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简介
《CALCULUS工科微积分(下册)(双语版)》讲授多元函数的微积分学。主要内容包括：第5章介绍向量代数和空间解析几何的基本知识。前者涉及向量的概念和向量的运算，而后者着重讨论空间平面、曲线和曲面的方程。
第6章讲授多元函数微分学的基本概念和偏导数的几何应用，重点将放在对二兀函数的研究上，相应的结果可以平行推广到二元以上的多元函数中。基本概念包括多元函数的定义、极限、连续性、偏导数和全微分。多元复合函数偏导数的运算法则如链式法则、全微分的形式不变形以及隐函数的微分法也将作为重点内容予以介绍。在几何应用部分，主要介绍空间曲线的切线方程、曲面的切平面方程以及解决多元函数极值问题的拉格朗日乘数法。
第7章讲授多元数量值函数的积分学。多元数量值函数的积分学是定积分的推广，其多样性的特点使得它较定积分有着更丰富的内容。本章将按照不同几何形体对应的不同积分，分别讨论二重积分、三重积分、对弧长的曲线积分及对面积的曲面积分的计算方法。最后，介绍向量值函数在几何、物理、力学等方面的应用。
第8章介绍向量值函数的曲线积分与曲面积分。本章除讨论第二型曲线、曲面积分的性质及计算外，还着重讨论各种积分之间的联系，这些联系体现在格林公式、高斯公式和斯托克斯公式中。最后介绍描述向量场特征的几个重要概念：散度与旋度。
在本册书的最后部分即第9章，介绍有关无穷级数的基本理论。本部分首先介绍常数项级数及其性质，重点讲授判别正项级数收敛的一些常用判别法，如比较判别法、根值判别法和比值判别法。然后，详细介绍有关幂级数的有关理论。最后着重讨论傅里叶级数的概念、收敛定理以及将函数展成傅里叶级数的方法。
目录
目录
5 Vector Algebra and Analytic Geometry in Space 向量代数与空间解析几何
5.0 Citing examples 引例
5.1 Vectors and its operations 向量及其运算
Key points of this section 本节重点
单词和短语
5.1.1 Concept of vectors 向量的概念
5.1.2 Linear operations of vectors 向量的线性运算
5.1.3 Scalar products of vectors(dot product,inner product) 向量的数量积(点积、内积)
5.1.4 Vector products of vectors(cross product,outer product) 向量的向量积(叉积、外积)
5.1.5 Mixed products of vectors 向量的混合积
5.1.6 Summarization of solving methods and typical examples 解题方法归纳与典型例题
5.2 Coordinates of points and coordinates of vectors 点的坐标与向量的坐标
Key points of this section 本节重点
单词和短语
5.2.1 Rectangular coordinate systems in space 空间直角坐标系
5.2.2 Coordinate representation of operati on sonvectors 向量运算的坐标表示
5.2.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
5.3 Planes and lines in space 空间的平面与直线
Key points of this section 本节重点
单词和短语
5.3.1 Plane 平面
5.3.2 Line 直线
5.3.3 The position relations among points,planes and lines 点、平面、直线的位置关系
5.3.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
5.4 Surface and curve 曲面与曲线
Key points of this section 本节重点
单词和短语
5.4.1 Equations of surfaces and curves 曲面、曲线的方程
5.4.2 Cylinder,surface of revolution and cone 柱面、旋转面和锥面
5.4.3 Quadric surface 二次曲面
5.4.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
Exercises习题
6 Differential Calculus of Multivariable Functions and its Applications 多元函数微分学及其应用
6.0 Citing examples 引例
6.1 The basic concepts of functions of several variables 多元函数的基本概念
Key points of this section 本节重点
单词和短语
6.1.1 n-dimensional point set n 维点集
6.1.2 Definition of function of several variables 多元函数的定义
6.1.3 Limit of function of two variables 二元函数的极限
6.1.4 Continuity of function of two variables 二元函数的连续性
6.1.5 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.2 Partial derivative and higher partial derivative 偏导数与高阶偏导数
Key points of this section 本节重点
单词和短语
6.2.1 Partial derivative 偏导数
6.2.2 Higher partial derivative 高阶偏导数
6.2.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.3 Total differential and its applications 全微分及其应用
Key points of this section 本节重点
单词和短语
6.3.1 Definition of total differential 全微分的概念
6.3.2 Relation between differentiability and partial derivability 可微与可偏导的关系
6.3.3 Geometric meaning of total differential 全微分的几何意义
6.3.4 Applications of total differential 全微分的应用
6.3.5 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.4 Differentiation of composite functions of several variables 多元复合函数的微分法
Key points of this section 本节重点
单词和短语
6.4.1 Chain rule 链式法则
6.4.2 Invariance of total differential form 全微分形式不变性
6.4.3 Derivative rule for implicit functions 隐函数的求导法则
6.4.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.5 Applications for partial derivatives in geometry 偏导数的几何应用
