本書是根據*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編寫的全英文教材，全書分為上、下兩冊。本書為下冊，主要包括空間解析幾何和向量代函數微積分及其應用，曲線積分與曲面積分和微分方程。本書對基本概念的敘述清晰準確，對基本理論的論述簡明易懂，例題習題的選配典型多樣，強調基本運算能力的培養及理論的實際應用。



本書可作為高等理工科院校非數學類專業本科生的教材，也可供其他專業選用和社會讀者閱讀。







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







References237

English is the most important language in international
academia.In order to strengthen academic exchange with western countries, many
universities in China pay more and more attention to the bilingual teaching in
classrooms in recent years.Considering the importance of advanced mathematics
and scarcity of bilingual mathematics textbook, we have written this book.



The main subject of this book is calculus.Besides, it also includes
differential equation, space analytic geometry, vector algebra and infinite series.This
book is divided into two volumes.The first volume contains calculus of
functions of a single variable and differential equation.The second volume
contains space analytic geometry and vector algebra,calculus of multivariate
function,curve integral and surface integral,infinite series.



We have attempted to give this book the following characteristics:



① The content of this book
is based on the Chinese textbook “advanced mathematics
(seventh edition)” which is written by department of
mathematics of Tongji University.The readers may read this book and use the
Chinese textbook “advanced mathematics” as a reference.It may help readers to understand the mathematical
contents and to improve the level of their English.



② In order to train the mathematical idea and ability
of the students, we use some modern idea, language and methods of
mathematics.We also bring in some mathematical symbol and logical symbol.



③ We pay more attention to the application of
mathematics in practical problems.We have added some other examples and
exercises in physics, chemistry, economics and even daily life.



④ Considering the different teaching requirements in
different schools, we mark some difficult sections and exercises by the symbol “*”.Teachers and students may choose suit
able contents as required.



In this volume, Chapter 8 is written by Xiaoming Guo, Chapter 9 and Chapter 12
are written by Bin Pan，Chapter 10 is written by
Jingxian Yu, Chapter 11 is written by Xiaoying Zhao.All the chapters are
checked and revised by Bin Pan.



We hope this book can bring readers some help in the studying and teaching of
bilingual mathematics.Due to the limit of our ability, it is impossible to
avoid some errors and unclear explanations.We would appreciate any constructive
criticisms and corrections from readers.







Authors



2019-5