高等数学

Chapter 1 Functions and limits11.1Mappings and functions

1.1.1 Sets

1.1.2 Mappings

1.1.3 Functions

Exercise 1 1141.2Limits of sequences

1.2.1 Concept of limits of sequences

1.2.2 Properties of convergent sequences

Exercise 1 2211.3Limits of functions

1.3.1 Definitions of limits of functions

1.3.2 The properties of functional limits

Exercise 1 3261.4Infinitesimal and infinity quantity

1.4.1 Infinitesimal quantity

1.4.2 Infinity quantity

Exercise 1 4291.5Rules of limit operations

Exercises 1 5341.6Principle of limit existence two important limits

Exercise 1 6391.7Comparing with two infinitesimals

Exercise 1 7421.8Continuity of functions and discontinuous points

1.8.1 Continuity of functions

1.8.2 Discontinuous points of functions

Exercise 1 8461.9Operations 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 9491.10Properties 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

Exercise

Chapter 2 Derivatives and differential552.1Concept 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 1622.2Fundamental 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 2692.3Higher order derivatives

Exercise 2 3732.4Derivation 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 4782.5The 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

Exercise

Chapter 3 Mean value theorems in differential calculus and

applications of derivatives873.1Mean value theorems in differential calculus

Exercise 3 1923.2L'Hospital's rule

Exercise 3 2963.3Taylor formula

Exercise 3 31003.4Monotonicity of functions and convexity of curves

3.4.1 Monotonicity of functions

3.4.2 Convexity of curves and inflection points

Exercise 3 41053.5Extreme values of functions，maximum and minimum

3.5.1 Extreme values of functions

3.5.2 Maximum and minimum of function

Exercise 3 51123.6Differentiation of arc and curvature

3.6.1 Differentiation of an arc

3.6.2 Curvature

Exercise 3

Exercise

Chapter 4Indefinite integral1204.1Concept 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 11254.2Integration by substitutions

4.2.1 Integration by substitution of the first kind

4.2.2 Integration by substitution of the second kind

Exercise 4 21334.3Integration by parts

Exercise 4 31374.4Integration 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

Exercise

Chapter 5 Definite integrals1435.1Concept 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 11485.2Fundamental 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 21545.3Integration 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 31605.4Improper integrals

5.4.1 Improper integrals on an infinite interval

5.4.2 Improper integrals of unbounded functions

Exercise 5 41655.5Tests for Convergence of improper integrals Γ function

5.5.1 Test for convergence of infinite integral

5.5.2Test for convergence of improper integrals of unbounded functions

5.5.3 Γ function

Exercise 5

Exercise

Chapter 6 Applications of definite integrals1736.1Method of elements for definite integrals1736.2The 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 21826.3The 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

Exercise

Chapter 7 Differential equations1897.1Differential equations and their solutions

Exercise 7 11917.2Separable equations

Exercise 7 21947.3Homogeneous equations

7.3.1 Homogeneous equations

7.3.2 Reduction to homogeneous equation

Exercise 7 31987.4A first order linear differential equations

7.4.1 Linear equations

7.4.2 Bernoulli's equation

Exercise 7 42017.5Reducible 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 52067.6second 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 62107.7Homogeneous linear differential equation with

constant coefficients

Exercise 7 72147.8Nonhomogeneous 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 82197.9Euler's differential equation

Exercise 7

Exercise

Reference

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.