Mathematical Analysis

Mathematics is an inevitable subject and it has significant role in many real life problems. In order to formulate and solve many engineering problems, one has to relate the problems with various Mathematical concepts. A thorough understanding of mathematics is an essential one to do such problems easily. In this course, I would like to discuss some basic concepts of mathematics under the following titles.1. Real Analysis2. Complex analysis3. Vector analysisThis is a 10 week course comprising 40 modules, assignments and tests under the above mentioned topics.The course is helpful to those students they are doing BSc Mathematics degree course under various universities of India.This course will help students from science stream for the systematic study of the basic concepts of metric space, complex analysis and vector analysis. By completing this course, the students will have a legible understanding of basic concepts of metric space, complex analysis and vector analysis which will be useful in many situations.

COURSE LAYOUT

Week 1 Day 1 Metric Space- open and closed sets Day 2 Boundedness, compactness and connectedness Day 3 Complete sets, dense sets and compact sets Day 4 Sequences in metric spaces Day 5 Interaction
Week 2 Day 1 limit and continuity Day 2 Isometry Mappings and Homeomorphisms Day 3 Extension Theorems Day 4 Equivalent Metrics and Subspaces Day 5 Interaction
Day 6 Assignment
Week 3 Day 1 Reimann Integral I Day 2 Reimann Integral II Day 3 Continuity of real valued functions
Day 4 Interaction
Day 5 Assignment Week 4
Day 1 Partial differentiation Day 2 Partial differentiation- Young’s theorem and Schwarz’s theorem Day 3 Partial differentiation- Implicit, Homogeneous and Composite functions Day 4 Interaction
Day 5 Assignment

Week 5 Day 1 Infinite series Day 2 Series of arbitrary terms
Day 3 Interaction Day 4 Assignment
Week 6 Day 1 Tests for positive term series Day 2 Double sequences and series Day 3 Convergence of double Series
Day 4 Interaction
Day 5 Assignment

Week 7
Day 1 Fourier series Day 2 Fourier series of functions having period 2C Day 3 Fourier series for even and odd periodic functions Day 4 Half Range Fourier Series Expansions
Day 5 Interaction
Day 6 Assignment

Week 8 Day 1 Power series solution of ordinary differential equations Day 2 Bessel functions Day 3 Legendre polynomials Day 4 Sturm-Liouville problem and Orthogonality properties of Bessel functions and Legendre polymonials
Day 5 Interaction
Day 6 Assignment

Week 9
Day 1 Derivative of complex valued functions Day 2 Cauchy Riemann equations
Day 3 Continuous functions Day 4 Mobius Transformation
Day 5 Interaction
Day 6 Assignment

Week 10 Day 1 Vector analysis I Day 2 Vector analysis II Day 3 Differentiation of vector valued function Day 4 Gradient, divergent and curl
Day 5 Interaction
Day 6 Assignment

Week 11 Day 1 Line integral of vector valued functions Day 2 Independence of path Day 3 Surface integral Day 4 Greens theorem
Day 5 Interaction

Week 12 Day 1 Volume integral and divergence theorem Day 2 Stokes theorem
Day 3 Interaction Day 4 Assignment