B数学(带目录)


一、运筹学基础及应用 Foundation and Application of Operations Research ............................... 1 二、运筹学 Operational Research ................................................................................................... 3 三、统计学及其应用软件 Statistics and Application Software ...................................................... 3 四、统计学 Statistics ....................................................................................................................... 4 五、概率论 Probability Theory ........................................................................................................ 4 六、概率论与数理统计 Probability Theory and Mathematical Statistics....................................... 4 七、高等数学 Advanced Mathematics ............................................................................................ 6 八、高等数学方法 Methods Used in Higher Mathematics ............................................................ 8 九、复变函数与积分函数 Complex Variables Functions and Integral Functions ........................... 8 十、实变函数 Real Variable Functions ............................................................................................ 9 十一、线性代数 Linear Algebra ...................................................................................................... 9 十二、高等代数(1)Advanced Algebra (1) ................................................................................ 10 十三、高等代数(2)Advanced Algebra (2) ................................................................................ 10 十四、抽象代数 Abstract Algebra ................................................................................................. 11 十五、数学分析(1)Mathematical Analysis (1) ......................................................................... 11 十六、数学分析 中 Mathematical Analysis Ⅱ ........................................................................... 12 十七、数学分析 下 Mathematical Analysis Ⅲ ........................................................................... 12 十八、离散数学 Discrete Mathematics ........................................................................................ 12 十九、微分几何 Differential Geometry ........................................................................................ 12 二十、解析几何 Analytic Geometry ............................................................................................. 13 二十一、几何与代数 下 Geometry and Algebra, Ⅱ .................................................................. 13 二十二、常微分方程 Ordinary Differential Equations.................................................................. 13 二十三、偏微分方程 Partial Differential Equations...................................................................... 14 二十四、数据结构与算法 Data Structures and Algorithms ......................................................... 14 二十五、数值分析 Numerical Analysis ......................................................................................... 14 二十六、数学实验与数学软件 Mathematical Experiments and Software .................................. 14

一、运筹学基础及应用 Foundation and Application of Operations Research
1. 绪论。(1)运筹学的起源、发展。(2)运筹学的研究方法。(3)运筹学的主要分支。 (4)运筹学与管理科学。 1. Introduction. (1) Historical origins and development of operations research. (2) Research approaches. (3) Main branches of operations research. (4) Operations research and management science. 2. 线性规划及单纯形法。 (1)一般线性规划问题的数学模型。 (2)线性规划问题的图解法。 (2)单纯形法原理。(3)单纯形法的计算步骤。 (4)单纯形法的进一步讨论。大 M 单纯 形法,两阶段法,退化问题,检验数的几种表示法。 (5)数据包络分析。 (6)计算机求解法。 2. Linear programming and simplex algorithm. (1) Mathematical models of normal linear programming problems. (2) Graphs of linear programming problems. (3) Principles of simplex algorithm. (4) Further discussion of simplex algorithm. M-method, two-phase, degeneracy, and denotations of test numbers. (5) Data Envelopment Analysis. (6) Computer solutions. 3. 线性规划的对偶理论与灵敏度分析。(1)对偶问题的提出。(2)原问题与对偶问题。 (3)对偶问题的基本性质。 (4)影子价格。(5)对偶单纯形法。 (6)灵敏度分析。对偶理 论在灵敏度分析上的应用。(7)参数线性规划。

3. The theory of duality and sensitivity analysis in linear programming. (1) Proposal of duality. (2) Original problem and duality problem. (3) Basic qualities of duality problems. (4) Shadow price. (5) Simplex algorithm in duality. (6) Sensitivity analysis. Application of duality theory in sensitivity analysis. (7) Parametric linear programming. 4. 运输和指派问题。(1)运输问题及数学模型。(2)表上作业法。 (3)产销不平衡的运 输问题及其应用。 (4)指派问题及求解方法。 4. Transport problems and assignment problems. (1) Transport problems and mathematical models. (2) Table-manipulation method. (3) Transport problems in unbalanced production-marketing and its application. (4) Assignment problems and the solutions. 5. 整数规划与分配问题。 (1)整数规划的特点及作用。 (2)分配问题与匈亚利法。 (3)分 支定界法。 (4)割平面法。 (5)应用举例。 5. Integer programming and assignment problem. (1) Characteristics and functions of integer programming. (2) Assignment problem and Hungarian algorithm. (3) The branch and bound methods. (4) The cutting plane algorithm. (5) Applications and instances. 6. 目标规划。 (1)问题的提出与目标规划的数学模型。 (2)目标规划的图解分析法。 (3 ) 用单纯型法求解目标规划。 (4)求解目标规划的层次算法。 6. Goal Programming. (1) Proposals and mathematical models. (2) Graph-analytic method in goal programming. (3) Application of simplex algorithm in goal programming. (4) Hierarchical algorithms of goal programming. 7. 图与网络分析。(1) 图的基本概念与模型。 (2)树图和图的最小部分树。(3)最短路 径问题。(4)网络的最大流。 (5)最小费用流。(6)应用举例。 7. Graph and network analysis. (1) Basic concepts of graphs. (2) Tree graph and the tree graph of the minimum part of the graph. (3) Shortest path problem. (4) Maximum network flow. (5) Minimum cost. (6) Applications and instances. 8. 计划评审方法和关键路线法动态规划。 (1)PERT 网络图。 (2)PERT 网络图的计算。 (3) 关键路线和网络计划优化。 8. Plan reviews and dynamic programming of critical path method. (1) PERT network planning. (2) Computation of PERT network planning. (3) Critical path and network planning optimization. 9. 动态规划。(1)多阶段决策问题。 (2)最优化原理与动态规划的数学模型。 (3)离散确 定性动态规划模型的求解。 (4)离散随机性动态规划模型的求解。(5)一般数学规划模型 的动态规划解法。 9. Dynamic programming. (1) Multi-stage decision process. (2) Principle of optimality and mathematical models of dynamic programming. (3) Solution of discrete determined dynamic programming models. (4) Solution of discrete random dynamic programming models. (5) Dynamic programming solution of normal mathematical programming models. 10. 存贮论。 (1)引言。存贮问题的产生,存贮模型的建立。 (2)经济订货批量的存贮模型。 (3)具有约束条件的存贮模型。 (4)具有价格折扣优惠的存贮模型。 (5)动态的存贮模型。 (6)单时期的随机存贮模型。 (7)多时期的随机存贮模型。 (8)确定性的多梯次存贮模型。 10. Inventory theory. (1) Introduction. The rise of inventory problems and the proposal of inventory models. (2) Inventory model with economic order quantity. (3) Inventory model with restraint conditions. (4) Inventory model with discounts. (5) Dynamic inventory model. (6) Single-period stochastic inventory model. (7) Multi-period stochastic inventory model. (8) Deterministic multi-echelon inventory model. 11. 排队论。(1)排队服务系统的基本概念。 (2)输入与服务时间的分布。 (3)生灭过程。

