UID:
almafu_9960119241702883
Umfang:
1 online resource (xvi, 525 pages) :
,
digital, PDF file(s).
ISBN:
1-316-04724-5
,
0-511-60956-6
Inhalt:
Drawing on a wide range of mathematical disciplines, including geometry, analysis, applied mathematics and algebra, this book presents an innovative synthesis of methods used to study problems of equivalence and symmetry which arise in a variety of mathematical fields and physical applications. Systematic and constructive methods for solving equivalence problems and calculating symmetries are developed and applied to a wide variety of mathematical systems, including differential equations, variational problems, manifolds, Riemannian metrics, polynomials and differential operators. Particular emphasis is given to the construction and classification of invariants, and to the reductions of complicated objects to simple canonical forms. This book will be a valuable resource for students and researchers in geometry, analysis, algebra, mathematical physics and other related fields.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
,
Cover -- Frontmatter -- Contents -- Preface -- Acknowledgments -- Introduction -- Geometric Foundations -- Manifolds -- Functions -- Submanifolds -- Vector Fields -- Lie Brackets -- The Differential -- Differential Forms -- Equivalence of Differential Forms -- Lie Groups -- Transformation Groups -- Invariant Subsets and Equations -- Canonical Forms -- Invariant Functions -- Lie Algebras -- Structure Constants -- The Exponential Map -- Subgroups and Subalgebras -- Infinitesimal Group Actions -- Classification of Group Actions -- Infinitesimal Invariance -- Invariant Vector Fields -- Lie Derivatives and Invariant Differential Forms -- The Maurer-Cartan Forms -- Representation Theory -- Representations -- Representations on Function Spaces -- Multiplier Representations -- Infinitesimal Multipliers -- Relative Invariants -- Classical Invariant Theory -- Jets and Contact Transformations -- Transformations and Functions -- Invariant Functions -- Jets and Prolongations -- Total Derivatives -- Prolongation of Vector Fields -- Contact Forms -- Contact Transformations -- Infinitesimal Contact Transformations -- Classification of Groups of Contact Transformations -- Differential Invariants -- Differential Invariants -- Dimensional Considerations -- Infinitesimal Methods -- Stabilization and Effectiveness -- Invariant Differential Operators -- Invariant Differential Forms -- Several Dependent Variables -- Several Independent Variables -- Symmetries of Differential Equations -- Symmetry Groups and Differential Equations -- Infinitesimal Methods -- Integration of Ordinary Differential Equations -- Characterization of Invariant Differential Equations -- Lie Determinants -- Symmetry Classification of Ordinary Differential Equations -- A Proof of Finite Dimensionality -- Linearization of Partial Differential Equations -- Differential Operators.
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Applications to the Geometry of Curves -- Symmetries of Variational Problems -- The Calculus of Variations -- Equivalence of Functionals -- Invariance of the Euler-Lagrange Equations -- Symmetries of Variational Problems -- Invariant Variational Problems -- Symmetry Classification of Variational Problems -- First Integrals -- The Cartan Form -- Invariant Contact Forms and Evolution Equations -- Equivalence of Coframes -- Frames and Coframes -- The Structure Functions -- Derived Invariants -- Classifying Functions -- The Classifying Manifolds -- Symmetries of a Coframe -- Remarks and Extensions -- Formulation of Equivalence Problems -- Equivalence Problems Using Differential Forms -- Coframes and Structure Groups -- Normalization -- Overdetermined Equivalence Problems -- Cartan's Equivalence Method -- The Structure Equations -- Absorption and Normalization -- Equivalence Problems for Differential Operators -- Fiber-preserving Equivalence of Scalar Lagrangians -- An Inductive Approach to Equivalence Problems -- Lagrangian Equivalence under Point Transformations -- Applications to Classical Invariant Theory -- Second Order Variational Problems -- Multi-dimensional Lagrangians -- Involution -- Cartan's Test -- The Transitive Case -- Divergence Equivalence of First Order Lagrangians -- The Intrinsic Method -- Contact Transformations -- Darboux' Theorem -- The Intransitive Case -- Equivalence of Nonclosed Two-Forms -- Prolongation of Equivalence Problems -- The Determinate Case -- Equivalence of Surfaces -- Conformal Equivalence of Surfaces -- Equivalence of Riemannian Manifolds -- The Indeterminate Case -- Second Order Ordinary Differential Equations -- Differential Systems -- Differential Systems and Ideals -- Equivalence of Differential Systems -- Vector Field Systems -- Frobenius' Theorem -- Vector Field Systems -- Differential Systems.
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Characteristics and Normal Forms -- The Technique of the Graph -- Global Equivalence -- The Cartan-Kähler Existence Theorem -- The Cauchy-Kovalevskaya Existence Theorem -- Necessary Conditions -- Sufficient Conditions -- Applications to Equivalence Problems -- Involutivity and Transversality -- Tables -- References -- Symbol Index -- Author Index -- Subject Index.
,
English
Weitere Ausg.:
ISBN 0-521-10104-2
Weitere Ausg.:
ISBN 0-521-47811-1
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9780511609565
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