Synopses & Reviews
In 1977 the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph.D. degree in Mathematics. Its purpose was to determine whether first-year students in the Ph.D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree. The purpose of this book is to publicize the material and aid in the preparation for the examination during the undergraduate years. The book is a compilation of over 1,250 problems which have appeared on the preliminary exams in Berkeley over the last twenty-five years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty. Tags with the exact exam year provide the opportunity to rehearse complete examinations. The appendix includes instructions on accessing electronic versions of the exams as well as a syllabus and statistics of passing scores. This new edition has been updated with the most recent exams, including exams given during the Fall 2003 semester. There are numerous new problems and solutions which were not included in previous editions.
Review
From the reviews of the third edition: "This new edition has been updated with the most recent exams ... . There are numerous new problems and solutions which were not included in previous editions. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph. D program. ... this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. ... Tags with the exact exam year provide the opportunity to rehearse complete examinations. ... This new edition has been updated with the most recent exams ... ." (Zentralblatt für Didaktik der Mathematik, November 2004) "The Mathematics department of the University of California, Berkeley, has set a written preliminary examination to determine whether first year Ph.D. students have mastered enough basic mathematics to succeed in the doctoral program. Berkeley Problems in Mathematics is a compilation of all the ... questions, together with worked solutions ... . All the solutions I looked at are complete ... . Some of the solutions are very elegant. ... This is an impressive piece of work and a welcome addition to any mathematician's bookshelf." (Chris Good, The Mathematical Gazette, 90:518, 2006) "During the last twenty-five years problems from written preliminary examinations that are required for the Ph.D. degree at the Mathematics Department of the University of California, Berkeley, have been assembled. ... The book is suited for students in mathematics, physics or engineering. Solutions are well explained, making the book valuable for self-study. The problems have a satisfactory high level, so the book is a rich resource of examples for lecturers as well, who need exercises ... . This book certainly is to be recommended." (Paula Bruggen, Bulletin of the Belgian Mathematical Society, 12:4, 2005)
Review
From the reviews of the third edition:
"This new edition has been updated with the most recent exams ... . There are numerous new problems and solutions which were not included in previous editions. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph. D program. ... this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. ... Tags with the exact exam year provide the opportunity to rehearse complete examinations. ... This new edition has been updated with the most recent exams ... ." (Zentralblatt für Didaktik der Mathematik, November 2004)
"The Mathematics department of the University of California, Berkeley, has set a written preliminary examination to determine whether first year Ph.D. students have mastered enough basic mathematics to succeed in the doctoral program. Berkeley Problems in Mathematics is a compilation of all the ... questions, together with worked solutions ... . All the solutions I looked at are complete ... . Some of the solutions are very elegant. ... This is an impressive piece of work and a welcome addition to any mathematician's bookshelf." (Chris Good, The Mathematical Gazette, 90:518, 2006)
"During the last twenty-five years problems from written preliminary examinations that are required for the Ph.D. degree at the Mathematics Department of the University of California, Berkeley, have been assembled. ... The book is suited for students in mathematics, physics or engineering. Solutions are well explained, making the book valuable for self-study. The problems have a satisfactory high level, so the book is a rich resource of examples for lecturers as well, who need exercises ... . This book certainly is to be recommended." (Paula Bruggen, Bulletin of the Belgian Mathematical Society, 12:4, 2005)
Synopsis
This book collects approximately nine hundred problems that have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions. Readers who work through this book will develop problem solving skills in such areas as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra.
Synopsis
This book is a compilation of approximately nine hundred problems, which have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty. This new edition contains approximately 120 new problems and 200 new solutions. It is an ideal means for students to strengthen their foundation in basic mathematics and to prepare for graduate studies.
