COURSES OF STUDIES
Courses of Studies ( Choice Based Credit System)
B.Sc. (Hons.) Mathematics
CORE COURSES
B.Sc.(Honours)-Mathematics
CREDIT : 06 each
KHALLIKOTE CLUSTER UNIVERSITY
BERHAMPUR,GANJAM, ODISHA-760001
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Semester – I
C-1.1: Calculus –I
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Curvature, Asymptotes, Singular points and Curve tracing.
Unit-II
Rectification, Quadrature, Volume and surface area of the solid surface of revolution.
Unit-III
Vector calculus, Limits, continuity and differentiability of vector function, Point functions (scalar and vector), Directional derivative, gradient and curl of point functions and their properties.
Unit-IV
Sphere, Cone and Cylinder.
Books Prescribed
1. Elementary Calculus by Panda and Satapathy.
Chapters: 1,2,5,6.
2. Analytical Geometry of Quadratic Surfaces by B.P.Acharya and D.P.Sahu, Kalyani Publishers
Chapters: 2, 3
Books for reference
1. Text book of Calculus, Part-II by Shantinarayan, S Chand & Co.
2. Text book of Calculus, Part-III by Shantinarayan, S Chand & Co.
3. Calculus by G.B.Thomas, Pearson Education, Delhi
4. Analytical Solid Geometry by S.Narayan and Mittal, S.Chand Co.
C-1.2: (Abstract Algebra) Algebra –I
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Group theory: Definition of a group, Some examples of groups, Some preliminaries lemmas, Subgroups, A counting principle, Normal subgroups and quotient groups, Homomorphism.
Unit-II
Ring theory: Definition and examples of rings, some special class of rings, Homomorphism, Ideals and quotient rings.
Unit-III
Theory of equation: Preliminary, Properties of equations, Descartes’ rule of signs, relation between roots and coefficients, Transformation of equations, Multiple roots, Sums of powers of roots, Reciprocal equations
Unit-IV
Cubic and Biquadratic equations: Solution of the cubic, Nature of the roots of a cubic, Expressing the cubic as a difference of two cubes, Solution by Symmetric functions of roots, Solution of the bi-quadratic, solution by redicals. Numerical solution of equations: Numerical equations, limits of the roots of equations, Integer roots, Newton’s method of approximation, Horner’s method,
Books Prescribed
1.Topics in algebra by I.N.Herstein, Wiley Eastern Limited, India, 1975
Chapters:2(up to 2.7), 3(up to 3.4)
2.Text book of algebra and theory of equation( Tenth edition) by Chandrika Prasad, Pothishala Private limited.
Chapters: XI, XII, XII .
Books for reference
1. A first course in Abstract Algebra by J.B.Fraleigh, Pearson, 2002.
2. An Introduction to theory of Groups by J.J.Rotman, Springer Verlag.
3. Contemporary Abstract Algebra by J.A.Gallian, Narosa Pub. House.
Semester – II
C-2.1: Analysis –I
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Field structure and order structure, Bounded and unbounded sets, (excluding Dedikinds form of completeness property), completeness in the set of real numbers, Absolute value of a real number. Neighborhood of a point, Interior point, Limit point, Open set, Closed set, Dense set, Perfect set, Bolzano-Weierstrass’s theorem, Countable and Uncountable sets.
Unit-II
Sequences, Limit points f a sequence, Limit inferior and superior, Convergent sequence, Non-convergent sequence, Cauchy’s general principle of convergence. Algebra of sequences, Some important theorems, Monotonic sequence.
Unit-III
Infinite series, Positive term series, Comparison test for positive term series, Cauchy’s root test, D’Alemberts root test, Raabe’s test, Logarithemic test, Integral test, Series with arbitrary terms, Rearrangement of the terms.
Unit-IV
Power series and Fourier series.
Books Prescribed
Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.
Chapters: 1,2,3,4,13,14.
Books for reference
1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.
2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak.
3. Elementary Analysis: The theory of calculus by K.A.Ross, Springer.
4. Introduction to Analysis by A.Mattuck, Prentice Hall.
C-2.2: Ordinary Differential Equation
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Basic concepts of differential equation, formation of differential equation, solution of first order first degree and first order, Equations first order but higher degree.
