**Manifolds and Differential Forms: MATH 3210**

Fall 2017

**Instructor: ** Xiaodong Cao, Office:521 Malott, Phone: 5-7431

**E-mail: **cao at math.cornell.edu

**Teaching
Assistant:** Ahmad Rafiqi (Email: ar776 at cornell.edu OH: Wed. 14:30-16:30,
111 MLT)

**Time
and Place: ** 10:10 am - 11:00 am, MWF, 203 MLT

**
Office Hour:** Tuesday 14-14:50, Thursday 14:00-14:50 or
by appointment.

**Text: **

**Prerequisites:** Calculus and Linear Algebra.

**Midterm
Exams: **The midterm exams are on **Friday, Sept. 29th ** and **Monday, Oct. 30th. ** *Make-ups will not be given for the midterm exams.*
Students can only be excused from the midterms because of serious illness or a
family emergency of comparable gravity. To be excused you will need a note from
your doctor or dean.

**Final
Exam: **The final exam (take-home) is due on **Friday, Dec. 8**.

**Homework:
**Homework will be assigned **on this page (see below)** every week and will be due on the
date stated on the homework. You must hand
in the homework at the beginning of class
each Friday. Late homework will **NOT** be accepted under any circumstances. **Please check everyweek befor you start!**

**Grading:
**The course grade is apportioned as follows:
Final exam 30%; the first midterm exam 25%;
the second midterm 25%; homework grades 20%.

**Academic honesty**: It is the obligation of each student to
understand the Cornell Code of Academic Integrity regarding academic honesty and to uphold
these standards. This states, "A Cornell student's submission of work for academic credit indicates that the work s the student's own. All outside assistance should be acknowledged, and the student's academic position truthfully
reported at all times." Students are encouraged
to talk about the problems, but should write
up the solutions individually. Students should
acknowledge the assistance of any book, software,
student or professor.

**Copyright **: Course materials posted on this website
or distributed in class are intellectual
property belonging to the author. Students
are not permitted to buy or sell any course
materials without the express permission
of the instructor. Such unauthorized behavior
constitutes academic misconduct.

**Disabilities**: Students with disabilities who will be
taking this course and may need disability-related
classroom accommodations are encouraged to
make an appointment to see the instructor
as soon as possible. Also, stop by the Office of Disability Services to register for support services.

Class |
Topic (Tentative) |
Read |
Exercises (Please check everyweek befor you start!) |
Due |

Aug. 23 | Overview, examples of manifolds and differential forms | Ch 1 | HW 1: 1.1, 1.2, 1.5, 1.7 | 9/1 |

Aug. 25 | Examples | 2.1, 2.2 | ||

Aug. 28 | Differential forms in R^n | Ch 2 | ||

Aug. 30 | Differential 1-forms and 2-forms in R^n | |||

Sept. 1 | Hodge operator, exterior product, Exterior derivative, closed and exact forms | HW 2: 2.3, 2.5, 2.6, 2.7(i), 2.8, 2.9, 2.12, 2.13 | 9/8 | |

Sept. 4 | No class | |||

Sept. 6 | Examples of operations on forms | |||

Sept. 8 | Pulling back forms (I) | Ch 3 | HW 3: Read 3.1; Page 28: 2.15, 2.18, 2.19, | 9/15 |

Sept. 11 | Pulling back forms (II) | Page 45: 3.7, 3.14, 3.15, 3.17, 3.18 | ||

Sept. 13 | Integration of 1-forms | Ch 4 | ||

Sept. 15 | Exact 1-forms, angle functions | HW 4: 4.1, 4.3, 4.4, 4.6, 4.7 (optional) | 9/22 | |

Sept. 18 | Examples | |||

Sept. 20 | Poincare Lemma | Ch 5 | ||

Sept. 22 | Poincare Lemma (II) | HW 5: 3.16, 4.7, 4.11 | 9/29 | |

Sept. 25 | Integration of forms over chains | |||

Sept. 27 | Stokes' theorem | |||

Sept. 29 | Prelim 1 | HW 6: 5.3, 5.5, read 5.6 | 10/6 | |

Oct. 2 | Smooth manifolds (I) | Ch 6 | ||

Oct. 4 | Smooth manifolds (II) | |||

Oct. 6 | Tagent space | HW 7: 6.1, 6.2, 6.3 | 10/13 | |

Oct. 9 | No class | |||

Oct. 11 | Forms on manifolds (I) | |||

Oct. 13 | Forms on manifolds (II) | HW 8: 6.6, 6.8, 6.14, read: 6.13 | 10/20 | |

Oct. 16 | Forms on manifolds (III) | |||

Oct. 18 | Lie Bracket | |||

Oct. 20 | Examples | Hw 9: 6.9, 6.10, 6.17, read 6.12, 6.16 | 10/27 | |

Oct. 23 | Volume forms | Ch 7 | ||

Oct. 25 | Partition of Unity | |||

Oct. 27 | Green's Theorem | Ch 8 | ||

Oct. 30 | Prelim 2 | |||

Nov. 1 | Manifolds with boundary | Ch 9 | ||

Nov. 3 | Integration on manifolds | HW 10: 7.4, 8.8, 9.1, 9.2, read: 7.8, 8.9 | 11/10 | |

Nov. 6 | Stoke's Theorem on manifolds | |||

Nov. 8 | Poincare Lemma (1) | |||

Nov. 10 | Poincare lemma (2) | HW 11: 9.4, 9.5, [DC] Page 70, 4.3, Extra credit: 4.5 | 11/17 | |

Nov. 13 | Structure equation in R^n | |||

Nov. 15 | Cartan's lemma | |||

Nov. 17 | Structure equation in R^3, curvatures | HW 12: TBA | 12/1 | |

Nov. 20 | Gauss curvature (I) | |||

Nov. 27 | Gauss curvature (II) | |||

Nov. 29 | Example | |||

Dec. 1 | Application | Ch 10 |