9780387952079

Berkeley Problems in Mathematics

by
  • ISBN13:

    9780387952079

  • ISBN10:

    0387952071

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2001-07-01
  • Publisher: SPRINGER VERLAG INC

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Summary

In 1977, the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first 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.

Table of Contents

Preface vii
I Problems 1(140)
Real Analysis
3(26)
Elementary Calculus
3(5)
Limits and Continuity
8(2)
Sequences, Series, and Products
10(3)
Differential Calculus
13(4)
Integral Calculus
17(4)
Sequences of Functions
21(5)
Fourier Series
26(2)
Convex Functions
28(1)
Multivariable Calculus
29(12)
Limits and Continuity
29(1)
Differential Calculus
30(7)
Integral Calculus
37(4)
Differential Equations
41(12)
First Order Equations
41(4)
Second Order Equations
45(2)
Higher Order Equations
47(1)
Systems of Differential Equations
48(5)
Metric Spaces
53(6)
Topology of Rn
53(3)
General Theory
56(1)
Fixed Point Theorem
57(2)
Complex Analysis
59(30)
Complex Numbers
59(2)
Series and Sequences of Functions
61(3)
Conformal Mappings
64(1)
Functions on the Unit Disc
65(2)
Growth Conditions
67(1)
Analytic and Meromorphic Functions
68(5)
Cauchy's Theorem
73(2)
Zeros and Singularities
75(3)
Harmonic Functions
78(1)
Residue Theory
79(6)
Integrals Along the Real Axis
85(4)
Algebra
89(22)
Examples of Groups and General Theory
89(2)
Homomorphisms and Subgroups
91(2)
Cyclic Groups
93(1)
Normality, Quotients, and Homomorphisms
94(2)
Sn, An, Dn,...
96(1)
Direct Products
97(1)
Free Groups, Generators, and Relations
98(1)
Finite Groups
99(1)
Rings and Their Homomorphisms
100(1)
Ideals
101(2)
Polynomials
103(3)
Fields and Their Extensions
106(2)
Elementary Number Theory
108(3)
Linear Algebra
111(30)
Vector Spaces
111(2)
Rank and Determinants
113(3)
Systems of Equations
116(1)
Linear Transformations
116(4)
Eigenvalues and Eigenvectors
120(5)
Canonical Forms
125(5)
Similarity
130(2)
Bilinear, Quadratic Forms, and Inner Product Spaces
132(3)
General Theory of Matrices
135(6)
II Solutions 141(372)
Real Analysis
143(72)
Elementary Calculus
143(15)
Limits and Continuity
158(5)
Sequences, Series, and Products
163(12)
Differential Calculus
175(9)
Integral Calculus
184(12)
Sequences of Functions
196(12)
Fourier Series
208(4)
Convex Functions
212(3)
Multivariable Calculus
215(28)
Limits and Continuity
215(2)
Differential Calculus
217(19)
Integral Calculus
236(7)
Differential Equations
243(26)
First Order Equations
243(9)
Second Order Equations
252(4)
Higher Order Equations
256(1)
Systems of Differential Equations
257(12)
Metric Spaces
269(14)
Topology of Rn
269(7)
General Theory
276(3)
Fixed Point Theorem
279(4)
Complex Analysis
283(108)
Complex Numbers
283(4)
Series and Sequences of Functions
287(6)
Conformal Mappings
293(4)
Functions on the Unit Disc
297(8)
Growth Conditions
305(4)
Analytic and Meromorphic Functions
309(12)
Cauchy's Theorem
321(9)
Zeros and Singularities
330(15)
Harmonic Functions
345(2)
Residue Theory
347(15)
Integrals Along the Real Axis
362(29)
Algebra
391(52)
Examples of Groups and General Theory
391(5)
Homomorphisms and Subgroups
396(3)
Cyclic Groups
399(1)
Normality, Quotients, and Homomorphisms
400(4)
Sn, An, Dn,...
404(2)
Direct Products
406(2)
Free Groups, Generators, and Relations
408(5)
Finite Groups
413(4)
Rings and Their Homomorphisms
417(3)
Ideals
420(4)
Polynomials
424(7)
Fields and Their Extensions
431(5)
Elementary Number Theory
436(7)
Linear Algebra
443(70)
Vector Spaces
443(6)
Rank and Determinants
449(5)
Systems of Equations
454(1)
Linear Transformations
455(10)
Eigenvalues and Eigenvectors
465(9)
Canonical Forms
474(13)
Similarity
487(4)
Bilinear, Quadratic Forms, and Inner Product Spaces
491(9)
General Theory of Matrices
500(13)
III Appendices 513(12)
A How to Get the Exams
515(6)
A.1 On-line
515(1)
A.2 Off-line, the Last Resort
515(6)
B Passing Scores
521(2)
C The Syllabus
523(2)
References 525(8)
Index 533

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