Key points of this section 本节重点
单词和短语
6.5.1 Tangent line and normal plane for spa cecurve 空间曲线的切线与法平面
6.5.2 Tangent plane and normal line of the surface 曲面的切平面与法线
6.5.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.6 Extremum of multivariable functions 多元函数的极值
Key points of this section 本节重点
单词和短语
6.6.1 Extreme value,global maximum,global minimum of multivariable functions 多元函数的极值及最大值、最小值
6.6.2 Conditional extremum Lagrange multiplier method 条件极值 拉格朗日乘数法
6.6.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.7 Directional derivative and gradient 方向导数与梯度
Key points of this section 本节重点
单词和短语
6.7.1 Directional derivative 方向导数
6.7.2 Gradient of scalar field 数量场的梯度
6.7.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
Exercises 习题
7 Integral Calculus of Multivariable Scalar Functions 多元数量值函数积分学
7.0 Citing examples 引例
7.1 Concept and properties of integral for multivariable scalar functions 多元数量值函数积分的概念与性质
Key points of this section 本节重点
单词和短语
7.1.1 The concept of integral of multivariable scalar functions 多元数量值函数积分的概念
7.1.2 Properties of integral of multivariable scalar functions 多元数量值函数积分的性质
7.1.3 The classifications of the integral for multivariable scalar functions 多元数量值函数积分的分类
7.2 Evaluation for double integral 二重积分的计算
Key points of this section 本节重点
单词和短语
7.2.1 The evaluation of double integral under Kectangular coordinate system 直角坐标系下二重积分的计算
7.2.2 The evaluation of double integral under polar coordinate system 极坐标系下二重积分的计算
7.2.3 Change of variables in double integral 二重积分的换元法
7.2.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
7.3 The evaluation of triple integral 三重积分的计算
Key points of this section 本节重点
单词和短语
7.3.1 The evaluation of triple integral under rcctangluar coordinate system 直角坐标系下三重积分的计算
7.3.2 Evaluation for triple integral in cylindrical and spherical coordinates 柱面坐标系与球面坐标系下三重积分的计算
7.3.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
7.4 The evaluations of line and surface integral for scalar value functions 数量值函数的曲线与曲面积分的计算
Key points of this section 本节重点
单词和短语
7.4.1 The evaluations of line integral with the first form 第一型曲线积分的计算
7.4.2 The evaluations of surface integral with the first form 第一型曲面积分的计算
7.4.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
7.5 Some typical applications of integrals of scalar valued functions in geometry and physics 数量值函数积分在几何、物理中的典型应用
Key points of this section 本节重点
单词和短语
Exercises 习题
8 Line Integrals and Surface Integrals of Vector Functions 向量值函数的曲线积分与曲面积分
8.0 Citing example 引例
8.1 Integrals of vector functi on sondirectional curves 向量值函数在有向曲线上的积分
Key points of this section 本节重点
单词和短语
8.1.1 Vector field 向量场
8.1.2 Concept of line integrals of the second form 第二型曲线积分的概念
8.1.3 Computation of the line integrals of the second form 第二型曲线积分的计算
8.1.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.2 Integrals of vector functi on sonoriented surface 向量值函数在有向曲面上的积分
Key points of this section 本节重点
单词和短语
8.2.1 Lateral of surface 曲面的侧
8.2.2 The concept of the surface integrals of the second form 第二型曲面积分的概念
8.2.3 Computation of the surface integrals of the second form 第二型曲面积分的计算
8.2.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.3 Relationships among iterated integral,line integral and surface integral 重积分、曲线积分、曲面积分之间的联系
Key points of this section 本节重点
单词和短语
