課程資訊
課程名稱
幾何學導論
Introduction to Geometry 
開課學期
108-1 
授課對象
理學院  數學系  
授課教師
崔茂培 
課號
MATH5356 
課程識別碼
221 U6580 
班次
 
學分
4.0 
全/半年
半年 
必/選修
必帶 
上課時間
星期三3,4(10:20~12:10)星期五3,4(10:20~12:10) 
上課地點
普503普503 
備註
此課程研究生選修不算學分。
限學士班學生
總人數上限:70人
外系人數限制:30人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1081MATH5356_ 
課程簡介影片
 
核心能力關聯
本課程尚未建立核心能力關連
課程大綱
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課程概述

Applying basic calculus and linear algebra to study the surface inside 3 dimensional space gives fruitful results and forms the basis of modern differential geometry, which, in turns, provide the framework of general relativity and quantum field theory.

Our course would concentrate on the concept of "curvature", "surface" and the interplay between them. We shall conduct a course from the local behavior of a surface to the global property of it. 

課程目標
1. Curvature of Curves.
2. Curvature of Surfaces.
3. Internal Curvature of Surfaces.
4. Curvature and Global Property of Surfaces 
課程要求
待補 
預期每週課後學習時數
 
Office Hours
 
參考書目
J. Oprea, Differential Geometry and its Applications, 2nd edition, 2007. 
指定閱讀
Elementary Differential Geometry ,Second Edition
by Andrew Pressley

It is available from Springer link (it can be downloaded from NTU IP)
https://link.springer.com/book/10.1007%2F978-1-84882-891-9
 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
Midterm 
35% 
 
2. 
Final 
35% 
 
3. 
Homework 
20% 
 
4. 
Quiz 
10% 
Average from best three quizzes 
 
課程進度
週次
日期
單元主題
第1週
9/11,9/13  9/11 Ch 1 . Curves in the plane and in space

Software to graph parametric curve

https://www.desmos.com/calculator/ksjcpazwa9

Software to draw level curve

https://www.desmos.com/calculator/scxe341uyn
13日 中秋節(放假日)
Elementary Differential Geometry ,Second Edition
by Andrew Pressley

It is available from Springer link (it can be downloaded from NTU IP)
https://link.springer.com/book/10.1007%2F978-1-84882-891-9  
第2週
9/18,9/20  9/18 More on Ch 1 . Curves in the plane and in space
& 2.1 Curvature

9/19 2.2 Plane curves 2.3 Space curves

I updated the lecture to fix some typos in the note.

HW1-1.pdf and HW1-2.pdf are the HW problems from the book. 
第3週
9/25,9/27  9/25 2.3 Space curves

9/27 2.3 Space curves 4.2 Smooth surfaces (Regular surfaces)
You can use
https://www.geogebra.org/3d
to graph the parametric surface. Please see the lecture note on Sep 27 for more detail. 
第4週
10/02,10/04, 10/05  10/2 4.1 What is a surface? 4.2 Smooth surfaces
10/4 4.1 What is a surface? 4.2 Smooth surfaces 4.4 Tangents and derivatives


10/5 Make up class for Oct 11
4.4 Tangents and derivatives
4.3 Smooth maps
4.5 Normals and orientability
6.1 First fundamental form


video of Mobius band at https://youtu.be/gibTQyDmQPQ

If you are interested learning more about topology, you can take a look at this book "Introduction to Topological Manifolds by John M. Lee."
https://link.springer.com/book/10.1007%2F978-1-4419-7940-7 You can download this book form NTU IP.
 
第5週
10/09,10/11  10/9 6.1 Lengths of curves on surfaces
6.2 Isometries of surfaces
11日 國慶紀念日調整放假 
第6週
10/16,10/18  0/16 6.3 Conformal mappings of surfaces
6.4 Equiareal maps and a theorem of Archimedes

10/18 7.1 The second fundamental form 7.2 The Gauss and Weingarten maps
 
第7週
10/23,10/25  10/23
Quiz on Oct. 23 TA session
7.2 The Gauss and Weingarten maps
7.3 Normal and geodesic curvatures
10/25
7.3 Normal and geodesic curvatures
8.1 Gaussian and mean curvatures
8.2 Principal curvatures of a surface  
第8週
10/30,11/01  10/30 8.2 Principal curvatures of a surface 7.4 Parallel transport and covariant derivative
9.1 Definition and basic properties

11/1 TA session 
第9週
11/06,11/08  11/06 9.2 Geodesic equations 9.5 Geodesic coordinates

11/08 First Midterm  
第10週
11/13,11/15  Midterm Break (No class)
15日 本校校慶(停課一天) 
第11週
11/20,11/22  11/20 10.1 The Gauss and Codazzi–Mainardi equations 10.2 Gauss’ remarkable theorem

11/22 13.1 Gauss–Bonnet for simple closed curves 13.2 Gauss–Bonnet for curvilinear polygons
 
第12週
11/27,11/29  11/27 13.3 Integration on compact surfaces 13.4 Gauss–Bonnet for compact surfaces

11/29 13.6 Holonomy and Gaussian curvature 13.7 Singularities of vector fields 
第13週
12/04,12/06  12/04,12/06 Abstract manifolds, Higher dimensional geometry 
第14週
12/11,12/13  12/11,12/13 Higher dimensional geometry 
第15週
12/18,12/20  12/25 12/27
Topics on discrete surface 
第16週
12/25,12/27  12/25 12/27
Topics on discrete surface 
第17週
1/01,1/03   1日 開國紀念日(放假日)
1/03 Topics on discrete surface 
第18週
1/6-1/10  Final exam on Jan 8th (Wednesday) 10:20 a.m. - 1 pm