(4)最简单排队系统模型。 (5)M/G/1 的排队系统。 (6)服务机构串联的排队系统。 ( 7) 具有优先服务权的排队模型。 (8)排队决策模型。 (9)排队系统的模拟。 11. Queuing theory. (1) Basic concepts of queuing service system. (2) Input and service time distribution. (3) Birth-and-death process. (4) A simplest queuing system model. (5) M/G/1 queuing system. (6) Service agencies tandem queuing system. (7) Priority queuing model. (8) Queuing decision model. (9) Simulation of queuing system. 12. 决策分析。 (1)引言。决策问题的背景,决策问题的提出。 (2)不确定型的决策分析。 (3)风险情况下的决策。 (4)贝叶斯决策。 (5)决策分析中的效用度量。 (6)Pareto 最优。 (7)层次分析法。 (8)多属性决策。 12. Decision analysis. (1) Introduction. Backgrounds and proposal of decision analysis. (2) Decision analysis of uncertainty. (3) Decision under risk. (4) Bayesian decision. (5) Utility measurement of decision analysis. (6) Pareto optimality. (7) The analytic hierarchy process. (8) The multiple attribute decision making. 13. 博弈论。 (1)引言。博弈论产生的背景,博弈模型的基本结构。 (2)完全信息静态博弈。 二人零和博弈模型,具有鞍点的博弈,混合策略,纳什均衡,用划线法求具有纯策略的纳什 均衡,混合策略下的纳什均衡。 (3)完全信息动态博弈。 (4)冲突分析简介。冲突分析,冲 突分析的简单模型。 13. Game theory. (1) Introduction. Background of the origin of game theory. The basic structure of game theory. (2) The static games of complete information. Two-person zero-sum game; fixed-point theorem; mixed strategy; Nash equilibrium; pure strategy Nash equilibrium by means of scoring method; mixed strategy Nash equilibrium. (3) The dynamic games of complete information. (4) An introduction to conflict analysis. Conflict analysis; simple model of conflict analysis.

二、运筹学 Operational Research
线性规划与单纯形法;整数规划;非线性规划;图与网络分析; Linear programming and simplex method, integer programming, nonlinear programming, graph and network analysis 运筹学/《运筹学》教程编写组编—3 版。北京:清华出版社,2005.6 Operational Research by Operational Research writing group, 3rd edition, Beijing: Tsinghua University Press, Jun. 2005.

三、统计学及其应用软件 Statistics and Application Software
集中趋势和离散趋势,方差、平均差、标方差,增长量,平均发展速度,区间估计 Central Tendency and Dispersion Tendency, Variance, Mean Deviation, Standard Variance, quantity of increase, average growth speed, interval estimation 时间序列,平均发展水平,相对数,时间序列,平均数 Time Series, average level of development, relative number, time series, 回归分析,利用回归方程对未来进行预测。方差分析。 Regression analysis, predicting future by using regression, analysis of variance 随机事件,事件间的关系与运算 Random event, relation and operation of events 概率,贝努里概型,随机变量,连续型分布 Probability, Bernoulli Scheme, random variable, and continuous distribution

实际操作: Practical lessons: 学习使用统计学软件 SPSS(全英文版)进行数据输入处理、数据分析,回归分析等应用。 Learn to use statistical software SPSS (all in English) process data input, data analysis, and regression analysis.

四、统计学 Statistics
本课程讲授统计学的基本理论,基本概念与基本运作。主要内容包括: This course teaches the basic theories, conceptions and operations of Statistics, including the following content: 统计调查方法 the methods of statistical investigation, 集中趋势离散趋势的 有关指标 related indexes on central tendency and dispersion tendency, 抽样评估 sampling evaluation, 相关分析与回归分析 correlation analysis and regression analysis, 时间数列 time series and 统计指数 statistical index.

五、概率论 Probability Theory
事件与概率 Events and probability ;条件概率与统计独立性 conditional probability and statistical independence;随机变量与分布函数 random variables and distribution functions;数 字特征与特征函数 numerical characteristics and characteristic functions;极限定 limit theorem. 概率论基础/李贤平编著。-北京:高等教育出版社,1997(2008 重印)Elementary Probability Theory by Li Xianping, Beijing: Higher Education Press, 1997(reprinting, 2008).

