeare’s Sister Chitra Banerjee Divakaruni: Indian Movie, New Jersey B. R. Ambedkar: Who are the Shudras? Prescribed Text: The Individual and Society: Essays Stories and Poems. New Delhi: Pearson, 2006

Page 17: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER III CORE COURSE

C305 – Theory of Real Functions (6 Credits) Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I

Limits of functions (휖 − 훿approach), sequential criterion for limits, divergence criteria. Limit theorems, one sided limits. Infinite limits and limits at infinity. Continuous functions, sequential criterion for continuity and discontinuity.

Unit-II

Algebra of continuous functions. Continuous functions on an interval, intermediate value theorem, location of roots theorem, preservation of intervals theorem. Uniform continuity, non-uniform continuity criteria, uniform continuity theorem, Monotone and Inverse Functions

Unit-III

Differentiability of a function at a point and in an interval, Caratheodory’s theorem, algebra of differentiable functions.Relative extrema, interior extremum theorem.L’Hospital Rules

Unit-IV Rolle’s theorem, Mean value theorem, intermediate value property of derivatives, Darboux’s theorem. Applications of mean value theorem to inequalities and approximation of polynomials, Taylor’s theorem to inequalities.Cauchy’s mean value theorem.

Unit-V Taylor’s theorem with Lagrange’s form of remainder, Taylor’s theorem with Cauchy’s form of remainder, application of Taylor’s theorem to convex functions, relative extrema. Taylor’s series and Maclaurin’s series expansions of exponential and trigonometric functions, 푙푛(1 + 푥), 1/푎푥 + 푏and(1 + 푥) . Books Recommended 1. R. Bartle and D.R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2003.(Chapter 4(4.1, 4.2, 4.3), 5 (excluding 5.5), 6) Books for Reference 1. G. Das and S. Pattanayak, Fundamentals of Mathematics Analysis, TMH Pub-lishing Co., Chapters:6(6.1-6.8), 7(7.1-7.7) 2. A. Mattuck, Introduction to Analysis, Prentice Hall, 1999. 3. K.A. Ross, Elementary Analysis: The Theory of Calculus, Springer, 2004. 4. S.R. Ghorpade and B.V. Limaye, A Course in Calculus and Real Analysis, Springer, 2006.

Page 18: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER III CORE COURSE

C306 – Group Theory-I (6 Credits) Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Symmetries of a square, Dihedral groups, definition and examples of groups including permutation groups and quaternion groups (illustration through matrices), elementary properties of groups. Subgroups and examples of subgroups, centralizer, normalizer, center of a group, product of two subgroups.

Unit-II Properties of cyclic groups, classification of subgroups of cyclic groups. Cycle notation for permutations, properties of permutations, even and odd permutations, alternating group

Unit-III Properties of cosets, Lagrange’s theorem and consequences including Fermat’s Little theorem.

Unit-IV

Normal subgroups, factor groups, Cauchy’s theorem for finite abelian groups.

Unit-V Group homomorphisms, properties of homomorphisms, Cayley’s theorem, properties of isomorphisms, First, Second and Third isomorphism theorems. Books Recommended 1. Joseph A. Gallian, Contemporary Abstract Algebra, 8th Ed., Cengage Learning India Private Limited, Delhi, 2013. (Chapter 1,2,3,4,5,6,7,9,10) Books for Reference 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995. 4. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

Page 19: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS)

SEMESTER III CORE COURSE

C307 – PDE and Systems of ODE (6 Credits) Full Mark 100 (Theory-70, Practical-30)

Theory (Mid Semester 20, End Semester 50)

Unit-I Partial Differential Equations – Basic concepts and Definitions, Mathematical Problems. First- Order Equations: Classification, Construction and Geometrical Interpretation. Method of Characteristics for obtaining General Solution of Quasi Linear Equations. Canonical Forms of First-order Linear Equations. Method of Separation of Variables for solving first order partial differential equations.

Unit-II Classification of Second-Order linear Equations: Second- Order equations in two independent variables, Canonical forms, equations with constant coefficients, general solutions.

Unit-III

The Cauchy problem and Wave Equations: The Cauchy problem, the Cauchy-Kowaleewskaya theorem, homogeneous Wave Equations, Initial Boundary Value Problems, Equations with non-homogeneous boundary conditions, Vibration of finite String with fixed ends, Non-Homogeneous Wave Equations.

Unit-IV Systems of linear differential equations, types of linear systems, differential operators, an operator method for linear systems with constant coefficients, Basic Theory of linear systems in normal form, homogeneous linear systems with constant coefficients: Two Equations in two unknown functions.

