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.

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 1 Functions and limits

 1.1 Mappings and functions

1.1.1 Sets

1.1.2 Mappings

1.1.3 Functions

Exercise 1-1

 1.2 Limits of sequences

1.2.1 Concept of limits of sequences

1.2.2 Properties of convergent sequences

Exercise 1-2

 1.3 Limits of functions

1.3.1 Definitions of limits of functions

1.3.2 The properties of functional limits

Exercise 1-3

 1.4 Infinitesimal and infinity quantity

1.4.1 Infinitesimal quantity

1.4.2 Infinity quantity

Exercise 1-4

 1.5 Rules of limit operations

Exercises 1-5

 1.6 Principle of limit existence two important limits

Exercise 1-6

 1.7 Comparing with two infinitesimals

Exercise 1-7

 1.8 Continuity of functions and discontinuous points

1.8.1 Continuity of functions

1.8.2 Discontinuous points of functions

Exercise 1-8

 1.9 Operations on continuous functions and the continuity of elementary functions

1.9.1 Continuity of the sum，difference，product and quotient of continuous functions

1.9.2 Continuity of inverse functions and composite functions

1.9.3 Continuity of elementary functions

Exercise 1-9

 1.10 Properties of continuous functions on a closed interval

1.10.1 Boundedness and maximum-minimum theorem

1.10.2 Zero point theorem and intermediate value theorem

*1.10.3 Uniform continuity

Exercise 1-10

 Exercise 1

Chapter 2 Derivatives and differential

 2.1 Concept of derivatives

2.1.1 Examples

2.1.2 Definition of derivatives

2.1.3 Geometric interpretation of derivative

2.1.4 Relationship between derivability and continuity

Exercise 2-1

 2.2 Fundamental Derivation Rules

2.2.1 Derivation rules for sum，difference，product and quotient of functions

2.2.2 The rules of derivative of inverse functions

2.2.3 The rules of derivative of composite functions（The Chain Rule）

2.2.4 Basic derivation rules and derivative formulas

Exercise 2-2

 2.3 Higher-order derivatives

Exercise 2-3

 2.4 Derivation of implicit functions and functions defined by parametric equations

2.4.1 Derivation of implicit functions

2.4.2 Derivation of a function defined by parametric equations

2.4.3 Related rates of change

Exercise 2-4

 2.5 The Differentials of functions

2.5.1 Concept of the differential

2.5.2 Geometric meaning of the differential

2.5.3 Formulas and rules on differentials

2.5.4 Application of the differential in approximate computation

Exercise 2-5

 Exercise 2

Chapter 3 Mean value theorems in differential calculus and applications of derivatives

 3.1 Mean value theorems in differential calculus

Exercise 3-1

 3.2 L’Hospital’s rule

Exercise 3-2

 3.3 Taylor formula

Exercise 3-3

 3.4 Monotonicity of functions and convexity of curves

3.4.1 Monotonicity of functions

3.4.2 Convexity of curves and inflection points

Exercise 3-4

 3.5 Extreme values of functions，maximum and minimum

3.5.1 Extreme values of functions

3.5.2 Maximum and minimum of function

Exercise 3-5

 3.6 Differentiation of arc and curvature

3.6.1 Differentiation of an arc

3.6.2 Curvature

Exercise 3-6

 Exercise 3

Chapter 4 Indefinite integral

 4.1 Concept and property of indefinite integral

4.1.1 Concept of antiderivative and indefinite integral

4.1.2 Table of fundamental indefinite integrals

4.1.3 Properties of the indefinite integral

Exercise 4-1

 4.2 Integration by substitutions

4.2.1 Integration by substitution of the first kind

4.2.2 Integration by substitution of the second kind

Exercise 4-2

 4.3 Integration by parts

Exercise 4-3

 4.4 Integration of rational function

4.4.1 Integration of rational function

4.4.2 Integration which can be transformed into the integration of rational function

Exercise 4-4

 Exercise 4

Chapter 5 Definite integrals

 5.1 Concept and properties of definite integrals

5.1.1 Examples of definite integral problems

5.1.2 The definition of define integral

5.1.3 Properties of definite integrals

Exercise 5-1

 5.2 Fundamental formula of calculus

5.2.1 The relationship between the displacement and the velocity

5.2.2 A function of upper limit of integral

5.2.3 Newton-Leibniz formula

Exercise 5-2

 5.3 Integration by substitution and parts for definite integrals

5.3.1 Integration by substitution for definite integrals

5.3.2 Integration by parts for definite integral

Exercise 5-3

 5.4 Improper integrals

5.4.1 Improper integrals on an infinite interval

5.4.2 Improper integrals of unbounded functions

Exercise 5-4

 5.5 Tests for Convergence of improper integrals Γ-function

5.5.1 Test for convergence of infinite integral

5.5.2 Test for convergence of improper integrals of unbounded functions

5.5.3 Γ-function

Exercise 5-5

 Exercise 5

Chapter 6 Applications of definite integrals

 6.1 Method of elements for definite integrals

 6.2 The applications of the definite integral in geometry

6.2.1 Areas of plane figures

6.2.2 The volumes of solid

6.2.3 Length of plane curves

Exercise 6-2

 6.3 The applications of the definite Integral in physics

6.3.1 Work done by variable force

6.3.2 Force by a liquid

6.3.3 Gravity

Exercise 6-3

 Exercise 6

Chapter 7 Differential equations

 7.1 Differential equations and their solutions

Exercise 7-1

 7.2 Separable equations

Exercise 7-2

 7.3 Homogeneous equations

7.3.1 Homogeneous equations

7.3.2 Reduction to homogeneous equation

Exercise 7-3

 7.4 A first-order linear differential equations

7.4.1 Linear equations

7.4.2 Bernoulli’s equation

Exercise 7-4

 7.5 Reducible second-order equations

7.5.1 y（n）=f（x）

7.5.2 y″=f（x，y′）

7.5.3 y″=f（y，y′）

Exercise 7-5

 7.6 second-order linear equations

7.6.1 Construction of solutions of second-order linear equation

7.6.2 The method of variation of parameters

Exercise 7-6

 7.7 Homogeneous linear differential equation with constant coefficients

Exercise 7-7

 7.8 Nonhomogeneous linear differential equation with constant coefficients

7.8.1 f（x）=eλxPm（x）

7.8.2 f（x）=eλxP（1）l（x）cosωx+P（2）n（x）sinωx

Exercise 7-8

 7.9 Euler’s differential equation

Exercise 7-9

 Exercise 7

Reference