微積分

前言

國(guó)內(nèi)出版的理工類非數(shù)學(xué)專業(yè)的微積分教材很多，其中不少是有一定特色的。特別是近幾年來隨著大學(xué)數(shù)學(xué)教學(xué)改革的不斷深入，反映在教材建設(shè)上，其成果還是比較突出的。但從我在教學(xué)和教改研究中所讀到的教材看，還存在著一些值得討論的問題。第一是教材雖多，但在總的體系結(jié)構(gòu)上大體雷同，受原蘇聯(lián)教材的影響還較重。當(dāng)然，這并不是說這種體系不好，而是太多差異不大的教材，不利于比較和促進(jìn)教材的建設(shè)工作。第二是教材的文風(fēng)都比較正統(tǒng)，語言不太生動(dòng)，有種使讀者，特別是數(shù)學(xué)基礎(chǔ)差一點(diǎn)的讀者望而生畏之感，也就是教材的可讀性方面值得改進(jìn)。第三是習(xí)題不夠豐富，題型的變化較少，應(yīng)用問題，特別是有真實(shí)數(shù)據(jù)的、符合我國(guó)實(shí)際的應(yīng)用問題很少。由Dale Varberg等編寫的《Calculus》第9版是一本在美國(guó)大學(xué)中使用面比較廣泛的微積分教材。該書與在美國(guó)采用更廣泛的微積分教材《Thomas’Calculus》比較，有不少共同之處，如重視應(yīng)用、便于自學(xué)、習(xí)題數(shù)量與內(nèi)容比較豐富等。而較大的差別是該教材比較強(qiáng)調(diào)數(shù)學(xué)的嚴(yán)謹(jǐn)性，例如在極限處理上，雖然也是主要講函數(shù)極限，但書中不但有嚴(yán)格的ε－δ定義，而且用較大的篇幅用其證明一些極限；許多定理都有較嚴(yán)謹(jǐn)?shù)淖C明。這一點(diǎn)與我國(guó)許多現(xiàn)行的理工科微積分教材比較類似，在美國(guó)也是另一種風(fēng)格的教材。本書強(qiáng)調(diào)應(yīng)用，習(xí)題數(shù)量多，類型多，重視不同數(shù)學(xué)學(xué)科之間的交叉，強(qiáng)調(diào)其實(shí)際背景，反映當(dāng)代科技發(fā)展。每章之后有附加內(nèi)容，包含利用圖形計(jì)算器或數(shù)學(xué)軟件計(jì)算的習(xí)題或帶研究性的小題目等。本教材的內(nèi)容有：一元微積分，包括函數(shù)、極限，函數(shù)連續(xù)性，倒數(shù)及其應(yīng)用，積分及其應(yīng)用，不定型的極限及廣義積分，級(jí)數(shù)、數(shù)值方法及逼近；多元微積分，包括空間解析幾何，向量，多元函數(shù)的導(dǎo)數(shù)與二重、三重積分，以及向量場(chǎng)的微積分；最后是微分方程?？傊?，這種基礎(chǔ)數(shù)學(xué)教材的影印出版，對(duì)于我們借鑒國(guó)外好的教學(xué)經(jīng)驗(yàn)，推動(dòng)我國(guó)的數(shù)學(xué)教學(xué)改革，特別是對(duì)當(dāng)前提倡的“雙語教學(xué)”工作，一定會(huì)起到很好的作用，收到良好的效果。

內(nèi)容概要

這是一本在美國(guó)大學(xué)中使用面比較廣泛的微積分教材。有重視應(yīng)用、便于自學(xué)、習(xí)題數(shù)量與內(nèi)容比較豐富等特點(diǎn)。而與其他美國(guó)教材的差別在于嚴(yán)謹(jǐn)性，本書許多定理都有較嚴(yán)謹(jǐn)?shù)淖C明，這一點(diǎn)與我國(guó)許多現(xiàn)行的理工科微積分教材比較類似。在美國(guó)也是另一種風(fēng)格的教材。
本書強(qiáng)調(diào)應(yīng)用，習(xí)題數(shù)量多，類型多，重視不同數(shù)學(xué)學(xué)科之間的交叉，強(qiáng)調(diào)其實(shí)際背景，反映當(dāng)代科技發(fā)展。每章之后有附加內(nèi)容，有利用圖形計(jì)算器或數(shù)學(xué)軟件計(jì)算的習(xí)題或帶研究性的小題目等。

