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KUMAUN UNIVERSITY, NAINITAL
Department of Mathematics
B. Sc. Mathematics
Semester system course structure:
_1. The course work shall be divided into six semesters with three papers in each semester.
- Each paper in a semester will be of_ 80 marks out of which 60 marks for theory and 20 marks are allotted for internal assessment (written test or assignments or both) 3. Each theory paper shall consists of section (A): 20% of total marks (objective, one word answer, fill in the blanks, true- false; all parts will be compulsory), section (B) : 40% of total marks (short answer) and section (C) : 40% of total marks (long answer). 4. Question paper shall cover the whole syllabus. 5. The duration of theory examination shall be 03 hrs.
B.A./B. Sc. Mathematics Course Structure (Semester System)
I Semester II Semester III Semester IV Semester V Semester VI Semester Elementary Algebra and Trigonometry
Group Theory Advanced Algebra
Vector spaces and Matrices
Linear Algebra Numerical Methods
Differential Calculus
Integral Calculus
Differential Equations
Real Analysis Complex Analysis Mathematical Statistics
Geometry and Vector Analysis
Analytical Geometry
Mechanics Mathematical Methods
Functions of several variables and Partial Differential Equations
Operations Research
B. Sc. Semester I
PAPER I: Elementary Algebra and Trigonometry: MM 60
Numbers: Natural numbers, Integers, Rational and Irrational numbers, Real numbers, Complex numbers, Mappings, Equivalence relation and partitions, Congruence modulo n. Roots of equations: Fundamental Theorem of Algebra, Relations between Roots and Coefficients, transformation of equations, Descartes rule of signs, Algebraic Solution of a Cubic equations (Carden method), Bi-quadratic Equation. Elementary Matrices: Symmetric, skew-symmetric, Hermitian and skew-Hermitian matrices; Elementary operations on matrices, adjoint and inverse of a matrix. Trigonometry: De Movire’s Theorem and its applications, Exponential, Logarithmic, Circular and hyperbolic functions together with their inverses, Gregory’s series, Summation of Trigonometric series.
Books Recommended:
- Leonard E. Dickson: First Course in the Theory of Equations.
- Burnside, William Snow,Panton and Arthur William: The Theory of Equations Vol I (1924).
- John Bird: Engineering Mathematics, Fifth edition.
- Rajendra Kumar Sharma, Sudesh Kumari Shah and Asha Gauri Shankar: Complex Numbers and the Theory of Equations, Anthom Press India
PAPER II: DIFFERENTIAL CALCULUS MM 60
Limit, Continuity and Differentiability: Functions of one variable, Limit of a function (ε-δ Definition), Continuity of a function, Properties of continuous functions, Intermediate value theorem, Classification of Discontinuities, Differentiability of a function, Rolle’s Theorem, Mean value theorems and their geometrical interpretations, Applications of mean value theorems. Successive Differentiation, Expansions of functions and Indeterminate forms: Successive Differentiation, nth^ Differential coefficient of functions, Leibnitz Theorem; Taylor’s Theorem, Maclaurin’s Theorem, Taylor’s and Maclaurin’s series expansions. Tangents and Normals: Geometrical meaning of 〱げ 〱け, Definition and equation of Tangent, Tangent at origin, Angle of intersection of two curves, Definition and equation of Normal, Cartesian subtangent and subnormal, Tangents and Normals of polar curves, Angle between radius vector and tangent, Perpendicular from pole to tangent, Pedal equation of curve, Polar subtangent and polar subnormal, Derivatives of arc (Cartesian and polar formula). Curvature and Asymptotes: Curvature, Radius of curvature; Cartesian, Polar and pedal formula for radius of curvature, Tangential polar form, Centre of curvature, Asymptotes of algebraic curves, Methods of finding asymptotes, Parallel asymptotes. Singular Points and Curve Tracing: Regular points and Singular Points of a curve, Point of inflection, Double Points, Cusp, Node and conjugate points, Curve tracing.
