目 录 第一部分 英汉微积分词汇 Part 1 English-Chinese Calculus Vocabulary
第一章 函数与极限 Chapter 1 function and Limit………………………………………………1 第二章 导数与微分 Chapter 2 Derivative and Differential………………………………………2 第三章 微分中值定理 Chapter 3 Mean Value theorem of differentials and the Application of Derivatives………………………………………3 第四章 不定积分 Chapter 4 Indefinite Integrals………………………………………………3 第五章 定积分 Chapter 5 Definite Integral…………………………………………………3 第六章 定积分的应用 Chapter 6 Application of the Definite Integrals……………………………4 第七章 空间解析几何与向量代数 Chapter 7 Space Analytic Geometry and Vector Algebra…………………4 第八章 多元函数微分法及其应用 Chapter 8 Differentiation of functions Several variables and Its Application………………………………………………5 第九章 重积分 Multiple Integrals………………………………………………6 第十章 曲线积分与曲面积分 Chapter 10 Line(Curve ) Integrals and Surface Integrals……………………6 第十一章 无穷级数 Chapter 11 Infinite Series……………………………………………………6 第十二章 微分方程 Chapter 12 Differential Equation……………………………………………7
第二部分 定理定义公式的英文表达
Part 2 English Expression for Theorem, Definition and Formula 第一章 函数与极限 Chapter 1 Function and Limit………………………………………………19 1.1 映射与函数(Mapping and Function ) ………………………………19 1.2 数列的极限(Limit of the Sequence of Number) ……………………20 1.3 函数的极限(Limit of Function) ……………………………………21 1.4 无穷小与无穷大(Infinitesimal and Infinity) ………………………23 1.5 极限运算法则(Operation Rule of Limit) ……………………………24 1.6 极限存在准则 两个重要的极限(Rule for the Existence of Limits Two Important Limits) …………………………25 1.7 无穷小的比较(The Comparison of infinitesimal) ……………………26 1.8 函数的连续性与间断点(Continuity of Function And Discontinuity Points) ……………………………………………28 1.9 连续函数的运酸与初等函数的连续性(Operation Of Continuous Functions and Continuity of Elementary Functions) …………………………………………………28 1.10 闭区间上连续函数的性质(Properties of Continuous Functions on a Closed Interval) …………………………30
第二章 导数与数分 Chapter2 Derivative and Differential……………………………………………31 2.1 导数的概念(The Concept of Derivative) ………………………………31 2.2 函数的求导法则(Rules for Finding Derivatives) ………………………33 2.3 高阶导数(Higher-order Derivatives) ……………………………………34 2.4 隐函数及由参数方程所确定的函数的导数相关变化率(Derivatives of Implicit Functions and Functions Determined by Parametric Equation and Correlative Change Rate) …34 2.5 函数的微分(Differential of a Function) ………………………35
第三章 微分中值定理与导数的应用 Chapter 3 Mean Value Theorem of Differentials and the Application of Derivatives………………………………………36 3.1 微分中值定理(The Mean Value Theorem) ……………………………36 3.2 洛必达法则(L’Hospital’s Rule) ……………………………………………38 3.3 泰勒公式(Taylor’s Formula) ……………………………………………41 3.4 函数的单调性和曲线的凹凸性(Monotonicity of Functions and Concavity of Curves) …………………………………43 3.5 函数的极值与最大最小值(Extrema, Maxima and Minima of Functions) ……………………………………………46 3.6 函数图形的描绘(Graphing Functions) ………………………………49 3.7 曲率(Curvature) ………………………………………………………50 3.8 方程的近似解(Solving Equation Numerically) ………………………53
第四章 不定积分 Chapter 4 Indefinite Integrals…………………………………………………54 4.1 不定积分的概念与性质(The Concept and Properties of Indefinite Integrals) ………………………………………54 4.2 换元积分法(Substitution Rule for Indefinite Integrals) ………………56 4.3 分部积分法(Integration by Parts) ………………………………………57 4.4 有理函数的积分(Integration of Rational Functions) …………………58
第五章 定积分 Chapter 5 Definite Integrals…………………………………………………61 5.1 定积分的概念和性质(Concept of Definite Integral and its Properties) ………………………………………………………61 5.2 微积分基本定理(Fundamental Theorem of Calculus) ………………67 5.3 定积分的换元法和分部积分法(Integration by Substitution and Definite Integrals by Parts) ……………………………………………69 5.4 反常积分(Improper Integrals) …………………………………………70
第六章 定积分的应用 Chapter 6 Applications of the Definite Integrals……………………………75 6.1 定积分的元素法(The Element Method of Definite Integra……………75 6.2 定积分在几何学上的应用(Applications of the Definite Integrals to Geometry) ………………………………………………76 6.3 定积分在物理学上的应用(Applications of the Definite Integrals to Physics) ……………………………………………………79
第七章 空间解析几何与向量代数 Chapter 7 Space Analytic Geometry and Vector Algebar……………………80 7.1 向量及其线性运算(Vector and Its Linear Operation) …………………80 7.2 数量积 向量积(Dot Product and Cross Product) ……………………86 7.3 曲面及其方程(Surface and Its Equation)………………………………89 7.4 空间曲线及其方程(The Curve in Three-space and Its Equation………91 7.5 平面及其方程（Plane in Space and Its Equation) ……………………93 7.6 空间直线及其方程（Lines in and Their Equations) ……………………95
第八章 多元函数微分法及其应用 Chapter 8 Differentiation of Functions of Several Variables and Its Application………………………………………99 8.1 多元函数的基本概念（The Basic Concepts of Functions of Several Variables) ………………………………………………………99 8.2 偏导数(Partial Derivative) ………………………………………………102 8.3 全微分(Total Differential) ………………………………………………103 8.4 链式法则（The Chain Rule) ……………………………………………104 8.5 隐函数的求导公式(Derivative Formula for Implicit Functions). ………104 8.6 多元函数微分学的几何应用(Geometric Applications of Differentiation of Ffunctions of Severalvariables) ……106 8.7方向导数与梯度(Directional Derivatives and Gradients) …………………107