普特南大学生数学竞赛(第2版)(英文版)/世界数学奥林匹克经典
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作者(美)拉兹万·吉尔卡,(美)蒂图·安德烈埃斯库
出版社世界图书出版有限公司北京分公司
ISBN9787519297794
出版时间2022-10
装帧平装
开本16开
定价168元
货号31585310
上书时间2024-05-26
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目录
1 Methods of Proof
1.1 Argumentby Contladiction
1.2 Mathematical Induction
1.3 ThePigeonholePrirciple
1.4 Ordered Sets and Extremal Elements
1.5 Invariants and Semiqnvariants
2 Algebra
2.1 Identities and Inequalities
2.1.1 Algebraic Identities
2.1.2 x2≥0
2.1.3 The Cauchy-Schwarz Inequality
2.1.4 The Triangle Inequality
2.1.5 The Arithmetic Mean-Geometric Mean Inequality
2.1.6 Sturm's Principle
2.1.7 Other Inequalities
2.2 Polynomials
2.2.1 A Warmup in One-Variable Polynomials
2.2.2 Polynomials in Several Variables
2.2.3 Quadratic Polynomials
2.2.4 Vi&e's Relations
2.2.5 The Derivative czf a Polynomial
2.2.6 The Location of the Zeros czf a Polynomial
2.2.7 Irreducible Polynomials
2.2.8 Chebyshev Polynomials
2.3 Linear Algebra
2.3.1 Operations with Matrices
2.3.2 Determinants
2.3.3 The Inverse of a Matrix
2.3.4 Systems of Linear Equations
2.3.5 Vector Spaces, Linear Combinations of Vectors, Bases
2.3.6 Linear Transformations, Eigenvalues, Eigenvectors
2.3.7 The Cayley-Hamilton and Perron-Frobenius Theorems
2.4 Abstract Algebra
2.4.1 Binary Operations
2.4.2 Groups
2.4.3 Rings
3 Real Analysis
3.1 Sequences and Series
3.1.1 Search for a Pattern
3.1.2 Linear Recursive Sequences
3.1.3 Limits of Sequences
3.1.4 More About Limits of Sequences
3.1.5 Series
3.1.6 Telescopic Series and Products
3.2 Continuity, Derivatives, and Integrals
3.2.1 Functions
3.2.2 Limits of Functions
3.2.3 Continuous Functions
3.2.4 The Intermediate Value Property
3.2.5 Derivatives and Their Applications
3.2.6 The Mean Value Theorem
3.2.7 Convex Functions
3.2.8 Indefinite Integrals
3.2.9 Definite Integrals
3.2.10 Riemann Sums
3.2.11 Inequalities for Integrals
3.2.12 Taylor and Fourier Series
3.3 Multivariable Differential and Integral Calculus
3.3.1 Partial Derivatives and Their Applications
3.3.2 Multivariable Integrals
3.3.3 The Many Versions of Stokes' Theorem
3.4 Equations with Functions as Unknowns
3.4.1 Functional Equations
3.4.2 Ordinary Differential Equations of the First Order
3.4.3 Ordinary Differential Equations of Higher Order
3.4.4 Problems Solved with Techniques of Differential Equations
4 Geometry and Trigonometry
4.1 Geometry
4.1.1 Vectors
4.1.2 The Coordinate Geometry of Lines and Circles
4.1.3 Quadratic and Cubic Curves in the Plane
4.1.4 Some Famous Curves in the Plane
4.1.5 Coordinate Geometry in Three and More Dimensions
4.1.6 Integrals in Geometry
4.1.7 Other Geometry Problems
4.2 Trigonometry
4.2.1 Trigonometric Identities
4.2.2 Euler's Formula
4.2.3 Trigonometric Substitutions
4.2.4 Telescopic Sums and Products in Trigonometry
5 Number Theory
5.1 Integer-Valued Sequences and Functions
5.1.1 Some General Problems
5.1.2 Fermat's Infinite Descent Principle
5.1.3 The Greatest Integer Function
5.2 Arithmetic
5.2.1 Factorization and Divisibility
5.2.2 Prime Numbers
5.2.3 Mcdular Arithmetic
5.2.4 Fermat's Little Theorem
5.2.5 Wilson's Theorem
5.2.6 Euler's Totient Function
5.2.7 The Chinese Remainder Theorem
5.2.8 Quadratic IntcgerRings
5.3 Diophantine Equations
5.3.1 Linear Dicphantine Equations
5.3.2 The Equation of Pythagoras
5.3.3 Pell's Equation
5.3.4 Other Diophantine Equations
6 Combinatories and Probability
6.1 Combinatorial Arguments in Set Theory
6.1.1 Combinatorics of Sets
6.1.2 Combinatorics of Numbers
6.1.3 Permutations
6.2 Combinatorial Geometry
6.2.1 Tessellations
6.2.2 Miscellaneous Combinatorial Geometry Problems
6.3 Graphs
6.3.1 Some Basic Graph The
内容摘要
普特南数学竞赛（TheWilliamLowellPutnamMathematicalCompetition）创办于1927年，是世界上最负盛名的大学生数学竞赛。普特南竞赛的桂冠获奖者（PutnamFellows）中包括后来的菲尔兹奖得主约翰·米尔诺（JohnMilnor）、
戴维·芒福德（DavidMumford）、丹尼尔·奎伦（DanielQuillen）和诺贝尔物理学奖获得者理查德·费曼（RichardFeynman）、
肯尼斯·威尔逊（KennethWilson）等。
通过尝试解答普特南数学竞赛试题，读者可以逐渐将其中学层面数学的解题思维转变为高等数学的解题思维，直至做数学研究的思维方式。本书中也包括不少国际数学奥林匹克竞赛（IMO）的试题，以及各国选拨奥数国家队的赛题，这些题目背后往往有着更为深刻的数学背景。本书以一种循序渐进的方式帮助学习者提升自己的数学能力。
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