作者簡(jiǎn)介

作者:(美)沃伯格、柏塞爾、里格登

書籍目錄

出版說明
序
Preface 
0 Preliminaries
  0.1 Real Numbe.Estimation，and Logic
  0.2 Inequalities and Absolute Values 
  0.3 The Rectangular Coordinate System
  0.4 Graphs of Equatio
  0.5 Functio and Their Graphs
  0.6 Operatio on Functio
  0.7 Trigonometric Functio
  0.8 Chapter Review
  Review and Preview Problems
1 Limits
  1.1 Introduction to Limits
  1.2 Rigorous Study of Limits
  1.3 Limit Theorems
  1.4 Limits Involving Trigonometric Functio
  1.5 Limits at Infinity；Infinite Limits
  1.6Continuity of Functio
  1.7Chapter Review
  Review and Preview Problems
2 The Derivative
  2.1 Two Problems with One Theme
  2.2 The Derivative
  2.3 Rules for Finding Derivatives
  2.4 Derivatives of Trigonometric Functio
  2.5 The Chain Rule
  2.6 Higher.Order Derivatives
  2.7 Implicit Differentiation
  2.8 Related Rates
  2.9 Differentials and Approximatio
  2.10 Chapter Review
  Review and Preview Problems
3 Applicatio of the Derivative
  3.1 Maxima and Minima
  3.2 Monotonicity and Concavity
  3.3 Local Extrema and Extrema on Open Intervals
  3.4 Practical Problems
  3.5 Graphing Functio Using Calculus
  3.6 The Mean Value Theorem for Derivatives
  3.7 Solving Equatio Numerically
  3.8 Antiderivatives
  3.9 Introduction to Differential Equatio 
  3.10 Chapter Review
  Review and Preview Problems
4 The Deftnite Integral
  4.1 Introduction to Area
  4.2 The Definite Integral
  4.3 The Fit Fundamental Theorem of Calculus
  4.4 The Second Fundamental Theorem of Calculus and the Method of
Substitution
  4.5 The Mean Value Theorem for Integrals and the Use of Symmetry
  4.6 Numerical Integration
  4.7 Chapter Review
  Review and Preview Problems
5 Applicatio of the Integral
  5.1 The Area of a Plane Region
  5.2 volumes of Solids：Slabs.Disks,Wlashe
  5.3 Volumes of Solids of Revolution：Shells
  5.4 Length of a Plane Curve
  5.5 Work and Fluid Force
  5.6 Moments and Center of Mass
  5.7 Probability and Random Variabtes
  5.8 Chapter Review322
  Review and Preview Problems
6 Tracendental Functio
  6.1 The Natural Logarithm Function
  6.2 Invee Functio and Their Derivatives
  6.3 The Natural Exponential Function
  6.4 General Exponential and Logarithmic Functio
  6.5 Exponential Growth and Decay
  6.6 Fit.Order Linear Differential Equatio
  6.7 Approximatio for Differential Equatio
  6.8 The Invee Trigonometric Functio and Their Derivatives
  6.9 The Hyperbolic Functio and Their Invees
  6.10 Chapter Review
  Review and Preview Problems
7 Techniques of Integration
  7.1 Basic Integration Rules
  7.2 Integration by Parts
  7.3 Some Trigonometric Integrals
  7.4 Rationalizing Substitutio
  7.5 Integration of Rational Functio Using Partial Fractio 
  7.6 Strategies for Integration 
  7.7 Chapter Review 
  Review and Preview Problems 
8 Indeterminate Forms and Improper
  Integrals
  8.1 Indeterminate Forms of Type 0/0