8.3.1 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.4 Conditions for line integral independent of path 曲线积分与路径无关的条件
Key points of this section 本节重点
单词和短语
8.4.1 Condition for lineinte gralindependent of path 曲线积分与路径无关的条件
8.4.2 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.5 Introduction to field theory 场论简介
Key points of this section 本节重点
单词和短语
8.5.1 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.6 Applications 应用
Exercises 习题
9 Infinitc Series 无穷级数
9.0 Citing examples 引例
9.1 Series of number terms 数项级数
Key points of this section 本节重点
单词和短语
9.1.1 Concepts and basicproperties of series of number terms 常数项无穷级数的概念及基本性质
9.1.2 Summary of solving methods and typical examples 解题方法归纳和典型例题
9.2 Tests for convergence or divergence of positive terms series 正项级数敛散性的判别法
Key points of this section 本节重点
单词和短语
9.2.1 Convergence or divergence for positive terms series 正项级数的收敛性定理
9.2.2 Comparison test 比较判别法
9.2.3 Ratio test 比值判别法
9.2.4 Roottest 根值判别法
9.2.5 Integraltest 积分判别法
9.2.6 Summarization of solving method and typical examples 解题方法归纳和典型例题
9.3 Tests for convergence or divergence of series with any terms 任意项级数敛散性的判别法
Key points of this section 本节重点
单词和短语
9.3.1 Tests for convergence or divergence of alternating series 交错级数收敛性的判别法
9.3.2 Absolute convergence and conditional convergence 绝对收敛与条件收敛
9.3.3 Summarization of solving method and typical examples 解题方法归纳与典型例题
9.4 Power series 幂级数
Key points of this section 本节重点
单词和短语
9.4.1 Basic conceptes of series with function terms 函数项级数的基本概念
9.4.2 Power series and its convergence domain 幂级数及其收敛域
9.4.3 Operations and properties of power series 幂级数的运算与性质
9.4.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
9.5 Fourier series 傅里叶级数
Key points of this section 本节重点
单词和短语
9.5.1 Fourier series of function with the period 2π 以2π 为周期的函数的傅里叶级数
9.5.2 Fourier series of function with the period 2l 以2l 为周期的函数的傅里叶级数
9.5.3 Fourier expansion of functions defined on[—l,l]在[—l,l] 上有定义的函数的傅里叶展开
9.5.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
Exercises 习题
References 参考文献
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5 Vector Algebra and Analytic Geometry in Space 向量代数与空间解析几何
5.0 Citing examples 引例
5.1 Vectors and its operations 向量及其运算
Key points of this section 本节重点
单词和短语
5.1.1 Concept of vectors 向量的概念
5.1.2 Linear operations of vectors 向量的线性运算
5.1.3 Scalar products of vectors(dot product,inner product) 向量的数量积(点积、内积)
5.1.4 Vector products of vectors(cross product,outer product) 向量的向量积(叉积、外积)
5.1.5 Mixed products of vectors 向量的混合积
5.1.6 Summarization of solving methods and typical examples 解题方法归纳与典型例题
5.2 Coordinates of points and coordinates of vectors 点的坐标与向量的坐标
Key points of this section 本节重点
单词和短语
5.2.1 Rectangular coordinate systems in space 空间直角坐标系
5.2.2 Coordinate representation of operati on sonvectors 向量运算的坐标表示
5.2.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
5.3 Planes and lines in space 空间的平面与直线
Key points of this section 本节重点
单词和短语
5.3.1 Plane 平面
5.3.2 Line 直线
5.3.3 The position relations among points,planes and lines 点、平面、直线的位置关系
5.3.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
5.4 Surface and curve 曲面与曲线
Key points of this section 本节重点
单词和短语
5.4.1 Equations of surfaces and curves 曲面、曲线的方程
5.4.2 Cylinder,surface of revolution and cone 柱面、旋转面和锥面
5.4.3 Quadric surface 二次曲面
5.4.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
Exercises习题
6 Differential Calculus of Multivariable Functions and its Applications 多元函数微分学及其应用
