# Calculus on Manifolds (MATH575) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Calculus on Manifolds MATH575 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English N/A Natural & Applied Sciences Master's Degree 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 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.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
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.
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.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
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.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
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.

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