Advanced mathematics.I

前言
Introduction
引言
Chapter 1 Functions，limits and continuity函数、极限与连续性
1.0 Cited examples引例
1.1 Functions函数
1.2 Limits极限
1.3 Properties and operations oflimits极限的运算与性质
1.4 Monotone boundedness principle and irrational number e单调有界原理与无理数e
1.5 Comparison between infinitesimals无穷小之间的比较
1.6 Continuity and discontinuity of function函数的连续性与间断
1.7 Properties of continuous functions on closed interval闭区间上连续函数的性质Introduction to Cauchy柯西简介
Exercises习题
Chapter 2 Differential calculus of one variable functions and its applications一元函数的微分学及其应用
2.0 Cited examples引例
2.1 Derivative导数
2.2 Rules of finding derivative求导法则
2.3 Derivatives ofhigher order and relative rates ofchange高阶导数与相关变化率
2.4 Differential and local linear approximation of functions函数的微分与线性逼近
2.5 Finding limits by using derivative——L'Hospital rule用导数求极限——罗必达法则
2.6 Mean value theorem of differential微分中值定理
2.7 Approximation to functions by using polynomial——Taylor formula用多项式逼近函数——泰勒公式
2.8 Properties of functions by derivatives用导数研究函数的性质
2.9 Curvature of plane curves平面曲线的曲率
Introduction to Lagrange 拉格朗日简介
Introduction to Taylor泰勒简介
Exercises习题
Chapter 3 Integral calculus of one variable functions and its application一元函数的积分及其应用
3.0 Cited examples引例
3.1 Concepts，properties and integrable rule of definite integral定积分的概念、性质与可积准则
3.2 Fundamental theorem of calculus微积分基本定理
3.3 Indefinite integral不定积分
3.4 Computation ofdefinite integral定积分的计算
3.5 Applications ofdefinite integral定积分的应用
3.6 Improper integral反常积分
Introduction to Newton牛顿简介
Introduction to Leibniz莱布尼兹简介
Introduction to Riemann黎曼简介
Exercises习题
Chapter 4 Differential Equations微分方程
4.0 Cited examples引例
4.1 Basic concepts ofdifferential equations微分方程的基本概念
4.2 Elementary integration methods for solving some simple differential equations解简单微分方程的初等积分法
4.3 Introduction to methods for establishing differential equations建立微分方程的方法简介
4.4 Higher—order differential equations高阶微分方程
Introduction to Euler欧拉简介
Exercises习题
References参考文献