Unit-V Matrices and Vectors, the matrix method forhomogeneous linear systems with constant coefficients: two equations in two unknown functions, the matrix method forhomogeneous linear systems with constant coefficients: n equations in two unknown functions. Practical-30 (End Semester evaluation)

List of Practicals (using any software) (i) Solution of Cauchy problem for first order PDE. (ii) Finding the characteristics for the first order PDE.

Page 20: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

(iii) Plot the integral surfaces of a given first order PDE with initial data.

(iv) Solution of wave equation − 푐 = 0 for the following associated conditions (a) 푢(푥, 0) = 휑(푥),푢 (푥, 0) = 휓(푥), 푥 ∈ ℝ, 푡 > 0. (b) 푢(푥, 0) = 휑(푥),푢 (푥, 표) = 휓(푥),푢(0, 푡) = 0, 푥 ∈ (0,∞), 푡 > 0 (c) 푢(푥, 0) = 휑(푥),푢 (푥, 0) = 휓(푥),푢 (0, 푡) = 0,푥 ∈ (0,∞), 푡 > 0 (d) 푢(푥, 0) = 휑(푥),푢 (푥, 0) = 휓(푥), 푢(0, 푡) = 0,푢(푙, 푡) = 0, 0 < 푥 < 푙, 푡 > 0

(v) Solution of wave equation − 푘 = 0 for the following associated conditions (a) 푢(푥, 0) = 휑(푥),푢(0, 푡) = 푎,푢(푙, 푡) = 푏, 0 < 푥 < 푙, 푡 > 0.

(b) 푢(푥, 0) = 휑(푥), 푥 ∈ ℝ, 0 < 푡 < 푇 (c) 푢(푥, 0) = 휑(푥),푢(0, 푡) = 푎, 푥 ∈ (0,∞), 푡 ≥ 0.

Books Recommended:

1. S. L. Ross, Differential Equations,3rd Edition, John Wiley & Sons, India, 2004. Chapters:7(7.1, 7.3-7.7).

2. TynMyint-U and LokenathDebnath, Linear Partial Differential Equations for Scientists and Engineers, 4th edition, Springer, Indian reprint, 2006. Chapters:1, 2, 4(4.1-4.4), 5(5.1-5.7)

Books for Reference 1.J.Sinha Roy and S. Padhy, A Course on Ordinary and Partial Di erential EquationPublishers, New Delhi, Ludhiana, 2012.Chapters:11, 12, 13(13.1-13.5), 15(15.1,15.5), 16(16.1, 16.1.1), 17(17.1, 17.2, 17.3). 2. Martha L Abell, James P Braselton, Differential equations with MATHEMATICA, 3rd Ed., Elsevier Academic Press, 2004.

Page 21: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS Semester – III

Generic Elective Course G303- Numerical Methods (6 Credit)

Full marks: 100 (Mid Term-20+ End Term-80)

(Use of non-programmable Scientific Calculator is allowed.)

Unit-I Algorithms, Convergence, Errors: Relative, Absolute, Round off, Truncation.

Unit-II Transcendental and Polynomial equations: Bisection method, Newton’s method, Secant method. Rate of convergence of these methods.

Unit-III System of linear algebraic equations: Gaussian Elimination and Gauss Jordan methods. Gauss Jacobi method, Gauss Seidel method and their convergence analysis.

Unit-IV

Interpolation: Lagrange and Newton’s methods. Error bounds.Finite difference operators. Gregory forward and backward difference interpolation.

Unit-V

Numerical Integration: Trapezoidal rule, Simpson’s rule, Simpsons 3/8th rule, Gauss- Legenre Rule, Midpoint rule, Composite Trapezoidal rule, Composite Simpson’s rule. Ordinary Differential Equations: Euler’s method. Runge-Kutta methods of orders two and four. Books Recommended: 1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, 5th Ed., New age International Publisher, India, 2007. Chapters: 1, 2(2.1-2.3, 2.5), 3(3.1-3.4), 4(4.1-4.4), 5(5.6-5.9), 6(6.3-6.4). Books for Reference: 1. John H. Mathews and Kurtis D. Fink, Numerical Methods using Matlab, 4th Ed., PHI Learning Private Limited, 2012. 2. C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Pearson Education, India, 2008. 3. Uri M. Ascher and Chen Greif, A First Course in Numerical Methods, 7th Ed., PHI Learning Private Limited, 2013. 4. Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India, 2007.

