Gautam Bharali

               Department of Mathematics

                 Indian Institute of Science

                 Bangalore 560012


Home Education Research Publications Abridged CV Miscellanea Teaching



  • Announcements on special measures related to COVID-19 concerning the planning of lectures/tutorials

    Lectures will be conducted online. The mode of online teaching (whether online real-time or via pre-recorded downloadable videos, etc.) will be determined by the quality of students' access to the internet.

    The first lecture will be on the date notified to you by the Institute, and will be from 12:00 noon to 1:00 p.m. on that date. The first lecture was held on November 11.

    The first lecture will be online, in real-time on Teams. This meeting will enable us to determine the modalities of online teaching for the remainder of the semester. For those who are unable to participate online due to internet-access issues: a recording of this (and every subsequent lecture) will be available for you to download and view offline.

    The link to the above-mentioned class will be available from this page (to access it, you must be logged into your IISc Microsoft Outlook account in a separate tab in the same browser); please click/tap on the "Course List" tab and scroll to the bottom where the UM courses are listed.

  • Meeting times

    Lectures: Monday, Wednesday and Friday 12:00 noon–1:00 p.m.

    Tutorials: Thursday 9:30–10:30 a.m.

  • Textbook

    Tom M. Apostol, Calculus, Vol. 1, 2nd edition, Wiley, India Edition, 2001

  • Tutorials (replace «...» by in the addresses below)


    Tutor: Gopal Datt (gopaldatt@«...»)


    Tutor: Pabitra Barman (pabitrab@«...»)


    Tutor: Shubham Rastogi (shubhamr@«...»)


    Tutor: Sivaram P. (sivaramp@«...»)

  • Documents

    Slides shown on Day 1: Microsoft Teams instructions and getting started

    Slides shown on Day 2: Brief description of student tasks and assessment

    Slides shown on Day 3: Where to find videos, documents, assignments, etc.

    Handout no. 1 : COMING SOON

  • Syllabus (tentative: the list below will grow as the semester progresses the topics below comprise the syllabus of the final exam)

    All numbers refer to sections in the textbook.

    Your lecture notes will cover all the material (except for a few topics assigned for self-study) in the syllabus. The chapters listed below provide more extensive explanations, and lots of exercises for you to work on.

    Basic set theory: I.2.1–I.2.5

    The natural numbers, Peano addition and multiplication

    Fields: Definition and examples, ordered fields

    The real line and the least upper-bound property: I.3.1–I.3.4, I.3.8–I.3.10

    Sequences and convergence: 10.2–10.4

    Infinite series and their convergence: 10.5–10.9

    Convergence tests for non-negative series: 10.11, 10.12, 10.14, and the criterion for summability of the pth powers

    The ratio test: 10.15, 10.16 (Note: Many similar tests given in Apostol's book, which are founded on the Comparison Test, are not in the syllabus this year.)

    The limit of a function: 3.1, 3.2

    Basic theorems on limits: Theorem 3.1 from Section 3.4, 3.6

    The topics above comprised the syllabus of the mid-term examination. They will also be a part of the syllabus of the final examination.

    Continuity: 3.3, 3.6–3.8

    The extreme-value theorem: 3.16

    Bolzano's Theorem, the intermediate-value theorem: 3.9, 3.10 (applications deferred until Jan. 15, 2021)

    The meaning of differentiability: 4.2, 4.3

    The meaning of differentiability: 4.2, 4.3

    Basic differential calculus: 4.4–4.6, 4.10, 4.13 (assigned for self-study)

    Rolle's Theorem, the mean-value theorem and their applications: 4.14–4.16 (the the second-derivative test and its consequences have been excluded due to the lack of time)

    Inverse functions and their derivatives: 3.12, 3.130, 6.20–6.22

    Brouwer's Fixed-point Theorem and other applications of the intermediate-value theorem: 3.11

    The Cartesian product of sets (the discussion on the n-dimensional Brouwer's Fixed-point Theorem is not a part of the syllabus; the theorem was merely introduced for perspective)

    Integration, motivation, step functions: 1.8–1.13, 1.15

    Integration, motivation, step functions: 1.8–1.13, 1.15

    Integration: 1.16, 1.17, 1.21, 1.24

    Uniform continuity

    Integrability of continuous functions: 3.18

    The first and second Fundamental Theorems of Calculus: 5.1, 5.3–5.5

    Primitives, Leibnizian notation: 5.3, 5.6, 5.8

    The substitution rule: 5.7, 5.8

    The logarithm and the exponential functions: 6.3, 6.7, 6.12, 6.15, 6.16

    Vector spaces and subspaces: 15.2–15.6

    Linear independence, bases and dimension: 15.7–15.9

    Linear transformations: 16.1, 16.4

    The null space and range of a linear transformation (definition and properties): 16.2, 16.4 (relevant exercises only)

    Algebra of linear transformations: 16.5, 16.8 (relevant exercises only)

    Matrix representations of linear transformations: 16.10

  • Announcements

    Feb. 8: The final exam is scheduled for February 18. Details like time, duration, etc., will follow and be announced on Teams.

    Dec. 19: The mid-term examination is scheduled for December 23 at 10:30 a.m. Please read the protocol for conduct of the examination, which is posted in Microsoft Teams.

    Nov. 17: From November 18 lectures will transition from the real-time mode to pre-recorded videos. I am exploring options of scheduling (after lab timings have stabilises) a interactive session (attendance optional) to supplement the videos.

    Oct. 17: Please see above for the list of special announcements on subjects such as the the date of the first lecture, how to attend, etc., etc.

  • Homework assignments

    Homework 11 [Click/tap here for hints/sketch of solutions to Homework 11 problems]

    Homework 10 [Click/tap here for hints/sketch of solutions to Homework 10 problems]

    Homework 9 [Click/tap here for hints/sketch of solutions to Homework 9 problems]

    Homework 8 [Click/tap here for hints/sketch of solutions to Homework 8 problems]

    Homework 7 [Click/tap here for hints/sketch of solutions to Homework 7 problems]

    Homework 6 [Click/tap here for hints/sketch of solutions to Homework 6 problems]

    Homework 5 [Click/tap here for hints/sketch of solutions to Homework 5 problems]

    Homework 4 [Click/tap here for hints/sketch of solutions to Homework 4 problems]

    Homework 3 [Click/tap here for hints/sketch of solutions to Homework 3 problems]

    Homework 2 [Click/tap here for hints/sketch of solutions to Homework 2 problems]

    Homework 1 [Click/tap here for hints/sketch of solutions to Homework 1 problems]


  • INTRODUCTION TO SEVERAL COMPLEX VARIABLES (MA328-329)  [experimentally as a "topics course" (MA329) in Autumn 2014, Autumn 2019 ]


  • UNDERGRADUATE ANALYSIS & LINEAR ALGEBRA (UM101)   [Autumn 2015, Autumn 2017]

  • COMPLEX ANALYSIS (MA224)  [ Spring 2016 ]


  • ANALYSIS–II: MEASURE AND INTEGRATION (MA222)  [Spring 2017, Spring 2020 ]

  • ANALYSIS–I (MA221)   [ Autumn 2018 ]

  • INTRODUCTION TO BASIC ANALYSIS (UM204)   [ Spring 2019 ]

Page last updated on February 16, 2021