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《高等數學(1)(英文版)》內容簡介：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 volumesThe first volume contains calculus of functions of a single variable and differential equationThe 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.
This book is divided into two volumesThe first volume contains calculus of functions of a single variable and differential equationThe 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.
目次
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 1923.2LHospitals 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 Bernoullis 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 82197.9Eulers differential equation
Exercise 7
Exercise
Reference
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 1923.2LHospitals 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 Bernoullis 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 82197.9Eulers differential equation
Exercise 7
Exercise
Reference
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