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 with the code ht1sm12.

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

Office : N23 in the mathematics building.

Office Hours : By appointment.

Classroom and timings : LH-4, Tue-Thur 3:30-5:00

Teaching Assistant : Mainak Bhowmik (mainakb@iisc.ac.in)

Tutorials : Fridays 4-5 PM in LH-4

The Grading policy: Quizzes-based-on-Homework (20%) - A quiz (of 20 minutes duration) based on (but not necessarily 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 **Either on 28 Sept (Thursday) or 29 Sept (Friday) from 3:00-5:00 in LH-4 depending on the institute holiday**. The syllabus is everything upto (but NOT including) integration.

The **Final** shall be held on **Dec 8 from 2-5 PM 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 |
31 July to 6 Aug |
Review of linear algebra (Tuesday notes), Review of Topology, differentiability (Thursday notes) |

2 |
7 Aug to 13 Aug |
Differentiability and its properties (Tuesday notes), Quotient rule and statement of chain rule (Thursday notes) |

3 |
14 Aug to 20 Aug |
No class on Tuesday, Chain rule proof and higher order derivatives (Thursday notes) |

4 |
21 Aug to 27 Aug |
Inverse function theorem (My take on the material covered by Ved Datar: This and this) |

5 |
28 Aug to 3 Sept |
Implicit function theorem (Tuesday notes), Surjective derivative theorem (Thursday notes) |

6 |
4 Sept to 10 Sept |
Injective derivative theorem and manifolds (Tuesday notes), Taylor's theorem (Thursday notes) |

7 |
11 Sept to 17 Sept |
Second derivative test (Tuesday notes), Riemann integration and measure zero sets (Thursday notes) |

8 |
18 Sept to 24 Sept |
No class on Tuesday, Lebesgue's theorem and Fubini's theorem (Thursday notes) |

9 |
25 Sept to 1 Oct |
Midterm week. Midterm on Friday or Thursday (whichever is not an institute holiday). |

10 |
2 Oct to 8 Oct |
Integrals over bounded sets (Tuesday notes), Improper integrals (Thursday notes) |

11 |
9 Oct to 15 Oct |
Partition-of-unity (Tuesday notes), Change of variables (Thursday notes) |

12 |
16 Oct to 22 Oct |
Volumes of parametrised manifolds (Tuesday notes), Manifolds-with-boundary (Thursday notes) |

13 |
23 Oct to 29 Oct |
No class on Tuesday, Orientability (Thursday notes) |

14 |
30 Oct to 5 Nov |
Examples of orientability (Tuesday notes), Tensors and forms (Thursday notes) |

14 |
6 Nov to 12 Nov |
Wedge product (Tuesday notes), Exterior derivative and pullbacks (Thursday notes) |

15 |
13 Nov to 19 Nov |
Forms on manifolds (Tuesday notes), Generalised Stokes' theorem (Thursday notes) |

16 |
20 Nov to 23 Nov |
Poincare lemma (Tuesday notes) |

Week |
Homework (subject to changes; please check regularly) |

1 |
HW 1 (Quiz on 11 August) |

2 |
HW 2 (Quiz on 18 August) |

3 |
HW 3 (Quiz on 1 Sept based on HW 3 and HW 4) |

4 |
HW 4 (Quiz on 1 Sept based on HW 3 and HW 4) |

5 |
HW 5 (Quiz on 15 Sept based on HW 5 and HW 6) |

6 |
HW 6 (Quiz on 15 Sept based on HW 5 and HW 6) |

7 |
HW 7 (Quiz on 22 Sept) |

8 |
HW 8 (Quiz on 6 Oct) |

9 |
Midterm week |

10 |
HW 9 (Quiz on 13 Oct) |

11 |
HW 10 (Quiz on 20 Oct) |

12 |
HW 11 (Quiz on 27 Oct) |

13 |
HW 12 (Quiz on 10 Nov based on HW 12 and HW 13) |

14 |
HW 13 (Quiz on 10 Nov based on HW 12 and HW 13) |

15 |
HW 14 (Quiz on 17 Nov) |

15 |
HW 15 (Quiz on 23 Nov DURING CLASS ROOM HOURS that is from 4-4:20) |