1977年，为考查一年级的博士研究生是否已经成功掌握为攻读数学博士学位所需的基本数学知识和技能，加州大学伯克利分校数学系设立了一项书面考试，作为获得博士学位的首要要求之一。该项考试自其创设以来，已成为研究生获得博士学位必须克服的一个主要障碍。本书的目的即为出版这些考试材料，以期对本科生准备该项考试有所帮助。《BR》 全书收录*近25年的1250余道伯克利数学考试试题，对所有计划攻读数学博士学位的学生，本书中的试题和解答都颇具价值，读者研读完本书，在诸如实分析、多变量微积分、微分方程、度量空间、复分析、代数学及线性代数等学科的解题能力都将得到提高。《BR》 这些问题按学科及难易程度编排，每道试题均注明相应的考试年月，读者可以依此方便地整理由各套试题。附录介绍如何得到电子版试题，考试大纲以及各次考试的及格线。《BR》 新版已包含直至2003秋季学期的*近考试试题和解答，增添了以前版本未收录的许多新的试题及题解。
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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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