6.0 Citing examples 引例
6.1 The basic concepts of functions of several variables 多元函数的基本概念
Key points of this section 本节重点
单词和短语
6.1.1 n-dimensional point set n 维点集
6.1.2 Definition of function of several variables 多元函数的定义
6.1.3 Limit of function of two variables 二元函数的极限
6.1.4 Continuity of function of two variables 二元函数的连续性
6.1.5 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.2 Partial derivative and higher partial derivative 偏导数与高阶偏导数
Key points of this section 本节重点
单词和短语
6.2.1 Partial derivative 偏导数
6.2.2 Higher partial derivative 高阶偏导数
6.2.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.3 Total differential and its applications 全微分及其应用
Key points of this section 本节重点
单词和短语
6.3.1 Definition of total differential 全微分的概念
6.3.2 Relation between differentiability and partial derivability 可微与可偏导的关系
6.3.3 Geometric meaning of total differential 全微分的几何意义
6.3.4 Applications of total differential 全微分的应用
6.3.5 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.4 Differentiation of composite functions of several variables 多元复合函数的微分法
Key points of this section 本节重点
单词和短语
6.4.1 Chain rule 链式法则
6.4.2 Invariance of total differential form 全微分形式不变性
6.4.3 Derivative rule for implicit functions 隐函数的求导法则
6.4.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.5 Applications for partial derivatives in geometry 偏导数的几何应用
Key points of this section 本节重点
单词和短语
6.5.1 Tangent line and normal plane for spa cecurve 空间曲线的切线与法平面
6.5.2 Tangent plane and normal line of the surface 曲面的切平面与法线
6.5.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.6 Extremum of multivariable functions 多元函数的极值
Key points of this section 本节重点
单词和短语
6.6.1 Extreme value,global maximum,global minimum of multivariable functions 多元函数的极值及最大值、最小值
6.6.2 Conditional extremum Lagrange multiplier method 条件极值 拉格朗日乘数法
6.6.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
6.7 Directional derivative and gradient 方向导数与梯度
Key points of this section 本节重点
单词和短语
6.7.1 Directional derivative 方向导数
6.7.2 Gradient of scalar field 数量场的梯度
6.7.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
Exercises 习题
7 Integral Calculus of Multivariable Scalar Functions 多元数量值函数积分学
7.0 Citing examples 引例
7.1 Concept and properties of integral for multivariable scalar functions 多元数量值函数积分的概念与性质
Key points of this section 本节重点
单词和短语
7.1.1 The concept of integral of multivariable scalar functions 多元数量值函数积分的概念
7.1.2 Properties of integral of multivariable scalar functions 多元数量值函数积分的性质
7.1.3 The classifications of the integral for multivariable scalar functions 多元数量值函数积分的分类
7.2 Evaluation for double integral 二重积分的计算
Key points of this section 本节重点
单词和短语
7.2.1 The evaluation of double integral under Kectangular coordinate system 直角坐标系下二重积分的计算
7.2.2 The evaluation of double integral under polar coordinate system 极坐标系下二重积分的计算
7.2.3 Change of variables in double integral 二重积分的换元法
7.2.4 Summarization of solving methods and typical examples 解题方法归纳与典型例题
7.3 The evaluation of triple integral 三重积分的计算
Key points of this section 本节重点
单词和短语
7.3.1 The evaluation of triple integral under rcctangluar coordinate system 直角坐标系下三重积分的计算
7.3.2 Evaluation for triple integral in cylindrical and spherical coordinates 柱面坐标系与球面坐标系下三重积分的计算
7.3.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
7.4 The evaluations of line and surface integral for scalar value functions 数量值函数的曲线与曲面积分的计算
Key points of this section 本节重点
