高等数学教学大纲（英文版）

1、Syllabus of Advanced Mathematics IICourse Name：Advanced Mathematics II Course Code： Credits： 9 Total Credit Hours： 144 Lecture Hours： 144 Experiment Hours： 0 Programming Hours： 0 Practice Hours：0Total Number of Experimental (Programming) Projects 0 , Where, Compulsory ( ), Optional ( ).School：School

2、 of Science Target Major：biological engineering, food science and engineering,chemistry engineering ,medicine and business、Course Nature & AimsBy this course, the students will learn basic theory and applications of differential and integral calculus, they will get a comprehensive understanding to t

3、he methods and ideas of calculus, and develop the ability of using calculus to solve some practical problem in engineering, business and social science. By the introduction of the history of calculus and many examples concerning the application of calculus, the students will develop lots of good hab

4、its such as thinking like a scientist, solving problems creatively, etc. Besides, the student will know some famous works of Chinese ancient mathematicians.、Course Objectives 1. Moral Education and Character Cultivation. In this course, students will learn scientific spirits and get some understandi

5、ng about Chinese ancient scientific works.2. Course ObjectivesThrough the study of this course, students qualities, skills, knowledge and abilities obtained are as follows：Objective 1. By this course, students will learn the basic knowledge and develop basic calculation skills about calculus of one

6、variable functions, calculus of functions of two or more variables, infinite series and ordinary differential equations, space analytic geometry.（Corresponding to Chapter1-10, supporting for graduation requirements index 1）Objective 2. This course will help students to develop ability in abstract th

7、inking and logical reasoning. （Corresponding to Chapter 1-10, supporting for graduation requirements index 2，3）Objective 3. This course will help students to develop students some skill of using what they learn to solving some problems in different fields.（Corresponding to Chapter 3,6,7,9, 10, suppo

8、rting for graduation requirements index2，3）3. Supporting for Graduation RequirementsThe graduation requirements supported by course objectives are mainly reflected in the graduation requirements indices 2,3, as follows:Supporting for Graduation RequirementsCourse ObjectivesGraduation RequirementsInd

9、ices and Contents Supporting for Graduation RequirementsTeaching TopicsLevel of Support IndicesContentsObjective 1Objective 1Grasp special knowledge for majorIndex 2Chapter 1-10,12HObjective 2,3Objective 2,3Grasp special methods for research and solving special problemsIndex 3Chapter 1-10,12M、Basic

10、Course Content Chapter 1 Functions and Limits (supporting course objectives 1,2,3)Mappings and functionsLimits of sequencesLimits of functionsinfinitely small and infinitely largeRules of limitsExisting rule of limits and two important limitsComparing two infinite small Continuous functions Rules an

11、d properties of continuous functionsProperties of continuous functions on closed intervalsTeaching Requirements: (1) Reviewing some knowledge about functions, strengthening the understanding about the concept of function, and knowing the properties of functions such as the properties of odd and even

12、, monotonity, periodicity and boundedness. (2) Understanding the concept of composition function，knowing the concept of inverse function. (3) Modeling practical problems by using appropriate functions. (4) Understanding the concept of limits, knowing the mathematical definition of limits. (5) Master

13、ing the basic calculation rule about limits, can calculate the limit of composition functions by substitution of variables. (6) Knowing the properties of limits (uniqueness, boundedness and sign preserving) and two rules about the existence of limits (sandwich rule and monotone bounded principle), u

14、sing the two important limit results to find the limits of some functions.(7) Knowing the concepts of infinitesimal, infinity, infinitesimal of higher order and equivalent infinitesimals, mastering the skill of finding limits by using the equivalent infinitesimals. (8) Understanding the concept of t

15、he continuity of function at a point and on an interval. (9) Knowing the concept of discontinuous points of functions, can discriminate the type of a discontinuous point. (10) Knowing the continuity of the elementary functions and the properties of continuous functions defined on closed intervals (M

16、aximum and Minimum theorem and Intermediate Value Theorem).Key Points：limit and continuityDifficult Points：The definitions of limitsChapter 2 Derivative and Differentials (supporting course objectives 1,2,3)2.1 Derivatives 2.2 rules of derivatives2.3 high order derivatives2.4 implicit functions, par

17、ametric equations and their derivatives2.5 differentials(1) Understanding the concept and the geometric meaning of derivatives, knowing the relationship between the existence of derivative and the continuity of functions at some point. (2) Knowing the practical meaning of derivative as the rate of v

