1、天津大学 高等数学 课程 教学大纲 课程代码： 2100004 2100005 课程名称： 高等数学 2A 高等数学 2B 学 时： 176 学 分： 11 学时分配： 授课： 176 上机： 0 实验： 0 实践： 0 实践（周）： 0 授课学院： 理学院 更新时间： 2011年 7月 适用专业： 本科工科专业 先修课程： 中学里的 初等数学课程 一、 课程的性质 与 目的 本课程的研究对象是函数（变化过程中量的依赖关系）。内容包括函数、极限、连续，一元函数微积分学，向 量代数与空间解析几何学，多元函数微分学，多元函数积分学，无穷级数（含 Fourier 级数）与常微分方程等。 要通过各个教
2、学环节逐步培养学生的抽象思维能力、逻辑推理能力、空间想象能力和自学能力，还要特别注意培养学生的熟练运算能力和综合运用所学知识去分析解决问题的能力。 二、 教学基本要求 通过本课程的学习，要使学生掌握微积分学的基本概念、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。 三、 教学内容 第一章 函数与极限 理解函数、复合函数及分段函数的概念；理解极限、左极限与右极 限的概念；理解无穷小、无穷大的概念；理解函数连续性的概念（含左连续与右连续）；掌握基本初等函数的性质及其图形；掌握极限的性质及四则运算法则；掌握极限存在的两个准则；掌握利用两个重要极限求极限的方法；掌握无穷
3、小的比较方法；了解函数的奇偶性、单调性、周期性和有界性；了解反函数及隐函数的概念，以及极限存在与左、右极限之间的关系；了解连续函数的性质和初等函数的连续性，了解闭区间上连续函数的性质（有界性、最大值最小值定理和介值定理）；会建立简单应用问题中的函数关系式；会利用极限存在的两个准则求极限；会用等价无穷小求极 限；会判别函数间断点的类型；会应用闭区间上连续函数的性质。 第二章 导数与微分 理解导数和微分的概念；理解导数与微分的关系；理解导数的几何意义；理解函数的可导性与连续性之间的关系；掌握导数的四则运算法则和复合函数的求导法则；掌握基本初等函数的导数公式；了解导数的物理意义；了解微分的四则运算法
4、则和一阶微分形式的不变性；了解微分在近似计算中的应用；了解高阶导数的概念；会求平面曲线的切线方程和法线方程；会用导数描述一些物理量；会求函数的微分；会求简单函数的 n 阶导数；会求分段函数的一阶、二阶导数；会求 隐函数和由参数方程所确定的函数的一阶、二阶导数，会求反函数的导数。 第三章 微分 中值定理与导数应用 理解函数的极值概念；掌握用导数判断函数的单调性和求函数极值的方法，掌握函数最大值和最小值的求法及其简单应用；掌握用洛必达法则求未定式极限的方法；了解柯西中值定理；了解曲率和曲率半径的概念；会用罗尔定理、拉格朗日中值定理和泰勒定理；会用导数判断函数图形的凹凸性和拐点，会求函数图形的水平、
5、铅直和斜渐近线，会描绘函数的图形；会计算曲率和曲率半径，会求两曲线的交角。 第四章 不定积分 理解原函数、不定积分 的概念；掌握不定积分性质；掌握不定积分的基本公式；掌握换元积分法与分部积分法；会求有理函数、三角函数有理式及简单无理函数的积分。 第五章 定积分及其应用 理解定积分的概念；理解变上限定积分定义的函数及其求导公式；掌握定积分的性质及定积分中值定理，掌握牛顿莱布尼茨公式；掌握定积分的换元积分法与分部积分法；了解广义积分的概念并会计算广义积分。掌握定积分的元素法；掌握用定积分表达和计算一些几何量与物理量（平面图形的面积、平面曲线的弧长、旋转体的体积及侧面积、平行截面面积为已知的立体体积
6、、变力作功、 引力、压力及函数的平均值等）。 第六章 微分方程 理解线性微分方程解的性质及解的结构定理；掌握变量可分离的方程及一阶线性方程的解法；掌握二阶常系数齐次线性微分方程的解法；了解微分方程及其解、阶、通解、初始条件和特解等概念；了解微分方程的幂级数解法；会解齐次方程、伯努利方程和全微分方程，会用简单的变量代换解某些微分方程；会用降价法解下列方程： )()( xfy n ， ),( yxfy 和 ),( yyfy 了；会解某些高于二阶的常系数齐次线性微分方程；会求自由项为多项式、指数函数、正弦函数、余弦函数，以及它们的和与积的二级常系数非齐次线性微分方程的特解和通解；会解欧拉方程，了解包
7、含两个未知函数的一阶常系数线性微分方程组；会用微分方程解决一些简单的应用问题。 第七章 向量代数与空间解析几何 理解空间直角坐标系，理解向量的概念及表示；理解曲面方程的概念；掌握向量的运算（线性运算、数量积、向量积、混合积），掌握单位向量、方向数与方向余弦、向量的坐标表达式，以及用坐标表达式 进行向量运算的方法；掌握平面方程和直线方程及其求法；了解两个向量垂直、平行的条件；了解常用二次曲面的方程及其图形；了解平面曲线的参数方程和一般方程；了解；会利用平面、直线的相互关系（平行、垂直、相交等）解决有关问题；会求以坐标轴为旋转轴的旋转曲面及母线平行于坐标轴的柱面方程；会求空间曲线在坐标面上的投影曲
8、线的方程。 第八章 多元函数微分学及其应用 理解多元函数的概念，理解二元函数的几何意义；理解多元函数偏导数和全微分的概念；理解方向导数与梯度的概念并掌握其计算方法；理解多元函数极值和条件 极值的概念；掌握多元复合函数偏导数的求法；掌握方向导数与梯度的计算方法；掌握多元函数极值存在的必要条件；了解二元函数的偏导数和全微分的概念，以及有界闭区域上连续函数的性质；了解全微分存在的必要条件和充分条件；了解全微分形式的不变性，了解全微分在近似计算中的应用；了解曲线的切线和法平面及曲面的切平面和法线的概念；了解二元函数极值存在的充分条件；会求全微分；会求隐函数（包括由方程组确定的隐函数）的偏导数；会求曲线
9、的切线和法平面及曲面的切平面和法线的方程；会求二元函数的极值，会用拉格朗日乘数法求条件极值；会求简单 多元函数的最大值和最小值，并会解决一些简单的应用问题。 第九章 重积分 理解二重积分、三重积分的概念；掌握二重积分（直角坐标、极坐标）的计算方法；了解重积分的性质，了解二重积分、三重积分的概念，了解二重积分的中值定理；会计算三重积分（直角坐标、柱面坐标、球面坐标）。 第十章 曲线积分与曲面积分 理解两类曲线积分的概念；掌握计算两类曲线积分的方法；掌握格林公式并会运用平面曲线积分与路径无关的条件；掌握计算两类曲面积分的方法；了解两类曲线积分的性质及两类曲线积分的关系；了解两类曲面积 分的概念，性
10、质及两类曲面积分的关系；了解高斯公式、斯托克斯公式；会用斯托克斯公式计算曲线积分；会用高斯公式计算曲面积分；会用重积分、曲线积分及曲面积分求一些几何量与物理量（平面图形的面积、体积、曲面面积、弧长、质量、重心、转动惯量、引力、功及流量等）；会计算散度与旋度。 第十一章 级数 理解数项级数收敛、发散以及收敛级数的和的概念；掌握级数的基本性质及收敛的必要条件；掌握几何级数与 p 级数的收敛与发散的条件；掌握正项级数的比较审敛法和比值审敛法；掌握交 错级数的莱布尼茨定理；掌握幂级数的收敛半径、收敛区间及收敛域的求法；掌握 xe ， xsin ， xcos ， )1ln( x 和 )1( x 的麦克劳
11、林展开式；了解任意项级数绝对收敛与条件收敛的概念，以及绝对收敛与条件收敛的关系；了解函数项级数的收敛域及和函数的概念；了解幂级数在其收敛区间内的一些基本性质；了解函数展开为泰勒级数的充分必要条件 ；了解幂级数在近似计算上的简单应用；了解傅里叶级数的概念和函数展开为傅里叶级数的狄利克雷定理；会用根值审敛法；会求一些幂级数在收敛区间内的和函数；会将一些简单函数间接展开成幂级数；会将定义在 , ll 上的函数展开为傅里叶级数，会将定义在 ,0 l 上的函数展开为正弦级数与余弦级数，会写出傅里叶级数的和函数的表达式。 四 、 学时分配 教学内容 授课 上机 实验 实践 实践(周 ) 第一章 函数与极限
12、 16 0 0 0 0 第二 章 导数与微分 14 0 0 0 0 第三章 微分 中值定理与导数应用 14 0 0 0 0 第四章 不定积分 12 0 0 0 0 第五章 定积分及其应用 20 0 0 0 0 第六章 微分方程 14 0 0 0 0 第七章 向量代数与空间解析几何 14 0 0 0 0 第八章 多元函数微分学及其应用 22 0 0 0 0 第九章 重积分 14 0 0 0 0 第十章 曲线积分与曲面积分 18 0 0 0 0 第十一章 级数 18 0 0 0 0 总计： 176 0 0 0 0 五、 评价与考核方式 课程采用闭卷笔试与平时成绩相结合的考核方式。 六 、 教材与主