六、概率论与数理统计 Probability Theory and Mathematical Statistics
《概率论与数理统计》 是高等学校经济管理类专业核心课程经济数学基础之一, 是研究随机 现象规律性的一门学科。通过本课程的学习,应使学生掌握概率论与数理统计的基本概念, 了解它的基本理论和方法, 从而使学生初步掌握处理随机现象的基本思想和方法, 培养学生 运用概率统计方法分析和解决实际问题的能力。 Probability Theory and Mathematical Statistics, which studies the regularity of random phenomenon, is part of economic mathematics, the core course for economy and management majors in higher education. The course is intended to acquaint the students with the basic concepts of probability theory and mathematical statistics, teach the students the principles and methodology, which equips the students with the necessary thinking and method to deal with random phenomenon and enables the students to apply the mathematical statistics to address practical problems. 该课程主要内容包括: This course mainly covers: 随机事件与概率 random events and their probability, 随机变量及其概率分布 random variable and its probability distribution, 多维随机 变量及其概率分布 multi-dimensional random variable and its probability distribution, 随机变量 的数字特征 the alphanumeric characters of random variable, 大数定律及中心极限定理 law of large numbers and central limit theorem, 数理统计的基本知识 the essential knowledge of mathematical statistics, 参数估计 parameter estimation, 假设检验 hypothesis testing, 回归分析 和方差分析 regression analysis and variance analysis. 1.随机事件及其概率。 (1)随机事件。 (2)随机事件的概率。 (3)古典概型与几何概型。 (4) 条件概率。 (5)事件的独立性。

1. Random event and its probability. (1) Random event. (2) Probability of random event. (3) Classical model of probability and geometric probability model. (4) Conditional probability. (5) Independence of event. 2.随机变量的分布与数字特征。 (1)随机变量及其分布。 (2)随机变量的数字特征。数学期 望、方差、标准差的概念,期望与方差的初等性质。 (3)常用的离散型分布。 (4)常用的连 续型分布。 (5)随机变量函数的分布。 2. Distribution of extraneous variables and the numerical characteristics. (1) Extraneous variables and its distribution. (2) The numerical characteristics of extraneous variables. Concepts of mathematical expectation, variance, and standard deviation; the elementary properties of expectation and variance. (3) Normal discrete distribution. (4) Normal continuous distribution. (5) Distribution of functions of random variables. 3. 随机向量。 (1)随机向量的分布。 (2)条件分布与随机变量的独立性。 (3)随机向量的函 数的分布与数学期望。 (4)随机向量的数字特征。协方差,协方差矩阵,相关系数,条件数 学期望,条件期望的预测含义。 (5)大数定律与中心极限定理。 3. Random vector. (1) Distribution of random vector. (2) Conditional distribution and the independence of random vector. (3) The distribution of functions of random vector and its mathematical expectation. (4) The numerical characteristics of random vector. Covariance, covariance matrix, correlation coefficient, conditional mathematical expectation, and the prediction meaning of conditional expectation. (5) Law of large numbers and central limit theorem. 4. 统计量及其分布。 (1)总体与样本。 (2)统计量。 (3)常用的统计分布。 (4)抽样分布。 4. Statistics and its distribution. (1) Population and samples. (2) Statistic. (3) Normal statistical distribution. (4) Sampling distribution. 5. 参数估计。 (1)点估计概述。点估计及其它的无偏性、有效性和相合性。 (2)参数的最大 似然估计与矩估计。 (3)置信区间。 5. Parametric estimation. (1) An introduction to point estimation. Point estimation and its unbiasedness, validity, and consistency (2) Maximum likelihood and method of moments of parameters (3) Confidence intervals 6.假设检验。 (1)假设检验概述。假设检验问题的提出,假设检验的基本思想和原理,假设 检验的一般步骤,检验的显著性水平与两类错误。 (2)单正态总体的参数假设检验。 (3)双 正态总体的参数假设检验。 (4)一般总体的参数假设检验。 (5)拟合优度 ? 检验与独立性
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检验。 6. Statistical hypothesis testing. (1) Introduction. The proposal of statistical hypothesis testing, its basic concepts and principles; the usual procedures; test of statistical significance and two types of mistakes. (2) The parametric hypothesis testing of single normal population. (3) The parametric hypothesis testing of double normal population. (4) The parametric hypothesis testing of normal population. (5) The testing of agreement ?
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and independence testing.

7.回归分析。 (1)一元线性回归模型及其参数估计。 (2)一元线性回归模型的检验。 (3)一 元线性回归的残差分析。 (4)一元线性回归的预测与控制。 (5)一元非线性问题的线性化。 (6)多元线性回归分析。回归系数的最小二乘估计,回归方程的显著性检验,多元线性回 归模型的预测。 7. Regression analysis. (1) Unary linear regression model and its parameter estimation. (2)

Testing of unary linear regression model. (3) Residual analysis of unary linear regression. (4) Prediction and control of unary linear regression. (5) Linearization of unary nonlinear equations. (6) Multiple linear regressions. Least square estimation of regression coefficient; significance test of regression equation; prediction of multiple linear regressions 龙永红主编: 《概率论与数理统计》 ,高等教育出版社,2001 年版。 Probability Theory and Mathematical Statistics, LONG Yonghong et al ed., Beijing: Higher Education Press, 2001.

