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) |
Week |
Homework (subject to changes; please check regularly) |
1 |
No HW |
2 |
|
3 |
|
4 |
|
5 |
|
6 |
|
7 |
|
8 |
Midterm week |
9 |
|
10 |
|
11 |
|
12 |
|
13 |
|
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 |