微分几何的讲义 王幼宁.pdf

目 录 第一章 预备知识 ·····························································································1······ §1 向量代数复习···················································································1 一．E3 中Descartes 直角坐标系 O-xyz 中的点与向量····················1· 二．向量空间R3 （起点自由） ······················································1 三．E3 中向量的乘积······································································2· 四．在初等几何中的应用例示 ························································3 §2 向量函数微积分 ···············································································4 一．E3 中实变向量函数 ···································································4 二．向量函数的极限、连续和微积分简介 ·····································5 三．常用几何条件的解析判定式 ····················································7 §3 标架和标架场 ···················································································9 一．E3 中的单位正交右手标架及其变换 ······································10 二．E3 中的刚体运动与等距变换··················································12 三．E3 中的正交标架场的运动公式··············································13 四．E3 中的仿射标架·····································································13 第二章 曲线的局部微分几何··········································································1·5· §1 参数化曲线与曲线的参数表示 ······················································15 一．E3 中参数化曲线的定义 ·························································15 二．正则曲线·················································································16 三．曲线的等价 ·············································································18 §2 曲线的弧长和弧长元素····················