# MA3402 Differential Forms on Manifolds - Autumn 2023

**Exam schedule:** https://docs.google.com/spreadsheets/d/1rqH5tGPJFwPNUnZ-l4B7FLo_6BjW3gPhjVYG5e53Smc/edit?usp=sharing

Talk to Abi if you would like to make any changes.

**Collection of possible exam question examples:** Exam Question Submissions

Schedule | Room | ||
---|---|---|---|

Lectures | Thursdays 14:15–16:00 | Realfagbygget R73 or Kjelhuset KJL4* | |

Fridays 14:15–16:00 | Sentralbygg2 Smia or Kjelhuset KJL4* | ||

*Note our room allocations are a bit random. Please make sure you check which room we're in here: https://www.ntnu.edu/studies/courses/MA3402/#tab=timeplan | |||

Exam: | Oral exam. Date to be confirmed | ||

Lecturer: | Abigail Linton (Abi) | ||

Office: | 1206 Sentralbygg 2 | ||

Email: | abigail [dot] linton [at] ntnu [dot] no |

## What this course is about

If TMA4192 Differential Topology is about learning to differentiate on manifolds, MA3204 is about how to integrate. This is an interactive course that will revisit what we learned in vector calculus but now on manifolds instead. In particular we will reformulate Stokes' and Green's Theorems, which have important applications in everything from topology to physics. We will also learn about de Rham cohomology and see that is surprisingly equivalent to the cohomology theories you will learn in Algebraic Topology I.

The course starts with developing the theory of forms on Euclidean space before applying the same techniques on manifolds. The main text for this course is An Introduction to Manifolds by Tu. It is available as a free pdf download from the SpringerLink website https://link.springer.com/book/10.1007/978-1-4419-7400-6 (if it doesn't work, make sure you are on campus or using the university VPN). We will not cover the entirety of the book.

## Prerequisites

It is *recommended* that students have taken TMA4190 Introduction to Topology and TMA4192 Differential Topology so that you have a feel for the objects we're working with, but this is **absolutely not a requirement**. You can learn what you need as we go. Many students have taken Differential Forms before taking Differential Topology and have managed fine. As with many topology topics, the willingness to ask questions and work to understand new ideas is almost more important than prerequisite knowledge.

Do not be afraid to ask for help! Many students take Differential Forms from different backgrounds, so be patient and help each other :). This course is designed so that we start building our intuition in Euclidean space first before moving to manifolds, so you will have time to learn about smooth manifolds before we use them. Students should be familiar with open and closed subsets of n-dimensional Euclidean space, smooth maps between open subsets of Euclidean space, derivatives of smooth maps between Euclidean space, the inverse function theorem. You could use Chapter 2 (sections 5-7) in Tu to both study and use as a reference throughout the course. An alternative reference is Chapter 1 §§1–5 and Chapter 2 §§1–2 in Guillemin and Pollack.

## Reference Group

Reference groups are a way for students to give feedback on the course. Get in touch with either me or the reference group with your feedback. Members: Carl Fredrik Andresen, Isak Drage, Maximilian Rønseth.

- First meeting: 19 Sep 2023
- Second meeting:
- Third meeting:

## Lecture Summary

This is not a hard schedule and is prone to change.

Week | Room | Topic | Reference | Homework |
---|---|---|---|---|

34.1 | R73 | Introduction and tangent vectors on Euclidean space | §2 | |

34.2 | Smia | Multicovectors | §§3.1-3.7 | Homework 1 |

35.1 | KJL4 | Differential forms on Euclidean space | §§3.8- 4.1 | |

35.2 | KJL4 | Differential forms on Euclidean space | §$4.1-4.4 | Homework 2 |

36.1 | KJL4 | *Applications to vector calculus (independent) | §§4.5-4.6 of [Tu] | Exercise 1 |

36.2 | KJL4 | *Exercise session (independent) | Solutions 1 | |

37.1 | R73 | Exterior derivative and manifolds recap | §4.4, §5 | |

37.2 | KJL4 | Smooth maps on manifolds | §6 | Homework 3 |

38.1 | R73 | Jacobians and Tangent space | §6.7, §§8.1-8.2 | |

38.2 | KJL4 | Differential of a map, Preimage Theorem | §§8.3-8.4, 8.9, 9.3-9.4 | Homework 4 + Exercises |

39.1 | KJL4 | Exercise session | ||

39.2 | KJL4 | Vector fields, differential 1-forms on manifolds | §§14.5, 17.0-17.2, 17.4-17.5 | Homework 5 |

