数学分析课程英文描述
数学分析 高等数学 微积分 英语课件 上海交通大学Chapter5a
i0 n
n
i1 n
n
Example
Ex. Determine a region whose area is equal to the given
limit
(1) lim
n
2 (5 2i )10
n n i1
n
n i
(2) lim
tan
n i1 4n 4n
Definition of definite integral
Ex. Use the definition of definite integral to prove that b f (x) c is integrable on [a,b], and find cdx. a
Interpretation of definite integral
b
If f (x) 0, the integral f (x)dx is the area under the a curve y=f(x) from a to b
Idea: first, divide the time interval [a,b] into n subintervals;
then, approximate the distance di in each subinterval [ti-1,ti]
by di¼(ti-ti-1)v(xi) since v(t) does not vary toonmuch and
lim
n
i1
Si
always exists and has same value.
The distance problem
Problem: find the distance traveled by an object during the time period [a,b], given the velocity function v=v(t).
数学分析课程简介
导言数学分析课程简介一、数学分析(mathematical analysis)简介:1.背景: 从切线、面积、计算sin、实数定义等问题引入.322.极限 ( limit ) ——变量数学的基本运算：3.数学分析的基本内容：数学分析以极限为基本思想和基本运算研究变实值函数.主要研究微分(differential)和积分(integration)两种特殊的极限运算,利用这两种运算从微观和宏观两个方面研究函数, 并依据这些运算引进并研究一些非初等函数. 数学分析基本上是连续函数的微积分理论.微积运算是高等数学的基本运算.数学分析与微积分(calculus)的区别.二、数学分析的形成过程：1.孕育于古希腊时期：在我国，很早就有极限思想. 纪元前三世纪,Archimedes就有了积分思想.2.十七世纪以前是一个漫长的酝酿时期,是微积分思想的发展、成果的积累时期.3.十七世纪下半叶到十九世纪上半叶——微积分的创建时期.4.十九世纪上半叶到二十世纪上半叶——分析学理论的完善和重建时期：三、数学分析课的特点:逻辑性很强, 很细致, 很深刻; 先难后易, 是说开头四章有一定的难度,倘能努力学懂前四章(或前四章的), 后面的学习就会容易一些; 只要在课堂上专心听讲, 一般是可以听得懂的, 但即便能听懂, 习题还是难以顺利完成. 这是因为数学分析技巧性很强, 只了解基本的理论和方法, 不辅以相应的技巧, 是很难顺利应用理论和方法的. 论证训练是数学分析课基本的,也是重要的内容之一, 也是最难的内容之一. 一般懂得了证明后, 能把证明准确、严密、简练地用数学的语言和符号书写出来，似乎是更难的一件事. 因此, 理解证明的思维方式, 学习基本的证明方法, 掌握叙述和书写证明的一般语言和格式, 是数学分析教学贯穿始终的一项任务.有鉴于此, 建议的学习方法是: 预习, 课堂上认真听讲, 必须记笔记,但要注意以听为主, 力争在课堂上能听懂七、八成. 课后不要急于完成作业, 先认真整理笔记, 补充课堂讲授中太简或跳过的推导, 阅读教科书, 学习证明或推导的叙述和书写. 基本掌握了课堂教学内容后, 再去做作业. 在学习中, 要养成多想问题的习惯.四、课堂讲授方法:1.关于教材及参考书:这是大学与中学教学不同的地方, 本课程主要从以下教科书中取材:[1]华东师范大学数学系编，数学分析(第三版)，高等教育出版社，2001；[2] 陈纪修於崇华等编，《数学分析》（第二版）高等教育出版社，2001[3]谢惠民，恽自求等数学分析习题课讲义，高等教育出版社，2003；[4]马振民，数学分析的方法与技巧选讲，兰州大学出版社，1999；[5]林源渠，方企勤数学分析解题指南，北京大学出版社，2003.2.本课程按[1]的逻辑顺序并在其中取材.本课程为适应教学改革的要求，只介绍数学分析最基本的内容，并加强实践环节，注重学生的创新能力的培养。
介绍数学的英语
介绍数学的英语Mathematics is the study of numbers, shapes, patterns, and their relationships. It is a field that deals with logical reasoning and problem-solving using numerical calculations, measurements, and mathematical models. Math is used extensively in various disciplines such as physics, engineering, finance, computer science, and many more.Here are 27 bilingual example sentences related to mathematics:1.数学是一门需要逻辑推理和问题解决的学科。
Mathematics is a discipline that requires logical reasoning and problem-solving.2.数学是一种描述和量化现实世界的语言。
Mathematics is a language that describes and quantifies the real world.3.我们使用数学来解决实际生活中的各种问题。
We use mathematics to solve various problems in everyday life.4.算数是数学的一个重要分支，涉及基本的加减乘除运算。
Arithmetic is an important branch of mathematics that involves basic operations like addition, subtraction, multiplication, and division.5.代数是研究数之间关系和未知量的分支。
数学分析课程汇报
1．加强政治思想和职业道德教育，培养教师具有对学生的高 度责任感，对教育事业的强烈事业心和献身精神。 2．建立一支对数学分析、高等数学内容领会深入、教育理 论扎实、教学经验丰富、教学效果好、教风严谨、勇于进行教学 改革的教学骨干队伍，争取教研室70%以上成为教学骨干。 3．拥有掌握本专业范围内容数学发展动态，具有本专业内 科研主攻方向，具有一定科研能力和水平的学术带头人，带动教 研室工作开展，并年均公开发表论文四篇以上。 4．优化教师结构，建立一个梯队状况良好、职称结构合理、 教学水平稳定、教学效果好、团结协作的教学群体，达到高、中、 初级教师人数比例1：2：2。中青年教师中100%达到硕士研究生 水平。
情境——问题教学模式
学生学习：质疑提问、自主合作探索
（基础）
（纽带）
（核心）
（归宿）
设置情境 （观察分析）
提出问题
解决问题
注重应用
（猜想探究）
（求解反驳）
（学做学用）
教师导学：启发诱导、矫正解惑讲授
情境—问题”校本化教学模式：
心理安全情境
教 师 启导
让学生敢问
（尝试参与）
学生主体情境
（激发内因）
(开启学生的思维，点拨学生的思路)
学生应采取以探究式为中心的自 主合作的学习方法。
(既要学好科学知识,又要掌握学习方法)
八、考核方式
本课程是一门考试课程，考核以笔试为主,闭 卷。主要考核学生对基础理论，基本概念的掌握 程度，以及学生逻辑推理能力和计算能力。成绩 考核评定按学校教务处的相关规定执行。
• 知识与技能目标——基石作用； • 过程与方法目标——桥梁作用； • 情感态度价值观目标——动力作用。
通过三维目标的整合，帮助学生学会学习、 在学习中体验思想方法对学习的作用，培养 其文化观与审美观。
高中数学课程描述(英文)
Mathematics Course DescriptionMathematics course in middle school has two parts: compulsory courses and optional courses. Compulsory courses content lots of modern mathematical knowledge and conceptions, such as calculus, statistics, analytic geometry, algorithm and vector. Optional courses are chosen by students which is according their interests.Compulsory Courses:Set TheoryCourse content:This course introduces a new vocabulary and set of rules that is foundational to the mathematical discussions. Learning the basics of this all-important branch of mathematics so that students are prepared to tackle and understand the concept of mathematical functions. Students learn about how entities are grouped into sets and how to conduct various operations of sets such as unions and intersections (i.e. the algebra of sets). We conclude with a brief introduction to the relationship between functions and sets to set the stage for the next stepKey Topics:➢The language of set theory➢Set membership➢Subsets, supersets, and equality➢Set theory and functionsFunctionsCourse content:This lesson begins with talking about the role of functions and look at the concept of mapping values between domain and range. From there student spend a good deal of time looking at how to visualize various kinds of functions using graphs. This course will begin with the absolute value function and then move on to discuss both exponential and logarithmic functions. Students get an opportunity to see how these functions can be used to model various kinds of phenomena. Key Topics:➢Single-variable functions➢Two –variable functions➢Exponential function➢ Logarithmic function➢Power- functionCalculusCourse content:In the first step, the course introduces the conception of limit, derivative and differential. Then students can fully understand what is limit of number sequence and what is limit of function through some specific practices. Moreover, the method to calculate derivative is also introduced to students.Key Topics:➢Limit theory➢Derivative➢DifferentialAlgorithmCourse content:Introduce the conception of algorithm and the method to design algorithm. Then the figures of flow charts and the conception of logical structure, like sequential structure, contracture of condition and cycle structure are introduced to students. Next step students can use theknowledge of algorithm to make simple programming language, during this procedure, student also approach to grammatical rules and statements which is as similar as BASIC language.Key Topics:➢Algorithm➢Logical structure of flow chart and algorithm➢Output statement➢Input statement➢Assignment statementStatisticsCourse content:The course starts with basic knowledge of statistics, such as systematic sampling and group sampling. During the lesson students acquire the knowledge like how to estimate collectivity distribution according frequency distribution of samples, and how to compute numerical characteristics of collectivity by looking at numerical characteristics of samples. Finally, the relationship and the interdependency of two variables is introduced to make sure that students mastered in how to make scatterplot, how