18、ariations, can use derivatives to express the rate of variations in science and technology. (3) Mastering the rules of computing the derivative of functions under rational operation and the chain rule for composite functions, mastering formulas of derivative of basic elementary functions. (4) Unders

19、tanding the concept of differential, knowing the locally linearization idea in differential, knowing the computation rules of differential and the invariance of the first differential forms. (5) Knowing higher order derivative, mastering the computation of the first and second order derivatives of e

20、lementary functions. (6) Mastering the skill of finding the first and second derivatives of implicit functions and functions described by parametric equations, knowing the relate rate of change in some practical problems. Key Points：differential rulesDifficult Points：concepts of derivatives and diff

21、erentialsChapter 3 Differential Mean-value Theorem and Applications of Derivatives (supporting course objectives 1,2,3)3.1 differential mean value theorems3.2 L Hospitals rule3.3 Taylor formulas3.4 montonicity and concavity3.5 minimum and maximum3.6 graphing functions3.7 curvatures3.8 approximate so

22、lutionsTeaching Requirements: (1) Understanding mean-value theorems (Rolles theorem and Lagranges theorem). Knowing Cauchys theorem, can use L Hospitals Rule to find the limits of indeterminate forms.(2) Knowing Taylors theorem and the idea of approximating a function by polynomials. (3) Understandi

23、ng the concept of local extreme, mastering the skill of studying the monotonicity and local extreme of functions by the derivative. Can find the global extreme in some practical problems. (4) Can study the convexity or concavity of the curve by using derivatives and find the inflection points, can g

24、raph some simple functions. (5) Knowing the concepts of curvature and the curvature radius, can compute out the curvature and the curvature radius. Key Points：Mean value theorems and their applications, optimization by differentialsDifficult Points：Taylors theorem and its applicationsChapter 4 Indef

25、inite Integral (supporting course objectives 1,2,3)4.1 indefinite integrals4.2 the substitution rule 4.3 integration by parts4.5 integration of rational functions 4.6 indefinite integration tables Teaching Requirements:(1) Understanding the concepts of anti-derivatives and indefinite integrals and t

26、heir properties. (2) Mastering the basic formula of indefinite integral.(3) Mastering the skill of integration by substitution and by parts.(4) Knowing the decomposition of the rational functions, can compute the integral of some simple rational functions, trigonometric functions and irrational func

27、tions. Key Points：indefinite integralsDifficult Points：integrating techniquesChapter 5 Definite Integral (supporting course objectives 1, 2, 3)5.1 definite integrals5.2 fundamental theorem of calculus5.3 the substitution rule and integration by parts for definite integrals5.4 improper integralsTeach

28、ing Requirements: (1) Understanding the concept of definite integral and its geometric meaning. (2) Knowing the properties of definite integral and integral mean value theorem. (3) Understanding function of integral with varying upper limit and its derivative. (4) Mastering the Newton-Leibnizs formu

29、la. (5) Mastering the methods of substitution and by parts about definite integral. (6) Knowing two types of improper integral and their convergence. Key Points：Fundamental theorem of calculus and applicationsDifficult Points：concepts of definite integrals and integrating techniquesChapter 6 Applica

30、tions of Definite Integral (supporting course objectives 1,2,3)6.1 infinitesimal 6.2 applications in geometry6.3 applications in physicsTeaching Requirements: (1) Mastering the infinitesimal element method of definite integral. (2) Can using infinitesimal element method to establish the integral exp

31、ression of some geometric and physics quantity. Key Points：geometric applications Difficult Points：infinitesimal methods Chapter 7 Differential Equation (supporting course objectives 1,2,3)7.1 concepts of differential equations7.2 separable differential equations 7.3 homogenous differential equation

32、s7.4 the first order linear differential equations7.5 reduced higher order differential equations7.6 higher order linear differential equations7.7 homogenous linear differential equations with constant coefficients7.8 non- homogenous linear differential equations with constant coefficients7.9 Euler

33、equations7.10 systems of linear differential equations with constant coefficientsTeaching Requirements:(1) Knowing the concepts of differential equation, solution and general solution of differential equation, the initial condition and special solution of differential equation. (2) Mastering the met

34、hods of solving separable differential equations and the first order linear differential equations. (3) Can solve the homogeneous equation, understand the idea of substitution of variable in solving differential equations. (4) Can solve the following three kinds of reducible high-order equations . (