40.1 | R73 | Pullback of 1-forms, examples | §§17.5-17.6 | exercise2.pdf |

40.2 | KJL4 | Differential k-forms | §§18.1-18.6 (not 18.3), 19.0-19.5 (not 19.2) | Homework 6 |

41.1 | KJL4 | Nowhere vanishing forms, contraction | §§19.5-19.7, 20.4 | |

41.2 | KJL4 | Orientation I | §§21.0-21.4 | Homework 7 |

42.1 | R73 | Orientation II | ||

42.2 | KJL4 | Exercise session | exercise2solutions.pdf | |

43.1 | KJL4 | Orientation examples, boundary | §§21.4-22.3 | |

43.2 | Berg 2 | Boundary orientation | §§22.5-22.6 | Homework 8 exercise3.pdf |

44.1 | R73 | Integration | §§23.1-23.4 | |

44.2 | KJL4 | *Stoke's Theorem and applications! | §§23.4-23.6 | *Independent work |

45.1 | KJL4 | Stoke's recap & Exercise session | ||

45.2 | KJL4 | De Rahm cohomology | §§24.0-24.2 | Homework 9 (edited) |

46.1 | R73 | Mayer-Vietoris | §§24.2-24.4, 26.2 | |

46.2 | KJL4 | Mayer-Vietoris examples | §§26.2, 27.4, 28.1 | Homework 10 (updated Q2) |

47.1 | KJL4 | Exercise session | Exercise 3 Solutions | |

47.2 | KJL4 | Cancelled - free time for revision and practice |

**Week 36 Independent Work:** Use the KJL4 and work together! I am away 7 & 8 September, but we can discuss some exercises and solutions when I get back. Good luck!

- Complete the exercises from lectures in Week 35 (
**Homework 2**). **Read Section 4.6**(and optionally 4.5) on your own and be able to explain the relationship between vector calculus and what we've learned so far about forms and the exterior product - you will need**Homework 2 Question (5)**.- Attempt
**Exercise Sheet 1**on your own/in groups. - Extra practice: There are extra exercises at the end of Section 4, choose your favourites and have a go!

Exercise sheets: exercise1.pdf exercise2.pdf exercise3.pdf

## Exams

**Exam schedule:** https://docs.google.com/spreadsheets/d/1rqH5tGPJFwPNUnZ-l4B7FLo_6BjW3gPhjVYG5e53Smc/edit?usp=sharing

Talk to Abi if you would like to make any changes.

**Collection of possible exam question examples:** Exam Question Submissions

Some guidance about the oral exam.

- Each exam will be approximately 30 minutes. There are two examiners, Abi (the lecturer) and a second.
- The exam will focus on results/proofs and exercises from the course. Practice exercises to get used to working with the key definitions.
- Practice your oral exam skills by discussing questions/ideas with other students - it is exactly the same process!
- Optionally come with your favourite result+proof, question+answer or exercise from the course and be prepared to talk about it for 5 minutes. The purpose of this isn't (necessarily!) to show off your creativity, but to showcase your talents! This way we won't miss out one of the topics that you are most comfortable with.
- It's ok to be nervous! Just focus on processing your thoughts aloud.

Please contact Abi if you have any specific requirements for the exam.

## References

Since there are many great books on differential forms and differential geometry, it is worth spending some time to flick through a few and find a style that works for you. Most of them will cover very similar material to what we will in lectures, though they may differ slightly in notation. As well as the campus library, you can find many books free online. We will mainly follow Tu.

Suggestions of books:

- [Tu] L. W. Tu, An Introduction to Manifolds, second edition, Springer Verlag, 2011.
- [Lee] J.M. Lee,
*Introduction to Smooth manifolds*, Springer-Verlag. - [GH] Guillemin and Haine,
*Differential Forms,*ISBN13: 9789813272774, World Scientific, 2019, Preprint here. - [GP] V. Guillemin and A. Pollack,
*Differential Topology*, Prentice Hall, 1974. - [M69] J. Milnor,
*Topology from the differentiable viewpoint*, The University Press of Virginia, 1969. - [M63] J. Milnor,
*Morse Theory*, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963. - [D] B. Dundas, Differential Topology. Version January 2013
- [S] D. Spivak,
*Calculus on Manifolds*, Addison-Wesley, 1965. - [Mu91] J.R. Munkres,
*Analysis on Manifolds*, Addison-Wesley, 1991. - [Mu75] J.R. Munkres,
*Topology: a first course*, Prentice-Hall, 1975. - [RS] Robbin and Salamon,
*Introduction to Differential Topology*, 2018. Preprint here