Gautam Bharali                Department of Mathematics                   Indian Institute of Science                   Bangalore 560012
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#### UM 202: MULTIVARIABLE CALCULUS & COMPLEX VARIABLES

• 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

GROUP I

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

GROUP II

Tutor: Samrat Sen (samrat12[you know what]math.iisc.ernet.in), 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

• 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