Pure Mathamatics

• Introduce students to the foundations of mathematics
• Familiarize students with the axiomatic method in Mathematics

• Apply propositional calculus to simplify and check the validity of an argument
• Prove various set theoretic identities and other properties involving set operations
• Classify various types of relations
• Analyze connections between various types of relations
• Classify various types of functions
• Prove elementary results involving functions and related concepts
• Simplify Boolean expressions using the axioms

• Propositional Logic: Connective; conjunction, disjunction, negation, implication; converse, inverse and contrapositive of a given statement; tautology, contradiction, direct proof, proof by contradiction, existential and universal quantifiers, arguments.
• Set Theory: Set operations, laws of the algebra of sets, Cartesian product, finite set, infinite set, indexed family of sets.
• Relation: Reflexivity, symmetry, transitivity and anti-symmetry of relations; equivalence relation, composition of relations, inverse relation, partially ordered set, totally ordered set,Zorn’s lemma.
• Function: Surjection, injection, bijection, composition of functions, inverse function, image and preimage of subsets, countable and uncountable sets.
• Boolean Algebra: Laws of Boolean algebra, duality, the principle of duality.

• Lecture demonstration by lecturer and tutorial discussion by instructors.

• In course Assessment          30%
• End of course examination  70%

• Set theory and related topics, Seymour Lipschitz, Tata McGraw-Hill, 1998.
• Set theory and logic, Robert R Stoll, Dover publications, 1979.
• Sets, functions and logic- an introduction to abstract mathematics, Keith Devlin, Chapman and Hall, 1992.
• The Foundation of Mathematics, Stewent I. and Tall D., Oxford University Press, 1997.s

• Introduce the algebraic structure of the real line and discuss its order and completeness properties.
• Provide a clear notion of concepts of limit as it is used in the context of the sequence and series
• Enable the students to test the convergence of series

• Recall the axioms governing the properties of Real number system
• Prove the elementary properties of Real number system
• Define various types of limits
• Justify limits from the first principle
• Define convergence sequence
• Apply comparison principle/ sandwich principle to test convergence
• Recall monotone convergence theorem
• Apply monotone convergence theorem to test the convergence of recursive sequence
• Apply Cauchy criterion to test convergence of sequences
• Define convergence series
• Develop tests to discuss the convergence of series

• Rationals and Reals: Algebraic and order properties of the real numbers, infimum and supremum, completeness properties of real numbers, dense property of rationals and irrationals.
• Sequence: Limits, Convergence, Sequence, Cauchy sequence, Monotone convergence theorem, Cauchy’s criterion for convergence, recurrence sequences.
• Series: Convergent series, Test for absolute convergence, Alternating series, Power series.

• Lectures,  Tutorials, Handouts, Problem solving, Use of e-resources

• In-course Assessments          30%
• End-of-course Examination  70%

• Introduction to Real Analysis, Bartle R. G. and Sherbert D. R., John Willy and Sons, 2000.
• Mathematical Analysis, Binmore. K. G., Cambridge University Press, 1983.
• Calculus, Robert T. Smith, Roland B. Minton, McGraw Hill Higher Education, 2008.

• To familiarize students with the basic concepts of theory of numbers
• To provide an introduction to some of the important algebraic structures in the modern algebra

• Apply divisibility properties of integers
• Prove the fundamental theorem of arithmetic and other results related to the prime numbers
• Solve system of linear congruences
• Identify groups and subgroups, in their variety
• Prove elementary results in group theory
• Identify rings and subrings, in their variety

• Number Theory: Historical introduction, integers, divisibility, Euclid’s algorithm, greatest common divisor, least common multiple, primes, fundamental theorem of arithmetic, linear diophantine equations, linear congruences, Chinese remainder theorem, Fermat’s little theorem, Wilson’s theorem.
• Group Theory: Binary operations, Definition and basic properties of a group, examples of group, subgroup, subgroup criterion, order of an element, cyclic group, cosets, Lagrange’s theorem.
• Ring Theory: Definition and basic properties of rings, subrings, ideals, integral domains, fields.

• Lecture demonstration by lecturer and tutorial discussion by instructors.

