Involutive Hyperbolic Differential Systems PDF Download
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Author: Deane Yang
Publisher: American Mathematical Soc.
ISBN: 0821824333
Category : Mathematics
Languages : en
Pages : 109
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Book Description
The Cartan-Kähler theorem demonstrates that certain real analytic exterior differential systems--ones which Cartan dubbed "involutive"--admit local real analytic solutions. The proof reduces the system to a set of determined systems of PDE's which are then solved using the Cauchy-Kovalevski theorem. The subtlety in the theorem and in the definition of involutivity lies in checking that the "solution" obtained from the determined systems does in fact satisfy the original differential system.
Author: Deane Yang
Publisher: American Mathematical Soc.
ISBN: 0821824333
Category : Mathematics
Languages : en
Pages : 109
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Book Description
The Cartan-Kähler theorem demonstrates that certain real analytic exterior differential systems--ones which Cartan dubbed "involutive"--admit local real analytic solutions. The proof reduces the system to a set of determined systems of PDE's which are then solved using the Cauchy-Kovalevski theorem. The subtlety in the theorem and in the definition of involutivity lies in checking that the "solution" obtained from the determined systems does in fact satisfy the original differential system.
Author: Phillip A. Griffiths
Publisher: Princeton University Press
ISBN: 9780691084305
Category : Mathematics
Languages : en
Pages : 244
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Book Description
The theory of exterior differential systems provides a framework for systematically addressing the typically non-linear, and frequently overdetermined, partial differential equations that arise in differential geometry. Adaptation of the techniques of microlocalization to differential systems have led to recent activity on the foundations of the theory; in particular, the fundamental role of the characteristic variety in geometric problems is now clearly established. In this book the general theory is explained in a relatively quick and concrete manner, and then this general theory is applied to the recent developments in the classical problem of isometric embeddings of Riemannian manifolds.
Author: Demeter Krupka
Publisher: Elsevier
ISBN: 0080556736
Category : Mathematics
Languages : en
Pages : 1243
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Book Description
This is a comprehensive exposition of topics covered by the American Mathematical Society’s classification “Global Analysis , dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics.This book provides a comprehensive coverage of modern global analysis and geometrical mathematical physics, dealing with topics such as; structures on manifolds, pseudogroups, Lie groupoids, and global Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential systems and distributions, and spectral theory); variational theory on manifolds, with applications to physics; function spaces on manifolds; jets, natural bundles and generalizations; and non-commutative geometry. - Comprehensive coverage of modern global analysis and geometrical mathematical physics- Written by world-experts in the field- Up-to-date contents
Author: Werner M. Seiler
Publisher: Springer Science & Business Media
ISBN: 3642012876
Category : Mathematics
Languages : en
Pages : 663
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Book Description
The book provides a self-contained account of the formal theory of general, i.e. also under- and overdetermined, systems of differential equations which in its central notion of involution combines geometric, algebraic, homological and combinatorial ideas.
Author: Thomas A. Ivey
Publisher: American Mathematical Soc.
ISBN: 1470409860
Category : Mathematics
Languages : en
Pages : 474
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Book Description
Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.
Author: Thomas Andrew Ivey
Publisher: American Mathematical Soc.
ISBN: 0821833758
Category : Mathematics
Languages : en
Pages : 394
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Book Description
This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems. It begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics.One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.
Author: Michiel Hazewinkel
Publisher: Springer Science & Business Media
ISBN: 9400903650
Category : Mathematics
Languages : en
Pages : 743
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Book Description
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.
Author: American Mathematical Society
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages :
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Book Description
Author: Tynysbek Sh. Kalmenov
Publisher: Springer
ISBN: 3319670530
Category : Mathematics
Languages : en
Pages : 485
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Book Description
This volume presents current research in functional analysis and its applications to a variety of problems in mathematics and mathematical physics. The book contains over forty carefully refereed contributions to the conference “Functional Analysis in Interdisciplinary Applications” (Astana, Kazakhstan, October 2017). Topics covered include the theory of functions and functional spaces; differential equations and boundary value problems; the relationship between differential equations, integral operators and spectral theory; and mathematical methods in physical sciences. Presenting a wide range of topics and results, this book will appeal to anyone working in the subject area, including researchers and students interested to learn more about different aspects and applications of functional analysis.
Author: Olle Stormark
Publisher: Cambridge University Press
ISBN: 9780521780889
Category : Mathematics
Languages : en
Pages : 604
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Book Description
Here is a lucid and comprehensive introduction to the differential geometric study of partial differential equations (PDE). The first book to present substantial results on local solvability of general and nonlinear PDE systems without using power series techniques, it describes a general approach to PDE systems based on ideas developed by Lie, Cartan and Vessiot. The central theme is the exploitation of singular vector field systems and their first integrals. These considerations naturally lead to local Lie groups, Lie pseudogroups and the equivalence problem, all of which are covered in detail. This book will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.