七、高等数学 Advanced Mathematics
高等数学课程是高等学校工科本科各专业学生的一门必修的重要基础理论课, 通过本课程的 学习,使学生获得:一元函数微积分学,空间解析几何与向量代数,多元函数微积分学,无 穷级数,常微分方程等方面的基本概念、基本理论和基本运算技能,为学习后继课程和进一 步获得数学知识奠定必要的数学基础, 在传授知识的同时, 通过各个教学环节逐步培养学生 具有抽象思维能力、逻辑推理能力、空间想象能力和自学能力,特别注意培养学生具有比较 熟练的运算能力和综合运用所学知识去分析问题和解决问题的能力。 As one of the compulsory basic theoretical courses, it enables students to obtain: the basic conceptions, theories and operation abilities on differential and integral calculus for function of one variable, analytic geometry and vectors, multivariate function differential, infinite series, ordinary differential equation and etc. Beside imparting knowledge and laying the necessary foundations of mathematics for further study, this course also gradually cultivates students the capacities of abstract thinking, logical thought, space imagination and self-learning, especially their proficient operational ability and the ability to analyze and solve problem with what they have learned synthetically. 高等数学课程体系目标是通过学习,拓宽基础和视野、培养能力和素质。使学生掌握高等数 学的基本理论、方法、思想和文化,培养学生的数学实验与数学建模能力,使学生获得抽象 思维能力、逻辑推理能力、归纳判断能力、空间想象能力、推导计算能力等,并具有综合运 用所学知识去发现问题、分析问题和解决问题的能力。 The objective of advanced mathematics curriculum system is to extend foundation, broaden horizon and cultivate abilities and qualities. This course helps students mastering the basic theories, methods, ideas and cultures of advanced mathematics, cultivating their ability of mathematic experiment and modeling, acquiring abstract thinking ability, logic reasoning ability, generalizing and judging ability, space imagination ability, deduction and calculation ability, etc. and possessing the capacity to applying what they have learned to finding, analyzing and solving problems. 该课程的主要内容有: 函数与极限, 一元函数的导数与微分, 中值定理(Middle Value Theorem) 与导数的应用,不定积分,定积分,定积分的应用,空间解析几何和向量代数,多元函数微 分法及其应用,重积分,曲线积分与曲面积分,无穷级数(Infinite Series),微分方程。 This course consists of several major parts, such as functions and limits, the derivative and differential of one-dimensional function, Middle Value Theorem and derivative application, indefinite integral, definite integral and its application, analytic geometry and vectors, multi-element function infinitesimal method and its application, integral, curve integral and surface integral, infinite series and differential equation. 高等数学课程体系目标是通过学习,拓宽基础和视野、培养能力和素质。使学生掌握高等数 学的基本理论、方法、思想和文化,培养学生的数学实验与数学建模能力,使学生获得抽象

思维能力、逻辑推理能力、归纳判断能力、空间想象能力、推导计算能力等,并具有综合运 用所学知识去发现问题、分析问题和解决问题的能力。 The objective of this course is to broaden students’ base and vision and develop their ability and personal quality. It devotes to enabling the students to master the basic theories, methods, ideas and culture of advanced mathematics, improving their Mathematics experiment and mathematical modeling ability, to equipping them with abilities of abstract thinking, logical reasoning, inducing and judging, spatial visualization, deducing and calculating, as well as the ability to discover, analyze and solve problems using knowledge they have learned. ? 函数与极限:映射与函数,数列的极限,函数的极限,无穷小与无穷大,极限运算法则,极 限存在准则、两个重要极限,无穷小的比较,函数的连续性与间断点,连续函数的运算与初 等函数的连续性,闭区间上连续函数的性质,总习题一 Function and limit: mapping and function, limit of series, limits of functions, infinitely small and infinitely great, algorithm of limit, rule of limit existence and two important limits, comparison of infinitesimals, continuity and discontinuous point of function, operation of continuous function and continuity of elementary function, properties of continuous function on a closed interval, Exercise 1 导数与微分:导数概念,函数的求导法则,高阶导数,隐函数及由参数方程所确定的函数的 导数、相关变化率,函数的微分,总习题二 Derivative and differential: concept of derivative, rules of finding derivative of function, higher order derivatives, implicit functions, the derivative of a function determined by its parametric equation, and relative rates of change, differential of function, Exercise 2 微分中值定理与导数的应用:微分中值定理,洛必达法则,泰勒公式,函数的单调性与曲线 的凹凸性,函数的极值与最大值最小值,函数图形的描绘,曲率,方程的近似解,总习题三 Differential mean value theorem and application of derivative: differential mean value theorem L'Hospital's rule, Taylor's formula, monotonicity of a function and convexity of curves, extreme value of a function and maximum value and minimal value, mapping the graph of a function, curvature, approximate solution to equations, Exercise 3 不定积分:不定积分的概念与性质,换元积分法,分部积分法,有理函数的积分,积分表的 使用,总习题四 Definite integral: concept and properties of definite integral, integration by substitution, integration by parts, residues of a rational function, how to use the table of integrals, Exercise 4 定积分: 定积分的概念与性质, 微积分基本公式, 定积分的换元法和分部积分法, 反常积分, 反常积分的审敛法、Γ 函数,总习题五 Definite integral: concept and properties of definite integral, fundamental formula of calculus, method of substitution for definite integrals and integration by parts, improper intergral, testing method of improper integral, Γ function, Exercise 5 定积分的应用:定积分的元素法,定积分在几何学上的应用,定积分在物理学上的应用,总 习题六 Application of definite integral: element method of definite integral, application of definite integral to geometry, application of definite integral to physics, Exercise 6 空间解析几何与向量代数:向量及其线性运算,数量积、向量积、混合积,曲面及其方程, 空间曲线及其方程,平面及其方程,空间直线及其方程,总习题七 Space analytic geometry and vector algebra: vector and its linear operation, dot product, vector product and mixed product, curved surface and its equation, plane and its equation, spatial lines and their equations, Exercise 7 多元函数微分法及其应用:多元函数的基本概念,偏导数,全微分,多元复合函数的求导法 则,隐函数的求导公式,多元函数微分学的几何应用,方向导数与梯度,多元函数的极值及 其求法,二元函数的泰勒公式,最小二乘法,总习题八 Multivariable differential calculus and its application: basic concept of multivariable differential calculus, partial derivative, Rules of finding derivative for multivariable function of functions, formula of finding derivation for implicit function, application multivariable differential calculus

