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 Coordinator
Course Lecturer(s)
Course Assistants
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; 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.
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
#	Program Qualifications / Competencies	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.

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