Exterior Differential Systems - Hubert L. Goldschmidt

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Sommaire:
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-K?hler Theory.- ?1. Integral Elements.- ?2. The Cartan-K?hler 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-Poincar? 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. PartialDifferential 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.