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  • 高等數學引論 第一卷
  • 點閱:1
  • 作者: 華羅庚著 , 蕭文杰譯
  • 出版社:高等教育出版社
  • 出版年:2016[民105]
  • ISBN:9787040314144
  • 格式:JPG
  • 附註:簡體字版
租期14天 今日租書可閱讀至2022-02-03

《高等數學引論(第一卷)(英文版)》是中國著名數學家華羅庚的一本著作,寫這本書的原因是華羅庚要為中國科學技術大學應用數學系的學生授課,在他的弟子王元協助下所寫成的一份講義。全書反映了作者的數學是一門有緊密內在聯系的學問,應將大學數學系的基礎課放在一起來講的教學思想,還包括了作者的要埋有伏筆、生書熟講,熟書生溫等教學技巧,書中還介紹了數學理論的不少應用。這使得本套書不同于許多現行的教科書,是一套有特色、高水平的高等數學教材。

  • Preface(第xv頁)
  • Translator’s note(第xviii頁)
  • Hua Loo-Keng and An Introduction toHigher Mathematics(第xix頁)
  • 1 Real and complex numbers(第1頁)
    • 1.1 Rational numbers(第1頁)
    • 1.2 The existence of irrational numbers(第3頁)
    • 1.3 A description of real numbers(第4頁)
    • 1.4 Limit(第7頁)
    • 1.5 The Bolzano–Weierstrass theorem(第12頁)
    • 1.6 Definitions for complex numbers and vectors(第15頁)
    • 1.7 Polar coordinates and multiplication(第19頁)
    • 1.8 De Moivre’s theorem(第22頁)
    • 1.9 Completeness of the complex numbers(第24頁)
    • 1.10 Introduction to quaternions(第27頁)
    • 1.11 Binary arithmetics(第28頁)
    • 1.12 Periodic decimals(第32頁)
    • 1.13 Rational approximations to real numbers(第33頁)
    • 1.14 Error terms(第39頁)
    • 1.15 Solutions to cubic and quartic equations(第43頁)
  • 2 Vector algebra(第50頁)
    • 2.1 Space coordinates and vectors(第50頁)
    • 2.2 Addition of vectors(第52頁)
    • 2.3 The decomposition of a vector(第54頁)
    • 2.4 Inner product(scalar product)(第55頁)
    • 2.5 Vector product(outer product)(第56頁)
    • 2.6 Multiple products(第58頁)
    • 2.7 Change of coordinates(第60頁)
    • 2.8 Planes(第63頁)
    • 2.9 Equation for a line in space(第66頁)
    • 2.10 Main formulae in spherical trigonometry(第67頁)
    • 2.11 Duality principle(第70頁)
    • 2.12 Right-angled and right-sided triangles(第71頁)
    • 2.13 Forces, systems and equivalent systems(第75頁)
    • 2.14 Combination of parallel forces(第76頁)
    • 2.15 Moments(第77頁)
    • 2.16 Couples(第78頁)
    • 2.17 Standard form for a system(第81頁)
    • 2.18 Equilibrium and its applications(第83頁)
  • 3 Functions and graphs(第88頁)
    • 3.1 Variables(第88頁)
    • 3.2 Functions(第89頁)
    • 3.3 Implicit functions(第90頁)
    • 3.4 Functions represented by graphs and tables(第91頁)
    • 3.5 Several elementary functions(第93頁)
    • 3.6 Functions with simple special properties(第96頁)
    • 3.7 Periodic functions(第98頁)
    • 3.8 Representations for a complex function(第100頁)
    • 3.9 Line of regression(第100頁)
    • 3.10 Lagrange’s interpolation formula(第105頁)
    • 3.11 Other interpolation formulae(第108頁)
    • 3.12 Experimental formulae(第110頁)
    • 3.13 Family of curves(第117頁)
  • 4 Limits(第119頁)
    • 4.1 Limits of sequences(第119頁)
    • 4.2 Sequences without limits(第121頁)
    • 4.3 Series(第123頁)
    • 4.4 Conditionally convergent series(第129頁)
    • 4.5 The method of Zu Chongzhi in calculating π(第132頁)
    • 4.6 Archimedes’ method for the area of a parabolic region(第135頁)
    • 4.7 Calculating pressure on a boundary(第137頁)
    • 4.8 The number e(第137頁)
    • 4.9 Taking limit in the continuum(第140頁)
    • 4.10 On several important limits(第142頁)
