厦门大学本科课程教学大纲

厦门大学本科课程教学大纲课程名称微分几何课程代码英文类别代号MATH授课对象本科生三年级第一学期适用年级2016级课程类型专业课程课程课型理论课总学分总学时授课讨论实验/上机实践其他36464先修课程数学分析、高等代数、解析几何、常微分方程一、课程简介本课程是面向数学系本科生开设的几何类课程之一。经典微分几何的研究对象是三维空间中的曲线和曲面，现代微分几何的研究对象是微分流形。本课程介绍经典的微分几何，主要内容包括：曲线的局部理论、曲面的局部理论、标架与曲面论基本定理、曲面的内蕴几何学。同时，本课程将介绍一些整体微分几何的内容，如平面曲线的旋转指标定理、曲面的整体Gauss-Bonnet公式等。二、培养目标理解和掌握经典微分几何，即三维欧氏空间中曲线和曲面的微分几何的基本概念与基本方法。培养学生的几何直观和图形想象能力，从具体到抽象的能力，同时使学生初步接触到现代微分几何的基本思想和基本方法，为进一步学习微分流形，黎曼几何，以及其他数学课程提供坚实的基础。三、教学方法课堂讲授、课后作业，教学和练习相结合；以板书为主，多媒体教学为辅。四、主要内容及学时安排 章（或节）主要内容学时安排第一章 欧氏空间1.微分几何的历史简介2.向量空间3.欧氏空间2第二章 曲线的局部理论第六章 平面曲线的整体性质1.曲线的概念2.空间曲线4.曲线基本定理5.平面曲线的整体性质6. 习题课8第三章 曲面的局部理论1. 曲面的概念2. 曲面的第一基本形式3. 曲面的第二基本形式4. 法曲率与Weingarten变换5. 主曲率与Gauss曲率6. 曲面的一些例子7. 习题课12第四章 标架与曲面论的基本定理1. 活动标架2. 自然标架的运动方程3. 曲面的结构方程、曲面的存在唯一性定理4. 正交活动标架5. 曲面的结构方程（外微分法）6. 习题课12第五章 曲面的内蕴几何1. 曲面的等距变换2. 曲面的协变微分3. 测地曲率与测地线4. 测地坐标系5 曲面上的Gauss-Bonnet公式6. 习题课14第七章 曲面的若干整体性质第八章 其他话题和复习1. 整体Gauss-Bonnet公式2.常平均曲率曲面的刚性3.极小曲面简介16合计64五、考核方式与要求总评成绩的登记方式是百分制，总评成绩由期中成绩、期末成绩（笔试+口试），平时作业、考勤成绩按30%、50%、20%组成。六、选用教材彭家贵, 陈卿编著微分几何 高等教育出版社，2002.七、参考书目与文献1. Montiel and Ros. Curves and Surfaces, 2nd edition, American Mathematical Society. 2009.2. Wilhelm Klingenberg著; A Course in Differential Geometry. Springer. 1978.八、课程网站等支持条件九、其它信息大纲制定者： 杨波 大纲审定者： 大纲制定时间：XMU Undergraduate Course SyllabusCourse nameFilled out by departmentDifferential GeometryCourse code Category codeMATHProgrammeSemesterSemester 1, Year 3Course typeBasic Common Courses General Education Courses Disciplinary General Courses Specialized CoursesTick a boxOther Teaching ProcessesCourse focusTick a boxLecture Experiment Skill-training PracticalCreditTotal learning hours=L+T+E+P+OTotal learning hoursLectureTutorialExperimentPracticalOthers364640000PrerequisitesMathematical Analysis, Linear Algebra, Elementary Geometry, Ordinary differential Equations1.Course descriptionThis course is one of geometry courses facing to undergraduate student in mathematics. The research subject for classical differential geometry is curves and surfaces in Euclidean 3-spaces and the research subject for modern differential geometry is general manifolds. This course will focus on the classical one. The content includes the local theory of curves and surfaces, frames and fundamental theorem for curves and surfaces, intrinsic geometry of surfaces. The method for this courses is based on calculus.2. Learning goalsUnderstand and master the classical differential geometry, that is the basic concepts and methods in geometry of curves and surfaces in three-dimensional Euclidean space, and use this to solve some geometric problems. Train the students their capability of geometric and graphic imagination, from the concrete to the abstract. Meanwhile let the students feel the basic ideas and methods of modern differential geometry and lay a solid foundation for further study of modern mathematics and its applications.3.Teaching approachesLectures, Exercises, combination of