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to geometry, directional derivative and gradient, extreme value of multivariable functions and its derivation, Taylor's formula for binary functions, least square method, Exercise 8 重积分:二重积分的概念与性质,二重积分的计算法,三重积分,重积分的应用,含参变量 的积分,总习题九 Multiple integral: concept and properties of double integral, operation of double integral, triple integral, application of multiple integral, integral including parameters, Exercise 9 曲线积分与曲面积分:对弧长的曲线积分,对坐标的曲线积分,格林公式及其应用,对面积 的曲面积分,对坐标的曲面积分,高斯公式、通量与散度,斯托克斯公式、环流量与旋度, 总习题十 Line integral and surface integral: line integral over arc length, line integral over coordinates, Green's theorem and its application, surface integral over areas, integral over coordinates, gauss formulas and flux and divergence, Stokes' formula, circulation and curl, Exercise 10 无穷级数:常数项级数的概念和性质,常数项级数的审敛法,幂级数,函数展开成幂级数, 函数的幂级数展开式的应用, 函数项级数的一致收敛性及一致收敛级数的基本性质, 傅里叶 级数,一般周期函数的傅里叶级数,总习题十一 Infinite series, concept and properties of constant term series, testing method of constant term series, power series, expansion of a function to a power series, Fourier series of general periodic functions, application of power series expansion, uniform convergence of function series and basic properties of uniform convergence series, Fourier series, Fourier series of General periodic function, Exercise 11. 微分方程:微分方程的基本概念,可分离变量的微分方程,齐次方程,一阶线性微分方程, 全微分方程,可降阶的高阶微分方程,高阶线性微分方程,常系数齐次线性微分方程,常系 数非齐次线性微分方程,欧拉方程,微分方程的幂级数解法,常系数线性微分方程组解法举 例,总习题十二 Differential equations: basic concept of differential equation, differential equation of separable variables, homogeneous equations, linear first-order differential equation, total differential equation, reducible higher differential equations, higher order linear differential equations, homogeneous linear differential equations with constant coefficients, nonhomogeneous linear differential equation with constant coefficients, Eulerian equation, power series solution of differential equation, examples of solution to system of linear differential equations with constant coefficients, Exercise 12

八、高等数学方法 Methods Used in Higher Mathematics
? ? ? ? ? ? ? ? ? ? ? 研究函数与极限的基本方法 Basic approaches for studying functions and limits 一元函数微分法及其应用 Single variable differential calculus and its application 一元函数积分法及其应用 Single variable integral calculus and its application 多元函数微分法及其应用 Multivariable differential calculus and its application 法及其应用 Multivariable integral calculus and its application 级数的判敛、求和及展开法 Convergence checking summation and expansion of series 几类常微分方程的求解法 Method of solving several types of ordinary differential equation 高等数学中的方法综述 An overview of methods used in higher mathematics 数学建模方法 Mathematical modeling method 数值计算方法 Numerical calculation Method 近代分析概念简介 Introduction to the concept of modern analysis

九、复变函数与积分函数 Complex Variables Functions and Integral Functions
本课程是理工科学生继高等数学后的又一门数学基础课。 本课程主要讲授复变函数与积分变 换的基本理论的方法。 能过本课程的学习, 学生不仅能够学到复变函数与积分变换的基本理 论和数学物理及工程技术中常用的数学方法,同时还可以巩固和复习高等数学的基础知识, 提高数学素养, 为学习有关的后续课程和进一步扩大数学知识面奠定必要的数学基础。 在培 养学生的抽象思维能力、 逻辑推理能力、 空间想象能力和科学计算能力等方面起着特殊重要

的作用。 The course is another basic mathematics course for the students’ majored science and engineering apart from the Advanced Mathematics, which mainly teaches the basic theoretical methods about the complex variables functions and integral transformation. Through the course, the students can learn the basic theory of the complex variables functions and integral transformation as well as common mathematics methods for the mathematical physics and engineering technology; in addition to the above, the students can also consolidate and review the basic knowledge of the Advanced Mathematics, improve the mathematical attainment, and lay a solid mathematical foundation for subsequent courses and further expanding the mathematics knowledge. Accordingly, this course is vital to train the students’ abstract thinking ability, logical reasoning ability, space imagination ability, and scientific computing ability. 本课程包括复数与复变函数、解析函数、复变函数的积分、级数、留数、傅里叶变换、拉普 拉斯共七章。 The course covers seven chapters, including the Plurality and Complex Variables Functions, Analytic Function, Integral of the Complex Variables Functions, Progression, Residue, Fourier Transform, and Laplace.

十、实变函数 Real Variable Functions
集合与点集;Lebesgue 测度;可测函数;Lebesgue 积分;微分与不定积分;Lebesgue 空间 Sets and point sets, Lebesgue measure, measurable functions, Lebesgue integral, differentials and indefinite integrals, Lebesgue spaces. 实变函数简明教程/邓东皋,常心怡编。-北京:高等教育出版社,2005.5(2006 重印) A Concise Course of Real Variable Functions by Deng Donggao & Chang Xinyi, Beijing: Higher Education Press, May 2005(reprinting, 2006).