to calculate regression line, and what is Method of Square.Key Topics:➢Systematic sampling➢Group sampling➢Relationship between two variables➢Interdependency of two variablesBasic Trigonometry ICourse content:This course talks about the properties of triangles and looks at the relationship that exists between their internal angles and lengths of their sides. This leads to discussion of the most commonly used trigonometric functions that relate triangle properties to unit circles. This includes the sine, cosine and tangent functions. Students can use these properties and functions to solve a number of issues.Key Topics:➢Common Angles➢The polar coordinate system➢Triangles properties➢Right triangles➢The trigonometric functions➢Applications of basic trigonometryBasic Trigonometry IICourse content:This course will look at the very important inverse trig functions such as arcsin, arcos, and arctan, and see how they can be used to determine angle values. Students also learn core trig identities such as the reduction and double angle identities and use them as a means for deriving proofs. Key Topics:➢Derivative trigonometric functions➢Inverse trig functions➢Identities●Pythagorean identities●Reduction identities●Angle sum/Difference identities●Double-angle identitiesAnalytic Geometry ICourse content:This course introduces analytic geometry as the means for using functions and polynomials to mathematically represent points, lines, planes and ellipses. All of these concepts are vital in student’s mathematical development since they are used in rendering and optimization, collision detection, response and other critical areas. Students look at intersection formulas and distance formulas with respect to lines, points, planes and also briefly talk about ellipsoidal intersections. Key Topics:➢Parametric representation➢Parallel and perpendicular lines➢Intersection of two lines➢Distance from a point to a line➢Angles between linesAnalytic Geometry IICourse content:Students look at how analytic geometry plays an important role in a number of different areas of class design. Students continue intersection discussion by looking at a way to detect collision between two convex polygons. Then students can wrap things up with a look at the Lambertian Diffuse Lighting model to see how vector dot products can be used to determine the lighting and shading of points across a surface.Key Topics:➢Reflections➢Polygon/polygon intersection➢LightingSequence of NumberCourse content:This course begin with introducing several conceptions of sequence of number, such as, term, finite sequence of number, infinite sequence of number, formula of general term and recurrence formula. Then, the conception of geometric sequence and arithmetic sequence is introduced to students. Through practices and mathematical games, students gradually understand and utilize the knowledge of sequence of number, eventually students are able to solve mathematical questions.Key Topics:➢Sequence of number➢Geometric sequence➢Arithmetic sequenceInequalityThis course introduces conception of inequality as well as its properties. In the following lessons students learn the solutions and arithmetic of one-variable quadratic inequality, two variables inequality, fundamental inequality as well how to solve simple linear programming problems.Key Topics:➢Unequal relationship and Inequality➢One-variable quadratic inequality and its solution➢Two-variable inequality and linear programming➢Fundamental inequalityVector MathematicsCourse content:After an introduction to the concept of vectors, students look at how to perform various important mathematical operations on them. This includes addition and subtraction, scalar multiplication, and the all-important dot and cross products. After laying this computational foundation, students engage in games and talk about their relationship with planes and the plane representation, revisit distance calculations using vectors and see how to rotate and scale geometry using vector representations of mesh vertices.Key Topics:➢Linear combinations➢Vector representations➢Addition/ subtraction➢Scalar multiplication/ division➢The dot product➢Vector projection➢The cross productOptional CoursesMatrix ICourse content:In this course, students are introduced to the concept of a matrix like vectors, matrices and so on. In the first two lessons, student look at matrices from a purely mathematical perspective. The course talks about what matrices are and what problems they are intended to solve and then looks at various operations that can be performed using them. This includes topics like matrix addition and subtraction and multiplication by scalars or by other matrices. At the end, students can conclude this course with an overview of the concept of using matrices to solve system of linear equations.Key Topics:➢Matrix relations➢Matrix operations●Addition/subtraction●Scalar multiplication●Matrix Multiplication●Transpose●Determinant●InversePolynomialsCourse content:This course begins with an examination of the algebra of polynomials and then move on to look at the graphs for various kinds of polynomial functions. The course starts with linear interpolation using polynomials that is commonly used to draw polygons on display. From there students are asked to look at how to take complex functions that would be too costly to compute in a relatively relaxed studying environment and use polynomials to approximate the behavior of the function to produce similar results. Students can wrap things up by looking at how polynomials can be used as means for predicting the future values of variables.Key Topics:➢Polynomial algebra ( single variable)●addition/subtraction●multiplication/division➢Quadratic equations➢Graphing polynomialsLogical Terms in MathematicsCourse content:This course introduces the relationships of four kinds of statements, necessary and sufficient conditions, basic logical conjunctions, existing quantifier and universal quantifier. By learning mathematical logic terms, students can be mastered in the usage of common logical terms and can self-correct logical mistakes. At the end of this course, students can deeply understand the mathematical expression is not only accurate but also concise.Key Topics:➢Statement and its relationship➢Necessary and sufficient conditions➢Basic logical conjunctions➢Existing quantifier and universal quantifierConic Sections and EquationCourse content:By using the knowledge of coordinate method which have been taught in the lesson of linear and circle, in this lesson students learn how to set an equation according the character of conic sections. Students is able to find out the property of conic sections during establishing equations. The aim of this course is to make students understand the idea of combination of number and shape by using the method of coordinate to solve simple geometrical problems which are related to conic sections.Key Topics:➢Curve and equation➢Oval➢Hyperbola➢Parabola。
