Expository Articles by Prof. S. Kumaresan
These are the articles (mainly expository in nature) written by me. They were used by me either as hand-outs in some of my courses or assigned as seminar material to students. While most of them are original, a few of them are my versions of others' work disseminated in a form suitable for my students.
If you use them in any way, please send me your comments. kumaresa at gmail
dot com.
All the articles given below are in pdf format. To download, please (left)click on the title of the article. To open pdf files, you may have to use pdf readers like xpdf or Acrobat reader which you can freely download from the internet.
- Set Theory
- Algebra
- Insolvability of Quintic
- Isometries of Rn and Sylvester Criterion for Positive Definite Matrices
- Burnside Theorem on Matrix Algebras and Its Application to Group Theory
- Classification of Real Division Algebras
- Outline of a Course in Field Theory
- Fundamental Theorem of Algebra via Linear Algebra
- Isometries of Rn
- Jordan Canonical Form
- Jordan Canonical Form
- Row Rank of a Matrix equals its Column Rank
- A Problem Course in Module Theory
- A Problem Course in Module Theory - Exercises
- Sign of a Permutation
- Polar Decomposition, Singular Values, Spectral Theorem
- How to work with quotient rings?
- Finite Dimensional Spectral Theory
- Structure of Linear Maps
- Sylow Theorem
- Sylvester Criterion for Positive Definiteness
- Analysis
- Arzela-Ascoli Theorem and Applications
- Picard's Little Theorem
- Bishop-Stone-Weierstrass Theorem
- Review of Differential Calculus
- Completeness of R
- Change of Variable Formula for Multiple Integrals
- LUB Property of Ordered Fields
- A Diagnostic Quiz in Complex Analysis
- Contraction Principle and its Applications
- Existence of Continuous Logarithms
- A Continuous but Nowhere Differentiable Function
- Decimal, Binary and Ternary Expansions
- Solution of Dirichlet Problem - Perron's Method
- Double Series
- The Fundamental Theorem of Algebra
- Fubini Theorem
- Fundamental Theorem of Calculus in HK-Integral
- Implicit Function Theorem for Banach Spaces
- Integers and Polynomials
- Lebesgue Integral without Measure Theory
- Similarity of Matrices via Module Theory
- Compact Subsets of Lp Spaces
- The Role of LUB Axiom in Real Analysis
- Minkowski Inequality
- Miscellaneous Results in Analysis
- A Conceptual Introduction to Several Variable Calculus
- Applications of Nested Interval Theorem
- Nested Interval Theorem and Its Applications
- The Role of LUB Axiom in Real Analysis
- Construction of real numbers
- Existence and Uniqueness of Solutions of ODE
- Little Picard Theorem
- Topics in Measure Theory
- An Outline of Real Analysis
- Riesz-Markov Representation Theorem
- Properties equivalent to the LUB property
- Stokes Theorem | A simple and intuitive proof
- Stokes Theorem ? An Outline of a Proof
- Stone-Weierstrass Theorem
- Uniform Convergence
- Every Real Sequence has a Monotone Subsequence
- Weierstrass Approximation Theorem
- Topology
- Every compact metric space is the continuous image of the cantor set
- Simple and Direct Proofs of Some Results in the Theory of Compact Spaces
- Discrete Subgroups of R^n
- Lebesgue Covering Lemma
- Vector Fields on Spheres and Brouwer's Fixed Point Theorem
- Characterization of Compact Metrizable Spaces via Metrics and Continuous Functions
- Existence of Smooth Functions and Partition of Unity
- Subspace Topology
- Topological Groups | via Problems
- Generating Topologies| A Unified View of Subspace, Product and Quotient Topologies
- Tychonoff?s Theorem
- Noncontractibility of the Circle
- Connected and Path Connected spaces
- Continuity of Roots of Polynomials
- R^m is homeomorphic to R^n iff m = n.
- Any compact hypersurface in Rn is a level surface
- Maps into the Punctured Plane
- Existence of Continuous Functions
- Winding Numbers of Loops
- Geometry
- Problem Set In Differential Geometry
- Frenet-Serret Formulas for Space Curves
- Implicit Function Theorem
- Inverse Mapping Theorem
- Gauss Golden Theorem - Isometry - Examples
- The Method of Lagrange Multipliers - A Geometric Treatment
- How to draw the graphs of functions on R2?
- Geometric Applications of Sturm Comparison Theorem
- Theory of Surfaces|A Rapid Introduction
- Harmonic Analysis


