# «B. A. (Hons.) / B. Sc. (Hons.) Mathematics Syllabus for Semester System First Semester Code Course Periods/ Week Credits BHM-101 Calculus 4 4 BHM-102 ...»

B. A. (Hons.) / B. Sc. (Hons.) Mathematics

Syllabus for Semester System

First Semester

Code Course Periods/ Week Credits

BHM-101 Calculus 4 4

BHM-102 Geometry of Two and Three Dimensions 4 4

Second Semester

Code Course Periods/ Week Credits

BHM-201 Differential Equations – I 4 4 BHM-202 Algebra and Complex Trigonometry 4 4 Third Semester Code Course Periods/ Week Credits BHM-301 Functions of Several Variables 4 4 BHM-302 Group Theory 4 4 BHM-303 Programming in C 3 2 Fourth Semester Code Course Periods/ Week Credits BHM-401 Ring Theory 4 4 BHM-402 Numerical Methods 4 4 BHM-403 Lab: Numerical Methods 2 2 Fifth Semester Code Course Periods/ Week Credits BHM-501 Analysis – I 4 4 BHM-502 Differential Equations – II 4 4 BHM-503 Metric Spaces 4 4 BHM-504 Linear Algebra 4 4 BHM-505 Probability and Statistics 4 4 Sixth Semester Code Course Periods/ Week Credits BHM-601 Analysis – II 4 4 BHM-602 Geometry of Curves and Surfaces 4 4 BHM-603 Complex Analysis 4 4 BHM-604 Operations Research and Linear Programming 4 4 BHM-605 Mechanics 4 4 BHM-606 Viva Voce 4 BHM-101: Calculus Unit I Ԑ-δ definition of the limit of a function, Algebra of limits, Continuity, Differentiability, Successive differentiation, Leibnitz theorem, Rolle ’s Theorem, Mean value theorems, Taylor’s and Mclauren’s series.

Unit II Indeterminate forms, Curvature, Cartesian, Polar and parametric formulae for radius of curvature, Partial derivatives, and Euler’s theorem on homogeneous functions.

Unit III Asymptotes, Test of concavity and convexity, Points of inflexion, Multiple points, Tracing of curves in Cartesian and polar coordinates.

Unit IV Reduction formulae, Quadrature, Rectification, Intrinsic equation, Volumes and surfaces of solids of revolution.

**Books Recommended:**

• Gorakh Prasad: Differential Calculus, Pothishalas Pvt Ltd, Allahabad.

• Gorakh Prasad: Integral Calculus, Pothishalas Pvt Ltd, Allahabad.

• Shanti Narayan: Differential Calculus, S. Chand & Co.

• Shanti Narayan: Integral Calculus, S. Chand & Co.

• Khalil Ahmad: Text Book of Calculus, World Education Publishers, 2012.

BHM-102: Geometry of Two and Three Dimensions Unit I General equation of second degree, Pair of lines, Parabola, Tangent, normal. Pole and polar and their properties. Ellipse, Hyperbola, Tangent, normal, pole and polar.

Conjugate diameters, Asymptotes, Conjugate hyperbola and rectangular hyperbola.

Unit II Polar equation of a conic, Polar equation of tangent, normal, polar and asymptotes, General equation of second degree, Tracing of parabola, Ellipse and hyperbola.

Unit III Equation of sphere, Tangent plane, Plane of contact and polar plane, Intersection of two spheres, radical plane, Coaxial spheres, Conjugate systems, Equation of a cone, Intersection of cone with a plane and a line, Enveloping cone, Right circular cone.

Unit IV Equation of cylinder, Enveloping and right circular cylinders, Equations of central conicoids, Tangent plane, Normal, Plane of contact and polar plane, Enveloping cone and enveloping cylinder, Conjugate diameters and diametral planes, Equations of paraboloids and its simple properties.

**Books Recommended:**

• Ram Ballabh: Text book of Coordinate Geometry.

• S. L. Loney: The elements of coordinate geometry, by Michigan Historical Reprint Series.

• Shanti Narayan, Analytical Solid Geometry, S. Chand and Company.

• P.K. Jain and Khalil Ahmad: Textbook of Analytical Geometry, New Age International (P) Ltd. Publishers.

BHM-201: Differential Equations – I Unit I Formulation of differential equations, Order and degree of a differential equation, equations of first order and first degree, solutions of equations in which variables are separable, Homogeneous equations, Linear equations and Bernoulli equations, Exact differential equations, integrating factors, Change of variables.

Unit II Equations of the first order and higher degree, Equations solvable for p, y and x, Clairaut equation, Lagrange’s equation, Trajectories.

