eBook ISBN:  9781470445409 
Product Code:  MMONO/132.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
eBook ISBN:  9781470445409 
Product Code:  MMONO/132.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 

Book DetailsTranslations of Mathematical MonographsVolume: 132; 1993; 251 ppMSC: Primary 58; Secondary 53; 81
This book provides a wide view of the calculus of variations as it plays an essential role in various areas of mathematics and science. Containing many examples, open problems, and exercises with complete solutions, the book would be suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods. The first part of the book is devoted to explaining the notion of (infinitedimensional) manifolds and contains many examples. An introduction to Morse theory of Banach manifolds is provided, along with a proof of the existence of minimizing functions under the PalaisSmale condition. The second part, which may be read independently of the first, presents the theory of harmonic maps, with a careful calculation of the first and second variations of the energy. Several applications of the second variation and classification theories of harmonic maps are given.
ReadershipGraduate students and research mathematicians. Containing many examples, open problems, and exercises with complete solutions, the book is suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods.

Table of Contents

Chapters

Chapter 1. Calculus of variations

Chapter 2. Manifolds

Chapter 3. Morse theory

Chapter 4. Harmonic mappings

Chapter 5. The second variation formula and stability

Chapter 6. Existence, construction, and classification of harmonic maps


Reviews

This book is organized in an elegant manner. The background and motives of the subjects concerned are clearly explained. A fundamental knowledge of differential geometry is also included and the exercises at the end of each chapter are useful for the readers to deepen their understandings. It is a good book for one who wants to know and to study the recent developments in the theories of harmonic maps and the variational methods.
Zentralblatt MATH 
This is probably the first textbook on the theory of harmonic maps ... It is ... an extremely welcome addition to the literature, both as an introductory text and as a reference for those details which are often taken for granted in research articles.
Mathematical Reviews


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This book provides a wide view of the calculus of variations as it plays an essential role in various areas of mathematics and science. Containing many examples, open problems, and exercises with complete solutions, the book would be suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods. The first part of the book is devoted to explaining the notion of (infinitedimensional) manifolds and contains many examples. An introduction to Morse theory of Banach manifolds is provided, along with a proof of the existence of minimizing functions under the PalaisSmale condition. The second part, which may be read independently of the first, presents the theory of harmonic maps, with a careful calculation of the first and second variations of the energy. Several applications of the second variation and classification theories of harmonic maps are given.
Graduate students and research mathematicians. Containing many examples, open problems, and exercises with complete solutions, the book is suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods.

Chapters

Chapter 1. Calculus of variations

Chapter 2. Manifolds

Chapter 3. Morse theory

Chapter 4. Harmonic mappings

Chapter 5. The second variation formula and stability

Chapter 6. Existence, construction, and classification of harmonic maps

This book is organized in an elegant manner. The background and motives of the subjects concerned are clearly explained. A fundamental knowledge of differential geometry is also included and the exercises at the end of each chapter are useful for the readers to deepen their understandings. It is a good book for one who wants to know and to study the recent developments in the theories of harmonic maps and the variational methods.
Zentralblatt MATH 
This is probably the first textbook on the theory of harmonic maps ... It is ... an extremely welcome addition to the literature, both as an introductory text and as a reference for those details which are often taken for granted in research articles.
Mathematical Reviews