Gautam Bharali

               Department of Mathematics

                 Indian Institute of Science

                 Bangalore 560012


Home Education Research Publications Abridged CV Miscellanea Teaching



  • Announcement on joining the UMA101 team (i.e., the Microsoft Teams app)

    Teams will, in most cases, be the primary medium for conveying important UMA101 announcements and for students to message me.

    You — i.e., registered UMA101 students — will be able to join the UMA101 team once you have your IISc e-mail accounts.

    You absolutely must join the UMA101 team once you receive the Join code for the team!

    The Join code will provided in class and will also be on the handout distributed during Lecture 2.

  • Meeting times

    Lectures: Monday, Wednesday and Friday 11:00–11:50 a.m.

    Tutorials: Tuesday 11:00–11:50 a.m. (with the exception of Tutorial 1)

    Office hour: Friday 6:30–7:30 p.m. (Room L19, Dept. of Mathematics)

  • Textbook

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

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

    Section A

    Tutor: Manpreet Singh (manpreets@«...»), Location: Lecture Hall 2, Physics Building (ground floor)

    Office hour: Wednesday 5:00–6:00 p.m. (Room N11, Dept. of Mathematics)

    Section B

    Tutor: Pintu Bhunia (pintubhunia@«...»), Location: Room MP30, ECE Department (ground floor of new building at the back)

    Office hour: Wednesday 6:00–7:00 p.m.

    Section C

    Tutor: Ravitheja Vangala (ravithejav@«...»), Location: Reading Room, Center for Ecological Science, Biological Sciences Building (3rd floor)

    Office hour: Wednesday 6:00–7:00 p.m. (Room L18, Dept. of Mathematics)

    Section D

    Tutor: Manoj Kumar (manojkumar1@«...»), Location: Room AE233, Aerospace Building

    Office hour: Thursday 5:30–6:30 p.m. (Room L23, Dept. of Mathematics)

    Section E

    Tutor: Arpita Mal (arpitamal@«...»), Location: Room AG04, New Chemical Sciences Building: IPC Wing (ground floor)

    Office hour: Friday 6:00–7:00 p.m. (Room N28, Dept. of Mathematics)

  • Documents

    Handout no. 1

  • 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 (excluding the limit comparison test), 10.14, the criterion for summability of the pth powers, 10.15, 10.16

    Absolute convergence: 10.18 (the definition and Theorem 10.18 only)

    The limit of a function: The sequential definition of the limit

    Basic theorems on limits: Uniqueness of limits, the limit of a scaling of a function, the limits of sums and products of functions

    The limit of a function: The "ε-δ" definition: 3.1, 3.2

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

    Continuity: 3.3, 3.6–3.8

    Bolzano's Theorem, the intermediate-value theorem, and applications: 3.9–3.11

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

    The extreme-value theorem for continuous functions: 3.16

    The meaning of differentiability: 4.2, 4.3

    Basic differential calculus: 4.4–4.6, 4.10, 4.13 (Section 4.12 assigned for self-study excluding the discussion on "implicit differentiation")

    Points of absolute/global and relative/local extremum: 4.13–4.15

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

    Inverse functions and their derivatives: 3.12, 3.13, 6.20–6.22

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

    Integration: 1.16, 1.17, 1.24

    Uniform continuity

    Integrability of continuous functions: 3.17, 3.18

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

    Primitives, Leibnizian notation: 5.3, 5.6

    Integration by parts: 5.9, excluding Theorem 5.5. Note: Section 5.10 is for self-study, as you have seen most of the problems in them in high school. For help and solved examples, see Section 5.9.

    The logarithm and the exponential functions: 6.3, 6.7, 6.12, 6.14, 6.16. Note: The material in Sections 6.14 and 6.16 was developed through homework problems.

    Vector spaces and subspaces: 15.2–15.6

    Linear independence, bases and dimension: 15.7–15.9

    Linear transformations: 16.1, 16.4

    Matrix representations of linear transformations: 16.10

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

    The null space and range of a linear transformation, injective linear transformations: 16.2, 16.7

  • Announcements

    Oct. 6: The October 10 tutorial is converted to a lecture (at the usual venue).

    Sep. 4: The UMA101 mid-semester examination will be on Sep. 27. Time and exam locations will be communicated by the UG Office.

    Aug. 13: The August 14 lecture is converted to a tutorial (the first tutorial of the semester).

    Aug. 2: The August 4 lecture stands cancelled due to the Freshers' Welcome. The tutorial hour on Tuesday, August 8, will be converted to a lecture to make up for the August 4 lecture.

    Aug. 2: Please see above for the special announcement on joining the UMA101 team (on Microsoft Teams).

  • Homework assignments

    Homework 14 [Click/tap here for hints/sketch of solutions to some Homework 14 problems]

    Homework 13 [Click/tap here for hints/sketch of solutions to some Homework 13 problems]

    Homework 12 [Click/tap here for hints/sketch of solutions to some Homework 12 problems]

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

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

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

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

    Homework 7

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

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

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

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

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

    Homework 1


  • ANALYSIS–I (MA221)   [ Autumn 2018 ]

  • INTRODUCTION TO BASIC ANALYSIS (UM204)   [ Spring 2019, Spring 2022]

  • ANALYSIS–II: MEASURE AND INTEGRATION (MA222)   [ Spring 2020, Spring 2023 ]





Page last updated on November 28, 2023