ECTS - Calculus on Manifolds
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) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer. |
| Course Lecturer(s) |
|
| Course Objectives | 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. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | 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. |
|---|---|
| Other Sources | 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
| Requirements | Number | Percentage of Grade |
|---|---|---|
| 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 |
|---|---|
| Percentage of Final Work | 40 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
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 | |||||
ECTS/Workload Table
| 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 |
| Total Workload | 35 | ||