13、要参考资料 教材 : 高等数学（上、下册）天津大学数学系编著，高等教育出版社 , 2010 年 8月 。 参考资料 : 1. 高等数学（上、下册）天津大学高等数学教研室编，天津大学出版社 , 2000年 8 月 。 2. 高等数学解题方法（上、下册）邱忠文、杨则燊主编， 天津大学出版社 ，2002 年 5 月 。 TU Syllabus for Advanced Mathematics Code: 2100004 2100005 Title: Advanced Mathematics 2A Advanced Mathematics 2B Semester Hours: 176 Credits:
14、 11 Semester Hour Structure Lecture： 176 Computer Lab： 0 Experiment： 0 Practice： 0 Practice (Week)： 0 Offered by: School of Science Date: July, 2011 for: engineering science undergraduate Prerequisite: Elementary Mathematics in middle school 1. Objective The ability of the abstract thought , logical
15、 reasoning, the imaginary in space and self-study for student will be trained during the process of teaching. The ability of adroit calculus , analyzing and solving the problem applying their knowledge also will be trained for special note. 2. Course Description This course considers the functions a
16、s the objective. The content includes functions, limits, and continuity; single variable calculus; vector algebra and spatial analytical geometry; multivariate calculus; infinite series (including Fourier series); ordinary differential equations. This course prepares students for subsequent courses
17、in mathematics, sciences, engineering, and management. Upon successful completion of this course, students will be able to master the fundamental concepts, theorems, and methods of calculus, and develop problem-solving skills using calculus to solve problems arising in other courses and in real life
18、. 3. Topics Chapter one: Function and Limits Comprehend the conception of the function, component function, piecewise function, limit, left limit, right limit, infinitesimal, infinity, continuity (includes left continuity and right continuity ). Master the properties of basic elementary functions an
19、d their graphs, the properties of the limits and their arithmetic calculations, the two rules for the existence of limit, the method of calculating the limits using the two important limits, the method for comparing the infinitesimal. Understand the parity, monotony, periodicity and boundedness, the
20、 conception of inverse function and implicit function, the relation between the existence of limit and left (right ) limit, the properties of continuous function, the continuity of elementary function, the properties of continuous functions on the closed intervals (boundedness, the theorem of the ma
21、ximum and minimum for continuous function on closed interval, intermediate value theorem). Be able to build the function analysis formula for simple application problem, calculate the limit applying the principles of existence of the limit and the equivalent infinitesimal, recognize the type of disc
22、ontinuity point of the function, apply the properties of the continuous functions on the closed intervals. Chapter two: Derivatives and differentials Comprehend the conception of the derivatives and differentials, the relations between derivatives and differentials, the geometry meaning of derivativ
23、es, the relations between differentiability and continuity. Master the arithmetic calculations of derivatives and the differential principles for component functions, the differential formulae for basic elementary functions. Understand the physical meaning of the derivatives, the arithmetic calculat
24、ions of the differential, the invariance of the differential form of first order, the applications of differential in the proximate calculation, the conception of high order derivatives. Be able to get the tangent and normal equations of the curves on the plane, describe the physical quantity using
25、the derivative, calculate the differentials of functions, calculate the high order differentials of functions, calculate the first and second order derivatives of the piecewise functions, calculate the first and second order derivatives of the functions determined by implicit functions and parameter
