1977年，為考查一年級的博士研究生是否已經(jīng)成功掌握為攻讀數(shù)學博士學位所需的基本數(shù)學知識和技能，加州大學伯克利分校數(shù)學系設(shè)立了一項書面考試，作為獲得博士學位的首要要求之一。該項考試自其創(chuàng)設(shè)以來，已成為研究生獲得博士學位必須克服的一個主要障礙。本書的目的即為出版這些考試材料，以期對本科生準備該項考試有所幫助�！禕R》 全書收錄*近25年的1250余道伯克利數(shù)學考試試題，對所有計劃攻讀數(shù)學博士學位的學生，本書中的試題和解答都頗具價值，讀者研讀完本書，在諸如實分析、多變量微積分、微分方程、度量空間、復分析、代數(shù)學及線性代數(shù)等學科的解題能力都將得到提高�！禕R》 這些問題按學科及難易程度編排，每道試題均注明相應(yīng)的考試年月，讀者可以依此方便地整理由各套試題。附錄介紹如何得到電子版試題，考試大綱以及各次考試的及格線�！禕R》 新版已包含直至2003秋季學期的*近考試試題和解答，增添了以前版本未收錄的許多新的試題及題解。
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Contents
Preface vii
I Problems 1
1 Real Analysis 3
1.1 Elementary Calculus 3
1.2 Limits and Continuity 8
1.3 Sequences,Series,and Products 10
1.4 Differential Calculus 14
1.5 Integral Calculus 18
1.6 Sequences of Functions 22
1.7 Fourier Series 27
1.8 Convex Functions 29
2 Multivariable Calculus 31
2.1 Limits and Continuity 31
2.2 Differential Calculus 32
2.3 Integral Calculus 40
3 Differential Equations 43
3.1 First Order Equations 43
3.2 Second Order Equations 47
3.3 Higher Order Equations 49
3.4 Systems of Differential Equations 50
4 Metric Spaces 57
4.1 Topology of Rn 57
4.2 General Theory 60
5 Complex Analysis 65
5.1 Complex Numbers 65
5.2 Series and Sequences of Functions 67
5.3 Conformal Mappings 70
5.4 Functions on the Unit Disc 71
5.5 Growth Conditions 74
5.6 Analytic and Meromorphic Functions 75
5.7 Cauchy's Theorem 80
5.8 Zeros and Singularities 82
5.9 Harmonic Functions 86
5.10 Residue Theory 87
5.11 Integrals Alongthe Real Axis 93
6 Algebra 97
6.1 Examples of Groups and General Theory 97
6.2 Homomorphisms and Subgroups 99
6.4 Normality,Quotients,and Homomorphisms 102
6.7 Free Groups,Generators,and Relations 106
6.9 Rings and Their Homomorphisms 109
6.12 Fields and Their Extensions 116
6.13 Elementary Number Theory 118
7 Linear Algebra 123
7.1 Vector Spaces 123
7.2 Rank and Determinants 125
7.3 Systems of Equations 129
7.4 Linear Transformations. 129
7.5 Eigenvalues and Eigenvectors 134
7.8 Bilinear,Quadratic Forms,and lnner Product Spaces 146
7.9 General Theory of Matrices 149
II Solutions 155
1 Real Analysis 157
1.1 Elementary Calculus 157
1.2 Limitsand Continuity 173
1.3 Sequences,Series,and Products 178
1.4 Differential Calculus 191
1.5 Integral Calculus 200
1.6 Sequences of Functions 213
1.7 Fourier Series 228
1.8 Convex Functions 232
2 Multivariable Calculus 235
2.1 Limitsand Continuity 235
2.2 Differential Calculus 237
2.3 Integral Calculus 258
3 Differential Equations 265
3.1 First Order Equations 265
3.2 Second Order Equations 274
3.3 Higher Order Equations 278
3.4 Systems of Differential Equations 280
4 Metric Spaces 289
4.1 Topology of Rn 289
4.2 General Theory 297
4.3 Fixed Point Theorem 300
5 Complex Analysis 305
5.1 Complex Numbers 305
5.2 Series and Sequences of Functions 309
5.3 Conformal Mappings 316
5.4 Functionson the Unit Disc 321
5.5 Growth Conditions 329
5.6 Analytic and Meromorphic Functions 334
5.7 Cauchy's Theorem 347
5.8 Zeros and Singularities 356
5.9 Harmonic Functions 372
5.10 Residue Theory 373
5.11 Integrals Along the Real Axis 390
6 Algebra 423
6.1 Examples of Groups and General Theory 423
6.2 Homomorphisms and Subgroups 429
6.3 Cyclic Groups 433
6.4 Normality,Quotients,and Homomorphisms 435
6.5 Sn,An,Dn,… 440
6.6 Direct Products 443
6.7 Free Groups,Generators,and Relations 445
6.8 Finite Groups 450
6.9 Rings and Their Homomorphisms 456
6.10 Ideals 460
6.11 Polynomials 463
6.12 Fields and Their Extensions 473
6.13 Elementary Number Theory 480
7 Linear Algebra 489
7.1 Vector Spaces 489
7.2 Rank and Determinants 495
7.3 Systems of Equations 501
7.4 Linear Transformations 503
7.5 Eigenvaluesand Eigenvectors 514
7.6 Canonical Forms 525
7.7 Similarity 540
7.8 Bilinear,Quadratic Forms,and lnner Product Spaces 545
7.9 General Theory of Matrices 553
Ⅲ Appendices 569
A How to Get the Exams 571
A.1 On-line 571
A.2 0ff-Iine,the Last Resort 571
B Passing Scores 577
C The Syllabus 579
References 581
Index 589
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