Page 22: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) Semester – III

Skill Enhancement Course SE302 – Computer Science (2 Credits)

Full marks: 50 (End Term evaluation)

UNIT-I Introduction: Introduction to computer system, uses, types. Data Representation: Number systems and character representation, binary arithmetic. Human Computer Interface: Types of software, Operating system as user interface, utility programs.

UNIT-II Devices: Input and output devices (with connections and practical demo), keyboard, mouse, joystick, scanner, OCR, OMR, bar code reader, web camera, monitor, printer, plotter. Memory: Primary, secondary, auxiliary memory, RAM, ROM, cache memory, hard disks, optical disks

UNIT-III Computer Organisation and Architecture: C.P.U., registers, system bus, main memory unit, cache memory, Inside a computer, SMPS, Motherboard, Ports and Interfaces, expansion cards, ribbon cables, memory chips, processors.

Reference Books:

1. Goel, Computer Fundamentals, Pearson Education, 2010. 2. P. Aksoy, L. DeNardis, Introduction to Information Technology, Cengage Learning, 2006 3. P. K.Sinha, P. Sinha, Fundamentals of Computers, BPB Publishers, 2007.

Page 23: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER IV

CORE COURSE C408 – Numerical Methods (6 Credits)

Full Mark 100 (Theory-70, Practical-30)

(Use of non-programmable Scientific Calculator is allowed) Theory (Mid Semester 20, End Semester 50)

Unit-I Algorithms, Convergence, Errors: Relative, Absolute, Round off, Truncation.

Unit-II Transcendental and Polynomial equations: Bisection method, Newton’s method, Secant method. Rate of convergence of these methods.

Unit-III System of linear algebraic equations: Gaussian Elimination and Gauss Jordan methods. Gauss Jacobi method, Gauss Seidel method and their convergence analysis.

Unit-IV

Interpolation: Lagrange and Newton’s methods. Error bounds.Finite difference operators. Gregory forward and backward difference interpolation.

Unit-V

Numerical Integration: Trapezoidal rule, Simpson’s rule, Simpsons 3/8th rule, Gauss- Legenre Rule, Midpoint rule, Composite Trapezoidal rule, Composite Simpson’s rule. Ordinary Differential Equations: Euler’s method. Runge-Kutta methods of orders two and four. Practical-30 (End Semester evaluation)

List of Practicals (Writing C-Programs for implementing the following methods) (i) Bisection Method. (ii) Newton Raphson Method. (iii) Secant Method. (iv) RegulaiFalsi Method. (v) Gauss-Jacobi Method. (vi) Gauss-Siedel Method. (vii) Lagrange Interpolation or Newton Interpolation. (viii) Simpson’s rule. (ix) Euler’s method of solution for IVP.

Page 24: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

Books Recommended: 1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, 5th Ed., New age International Publisher, India, 2007. Chapters: 1, 2(2.1-2.3, 2.5), 3(3.1-3.4), 4(4.1-4.4), 5(5.6-5.9), 6(6.3-6.4). Books for Reference: 1. John H. Mathews and Kurtis D. Fink, Numerical Methods using Matlab, 4th Ed., PHI Learning Private Limited, 2012. 2. C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Pearson Education, India, 2008. 3. Uri M. Ascher and Chen Greif, A First Course in Numerical Methods, 7th Ed., PHI Learning Private Limited, 2013. 4. Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India, 2007.

Page 25: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER IV

CORE COURSE C409 – Riemann Integration and Series of Functions (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Riemann integration; inequalities of upper and lower sums; Riemann conditions of integrability. Riemann sum and definition of Riemann integral through Riemann sums; equivalence of two definitions; Riemann integrability of monotone and continuous functions, Properties of the Riemann integral; definition and integrability of piecewise continuous and monotone functions. Intermediate Value theorem for Integrals; Fundamental theorems of Calculus.

Unit-II Generalized Riemann Integral, Improper integrals, Lebesgue Integrals, Convergence of Beta and Gamma functions.

Unit-III Pointwise and uniform convergence of sequence of functions. Theorems on continuity, derivability and integrability of the limit function of a sequence of functions.

Unit-IV Series of functions; Theorems on the continuity and derivability of the sum function of a series of functions; Cauchy criterion for uniform convergence and Weierstrass M-Test.

Unit-V Limit superior and Limit inferior. Power series, radius of convergence, Cauchy Hadamard Theorem, Differentiation and integration of power series; Abel’s Theorem; Weierstrass Approximation Theorem. Books Recommended

1. R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002. (Chapter 7,8, 9(9.4),10)

2. S.C Malik and Savita Arora, Mathematical Analysis, 4th Ed., New Age International Publishers, 2014 (Chapter 11, Appendix I)

Books for Reference 1. G. Das and S. Pattanayak-Fundamentals of Mathematics Analysis, TMH Publishing Co., Chapters:8, 9. 2. Charles G. Denlinger, Elements of Real Analysis, Jones & Bartlett (Student Edition), 2011. 3. K.A. Ross, Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004.

