Berkeley Problems in Mathematics [Paperback]

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.

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.ContentsPrefaceI Problems1 Real Analysis1.1 Elementary Calculus1.2 Limitsand Continuity1.3 Sequences, Series, and Products1.4 Differential Calculus1.5 Integral Calculus1.6 Sequences of Functions1.7 Fourier Series1.8 Convex Functions 2 Multivariable Calculus2.1 Limitsand Continuity2.2 Differential Calculus2.3 Integral Calculus 3 Differential Equations3.1 First Order Equations3.2 SecondOrder Equations3.3 Higher Order Equations3.4 Systems of Differential Equations 4 Metric Spaces4.1 Topology of Rn4.2 General Theory4.3 Fixed Point Theorem 5 Complex Analysis5.1 Complex Numbers5.2 Series and Sequences of Functions5.3 Conformal Mappings5.4 Functions on the Unit Disc5.5 Growth Conditions5.6 Analytic and Meromorphic Functions5.7 Cauchys Theorem5.8 Zeros and Singularities5.9 Harmonic Functions5.10 Rlƒ*