The Department of Mathematics offers Undergraduate course – B.sc Mathematics, a six-semester programme from 2009 onwards. The department also provides Mathematics complementary subject for B.sc Physics, B.sc Chemistry, and B.sc Computer science programmes. The UG course has been organized by the instruction of the CCSS(Choice Based Credit Semester System) 2009 by Calicut University. The programmes (programme means course) and courses (course means paper) opted by the department under the Calicut university CCSS in 6 semesters (3 years) are:
1.CORE COURSES (CODE B)
Mathematics B.sc degree with Statistics and Computer Applications as complementary subjects with 120 credits-3200 marks in 6 semesters.
2.COMPLEMENTARY COURSE (CODE C)
Complementary courses for B.sc Physics, B.sc Chemistry and B.sc Computer Science.
3. OPEN COURSE (CODE D)
Open course ”Mathematics for Natural Sciences“ for 5th semester students of the other departments.
Course outcome – 2014 ONWARDS
1 MTS1B01 BASIC LOGIC AND NUMBER THEORY
On Completion of this course the students will be able to:
Prove results involving divisibility, greatest common divisor, least common multiple and a few applications.
Understand the theory and method of solutions of linear Diophantine equations
Understand the theory of congruence and a few applications. Solve linear congruent equations.
Learn three classical theorems viz. Wilson’s theorem, Fermat’s little theorem and Euler’s theorem and a few important consequences
MAT2B02: CALCULUS
On successful completion of the course, Students will be able to:
Learn The mean value theorem, The first derivative test for local
Extremum values, Graphing with y’ and y” , Limit as x Assymptotes and dominant terms
and the mean value theorem, The fundamental theorem, Substitution in Definite Integrals.
Areas between curves, Finding volumes by slicing, Volumes of solids of revolution (Disk method
. • Learn easier & systematic way to find, Lengths of plane curves, Areas of surface of revolution Moments and centres of mass,
MAT3B03: CALCULUS AND ANALYTIC GEOMETRY
After completing the course students are expected to be able to:
To able to calculate limits in inderminate forms by a repeated use of L’ Hospital rule and to Determine Hyperbolic functions
Understand the basic principles Infinite Series , Integral test for series ,series of non negative terms, Alternating series, Absolute and conditional convergence.
Describe fundamental properties of Power series, Taylor and Maclaurin’s and Convergence
Explain the fundamental concepts of Conic section and quadratic equations,Parametrisation ,Polar coordinates, and Graphing
MAT4B04: THEORY OF EQUATIONS, MATRICES AND VECTOR CALCULUS
On successful completion of the course, students will be able to:
learn about Polynomial Equations and Fundamental Theorem of Algebra Relations between roots and co‐efficients of a polynomial equation and computation of symmetric functions of roots.
Methods for solving Reciprocal equation and method of finding its roots.Analytical methods for solving polynomial equations of order up to four ‐Cardano’s method Ferrari’s method .Descartes’ rule of signs.
Students will understand the fundamentals of Rank of a matrix – reduction to normal form, and echelon form.
Computing the inverse of a non singular matrix using elementary row transformation.
Solve a homogeneous linear system by the eigenvalue method of a matrix, andCayley‐Hamilton theorem.
Students Understand the basic principles Lines and planes in space.
