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7 T J0 A, E+ H# s给你个高等数学:
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' X# X; @2 I: B t课程简介:高等数学# `3 l3 S9 ]' P) O
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通过本课程的学习，使学生获得：函数、极限、连续；一元函数微积分学；多元函数微积分学；常微分方程；概率论初步；线性代数基础等方面的基本概念、基本理论和基本运算技能。
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$ S# }) B0 b+ [. P) \9 j g; m0 w, |教学总时数为108学时（4学分）' l* ~1 G6 `' K( {8 w
0 V j; N5 m& x0 p, g课程列表:- S6 M8 o& A/ q$ L
第1章:函数、极限、连续
. x& B8 s* s! B( h: }' Y7 k( `! b/ X大学数学与初等数学的根本区别在于大学数学研究的主体是变量关系。本节要求学生了解和掌握函数的基本概念、函数关系的构建以及函数的特性。了解和掌握极限和连续的基本思想和基本概念。极限作为研究变量变化趋势的工具是高等数学诸多知识的基础。# J, u. B; v. A$ K
+ c6 r% I x2 F7 [/ D; r第2章:导数与微分 m, T" @# I1 r& h6 q
理解导数的概念（包括左、右导数）导数的几何意义和物理意义，函数的可导性与连续性之间关系。掌握导数的四则运算法则和复合函数的求导法，掌握函数的导数计算。了解微分的概念和四则运算。学会用导数描述一些简单的物理和经济现象。
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第3章:中值定理与导数的应用 N- p7 k9 j6 K; U2 V* r7 l3 |
理解和掌握中值定理的基本思想和基本原理。学会用导数知识讨论函数的基本性质。理解和掌握一元函数的优化思想。了解曲率、曲率半径的概念，并会计算。了解求方程近似解的二分法和切线法。
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, f& z/ d8 w( t( Z( g& R0 T. `3 t第4章:不定积分
6 V( p, W5 ~# }$ F) r4 Q( K q理解原函数概念，理解不定积分的概念及性质。掌握不定积分的基本公式、换元法、分部积分法，了解和掌握定积分换元积分法、分部积分法及其所包含的数学原理。2 r) r/ y7 s4 x+ i( y5 ^
2 h+ E" n* @* A% w8 u+ v第5章:定积分及其应用( I6 a V7 ~" b& Y6 w
理解定积分的基本概念，掌握牛顿-莱布尼兹公式。了解和掌握定积分所包含的“以直代曲”的哲学基本原理。会用定积分的基本概念建立物理和经济中的一些数学模型。
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第6章:向量代数和空间解析几何
% k6 q* ^1 i8 P6 ^* L学会把几何问题归结为代数问题解决、反过来通过代数问题的研究来发现新的几何结果。本节介绍向量和向量的运算，用向量这工具讨论平面、直线的问题，介绍一些常见的曲面与曲线，为学习多元函数微积分做准备。* b- W: t. M; D$ C
. e& P. t6 Q. _2 W% C* i" \) \第7章:多元函数微分学
& I+ ~9 M9 Z% ^; `6 y3 H理解和掌握多元函数、极限与连续的概念，理解偏导数与全微分的概念。理解方向导数与梯度的概念并掌握其计算方法。掌握多元函数偏导数的计算及其应用。学会用类比的方法研究多元与一元函数之间关系。掌握多元函数的最优化方法及其在经济、物理等领域中的数学建模。6 l; }9 e \2 w
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第8章:重积分/ @$ p( O; N4 v& Q; ^ s- F
理解二重积分、三重积分的概念，了解重积分的性质。掌握二重积分(直角坐标、极坐标)的计算方法，会计算三重积分(直角坐标、柱面坐标、球面坐标)。会用重积分建立相应的数学模型并进行计算。
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1 e9 y* ^1 j% P第9章:曲线积分与曲面积分# I# y3 C. @8 L q) h
理解两类曲线和曲面积分的概念，了解两类曲线和曲面积分的性质，了解两类曲线积分的关系以及两类曲面积分的关系。掌握计算两类曲线和曲面积分的方法。通过两类曲线和曲面积分学会解决物理学中的一些问题。$ {( m6 N4 F" `$ [/ |: Z5 M
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第10章:微分方程
+ n. B# L5 ?$ _1 q9 _了解微分方程及解、通解、初始条件和特解等概念。掌握和理解线性微分方程的一些基本原理，并学会用这些原理求线性微分方程的解。会应用微分的思想将一个相应的实际问题建立其数学模型并求解。% c. M( S I- ~) I0 w: [
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第11章:级数
; E! ?5 i6 T' H1 z理解和掌握级数收敛与发散的概念、收敛级数和的概念，理解级数的有限和与无限和之间的关系。掌握级数审敛的判别法则。了解无穷级数绝对收敛与条件收敛的概念，及二者之间的关系。掌握函数的幂级数和傅立叶级数展开来研究函数性质的方法。6 @, k! ]% h- i- T% J- m
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, W4 L; q s: tCourse description: Advanced Mathematics+ ^ P/ l- c7 J. _
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The Advanced Mathematics covers differentiation and integration of functions, with applications. Topics include: Concepts of function, limits, and continuity；Differentiation rules, application to graphing, rates, approximations, and extremum problems；Definite and indefinite integration；Fundamental theorem of calculus；Applications of integration to geometry and science；Elementary functions；Techniques of integration；Approximation of definite integrals, improper integrals, and L'Hospital's rule；Differential Equations； Mathematical Statistics；Series.8 n4 D% z. f" y8 H# n
- z0 y8 J5 L9 S+ _; M$ vThe total teaching process involves 108periods (4Credits)./ }+ h. ] o: A
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Lecture Subject Outline:; [" Z3 Z- f1 X; p8 a$ Y$ C
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Chapter 1: Function, the Limit, Continuity
* u( C7 C& t) C5 z- Z3 KThis section requires students to understand and grasp the basic concepts and features of function, and how to build the function. Understand and master the concept and thinking of limit and continuous. Limit used as a research tool to study the trend of variables changing is the base of many areas in advanced mathematics.