    • 4.11 Some examples(第144頁)
    • 4.12 Orders of infinity(第146頁)
    • 4.13 The symbols ∼, O and o(第147頁)
    • 4.14 Continuous functions(第150頁)
    • 4.15 Types of discontinuities(第153頁)
    • 4.16 Some fundamental properties of continuous functions(第154頁)
    • 4.17 The Heine–Borel theorem(第156頁)
  • 5 The differential calculus(第158頁)
    • 5.1 The notion of the derivative(第158頁)
    • 5.2 Geometric interpretation of the derivative(第160頁)
    • 5.3 Derivatives of sums and products(第162頁)
    • 5.4 Derivatives of elementary functions(第163頁)
    • 5.5 Derivatives of composite functions(第165頁)
    • 5.6 The hyperbolic functions(第168頁)
    • 5.7 Formulae for differentiation(第170頁)
    • 5.8 Examples(第171頁)
    • 5.9 Differentials(第176頁)
    • 5.10 Error estimates(第178頁)
    • 5.11 Higher derivatives(第182頁)
    • 5.12 Leibniz’s formula(第186頁)
    • 5.13 Higher differentials(第188頁)
    • 5.14 Differences in functions(第192頁)
  • 6 Applications of the derivative(第194頁)
    • 6.1 Ups and downs along a curve(第194頁)
    • 6.2 Maxima and minima(第196頁)
    • 6.3 Fermat’s theorem(第203頁)
    • 6.4 Mean-value formula(第205頁)
    • 6.5 Convexity and points of inflection(第209頁)
    • 6.6 Asymptotes(第214頁)
    • 6.7 Essential points in curve sketching(第218頁)
    • 6.8 Sketching parametric curves(第225頁)
    • 6.9 Tangents and normals(第227頁)
    • 6.10 Integration formulae(第230頁)
    • 6.11 Implicit differentiation(第234頁)
    • 6.12 The indeterminate form 0/0(第236頁)
    • 6.13 The indeterminate form∞/∞(第239頁)
    • 6.14 Other indeterminate forms(第242頁)
  • 7 Taylor expansions(第245頁)
    • 7.1 Taylor’s formula for a polynomial(第245頁)
    • 7.2 Taylor expansions for functions(第246頁)
    • 7.3 Taylor series and remainder terms(第247頁)
    • 7.4 The expansion for ex(第250頁)
    • 7.5 Expansions for sin x and cos x(第251頁)
    • 7.6 The binomial expansion(第255頁)
    • 7.7 The expansion for log(1 + x)(第257頁)
    • 7.8 The expansion for arctan x(第259頁)
    • 7.9 Power series and radius of convergence(第261頁)
    • 7.10 Arithmetic operations on power series(第263頁)
    • 7.11 Differentiation and integration of power series(第265頁)
    • 7.12 Uniqueness theorem and inverse functions(第267頁)
    • 7.13 Kummer’s test and Gauss’ test(第269頁)
    • 7.14 Hypergeometric series(第271頁)
    • 7.15 Power series solutions of differential equations(第277頁)
  • 8 Approximate solutions to equations(第283頁)
    • 8.1 Introduction(第283頁)
    • 8.2 Graphical methods(第284頁)
    • 8.3 Method of successive substitutions(第285頁)
    • 8.4 Interpolation method(第289頁)
    • 8.5 Newton’s method(第291頁)
    • 8.6 A combination of methods(第294頁)
    • 8.7 Digital refinement method(第296頁)
    • 8.8 Lobachevskiy method(第298頁)
    • 8.9 Theorems on real roots(第300頁)
    • 8.10 Sturm’s theorem(第303頁)
  • 9 Indefinite integrals(第306頁)
    • 9.1 Change of variables(第306頁)
    • 9.2 Integration by parts(第308頁)
    • 9.3 Partial fractions(第312頁)
    • 9.4 Integration of rational functions(第313頁)
    • 9.5 Ostrogradskiy method(第316頁)