35、5) Understanding the structure of solutions of second linear differential equations. (6) Mastering the methods of solving homogenous second linear differential equations with constant coefficients, knowing the methods of solving homogenous high-order linear differential equations with constant coeff

36、icients.(7) Can solve non-homogenous second linear differential equations with constant coefficients with the forced term . (8) Can establish differential equation models, and use them to solve some practical problems. Key Points：solving differential equationsDifficult Points：modeling by differentia

37、l equationsChapter 8 Space Analytic Geometry and Vector Algebra (supporting course objectives 1,2,3)8.1 vectors and linear operations8.2 dot product and cross product8.3 planes8.6 lines8.7 surfaces8.8 curves in spaceTeaching Requirements: (1) Understanding three dimensional rectangular coordinate sy

38、stems, understanding the concept of vectors and their expression. (2) Mastering the operations (linear operation, dot product and cross product) of vectors, knowing the conditions for perpendicular and parallel of vectors .(3) Mastering unit vector, direction cosines, and the expression of vectors i

39、n coordinate systems and their operations. (4) Can write equations of planes and straight lines, knowing how to find the position relation of planes and straight lines. (5) Understanding equations of the quadratic surface and the space curve. (6) Knowing the graph of some ordinary quadratic surface,

40、 knowing surfaces generated by rotating a curve around an axis and the cylindrical surface with generatrix parallel to an axis. (7) Knowing the parameter equations and general equations of space curves. (8) Knowing the projection on a coordinate plane of intersection curve of two surfaces. Key Point

41、s：equations of planes, lines, surfaces and curves in spaceDifficult Points：equations of planes, lines, surfaces and curves in spaceChapter 9 Differential Calculus of Multivariable Functions (supporting course objectives 1,2,3)9.1 functions with more variables9.2 partial derivatives9.3 total differen

42、tials9.4 chain rules for functions with more variables9.5 implicits functions9.6 applications in geometry9.7 directional derivatives and gradients9.8 maximum and minimumTeaching Requirements:(1) Understanding the concepts of function with two variables and more variables. (2) Knowing the limits and

43、continuity of function with two variables, knowing the properties of continuous functions defined on bounded closed domains. (3) Understanding the concepts of partial derivative and total differential of function with two variables, knowing the necessary condition and sufficient condition for the ex

44、istence of total differentials. (4) Knowing the concept and computation of single variable vector-value functions.(5) Knowing the concepts and computations of directional derivative and gradient.(6) Mastering the chain rule of partial derivative of composite functions, the computation of the second

45、derivative of composite functions.(7) Can compute the first partial derivative of implicit function. (8) Knowing the tangent line of curves and the tangent planes of surface, and knowing how to write their equations. (9) Understanding the local extreme and conditional local extreme of functions of t

46、wo variables, and can solve the local extreme, knowing the Lagrange multiplier method for conditional local extreme, can solve some practical problems about the global extreme.Key Points：differential rules and optimization of function with more variablesDifficult Points：concepts of partial derivativ

47、es and differentialsChapter 10 Multivariable Integration (supporting course objectives 1,2,3)10.1 double integrals10.2 evaluating double integrals10.3 triples integrals10.4 applicationsTeaching Requirements: (1) Understanding the concepts and properties of double and triple integrals. (2) Mastering

48、the computation of double integral under both rectangular coordinate system and polar coordinate system, computation of triple integrals under rectangular coordinate system, cylindrical coordinate system and sphere coordinate system. (3) Knowing the infinitesimal element method of multivariable inte

49、gration, can establish the multivariable integral expression of some geometry and physics quantities. Key Points：evaluating double integrals and triple integralsDifficult Points：integrating methods、Table of Credit Hour DistributionTeaching ContentIdeological and Political Integrated Lecture HoursExp

50、eriment HoursPractice HoursProgramming HoursSelf-study HoursExercise ClassDiscussion HoursChapter 1 Functions and LimitsScientific spirit202Chapter 2 Derivatives andDifferentialsScientific spirit122Chapter 3Mean Value Theorems and Applications of DerivativesScientific spirit122Chapter 4 Indefinite I

51、ntegralsScientific spirit102Chapter 5 Definite IntegralScientific spirit102Chapter 6 Applicationsof Definite IntegralChineseAncient scientific works 62Chapter7 Differential EquationsScientific spirit122Chapter 8Space Analytic Geometry and Vector AlgebraScientific spirit102Chapter 9Multivariable Differential Calculus andTheir ApplicationsScientific spirit102Chapter10 Multiv