MAT5B05: VECTOR CALCULUS
After studying this course,student should be able to
To know the Functions of several variables ,Limits and Continuity , Partial derivatives , Differentiability Give the definition of concepts related to linearization and differentials,Chain rule, Partial derivatives with constrained variables
Give the difination of concepts related to Multivariable functions and Partial Derivatives Directional derivatives, gradient vectors and tangent planes , and saddle points,
Explain the concept of Lagrange multipliers , Taylor’s formula, Double Integrals , Double integrals Triple integrals in Rectangular Coordinates , in cylindrical and spherical coordinates, Students will be able to demonstrate basic knowledge of key topics in
Vector calculus motivates the study of vector differentiation and integration.Path independence, potential functions and conservative fields
Give the essence of the proof of Green’s theorem , Stokes’
Theorem (statement only) and Divergence theorem in the plane ,
Apply problem-solving of Surface area and surface integrals,
MAT5B06 : ABSTRACT ALGEBRA
After studying this course, student should be able to
emphasizes the concept of Binary operations; Isomorphic binary structures and Groups
Explain the concept of Cyclic groups; Groups and permutations; Cosets and Theorem of Lagrange; Homomorphism Explain the fundamental concepts of advanced algebra – Rings and Fields,Integral Domains and The Field of Quotients of an Integral Domain
MAT5B07 : BASIC MATHEMATICAL ANALYSIS
On successful completion of the course, students will be able to:
A quick review of sets and functions ,Mathematical induction ,Finite and infinite sets Real Numbers ,The algebraic property of real numbers Describe fundamental properties of the Absolute value and real line ,The completeness property of R ,Applications of supremum property Intervals, Nested interval property and uncountability of R Sequence of l numbers,
Give the definition of concepts Sequence and their limits, Limit theorems, Monotone sequences Subsequence Read analyze and write logical arguments to Bolzano – Weirstrass theorem, Cauchy criterio,Properly divergent sequences. Explain the concept of Complex Numbers Complex conjugates; Exponential form; Product and powers in exponential form; Arguments of products and quotients; Roots of complex numbers; Regions in the complex plane. Students will be able to understand the Functions of complex variable, Limits, Theorems on limits, Limits involving the points at infinity and Continuity
)MAT5B08: DIFFERENTIAL EQUATIONS:
After studying this course
The main aim is to introduce the students to the technique of solving various problems of engineering and science Distinguish between linear, nonlinear, partial and ordinary differential equations. The existence and uniqueness theorem (proof omitted)
Solve basic application problems described by second order linear differential equations with constant coefficients. Obtain an approximate set of solution function values to a second order boundary value problem using Linear independence and Wronskian, Complex roots of characteristic equations, Repeated roots; Reduction of order, Non homogeneous equations; Method Undetermined coefficients,
Solve a homogeneous linear system
Obtain an approximate set of solution function values to a second order. boundary value problem Variation of parameters, Mechanical and Electrical vibrations Separation of variables; Heat conduction in a rod , The Wave equation:Vibrations of an elastic string
Find Laplace Transforms Definition of Laplace Transforms , Solution of Initial Value Problem , Step functions , Impulse functions, The Convolution Integral) Partial Differential Equations and Fourier Series Two point Boundary value problems , Fourier Series, The Fourier Convergence Theorem
(OPEN COURSE) MAT5D02: MATHEMATICS FOR NATURAL SCIENCES
students not having Mathematics as Core Course will be able to:
Explain the relationship between The idea of sets.Operations,Relations and functions, Variables and graphs To know more about the Frequency distributions ,The Mean, Median, Mode and other measures of central tendency To able to evaluate problems on Dispersion ,The Range, The Mean deviation, The Semi‐inter quartile range, Then 10‐90 Percentile range. To inculcate to solve The standard deviation problems. Properties of standard deviation. The variance. Short methods of computing standard deviation Appreciate how abstract ideas and regions methods in Moments, Moments for grouped data, relation between moments, Computation of moment for grouped data. Skewness and Kurtosis.
MAT6B09: REAL ANALYSIS
After studying this course the students supposed to
Describe fundamental properties of the real numbers that lead to the formal development of Continuous functions on intervals , Uniform continuity Appreciate how abstract ideas in Riemann Integral Riemann Integrable Functions , The fundamental theorem Substitution theorem and application, Approximate Integration Demonstrate an understanding of Sequence and series of functions Pointwise and uniform convergence, Interchange of limit and continuity, Series of functions . Comprehend regions arguments developing the Improper Integrals ,integrals of the first kind, Improper integrals of the second kind,Cauchy Principal value,Improper Integrals of the third kind. Appreciate how abstract ideas and regions methods in Beta and Gamma functions Relation between Beta and Gamma Functions
MAT6B10 : COMPLEX ANALYSIS
After studying this course the students will able to