    • 9.6 Integration of certain functions with roots(第318頁)
    • 9.7 Integration of(第322頁)
    • 9.8 Abelian integrals(第324頁)
    • 9.9 Integrals not representable by ‘known’ functions(第327頁)
    • 9.10 Differential equations, variables separable(第328頁)
    • 9.11 Homogeneous differential equations(第330頁)
    • 9.12 Integrating factor method(第332頁)
    • 9.13 First order linear equations(第337頁)
    • 9.14 Second order linear equations(第341頁)
    • 9.15 Linear equations with constant coefficients(第344頁)
  • 10 Definite integrals(第347頁)
    • 10.1 Area determination(第347頁)
    • 10.2 The notion of a definite integral(第350頁)
    • 10.3 Properties of integrable functions(第354頁)
    • 10.4 Fundamental properties of definite integrals(第355頁)
    • 10.5 Mean-value theorem and the fundamental theorem of calculus(第359頁)
    • 10.6 The second mean-value theorem(第362頁)
    • 10.7 Examples(第364頁)
    • 10.8 Integration by substitution(第367頁)
    • 10.9 Integration by parts(第371頁)
    • 10.10 Improper integrals(第374頁)
    • 10.11 Applications of the definite integral(第377頁)
    • 10.12 Integration by special techniques(第378頁)
    • 10.13 Applications of area consideration(第384頁)
    • 10.14 Euler’s summation formula(第388頁)
    • 10.15 Trapezium, rectangle and Simpson’s rules(第391頁)
  • 11 Applications of integration(第401頁)
    • 11.1 The length of a curve(第401頁)
    • 11.2 Area(第406頁)
    • 11.3 Volumes from cross sections(第409頁)
    • 11.4 Surface area of revolution(第412頁)
    • 11.5 Area of a vertical screen(第415頁)
    • 11.6 Centroid(第417頁)
    • 11.7 Moment of inertia(第421頁)
    • 11.8 Fluid pressure(第422頁)
    • 11.9 Work(第424頁)
  • 12 Functions of several variables(第427頁)
    • 12.1 Variables(第427頁)
    • 12.2 The n dimensional space(第428頁)
    • 12.3 Neighbourhoods(第430頁)
    • 12.4 Regions(第431頁)
    • 12.5 Limit and continuity(第433頁)
    • 12.6 Functions continuous in a region(第436頁)
    • 12.7 Partial derivatives and total differentials(第438頁)
    • 12.8 Homogeneous functions(第441頁)
    • 12.9 Tangent plane(第442頁)
    • 12.10 Directional derivatives(第444頁)
    • 12.11 Higher partial derivatives(第445頁)
    • 12.12 Implicit functions(第449頁)
    • 12.13 Taylor expansions(第451頁)
    • 12.14 Maxima and minima(第452頁)
    • 12.15 Maxima and minima for implicit functions(第458頁)
    • 12.16 Change of coordinate systems(第460頁)
    • 12.17 Coordinate systems in space(第463頁)
  • 13 Sequences, series and integrals of functions(第467頁)
    • 13.1 Uniform convergence of sequences(第467頁)
    • 13.2 Sequences of derivatives and integrals(第470頁)
    • 13.3 Bounded convergence(第472頁)
    • 13.4 Uniform convergence of series(第476頁)
    • 13.5 Certain criteria for uniform convergence(第480頁)
    • 13.6 Abel and Dirichlet tests for uniform convergence(第481頁)
    • 13.7 Abel’s theorem and Tauber’s theorem(第483頁)
    • 13.8 An iterative method for implicit functions(第485頁)
    • 13.9 Infinite products(第487頁)
    • 13.10 Conditions for the convergence of an infinite product(第489頁)
    • 13.11 The logarithm of an infinite product(第491頁)