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Chapter 2: Derivative and differential
?5 P4 ?# Z' k' K, h* oUnderstand the concept of derivative (including left and right derivative), the significance of the geometric and physical meaning, and the relationship between the function derivative and continuity. Master the four algorithms of derivatives and the derivation of composite function. Master how to calculate the derivative of a function. Understand the concept of differential and its four operations. Learn how to use some simple derivative to describe the physical and economic phenomena.1 f. M' X4 |9 A6 }
! [3 x$ X/ i5 i- W0 dChapter 3: Mean Value Theorem and Derivative Application
$ h u6 l8 E1 {- D: nUnderstand and master the basic idea of Mean Value Theorem and basic principles. Learn how to use the derivative function of the basic knowledge of the nature of the discussion. Understanding and mastering one single variable function optimization thinking. Understand curvature, the concept of the radius of curvature, and how to calculate. Understand the equation for the approximate solution of dichotomy and Tangent.+ x/ {# \+ F7 K5 f0 j+ ?
2 l8 s& \2 N7 Y8 o2 l \ BChapter 4: Indefinite Integral
6 C; k, b4 f* w* y5 JUnderstand the concept of the original function, and the concept and nature of indefinite integral. Grasp the basic formula, Element Method, Division integral method of Indefinite Integral. Understand and master the Integration by substitution, and Integration by Parts, and all the mathematical theories contained.
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Chapter 5: Definite integral and its application: J3 `' \0 v5 ` V4 m9 f1 W" l
Understand the basic concept of integral and master Newton - Leibniz formula. Learn using integral to set up the Mathematical Modeling in physics and economics.
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Chapter 6: Vector algebra and space analytic geometry
# S, x& c. [' k/ R' tLearn how to solve the problem via boiling geometry problem down to algebraic, in turn, through studying algebra problems and discover new geometric results. And this section presents vector and vector calculus; vector used as a tool to discuss planar, linear problems and introduce some of the most common Curve and Curve surface, to get ready for the multi-function study of calculus. ; d) I5 k; R H) W6 Z/ l3 o( x
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Chapter 7: Multiple function of differential* ^# D, j9 t$ l0 ]! N/ o
Understanding and grasp the concept of the Multi-function, and the Limit and the Continuity of multi-function. Understand the concept of partial derivative and total differential. Understand the concept of direction Derivative and the gradient and master its calculation method. Master the application and the calculation of multiple-function’s partial derivative. Learn how to use the analog method study the relationship between the multiple-function and single variable function. Grasp the optimization method of multi-function, and its mathematical modeling in the field of physics and economy.
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Chapter 8: Multiple Integral9 |9 m/ E9 B6 Q+ p$ _7 U
Understand the concept of double integral and triple integral, and the nature of re-integration. Master the calculation method of Double Integral (Cartesian coordinates, and polar coordinates) as well as that of triple integral (Cartesian coordinates, and cylindrical coordinates, and spherical coordinates).Learn how to establish mathematical model via Multiple integral and grasp its calculation. # i/ J" J5 H: R+ a! d
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Chapter 9: Curve Integral and Surface Integral7 _. W: x. [& g# d' v2 u- I9 U% r
Understand two kinds of the concept of curve and surface integral; understand two kinds of nature of curve and surface integration, understanding the relationship between the two types of curve integration and the relationship between the two surface integrations. Master the calculation method of those two types of curve and surface of the integral. Through studying two types of curve and surface integral, learn to solve some of the problems in physics.* n8 |4 d' @3 q
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Chapter 10: differential equation
# t9 |8 l% I: W3 _' N! E* GUnderstand the differential equations and solutions, general solution, the initial conditions and concepts such as special solution. Master and understand some of the basic linear differential equation theory, and learn how to use these principles to get the solution of linear differential equations. Differential thinking will be applied to set up a corresponding mathematical model for some practical problems and solve it.: z- R, U6 Y0 e6 @- X; D
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Chapter 11: Series
1 _5 H% i$ a2 J* }( o$ }( a" PUnderstand and grasp the concept of Series’ convergence and divergence, and convergence series. Understand the relationship between the limited series and the infinite series. Master the Convergence discriminant rules of series. Understand the concept of the infinite series’ absolute convergence and conditional convergence, and the relationship between them. Master how to use function’s power series and Fourier series expansion to study the nature of function. |
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