Draft syllabus for b. A/B. Sc. (Honours) in mathematics under Choice Based Credit System (cbcs) Effective from the academic session 2017-2018 sidho-kanho-birsha university purulia-723104 West Bengal



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Draft

SYLLABUS FOR B.A/B.SC. (HONOURS)

IN

MATHEMATICS

Under Choice Based Credit System (CBCS)

Effective from the academic session 2017-2018

logo new

SIDHO-KANHO-BIRSHA UNIVERSITY PURULIA-723104

West Bengal

B.A/B.Sc MATHEMATICS HONOURS COURSE STRUCTURE



Semester

Core Course (14)


Discipline Specific Elective (4)


Generic Elective (4)


Skill Enhancement Course (2)


Ability Enhancement Course (2)

I

CC1

CC2





GE1




Env Sc

II

CC3

CC4





GE2




Eng/MIL

III

CC5

CC6


CC7




GE3

SE1




IV

CC8

CC9


CC10




GE4

SE2




V

CC11

CC12


DS1

DS2











VI

CC13

CC14


DS3

DS4











Core Subjects Syllabus



CC1 – Calculus, Geometry & Differential Equation

CC2 – Algebra

CC3 – Real Analysis

CC4 – Differential Equations and Vector Calculus

CC5 – Theory of Real Functions & Introduction to Metric Space

CC6 – Group Theory-I

CC7 –Dynamics of Particle and Integral transform

CC8 – Riemann Integration and Series of Functions

CC9 – Multivariate Calculus and Partial Differential Equation

CC10 – Ring Theory and Linear Algebra-I

CC11 – Metric Spaces and Complex Analysis

CC12 – Group Theory-II, Ring theory-II, Linear Algebra-II

CC13 – Numerical Methods & Computer Programming

CC14 – Computer Aided Numerical & Statistical Practical (P)
Department Specific Electives Subjects

DS1 – Linear Programming

DS2 – Probability and Statistics

DS3 – Number Theory

DS4 – Mechanics

DS5 – Differential Geometry
Skill Enhancement Subjects

SE1– Logic and Sets

SE2– Object Oriented Programming in C++

SE3– Graph Theory

SE4– Operating System: Linux

Generic Elective Subjects (for other courses)



GE1–Calculus, Geometry & Differential Equation

GE2– Algebra

GE3– Differential Equations and Vector Calculus

GE4– Numerical Methods & Computer Programming

Ability Enhancement Course



AEL1-

AEE1-

Detailed Syllabus

CC1 – Calculus, Geometry & Differential Equation [Credit: 1+5]

Unit -1

Hyperbolic functions, higher order derivatives, Leibnitz rule of successive differentiation and its applications, concavity and inflection points, envelopes, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, L’Hospital’s rule, applications in business, economics and life sciences.



Unit-2

Reduction formulae, derivations and illustrations of reduction formulae, parametric equations, parametrizing a curve, arc length, arc length of parametric curves, area of surface of revolution.

Techniques of sketching conics.

Unit -3

Reflection properties of conics, translation and rotation of axes and second degree equations, classification of conics using the discriminant, polar equations of conics.

Spheres. Cylindrical surfaces. Central conicoids, paraboloids, plane sections of conicoids, Generating lines, classification of quadrics, Illustrations of graphing standard quadric surfaces like cone, ellipsoid.

Unit-4

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.



Graphical Demonstration (Teaching Aid)

1. Plotting of graphs of function eax + b, log(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b), |ax + b| and to illustrate the effect of a and b on the graph.



2. Plotting the graphs of polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them.

3. Sketching parametric curves (Eg. Trochoid, cycloid, epicycloids, hypocycloid).

4. Obtaining surface of revolution of curves.

5. Tracing of conics in Cartesian coordinates/polar coordinates.

6. Sketching ellipsoid, hyperboloid of one and two sheets, elliptic cone, elliptic, paraboloid, and hyperbolic paraboloid using Cartesian coordinates.

Reference Books

  1. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005.

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

  3. H. Anton, I. Bivens and S. Davis, Calculus, 7th Ed., John Wiley and Sons (Asia) P. Ltd., Singapore, 2002.

  4. R. Courant and F. John, Introduction to Calculus and Analysis (Volumes I & II), Springer- Verlag, New York, Inc., 1989.

  5. S.L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, India, 2004.

  6. Murray, D., Introductory Course in Differential Equations, Longmans Green and Co.

  7. G.F.Simmons, Differential Equations, Tata Mcgraw Hill.

  8. T. Apostol, Calculus, Volumes I and II.

  9. S. Goldberg, Calculus and mathematical analysis.

  10. S.C. Malik and S. Arora, Mathematical analysis.

  11. Shantinarayan, Mathematical analysis.

  12. J.G. Chakraborty & P.R.Ghosh, Advanced analytical geometry.

  13. S.L.Loney, Coordinate geometry.

CC2 – Algebra [Credit: 1+5]

Unit -1 [Credit: 3]

Polar representation of complex numbers, n-th roots of unity, De Moivre’s theorem for rational indices and its applications.