Books Recommended:
- M. Ray: Differential Calculus, Shiva Lal Agarwal and Co., Agra
- H. S. Dhami: Differential Calculus, New Age International, New Delhi
- T. M. Apostol: Calculus, John Willey and Sons, New York
- S. Lang: A First Course in Calculus, Addison Wesley Publishing Co., Philippines
- Gorakh Prasad: Differential Calculus, Pothishala publication, Allahabad.
Definite Integrals: Integral as a limit of sum, Properties of Definite integrals, Fundamental theorem of integral calculus, Summation of series by integration, Infinite integrals, Differentiation and integration under the integral sign. Functions Defined by Infinite Integrals: Beta function, Properties and various forms, Gamma function, Recurrence formula and other relations, Relation between Beta and Gamma function, Evaluation of integrals using Beta and Gamma functions. Multiple Integrals: Double integrals, Repeated integrals, Evaluation of Double integrals, Double integral in polar coordinates, Change of variables and Introduction to Jacobians, Change of order of integration in Double integrals, Triple integrals, Evaluation of Triple integrals, Drichlet’s theorem and its Liovelle’s extension. Geometrical Applications of Definite Integrals: Area bounded by curves (quadrature), Rectification (length of curves), Volumes and Surfaces of Solids of revolution.
Books Recommended:
- M. Ray: Integral Calculus, Shiva Lal Agarwal and Co., Agra
- H. S. Dhami: Integral Calculus, New Age International, New Delhi
- T. M. Apostol: Calculus, John Willey and Sons, New York
- S. Lang: A First Course in Calculus, Addison Wesley Publishing Co., Philippines
- Gorakh Prasad: Integral Calculus, Pothishala Publication, Allahabad
PAPER III: ANALYTICAL GEOMETRY MM 60
System of co-ordinates: Curvilinear coordinates, Spherical and Cylindrical coordinates. The Sphere: Definition and equation of a sphere, Plane section of a sphere, Intersection of two spheres, Intersection of a sphere and a line, Power of a point, tangent plane, Plane of contact, Polar plane, Pole, Angle of Intersection of two spheres, Radical plane, Co-axial system of spheres. Cone and Cylinder: Definition and equation of a cone, Vertex , Guiding curve, Generators, Three mutually perpendicular generators, Intersection of a line with a cone, Tangent line and tangent plane, Reciprocal cone, Right circular cone, Definition and equation of a cylinder, Right circular cylinder, Enveloping cylinder. Conicoids: General equation of second degree, Central conicoids, Tangent plane, Director sphere, Normal, Plane of contact, Polar plane, Conjugate plane and conjugate points.
Books Recommended:
- Shanti Narayan: A Text book of Analytical Geometry, S. Chand, & company, New Delhi.
- H. Burchared Fine and E. D. Thompson: Coordinate Geometry, The Macmillan company.
- P. K. Jain and Khalil Ahmed: A textbook of Analytical Geometry, New Age, Delhi.
B. Sc. Semester III
PAPER I: ADVANCED ALGEBRA MM 60
Rings : Rings, Various types of rings, Rings with unity, Rings without zero divisors, Properties of rings, Sub rings. Ideals: Ideals, Quotient rings, Principal ideals, Maximal ideals, Prime ideals, Principal ideal domains, Characteristic of a ring.
Integral domains and Fields : Integral domain, Field, Skew field etc., Field of quotients of an integral domain, Embedding of an integral domain in a field, Factorization in an integral domain, Divisibility, Units, Associates, Prime and irreducible elements, Unique Factorisation Domain, Euclidean rings. Polynomial rings : Polynomials over a ring, Degree of a polynomial, Zero, Constant and monic polynomials, Equality of polynomials, Addition and multiplication of polynomials, Polynomial rings, Embedding of a ring R into R[x], Division algorithm, Euclidean algorithm, Units and associates in polynomials, Irreducible polynomials.
Books recommended
- I. N. Herstein: Topics in Algebra. Wiley Eastern Ltd.
- N. Jacobson: Basic Algebra Vol I & II. Hindustan Publishing Co.
- Joseph A. Gallian: Contemporary Abstract Algebra. Narosa Publishing House.
- Shanti Narayan: Textbook of Modern Abstract Algebra. S Chand & Co.
