Calculus on Manifolds (MATH575) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Calculus on Manifolds MATH575 Area Elective 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English Elective Courses Ph.D. Face To Face Lecture, Discussion, Question and Answer. The aim of this course is to extend the notions of differentiation and integration, which are taught in the undergraduate program, to manifolds, and to explore the relation of these notions with geometry. The students who succeeded in this course; At the end of the course students are expected to Understand the fundamental notions of derivative, integration and tangent vectors on a Euclidean space Learn manifolds, tangent space, submanifolds, vector fields and differential forms Learn the notions of orientation and integration of manifolds (with or without boundary) and Stokes’ theorem and some of its applications. Euclidean spaces, manifolds, the tangent spaces, vector fields, differential forms, integration on manifolds, Stokes? theorem.

Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Smooth functions on a Euclidean space, Tangent vectors in R^n pp. 3-5, pp. 10-16
2 Exterior algebra of Multicovectors pp. 18-31
3 Differential forms on R^n pp. 34-44
4 Manifolds pp. 48-53
5 Smooth maps on a manifold pp. 59-68
6 Tangent space pp. 86-96
7 Submanifolds pp. 100-106
8 Midterm
9 The rank of a smooth map pp. 115-125
10 The tangent bundles, vector fields pp. 129-137, pp. 149-159
11 Vector fields (cont. ), Differential 1-forms pp. 190-197
12 Differential k-forms, The exterior derivative pp. 200-206, pp. 210-216
13 Orientations pp. 236-245
14 Manifolds with boundary pp. 248-255
15 Integration on a manifold, Stokes’ theorem pp. 260-271
16 Final Exam

Sources

Course Book 1. L. W. Tu, An Introduction to Manifolds, 2nd edition, Springer, 2011. 2. M. Spivak, Calculus on Manifolds, 24th edition, Addison-Wesley Publishing Company, 1995 . 3. J. M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer, 2013 4. N. Hitchin, Differentiable Manifolds, Lecture Notes

Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 3 30
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 1 30
Final Exam/Final Jury 1 40
Toplam 5 100
 Percentage of Semester Work 60 40 100

Course Category

Core Courses X

The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Ability to carry out advanced research activities, both individual and as a member of a team
2 Ability to evaluate research topics and comment with scientific reasoning
3 Ability to initiate and create new methodologies, implement them on novel research areas and topics
4 Ability to produce experimental and/or analytical data in systematic manner, discuss and evaluate data to lead scintific conclusions
5 Ability to apply scientific philosophy on analysis, modelling and design of engineering systems
6 Ability to synthesis available knowledge on his/her domain to initiate, to carry, complete and present novel research at international level
7 Contribute scientific and technological advancements on engineering domain of his/her interest area
8 Contribute industrial and scientific advancements to improve the society through research activities

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class
Presentation/Seminar Prepration
Project
Report
Homework Assignments 3 5 15
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 1 10 10
Prepration of Final Exams/Final Jury 1 10 10