Page 26: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER IV

CORE COURSE C410 – Ring Theory and Linear Algebra I (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Definition and examples of rings, properties of rings, subrings, integral domains and fields, characteristic of a ring.

Unit-II

Ideal, ideal generated by a subset of a ring, factor rings, operations on ideals, prime and maximal ideals.

Unit-III Ring homomorphisms, properties of ring homomorphisms, Isomorphism theorems I, II and III, field of quotients.

Unit-IV Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces.

Unit-V

Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representation of a linear transformation, algebra of linear transformations. Isomorphisms, Isomorphism theorems, invertibility and isomorphisms, change of coordinate matrix. Books Recommended 1.Joseph A. Gallian, Contemporary Abstract Algebra, 8th Ed., Cengage Learning India Private Limited, Delhi, 2013. (Chapter 12,13,14,15) 2.Kenneth Hoffman, Ray Alden Kunze, Linear Algebra, 2nd Ed., Prentice-Hall of India Pvt. Ltd., 1971.(Chapter 2(2.1-2.3),3 (3.1-3.4)) Books for Reference 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th Ed., Prentice- Hall of India Pvt. Ltd., New Delhi, 2004. 4. S. Lang, Introduction to Linear Algebra, 2nd Ed., Springer, 2005. 5. Gilbert Strang, Linear Algebra and its Applications, Thomson, 2007. 6. S. Kumaresan, Linear Algebra- A Geometric Approach, Prentice Hall of India,1999. 7. D.A.R. Wallace, Groups, Rings and Fields, Springer Verlag London Ltd., 1998.

Page 27: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS Semester – IV

Generic Elective Course G404 - Linear Programming (6 Credit)

Full marks: 100 (Mid Term-20+ End Term-80)

Unit-I Introduction to linear programming problem, Theory of simplex method, optimality and unboundedness, the simplex algorithm, simplex method in tableau format.

Unit-II

Introduction to artificial variables, two‐phase method, Big‐M method and their comparison.

Unit-III Duality, formulation of the dual problem, primal‐dual relationships, economic interpretation of the dual.

Unit-IV

Transportation problem and its mathematical formulation, northwest‐corner method least cost method and Vogel approximation method for determination of starting basic solution, algorithm for solving transportation problem, assignment problem and its mathematical formulation, Hungarian method for solving assignment problem.

Unit-V

Game theory: formulation of two person zero sum games, solving two person zero sum games, games with mixed strategies, graphical solution procedure, linear programming solution of games. Books Recommended

1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear Programming and Network Flows,2nd Ed., John Wiley and Sons, India, 2004. Chapters:3(3.2-3.3, 3.5-3.8), 4(4.1-4.4), 6(6.1-6.3).

2. F.S. Hillier and G.J. Lieberman, Introduction to Operations Research, 9th Ed., Tata McGraw Hill,Singapore, 2009. Chapter:14

3. Hamdy A. Taha, Operations Research, An Introduction, 8th Ed., PrenticeHall India, 2006. Chapter:5(5.1, 5.3, 5.4).

Page 28: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) Semester – IV

Skill Enhancement Course SE403 – PROGRAMMING IN C (2 Credits)

Full marks: 50 (End Term evaluation)

UNIT-I Overview of C, Constants, variables and key words, data types, Rules for constructing variables, operations and Expressions, Managing Input and output operation. Writing simple programs.

UNIT-II Decision making and branching.The if statement, if-else statement, nested if, use of logical operators, hierarchy of logical operators, the conditional operator. The Case Control Structure,Decisions Using switch, switch Versus if-else, The goto Keyword. The Loop Control Structure, Loops ,Thewhile Loop ,The for Loop ,Nesting of Loops Multiple Initializations in the for Loop ,The break Statement, The continue Statement The do-while Loop

UNIT-III What is a Function ,Why Use Functions, Passing Values between Functions, Scope Rule of Functions, Calling Convention, Advanced Features of Functions, Function Declaration and Prototypes, Call by Value and Call by Reference, Arrays, A Simple Program Using Array, More on Arrays,Array Initialization, use of numerical array, use of character array, Bounds Checking, Passing Array Elements to a Function Two Dimensional Arrays, Initializing a 2-Dimensional Array. Books Recommended

Programming in ANSI C(6th Edition), E.Balaguruswamy (Tata Mc. Graw Hill) Books for Reference

Yashawant Kanetkar, Let Us C, 13th Ed., Bpb Publishers, 2013, New-Delhi.