- R. S. Aggarwal: A Textbook on Modern Algebra. S Chand & Co.
PAPER II: DIFFERENTIAL EQUATIONS MM 60
Differential equations: Introduction of Differential equations, Order and Degree of Differential Equations, Complete primitive (general solution, particular solution and singular solutions), Existence and uniqueness of the solution dy/dx= f(x,y). First Order Differential Equations: Differential equations of first order and first degree, Separation of variables, Homogeneous Equations, Exact Equations, Integrating Factor, Linear Equation, Equation of First order but not of first degree, Various methods of solution, Clairaut’s form, Singular solutions, Trajectory, Orthogonal Trajectory, Self-Orthogonal family of Curves. Linear Differential Equations: Linear equations with constant coefficients, Complementary function, Particular integral, Working rule for finding solution, Homogeneous linear equations. Miscellaneous Equations: Simultaneous differential equations, Differential equations of the form dx/P= dy/Q= dz/R where P, Q, R are functions of x, y, z. Exact differential equations, Total differential equations, Series solutions of differential equations, Linear differential equations of second order with variable coefficients. Applications: Initial and boundary value problems, Simple applications of differential equations of first order.
Books Recommended:
- Earl A. Coddington and Norman Levinson: Theory of Ordinary Differential Equations, Tata McGraw-Hill Publishing Company (1998).
- Shepley L. Ross: Differential Equations, Wiley (1984).
- Ravi P. Agarwal: Ordinary and Partial Differential Equations.
- L. Elsgolts: Differential Equations and Calculus of Variations, Mir Publishers, 1970.
- M D Raisinghania: Ordinary & Partial Differential Equation.
PAPER III: MECHANICS MM 60
Rectilinear motion: Newton’s Laws of Motion, velocity and acceleration, motion under constant acceleration, motion under inverse square law, rectilinear motion with variable acceleration, Simple Harmonic Motion.
Absolute and uniform convergence, Weierstrass M-Test, Infinite integral depending on a parameter. Sequence and Series: Sequences, theorems on limit of sequences, Cauchy’s convergence criterion, infinite series, series of non-negative terms, Absolute convergence, tests for convergence, comparison test, Cauchy’s root Test, ratio Test, Rabbe’s, Logarithmic test, De Morgan’s Test, Alternating series, Leibnitz’s theorem. Uniform Convergence: Point wise convergence, Uniform convergence, Test of uniform convergence, Weierstrass M-Test, Abel’s and Dritchlet’s test, Convergence and uniform convergence of sequences and series of functions.
Books Recommended:
- Walter Rudin: Principle of Mathematical Analysis (3rd edition) McGraw-Hill Kogakusha, 1976, International Student Edition.
- K. Knopp: Theory and Application of Infinite Series.
- T. M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi, 1985.
PAPER III: MATHEMATICAL METHODS MM 60
Integral Transforms : Definition, Kernel. Laplace Transforms : Definition, Existence theorem, Linearity property, Laplace transforms of elementary functions, Heaviside Step and Dirac Delta Functions, First Shifting Theorem, Second Shifting Theorem, Initial-Value Theorem, Final-Value Theorem, The Laplace Transform of derivatives, integrals and Periodic functions. Inverse Laplace transforms : Inverse Laplace transforms of simple functions, Inverse Laplace transforms using partial fractions, Convolution, Solutions of differential and integro-differential equations using Laplace transforms. Dirichlet’s condition, Fourier Transforms : Fourier Complex Transforms, Fourier sine and cosine transforms, Properties of FourierTransforms, Inverse Fourier transforms.
Books Recommended:
- Murry R. Spiegal: Laplace Transform (SCHAUM Outline Series), McGraw-Hill
- J. F. James: A student’s guide to Fourier transforms, Cambridge University Press.
- Ronald N. Bracewell: The Fourier transforms and its applications, Mcgraw Hill.
