Problems And Solutions In Mathematics (2nd Edition)

by Tatsien Li (Editor)

Overview

This book contains a selection of more than 500 mathematical problems and their solutions from the PhD qualifying examination papers of more than ten famous American universities. The mathematical problems cover six aspects of graduate school mathematics: Algebra, Topology, Differential Geometry, Real Analysis, Complex Analysis and Partial Differential Equations. While the depth of knowledge involved is not beyond the contents of the textbooks for graduate students, discovering the solution of the problems requires a deep understanding of the mathematical principles plus skilled techniques. For students, this book is a valuable complement to textbooks. Whereas for lecturers teaching graduate school mathematics, it is a helpful reference.

Product Details

ISBN-13:	9789814304962
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/04/2011
Series:	Major American Universities Ph.d. Qualifying Questions And Solutions - Mathematics
Edition description:	Second Edition
Pages:	804
Product dimensions:	6.10(w) x 9.00(h) x 1.70(d)

Table of Contents

Preface v

Preface to the Second Edition vii

Part 1 Algebra 1

Section 1 Linear Algebra 3

Section 2 Group Theory 33

Section 3 Ring Theory 55

Section 4 Field and Galois Theory 73

Part 2 Topology 119

Section 1 Point Set Topology 121

Section 2 Homotopy Theory 143

Section 3 Homology Theory 171

Part 3 Differential Geometry 215

Section 1 Differential Geometry of Curves 217

Section 2 Differential Geometry of Surfaces 241

Section 3 Differential Geometry of Manifolds 287

Part 4 Real Analysis 347

Section 1 Measurability and Measure 349

Section 2 Integral 389

Section 3 Space of Integrable Functions 433

Section 4 Differential 459

Section 5 Miscellaneous Problems 487

Part 5 Complex Analysis 503

Section 1 Analytic and Harmonic Functions 505

Section 2 Geometry of Analytic Functions 539

Section 3 Complex Integration 563

Section 4 The Maximum Modulus and Argument Principles 615

Section 5 Series and Normal Families 643

Part 6 Partial Differential Equations 675

Section 1 General Theory 677

Section 2 Elliptic Equations 697

Section 3 Parabolic Equations 729

Section 4 Hyperbolic Equations 755

Abbreviations of Universities in this Book 793