國立中央大學一百零五學年度下學期幾何學ㄧ課程網頁


宣佈事項：

 6/14上課時間舉行第三次段考，考試範圍：Section2.3-2.4, Section 3.1

 5/10上課時間舉行第二次段考，考試範圍：Section1.1-1.3, Section 2.1-2.2

3/22上課時間舉行第一次段考


考試 : 段考一(含解答), 段考二(含解答), 期末考(含解答)

作業：作業1 (due 3/2), 作業2 (due 3/9), 作業3 (due 3/16), 作業4 (due 4/6), 作業5 (due 4/20), 作業6 (due 4/27), 作業7 (due 5/4), 作業8 (due 5/25), 作業9 (due 6/1), 作業10 (due 6/8)
 
 
課程進度 :
 
Part I : Analytic Geometry:
Week1:
2/15: Conic sections, focus-directrix definition of the non-degenerate conics (parabola, ellipse)
         注：課本第15頁Example 3要加上條件 "...is independent of the choice of P and Q if the slopes of and OM and PQ are defined:"    
2/16: Focus-directrix definition of the non-degenerate conics (hyperbola), recognizing conics
Week2:
2/22: Recognizing conics (continued), Geometry and transformations (Euclidean geometry)
2/23: Euclidean geometry (continued)
Week3:
3/1: Affine geometry
3/2: Projective plane
Week4:
3/8: Projective transformation, Fundamental theorem of projective geometry
3/9: Fundamental theorem of projective geometry (continued), Cross ratio
Week5:
3/15: Cross ratio, projective conics
3/16: Kleinian view of geometry, review for midterm 1
Week6:
3/22: Midterm 1

Part II : Differential Geometry
3/23: Curves, Examples
Week7:
3/29: Arclength parametrization, Frenet frame for arclength-parametrized curves
3/30: Frenet frame for non-arclength-parametrized curves
Week8:
4/5: No class (School holiday)
4/6: Frenet frame for non-arclength-parametrized curves(continued)
Week9:
4/12: Fundamental theorem of curve theory, Hopf Umlaufsatz (Hopf rotation theorem)
4/13: Hopf Umlaufsatz (continued)
Week10:
4/19: Parametrized surfaces, First fundamental form
4/20: Shape operator
Week11:
4/26: Shape operator (continued), Second fundamental form
4/27: Second fundamental form (continued)
Week12:
5/3: Codazzi and Gauss equations
5/4: Review for midterm2
Week13:
5/10: Midterm 2
5/11: Fundamental Theorem of Surface
Week14:
5/17: Covariant differentiation, Parallel translation
5/18: Parallel translation (continued)
Week15:
5/24: Geodesic, Clairaut's relation
5/25: Geodesic(continued), Holonomy
Week16:
5/31: Holonomy and the Gauss-Bonnet Theorem
6/1: Holonomy and the Gauss-Bonnet Theorem (continued)
Week17:
6/7: The Gauss-Bonnet Theorem (continued)
6/8: Review for Final
Week18:
6/14: Final
Office hour and Office:

黃榮宗老師：星期三下午1點至3點或預約, 鴻經館418, 分機：65152

廖欣瑩助教：預約，鴻經館417, 分機：65134

課本：

1. 段考一： Geometry by D. A. Brannan, M. F. Esplen, J. J. Gray, 2nd edition, 上課講義請至LMS系統下載
2. 段考二及期末考： Differential Geometry: A First Course in Curves and Surfaces, Lectures notes by Professor Theodore Shifrin, Link

參考書：

1. 邱鴻麟老師講義  
2. Elementary Differential Geometry, by Andrew Pressley, 2nd edition. Springer
授課大綱及建議習題：

Part I: Analytic Geometry (Geometry by D. A. Brannan, M. F. Esplen, J. J. Gray, 2nd edition)

1. Conic Section: P.14 Problem 7; P.16 Problem 8; P.18 Problem 9; P.38 Problem 1; P.42 Problem 2, Problem 3; P.52 Section 1,1 #4, #5; P. 55 Section 1.3 #1
2. Affine Geometry: Section 2.1 Problem 3~8; Section 2.2 Probelm 1~3; Section 2.3 Probelm 1~5;
                             Exercises: Section 2.1 #2~4, #5; Section 2.2 #1~4, Section 2.3 #1~6, Section 2.5 #2
3. Projective Geometry: Section 3.2, Problem 1~10, Section 3.3 Problem 1~9, Section Problem 1, 2, 3, 7
                             Exercises: Section 3.2 #1~6, Section 3.3 #1~6, Section 3.5 #1~5

Part II: Differential Geometry (Lecture notes by Professor Theodore Shifrin)

Chapter 1. Curves:
    1. Examples, Arclength Parametrization: 1, 2, 3, 4, 5, 6, 7, 8, 12
    2. Local Theory: Frenet Frame: 1, 2, 3, 4, 5, 6, 7, 8, 10, 11
    3. Some Global Results : 1, 3, 4
Chapter 2. Surfaces: Local theory
    1. Parametrized Surfaces and the First Fundamental Form: 1, 2, 3, 4, 5, 6, 7, 8, 11(a), 15, 16(a)
    2. The Gauss Map and the Second Fundamental Form: 1, 2, 3, 4(Find the principal curvatures and the principal directions only), 6, 7, 10(b), 13
    3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory: 1, 2, 3, 4, 5, 7, 9, 12, 18(a)(c)(d), 19
    4. Covariant Differentiation, Parallel Translation, and Geodesics: 1, 2, 3, 4, 6, 7, 9, 10, 13(a)(b), 19
3. Surfaces: Global theory
    1. Holonomy and the Gauss-Bonnet Theorem: 1, 2, 3, 6, 8(a)(b)(c)(d), 9(a)(b)(c)(d), 11(a)(b)


評量配分比重：
  
作業20%+第一次期中考25%+第二次期中考25%+期末考30%

舊課程網頁：

一百零三學年度下學期幾何學ㄧ課程網頁