• In course Assessment           30%
• End of course examination  70%

• Elementary Number Theory, Underwood Dudley, Dover publications, 2nd edition, 2008.
• Elementary Number Theory, David M Burton, Tata-McGraw Hill Edition, 6th edition, 2015.
• Contemporary Abstract Algebra, Joseph A. Gallian, 8th edition, 2012.
• Abstract Algebra, John A. Beachy, William D. Blair, Third Edition, 2006.

• Introduce the concepts of continuity and differentiability of functions
• Enable the students to locate the roots of continuous functions
• Familiarize expansion theorems and their applications

• Define the limit and continuity of a function
• Solve problems concerning limit and continuity
• Apply the sequence characterization of the limit of a function
• Discuss differentiability and the properties of differentiable functions
• Apply Mean Value and Rolle’s Theorems to determine roots and obtain inequalities
• Prove Taylor’s expansion and L’ Hospital rule
• Solve limit problems using L’ Hospital’s rule

• Continuous Functions: Definitions and properties, continuous functions on closed intervals, Intermediate value theorem and extreme value theorem.
• Differentiation and its Simple Properties: Rolle’s theorem, Mean value theorem and applications, Taylor’s theorem, L’ Hospital rule.

• Lectures,  Tutorials, Handouts, Problem solving, Use of e-resources

• In-course Assessments          30%
• End-of-course Examination  70%

• Advanced Calculus with Applications, Dehilo. N. J.
• Calculus: One-variable Calculus with an Introduction to Linear Algebra, Tom M. Apostol, Volume 1, Wiley India Pvt. Limited, 1991.

Hourly Breakdown

• Introduce the principal topics of linear algebra through a set of theorems, illustrative examples and problems
• Develop, in students, the ability to understand and manipulate the objects of linear algebra

• Identify vector spaces and their subspaces
• Identify a basis for given vector space
• Determine the dimension of any finite dimensional vector space
• Prove the Steinitz replacement theorem and its corollaries.
• Establish the theorems regarding the dimension of vector spaces and subspaces
• Identify the linear transformations and their rank and nullity
• Prove the rank- nullity theorem
• Apply rank-nullity theorem in proving certain results
• Prove basic results of matrices regarding the addition and multiplication of matrices
• Prove the row rank is equal to the column rank, for a given matrix
• Identify the rank of given matrix by using elementary operations
• Find the inverse of given matrix by using elementary operations
• Establish the necessary and sufficient condition for the consistency of given linear system
• Find the general solution of a given linear system by using the Gaussian elimination method

Linear transformations: linear transformation, image and kernel of a linear transformation, the rank-nullity theorem, algebra of linear transformations, matrix representation of a linear transformation, isomorphism.

Matrices: matrix addition , matrix multiplication ,elementary row operations, echelon matrices, row reduced echelon matrices, transpose, trace and diagonal of a matrix, special kinds of matrices , row space and column space of a matrix, row rank and column rank of a matrix, elementary matrices, normal form of a matrix, invertible  matrices.

System of linear equations: homogeneous and non-homogeneous systems, conditions for consistency, Gaussian elimination method.

•  Lectures,  Tutorials, Handouts, Problem solving, Use of e-resources

• In-course Assessments 30%
• End-of-course Examination 70%

• Stephen.H.Friedberg et al., Linear Algebra, Prentice Hall, 2003.
• T.S.Blyth, E.F.Robertson,Basic Linear Algebra, Springer, 2005.
• S.Axler, Linear Algebra done right, Springer, 1997.
• Kenneth Hoffman, Ray Kunze, Linear Algebra, Prentice Hall, 1999.
• Serge Lang, Linear Algebra, Springer, 1987.
• P.B.Bhattacharya et al., First course in Linear Algebra, Wiley, 1991.

Hourly Breakdown

• Introduce the concepts of limits, continuity and differentiability of functions of several variables
• Enable the students to solve the extreme value problems

• Define the limit and continuity of functions of several variables
• Justify limits and continuity from the first principles
• Apply the sequence characterization of continuity of a function
• Define the partial derivatives, directional derivatives and total derivative
• Discuss the continuity and differentiability
• Apply the chain rule to find the derivative of composite function of several variables
• Prove Mean value theorem and Taylor’s theorem for functions of several variables
• Recall the necessary conditions for minimization problems
• Apply the second derivative test to find local extremum
• Solve extreme value problems with constraints.