数学分析课程简介
数学分析课程简介课程编码:21090031-21090033课程名称：数学分析英文名称：Mathematical Analysis课程类别：学科基础课程课程简介：数学分析俗称：“微积分”，创建于17世纪，直到19世纪末及20世纪初才发展为一门理论体系完备，内容丰富，应用十分广泛的数学学科。
数学分析课是各类大学数学与应用数学专业、信息与计算科学专业最主要的专业基础课。
是进一步学习复变函数论、微分方程、微分几何、概率论、实变分析与泛函分析等后继课程的阶梯，是数学类硕士研究生的必考基础课之一。
本课程基本的内容有：极限理论、一元函数微积分学、级数理论、多元函数微积分学等方面的系统知识，用现代数学工具——极限的思想与方法研究函数的分析特性——连续性、可微性、可积性。
极限方法是贯穿于全课程的主线。
课程的目的是通过三个学期学习和系统的数学训练，使学生逐步提高数学修养，特别是分析的修养，积累从事进一步学习所需要的数学知识，掌握数学的基本思想和方法，培养与锻炼学生的数学思维素质，提高学生分析与解决问题的能力。
教材名称：数学分析教材主编：华东师范大学主编（第四版）出版日期：2010年6月第四版出版社：高等教育出版社《数学分析1》课程教学大纲（2010级执行）课程代号：21090031总学时：80学时（讲授58学时，习题22学时）适用专业：数学与应用数学、信息与计算科学先修课程：本课程不需要先修课程，以高中数学为基础一、本课程地位、性质和任务本课程是本科数学与应用数学专业、信息与计算科学专业的一门必修的学科基础课程。
通过本课程的教学，使学生掌握数学分析的基本概念、基本理论、思想方法，培养学生解决实际问题的能力和创新精神，为学习后继课程打下基础。
二、课程教学的基本要求重点：极限理论；一元函数微分学及贯穿整个课程内容的无穷小分析的方法。
基本要求：掌握极限、函数连续性、可微等基本概念；掌握数列极限、函数极限；闭区间连续函数性质；熟练掌握函数导数、微分的计算及应用；掌握微分中值定理及其应用。
(完整版)部分基础课中英文描述
课程名称：高等数学课程类别：学科基础课课程简介：该课程主要包括函数、极限、连续、导数、微分、中值定理、导数的应用、微分学在经济学中的应用、不定积分、定积分及在经济学中的应用、无穷级数、多元函数微积分（偏导、全微分、复合函数与隐函数的微分法、极值应用、二重积分、多元函数极值）、微分方程（一、二阶微分方程及高阶线性微积分方程等），差分方程简介，及在几何、经济等方面的应用。
通过本课程的学习，使学生获得必要的微积分知识，学习从数量的角度观察问题，分析问题，培养抽象思维和逻辑推理能力。
Course Name：Advanced MathematicsCourse Description：This subject introduces mainly function, limit, sequence, derivative, differential, theorem of mean, application of derivate, application of differential in the economics, indefinite integration, definite integration and its application in Economics, infinite series, multi-variables function calculus(partial derivative, total differential, composite function and differential method of implicit function, application of extreme, double integral, extreme value of multi-variable function), differential equation(first and second order differential equation and higher order linear equation), brief introduction to difference equation and its application in geometry and economy. By learning this subject, students could understand essential knowledge of Calculus, observe and analyze questions from quantity.课程名称：大学物理课程简介：本课程主要包括热学与力学两个部分，热学研究的是自然界中物质与冷热有关的性质及这些性质变化的规律。
数学分析英文词汇
数学分析中的英语单词第一章real number and its properties 实数及其性质 rational number 有理数 irrational number 无理数 definition 定义 proposition 命题plus 加minus 减multiplied by/times 乘over/is to/divided by 除absolute value and inequality 绝对值与不等式 triangle inequality 三角不等式 inverse triangle inequality 反三角不等式 Bernoulli inequality 伯努利不等式 principles of supremum and infimum 确界原理open interval 开区间closed interval 闭区间semi-open interval 半开区间semi-closed interval 半闭区间finite interval 有限区间 infinite interval 无限区间 neighborhood 领域deleted neighborhood 去心邻域sum 和difference 差product 积quotient 商number axis/number line 数轴 closeness 封闭性 Archimedean property 阿基米德性质 density 稠密行upper and lower bounds 上界和下界 bounded set 有界集 unbounded set 无界集 existence domian 存在域 supremum 上确界 infimum 下确界order-complete set 有序完备集 completeness of real numbers 实数的完备性 complete ordered field 全序域 axiom of completeness 完备性公理 Dedekind cut 戴德金分割 Dedekind property 戴德金性质 constant and variable quantities 常量与变量 definition of function 函数的定义域 domain 定义域 range 值域 independent variable 自变量 dependent variable 因变量 intermediate variable 中间量 elementary function 初等函数 constant function 常量函数 power function 幂函数 exponential function 指数函数 logarithmic function 对数函数 trigonometric function 三角函数 inverse trigonometric function 反三角函数 compound function 复合函数 mapping 映射inverse mapping 逆映射 image 像primary image 原像 piecewise function 分段函数sign function 符号函数 Dirichlet function 狄利克雷函数 Riemann function 黎曼函数 bounded function 有界函数 monotone function 单调函数 monotone increasing function 单调增函数 strictly monotone function 严格单调函数 odd(even) function 奇函数（偶函数） minimal positive period 最小正周期 absolute value function 绝对值函数 identity function 恒等函数 polynomial function 多项式函数linear function 线性函数 quadratic function 二次函数rational function 有理函数 hyperbolic sine 双曲正弦 hyperbolic cosine 双曲余弦 trigonometric identity 三角恒等式odd-even identity 奇偶恒等式 cofunction identity 余函数恒等式 Pythagorean identity 毕达哥拉斯恒等式 half-angle identity 半角恒等式 product identity 积恒等式sum identity 和恒等式 addition identity 加法恒等式 double-angle identity 倍角恒等式第二章 数列极限limit of sequence 数列极限 divergent sequence 发散数列 infinitesimal sequence 无穷小数列 convergent sequence 收敛数列 uniqueness theorem 唯一性定理boundedness theorem 有界性定理 inheriting order properties 保序性inheriting inequality 保不等式 subsequence 子列strictly increasing 严格递增 monotone increasing sequence 单调递增序列 monotone decreasing sequence 单调递减序列 necessary condition 必要条件sufficient condition 充分条件squeeze principle 夹逼定理Cauchy convergence criterion 柯西收敛准则第三章 函数极限limit of function 函数极限infinite limit 无穷极限one-sided limit 单侧极限right(left)limit/right(left) hand limit 右（左）极限 property of limit of function 函数极限的性质 local boundedness 局部有界性Heine theorem 海涅定理infinity 无穷大量order of infinitesimal 无穷小量的阶 infinitesimal of higher(lower) order 高（低）阶无穷小量 infinitesimal of same order 同阶无穷小量 equivalent infinitesimal 等价无穷小量k-order infinitesimal k-阶无穷小量 vertical asymptote 垂直渐近线oblique asymptote 斜渐进线 horizontal asymptote 水平渐近线第四章函数的连续性increment of independent variable 自变量的增量 increment of function 函数的增量right(left) continuous 右(左)连续discontinuity point and its classification 间断点及其分类 removable discontinuity 可去间断点jump discontinuity 跳跃间断点 discontinuity of the first kind 第一类间断点 discontinuity of the second kind 第二类间断点local properties of continuous function 连续函数的局部性质 composition properties of continuous function 连续函数的复合性质闭区间上连续函数的性质 properties of continuous function over closedintervalextreme value theorem 极值定理maximum and minimum value theorem 最大值和最小值定理 intermediate value theorem 介值性定理zero-point theorem 零点定理uniform continuity theorem 一致连续定理continuity of inverse function 反函数的连续性local inheriting order property 局部保号性continuity of elementary function 初等函数的连续性第五章 导数和微分finite increment formula 有限增量公式rate of change 变化率difference quotient 差商left(right) derivative 左（右）导数 derivative function 导函数derivable function 可导函数geometric meaning of derivative 导数的集合意义Fermat theorem 费马定理Darboux theorem 达布定理intermediate value theorem of derivative function 导函数的介值定理 algebra of derivatives 导数的四则运算 derivative of sum 和的导数derivatives of difference 差的导数derivative of product 积的导数derivative of quotient 商的导数 derivative of inverse function 反函数的导数 maximum value 最大值minimum value 最小值 derivative of composite function 复合函数的导数 logarithmic derivative 对数求导法 parametric equation of circle 圆的参数方程 parametric equation of ellipse 椭圆的参数方程 parametric equation of cycloid 摆线的参数方程 parametric equation of asteroid 星形线的参数方程 second derivative 二阶导数third derivative 三阶导数n-th derivative n阶导数Leibniz formula 莱布尼茨公式 acceleration 加速度physical interpretation 物理解释 concept of differential 微分的概念 differentiable function 可微函数linear principal part 线性主部 differential of independent variable 自变量的微分 operational rules of differential 微分的运算法则 