MA 200 - Multivariable calculus - Autumn 2022 - Vamsi Pingali


The texts we will be referring to are as follows :
1. Munkres, Analysis on manifolds (Primary text).
2. Spivak, Calculus on manifolds.
3. Rudin, Principle of Mathematical Analysis.
4. J. H. Hubbard and B.B. Hubbard, Vector Calculus, Linear algebra and differential forms.

The course description (along with pre-requisites) can be found on this webpage. Please join the MS Team (the link will be on the intranet).

Instructor: Vamsi Pritham Pingali, vamsipingali@iisc.ac.in.

Office : N23 in the mathematics building.

Office Hours : By appointment.

Classroom and timings : LH-4, Tue, Thu 2-3:30

Teaching Assistants : Arindam Mandal (arindamm@iisc.ac.in)

Tutorials : Saturdays, 11-12.

The Grading policy: Quizzes-based-on-Homework (20%) - A quiz based on (but not identically the same as) the HW will be conducted by the TA during tutorial session every week or so (the best n-1 out of n such quizzes will be considered for averaging), Midterm (30%), and Final (50%).

Exams:

The Midterm shall be held on 20 Sept (Tue) from 2-4 in LH-4. The syllabus is everything until (and including) Lagrange's multipliers and the injective derivative theorem (that is, all the lectures, until Sept 16).

The Final shall be held on 6 Dec (Tue) from 2-5 in LH-4. The syllabus is everything taught in the course.

Ethics: Read the information on the IISc student ethics page. In short, cheating is a silly thing. Don't do it.

Here is the tentative schedule. (It is subject to changes and hence visiting this webpage regularly is one of the best ideas in the history of best ideas.)

Week

 Dates

 Syllabus covered

1

1 Aug to 7 Aug

Review of linear algebra (Tuesday notes), Review of the topology of Euclidean space (Thursday notes)

2

8 Aug to 14 Aug

No class on Tuesday, Differentiability (Thursday notes)

3

15 Aug to 21 Aug

C^1 implies differentiability (Tuesday notes), Properties of derivatives (Thursday notes)

4

22 Aug to 28 Aug

Chain rule (Tuesday notes), Clairaut's theorem and applications of the chain rule (Thursday notes)

5

29 Aug to 4 Sep

Inverse function theorem (Tuesday notes, Thursday notes)

6

5 Sep to 11 Sep

Implicit function theorem (Tuesday notes, Thursday notes)

7

12 Sep to 18 Sep

Global extrema (Tuesday notes), Lagrange's multipliers (Thursday notes)

8

19 Sep to 25 Sep

Midterm week (Midterm on Sept 20)

9

25 Sep to 2 Oct

Manifolds, Taylor's theorem in one variable (Tuesday notes), Taylor's theorem in multivariable calculus (Thursday notes)

10

4 Oct to 9 Oct

Definition of Riemann integration (Tuesday notes), Measure zero and Lebesgue's theorem (Thursday notes)

11

10 Oct to 16 Oct

Fubini's theorem (Tuesday notes), Rectifiable sets and Improper integrals (Thursday notes)

12

17 Oct to 23 Oct

Partitions of unity (Tuesday notes), Change of variables (Thursday notes)

13

24 Oct to 30 Oct

Volumes of parametrised manifolds, definition of manifolds-with-boundary (Tuesday notes), Integration of functions over manifolds-with-boundary (Thursday notes)

14

31 Oct to 6 Nov

No class on Tuesday, Orientability of manifolds (Thursday notes)

15

7 Nov to 13 Nov

No class on Tuesday, Wedge product done naively (Thursday notes)

16

14 Nov to 20 Nov

Wedge product done correctly, Exterior derivative (Tuesday notes), Pullback, integrating forms in R^n, smooth forms on manifolds (Thursday notes)

17

21 Nov to 27 Nov

Generalised Stokes' theorem (Tuesday notes)



HW is like a box of chocolates, you never know what you are gonna get...

Week

 Homework (subject to changes; please check regularly)

1

No HW

2

HW 1 (Quiz on 20 August 2022)

3

HW 2 (Quiz on 27 August 2022)

4

HW 3 (Quiz based on this and HW 4 on 10 September 2022)

5

HW 4 (Quiz based on this and HW 3 on 10 September 2022)

6

HW 5 (Quiz on 17 September 2022)

7

HW 6 (No quiz based on this HW)

8

Midterm week

9

HW 7 (Quiz on 8th October 2022)

10

HW 8 (Quiz on 19th October 2022)

11

HW 9 (Quiz on 31st October 2022 based on HW9 and HW10)

12

HW 10 (Quiz on 31st October 2022 based on HW9 and HW10)

13

HW 11 (Quiz on 7th Nov 2022)

14

HW 12 (Quiz on 14th Nov 2022, This HW has been corrected. Sorry for the mistake!)

15

No HW

16

No HW

17

No HW