MAIR34 REAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS

No of Credits: 3

Prerequisites: MAIR11, MAIR21

Course Type: GIR (General Institute Requirements)

Learning Objectives: Objective of the course is to introduce and study

1. the system of real numbers and their properties like GLB, LUB properties, Archimedian property, density, countability and uncountability

2. the limit of a function at a point, the limit of a sequence, and the Cauchy criterion in a metric space settings, limit points, compactness, Bolzano-Weierstrass and Heine-Borel properties.

3. the notions of continuity and differentiability of functions several variables, mean value theorems for these functions

4. the Riemann integrable functions of real valued functions

5. sequence and series of functions, and their (pointwise and uniform) convergences

6. how PDE arise, solutions of certain first order PDE, seperation of variables method, and its use in the solutions of Laplace and Helmholtz equations.

Syllabus: Real number system. Sets, relations and functions. Properties of real numbers.

Sequences. Cauchy sequences. Bolzano-Weierstrass and Heine-Borel properties.

Functions of real variables. Limits, continuity and differentiability. Taylor’s formula. Implicit and inverse function theorems. Extrema of functions.

Reimann integral. Mean value theorems. Differentiation under integral sign. Improper and multiple integrals. Change-of-variables formula. Sequences and series of functions. Pointwise and uniform convergence. Power series and Taylor series.

Laplace and Helmholtz equations. Boundary and initial value problems. Solution by separation of variables and eigen function expansion.

Learning Outcomes: On completion of the course, students should be able to

1. discuss the convergence of sequences of real numbers.

2. define the limit of a function , the limit of a sequence, and use the Cauchy criterion

3. prove theorems about limit of sequences and functions

4. define continuity/uniform continuity of a function and prove a theorem about continuous functions

5. discuss convergence of sequence and series of functions.

6. define Riemann integrable functions through Riemann sums

7. solve the Laplace and Helmoltz equations

Reference Books:

1. Guenther, R. B. & Lee, J. W., Partial Differential Equations of Mathematical Physics and Integral Equations, Courier Corporation, 1996.

2. Mattuck, A., Introduction to Analysis, Prentice-Hall,1999

3. Kreyszig, E., Advanced Engineering Mathematics, 10^th edn, John Wiley Sons, 2010.

4. W.R. Parzynski & P.W. Zipse, Introduction to Mathematical Analysis, McGraw-Hill, (1/e), 1987.

5. G.B.Gustafson & C.H. Wilcox, Analytical and Computational Methods of Advanced Engineering Mathematics, Springer, 1998

6. Walter Rudin, Real and Complex Analysis, Tata McGraw-Hill,2006

Homework Problems: Solve the following exercises in the attached notes. These numbers may be different from the previous notes given to you.

Chapter 1. Ex. 1.3, 1.4, 1.5, 1.8, 1.24, 1.28, 1.29, 1.31

Chapter 2: Ex. 2.3, 2.4, 2.5, 2.20, 2.24

Chapter 3: Ex. 3.8, 3.14, 1.15, 3.17, 3.19, 3.20, 3.22

The course is now completed. The PDF file of the notes made available here is removed now. Some of you had these notes during the class and minimized the notes taking in the class. Hope you listened and understood the lectures.