十一、线性代数 Linear Algebra
本课程为数学的一个分支,研究对象是向量,向量空间(或称线性空间) ,线性变换和有限 维的线性方程组。 向量空间是现代数学的一个重要课题; 因而线性代数被广泛地应用于抽象 代数和泛函分析中;通过解析几何,线性代数得以被具体表示。 As a branch of mathematics, the study objects of this course are vector, vector space, linear transformation and linear equations in finite-dimensional. Vector space is one of the significant topics in modern mathematics; linear algebraic is widely applicable in abstract algebraic and functional analysis; the analytic geometry embodies the linear algebraic. 课程分为两部分: (一)习题解答与注释部分结合教学作了大量的注释,让学生深刻领会课 程中的基本概念的准确含义,开阔解题思路,掌握解题方法,避免在容易发生错误的环节上 出现问题,从而提高解题能力,培养良好的数学思维。 (二)参考题部分涉及难度略大一点 且有参考意义的题目,目的是给学生提供自学材料,也为教师在复习、考试环节的命题工作 提供一些参考资料。 Two parts are involved in this course: 1) Combined with the teaching practice, there is a large quantity of notes in problems and solutions section and comments section. So that students are able to grasp the exact meaning of basic conceptions in the course, blaze new trails for solutions and master the solving methods. Consequently, they could take actions to protect against problems where they are liable to make mistakes, improve their capacity of solving problems and cultivate a good mathematical thinking. 2) Questions to consider section consists of some questions with

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higher difficulty and guiding meaning, in order to offer students some self-learning materials, and to provide worthy reference information for teachers for their review and preparation to set papers for the examination. 行列式: 二阶和三阶行列式, n 阶行列式定义,行列式性质,行列式按行展开定理,克莱姆法 则,习题一 Determinant: the second-order determinant, the third-order determinant, the definition of the nth-order determinant, properties of determinants, Cramer's rule, theorem of expansion of a determinant in row, Exercise 1 矩阵:高斯消元法与矩阵的初等变化,矩阵的运算,特殊矩阵,逆矩阵,分块矩阵,利用初 等变换求矩阵,矩阵的秩 Matrix: Gaussian elimination and elementary transformation of matrix, matrix operation, special matrices, inverse matrix, partitioned matrix, elementary transformation, solution of matrix by elementary transformation. rank of matrix 线性方程组:n 维向量及其线性运算,向量组的线性相关性,向量组的秩,矩阵的秩与向量 组秩的关系,齐次线性方程,线性方程组解的结构,习题三 System of linear equations: n dimensional vector and its linear operation, linear correlation of vector group, rank of vector group, correlation between rank of matrix and rank of vector group 矩阵的对角化:矩阵的特征值和特征向量,相似矩阵和矩阵对角化,向量的内积和施密特正 交化,实对称矩阵的对角化 Diagonalization of matrix: eigenvalue and eigenvector of matrix, similar matrix and diagonalization of matrix, inner product of vector and schmidt orthogonalization 二次型:二次型极其矩阵表示,化二次型为标准型,惯性定理与正定二次型,习题五 Quadratic form: quadratic form and its matrix representation, converting quadratic form into standard form, inertial theorem and positive definite quadratic form, Exercise 5 线性空间与线性变换:线性空间的定义与性质,基、坐标及其变换,线性空间的子空间,线 性变换,线性变换的矩阵表示 Linear space and linear transformation: definition and properties of linear space, base, coordinate and their transformation, subspace of linear space, linear transformation, matrix representation of linear transformation 欧式空间与酉空间:向量的内积与欧式空间,标准正交基,正交变换,向量到子空间的距离. 最小二乘法,酉空间介绍 Euclid space and unitary space: inner product of vector and Euclid space, orthonormal basis. orthogonal transformation, distance from vector to subspace, method of the least squares, introduction to unitary space

十二、高等代数(1)Advanced Algebra (1)
高等代数,一般包括两部分:线性代数初步线性代数课本、多项式代数。高等代数在初等代 数的基础上研究对象进一步的扩充, 引进了许多新的概念以及与通常很不相同的量, 比如最 基本的有集合、向量和向量空间等。这些量具有和数相类似的运算的特点,不过研究的方法 和运算的方法都更加繁复。 Ordinarily, Advanced Algebra includes two parts: Elementary Linear Algebra and Polynomial Algebra. Advanced Algebra further enlarges the objects of study in the basis of Elementary Algebra, brings in many new concepts and quantities. The quantities are different from the general ones, like the most essential set, vector, and vector space, etc. These quantities have similar operational features like numbers, but more complicated investigative and operational methods.

十三、高等代数(2)Advanced Algebra (2)
高等代数是大学数学专业开设的专业课, 线性代数是大学中除了数学专业以外的理科, 工科 和部分医科专业开设的课程。高等代数是代数学发展到高级阶段的总称,它包括许多分支。 现在大学里开设的高等代数,一般包括两部分:线性代数初步, 多项式代数。高等代数在

初等代数的基础上研究对象进一步的扩充,引进了许多新的概念以及与通常很不相同的量, 比如最基本的有集合、向量和向量空间等。这些量具有和数相类似的运算的特点,不过研究 的方法和运算的方法都更加繁复。 Advanced Algebra is a specialized course opened for students whose major is mathematics. Linear Algebra is opened for departments except mathematics, like science, engineering and part of the medical majors. Advanced Algebra is a generic term for algebra that develops to a high-grade stage. It includes many branches. Generally, the course opened in college nowadays has two parts: Elementary Linear Algebra and Polynomial Algebra. Advanced Algebra further enlarges the objects of study in the basis of Elementary Algebra, brings in many new concepts and quantities. The quantities are different from the general ones, like the most essential set, vector, and vector space, etc. These quantities have similar operational features like numbers, but the investigative and operational methods are more complicated.