Explain Analytic Functions ,Cauchy‐Riemann Equations, Polar coordinates, , Harmonic functions) Demonstrate an understanding of Elementary functions ,The exponential function, Logarithmic function, Complex exponents,Trigonometric functions, Hyperbolic functions, Inverse Trigonometric and Hyperbolic functions. Compare the differences between Derivatives of functions ω(t); Indefinite integral of ω(t); Contours, Contour integrals, Anti derivatives, Cauchy‐Goursat theorem (without proof), Simply and multiply connected domains, Cauchy’s integral formula and its extension, Liouville’s theorem and fundamental theorem of algebra, Maximum modulus principle. A quick review of convergence of sequence and series of complex numbers.Taylor series, Laurents series (without proof), Applications. Power series: Absolute and uniform convergence. Continuity of sum of power series, Differentiation and integration of power series, Multiplication and division power series. Isolated singular points, Residues, Cauchy’s residue theorem, Residue at infinity,Three types of isolated singular points, Residues at poles, Zeroes of analytic functions, Zeroes and poles. Applications of residues, Evaluation of improper integrals, Jordan’s Lemma (statement only),
MAT6B11 : NUMERICAL METHODS
After studying this course the students will able to
Solve an algebraic or transcendental equation using an appropriate numerical method Solution of Algebraic and Transcendental Equation
Solve a linear system of equations using Interpolation
Calculate a definite integral using an appropriate numerical method Numerical Differentiation and Integration Develop a working knowledge of concepts and methods related to Matrices and Linear Systems of equations ,Gauss elimination Gauss‐Jordan Method, the inverse LU Decomposition, Decomposition Solution of Linear Systems – Iterative methods, The eigen value problem , Eigen values of Symmetric Tridiazonal matrix Develop a working knowledge Numerical Solutions of Ordinary Differential Equations Taylor’s series , Picard’s method of successive approximations ,Euler’s method Modified Runge‐Kutta method Predictor‐Corrector Methods, Adams‐Moulton Method, Milne’s method
MAT6B12 : NUMBER THEORY AND LINEAR ALGEBRA
Elementary Number Theory is the study of the basic structure and properties of integers. Learning Number Theory helps improving one’s ability of mathematical thinking. Successful completion of this course will enable you to: Prove results involving Divisibility theory in the integers – the division algorithm, the greatest common divisor, the Euclidean algorithm,
Find integral solutions to specified Diophantine equation ax + by = c. Primes and their distribution. The fundamental theorem of arithmetic.The sieve of Eratosthenes.
Solve systems of linear congruences The theory of congruences.Basic properties of congruence. Chinese remainder theorem. Apply Euler-Fermat’s Theorem to prove relations involving prime numbers; Fermat’s little theorem and pseudo primes
Apply the Wilson’s theorem. Euler’s generalization of Fermat’s theorem.Properties of the phi‐function.Explain the fundamental concepts of advanced Vector spaces‐
‐ Linear transformations,
(ELECTIVE COURSE)
MAT6B13(E02) : LINEAR PROGRAMMING
Upon successful completion of this course, the student will be able to;
Linear programming is used for defence capability acquisition decision making.
It is used to find optimal or near optimal solutions to complex decision making problems. •
It is used in finding maximum (of profit or yield) in real-world objective. •
It is used in finding minimum (of loss or cost) in real-world objective
Formulation, Convex sets, General LLP Simplex Method, Duality
The transportation problem, The assignment Problems
It is used in data envelopment. and has strong ties to computer science and analytics
Course outcome – 2019 onwards
MTS1B01 Basic logic and Number theory
Logic, the study of principles of techniques and reasoning, is fundamental to every branch of learning. Besides, being the basis of all mathematical reasoning, it is required in the field of computer science for developing programming languages and also to check the correctness of the programmes. Electronic engineers apply logic in the design of computer chips. The first module discusses the fundamentals of logic, its symbols and rules. This enables one to think systematically,to express ideas in precise and concise mathematical terms and also to make valid arguments. How to use logic to arrive at the correct conclusion in the midst of confusing and contradictory statements is also illustrated.The classical number theory is introduced and some of the very fundamental results are discussed in other modules. It is hoped that the method of writing a formal proof, using proof methods discussed in the first module, is best taught in a concrete setting, rather than as an abstract exercise in logic. Number theory, unlike other topics such as geometry and analysis, doesn’t suffer from too much abstraction and the consequent difficulty in conceptual understanding. Hence, it is an ideal topic for a beginner to illustrate how mathematicians do their normal business. By the end of the course, the students will be able to enjoy and master several techniques of problem solving such as recursion, induction etc., the importance of pattern recognition in mathematics, the art of conjecturing and a few applications of number theory. Enthusiastic students will have acquired knowledge to read and enjoy on their own a few applications of number theory in the field of art, geometry and coding theory. Successful completion of the course enables students to
Prove results involving divisibility, greatest common divisor, least common multiple and a few applications.