Differentiation of functions of several variables: Partial derivatives, Directional derivatives, Total derivative, Elementary properties of derivatives, Relation between directional derivative and total derivative, Derivative matrix, Jacobian, Chain rule and applications, Mean value theorem, Partial derivatives and differentiability, Power series, Radius of convergence, Taylor’s expansion and applications

Extremum Problems: Local maximum and minimum, Global maximum and minimum, Hessian matrix, Second derivative test for local extremum, Extreme problems with constraints.

• Lectures, Tutorial discussions, Handouts, Self-learning guides and E-resources

• In-course assessment           30%
• End of course Examination 70%

• Tom M. Apostol, Mathematical Analysis, Addison Wesley Publishing Company, 1981.
• Bartle R.G. and Sherbet. D.R., Introduction to Real Analysis, John Willy and Sons, 2000.
• Gerald B. Folland, Advanced Calculus, Pearson, 2002.
• Wilfred Kaplan, Advanced Calculus, Addison Wesley Publishing Company, 2003.

Hourly Breakdown

• Introduce the fundamental concepts of metric space
• Provide a clear notion of concepts of Riemann Integral for functions of real variable
• Acquaint with uniform convergence and theorems regarding the preservation of continuity, differentiability and integrability
• Enable the students to perform the integration of singular functions on unbounded domains

• Explain the fundamental concepts of metric space
• Discuss convergence, completeness and continuity of functions in metric space
• Recall interior point, closure points and frontier points and their relationships
• Illustrate the convergence and continuity properties including uniform and Lipschitzcontinuity
• Determine the Riemann Integrability
• Prove a basic theorems concerning Riemann integration
• Compare the difference between pointwise and uniform convergence of sequences and series of functions
• Apply the tests for uniform convergence
• Elaborate the effect of uniform convergence with respect to continuity, differen-tiability and integrability
• Evaluate the integral of singular functions on unbounded domains
• Apply the tests for convergence of improper integrals

Riemann integration: Riemann Sums, Riemann Integrals, Criterion of Integrability, Properties of Riemann Integral, Indefinite Integral, Fundamental Theorem of Calculus and Mean Value Theorems.

Uniform convergence of sequences and series of functions: Pointwise and Uniform Convergence of sequence of functions, Cauchy Criterion for Uniform convergence, -Test, Uniform Convergence and Continuity, Uniform Convergence of series of functions, Uniform Convergence and Integration, Uniform Convergence and Differentiation, Weierstrass M-Test, Dirichlet’s Test, and Abel’s Test.

Improper Integrals: Three kinds of Improper Integrals, Evaluation of Integrals, Absolute and Conditionally Convergence, Tests for Absolutely Convergence, Cauchy Criterion for Integrals, Dirichlet’s Test,-Test, Abel’s Test and Uniform Convergence and Integrals.

• Lectures,  Tutorials, Handouts, Problem solving, Use of e-resources, Group Discussion.

• In-course Assessments                 30%
• End-of-course Examination         70%

• O’Searcoid, Mícheál, Metric Spaces, Springer Undergraduate Mathematics Series, 2006.
• Tom M. Apostol, Calculus Volume-I, John Wiley & Sons, Inc., 1967.
• Tom M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.

Hourly Breakdown

• Introduce some additional topics of linear algebra, namely determinants and diagonalization
• Acquaint students with the basic concepts of analytic geometry
• Illustrate the underlying applications of linear algebra techniques in the analytic geometry

• Prove various properties of determinants
• Find determinants using Laplace’s expansion
• Establishvarious properties of theadjoint of given matrix
• Find eigenvalues and the associated eigenvectors of given matrix or linear transformation
• Determine the diagonalizability of given matrix or linear transformation
• Prove necessary and sufficient condition for diagonalizability of matrices and   linear transformations
• Prove the Cayley-Hamilton theorem
• Apply the Cayley -Hamilton theorem to compute the inverse of given matrix
• Compute minimum polynomial of given matrix or linear transformation to determine itsdiagonalizability
• Recall the metric properties of plane
• Recall the different forms of straight lines
• Classify the given second degree equation in the two dimensional coordinate system
• Solve problems in system of circles
• Define the types of conic sections
• Classify the types of conic sections
• Solve geometric problems concerning parabola, ellipse and hyperbola

Diagonalizability: eigenvalues and eigenvectors, characteristic polynomial, geometric and algebraic multiplicities of eigenvalues, test for diagonalizability, the Cayley-Hamilton theorem, minimum polynomial, quadratic forms and change of variables.