invariance of differential form 微分形式的不变性 geometric meaning of differential 微分的几何意义 higher-order differential 高阶微分第六章 微分中值定理Rolle mean value theorem 罗尔中值定理 Lagrange mean value theorem 拉格朗日中值定理 Cauchy mean value theorem 柯西中值定理 Taylor theorem 泰勒定理L’Hospital rule 洛必达法则limit of indeterminate form of type 0/0 0/0型不定式极限 indeterminate form of type ∞/∞ ∞/∞型不定式other indeterminate forms 其他类型不定式Taylor formula with Peano remainder 带有皮亚诺型余项的泰勒公式Taylor coefficient 泰勒系数Taylor polynomial 泰勒多项式remainder of Taylor formula 带有拉格朗日型余项的泰勒公式Taylor formula with Lagrange remainder 带有拉格朗日型余项的麦克劳林公式extreme value of function 函数的极值test of extreme value 极值判断the first sufficient condition of extreme value 极值的第一充分条件 convexity and inflection point of function 函数的凸性和拐点 convex function 凸函数concave function 凹函数strictly convex function 严格凸函数strictly concave function 严格凹函数Jenson inequality 琴生不等式第七章实数的完备性实数击完备性的基本定理 fundamental theorems of completeness in the setof real numbersnested interval theorem 闭区间套定理Cauchy convergence criterion 柯西收敛准着 accumulation theorem 聚点定理finite covering theorem 有限覆盖定理upper and lower limits 上极限和下极限第八章 不定积分indefinite integral 不定积分primitive function 原函数integrand function 被积函数integral sign 积分符号expression of integrand 积分表达式integral constant 积分常数table of basic integrals 基本积分表 geometric meaning of indefinite 不定积分的几何意义 integral curve 积分曲线initial condition 初始条件 integration by parts 分部积分法 integration by substitutions 换元积分法formula of substitution of the first kind 第一换元公式 indefinite integral of rational function 有理函数的不定积分 proper fraction 真分式improper fraction 假分式 decomposition into partial fractions 部分分式分解 method of undermined coefficients 待定系数法第九章 定积分definite integral 定积分curvilinear trapezoid 曲边梯形dividing/partition 分割norm/modulus 模Riemann sum 黎曼和Riemann integral 黎曼积分 integrability in the sense of Riemann 黎曼可积interval of integration 积分区间upper and lower limits 上限和下限 geometric meaning of definite integral 定积分的几何意义 upper sum 上和lower sum 下和Darboux upper sum 达布上和Darboux lower sum 达布下和Newton-Leibniz formula 牛顿-莱布尼茨公式 necessary condition for integrability 可积的必要条件 necessary and sufficient conditions for integrability 可积的充要条件 integrable function class 可积函数类linear property of definite integral 定积分的线性性质 additive with respect to integral interval/additivityover integral interval积分区间的可加性 mean value theorem of integral 积分中值定理 average value 平均值integral with variant upper limit 变上限积分 existence of primitive function 原函数的存在性 integration by substitution 换元积分法 integration by parts 分部积分法integral form of reminder of Taylor formula 泰勒公式的积分类型 第十章 定积分的应用area of plane figure 平面图形的面积method of finding volume of a solid from the knows area of parallel sections 由平行截面面积求体积的方法arc length and curvature of plane curve 平面的曲线的弧长与曲率 smooth curve 光滑曲线differential of arc 弧微分circle of curvature 曲率圆radius of curvature 曲率半径area of revolution surface 旋转曲面的面积some applications of definite integral in physics 定积分在物理中的某些应用gravitation 引力work 功approximate computation of definite integral 定积分的近似计算 trapezoidal method 梯形法parabola method 抛物线法第十一章反常积分notion of improper integral 反常积分的概念 improper integral on infinite interval 无穷区间上的反常积分 improper integral of unbounded function 无界函数的反常积分 property of infinite integral and test of 无穷积分的性质和收敛判convergence 别absolutely convergent 绝对收敛comparison test 比较收敛法Cauchy test 柯西判别法Dirichlet test 狄利克雷判别法Abel test 阿贝尓判别法第十二章数项级数series with number terms 数项级数infinite series 无穷级数convergence of series 级数的收敛性 sequence of partial sum 部分和序列geometric series 几何级数harmonic series 调和级数Cauchy convergence criterion for series 级数收敛的柯西准则 series with positive terms 正项级数necessary condition for convergence 收敛的必要条件root test 根式判别法D’alembert(ratio) test 达朗贝尔判别法（比式判别法）limit form of ratio test 比式判别法的极限形式 Gauss test 高斯判别法integral test 积分判别法Raabe test 拉贝尓判别法p-series p级数series with arbitraty terms 一般项级数alternating series 交错级数Leibnitz test 莱布尼茨判别法 rearrangement of series 级数的重排Abel test 阿贝尓判别法Dirichlet test 狄利克雷判别法第十章 函数列与函数项级数convergent at x"在点x"收敛convergence domain/region of convergence 收敛域sum function 和函数limit function 极限函数series of functions 函数项级数uniform convergence 一致收敛test of uniform convergence 一致收敛判别法魏尔斯特拉判别法 Weierstrass’s test/Weierstrass uniformconvergence criterion/Weierstrass m-test foruniform convergenceuniform boundedness 一致有界Dirichlet test 狄利克列判别法第十四章幂级数power series 幂级数interval of convergence 收敛区间radius of convergence 收敛半径Abel theorem 阿贝尓定理operations of power series 幂级数的运算taylor series 泰勒级数expansion of power series of elementary function 初等函数的幂级数展开式 exponential function of complex variable 复变量的指数函数Euler formula 欧拉公式第十五章傅里叶级数Fourier series 傅里叶级数 trigonometric series 三角级数system of orthogonal functions 正交函数系simple harmonic vibration 简谐振动Fourier series for function of period 2π以2π为周期的函数的傅立叶级数angular frequency 角频率piecewise smooth 分段光滑Fourier coefficient 傅立叶系数 convergence theorem 收敛定理Fourier series of even and odd functions 奇函数与偶函数的傅立叶级数amplitude 振幅sine series 正弦级数cosine series 余弦级数periodic extension 周期延拓第十六章多元函数的极限与连续functions of several variables 多元函数plane point set 平面点集coordinate plane 坐标平面interior point 内点outer point 外点boundary point 界点boundary 边点isolated point 孤立点open set 开集closed set 闭集connectedness 连通性connected open set 连通开集open domain(region) 开域closed domain(region) 闭域bounded point set 有界点集unbounded point set 无界点集nested closed domain theorem 闭域套定理function of two variables 二元函数n-dimensional vector space n-维向量空间improper limit 非正常极限double limit 二重极限properties of continuous functions on bounded closed region 有界闭区域上连续函数的性质repeated limits 累次极限 total increment 全增量partial increment 偏增量第十七章多元函数微分学total differential 全微分partial derivative 偏导数continuously differentiate 连续可微tangent plane of surface 曲面的切平面normal plane of curve 曲线的法平面 differentiation of composite function 复合函数微分法chain rule for functions of several variables 多元函数的链式法则 invariance of differential form of first order 一阶微分形式的不变性 directional derivative and gradient 方向导数与梯度 differentiability 可微性Taylor formula 泰勒公式problem of extreme value 极值问题partial derivative of higher order 高阶偏导数mixed partial derivative 混合偏导数第十八章隐函数定理及其应用existence and uniqueness theorem of implicit隐函数存在唯一性定理 functionssystem of implicit functions 隐函数组functional determinant (Jacobian determinant) 函数行列式（雅可比行列式）system of inverse functions 反函数组coordinate transformation 坐标变换geometrical application 几何应用tangent line and normal line of plane curve 平面曲线的切线与法线 tangent line and normal plane of space curve 空间曲线的切线与法平面 conditional extremum 条件极值Lagrange multiplier method 拉格朗日乘数法第十九章 含参变量积分proper integral with parameter 含参量的正常积分 improper integral with parameter 含参量的反常积分Euler integral 欧拉积分 Weierstrass test 魏尔斯特拉判别法 Dicichlet test 狄利克雷判别法 improper integral with infinite bound 无穷限的反常积分 improper integral with unbounded 无界函数的反常积分 Gamma functionBeta function第二十章inner area 内面积outer area 外面积cylindrical body with tip surface 曲顶柱体fineness 细度integral region 积分区域double integral 二重积分x-type region x型区域y-type region y型区域triple integral 三重积分change of variable in triple integral 三重积分换元法 transformation of cylindrical 柱面坐标变换 transformation of spherical coordinates 球面坐标变换area of surface 曲面面积第十一章曲线积分第一型曲线积分 curvilinear integrals of the first kind/line integrationof 1-form第二型曲线积分 curvilinear integrals of the first kind/line integrationof 2-formrelation between two classes of curvilinear两次曲线积分的联系 integralsGreen formula 格林公式 independence of curvilinear integrals with曲线积分与路径无关性 pathsimple connected region 单连通区域complex connected region 复连通区域第二十二章曲面积分surface integral of the first kind 第一型曲面积分 surface integral of the second kind 第二型曲面积分 unilateral surface 单侧曲面bilateral surface/two sided face 双侧曲面right-hand rule 右手法则relation between two classes of surface integrals 两类曲面积分的关系 Gauss formula 高斯公式Stokes formula 斯托克斯公式 introduction to field 场论初步field of vectors 向量场gradient field 梯度场gravitation field 引力场divergence field 散度场rotation field 旋度场circulation 环流量。