  8.2 Other Indeterminate Forms
  8.3 Improper Integrals: Infinite Limits of Integration
  8.4 Improper Integrals: Infinite Integrands
  8.5 Chapter Review
  Review and Preview Problems
9 Infinite Series
  9.1 Infinite Sequences
  9.2 Infinite Series
  9.3 Positive Series: The Integral Test
  9.4 Positive Series: Other Tests
  9.5 Alternating Series, Absolute Convergence, and Conditional
Convergence
  9.6 Power Series
  9.7 Operatio on Power Series
  9.8 Taylor and Maclaurin Series
  9.9 The Taylor Approximation to a Function
  9.10 Chapter Review
  Review and Preview Problems
10 Conics and Polar Coordinates
  10.1 The Parabola
  10.2 Ellipses and Hyperbolas
  10.3 Tralation and Rotation of Axes
  10.4 Parametric Representation of Curves in the Plane
  10.5 The Polar Coordinate System
  10.6 Graphs of Polar Equatio
  10.7 Calculus in Polar Coordinates
  10.8 Chapter Review
  Review and Preview Problems
11 Geometry in Space and Vecto
  11.1 Cartesian Coordinates in Three-Space
  11.2 Vecto
  11.3 The Dot Product
  11.4 The Cross Product
  11.5 Vector-Valued Functio and Curvilinear Motion
  11.6 Lines and Tangent Lines in Three-Space
  11.7 Curvature and Components of Acceleration
  11.8 Surfaces in Three-Space
  11.9 Cylindrical and Spherical Coordinates
  11.10 Chapter Review
  Review and Preview Problems
12 Derivatives for Functio of Two or More Variables
  12.1 Functio of Two or More Variables
  12.2 Partial Derivatives
  12.3 Limits and Continuity
  12.4 Differentiability
  12.5 Directional Derivatives and Gradients
  12.6 The Chain Rule
  12.7 Tangent Planes and Approximatio
  12.8 Maxima and Minima
  12.9 The Method of Lagrange Multiplie
  12.10 Chapter Review
  Review and Preview Problems
13 Multiple Integrals
  13.1 Double Integrals over Rectangles
  13.2 Iterated Integrals
  13.3 Double Integrals over Nonrectangular Regio
  13.4 Double Integrals in Polar Coordinates
  13.5 Applicatio of Double Integrals
  13.6 Surface Area
  13.7 Triple Integrals in Cartesian Coordinates
  13.8 Triple Integrals in Cylindrical and Spherical Coordinates
  13.9 Change of Variables in Multiple Integrals
  13.10 Chapter Review
  Review and Preview Problems
14 Vector Calculus
  14.1 Vector Fields
  14.2 Line Integrals
  14.3 Independence of Path
  14.4 Green's Theorem in the Plane
  14.5 Surface Integrals
  14.6 Gauss's Divergence Theorem
  14.7 Stokes's Theorem
  14.8 Chapter Review
Appendix
  A.1 Mathematical Induction
  A.2 Proofs of Several Theorems
教輔材料說明
教輔材料申請(qǐng)表

章節(jié)摘錄

插圖：

編輯推薦

沃伯格、柏塞爾、里格登編寫的《微積分(英文版原書第9版)》是一本在美國(guó)大學(xué)中使用面比較廣泛的微積分教材。教材共分14章，內(nèi)容有：一元微積分，包括函數(shù)、極限，函數(shù)連續(xù)性，倒數(shù)及其應(yīng)用，積分及其應(yīng)用，不定型的極限及廣義積分，級(jí)數(shù)、數(shù)值方法及逼近；多元微積分，包括空間解析幾何，向量，多元函數(shù)的導(dǎo)數(shù)與二重、三重積分，以及向量場(chǎng)的微積分；最后是微分方程。每章之后有附加內(nèi)容，包含利用圖形計(jì)算器或數(shù)學(xué)軟件計(jì)算的習(xí)題或帶研究性的小題目等。

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