Ders Bilgileri

#### Ders Tanımı

Ders Kodu Yarıyıl T+U Saat Kredi AKTS
DIFFERENTIAL GEOMETRY I MAT 303 5 3 + 1 4 6
Ön Koşul Dersleri Students are assumed to be familiar with Analytic Geometry and Linear Algebra
 Dersin Dili Türkçe Dersin Seviyesi Lisans Dersin Türü ZORUNLU Dersin Koordinatörü Prof.Dr. MEHMET ALİ GÜNGÖR Dersi Verenler Prof.Dr. MURAT TOSUN Dr.Öğr.Üyesi HİDAYET HÜDA KÖSAL Dersin Yardımcıları Research assistants of geometry Dersin Kategorisi Dersin Amacı This courses aim is to give the fundamental concepts of the differential geometry. To acquaint students with topological manifolds and Euclidean Space as a manifold in this space. Also teaching tangent vectors, tangent space, and the space of vector fields, directional derivative, cotangent spaces and 1-forms. Acquainting students with curve theory, Frenet vectors and varieties of curve. Learning this theory from the technical aspect and showing that how any problem can be solved by this way. Dersin İçeriği Euclidean Space, Differentiable Functions, Tangent Spaces, Vector Field, Derivative, Transformation, Covariant Derivative, Lie Operator, Lie Algebra, Cotangent Vectors, Cotangent Spaces and 1-forms, Gradient, Divergence and Curl Functions, The Differential of Transformation, Submanifolds, Tensors and Tensor Spaces, Analysis of the curves, Frenet Formulas, Osculating Sphere, Spherical Curves, Bertrand curves, Helix, Evolutes and Involutes.
 Dersin Öğrenme Çıktıları Öğretim Yöntemleri Ölçme Yöntemleri 1 - He/She defines the basic concepts of differential geometry, 1 - 2 - 3 - A - C - 2 - He/She relates mathematics and fundamental sciences to discipline of differential geometry, 2 - 3 - 4 - 15 - A - C - 3 - He/She compares the structure of affine space with structure of Euclidean space, 2 - 3 - 4 - A - C - 4 - He/She decides to the Euclidean space is a topologic space, 1 - 2 - 3 - A - C - 5 - He/She solves the problems related to manifolds, 2 - 4 - 15 - A - C - 6 - He/She adapts concepts of directional derivative and differentiation from analysis courses to directional derivative along a vector and differentiation on manifolds, 1 - 2 - 3 - 4 - 15 - A - C - 7 - He/She adapts functions of gradient divergence and rotational from analysis courses to functions on manifolds, 1 - 2 - 3 - 4 - 15 - A - C - 8 - He/She defines and categorizes the tensors. 1 - 2 - 3 - A - C - 9 - He/She defines the concept of the curve, 1 - 2 - 3 - A - C - 10 - He/She constructs the Frenet frame of the curve, 1 - 4 - 6 - A - C - 11 - He/She formulates the curvatures of the curve, 1 - 4 - 6 - A - C - 12 - He/She categorizes the tangent spaces at a point of the curve, 1 - 2 - 3 - A - C - 13 - He/She calculates algebraic invariants of the curve, 1 - 2 - 3 - 4 - 15 - A - C - 14 - He/She defines and characterizes the types of the curves, 1 - 2 - 3 - A - C -
 Öğretim Yöntemleri: 1:Lecture 2:Question-Answer 3:Discussion 4:Drilland Practice 15:Problem Solving 6:Motivations to Show Ölçme Yöntemleri: A:Testing C:Homework

#### Ders Akışı

Hafta Konular ÖnHazırlık
1 Affine Space, Euclidean Space, Euclidean Frame, Topological Manifolds, Differentiable Manifold Concept
2 Tangent Vectors, Tangent Space, Vector Fields, Directional derivative, Covariant Derivative
3 Integral curves, Lie Operators, Lie Algebra
4 Cotangent Vectors, Cotangent Spaces and 1-forms
5 Gradient, Divergence and Curl Functions, the Differentiation of Transformation
6 Submanifolds
7 Tensors and Tensor Spaces
8 Exterior Product Space, Vector Product
9 Midterm
10 Analysis of the curve, Reparametrization, Velocity vector of the curves, Serret-Frenet Formulas
11 Curvatures, Circle of Curvature, Sphere of the curvature, Osculating Sphere
12 Spherical Curves
13 Inclination Lines
14 Evolutes and Involutes, Bertrand curves, Spherical indications of a curve

#### Kaynaklar

Ders Notu 1. Hacısalihoğlu, H. H., Diferensiyel Geometri, Cilt I, Ankara Üniversitesi, Fen Fakültesi Matematik Bölümü, 1998.
2. Hacısalihoğlu, H. H., Çözümlü Diferensiyel Geometri Problemleri, Cilt I, Ankara Üniversitesi, Fen Fakültesi Matematik Bölümü, 1995.
Ders Kaynakları 1. Sabuncuoğlu, Arif. Diferensiyel Geometri, Nobel Yayınları, Ankara, 2001.
2. ONeill, B., Elementary Differential Geometry, Academic Press, New York, 1966.
3. Lipschutz, M. M., Theory and problems of Differential Geometry, Schaums Outline Series, McGraw-Hill, New York, 1969

#### Dersin Program Çıktılarına Katkısı

No Program Öğrenme Çıktıları KatkıDüzeyi
1 2 3 4 5
1 He/ she has the ability to use the related materials about mathematics, constructed on competency, achieved in secondary education and also has the further knowledge equipment. X
2 Evaluating the fundamental notions, theories and data with academic methods, he/ she determines and analyses the encountered problems and subjects, exchanges ideas, improves suggestions propped up proofs and inquiries. X
3 He/ she has the competency of executing the further studies of undergraduate subjects independently or with shareholders. X
4 He/ she follows up the knowledge of mathematics and has the competency of getting across with his (or her) professional colleagues within a foreign language. X
5 He/ she has the knowledge of computer software information as a mathematician needs. X
6 He/ she has scientific and ethic assets in the phases of congregating, annotating and announcing the knowledge about mathematics. X
7 He/ she has the ability to make the mathematical models of contemporary problems and solving them. X
8 He/ she uses the ability of abstract thinking. X

#### Değerlendirme Sistemi

YARIYIL İÇİ ÇALIŞMALARI SIRA KATKI YÜZDESİ
AraSinav 1 70
KisaSinav 1 10
KisaSinav 2 10
Odev 1 10
Toplam 100
Yıliçinin Başarıya Oranı 50
Finalin Başarıya Oranı 50
Toplam 100

#### AKTS - İş Yükü

Etkinlik Sayısı Süresi(Saat) Toplam İş yükü(Saat)
Course Duration (Including the exam week: 16x Total course hours) 16 4 64
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 11 11
Quiz 2 8 16
Assignment 1 8 8
Final examination 1 15 15
Toplam İş Yükü 162
Toplam İş Yükü /25(s) 6.48
Dersin AKTS Kredisi 6.48
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