Math 457 ~ CALCULUS ON MANIFOLDS  (Fall 2021)

Schedule:  (face-to-face and synchronic lectures)                                                            
Mondays    13:40-15:30, room: M-105

Wednesdays  12:40-13:30, room: M-105

Zoom Link for Official Lecture Meetings:

https://zoom.us/j/6558344427?pwd=Y2ZBNFpGbkl4ZkFSNTB1KzRETzgzQT09

(NO Password required if you use the link above)

(recorded lectures will be posted online HERE) 

Instructor: M. Fırat Arıkan [office: M-130, email: farikan(at)metu.edu.tr]

Office Hours will be held online as follows:

Monday 11:00 -11:40

Wednesday 10:00 -10:40

Office Hour Zoom Link:

https://zoom.us/j/6558344427?pwd=Y2ZBNFpGbkl4ZkFSNTB1KzRETzgzQT09

(NO Password required if you use the link above)

Announcements: All course announcements will be posted here!

*** SUGGESTED EXERCISES are available in METUClass***

Course Content:

Review of differentiation, inverse and implicit function theorems, integration on subsets of Euclidean space, tensors, differential forms, integration on chains, integration on manifolds. Stokes` theorem, and some applications.

Syllabus:

Week 1-2          Introduction and Preliminaries: Differentiability, Inverse Function Theorem

Week 3-4          Manifolds, Rank Theorems, Tangent Space, Vector Fields

Week 5-6          Differential Forms and operations on them

Week 7-9          Integration of Differential Forms, Volumes of discs and spheres

Week 10           Stokes’ Theorem and some applications

Week 11-14      De Rham Cohomology and Applications: Degree Theory, Gauss-Bonnet Theorem, Rotation and Linking number (as time permits)

Prerequisite Course: One of the courses 252, 262, 264, or 272 should be completed before taking MATH 457.

Textbook: We will mainly (but not exactly) follow the related sections of the following books: ''Calculus on Manifolds'' by Michael Spivak, and ''Differentiable Manifolds'' by Nigel Hitchin.

Exams and Grading:
 
 There will be one Midterm (M) and one Final Exam (F) [and an Oral Exam (OE) if a student choose Option-2 below].

Option-1: The student takes all exams in class (face-to-face)

Grade distribution: M (%45)+ F(%55) = Total (%100)   

Option-2: The student takes all exams online (via recorded Zoom meeting)

Grade distribution: M (%35)+ F(%45) + OE(%20) = Total (%100)  

The letter grade will be assigned by the instructor according to the distribution of weighted total grades.
 
Midterm :   Time:    To be announced later
                        Topics included:   To be announced later
 
Final Exam:   Time:   To be announced later
                         Topics included: Everything seen during the semester.