Understand the theory and method of solutions of LDE.
Understand the theory of congruence and a few applications.
Solve linear congruent equations.
Learn three classical theorems viz. Wilson’s theorem, Fermat’s
MTS2B02 Calculus of single variable 1
The mathematics required for viewing and analyzing the physical world around us is contained in calculus. While Algebra and Geometry provide us very useful tools for expressing the relationship between static quantities, the concepts necessary to explore the relationship between moving/changing objects are provided in calculus. The objective of the course is to introduce students to the fundamental ideas of limit, continuity and differentiability and also to some basic theorems of differential calculus. It is also shown how these ideas can be applied in the problem of sketching of curves and in the solution of some optimization problems of interest in real life. This is done in the first two modules.
The next two modules deal with the other branch of calculus viz. integral calculus. Historically, it is motivated by the geometric problem of finding out the area of a planar region. The idea of definite integral is defined with the notion of limit. A major result is the Fundamental Theorem of Calculus, which not only gives a practical way of evaluating the definite integral but establishes the close connection between the two branches of Calculus. The notion of definite integral not only solves the area problem but is useful in finding out the arc length of a plane curve, volume and surface areas of solids and so on. The integral turns out to be a powerful tool in solving problems in physics, chemistry, biology, engineering, economics and other fields. Some of the applications are included in the syllabus.
MTS3B03 Calculus of single variable 2
Using the idea of definite integral developed in previous semester, the natural logarithm function is defined and its properties are examined. This allows us to define its inverse function namely the natural exponential function and also the general exponential function. Exponential functions model a wide variety of phenomenon of interest in science, engineering, mathematics and economics. They arise naturally when we model the growth of a biological population, the spread of a disease, the radioactive decay of atoms, and the study of heat transfer problems and so on. We also consider certain combinations of exponential functions namely hyperbolic functions that also arise very frequently in applications such as the study of shapes of cables hanging under their own weight. After this, the students are introduced to the idea of improper integrals, their convergence and evaluation. This enables to study a related notion of convergence of a series, which is practically done by applying several different tests such as integral test, comparison test and so on. As a special case, a study on power series- their region of convergence, differentiation and integration etc.,- is also done. A detailed study of plane and space curves is then taken up. The students get the idea of parametrization of curves, they learn how to calculate the arc length, curvature etc. Using parametrization and also the area of surface of revolution of a parametrized plane curve. Students are introduced into other coordinate systems which often simplify the equation of curves and surfaces and the relationship between various coordinate systems are also taught. This enables them to directly calculate the arc length and surface areas of revolution of a curve whose equation is in polar form. At the end of the course, the students will be able to handle vectors in dealing with the problems involving geometry of lines, curves, planes and surfaces in space and have acquired the ability to sketch curves in plane and space given in vector valued form
MTS4B04 linear algebra
An introductory treatment of linear algebra with an aim to present the fundamentals in the clearest possible way is intended here. Linear algebra is the study of linear systems of equations, vector spaces, and linear transformations. Virtually every area of mathematics relies on or extends the tools of linear algebra. Solving systems of linear equations is a basic tool of many mathematical procedures used for solving problems in science and engineering. A number of methods for solving a system of linear equations are discussed. In this process, the student will become competent to perform matrix algebra and also to calculate the inverse and determinant of a matrix. Another advantage is that the student will come to understand the modern view of a matrix as a linear transformation. The discussion necessitates the introduction of central topic of linear algebra namely the concept of a vector space. The familiarity of the students with planar vectors and their algebraic properties under vector addition and scalar multiplication will make them realize that the idea of a general vector space is in fact an abstraction of what they already know. Several examples and general properties of vector spaces are studied. The idea of a subspace, spanning vectors, basis and dimension are discussed and fundamental results in these areas are explored. This enables the student to understand the relationship among the solutions of a given system of linear equations and some important subspaces associated with the coefficient matrix of the system.After this, some basic matrix transformations in the vector spaces ℝ2 ℝ3 ,having interest in the field of computer graphics, engineering and physics are studied by specially pinpointing to their geometric effect. Just like choosing an appropriate coordinate system greatly simplifies a problem at our hand as we usually see in analytic geometry and calculus, a right choice of the basis of the vector space ℝ greatly simplifies the analysis of a matrix operator on it. With this aim in mind, a study on eigenvalues and eigenvectors of a given matrix (equivalently, that of the corresponding matrix operator) is taken up. Practical method of finding out the eigenvalues from the characteristic equation and the corresponding eigenvectors are also discussed. A bonus point achieved during this process is a test for the invertibility of a square matrix. As diagonal matrices are the matrices with simplest structure, the idea of diagonalization of a matrix (and hence the diagonalization of a matrix operator) is introduced and students learn a few fundamental results involving diagonalization and eigenvalues which enable them to check whether diagonalization is possible. They realise that there
