ANNOUNCEMENTS
Midterm I Solutions Midterm II Solutions Final Exam Solutions
Course information
| Instructor: | Ibrahim Unal |
| Classes: | Tu 10:40-12:30, Th 08:40-10:30 |
| Office-Hours: | Th 10:40-12:30 |
| Teaching Assistant | Deniz Genlik [Office Hours: F 15:40-17:30] |
| Textbook | Barrett O'neill "Elementary Differential Geometry" 2nd Ed. |
Course Content
Curves in 3 space: Local Theory of curves. Frenet formulas and Fundamental Theorem.Regular surfaces, definition and examples. Inverse image of regular values. Change of parameters, differentiable functions on surfaces The tangent plane; The differential of a map, vector fields, the first fundamental form. Gauss map, second fundamental form, normal curvature,principal curvature and principal directions, asymptotic directions. Gauss map in local coordinates. Covariant derivative, geodesics.
Grading
| Midterm 1 | 35% | March 29th, Thursday at 17:40 |
| Midterm 2 | 35% | April 26th, Thursday at 17:40 |
| Final | 50% | June 1st, Friday at 09:30 |
Schedule
| Weeks | Topics | Problems |
| 1 | 1.1 Euclidean Space 1.2 Tangent Vectors 1.3 Directional Derivatives 1.4 Curves in R^3 |
1.1: 3,4 1.2: 3(d,e), 5(b) 1.3. 1(a),3(c,f)4,5 |
| 2 | 1.5 1-Forms 1.6 Differential Forms 1.7 Mappings 2.1 Dot Product |
1.5: 1(c),3,4(b,c),6(c),7,10 1.6: 1-9 1.7: 3,4,6,7,9,10 |
| 3 | 2.2 Curves 2.3 The Frenet Formulas 2.4 Arbitrary Speed Curves |
1.4: 4,6,7,9 2.1: 5,11,12 2.2: 3,5,6,8,10,11 2.3: 1,2,6,7,10,11 |
| 4 | Planar Curves (4.7 of E. Bloch) 2.5 Covariant Derivatives 2.6 Frame Fields |
2.4: 1,2,3,5,7,12,16,17,18 2.5: 1(b),2(c,d,e),3,5 2.6: 1,2(c) |
| 5 | 3.1 Isometries of R^3 3.2 The Tangent Map of an Isometry 3.3 Orientation 3.4 Euclidean Geometry |
3.1: 4,6,9 3.2: 3,4 3.3: 3,4,5 3.4: 1(b),2,4,5 |
| 6 | 3.5 Conguence of Curves + Fundamental Theorem of Curves (4.6 of E. Bloch) | 3.5: 1,3,6,7
|
| 7 | 4.1 Surfaces in R^3 4.2 Patch Computations |
4.1: 1,4,5,8,9,10,11 4.2: 1,2,3,5,7,8,9(a,b),10(b),11(c) |
| 8 | 4.3 Differentiable Functions and Tangent Vectors 5.1 The Shape Operator of M in R^3 + Gauss map
|
4.3: 1(b),2,3,4,5,6(b),7,12 5.1: 3(c,d),4,5,7,9 |
| 9 | 5.2 Normal Curvature 5.3 Gaussian Curvature 5.4 Computational Techniques |
5.2: 1 5.3: 1-4,7 5.4: 1-3,5,7-15 |
| 10 | 5.5 The Implicit Case |
|
| 11 | Isometries and Theorema Egregium | 6.4:1,8,9,14 |
| 12 | 5.6 Special Curves in Surface: Geodesics | 5.6:3,17a,19 |
| 13 | 6.3 Some Global Theorems | 6.3: 1,3 |
| 14 | Gauss-Bonnet Theorem* |
Additional Sources
A First Course in Geometric Topology and Differential Geometry, Ethan D. BlochProf. Cem Tezer's lecture notes