Page 29: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER V

CORE COURSE C511– Multivariate Calculus (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Functions of several variables, limit and continuity of functions of two variables Partial differentiation, total differentiability and differentiability, sufficient condition for differentiability. Chain rule for one and two independent parameters, directional derivatives, the gradient, maximal and normal property of the gradient, tangent planes

Unit-II

Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems, Definition of vector field, divergence and curl

Unit-III Double integration over rectangular region, double integration over non-rectangular region, Double integrals in polar co-ordinates, Triple integrals, Triple integral over a parallelepiped and solid regions. Volume by triple integrals, cylindrical and spherical co-ordinates. Change of variables in double integrals and triple integrals.

Unit-IV

Line integrals, Applications of line integrals: Mass and Work. Fundamental theorem for line integrals, conservative vector fields, independence of path.

Unit-V Green’s theorem, surface integrals, integrals over parametrically defined surfaces. Stoke’s theorem, The Divergence theorem. Books Recommended:

1. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) Pvt. Ltd.(Pearson Education), Delhi, 2007. Chapters: 11(11.1(Pages: 541-543), 11.2-11.6, 11.7(Pages:598-605), 11.8(Pages:610-614)), 12 (12.1, -12.3, 12.4(Pages:652-660), 12.5, 12.6), 13 (13.2,13.3,13.4(Pages:712-716), 13.5(Pages:723-726; 729-730), 13.6 (Pages:733-737), 13.7(Pages:742-745)).

Books for Reference: 1. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005. 2. E. Marsden, A.J. Tromba and A. Weinstein, Basic Multivariable Calculus, Springer (SIE), Indianreprint, 2005. 3. James Stewart, Multivariable Calculus, Concepts and Contexts, 2nd Ed., Brooks /Cole, Thomson Learning, USA, 2001.

Page 30: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER V

CORE COURSE C512 – Group Theory-II (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Automorphism, inner automorphism, automorphism groups, automorphism groups of finite and infinite cyclic groups, applications of factor groups to automorphism groups, Characteristic subgroups, Commutator subgroup and its properties.

Unit-II

External direct products, Properties of external direct products, the group of units modulo 푛 as an external direct product, internal direct products, Fundamental Theorem of finite abelian groups.

Unit-III

Group actions, stabilizers and kernels, permutation representation associated with a given group action

Unit-IV

Applications of group actions: Generalized Cayley’s theorem, Index theorem. Groups acting on themselves by conjugation, class equation and consequences, conjugacy in 푆 , p-groups

Unit-V

Sylow’s theorems and consequences, Cauchy’s theorem, Simplicity of An for n ≥ 5, non-simplicity tests. Books Recommended

1. Joseph A. Gallian, Contemporary Abstract Algebra, 8th Ed., Cengage Learning India Private Limited, Delhi, 2013. (Chapter 6(6.2-6.5), 8), 11, 24,25,29

Books for Reference 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2004. 4. J.R. Durbin, Modern Algebra, John Wiley & Sons, New York Inc., 2000. 5. D. A. R. Wallace, Groups, Rings and Fields, Springer Verlag London Ltd., 1998.

Page 31: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER V

Discipline Specific Elective Course D501A – Number Theory (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Linear Diophantine equation, Goldbach conjecture, linear congruences, complete set of residues, Chinese Remainder theorem, Fermat’s Little theorem, Wilson’s theorem, statement of prime number theorem

Unit-II Number theoretic functions, sum and number of divisors (statement of theorems with applications), totally multiplicative functions, the Mobius Inversion formula.

Unit-III

The greatest integer function, Euler’s phi‐function, Euler’s theorem, reduced set of residues, some properties of Euler’s phi-function.

Unit-IV Order of an integer modulo n, primitive roots for primes, the equation 푥 + 푦 = 푧 , Fermat’s Last theorem ( the equation 푥 + 푦 = 푧 ).

Unit-V

The Legendre symbol and its properties, quadratic reciprocity, quadratic congruences with composite moduli. Books Recommended 1. David M. Burton, Elementary Number Theory, 6th Ed., Tata McGraw‐Hill, Indian reprint, 2007. Chapter 2(2.5), 3 (3.3), 4(4.2, 4.4)), 5 (5.2,5.3), 6(6.1,6.2,6.3), 7(7.2,7.3,7.4), 8(8.1,8.2),9 (9.2, 9.3, 9.4), 16(16.4) Books for Reference 1. Neville Robinns, Beginning Number Theory, 2nd Ed., Narosa Publishing House Pvt. Ltd., Delhi, 2007.

