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 | Natural & Applied Sciences Master's Degree |
| 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 | |
|---|---|
| Major Area Courses | |
| Supportive Courses | X |
| 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 | Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 4 | Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary. | X | ||||
| 5 | Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 8 | Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 11 | Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. | X | ||||
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 | ||