十四、抽象代数 Abstract Algebra
它的概念与思想渗透到所有的数学分支,而其理论与方法在统计学、信息论、计算机科学、 近代物理、 化学以及其他许多科学与工程领域中都有广泛而深入的应用, 是理工类和其它相 关专业高要求的研究生应具备的数学基础。本课程主要讨论群论,环论和域论。 Its concept and thought permeate all branches of mathematics while its theories and methods have been widespread applied to statistics, informatics, computer science, modern physics and chemistry, and many other scientific and engineering fields, which are considered mathematical foundations for high-demanding postgraduates of relevant majors to lay stress on. And this course mainly talks about group theory, ring theory and field theory. 通过本课程的学习,让学生能够系统地理解和掌握近世代数的基本概念、理论与方法,提高 学生的数学素养,抽象思维能力得到系统的训练和提高。学会运用近世代数的理论、方法及 技巧分析解决科学技术和工程领域中遇到的有关问题。 Learning this course can help students to systematically understand and grasp basic concepts, theories and methods of modern mathematics; improve their mathematical qualities; systematically train and improve abstract thinking ability. And it will also contribute to students’ analyzing and solving relevant problems met with in scientific technology and engineering fields with theories, methods and techniques of modern algebra. 群得基本知识;环和域的基本知识;多项式和有理函数;向量空间;域的扩张;有限域; Galois 理论初步 Elementary knowledge of groups, elementary knowledge of rings and fields, polynomials and rational functions, vector spaces, expansion of fields, finite fields, preliminary introduction of Galois theory. 近世代数讲义/杨劲根编著。-北京:科学出版社,2009 Modern Algebra Lectures by Yang Jingen, Beijing: Science Press, 2009.

十五、数学分析(1)Mathematical Analysis (1)
是数学专业的必修课程之一,数学分析的基础是实数理论。实数系最重要的特征是连续性, 有了实数的连续性,才能讨论极限,连续,微分和积分。正是在讨论函数的各种极限运算的 合法性的过程中, 人们逐渐建立起严密的数学分析理论体系。 数学中的分析分支是专门研究 实数与复数及其函数的数学分支。它的发展由微积分开始,并扩展到函数的连续性、可微分

及可积分等各种特性。这些特性,有助我们应用在对物理世界的研究,研究及发现自然界的 规律。 As one of the required courses for mathematical major, Mathematical Analysis is based on real number theory. The most important feature of real number system is the continuity. Only on condition that continuity exists, can we discuss limit, continuation, differential and integration. It is in the process of discussing the validity of various functional limit operations that the strict mathematical analysis theory system gradually been set up. The analysis branch in mathematics specializes in researching real number, complex number and its functions. It develops from the differential and integral calculus and expands to the continuity, differentiable, integrable features of function. These features can help us in researching the physics world and discovering the law in natural world.

十六、数学分析 中 Mathematical Analysis Ⅱ
数项级数;广义积分;函数项级数;幂级数;傅里叶 Numerical series, improper integrals, function series, powder series, Fourier series, the limits and continuity of multivariate functions, partial derivatives and complete differentials, implicit function theorem, extremes and conditional extremes. 数学分析简明教程.上册/邓东皋。尹小玲编著 2 版。--北京:高等教育出版社 2006.3 A Concise Course of Mathematical Analysis Ⅰ by Deng Donggao & Yin Xiaoling, 2nd edition, Beijing: Higher Education Press, Mar. 2006.

十七、数学分析 下 Mathematical Analysis Ⅲ
含参变量的积分;重积分;曲线积分与曲面积分;各种积分间的联系与场论初步 Integrals with parameter, multiple integrals, line integrals and surface integrals, relation between various integrals and preliminary introduction of field theory 数学分析简明教程.上册/邓东皋。小玲编著 2 版。--北京:高等教育出版社 2006.3.7 A Concise Course of Mathematical Analysis Ⅰ by Deng Donggao & Yin Xiaoling, 2nd edition, Beijing: Higher Education Press, Mar. 7th, 2006.

十八、离散数学 Discrete Mathematics
数理逻辑;集合论;代数系统;图论;计算机科学中的应用 Mathematical logic, set theory, algebraic system, graph theory, application in computer science 离散数学 左孝凌 李为 刘永才编著上海科学技术文献出版社 2008.6 第 57 次印刷 Discrete Mathematics by Zuo Xiaoling, Li Wei & Liu Yongcai, 57th printing, Shanghai Science and Technology Publishing House, Jun. 2008.

十九、微分几何 Differential Geometry
曲线;正则曲面;Gauss 映照的几何学;全面的内蕴几何 Curves, regular surfaces, Gauss mapping of the geometry, full intrinsic geometry. 曲线与曲面的微分几何/杜卡莫著;田畴等译。--北京:机械工业出版社,2005.1 Differential Geometry of Curves and Surfaces by Du Kamo & Tian Chou, Beijing: China Machine Press, Jan. 2005.

