本書(shū)是根據(jù)教育部非數(shù)學(xué)專(zhuān)業(yè)數(shù)學(xué)基礎(chǔ)課教學(xué)指導(dǎo)分委員會(huì)制定的工科類(lèi)本科數(shù)學(xué)基礎(chǔ)課程教學(xué)基本要求編寫(xiě)的全英文教材，全書(shū)分為上、下兩冊(cè)。本書(shū)為下冊(cè)，主要包括空間解析幾何和向量代數(shù)，多元函數(shù)微積分及其應(yīng)用，曲線積分與曲面積分和微分方程。本書(shū)對(duì)基本概念的敘述清晰準(zhǔn)確，對(duì)基本理論的論述簡(jiǎn)明易懂，例題習(xí)題的選配典型多樣，強(qiáng)調(diào)基本運(yùn)算能力的培養(yǎng)及理論的實(shí)際應(yīng)用。 本書(shū)可作為高等理工科院校非數(shù)學(xué)類(lèi)專(zhuān)業(yè)本科生的教材，也可供其他專(zhuān)業(yè)選用和社會(huì)讀者閱讀。The aim of this book is to meet the requirement of bilingual teaching of advanced mathematics. The selection of the contents is in accordance with the fundamental requirements of teaching issued by the Ministry of Education of China. And base on the property of our university， we select some examples about petrochemical industry. These examples may help readers to understand the application of advanced mathematics in petrochemical industry.Moreover，through the teaching experience，in this edition，we begin with a pretest to assess the necessary mathematical ability. This book is divided into two volumes.This volume contains space analytic geometry and vector algebra，calculus of multivariate function，curve integral and surface integral，infinite series.We select the examples and exercises carefully，emphasizing the cultivation of basic computing skills and the practical application of the theory. This book may be used as a textbook for undergraduate students in the science and engineering schools whose majors are not mathematics， and may also be suitable to the readers at the same level.
Chapter 8 Vector algebra and analytic geometry of space1 8.1Vectors and their linear operations1 8.1.1The concept of vector1 8.1.2Vector linear operations2 8.1.3Three-dimensional rectangular coordinate system6 8.1.4Component representation of vector linear operations8 8.1.5Length，direction angles and projection of a vector9 Exercises 8-1 12 8.2Multiplicative operations on vectors12 8.2.1The scalar product(dot product，inner product)of two vectors13 8.2.2The vector product(cross product，outer product)of two vectors15 *8.2.3The mixed product of three vectors17 Exercises 8-2 19 8.3Surfaces and their equations19 8.3.1Definition of surface equations19 8.3.2Surfaces of revolution21 8.3.3Cylinders22 8.3.4Quadric surfaces24 Exercises 8-3 26 8.4Space curves and their equations27 8.4.1General form of equations of space curves27 8.4.2Parametric equations of space curves28 *8.4.3Parametric equations of a surface29 8.4.4Projections of space curves on coordinate planes30 Exercises 8-4 31 8.5Plane and its equation32 8.5.1Point-normal form of the equation of a plane32 8.5.2General form of the equation of a plane33 8.5.3The included angle between two planes34 Exercises 8-5 36 8.6Straight line in space and its equation36 8.6.1General form of the equations of a straight line36 8.6.2Parametric equations and symmetric form equations of a straight line37 8.6.3The included angel between two lines38 8.6.4The included angle between a line and a plane38 8.6.5Some examples39 Exercises 8-6 41 Exercises 8 42
Chapter 9 The multivariable differential calculus and its applications44 9.1Basic concepts of multivariable functions44 9.1.1Planar sets n-dimensional space44 9.1.2The concept of a multivariable function47 9.1.3Limits of multivariable functions49 9.1.4Continuity of multivariable functions51 Exercises 9-1 52 9.2Partial derivatives53 9.2.1Definition and computation of partial derivatives53 9.2.2Higher-order partial derivatives57 Exercises 9-2 59 9.3Total differentials60 9.3.1Definition of total differential60 9.3.2Applications of the total differential to approximate computation63 Exercises 9-3 64 9.4Differentiation of multivariable composite functions65 9.4.1Composition of functions of one variable and multivariable functions65 9.4.2Composition of multivariable functions and multivariable functions66 9.4.3Other case66 Exercises 9-4 70 9.5Differentiation of implicit functions71 9.5.1Case of one equation71 9.5.2Case of system of equations73 Exercises 9-5 75 9.6Applications of differential calculus of multivariable functions in geometry76 9.6.1Derivatives and differentials of vector-valued functions of one variable77 9.6.2Tangent line and normal plane