Unit-II
Linear equation with constant coefficient, Differential equation with Variable Coefficients
Unit-III
Series solutions and Special function
Unit-IV
Laplace Transformation.
Books Prescribed
A Course of ordinary and partial differential equation by J.Sinha Roy and S.Padhy, Kalyani Publisher
Chapters: 1,2(2.1-2.7),3,4(4.1-4.7),5,7(7.1-7.4),9(9.1-9.5,9.10,9.11,9.13)
Books for reference
1. Text book of Differential equation by N.M.Kapur,
2. Advanced Engineering Mathematics by Erwin Kreyzig, John-Wiley
3. Differential Equation and their Apllications by Martin Braun, Springer International.
Semester – III
C-3.1: Theory of Real functions(Analysis –II)
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Functions of single variable, Limits, Continuous functions, Continuous functions on closed intervals, Uniform continuity.
Unit-II
Derivatives, Increasing and decreasing functions, Darboux’s theorem, Rolle’s theorem, Langrage’s Mean value theorem, Cauchy’s Mean value theorem, Higher order derivatives.
Unit-III
Power series, Exponential functions, Logarithmic functions, Trigonometric functions, Functions of Bounded variation.
Unit-IV
Uniform convergence, Point-wise convergence, Uniform convergence on an interval, Test of uniform convergence, Weierstrass Approximation theorem.
Books Prescribed
Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.
Chapters: 5,6,8(except 8.7),12(except 12.4).
Books for reference
1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.
2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak.
3. Elementary Analysis: The theory of calculus by K.A.Ross, Springer.
4. Introduction to Analysis by A.Mattuck, Prentice Hall.
C-3.2: (Linear Algebra) Algebra –II
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I .
Vector space: Definitions and examples, Subspaces, Span of a set, More about subspaces, Linear dependence and independence, Dimension and Basis.
Unit-II
Linear transformation: Definition and examples, Range and kernel of a linear map, Rank and nullity, The space L(U,V), Composition of linear maps, Matrix Associated with a linear map, Linear map associated with a matrix, Linear operation, Matrix multiplication, Rank and nullity of a matrix, Transpose of a matrix.
Unit-III
Elementary row operations, System of linear equations, Matrix inversion, Determinants, Minor and cofactors, Rank of a matrix, Product of determinants, Application to linear equations, Eigen value and Eigen vectors.
Unit-IV
Similarity of Matrices, Inner Product space, Orthogonal and Unitary matrices, Application to reduction to Quadrics.
Books Prescribed
An Introduction to Linear Algebra by V. Krishnamurthy, V.P.Mainra ad J.L.Arora, Affiliated East-West Press Pvt. Ltd.
Chapters: 3,4,5,6,7.
Books for reference
1. Linear Algebra- A Geometric Approach by S.Kumarsen, Prentice Hall.
2. Linear Algebra by Hoffman and Kunze, Prentice Hall
C-3.3: Partial Differential Equation
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Simultaneous Differential equations: Simultaneous equations with constant coefficients, Simultaneous equations with variable coefficients, Method of solution of equation in symmetrical form, Method of introduction of new variable.
Total differential equations: Condition of integrability, Method of obtaining primitive, Solution by inspection, Homogeneous equations,
Unit-II
Partial differential equation of first order: Classification of integrals, Complete, Singular and General integrals and their geometrical interpretations, Formation of partial differential equations, Linear partial differential equations and its solutions by Lagrange’s method, Non-linear partial differential equations and its solutions by Charpit’s method, Standard Forms: I,II,III,IV of Non-linear partial differential equations and their solutions.
Unit-III
Linear partial differential equations with constant coefficients: Solutions of Homogeneous linear equations with constant coefficients, Methods for finding complementary functions and Particular integrals of Linear partial differential equations with constant coefficients.
Linear partial differential equations order two with constant coefficients: Methods of solutions of Linear partial differential equations with constant coefficients of Type I,II,III,IV. Laplaces transformation for reducing Linear partial differential equations with constant coefficients to standard types.
