Problems and Solutions in Mathematics

Ji-Xiu Chen, Li Ta-Tsien, Ji-Xiu Chen, Li Ta-Tsien, Jiang Guo-Ying, Pan Yang-Lian, Qin Tie-Hu, Tong Yu-Sun, Wu Quan-Shui, Xu Shen-Zhi9789810234799, 9810234791, 9810234805

This text contains more than 500 mathematical problems and their solutions from the PhD qualifying examination papers of more than 10 famous American universities. The problems cover six aspects of graduate school mathematics: algebra, differential geometry, topology, real analysis, complex analysis and partial differential equations. The depth of knowledge involved is not beyond the contents of the textbooks for graduate students, while solution of the problems requires deep understanding of the mathematical principles and skilled techniques.

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Table of contents :
PREFACE……Page 6
Part I Algebra……Page 10
Section 1 Linear Algebra……Page 12
SECTION 2 GROUP THEORY……Page 35
SECTION 3 RING THEORY……Page 53
SECTION 4 FIELD AND GALOIS THEORY……Page 68
Part II Topology……Page 90
SECTION 1 POINT SET TOPOLOGY……Page 92
SECTION 2 HOMOTOPY THEORY……Page 108
SECTION 3 HOMOLOGY THEORY……Page 127
Part III Differential Geometry……Page 160
SECTION 1 DIFFERENTIAL GEOMETRY OF CURVES……Page 162
SECTION 2 DIFFERENTIAL GEOMETRY OF SURFACES……Page 180
SECTION 3 DIFFERENTIAL GEOMETRY OF MANIFOLD……Page 203
Part IV Real Analysis……Page 238
SECTION 1 MEASURABLITY AND MEASURE……Page 240
SECTION 2 INTEGRAL……Page 265
SECTION 3 SPACE OF INTEGRABLE FUNCTIONS……Page 292
SECTION 4 DIFFERENTIAL……Page 311
SECTION 5 MISCELLANEOUS PROBLEMS……Page 331
Part V Complex Analysis……Page 342
SECTION 1 ANALYTIC AND HARMONIC FUNCTIONS……Page 344
SECTION 2 GEOMETRY OF ANALYTIC FUNCTIONS……Page 369
SECTION 3 COMPLEX INTEGRATION……Page 386
SECTION 4 THE MAXIMUM MODULUS AND ARGUMENT PRINCIPLES……Page 422
SECTION 5 SERIES AND NORMAL FAMILIES……Page 442
Part VI Partial Differential Equations……Page 464
SECTION 1 GENERAL THEORY……Page 466
SECTION 2 ELLIPTIC EQUATIONS……Page 481
SECTION 3 PARABOLIC EQUATIONS……Page 505
SECTION 4 HYPERBOLIC EQUATIONS……Page 522
Abbreviations of Universities in This Book……Page 548