Theory of equations: Relation between roots and coefficients, Transformation of equation, Descartes rule of signs, Cubic and biquadratic equations, special roots, reciprocal equation, binomial equation.

Inequality: The inequality involving AM≥GM≥HM, Cauchy-Schwartz inequality.

Equivalence relations and partitions, Functions, Composition of functions, Invertible functions, One to one correspondence and cardinality of a set. Well-ordering property of positive integers, Division algorithm, Divisibility and Euclidean algorithm. Congruence relation between integers. Principles of Mathematical Induction, statement of Fundamental Theorem of Arithmetic.

Unit -2 [Credit: 2]

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.



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, rank of a matrix, Eigen values, Eigen Vectors and Characteristic Equation of a matrix. Cayley-Hamilton theorem and its use in finding the inverse of a matrix.

Reference Books

  1. Titu Andreescu and Dorin Andrica, Complex Numbers from A to Z, Birkhauser, 2006.

  2. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rd Ed., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2005.

  3. David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education Asia, Indian Reprint, 2007.

  4. K.B. Dutta, Matrix and linear algebra.

  5. K. Hoffman, R. Kunze, Linear algebra.

  6. W.S. Burnstine and A.W. Panton, Theory of equations.

  7. S.K,Mapa, Higher Algebra (Classical).

  8. S.K,Mapa, Higher Algebra (Linear and Abstract).

  9. Friedberg, Insel and Spence, Linear Algebra.



CC3 – Real Analysis [Credit: 1+5]
Review of Algebraic and Order Properties of R, ε-neighbourhood of a point in R. Idea of countable sets, uncountable sets and uncountability of R. Bounded above sets, Bounded below sets, Bounded Sets, Unbounded sets. Suprema and Infima. Completeness Property of R and its equivalent properties. The Archimedean Property, Density of Rational (and Irrational) numbers in R, Intervals. Limit points of a set, Isolated points, Open set, closed set, derived set, Illustrations of Bolzano- Weierstrass theorem for sets, compact sets in R, Heine-Borel Theorem.
Sequences, Bounded sequence, Convergent sequence, Limit of a sequence, lim inf, lim sup. 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.
Infinite series, convergence and divergence of infinite series, Cauchy Criterion, Tests for convergence: Comparison test, Limit Comparison test, Ratio Test, Cauchy’s nth root test, Raabe’s test, Gauss’s test, Cauchy’s condensation test, Integral test. Alternating series, Leibniz test. Absolute and Conditional convergence.
Graphical Demonstration (Teaching Aid)
1. Plotting of recursive sequences.

2. Study the convergence of sequences through plotting.

3. Verify Bolzano-Weierstrass theorem through plotting of sequences and hence identify convergent subsequences from the plot.

4. Study the convergence/divergence of infinite series by plotting their sequences of partial sum.

5. Cauchy's root test by plotting nth roots.

6. Ratio test by plotting the ratio of nth and (n+1)th term.


Reference Books

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

  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.

  5. Tom M. Apostol, Mathematical Analysis, Narosa Publishing House

  6. Courant and John, Introduction to Calculus and Analysis, Vol I, Springer

  7. W. Rudin, Principles of Mathematical Analysis, Tata McGraw-Hill

  8. Terence Tao, Analysis I, Hindustan Book Agency, 2006

  9. S. Goldberg, Calculus and mathematical analysis.

  10. S.K.Mapa, Real analysis.



CC4 – Differential Equations and Vector Calculus [Credit: 1+5]
Unit 1 [Credit: 3]

Lipschitz condition and Picard’s Theorem (Statement only). 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 equations of higher order with constant coefficients, Euler’s equation, method of undetermined coefficients, method of variation of parameters.


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.
Equilibrium points, Interpretation of the phase plane

Power series solution of a differential equation about an ordinary point, solution about a regular singular point.


Unit 2 [Credit: 2]

Triple product, introduction to vector functions, operations with vector-valued functions, limits and continuity of vector functions, differentiation and line integration of vector functions, Surface and volume integration [Gauss’s theorem, Green’s theorem, Stoke’s theorem (proof not required)].


Graphical Demonstration (Teaching Aid)

1. Plotting of family of curves which are solutions of second order differential equation.

2. Plotting of family of curves which are solutions of third order differential equation.
Reference Books

  1. 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.

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

  3. S.L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, India, 2004.

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

  5. Murray, D., Introductory Course in Differential Equations, Longmans Green and Co.

  6. Boyce and Diprima, Elementary Differential Equations and Boundary Value Problems, Wiley.

  7. G.F.Simmons, Differential Equations, Tata Mc Graw Hill

  8. Marsden, J., and Tromba, Vector Calculus, McGraw Hill.

  9. Maity, K.C. and Ghosh, R.K. Vector Analysis, New Central Book Agency (P) Ltd. Kolkata (India).

  10. M.R. Speigel, Schaum’s outline of Vector Analysis

  11. P.R.Ghosh and J.G.Chakraborty, Vector Calculus.


CC5 – Theory of Real Functions & Introduction to Metric Space [Credit: 1+5]
Unit 1 [Credit: 4]

Limits of functions (ε - δ approach), 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. 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.


