高等數學

作者：東南大學大學數學教研室編

出版社：東南大學出版社

圖書書號：ISBN 978-7-5641-5482-0

出版日期：2015年1月

開本：B5

圖書裝訂：平裝

版次：2015年1月第1版第1次印刷

印張：21

字數：411

上架時間：2015-06-10

圖書點擊數：822

價格：￥43元

前言

本書是為響應東南大學國際化需要，根據國家教育部非數學專業數學基礎課教學指導分委員會制定的工科類本科數學基礎課程教學基本要求，並結合東南大學數學系多年教學改革實踐經驗編寫的全英文教材。全書分為上、下兩冊，內容包括極限、一元函數微分學、一元函數積分學、常微分方程、級數、向量代數與空間解析幾何、多元函數微分學、多元函數積分學、向量場的積分、複變函數等十個章節。

Chapter 5Infinite Series1

§5.1Infinite Series 1

§5.1.1The Concept of Infinite Series1

§5.1.2Conditions for Convergence3

§5.1.3Properties of Series5

Exercise 5.18

§5.2Tests for Convergence of Positive Series9

Exercise 5.218

§5.3Alternating Series, Absolute Convergence, and Conditional Convergence19

§5.3.1Alternating Series19

§5.3.2Absolute Convergence and Conditional Convergence21

Exercise 5.323

§5.4Tests for Improper Integrals24

§5.4.1Tests for the Improper Integrals: Infinite Limits of Integration24

§5.4.2Tests for the Improper Integrals: Infinite Integrands 26

§5.4.3The Gamma Function28

Exercise 5.430

§5.5Infinite Series of Functions31

§5.5.1General Definitions31

§5.5.2Uniform Convergence of Series32

§5.5.3Properties of Uniformly Convergent Functional Series

34

Exercise 5.536

§5.6Power Series37

§5.6.1The Radius and Interval of Convergence37

§5.6.2Properties of Power Series41

§5.6.3Expanding Functions into Power Series45

Exercise 5.655

§5.7Fourier Series56

§5.7.1The Concept of Fourier Series56

§5.7.2Fourier Sine and Cosine Series62

§5.7.3Expanding Functions with Arbitrary Period65

Exercise 5.768

Review and Exercise69

Chapter 6Vectors and Analytic Geometry in Space72

§6.1Vectors72

§6.1.1Vectors72

§6.1.2Linear Operations on Vectors73

§6.1.3Dot Products and Cross Product75

Exercise 6.179

§6.2Operations on Vectors in Cartesian Coordinates in Three Space

80

§6.2.1Cartesian Coordinates in Three Space80

§6.2.2Operations on Vectors in Cartesian Coordinates84

Exercise 6.288

§6.3Planes and Lines in Space89

§6.3.1Equations for Plane89

§6.3.2Lines92

§6.3.3Some Problems Related to Lines and Planes95

Exercise 6.3100

§6.4Curves and Surfaces in Space101

§6.4.1Sphere and Cylinder101

§6.4.2Curves in Space103

§6.4.3Surfaces of Revolution105

§6.4.4Quadric Surfaces106

Exercise 6.4109

Exercise Review110

Chapter 7Multivariable Functions and Partial Derivatives113

§7.1Functions of Several Variables113

Exercise 7.1116

§7.2Limits and Continuity116

Exercise 7.2120

§7.3Partial Derivative121

§7.3.1Partial Derivative121

§7.3.2Second Order Partial Derivatives123

Exercise 7.3126

§7.4Differentials128

Exercise 7.4132

§7.5Rules for Finding Partial Derivative133

§7.5.1The Chain Rule133

§7.5.2Implicit Differentiation137

Exercise 7.5140

§7.6Direction Derivatives, Gradient Vectors142

§7.6.1Direction Derivatives142

§7.6.2Gradient Vectors144

Exercise 7.6146

§7.7Geometric Applications of Differentiation of Functions of Several Variables147

§7.7.1Tangent Line and Normal Plan to a Curve147

§7.7.2Tangent Plane and Normal Line to a Surface149

Exercise 7.7152

§7.8Taylor Formula for Functions of Two Variables and Extreme Values

153

§7.8.1Taylor Formula for Functions of Two Variables153

§7.8.2Extreme Values155

§7.8.3Absolute Maxima and Minima on Closed Bounded Regions

160

§7.8.4Lagrange Multipliers161

Exercise 7.8164

Exercise Review166

Chapter 8Multiple Integrals172

§8.1Concept and Properties of Multiple Integrals172

§8.2Evaluation of Double Integrals174

§8.2.1Double Integrals in Rectangular Coordinates174

§8.2.2Double Integrals in Polar Coordinates178

§8.2.3Substitutions in Double Integrals182

Exercise 8.2185

§8.3Evaluation of Triple Integrals188

§8.3.1Triple Integrals in Rectangular Coordinates188

§8.3.2Triple Integrals in Cylindrical and Spherical Coordinates

192

Exercise 8.3196

§8.4Evaluation of Line Integral with Respect to Arc Length197

Exercise 8.4199

§8.5Evaluation of Surface Integrals with Respect to Area200

§8.5.1Surface Area200

§8.5.2Evaluation of Surface Integrals with Respect to Area202

Exercise 8.5204

§8.6Application for the Integrals205

Exercise 8.6208

Review and Exercise209

Chapter 9Integration in Vectors Field213

§9.1Vector Fields213

Exercise 9.1215

§9.2Line Integrals of the Second Type216

§9.2.1The Concept and Properties of the Line Integrals of the Second Type216

§9.2.2Calculation218

§9.2.3The Relation between the Two Line Integrals221

Exercise 9.2221

§9.3Green Theorem in the Plane222

§9.3.1Green Theorem223

§9.3.2Path Independence for the Plane Case228

Exercise 9.3232

§9.4The Surface Integral for Flux234

§9.4.1Orientation234

§9.4.2The Conception of the Surface Integral for Flux235

§9.4.3Calculation237

§9.4.4The Relation between the Two Surface Integrals240

Exercise 9.4241

§9.5Gauss Divergence Theorem242

Exercise 9.5246

§9.6Stoke Theorem247

§9.6.1Stoke Theorem247

§9.6.2Path Independence in Threespace251

Exercise 9.6252

Review and Exercise252

Chapter 10Complex Analysis255

§10.1Complex Numbers255

Exercise 10.1257

§10.2Complex Functions 259

§10.2.1Complex Valued Functions259

§10.2.2Limits259

§10.2.3Continuity261

Exercise 10.2263

§10.3Differential Calculus of Complex Functions264

§10.3.1Derivatives264

§10.3.2Analytic Functions268

§10.3.3Elementary Functions272

Exercise 10.3276

§10.4Complex Integration279

§10.4.1Complex Integration279

§10.4.2CauchyGoursat Theorem and Deformation Theorem

282

§10.4.3Cauchy Integral Formula and Cauchy Integral Formula for Derivatives289

Exercise 10.4292

§10.5Series Expansion of Complex Function 295

§10.5.1Sequences of Functions296

§10.5.2Taylor Series297

§10.5.3Laurent Series299

Exercise 10.5304

§10.6Singularities and Residue307

§10.6.1Singularities and Poles307

§10.6.2Cauchy Residue Theorem311

§10.6.3Evaluation of Real Integrals316

Exercise 10.6320

Exercise Review323