Unit III Linear differential equations with constant coefficient, Complementary function and particular integral. Particular integral of the forms, sinax, cosax and, Homogeneous linear equations.

Unit IV Linear differential equations of second order, Complete solution in terms of known integral belonging to the complementary function, Normal form, Change of independent variable, Method of undetermined coefficients, Method of variation of parameters, Simultaneous equations with constant coefficients, Simultaneous equations of form.

**Books Recommended:**

• C. H. Edwards and D. E. Penny, Differential Equations and Boundary Value Problems: Computing and Modelling, Pearson education, India 2005.

• Dennis G. Zill, A first course in differential equations,

• S. L. Ross: Differential equations, John Wiley and Sons, 2004.

• Zafar Ahsan: Text Book of Differential Equations and their Applications, Prentice Hall of India.

• Khalil Ahmad: Text Book of Differential Equations, World Education Publishers, 2012.

BHM-202: Algebra and Complex Trigonometry Unit I Relations, Types of relation, Equivalence relations, Partitions, Congruent modulo n, symmetric and skew symmetric matrices, Hermitian and skew Hermitian matrices, Elementary row operations, Elementary matrices and their properties, Singular and nonsingular matrices and their properties.

Unit II Row rank and column rank, Equivalent matrices and their properties, Similar matrices, Equivalence of row and column ranks, Row echelon and reduced row echelon forms of matrix and their properties.

Unit III Eigen values and eigen vectors, Characteristic equation, Cayley Hamilton Theorem and its application in finding the inverse of a matrix. Application of matrices to a system of linear (both homogeneous ad non-homogenious) equations. Theorem on consistencies of a system of linear equations.

Unit IV De Moivre’s theorem and its application. Circular and Hyperbolic functions. Inverse circular and hyperbolic functions. Expansion of trigonometric functions in terms of power and multiple. Separation of real and imaginary parts of logarithmic, trigonometric and inverse trigonometric functions. Summation of series including C+iS method.

**Books Recommended:**

• I. N. Herstein: Topics in Algebra, Wiley; 2nd edition (June 20, 1975).

• P.B. Bhattacharya, S. K. Jain and S. R. Nagpaul: First course in

**Abstract**

Algebra.

• K. B. Dutta: Matrix and Linear Algebra.

• J. Finkbecner: Matrix theory.

• Ushri Dutta, A.S.Muktibodh and S.D. Mohagaonkar: Algebra and Trigonometry, PHI India BHM-301: Functions of Several Variables Unit I Functions of several variables, Domain and range, Level curves and level surfaces, Limits and continuity, Partials derivatives, Total differential, Fundamental lemmas, differential of functions of n variables and of vector functions, The Jacobian matrix, derivatives and differentials of composite functions, The general chain rule.

Unit II Implicit functions, invers functions, Curvilinear coordinates, Geometricals applications, The directional derivatives, Partial derivatives of higher order, Higher derivatives of composite functions, The Laplacian in polar, Cylindrical and spherical coordinates, Higher derivatives of implicit functions, Maxima and minima of functions of several variables, Lagrange Multiplier method.

Unit II Vector fields and scalar fields, the gradient field, Divergence and curl of a vector field, Combined operations, Irrotational and solenoidal fields, double, triple and multiple integrals, Change of variable in integrals, Surface area.

Unit III Line integrals, Integrals with respect to arc length, Basic properties of line integrals, Green’s theorem, Simply connected domains, Extension of results to multiply connected domains, Surfaces in space, Orientability, Surface integrals, Divergence theorem and Stoke’s theorem, Integrals independent of path.

**Books Recommended:**

• Wilfred Kaplan, Advanced Calculus, Addison- Wesley Publishing Company, 1973.

• E. Swokowski: Calculus with Analytical Geometry, Prindle, Weber & Schmidt, 1984.

• E. Kreyszig: Advanced Engineering Mathematics, John Wiley and Sons, 1999.

• David Widder: Advanced Calculus, Prentice- Hall of India, 1999,

• S.C. Malik and Savita Arora: Mathematical Analysis, New Age International (P) Ltd. Publishers, 1996.

BHM-302: Group Theory Unit I Sets, relations, functions, binary operations, Definition of groups with examples and its elementary properties, subgroups, order of an element of a group, cyclic groups, coset decomposition, Lagrange’s theorem and its consequences, normal subgroup and factor groups. Various types of groups up to order 8.

Unit II Group Homomorphism, Isomorphism, kernel of a homomorphism, The homomorphism theorems, The isomorphism theorems, Permutation groups, Even and odd permutations, Alternating groups, Cayley’s theorem, and Regular permutation group.

