The aim of this book is to meet the requirements of teaching Calculus in English or inbilingual education according to the customs of teaching and the present domesticconditions. It is divided into two volumes. The first volume contains calculus of singlevariable, simple differential equations and infinite series; the second volume containsvector algebra, analytic geometry in space, multivariable calculus and Linear ordinarydifferential equations.The selection of the contents is in accordance with the fundamental requirements of teachingissued by the Ministry of Education of China, and is based on the accomplishments of reformin teaching during the past ten years. The arrangement and explanation of the main contentsin this book are approximately the same as the published Chinese version with the same titleand edited in chief by the first two authors. It may help readers to understand themathematics and to improve the level of their English by reading one of them and using theother one as a reference.This book may be used as a textbook for undergraduate students in the science andengineering schools whose majors are not mathematics, and may also be suitable to thereaders at the same level.

Chapter 5 Vector Algebra and Analytic Geometry in Space
  5.1 Vectors and Their Linear Operations
   5.1.1 The concept of vector
   5.1.2 Linear operations on vectors
   5.1.3 Projection of vectors
   5.1.4 Rectangular coordinate systems in space and components of vectors
  Exercises 5.1
  5.2 M ultiplicative Operations on Vectors
   5.2.1 The scalar product(dot product,inner product) of two vectors
   5.2.2 The vector product (cross product,outer product) of two vectors
   5.2.3 The mixed product of three vectors
  Exercises 5.2
  5.3 Planes and Lines in Space
   5.3.1 Equations of planes
   5.3.2 Position relationships between two planes
   5.3.3 Equations of straight lines in space
   5.3.4 Position relationships between two lines
   5.3.5 Position relationships between a line and a plane
   5.3.6 Distance from a point to a plane (line)
  Exercises 5.3
  5.4 Surfaces and Space Curves
   5.4.1 Equations of surfaces
   5.4.2 Quadric surfaces
   5.4.3 Equations of space curves
  Exercises 5.4
Chapter 6 The M ultivariable Differential Calculus and its Applications
  6.1 Limits and Continuity of M ultivariable Functions
   6.1.1 Primary knowledge of point sets in the space Rn
   6.1.2 The concept of a multivariable function
   6.1.3 Limit and continuity of multivariable functions
  Exercises 6.1
  6.2 Partial Derivatives and Total Differentials of M ultivariable Functions
   6.2.1 Partial derivatives
   6.2.2 Total differentials
   6.2.3 Higher-order partial derivatives
   6.2.4 Directional derivatives and the gradient
  Exercises 6.2
  6.3 Differentiation of M ultivariable Composite Functions and Implicit Functions
   6.3.1 Partial derivatives and total differentials of multivariable composite functions
   6.3.2 Differentiation of implicit functions defined by one equation
   6.3.3 Differentiation of implicit functions determined by more than one equation
  Exercises 6.3
  6.4 Extreme Value Problems for M ultivariable Functions
   6.4.1 Unrestricted extreme values
   6.4.2 Global maxima and minima
   6.4.3 Extreme values with constraints; The method of Lagrange multipliers
  Exercises 6.4
  *6.5 Taylor's Formula for Functions of Two Variables
   6.5.1 Taylor's formula for functions of two variables
   6.5.2 Proof of the sufficient condition for extreme values of function of two variables
  Exercises 6.5
  6.6 Derivatives and Differentials of Vector-valued Functions
   6.6.1 Derivatives and differentials of vector-valued functions of one variable
   6.6.2 Derivatives and differentials of vector-valued functions of two variables
   6.6.3 Rules for differential operations
  Exercises 6.6
  6.7 Applications of Differential Calculus of M ultivariable Functions in Geometry
   6.7.1 Tangent line and normal plane to a space curve
   6.7.2 Arc length
   6.7.3 Tangent planes and normal lines of surfaces
   6.7.4 Curvature
   *6.7.5 The Frenet frame
   *6.7.6 Torsion
  Exercises 6.7
  Synthetic exercises
Chapter 7 The Integral Calculus of M ultivariable Scalar Functions and Its Applications
  7.1 The Concept and Properties of the Integral of a M ultivariable Scalar Function
   7.1.1 Computation of mass of an object
   7.1.2 The concept of the integral of a multivariable scalar function
   7.1.3 Properties of integrals of multivariable scalar functions
  Exercises 7.1
  7.2 Computation of Double Integrals
   7.2.1 Geometric meaning of the double integral
   7.2.2 Computation methods for double integrals in rectangular coordinates
   7.2.3 Computation of double integrals in polar coordinates
   *7.2.4 Integration by substitution for double integrals in general
  Exercises 7.2
  7.3 Computation of Triple Integrals
   7.3.1 Reduction of a triple integral to an iterated integral consisting of a single integral and a double integral
   7.3.2 Computation of triple integrals in cylindrical and spherical coordinates
   *7.3.3 Computation of triple integrals by general substitutions
  Exercises 7.3
  7.4 Applications of M ultiple Integrals
   7.4.1 The method of elements for multiple integrals
   7.4.2 Examples of applications
  Exercises 7.4
  7.5 Line and Surface Integrals of the First Type
   7.5.1 Line integrals of the first type
   7.5.2 Surface integrals of the first type
  Exercises 7.5
  Synthetic exercises
Chapter 8 The Integral Calculus of M ultivariable Vectorvalued Functions and its Applications in the Theory of Fields
  8.1 Line and Surface Integrals of the Second Type
   8.1.1 The concept of field
   8.1.2 Line integrals of the second type
   8.1.3 Surface integrals of the second type
  Exercises 8.1
  8.2 The Relations Between Different Kinds of Integrals and their Applications to Fields
   8.2.1 Green's formula
   8.2.2 The conditions for a planar line integral to have independence of path
   8.2.3 Stokes'ormula and the curl of a vector
   8.2.4 Gauss'ormula and divergence
   8.2.5 Some important particular vector fields
  Exercises 8.2
Chapter 9 Linear Ordinary Differential Equations
  9.1 Linear Differential Equations of Higher Order
   9.1.1 Some examples of linear differential equation of higher order
   9.1.2 Structure of solutions of linear differential equations
   9.1.3 Solution of higher-order homogeneous linear differential equations with constant coefficients
   9.1.4 Solution of higher-order nonhomogeneous linear differential equations with constant coefficients
   9.1.5 Solution of higher-order linear differential equations with variable coefficients
  Exercises 9.1
  *9.2 Linear Systems of Differential Equations
   9.2.1 Basic concepts of linear systems of differential equations
   9.2.2 The structure of solutions of a linear system of equations
   9.2.3 Solution of a homogeneous system of linear equations with constant coefficients
   9.2.4 Solution of nonhomogeneous systems of linear equations with constant coefficients
   9.2.5 Some applications of systems of linear equations
  *Exercises 9.2
  Synthetic exercise
Appendix A Basic Properties of M atrices and Determinants
  A.1 M atrices
   A.1.1 Elementary concepts of matrices
   A.1.2 Operations on matrices
   A.1.3 Representation of the product of two matrices by rows(columns)
  Exercises A.1
  A.2 Determinants and Cramer's Rule
   A.2.1 Definition and properties of determinants
   A.2.2 Cramer's rule
  Exercises A.2
Appendix B Answers and Hints for Exercises