二十、解析几何 Analytic Geometry
解析几何包括平面解析几何和立体解析几何两部分。 平面解析几何通过平面直角坐标系, 建 立点与实数对之间的一一对应关系, 以及曲线与方程之间的一一对应关系, 运用代数方法研 究几何问题, 或用几何方法研究代数问题。 解析几何的建立第一次真正实现了几何方法与代 数方法的结合,使形与数统一起来,这是数学发展史上的一次重大突破。 Analytic Geometry includes Plane Analytic Geometry and Solid Analytic Geometry. By using plane rectangular coordinate system, Plane Analytic Geometry sets up one-to-one corresponding relationship between points and real number pairs, between curves and equations. Geometry questions are solved through algebraic approaches, and vise versa. Because of Analytic Geometry, the geometry and algebraic approaches are combined together for the first time, thus shapes and numbers are unified. This is an important breakthrough in mathematics development history.

二十一、几何与代数 下 Geometry and Algebra, Ⅱ
向量平面与直线;二次曲面与坐标变换;线性空间与线性映射;矩阵线性方程与行列式;多 项式;线性变换;双线性型与欧式空间;仿射空间与射影空间 Vector planes and straight lines, quadratic surfaces and coordinate transformation, linear spaces and linear mapping, matrix linear equations and determinants, polynomials, linear transformation, bilinear form and Euclidean space, affine space and projective space. 几何与代数 胡国权编著。北京:科学出版社 2006. Geometry and Algebra by Hu Guoquan, Beijing: Science Press, Sep. 2006.

二十二、常微分方程 Ordinary Differential Equations
凡是表示未知函数的导数以及自变量之间的关系的方程, 就叫做微分方程。 常微分方程的概 念、解法、和其它理论很多,比如,方程和方程组的种类及解法、解的存在性和唯一性、奇 解、定性理论等等,现在,常微分方程在很多学科领域内有着重要的应用,自动控制、各种 电子学装置的设计、弹道的计算、飞机和导弹飞行的稳定性的研究、化学反应过程稳定性的 研究等。这些问题都可以化为求常微分方程的解,或者化为研究解的性质的问题。 Differential Equations are equations that show the relationship between derivative and independent variable of unknown functions. There are many concepts, solutions and other theories in Ordinary Differential Equations, such as the types and solutions of equations and equation sets, existence and uniqueness of solutions, singular solutions, and qualitative theory, etc. Nowadays, Ordinary Differential Equations plays an important role in many academic ambits like automation, design of various electronic devices, ballistic calculation, research on the stability of flights by airplanes and missiles, research on stability of chemical reaction processes, etc. All of the above mentioned problems can be transformed into solutions of Ordinary Differential Equations or into research on the natures of the solution. 一阶微分方程的初等解法; 一阶微分方程的解的存在定理; 高阶微分方程; 线性微分方程组; 非线性微分方程;一阶线性偏微分方程; Elementary solution of first-order differential equations, theorem of existence of solutions of first-order differentia equations, higher differential equations, systems of linear differential equations, nonlinear differential equations, first-order linear partial differential equation. 常微分方程 王高雄 周之铭 王寿松 编著—3 版。-北京 高等教育出版社 2006.7(2009 重 印)

Ordinary Differential Equations by Wang Gaoxiong, Zhou Zhiming & Wang Shousong, 3rd edition, Beijing: Higher Education Press, Jul. 2006(reprinting in 2009).

二十三、偏微分方程 Partial Differential Equations
波动方程; 热传导方程; 调和方程; 二阶线性偏微分方程的分类与总结; 一阶偏微分方程组; 广义解与广义函数解;偏微分方程的数值解 Wave equations, heat conduction equations, harmonic equations, classification and summarization of second-order linear partial differential equations, systems of first-order partial differential equations, generalized solutions and generalized function solutions, numerical solutions of partial differential equations. 数学物理方程(第二版)谷超豪 李大潜 陈恕行 郑宋穆 谭永基编著 高等教育出版社 2001.9 Mathematical Physics Equations by Gu Chaohao, Li Daqian, Chen Shuxing, Zheng Songmu & Tan Yongji, 2nd edition, Higher Education Press, Sep. 2001

二十四、数据结构与算法 Data Structures and Algorithms
线性表;栈和队列;串;数组和广义表;递归;树和二叉树;图;查找;内排序; Linear list, stacks and queues, strings, arrays and generalized list, recursion, trees and binary trees, graphs, searching, internal sorting 数据结构教程/李春葆等编著。-北京:清华大学出版社,2009.3 A Course of Data Structures by Li Chunbao, Beijing: Tsinghua University Press, Mar. 2009.

二十五、数值分析 Numerical Analysis
插值法;函数逼近与快速傅里叶变换;数值积分与数值微分;解线性方程组的直接方法;解 线性方程组的迭代法 Interpolation method, approximation of function and fast Fourier transform, numerical integration and numerical differentiation, direct method of system of linear equations, iterative method of system of linear equations. 数值分析/李庆扬,王能超,易大义编。-北京:清华大学出版社,2008.12 Numerical Analysis by Li Qingyang, Wang Nengchao & Yi Dayi, Beijing: Tsinghua University Press, Dec. 2008.

二十六、数学实验与数学软件 Mathematical Experiments and Software
微积分基础; 怎样计算 π; 最佳分数近似值; 数列与级数; 素数; 概率; 几何变换; 迭代(一)—— 方程求解;迭代(二)——分形;迭代(三)——混沌 Elementary calculus, how to calculate π, best fraction approximation, progressions and series, prime numbers, probability, geometric transformation, iteration Ⅰ —equation solutions, iteration Ⅱ —fractal, iteration Ⅲ —chaos. 数学实验/李尚志,高等教育出版社, 2004 Mathematical Experiments by Li Shangzhi, Higher Education Press, 2004


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