本书收录最近25年的1250余道伯克制数学考试试题，对所有计划攻读数学博士学位的学生，本书中的试题和解答都颇具价值；读者研读完本书，在诸如实分析、多变量微积分、微分方程、度量空阔、复分析、代数学及线性代数等学科的解题能力都将得到提高。

Preface

I Problems

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 Muitivariable Calculus

2.1  Limits and 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

  Metric Spaces

  4.1   Topology of Rn

  4.2  General Theory

  4.3  Fixed Point Theorem

  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

  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  Rank and 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

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

Multivariable Calculus

2.1  Limits and 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

Metric Spaces

4.1   Topology cf Rn

4.2  General Theory

4.3  Fixed Point Theorem

Complex Analysis

5.1  Complex Numbers

5.2  Series and Sequences of Functions

5.3  Ccnformal 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

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. l 3  Elementary Number Theory

7 Linear Algebra

7.1  Vector Spaces

7.2  Rank and 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