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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.

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
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