Unit III Definition and example of Automorphism, inner automorphism, automorphism group of finite and infinite cyclic groups, conjugacy relation, normalizer and centre, External direct products, definition and examples of internal direct products.

Unit IV Class equation of a finite group and its applications, structure of finite abelian groups, Cauchy’s theorem, Sylow’s theorem and consequences. Definition and example of Simple groups, non-simplicity test.

**Books Recommended:**

• N. Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi.

• Joseph A. Gallian, Contemporary Abstract Algebra (4th Ed), Narosa Publishing House, New Delhi.

• N. Jacobson, Basic Algebra Vol. I & II, W. H. Freeman.

• N S Gopalakrishan, University Algebra, New Age International (P) Limited, New Delhi.

• Surjeet Singh and Qazi Zameeruddin, Modern Algebra, Vikas Publishing House Pvt.

Ltd., New Delhi.

BHM-303: Programming in C Unit-I Number system, Programming languages, Types of programming languages, compiler, interpreter, algorithms and flowcharts.

Unit-II Character set, Identifiers and Keywords, Data Types, Constants, Variables and basic structure of C programming, Declarations, Operators & Expressions, Statements, Preprocessor directives, Storage classes.

Unit-III Basic Input and Output, Control Statements, Loops Statements, switch, break, Continue statements. Function prototyping, Library functions, user define functions, Passing arguments to a function, Recursion.

Unit-IV Defining array, passing arrays to functions, Introduction to multidimensional arrays, strings. Pointers Declarations, Call by value and call by reference, pointer to array, Structures and Unions, File handling.

**Books Recommended:**

• E. Balagurusamy, Programming in Ansi C, McGraw-Hill Education

• R.K. Jain, Flowcharts,

• Y. Kanitkar, Let Us C, BPB Publications BHM-401: Ring Theory Unit I Rings and their elementary properties, Integral domain, Field. Subrings, Ideals and their properties, Field of quotients, Quotient rings.

Unit II Homomorphism of rings and its properties, Kernel of a homomorphism, Natural homomorphism, Isomorphism and related theorems, Euclidian rings, Unique factorization theorem.

Unit III Rings of polynomials over a field F, Properties of F[X], Rings of Gaussian integers, Rings of polynomials over rational field. Primitive polynomials and their properties.

Gauss’ Lemma and Eienstien’s criterion for irreducibility.

Unit IV Polynomial rings over commutative rings, unique factorization domain and its properties.

**Books Recommended:**

• I. N. Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi.

• N. Jacobson: Basic Algebra, Volume I and II. W. H. Freeman and Co.

• Surjeet Singh and Qazi Zameeruddin: Modern Algebra, Vikas Publication.

• J.A. Gallian, Contemporary Abstract Algebra, Narosa Publication.

BHM-402: Numerical Methods Unit1 Solution of algebraic and transcendental equations: Bisection method, False position method, Fixed-point iteration method, Newton’s method and its convergence, Chebyshev method. Solution of system of non-linear equations by Iteration and Newton-Raphson method. Program in C for Bisection method, False position method and Newton’s method.

Unit 2 Finite difference operators and finite differences; Interpolation and interpolation formulae: Newton’s forward and backward difference, Central difference: Sterling’s and Bessel’s formula, Lagrange’s interpolation formula and Newton’s divided difference interpolation formula, Hermite interpolation. Program in C for Newton’s forward and backward formula, Newton’s divided difference formula.

Unit 3 Direct methods to solve system of linear equations: Gauss elimination method, GaussJordan method, LU decomposition; Indirect methods: Gauss-Jacobi and Gauss-Seidal methods. The algebraic eigen value problems by Householder and Power method.

Algorithms and program in C for Gauss-Jacobi and Gauss-Seidal method.

Unit 4 Numerical differentiation and Numerical integration by Newton cotes formulae, Trapezoidal rule, Simpson’s rule, Romberg formula and their error estimation.

Numerical solution of ordinary differential equations by Euler’s method, Picard’s method, Taylor series and Runge-Kutta methods. Program in C for Trapezoidal and Simpson’s rule.

**Books Recommended:**

• B. Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India,

• M. K. Jain, S. R. K. Iyengar and R. K. Jain, Numerical Methods for Scientific and Engineering Computation, New age International Publisher, India, 5th edition, 2007

• C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Pearson Education, India,7th edition, 2008.

• M. Pal : Numerical Analysis for scientific and engineering computation, Narosa Publication

• N. Ahmad, Fundamental Numerical Analysis with error estimation, Anamaya Publisher.

BHM-403: Lab: Programming in C and Numerical Methods