are matrices that cannot be diagonalized and even learn to check it. Also they are taught a well defined procedure for diagonalizing a given matrix, if this is actually the case. The topic is progressed further to obtain the ultimate goal of spectral decomposition of a symmetric matrix. In this process , students realise that every symmetric matrix is diagonalizable and that this diagonalization can be done in a special way ie., by choosing an orthogonal matrix to perform the diagonalization. This is known as orthogonal diagonalization. Students also learn that only symmetric matrices with real entries can be orthogonally diagonalized and using Gram-Schmidt process a well defined procedure for writing such a diagonalization is also taught. In short, the course gives the students an opportunity to learn the fundamentals of linear algebra by capturing the ideas geometrically, by justifying them algebraically and by preparing them to apply it in several different fields such as data communication, computer graphics, modelling etc.
MTS5B05 Theory of equations and Abstract algebra
Theory of equations is an important part of traditional algebra course and it mainly deals with polynomial equations and methods of finding their algebraic solution or solution by radicals. This means we seek a formula for solutions of polynomial equations in terms of coefficients of polynomials that involves only the operations of addition, subtraction, multiplication, division and taking roots. A well knitted formula for the solution of a quadratic polynomial equation is known to us from high school classes and is not difficult to derive. However, there is an increasing difficulty to derive such a formula for polynomial equations of third and fourth degree. One of our tasks in this learning process is to derive formulae for the solutions of third and fourth degree polynomial equations given by Carden and Ferrari respectively. In the mean time, we shall find out the relationship between the roots and coefficients of an ℎ degree polynomial and an upper and lower limit for the roots of such a polynomial. This often help us to locate the region of solutions for a general polynomial equation. Methods to find out integral and rational roots of a general ℎ degree polynomial with rational coefficients are also devised. However, all efforts to find out an algebraic solution for general polynomial equations of degree higher than fourth failed or didn’t work. This was not because one failed to hit upon the right idea, but rather due to the disturbing fact that there was no such formula. Credit goes to the brilliant mathematician Evariste Galois for proving this fact and he developed an entire theory that connected the solvability by radicals of a polynomial equation with the permutation group of its roots. The theory now known as Galois theory solves the famous
problem of insolvability of quintic. A study on symmetric functions now becomes inevitable. One can now observe the connection emerging between classical algebra and modern algebra. The last three modules are therefore devoted to the discussion on basic ideas and results of abstract algebra. Students understand the abstract notion of a group, learn several examples, are taught to check whether an algebraic system forms a group or not and are introduced to some fundamental results of group theory. The idea of structural similarity, the notion of cyclic group, permutation group , various examples and very fundamental results in the areas are also explored.
MTS5BO6 Basic analysis
In this course, basic ideas and methods of real and complex analysis are taught. Real analysis is a theoretical version of single variable calculus. So many familiar concepts of calculus are reintroduced but at a much deeper and more rigorous level than in a calculus course. At the same
time there are concepts and results that are new and not studied in the calculus course but very much needed in more advanced courses. The aim is to provide students with a level of mathematical sophistication that will prepare them for further work in mathematical analysis and other fields of knowledge, and also to develop their ability to analyse and prove statements of mathematics using logical arguments. The course will enable the students
to learn and deduce rigorously many properties of real number system by assuming a few fundamental facts about it as axioms. In particular they will learn to prove Archimedean property, density theorem, existence of a positive square root for positive numbers and so on and the learning will help them to appreciate the beauty of logical arguments and embolden them to apply it in similar and unknown problems.
to know about sequences ,their limits, several basic and important theorems involving sequences and their applications . For example, they will learn how monotone convergence theorem can be used in establishing the divergence of the harmonic series, how it helps in the calculation of square root of positive numbers and how it establishes the existence of the transcendental number e (Euler constant).
to understand some basic topological properties of real number system such as the concept of open and closed sets, their properties, their characterization and so on.
to get a rigorous introduction to algebraic, geometric and topological structures of complex number system, functions of complex variable, their limit and continuity and so on. Rich use of geometry, comparison between real and complex calculus-areas where they agree and where they differ, the study of mapping properties of a few important complex functions exploring the underlying geometry etc. will demystify student’s belief that complex variable theory is incomprehensible.