数学分析12教学大纲
《数学分析12》课程教学大纲一课程说明1.课程基本情况课程名称：数学分析12英文名称：Mathematical Analysis课程编号：2411204开课专业：数学与应用数学专业开课学期：第2学期学分/周学时：6/6课程类型：专业基础课2．课程性质（本课程在该专业的地位作用）《数学分析12》是数学专业的基础学科，是数学与应用数学、信息与计算科学、统计学三个专业的一门重要的核心课程，以不定积分、定积分、无穷级数、反常积分、傅立叶级数与傅立叶变换为基本内容，是学生学习分析学系列课程及其后继课程的重要基础，在第2学期开设。
本课程的教学，对锻炼和提高学生的思维能力，培养学生掌握分析问题和解决问题的思想方法有重要的意义，它不仅关系到能否学好后续课程，对学生未来的发展也将产生重大影响。
3．本课程的教学目的和任务本课程是进一步学习复变函数论、微分方程、微分几何、实变函数论、概率论、拓扑学、泛函分析等后继课程的阶梯，也为深入理解中学数学打下必要的基础。
与中学数学的许多内容，如实数系、函数、方程、不等式、极值、面积、体积、弧长等有着密切的联系。
通过本课程的学习，使学生掌握不定积分、定积分、无穷级数、反常积分、傅立叶级数与傅立叶变换等基本内容，为学习数学分析3及分析学系列课程（复变函数、实变函数、微分方程、泛函分析等）及其后继课程打好基础，并自然地渗透对学生进行逻辑和数学抽象的特殊训练，达到如下目的：1、通过对贯穿数学分析始终的极限思想和方法的教学，使学生弄清不变与变，有限与无限，特殊与一般的辩证关系，进一步培养他们的辩证唯物主义观；2、使学生正确理解数学分析的基本概念，牢固地掌握数学分析中的基本理论和基本方法，逐步提高他们抽象思维和逻辑推理的能力，培养他们熟练的演算技能和初步应用的能力，为进一步学习其它课程打下基础。
4．本课程与相关课程的关系、教材体系特点及具体要求本课程是高等院校数学系的数学与应用数学专业的一门重要基础课，它的任务是使学生获得不定积分、定积分、无穷级数、反常积分、傅立叶级数与傅立叶变换等方面的系统知识。
数学分析英文词汇
数学分析中的英语单词第一章real number and its properties 实数及其性质 rational number 有理数 irrational number 无理数 definition 定义 proposition 命题plus 加minus 减multiplied by/times 乘over/is to/divided by 除absolute value and inequality 绝对值与不等式 triangle inequality 三角不等式 inverse triangle inequality 反三角不等式 Bernoulli inequality 伯努利不等式 principles of supremum and infimum 确界原理open interval 开区间closed interval 闭区间semi-open interval 半开区间semi-closed interval 半闭区间finite interval 有限区间 infinite interval 无限区间 neighborhood 领域deleted neighborhood 去心邻域sum 和difference 差product 积quotient 商number axis/number line 数轴 closeness 封闭性 Archimedean property 阿基米德性质 density 稠密行upper and lower bounds 上界和下界 bounded set 有界集 unbounded set 无界集 existence domian 存在域 supremum 上确界 infimum 下确界order-complete set 有序完备集 completeness of real numbers 实数的完备性 complete ordered field 全序域 axiom of completeness 完备性公理 Dedekind cut 戴德金分割 Dedekind property 戴德金性质 constant and variable quantities 常量与变量 definition of function 函数的定义域 domain 定义域 range 值域 independent variable 自变量 dependent variable 因变量 intermediate variable 中间量 elementary function 初等函数 constant function 常量函数 power function 幂函数 exponential function 指数函数 logarithmic function 对数函数 trigonometric function 三角函数 inverse trigonometric function 反三角函数 compound function 复合函数 mapping 映射inverse mapping 逆映射 image 像primary image 原像 piecewise function 分段函数sign function 符号函数 Dirichlet function 狄利克雷函数 Riemann function 黎曼函数 bounded function 有界函数 monotone function 单调函数 monotone increasing function 单调增函数 strictly monotone function 严格单调函数 odd(even) function 奇函数（偶函数） minimal positive period 最小正周期 absolute value function 绝对值函数 identity function 恒等函数 polynomial function 多项式函数linear function 线性函数 quadratic function 二次函数rational function 有理函数 hyperbolic sine 双曲正弦 hyperbolic cosine 双曲余弦 trigonometric identity 三角恒等式odd-even identity 奇偶恒等式 cofunction identity 余函数恒等式 Pythagorean identity 毕达哥拉斯恒等式 half-angle identity 半角恒等式 product identity 积恒等式sum identity 和恒等式 addition identity 加法恒等式 double-angle identity 倍角恒等式第二章 数列极限limit of sequence 数列极限 divergent sequence 发散数列 infinitesimal sequence 无穷小数列 convergent sequence 收敛数列 uniqueness theorem 唯一性定理boundedness theorem 有界性定理 inheriting order properties 保序性inheriting inequality 保不等式 subsequence 子列strictly increasing 严格递增 monotone increasing sequence 单调递增序列 monotone decreasing sequence 单调递减序列 necessary condition 必要条件sufficient condition 充分条件squeeze principle 夹逼定理Cauchy convergence criterion 柯西收敛准则第三章 函数极限limit of function 函数极限infinite limit 无穷极限one-sided limit 单侧极限right(left)limit/right(left) hand limit 右（左）极限 property of limit of function 函数极限的性质 local boundedness 局部有界性Heine theorem 海涅定理infinity 无穷大量order of infinitesimal 无穷小量的阶 infinitesimal of higher(lower) order 高（低）阶无穷小量 infinitesimal of same order 同阶无穷小量 equivalent infinitesimal 等价无穷小量k-order infinitesimal k-阶无穷小量 vertical asymptote 垂直渐近线oblique asymptote 斜渐进线 horizontal asymptote 水平渐近线第四章函数的连续性increment of independent variable 自变量的增量 increment of function 函数的增量right(left) continuous 右(左)连续discontinuity point and its classification 间断点及其分类 removable discontinuity 可去间断点jump discontinuity 跳跃间断点 discontinuity of the first kind 第一类间断点 discontinuity of the second kind 第二类间断点local properties of continuous function 连续函数的局部性质 composition properties of continuous function 连续函数的复合性质闭区间上连续函数的性质 properties of continuous function over closedintervalextreme value theorem 极值定理maximum and minimum value theorem 最大值和最小值定理 intermediate value theorem 介值性定理zero-point theorem 零点定理uniform continuity theorem 一致连续定理continuity of inverse function 反函数的连续性local inheriting order property 局部保号性continuity of elementary function 初等函数的连续性第五章 导数和微分finite increment formula 有限增量公式rate of change 变化率difference quotient 差商left(right) derivative 左（右）导数 derivative function 导函数derivable function 可导函数geometric meaning of derivative 导数的集合意义Fermat theorem 费马定理Darboux theorem 达布定理intermediate value theorem of derivative function 导函数的介值定理 algebra of derivatives 导数的四则运算 derivative of sum 和的导数derivatives of difference 差的导数derivative of product 积的导数derivative of quotient 商的导数 derivative of inverse function 反函数的导数 maximum value 最大值minimum value 最小值 derivative of composite function 复合函数的导数 logarithmic derivative 对数求导法 parametric equation of circle 圆的参数方程 parametric equation of ellipse 椭圆的参数方程 parametric equation of cycloid 摆线的参数方程 parametric equation of asteroid 星形线的参数方程 second derivative 二阶导数third derivative 三阶导数n-th derivative n阶导数Leibniz formula 莱布尼茨公式 acceleration 加速度physical interpretation 物理解释 concept of differential 微分的概念 differentiable function 可微函数linear principal part 线性主部 differential of independent variable 自变量的微分 operational rules of differential 微分的运算法则 invariance of differential form 微分形式的不变性 geometric meaning of differential 微分的几何意义 higher-order differential 高阶微分第六章 微分中值定理Rolle mean value theorem 罗尔中值定理 Lagrange mean value theorem 拉格朗日中值定理 Cauchy mean value theorem 柯西中值定理 Taylor theorem 泰勒定理L’Hospital rule 洛必达法则limit of indeterminate form of type 0/0 0/0型不定式极限 indeterminate form of type ∞/∞ ∞/∞型不定式other