单词和短语
7.4.1 The evaluations of line integral with the first form 第一型曲线积分的计算
7.4.2 The evaluations of surface integral with the first form 第一型曲面积分的计算
7.4.3 Summarization of solving methods and typical examples 解题方法归纳与典型例题
7.5 Some typical applications of integrals of scalar valued functions in geometry and physics 数量值函数积分在几何、物理中的典型应用
Key points of this section 本节重点
单词和短语
Exercises 习题
8 Line Integrals and Surface Integrals of Vector Functions 向量值函数的曲线积分与曲面积分
8.0 Citing example 引例
8.1 Integrals of vector functi on sondirectional curves 向量值函数在有向曲线上的积分
Key points of this section 本节重点
单词和短语
8.1.1 Vector field 向量场
8.1.2 Concept of line integrals of the second form 第二型曲线积分的概念
8.1.3 Computation of the line integrals of the second form 第二型曲线积分的计算
8.1.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.2 Integrals of vector functi on sonoriented surface 向量值函数在有向曲面上的积分
Key points of this section 本节重点
单词和短语
8.2.1 Lateral of surface 曲面的侧
8.2.2 The concept of the surface integrals of the second form 第二型曲面积分的概念
8.2.3 Computation of the surface integrals of the second form 第二型曲面积分的计算
8.2.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.3 Relationships among iterated integral,line integral and surface integral 重积分、曲线积分、曲面积分之间的联系
Key points of this section 本节重点
单词和短语
8.3.1 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.4 Conditions for line integral independent of path 曲线积分与路径无关的条件
Key points of this section 本节重点
单词和短语
8.4.1 Condition for lineinte gralindependent of path 曲线积分与路径无关的条件
8.4.2 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.5 Introduction to field theory 场论简介
Key points of this section 本节重点
单词和短语
8.5.1 Summarization of solving method and typical examples 解题方法归纳与典型例题
8.6 Applications 应用
Exercises 习题
9 Infinitc Series 无穷级数
9.0 Citing examples 引例
9.1 Series of number terms 数项级数
Key points of this section 本节重点
单词和短语
9.1.1 Concepts and basicproperties of series of number terms 常数项无穷级数的概念及基本性质
9.1.2 Summary of solving methods and typical examples 解题方法归纳和典型例题
9.2 Tests for convergence or divergence of positive terms series 正项级数敛散性的判别法
Key points of this section 本节重点
单词和短语
9.2.1 Convergence or divergence for positive terms series 正项级数的收敛性定理
9.2.2 Comparison test 比较判别法
9.2.3 Ratio test 比值判别法
9.2.4 Roottest 根值判别法
9.2.5 Integraltest 积分判别法
9.2.6 Summarization of solving method and typical examples 解题方法归纳和典型例题
9.3 Tests for convergence or divergence of series with any terms 任意项级数敛散性的判别法
Key points of this section 本节重点
单词和短语
9.3.1 Tests for convergence or divergence of alternating series 交错级数收敛性的判别法
9.3.2 Absolute convergence and conditional convergence 绝对收敛与条件收敛
9.3.3 Summarization of solving method and typical examples 解题方法归纳与典型例题
9.4 Power series 幂级数
Key points of this section 本节重点
单词和短语
9.4.1 Basic conceptes of series with function terms 函数项级数的基本概念
9.4.2 Power series and its convergence domain 幂级数及其收敛域
9.4.3 Operations and properties of power series 幂级数的运算与性质
9.4.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
9.5 Fourier series 傅里叶级数
Key points of this section 本节重点
单词和短语
9.5.1 Fourier series of function with the period 2π 以2π 为周期的函数的傅里叶级数
9.5.2 Fourier series of function with the period 2l 以2l 为周期的函数的傅里叶级数
9.5.3 Fourier expansion of functions defined on[—l,l]在[—l,l] 上有定义的函数的傅里叶展开
9.5.4 Summarization of solving method and typical examples 解题方法归纳与典型例题
Exercises 习题
References 参考文献
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