The main subject of this book is calculusBesides，it also includes differential equation，analytic geometry in space，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 vector algebra and analytic geometry in space，multivariable calculus and infinite series.

本书为《高等数学(Ⅱ)(英文版)》，由陈明明、郭振宇、于晶贤、李金秋编著。

《高等数学(Ⅱ)(英文版)》内容如下：

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.

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 vector algebra and analytic geometry in space, multivariable calculus and infinite series.

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 Space

 8.1 Vectors and their linear operations

8.1.1 The concept of vector

8.1.2 Vector linear operations

8.1.3 Three-dimensional rectangular coordinate system

8.1.4 Component representation of vector linear operations

8.1.5 Length，direction angles and projection of a vector

 Exercise 8-1

 8.2 Multiplicative operations on vectors

8.2.1 The scalar product（dot product，inner product）of two vectors

8.2.2 The vector product（cross product，outer product）of two vectors

*8.2.3 The mixed product of three vectors

 Exercise 8-2

 8.3 Surfaces and their equations

8.3.1 Definition of surface equations

8.3.2 Surfaces of revolution

8.3.3 Cylinders

8.3.4 Quadric surfaces

 Exercise 8-3

 8.4 Space curves and their equations

8.4.1 General form of equations of space curves

8.4.2 Parametric equations of space curves

*8.4.3 Parametric equations of a surface

8.4.4 Projections of space curves on coordinate planes

 Exercise 8-4

 8.5 Plane and its equation

8.5.1 Point-normal form of the equation of a plane

8.5.2 General form of the equation of a plane

8.5.3 The included angle between two planes

 Exercise 8-5

 8.6 Straight line in space and its equation

8.6.1 General form of the equations of a straight line

8.6.2 Parametric equations and symmetric form equations of a straight line

8.6.3 The included angel between two lines

8.6.4 The included angle between a line and a plane

8.6.5 Some examples

 Exercise 8-6

Exercise 8

Chapter 9 The multivariable differential calculus and its applications44

 9.1 Basic concepts of multivariable functions

9.1.1 Planar sets n-dimensional space

9.1.2 The concept of a multivariable function

9.1.3 Limits of multivariable functions

9.1.4 Continuity of multivariable functions

 Exercise 9-1

 9.2 Partial derivatives

9.2.1 Definition and computation of partial derivatives

9.2.2 Higher-order partial derivatives

 Exercise 9-2

 9.3 Total differentials

9.3.1 Definition of total differential

9.3.2 Applications of the total differential to approximate computation

 Exercise 9-3

 9.4 Differentiation of multivariable composite functions

9.4.1 Composition of functions of one variable and multivariable functions

9.4.2 Composition of multivariable functions and multivariable functions

9.4.3 Other case

 Exercise 9-4

 9.5 Differentiation of implicit functions

9.5.1 Case of one equation

9.5.2 Case of system of equations

 Exercise 9-5

 9.6 Applications of differential calculus of multivariable functions in geometry

9.6.1 Derivatives and differentials of vector-valued functions of one variable

9.6.2 Tangent line and normal plane to a space curve

9.6.3 Tangent plane and normal line of surfaces

 Exercise 9-6

 9.7 Directional derivatives and gradient

9.7.1 Directional derivatives

9.7.2 Gradient

 Exercise 9-7

 9.8 Extreme value problems for multivariable functions

9.8.1 Unrestricted extreme values and global maxima and minima

9.8.2 Extreme values with constraints the method of Lagrange multipliers

 Exercise 9-8

 9.9 Taylor formula for functions of two variables

9.9.1 Taylor formula for functions of two variables

9.9.2 Proof of the sufficient condition for extreme values of function of two variables

 Exercise 9-9

Exercise 9

Chapter 10 Multiple Integrals

 10.1 The concept and properties of double integrals

10.1.1 The concept of double integrals

10.1.2 Properties of double Integrals

 Exercise 10-1

 10.2 Computation of double integrals

10.2.1 Computation of double integrals in rectangular coordinates

10.2.2 Computation of double integrals in polar coordinates

*10.2.3 Integration by substitution for double integrals

 Exercise 10-2

 10.3 Triple integrals

10.3.1 Concept of triple integrals

10.3.2 Computation of triple integrals

 Exercise 10-3

 10.4 Application of multiple integrals

10.4.1 Area of a surface

10.4.2 Center of mass

10.4.3 Moment of inertia

10.4.4 Gravitational force

 Exercise 10-4

 10.5 Integral with parameter

 Exercise 10-5

Exercise 10

Chapter 1 1Line and Surface Integrals

 11.1 Line integrals with respect to arc lengths

11.1.1 The concept and properties of the line integral with respect to arc lengths

11.1.2 Computation of line integral with respect to arc lengths

 Exercise 11-1

 11.2 Line integrals with respect to coordinates

11.2.1 The concept and properties of the line integrals with respect to coordinates

11.2.2 Computation of line integrals with respect to coordinates

11.2.3 The relationship between the two types of line integral

 Exercise 11-2

 11.3 Green’s formula and the application to fields

11.3.1 Green’s formula

11.3.2 The conditions for a planar line integral to have independence of path

11.3.3 Quadrature problem of the total differential

 Exercise 11-3

 11.4 Surface integrals with respect to acreage

11.4.1 The concept and properties of the surface integral with respect to acreage

11.4.2 Computation of surface integrals with respect to acreage

 Exercise 11-4

 11.5 Surface integrals with respect to coordinates

11.5.1 The concept and properties of the surface integrals with respect to coordinates

11.5.2 Computation of surface integrals with respect to coordinates

11.5.3 The relationship between the two types of surface integral

 Exercise 11-5

 11.6 Gauss’ formula

11.6.1 Gauss’ formula

*11.6.2 Flux and divergence

 Exercise 11-6

 11.7 Stokes formula

11.7.1 Stokes formula

11.7.2 Circulation and rotation

 Exercise 11-7

Exercise 11

Chapter 12 Infinite Series

 12.1 Concepts and properties of series with constant terms

12.1.1 Concepts of series with constant terms

12.1.2 Properties of convergence with series

*12.1.3 Cauchy’s convergence principle

 Exercise 12-1

 12.2 Convergence tests for series with constant terms

12.2.1 Convergence tests for series of positive terms

12.2.2 Alternating series and Leibniz’s test

12.2.3 Absolute and conditional convergence

 Exercise 12-2

 12.3 Power series

12.3.1 Concepts of series of functions

12.3.2 Power series and convergence of power series

12.3.3 Operations on power series

 Exercise 12-3

 12.4 Expansion of functions in power series

 Exercise 12-4

 12.5 Application of expansion of functions in power series

12.5.1 Approximations by power series

12.5.2 Power series solutions of differential equation

12.5.3 Euler formula

 Exercise 12-5

 12.6 Fourier series

12.6.1 Trigonometric series and orthogonality of the system of trigonometric functions

12.6.2 Expand a function into a Fourier series

12.6.3 Expand a function into the sine series and cosine series

 Exercise 12-6

 12.7 The Fourier series of a function of period 21

 Exercise 12-7

Exercise 12

Reference