內容大鋼
本書是根據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數，多元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.
目錄
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
Exercises 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
Exercises 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
Exercises 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
Exercises 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
Exercises 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
Exercises 8-6
Exercises 8
Chapter 9 The multivariable differential calculus and its applications
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
Exercises 9-1
9.2 Partial derivatives
9.2.1 Definition and computation of partial derivatives
9.2.2 Higher-order partial derivatives
Exercises 9-2
9.3 Total differentials
9.3.1 Definition of total differential
9.3.2 Applications of the total differential to approximate computation
Exercises 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
Exercises 9-4
9.5 Differentiation of implicit functions
9.5.1 Case of one equation
9.5.2 Case of system of equations
Exercises 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
Exercises 9-6
9.7 Directlorial derivatives and gradient
9.7.1 Directlorial derivatives
9.7.2 Gradient
Exercises 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
Exercises 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
Exercises 9-9
Exercises 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
Exercises 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
Exercises 10-2
10.3 Triple integrals
10.3.1 Concept of triple integrals
10.3.2 Computation of triple integrals
Exercises 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
Exercises 10-4
*10.5 Integral with parameter
*Exercises 10-5
Exercises 10
Chapter 11 Line 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
Exercises 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
Exercises 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
Exercises 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
Exercises 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
Exercises 11-5
11.6 Gauss'formula
11.6.1 Gauss'formula
*11.6.2 Flux and divergence
Exercises 11-6
11.7 Stokes formula
11.7.1 Stokes formula
11.7.2 Circulation and rotation
Exercises 11-7
Exercises 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
Exercises 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
Exercises 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
Exercises 12-3
12.4 Expansion of functions in power series
Exercises 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
Exercises 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
Exercises 12-6
12.7 The Fourier series of a function of period 21
Exercises 12-7
Exercises 12
References
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