《高等数学》教学大纲(Syllabus of advanced mathematics)

《高等数学》教学大纲（Syllabus of advanced mathematics）People do a Book slaves,then living with dead......Put the book as a tool,the books of knowledge will live.It's alive. --Hua LuogengSyllabus of advanced mathematicsAdvanced MathematicsCourse Code:070A1012for professional:tube each professional class Polytechnic:186Credits:12Content introductionThe research object of this course is a function(dependence change process quantity).The content includes the function, limit,continuity,unary function calculus,vector algebra and space analytic geometry,multivariate function differential, multi function calculus,infinite s eries(Fourier Series)and ordinary differential equations etc..Two,the purpose and task of this courseThrough the study of this course,we should make students master the basic concepts,basic theory and basic operation skills of calculus,so as to lay the necessary mathematical foundation for learning subsequent courses and further acquiring mathematical knowledge.Through each teaching link to cultivate students'abstract thinking ability,logical reasoning ability,spatial imagination ability andself-learning ability,but also pay special attention to thecultivation of students'skilled operation ability and comprehensive use of the knowledge to the ability to analyze and solve problems.Three,the relationship between this course and other coursesThis course is the first basic course in science,engineering, management and other related professional.The study of this course is closely related to the learning of students' subsequent courses,which is related to the determination of students'learning goals and the future trend of students.This course after the end of the study,as a starting point,students can enter the relevant courses.This course is a basic course of the University for four years to learn must learn.The course is basic and theoretical,and closely related to the related courses.It is the subject of national postgraduate entrance examination,which is related to the cultivation of students'comprehensive ability.This course directly affects the school teaching level.Four,the basic requirements of the courseA basic understanding of the basic theory of calculus;fully understand the background of thought and mathematical thought of calculus.Master the basic methods,means,skills of calculus,and have certain ability of analysis and demonstration and strong calculation ability.The thought method can skillfully applied calculus to solve applicationproblems.Five,the course content and the distribution of class hours Teaching content of theoryChapter1function and limit(14hours)Understanding the concept of function,compound function and piecewise function;understanding of the concept of limit, limit on the left and right limit;understand the concept of infinity and infinitesimal;understanding the concept of function continuity(including left and right continuous continuous);graphic properties and grasp the basic elementary function;and the properties of palm grip limit four operations rule;master two criteria on the existence of limit;grasp the method of two important limits;grasp the method of infinitesimal;understand the function of parity,monotonicity, periodicity and boundedness;understand the concept of inverse function and implicit function,and the relationship between the existence of the limit and the left and right limit understanding;the nature of continuous function and continuity of elementary functions,understand the nature of continuous function on a closed interval(boundedness,maximum and minimum value theorem and intermediate value theorem);a simple stress By using the functional relation in the problem, the limit of the two criteria of the limit can be used;the limit of equivalent infinitesimal is used;the type of the discontinuous point of the function is distinguished;the properties of the continuous function on the closed interval can be applied.The second chapter,derivative and differential(12hours)To understand the concept of derivative and differential; understand the relation between derivative and differential geometry;understanding of derivative;understand the function of the relationship between the derivation and continuity; derivation rule master derivative four arithmetic operations and composite function;derivative formula master basic elementary function;understand the physical meaning of derivative;understanding four differential algorithm andfirst-order differential form invariance;understand the application of differential in approximation;understand the concept of high order derivative;tangent equation and normal equation for the plane curve will;Describe some physical quantity with the derivative; differential will function;n derivative will ask simple function;one order and two order derivative will ask piecewise function;one order and two order derivative function for implicit function and is determined by the parameter equation of the derivative of the inverse function of the will.The third chapter is the application of mean value theorem and the derivative(12hours)The extreme value of understanding the function of judgment; grasp the monotonicity of the function and method of the extremal function with the derivative,grasp the function of maximum value and minimum value method and simple method for application;grasp the limits with the Cauchy theorem ofl'Hopital's rule;understand;understand the concept of curvature and radius of curvature;will use Rolle's theorem, the Lagrange theorem and Taylor theorem;judgment function graph convexity and inflection point with derivative f unction graph will ask the level,vertical and oblique asymptote,will describe the function of the graphics;calculate the curvature and radius of curvature,angle will seek two curve.The fourth chapter