Unit-IV
Monge’s method of solving non-linear partial differential equations of order two of the form Rr + Ss + Tt = V.
Application of partial differential equation: Mathematical model for Wave equation, Solution of wave equation by separation of variables method using Fourier series, D’Alembert’s solutions of wave equation, Solution of Heat equation by Fourier series.
Books Prescribed
1.Text book of Differential equation by N.M.Kapur,
Chapters: 8,9(up to 9.7),10,11(up to 11.8),12(up to 12.9).
2.Advanced Engineering Mathematics by Erwin Kreyzig, John-Wiley .
Chapter: 11(11.2,11.3,11.4,11.5.)
Books for reference
1. A Course of ordinary and partial differential equation by J.Sinha Roy and S.Padhy, Kalyani Publisher.
2. Differential Equations by S.L.Ross, John Wiley & Sons.
3. Linear Partial Differential Equations For Scientist and Engineers by Tyn Myint-U and L.Debnath, Springer.
Semester – IV
C-4.1: Numerical Methods
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Source of Error, Significant figures: Absolute, Relative and Percentage errors, Generation and propagation of Round of errors, Significance errors, Estimation of errors, Power series for evaluating Transcendental functions, Binary, octal and hexadecimal number system, Floating point Arithmetic. Solution of system of Linear equations: Gaussian elimination method, Gauss-Jordan elimination method, Gauss elimination method to compute the inverse of a matrix, Method of Matrix Factorization, Solution of Tri-Diagonal systems Method of Iteration(Jacobi and Gauss-Seidel)
Unit-II
Interpolation: Introduction, Lagrangian Interpolating formula, Error in the Interpolating Polynomial, Advantage and disadvantage of Lagrangiian interpolation, Divided differences and their properties, Newton’s fundamental Interpolation formula, Equivalence of Lagrangian and Newtonian Interpolation, Finite difference, Forward and Backward difference and their relationship, Symbolic operators and their relation, Differences of a polynomial, Factorial Power functions, Difference of a Factorial Power function, Newton’s Forward and Backward Interpolation Formulas and errors, Gaussian, Stirling’s, and Bessel’s Interpolation formulas.
Unit-III
Numerical Differentiation and Integration: Errors in Numerical differentiation, Differentiation based on Newton’s forward and backward interpolation formula, Differentiation based on Strling’s formula, Integration of Lagrangian Interpolation polynomial, Newton’s-Cote’s Quadrature formula, Trapezoidal and Simpson’s one-third formulas and their errors, Derivation of Trapezoidal and Simpson’s one-third formula using Newton’ forward difference formula, Trapezoidal, Simpsons’s one-third and Weddle’s rule of Integration,
Numerical Solutions of ordinary differential equations: Picard’s method of successive approximation, Euler’s method, Taylor’s series method and Runga-Kutta methods of solving ordinary differential equations.
Unit-IV
Solutions of Algebraic and Transcendental equations: Methods of finding out crude approximation of a real root, Method of bisection, Method of iteration, Regula-Falsi method, Newton-Raphson method, Convergence of Newton-Raphson method, Curve fitting: Method of least square, Curve fitting with an exponential curve, Fitting a trigonometric function.
Books Prescribed
Introductory Numerical Analysis by N.Datta and R.N.Jana, Shreedhar Prakashani, Calcutta.
Chapters: 1,2(up to 2.19),3(3.1-3.4, 3.7-3.15), 4(up to 4.7), 5, 6(up to 6.5), 7(7.1-7.4).
Books for reference
1. Numerical methods for Scientific and Engineering Computations by Jain Iyengar and Jain, New Age International
2. A course on Numerical Analysis by B.P.Acharya and R.N.Das, Kalyani Publisher.
3. A friendly Introduction to Numerical Analysis by Brian Braie, Pearson Educaton.
C-4.2: Riemann Integration and Complex Analysis(Analysis-III)
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Riemann Integrals: Definitions of the integral, Refinement of partitions, Darboux’s theorem, Conditions of integrability, Integrability of sum and difference of integrable functions, Integral as limit of sum, Some integrable functions, Integration and differentiation, Fundamental theorem of calculus.