    • 13.12 Uniform convergence of products(第494頁)
    • 13.13 Integrals with parameters(第497頁)
    • 13.14 Differentiation under the integral sign(第502頁)
    • 13.15 Repeated integrals(第504頁)
    • 13.16 Integrals with parametric limits(第511頁)
    • 13.17 Double sequences(第512頁)
    • 13.18 Double series(第513頁)
    • 13.19 Products of series(第521頁)
    • 13.20 Power series with several variables(第524頁)
    • 13.21 Series solutions of implicit functions(第525頁)
    • 13.22 Existence and uniqueness of solutions to ordinary differential equations(第529頁)
    • 13.23 Existence and uniqueness of solutions to integral equations(第532頁)
    • 13.24 Existence and uniqueness of solutions to systems of differential equations(第534頁)
    • 13.25 The fundamental principle of a contraction mapping(第537頁)
    • 13.26 Use of power series for solutions to differential equations(第538頁)
    • 13.27 Systems of differential equations(第540頁)
    • 13.28 Partial differential equations(第541頁)
  • 14 Properties of differentials of curves(第545頁)
    • 14.1 Vector derivatives(第545頁)
    • 14.2 Movement along a plane(第548頁)
    • 14.3 Curvature of a plane curve(第549頁)
    • 14.4 The intrinsic equation of a curve(第551頁)
    • 14.5 Circles of curvature and evolutes(第554頁)
    • 14.6 The general first order differential equation(第557頁)
    • 14.7 Envelope curves(第560頁)
    • 14.8 Pursuit problems(第562頁)
    • 14.9 Fundamentals for curves in space(第566頁)
    • 14.10 Representations in rectangular coordinates(第568頁)
    • 14.11 Spirals(第571頁)
    • 14.12 Uniqueness theorem for curves in space(第573頁)
    • 14.13 Circles and spheres of curvature(第576頁)
    • 14.14 Families of surfaces and curves and their envelopes(第577頁)
  • 15 Multiple integrals(第581頁)
    • 15.1 Definition for multiple integrals(第581頁)
    • 15.2 Regions with rectifiable areas(第585頁)
    • 15.3 Change of coordinates in double integrals(第587頁)
    • 15.4 Fundamental properties of multiple integrals(第592頁)
    • 15.5 Triple integrals(第594頁)
    • 15.6 Moments(第599頁)
    • 15.7 Area of curved surfaces(第602頁)
    • 15.8 Attraction of particles toward a point(第606頁)
    • 15.9 Direct area estimation(第610頁)
    • 15.10 Direct volume estimation(第613頁)
    • 15.11 Determination of area representations(第622頁)
  • 16 Line integrals and surface integrals(第631頁)
    • 16.1 Definition for a line integral(first type)(第631頁)
    • 16.2 Definition for a line integral(second type)(第634頁)
    • 16.3 Area determination from line integrals(第639頁)
    • 16.4 Green’s formula and Ostrogradsky’s formula(第642頁)
    • 16.5 Stokes’ formula(第644頁)
    • 16.6 Path-independent line integrals(第648頁)
    • 16.7 Multi-connected regions(第651頁)
    • 16.8 Path-independent integrals in space(第654頁)
    • 16.9 Fluid under steady flow(第655頁)
  • 17 Potential fields and vector fields(第658頁)
    • 17.1 Definitions(第658頁)
    • 17.2 Properties of the div, grad and curl operations(第660頁)
    • 17.3 Combined operations of div, grad and curl(第661頁)
    • 17.4 The geometric interpretation of grad(第662頁)