26、 equations, calculate the derivatives of the inverse functions. Chapter three: The mean value theorem and applications of differentiation Comprehend the conception of the extreme value. Master the method of judging the monotony and calculating the extreme values of the functions, master the method o
27、f the method of calculating the maximum and minimum of the functions and their applications, master the method of calculating the limits of indeterminate forms using the L Hospital Principle. Understand Cauchy Mean Value Theorem and the conception of radius of curvature. Be able to apply Rolle Theor
28、em, Lagrange Theorem and Taylor Therorem, judge the convexity and concavity and the inflection of the graph using the derivatives, seek the horizontal,vertical and oblique asymptote, draw the graphs of the functions, calculate the curvature and the radius of the curvature, calculate the angle of the
29、 two curves. Chapter four : indefinite integrals Comprehend the conception of the object functions, indefinite integrals. Master the properties of the indefinite integrals, the basic formulae of the indefinite integrals, the method of integration by substitute and integration by parts for indefinite
30、 integral. Be able to calculate the integral of the rational functions , calculate the rational formulae of the trigonometric functions and the simple irrational functions. Chapter five : Definite Integral and its Applications Comprehend the conception of the definite integral, the functions defined
31、 by definite integral with a variable upper limit and their derivatives formulae. Master the properties of the definite integrals and integral mean value theorem and the Newton- Lebnitze formula, the method of integration by substitute and integration by parts for definite integral, the element meth
32、od of the definite integral, express and calculate some geometrical quantity, physical quantity(the area of the plane figure, the arc length of the plane curve, the volume and the lateral area of the revolving solid, the volume of the solid whose parallel sections area are given, questions of the ac
33、ting of variable force, gravitation, pressure and the mean value of the functions, and etc) using the definite integral, Understand the conception of the improper definite integral and can calculate it. Chapter Six : Differential Equations Comprehend the properties of the solution for the linear dif
34、ferential equations and the theorems of the construction of their solutions. Master the method of solving the separated variable, first order linear differential equations and second order homogeneous linear differential equations with constant coefficient, Understand the conceptions of the solution
35、, order, general solution, initial conditions and special solutions, and the method of solving the differential equations using the power series. Be able to solve the homogeneous differential equations, Bernoulli differential equations, total differential equations, some differential equations using
36、 simple variable substitute, the differential equations )()( xfy n ， ),( yxfy and ),( yyfy by degradation, some higher than two orders homogeneous linear differential equations with constant coefficient, Euler equations. Be able to seek the special and general solutions for some non-homogeneous line
37、ar differential equations with free term which has the form of polynomial, exponent functions, sine functions, cosine functions, their sum or difference. Be able to solve some simple application problems applying the differential equations. Understand the first order linear differential equations ar
