Problems and solutions in mathematics (PhD qualifying questions)

✍ Scribed by Ji-Xiu Chen, Jiang Guo-Ying, Pan Yang-Lian, Qin Tie-Hu, Tong Yu-Sun, Wu Quan-Shui, Xu Shen-Zhi, Ta-Chien Li

• Publisher: WS
• Year: 1999
• Tongue: English
• Leaves: 549
• Category: Library

✦ Synopsis

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.

✦ 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