Differentiability of a function at a point and in an interval, Caratheodory’s theorem, algebra of differentiable functions. Relative extrema, interior extremum theorem. 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.
Cauchy’s mean value theorem. 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. Application of Taylor’s theorem to inequalities.
Unit 2 [Credit: 1]

Metric spaces: Definition and examples. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, subspaces, dense sets, separable spaces.


Reference Books

  1. R. Bartle and D.R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2003.

  2. K.A. Ross, Elementary Analysis: The Theory of Calculus, Springer, 2004.

  3. A, Mattuck, Introduction to Analysis, Prentice Hall, 1999.

  4. S.R. Ghorpade and B.V. Limaye, a Course in Calculus and Real Analysis, Springer, 2006.

  5. Tom M. Apostol, Mathematical Analysis, Narosa Publishing House

  6. Courant and John, Introduction to Calculus and Analysis, Vol II, Springer

  7. W. Rudin, Principles of Mathematical Analysis, Tata McGraw-Hill

  8. Terence Tao, Analysis II, Hindustan Book Agency, 2006

  9. Satish Shirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag, London, 2006

  10. S. Kumaresan, Topology of Metric Spaces, 2nd Ed., Narosa Publishing House, 2011.

  11. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 2004.

CC6 – Group Theory 1 [Credit: 1+5]

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

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

Properties of cyclic groups, classification of subgroups of cyclic groups. Cycle notation for permutations, properties of permutations, even and odd permutations, alternating group, properties of cosets, Lagrange’s theorem and consequences including Fermat’s Little theorem.

External direct product of a finite number of groups, normal subgroups, factor groups, Cauchy’s theorem for finite abelian groups.

Group homomorphisms, properties of homomorphisms, Cayley’s theorem, properties of isomorphisms. First, Second and Third isomorphism theorems, Automorphism.



Reference Books

  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 A. Gallian, Contemporary Abstract Algebra, 4th Ed., 1999.

  4. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., 1995.

  5. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

  6. D.S. Malik, John M. Mordeson and M.K. Sen, Fundamentals of abstract algebra.

  7. Sen, Ghosh, Mukhopadhaya, Abstract Algebra.

CC7 –Dynamics of particle and Integral transform [Credit: 1+5]

Unit 1 [Credit: 3]

Central force. Constrained motion, varying mass, tangent and normal components of acceleration, motion of a particle in polar coordinate system, modelling ballistics and planetary motion, Kepler's second law.

Unit-2 [Creidt: 2]

Fourier integral theorem, Definition of Fourier Transforms, Algebraic and analytic properties of Fourier Transform, Fourier sine and cosine Transforms, Fourier Transforms of derivatives, Fourier Transforms of some useful functions, Inversion formula of Fourier Transforms, Convolution Theorem, Parseval’s relation, Applications of Fourier transforms in solving ordinary and partial differential equations.

Definition and properties of Laplace transforms, Sufficient conditions for the existence of Laplace Transform, Laplace Transform of some elementary functions, Laplace Transforms of the derivatives, Initial and final value theorems, Convolution theorems, Inverse of Laplace Transform, Application to Ordinary and Partial differential equations



Reference Books



  1. Loney, S. L., An Elementary Treatise on the Dynamics of particle and of Rigid Bodies, Loney Press

  2. Sneddon, I.N., Fourier Transforms, McGraw-Hill Pub, 1995.

  3. Sneddon, I.N., Use of Integral Transforms, McGraw-Hill Pub.

  4. Andrews, L.C., Shivamoggi, B., Integral Transforms for Engineers, PHI.

  5. Debnath, L., Bhatta,D., Integral Transforms and Their Applications, CRC Press, 2007.


CC8 – Riemann Integration and Series of Functions [Credit: 1+5]

Riemann integration: inequalities of upper and lower sums, Darbaux integration, Darbaux theorem, 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 theorem of Integral Calculus.

Improper integrals. Convergence of Beta and Gamma functions.

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

Fourier series: Definition of Fourier coefficients and series, Reimann Lebesgue lemma, Bessel's inequality, Parseval's identity, Dirichlet's condition.

Examples of Fourier expansions and summation results for series.

Power series, radius of convergence, Cauchy Hadamard Theorem.

Differentiation and integration of power series; Abel’s Theorem; Weierstrass Approximation Theorem.

Reference Books


  1. K.A. Ross, Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004.

  2. R.G. Bartle D.R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002.

  3. Charles G. Denlinger, Elements of Real Analysis, Jones & Bartlett (Student Edition), 2011.

  4. S. Goldberg, Calculus and mathematical analysis.

  5. Santi Narayan, Integral calculus.

  6. T. Apostol, Calculus I, II.

CC9 – Multivariate Calculus and Partial differential equation [Credit: 1+5]

Unit 1 [Credit: 2]

Functions of several variables, limit and continuity of functions of two or more 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, Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems

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-2 [Credit: 3]

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. Solution by Lagrange’s and Charpit’s method.

Derivation of Heat equation, Wave equation and Laplace equation. Classification of second order linear equations as hyperbolic, parabolic or elliptic. Reduction of second order Linear Equations to canonical forms.




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