数学分析课程描述

定位：数学分析课程是实现数学专业培养目标的最重要基础课之一，它所体现的数学思想、数学方法是进行数学研究的基石。通过数学分析课程教学使学生受到最基本的数学专业训练。数学分析课程成败直接影响着学生其它专业后继课程的学习和分析问题、解决问题能力的提高。

课程目标：培养学生掌握数学分析的基本理论、基础知识与基本方法，为专业后续课程的学习和以后进行数学研究打下坚实的基础。

极限理论：包括实数集与函数、数列极限、函数极限、函数连续性、实数完备性等内容，对应学时为66学时

一元函数微积分：包括导数与微分、微分中值定理及其应用、不定积分、定积分、定积分的应用、反常积分等内容，对应学时为96学时

级数理论：包括数项级数、函数列与函数项级数、幂级数、傅里叶级数等内容，对应学时为54学时；

多元函数微积分：包括多元函数极限与连续、多元函数微分学、隐函数定理及其应用、含参量积分、曲线积分、重积分、曲面积分等内容，对应学时为96学时。

重点：极限的概念、极限交换次序的有关理论、连续性的概念、闭区间上连续函数性质、导数与微分的概念、微分中值定理、实数完备性理论、不定积分、定积分有关理论、反常积分的含义与性质、级数敛散性的概念与性质、全微分的概念、隐函数定理、含参量积分性质、线面积分与重积分的计算。

难点：极限的定义、一致连续的概念、复合函数求导、辅助函数解决问题的方法、实数完备性理论、第二换元公式、可积条件、微元法、反常积分敛散性判别、级数敛散性的判别、二元函数极限的讨论、含参量反常积分一致收敛性的判定、各种积分计算。

解决办法：分散难点、数形结合、形象思维、逻辑思维训练与讲练结合训练等。

Mathematical Analysis

Course Description

Mathematical Analysis Course is the most important foundation course to train the professionals in mathematics. The mathematical thinking and mathematical method incorporated in this course are the cornerstones for mathematical research. This course provides the students with the most basic professional mathematical training. Whether the students can successfully complete this course or not will directly affect their study of other special successive courses and the analytical ability and problem-solving ability.

Course Objective: The students are trained to command the basic theory, basic knowledge and basic method of mathematical analysis to lay the solid foundation for their successive course study and their future mathematical research.

Limit Theory: It covers set of real number and function, sequence limit, function limit, continuity of function, completeness of real number and etc with the total study hours of 66.

Differential and Integral Calculus of Function of Single Variable: It covers derivative and differential, differential mean value theorem and its application, indefinite integral, definite integral, application of definite integral, improper integral and etc with the total study hours of 96.

Series Theory: It covers numerical series, function sequence and function series, power series, Fourier series and etc with the total study hours of 54.

Differential and Integral Calculus of Function of Several Variables: It covers limit and continuity of function of several variables, differential calculus of function of several variables, implicit function theorem and its application, integral with parameters, curvilinear integral, multi integral, surface integral and etc with the total study hours of 96.

Key Study Areas: Limit conception, the repeated exchange limit theorem, continuity concept, continuous function on closed interval, concept of derivative and differential, differential mean value theorem, theory of completeness of real number, theory of infinite integral and finite integral, the definition and properties of improper integral, concept and properties of series convergence, the concept of total differential, implicit integral theorem, properties of integral with parameters, calculation of surface integral and multi integral.

Difficult Areas: Definition of limit, the concept of uniform continuity, the derivative of composite function, solution to auxiliary function, theory of completeness of real number, the second element substitution formula, integrability condition, micro-element method, the criterion of convergence of improper integral, series convergence criterion, discussion of limit of binary function, criterion of uniform convergence of improper integral with parameter and the calculation of different integrals.

Solution: Spreading the difficult areas, thinking in combination of mathematics and images, thinking in image, training the logical thinking and combining class study with practice.