Analytic geometry in two dimension: transformation of coordinates, pair of straight lines, circle, conic sections; parabola, ellipse, hyperbola, tangents, normals, pole and polar, classification of conics.

• Lectures,  Tutorials, Handouts, Problem solving, Use of e-resources

• In-course Assessments         30%
• End-of-course Examination 70%

• Stephen.H.Friedberg et al., Linear Algebra, Prentice Hall, 2003.
• T.S.Blyth, E.F.Robertson, Basic Linear Algebra, Springer, 2005.
• Chatterjee.D,Analytical geometry of two dimension,Narosa Publishers, 2008.
• A.N.Das, Analytical geometry of two and three dimensions, New Central Book Agency, 2009.
• Roger Fenn, Geometry, Springer, 2000.

Hourly Breakdown

Theory

Practical

Independent Learning

45

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105

• Extend and deepen the understanding of algebraic structures: groups, rings and fields.

• Define basic notions in group theory
• Prove the subgroup criterion
• Apply subgroup criterion to determine subgroups and to derive other results
• Recall properties of normal subgroups and the factor groups
• Prove various properties of group homomorphisms and the classical isomorphism theorems
• Discuss various results regarding special subgroups: center, centralizer, normalizer,    commmutator subgroup
• Identify explicitly the symmetric groups and alternating groups of small order, and their subgroups, conjugacy classes
• Define fundamental notions in the theory of rings
• Prove elementary properties of rings, subrings, and ideals
• Discuss various results regarding the operations on ideals
• Prove various properties of ring homomorphisms
• Prove division algorithm for polynomials, and its consequences
• Verify irreducibility of given polynomial using various tests
• Define basic notions in the theory of fields
• Prove the fundamental theorem of field theory
• Prove various results regarding finite and algebraic extensions
• Find the basis and degree of given field extension
• Discuss the impossibility of finding a general construction for trisecting an angle, duplicating a cube or squaring a circle

• Groups: Definition and examples, subgroups, subgroup generated by a set, cosets , cyclic group, normal subgroup, factor groups, homomorphism, the isomorphism theorems, the correspondence theorem, center of a group, centralizer , normalizer, commutator subgroup, conjugacy class, permutation group, symmetric group, alternating group
• Rings:  Definition and examples, integral domain, characteristic of a ring, subring , ideals, operations on  ideals , factor ring , ring homomorphisms , polynomial ring, minimal polynomial, division algorithm, irreducible and reducible polynomials, tests for irreducibility.
• Fields: Definition and examples, field extensions, algebraic and transcendental elements. The fundamental theorem of field theory (The Kronecker’s theorem), finite and algebraic extensions, geometric constructions.

• Lectures,  Tutorials, Handouts, Problem solving,  e-resources

• In-course Assessments 30%
• End-of-course Examination70%

• Joseph. A. Gallian ,  Contemporary Abstract Algebra, Cengage, 2012.
•  David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley, 2006.
• John A Beachy , William. D. Blair, Abstract Algebra, Waveland, 2006.
• Joseph Rotman, A First Course In Abstract Algebra, Pearson, 2005.

Theory

Practical

Independent Learning

45

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105

• Provide the fundamental concepts of complex variable theory.
• Introduce the skill of contour integration to evaluate complicated integrals via Cauchy’s integral theorem or residue calculus.