- J. H. Davis: Methods of Applied Mathematics with a MATLAB Overview, Birkhäuser, Inc.,Boston, MA,
B. Sc. Semester V
PAPER I: LINEAR ALGEBRA MM 60
Linear Transformations: Linear transformations, rank and nullity, Linear operators, Algebra of linear transformations, Invertible linear transformations, isomorphism; Matrix of a linear transformation, Matrix of the sum and product of linear transformations, Change of basis, similarity of matrices. Linear Functionals: Linear functional, Dual space and dual basis, Double dual space, Annihilators, hyperspace; Transpose of a linear transformation. Eigen vectors and Eigen values: Eigen vectors and Eigen values of a matrix, product of characteristic roots of a matrix and basic results on characteristic roots, nature of the characteristic roots of Hermitian, skew-Hermitian, unitary and orthogonal matrices, characteristic equation of a matrix, Cayley-Hamilton theorem and its use in finding inverse of a matrix.
Bilinear forms: Bilinear forms, symmetric and skew-symmetric bilinear forms, quadratic form associated with a bilinear form.
Books Recommended:
- Hadley: Linear Algebra.
- Hoffman and Kunze: Linear Algebra, Prentice Hall of India, New Delhi, 1972.
- H. Helson: Linear Algebra, Hindustan Book Agency, New Delhi, 1994.
- K. B. Dutta: Matrix and Linear Algebra, Prentice Hall of India.
- S. Lang: Linear Algebra, Springer
PAPER II: COMPLEX ANALYSIS MM 60
Complex Variables: Functions of a complex variable; Limit, continuity and differentiability. Analytic functions: Analytic functions, Cauchy and Riemann equations, Harmonic functions. Complex Integration: Complex integrals, Cauchy's theorem, Cauchy's integral formula, Morera’s Theorem, Liouville’s Theorem, Taylor's series, Laurent's series, Poles and singularities. Residues: Residues, the Residue theorem, the principle part of a function, Evaluation of Improper real integrals.
Books Recommended:
- J. B. Conway: Functions of One Complex Variable, Narosa Publishing House, 1980.
- E. T. Copson: Complex Variables, Oxford University Press.
- L. V. Ahlfors: Complex Analysis, McGraw-Hill, 1977.
- D. Sarason: Complex Function Theory, Hindustan Book Agency, Delhi, 1994..
- P. R. Halmos: Naive Set Theory, Van Nostrand, 1960.
PAPER III: FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL
DIFFERENTIAL EQUATIONS MM 60
Functions of several variables : Limit, continuity and differentiability of functions of several variables. Partial Derivatives : Partial derivatives and their geometrical interpretation, differentials, derivatives of composite and implicit functions, Jacobians, Chain rule, Euler’s theorem on homogeneous functions, harmonic functions, Taylor’s expansion of functions of several variables. Maxima and Minima : Maxima and minima of functions of several variables – Lagrange’s method of multipliers. Partial differential equations : Partial differential equations of first order, Charpit’s method, Linear partial differential equations with constant coefficients. First-order linear, quasi-linear PDE's using the method of characteristics. Partial differential equations of 2nd-order: Classification of 2nd-order linear equations in two independent variables: hyperbolic, parabolic and elliptic types (with examples).
Books Recommended:
- W. Fleming: Functions of several variables, Springer
- R P Agrawal: Ordinary and Partial Differential Equations, Springer
- K Sankar Rao: Partial Diffrential Equations, PHI
B. Sc. Semester VI
method, Big M Method and Two phase simplex method, Degeneracy in LPP. Duality in LPP, Duality and simplex method, Dual simplex method. Transportation and assignment Models : Formulation of TP, Transportation Table, Finding initial basic feasible solution, Test of optimality, Degeneracy, MODI method, Stepping Stone method, Solutions of Assignment problems, Hungarian method.
Recommended Books:
- G. Hadley, Linear Programming, Narosa Publishing House, 1995.
- S. I. Gass, Linear Programming: Methods and Applications (4th^ edition) McGraw-Hill, New York, 1975.
- KantiSwaroop, P.K. Gupta and Man Mohan, Operations Research, Sultan Chand & Sons, New Delhi, 1998.
- Hamdy A. Taha, Operations Research, Prentice-Hall of India, 1997.