Page 32: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER V

Discipline Specific Elective Course D501B – Mechanics (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Moment of a force about a point and an axis, couple and couple moment, Moment of a couple about a line, resultant of a force system, distributed force system, free body diagram, free body involving interior sections, general equations of equilibrium, two point equivalent loading, problems arising from structures, static indeterminacy.

Unit-II

Laws of Coulomb friction, application to simple and complex surface contact friction problems, transmission of power through belts, screw jack, wedge, first moment of an area and the centroid, other centers,

Unit-III

Theorem of Pappus-Guldinus, second moments and the product of area of a plane area, transfer theorems, relation between second moments and products of area, polar moment of area, principal axes.

Unit-IV Conservative force field, conservation for mechanical energy, work energy equation, kinetic energy and work kinetic energy expression based on center of mass, moment of momentum equation for a single particle and a system of particles, translation and rotation of rigid bodies

Unit-V Chasles’ theorem, general relationship between time derivatives of a vector for different references, relationship between velocities of a particle for different references, acceleration of particle for different references. Book Recommended: 1. I.H. Shames and G. Krishna Mohan Rao, Engineering Mechanics: Statics and Dynamics, (4th Ed.), Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi, 2009. Chapters:3, 4, 5, 6(6.1-6.7), 7, 11, 12(12.5, 12.6), 13. Books for Reference: 1. R.C. Hibbeler and Ashok Gupta, Engineering Mechanics: Statics and Dynamics, 11th Ed., DorlingKindersley (India) Pvt. Ltd. (Pearson Education), Delhi. 2. Grant R Fowles, Analytical Mechanics, Cengage Learning India Pvt. Ltd.

Page 33: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER V

Discipline Specific Elective Course D502A – Probability and Statistics (6 Credits) Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I

Sample space, probability axioms, real random variables (discrete and continuous), cumulative distribution function, probability mass/density functions, mathematical expectation, moments, moment generating function, characteristic function,

Unit-II

discrete distributions: uniform, binomial, Poisson, geometric, negative binomial, continuous distributions: uniform, normal, exponential.

Unit-III

Joint cumulative distribution function and its properties, joint probability density functions, marginal and conditional distributions, expectation of function of two random variables, conditional expectations, independent random variables, bivariate normal distribution, correlation coefficient, joint moment generating function (jmgf) and calculation of covariance (fromjmgf), linear regression for two variables.

Unit-IV

Chebyshev’s inequality, statement and interpretation of (weak) law of large numbers and strong law of large numbers, Central Limit theorem for independent and identically distributed random variables with finite variance.

Unit-V Markov Chains, Chapman-Kolmogorov equations, classification of states. Books Recommended

1. Robert V. Hogg, Joseph W. McKean and Allen T. Craig, Introduction to Mathematical Statistics,Pearson Education, Asia, 2007. Chapters: 1(1.1, 1.3. 1.5-1.9), 2(2.1, 2.3-2.5).

2. Irwin Miller and Marylees Miller, John E. Freund, Mathematical Statistics with Applications, 7thEd., Pearson Education, Asia, 2006. Chapters: 4, 5(5.1-5.5, 5.7), 6(6.2,6.3, 6.5-6.7), 14(14.1,14.2)

3. Sheldon Ross, Introduction to Probability Models, 9th Ed., Academic Press, Indian Reprint, 2007.Chapters:2(2.7), 4(4.1-4.3).

Books for Reference 1. Alexander M. Mood, Franklin A. Graybill and Duane C. Boes, Introduction to the

Theory of Statistics, 3rd Ed., Tata McGraw- Hill, Reprint 2007

Page 34: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER V

Discipline Specific Elective Course D502B– Differential Geometry (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Theory of Space Curves: Space curves, Planer curves, Curvature, torsion and Serret-Frenet

formulae. Osculating circles, Osculating circles and spheres. Existence of space curves. Evolutes and involutes of curves.

Unit-II

Theory of Surfaces: Parametric curves on surfaces. Direction coefficients. First and second Fundamental forms.Principal and Gaussian curvatures.Lines of curvature, Euler’s theorem. Rodrigue’s formula, Conjugate and Asymptotic lines. Developables: Developable associated with space curves and curveson surfaces, Minimal surfaces.

