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Berkeley Problems in Mathematics / Edition 3
     

Berkeley Problems in Mathematics / Edition 3

by Paulo Ney de Souza, Jorge-Nuno Silva
 

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ISBN-10: 0387008926

ISBN-13: 9780387008929

Pub. Date: 01/28/2004

Publisher: Springer New York

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

Overview

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.

Product Details

ISBN-13:
9780387008929
Publisher:
Springer New York
Publication date:
01/28/2004
Series:
Problem Books in Mathematics Series
Edition description:
3rd ed. 2004
Pages:
593
Sales rank:
1,256,362
Product dimensions:
9.21(w) x 6.14(h) x 1.23(d)

Table of Contents

Prefacevii
IProblems1
1Real Analysis3
1.1Elementary Calculus3
1.2Limits and Continuity8
1.3Sequences, Series, and Products10
1.4Differential Calculus14
1.5Integral Calculus18
1.6Sequences of Functions22
1.7Fourier Series27
1.8Convex Functions29
2Multivariable Calculus31
2.1Limits and Continuity31
2.2Differential Calculus32
2.3Integral Calculus40
3Differential Equations43
3.1First Order Equations43
3.2Second Order Equations47
3.3Higher Order Equations49
3.4Systems of Differential Equations50
4Metric Spaces57
4.1Topology of R[superscript n]57
4.2General Theory60
4.3Fixed Point Theorem62
5Complex Analysis65
5.1Complex Numbers65
5.2Series and Sequences of Functions67
5.3Conformal Mappings70
5.4Functions on the Unit Disc71
5.5Growth Conditions74
5.6Analytic and Meromorphic Functions75
5.7Cauchy's Theorem80
5.8Zeros and Singularities82
5.9Harmonic Functions86
5.10Residue Theory87
5.11Integrals Along the Real Axis93
6Algebra97
6.1Examples of Groups and General Theory97
6.2Homomorphisms and Subgroups99
6.3Cyclic Groups102
6.4Normality, Quotients, and Homomorphisms102
6.5S[superscript n], A[superscript n], D[superscript n], ...104
6.6Direct Products106
6.7Free Groups, Generators, and Relations106
6.8Finite Groups107
6.9Rings and Their Homomorphisms109
6.10Ideals111
6.11Polynomials112
6.12Fields and Their Extensions116
6.13Elementary Number Theory118
7Linear Algebra123
7.1Vector Spaces123
7.2Rank and Determinants125
7.3Systems of Equations129
7.4Linear Transformations129
7.5Eigenvalues and Eigenvectors134
7.6Canonical Forms138
7.7Similarity144
7.8Bilinear, Quadratic Forms, and Inner Product Spaces146
7.9General Theory of Matrices149
IISolutions155
1Real Analysis157
1.1Elementary Calculus157
1.2Limits and Continuity173
1.3Sequences, Series, and Products178
1.4Differential Calculus191
1.5Integral Calculus200
1.6Sequences of Functions213
1.7Fourier Series228
1.8Convex Functions232
2Multivariable Calculus235
2.1Limits and Continuity235
2.2Differential Calculus237
2.3Integral Calculus258
3Differential Equations265
3.1First Order Equations265
3.2Second Order Equations274
3.3Higher Order Equations278
3.4Systems of Differential Equations280
4Metric Spaces289
4.1Topology of R[superscript n]289
4.2General Theory297
4.3Fixed Point Theorem300
5Complex Analysis305
5.1Complex Numbers305
5.2Series and Sequences of Functions309
5.3Conformal Mappings316
5.4Functions on the Unit Disc321
5.5Growth Conditions329
5.6Analytic and Meromorphic Functions334
5.7Cauchy's Theorem347
5.8Zeros and Singularities356
5.9Harmonic Functions372
5.10Residue Theory373
5.11Integrals Along the Real Axis390
6Algebra423
6.1Examples of Groups and General Theory423
6.2Homomorphisms and Subgroups429
6.3Cyclic Groups433
6.4Normality, Quotients, and Homomorphisms435
6.5S[subscript n], A[subscript n], D[subscript n], ...440
6.6Direct Products443
6.7Free Groups, Generators, and Relations445
6.8Finite Groups450
6.9Rings and Their Homomorphisms456
6.10Ideals460
6.11Polynomials463
6.12Fields and Their Extensions473
6.13Elementary Number Theory480
7Linear Algebra489
7.1Vector Spaces489
7.2Rank and Determinants495
7.3Systems of Equations501
7.4Linear Transformations503
7.5Eigenvalues and Eigenvectors514
7.6Canonical Forms525
7.7Similarity540
7.8Bilinear, Quadratic Forms, and Inner Product Spaces545
7.9General Theory of Matrices553
IIIAppendices569
AHow to Get the Exams571
A.1On-line571
A.2Off-line, the Last Resort571
BPassing Scores577
CThe Syllabus579
References581
Index589

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