Gautam Bharali

               Department of Mathematics

                  Indian Institute of Science

                  Bangalore 560012


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  • Meeting times

    Lectures: Tuesday, Thursday and Friday, 8:00–9:00 a.m.

    Tutorials: Tuesday, 6:30–7:30 p.m.

  • Recommended books

    Tom M. Apostol, Calculus, Vol. II, 2nd edition, Wiley, India Edition, 2001, 2007

    Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill International Editions (will be used primarily as a source of exercises and examples)

    Theodore W. Gamelin, Complex Analysis, Springer UTM, Springer International Edition, 2006

  • Tutorials


    Tutor: Anwoy Maitra (maitra12[you know what], Location: Lecture Hall 4, Dept. of Mathematics


    Tutor: Samrat Sen (samrat12[you know what], Location: Lecture Hall 1 Lecture Hall 5, Dept. of Mathematics

  • Documents

    A note on a theorem about countable sets

    Handout no. 1: Course information

  • Syllabus

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

    Norms on vector spaces, metrics, metrics induced by norms

    The Cauchy–Schwarz inequality

    (Open) balls, interior points, open sets: Section 8.2 of Apostol, Volume II

    Sequences and convergence, limit points of a set, limits versus limit points

    Review of basic set theory and countable sets, the definition of uncountability

    Dense subsets, countable dense subsets of Rn

    Description of the open subsets of the real line

    Limits of the values of vector-valued functions; CAUTION: Treatment of this is somewhat non-rigorous in Apostol!

    Continuity of sums, products, etc., of continuous functions, examples: Sections 8.4 and 8.5 of Apostol, Volume II

    The concepts of compactness and uniform continuity

    Properties of compact sets

    The Heine–Borel theorem

    Cauchy sequences and the completeness of Euclidean space

    Differentiability in Rn: Sections 8.11 and 8.18 of Apostol, Volume II

    The Chain Rule

    The relation between continuity of partial derivaties and differentiability: Sections 8.13 and relevant exercises in Section 8.14 of Apostol, Volume II

    Taylor's Theorem

    Critical points, points of local extremum, non-degenerate critical points and the Hessian: Sections 9.11–9.13 of Apostol, Volume II

    Step functions: Sections 11.2 and 11.31 of Apostol, Volume II

    Multiple integrals: Sections 11.3–11.6, Section 11.31 and relevant exercises from Sections 11.9 and 11.15 of Apostol, Volume II

    The change-of-variables formula, plane polar and spherical polar coordinates

    Parametric manifolds and smooth embedded manifolds

    Brief survey of the exterior product of finite-dimensional vector spaces

    Differential forms on open sets of Rn

    The integral of a differential form, and its independence of parametrization

    Orientation: The recommended sources, for the detail in which this was covered, are your lecture notes.

    Stokes' Theorem for regular parametric manifolds

    Special cases of Stokes' Theorem: The theorem in Apostol, Section 12.11; Green's Theorem

    The notion of C-differentiability and the Cauchy–Riemann condition: Gamelin, Sections II.2, II.3

    Polynomials, rational functions and power series

    The holomorphic sine, cosine and exponential functions: Gamelin, Sections I.5, I.8

  • Announcements

    Apr. 10: The third (and last) make up lecture will be from 4:00 to 5:00 p.m. tomorrow in our usual classroom.

    Apr. 10: The final exam is scheduled for April 26, 9:30 a.m.–1:00 p.m. Location to be announced.

    Mar. 25: The next two lectures, i.e., on March 26 and 27, are suspended as I will be away for a conference.

    Mar. 20: The next make-up lecture will be on April 4 from 10:00 to 11:00 a.m. at the usual location.

    Mar. 13: The location of the first of the three make-up lectures (scheduled for 4:00–5:00 p.m. tomorrow) is Lecture Hall 5, Department of Mathematics.

    Feb. 23: This is a reminder that the mid-term examination is at 10:00 a.m. on Feb. 25. Venue: Room G-01 in the old physics building.

    Feb. 16: There will be no tutorial on Feb. 17 on account of Mahashivaratri. To make up for this, there will be a tutorial session on Feb. 19, 6:30–7:30 p.m. Location: to be announced soon.

    Jan. 20: There has been a change in the room assigned to Tutorial Group II. Please see above for the new room assignment. (Please ignore this announcement.)

  • Homework assignments

    Homework 11

    Homework 10

    Homework 9

    Homework 8

    Homework 7

    Homework 6

    Homework 5

    Homework 4

    Homework 3

    Homework 2

    Homework 1, part 2

    Homework 1, part 1

  • Quiz solutions

    The solution to Quiz 6

    The solution to Quiz 5

    The solution to Quiz 4

    The solution to Quiz 3

    The solution to Quiz 2

    The solution to Quiz 1

Page last updated on April 18, 2015