Unit-III

Geodesics: Canonical geodesic equations. Nature of geodesics on a surface of revolution. Clairaut’s theorem.Normal property of geodesics.Torsion of a geodesic.Geodesic curvature.Gauss-Bonnet theorem.Surfaces of constant curvature.Conformal mapping.Geodesic mapping.Tissot’s theorem.

Unit-IV Tensors: Summation convention and indicial notation, Coordinate transformation and Jacobian, Contra-variant and Covariant vectors, Tensors of different type, Algebra of tensors and contraction, Metric tensor and 3-index Christoffel symbols

Unit-V Parallel propagation of vectors, Covariant and intrinsic derivatives, Curvature tensor and its properties, Curl, Divergence and Laplacian operators in tensor form, Physical components. Books Recommended 1. T.J. Willmore, An Introduction to Differential Geometry, Dover Publications, 2012. Books for Reference 1. B. O'Neill, Elementary Differential Geometry, 2nd Ed., Academic Press, 2006. 2. C.E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge University Press 2003. 3. D.J. Struik, Lectures on Classical Differential Geometry, Dover Publications, 1988. 4. S. Lang, Fundamentals of Differential Geometry, Springer, 1999. 5. B. Spain, Tensor Calculus: A Concise Course, Dover Publications, 2003.

Page 35: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS)

Semester – V Skill Enhancement Course

SE504 – Logic and Sets (2 Credits) Full marks: 50 (End Term evaluation)

UNIT-I

Introduction, propositions, truth table, negation, conjunction and disjunction. Implications, biconditional propositions, converse, contra positive and inverse propositions and precedence of logical operators.Propositional equivalence: Logical equivalences. Predicates and quantifiers: Introduction, Quantifiers, Binding variables and Negations.

UNIT-II Sets, subsets, Set operations and the laws of set theory and Venn diagrams. Examples of finite and infinite sets. Finite sets and counting principle. Empty set, properties of empty set. Standard set operations. Classes of sets. Power set of a set. Difference and Symmetric difference of two sets. Set identities, Generalized union and intersections.

UNIT-III Relation: Product set, Composition of relations, Types of relations, Partitions, Equivalence Relations with example of congruence modulo relation, Partial ordering relations, n-ary relations. Books Recommended

R.P. Grimaldi, Discrete Mathematics and Combinatorial Mathematics, Pearson Education, 1998.

Books for Reference

1. P.R. Halmos, Naive Set Theory, Springer, 1974.

2. E. Kamke, Theory of Sets, Dover Publishers, 1950.

Page 36: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER VI

CORE COURSE C613-Metric Spaces and Complex Analysis (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Metric spaces: definition and examples. Sequences in metric spaces, Cauchy sequences. Complete Metric Spaces. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, Cantor’s theorem. Subspaces, dense sets, separable spaces.

Unit-II Continuous mappings, sequential criterion and other characterizations of continuity. Uniform continuity. Homeomorphism, Contraction mappings, Banach Fixed point Theorem. Connectedness, connected subsets of R.

Unit-III Functions of a complex variable, mappings, Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function

Unit-IV Derivatives of functions, Definite integrals of functions.Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals.Cauchy-Goursat theorem, Cauchy integral formula.Liouville’s theorem and the fundamental theorem of algebra.

Unit-V

Convergence of sequences and series, Taylor series and its examples.Laurent series and its examples, absolute and uniform convergence of power series. Books Recommended 1.S. Kumaresan, Topology of Metric Spaces, 2nd Ed., Narosa Publishing House, 2011. (Chapter 1(1.1,1.2), 2(2.1 to 2.5), 3(3.1 to 3.4),5(5.1), 6(6.4)) 2. James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 8th Ed., McGraw – Hill International Edition, 2009.(Chapter 2 (12-26), Chapter 3 (29-34), Chapter 4(37-41, 43,44,46-53), Chapter 5 (55-57, 59,60, 62-63)) Books for Reference 1. Satish Shirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag, London, 2006. 2. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 2004. 3. Joseph Bak and Donald J. Newman, Complex Analysis, 2nd Ed., Undergraduate Texts in Mathematics, Springer-Verlag New York, Inc., NewYork, 1997.

Page 37: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER VI

CORE COURSE C614-Ring Theory and Linear Algebra II (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Polynomial rings over commutative rings, division algorithm and consequences, principal ideal domains,

Unit-II Factorization of polynomials, reducibility tests, irreducibility tests, Eisenstein criterion, unique factorization in Z[x]. Divisibility in integral domains, irreducibles, primes, unique factorization domains, Euclidean domains.