• Express complex numbers algebraically and geometrically
• Explain the convergence properties of a power series
• Find the Radius of Convergence of power series
• Apply the concept and consequences of analyticity and the Cauchy-Riemann equations
• Find the harmonic conjugate to a harmonic function
• Evaluate complex contour integrals using Cauchy integral theorem and the Cauchy integral formula
• Calculate Laurent series for analytic functions with the classification of singularities
• Find the number of zeroes and poles using the argument principle andRouche’s theorem
• Compute the residue of a function
• Utilize the residue theorem to evaluate a contour integral or an integral over the real line

Algebraic Aspects: Definition of Complex Numbers, Cartesian form and polar form, Euler’s formulae, de Moivre’s theorem, Relation between trigonometric functions and hyperbolic functions, Complex Logarithms.

Geometric Aspects: The Argand diagram, representation, locus, straight lines in terms of complex numbers, circles in terms of complex numbers, joint family of circles and straight lines, two families of circles, representation of complex numbers by points of a sphere

Complex Power Series:

Radius of Convergence, Properties of Power Series,Hadamard’s formula

Differentiation:

Function of a complexvariable, Limits, Continuity, Derivatives, Cauchy Riemann equations, Sufficientconditions, Analytic functions: Entire functions, Harmonic functions, and Harmonic conjugate.

Contour Integrals :

Smooth and piecewise smooth paths, Integration of continuous functions along smooth contours, Estimation Lemma, Fundamental theorem of Calculus, Cauchy’s theorem, Cauchy -Goursat’s theorem, Deformation theorem, Cauchy’s integral formula, Morera’s theorem, Cauchy’s estimates, Liouville’s theorem, fundamental theorem of algebra, Taylor’s expansion, Laurent’s expansion.

Residue Theorem:

Zeros and singularities of complex variable functions, Residue theorem, Evaluation of the certain real integrals, Argument principle, Rouche’s theorem, Application of the definite integrals.

• Lectures,  Tutorials, Handouts, Problem solving,  e-resources

• In-course Assessments 30%
•  End-of-course Examination 70%

• James.W.B and Rel.V.C., Complex Variables and Applications, 8^thedition, McGraw-Hill, 2009
• Ian Stewart and David Tall, Complex Analysis, 2^nd edition, Cambridge University Press, 2018
• Shanti. N. and Mittal. P.K .,Theory of functions of a complex variable, 2005
• Saff.E.B. and Snider.A.D., Fundamentals of Complex Analysis with  Applications to Engineering, Science and Mathematics, 3^rd edition, Prentice Hall, 2003

Theory

Practical

Independent Learning

45

–

105

• Develop an understanding of the core ideas and concepts of fundamental and advanced counting techniques.
• Recognize the power of abstraction and generalization, and apply logical reasoning to investigate some combinatorial problems.
• Provide a clear notion of concepts in graphs and solving methods for practical problems using graph theory

• Solve common counting problems using elementary counting techniques involving the multiplication rule, permutations, and combinations.
• Apply the Principle of inclusion and exclusion to solve variety of counting problems.
•  Use the recurrence relations to model a wide variety of counting problems and solve them.
• Apply generating functions to many types of counting problems subject to variety of constraints, and solve recurrence relations.
•  Identify which type of counting technique is the best for given counting problem.
•  Investigate important classes of graph theoretic problems.
• Formulate and prove central theorems about trees, matching, connectivity, planar graph, and coloring.
•  Utilize graph theory as a modeling tool.

• Basic Counting Techniques: Fundamental principles of counting: Sum, product, subtraction, and division rules. permutations and  combinations, the Binomial Theorem, Pascal’s identity and Triangle, Vandermonde’s Identity, permutations and combinations with limited and unlimited repetition, multinomial theorem and its applications, generating permutations and combinations. The Pigeonhole Principle and some of its interesting applications.
• Advanced Counting Techniques:
  • Inclusion-Exclusion: The principle of inclusion-exclusion, an alternative form of inclusion-exclusion, the Sieve of Eratosthenes, derangements
  • Recurrence Relations: Modeling with recurrence relations, solving linear homogeneous recurrence relations with constant coefficients, linear non homogeneous recurrence relations with constant coefficients, divide-and-conquer algorithms and recurrence relations.
  • Generating Functions: Extended binomial theorem, counting problems and generating Functions, using generating functions to solve recurrence relations. Exponential generating functions and its applications on solving some permutation problems.
• Graph Theory: Graphs models, graph terminology and special types of graphs, representing graphs, isomorphism of graphs, paths, connectedness in directed and undirected graphs, Euler paths and circuits, Hamilton paths and circuits, shortest path problem, traveling salesman problem.
• Planar graphs: Planar graphs, Euler’s Formula, graph coloring, the four color theorem, and applications of graph coloring.
• Trees: Properties of Trees, Applications of Trees, Binary Search Trees, Decision Trees, spanning tree, Depth-First Search, Breadth-First Search, backtracking, minimum spanning trees.

