Manifolds and Differential Forms: MATH 3210
Fall 2024
Instructor: Xiaodong Cao, Office:521 Malott
E-mail: xiaodongcao at cornell.edu
Teaching Assistant: Isaac Broudy (Email: iab38 at cornell.edu OH: 3-4PM Friday, 218 MLT)
Time and Place: 11:15 am - 12:05 pm, MWF, 203 MLT
Office Hour: Thursday 14:00-14:50 or by appointment.
Text: Differential forms , by Victor Guillemin and Peter Haine, [2019].
Other References: Manifolds and differential forms , lecture notes by Reyer Sjamaar, [2015]; Differential Forms and Applications by Manfredo P. do Carmo , [1994]; An Introduction to Manifolds by Loring W. Tu, [2011]; Vector Calculus, Linear Algebra and Differential Forms by John H. Hubbard and Barbara Burke Hubbard.
Prerequisites: Calculus and Linear Algebra.
Midterm Exams: The midterm exams are on Friday, Oct. 9th and Monday, Nov. 11th. 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 and Project: The final exam (take-home) and project are due on Wednesday, Dec. 18.
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 turn in the homework in Gradescope 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 20%; the first midterm exam 30%; the second midterm 30%; homework grades 20%. OR Final exam 20%; the first midterm exam 20%; the second midterm 20%; homework grades 20%, final project 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.
| Week of | Topic (Tentative) | Read | Exercises (Please check everyweek befor you start!) | Due |
| Aug. 26 | Overview, examples of manifolds and differential forms, review of Linear Algebra | 1.1-1.2 | HW 0: Ex. 1.1 (Extra credit: 1 pt) | 8/30 |
| Sept. 2 | Tensors and operations | 1.3-1.4 | HW 1: Ex. 1.2: iv, v, vi; | 9/6 |
| Sept. 9 | Differential forms and operations (I) | 1.5-1.6 | HW 2: 1.3: v (only do k=2), vi, vii | 9/13 |
| Sept. 16 | Differential forms and operations (II) | 1.7-1.9 | HW 3: 1.4: 4, 6, 7, 9; 1.5: 4 | 9/20 |
| Sept. 23 | Examples of operations on forms, vector fields, 1-forms | 2.1-2.3 | HW 4: 1.5: 10; 1.6: 3, 5 | 9/27 |
| Sept. 30 | Integral curves, differential k-forms | HW 5: 1.7: 5, 6; 1.8: 4, 5; 1.9: 3, 5. | 10/4 | |
| Oct. 7 | Exterior differential, interior product, pullback on forms | HW 6: 2.1: 8, 9; 2.2: 4; 2.3: 1, 2 | 10/11 | |
| Oct. 14 | Fall Break, divergence, curl and gradient | HW 7: PDF | 11/3 | |
| Oct. 21 | Examples, gradient, divergence, curl | HW 8: [Sjamaar] Page 29: 2.18; Page 58: 4.4, 4.5 | 11/10 | |
| Oct. 28 | Poincare Lemma (I) | HW 9: | ||
| Nov. 4 | Hodge Star Operator | HW 10: | ||
| Nov. 11 | Manifolds | HW 11: | ||
| Nov. 18 | HW 12: | |||
| Nov. 25 | Thanks giving | |||
| Dec. 2 | Smooth manifolds (I) | |||
| Tagent space | Final (and?) Project | 12/18 | ||
| Immersion, embedding | ||||
| Stokes' theorem | ||||
| Smooth manifolds (II) | ||||
| Lie Bracket | ||||
| Forms on manifolds (I) | ||||
| Forms on manifolds (II) | ||||
| Forms on manifolds (III) | ||||
| Integration on manifolds | ||||
| Partition of Unity | ||||
| Green's Theorem, Stoke's Theorem (I) | ||||
| Stoke's Theorem (II), Divengence Theorem | ||||
| Examples | ||||
| Prelim 2 | ||||
| Poincare Lemma (I) | ||||
| Poincare lemma (II) | ||||
| Examples | ||||
| Laplace operator | ||||
| Structure equation in R^n (I) | ||||
| Structure equation in R^n (II) | ||||
| Cartan's lemma | ||||
| Thanksgiving--No class Structure equation in R^3, curvatures | ||||
| Snow day, NO class | ||||
| Gauss curvature and Mean Curvature | ||||
| Example | ||||
| Application to geometry and topology | Hodge operator, exterior product, Exterior derivative, closed and exact |