Exterior Differential Systems

Overview

This book gives a treatment of exterior differential systems including both the general theory and various applications. Topics include: a review of exterior algebra, simple exterior differential systems, the generation of integral manifolds through the solution of a succession of initial- value problems, involution, linear differential systems, tableau and torsion, the characteristic variety of a differential system, prolongation, the Algebra of a linear Pfaffian system, and an introduction to Spencer Theory. ...
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Paperback (Softcover reprint of the original 1st ed. 1991)
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Overview

This book gives a treatment of exterior differential systems including both the general theory and various applications. Topics include: a review of exterior algebra, simple exterior differential systems, the generation of integral manifolds through the solution of a succession of initial- value problems, involution, linear differential systems, tableau and torsion, the characteristic variety of a differential system, prolongation, the Algebra of a linear Pfaffian system, and an introduction to Spencer Theory. Much emphasis is placed on the general theory while many examples are given.
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Editorial Reviews

Booknews
A treatment of both the general theory and various applications of exterior differential systems, i.e. systems of equations on a manifold defined by equating to zero a number of exterior differential forms. The volume particularly studies the integral manifolds, that is, submanifolds satisfying all the equations of the system. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9781461397168
  • Publisher: Springer New York
  • Publication date: 12/14/2011
  • Series: Mathematical Sciences Research Institute Publications Series, #18
  • Edition description: Softcover reprint of the original 1st ed. 1991
  • Edition number: 1
  • Pages: 475
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.98 (d)

Table of Contents

I. Preliminaries.- §1. Review of Exterior Algebra.- §2. The Notion of an Exterior Differential System.- §3. Jet Bundles.- II. Basic Theorems.- §1. Probenius Theorem.- §2. Cauchy Characteristics.- §3. Theorems of Pfaff and Darboux.- §4. Pfaffian Systems.- §5. Pfaffian Systems of Codimension Two.- III. Cartan-Kähler Theory.- §1. Integral Elements.- §2. The Cartan-Kähler Theorem.- §3. Examples.- IV. Linear Differential Systems.- §1. Independence Condition and Involution.- §2. Linear Differential Systems.- §3. Tableaux.- §4. Tableaux Associated to an Integral Element.- §5. Linear Pfaffian Systems.- §6. Prolongation.- §7. Examples.- §8. Families of Isometric Surfaces in Euclidean Space.- V. The Characteristic Variety.- §1. Definition of the Characteristic Variety of a Differential System.- §2. The Characteristic Variety for Linearc Pfaffian Systems; Examples.- §3. Properties of the Characteristic Variety.- VI. Prolongation Theory.- §1. The Notion of Prolongation.- §2. Ordinary Prolongation.- §3. The Prolongation Theorem.- §4. The Process of Prolongation.- VII. Examples.- §1. First Order Equations for Two Functions of Two Variables.- §2. Finiteness of the Web Rank.- §3. Orthogonal Coordinates.- §4. Isometric Embedding.- VIII. Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems.- §1. Involutive Tableaux.- §2. The Cartan-Poincaré Lemma, Spencer Cohomology.- §3. The Graded Module Associated to a Tableau; Koszul Homology.- §4. The Canonical Resolution of an Involutive Module.- §5. Localization; the Proofs of Theorem 3.2 and Proposition 3.8.- §6. Proof of Theorem 3.8 in Chapter V; Guillemin’s Normal Form.- §7. The Graded Module Associated to a Higher Order Tableau.- IX. Partial Differential Equations.- §1. An Integrability Criterion.- §2. Quasi-Linear Equations.- §3. Existence Theorems.- X. Linear Differential Operators.- §1. Formal Theory and Complexes.- §2. Examples.- §3. Existence Theorems for Elliptic Equations.
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