Table of Contents
Contents
Preface
I Problems
1 Real Analysis
1.1 Elementary Calculus
1.2 Limitsand Continuity
1.3 Sequences, Series, and Products
1.4 Differential Calculus
1.5 Integral Calculus
1.6 Sequences of Functions
1.7 Fourier Series
1.8 Convex Functions
2 Multivariable Calculus
2.1 Limitsand Continuity
2.2 Differential Calculus
2.3 Integral Calculus
3 Differential Equations
3.1 First Order Equations
3.2 SecondOrder Equations
3.3 Higher Order Equations
3.4 Systems of Differential Equations
4 Metric Spaces
4.1 Topology of Rn
4.2 General Theory
4.3 Fixed Point Theorem
5 Complex Analysis
5.1 Complex Numbers
5.2 Series and Sequences of Functions
5.3 Conformal Mappings
5.4 Functions on the Unit Disc
5.5 Growth Conditions
5.6 Analytic and Meromorphic Functions
5.7 Cauchy's Theorem
5.8 Zeros and Singularities
5.9 Harmonic Functions
5.10 Residue Theory
5.11 Integrals Along the Real Axis
6 Algebra
6.1 Examples of Groups and General Theory
6.2 Homomorphisms and Subgroups
6.3 Cyclic Groups
6.4 Normality, Quotients, and Homomorphisms
6.5 Sn, An , Dn, ..
6.6 Direct Products
6.7 Free Groups, Generators, and Relations
6.8 Finite Groups
6.9 Ringsand Their Homomorphisms
6.10 Ideals
6.11 Polynomials
6.12 Fields and Their Extensions
6.13 Elementary Number Theory
7 Linear Algebra
7.1 Vector Spaces
7.2 Rankand Determinants
7.3 Systems of Equations
7.4 Linear Transformations
7.5 Eigenvalues and Eigenvectors
7.6 Canonical Forms
7.7 Similarity
7.8 Bilinear, Quadratic Forms, and Inner Product Spaces
7.9 General Theory ofMatrices
II Solutions
1 Real Analysis
1.1 Elementary Calculus
1.2 Limits and Continuity
1.3 Sequences, Series, and Products
1.4 Differential Calculus
1.5 Integral Calculus
1.6 Sequences of Functions
1.7 Fourier Series
1.8 Convex Functions
2 Multivariable Calculus
2.1 Limitsand Continuity
2.2 Differential Calculus
2.3 Integral Calculus
3 Differential Equations
3.1 First Order Equations
3.2 Second Order Equations
3.3 Higher Order Equations
3.4 Systems of Differential Equations
4 Metric Spaces
4.1 Topology of Rn
4.2 General Theory
4.3 Fixed Point Theorem
5 Complex Analysis
5.1 Complex Numbers
5.2 Series and Sequences of Functions
5.3 Conformal Mappings
5.4 Functions on the Unit Disc
5.5 Growth Conditions
5.6 Analytic and Meromorphic Functions
5.7 Cauchy's Theorem
5.8 Zeros and Singularities
5.9 Harmonic Functions
5.10 Residue Theory
5.11 Integrals Along the Real Axis
6 Algebra
6.1 Examples of Groups and General Theory
6.2 Homomorphisms and Subgroups
6.3 Cyclic Groups
6.4 Normality, Quotients, and Homomorphisms
6.5 Sn, An , Dn, ..
6.6 Direct Products
6.7 Free Groups, Generators, and Relations
6.8 Finite Groups
6.9 Rings and Their Homomorphisms
6.10 Ideals
6.11 Polynomials
6.12 Fields and Their Extensions
6.13 Elementary Number Theory
7 Linear Algebra
7.1 Vector Spaces
7.2 Rankand Determinants
7.3 Systems of Equations
7.4 Linear Transformations
7.5 Eigenvalues and Eigenvectors
7.6 Canonical Forms
7.7 Similarity
7.8 Bilinear, Quadratic Forms, and Inner Product Spaces
7.9 General Theory of Matrices
III Appendices
A How to Get the Exams
A.1 On-line
A.2 Off-line, the Last Resort
B Passing Scores
C The Syllabus
References
Index