• Lectures, demonstration, tutorial discussion, handouts, problem solving and e-resources

• In course Assessments 30%

• End of course Examination  70%

• Oscar Levin , Discrete Mathematics – An Open Introduction, 3rd Edition, 2019
• Kenneth H. Rosen, Discrete Mathematics and its Applications, 7^th edition, McGraw Hill, 2012.
• Bernard Kolman, Robert C. Busby, S. C. Ross, Discrete Mathematical Structures, 4^th edition, Prentice Hall, 2001
• Grimaldi R.P, Discrete and combinatorial mathematics: an applied introduction, 5^th edition, Pearson education Inc, 2004
• Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill (2007).

• Introduce advanced concepts of Euclidean geometry
• Introduce the fundamentals of solid geometry
• Provide profound knowledge in conic sections
• Inroduce the concepts of space curves in differential geometry

• Recall nine point circle Theorem and Napoleon’s Theorem
• Discuss the configuration of lines and planes
• Determine the angle and shortest distance between two straight lines
• Explain the condition for common line of intersection of planes
• Determine the volume of a tetrahedron formed by four planes
• Discuss the types of conic sections and their properties
• Apply various techniques to derive the equations of tangent and normal of conic sections
• Outline the special properties of conic sections
• Discuss the properties of conjugate diameters of ellipse and hyperbola
• Define the arc length, tangent, normal, binormal, curvature and torsion for a curve in a space
• Construct the vector equation of  the normal plane
• Explain the three fundamental planes

• Euclidean Geometry – Euclidean axioms for the plane, Angles and lines, Triangles, General polygons, Congruences and similarities, Isosceles triangles, Circles, Metric properties of triangles
• Solid Geometry – Points and coordinates, Scalar product, Cross product, Planes, Lines in space, Isometries of space, Projections, Polyhedra, Distance between two points, Projection of a segment, Direction cosines, Direction ratios, Angle between two straight lines, Equation of a plane through three given points, Normal form of a plane, Angle between two planes, Equation of bisectors of the angle between two planes, Plane through the line of intersection of two planes, Symmetrical form, Distance of a point from a line, Expression of line of intersection of two planes in symmetrical form, Intersection of a straight line and a plane, Skew lines and the shortest distance between them, Equations of two skew lines, A straight line intersecting other given lines, Area of a triangle and volume of a tetrahedron, Tetrahedron formed by four given planes
• Analytical Geometry
  • Transformations: Transformation of coordinates, Rotation of Axes, Invariants under orthogonal transformation
  • Conic sections: Definition of conic, General equations of parabola, ellipse and hyperbola, Equation of tangent and normal, Parametric equations, Auxiliary circles and eccentric angles, Conjugate hyperbola, Diameters of conics, conjugate diameters and principal diameters, Properties of diameters of a conic, Properties of conjugate diameters of ellipse and hyperbola, Equi-conjugate diameters.
• Differential Geometry – Curves in plane and space, Arc length, Tangent to a curve, Vector equation of  the normal plane, Osculating plane, Curvature, binormal and torson of a space curve, Frenet equations, Three fundamental planes

• Lectures demonstration,  tutorial discussion, handouts and e-resources

• In-course assessment                                              30%
• End of course Examination                                    70%

• Roger Fenn, Geometry, Springer, 2009.
• A.N.Das, Analytical Geometry of Two and Three Dimensions, New Central Book Agency (P) Ltd., 2009.
• P.K.Jain, A Textbook of Analytical Geometry Of Three DimensionsNew Age Internationals, 2005
• Maxwell, E.A., Elementary Coordinate Geometry, Oxford University press ,1962.
• Bansi Lal, Three Dimensional Differential Geometry, Atmaram, 1969.