AdvancedMathematics(Ⅰ)高等数学(上)

本书是根据国家教育部非数学专业数学基础课教学指导分委员会制定的工科类本科数学基础课程教学基本要求编写的全英文教材，全书分为上、下两册，此为上册，主要包括函数与极限，一元函数微积分及其应用和无穷级数三部分。本书对基本概念的叙述清晰准确，对基本理论的论述简明易懂，例题习题的选配典型多样，强调基本运算能力的培养及理论的实际应用。 本书可作为高等理工科院校非数学类专业本科生的教材，也可供其他专业选用和社会读者阅读。

chapter 0 preliminary knowledge

  0.1 polar coordinate system

  　0.1.1 plotting points with polar coordinates

 　 0.1.2 converting between polar and cartesiancoordinates

  0.2 complex numbers

　  0.2.1 the definition of the complex number

  　0.2.2 the complex plane

　  0.2.3 absolute value,conjugation and distance

  　0.2.4 polar form of complex numbers

chapter 1 theoretical basis of calculus

  1.1 sets and functions

　  1.1.1 sets and their operations

　  1.1.2 mappings and functions

  　1.1.3 the primary properties of functions

  　1.1.4 composition of functions

　  1.1.5 elementary functions and hyperbolic functions

  　1.1.6 modeling our real world

　  exercises 1.1

  1.2 limits of sequences of numbers

  　1.2.1 the sequence

　  1.2.2 convergence of a sequence

  　1.2.3 calculating limits of sequences

　  exercises 1.2

  1.3 limits of functions

  　1.3.1 speed and rates of change

　  1.3.2 the concept of limit of a function

  　1.3.3 properties and operation rules of functionallimits

　  1.3.4 two important limits

  　exercises 1.3

  1.4 infinitesimal and infinite quantities

　  1.4.1 infinitesimal quantities and their order

  　1.4.2 infinite quantities

　  exercises 1.4

  1.5 continuous functions

  　1.5.1 continuous function and discontinuous points

　  1.5.2 operations on continuous functions and the continuityof elementary functions

　  1.5.3 properties of continuous functions on a closedinterval

  　exercises 1.5

chapter 2 derivative and differential

  2.1 concept of derivatives

　  2.1.1 introductory examples

  　2.1.2 definition of derivatives

　  2.1.3 geometric interpretation of derivative

  　2.1.4 relationship between derivability andcontinuity

　  exercises 2.1

  2.2 rules of finding derivatives

  　2.2.1 derivation rules of rational operations

　  2.2.2 derivative of inverse functions

  　2.2.3 derivation rules of composite functions

　  2.2.4 derivation formulas of fundamental elementaryfunctions

  　exercises 2.2

  2.3 higher-order derivatives

　  exercises 2.3

  2.4 derivation of implicit functions and parametricequations,related rates

  　2.4.1 derivation of implicit functions

　  2.4.2 derivation of parametric equations

  　2.4.3 related rates

　  exercises 2.4

  2.5 differential of the function

  　2.5.1 concept of the differential

　  2.5.2 geometric meaning of the differential

  　2.5.3 differential rules of elementary functions

　  exercises 2.5

  2.6 differential in linear approximate computation

  　exercises 2.6

chapter 3 the mean value theorem and applications ofderivatives

  3.1 the mean value theorem

　  3.1.1 rolle's theorem

　  3.1.2 lagrange's theorem

  　3.1.3 cauchy s theorem

　  exercises 3.1

  3.2 l'hospital's rule

　  exercises 3.2

  3.3 taylor's theorem

  　3.3.1 taylor's theorem

　  3.3.2 applications of taylor's theorem

  　exercises 3.3

  3.4 monotonicity and convexity of functions

　  3.4.1 monotonicity of functions

  　3.4.2 convexity of functions,inflections

　  exercises 3.4

  3.5 local extreme values,global maxima and minima

  　3.5.1 local extreme values

　  3.5.2 global maxima and minima

　  exercises 3.5

  3.6 graphing functions using calculus

　  exercises 3.6

chapter 4 indefinite integrals

  4.1 concepts and properties of indefinite integrals

　  4.1.1 antiderivatives and indefinite integrals

　  4.1.2 properties of indefinite integrals

　  exercises 4.1

  4.2 integration by substitution

  　4.2.1 integration by the first substitution

　  4.2.2 integration by the second substitution

  　exercises 4.2

  4.3 integration by parts

　  exercises 4.3

  4.4 integration of rational fractions

　  4.4.1 integration of rational fractions

　  4.4.2 antiderivatives not expressed by elementaryfunctions

  　exercises 4.4

chapter 5 definite integrals

  5.1 concepts and properties of definite integrals

　  5.1.1 instances of definite integral problems

　  5.1.2 the definition of definite integral

  　5.1.3 properties of definite integrals

　  exercises 5.1

  5.2 the fundamental theorems of calculus

  　exercises 5.2

  5.3 integration by substitution and by parts in definiteintegrals

　  5.3.1 substitution in definite integrals

　  5.3.2 integration by parts in definite integrals

  　exercises 5.3

  5.4 improper integral

　  5.4.1 integration on an infinite interval

  　5.4.2 improper integrals with infinitediscontinuities

　  exercises 5.4

  5.5 applications of definite integrals

  　5.5.1 method of setting up elements of integration

　  5.5.2 the area of a plane region

　  5.5.3 the arc length of a curve

　  5.5.4 the volume of a solid

　  5.5.5 applications of definite integral in physics

　  exercises 5.5

chapter 6 infinite series

  6.1 concepts and properties of series with constantterms

　  6.1.1 examples of the sum of an infinite sequence)

  　6.1.2 concepts of series with constant terms

　  6.1.3 properties of series with constant terms

  　exercises 6.1

  6.2 convergence tests for series with constant terms

　  6.2.1 convergence tests of series with positive terms

  　6.2.2 convergence tests for alternating series

　  6.2.3 absolute and conditional convergence

  　exercises 6.2

  6.3 power series

　  6.3.1 functional series

  　6.3.2 power series and their convergence

　  6.3.3 operations of power series

  　exercises 6.3

  6.4 expansion of functions in power series

　  6.4.1 taylor and maclaurin series

  　6.4.2 expansion of functions in power series

　  6.4.3 applications of power series expansion offunctions

  　exercises 6.4

  6.5 fourier series

　  6.5.1 orthogonality of the system of trigonometricfunctions

  　6.5.2 fourier series

　  6.5.3 convergence of fourier series

  　6.5.4 sine and cosine series

　  exercises 6.5

  6.6 fourier series of other forms

　  6.6.1 fourier expansions of periodic functions with period2l

  　6.6.2 complex form of fourier series

　  exercises 6.6

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