    • 17.5 Formulae of Ostrogradsky–Gauss and Stokes(第665頁)
    • 17.6 The nabla operator(第668頁)
    • 17.7 Curvilinear coordinates and substitutions(第670頁)
    • 17.8 Plane fields(第675頁)
    • 17.9 Applications to fluid mechanics(第681頁)
    • 17.10 Sound propagation(第688頁)
    • 17.11 Heat transfer(第689頁)
  • 18 Properties of differentials of surfaces(第692頁)
    • 18.1 Algebraic tools(第692頁)
    • 18.2 The first differential form of Gauss(第694頁)
    • 18.3 The second differential form of Gauss(第698頁)
    • 18.4 Curvatures of curves on curved surfaces(第700頁)
    • 18.5 Classification of points into types(第702頁)
    • 18.6 Lines of curvature(第703頁)
    • 18.7 Euler’s formula(第706頁)
    • 18.8 Olinde Rodrigues’ formula(第707頁)
    • 18.9 Dupin’s theorem(第709頁)
    • 18.10 Geometric interpretation of the Gaussian curvature(第710頁)
    • 18.11 Geometric interpretation of the mean curvature(第712頁)
    • 18.12 Active coordinates(第713頁)
    • 18.13 Developable surfaces(第716頁)
    • 18.14 Families of surfaces and partial differential equations(第717頁)
    • 18.15 The first fundamental form(第721頁)
    • 18.16 Tensors(第723頁)
    • 18.17 A fundamental equation – the Gauss equation(第726頁)
    • 18.18 A fundamental equation – the Weingarten equation(第729頁)
    • 18.19 The equations of Gauss and Codazzi(第730頁)
    • 18.20 Curvature tensors(第731頁)
  • 19 Fourier series(第734頁)
    • 19.1 The orthogonality of trigonometric functions(第734頁)
    • 19.2 Several trigonometric sums(第736頁)
    • 19.3 The Dirichlet integral(第738頁)
    • 19.4 Mean-square errors and Bessel’s inequality(第739頁)
    • 19.5 A criterion for convergence(第742頁)
    • 19.6 Expansions in the interval(0, π)(第746頁)
    • 19.7 The Gibbs phenomenon(第750頁)
    • 19.8 Summability(第752頁)
    • 19.9 Parseval’s identity(第755頁)
    • 19.10 Integrating Fourier series term-by-term(第756頁)
    • 19.11 Properties of Fourier coefficients(第758頁)
    • 19.12 Other forms of Fourier series(第760頁)
    • 19.13 Practical harmonic analysis – finite harmonic analysis(第761頁)
    • 19.14 Fourier integrals(第767頁)
    • 19.15 The Fourier transform(第768頁)
    • 19.16 Poisson summation formula(第770頁)
    • 19.17 The complex Fourier transform(第772頁)
    • 19.18 Some other transforms(第773頁)
  • 20 Systems of ordinary differential equations(第776頁)
    • 20.1 Reduction of any system of differential equations to a first order system(第776頁)
    • 20.2 Systems of ordinary differential equations(第777頁)
    • 20.3 Equation of motion for a particle(第781頁)
    • 20.4 Orbital equation for an artificial satellite(第784頁)
    • 20.5 Orbits – the first and second escape velocities(第788頁)
    • 20.6 The third escape velocity(第791頁)
    • 20.7 Systems of particles – the n-body problem(第793頁)
    • 20.8 Lagrange’s linear equation(第796頁)
    • 20.9 A general solution to a linear equation(第802頁)
    • 20.10 A method for solutions to first order partial differentialequations – Charpit’s method(第803頁)
    • 20.11 Special examples for the method in Section 20.10(第805頁)
  • Index(第809頁)
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