COURSE SYLLABUS

DIFFERENTIAL GEOMETRY I

1	Course Title:	DIFFERENTIAL GEOMETRY I
2	Course Code:	MAT3015
3	Type of Course:	Compulsory
4	Level of Course:	First Cycle
5	Year of Study:	3
6	Semester:	5
7	ECTS Credits Allocated:	6
8	Theoretical (hour/week):	2
9	Practice (hour/week) :	2
10	Laboratory (hour/week) :	0
11	Prerequisites:	MAT 2013 Analytic Geometry I, MAT 2013 Analytic Geometry II
12	Recommended optional programme components:	None
13	Language:	Turkish
14	Mode of Delivery:	Face to face
15	Course Coordinator:	Prof. Dr. Kadri Arslan
16	Course Lecturers:
17	Contactinformation of the Course Coordinator:	arslan@uludag.edu.tr (0 224) 294 17 75 Uludağ Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü
18	Website:
19	Objective of the Course:	The purpose of this course to teach the basic concepts of differential geometry undergraduate level students. Especially some concepts of Euclidean space was introduced. Such as tangent vectors, tangent space, vector space, space of vector fields, directional derivative, cotangent space, 1-form are introduced. However, the course aims are to examine and curves, velocity vector of the curve, and the Serret-Frenet curvatures and Serret-Frenet formulas of the curves in Euclidean spaces.
20	Contribution of the Course to Professional Development

Learning Outcomes:

1	He/She defines the basic concepts of differential geometry.;
2	He/She relates mathematics and fundamental sciences to discipline of differential geometry,;
3	He/She compares the structure of affine space with structure of Euclidean space,;
4	He/She decides to the Euclidean space is a topologic space,;
5	He/She adapts concepts of directional derivative and differentiation from analysis courses to directional derivative along a vector and differentiation on manifolds,;
6	He/She adapts functions of gradient divergence and rotational from analysis courses to functions on manifolds,;
7	He/She defines the concept of the curve,;
8	He/She constructs the Frenet frame of the curve,;
9	He/She formulates the curvatures of the curve,;
10	He/She defines and characterizes the types of the curves,;

22	Course Content:

Week	Theoretical	Practical
1	The concepts of differentiable functions, Euclidean space, Euclidean coordinates, the Euclidean frame are handled.	Some examples of a differentiable functionsare given
2	Tangent vectors, tangent space, vector fields are considered.	Some examples of a tangent vectors and vector fields are given
3	The directional derivative of a function is given.	Some examples of a directional derivative are given
4	Curves, the parameters, arc length of the curve are discussed.	Some examles of arc length of the curve are given
5	Serret-Frenet formulas, and curvatures are analyzed.	Some examles of Serret-Frenet curvatures are given
6	Osculator planes of the curve, the circle of curvature, curvature of the sphere, osculator sphere are discussed.	Some examles of osculator planes of the curve are given
7	Spherical curves and lines of curvatures are characterized.	Some examles of lines of curvatures are given
8	Repeating courses and midterm exam	The classification of curves are given.
9	Integral curves of a curve are discussed.	Some examles of integral curves are given
10	Evolute and involute, Bertrand curve, indicatrix of a curve are analyzed.	Some examles of evolute and involutes are given
11	Helices, and some special curves are discussed.	Some examles of some special curves are given
12	Transformations and isometries of Euclidean spaces and orientation are discussed.	Some examles of isometry and orientation are given
13	Diffrential map, Jacobien matrix of diffrential map and covariant derivatives are analyzed.	Some examles of covariant derivatives are given
14	Lie bracket operator, 1-forms, gradient, divergence and rotational functions are handled.	Some examles of gradient, divergence and rotational of the functions are given

23	Textbooks, References and/or Other Materials:	O’Neill, B., Elementary Differential Geometry, Academic Press, New York, 1966. G . Gray, A. “Modern Differential Geometry of Curves and Surfaces”. CRC Press, Boca Raton Ann Abor London Tokyo, 1993. Andrew Pressley, Elemantary Differential Geometry, Springer-Verlag London Limited, Great Britain, 2001.
24	Assesment

TERM LEARNING ACTIVITIES	NUMBER	PERCENT
Midterm Exam	1	40
Quiz	0	0
Homeworks, Performances	0	0
Final Exam	1	60
Total	2	100
Contribution of Term (Year) Learning Activities to Success Grade		40
Contribution of Final Exam to Success Grade		60
Total		100
Measurement and Evaluation Techniques Used in the Course
Information

ECTS / WORK LOAD TABLE

Activites	NUMBER	TIME [Hour]	Total WorkLoad [Hour]
Theoretical	14	2	28
Practicals/Labs	14	2	28
Self Study and Preparation	10	4	40
Homeworks, Performances	0	0	0
Projects	0	0	0
Field Studies	0	0	0
Midtermexams	1	2	2
Others	2	40	80
Final Exams	1	2	2
Total WorkLoad			180
Total workload/ 30 hr			6
ECTS Credit of the Course			6

CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS

LO: Learning Objectives

PQ: Program Qualifications

Contribution Level:

1 Very Low

2 Low

3 Medium

4 High

5 Very High