38、ray with constant coefficient including two unknown function. Chapter seven: Vector Algebra and Spatial Analytical Geometry Comprehend the conception of rectangular coordinates system in space, the vector, the equations of the curve. Master the operation of the vector (linear operation, scalar produ
39、ct, vector product, mixed product), the coordinate expression of vectors, unit vectors, direction numbers, and direction cosine, the method of vector operations using the coordinate expression, the plane equations, the line equation and their solving method. Understand the conditions under which two
40、 vectors can be perpendicular and parallel, the equations of conicoid and their graphs, the parameter equations of plane curve and general equations. Be able to solve some problems using the relations between planes and lines, solve the equations of the rotating surface rotating around the coordinat
41、e axis, solve the cylindrical equations whose generatrix parallel to the coordinate axis, solve the projection curve equations on the plane associated with the spatial curve. Chapter eight: Differential of Multivariate Functions and Their Applications Comprehend the conceptions of multivariate funct
42、ions, the geometrical meaning of the two variable functions, the conception of the partial derivatives and total differentials of the multivariate functions, the conception of the direction derivative, gradient and their computation, the conception of the extreme value and the extremum with a condit
43、ion of the multivariate functions. Master the method of calculating the partial differential of the multivariate component functions, the method of calculating direction derivative and gradient, the necessary condition of the existence of the extremum for multivariate functions Understand the concep
44、tion of the partial differential and total differential for the two variable function, the properties of the continuous functions on the bounded closed domain, the necessary and sufficient conditions for the existence of the total differential, the invariance of the total differential form, the appl
45、ications of total differential in the proximate calculation, the conception of tangent, the normal plane of the curve, the conception of the tangent plane and the normal line of the surface, the sufficient necessary of the existence of the extremum for two variables function. Be able to calculate th
46、e total differential, the partial derivative of the implicit function, calculate the equations of tangent, the normal plane of the curve, the equations of the tangent plane and the normal line of the surface, the extremum of the two variable functions, the extremum with a condition using the Lagrang
47、es method of multipliers, the maximum and minimum of the simple multivariate functions, solve some simple application questions. Chapter nine: Multiple Integral Comprehend the conception of the double integration and triple integration. Master the method of calculating the double integration in rect
48、angular coordinates and polar coordinates. Understand the properties of the multiple integration, the conception of the double integration and triple integration, intermediate value theorem of double integration. Be able to calculate the triple integration in rectangular coordinates, cylindrical coo
49、rdinates and spherical coordinates. Chapter ten: Curvilinear integral and Surface integral Comprehend the conception of the two classes of the curvilinear integral. Master the method of calculating the two classes of the curvilinear integral, Green Formula and the condition of plane curve integral independent of the integral path, the method of calculating the two classes of the surface integral. Understand the properties and the relations of the two classes of curvilinear inte