is indefinite integral(8hours)Understanding the original function,the concept of indefinite integral grasp;indefinite integral properties;master the basic formula of indefinite integral;grasp the changing integral method and integral method;integral rational function,will seek the rational formula of trigonometric function and simple irrational function.The fifth chapter is definite integral(8hours)Understanding the concept of definite integral;understand the function of variable upper limit definite integral definition and derivation;grasp the nature of definite integral and definite integral mean value theorem,grasp the Newton Leibniz formula;master element integral method and integral method of definite integral;understand the concept of generalized integral and generalized integral calculation.The application of definite integral in the sixth chapter(6 hours)Master the definite integral element method;master by integralexpression and calculation of some geometrical and physical quantities(plane graphics area,the length of plane curve, volume and side area,the rotating body parallel section area, average volume of a solid known variable work,gravity, pressure and function value etc.).The seventh chapter of space analytic geometry and vector algebra(12hours)To understand the spatial Cartesian coordinate system,and understand the concept of vector representation;concept surface equation;master vector(linear operation,the number of product,vector product,mixed product),grasp the coordinate expression of unit vector and direction number and direction cosine,vector,and method of coordinate expression vector operations;control plane equation and linear equation and its solution;to understand the two vector vertical and parallel conditions;understand the equation and the common figures of two surface;understand the plane curve parameter equation and general equation;understand;use the relationship between plane and straight line(parallel, vertical,intersecting etc.)to solve the problem will ask for the cylinder;equation is parallel to the axis of the rotation axis of rotation surface and bus to the axis;projection curve will space curve on coordinate plane equation.The eighth chapter of multivariate function differential method and its application(18hours)To understand the concept of multivariate function,geometric meaning of function of two variables;understanding multifunction partial derivative and differential concept;concept of directional derivative and gradient and master the calculation method;understand the concept of multiple function extremum and conditional extremum;grasp the method for multiple function partial derivative;master the calculation method of directional derivative and gradient; master the necessary condition for the existence of extremum of multivariate function;understand the partial derivative function of two variables and total differential concept,and the bounded property of continuous function in closed area; understand the necessary conditions and sufficient conditions for the existence of differential;understand the invariance of differential form,understand the application of differential in approximation;understand the concept of curve and the tangent plane and the tangent plane and surface normals; understand the sufficient conditions for the existence of two yuan would blame the extremum of function;differential;Will find the implicit function(including the implicit function determined by equations)of the partial derivative;will find the curve and the tangent plane and the surface of the tangent plane and normal equation;extreme for two yuan function,using Lagrange multiplier method;will ask simple multivariate function of the maximum and minimum,And it will solve some simple application problems.The ninth chapter is the multiple integral(10hours)Understanding the concept of double integral,triple integral three;grasp the double integral(Cartesian coordinates and polar coordinates)calculation method;understand the natureof the double integral concept of double integral,triple integral three,understand the nature of the double integral to the mean value theorem of double integral;triple integral calculation three(rectangular coordinate and cylindrical coordinate and spherical coordinates).The tenth chapter of curvilinear integral and surface integral (14hours)Understand the concept of two kinds of curve integral;master the calculation method of two kinds of curve integral;master Green formula and use plane curve integral is path independent of the conditions;grasp the method of calculation of two types of surface integral relations;to understand the nature of two curvilinear integral and two curvilinear integral;understand the concept of two types of surface integral the nature of the relationship,and two kinds of surface integral;understand the Gauss formula,Stokes Gong;;calculate the surface integral with Gauss formula;with triple integral,curve integral and surface integral for some geometrical and physical quantities (graphic area,volume,surface area,arc length,mass,center of gravity and moment of inertia gravity power and flow rate); calculate the divergence and curl.The eleventh chapter infinite series(14hours)Understanding of constant series convergence,divergence and convergence of a series of necessary conditions and concepts; grasp the basic properties and convergence series;grasp the convergence and divergence of geometric series and series of conditions;grasp the positive series comparison test and ratiotest;master class number theorem Leibniz staggered;master method the radius of convergence,convergence and convergence interval of power series;master,McLaughlin,expansion and understanding;concept of absolute convergence and conditional convergence of a series of arbitrary items,as well as the relationship