Unit-II
Improper integrals: Integration of unbounded functions with finite limits of integration, Comparison tests for convergence, Infinite range of integration, Beta and Gamma functions and their convergences.
Unit-III
Complex numbers, Complex plane, Polar form of complex numbers, powers and roots, Derivative, analytic functions, Cauchy-Riemann equation, Laplace’s equation.
Unit-IV
Geometry of Analytic functions: Conformal mapping, Exponential, Trigonometric, Hyperbolic, Logarithmic functions, Linear fractional transformations.
Books Prescribed
1.Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.
Chapters: 9,11.
3. Advanced Engineering Mathematics by Erwin Kreyzig, John-Wiley
Chapter: 12.
Books for reference
1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.
2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak.
3. Complex Variables and Applications by J.W.Brown and R.V.Churchill, McGraw Hill International Edition.
4. Functions of one complex variable by J.B.Conway, Springer.
C-4.3: (ANSI- C) Computer fundamentals
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Overview of C; Costants, Variables and Data types; Operators and expressions.
Unit-II
Decision making and branching, Decision making and looping.
Unit-III
Arrays, Character Arrays and Strings.
Unit-IV
User-defined Functions.
Books Prescribed
Programming in ANSI C, Third Edition, by Balagurusamy,Tata McGraw-Hill Publishing company Limited.
Chapters:1,2,3,5,6,7,8,9.
Books for reference
Applications Programming in ANSI C by R. Johnsonbaugh and L. Ljoie, Pearson Education.
Semester – V
C-5.1: Multivariate Calculus(Calculus –II)
(Total Marks; 80+20)
5 Lectures, 1 Tutorial per week
Unit-I
Functions of several variables, Explicit and implicit functions, Continuity, Partial derivatives, Differentiability, Partial derivative and differentials of higher order, Functions of functions, Change of variables.
Unit-II
Taylors theorem of function of two variables, Extreme values(Maxima and minima), Jacobians, Stationary values under subsidiary conditions.
Unit-III
Integration on .
Unit-IV
Integration on .
Books Prescribed
Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.
Chapters: 15,16,17,18.
Books for reference
1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.
2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak
3. Calculus by G.B.Thomas and R.L.Finney, Pearson Education.
C-5.2: Practical
(Total Marks: 100)
2 Practical classes per week
A student is required to perform Four experiments two from each Group. The duration of Practical examination is six hours. A student has to perform at least 75% of the number of prescribed experiments.
Two experiments from Group-A 30 Marks
Two experiments from Group-A 30 Marks
Practical record( 10+10) 20 Marks
Viva-voce ( 10+10) 20 Marks
A student shall be required to maintain two records, one for each group, certified by the teacher and produce them at the time of examination.
List of Experiments
Group – A
i. Graphical solution of Linear Programming problems.
ii. Solution of Linear Programming problems by simplex method.
iii. Solution of Transportation problems.
iv. Solution of Assignment problems by Hungarian method.
v. Tracing of Curves: Catenaries, Beroulli’s Lemniscates, Cissoids, Asteroid, Cardioids and Descarte’s Folium.
vi. Estimation of a function using Lagrange’s, Newton’s forward, Newton’s backward Interpolation formulas.
vii. Numerical solutions of transcendental equations by Bisection method, Regula-Falsi method, Newton-Raphson method and Iteration methods.
viii. Numerical Integration by Composite Trapezoidal and Composite Simpson’s one-third rule.
ix. Fitting of exponential and Logarithmic function.
x. To find the correlation coefficient between tw variable and also the line of regression.
Group – B( using any soft-ware)
Lab-work to be performed on a computer
i. Writing a program for searching primes less than or equal to a specific number N.
ii. Writing a program for arranging a given set of numbers in a specific order.
iii. Writing a program for the Numerical solutions of transcendental equations by Bisection method, Regula-Falsi method, Newton-Raphson method
iv. Writing a program to solve quadratic equation