MA3010: Real Analysis
Syllabus: Real number system: Field properties, ordered properties, completeness axiom, Archimedean property, subsets of R, infimum, supremum, extended real numbers. Finite, countable and uncountable sets, decimal expansion. Sequences of real numbers, Subsequences, Monotone sequences, Limit infimum, Limit Supremum, Convergence of Sequences.
Metric spaces, limits in metric spaces. Functions of single real variable, Limits of functions, Continuity of functions, Uniform continuity, Continuity and compactness, Continuity and connectedness, Monotonic functions, Limit at infinity. Differentiation, Properties of derivatives, Chain rule, Rolle's theorem, Mean-value theorems, L'Hospital's rule, Derivatives of higher order, Taylor's theorem. Definition and existence of Riemann integral, properties, Differentiation and integration.
Revision of Series, Sequences and Series of functions, Pointwise and uniform convergence, Uniform convergence of continuous functions, Uniform convergence and differentiability, Equicontinuity, Pointwise and uniform boundedness, Ascoli's theorem, Weierstrass approximation theorem, Fourier series
References:
1. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill international editions (Math Series), 3rd Edition, 1976.
2. S.R. Ghorpade and B.V. Limaye. A course in calculus and real analysis. Undergraduate Texts in Mathematics. Springer, New York, Springer International Ed., New Delhi, 2006.
3. R.R.Goldberg, Methods of Real Analysis, Oxford & IBH Publishing Co. Pvt Ltd, 1970.
4. Kenneth A. Ross, Elements of Analysis: The Theory of Calculus, Springer Verlag, UTM, 1980
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MA1140 Elementary Linear Algebra
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MA5050: Mathematical Methods
Syllabus: Integral Transforms: Laplace transforms: Definitions - properties - Laplace transforms of some elementary functions - Convolution Theorem - Inverse Laplace transformation - Applications.
Fourier transforms: Definitions - Properties - Fourier transforms of some elementary functions - Convolution theorems - Fourier transform as a limit of Fourier Series - Applications to PDE.
Integral Equations: Volterra Integral Equations: Basic concepts - Relationship between Linear differential equations and Volterra integral equations - Resolvent Kernel of Volterra Integral equation - Solution of Integral equations by Resolvent Kernel - The Method of successive approximations - Convolution type equations, solution of integral differential equations with the aid of Laplace transformation.
Fredholm Integral equations: Fredholm equations of the second kind, Fundamentals - Iterated Kernels, Constructing the resolvent Kernel with the aid of iterated Kernels - Integral equations with degenerate Kernels - Characteristic numbers and eigen functions, solution of homogeneous integral equations with degenerate Kernel - non homogeneous symmetric equations - Fredholm alternative.
Calculus of Variations: Extrema of Functionals: The variation of a functional and its properties - Euler's equation - Field of extremals - sufficient conditions for the Extremum of a Functional conditional Extremum Moving boundary problems - Discontinuous problems - one sided variations - Ritz method.
References:
J W Brown and R V Churchill: Fourier Series and Boundary Value Problems, McGraw Hill, 8th Edition, 2011.
A Chakraborty: Applied Integral Equations, Tata McGraw Hill, 2008.
F G Tricomi: Integral Equations, Dover Publications, 1985.
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MA2150: Introduction to Metric Spaces (July-Nov. 2024)
Syllabus: Definition and examples of open balls and open sets, sequence in metric space; Cauchy sequence, convergence, bounded, dense sets, continuous functions and related properties, other topological properties.
Ref:
S. Kumaresan. Topology of Metric Spaces. Second Edition, Narosa Publishing house, New Delhi, 2005. xii+152 pp. ISBN: 81-7319-656-7
Mícheál O'Searcoid. Metric spaces. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2007. xx+304 pp
Assignment: Click here
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MA6220: Distribution Theory and Sobolev Spaces (Jan-Apr 2024)
Distributions: Test functions and Distributions, Convolution of Distributions, Fundamental solutions, The Fourier transforms, The Schwartz space S, Tempered Distributions. Sobolev spaces: Definition and basic properties, Approximations by smooth functions, Traces, Sobolev inequalities, Compactness, Dual spaces, Fractional order spaces, and trace spaces. Weak solutions of elliptic boundary value problems: Definitions of weak solutions, Existence, The Lax-Milgram theorem, Regularity, Galerkin method, Maximum principle, eigenvalue problems, Introduction to finite element methods.
Ref:
1. S. Kesavan, Topics in Functional Analysis and Applications
2. Robert A. Adams, John J. F. Fournier, Sobolev spaces
3. Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations PDF
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MA 5030 Partial Differential Equations July-Nov 2023
Ref:
F. John, Partial Differential Equations, 3rd Edition, Narosa, 1979
L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 1998
A. K. Nandakumaran and P. S. Datti, Partial Differential Equations: Classical theory with a modern touch, 2020
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MA 4030 Ordinary Differential Equations Jan - April 2023
References: 1. Ordinary Differential Equations: Principles and Applications by A. K. Nandakumaran, P. S. Datti, Raju K. George
2. An Introduction to Ordinary Differential Equations by Earl A. Coddington
Assignment 1 Click here
Assignment 2 Click here
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MA 1220 Calculus II Dec 2022 - Jan 2023
Syllabus
Integral Calculus: Definite Integrals as a limit of sums, Applications of integration to the area, volume, surface area, and improper integrals. Functions of several variables: Continuity and differentiability, mixed partial derivatives, local maxima and minima for functions of two variables, Lagrange multipliers.
Textbook:
"Thomas' Calculus"
Homework: Click here
Assignment: Click here Please submit the assignment by Wednesday 21/12/2022 in the Class.
End sem : 3 rd January 2023, Tuesday 17:30 - 19:30
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MA 1000 Math foundation Oct 2022-Feb 2023
Syllabus
Statements, Logic, Proofs in Mathematics, Sets, Functions, Relations, Equivalence Relations, Partition of a Set, The Induction Principles, The Well ordering principles, Countability of Sets (finite and countable), Order Relations, Posets, Axioms of Choice
Textbook
"A Foundation Course in Mathematics" by Ajit Kumar, B. K. Sarma, and S. Kumaresan
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MA 1230 Series of functions June-July 2022