teaching and exercising; Mainly blackboard-based teaching and multimedia teaching as supplement.4. Content outline of the courseChapter(Section)ContentLearning hoursChapter 1Euclidean spaces1. History of differential geometry2. Vector spaces3. Euclidean spaces2Chapter 2 Local theory of curvesChapter 6 Global property for planar curves1. Concept of curves2. Curves in 3-space3. Fundamental theorem for curves4. Global property for planar curves5. Exercise8Chapter 3 Local theory of surfaces1. Concept of surfaces2. First fundamental form3. Second fundamental form4. Normal curvature,Weingarten map5. Principal curvature, Gauss curvature6. Examples of surfaces7. Exercise12Chapter 4 Frames and fundamental theorem for curves and surfaces1. Frames2. Basic equations for frames3. Structure equations, Fundamental theorem for surfaces4. Orthonormal frames5. Exterior differential6. Exercise12Chapter 5intrinsic geometry of surfaces1. Isometry between surfaces2. Covariant derivative3. Geodesic, geodesic curvature4. Geodesic coordinates5. Gauss-Bonnet theorem6. Exercise14Chapter 7Global properties for surfacesChapter 9Other topics and Review1. Global Gauss-Bonnet formula2. Rigidity for Constant mean curvature surfaces3. An introduction to minimal surfaces16Total645. Assessment methods and requirementse.g. grading plan, assessment criteria, etc.Hundred mark system Attendance and Exercises: 15% Midterm examination: 35% Final examination: 50%6.TextbooksPENG Jiagui and CHEN Qing, Differential Geometry, High Education Press, 20027.References1. Montiel and Ros. Curves and Surfaces, 2nd edition, American Mathematical Society. 2009.2. Wilhelm Klingenberg; A Course in Differential Geometry. Springer. 1978.8.Website9.OthersFilled out by: Chao XIA Approved by: Date: 厦门大学本科课程大纲填写说明(Notes)1 须同时填写课程大纲中文版和英文版。2 课程名称必须准确、规范。3 课程代码：非任课教师填写。该课程在教务系统生成后，由学院代为填写。4 授课对象填写专业。5 适用年级填写可修读本课程的时间，如本科三年级第一学期。6 课程类型指公共基本课程、通识教育课程、学科通修课程、专业（或专业方向）课程、其他教学环节。7 课程课型指理论课、实验课、技能课、实践课。8 总学时=授课学时+讨论学时+实验学时+上机学时+其他学时9 先修课程是与该课程具有严格的前后逻辑关系，非先修课程则无法学习该课程。10.培养目标不少于150字。11.考核方式包括成绩登记方式、成绩组成、考核标准等。成绩登记方式包括百分制、通过/不通过等。成绩组成指各种考核方式占比。考核标准指衡量各项考评指标得分的基准。12.选用教材和主要参考书要求注明作者、书目、出版社、出版年份。例如，“丹利维尔：民主、官僚制组织和公共选择，中国青年出版社，2001年。”13.其它信息指课堂规范要求等，如课上禁止使用手机、缺勤要求等。14.课程英文类别代号:英文类别代号代号英文说明代号中文说明ANTH:Anthropology人类学类课程；ARCH:Architecture建筑类课程；ARTS:Arts艺术类；AUTO:Automation自动化类课程；BIOL:Biology生物科学类课程；BUSI:Business Administration工商管理类课程；CHEE:Chemical Engineering化工类课程；CHEM:Chemistry化学类课程；CHIN:Chinese中国语言文学类课程；CIVL:Civil Engineering土建类课程；CSCI:Computer Science计算机科学类课程；ECON:Economics经济学类课程；EENG:Electronic Engineering电子工程类课程；ELIN:Electrical Information电气信息类课程；ELIS:Electronic Information Science电子信息科学类课程；ENGL:English英国语言文学类课程；ENVS:Environmental Science环境科学类课程；FREN:French法国语言文学类课程；GERM:German德国语言文学类课程；HIST:History历史学类课程；JAPA:Japanese日本语言文学类课程；JOUR:Journalism新闻传播类课程；LAWS:Laws法学类课程；MATH:Mathematics数学类课程；MATL:Material材料类课程；MECH:Machinery机械类课程；MECM:Clinical Medical临床医学类课程；MEDN:Nursi