between the absolute convergence and conditional convergence;understand the concept of function series convergence domain and function;to understand some of the basic properties of power series in the convergence interval; understand the function necessary and sufficient conditions for the Taylor series;understand the power series in a simple approximate calculation of the understanding of the Fu Liye series;The concept and function to the de Lickley theorem of Fourier series;will law root;will seek some power series in the convergence region and function;some simple function will be indirectly expanded in power series;will be defined in the function is expanded into Fourier series,will be defined in the function sine and cosine series series,Fourier series and the expression will write function.The twelfth chapter of ordinary differential equations(12 hours)Understanding the properties of solutions of linear differential equations and structure theorems of solutions; grasp the solution variable separation equation and linear equation;grasp the two order constant coefficient homogeneous linear differential equation solution;differential equation and its solution,in order to understand the general solution, and the initial condition and the solution concept;understand the power series solution of differential equation the will;solutions of homogeneous equations,Bernoulli equation and differential equation,the solution will be somedifferential equations with simple variable substitution;will solve the following equations by reduction method,and the solution will be:;some higher than two order linear homogeneous differential equation with constant coefficients;be polynomial, exponential function,sine function and cosine function for the free term is two,and their sum and product of constant coefficient non homogeneous linear differential equations special solution and general solution of Euler equations;will, will first order solution contains two unknown functions with constant coefficients Linear differential equations;using differential equations(or equations)to solve some simple application problems.Practice teaching content1.Exercise classThe first chapter arranges three exercise classes,the sixth, seventh chapter arranges an exercise class,and the remaining chapters arrange two exercise classes each chapter,A total of23times,accounting for46hours.Six,teaching materials and reference books"Higher mathematics"(upper and lower)Mathematics Education Department of Tongji University,higher education press.The"guiding learning of higher mathematics Harbin Universityof Science and Technology mathematics"(upper and lower)."Higher mathematics"(upper,middle andlower volumes)edited by Wen Li,Cambridge University Press.Seven,the teaching methods of this courseThe characteristics of this course are theoretical, ideological,and related basic courses and specialized courses more contacts,should pay attention to inspire and guide students to master the important concept of ideological background teaching,ideological essence of important concepts, avoid rote learning.To be good at the relevant disciplines or life often encountered in terms of the concept of concepts and calculus combined,so that students understand the necessity of learning calculus.Pay attention to various aspects of teaching(theoretical teaching,exercise class,homework, counseling)the organic links,especially to strengthen the operation and guidance link,deepen the students' understanding of the content of classroom teaching,improve the ability of analyzing and solving problems and operation ability. In teaching,the relationship between learning mathematics and learning specialized courses is introduced to students in a planned and purposeful way.Learning advanced mathematics is the key subject to obtain further learning opportunities.Due to the characteristics of the subject,the teaching should highlight the central position of the teacher,and through the efforts of teachers,fully mobilize the students'interest in learning.Eight.Distribution of teaching hoursChapterClassroom teaching Exercise class SubtotalChapter one FourteenSixTwentySecond chapters TwelveFourSixteenThird chapters TwelveFourSixteenFourth chapters EightFourTwelveFifth chapters EightFourTwelveSixth chapters SixTwoEightSeventh chapters TwelveTwoFourteenEighth chapters EighteenFourTwenty-twoNinth chapters TenFourFourteenTenth chapters FourteenFourEighteenEleventh chapters FourteenFourEighteenTwelfth chaptersTwelveFourSixteenTotalOne hundred and fortyForty-sixOne hundred and eighty-sixNine,should pay attention to the implementation of the outlineThis syllabus is based on the basic requirements of the undergraduate basic course issued by the Higher Education Department of the State Education Commission,and is formulated in accordance with the teaching plan of our university.In the process of development,the demand for mathematics knowledge based on the management profession is taken into consideration, and the physics specialty and the relatively few professional needs of mathematics are dealt with separately.The syllabus specifies four aspects:understanding,mastering,understanding and meeting,and should pay attention to the implementation.The ratio of hours to hours is1:2.Exercise course is an important teaching link to complete the basic requirements of teaching.Exercise class hours should not be less than the total class hours of1/6,and the small class is appropriate.In the teaching process,teachers should according to the students'situation,according to the requirements of the outline,in each part in the review guide book for students to indicate different grades ofextra-curricular self-study content.People do a Book slaves,then living with dead......Put the book as a tool,the books of knowledge will live.It's alive. --Hua Luogeng。