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AuthorBryant, Robert L. author
TitleExterior Differential Systems [electronic resource] / by Robert L. Bryant, S. S. Chern, Robert B. Gardner, Hubert L. Goldschmidt, P. A. Griffiths
ImprintNew York, NY : Springer New York, 1991
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Descript VII, 475 p. online resource


This book gives a treatment of exterior differential systems. It will inยญ clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of indeยญ pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepenยญ dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object


I. Preliminaries -- ยง1. Review of Exterior Algebra -- ยง2. The Notion of an Exterior Differential System -- ยง3. Jet Bundles -- II. Basic Theorems -- ยง1. Probenius Theorem -- ยง2. Cauchy Characteristics -- ยง3. Theorems of Pfaff and Darboux -- ยง4. Pfaffian Systems -- ยง5. Pfaffian Systems of Codimension Two -- III. Cartan-Kรคhler Theory -- ยง1. Integral Elements -- ยง2. The Cartan-Kรคhler Theorem -- ยง3. Examples -- IV. Linear Differential Systems -- ยง1. Independence Condition and Involution -- ยง2. Linear Differential Systems -- ยง3. Tableaux -- ยง4. Tableaux Associated to an Integral Element -- ยง5. Linear Pfaffian Systems -- ยง6. Prolongation -- ยง7. Examples -- ยง8. Families of Isometric Surfaces in Euclidean Space -- V. The Characteristic Variety -- ยง1. Definition of the Characteristic Variety of a Differential System -- ยง2. The Characteristic Variety for Linearc Pfaffian Systems; Examples -- ยง3. Properties of the Characteristic Variety -- VI. Prolongation Theory -- ยง1. The Notion of Prolongation -- ยง2. Ordinary Prolongation -- ยง3. The Prolongation Theorem -- ยง4. The Process of Prolongation -- VII. Examples -- ยง1. First Order Equations for Two Functions of Two Variables -- ยง2. Finiteness of the Web Rank -- ยง3. Orthogonal Coordinates -- ยง4. Isometric Embedding -- VIII. Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems -- ยง1. Involutive Tableaux -- ยง2. The Cartan-Poincarรฉ Lemma, Spencer Cohomology -- ยง3. The Graded Module Associated to a Tableau; Koszul Homology -- ยง4. The Canonical Resolution of an Involutive Module -- ยง5. Localization; the Proofs of Theorem 3.2 and Proposition 3.8 -- ยง6. Proof of Theorem 3.8 in Chapter V; Guilleminโ{128}{153}s Normal Form -- ยง7. The Graded Module Associated to a Higher Order Tableau -- IX. Partial Differential Equations -- ยง1. An Integrability Criterion -- ยง2. Quasi-Linear Equations -- ยง3. Existence Theorems -- X. Linear Differential Operators -- ยง1. Formal Theory and Complexes -- ยง2. Examples -- ยง3. Existence Theorems for Elliptic Equations

Mathematics Manifolds (Mathematics) Complex manifolds Mathematics Manifolds and Cell Complexes (incl. Diff.Topology)


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