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, 9789810234805, 9789812385406, 9810234791, 9810234805

Covers 6 aspects of graduate school mathematics: Algebra, Topology, Differential Geometry, Real Analysis, Complex Analysis & Partial Differential Equations. Contains a selection of more than 500 problems & solutions based on the Ph.D. qualifying test papers of a decade of influential universities in North America. Paper. DLC: Mathematics – Problems, exercises, etc.

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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