indeterminate forms 其他类型不定式Taylor formula with Peano remainder 带有皮亚诺型余项的泰勒公式Taylor coefficient 泰勒系数Taylor polynomial 泰勒多项式remainder of Taylor formula 带有拉格朗日型余项的泰勒公式Taylor formula with Lagrange remainder 带有拉格朗日型余项的麦克劳林公式extreme value of function 函数的极值test of extreme value 极值判断the first sufficient condition of extreme value 极值的第一充分条件 convexity and inflection point of function 函数的凸性和拐点 convex function 凸函数concave function 凹函数strictly convex function 严格凸函数strictly concave function 严格凹函数Jenson inequality 琴生不等式第七章实数的完备性实数击完备性的基本定理 fundamental theorems of completeness in the setof real numbersnested interval theorem 闭区间套定理Cauchy convergence criterion 柯西收敛准着 accumulation theorem 聚点定理finite covering theorem 有限覆盖定理upper and lower limits 上极限和下极限第八章 不定积分indefinite integral 不定积分primitive function 原函数integrand function 被积函数integral sign 积分符号expression of integrand 积分表达式integral constant 积分常数table of basic integrals 基本积分表 geometric meaning of indefinite 不定积分的几何意义 integral curve 积分曲线initial condition 初始条件 integration by parts 分部积分法 integration by substitutions 换元积分法formula of substitution of the first kind 第一换元公式 indefinite integral of rational function 有理函数的不定积分 proper fraction 真分式improper fraction 假分式 decomposition into partial fractions 部分分式分解 method of undermined coefficients 待定系数法第九章 定积分definite integral 定积分curvilinear trapezoid 曲边梯形dividing/partition 分割norm/modulus 模Riemann sum 黎曼和Riemann integral 黎曼积分 integrability in the sense of Riemann 黎曼可积interval of integration 积分区间upper and lower limits 上限和下限 geometric meaning of definite integral 定积分的几何意义 upper sum 上和lower sum 下和Darboux upper sum 达布上和Darboux lower sum 达布下和Newton-Leibniz formula 牛顿-莱布尼茨公式 necessary condition for integrability 可积的必要条件 necessary and sufficient conditions for integrability 可积的充要条件 integrable function class 可积函数类linear property of definite integral 定积分的线性性质 additive with respect to integral interval/additivityover integral interval积分区间的可加性 mean value theorem of integral 积分中值定理 average value 平均值integral with variant upper limit 变上限积分 existence of primitive function 原函数的存在性 integration by substitution 换元积分法 integration by parts 分部积分法integral form of reminder of Taylor formula 泰勒公式的积分类型 第十章 定积分的应用area of plane figure 平面图形的面积method of finding volume of a solid from the knows area of parallel sections 由平行截面面积求体积的方法arc length and curvature of plane curve 平面的曲线的弧长与曲率 smooth curve 光滑曲线differential of arc 弧微分circle of curvature 曲率圆radius of curvature 曲率半径area of revolution surface 旋转曲面的面积some applications of definite integral in physics 定积分在物理中的某些应用gravitation 引力work 功approximate computation of definite integral 定积分的近似计算 trapezoidal method 梯形法parabola method 抛物线法第十一章反常积分notion of improper integral 反常积分的概念 improper integral on infinite interval 无穷区间上的反常积分 improper integral of unbounded function 无界函数的反常积分 property of infinite integral and test of 无穷积分的性质和收敛判convergence 别absolutely convergent 绝对收敛comparison test 比较收敛法Cauchy test 柯西判别法Dirichlet test 狄利克雷判别法Abel test 阿贝尓判别法第十二章数项级数series with number terms 数项级数infinite series 无穷级数convergence of series 级数的收敛性 sequence of partial sum 部分和序列geometric series 几何级数harmonic series 调和级数Cauchy convergence criterion for series 级数收敛的柯西准则 series with positive terms 正项级数necessary condition for convergence 收敛的必要条件root test 根式判别法D’alembert(ratio) test 达朗贝尔判别法（比式判别法）limit form of ratio test 比式判别法的极限形式 Gauss test 高斯判别法integral test 积分判别法Raabe test 拉贝尓判别法p-series p级数series with arbitraty terms 一般项级数alternating series 交错级数Leibnitz test 莱布尼茨判别法 rearrangement of series 级数的重排Abel test 阿贝尓判别法Dirichlet test 狄利克雷判别法第十章 函数列与函数项级数convergent at x"在点x"收敛convergence domain/region of convergence 收敛域sum function 和函数limit function 极限函数series of functions 函数项级数uniform convergence 一致收敛test of uniform convergence 一致收敛判别法魏尔斯特拉判别法 Weierstrass’s test/Weierstrass uniformconvergence criterion/Weierstrass m-test foruniform convergenceuniform boundedness 一致有界Dirichlet test 狄利克列判别法第十四章幂级数power series 幂级数interval of convergence 收敛区间radius of convergence 收敛半径Abel theorem 阿贝尓定理operations of power series 幂级数的运算taylor series 泰勒级数expansion of power series of elementary function 初等函数的幂级数展开式 exponential function of complex variable 复变量的指数函数Euler formula 欧拉公式第十五章傅里叶级数Fourier series 傅里叶级数 trigonometric series 三角级数system of orthogonal functions 正交函数系simple harmonic vibration 简谐振动Fourier series for function of period 2π以2π为周期的函数的傅立叶级数angular frequency 角频率piecewise smooth 分段光滑Fourier coefficient 傅立叶系数 convergence theorem 收敛定理Fourier series of even and odd functions 奇函数与偶函数的傅立叶级数amplitude 振幅sine series 正弦级数cosine series 余弦级数periodic extension 周期延拓第十六章多元函数的极限与连续functions of several variables 多元函数plane point set 平面点集coordinate plane 坐标平面interior point 内点outer point 外点boundary point 界点boundary 边点isolated point 孤立点open set 开集closed set 闭集connectedness 连通性connected open set 连通开集open domain(region) 开域closed domain(region) 闭域bounded point set 有界点集unbounded point set 无界点集nested closed domain theorem 闭域套定理function of two variables 二元函数n-dimensional vector space n-维向量空间improper limit 非正常极限double limit 二重极限properties of continuous functions on bounded closed region 有界闭区域上连续函数的性质repeated limits 累次极限 total increment 全增量partial increment 偏增量第十七章多元函数微分学total differential 全微分partial derivative 偏导数continuously differentiate 连续可微tangent plane of surface 曲面的切平面normal plane of curve 曲线的法平面 differentiation of composite function 复合函数微分法chain rule for functions of several variables 多元函数的链式法则 invariance of differential form of first order 一阶微分形式的不变性 directional derivative and gradient 方向导数与梯度 differentiability 可微性Taylor formula 泰勒公式problem of extreme value 极值问题partial derivative of higher order 高阶偏导数mixed partial derivative 混合偏导数第十八章隐函数定理及其应用existence and uniqueness theorem of implicit隐函数存在唯一性定理 functionssystem of implicit functions 隐函数组functional determinant (Jacobian determinant) 函数行列式（雅可比行列式）system of inverse functions 反函数组coordinate transformation 坐标变换geometrical application 几何应用tangent line and normal line of plane curve 平面曲线的切线与法线 tangent line and normal plane of space curve 空间曲线的切线与法平面 conditional extremum 条件极值Lagrange multiplier method 拉格朗日乘数法第十九章 含参变量积分proper integral with parameter 含参量的正常积分 improper integral with parameter 含参量的反常积分Euler integral 欧拉积分 Weierstrass test 魏尔斯特拉判别法 Dicichlet test 狄利克雷判别法 improper integral with infinite bound 无穷限的反常积分 improper integral with unbounded 无界函数的反常积分 Gamma functionBeta function第二十章inner area 内面积outer area 外面积cylindrical body with tip surface 曲顶柱体fineness 细度integral region 积分区域double integral 二重积分x-type region x型区域y-type region y型区域triple integral 三重积分change of variable in triple integral 三重积分换元法 transformation of cylindrical 柱面坐标变换 transformation of spherical coordinates 球面坐标变换area of surface 曲面面积第十一章曲线积分第一型曲线积分 curvilinear integrals of the first kind/line integrationof 1-form第二型曲线积分 curvilinear integrals of the first kind/line integrationof 2-formrelation between two classes of curvilinear两次曲线积分的联系 integralsGreen formula 格林公式 independence of curvilinear integrals with曲线积分与路径无关性 pathsimple connected region 单连通区域complex connected region 复连通区域第二十二章曲面积分surface integral of the first kind 第一型曲面积分 surface integral of the second kind 第二型曲面积分 unilateral surface 单侧曲面bilateral surface/two sided face 双侧曲面right-hand rule 右手法则relation between two classes of surface integrals 两类曲面积分的关系 Gauss formula 高斯公式Stokes formula 斯托克斯公式 introduction to field 场论初步field of vectors 向量场gradient field 梯度场gravitation field 引力场divergence field 散度场rotation field 旋度场circulation 环流量。
