高等數學(上)（簡體書）

《普通高等教育"十二五"規劃教材:高等數學(上冊)》可作為普通高等院校學生高等數學課程的教材，也可作為科技英語專業相關課程的教材和參考書。

前言
Introduction
引言
Chapter 1 Functions，limits and continuity函數、極限與連續性
1.0 Cited examples引例
1.1 Functions函數
1.2 Limits極限
1.3 Properties and operations oflimits極限的運算與性質
1.4 Monotone boundedness principle and irrational number e單調有界原理與無理數e
1.5 Comparison between infinitesimals無窮小之間的比較
1.6 Continuity and discontinuity of function函數的連續性與間斷
1.7 Properties of continuous functions on closed interval閉區間上連續函數的性質Introduction to Cauchy柯西簡介
Exercises習題
Chapter 2 Differential calculus of one variable functions and its applications一元函數的微分學及其應用
2.0 Cited examples引例
2.1 Derivative導數
2.2 Rules of finding derivative求導法則
2.3 Derivatives ofhigher order and relative rates ofchange高階導數與相關變化率
2.4 Differential and local linear approximation of functions函數的微分與線性逼近
2.5 Finding limits by using derivative——L'Hospital rule用導數求極限——羅必達法則
2.6 Mean value theorem of differential微分中值定理
2.7 Approximation to functions by using polynomial——Taylor formula用多項式逼近函數——泰勒公式
2.8 Properties of functions by derivatives用導數研究函數的性質
2.9 Curvature of plane curves平面曲線的曲率
Introduction to Lagrange 拉格朗日簡介
Introduction to Taylor泰勒簡介
Exercises習題
Chapter 3 Integral calculus of one variable functions and its application一元函數的積分及其應用
3.0 Cited examples引例
3.1 Concepts，properties and integrable rule of definite integral定積分的概念、性質與可積準則
3.2 Fundamental theorem of calculus微積分基本定理
3.3 Indefinite integral不定積分
3.4 Computation ofdefinite integral定積分的計算
3.5 Applications ofdefinite integral定積分的應用
3.6 Improper integral反常積分
Introduction to Newton牛頓簡介
Introduction to Leibniz萊布尼茲簡介
Introduction to Riemann黎曼簡介
Exercises習題
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解簡單微分方程的初等積分法
4.3 Introduction to methods for establishing differential equations建立微分方程的方法簡介
4.4 Higher—order differential equations高階微分方程
Introduction to Euler歐拉簡介
Exercises習題
References參考文獻

4.1 Basic concepts of differential equations
微分方程的基本概念
Differential equation: An equation containing the derivatives of one or moreunknown functions with respect to one or more independent variables is said to be adifferential equation (DE).
Classification by type: If the unknown functions in a differential equation are allfunctions with one independent variable, it is called an ordinary differential equation(ODE). An equation involving the partial derivatives of one or more unknown functionsof two or more independent variables is said to be a partial differential equation (PDE).
Classification by order: The order of a differential equation (either ODE or PDE)is the order of the highest derivative in the equation.
Classification by linearity: If the power of the unknown functions and theirderivatives in a differential equation are all linear, the equation is said to be a lineardifferential equation. Otherwise, it is called a nonlinear one.
Solutions of an ODE: If a solution of a ODE has independent constants, and if thenumber of the constants are equal to the order of the ODE, then it is said to be a generalsolution of the ODE. If the constants in a general solution are determined, the solution iscalled a particular solution.
Explicit and implicit solutions: A solution in which the dependent variable isexpressed solely in terms of the independent variable and constants is said to be anexplicit solution. A relation G(x,y) = 0 is said to be an implicit solution of an ordinarydifferential equation, provided there exists at least one function φ that satisfies therelation as well as the differential equation.