Berkeley Problems in Mathematics by Paulo Ney de Souza, Jorge-Nuno Silva , Springer

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Springer Berkeley Problems in Mathematics by Paulo Ney de Souza, Jorge-Nuno Silva

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._x000D_ Table of contents :- _x000D_ Contents_x000D_ Preface _x000D_ I Problems _x000D_ 1 Real Analysis _x000D_ 1.1 Elementary Calculus _x000D_ 1.2 Limitsand Continuity _x000D_ 1.3 Sequences, Series, and Products _x000D_ 1.4 Differential Calculus _x000D_ 1.5 Integral Calculus _x000D_ 1.6 Sequences of Functions _x000D_ 1.7 Fourier Series _x000D_ 1.8 Convex Functions 2 Multivariable Calculus _x000D_ 2.1 Limitsand Continuity _x000D_ 2.2 Differential Calculus _x000D_ 2.3 Integral Calculus 3 Differential Equations _x000D_ 3.1 First Order Equations _x000D_ 3.2 SecondOrder Equations _x000D_ 3.3 Higher Order Equations _x000D_ 3.4 Systems of Differential Equations 4 Metric Spaces _x000D_ 4.1 Topology of Rn _x000D_ 4.2 General Theory _x000D_ 4.3 Fixed Point Theorem 5 Complex Analysis _x000D_ 5.1 Complex Numbers _x000D_ 5.2 Series and Sequences of Functions _x000D_ 5.3 Conformal Mappings _x000D_ 5.4 Functions on the Unit Disc _x000D_ 5.5 Growth Conditions _x000D_ 5.6 Analytic and Meromorphic Functions _x000D_ 5.7 Cauchy's Theorem _x000D_ 5.8 Zeros and Singularities _x000D_ 5.9 Harmonic Functions _x000D_ 5.10 Residue Theory _x000D_ 5.11 Integrals Along the Real Axis 6 Algebra _x000D_ 6.1 Examples of Groups and General Theory _x000D_ 6.2 Homomorphisms and Subgroups _x000D_ 6.3 Cyclic Groups _x000D_ 6.4 Normality, Quotients, and Homomorphisms _x000D_ 6.5 Sn, An , Dn, .._x000D_ 6.6 Direct Products _x000D_ 6.7 Free Groups, Generators, and Relations _x000D_ 6.8 Finite Groups _x000D_ 6.9 Ringsand Their Homomorphisms _x000D_ 6.10 Ideals _x000D_ 6.11 Polynomials _x000D_ 6.12 Fields and Their Extensions _x000D_ 6.13 Elementary Number Theory 7 Linear Algebra _x000D_ 7.1 Vector Spaces _x000D_ 7.2 Rankand Determinants _x000D_ 7.3 Systems of Equations _x000D_ 7.4 Linear Transformations _x000D_ 7.5 Eigenvalues and Eigenvectors _x000D_ 7.6 Canonical Forms _x000D_ 7.7 Similarity _x000D_ 7.8 Bilinear, Quadratic Forms, and Inner Product Spaces _x000D_ 7.9 General Theory ofMatrices II Solutions _x000D_ 1 Real Analysis _x000D_ 1.1 Elementary Calculus _x000D_ 1.2 Limits and Continuity _x000D_ 1.3 Sequences, Series, and Products _x000D_ 1.4 Differential Calculus _x000D_ 1.5 Integral Calculus _x000D_ 1.6Sequences of Functions _x000D_ 1.7 Fourier Series _x000D_ 1.8 Convex Functions 2 Multivariable Calculus _x000D_ 2.1 Limitsand Continuity _x000D_ 2.2 Differential Calculus _x000D_ 2.3 Integral Calculus _x000D_ 3 Differential Equations _x000D_ 3.1 First Order Equations _x000D_ 3.2 Second Order Equations _x000D_ 3.3 Higher Order Equations _x000D_ 3.4 Systems of Differential Equations 4 Metric Spaces _x000D_ 4.1 Topology of Rn _x000D_ 4.2 General Theory _x000D_ 4.3 Fixed Point Theorem 5 Complex Analysis _x000D_ 5.1 Complex Numbers _x000D_ 5.2 Series and Sequences of Functions _x000D_ 5.3 Conformal Mappings _x000D_ 5.4 Functions on the Unit Disc _x000D_ 5.5 Growth Conditions _x000D_ 5.6 Analytic and Meromorphic Functions _x000D_ 5.7 Cauchy's Theorem _x000D_ 5.8 Zeros and Singularities _x000D_ 5.9 Harmonic Functions _x000D_ 5.10 Residue Theory _x000D_ 5.11 Integrals Along the Real Axis 6 Algebra _x000D_ 6.1 Examples of Groups and General Theory _x000D_ 6.2 Homomorphisms and Subgroups _x000D_ 6.3 Cyclic Groups _x000D_ 6.4 Normality, Quotients, and Homomorphisms _x000D_ 6.5 Sn, An , Dn, .._x000D_ 6.6 Direct Products _x000D_ 6.7 Free Groups, Generators, and Relations _x000D_ 6.8 Finite Groups _x000D_ 6.9 Rings and Their Homomorphisms _x000D_ 6.10 Ideals _x000D_ 6.11 Polynomials _x000D_ 6.12 Fields and Their Extensions _x000D_ 6.13 Elementary Number Theory 7 Linear Algebra _x000D_ 7.1 Vector Spaces _x000D_ 7.2 Rankand Determinants _x000D_ 7.3 Systems of Equations _x000D_ 7.4 Linear Transformations _x000D_ 7.5 Eigenvalues and Eigenvectors _x000D_ 7.6 Canonical Forms _x000D_ 7.7 Similarity _x000D_ 7.8 Bilinear, Quadratic Forms, and Inner Product Spaces _x000D_ 7.9 General Theory of Matrices III Appendices _x000D_ A How to Get the Exams _x000D_ A.1 On-line _x000D_ A.2 Off-line, the Last Resort _x000D_ B Passing Scores _x000D_ C The Syllabus _x000D_ References _x000D_ Index_x000D_