MA642 Real Analysis-1, during July-Nov 2025

Preamble: This course offers a rigorous and elegant introduction to real analysis, laying the foundation for advanced mathematics. It begins with the structure of the real number system and the theory of countability, then develops the topology of metric spaces-convergence, compactness, and continuity. A central focus is placed on sequences and series of functions, culminating in powerful results like Ascoli's and Weierstrass's theorems. The course progresses to the differential and integral calculus of functions of one and several variables, covering key theorems and techniques in optimization and multivariable analysis. Classical results such as Fubini's theorem and the theorems of Green, Gauss, and Stokes are treated with full mathematical rigor, emphasizing both theory and application.

 Syllabus: Completeness properties of real numbers, countable and uncountable sets, cardinality. Norms and metrics: Metric spaces, convergence of sequences, completeness, connectedness and sequential compactness; Continuity and uniform continuity; sequences and series of functions, uniform convergence, equicontinuity, Ascoli's theorem, Weierstrass approximation theorem, power series. Calculus of functions of a real variable: Differentiability, Mean value theorems, Taylor's theorem. Calculus of functions of several real variables: Partial and directional derivatives, differentiability, Chain Rule, Taylor's theorem, Maxima and Minima, Lagrange multipliers, Inverse function theorem, Implicit function theorem. Multiple Integration: Fubini's Theorem, Line integrals, Surface integrals, Green, Gauss and Stokes theorems.


Texts/References:
1. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Edition, W. H. Freeman, 1993.
2. P. M. Fitzpatrick, Advanced Calculus, 2nd Edition, AMS, Indian Edition, 2010.
3. N. L. Carothers, Real Analysis, Cambridge University Press, Indian Edition, 2009.
4. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw Hill, 1976.
5. Elias M. Stein and Rami Shakarchi, Real Analysis - Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis), 2005.

Course policy (click here)

Classroom and slot: 2203, Slot-B1 (Mon 16:00-16:55, Tue 15:00-15:55, Wed 14:00-14:55, Fri 17:00-17:55)


Lecture Notes:  Note that this course closely resembles MA642 (Real Analysis - I, 2023). For reference, you may consult the previous course materials available at: https://fac.iitg.ac.in/rksri/MA642_2023.htm.

Assignments: Assignment 1

 



Exams:

Class Discipline:

Students must ensure that mobile phones are switched off or set to silent mode and kept securely inside their bags, which must remain under the desk throughout the class. Only a notebook and a pen are permitted on the desk - no other items are allowed. Any student found with a mobile phone or any electronic device on the desk, regardless of intent, will be required to leave the classroom immediately and the incident will be reported to the Academic Section without exception.