Berkeley Problems in Mathematics (Problem Books in Mathematics)

In 1977 the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the fist 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 since a) students are already deeply involved with the material and b) they will be prepared to take the exam within the first month of the graduate program rather than in the middle or end of the first year. The 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, multi-variable 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, statistics of passing scores, and a Bibliography used throughout the solutions. This volume is an ideal means for students to strengthen their foundation in basic mathematics and to prepare for graduate studies.

From the reviews of the third edition:

"Problems that appeared in the preliminary exams ... are treated in this very interesting book. It is extremely helpful that the problems appear by subject ... . It is also convenient for the solver that in each chapter the problems are ordered in an increasing level of difficulty. A very clever idea is included in the appendix: the method to access the electronic versions of the exams and some relevant statistics." (Panayiotis Vlamos, Zentralblatt MATH, Vol. 1041 (16), 2004)

"Over 1250 problems that appeared in the preliminary exams of the Mathematics Department at the University of California, Berkeley, over the last 25 years are treated in this very interesting book. It is extremely helpful that the problems appear by subject ... . It is also convenient for the solver that in each chapter the problems are ordered in an increasing level of difficulty. A very clever idea is included in the appendix: the method to access ... ." (Panayiotis Vlamos, Zentralblatt MATH, Vol. 1041 (16))