高中数学课程描述(英文)
Mathematics Course DescriptionMathematics course in middle school has two parts: compulsory courses and optional courses. Compulsory courses content lots of modern mathematical knowledge and conceptions, such as calculus, statistics, analytic geometry, algorithm and vector. Optional courses are chosen by students which is according their interests.Compulsory Courses:Set TheoryCourse content:This course introduces a new vocabulary and set of rules that is foundational to the mathematical discussions. Learning the basics of this all-important branch of mathematics so that students are prepared to tackle and understand the concept of mathematical functions. Students learn about how entities are grouped into sets and how to conduct various operations of sets such as unions and intersections (i.e. the algebra of sets). We conclude with a brief introduction to the relationship between functions and sets to set the stage for the next stepKey Topics:➢The language of set theory➢Set membership➢Subsets, supersets, and equality➢Set theory and functionsFunctionsCourse content:This lesson begins with talking about the role of functions and look at the concept of mapping values between domain and range. From there student spend a good deal of time looking at how to visualize various kinds of functions using graphs. This course will begin with the absolute value function and then move on to discuss both exponential and logarithmic functions. Students get an opportunity to see how these functions can be used to model various kinds of phenomena. Key Topics:➢Single-variable functions➢Two –variable functions➢Exponential function➢ Logarithmic function➢Power- functionCalculusCourse content:In the first step, the course introduces the conception of limit, derivative and differential. Then students can fully understand what is limit of number sequence and what is limit of function through some specific practices. Moreover, the method to calculate derivative is also introduced to students.Key Topics:➢Limit theory➢Derivative➢DifferentialAlgorithmCourse content:Introduce the conception of algorithm and the method to design algorithm. Then the figures of flow charts and the conception of logical structure, like sequential structure, contracture of condition and cycle structure are introduced to students. Next step students can use the knowledge of algorithm to make simple programming language, during this procedure, student also approach to grammatical rules and statements which is as similar as BASIC language.Key Topics:➢Algorithm➢Logical structure of flow chart and algorithm➢Output statement➢Input statement➢Assignment statementStatisticsCourse content:The course starts with basic knowledge of statistics, such as systematic sampling and group sampling. During the lesson students acquire the knowledge like how to estimate collectivity distribution according frequency distribution of samples, and how to compute numerical characteristics of collectivity by looking at numerical characteristics of samples. Finally, the relationship and the interdependency of two variables is introduced to make sure that students mastered in how to make scatterplot, how to calculate regression line, and what is Method of Square.Key Topics:➢Systematic sampling➢Group sampling➢Relationship between two variables➢Interdependency of two variablesBasic Trigonometry ICourse content:This course talks about the properties of triangles and looks at the relationship that exists between their internal angles and lengths of their sides. This leads to discussion of the most commonly used trigonometric functions that relate triangle properties to unit circles. This includes the sine, cosine and tangent functions. Students can use these properties and functions to solve a number of issues.Key Topics:➢Common Angles➢The polar coordinate system➢Triangles properties➢Right triangles➢The trigonometric functions➢Applications of basic trigonometryBasic Trigonometry IICourse content:This course will look at the very important inverse trig functions such as arcsin, arcos, and arctan, and see how they can be used to determine angle values. Students also learn core trig identities such as the reduction and double angle identities and use them as a means for deriving proofs. Key Topics:➢Derivative trigonometric functions➢Inverse trig functions➢Identities●Pythagorean identities●Reduction identities●Angle sum/Difference identities●Double-angle identitiesAnalytic Geometry ICourse content:This course introduces analytic geometry as the means for using functions and polynomials to mathematically represent points, lines, planes and ellipses. All of these concepts are vital in student’s mathematical development since they are used in rendering and optimization, collision detection, response and other critical areas. Students look at intersection formulas and distance formulas with respect to lines, points, planes and also briefly talk about ellipsoidal intersections. Key Topics:➢Parametric representation➢Parallel and perpendicular lines➢Intersection of two lines➢Distance from a point to a line➢Angles between linesAnalytic Geometry IICourse content:Students look at how analytic geometry plays an important role in a number of different areas of class design. Students continue intersection discussion by looking at a way to detect collision between two convex polygons. Then students can wrap things up with a look at the Lambertian Diffuse Lighting model to see how vector dot products can be used to determine the lighting and shading of points across a surface.Key Topics:➢Reflections➢Polygon/polygon intersection➢LightingSequence of NumberCourse content:This course begin with introducing several conceptions of sequence of number, such as, term, finite sequence of number, infinite sequence of number, formula of general term and recurrence formula. Then, the conception of geometric sequence and arithmetic sequence is introduced to students. Through practices and mathematical games, students gradually understand and utilizethe knowledge of sequence of number, eventually students are able to solve mathematical questions.Key Topics:➢Sequence of number➢Geometric sequence➢Arithmetic sequenceInequalityThis course introduces