Manifolds and Differential Forms: MATH 3210
Fall 2019

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

E-mail: xiaodongcao at cornell.edu

Teaching Assistant: Daoji Huang (Email: dh539 at cornell.edu OH: Wed. 1:30-3:30 532 MLT)

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

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

Text:  Manifolds and differential forms , lecture notes by Reyer Sjamaar, [2015]. You can download them at the address " http://http://www.math.cornell.edu/~sjamaar/manifolds/" .Differential Forms and Applications by Manfredo P. do Carmo , [1994];

Other References:  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. 4th  and Friday, Nov. 8th.   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 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 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.


 

Schedule of Lectures

Class Topic (Tentative) Read   Exercises (Please check everyweek befor you start!) Due
Aug. 30 Overview, examples of manifolds and differential forms Ch 1 HW 1: 1.1, 1.2, 1.5, 1.7 9/9
Sept. 4 Examples   2.1, 2.2  
Sept. 6 Differential forms in R^n Ch 2    
Sept. 9 Differential 1-forms and 2-forms in R^n      
Sept. 11 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 9/16
Sept. 13 Examples of operations on forms      
Sept. 16 Pulling back forms (I)      
Sept. 18 Pulling back forms (II) Ch 3 HW 3: Read 3.1; Page 28: 2.15, 2.18, 2.19, 9/23
Sept. 20 Integration of 1-forms   Page 45: 3.7, 3.14, 3.15, 3.17, 3.18  
Sept. 23 Angle functions, Poincare Lemma (0) Ch 4    
Sept. 25 Examples, gradient, divergence, curl   HW 4: 4.1, 4.3, 4.4, 4.6, 4.7 (optional) 9/30
Sept. 27 Poincare Lemma (I)      
Sept. 30 Integration of forms over chains Ch 5    
Oct. 2 Stokes' theorem   HW 5: 3.16, 4.7, 4.11 10/7
Oct. 4 Prelim 1      
Oct. 7 Smooth manifolds (I)      
Oct. 9 Tagent space   HW 6: 5.3, 5.5, read 5.6 10/18
Oct. 11 Immersion, embedding Ch 6    
Oct. 14 NO CLASS      
Oct. 16 Smooth manifolds (II)   HW 7: 6.1, 6.2, 6.3 10/25
Oct. 18 Lie Bracket      
Oct. 21 Forms on manifolds (I)      
Oct. 23 Forms on manifolds (II)   HW 8: 6.6, 6.8, 6.14, read: 6.13 11/1
Oct. 25 Forms on manifolds (III)      
Oct. 28 Integration on manifolds      
Oct. 30 Partition of Unity   Hw 9: 6.9, 6.10, 6.17, read 6.12, 6.16 11/8
Nov. 1 Green's Theorem, Stoke's Theorem (I) Ch 7    
Nov. 4 Stoke's Theorem (II), Divengence Theorem      
Nov. 6 Examples Ch 8    
Nov. 8 Prelim 2      
Nov. 11 Poincare Lemma (I) Ch 9    
Nov. 13 Poincare lemma (II)   HW 10: 7.4, 8.8, 9.1, 9.2, read: 7.8, 8.9 11/15
Nov. 15 Examples      
Nov. 18 Laplace operator      
Nov. 20 Structure equation in R^n (I)   HW 11: 9.4, 9.5, [DC] Page 69-71, 4.1, 4.3, Extra credit: 4.5 11/22
Nov. 22 Structure equation in R^n (II)      
Nov. 25 Cartan's lemma      
Nov. 27-9 Thanksgiving--No class Structure equation in R^3, curvatures      
Dec. 2 Snow day, NO class      
Dec. 4 Gauss curvature and Mean Curvature   Final 12/18
Dec. 6 Example      
Dec. 9 Application to geometry and topology Ch 10