to a space curve80 9.6.3Tangent plane and normal line of surfaces82 Exercises 9-6 85 9.7Directlorial derivatives and gradient85 9.7.1Directlorial derivatives85 9.7.2Gradient88 Exercises 9-7 91 9.8Extreme value problems for multivariable functions92 9.8.1Unrestricted extreme values and global maxima and minima92 9.8.2Extreme values with constraints the method of Lagrange multipliers96 Exercises 9-8 99 9.9Taylor formula for functions of two variables100 9.9.1Taylor formula for functions of two variables100 9.9.2Proof of the sufficient condition for extreme values of function of two variables101 Exercises 9-9 102 Exercises 9 102
Chapter 10 Multiple integrals105 10.1The concept and properties of double integrals105 10.1.1The concept of double integrals105 10.1.2Properties of Double Integrals108 Exercises 10-1 109 10.2Computation of double integrals110 10.2.1Computation of double integrals in rectangular coordinates110 10.2.2Computation of double integrals in polar coordinates115 *10.2.3Integration by substitution for double integrals119 Exercises 10-2 123 10.3Triple integrals126 10.3.1Concept of triple integrals126 10.3.2Computation of triple integrals127 Exercises 10-3 132 10.4Application of multiple integrals134 10.4.1Area of a surface134 10.4.2Center of mass136 10.4.3Moment of inertia138 10.4.4Gravitational force139 Exercises 10-4 140 *10.5Integral with parameter142 *Exercises 10-5 145 Exercises 10 146
Chapter 11 Line and surface integrals148 11.1Line integrals with respect to arc lengths148 11.1.1The concept and properties of the line integral with respect to arc lengths148 11.1.2Computation of line integral with respect to arc lengths149 Exercises 11-1 152 11.2Line integrals with respect to coordinates152 11.2.1The concept and properties of the line integrals with respect to coordinates152 11.2.2Computation of line integrals with respect to coordinates155 11.2.3The relationship between the two types of line integral158 Exercises 11-2 158 11.3Green’s formula and the application to fields159 11.3.1Green’s formula159 11.3.2The conditions for a planar line integral to have independence of path163 11.3.3Quadrature problem of the total differential165 Exercises 11-3 169 11.4Surface integrals with respect to acreage170 11.4.1The concept and properties of the surface integral with respect to acreage170 11.4.2Computation of surface integrals with respect to acreage171 Exercises 11-4 173 11.5Surface integrals with respect to coordinates174 11.5.1The concept and properties of the surface integrals with respect to coordinates174 11.5.2Computation of surface integrals with respect to coordinates177 11.5.3The relationship between the two types of surface integral180 Exercises 11-5 181 11.6Gauss’formula181 11.6.1Gauss’formula181 *11.6.2Flux and divergence184 Exercises 11-6 185 11.7Stokes formula186 11.7.1Stokes formula186 11.7.2Circulation and rotation187 Exercises 11-7 188 Exercises 11 188
Chapter 12 Infinite series191 12.1Concepts and properties of series with constant terms191 12.1.1Concepts of series with constant terms191 12.1.2Properties of convergence with series193 *12.1.3Cauchy’s convergence principle195 Exercises 12-1 196 12.2Convergence tests for series with constant terms197 12.2.1Convergence tests for series of positive terms197 12.2.2Alternating series and Leibniz’s test202 12.2.3Absolute and conditional convergence203 Exercises 12-2 204 12.3Power series205 12.3.1Concepts of series of functions205 12.3.2Power series and convergence of power series206 12.3.3Operations on power series211 Exercises 12-3 212 12.4Expansion of functions in power series213 Exercises 12-4 219 12.5Application of expansion of functions in power series219 12.5.1Approximations by power series219 12.5.2Power series solutions of differential equation221 12.5.3Euler formula222 Exercises 12-5 223 12.6Fourier series223 12.6.1Trigonometric series and orthogonality of the system of trigonometric functions223 12.6.2Expand a function into a Fourier series225 12.6.3Expand a function into the sine series and cosine series229 Exercises 12-6 232 12.7The Fourier series of a function of period 2l 233 Exercises 12-7 235 Exercises 12 235
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