Department of Mathematicscoretheory
REAL ANALYSIS
MAT 2236
Syllabus
- 01Real Number System: Algebra of Real Numbers, Upper and Lower Bounds, LUB and GLB property and its applications, Absolute Value and Triangle Inequality
- 02Sequences and Their Convergence: Cauchy Sequences, Monotone Sequences, Sandwich Lemma, Some important Limits, Sequences Diverging to +, -, Subsequences, Sequences Defined Recursively
- 03Continuity: Continuous Functions, definition of continuity, Intermediate Value Theorem, Extreme Value Theorem, Monotone Functions, Limits, Uniform Continuity
- 04Differentiation: Differentiability of functions, Mean Value Theorems, L'Hospital Rules, Higher Order Derivatives, Taylor's Theorem, Convex functions
- 05Functions of Two variables
- 06Infinite Series: Convergence, Abel's Summation by parts, Cauchy product of two infinite Series
- 07Reimann Integration: Darboux Integrability, Properties of the integral, Fundamental Theorem of Calculus
- 08Introduction to Metric spaces, examples, open sets, closed sets, limit point, continuity
References
- Ajit Kumar and Kumaresan, A Basic Course in Real Analysis, CRC Press. 2014.
- Tom M. Apostol: Mathematical Analysis, Second Edition, Addison Wesley Publishing Company, 1974.
- W. Rudin, Principles of Mathematical Analysis, 3rd Edition (2017).
- Sudhir R. Ghorpade and B. V. Limaye, A Course in Calculus and Real Analysis (Undergraduate Texts in Mathematics) Hardcover – 2006 (Springer).
- Sudhir R. Ghorpade and B. V. Limaye, A Course in Multivariable Calculus and Analysis, 2010 (Springer)
Credits Structure
3Lecture
1Tutorial
0Practical
4Total