Unit-III Dual spaces, dual basis, double dual, Statement of theorems (without proof and its applications to problem) related totranspose of a linear transformation and its matrix in the dual basis, annihilators, Eigen spaces of a linear operator, diagonalizability, statement of Cayley-Hamilton theorem and its applications to problem, the minimal polynomial for a linear operator.

Unit-IV

Inner product spaces and norms, Gram-Schmidt orthogonalisation process, orthogonal complements, Bessel’s inequality, the adjoint of a linear operator

Unit-V Normal and self-adjoint operators, Orthogonal projections and Spectral theorem. Books Recommended 1.Joseph A. Gallian, Contemporary Abstract Algebra, 8th Ed., Cengage Learning India Private Limited, Delhi, 2013. (Chapter 16,17,18) 2.Kenneth Hoffman, Ray Alden Kunze, Linear Algebra, 2nd Ed., Prentice-Hall of India Pvt. Ltd., 1971.(Chapter 3(3.5,3.6,3.7), 6(6.2, 6.3), 8 (8.2-8.3, 8.5), 9(9.5(Spectral theorem only)) Books for Reference 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th Ed., Prentice- Hall of India Pvt. Ltd., New Delhi, 2004. 4. S. Lang, Introduction to Linear Algebra, 2nd Ed., Springer, 2005. 5. Gilbert Strang, Linear Algebra and its Applications, Thomson, 2007. 6. S. Kumaresan, Linear Algebra- A Geometric Approach, Prentice Hall of India, 1999. 7. S.H. Friedberg, A.L. Insel and L.E. Spence, Linear Algebra, Prentice Hall of India Pvt. Ltd., 2004.

Page 38: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER VI

Discipline Specific Elective Course D603A – Linear Programming (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I Introduction to linear programming problem, Theory of simplex method, optimality and unboundedness, the simplex algorithm, simplex method in tableau format.

Unit-II

Introduction to artificial variables, two‐phase method, Big‐M method and their comparison.

Unit-III Duality, formulation of the dual problem, primal‐dual relationships, economic interpretation of the dual.

Unit-IV

Transportation problem and its mathematical formulation, northwest‐corner method least cost method and Vogel approximation method for determination of starting basic solution, algorithm for solving transportation problem, assignment problem and its mathematical formulation, Hungarian method for solving assignment problem.

Unit-V Game theory: formulation of two person zero sum games, solving two person zero sum games, games with mixed strategies, graphical solution procedure, linear programming solution of games. Books Recommended

1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear Programming and Network Flows,2nd Ed., John Wiley and Sons, India, 2004. Chapters:3(3.2-3.3, 3.5-3.8), 4(4.1-4.4), 6(6.1-6.3).

2. F.S. Hillier and G.J. Lieberman, Introduction to Operations Research, 9th Ed., Tata McGraw Hill,Singapore, 2009. Chapter:14

3. Hamdy A. Taha, Operations Research, An Introduction, 8th Ed., PrenticeHall India, 2006. Chapter:5(5.1, 5.3, 5.4).

Page 39: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER VI

Discipline Specific Elective Course D603B – Theory of Equations (6 Credits)

Full Mark 100 (Mid Semester 20, End Semester 80)

Unit-I General properties of polynomials, Graphical representation of a polynomial, maximum and minimum values of a polynomials, General properties of equations, Descarte’s rule of signs positive and negative rule,

Unit-II

Relation between the roots and the coefficients of equations.Symmetric functions, Applications of symmetric function of the roots, Transformation of equations. Solutions of reciprocal and binomial equations.

Unit-III Algebraic solutions of the cubic and biquadratic.Properties of the derived functions.Symmetric functions of the roots, Newton’s theorem on the sums of powers of roots, homogeneous products, limits of the roots of equations.

Unit-IV

Separation of the roots of equations, Strums theorem, Applications of Strum’s theorem,

Unit-V Conditions for reality of the roots of an equation and biquadratic. Solution of numerical equations. Books Recommended 1. W.S. Burnside and A.W. Panton, The Theory of Equations, Dublin University Press, 1954.

Books for Reference C. C. MacDuffee, Theory of Equations, John Wiley & Sons Inc., 1954.

Page 40: Mathematics (Hons.) - Ravenshaw University · Mathematics (Hons.) Syllabus Choice Based Credit System (CBCS) w.e.f. ACADEMIC SESSION 2016-17 SCHOOL OF MATHEMATICAL SCIENCES RAVENSHAW

MATHEMATICS (HONOURS) SEMESTER VI

Discipline Specific Elective Course D604 – Project (6 Credit)

Full marks: 100 (End Term Evaluation)