conception of inequality as well as its properties. In the following lessons students learn the solutions and arithmetic of one-variable quadratic inequality, two variables inequality, fundamental inequality as well how to solve simple linear programming problems.Key Topics:➢Unequal relationship and Inequality➢One-variable quadratic inequality and its solution➢Two-variable inequality and linear programming➢Fundamental inequalityVector MathematicsCourse content:After an introduction to the concept of vectors, students look at how to perform various important mathematical operations on them. This includes addition and subtraction, scalar multiplication, and the all-important dot and cross products. After laying this computational foundation, students engage in games and talk about their relationship with planes and the plane representation, revisit distance calculations using vectors and see how to rotate and scale geometry using vector representations of mesh vertices.Key Topics:➢Linear combinations➢Vector representations➢Addition/ subtraction➢Scalar multiplication/ division➢The dot product➢Vector projection➢The cross productOptional CoursesMatrix ICourse content:In this course, students are introduced to the concept of a matrix like vectors, matrices and so on. In the first two lessons, student look at matrices from a purely mathematical perspective. The course talks about what matrices are and what problems they are intended to solve and then looks at various operations that can be performed using them. This includes topics like matrix addition and subtraction and multiplication by scalars or by other matrices. At the end, students can conclude this course with an overview of the concept of using matrices to solve system of linear equations.Key Topics:➢Matrix relations➢Matrix operations●Addition/subtraction●Scalar multiplication●Matrix Multiplication●Transpose●Determinant●InversePolynomialsCourse content:This course begins with an examination of the algebra of polynomials and then move on to look at the graphs for various kinds of polynomial functions. The course starts with linear interpolation using polynomials that is commonly used to draw polygons on display. From there students are asked to look at how to take complex functions that would be too costly to compute in a relatively relaxed studying environment and use polynomials to approximate the behavior of the function to produce similar results. Students can wrap things up by looking at how polynomials can be used as means for predicting the future values of variables.Key Topics:➢Polynomial algebra ( single variable)●addition/subtraction●multiplication/division➢Quadratic equations➢Graphing polynomialsLogical Terms in MathematicsCourse content:This course introduces the relationships of four kinds of statements, necessary and sufficient conditions, basic logical conjunctions, existing quantifier and universal quantifier. By learning mathematical logic terms, students can be mastered in the usage of common logical terms and can self-correct logical mistakes. At the end of this course, students can deeply understand the mathematical expression is not only accurate but also concise.Key Topics:➢Statement and its relationship➢Necessary and sufficient conditions➢Basic logical conjunctions➢Existing quantifier and universal quantifierConic Sections and EquationCourse content:By using the knowledge of coordinate method which have been taught in the lesson of linear and circle, in this lesson students learn how to set an equation according the character of conic sections. Students is able to find out the property of conic sections during establishing equations. The aim of this course is to make students understand the idea of combination of number and shape by using the method of coordinate to solve simple geometrical problems which are related to conic sections.Key Topics:➢Curve and equation ➢Oval➢Hyperbola➢Parabola。
数学分析专题选讲课程教学大纲
《数学分析专题选讲》课程教学大纲一、课程基本信息课程中文名称:数学分析专题选讲课程英文名称:Selective Lectures of Mathematic Analysis课程类别:选修课使用专业:数学与应用数学专业、计算与信息科学专业、物理学、计算机科学等开设学时：24学时使用年级:20XX级、20XX级预修课程:数学分析或高等数学一并修课程:课程简介:数学分析专题系统拓展和加深讲授极限理论, 函数的连续性, 微分中值定理的及其应用, 一元函数积分学, 数值级数与无穷积分, 多元函数微分学,函数级数与含参变量的无穷积分, 多元函数积分学这八个专题的核心内容.建议教材: 自编讲义参考书:[1].毛羽辉编著《数学分析选论》,北京:科学出版社(第二版).[2].胡小敏李承家编著《数学分析考研教案》,西安:西北工业大学出版社(第二版).[3].王戈平编《数学分析选讲》,西安:中国矿业大学出版社.[4].裘兆泰王承国章仰文编《数学分析学习指导》,北京:科学出版社.[5].孙本旺汪浩《数学分析中的典型例题和方法》,长沙:湖南科学技术出版社.[6].周中群主编《数学分析方法选讲》,重庆:西南师范大学出版社.[7].刘玉琏扬奎元吕风编《数学分析讲义学习指导书》(上),北京:高等教育出版社(第二版).[8].刘玉琏扬奎元吕风编《数学分析讲义学习指导书》(下),北京:高等教育出版社(第二版).[9].谢惠民恽自求易法槐钱定边编《数学分析习题课讲义》(上),北京: 高等教育出版社.[10].谢惠民恽自求易法槐钱定边编《数学分析习题课讲义》(下),北京: 高等教育出版社.[11].钱吉林编《数学分析解题精粹》,武汉:崇文书局.[12].牟俊霖李青吉《洞穿考研数学》,北京:航空工业出版社.二、课程性质、目的及总体要求课程的基本特性:数学分析专题选讲是数学与应用数学专业,计算与信息科学专业重要的选修课,它是学生进一步学习分析数学的分支和科学研究必不可少的专业基础知识, 同时也可使其他理科专业学生进一步了解微积分学知识，是报考对数学要求较高的硕士学位研究生同学的必修课程.课程的教学目标:该课程主要系统拓展和加深学习极限理论, 函数的连续性, 微分中值定理的及其应用, 一元函数积分学,数值级数与无穷积分, 多元函数微分学, 函数级数与含参变量的无穷积分, 多元函数积分学这八个专题的核心内容.课程的总体要求:通过本课程的学习,主要要求学生系统拓展和加深极限理论, 函数的连续性, 微分中值定理的极其应用, 一元函数积分学,数值级数与无穷积分, 多元函数微分学, 函数级数与含参变量的无穷积分, 多元函数积分学的基本技能、基本思想和方法,主要培养学生分析论证问题的能力、抽象思维能力和科学研究的初步能力.三、章节教学内容与要求（进度表）第一章极限理论的应用(6学时)总的要求:极限理论是数学分析的基础理论,它是学习微分理论、积分理论、级数理论等的奠基理论,极限理论的基本思想和方法贯穿于数学分析始终.在本章中,主要进一步学习解决极限问题的若干基本方法.通过学习,主要培养学生分析论证问题的能力、抽象思维能力和解决实际问题的能力,培养学生科学研究的初步能力。
数学分析中的英文单词和短语
数学分析中的英文单词和短语第一章实数集与函数第二章数列极限Chapter 2 Limits of Sequences第三章函数极限Chapter 3 Limits of Functions第四章函数的连续性Chapter 4 Continuity of Functions第六章 微分中值定理及其应用Chapter 6 Mean Value Theorems of Differentials and their Applications第七章 实数的完备性Chapter 7 Completeness of Real Numbers第八章 不定积分Chapter 8 Indefinite Integrals第九章 定积分Chapter 9 Definite Integrals第十章定积分的应用Chapter 10 Applications of Definite Integrals第十一章反常积分Chapter 11 Improper Integrals 第十二章数项级数Chapter 12 Series of Number Terms第十三章函数列与函数项级数Chapter 13 Sequences of Functions andSeries of Functions第十四章 幂级数Chapter 14 Power Series第十五章 傅里叶级数Chapter 15 Fourier Series第十六章 多元函数的极限与连续Chapter 16 Limits and Continuity of Functions of Several Variavles第十七章多元函数微分学Chapter 17 Differential Calculus of Functions of Several Variables第十八章隐函数定理及其应用Chapter 18 Implicit Funciton Theorems and their Applications第十九章含参量积分Chapter 19 Integrals with Parameters第二十章重积分Chapter 20 Multiple Integrals第二十一章曲线积分Chapter 21 Curvilinear Integrals 第二十二章曲面积分Chapter 22 Surface Integrals。
数学分析(中文)
数学分析
课程代码: 82135030
课程名称: 数学分析
英文名称: Mathematical analysis
学分：17 开课学期：第1、2、3、4学期
授课对象：数学与统计学院各专业先修课程：初等数学
课程主任：陈绍著、教授、硕士
课程简介：
数学分析是伴随着牛顿力学的产生而发展起来的一门数学学科，是现代科学的基石，是综合大学数学专业的重要基础课，它直接影响到许多后续专业课程的学习，通过本课程的学习，使学生初步掌握数学分析的基本理论和基本概念，掌握数学分析中基本的论证方法，较熟悉地获得本课程所要求的基本运算能力，逐步培养学生分析问题和解决问题的能力，培养学生抽象思维和逻辑思维能力，为学习相关专业课程及以后实际应用提供必要基础。
课程考核：
课程最终成绩=平时成绩*30%+期末考试成绩*70%；
平时成绩由出勤率、作业、小论文的完成情况决定；
期末考试采取闭卷考试。
指定教材：
【1】陈纪修，於崇华，金路。
《数学分析(上、下册)》，北京：高等教育出版社，2004，第二版。
参考书目：
【1】陈传璋，《数学分析(上、下册)》，北京：高等教育出版社，1998年10月，第二版。
【2】华东师范大学，《数学分析》，北京：高等教育出版社，1998年10月，第二版。