《高等數(shù)學(英文版 套裝1-2冊)》分上、下兩冊出版.上冊共七章,著重介紹一元微積分學的基礎理論知識,內容包括函數(shù)、極限、函數(shù)連續(xù)性,導數(shù)、微分及其應用,不定積分、定積分及其應用;下冊共六章,著重介紹多元微分學的基礎理論知識.內容包括無窮級數(shù)、向量代數(shù)與空間解析幾何,多元函數(shù)、極限及其連續(xù)性,多元函數(shù)的微分及應用,重積分、曲線積分、曲面積分及常微分方程。
《高等數(shù)學(英文版 套裝1-2冊)》是基于多年教學經(jīng)驗,兼顧國內工科類本科數(shù)學基礎要求和海外學習的雙重需要編寫而成的,與經(jīng)典的中文微積分教材相比,《高等數(shù)學(英文版 套裝1-2冊)》適當降低了難度,突出了微積分學和后續(xù)應用型課程中常用的計算和證明方法.在保證教材內容符合學科要求且不低于本科階段微積分課程教學標準的前提下,力求語言精準、簡練,以適應我國學生的外語水平和學習特點。
《高等數(shù)學(英文版 套裝1-2冊)》適于作為工科院校的國際班、雙語教學班的高等數(shù)學教材和參考書。
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微積分歷來是大學數(shù)學最重要的組成部分,是工科院校非數(shù)學專業(yè)學生必修的一門數(shù)學基礎課程.本課程是運用數(shù)學概念、理論或方法去研究現(xiàn)實世界的空間形式和數(shù)量關系.通過本課程的學習,培養(yǎng)學生綜合分析、解決問題等邏輯思維能力,使其學會將問題化難為易、化繁為簡,激發(fā)其創(chuàng)新意識.本教材分上、下兩冊出版.上冊共七章,著重介紹一元微積分學的基礎理論知識.內容包括函數(shù)、極限、函數(shù)連續(xù)性,導數(shù)、微分及其應用,不定積分、定積分及其應用;下冊共六章,著重介紹多元微分學的基礎理論知識,內容包括無窮級數(shù)、向量代數(shù)與空間解析幾何,多元函數(shù)、極限及其連續(xù)性,多元函數(shù)的微分及應用,重積分、曲線積分、曲面積分及常微分方程。
為了使我國的高等教育盡快與國際接軌,國家教育部出臺了一系列倡導高校開設英語授課或雙語教學的國際班的相關政策.目前,大部分高校多采用將母語外的另一種外國語言(主要指英語)直接應用于非語言類課程教學,并使外語與學科知識同步獲取的一種教學模式.但是,由于國內外高校授課方式的差異,直接使用外文原版教材根本無法達到國際班的教學目的.也就是,國際班的教學內容及教學方式仍處于探索階段.鑒于此,我們兼顧中文教材的理論嚴謹性和外文原版教材的重實際應用,適當降低了中文教材的難度,突出了微積分學中實用的計算和證明方法,力求語言簡練,通俗易懂,編制了適用于理工科本科國際班的高等數(shù)學教材。
在本書的編寫過程中,我們嚴格遵循從直觀到抽象、由淺入深、由易到難等循序漸進的原則,概念清晰,內容簡練,語言通俗易懂,便于自學與教學.上、下冊內容,各需60學時,即可完成全部教學內容。
本書的編寫得到了“十二五”期間北京科技大學教材建設經(jīng)費資助,在此表示感謝,北京科技大學汪飛星教授與北京理工大學蔣立寧教授審閱了全部書稿,并提出了許多中肯的意見和建議,倫敦國王學院SamBeatson博士和香港大學SiumingYiu副教授分別對書稿上冊和下冊進行了語言潤色與修改,編者向以上同志致以最誠摯的謝意。
Contents
Chapter lPreliruinaries 1
I.I Some Set Theory Notation for the Study of Calculus 1
1.1.1 Definition of Set 1
1.1.2 Descriptions of set 1
1.1.3 Set Operations 2
1.1.4 Interval 3
1.1.5 Neighbourhood 3
1.2 The Rectangular Coordinate System 3
1.2.1 Cartesian Coordinates 3
1.2.2 Distance Formula 4
1.2.3 The Equation of a Circle 4
1.3 The Straight Line 6
1.3.1 The Slope of' a Line 6
1.3.2 The Equation of a Line 6
1.4 Graphs of Equations 7
1.4.1 The Graphing Procedure 7
1.4.2 Symmetry of a Graph 7
1.4.3 Intercepts 9
1.4.4 Problems for Chapter 10
Chapter 2Functions and Limits 12
2.1 Functions 12
2.1.1 Definition of Function 12
2.1.2 Properties of Functions 14
2.1.3 0perations on Functions 16
2.1.4 Elementary Functions 18
2.1.5 Problems for Section 2.1 19
2.2 Limits 20
2.2.1 Introduction to Limits 20
2.2.2 Definition of Limit 99
2.2.3 0perations on Limits 25
2.2.4 Limits at Infinity and Infinite Limits 29
2.2.5 Infinitely Small Quantity (or Infinitesimal) 33
2.2.6 Problems for Section 2.2 35
2.3 Continuity of Functions 36
2.3.1 Definition of Continuity 36
2.3.2 Continuity under Function Operations 38
2.3.3 Continuity of Elementary Functions 38
2.3.4 Intermediate Value Theorem 39
2.3.5 Problems for Section 2.3 39
2.4 Chapter Review 40
2.4.1 Drills 40
2.4.2 Sample Test Problems 41
Chapter 3 Differentiation 43
3.1 Derivatives 43
3.1.1 Two Problems with One Theme 43
3.1.2 Definition 45
3.1.3 Rules for Finding Derivatives 45
3.1.4 Problems for Section 3.1 52
3.2 Higher-Order Derivatives 53
3.2.1 Definition 53
3.2.2 Sum, Difference and Product Rules 55
3.2.3 Problems for Section 3.2 56
3.3 Implicit Differentiation 57
3.3.1 Guidelines for implicit Differentiation 57
3.3.2 Related Rates 59