MTS5BO7 Numerical analysis
The goal of numerical analysis is to provide techniques and algorithms to find approximate numerical solution to problems in several areas of mathematics where it is impossible or hard to find the actual/closed form solution by analytical methods and also to make an error analysis to ascertain the accuracy of the approximate solution. The subject addresses a variety of questions ranging from the approximation of functions and integrals to the approximate solution of algebraic, transcendental, differential and integral equations, with particular emphasis on the stability, accuracy, efficiency and reliability of numerical algorithms. The course enables the students to
Understand several methods such as bisection method, fixed point iteration method, regula falsi method etc. to find out the approximate numerical solutions of algebraic and transcendental equations with desired accuracy.
Understand the concept of interpolation and also learn some well known interpolation techniques.
Understand a few techniques for numerical differentiation and integration and also realize their merits and demerits.
Find out numerical approximations to solutions of initial value problems and also to understand the efficiency of various methods
MTS5BO8 Linear programming
Linear programming problems are having wide applications in mathematics, statistics, computer science, economics, and in many social and managerial sciences. For mathematicians it is a sort of mathematical modelling process, for statisticians and economists it is useful for planning many economic activities such as transport of raw materials and finished products from one place to another with minimum cost and for military heads it is useful for scheduling the training activities and deployment of army personnel. The emphasis of this course is on nurturing the linear programming skills of students via. the algorithmic solution of small-scale problems, both in the
general sense and in the specific applications where these problems naturally occur. On successful completion of this course, the students will be able to
solve linear programming problems geometrically
understand the drawbacks of geometric methods
solve LP problems more effectively using Simplex algorithm via. the use of condensed tableau of A.W. Tucker
convert certain related problems, not directly solvable by simplex method, into a form that can be attacked by simplex method.
understand duality theory, a theory that establishes relationships between linear programming problems of maximization and minimization
understand game theory
solve transportation and assignment problems by algorithms that take advantage of the simpler nature of these problems
MTS5B09 Introduction to Geometry
Geometry is, basically, the study concerned with questions of shape, size, and relative position of planar and spatial objects. The classical Greek geometry, also known as Euclidean geometry after the work of Euclid, was once regarded as one of the highest points of rational thought, contributing to the thinking skills of logic, deductive reasoning and skills in problem solving. In the early 17 th century, the works of Rene Descartes and Pierre de Fermat put the foundation stones for the creation of analytic geometry where the idea of a coordinate system was introduced to simplify the treatment of geometry and to solve a wide variety of geometric problems. Desargues, a contemporary of Descartes was fascinated towards the efforts of artists/painters to give a realistic view of their art works/paintings usually done on a flat surface such as canvas or paper. To get a realistic view of a three dimensional object/scene depicted on a flat
surface, a right impression of height, width, depth and position in relation to each other of the objects in the scene is required. This idea is called perspective in art. If two artists make perspective drawings of the same object, their drawings shall not be identical but there shall be certain properties of these drawings that remain the same or that remain invariant. The study of such invariant things crystallised into what is now called projective geometry. Now days, it plays a major role in computer graphics and in the design of camera models. Another development is the evolution of affine geometry. In simple terms, if we look at the shadows of a rectangular window on the floor under sunlight, we could see the shadows not in perfect rectangular form but often in the shape of a parallelogram. The size of shadows also changes with respect to the position of the sun. Hence, neither length nor angle is invariant in the transformation process. However, the opposite sides of the images are always parallel. So this transformation keeps parallelism intact. The investigation of invariants of all shadows is the basic problem of affine geometry. Towards the end of nineteenth century, there were several different geometries: Euclidean, affine, projective, inversive, spherical, hyperbolic, and elliptic to name a few. It was the idea of Felix Klein to bring the study of all these different geometries into a single platform. He viewed each geometry as a space together with a group of transformations of that space and regarded those
properties of figures left unaltered by the group as geometrical properties. In this course, it is intended to take up a study of a few geometries based on the philosophy of Klein. Upon successful completion of the course, students will be able to
Understand several basic facts about parabola, hyperbola and ellipse (conics) such as their equation in standard form, focal length properties, and reflection properties, their tangents and normal. Recognise and classify conics. Understand Kleinian view of Euclidean geometry. Understand affine transformations, the inherent group structure, the idea of parallel projections and the basic properties of parallel projections.