3.3.3 Problems for Section 3.3 61
3.4 Differentials and Approximations 62
3.4.1 Definition of Differential 62
3.4.2 Differential Rules63
3.4.3 Approximations 64
3.4.4 Problems for Section 3.4 67
3.5 Chapter Review 68
3.5.1 Drills 68
3.5.2 Sample Test Problems 69
Chapter 4 Applications of Differentiation 70
4.1 Maxima and Minima 70
4.1.1 Extrema on an Interval 70
4.1.2 Problems for Section 4.1 74
4.2 Monotonicity and Concavity 75
4.2.1 The First Derivative and Monotonicity 75
4.2.2 The Second Derivative and Concavity 77
4.2.3 Problems for Section 4.2 81
4.3 Local Maxima and Minima 82
4.3.1 Definition 82
4.3.2 Tests for Local Maxima and Minima 82
4.3.3 More Maxima and Minima Problems 84
4.3.4 Problems for Section 4.3 87
4.4 Sophisticated Graphing 88
4.4.1 Asymptote 88
4.4.2 Sophisticated Graphing 90
4.4.3 Problems for Section 4.4 94
4.5 The Mean Value Theorem 94
4.5.1 Rolle's Theorem 94
4.5.2 Lagrange's Mean Value Theorem 96
4.5.3 Cauchy's Mean Value Theorem 99
4.5.4 Problems for Section 4.5 99
4.6 L'Hopital's Rule 100
4.6.1 Indeterminate Forms of Type 100
4.6.2 Indeterminate Forms of Type里 101
4.6.3 0ther Indeterminate Forms 102
4.6.4 Problems for Section 4.6 103
4.7 Chapter Review 104
4.7.1 Drills 104
4.7.2 Sample Test Problems 106
Chapter5 Indefinite Integrals 108
5.1 Definition and Properties of Indefinite Integrals 108
5.1.1 Definition of Indefinite Integrals 108
5.1.2 Standard Integral Forms 110
5.1.3 Properties of' Indefinite Integrals 111
5.1.4 Problems for Section 5.1 113
5.2 Integration by Substitution 114
5.2.1 The Mehod of Substitution 114
5.2.2 Rationalizing Substitutions 121
5.2.3 0ther Substitutions (Inverse) 125
5.2.4 Problems for Section 5.2 126
5.3 Integration by Parts 127
5.3.1 Problems for Section 5.3 132
5.4 Integration of Rational Functions 132
5.4.1 Partial Fraction Decompositions 133
5.4.2 Integration of Rational Functions Using Partial Fractions 135
5.4.3 Problems for Section 5.4 137
5.5 Chapter Review 137
5.5.1 Drills 137
5.5.2 Sample Test Problems 138
Chapter6 Definite Integrals 140
6.1 Definition and Properties of Definite Integrals 140
6.1.1 Two Problems 140
6.1.2 Definition of Definite Integrals 143
6.1.3 Existence of' Definite Integrals 145
6.1.4 Geometric Interpretation 145
6.1.5 Properties of Definite Integrals 149
6.1.6 Problems for Section 6.1 152
6.2 Fundamental Theorem of Calculus 153
6.2.1 Problems for Section 6.2 157
6.3 Evaluation of Definite Integrals 158
6.3.1 Integration by Substitution 158
6.3.2 Integration by Parts 161
6.3.3 Problems for Section 6.3 163
6.4 hnproper Integrals 164
6.4.1 Infinite Limits of Integration 164
6.4.2 Infinite Integrands 166
6.4.3 Problems for Section 6.4 168
6.5 Chapter Review 169
6.5.1 Drills 169
6.5.2 Sample Test Problems 169
Chapter7 Applications of Integration 172
7.1 The Area of a Plane Region 172
7.1.1 Problems for Section 7.1 175
7.2 Volumes of Solids: Slabs, Disks and Washers 175
7.2.1 Volume of a Cylinder and a Solid 175
7.2.2 Solids of' Revolution: Disk Method 177
7.2.3 0ther Examples 180
7.2.4 Problems for Section 7.2 181
7.3 Volumes of Solids of Revolution: Shells 182
7.3.1 Problems for Section 7.3 185
7.4 Length of a Plane Curve 185
7.4.1 Problems for Section 7.4 189