Understand the fundamental theorem of affine geometry, its use in the proof of Median theorem, Ceva’s theorem, Menelaus’ theorem etc. Understand which conics are affine-congruent to each other Realise the basic difference in identifying two geometric objects in Euclidean and affine geometries. Understand Kleinian view of projective geometry
Understand the idea of homogeneous coordinate of a point in projective plane and write down the equation of a line in projective plane passing through two homogeneous coordinate Know collinearity property and incidence property in projective plane.
Check whether a transformation is indeed projective and also to find the composite and inverse of projective transformations. Identify some projective properties
Write down the projective transformation that maps a given set of four points to another set of four points. Appreciate the advantage of interpreting a Euclidean theorem as a projective theorem by learning a simpler proof for Desargues and Pappu’s theorem.
Understand the concept of cross ratio and calculate it
Find an application of cross ratio in the context of aerial photography.
MTS6B010 Real Analysis
The course is built upon the foundation laid in Basic Analysis course of fifth
semester. The course thoroughly exposes one to the rigour and methods of an analysis course. One has to understand definitions and theorems of text and study examples well to acquire skills in various problem solving techniques. The course will teach one how to combine different definitions, theorems and techniques to solve problems one has never seen before. One shall acquire ability to realise when and how to apply a particular theorem and how to avoid common errors and pitfalls. The course will prepare students to formulate and present the ideas of mathematics and to communicate them elegantly. On successful completion of the course, students will be able to State the definition of continuous functions, formulate sequential criteria for continuity and prove or disprove continuity of functions using this criteria. Understand several deep and fundamental results of continuous functions on intervals such as boundedness theorem, maximum-minimum theorem, intermediate value theorem, preservation of interval theorem and so on. Realise the difference between continuity and uniform continuity and equivalence of these ideas for functions on closed and bounded interval. Understand the significance of uniform continuity in continuous extension theorem. Develop the notion of Riemann integrability of a function using the idea of tagged partitions and calculate the integral value of some simple functions using the definition. Understand a few basic and fundamental results of integration theory. Formulate Cauchy criteria for integrability and a few applications of it. In particular they learn to use Cauchy criteria in proving the non integrability of certain functions. Understand classes of functions that are always integrable
Understand two forms of fundamental theorem of calculus and their significance in the practical problem of evaluation of an integral.
Find a justification for ‘change of variable formula’ used in the practical problem of evaluation of an integral. Prove convergence and divergence of sequences of functions and series Understand the difference between pointwise and uniform convergence of sequences and series of functions Answer a few questions related to interchange of limits.
Learn and find out examples/counter examples to prove or disprove the validity of several mathematical statements that arise naturally in the process/context of learning.
Understand the notion of improper integrals, their convergence, principal value and evaluation. Learn the properties of and relationship among two important improper integrals namely beta and gamma functions that frequently appear in mathematics, statistics, science and engineering.
MTS6B011 Complex Analysis
The course is aimed to provide a thorough understanding of complex function theory. It is intended to develop the topics in a fashion analogous to the calculus of real functions. At the same time differences in both theories are clearly emphasised. When real numbers are replaced by complex numbers in the definition of derivative of a function, the resulting complex differentiable functions (more precisely analytic functions) turn out to have many remarkable properties not possessed by their real analogues. These functions have numerous applications in several areas of mathematics such as differential equations, number theory etc. and also in science and engineering. The focus of the course is on the study of analytic functions and their basic behaviour with respect tothe theory of complex calculus. The course enables students
to understand the difference between differentiability and analyticity of a complex function and construct examples.
to understand necessary and sufficient condition for checking analyticity.
to know of harmonic functions and their connection with analytic functions
to know a few elementary analytic functions of complex analysis and their properties.
to understand definition of complex integral, its properties and evaluation.
to know a few fundamental results on contour integration theory such as Cauchy’s theorem, Cauchy-Goursat theorem and their applications.
to understand and apply Cauchy’s integral formula and a few consequences of it such as Liouville’s theorem, Morera’s theorem and so forth in various situations.
to see the application of Cauchy’s integral formula in the derivation of power series expansion of an analytic function. to know a more general type of series expansion analogous to power series expansion viz. Laurent’s series expansion for functions having singularity. to understand how Laurent’s series expansion lead to the concept of residue, which in turn provide another fruitful way to evaluate complex integrals and, in some cases, even real integrals. to see another application of residue theory in locating the region of zeros of an analytic function.
MTS6B012 Calculus of Multivariable
The intention of the course is to extend the immensely useful ideas and notions such as limit, continuity, derivative and integral seen in the context of function of single variable to function of several variables. The corresponding results will be the higher dimensional analogues of what we learned in the case of single variable functions. The results we develop in the course of calculus of multivariable is extremely useful in several areas of science and technology as many functions that arise in real life situations are functions of multivariable. The successful completion of the course will enable the student to
Understand several contexts of appearance of multivariable functions and their representation using graph and contour diagrams. Formulate and work on the idea of limit and continuity for functions of several variables. Understand the notion of partial derivative, their computation and interpretation. Understand chain rule for calculating partial derivatives. Get the idea of directional derivative, its evaluation, interpretation, and relationship with partial derivatives. Understand the concept of gradient, a few of its properties, application and interpretation. Understand the use of partial derivatives in getting information of tangent plane and normal line. Calculate the maximum and minimum values of a multivariable function using second derivative test and Lagrange multiplier method. Find a few real life applications of Lagrange multiplier method in optimization problems. Extend the notion of integral of a function of single variable to integral of functions of two and three variables. Address the practical problem of evaluation of double and triple integral using Fubini’s theorem and change of variable formula. Realise the advantage of choosing other coordinate systems such as polar, spherical, cylindrical etc. in the evaluation of double and triple integrals .
See a few applications of double and triple integral in the problem of finding out surface area ,mass of lamina, volume, centre of mass and so on. Understand the notion of a vector field, the idea of curl and divergence of a vector field, their evaluation and interpretation.
Understand the idea of line integral and surface integral and their evaluations.
Learn three major results viz. Green’s theorem, Gauss’s theorem and Stokes’ theorem of multivariable calculus and their use in several areas and directions.
MTS6B013 Differential equations
Differential equations model the physical world around us. Many of the laws or principles governing natural phenomenon are statements or relations involving rate at which one quantity changes with respect to another. The mathematical formulation of such relations (modelling) often results in an equation involving derivative (differential equations). The course is intended to find out ways and means for solving differential equations and the topic has wide range of applications in physics, chemistry, biology, medicine, economics and engineering. On successful completion of the course, the students shall acquire the following skills/knowledge. Students could identify a number of areas where the modelling process results in a differential equation. They will learn what an ODE is, what it means by its solution, how to classify DEs, what it means by an IVP and so on.
They will learn to solve DEs that are in linear, separable and in exact forms and also to analyse the solution. They will realise the basic differences between linear and non linear DEs and also basic results that guarantees a solution in each case.
They will learn a method to approximate the solution successively of a first order IVP.
They will become familiar with the theory and method of solving a second order linear homogeneous and nonhomogeneous equation with constant coefficients.
They will learn to find out a series solution for homogeneous equations with variable coefficients near ordinary points. Students acquire the knowledge of solving a differential equation using Laplace method which is especially suitable to deal with problems arising in engineering field. Students learn the technique of solving partial differential equations using the method of separation of variables
On completion of the Programme student will be able to create a Evaluate hypothesis, theories, methods and evidence within their proper contexts
Solve complex problems by critical understanding, analysis and synthesis
Demonstrate engagement with current research and development in the subject
Provide a systematic understanding of core mathematical concepts, principles and theories along with their applications Communicate effectively by oral, written, computing and graphical means
Recognize the need to encourage in lifelong learning through continuing education and research
Career opportunities exist in teaching in schools and college (after M.Sc B.Ed/NET/SET/Ph.D) where any of this subject is an important discipline in IAS and other state and central govt,services where knowledge of Mathematics, Statistics, Computer applications with other science subject is an advantage
